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*J. Chem. Theory Comput.*2023, 19, 13, 3982-3995

# Vertical Ionization Potentials and Electron Affinities at the Double-Hybrid Density Functional LevelClick to copy article linkArticle link copied!

- Dávid Mester
*****Dávid MesterDepartment of Physical Chemistry and Materials Science, Faculty of Chemical Technology and Biotechnology, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, HungaryELKH-BME Quantum Chemistry Research Group, Műegyetem rkp. 3., H-1111 Budapest, HungaryMTA-BME Lendület Quantum Chemistry Research Group, Műegyetem rkp. 3., H-1111 Budapest, Hungary*****E-mail: [email protected]More by Dávid Mester - Mihály Kállay
*****Mihály KállayDepartment of Physical Chemistry and Materials Science, Faculty of Chemical Technology and Biotechnology, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, HungaryELKH-BME Quantum Chemistry Research Group, Műegyetem rkp. 3., H-1111 Budapest, HungaryMTA-BME Lendület Quantum Chemistry Research Group, Műegyetem rkp. 3., H-1111 Budapest, Hungary*****E-mail: [email protected]More by Mihály Kállay

## Abstract

The double-hybrid (DH) time-dependent density functional theory is extended to vertical ionization potentials (VIPs) and electron affinities (VEAs). Utilizing the density fitting approximation, efficient implementations are presented for the genuine DH ansatz relying on the perturbative second-order correction, while an iterative analogue is also elaborated using our second-order algebraic-diagrammatic construction [ADC(2)]-based DH approach. The favorable computational requirements of the present schemes are discussed in detail. The performance of the recently proposed spin-component-scaled and spin-opposite-scaled (SOS) range-separated (RS) and long-range corrected (LC) DH functionals is comprehensively assessed, while popular hybrid and global DH approaches are also discussed. For the benchmark calculations, up-to-date test sets are selected with high-level coupled-cluster references. Our results show that the ADC(2)-based SOS-RS-PBE-P86 approach is the most accurate and robust functional. This method consistently outperforms the excellent SOS-ADC(2) approach for VIPs, although the results are somewhat less satisfactory for VEAs. Among the genuine DH functionals, the SOS-ωPBEPP86 approach is also recommended for describing ionization processes, but its performance is even less reliable for electron-attached states. In addition, surprisingly good results are attained by the LC hybrid ωB97X-D functional, where the corresponding occupied (unoccupied) orbital energies are retrieved as VIPs (VEAs) within the present formalism.

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### License Summary*

You are free to share(copy and redistribute) this article in any medium or format and to adapt(remix, transform, and build upon) the material for any purpose, even commercially within the parameters below:

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*Disclaimer

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### License Summary*

You are free to share(copy and redistribute) this article in any medium or format and to adapt(remix, transform, and build upon) the material for any purpose, even commercially within the parameters below:

Creative Commons (CC): This is a Creative Commons license.

Attribution (BY): Credit must be given to the creator.

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## 1. Introduction

*n*-electron system can also be interpreted as the negative of the exact ionization potential; (12−14) however, these values are often underestimated in comparison with the experimental results. (12,15) Consequently, the use of the more advanced wave function-based methods is desired, but a good alternative could also be offered by the efficient double-hybrid (DH) DFT approaches, which combine KS-DFT with second-order wave function approximations. (16−21)

*n*-electron reference system. For instance, ionization processes can be studied by diagonalizing the corresponding operator in a basis of determinants containing

*n*– 1 electrons, (27−31) while electron-attached states correspond to the diagonal representation in the

*n*+ 1 particle space. (32,33)

*n*-electron system.

## 2. Theoretical Overview

### 2.1. Double-Hybrid Density Functional Theory for Excitations

**A**

^{DH}denotes the corresponding DH Jacobian,

**c**is the singles excitation vector, and ω

^{TDA}is the TDA excitation energy. Using the spatial-orbital representation, the elements of the Jacobian are defined by

*i*,

*j*, ... (

*a*,

*b*, ...) denote occupied (unoccupied) molecular orbitals, and ε

_{a}and ε

_{i}are the corresponding orbital energies. (

*ia*|

*jb*) is a two-electron repulsion integral in Mulliken’s convention, whereas (

*ia*|

*f*

_{X}|

*jb*) and (

*ia*|

*f*

_{C}|

*jb*) are the integrals of the exchange and correlation kernels, respectively. The above expression contains two adjustable parameters: the ratio of the Hartree–Fock (HF) and DFT contributions to the exchange energy is handled by α

_{X}, while the DFT correlation part is scaled by 1 – α

_{C}. The excitation energies obtained in this way have just hybrid quality. With the solution of eq 1 at hand, in the second step, the second-order correction is calculated perturbatively relying on the configuration interaction singles (CIS) (73) with perturbative doubles [CIS(D)] (74) approach. Accordingly, the improved excitation energy at the DH level is obtained as

^{(D)}is the perturbative correction. The thorough theoretical background of genuine excited-state DH approaches has been presented in excellent papers, (71,75−77) while the efficient calculation of the second-order terms invoking the density fitting (DF) approximation has been detailed in our previous work. (78)

^{ADC(2)}denotes the ADC(2) excitation energy. The corresponding Jacobian can be split into two parts as

**A**

^{CIS}is the CIS Jacobian, and all of the terms including second-order contributions are collected into matrix

**A**

^{[2]}.

*a posteriori*to the TDA excitation energy, these excitations are treated iteratively in this ansatz. Thus, an improvement in the calculated excitation energies is expected. Additionally, the transition properties, such as oscillator strengths, can be calculated at a higher level using the presented approach. Needless to say, the computational cost also increases since the ADC(2)-based approach is iterative.

*N*

^{4}invoking the DF approximation for the electron-repulsion integrals and Laplace transform-based techniques, (88) whereas the scaling of the original and spin-component-scaled (SCS) variants are

*N*

^{5}, where

*N*is a measure of the system size.

### 2.2. VIP and VEA Calculations for CIS(D)

*t*

_{ij}

^{ab}, can be obtained by acting the ${\hat{T}}_{2}$ operator on the HF reference determinant, Φ

_{0}:

_{ij}

^{ab}is the corresponding double substitution, and the orbital energy differences

*D*

_{ij}

^{ab}are constructed as ε

_{i}+ ε

_{j}– ε

_{a}– ε

_{b}. Using the DF approximation, the four-center quantities can be recast as

*P*and

*Q*stand for the elements of the auxiliary basis, whereas

*I*

_{ia}

^{P}and

*V*

_{PQ}are three- and two-center Coulomb integrals, respectively, and

*V*

_{PQ}

^{–1}is a simplified notation for the corresponding element of the inverse of the two-center Coulomb integral matrix. Usually, the matrix

**K**with elements

*K*

_{ia,jb}= (

*ia*|

*jb*) is factorized as

**K**=

**IV**

^{–1/2}

**V**

^{–1/2}

**I**

^{T}=

**JJ**

^{T}. Using the latter notation, the MP2 correlation energy can be expressed in the following simple form:

*Y*

_{ia}

^{Q}is defined by the contraction of the three-center integrals and the antisymmetrized amplitudes.

*k*̅th orbital can be interpreted as a CIS(D) (74) calculation using the corresponding ionized determinant:

*V*

_{ij}

^{a}can be expressed as

*n*.

*N*

^{4}. Since one of the occupied indices on the right-hand side of eq 16 is restricted to the ionized orbital, the computation of

*V*

_{ij}

^{a}scales as

*N*

_{occ}

^{2}

*N*

_{unocc}

*N*

_{aux}, where

*N*

_{occ},

*N*

_{unocc}, and

*N*

_{aux}are the number of occupied, unoccupied, and auxiliary orbitals, respectively. The rate-determining step of eq 17 is the evaluation of intermediate

**Y**, which also requires only a fourth-power scaling operation due to the restriction. This means that the cost of the perturbative second-order correction is comparable to a single iteration in a ground-state HF calculations. In addition, the unoccupied–unoccupied block of the three-center integrals is not required for the calculations, which is not true for standard CIS(D) calculations.

*c*̅th orbital is calculated as

*n*+ 1-electron state is expressed as

*V*

_{i}

^{ab}can be written in the

*X*can be calculated as

*V*

_{i}

^{ab}and ${Y}_{i\overline{c}}^{Q}$, scale as

*N*

^{4}.

### 2.3. VIP and VEA Calculations for ADC(2)

_{n–1}=

*∑*

_{i}

*c*

_{i}Φ

_{i}≠

*∑*

_{i}δ

_{ij}Φ

_{i}. Accordingly, the elements of intermediate

*V*

_{ij}

^{a}can be expressed as

**Y**, but the restriction regarding the ionized orbital cannot be applied in this case. This implies that the most expensive step is proportional to

*N*

^{5}; however, it has to be carried out only once, regardless of the number of ionized states. The iterative procedure still scales as

*N*

^{4}since the intermediate in parentheses can be evaluated before it. In addition, the demanding

*J*

_{ab}

^{Q}-type integrals are not required for the calculations.

*V*

_{i}

^{ab}is calculated as

_{C}. The working equations to calculate pole strengths for ADC(2) can be found in ref (38).

## 3. Computational Details

### 3.1. Calculation of the Numerical Results

*X*Z, where

*X*= D and T) (99,100) and their diffuse function augmented variants (aug-cc-pV

*X*Z) (101) were employed for the calculations, and the DF approximation was utilized for both the ground and attached/detached states. For this purpose, the corresponding auxiliary bases of Weigend and co-workers (102−104) were employed. The frozen core approximation was employed in a similar manner to the original benchmark studies (see Sect. 3.2) in the post-KS/HF steps. The convergence threshold for the energies was set to 10

^{–6}E

_{h}, while the default adaptive integration grid of the Mrcc package was used for the XC contributions. (105)

Functional | Exchange | Correlation | Class | Number of parameters | References |
---|---|---|---|---|---|

SCS-RS-PBE-P86 | PBE | P86 | RS DH | 4 | (90) |

SOS-RS-PBE-P86 | PBE | P86 | RS DH | 3 | (90) |

SCS-ωPBEPP86 | PBE | P86 | LC DH | 7 | (77) |

SOS-ωPBEPP86 | PBE | P86 | LC DH | 5 | (77) |

DSD-PBEP86 | PBE | P86 | global DH | 4 | (113) |

PBE0-2 | PBE | PBE | global DH | 2 | (114) |

SOS-PBE0-2 | PBE | PBE | global DH | 3 | (115) |

PBE-QIDH | PBE | PBE | global DH | 2 | (116) |

B2GPPLYP | B88 | LYP | global DH | 2 | (117) |

ωB97X-D | B97 | B97 | LC hybrid | 18 | (118) |

CAM-B3LYP | B88 | LYP | LC hybrid | 3 | (119) |

PBE0 | PBE | PBE | global hybrid | 1 | (120) |

### 3.2. The Benchmark Sets

## 4. Results and Discussion

### 4.1. The Acceptor Test Set

### 4.2. The Bartlett/Ortiz Test Set

### 4.3. Nucleobases

### 4.4. Overall Performance

MAE | Scaling | |||||
---|---|---|---|---|---|---|

Class | Method | VIP | VEA | Number of parameters | Ground state | VIP/VEA |

wave function-based | CCSD | 0.153 | 0.051 | – | iterative N^{6} | iterative N^{5} |

SOS-ADC(2) | 0.211 | 0.135 | 1 | perturbative N^{4} | iterative N^{4} | |

SOS-CIS(D) | 0.325 | 0.146 | 1 | perturbative N^{4} | perturbative N^{4} | |

ADC(2) | 0.585 | 0.431 | – | perturbative N^{5} | iterative N^{4} | |

CIS(D) | 0.834 | 0.492 | – | perturbative N^{4} | perturbative N^{4} | |

ADC(2)-based DH | SCS-RS-PBE-P86 | 0.212 | 0.458 | 4 | perturbative N^{5} | iterative N^{4} |

SOS-RS-PBE-P86 | 0.182 | 0.354 | 3 | perturbative N^{4} | iterative N^{4} | |

SOS-PBE0-2 | 0.300 | 0.506 | 3 | perturbative N^{4} | iterative N^{4} | |

CIS(D)-based DH | SCS-RS-PBE-P86 | 0.328 | 0.511 | 4 | perturbative N^{4} | perturbative N^{4} |

SOS-RS-PBE-P86 | 0.306 | 0.396 | 3 | perturbative N^{4} | perturbative N^{4} | |

SCS-ωPBEPP86 | 0.256 | 0.683 | 7 | perturbative N^{4} | perturbative N^{4} | |

SOS-ωPBEPP86 | 0.226 | 0.552 | 5 | perturbative N^{4} | perturbative N^{4} | |

SOS-PBE0-2 | 0.352 | 0.518 | 3 | perturbative N^{4} | perturbative N^{4} | |

LC hybrid | ωB97X-D | 0.457 | 0.406 | 18 | – | – |

CAM-B3LYP | 0.924 | 1.121 | 3 | – | – |

## 5. Conclusions

*N*

^{5}. However, it has to be carried out only once, regardless of the number of tackled states, and the iterative procedure still scales as

*N*

^{4}. For a detailed comparison, state-of-the-art RS-DH and LC-DH methods, including spin-scaling techniques, were selected, as well as robust and popular hybrid and global DH approaches were also included. To assess the performance of the functionals, comprehensive benchmark calculations were carried out on the best available test sets. A total of 224 electron-detached and 54 electron-attached states were examined, with references provided by higher-order coupled-cluster calculations that include triple excitations.

## Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.3c00363.

Computed vertical ionization potentials and electron affinities (XLSX)

## Terms & Conditions

Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page/copyright/permissions.html.

## Acknowledgments

The work of D.M. is supported by the NKFIH PD142372 grant and the ÚNKP-22-4-II-BME-157 New National Excellence Program of the Ministry for Culture and Innovation from the source of the National Research, Development, and Innovation Fund. M.K. is grateful for the financial support from the National Research, Development, and Innovation Office (NKFIH, Grant No. KKP126451). The research reported in this paper is part of project BME-EGA-02, implemented with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021 funding scheme. The computing time granted on the Hungarian HPC Infrastructure at NIIF Institute, Hungary is gratefully acknowledged.

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(American Chemical Society)The mechanisms of action of low-energy electrons (LEEs) generated in large quantities by ionizing radiation constitute an essential element of our understanding of early events in radiolysis and radiobiol. We present the 2-20 eV electron energy dependence of the yields of base damage (BD), BD-related crosslinks (CLs), and non-double-strand break (NDSB) clustered damage induced in DNA. These new yield functions are generated by the impact of LEEs on plasmid DNA films. The damage is analyzed by gel electrophoresis with and without enzyme treatment. Maxima at 5 and 10 eV in BDs and BD-related CLs yield functions, and two others, at 6 and 10 eV, in those of NDSB clustered damage are ascribed to core-excited transient anions that decay into bond-breaking channels. The mechanism causing all types of DNA damages can be attributed to the capture of a single electron by a base followed by multiple different electron transfer pathways.**9**Mukherjee, M.; Tripathi, D.; Brehm, M.; Riplinger, C.; Dutta, A. K. Efficient EOM-CC-based Protocol for the Calculation of Electron Affinity of Solvated Nucleobases: Uracil as a Case Study.*J. Chem. Theory Comput.*2021,*17*, 105, DOI: 10.1021/acs.jctc.0c00655Google Scholar9https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXislWhsbrK&md5=8e321848942809c54c1df1acb5aa313dEfficient EOM-CC-based Protocol for the Calculation of Electron Affinity of Solvated Nucleobases: Uracil as a Case StudyMukherjee, Madhubani; Tripathi, Divya; Brehm, Martin; Riplinger, Christoph; Dutta, Achintya KumarJournal of Chemical Theory and Computation (2021), 17 (1), 105-116CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present an explicit solvation protocol for the calcn. of electron affinity values of the solvated nucleobases. The protocol uses a quantum mechanics/mol. mechanics (QM/MM) approach based on the newly implemented domain-based pair natural orbital EOM-CCSD (equation-of-motion coupled-cluster single-double) method. The stability of the solvated nucleobase anion is sensitive to the local distribution of the water mols. around the nucleobase, and the calcd. electron affinity values converge slowly with respect to the no. of snapshots and the size of the water box. The use of nonpolarizable water mols. leads to an overestimation of the electron affinity and makes the result sensitive to the size of the QM region in the QM/MM calcn. The electron affinity values, although sensitive to the size of the basis set, lead to an almost const. blue shift of the electron affinity upon the increase in the basis set. The present protocol allows for a controllable description of the various parameters affecting the electron affinity value, and the calcd. adiabatic electron affinity values are in excellent agreement with exptl. results.**10**Verma, P.; Ghosh, D.; Dutta, A. K. Electron Attachment to Cytosine: The Role of Water.*J. Phys. Chem. A*2021,*125*, 4683, DOI: 10.1021/acs.jpca.0c10199Google Scholar10https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXht1Srt7nJ&md5=c95143e520d9d09017588b3c5b87190cElectron Attachment to Cytosine: The Role of WaterVerma, Pooja; Ghosh, Debashree; Dutta, Achintya KumarJournal of Physical Chemistry A (2021), 125 (22), 4683-4694CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)We present an EOM-CCSD-based quantum mech./mol. mech. (QM/MM) study on the electron attachment process to solvated cytosine. The electron attachment in the bulk solvated cytosine occurs through a doorway mechanism, where the initial electron is localized on water. The electron is subsequently transferred to cytosine by the mixing of electronic and nuclear degrees of freedom, which occurs on an ultrafast time scale. The bulk water environment stabilizes the cytosine-bound anion by an extensive hydrogen-bond network and drastically enhances the electron transfer rate from that obsd. in the gas phase. Microhydration studies cannot reproduce the effect of the bulk water environment on the electron attachment process, and one needs to include a large no. of water mols. in the calcn. to obtain converged results. The predicted adiabatic electron affinity and electron transfer rate obtained from our QM/MM calcns. are consistent with the available exptl. results.**11**Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects.*Phys. Rev.*1965,*140*, A1133, DOI: 10.1103/PhysRev.140.A1133Google ScholarThere is no corresponding record for this reference.**12**Zhan, C.-G.; Nichols, J. A.; Dixon, D. A. Ionization Potential, Electron Affinity, Electronegativity, Hardness, and Electron Excitation Energy: Molecular Properties from Density Functional Theory Orbital Energies.*J. Phys. Chem. A*2003,*107*, 4184, DOI: 10.1021/jp0225774Google Scholar12https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXjt1Shtbk%253D&md5=71feb2642a03fc6ada0177fd5e1f153dIonization Potential, Electron Affinity, Electronegativity, Hardness, and Electron Excitation Energy: Molecular Properties from Density Functional Theory Orbital EnergiesZhan, Chang-Guo; Nichols, Jeffrey A.; Dixon, David A.Journal of Physical Chemistry A (2003), 107 (20), 4184-4195CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)Representative at. and mol. systems, including various inorg. and org. mols. with covalent and ionic bonds, have been studied by using d. functional theory. The calcns. were done with the commonly used exchange-correlation functional B3LYP followed by a comprehensive anal. of the calcd. highest-occupied and lowest-unoccupied Kohn-Sham orbital (HOMO and LUMO) energies. The basis set dependence of the DFT results shows that the economical 6-31+G* basis set is generally sufficient for calcg. the HOMO and LUMO energies (if the calcd. LUMO energies are neg.) for use in correlating with mol. properties. The directly calcd. ionization potential (IP), electron affinity (EA), electronegativity (χ), hardness (η), and first electron excitation energy (τ) are all in good agreement with the available exptl. data. A generally applicable linear correlation relationship exists between the calcd. HOMO energies and the exptl./calcd. IPs. We have also found satisfactory linear correlation relationships between the calcd. LUMO energies and exptl./calcd. EAs (for the bound anionic states), between the calcd. av. HOMO/LUMO energies and χ values, between the calcd. HOMO-LUMO energy gaps and η values, and between the calcd. HOMO-LUMO energy gaps and exptl./calcd. first excitation energies. By using these linear correlation relationships, the calcd. HOMO and LUMO energies can be employed to semiquant. est. ionization potential, electron affinity, electronegativity, hardness, and first excitation energy.**13**Perdew, J. P.; Levy, M. Comment on “Significance of the highest occupied Kohn–Sham eigenvalue.*Phys. Rev. B*1997,*56*, 16021, DOI: 10.1103/PhysRevB.56.16021Google Scholar13https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXotFWntbg%253D&md5=174bb39db5f10f6c0862d5837d672ba0Comment on "Significance of the highest occupied Kohn-Sham eigenvalue"Perdew, John P.; Levy, MelPhysical Review B: Condensed Matter (1997), 56 (24), 16021-16028CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)With more explanation than usual and without appeal to Janak's theorem, we discuss the statement and proof of the ionization potential theorems for the exact Kohn-Sham d.-functional theory of a many-electron system: (1) For any av. electron no. N between the integers Z - 1 and Z, and thus for N → Z from below, the highest occupied or partly occupied Kohn-Sham orbital energy is minus the ionization energy of the Z-electron system. (2) For Z - 1 < N < Z, the exact Kohn-Sham effective potential vs(r) tends to zero as |r| → ∞. We then argue that an objection to these theorems [L. Kleinman, Phys. Rev. B 56, 12042 (1997)] overlooks a crucial step in the proof of theorem (2): The asymptotic exponential decay of the exact electron d. of the Z-electron system is controlled by the exact ionization energy, but the decay of an approx. d. is not controlled by the approx. ionization energy. We discuss relevant evidence from the numerical construction of the exact Kohn-Sham potential. In particular, we point out a model two-electron problem for which the ionization potential theorems are exactly confirmed. Finally, we comment on related issues: the self-interaction correction, the discontinuity of the exact Kohn-Sham potential as N passes through the integer Z, and the generalized sum rule on the exchange-correlation hole.**14**Levy, M.; Perdew, J. P.; Sahni, V. Exact differential equation for the density and ionization energy of a many-particle system.*Phys. Rev. A*1984,*30*, 2745, DOI: 10.1103/PhysRevA.30.2745Google ScholarThere is no corresponding record for this reference.**15**Zhang, G.; Musgrave, C. B. Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations.*J. Phys. Chem. A*2007,*111*, 1554, DOI: 10.1021/jp061633oGoogle Scholar15https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXht1yrs7w%253D&md5=707bb4d5e592c5592f93045e5cef67ddComparison of DFT Methods for Molecular Orbital Eigenvalue CalculationsZhang, Gang; Musgrave, Charles B.Journal of Physical Chemistry A (2007), 111 (8), 1554-1561CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)We report how closely the Kohn-Sham HOMO and LUMO eigenvalues of 11 d. functional theory (DFT) functionals, resp., correspond to the neg. ionization potentials (-IPs) and electron affinities (EAs) of a test set of mols. We also report how accurately the HOMO-LUMO gaps of these methods predict the lowest excitation energies using both time-independent and time-dependent DFT (TD-DFT). The 11 DFT functionals include the local spin d. approxn. (LSDA), five generalized gradient approxn. (GGA) functionals, three hybrid GGA functionals, one hybrid functional, and one hybrid meta GGA functional. We find that the HOMO eigenvalues predicted by KMLYP, BH and HLYP, B3LYP, PW91, PBE, and BLYP predict the -IPs with av. abs. errors of 0.73, 1.48, 3.10, 4.27, 4.33, and 4.41 eV, resp. The LUMOs of all functionals fail to accurately predict the EAs. Although the GGA functionals inaccurately predict both the HOMO and LUMO eigenvalues, they predict the HOMO-LUMO gap relatively accurately (∼0.73 eV). On the other hand, the LUMO eigenvalues of the hybrid functionals fail to predict the EA to the extent that they include HF exchange, although increasing HF exchange improves the correspondence between the HOMO eigenvalue and -IP so that the HOMO-LUMO gaps are inaccurately predicted by hybrid DFT functionals. We find that TD-DFT with all functionals accurately predicts the HOMO-LUMO gaps. A linear correlation between the calcd. HOMO eigenvalue and the exptl. -IP and calcd. HOMO-LUMO gap and exptl. lowest excitation energy enables us to derive a simple correction formula.**16**Grimme, S. Semiempirical hybrid density functional with perturbative second-order correlation.*J. Chem. Phys.*2006,*124*, 034108, DOI: 10.1063/1.2148954Google Scholar16https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XptVGnuw%253D%253D&md5=e0e89576e15f6a7c9fb40756b601dc66Semiempirical hybrid density functional with perturbative second-order correlationGrimme, StefanJournal of Chemical Physics (2006), 124 (3), 034108/1-034108/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A new hybrid d. functional for general chem. applications is proposed. It is based on a mixing of std. generalized gradient approxns. (GGAs) for exchange by Becke (B) and for correlation by Lee, Yang, and Parr (LYP) with Hartree-Fock (HF) exchange and a perturbative second-order correlation part (PT2) that is obtained from the Kohn-Sham (GGA) orbitals and eigenvalues. This virtual orbital-dependent functional contains only two global parameters that describe the mixt. of HF and GGA exchange (ax) and of the PT2 and GGA correlation (c), resp. The parameters are obtained in a least-squares-fit procedure to the G2/97 set of heat of formations. Opposed to conventional hybrid functionals, the optimum ax is found to be quite large (53% with c = 27%) which at least in part explains the success for many problematic mol. systems compared to conventional approaches. The performance of the new functional termed B2-PLYP is assessed by the G2/97 std. benchmark set, a second test suite of atoms, mols., and reactions that are considered as electronically very difficult (including transition-metal compds., weakly bonded complexes, and reaction barriers) and comparisons with other hybrid functionals of GGA and meta-GGA types. According to many realistic tests, B2-PLYP can be regarded as the best general purpose d. functional for mols. (e.g., a mean abs. deviation for the two test sets of only 1.8 and 3.2 kcal/mol compared to about 3 and 5 kcal/mol, resp., for the best other d. functionals). Very importantly, also the max. and minium errors (outliers) are strongly reduced (by about 10-20 kcal/mol). Furthermore, very good results are obtained for transition state barriers but unlike previous attempts at such a good description, this definitely comes not at the expense of equil. properties. Preliminary calcns. of the equil. bond lengths and harmonic vibrational frequencies for diat. mols. and transition-metal complexes also show very promising results. The uniformity with which B2-PLYP improves for a wide range of chem. systems emphasizes the need of (virtual) orbital-dependent terms that describe nonlocal electron correlation in accurate exchange-correlation functionals. From a practical point of view, the new functional seems to be very robust and it is thus suggested as an efficient quantum chem. method of general purpose.**17**Goerigk, L.; Grimme, S. Double-hybrid density functionals.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2014,*4*, 576, DOI: 10.1002/wcms.1193Google Scholar17https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhvVelu7jI&md5=c2c8a4d2d17cea5bc4a9c559d42742c8Double-hybrid density functionalsGoerigk, Lars; Grimme, StefanWiley Interdisciplinary Reviews: Computational Molecular Science (2014), 4 (6), 576-600CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)Double-hybrid d. functionals (DHDFs) are reviewed in this study. In DHDFs parts of conventional d. functional theory (DFT) exchange and correlation are replaced by contributions from nonlocal Fock-exchange and second-order perturbative correlation. The latter portion is based on the well-known MP2 wave-function approach in which, however, Kohn-Sham orbitals are used to calc. its contribution. First, related methods preceding this idea are reviewed, followed by a thorough discussion of the first modern double-hybrid B2-PLYP. Parallels and differences between B2-PLYP and its various successors are then outlined. This discussion is rounded off with representative thermochem. examples demonstrating that DHDFs belong to the most robust and accurate DFT approaches currently available. This anal. also presents hitherto unpublished results for recently developed DHDFs. Finally, how double-hybrids can be combined with linear-response time-dependent DFT is also outlined and the value of this approach for electronically excited states is shown. WIREs Comput Mol Sci 2014, 4:576-600. doi: 10.1002/wcms.1193 For further resources related to this article, please visit the . Conflict of interest: The authors have declared no conflicts of interest for this article.**18**Brémond, É.; Ciofini, I.; Sancho-García, J. C.; Adamo, C. Nonempirical Double-Hybrid Functionals: An Effective Tool for Chemists.*Acc. Chem. Res.*2016,*49*, 1503, DOI: 10.1021/acs.accounts.6b00232Google Scholar18https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xht1ynsb7O&md5=f2722f267174ead8e7ab3e5daf92a76bNonempirical Double-Hybrid Functionals: An Effective Tool for ChemistsBremond, Eric; Ciofini, Ilaria; Sancho-Garcia, Juan Carlos; Adamo, CarloAccounts of Chemical Research (2016), 49 (8), 1503-1513CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)A review. D. functional theory (DFT) emerged in the last two decades as the most reliable tool for the description and prediction of properties of mol. systems and extended materials, coupling in an unprecedented way high accuracy and reasonable computational cost. This success rests also on the development of more and more performing d. functional approxns. (DFAs). Indeed, the Achilles' heel of DFT is represented by the exchange-correlation contribution to the total energy, which, being unknown, must be approximated. Since the beginning of the 1990s, global hybrids (GH) functionals, where an explicit dependence of the exchange-correlation energy on occupied Kohn-Sham orbitals is introduced thanks to a fraction of Hartree-Fock-like exchange, imposed themselves as the most reliable DFAs for chem. applications. However, if these functionals normally provide results of sufficient accuracy for most of the cases analyzed, some properties, such as thermochem. or dispersive interactions, can still be significantly improved. A possible way out is represented by the inclusion, into the exchange-correlation functional, of an explicit dependence on virtual Kohn-Sham orbitals via perturbation theory. This leads to a new class of functionals, called double-hybrids (DHs). In this Account, we describe our nonempirical approach to DHs, which, following the line traced by the Perdew-Burke-Ernzerhof approach, allows for the definition of a GH (PBE0) and a DH (QIDH) model. In such a way, a whole family of nonempirical functionals, spanning on the highest rungs of the Perdew's quality scale, is now available and competitive with other-more empirical-DFAs. Discussion of selected cases, ranging from thermochem. and reactions to weak interactions and excitation energies, not only show the large range of applicability of nonempirical DFAs, but also underline how increasing the no. of theor. constraints parallels with an improvement of the DFA's numerical performances. This fact further consolidates the strong theor. framework of nonempirical DFAs.Finally, even if nonempirical DH approaches are still computationally expensive, relying on the fact that they can benefit of all tech. enhancements developed for speeding up post-Hartree-Fock methods, there is substantial hope for their near future routine application to the description and prediction of complex chem. systems and reactions.**19**Martin, J. M. L.; Santra, G. Empirical Double-Hybrid Density Functional Theory: A ‘Third Way’ in Between WFT and DFT.*Isr. J. Chem.*2020,*60*, 787, DOI: 10.1002/ijch.201900114Google Scholar19https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXitlyhsrnM&md5=182610d9f5560d261abf36f80a6d9d2eEmpirical Double-Hybrid Density Functional Theory: A 'Third Way' in Between WFT and DFTMartin, Jan M. L.; Santra, GolokeshIsrael Journal of Chemistry (2020), 60 (8-9), 787-804CODEN: ISJCAT; ISSN:0021-2148. (Wiley-VCH Verlag GmbH & Co. KGaA)A review. Double hybrid d. functional theory arguably sits on the seamline between wavefunction methods and DFT: it represents a special case of Rung 5 on the "Jacob's Ladder" of John P. Perdew. For large and chem. diverse benchmarks such as GMTKN55, empirical double hybrid functionals with dispersion corrections can achieve accuracies approaching wavefunction methods at a cost not greatly dissimilar to hybrid DFT approaches, provided RI-MP2 and/or another MP2 acceleration techniques are available in the electronic structure code. Only a half-dozen or fewer empirical parameters are required. For vibrational frequencies, accuracies intermediate between CCSD and CCSD(T) can be achieved, and performance for other properties is encouraging as well. Organometallic reactions can likewise be treated well, provided static correlation is not too strong. Further prospects are discussed, including range-sepd. and RPA-based approaches.**20**Su, N. Q.; Xu, X. The XYG3 Type of Doubly Hybrid Density Functionals.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2016,*6*, 721, DOI: 10.1002/wcms.1274Google Scholar20https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhslKgsbfP&md5=be9356f372be073ff2a1794c47a9fc11The XYG3 type of doubly hybrid density functionalsSu, Neil Qiang; Xu, XinWiley Interdisciplinary Reviews: Computational Molecular Science (2016), 6 (6), 721-747CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)Doubly hybrid (DH) functionals have emerged as a new class of d. functional approxns. (DFAs), which not only have a nonlocal orbital-dependent component in the exchange part, but also incorporate the information of unoccupied orbitals in the correlation part, being at the top rung of Perdew's view of Jacob's ladder in DFAs. This review article focuses on the XYG3 type of DH (xDH) functionals, which use a low rung functional to perform the self-consistent-field calcn. to generate orbitals and densities, with which a top rung DH functional is used for final energy evaluation. We will discuss the theor. background of the xDH functionals, briefly reviewing the adiabatic connection formalism, coordinate scaling relations, and Goerling-Levy perturbation theory. General performance of the xDH functionals will be presented for both energies and structures. In particular, we will present the fractional charge behaviors of the xDH functionals, examg. the self-interaction errors, the delocalization errors and the deviation from the linearity condition, as well as their effects on the predicted ionization potentials, electron affinities and fundamental gaps. This provides a theor. rationale for the obsd. good performance of the xDH functionals. WIREs Comput Mol Sci 2016, 6:721-747. doi: 10.1002/wcms.1274 For further resources related to this article, please visit the .**21**Sancho-García, J. C.; Adamo, C. Double-hybrid density functionals: Merging wavefunction and density approaches to get the best of both worlds.*Phys. Chem. Chem. Phys.*2013,*15*, 14581, DOI: 10.1039/c3cp50907aGoogle Scholar21https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXht1KhtbvF&md5=4216f41fe053cdc2348840cbd2567f0cDouble-hybrid density functionals: merging wavefunction and density approaches to get the best of both worldsSancho-Garcia, J. C.; Adamo, C.Physical Chemistry Chemical Physics (2013), 15 (35), 14581-14594CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)A review. We review why and how double-hybrid d. functionals have become new leading actors in the field of computational chem., thanks to the combination of an unprecedented accuracy together with large robustness and reliability. Similar to their predecessors, the widely employed hybrid d. functionals, they are rooted in the Adiabatic Connection Method from which they emerge in a natural way. We present recent achievements concerning applications to chem. systems of the most interest, and current extensions to deal with challenging issues such as non-covalent interactions and excitation energies. These promising methods, despite a slightly higher computational cost than other typical d.-based models, are called to play a key role in the near future and can thus pave the way towards new discoveries or advances.**22**Geertsen, J.; Rittby, M.; Bartlett, R. J. The equation-of-motion coupled-cluster method: Excitation energies of Be and CO.*Chem. Phys. Lett.*1989,*164*, 57, DOI: 10.1016/0009-2614(89)85202-9Google Scholar22https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3cXkt12nsbw%253D&md5=00bf6668a536c3c45dc538396deb5ba7The equation-of-motion coupled-cluster method: excitation energies of beryllium and carbon monoxideGeertsen, Jan; Rittby, Magnus; Bartlett, Rodney J.Chemical Physics Letters (1989), 164 (1), 57-62CODEN: CHPLBC; ISSN:0009-2614.The equation-of-motion coupled-cluster (EOM-CC) method for the calcn. of excitation energies is presented. The procedure is based upon representing an excited state as an excitation from a coupled-cluster ground state and the excitation energies are obtained by solving a non-Hermitian eigenvalue problem. Numerical applications are reported for Be and CO, and the compared to full CI, Fock space multi-ref. coupled-cluster, multi-ref. MBPT, and propagator results.**23**Stanton, J. F.; Bartlett, R. J. The equation of motion coupled-cluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited state properties.*J. Chem. Phys.*1993,*98*, 7029, DOI: 10.1063/1.464746Google Scholar23https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXksFKgu78%253D&md5=bb8b7c7ea2e69d1272a8e98ee83d9be7The equation-of-motion, coupled-cluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited-state propertiesStanton, John F.; Bartlett, Rodney J.Journal of Chemical Physics (1993), 98 (9), 7029-39CODEN: JCPSA6; ISSN:0021-9606.A comprehensive overview of the equation of motion coupled-cluster (EOM-CC) method and its application to mol. systems is presented. By exploiting the biorthogonal nature of the theory, it is shown that excited-state properties and transition strengths can be evaluated via a generalized expectation-value approach that incorporates both the bra and ket state wave functions. Reduced d. matrixes defined by this procedure are given by closed form expressions. For the root of the EOM-CC effective Hamiltonian that corresponds to the ground state, the resulting equations are equiv. to the usual expressions for normal single-ref. CC d. matrixes. Thus, the method described in this paper provides a universal definition of coupled-cluster d. matrixes, providing a link between EOM-CC and traditional ground state CC theory. Excitation energy, oscillator strength, and property calcns. are illustrated by means of several numerical examples, including comparisons with full CI calcns. and a detailed study of the 10 lowest electronically excited states of the cyclic isomer of C4.**24**Watts, J. D.; Bartlett, R. J. The inclusion of connected triple excitations in the equation-of-motion coupled-cluster method.*J. Chem. Phys.*1994,*101*, 3073, DOI: 10.1063/1.467620Google Scholar24https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXlvVOmu7c%253D&md5=850291febfd7c61d08ef73f2ed7ba367The inclusion of connected triple excitations in the equation-of-motion coupled-cluster methodWatts, John D.; Bartlett, RodneyJournal of Chemical Physics (1994), 101 (4), 3073-8CODEN: JCPSA6; ISSN:0021-9606.The implementation of connected triple excitations in the equation-of-motion (EOM) coupled-cluster (CC) method for excitation energies is reported for the first time. The ref. state is described by the complete CC singles, doubles, and triples (CCSDT) method. Excited states are generated from the ref. state wave function by the action of a linear excitation operator including single, double, and triple excitations. The excited state wave functions and energies are obtained by diagonalizing the effective Hamiltonian e-THeT, where T is the cluster operator for the ref. state, in the space of singly, doubly, and triply excited determinants. Comparison is made with full CI excitation energies for several examples (CH+, Be, SiH2, and CH2). These show that EOM-CCSDT is able to describe states which are doubly excited relative to the ref. state, as well as singly excited states. Calcns. of several excitation energies of BH using an extended basis set are also reported, and show good agreement with expt.**25**Bartlett, R. J. Coupled-cluster theory and its equation-of-motion extensions.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2012,*2*, 126, DOI: 10.1002/wcms.76Google Scholar25https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhvFGls7c%253D&md5=b75a016dfe3ed5488b83de78ceabd0c7Coupled-cluster theory and its equation-of-motion extensionsBartlett, Rodney J.Wiley Interdisciplinary Reviews: Computational Molecular Science (2012), 2 (1), 126-138CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)A review. Coupled-cluster theory offers today's ref. quantum chem. method for most of the problems encountered in electronic structure theory. It has been instrumental in establishing the now well-known paradigm of converging, many-body methods, MBPT2 < CCD < CCSD < MBPT4 < CCSD(T) < CCSDT-n < CCSDT < CCSDT(Q) < CCSDTQ-n < CCSDTQ < fullCI. Many-body perturbation theory (MBPT) for second, MBPT2, and fourth-order MBPT4; and coupled-cluster (CC) theory for different categories of excitations, singles, doubles, triples, quadruples (SDTQ). Although built on the same basic concept as CI (CI), many-body methods fundamentally improve upon CI approxns. by introducing the property of size extensivity, meaning that contrary to any truncated CI all terms properly scale with the no. of electrons in the problem. This fundamental aspect of many-electron methods leads to the exceptional performance of CC theory and its finite-order MBPT approxns. plus its equation-of-motion extensions for excited, ionized, and electron attached states. This brief overview will describe formal aspects of the theory which should be understood by perspective users of CC methods. We will also comment on some current developments that are improving the theory's accuracy or applicability.**26**Sneskov, K.; Christiansen, O. Excited state coupled cluster methods.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2012,*2*, 566, DOI: 10.1002/wcms.99Google Scholar26https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhtlWiu7fN&md5=96241519af990fde76ce3473a0b87231Excited state coupled cluster methodsSneskov, Kristian; Christiansen, OveWiley Interdisciplinary Reviews: Computational Molecular Science (2012), 2 (4), 566-584CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)We review coupled cluster (CC) theory for electronically excited states. We outline the basics of a CC response theory framework that allows the transfer of the attractive accuracy and convergence properties assocd. with CC methods over to the calcn. of electronic excitation energies and properties. Key factors affecting the accuracy of CC excitation energy calcns. are discussed as are some of the key CC models in this field. To aid both the practitioner as well as the developer of CC excited state methods, we also briefly discuss the key computational steps in a working CC response implementation. Approaches aimed at extending the application range of CC excited state methods either in terms of mol. size and phenomena or in terms of environment (soln. and proteins) are also discussed.**27**Nooijen, M.; Bartlett, R. J. Similarity transformed equation-of-motion coupled-cluster theory: Details, examples, and comparisons.*J. Chem. Phys.*1997,*107*, 6812, DOI: 10.1063/1.474922Google Scholar27https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXmsFOgsrk%253D&md5=0358e456ea5cd5607529aa3d2c874ce1Similarity transformed equation-of-motion coupled-cluster theory: Details, examples, and comparisonsNooijen, Marcel; Bartlett, Rodney J.Journal of Chemical Physics (1997), 107 (17), 6812-6830CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The similarity transformed equation-of-motion coupled-cluster (STEOM-CC) method is presented in full detail. Comparisons are made with the Fock space coupled-cluster (FSCC) method and the equation-of-motion coupled-cluster (EOM-CC) scheme. The role of implicit triple excitations and, relatedly, charge transfer separability in STEOM is discussed. The dependence on the choice of active space in STEOM is addressed and criteria for the selection of the active space are given. The evaluation of properties within STEOM is outlined and a large no. of illustrative examples of STEOM is presented.**28**Stanton, J. F.; Gauss, J. Perturbative treatment of the similarity transformed Hamiltonian in equation-of-motion coupled-cluster approximations.*J. Chem. Phys.*1995,*103*, 1064, DOI: 10.1063/1.469817Google Scholar28https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXmvFCktrk%253D&md5=7a402877a01649f980a13981f512dad3Perturbative treatment of the similarity transformed Hamiltonian in equation-of-motion coupled-cluster approximationsStanton, John F.; Gauss, JuergenJournal of Chemical Physics (1995), 103 (3), 1064-76CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A series of size-consistent approxns. to the equation-of-motion coupled cluster method in the singles and doubles approxn. (EOM-CCSD) are developed by subjecting the similarity transformed Hamiltonian ‾H = exp(-T)H exp(T) to a perturbation expansion. Attention is directed to N and N - 1 electron final state realizations of the method defined by truncation of ‾H at second order. Explicit spin-orbital equations for the energy and its first deriv. are documented for both approaches [EOMEE-CCSD(2) and EOMIP-CCSD(2), resp.], and have been implemented in a large-scale quantum chem. program. Vertical ionization potentials calcd. by EOMIP-CCSD(2) are shown to be equiv. to those of an approach presented recently by Nooijen and Snijders [J. Chem. Phys. 102, 1681,(1995)]. Applications of both EOMIP-CCSD(2) and EOMEE-CCSD(2) provide results for final state properties that compare favorably with those obtained in full EOM-CCSD calcns. Anal. of the computational aspects of the approx. and full EOM-CCSD methods shows that the cost of EOMIP-CCSD(2) energy and gradient calcns. scales in proportion to the fifth power of the basis set size, a significant savings over the sixth power dependence of EOMIP-CCSD. This feature is of great practical importance, as it shows that this N - 1 electron final state approach has a large domain of applicability and is therefore likely to become a valuable tool for application calcns. On the other hand, the same cannot be said for EOMEE-CCSD(2) since its asymptotic computational dependence and storage requirements are the same as the full EOMEE-CCSD method.**29**Stanton, J. F.; Gauss, J. Analytic energy derivatives for ionized states described by the equation-of-motion coupled cluster method.*J. Chem. Phys.*1994,*101*, 8938, DOI: 10.1063/1.468022Google Scholar29https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXjt1yitL8%253D&md5=86ccf94994d88b83e0eedafd6ca3ae97Analytical energy derivatives for ionized states described by the equation-of-motion coupled cluster methodStanton, John F.; Gauss, JuergenJournal of Chemical Physics (1994), 101 (10), 8938-44CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The theory for analytic energy derivs. of excited electronic states described by the equation-of-motion coupled cluster (EOM-CC) method has been generalized to treat cases in which ref. and final states differ in the no. of electrons. While this work specializes to the sector of Fock space that corresponds to ionization of the ref., the approach can be trivially modified for electron attached final states. Unlike traditional coupled cluster methods that are based on single determinant ref. functions, several electronic configurations are treated in a balanced way by EOM-CC. Therefore, this quantum chem. approach is appropriate for problems that involve important nondynamic electron correlation effects. Furthermore, a fully spin adapted treatment of doublet electronic states is guaranteed when a spin restricted closed shell ref. state is used-a desirable feature that is not easily achieved in std. coupled cluster approaches. The efficient implementation of analytic gradients reported here allows this variant of EOM-CC theory to be routinely applied to multidimensional potential energy surfaces for the first time. Use of the method is illustrated by an investigation of the formyloxyl radical (HCOO), which suffers from notorious symmetry breaking effects.**30**Pieniazek, P. A.; Bradforth, S. E.; Krylov, A. I. Charge localization and Jahn–Teller distortions in the benzene dimer cation.*J. Chem. Phys.*2008,*129*, 074104, DOI: 10.1063/1.2969107Google Scholar30https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXhtVamtL3N&md5=e905d21ec9bbe491d0b3909566a139beCharge localization and Jahn-Teller distortions in the benzene dimer cationPieniazek, Piotr A.; Bradforth, Stephen E.; Krylov, Anna I.Journal of Chemical Physics (2008), 129 (7), 074104/1-074104/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Jahn-Teller (JT) distortions and charge localization in the benzene dimer cation are analyzed using the equation-of-motion coupled cluster with single and double substitutions for ionization potential (EOM-IP-CCSD) method. Ionization of the dimer changes the bonding from noncovalent to covalent and induces significant geometrical distortions, e.g., shorter interfragment distance and JT displacements. Relaxation along interfragment coordinates lowers the energy of the t-shaped and displaced sandwich isomers by 0.07 and 0.23 eV, resp., whereas JT displacements result in addnl. 0.18 and 0.23 eV. Energetically, the effect of JT distortion on the dimer is similar to the monomer where JT relaxation lowers the energy by 0.18 eV. While the change in the interfragment distance has dramatic spectroscopic consequences, the JT distortion causes only a small perturbation in the electronic spectra. The two geometrical relaxations in the t-shaped isomer lead to opposing effects on hole localization. Intermol. relaxation leads to an increased delocalization, whereas JT ring distortion localizes the charge. In the sandwich isomers, breaking the symmetry by ring rotation does not induce considerable charge localization. The optimization and property calcns. were performed using a new implementation of EOM-IP-CCSD energies and gradients in the Q-CHEM electronic structure package. (c) 2008 American Institute of Physics.**31**Musiał, M.; Kucharski, S. A.; Bartlett, R. J. Equation-of-motion coupled cluster method with full inclusion of the connected triple excitations for ionized states: IP-EOM-CCSDT.*J. Chem. Phys.*2003,*118*, 1128, DOI: 10.1063/1.1527013Google Scholar31https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXhvFWgug%253D%253D&md5=9697f8157100ea77f8d6f614750c83d9Equation-of-motion coupled cluster method with full inclusion of the connected triple excitations for ionized states: IP-EOM-CCSDTMusial, Monika; Kucharski, Stanislaw A.; Bartlett, Rodney J.Journal of Chemical Physics (2003), 118 (3), 1128-1136CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The equation-of-motion (EOM) coupled cluster (CC) method with full inclusion of the connected triple excitations for ionization energies has been formulated and implemented. Using proper factorization of the three- and four-body parts of the effective Hamiltonian, an efficient computational procedure has been proposed for IP-EOM-CCSDT which at the EOM level requires no-higher-than nocc3nvir4 scaling. The method is calibrated by the evaluation of the valence vertical ionization potentials for CO, N2, and F2 mols. for several basis sets up to 160 basis functions. At the basis set limit, errors vary from 0.0 to 0.2 eV, compared to "exptl." vertical ionization potentials.**32**Nooijen, M.; Bartlett, R. J. Equation of motion coupled cluster method for electron attachment.*J. Chem. Phys.*1995,*102*, 3629, DOI: 10.1063/1.468592Google Scholar32https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXjvFGqu74%253D&md5=8ad0ce69f40ddcaa984fe7c304af5e28Equation of motion coupled cluster method for electron attachmentNooijen, Marcel; Bartlett, Rodney J.Journal of Chemical Physics (1995), 102 (9), 3629-47CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The electron attachment equation of motion coupled cluster (EA-EOMCC) method is derived which enables detn. of the various bound states of an (N+1)-electron system and the corresponding energy eigenvalues relative to the energy of an N-electron CCSD ref. state. Detailed working equations for the EA-EOMCC method are derived using diagrammatic techniques for both closed-shell and open-shell CCSD ref. states based upon a single determinant. The EA-EOMCC method is applied to a variety of different problems, the main purpose being to establish its prospects and limitations. The results from EA-EOMCC calcns. are compared to other EOMCC approaches, starting from different ref. states, as well as other theor. methods and exptl. values, where available. We have investigated electron affinities for a wide selection of both closed-shell and open-shell systems. Excitation spectra of atoms and mols. with an odd no. of electrons are obtained, taking the closed-shell ground state of the ion as a ref. in the EA-EOMCC calcn. Finally we consider excitation spectra of some closed-shell systems, and find in particular that the electron attachment approach is capable of yielding accurate triplet excitation energies in an efficient way.**33**Musiał, M.; Bartlett, R. J. Equation-of-motion coupled cluster method with full inclusion of connected triple excitations for electron-attached states: EA-EOM-CCSDT.*J. Chem. Phys.*2003,*119*, 1901, DOI: 10.1063/1.1584657Google Scholar33https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXltlyqsrw%253D&md5=5c6a045a2bf6bf07f5d3674930f03e8cEquation-of-motion coupled cluster method with full inclusion of connected triple excitations for electron-attached states: EA-EOM-CCSDTMusial, Monika; Bartlett, Rodney J.Journal of Chemical Physics (2003), 119 (4), 1901-1908CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We extend the full triples equation-of-motion (EOM) coupled cluster (CC) method to electron attached states. Proper factorization of the three- and four-body parts of the effective Hamiltonian makes it possible to achieve for the EA-EOM part a scaling no higher than nocc2nvir5. The method is calibrated by the evaluation of the valence vertical electron affinities for the C2 and O3 mols. for several basis sets up to 160 basis functions. For C2, EA-EOM-CCSDT gives 3.24 eV at the extrapolated basis limit, while the exptl. adiabatic EA is equal to 3.27 ± 0.008 eV. For O3 the agreement is ∼1.9 eV compared to an adiabatic value of 2.1 eV.**34**Schirmer, J. Beyond the random-phase approximation: A new approximation scheme for the polarization propagator.*Phys. Rev. A*1982,*26*, 2395, DOI: 10.1103/PhysRevA.26.2395Google Scholar34https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL38Xmt1eisbk%253D&md5=491a01718b7aeaff6b8eec59b5a201d9Beyond the random-phase approximation: A new approximation scheme for the polarization propagatorSchirmer, JochenPhysical Review A: Atomic, Molecular, and Optical Physics (1982), 26 (5), 2395-416CODEN: PLRAAN; ISSN:0556-2791.Within the framework of the many-body Green's-function method, a new approach is given to the polarization propagator for finite Fermi systems. This approach makes explicit use of the diagrammatic perturbation expansion for the polarization propagator, and reformulates the exact summation in terms of a simple algebraic scheme, referred to as the algebraic diagrammatic construction (ADC). The ADC defines in a natural way a set of approxn. schemes (nth-order ADC schemes) which represent infinite partial summations exact up to nth order of perturbation theory. In contrast to the random-phase-approxn. (RPA)-like schemes, the corresponding math. procedures are essentially Hermitian eigenvalue problems in limited configuration spaces of unperturbed excited configurations. Explicit equations for the 1st- and 2nd-order ADC schemes are derived. These schemes are thoroughly discussed and compared with the Tamm-Dancoff approxn. and RPA schemes.**35**Schirmer, J.; Cederbaum, L. S.; Walter, O. New approach to the one-particle Green’s function for finite Fermi systems.*Phys. Rev. A*1983,*28*, 1237, DOI: 10.1103/PhysRevA.28.1237Google Scholar35https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3sXlt1ymsrk%253D&md5=98a05eac7efa66da70a417ae38e94f7fNew approach to the one-particle Green's function for finite Fermi systemsSchirmer, J.; Cederbaum, L. S.; Walter, O.Physical Review A: Atomic, Molecular, and Optical Physics (1983), 28 (3), 1237-59CODEN: PLRAAN; ISSN:0556-2791.A new approach to the one-particle Green's functions G for finite electronic systems is presented. This approach is based on the diagrammatic perturbation expansions of the Green's function and of the dynamic self-energy part M related to G via the Dyson equation. The exact summation of the latter expansion is reformulated in terms of a simple algebraic form referred to as algebraic diagrammatic construction (ADC). The ADC defines in a systematic way a set of approxn. schemes (nth-order ADC schemes) that represent infinite partial summations for M and (via the Dyson equation) for G being complete through nth order of perturbation theory. The corresponding math. procedures are essentially Hermitian eigenvalue problems in restricted configuration spaces of unperturbed ionic configurations. Explicit equations for the second-, third-, and fourth-order ADC schemes are derived and analyzed.**36**Schirmer, J.; Trofimov, A. B.; Stelter, G. A non-Dyson third-order approximation scheme for the electron propagator.*J. Chem. Phys.*1998,*109*, 4734, DOI: 10.1063/1.477085Google Scholar36https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXlslyrs70%253D&md5=84afab4f57ce9e4a20c40730b51f0b51A non-Dyson third-order approximation scheme for the electron propagatorSchirmer, J.; Trofimov, A. B.; Stelter, G.Journal of Chemical Physics (1998), 109 (12), 4734-4744CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An efficient third-order propagator method to compute ionization potentials and electron affinities of atoms and mols. is presented. The development is based on the algebraic diagrammatic construction (ADC) representing a specific reformulation of the diagrammatic perturbation series of the electron propagator G(ω). In contrast with previous approxn. schemes, relying on the Dyson equation and approxns. for the self-energy part, the ADC procedure here is applied directly to the (N.-+.1)-electron parts G-(ω) and G+(ω), resp., of the electron propagator. This leads to decoupled secular equations for the ionization energies ((N-1)-electron part) and electron affinities ((N+1)-electron part), resp. In comparison with the Dyson-type approach, there is a substantial redn. of the secular matrix dimension opposed by a small addnl. expense in computing some second- and third-order contributions to the secular matrix elements. The relationship of the non-Dyson ADC(3) method to coupled cluster methods is outlined.**37**Trofimov, A. B.; Schirmer, J. Molecular ionization energies and ground- and ionic-state properties using a non-Dyson electron propagator approach.*J. Chem. Phys.*2005,*123*, 144115, DOI: 10.1063/1.2047550Google Scholar37https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXhtFCmsrrM&md5=cb9ab355f330f71830544839568f2a82Molecular ionization energies and ground- and ionic-state properties using a non-Dyson electron propagator approachTrofimov, A. B.; Schirmer, J.Journal of Chemical Physics (2005), 123 (14), 144115/1-144115/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An earlier proposed propagator method for the treatment of mol. ionization is tested in first applications. The method referred to as the non-Dyson third-order algebraic-diagrammatic construction [nD-ADC(3)] approxn. for the electron propagator represents a computationally promising alternative to the existing Dyson ADC(3) method. The advantage of the nD-ADC(3) scheme is that the (N±1)-electronic parts of the one-particle Green's function are decoupled from each other and the corresponding equations can be solved sep. For a test of the method the nD-ADC(3) results for the vertical ionization transitions in C2H4, CO, CS, F2, H2CO, H2O, HF, N2, and Ne are compared with available exptl. and theor. data including results of full CI (FCI) and coupled cluster computations. The mean error of the nD-ADC(3) ionization energies relative to the exptl. and FCI results is about 0.2 eV. The nD-ADC(3) method, scaling as n5 with the no. of orbitals, requires the soln. of a relatively simple Hermitian eigenvalue problem. The method renders access to ground-state properties such as dipole moments. Moreover, also one-electron properties of (N±1) electron states can now be studied as a consequence of a specific intermediate-state representation (ISR) formulation of the nD-ADC approach. Corresponding second-order ISR equations are presented.**38**Dempwolff, A. L.; Schneider, M.; Hodecker, M.; Dreuw, A. Efficient implementation of the non-Dyson third-order algebraic diagrammatic construction approximation for the electron propagator for closed- and open-shell molecules.*J. Chem. Phys.*2019,*150*, 064108, DOI: 10.1063/1.5081674Google Scholar38https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXjtFGnsrc%253D&md5=8cfceddce18399114cc4ec211aa488beEfficient implementation of the non-Dyson third-order algebraic diagrammatic construction approximation for the electron propagator for closed- and open-shell moleculesDempwolff, Adrian L.; Schneider, Matthias; Hodecker, Manuel; Dreuw, AndreasJournal of Chemical Physics (2019), 150 (6), 064108/1-064108/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A novel efficient implementation of the non-Dyson algebraic diagrammatic construction (ADC) scheme of the (N - 1)-part of the electron propagator up to third order of perturbation theory is presented. Due to the underlying spin-orbital formulation, for the first time, the computation of ionization potentials of open-shell radicals is thus possible via non-Dyson ADC schemes. Thorough evaluation of the accuracy, applicability, and capabilities of the new method reveals a mean error of 0.15 eV for closed- as well as open-shell atoms and mols. (c) 2019 American Institute of Physics.**39**Banerjee, S.; Sokolov, A. Y. Third-order algebraic diagrammatic construction theory for electron attachment and ionization energies: Conventional and Green’s function implementation.*J. Chem. Phys.*2019,*151*, 224112, DOI: 10.1063/1.5131771Google Scholar39https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXitlyhs7vO&md5=4634701de50853ee43f0f8bf4d137e94Third-order algebraic diagrammatic construction theory for electron attachment and ionization energies: Conventional and Green's function implementationBanerjee, Samragni; Sokolov, Alexander Yu.Journal of Chemical Physics (2019), 151 (22), 224112/1-224112/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present implementation of second- and third-order algebraic diagrammatic construction (ADC) theory for efficient and accurate computations of mol. electron affinities (EA), ionization potentials (IP), and densities of states [EA-/IP-ADC(n), n = 2, 3]. Our work utilizes the non-Dyson formulation of ADC for the single-particle propagator and reports working equations and benchmark results for the EA-ADC(2) and EA-ADC(3) approxns. We describe two algorithms for solving EA-/IP-ADC equations: (i) conventional algorithm that uses iterative diagonalization techniques to compute low-energy EA, IP, and d. of states and (ii) Green's function algorithm (GF-ADC) that solves a system of linear equations to compute d. of states directly for a specified spectral region. To assess the accuracy of EA-ADC(2) and EA-ADC(3), we benchmark their performance for a set of atoms, small mols., and five DNA/RNA nucleobases. As our next step, we demonstrate the efficiency of our GF-ADC implementation by computing core-level K-, L-, and M-shell ionization energies of a zinc atom without introducing the core-valence sepn. approxn. Finally, we use EA- and IP-ADC methods to compute the bandgaps of equally spaced hydrogen chains Hn with n up to 150, providing their ests. near thermodn. limit. Our results demonstrate that EA-/IP-ADC(n) (n = 2, 3) methods are efficient and accurate alternatives to widely used electronic structure methods for simulations of electron attachment and ionization properties. (c) 2019 American Institute of Physics.**40**Hodecker, M.; Dempwolff, A. L.; Schirmer, J.; Dreuw, A. Theoretical analysis and comparison of unitary coupled-cluster and algebraic-diagrammatic construction methods for ionization.*J. Chem. Phys.*2022,*156*, 074104, DOI: 10.1063/5.0070967Google Scholar40https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XktVGgtb0%253D&md5=41cafc86bcb38668ee0419f245f6a837Theoretical analysis and comparison of unitary coupled-cluster and algebraic-diagrammatic construction methods for ionizationHodecker, Manuel; Dempwolff, Adrian L.; Schirmer, Jochen; Dreuw, AndreasJournal of Chemical Physics (2022), 156 (7), 074104CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)This article describes a novel approach for the calcn. of ionization potentials (IPs), or, more generally, electron-detachment energies, based on a unitary coupled-cluster (UCC) parameterization of the ground-state wave function. Explicit working equations for a scheme referred to as IP-UCC3 are given, providing electron-detachment energies and spectroscopic amplitudes of electron-detached states dominated by one-hole excitations correct through third order. In the derivation, an expansion of the UCC transformed Hamiltonian involving Bernoulli nos. as expansion coeffs. is employed. Both the secular matrix and the effective transition moments are shown to be essentially equiv. to the strict third-order algebraic-diagrammatic construction scheme for the electron propagator (IP-ADC). Interestingly, due to the Bernoulli expansion, neglecting triple substitutions in the UCC expansion manifold does not affect the third-order consistency of the IP-UCC effective transition moments. Finally, the equivalence between ADC and UCC excited-state schemes is shown to not hold in fourth or higher order due to a different treatment of the correlated excited-state basis. (c) 2022 American Institute of Physics.**41**Dempwolff, A. L.; Hodecker, M.; Dreuw, A. Vertical ionization potential benchmark for unitary coupled-cluster and algebraic-diagrammatic construction methods.*J. Chem. Phys.*2022,*156*, 054114, DOI: 10.1063/5.0079047Google Scholar41https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XivV2isbY%253D&md5=10d48d54a39c35dc5a41c4dcf9edc85bVertical ionization potential benchmark for unitary coupled-cluster and algebraic-diagrammatic construction methodsDempwolff, Adrian L.; Hodecker, Manuel; Dreuw, AndreasJournal of Chemical Physics (2022), 156 (5), 054114CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The performance of several methods for the calcn. of vertical ionization potentials (IPs) or, more generally, electron-detachment energies based on unitary coupled-cluster (UCC) theory and the algebraic-diagrammatic construction (ADC) scheme is evaluated with respect to benchmark data computed at the level of equation-of-motion coupled-cluster theory, including single, double, and triple excitations (IP-EOM-CCSDT). Based on a statistical evaluation of about 200 electron-detached states of 41 mols., the second-order methods IP-ADC(2) and IP-UCC2 show modest accuracies with IP-EOM-CCSDT as ref., exposing a mean signed error and a std. deviation of the error of -0.54 ± 0.50 and -0.49 ± 0.54 eV, resp., accompanied by a mean abs. error (MAE) of 0.61 and 0.58 eV, resp. The strict third-order IP-ADC method demonstrates an accuracy of 0.26 ± 0.35 eV (MAE = 0.35 eV), while the IP-UCC3 method is slightly more accurate with 0.24 ± 0.26 eV (MAE = 0.29 eV). Employing the static self-energy computed using the Dyson expansion method (DEM) improves the IP-ADC(3) performance to 0.27 ± 0.28 eV, with the mean abs. error of this method being 0.32 eV. However, employing the simpler improved fourth-order scheme Σ(4+) for the static self-energy provides almost identical results as the DEM. Based on the quality of the present benchmark results, it therefore appears not necessary to use the computationally more demanding DEM. (c) 2022 American Institute of Physics.**42**Dempwolff, A. L.; Paul, A. C.; Belogolova, A. M.; Trofimov, A. B.; Dreuw, A. Intermediate state representation approach to physical properties of molecular electron-detached states. I. Theory and implementation.*J. Chem. Phys.*2020,*152*, 024113, DOI: 10.1063/1.5137792Google Scholar42https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXpt1KntQ%253D%253D&md5=5643d2d8b6776767051d1e2952f4d973Intermediate state representation approach to physical properties of molecular electron-detached states. I. Theory and implementationDempwolff, Adrian L.; Paul, Alexander C.; Belogolova, Alexandra M.; Trofimov, Alexander B.; Dreuw, AndreasJournal of Chemical Physics (2020), 152 (2), 024113CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The third-order non-Dyson algebraic-diagrammatic construction approach to the electron propagator [IP-ADC(3)] is extended using the intermediate state representation (ISR) formalism, allowing the wave functions and properties of mol. states with detached electron to be studied. The second-order ISR equations [ISR(2)] for the one-particle (transition) d. matrix have been derived and implemented in the Q-CHEM program. The approach is completely general and enables evaluation of arbitrary one-particle operators and interpretation of electron detachment processes in terms of d.-based quantities. The IP-ADC(3)/ISR(2) equations were implemented for Ŝz-adapted intermediate states, allowing open-shell mols. to be studied using UHF refs. As a first test for computations of ground state properties, dipole moments of various closed- and open-shell mols. have been computed by means of electron detachment from the corresponding anions. The results are in good agreement with exptl. data. The potential of IP-ADC(3)/ISR(2) for the interpretation of photoelectron spectra is demonstrated for the galvinoxyl free radical. (c) 2020 American Institute of Physics.**43**Dempwolff, A. L.; Paul, A. C.; Belogolova, A. M.; Trofimov, A. B.; Dreuw, A. Intermediate state representation approach to physical properties of molecular electron-detached states. II. Benchmarking.*J. Chem. Phys.*2020,*152*, 024125, DOI: 10.1063/1.5137794Google Scholar43https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXht1Gis74%253D&md5=4efd24e4bf7a51c5b5e77a48abe7e70fIntermediate state representation approach to physical properties of molecular electron-detached states. II. BenchmarkingDempwolff, Adrian L.; Paul, Alexander C.; Belogolova, Alexandra M.; Trofimov, Alexander B.; Dreuw, AndreasJournal of Chemical Physics (2020), 152 (2), 024125CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The third-order algebraic-diagrammatic construction method for studies of electron detachment processes within the electron propagator framework [IP-ADC(3)] was extended to treat the properties of mol. states with a detached electron using the intermediate state representation (ISR) formalism. The second-order ISR(2) equations for the one-particle (transition) d. matrix have been derived and implemented as an extension of the IP-(U)ADC(3) method available in the Q-CHEM program. As a first systematic test of the present IP-(U)ADC(3)/ISR(2) method, the dipole moments of various electronic states of closed- and open-shell mols. have been computed and compared to full CI (FCI) results. The present study employing FCI benchmarks also provides the first rigorous ests. for the accuracy of electron detachment energies obtained using the IP-ADC(3) method. (c) 2020 American Institute of Physics.**44**Dempwolff, A. L.; Belogolova, A. M.; Trofimov, A. B.; Dreuw, A. Intermediate state representation approach to physical properties of molecular electron-attached states: Theory, implementation, and benchmarking.*J. Chem. Phys.*2021,*154*, 104117, DOI: 10.1063/5.0043337Google Scholar44https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXmtlGjtLo%253D&md5=66242d27d374e13ea6608cfc2c66eee8Intermediate state representation approach to physical properties of molecular electron-attached states: Theory, implementation, and benchmarkingDempwolff, Adrian L.; Belogolova, Alexandra M.; Trofimov, Alexander B.; Dreuw, AndreasJournal of Chemical Physics (2021), 154 (10), 104117CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Computational schemes for comprehensive studies of mol. electron-attached states and the calcn. of electron affinities (EAs) are formulated and implemented employing the intermediate state representation (ISR) formalism and the algebraic-diagrammatic construction approxn. for the electron propagator (EA-ADC). These EA-ADC(n)/ISR(m) schemes allow for a consistent treatment of not only electron affinities and pole strengths up to third-order of perturbation theory (n = 3) but also one-electron properties of electron-attached states up to second order (m = 2). The EA-ADC/ISR equations were implemented in the Q-CHEM program for Ŝz-adapted intermediate states, allowing also open-shell systems to be studied using UHF refs. For benchmarking of the EA-(U)ADC/ISR schemes, EAs and dipole moments of various electron-attached states of small closed- and open-shell mols. were computed and compared to full CI data. As an illustrative example, EA-ADC(3)/ISR(2) has been applied to the thymine-thymine (6-4) DNA photolesion. (c) 2021 American Institute of Physics.**45**Ortiz, J. V. Electron propagator theory: an approach to prediction and interpretation in quantum chemistry.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2013,*3*, 123, DOI: 10.1002/wcms.1116Google Scholar45https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXmtFaiurY%253D&md5=de0c9211fddba230feff0336718bd7a1Electron propagator theory: an approach to prediction and interpretation in quantum chemistryOrtiz, Joseph VincentWiley Interdisciplinary Reviews: Computational Molecular Science (2013), 3 (2), 123-142CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)A review. Electron propagator theory provides a practical means of calcg. electron binding energies, Dyson orbitals, and ground-state properties from first principles. This approach to ab initio electronic structure theory also facilitates the interpretation of its quant. predictions in terms of concepts that closely resemble those of one-electron theories. An explanation of the phys. meaning of the electron propagator's poles and residues is followed by a discussion of its couplings to more complicated propagators. These relationships are exploited in superoperator theory and lead to a compact form of the electron propagator that is derived by matrix partitioning. Expressions for ref.-state properties, relationships to the extended Koopmans's theorem technique for evaluating electron binding energies, and connections between Dyson orbitals and transition probabilities follow from this discussion. The inverse form of the Dyson equation for the electron propagator leads to a strategy for obtaining electron binding energies and Dyson orbitals that generalizes the Hartree-Fock equations through the introduction of the self-energy operator. All relaxation and correlation effects reside in this operator, which has an energy-dependent, nonlocal form that is systematically improvable. Perturbative arguments produce several, convenient (e.g. partial third order, outer valence Green's function, and second-order, transition-operator) approxns. for the evaluation of valence ionization energies, electron affinities, and core ionization energies. Renormalized approaches based on Hartree-Fock or approx. Brueckner orbitals are employed when correlation effects become qual. important. Ref.-state total energies based on contour integrals in the complex plane and gradients of electron binding energies enable exploration of final-state potential energy surfaces.**46**Ortiz, J. V. Partial third-order quasiparticle theory: Comparisons for closed-shell ionization energies and an application to the borazine photoelectron spectrum.*J. Chem. Phys.*1996,*104*, 7599, DOI: 10.1063/1.471468Google Scholar46https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28XivFGrt7k%253D&md5=bdaf740c62b87ccd783585b50ae231fbPartial third-order quasiparticle theory: comparisons for closed-shell ionization energies and an application to the Borazine photoelectron spectrumOrtiz, J. V.Journal of Chemical Physics (1996), 104 (19), 7599-7605CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Valence ionization energies of a set closed-shell mols. calcd. in a partial 3rd-order (P3) quasiparticle approxn. of the electron propagator have an av. abs. error of 0.19 eV. Diagonal elements of the self-energy matrix include all 2nd-order and some 3rd-order self-energy diagrams. Because of its 5th power dependence on basis set size and its independence from electron repulsion integrals with four virtual indexes, this method has considerable potential for large mols. Formal and computational comparisons with other electron propagator techniques illustrate the advantages of the P3 procedure. Addnl. applications to benzene and borazine display the efficacy of the P3 propagator in assigning photoelectron spectra. In the borazine spectrum, 2E' and 2A2' final states are responsible for an obsd. feature at 14.76 eV. Another peak at 17.47 eV is assigned to a 2E' final state.**47**Corzo, H. H.; Galano, A.; Dolgounitcheva, O.; Zakrzewski, V. G.; Ortiz, J. V. NR2 and P3+: Accurate, Efficient Electron-Propagator Methods for Calculating Valence, Vertical Ionization Energies of Closed-Shell Molecules.*J. Phys. Chem. A*2015,*119*, 8813, DOI: 10.1021/acs.jpca.5b00942Google Scholar47https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXht1KmsbvJ&md5=a47a13b49be29fb616c6b71fdddd42c1NR2 and P3+: Accurate, Efficient Electron-Propagator Methods for Calculating Valence, Vertical Ionization Energies of Closed-Shell MoleculesCorzo, H. H.; Galano, Annia; Dolgounitcheva, O.; Zakrzewski, V. G.; Ortiz, J. V.Journal of Physical Chemistry A (2015), 119 (33), 8813-8821CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)Two accurate and computationally efficient electron-propagator (EP) methods for calcg. the valence, vertical ionization energies (VIEs) of closed-shell mols. have been identified through comparisons with related approxns. VIEs of a representative set of closed-shell mols. were calcd. with EP methods using 10 basis sets. The most easily executed method, the diagonal, second-order (D2) EP approxn., produces results that steadily rise as basis sets are improved toward values based on extrapolated coupled-cluster singles and doubles plus perturbative triples calcns., but its mean errors remain unacceptably large. The outer valence Green function, partial third-order and renormalized partial third-order methods (P3+), which employ the diagonal self-energy approxn., produce markedly better results but have a greater tendency to overestimate VIEs with larger basis sets. The best combination of accuracy and efficiency with a diagonal self-energy matrix is the P3+ approxn., which exhibits the best trends with respect to basis-set satn. Several renormalized methods with more flexible nondiagonal self-energies also have been examd.: the two-particle, one-hole Tamm-Dancoff approxn. (2ph-TDA), the third-order algebraic diagrammatic construction or ADC(3), the renormalized third-order (3+) method, and the nondiagonal second-order renormalized (NR2) approxn. Like D2, 2ph-TDA produces steady improvements with basis set augmentation, but its av. errors are too large. Errors obtained with 3+ and ADC(3) are smaller on av. than those of 2ph-TDA. These methods also have a greater tendency to overestimate VIEs with larger basis sets. The smallest av. errors occur for the NR2 approxn.; these errors decrease steadily with basis augmentations. As basis sets approach satn., NR2 becomes the most accurate and efficient method with a nondiagonal self-energy.**48**Ortiz, J. V. An efficient, renormalized self-energy for calculating the electron binding energies of closed-shell molecules and anions.*Int. J. Quantum Chem.*2005,*105*, 803, DOI: 10.1002/qua.20664Google Scholar48https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXht1SiurzI&md5=a6ef2e2bf7f9dee84c7b36ffec7b1002An efficient, renormalized self-energy for calculating the electron binding energies of closed-shell molecules and anionsOrtiz, J. V.International Journal of Quantum Chemistry (2005), 105 (6), 803-808CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)The energy-dependent, nonlocal correlation potential known as the self-energy that appears in the Dyson equation has a pole and residue structure that enables renormalizations of its low-order, perturbative contributions to be estd. The partial third-order (P3) approxn. has been extensively applied to the ionization energies of closed-shell, org. mols. and is the most successful example of a low-order, self-energy method. A renormalization based on the P3 self-energy ests. higher-order contributions by scaling low-order terms that chiefly describe final-state relaxation. The resulting P3 + self-energy retains the accuracy and efficiency of the P3 approxn., but also improves the latter method's performance with respect to the calcn. of anion electron detachment energies without the introduction of adjustable parameters. An application to an anion that previously has yielded only to more intricate treatments of electron correlation demonstrates the power of this simple, new approxn.**49**Ortiz, J. V. A nondiagonal, renormalized extension of partial third-order quasiparticle theory: Comparisons for closed-shell ionization energies.*J. Chem. Phys.*1998,*108*, 1008, DOI: 10.1063/1.475463Google Scholar49https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXitVSgsA%253D%253D&md5=5bffa8187f2aceebad008ded498bfc74A nondiagonal, renormalized extension of partial third-order quasiparticle theory: comparisons for closed-shell ionization energiesOrtiz, J. V.Journal of Chemical Physics (1998), 108 (3), 1008-1014CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Valence ionization energies of a set closed-shell mols. calcd. in a nondiagonal, renormalized approxn. of the electron propagator have an av. abs. error of 0.17 eV. This procedure extends the partial third order, quasiparticle approxn. of J. Chem. Phys. 104, 7599 (1996) that has proven successful in many applications. Elements of the self-energy matrix include all second-order and many higher-order terms. Because of its fifth power dependence on basis set size and its independence from electron repulsion integrals with four virtual orbital indexes, this method has considerable promise for large mols. Formal and computational comparisons with renormalized electron propagator techniques that are complete through third-order illustrate the advantages of this procedure.**50**Opoku, E.; Pawłowski, F.; Ortiz, J. V. A new generation of diagonal self-energies for the calculation of electron removal energies.*J. Chem. Phys.*2021,*155*, 204107, DOI: 10.1063/5.0070849Google Scholar50https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXis1KisLrE&md5=6395803c7d665e06864cc840e89c5452A new generation of diagonal self-energies for the calculation of electron removal energiesOpoku, Ernest; Pawlowski, Filip; Ortiz, J. V.Journal of Chemical Physics (2021), 155 (20), 204107CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A new generation of diagonal self-energy approxns. in ab initio electron propagator theory for the calcn. of electron removal energies of mols. and mol. ions has been derived from an intermediately normalized, Hermitized super-operator metric. These methods and widely used antecedents such as the outer valence Green's function and the approx. renormalized partial third order method are tested with respect to a dataset of vertical ionization energies generated with a valence, triple-ζ, correlation-consistent basis set and a converged series of many-body calcns. whose accuracy approaches that of full CI. Several modifications of the diagonal second-order self-energy, a version of G0W0 theory based on Tamm-Dancoff excitations and several non-diagonal self-energies are also included in the tests. All new methods employ canonical Hartree-Fock orbitals. No adjustable or empirical parameters appear. A hierarchy of methods with optimal accuracy for a given level of computational efficiency is established. Several widely used diagonal self-energy methods are rendered obsolete by the new hierarchy whose members, in order of increasing accuracy, are (1) the opposite-spin non-Dyson diagonal second-order or os-nD-D2, (2) the approx. renormalized third-order quasiparticle or Q3+ , (3) the renormalized third-order quasiparticle or RQ3, (4) the approx. renormalized linear third-order or L3+ , and (5) the renormalized linear third-order or RL3 self-energies. (c) 2021 American Institute of Physics.**51**Opoku, E.; Pawłowski, F.; Ortiz, J. V. Electron Propagator Theory of Vertical Electron Detachment Energies of Anions: Benchmarks and Applications to Nucleotides.*J. Phys. Chem. A*2023,*127*, 1085, DOI: 10.1021/acs.jpca.2c08372Google Scholar51https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3sXhtlGgu78%253D&md5=2529454b60cc520fbd4334e1529b5e7dElectron Propagator Theory of Vertical Electron Detachment Energies of Anions: Benchmarks and Applications to NucleotidesOpoku, Ernest; Pawlowski, Filip; Ortiz, J. V.Journal of Physical Chemistry A (2023), 127 (4), 1085-1101CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)A new generation of ab initio electron-propagator self-energy approxns. that is free of adjustable parameters is tested on a benchmark set of 55 vertical electron detachment energies of closed-shell anions. Comparisons with older self-energy approxns. indicate that several new methods that make the diagonal self-energy approxn. in the canonical Hartree-Fock orbital basis provide superior accuracy and computational efficiency. These methods, their acronyms, mean abs. errors (in eV) and arithmetic bottlenecks expressed in terms of occupied (O) and virtual (V) orbitals are the opposite-spin, non-Dyson, diagonal second-order method (os-nD-D2, 0.2, OV2), the approx. renormalized quasiparticle third-order method (Q3+ , 0.15, O2V3) and the approx. renormalized, non-Dyson, linear, third-order method (nD-L3+ , 0.1, OV4). The BD-T1 (Brueckner Doubles with Triple field operators) nondiagonal electron-propagator method provides such close agreement with coupled-cluster single, double and perturbative triple replacement total energy differences that it may be used as an alternative means of obtaining std. data. The new methods with diagonal self-energy matrixes are the foundation of a composite procedure for estg. basis-set effects. This model produces accurate predictions and clear interpretations based on Dyson orbitals for the photoelectron spectra of the nucleotides found in DNA.**52**Gilbert, A. T. B.; Besley, N. A.; Gill, P. M. W. Self-Consistent Field Calculations of Excited States Using the Maximum Overlap Method (MOM).*J. Phys. Chem. A*2008,*112*, 13164, DOI: 10.1021/jp801738fGoogle Scholar52https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXhtValurbL&md5=3baaf7b15c1c6fcd86bc3c071deacfadSelf-Consistent Field Calculations of Excited States Using the Maximum Overlap Method (MOM)Gilbert, Andrew T. B.; Besley, Nicholas A.; Gill, Peter M. W.Journal of Physical Chemistry A (2008), 112 (50), 13164-13171CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)We present a simple algorithm, which we call the max. overlap method (MOM), for finding excited-state solns. to SCF equations. Instead of using the aufbau principle, the algorithm maximizes the overlap between the occupied orbitals on successive SCF iterations. This prevents variational collapse to the ground state and guides the SCF process toward the nearest, rather than the lowest energy, soln. The resulting excited-state solns. can be treated in the same way as the ground-state soln. and, in particular, derivs. of excited-state energies can be computed using ground-state code. We assess the performance of our method by applying it to a variety of excited-state problems including the calcn. of excitation energies, charge-transfer states, and excited-state properties.**53**Bagus, P. S. Self-Consistent-Field Wave Functions for Hole States of Some Ne-Like and Ar-Like Ions.*Phys. Rev.*1965,*139*, A619, DOI: 10.1103/PhysRev.139.A619Google Scholar53https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaF2MXktlGgu7w%253D&md5=c09be4ef8d7fb5554f0bb12eba07b08aSelf-consistent-field wave functions for hole states of some Ne-like and Ar-like ionsBagus, P. S.Physical Review (1965), 139 (3A), 619-34CODEN: PHRVAO; ISSN:0031-899X.Analytic self-consistent-field (SCF) wave functions were computed for the ground states of the closed-shell at. systems F-, Ne, and Na+ and Cl-, Ar, and K+; and for those ground and excited states of the open-shell systems which are obtained by removing a single electron from any one of the occupied shells of these closed-shell systems. Details of the calcn. of the functions are presented, with emphasis on a justification of the procedures used for the calcns. for excited states. A high accuracy is obtained; the calcns. for the closed-shell systems give the most accurate analytic SCF wave functions which have yet been reported. Ionization potentials are compared with exptl. values. Computed ionization potentials for the removal of a 2s electron from Cl-, Ar, and K+, for which no direct exptl. data are available, are estd. to be accurate to 1%. The removal of an electron from the outermost s shell increases the correlation energy, in contradiction to the predictions of a recently proposed semi- empirical scheme for estg. the correlation energy. For example, the magnitude of the correlation energy of the lowest 2S state of Ar+ is ∼4 ev. greater than the magnitude of the correlation energy of neutral Ar. The effect of the nonzero off-diagonal Lagrangian multipliers is important for the inner shell hole states.**54**Triguero, L.; Pettersson, L. G. M.; Ågren, H. Calculations of near-edge x-ray-absorption spectra of gas-phase and chemisorbed molecules by means of density-functional and transition-potential theory.*Phys. Rev. B*1998,*58*, 8097, DOI: 10.1103/PhysRevB.58.8097Google Scholar54https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXmtVCnt7k%253D&md5=9b70092828315cc859abab4d13173b5bCalculations of near-edge x-ray-absorption spectra of gas-phase and chemisorbed molecules by means of density-functional and transition-potential theoryTriguero, L.; Pettersson, L. G. M.; Agren, H.Physical Review B: Condensed Matter and Materials Physics (1998), 58 (12), 8097-8110CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)The authors explore the utility of d.-functional theory (DFT) in conjunction with the transition-potential (TP) method to simulate x-ray-absorption spectra. Calcns. on a set of small C-contg. mols. and chemisorbed species show that this provides a viable option for obtaining excitation energies and oscillator strengths close to the exptl. accuracy of core-valence transitions. Systematic variations in energy positions and intensities of the different spectra in the test series were studied, and comparison is made with respect to the static exchange-, SCF, and explicit electron-correlation methods. The choice between std. exchange-correlation functionals is of little consequence for the valence resonant, here π*, parts of the x-ray-absorption spectra, while the long-range behavior of presently available functionals is found not to be completely satisfactory for Rydberg-like transitions. Implementing a basis set augmentation technique, DFT methods still account well for most of the salient features in the near-edge x-ray-absorption spectra, save for the multielectron transitions in the near continuum, and for some loss of Rydberg structure. For clusters modeling surface adsorbates, the DFT transition potential method reproduces well the spectral compression and intensity redn. for the valence level absorption compared to the free phase, provided fairly large clusters are taken into account. While for near-edge x-ray-absorption fine-structure (NEXAFS) spectra of free mols. the DFT-TP and Hartree-Fock/static exchange methods have complementary advantages, the DFT-TP method is clearly to be preferred when using clusters to simulate NEXAFS spectra of surface adsorbates.**55**Lee, J.; Small, D. W.; Head-Gordon, M. Excited states via coupled cluster theory without equation-of-motion methods: Seeking higher roots with application to doubly excited states and double core hole states.*J. Chem. Phys.*2019,*151*, 214103, DOI: 10.1063/1.5128795Google Scholar55https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXit12iurvL&md5=df69beffae8f965202030d76eba217a0Excited states via coupled cluster theory without equation-of-motion methods: Seeking higher roots with application to doubly excited states and double core hole statesLee, Joonho; Small, David W.; Head-Gordon, MartinJournal of Chemical Physics (2019), 151 (21), 214103/1-214103/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In this work, we revisited the idea of using the coupled-cluster (CC) ground state formalism to target excited states. Our main focus was targeting doubly excited states and double core hole states. Typical equation-of-motion (EOM) approaches for obtaining these states struggle without higher-order excitations than doubles. We showed that by using a non-Aufbau determinant optimized via the max. overlap method, the CC ground state solver can target higher energy states. Furthermore, just with singles and doubles (i.e., CCSD), we demonstrated that the accuracy of ΔCCSD and ΔCCSD(T) (triples) far surpasses that of EOM-CCSD for doubly excited states. The accuracy of ΔCCSD(T) is nearly exact for doubly excited states considered in this work. For double core hole states, we used an improved ansatz for greater numerical stability by freezing core hole orbitals. The improved methods, core valence sepn. (CVS)-ΔCCSD and CVS-ΔCCSD(T), were applied to the calcn. of the double ionization potential of small mols. Even without relativistic corrections, we obsd. qual. accurate results with CVS-ΔCCSD and CVS-ΔCCSD(T). Remaining challenges in ΔCC include the description of open-shell singlet excited states with the single-ref. CC ground state formalism as well as excited states with genuine multireference character. The tools and intuition developed in this work may serve as a stepping stone toward directly targeting arbitrary excited states using ground state CC methods. (c) 2019 American Institute of Physics.**56**Meissner, L.; Balková, A.; Bartlett, R. J. Multiple solutions of the single-reference coupled-cluster method.*Chem. Phys. Lett.*1993,*212*, 177, DOI: 10.1016/0009-2614(93)87127-OGoogle Scholar56https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXmtFSqurc%253D&md5=0546e409176ee2281807fbdcbaad2b83Multiple solutions of the single-reference coupled-cluster methodMeissner, Leszek; Balkova, Anna; Bartlett, Rodney J.Chemical Physics Letters (1993), 212 (1-2), 177-84CODEN: CHPLBC; ISSN:0009-2614.The nonlinear coupled-cluster (CC) equations possess several solns. They describe various excited states as long as they contain a contribution from the ref. function. The authors study the addnl. solns. of the single-ref. CC equations and demonstrate that unlike the ground state, the approx. soln. for excited states in general does not satisfy the cluster condition nor the std. proof of extensivity. For such states that makes the CC approxn. poorer than that for the analogous CI.**57**Zheng, X.; Cheng, L. Performance of Delta-Coupled-Cluster Methods for Calculations of Core-Ionization Energies of First-Row Elements.*J. Chem. Theory Comput.*2019,*15*, 4945, DOI: 10.1021/acs.jctc.9b00568Google Scholar57https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhsV2jsbzO&md5=9552d88cc44e3b3aff173462e03ea4edPerformance of Delta-Coupled-Cluster Methods for Calculations of Core-Ionization Energies of First-Row ElementsZheng, Xuechen; Cheng, LanJournal of Chemical Theory and Computation (2019), 15 (9), 4945-4955CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A thorough study of the performance of delta-coupled-cluster (ΔCC) methods for calcns. of core-ionization energies for elements of the first long row of the periodic table is reported. Inspired by the core-valence sepn. (CVS) scheme in response theories, a simple CVS scheme of excluding the vacant core orbital from the CC treatment has been adopted to solve the convergence problem of the CC equations for core-ionized states. Dynamic correlation effects have been shown to make important contributions to the computed core-ionization energies, esp. to chem. shifts of these quantities. The max. abs. error (MaxAE) and std. deviation (SD) of delta-Hartree-Fock results for chem. shifts of core-ionization energies with respect to the corresponding exptl. values amt. to more than 1.7 and 0.6 eV, resp. In contrast, the inclusion of electron correlation in ΔCC singles and doubles augmented with a noniterative triples correction [ΔCCSD(T)] method significantly reduces the corresponding deviations to around 0.3 and 0.1 eV. With the consideration of basis set effects and the corrections to the CVS approxn., ΔCCSD(T) has been shown to provide highly accurate results for abs. values of core-ionization energies, with a MaxAE of 0.22 eV and SD of 0.13 eV. To further demonstrate the usefulness of ΔCCSD(T), calcns. of carbon K-edge ionization energies of Et trifluoroacetate, a mol. of significant interest to the study of X-ray spectroscopy and dynamics, are reported.**58**Hirata, S.; Hermes, M. R.; Simons, J.; Ortiz, J. V. General-Order Many-Body Green’s Function Method.*J. Chem. Theory Comput.*2015,*11*, 1595, DOI: 10.1021/acs.jctc.5b00005Google Scholar58https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXjtF2lsr0%253D&md5=220dfeed02c0268f49d7941bd2ad0e73General-Order Many-Body Green's Function MethodHirata, So; Hermes, Matthew R.; Simons, Jack; Ortiz, J. V.Journal of Chemical Theory and Computation (2015), 11 (4), 1595-1606CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Electron binding energies are evaluated as differences in total energy between the N- and (N ± 1)-electron systems calcd. by the nth-order Moller-Plesset perturbation (MPn) theory using the same set of orbitals. The MPn energies up to n = 30 are, in turn, obtained by the determinant-based method of Knowles et al. (Chem. Phys. Lett.1985, 113, 8-12). The zeroth- through third-order electron binding energies thus detd. agree with those obtained by solving the Dyson equation in the diagonal and frequency-independent approxns. of the self-energy. However, as n → ∞, they converge at the exact basis-set solns. from the Dyson equation with the exact self-energy, which is nondiagonal and frequency-dependent. This suggests that the MPn energy differences define an alternative diagrammatic expansion of Koopmans-like electron binding energies, which takes into account the perturbation corrections from the off-diagonal elements and frequency dependence of the irreducible self-energy. Our anal. shows that these corrections are included as semireducible and linked-disconnected diagrams, resp., which are also found in a perturbation expansion of the electron binding energies of the equation-of-motion coupled-cluster methods. The rate of convergence of the electron binding energies with respect to n and its acceleration by Pade approximants are also discussed.**59**Hirata, S.; Doran, A. E.; Knowles, P. J.; Ortiz, J. V. One-particle many-body Green’s function theory: Algebraic recursive definitions, linked-diagram theorem, irreducible-diagram theorem, and general-order algorithms.*J. Chem. Phys.*2017,*147*, 044108, DOI: 10.1063/1.4994837Google Scholar59https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXht1GjtbfO&md5=a3f66c1ad67ee4656e9010e7089656e8One-particle many-body Green's function theory: Algebraic recursive definitions, linked-diagram theorem, irreducible-diagram theorem, and general-order algorithmsHirata, So; Doran, Alexander E.; Knowles, Peter J.; Ortiz, J. V.Journal of Chemical Physics (2017), 147 (4), 044108/1-044108/31CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A thorough anal. and numerical characterization of the whole perturbation series of one-particle many-body Green's function (MBGF) theory is presented in a pedagogical manner. Three distinct but equiv. algebraic (first-quantized) recursive definitions of the perturbation series of the Green's function are derived, which can be combined with the well-known recursion for the self-energy. Six general-order algorithms of MBGF are developed, each implementing one of the three recursions, the ΔMPn method (where n is the perturbation order) [S. Hirata et al., J. Chem. Theory Comput. 11, 1595 (2015)], the automatic generation and interpretation of diagrams, or the numerical differentiation of the exact Green's function with a perturbation-scaled Hamiltonian. They all display the identical, nondivergent perturbation series except ΔMPn, which agrees with MBGF in the diagonal and frequency-independent approxns. at 1 ≤ n ≤ 3 but converges at the full-configuration-interaction (FCI) limit at n = ∞ (unless it diverges). Numerical data of the perturbation series are presented for Koopmans and non-Koopmans states to quantify the rate of convergence towards the FCI limit and the impact of the diagonal, frequency-independent, or ΔMPn approxn. The diagrammatic linkedness and thus size-consistency of the one-particle Green's function and self-energy are demonstrated at any perturbation order on the basis of the algebraic recursions in an entirely time-independent (frequency-domain) framework. The trimming of external lines in a one-particle Green's function to expose a self-energy diagram and the removal of reducible diagrams are also justified math. using the factorization theorem of Frantz and Mills. Equivalence of ΔMPn and MBGF in the diagonal and frequency-independent approxns. at 1 ≤ n ≤ 3 is algebraically proven, also ascribing the differences at n = 4 to the so-called semi-reducible and linked-disconnected diagrams. (c) 2017 American Institute of Physics.**60**Goerigk, L.; Hansen, A.; Bauer, C.; Ehrlich, S.; Najibi, A.; Grimme, S. A look at the density functional theory zoo with the advanced GMTKN55 database for general main group thermochemistry, kinetics and noncovalent interactions.*Phys. Chem. Chem. Phys.*2017,*19*, 32184, DOI: 10.1039/C7CP04913GGoogle Scholar60https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhslajtLnF&md5=f9393c9e3907336c4da053743797f8dfA look at the density functional theory zoo with the advanced GMTKN55 database for general main group thermochemistry, kinetics and noncovalent interactionsGoerigk, Lars; Hansen, Andreas; Bauer, Christoph; Ehrlich, Stephan; Najibi, Asim; Grimme, StefanPhysical Chemistry Chemical Physics (2017), 19 (48), 32184-32215CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)We present the GMTKN55 benchmark database for general main group thermochem., kinetics and noncovalent interactions. Compared to its popular predecessor GMTKN30, it allows assessment across a larger variety of chem. problems - with 13 new benchmark sets being presented for the first time - and it also provides ref. values of significantly higher quality for most sets. GMTKN55 comprises 1505 relative energies based on 2462 single-point calcns. and it is accessible to the user community via a dedicated website. Herein, we demonstrate the importance of better ref. values, and we re-emphasize the need for London-dispersion corrections in d. functional theory (DFT) treatments of thermochem. problems, including Minnesota methods. We assessed 217 variations of dispersion-cor. and -uncorrected d. functional approxns., and carried out a detailed anal. of 83 of them to identify robust and reliable approaches. Double-hybrid functionals are the most reliable approaches for thermochem. and noncovalent interactions, and they should be used whenever tech. feasible. These are, in particular, DSD-BLYP-D3(BJ), DSD-PBEP86-D3(BJ), and B2GPPLYP-D3(BJ). The best hybrids are ωB97X-V, M052X-D3(0), and ωB97X-D3, but we also recommend PW6B95-D3(BJ) as the best conventional global hybrid. At the meta-generalized-gradient (meta-GGA) level, the SCAN-D3(BJ) method can be recommended. Other meta-GGAs are outperformed by the GGA functionals revPBE-D3(BJ), B97-D3(BJ), and OLYP-D3(BJ). We note that many popular methods, such as B3LYP, are not part of our recommendations. In fact, with our results we hope to inspire a change in the user community's perception of common DFT methods. We also encourage method developers to use GMTKN55 for cross-validation studies of new methodologies.**61**Goerigk, L.; Grimme, S. A general database for main group thermochemistry, kinetics, and noncovalent interactions – Assessment of common and reparameterized (meta-)GGA density functionals.*J. Chem. Theory Comput.*2010,*6*, 107, DOI: 10.1021/ct900489gGoogle Scholar61https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhsVClurvM&md5=6452b32bf508de27fb37c221b8fdfdd4A General Database for Main Group Thermochemistry, Kinetics, and Noncovalent Interactions - Assessment of Common and Reparameterized (meta-)GGA Density FunctionalsGoerigk, Lars; Grimme, StefanJournal of Chemical Theory and Computation (2010), 6 (1), 107-126CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a quantum chem. benchmark database for general main group thermochem., kinetics, and noncovalent interactions (GMTKN24). It is an unprecedented compilation of 24 different, chem. relevant subsets that either are taken from already existing databases or are presented here for the first time. The complete set involves a total of 1.049 at. and mol. single point calcns. and comprises 731 data points (relative chem. energies) based on accurate theor. or exptl. ref. values. The usefulness of the GMTKN24 database is shown by applying common d. functionals on the (meta-)generalized gradient approxn. (GGA), hybrid-GGA, and double-hybrid-GGA levels to it, including an empirical London dispersion correction. Furthermore, we refitted the functional parameters of four (meta-)GGA functionals based on a fit set contg. 143 systems, comprising seven chem. different problems. Validation against the GMTKN24 and the mol. structure (bond lengths) databases shows that the reparameterization does not change bond lengths much, whereas the description of energetic properties is more prone to the parameters' values. The empirical dispersion correction also often improves for conventional thermodn. problems and makes a functional's performance more uniform over the entire database. The refitted functionals typically have a lower mean abs. deviation for the majority of subsets in the proposed GMTKN24 set. This, however, is also often accompanied at the expense of poor performance for a few other important subsets. Thus, creating a broadly applicable (and overall better) functional by just reparameterizing existing ones seems to be difficult. Nevertheless, this benchmark study reveals that a reoptimized (i.e., empirical) version of the TPSS-D functional (oTPSS-D) performs well for a variety of problems and may meet the stds. of an improved functional. We propose validation against this new compilation of benchmark sets as a definitive way to evaluate a new quantum chem. method's true performance.**62**Curtiss, L. A.; Raghavachari, K.; Trucks, G. W.; Pople, J. A. Gaussian-2 theory for molecular energies of first-and second-row compounds.*J. Chem. Phys.*1991,*94*, 7221, DOI: 10.1063/1.460205Google Scholar62https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3MXksFOlsr4%253D&md5=2de9eaceadf097004bde9659ee043f42Gaussian-2 theory for molecular energies of first- and second-row compoundsCurtiss, Larry A.; Raghavachari, Krishnan; Trucks, Gary W.; Pople, John A.Journal of Chemical Physics (1991), 94 (11), 7221-30CODEN: JCPSA6; ISSN:0021-9606.The Gaussian-2 theor. procedure (G2 theory), based on ab-initio MO theory, for calcn. of mol. energies (atomization energies, ionization potentials, electron affinities, and proton affinities) of compds. contg. first- (Li-F) and second-row atoms (Na-Cl) is presented. This new theor. procedure adds three features to G1 theory (P., et al., 1989; C., et al., 1990), including a correction for nonadditivity of diffuse-sp and 2df basis-set extensions, a basis-set extension contg. a third d-function on nonhydrogen atoms and a second p-function on hydrogen atoms, and a modification of the higher level correction. G2 theory is a significant improvement over G1 theory, because it eliminates a no. of deficiencies present in G1 theory. Of particular importance is the improvement in atomization energies of ionic mols. such as LiF and hydrides such as C2H6,NH3, N2H4, H2O2, and CH3SH. The av. abs. deviation from expt. of atomization energies of 39 first-row compds. is reduced from 1.42 to 0.92 kcal/mol. In addn., G2 theory gives improved performance for hypervalent species and electron affinities of second-row species (the av. deviation from expt. of electron affinities of second-row species is reduced from 1.94 to 1.08 kcal/mol). Finally, G2 atomization energies for another 43 mols., not previously studied with G1 theory, many of which have uncertain exptl. data, are presented and differences with expt. are assessed.**63**Śmiga, S.; Franck, O.; Mussard, B.; Buksztel, A.; Grabowski, I.; Luppi, E.; Toulouse, J. Self-consistent double-hybrid density-functional theory using the optimized-effective-potential method.*J. Chem. Phys.*2016,*145*, 144102, DOI: 10.1063/1.4964319Google Scholar63https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xhs1yjurzJ&md5=39d1fb62cea025accba2744dcb59e1f9Self-consistent double-hybrid density-functional theory using the optimized-effective-potential methodSmiga, Szymon; Franck, Odile; Mussard, Bastien; Buksztel, Adam; Grabowski, Ireneusz; Luppi, Eleonora; Toulouse, JulienJournal of Chemical Physics (2016), 145 (14), 144102/1-144102/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We introduce an orbital-optimized double-hybrid (DH) scheme using the optimized-effective-potential (OEP) method. The orbitals are optimized using a local potential corresponding to the complete exchange-correlation energy expression including the second-order Moller-Plesset correlation contribution. We have implemented a one-parameter version of this OEP-based self-consistent DH scheme using the BLYP d.-functional approxn. and compared it to the corresponding non-self-consistent DH scheme for calcns. on a few closed-shell atoms and mols. While the OEP-based self-consistency does not provide any improvement for the calcns. of ground-state total energies and ionization potentials, it does improve the accuracy of electron affinities and restores the meaning of the LUMO orbital energy as being connected to a neutral excitation energy. Moreover, the OEP-based self-consistent DH scheme provides reasonably accurate exchange-correlation potentials and correlated densities. (c) 2016 American Institute of Physics.**64**Śmiga, S.; Grabowski, I.; Witkowski, M.; Mussard, B.; Toulouse, J. Self-Consistent Range-Separated Density-Functional Theory with Second-Order Perturbative Correction via the Optimized-Effective-Potential Method.*J. Chem. Theory Comput.*2020,*16*, 211, DOI: 10.1021/acs.jctc.9b00807Google Scholar64https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BB3MfotVCrsA%253D%253D&md5=e2e43924a3c8e3a5668b8e023cafe47bSelf-Consistent Range-Separated Density-Functional Theory with Second-Order Perturbative Correction via the Optimized-Effective-Potential MethodSmiga Szymon; Grabowski Ireneusz; Witkowski Mateusz; Mussard Bastien; Toulouse JulienJournal of chemical theory and computation (2020), 16 (1), 211-223 ISSN:.We extend the range-separated double-hybrid RSH+MP2 method (Angyan, J. G.; et al. Phys. Rev. A2005, 72, 012510), combining long-range HF exchange and MP2 correlation with a short-range density functional to a fully self-consistent version using the optimized-effective-potential technique in which the orbitals are obtained from a local potential including the long-range HF and MP2 contributions. We test this approach, that we name RS-OEP2, on a set of small closed-shell atoms and molecules. For the commonly used value of the range-separation parameter μ = 0.5 bohr(-1), we find that self-consistency does not seem to bring any improvement for total energies, ionization potentials, and electronic affinities. However, contrary to the non-self-consistent RSH+MP2 method, the present RS-OEP2 method gives a LUMO energy which physically corresponds to a neutral excitation energy and gives local exchange-correlation potentials which are reasonably good approximations to the corresponding Kohn-Sham quantities. At a finer scale, we find that RS-OEP2 gives largely inaccurate correlation potentials and correlated densities, which points to the need of further improvement of this type of range-separated double hybrids.**65**Cohen, A. J.; Mori-Sánchez, P.; Yang, W. Second-Order Perturbation Theory with Fractional Charges and Fractional Spins.*J. Chem. Theory Comput.*2009,*5*, 786, DOI: 10.1021/ct8005419Google Scholar65https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXjtlOhtLw%253D&md5=cd0e821eee7816ae162f2bb3d14ba2c6Second-Order Perturbation Theory with Fractional Charges and Fractional SpinsCohen, Aron J.; Mori-Sanchez, Paula; Yang, WeitaoJournal of Chemical Theory and Computation (2009), 5 (4), 786-792CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The behavior of MP2 for fractional occupations is investigated. The consideration of fractional charge behavior gives a simple derivation of an expression for the chem. potential (or the deriv. of energy with respect to the no. of electrons) of MP2. A generalized optimized effective potential formalism (OEP) has been developed in which the OEP is a nonlocal potential, which can be applied to explicit functionals of the orbitals and eigenvalues and also facilitates the evaluation of the chem. potential. The MP2 deriv. improves upon the corresponding Koopmans' theorem in Hartree-Fock theory for the ionization energy and also gives a good est. of the electron affinity. In strongly correlated systems with degeneracies and fractional spins, MP2 diverges, and another cor. second-order perturbative method ameliorates this failure for the energy but still does not recapture the correct behavior for the energy derivs. that yield the gap. Overall we present a view of wave function based methods and their behavior for fractional charges and spins that offers insight into the application of these methods to challenging chem. problems.**66**Su, N. Q.; Yang, W.; Mori-Sánchez, P.; Xu, X. Fractional Charge Behavior and Band Gap Predictions with the XYG3 Type of Doubly Hybrid Density Functionals.*J. Phys. Chem. A*2014,*118*, 9201, DOI: 10.1021/jp5029992Google Scholar66https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXotlWjur4%253D&md5=7b62ae2403f6c31d8b4e8c620e128097Fractional Charge Behavior and Band Gap Predictions with the XYG3 Type of Doubly Hybrid Density FunctionalsSu, Neil Qiang; Yang, Weitao; Mori-Sanchez, Paula; Xu, XinJournal of Physical Chemistry A (2014), 118 (39), 9201-9211CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)In this work, we examine the fractional charge behaviors of doubly hybrid (DH) functionals. By plotting the ground-state energies E and energy derivs. for atoms and mols. with fractional electron nos. N, we directly quantify the delocalization errors of some representative DH functionals such as B2PLYP, XYG3, and XYGJ-OS. Numerical assessments on ionization potentials (IPs), electron affinities (EAs), and fundamental gaps, from either integer no. calcns. or energy deriv. calcns., are provided. It is shown that the XYG3 type of DH functionals gives good agreement between their energy derivs. and the exptl. IPs, EAs, and gaps, as expected from their nearly straight line fractional charge behaviors.**67**Mussard, B.; Toulouse, J. Fractional-charge and fractional-spin errors in range-separated density-functional theory.*Mol. Phys.*2017,*115*, 161, DOI: 10.1080/00268976.2016.1213910Google Scholar67https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xht1yqsbbO&md5=7d6e54d5e0aa61f9d751af2846eac407Fractional-charge and fractional-spin errors in range-separated density-functional theoryMussard, Bastien; Toulouse, JulienMolecular Physics (2017), 115 (1-2), 161-173CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)We investigate fractional-charge and fractional-spin errors in range-sepd. d.-functional theory (DFT). Specifically, we consider the range-sepd. hybrid (RSH) method which combines long-range Hartree-Fock (HF) exchange with a short-range semilocal exchange-correlation d. functional, and the RSH+MP2 method which adds long-range, second-order Moller-Plesset (MP2) correlation. Results on atoms and mols. show that the fractional-charge errors obtained in RSH are much smaller than in the std. Kohn-Sham (KS) scheme applied with semilocal or hybrid approxns., and also generally smaller than in the std. HF method. The RSH+MP2 method tends to have smaller fractional-charge errors than std. MP2 for the most diffuse systems, but larger fractional-charge errors for the more compact systems. Even though the individual contributions to the fractional-spin errors in the H atom coming from the short-range exchange and correlation d.-functional approxns. are smaller than the corresponding contributions for the full-range exchange and correlation d.-functional approxns., RSH gives fractional-spin errors that are larger than in the std. KS scheme and only slightly smaller than in std. HF. Adding long-range MP2 correlation only leads to infinite fractional-spin errors. This work clarifies the successes and limitations of range-sepd. DFT approaches for eliminating self-interaction and static-correlation errors.**68**Su, N. Q.; Xu, X. Insights into Direct Methods for Predictions of Ionization Potential and Electron Affinity in Density Functional Theory.*J. Phys. Chem. Lett.*2019,*10*, 2692, DOI: 10.1021/acs.jpclett.9b01052Google Scholar68https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXovVKnsbc%253D&md5=7556f302c3c7856511cefeb2290a20f8Insights into Direct Methods for Predictions of Ionization Potential and Electron Affinity in Density Functional TheorySu, Neil Qiang; Xu, XinJournal of Physical Chemistry Letters (2019), 10 (11), 2692-2699CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)Vertical ionization potential (IP) and electron affinity (EA) are fundamental mol. properties, while the Δ method and the direct method are the widely used approaches to compute these properties. The Δ method is calcd. by taking the total energy difference of the initial and final states, whose reliability is seriously affected by the issue assocd. with the imbalanced treatment of these two states. The direct method based on the derivs. involving only one single-state calcn. can yield a quasiparticle spectrum whose accuracy, on the other hand, is mostly affected by the levels of approx. mol. structure theories. Because of the aforementioned issues, EA prediction can be particularly problematic. Here we present, for the first time, an analytic theory on the derivation and realization of generalized Kohn-Sham (KS) eigenvalues of doubly hybrid (DH) functionals that depend on both occupied and unoccupied orbitals. The method based on the KS eigenvalues of neutral systems, termed the NKS method, is found to suffer little from the imbalance issue, while it is only the NKS method that can offer accurate EA prediction from a good functional approxn., such as the XYG3 type of DH functionals. Being less sensitive to the size of basis sets, the NKS method is of great significance for its application to large systems. The insights gained in this work are useful for the calcn. of properties assocd. with small energy differences while emphasizing the importance of the development of generalized functionals that rely on both occupied and unoccupied orbitals.**69**Beste, A.; Vázquez-Mayagoitia, Á.; Ortiz, J. V. Direct ΔMBPT(2) method for ionization potentials, electron affinities, and excitation energies using fractional occupation numbers.*J. Chem. Phys.*2013,*138*, 074101, DOI: 10.1063/1.4790626Google Scholar69https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXis1Wrtbo%253D&md5=621540912c7873be7720dd6baa5aef35Direct ΔMBPT(2) method for ionization potentials, electron affinities, and excitation energies using fractional occupation numbersBeste, Ariana; Vazquez-Mayagoitia, Alvaro; Ortiz, J. V.Journal of Chemical Physics (2013), 138 (7), 074101/1-074101/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A direct method (D-ΔMBPT(2)) to calc. 2nd-order ionization potentials (IPs), electron affinities (EAs), and excitation energies is developed. The ΔMBPT(2) method is defined as the correlated extension of the ΔHF method. Energy differences are obtained by integrating the energy deriv. with respect to occupation nos. over the appropriate parameter range. This is made possible by writing the 2nd-order energy as a function of the occupation nos. Relaxation effects are fully included at the SCF level. This is in contrast to linear response theory, which makes the D-ΔMBPT(2) applicable not only to single excited but also higher excited states. We show the relationship of the D-ΔMBPT(2) method for IPs and EAs to a 2nd-order approxn. of the effective Fock-space coupled-cluster Hamiltonian and a 2nd-order electron propagator method. We also discuss the connection between the D-ΔMBPT(2) method for excitation energies and the CIS-MP2 method. Finally, as a proof of principle, we apply our method to calc. ionization potentials and excitation energies of some small mols. For IPs, the ΔMBPT(2) results compare well to the 2nd-order soln. of the Dyson equation. For excitation energies, the deviation from equation of motion coupled cluster singles and doubles increases when correlation becomes more important. When using the numerical integration technique, we encounter difficulties that prevented us from reaching the ΔMBPT(2) values. Most importantly, relaxation beyond the Hartree-Fock level is significant and needs to be included in future research. (c) 2013 American Institute of Physics.**70**Gu, Y.; Xu, X. Extended Koopmans’ theorem in the adiabatic connection formalism: Applied to doubly hybrid density functionals.*J. Chem. Phys.*2020,*153*, 044109, DOI: 10.1063/5.0010743Google Scholar70https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhsVOqtLrK&md5=613aa454c3423f076a2da0dd1ba9c2ceExtended Koopmans' theorem in the adiabatic connection formalism: Applied to doubly hybrid density functionalsGu, Yonghao; Xu, XinJournal of Chemical Physics (2020), 153 (4), 044109CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A rigorous framework that combines the extended Koopmans' theorem (EKT) with the adiabatic connection (AC) formalism of d. functional theory is developed here, namely, EKT-AC, to calc. the vertical ionization potentials (IPs) of mol. systems. When applied to the doubly hybrid d. functional approxns. (DH-DFAs), the EKT-DH approach is established for the B2PLYP-type DHs with one-parameter and two-parameters, as well as the XYG3-type DHs. Based on EKT-DH, an approxn. of the KT type is introduced, leading to the KT-DH approach. The IP-condition that the calcd. vertical IPs with EKT-DH or KT-DH are to reproduce the exptl. IPs closely is applied to investigate the commonly used DH-DFAs for such a purpose and is utilized as a principle for DH-DFA developments. Considering the systematic improvements, as well as its numeric stability, we recommend the KT-B2GPPLYP approach as a pragmatic way for vertical IP calcns. (c) 2020 American Institute of Physics.**71**Grimme, S.; Neese, F. Double-hybrid density functional theory for excited electronic states of molecules.*J. Chem. Phys.*2007,*127*, 154116, DOI: 10.1063/1.2772854Google Scholar71https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXht1ejs7fP&md5=99b6be2c14771032a2aa44c9827d64afDouble-hybrid density functional theory for excited electronic states of moleculesGrimme, Stefan; Neese, FrankJournal of Chemical Physics (2007), 127 (15), 154116/1-154116/18CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Double-hybrid d. functionals are based on a mixing of std. generalized gradient approxns. (GGAs) for exchange and correlation with Hartree-Fock (HF) exchange and a perturbative second-order correlation part (PT2) that is obtained from the Kohn-Sham (GGA) orbitals and eigenvalues. This virtual orbital-dependent functional (dubbed B2PLYP) contains only two empirical parameters that describe the mixt. of HF and GGA exchange (ax) and of the PT2 and GGA correlation (ac), resp. Extensive testing has recently demonstrated the outstanding accuracy of this approach for various ground state problems in general chem. applications. The method is extended here without any further empirical adjustments to electronically excited states in the framework of time-dependent d. functional theory (TD-DFT) or the closely related Tamm-Dancoff approxn. (TDA-DFT). In complete analogy to the ground state treatment, a scaled second-order perturbation correction to CI with singles (CIS(D)) wave functions developed some years ago by Head-Gordon et al. is computed on the basis of d. functional data and added to the TD(A)-DFT/GGA excitation energy. The method is implemented by applying the resoln. of the identity approxn. and the efficiency of the code is discussed. Extensive tests for a wide variety of mols. and excited states (of singlet, triplet, and doublet multiplicities) including electronic spectra are presented. In general, rather accurate excitation energies (deviations from ref. data typically <0.2 eV) are obtained that are mostly better than those from std. functionals. Still, systematic errors are obtained for Rydberg (too low on av. by about 0.3 eV) and charge-transfer transitions but due to the relatively large ax parameter (0.53), B2PLYP outperforms most other functionals in this respect. Compared to conventional HF-based CIS(D), the method is more robust in electronically complex situations due to the implicit account of static correlation effects by the GGA parts. The (D) correction often works in the right direction and compensates for the overestimation of the transition energy at the TD level due to the elevated fraction of HF exchange in the hybrid GGA part. Finally, the limitations of the method are discussed for challenging systems such as transition metal complexes, cyanine dyes, and multireference cases.**72**Hirata, S.; Head-Gordon, M. Time-dependent density functional theory within the Tamm–Dancoff approximation.*Chem. Phys. Lett.*1999,*314*, 291, DOI: 10.1016/S0009-2614(99)01149-5Google Scholar72https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXnsl2rsr0%253D&md5=1b62f410de6c2a2193f1011d42f389c5Time-dependent density functional theory within the Tamm-Dancoff approximationHirata, S.; Head-Gordon, M.Chemical Physics Letters (1999), 314 (3,4), 291-299CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)A computationally simple method for mol. excited states, namely, the Tamm-Dancoff approxn. to time-dependent d. functional theory, is proposed and implemented. This method yields excitation energies for several closed- and open-shell mols. that are essentially of the same quality as those obtained from time-dependent d. functional theory itself, when the same exchange-correlation functional is used.**73**Foresman, J. B.; Head-Gordon, M.; Pople, J. A.; Frisch, M. J. Toward a systematic molecular orbital theory for excited states.*J. Phys. Chem.*1992,*96*, 135, DOI: 10.1021/j100180a030Google Scholar73https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK38Xks1GhsA%253D%253D&md5=fc67a2af6aef6358d605cefe654b991aToward a systematic molecular orbital theory for excited statesForesman, James B.; Head-Gordon, Martin; Pople, John A.; Frisch, Michael J.Journal of Physical Chemistry (1992), 96 (1), 135-49CODEN: JPCHAX; ISSN:0022-3654.The methodol. and computational considerations necessary for the detn. of the ab initio energy, wave function, and gradient of a mol. in an electronically excited state using MO theory are discussed. A fundamental level of theory is reexamd. which was employed several years ago for the interpretation of the electronic spectra of simple org. mols.: CI (CI) among all singly substituted determinants using a Hartree-Fock ref. state. Several new enhancements to this general theory are given. First, it is shown how the "CI-singles" wave function can be used to compute efficiently the analytic first deriv. of the energy in order to obtain accurate properties and optimized geometries for a wide range of mols. in their excited states. Secondly, a computer program is described which allows these computations to be done in a "direct" fashion, with no disk storage required for the two-electron repulsion integrals. This allows investigations of systems with large nos. of atoms (or large nos. of basis functions). Thirdly it is shown how the CI-singles approxn. can be cor. via second-order Moeller-Plesset perturbation theory to produce a level of theory for excited states which further includes some effects of electronic correlation. The relative success of the model as a function of basis set indicates that a judicious choice of basis set is needed in order to evaluate its performance adequately. Application of the method to the excited states of formaldehyde, ethylene, pyridine, and porphin demonstrates the utility of CI-singles theory.**74**Head-Gordon, M.; Rico, R. J.; Oumi, M.; Lee, T. J. A doubles correction to electronic excited states from configuration interaction in the space of single substitutions.*Chem. Phys. Lett.*1994,*219*, 21, DOI: 10.1016/0009-2614(94)00070-0Google Scholar74https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXis12htb8%253D&md5=7c5a5fc22d8498896b163b9d621923a3A doubles correction to electronic excited states from configuration interaction in the space of single substitutionsHead-Gordon, Martin; Rico, Rudolph J.; Oumi, Manabu; Lee, Timothy J.Chemical Physics Letters (1994), 219 (1-2), 21-9CODEN: CHPLBC; ISSN:0009-2614.A perturbative correction to the method of CI with single substitutions (CIS) is presented. This CIS(D) correction approx. introduces the effect of double substitutions which are absent in CIS excited states. CIS(D) is a second-order perturbation expansion of the coupled-cluster excited state method, restricted to single and double substitutions, in a series in which CIS is zeroth order, and the first-order correction vanishes. CIS(D) excitation energies are size consistent and the calculational complexity scales with the fifth power of mol. size, akin to second-order Moeller-Plesset theory for the ground state. Calcns. on singlet excited states of ethylene, formaldehyde, acetaldehyde, butadiene and benzene show that CIS(D) is a uniform improvement over CIS. CIS(D) appears to be a promising method for examg. excited states of large mols., where more accurate methods are not feasible.**75**Ottochian, A.; Morgillo, C.; Ciofini, I.; Frisch, M. J.; Scalmani, G.; Adamo, C. Double hybrids and time-dependent density functional theory: An implementation and benchmark on charge transfer excited states.*J. Comput. Chem.*2020,*41*, 1242, DOI: 10.1002/jcc.26170Google Scholar75https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXmvVSisr0%253D&md5=43612d833b42b19a6c615db08c13577fDouble hybrids and time-dependent density functional theory: An implementation and benchmark on charge transfer excited statesOttochian, Alistar; Morgillo, Carmela; Ciofini, Ilaria; Frisch, Michael J.; Scalmani, Giovanni; Adamo, CarloJournal of Computational Chemistry (2020), 41 (13), 1242-1251CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)In this paper we present the implementation and benchmarking of a Time Dependent D. Functional Theory approach in conjunction with Double Hybrid (DH) functionals. We focused on the anal. of their performance for through space charge-transfer (CT) excitations which are well known to be very problematic for commonly used functionals, such as global hybrids. Two different families of functionals were compared, each of them contg. pure, hybrid and double-hybrid functionals. The results obtained show that, beside the robustness of the implementation, these functionals provide results with an accuracy comparable to that of adjusted range-sepd. functionals, with the relevant difference that for DHs no parameter is tuned on specific compds. thus making them more appealing for a general use. Furthermore, the algorithm described and implemented is characterized by the same computational cost scaling as that of the ground state algorithm employed for MP2 and double hybrids.**76**Schwabe, T.; Goerigk, L. Time-Dependent Double-Hybrid Density Functionals with Spin-Component and Spin-Opposite Scaling.*J. Chem. Theory Comput.*2017,*13*, 4307, DOI: 10.1021/acs.jctc.7b00386Google Scholar76https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXht1Ghur7P&md5=a399974d4fa85eae2c3dfa45d591959bTime-Dependent Double-Hybrid Density Functionals with Spin-Component and Spin-Opposite ScalingSchwabe, Tobias; Goerigk, LarsJournal of Chemical Theory and Computation (2017), 13 (9), 4307-4323CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)For the first time, we combine time-dependent double-hybrid d. functional approxns. (TD-DHDFAs) for the calcn. of electronic excitation energies with the concepts of spin-component and spin-opposite scaling (SCS/SOS) of electron-pair contributions to their nonlocal correlation components. Different flavors of this idea, ranging from std. SCS parameters to fully fitted parameter sets, are presented and tested on six different parent DHDFAs. For cross-validation, we assess those methods on three benchmark sets that cover small- to medium-sized chromophores (up to 78 atoms) and different excitation types. For this purpose, we also introduce new CC3 ref. values for the popular Gordon benchmark set that we recommend using in future studies. Our results confirm that already the (unscaled) parent TD-DHDFAs are accurate and outperform some wave function methods. Further introduction of SCS/SOS eliminates extreme outliers, reduces deviation spans from ref. values by up to 0.5 eV, aligns the performance of the Tamm-Dancoff approxn. (TDA) to that of full TD calcns., and also enables a more balanced description of different excitation types. The best-performing TD-based methods in our cross validation have mean abs. deviations as low as 0.14 eV compared to the time- and resource-intensive CC3 approach. A very important finding is that we also obtained SOS variants with excellent performance, contrary to wave function based methods. This opens a future pathway to highly efficient methods for the optimization of excited-state geometries, particularly when paired with computing strategies such as the Laplace transform. We recommend our SCS- and SOS-based variants for further testing and subsequent applications.**77**Casanova-Páez, M.; Goerigk, L. Time-Dependent Long-Range-Corrected Double-Hybrid Density Functionals with Spin-Component and Spin-Opposite Scaling: A Comprehensive Analysis of Singlet–Singlet and Singlet–Triplet Excitation Energies.*J. Chem. Theory Comput.*2021,*17*, 5165, DOI: 10.1021/acs.jctc.1c00535Google Scholar77https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXhsF2gurvL&md5=bfec26ba2c0635232c3a0c2ac4f894f7Time-Dependent Long-Range-Corrected Double-Hybrid Density Functionals with Spin-Component and Spin-Opposite Scaling: A Comprehensive Analysis of Singlet-Singlet and Singlet-Triplet Excitation EnergiesCasanova-Paez, Marcos; Goerigk, LarsJournal of Chemical Theory and Computation (2021), 17 (8), 5165-5186CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Following the work on spin-component and spin-opposite scaled (SCS/SOS) global double hybrids for singlet-singlet excitations by Schwabe and Goerigk [J. Chem. Theory Comput., 20217, 13, 4307] and our own works on new long-range cor. (LC) double hybrids for singlet-singlet and singlet-triplet excitations [J. Chem. Theory Comput., 2019, 15, 4735], we present new LC double hybrids with SCS/SOS that demonstrate further improvement over previously published results and methods. We introduce new unscaled and scaled versions of different global and LC double hybrids based on Becke88 or PBE exchange combined with LYP, PBE, or P86 correlation. For singlet-singlet excitations, we cross-validate them on six benchmark sets that cover small to medium-sized chromophores with different excitation types (local-valence, Rydberg, and charge transfer). For singlet-triplet excitations, we perform the cross-validation on three different benchmark sets following the same anal. as in our previous work in 2020. In total, 203 excitations are analyzed. Our results confirm and extend those of Schwabe and Goerigk regarding the superior performance of SCS and SOS variants compared to their unscaled parents by decreasing mean abs. deviations, root-mean-square deviations, or error spans by more than half and bringing abs. mean deviations closer to zero. Our SCS/SOS variants are shown to be highly efficient and robust for the computation of vertical excitation energies, which even outperform specialized double hybrids that also contain an LC in their perturbative part. In particular, our new SCS/SOS-ωPBEPP86 and SCS/SOS-ωB88PP86 functionals are four of the most accurate and robust methods tested in this work, and we fully recommend them for future applications. However, if the relevant SCS and SOS algorithms are not available to the user, we suggest ωPBEPP86 as the best unscaled method in this work.**78**Mester, D.; Kállay, M. A simple range-separated double-hybrid density functional theory for excited states.*J. Chem. Theory Comput.*2021,*17*, 927, DOI: 10.1021/acs.jctc.0c01135Google Scholar78https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXislWktA%253D%253D&md5=5322233ba064d903c9bb12fc3ec76687A Simple Range-Separated Double-Hybrid Density Functional Theory for Excited StatesMester, David; Kallay, MihalyJournal of Chemical Theory and Computation (2021), 17 (2), 927-942CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A simple and robust range-sepd. (RS) double-hybrid (DH) time-dependent d. functional approach is presented for the accurate calcn. of excitation energies of mols. within the Tamm-Dancoff approxn. The scheme can be considered as an excited-state extension of the ansatz proposed by Toulouse and co-workers [J. Chem. Phys. 2018, 148, 164105], which is based on the two-parameter decompn. of the Coulomb potential, for which both the exchange and correlation contributions are range-sepd. A flexible and efficient implementation of the new scheme is also presented, which facilitates its extension to any combination of exchange and correlation functionals. The performance of the new approxn. is tested for singlet excitations on several benchmark compilations and thoroughly compared to that of representative DH, RS hybrid, and RS DH functionals. The one-electron basis set dependence and computation times are also assessed. Our results show that the new approach improves on std. DHs in most cases, and it can provide a more robust and accurate alternative. In addn., on av., it noticeably surpasses the existing RS hybrid and RS DH functionals.**79**Schirmer, J.; Trofimov, A. B. Intermediate state representation approach to physical properties of electronically excited molecules.*J. Chem. Phys.*2004,*120*, 11449, DOI: 10.1063/1.1752875Google Scholar79https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXkvVansr0%253D&md5=e865bbb6ea1276e663b4cc4212e608bfIntermediate state representation approach to physical properties of electronically excited moleculesSchirmer, J.; Trofimov, A. B.Journal of Chemical Physics (2004), 120 (24), 11449-11464CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Propagator methods provide a direct approach to energies and transition moments for (generalized) electronic excitations from the ground state, but they do not usually allow 1 to det. excited state wave functions and properties. Using a specific intermediate state representation (ISR) concept, the authors here show how this restriction can be overcome in the case of the algebraic-diagrammatic construction (ADC) propagator approach. In the ISR reformulation of the theory the basic ADC secular matrix is written as a representation of the Hamiltonian (or the shifted Hamiltonian) in terms of explicitly constructible states, referred to as intermediate (or ADC) states. Similar intermediate state representations can be derived for operators other than the Hamiltonian. Together with the ADC eigenvectors, the intermediate states give rise to an explicit formulation of the excited wave functions and allow 1 to calc. phys. properties of excited states as well as transition moments for transitions between different excited states. As for the ground-state excitation energies and transition moments, the ADC excited state properties are size consistent so that the theory is suitable for applications to large systems. The established hierarchy of higher-order [ADC(n)] approxns., corresponding to systematic truncations of the IS configuration space and the perturbation-theor. expansions of the ISR matrix elements, can readily be extended to the excited state properties. Explicit ISR matrix elements for arbitrary 1-particle operators were derived and coded at the 2nd-order [ADC(2)] level of theory. As a 1st computational test of the method the authors have carried out ADC(2) calcns. for singlet and triplet excited state dipole moments in H2O and HF, where comparison to full CI results can be made. The potential of the ADC(2) method is further demonstrated in an exploratory study of the excitation energies and dipole moments of the low-lying excited states of para-nitroaniline. Four triplet states, T1-T4, and 2 singlet states, S1 and S2, lie (vertically) below the prominent charge transfer (CT) excitation, S3. The dipole moment of the S3 state (17.0 D) is distinctly larger than that of the corresponding T3 triplet state (11.7 D).**80**Wormit, M.; Rehn, D. R.; Harbach, P. H. P.; Wenzel, J.; Krauter, C. M.; Epifanovsky, E.; Dreuw, A. Investigating Excited Electronic States using the Algebraic Diagrammatic Construction (ADC) Approach of the Polarisation Propagator.*Mol. Phys.*2014,*112*, 774, DOI: 10.1080/00268976.2013.859313Google Scholar80https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXhslWjur3L&md5=a904ad56ad4361ea8dea05269493b62dInvestigating excited electronic states using the algebraic diagrammatic construction (ADC) approach of the polarisation propagatorWormit, Michael; Rehn, Dirk R.; Harbach, Philipp H. P.; Wenzel, Jan; Krauter, Caroline M.; Epifanovsky, Evgeny; Dreuw, AndreasMolecular Physics (2014), 112 (5-6), 774-784CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)The development of reliable theor. methods and the provision of efficient computer programs for the investigation of optical spectra and photochem. of large mols. in general is one of the most important tasks of contemporary theor. chem. Here, we present an overview of the current features of our implementation of the algebraic diagrammatic construction scheme of the polarisation propagator, which is a versatile and robust approach for the theor. investigation of excited states and their properties.**81**Véril, M.; Scemama, A.; Caffarel, M.; Lipparini, F.; Boggio-Pasqua, M.; Jacquemin, D.; Loos, P.-F. QUESTDB: A database of highly accurate excitation energies for the electronic structure community.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2021,*11*, e1517 DOI: 10.1002/wcms.1517Google Scholar81https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XitFajs7w%253D&md5=d2a4214e0bb370d5a06a558baacda10fQUESTDB and A database of highly accurate excitation energies for the electronic structure communityVeril, Mickael; Scemama, Anthony; Caffarel, Michel; Lipparini, Filippo; Boggio-Pasqua, Martial; Jacquemin, Denis; Loos, Pierre-FrancoisWiley Interdisciplinary Reviews: Computational Molecular Science (2021), 11 (5), e1517CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)We describe our efforts of the past few years to create a large set of more than 500 highly accurate vertical excitation energies of various natures (π → π*, n → π*, double excitation, Rydberg, singlet, doublet, triplet, etc.) in small- and medium-sized mols. These values have been obtained using an incremental strategy which consists in combining high-order coupled cluster and selected CI calcns. using increasingly large diffuse basis sets in order to reach high accuracy. One of the key aspects of the so-called QUEST database of vertical excitations is that it does not rely on any exptl. values, avoiding potential biases inherently linked to expts. and facilitating theor. cross comparisons. Following this composite protocol, we have been able to produce theor. best ests. (TBEs) with the aug-cc-pVTZ basis set for each of these transitions, as well as basis set cor. TBEs (i.e., near the complete basis set limit) for some of them. The TBEs/aug-cc-pVTZ have been employed to benchmark a large no. of (lower-order) wave function methods such as CIS(D), ADC(2), CC2, STEOM-CCSD, CCSD, CCSDR(3), CCSDT-3, ADC(3), CC3, NEVPT2, and so on (including spin-scaled variants). In order to gather the huge amt. of data produced during the QUEST project, we have created a website () where one can easily test and compare the accuracy of a given method with respect to various variables such as the mol. size or its family, the nature of the excited states, the type of basis set, and so on. We hope that the present review will provide a useful summary of our effort so far and foster new developments around excited-state methods. This article is categorized under:Electronic Structure Theory > Ab Initio Electronic Structure Methods.**82**Mester, D.; Kállay, M. Combined density functional and algebraic-diagrammatic construction approach for accurate excitation energies and transition moments.*J. Chem. Theory Comput.*2019,*15*, 4440, DOI: 10.1021/acs.jctc.9b00391Google Scholar82https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhtlSnt7nO&md5=0e3d5d10bd97a7b51865aa3531938892Combined Density Functional and Algebraic-Diagrammatic Construction Approach for Accurate Excitation Energies and Transition MomentsMester, David; Kallay, MihalyJournal of Chemical Theory and Computation (2019), 15 (8), 4440-4453CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A composite of time-dependent d. functional theory (TDDFT) and the second-order algebraic-diagrammatic construction [ADC(2)] approach is presented for efficient calcn. of spectral properties of mols. Our method can be regarded as a new excited-state double-hybrid (DH) approach or a dressed TDDFT scheme, but it can also be interpreted as an empirically tuned ADC(2) model. Several combinations of exchange-correlation functionals and spin-scaling schemes are explored. Our best-performing method includes the Perdew, Burke, and Ernzerhof exchange and Perdew's 1986 correlation functional and employs the scaled-opposite-spin approxn. for the higher-order terms. The computation time of the new method scales as the fourth power of the system size, and an efficient cost-redn. approach is also presented, which further speeds up the calcns. Our benchmark calcns. show that the proposed model outperforms not only the existing DH approaches and ADC(2) variants but also the considerably more expensive coupled-cluster methods.**83**Grimme, S. Improved second-order Møller–Plesset perturbation theory by separate scaling of parallel- and antiparallel-spin pair correlation energies.*J. Chem. Phys.*2003,*118*, 9095, DOI: 10.1063/1.1569242Google Scholar83https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXjs1Gktbk%253D&md5=4857ad3bcf3e894bb7b94a4d3cf86fc1Improved second-order Moller-Plesset perturbation theory by separate scaling of parallel- and antiparallel-spin pair correlation energiesGrimme, StefanJournal of Chemical Physics (2003), 118 (20), 9095-9102CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A simple modification of the second-order Moller-Plesset perturbation theory (MP2) to improve the description of mol. ground state energies is proposed. The total MP2 correlation energy is partitioned into parallel- and antiparallel-spin components which are sep. scaled. The two parameters (scaling factors), whose values can be justified by basic theor. arguments, were optimized on a benchmark set of 51 reaction energies composed of 74 first-row mols. The new method performs significantly better than std. MP2: the rms [mean abs. error (MAE)] deviation drops from 4.6 (3.3) to 2.3 (1.8) kcal/mol. The max. error is reduced from 13.3 to 5.1 kcal/mol. Significant improvements are esp. obsd. for cases which are usually known as MP2 pitfalls while cases already described well with MP2 remain almost unchanged. Even for 11 atomization energies not considered in the fit, uniform improvements [MAE: 8.1 kcal/mol (MP2) vs. 3.2 kcal/mol (new)] were found. The results are furthermore compared with those from d. functional theory (DFT/B3LYP) and quadratic CI [QCISD/QCISD(T)] calcns. Also for difficult systems including strong (nondynamical) correlation effects, the improved MP2 method clearly outperforms DFT/B3LYP and yields results of QCISD or sometimes QCISD(T) quality. Preliminary calcns. of the equil. bond lengths and harmonic vibrational frequencies for ten diat. mols. also show consistent enhancements. The uniformity with which the new method improves upon MP2, thereby rectifying many of its problems, indicates significant robustness and suggests it as a valuable quantum chem. method of general use.**84**Jung, Y.; Lochan, R. C.; Dutoi, A. D.; Head-Gordon, M. Scaled opposite-spin second order Møller–Plesset correlation energy: An economical electronic structure method.*J. Chem. Phys.*2004,*121*, 9793, DOI: 10.1063/1.1809602Google Scholar84https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXpslCitb0%253D&md5=7423ee78fb73040427b9762bfef69705Scaled opposite-spin second order Moller-Plesset correlation energy: An economical electronic structure methodJung, Yousung; Lochan, Rohini C.; Dutoi, Anthony D.; Head-Gordon, MartinJournal of Chemical Physics (2004), 121 (20), 9793-9802CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A simplified approach to treating the electron correlation energy is suggested in which only the α-β component of the second order Moller-Plesset energy is evaluated, and then scaled by an empirical factor which is suggested to be 1.3. This scaled opposite-spin second order energy (SOS-MP2), where MP2 is Moller-Plesset theory, yields results for relative energies and deriv. properties that are statistically improved over the conventional MP2 method. Furthermore, the SOS-MP2 energy can be evaluated without the fifth order computational steps assocd. with MP2 theory, even without exploiting any spatial locality. A fourth order algorithm is given for evaluating the opposite spin MP2 energy using auxiliary basis expansions, and a Laplace approach, and timing comparisons are given.**85**Kozuch, S.; Gruzman, D.; Martin, J. M. L. DSD-BLYP: A General Purpose Double Hybrid Density Functional Including Spin Component Scaling and Dispersion Correction.*J. Phys. Chem. C*2010,*114*, 20801, DOI: 10.1021/jp1070852Google Scholar85https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXht12jur3K&md5=42ee3607f6a9a26a3fc5353e295fb564DSD-BLYP: A General Purpose Double Hybrid Density Functional Including Spin Component Scaling and Dispersion CorrectionKozuch, Sebastian; Gruzman, David; Martin, Jan M. L.Journal of Physical Chemistry C (2010), 114 (48), 20801-20808CODEN: JPCCCK; ISSN:1932-7447. (American Chemical Society)We present a general purpose double-hybrid DFT parametrization based on the BLYP functional, spin-component scaled (SCS) MP2-like correlation and a dispersion correction, called DSD-BLYP. Six training sets were used, including main group and transition state thermochem., kinetics, and dispersion forces. This new parametrization is usually 10-15% more accurate than the already exceptional B2GP-PLYP double hybrid, at the same computational cost. Its principal benefit is greater robustness for systems with significant nondynamical correlation. If a scaling factor is included in the harmonic frequency calcns., B2GP-PLYP was found to give very accurate results for kinetics, thermochem., and frequencies.**86**Rhee, Y. M.; Head-Gordon, M. Scaled Second-Order Perturbation Corrections to Configuration Interaction Singles: Efficient and Reliable Excitation Energy Methods.*J. Phys. Chem. A*2007,*111*, 5314, DOI: 10.1021/jp068409jGoogle Scholar86https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXlsFOrs78%253D&md5=71e09c1286af980c5be83bf70b8fe089Scaled Second-Order Perturbation Corrections to Configuration Interaction Singles: Efficient and Reliable Excitation Energy MethodsRhee, Young Min; Head-Gordon, MartinJournal of Physical Chemistry A (2007), 111 (24), 5314-5326CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)Two modifications of the perturbative doubles correction to CI with single substitutions (CIS(D)) are suggested, which are excited state analogs of ground state scaled second-order Moller-Plesset (MP2) methods. The first approach employs two parameters to scale the two spin components of the direct term of CIS(D), starting from the two-parameter spin-component scaled (SCS) MP2 ground state, and is termed SCS-CIS(D). An efficient resoln.-of-the-identity (RI) implementation of this approach is described. The second approach employs a single parameter to scale only the opposite-spin direct term of CIS(D), starting from the one-parameter scaled opposite-spin (SOS) MP2 ground state, and is called SOS-CIS(D). By utilizing auxiliary basis expansions and a Laplace transform, a fourth-order algorithm for SOS-CIS(D) is described and implemented. The parameters that describe SCS-CIS(D) and SOS-CIS(D) are optimized based on a training set that includes valence excitations of various org. mols. and Rydberg transitions of water and ammonia, and they significantly improve upon CIS(D) itself. The accuracy of the two methods is found to be comparable. This arises from a strong correlation between the same-spin and the opposite-spin portions of the excitation energy terms. The methods are successfully applied to the zincbacteriochlorin-bacteriochlorin charge-transfer transition, for which time-dependent d. functional theory, with presently available exchange-correlation functionals, is known to fail. The methods are also successfully applied to describe various electronic transitions outside of the training set. The efficiency of the SOS-CIS(D) and the auxiliary basis implementation of CIS(D) and SCS-CIS(D) are confirmed with a series of timing tests.**87**Hellweg, A.; Grün, S. A.; Hättig, C. Benchmarking the performance of spin-component scaled CC2 in ground and electronically excited states.*Phys. Chem. Chem. Phys.*2008,*10*, 4119, DOI: 10.1039/b803727bGoogle Scholar87https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXot1enu7s%253D&md5=eb804f6298d17d5a6b3432564d370ab9Benchmarking the performance of spin-component scaled CC2 in ground and electronically excited statesHellweg, Arnim; Gruen, Sarah A.; Haettig, ChristofPhysical Chemistry Chemical Physics (2008), 10 (28), 4119-4127CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)A generalization of the spin-component scaling and scaled opposite-spin modifications of second-order Moller-Plesset perturbation theory to the approx. coupled-cluster singles-and-doubles model CC2 (termed SCS-CC2 and SOS-CC2) is discussed and a preliminary implementation of ground and excited state energies and analytic gradients is reported. The computational results for bond distances, harmonic frequencies, adiabatic and 0-0 excitation energies are compared with exptl. results to benchmark their performance. Both variants of the spin-scaling increase the robustness of CC2 against strong correlation effects and lead for this method even to somewhat larger improvements than those obsd. for second-order Moller-Plesset perturbation theory. The spin-component scaling also enhances systematically the accuracy of CC2 for 0-0 excitation energies for π → π* and n → π* transitions, if geometries are detd. at the same level.**88**Winter, N. O. C.; Hättig, C. Scaled opposite-spin CC2 for ground and excited states with fourth order scaling computational costs.*J. Chem. Phys.*2011,*134*, 184101, DOI: 10.1063/1.3584177Google Scholar88https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXmtVajtb4%253D&md5=7291417f6c3bce6fec1f54a489511024Scaled opposite-spin CC2 for ground and excited states with fourth order scaling computational costsWinter, Nina O. C.; Haettig, ChristofJournal of Chemical Physics (2011), 134 (18), 184101/1-184101/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An implementation of scaled opposite-spin CC2 (SOS-CC2) for ground and excited state energies is presented that requires only fourth order scaling computational costs. The SOS-CC2 method yields results with an accuracy comparable to the unscaled method. Furthermore the time-detg. fifth order scaling steps in the algorithm can be replaced by only fourth order scaling computational costs using a "resoln. of the identity" approxn. for the electron repulsion integrals and a Laplace transformation of the orbital energy denominators. This leads to a significant redn. of computational costs esp. for large systems. Timings for ground and excited state calcns. are shown and the error of the Laplace transformation is investigated. An application to a chlorophyll mol. with 134 atoms results in a speed-up by a factor of five and demonstrates how the new implementation extends the applicability of the method. A SOS variant of the algebraic diagrammatic construction through second order ADC(2), which arises from a simplification of the SOS-CC2 model, is also presented. The SOS-ADC(2) model is a cost-efficient alternative in particular for future extensions to spectral intensities and excited state structure optimizations. (c) 2011 American Institute of Physics.**89**Casanova-Páez, M.; Dardis, M. B.; Goerigk, L. ωB2PLYP and ωB2GPPLYP: The First Two Double-Hybrid Density Functionals with Long-Range Correction Optimized for Excitation Energies.*J. Chem. Theory Comput.*2019,*15*, 4735, DOI: 10.1021/acs.jctc.9b00013Google Scholar89https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhtlKltrjJ&md5=f8df0a7bf6e079b7c40edd43f3cb642eωB2PLYP and ωB2GPPLYP: The First Two Double-Hybrid Density Functionals with Long-Range Correction Optimized for Excitation EnergiesCasanova-Paez, Marcos; Dardis, Michael B.; Goerigk, LarsJournal of Chemical Theory and Computation (2019), 15 (9), 4735-4744CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Double-hybrid d. functionals are currently the most accurate d. functionals for ground-state properties and electronic excitations. Nevertheless, the lack of a long-range correction scheme makes them unreliable when it comes to long-range excitations. For this reason, we propose the first two time-dependent double-hybrid functionals with correct asymptotic long-range behavior named ωB2PLYP and ωB2GPPLYP. Herein, we demonstrate their excellent performance and show that they are the most accurate and robust time-dependent d. functional theory methods for electronic excitation energies. They provide a balanced description of local-valence, Rydberg, and charge-transfer states. They are also able to tackle the difficult first two transitions in polycyclic arom. hydrocarbons and show very promising results in a preliminary study on transition-metal compds., exemplified for titanium dioxide clusters. This work shows that double hybrids can be systematically improved also for excitation energies, and further work in this field is warranted.**90**Mester, D.; Kállay, M. Spin-Scaled Range-Separated Double-Hybrid Density Functional Theory for Excited States.*J. Chem. Theory Comput.*2021,*17*, 4211, DOI: 10.1021/acs.jctc.1c00422Google Scholar90https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXhtlCqu7fP&md5=5dfa9ca2e71c4abab4820a1c56ca71f8Spin-Scaled Range-Separated Double-Hybrid Density Functional Theory for Excited StatesMester, David; Kallay, MihalyJournal of Chemical Theory and Computation (2021), 17 (7), 4211-4224CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Our recently presented range-sepd. (RS) double-hybrid (DH) time-dependent d. functional approach [J. Chem. Theory Comput.17, 927 (2021)] is combined with spin-scaling techniques. The proposed spin-component-scaled (SCS) and scaled-opposite-spin (SOS) variants are thoroughly tested for almost 500 excitations including the most challenging types. This comprehensive study provides useful information not only about the new approaches but also about the most prominent methods in the DH class. The benchmark calcns. confirm the robustness of the RS-DH ansatz, while several tendencies and deficiencies are pointed out for the existing functionals. Our results show that the SCS variant consistently improves the results, while the SOS variant preserves the benefits of the original RS-DH method reducing its computational expenses. It is also demonstrated that, besides our approaches, only the nonempirical functionals provide balanced performance for general applications, while particular methods are only suggested for certain types of excitations.**91**Mester, D.; Kállay, M. Accurate Spectral Properties within Double-Hybrid Density Functional Theory: A Spin-Scaled Range-Separated Second-Order Algebraic-Diagrammatic Construction-Based Approach.*J. Chem. Theory Comput.*2022,*18*, 865, DOI: 10.1021/acs.jctc.1c01100Google Scholar91https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XhtVSlt7c%253D&md5=8cab654bf1030d77b6a385d22c302e21Accurate Spectral Properties within Double-Hybrid Density Functional Theory: A Spin-Scaled Range-Separated Second-Order Algebraic-Diagrammatic Construction-Based ApproachMester, David; Kallay, MihalyJournal of Chemical Theory and Computation (2022), 18 (2), 865-882CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Our second-order algebraic-diagrammatic construction [ADC(2)]-based double-hybrid (DH) ansatz [J. Chem. Theory Comput.15, 4440 (2019)] is combined with range-sepn. techniques. In the present scheme, both the exchange and the correlation contributions are range-sepd., while spin-scaling approaches are also applied. The new methods are thoroughly tested for the most popular benchmark sets including 250 singlet and 156 triplet excitations, as well as 80 oscillator strengths. It is demonstrated that the range sepn. for the correlation contributions is highly recommended for both the genuine and the ADC(2)-based DH approaches. Our results show that the latter scheme slightly but consistently outperforms the former one for single excitation dominated transitions. Furthermore, states with larger fractions of double excitations are assessed as well, and challenging charge-transfer excitations are also discussed, where the recently proposed spin-scaled long-range cor. DHs fail. The suggested iterative fourth-power scaling RS-PBE-P86/SOS-ADC(2) method, using only three adjustable parameters, provides the most robust and accurate excitation energies within the DH theory. In addn., the relative error of the oscillator strengths is reduced by 65% compared to the best genuine DH functionals.**92**Śmiga, S.; Grabowski, I. Spin-Component-Scaled ΔMP2 Parametrization: Toward a Simple and Reliable Method for Ionization Energies.*J. Chem. Theory Comput.*2018,*14*, 4780, DOI: 10.1021/acs.jctc.8b00638Google Scholar92https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhtlyktLfM&md5=1eddf80e99291290a29224478cd2ec04Spin-Component-Scaled ΔMP2 Parametrization: Toward a Simple and Reliable Method for Ionization EnergiesSmiga, Szymon; Grabowski, IreneuszJournal of Chemical Theory and Computation (2018), 14 (9), 4780-4790CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A practical, accurate, and cost- and implementation-free method (ΔMP2-SOS(IP)) for the calcn. of vertical ionization potentials is proposed. The simple method is based on a single-step, a diagonal, frequency-independent approxn. to the second-order self-energy expression combined with the spin-component-scaled technique. The search for an optimal scaling factor is performed for a set of 50 moderately sized mols., and the quality of the method is addnl. assessed for a benchmark set of 24 org. acceptor mols. The proposed ΔMP2-SOS(IP) method provides the best results of valence ionization energies as compared to the several std. self-consistent variants of the electron propagator methods at the second and higher orders (EP2, SCS-EP2, EP3, OVGF) with almost CCSD(T) or IP-EOM-CCSD accuracy and the cost of only a single opposite-spin ΔMP2-type calcn. (O(N3)). For core ionization energies, our new methods outperform the std. ΔMP2 results due to a better balanced treatment of the correlation and relaxation term in the second-order self-energy.**93**Pickup, B. T.; Goscinski, O. Direct calculation of ionization energies.*Mol. Phys.*1973,*26*, 1013, DOI: 10.1080/00268977300102261Google Scholar93https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE3sXlsV2gtLw%253D&md5=9fcb0beae9c8316d891a5d2e4760a742Direct calculation of ionization energies. I. Closed shellsPickup, B. T.; Goscincki, O.Molecular Physics (1973), 26 (4), 1013-35CODEN: MOPHAM; ISSN:0026-8976.The advantages of the propagator formalism, as a direct method of calcg. ionization energies, are stressed. The propagator equations are derived for closed-shell systems using an operator method instead of the usual diagrammatic derivations. The equations enable the development of an interpretation of the ionization energies in terms of conceptually simple quantities, such as pair correlation energies and assocd. relaxation effects, and retain the idea of orbital ionization. Infinite summations appearing in the self-energy terms are replaced by finite expressions involving functions satisfying uncoupled inhomogeneous differential equations. Certain high-order propagator equations are derived, and a connection with the Bethe-Goldstone formulation of pair correlation is made. Several computational procedures are advocated as forming the basis for balanced calcns. of at. and mol. ionization energies.**94**Vidal, M. L.; Krylov, A. I.; Coriani, S. Dyson orbitals within the fc-CVS-EOM-CCSD framework: theory and application to X-ray photoelectron spectroscopy of ground and excited states.*Phys. Chem. Chem. Phys.*2020,*22*, 2693, DOI: 10.1039/C9CP03695DGoogle Scholar94https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhvFKgs73O&md5=e5178ffb5ce5a1412e72ce3f09878ec7Dyson orbitals within the fc-CVS-EOM-CCSD framework: theory and application to X-ray photoelectron spectroscopy of ground and excited statesVidal, Marta L.; Krylov, Anna I.; Coriani, SoniaPhysical Chemistry Chemical Physics (2020), 22 (5), 2693-2703CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)We report on the implementation of Dyson orbitals within the recently introduced frozen-core (fc) core-valence sepd. (CVS) equation-of-motion (EOM) coupled-cluster singles and doubles (CCSD) method, which enables efficient and reliable characterization of core-level states. The ionization potential (IP) variant of fc-CVS-EOM-CCSD, in which the EOM target states have one electron less than the ref., gives access to core-ionized states thus enabling modeling of X-ray photoelectron spectra (XPS) and its time-resolved variant (TR-XPS). Dyson orbitals are reduced quantities that can be interpreted as correlated states of the ejected/attached electron; they enter the expressions of various exptl. relevant quantities. In the context of photoelectron spectroscopy, Dyson orbitals can be used to est. the strengths of photoionization transitions. We illustrate the utility of Dyson orbitals and fc-CVS-EOM-IP-CCSD by calcg. XPS of the ground state of adenine and TR-XPS of the excited states of uracil.**95**Díaz-Tinoco, M.; Corzo, H. H.; Pawłowski, F.; Ortiz, J. V. Do Dyson Orbitals resemble canonical Hartree–Fock orbitals?.*Mol. Phys.*2019,*117*, 2275, DOI: 10.1080/00268976.2018.1535142Google Scholar95https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhvFajt7%252FI&md5=fd0e6fc4899694f1c3dc8c9ed858f09cDo Dyson Orbitals resemble canonical Hartree-Fock orbitals?Diaz-Tinoco, Manuel; Corzo, Hector H.; Pawlowski, Filip; Ortiz, J. V.Molecular Physics (2019), 117 (17), 2275-2283CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)A review. Dyson orbitals are overlaps between states with N and N±1 electrons and provide conceptual links between transition probabilities of electron detachment or attachment, d. matrixes, total energies and general principles of chem. bonding. Canonical, Hartree-Fock orbitals are compared with Dyson orbitals obtained with electron-propagator calcns. that retain all elements of the self-energy matrix, wherein all orbital-relaxation and electron-correlation corrections to Koopmans results reside. For valence ionization energies and electron affinities of representative closed-shell mols., canonical, Hartree-Fock orbitals usually are excellent approxns. to Dyson orbitals, although there are some notable cases where the resemblance is not as strong. Numerical relationships between pole strengths and the Koopmans contributions to Dyson orbitals also are inferred from the data.**96**Ortiz, J. V. Dyson-orbital concepts for description of electrons in molecules.*J. Chem. Phys.*2020,*153*, 070902, DOI: 10.1063/5.0016472Google Scholar96https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhs1ert7%252FL&md5=cec497ef333e96a3120a0e62753a604aDyson-orbital concepts for description of electrons in moleculesOrtiz, J. V.Journal of Chemical Physics (2020), 153 (7), 070902CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Dyson orbitals, their electron-binding energies, and probability factors provide descriptions of electrons in mols. that are exptl. verifiable and that generalize qual. useful concepts of uncorrelated, mol.-orbital theory to the exact limit of Schrodinger's time-independent equation. Dyson orbitals are defined as overlaps between initial, N-electron states and final states with N±1 electrons and therefore are useful in the prediction and interpretation of many kinds of spectroscopic and scattering expts. They also are characteristic of N-electron initial states and may be used to construct electron densities, one-electron properties, and total energies with correlated Aufbau procedures that include probability factors between zero and unity. Relationships with natural orbitals, Kohn-Sham orbitals, and Hartree-Fock orbitals facilitate insights into the descriptive capabilities of Dyson orbitals. Electron-propagator approxns. that employ the Dyson quasiparticle equation or super-operator secular equations enable direct detn. of Dyson orbitals and obviate the need for many-electron wavefunctions of initial or final states. Numerical comparisons of the amplitudes and probability factors of Dyson orbitals calcd. with several self-energy approxns. reveal the effects of electron correlation on these uniquely defined, one-electron wavefunctions. (c) 2020 American Institute of Physics.**97**Kállay, M.; Nagy, P. R.; Mester, D.; Gyevi-Nagy, L.; Csóka, J.; Szabó, P. B.; Rolik, Z.; Samu, G.; Csontos, J.; Hégely, B.; Ganyecz, Á.; Ladjánszki, I.; Szegedy, L.; Ladóczki, B.; Petrov, K.; Farkas, M.; Mezei, P. D.; Horváth, R. A.*Mrcc, a quantum chemical program suite*. See https://www.mrcc.hu/ (accessed March 1, 2023).Google ScholarThere is no corresponding record for this reference.**98**Butscher, W.; Kammer, W. E. Modification of Davidson’s Method for the Calculation of Eigenvalues and Eigenvectors of Large Real-Symmetric Matrices: “Root Homing Procedure”.*J. Comput. Phys.*1976,*20*, 313, DOI: 10.1016/0021-9991(76)90084-XGoogle ScholarThere is no corresponding record for this reference.**99**Dunning, T. H., Jr. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen.*J. Chem. Phys.*1989,*90*, 1007, DOI: 10.1063/1.456153Google Scholar99https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1MXksVGmtrk%253D&md5=c6cd67a3748dc61692a9cb622d2694a0Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogenDunning, Thom H., Jr.Journal of Chemical Physics (1989), 90 (2), 1007-23CODEN: JCPSA6; ISSN:0021-9606.Guided by the calcns. on oxygen in the literature, basis sets for use in correlated at. and mol. calcns. were developed for all of the first row atoms from boron through neon, and for hydrogen. As in the oxygen atom calcns., the incremental energy lowerings, due to the addn. of correlating functions, fall into distinct groups. This leads to the concept of correlation-consistent basis sets, i.e., sets which include all functions in a given group as well as all functions in any higher groups. Correlation-consistent sets are given for all of the atoms considered. The most accurate sets detd. in this way, [5s4p3d2f1g], consistently yield 99% of the correlation energy obtained with the corresponding at.-natural-orbital sets, even though the latter contains 50% more primitive functions and twice as many primitive polarization functions. It is estd. that this set yields 94-97% of the total (HF + 1 + 2) correlation energy for the atoms neon through boron.**100**Woon, D. E.; Dunning, T. H., Jr. Gaussian basis sets for use in correlated molecular calculations. III. The atoms aluminum through argon.*J. Chem. Phys.*1993,*98*, 1358, DOI: 10.1063/1.464303Google Scholar100https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXhtlans7Y%253D&md5=3f2e6860ac29511cb96da63f31bdc1eeGaussian basis sets for use in correlated molecular calculations. III. The atoms aluminum through argonWoon, David E.; Dunning, Thom H., Jr.Journal of Chemical Physics (1993), 98 (2), 1358-71CODEN: JCPSA6; ISSN:0021-9606.Correlation consistent and augmented correlation consistent basis sets were detd. for the second row atoms. The methodol., originally developed for the first row atoms (T. H. D., Jr., 1989) is first applied to S. The exponents for the polarization functions (dfgh) are systematically optimized for a correlated wave function (HF+1+2). The (sp) correlation functions are taken from the appropriate HF primitive sets; these functions differ little from the optimum functions. Basis sets of double zeta [4s3p1d], triple zeta [5s4p2d1f], and quadruple zeta [6s5p3d2f1g] quality are defined. Each of these sets is then augmented with diffuse functions to better describe electron affinities and other mol. properties: s and p functions were obtained by optimization for the anion HF energy, while an addnl. polarization function for each symmetry present in the std. set was optimized for the anion HF+1+2 energy. The results for S are then used to assist in detg. double zeta, triple zeta, and quadruple zeta basis sets for the remainder of the second row of the p block.**101**Kendall, R. A.; Dunning, T. H., Jr.; Harrison, R. J. Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions.*J. Chem. Phys.*1992,*96*, 6796, DOI: 10.1063/1.462569Google Scholar101https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK38XktFClurw%253D&md5=948a06eee10604a8fa37eae2b2ada4beElectron affinities of the first-row atoms revisited. Systematic basis sets and wave functionsKendall, Rick A.; Dunning, Thom H., Jr.; Harrison, Robert J.Journal of Chemical Physics (1992), 96 (9), 6796-806CODEN: JCPSA6; ISSN:0021-9606.The authors describe a reliable procedure for calcg. the electron affinity of an atom and present results for H, B, C, O, and F (H is included for completeness). This procedure involves the use of the recently proposed correlation-consistent basis sets augmented with functions to describe the more diffuse character of the at. anion coupled with a straightforward, uniform expansion of the ref. space for multireference singles and doubles configuration-interaction (MRSD-CI) calcns. A comparison is given with previous results and with corresponding full CI calcns. The most accurate EAs obtained from the MRSD-CI calcns. are (with exptl. values in parentheses): H 0.740 eV (0.754), B 0.258 (0.277), C 1.245 (1.263), O 1.384 (1.461), and F 3.337 (3.401). The EAs obtained from the MR-SDCI calcns. differ by less than 0.03 eV from those predicted by the full CI calcns.**102**Weigend, F.; Köhn, A.; Hättig, C. Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations.*J. Chem. Phys.*2002,*116*, 3175, DOI: 10.1063/1.1445115Google Scholar102https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XhtlSiu7k%253D&md5=0130fa656254a693e80d4be6b0f442b8Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculationsWeigend, Florian; Kohn, Andreas; Hattig, ChristofJournal of Chemical Physics (2002), 116 (8), 3175-3183CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The convergence of the second-order Moller-Plesset perturbation theory (MP2) correlation energy with the cardinal no. X is investigated for the correlation consistent basis-set series cc-pVXZ and cc-pV(X+d)Z. For the aug-cc-pVXZ and aug-cc-pV(X+d)Z series the convergence of the MP2 correlation contribution to the dipole moment is studied. It is found that, when d-shell electrons cannot be frozen, the cc-pVXZ and aug-cc-pVXZ basis sets converge much slower for third-row elements then they do for first- and second-row elements. Based on the results of these studies criteria are deduced for the accuracy of auxiliary basis sets used in the resoln. of the identity (RI) approxn. for electron repulsion integrals. Optimized auxiliary basis sets for RI-MP2 calcns. fulfilling these criteria are reported for the sets cc-pVXZ, cc-pV(X+d)Z, aug-cc-pVXZ, and aug-cc-pV(X+d)Z with X=D, T, and Q. For all basis sets the RI error in the MP2 correlation energy is more than two orders of magnitude smaller than the usual basis-set error. For the auxiliary aug-cc-pVXZ and aug-cc-pV(X+d)Z sets the RI error in the MP2 correlation contribution to the dipole moment is one order of magnitude smaller than the usual basis set error. Therefore extrapolations towards the basis-set limit are possible within the RI approxn. for both energies and properties. The redn. in CPU time obtained with the RI approxn. increases rapidly with basis set size. For the cc-pVQZ basis an acceleration by a factor of up to 170 is obsd.**103**Weigend, F.; Häser, M.; Patzelt, H.; Ahlrichs, R. RI-MP2: optimized auxiliary basis sets and demonstration of efficiency.*Chem. Phys. Lett.*1998,*294*, 143, DOI: 10.1016/S0009-2614(98)00862-8Google Scholar103https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXlvVGjtrc%253D&md5=305af6c7dc6ec9a83cb2d4760adb9f3bRI-MP2: optimized auxiliary basis sets and demonstration of efficiencyWeigend, Florian; Haser, Marco; Patzelt, Holger; Ahlrichs, ReinhartChemical Physics Letters (1998), 294 (1,2,3), 143-152CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)Applications of the RI-MP2 method require high-quality auxiliary basis sets employed to approx. charge distributions. A variational procedure is proposed and applied to optimize auxiliary bases for main group and transition metal atoms which are tested for more than 350 mols. The RI approxn. affects mol. MP2 energies by less than 60 μEh per atom and equil. distances by less than 0.2 pm. We further comment on the extension from RHF to UHF and the exploitation of mol. symmetry. Applications to (Cu2S)n clusters and hydrocarbons CnH2n+2 document a significant redn. of computation times which allows for calcns. with up to 1000 basis functions in C1 symmetry.**104**Weigend, F. Hartree–Fock Exchange Fitting Basis Sets for H to Rn.*J. Comput. Chem.*2008,*29*, 167, DOI: 10.1002/jcc.20702Google Scholar104https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BD2sjktVOjtw%253D%253D&md5=e16d82b5b49d9c043fc719723cfd69aeHartree-Fock exchange fitting basis sets for H to RnWeigend FlorianJournal of computational chemistry (2008), 29 (2), 167-75 ISSN:0192-8651.For elements H to Rn (except Lanthanides), a series of auxiliary basis sets fitting exchange and also Coulomb potentials in Hartree-Fock treatments (RI-JK-HF) is presented. A large set of small molecules representing nearly each element in all its common oxidation states was used to assess the quality of these auxiliary bases. For orbital basis sets of triple zeta valence and quadruple zeta valence quality, errors in total energies arising from the RI-JK approximation are below approximately 1 meV per atom in molecular compounds. Accuracy of RI-JK-approximated HF wave functions is sufficient for being used for post-HF treatments like Moller-Plesset perturbation theory, MP2. Compared to nonapproximated treatments, RI-JK-HF leads to large computational savings for quadruple zeta valence orbital bases and, in case of small to midsize systems, to significant savings for triple zeta valence bases.**105**Krack, M.; Köster, A. M. An adaptive numerical integrator for molecular integrals.*J. Chem. Phys.*1998,*108*, 3226, DOI: 10.1063/1.475719Google Scholar105https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXhtFChs7s%253D&md5=d724deb8144792b35e82a98c5dbdb5d5An adaptive numerical integrator for molecular integralsKrack, Matthias; Koster, Andreas M.Journal of Chemical Physics (1998), 108 (8), 3226-3234CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a new numerical integrator for mol. integrals that generates automatically an adaptive mol. grid. The tolerance of the numerical integration is the only input parameter of the integrator besides the at. coordinates and the at. nos. The adaptive numerical integrator was successfully implemented in our new d.-functional theory method (DFT method) ALLCHEM using the self consistent field (SCF) procedure. The accuracy of the numerical integration is superior to pruned fixed grids for a given no. of grid points. The adaptive grid generator allows a very efficient optimization of the grid for an individual mol. system. It is shown that the grid accuracy increases monotonically, if the given tolerance of the numerical integration is decreased. In this way it is possible to obtain results of high numerical precision. For a given tolerance of the numerical integration the adaptive grid generator automatically adjusts the grid to the basis set and the mol. structure. This feature increases the reliability of the numerical integration considerably and the no. of expensive ref. grid calcns. can be reduced. The time for the grid generation is small compared to the total computation time for a SCF calcn.**106**Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple.*Phys. Rev. Lett.*1996,*77*, 3865, DOI: 10.1103/PhysRevLett.77.3865Google Scholar106https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28XmsVCgsbs%253D&md5=55943538406ee74f93aabdf882cd4630Generalized gradient approximation made simplePerdew, John P.; Burke, Kieron; Ernzerhof, MatthiasPhysical Review Letters (1996), 77 (18), 3865-3868CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)Generalized gradient approxns. (GGA's) for the exchange-correlation energy improve upon the local spin d. (LSD) description of atoms, mols., and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental consts. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential.**107**Becke, A. D. Density-functional exchange-energy approximation with correct asymptotic-behavior.*Phys. Rev. A*1988,*38*, 3098, DOI: 10.1103/PhysRevA.38.3098Google Scholar107https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1cXmtlOhsLo%253D&md5=d4d219c134a5a90f689a8abed04d82ccDensity-functional exchange-energy approximation with correct asymptotic behaviorBecke, A. D.Physical Review A: Atomic, Molecular, and Optical Physics (1988), 38 (6), 3098-100CODEN: PLRAAN; ISSN:0556-2791.Current gradient-cor. d.-functional approxns. for the exchange energies of at. and mol. systems fail to reproduce the correct 1/r asymptotic behavior of the exchange-energy d. A gradient-cor. exchange-energy functional is given with the proper asymptotic limit. This functional, contg. only one parameter, fits the exact Hartree-Fock exchange energies of a wide variety of at. systems with remarkable accuracy, surpassing the performance of previous functionals contg. two parameters or more.**108**Lee, C.; Yang, W.; Parr, R. G. Development of the Colle–Salvetti correlation-energy formula into a functional of the electron density.*Phys. Rev. B*1988,*37*, 785, DOI: 10.1103/PhysRevB.37.785Google Scholar108https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1cXktFWrtbw%253D&md5=ee7b59267a2ff72e15171a481819ccf8Development of the Colle-Salvetti correlation-energy formula into a functional of the electron densityLee, Chengteh; Yang, Weitao; Parr, Robert G.Physical Review B: Condensed Matter and Materials Physics (1988), 37 (2), 785-9CODEN: PRBMDO; ISSN:0163-1829.A correlation-energy formula due to R. Colle and D. Salvetti (1975), in which the correlation energy d. is expressed in terms of the electron d. and a Laplacian of the 2nd-order Hartree-Fock d. matrix, is restated as a formula involving the d. and local kinetic-energy d. On insertion of gradient expansions for the local kinetic-energy d., d.-functional formulas for the correlation energy and correlation potential are then obtained. Through numerical calcns. on a no. of atoms, pos. ions, and mols., of both open- and closed-shell type, it is demonstrated that these formulas, like the original Colle-Salvetti formulas, give correlation energies within a few percent.**109**Perdew, J. P. Density-functional approximation for the correlation energy of the inhomogeneous electron gas.*Phys. Rev. B*1986,*33*, 8822, DOI: 10.1103/PhysRevB.33.8822Google Scholar109https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC2sfgsFSktA%253D%253D&md5=fb343a074cf09acda3e96d7f13ec2c7eDensity-functional approximation for the correlation energy of the inhomogeneous electron gasPerdewPhysical review. B, Condensed matter (1986), 33 (12), 8822-8824 ISSN:0163-1829.There is no expanded citation for this reference.**110**Becke, A. D. Density-functional thermochemistry. V. Systematic optimization of exchange-correlation functionals.*J. Chem. Phys.*1997,*107*, 8554, DOI: 10.1063/1.475007Google Scholar110https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXntFGiuro%253D&md5=e9e466d42d8ea239be08b3a1ede19ae7Density-functional thermochemistry. V. Systematic optimization of exchange-correlation functionalsBecke, Axel D.Journal of Chemical Physics (1997), 107 (20), 8554-8560CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A systematic procedure for refining gradient corrections in Kohn-Sham exchange-correlation functionals is presented. The procedure is based on least-squares fitting to accurate thermochem. data. In this first application of the method, we use the G2 test set of Pople and co-workers to generate what we believe to be an optimum GGA/exact-exchange d.-functional theory (i.e., generalized gradient approxn. with mixing of exactly computed exchange).**111**Lehtola, S.; Steigemann, C.; Oliveira, M. J. T.; Marques, M. A. L. Recent developments in Libxc – A comprehensive library of functionals for density functional theory.*SoftwareX*2018,*7*, 1, DOI: 10.1016/j.softx.2017.11.002Google ScholarThere is no corresponding record for this reference.**112**https://www.tddft.org/programs/libxc/ (accessed May 1, 2023).Google ScholarThere is no corresponding record for this reference.**113**Kozuch, S.; Martin, J. M. L. DSD-PBEP86: in search of the best double-hybrid DFT with spin-component scaled MP2 and dispersion corrections.*Phys. Chem. Chem. Phys.*2011,*13*, 20104, DOI: 10.1039/c1cp22592hGoogle Scholar113https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhsVCgurbF&md5=c16e37716e24bc5aa979fe3d506a92f5DSD-PBEP86: in search of the best double-hybrid DFT with spin-component scaled MP2 and dispersion correctionsKozuch, Sebastian; Martin, Jan M. L.Physical Chemistry Chemical Physics (2011), 13 (45), 20104-20107CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)Spin-component scaled double hybrids including dispersion correction were optimized for many exchange and correlation functionals. Even DSD-LDA performs surprisingly well. DSD-PBEP86 emerged as a very accurate and robust method, approaching the accuracy of composite ab initio methods at a fraction of their computational cost.**114**Chai, J.-D.; Mao, S.-P. Seeking for reliable double-hybrid density functionals without fitting parameters: The PBE0-2 functional.*Chem. Phys. Lett.*2012,*538*, 121, DOI: 10.1016/j.cplett.2012.04.045Google Scholar114https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XnslKhtL4%253D&md5=dec5a978da0c4abb00356eacab979aaaSeeking for reliable double-hybrid density functionals without fitting parameters: The PBE0-2 functionalChai, Jeng-Da; Mao, Shan-PingChemical Physics Letters (2012), 538 (), 121-125CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)Without the use of any empirical fitting to exptl. or high-level ab initio data, we present a double-hybrid d. functional approxn. for the exchange-correlation energy, combining the exact Hartree-Fock exchange and second-order Moller-Plesset (MP2) correlation with the Perdew-Burke-Ernzerhof (PBE) functional. This functional, denoted as PBE0-2, is shown to be accurate for a wide range of applications, when compared with other functionals and the ab initio MP2 method. The qual. failures of conventional d. functional approxns., such as self-interaction error and noncovalent interaction error, are significantly reduced by PBE0-2.**115**Alipour, M. On the opposite-spin to same-spin ratio of absolute and interaction MP2 correlation energy in parameter-free spin-opposite-scaled double hybrids.*Chem. Phys. Lett.*2017,*684*, 423, DOI: 10.1016/j.cplett.2017.07.023Google Scholar115https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhtFynsLjN&md5=2ac6a347bf46db47b308d2122890fe0cOn the opposite-spin to same-spin ratio of absolute and interaction MP2 correlation energy in parameter-free spin-opposite-scaled double hybridsAlipour, MojtabaChemical Physics Letters (2017), 684 (), 423-426CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)The applicability of the two types of opposite-spin to same-spin ratio of second-order Moller-Plesset correlation energy, namely the ratio of opposite-spin to same-spin of the abs. correlation energy and the most recently proposed ratio of opposite-spin to same-spin of the interaction correlation energy, in spin-opposite-scaled parameter-free double-hybrid d. functionals has been evaluated. Considering the interaction energies and hydrogen transfer barrier heights benchmark sets and comparing various functionals, it is shown that using the ratio related to the abs. correlation energy in double-hybrid calcns. provides lower deviations than that of the same ratio for the interaction correlation energy.**116**Brémond, É.; Sancho-García, J. C.; Pérez-Jiménez, Á. J.; Adamo, C. Double-hybrid functionals from adiabatic-connection: The QIDH model.*J. Chem. Phys.*2014,*141*, 031101, DOI: 10.1063/1.4890314Google Scholar116https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhtFOkt7jL&md5=4b517192cbedb346344b06b1d0c6ae2eCommunication: Double-hybrid functionals from adiabatic-connection: The QIDH modelBremond, Eric; Sancho-Garcia, Juan Carlos; Perez-Jimenez, Angel Jose; Adamo, CarloJournal of Chemical Physics (2014), 141 (3), 031101/1-031101/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A new approach stemming from the adiabatic-connection (AC) formalism is proposed to derive parameter-free double-hybrid (DH) exchange-correlation functionals. It is based on a quadratic form that models the integrand of the coupling parameter, whose components are chosen to satisfy several well-known limiting conditions. Its integration leads to DHs contg. a single parameter controlling the amt. of exact exchange, which is detd. by requiring it to depend on the wt. of the MP2 correlation contribution. Two new parameter-free DHs functionals are derived in this way, by incorporating the non-empirical PBE and TPSS functionals in the underlying expression. Their extensive testing using the GMTKN30 benchmark indicates that they are in competition with state-of-the-art DHs, yet providing much better self-interaction errors and opening a new avenue towards the design of accurate double-hybrid exchange-correlation functionals departing from the AC integrand. (c) 2014 American Institute of Physics.**117**Karton, A.; Tarnopolsky, A.; Lamère, J.-F.; Schatz, G. C.; Martin, J. M. L. Highly Accurate First-Principles Benchmark Data Sets for the Parametrization and Validation of Density Functional and Other Approximate Methods. Derivation of a Robust, Generally Applicable, Double-Hybrid Functional for Thermochemistry and Thermochemical Kinetics.*J. Phys. Chem. A*2008,*112*, 12868, DOI: 10.1021/jp801805pGoogle Scholar117https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXhtVSnt7vI&md5=b20615bf75cfd7c55daa27a2a7aec3dcHighly Accurate First-Principles Benchmark Data Sets for the Parametrization and Validation of Density Functional and Other Approximate Methods. Derivation of a Robust, Generally Applicable, Double-Hybrid Functional for Thermochemistry and Thermochemical KineticsKarton, Amir; Tarnopolsky, Alex; Lamere, Jean-Francois; Schatz, George C.; Martin, Jan M. L.Journal of Physical Chemistry A (2008), 112 (50), 12868-12886CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)We present a no. of near-exact, nonrelativistic, Born-Oppenheimer ref. data sets for the parametrization of more approx. methods (such as DFT functionals). The data were obtained by means of the W4 ab initio computational thermochem. protocol, which has a 95% confidence interval well below 1 kJ/mol. Our data sets include W4-08, which are total atomization energies of over 100 small mols. that cover varying degrees of non-dynamical correlations, and DBH24-W4, which are W4 theory values for Truhlar's set of 24 representative barrier heights. The usual procedure of comparing calcd. DFT values with exptl. atomization energies is hampered by comparatively large exptl. uncertainties in many exptl. values and compds. errors due to deficiencies in the DFT functional with those resulting from neglect of relativity and finite nuclear mass. Comparison with accurate, explicitly nonrelativistic, ab initio data avoids these issues. We then proceed to explore the performance of B2x-PLYP-type double hybrid functionals for atomization energies and barrier heights. The optimum hybrids for hydrogen-transfer reactions, heavy-atoms transfers, nucleophilic substitutions, and unimol. and recombination reactions are quite different from one another: out of these subsets, the heavy-atom transfer reactions are by far the most sensitive to the percentages of Hartree-Fock-type exchange y and MP2-type correlation x in an (x,y) double hybrid. The (42,72) hybrid B2K-PLYP, as reported in a preliminary communication, represents the best compromise between thermochem. and hydrogen-transfer barriers, while also yielding excellent performance for nucleophilic substitutions. By optimizing for best overall performance on both thermochem. and the DBH24-W4 data set, however, we find a new (36,65) hybrid which we term B2GP-PLYP. At a slight expense in performance for hydrogen-transfer barrier heights and nucleophilic substitutions, we obtain substantially better performance for the other reaction types. Although both B2K-PLYP and B2GP-PLYP are capable of 2 kcal/mol quality thermochem., B2GP-PLYP appears to be the more robust toward non-dynamical correlation and strongly polar character. We addnl. find that double-hybrid functionals display excellent performance for such problems as hydrogen bonding, prototype late transition metal reactions, pericyclic reactions, prototype cumulene-polyacetylene system, and weak interactions.**118**Chai, J.-D.; Head-Gordon, M. Long-range corrected hybrid density functionals with damped atom–atom dispersion corrections.*Phys. Chem. Chem. Phys.*2008,*10*, 6615, DOI: 10.1039/b810189bGoogle Scholar118https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXhtlCksbfO&md5=c7848f8bf050e11972d438aaebd68fdfLong-range corrected hybrid density functionals with damped atom-atom dispersion correctionsChai, Jeng-Da; Head-Gordon, MartinPhysical Chemistry Chemical Physics (2008), 10 (44), 6615-6620CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)We report re-optimization of a recently proposed long-range cor. (LC) hybrid d. functional [J.-D. Chai and M. Head-Gordon, J. Chem. Phys., 2008, 128, 084106] to include empirical atom-atom dispersion corrections. The resulting functional, ωB97X-D yields satisfactory accuracy for thermochem., kinetics, and non-covalent interactions. Tests show that for non-covalent systems, ωB97X-D shows slight improvement over other empirical dispersion-cor. d. functionals, while for covalent systems and kinetics it performs noticeably better. Relative to our previous functionals, such as ωB97X, the new functional is significantly superior for non-bonded interactions, and very similar in performance for bonded interactions.**119**Yanai, T.; Tew, D. P.; Handy, N. C. A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP).*Chem. Phys. Lett.*2004,*393*, 51, DOI: 10.1016/j.cplett.2004.06.011Google Scholar119https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXlsFKgtbs%253D&md5=75f311240ff8ebedb174757f3eedbf3eA new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)Yanai, Takeshi; Tew, David P.; Handy, Nicholas C.Chemical Physics Letters (2004), 393 (1-3), 51-57CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)A new hybrid exchange-correlation functional named CAM-B3LYP is proposed. It combines the hybrid qualities of B3LYP and the long-range correction presented by Tawada et al. [J. Chem. Phys., in press]. We demonstrate that CAM-B3LYP yields atomization energies of similar quality to those from B3LYP, while also performing well for charge transfer excitations in a dipeptide model, which B3LYP underestimates enormously. The CAM-B3LYP functional comprises of 0.19 Hartree-Fock (HF) plus 0.81 Becke 1988 (B88) exchange interaction at short-range, and 0.65 HF plus 0.35 B88 at long-range. The intermediate region is smoothly described through the std. error function with parameter 0.33.**120**Perdew, J. P.; Ernzerhof, M.; Burke, K. Rationale for mixing exact exchange with density functional approximations.*J. Chem. Phys.*1996,*105*, 9982, DOI: 10.1063/1.472933Google Scholar120https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28XnsFahtbg%253D&md5=cb0b0c07f3fde8c429bfe9fa8a1f2a4aRationale for mixing exact exchange with density functional approximationsPerdew, John P.; Ernzerhof, Matthias; Burke, KieronJournal of Chemical Physics (1996), 105 (22), 9982-9985CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)D. functional approxns. for the exchange-correlation energy ExcDFA of an electronic system are often improved by admixing some exact exchange Ex: Exc ≈ ExcDFA + (1/n)(Ex - ExDFA). This procedure is justified when the error in ExcDFA arises from the λ = 0 or exchange end of the coupling-const. integral ∫01dλ Exc,λDFA. We argue that the optimum integer n is approx. the lowest order of Goerling-Levy perturbation theory which provides a realistic description of the coupling-const. dependence Exc,λ in the range 0 ≤ λ ≤ 1, whence n ≈ 4 for atomization energies of typical mols. We also propose a continuous generalization of n as an index of correlation strength, and a possible mixing of second-order perturbation theory with the generalized gradient approxn.**121**Liang, J.; Feng, X.; Hait, D.; Head-Gordon, M. Revisiting the Performance of Time-Dependent Density Functional Theory for Electronic Excitations: Assessment of 43 Popular and Recently Developed Functionals from Rungs One to Four.*J. Chem. Theory Comput.*2022,*18*, 3460, DOI: 10.1021/acs.jctc.2c00160Google Scholar121https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38Xht1WltLnI&md5=ce4df074cef538883bf95d89de45c766Revisiting the Performance of Time-Dependent Density Functional Theory for Electronic Excitations: Assessment of 43 Popular and Recently Developed Functionals from Rungs One to FourLiang, Jiashu; Feng, Xintian; Hait, Diptarka; Head-Gordon, MartinJournal of Chemical Theory and Computation (2022), 18 (6), 3460-3473CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)In this paper, the performance of more than 40 popular or recently developed d. functionals is assessed for the calcn. of 463 vertical excitation energies against the large and accurate QuestDB benchmark set. For this purpose, the Tamm-Dancoff approxn. offers a good balance between computational efficiency and accuracy. The functionals ωB97X-D and BMK are found to offer the best performance overall with a root-mean square error (RMSE) of around 0.27 eV, better than the computationally more demanding CIS(D) wave function method with a RMSE of 0.36 eV. The results also suggest that Jacob's ladder still holds for time-dependent d. functional theory excitation energies, though hybrid meta generalized-gradient approxns. (meta-GGAs) are not generally better than hybrid GGAs. Effects of basis set convergence, gauge invariance correction to meta-GGAs, and nonlocal correlation (VV10) are also studied, and practical basis set recommendations are provided.**122**Richard, R. M.; Marshall, M. S.; Dolgounitcheva, O.; Ortiz, J. V.; Brédas, J.-L.; Marom, N.; Sherrill, C. D. Accurate Ionization Potentials and Electron Affinities of Acceptor Molecules I. Reference Data at the CCSD(T) Complete Basis Set Limit.*J. Chem. Theory Comput.*2016,*12*, 595, DOI: 10.1021/acs.jctc.5b00875Google Scholar122https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xis1Kjsg%253D%253D&md5=272d81e4e0ecf61286a8492605953c70Accurate Ionization Potentials and Electron Affinities of Acceptor Molecules I. Reference Data at the CCSD(T) Complete Basis Set LimitRichard, Ryan M.; Marshall, Michael S.; Dolgounitcheva, O.; Ortiz, J. V.; Bredas, Jean-Luc; Marom, Noa; Sherrill, C. DavidJournal of Chemical Theory and Computation (2016), 12 (2), 595-604CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)In designing org. materials for electronics applications, particularly for org. photovoltaics (OPV), the ionization potential (IP) of the donor and the electron affinity (EA) of the acceptor play key roles. This makes OPV design an appealing application for computational chem. since IPs and EAs are readily calculable from most electronic structure methods. Unfortunately reliable, high-accuracy wave function methods, such as coupled cluster theory with single, double, and perturbative triples [CCSD(T)] in the complete basis set (CBS) limit are too expensive for routine applications to this problem for any but the smallest of systems. One soln. is to calibrate approx., less computationally expensive methods against a database of high-accuracy IP/EA values; however, to our knowledge, no such database exists for systems related to OPV design. The present work is the first of a multipart study whose overarching goal is to det. which computational methods can be used to reliably compute IPs and EAs of electron acceptors. This part introduces a database of 24 known org. electron acceptors and provides high-accuracy vertical IP and EA values expected to be within ±0.03 eV of the true non-relativistic, vertical CCSD(T)/CBS limit. Convergence of IP and EA values toward the CBS limit is studied systematically for the Hartree-Fock, MP2 correlation, and beyond-MP2 coupled cluster contributions to the focal point ests.**123**Ranasinghe, D. S.; Margraf, J. T.; Perera, A.; Bartlett, R. J. Vertical valence ionization potential benchmarks from equation-of-motion coupled cluster theory and QTP functionals.*J. Chem. Phys.*2019,*150*, 074108, DOI: 10.1063/1.5084728Google Scholar123https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXjsF2rtr0%253D&md5=d767ed147fcd273a0266a32f4990e13aVertical valence ionization potential benchmarks from equation-of-motion coupled cluster theory and QTP functionalsRanasinghe, Duminda S.; Margraf, Johannes T.; Perera, Ajith; Bartlett, Rodney J.Journal of Chemical Physics (2019), 150 (7), 074108/1-074108/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The ionization potential (IP) of a mol. quantifies the energy required to remove an electron from the system. As such, it is a fundamental quantity in the context of redox chem., charge transfer, and mol. electronics. The accurate theor. prediction of this property is therefore highly desirable for virtual materials design. Furthermore, vertical IPs are of interest in the development of many-body Green's function methods like the GW formalism, as well as d. functionals and semiempirical methods. In this contribution, we report over 1468 vertical valence IPs calcd. with the IP variant of equation-of-motion coupled cluster theory with singles and doubles (IP-EOM-CCSD) covering 155 mols. The purpose of this is two-fold: First, the quality of the predicted IPs is compared with respect to expts. and higher-order coupled cluster theory. This confirms the overall high accuracy and robustness of this method, with some outliers which are discussed in detail. Second, a large set of consistent theor. ref. values for vertical valence IPs are generated. This addresses a lack of reliable ref. data for lower-lying valence IPs, where exptl. data are often unavailable or of dubious quality. The benchmark set is then used to assess the quality of the eigenvalues predicted by different d. functional approxns. (via Bartlett's IP-eigenvalue theorem) and the extended Koopmans' theorem approach. The QTP family of functionals are found to be remarkably accurate, low-cost alternatives to IP-EOM-CCSD. (c) 2019 American Institute of Physics.**124**Alag, A. S.; Jelenfi, D. P.; Tajti, A.; Szalay, P. G. Accurate Prediction of Vertical Ionization Potentials and Electron Affinities from Spin-Component Scaled CC2 and ADC(2) Models.*J. Chem. Theory Comput.*2022,*18*, 6794, DOI: 10.1021/acs.jctc.2c00624Google ScholarThere is no corresponding record for this reference.**125**Śmiga, S.; Siecińska, S.; Grabowski, I. From simple molecules to nanotubes. Reliable predictions of ionization potentials from the ΔMP2-SCS methods.*New J. Phys.*2020,*22*, 083084, DOI: 10.1088/1367-2630/abaa00Google Scholar125https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXit1yltL%252FI&md5=b6f03b792c48bc7457ad96abb2fdedd3From simple molecules to nanotubes. Reliable predictions of ionization potentials from the ΔMP2-SCS methodsSmiga, Szymon; Siecinska, Sylwia; Grabowski, IreneuszNew Journal of Physics (2020), 22 (Aug.), 083084CODEN: NJOPFM; ISSN:1367-2630. (IOP Publishing Ltd.)The vertical ionization potentials for systems of various sizes, ranging from simple mols., DNA/RNA bases, donor and acceptor org. mol. systems as well as nanotubes are calcd. using the ΔMP2-SCS family of methods. We have shown that for all investigated cases, the ΔMP2-SCS methods, being almost cost free single-step post-Hartree-Fock calcn., provide very accurate vertical ionization potentials comparable in quality with state-of-art outer valence Green's function methods. Moreover, we show that a combination of the ΔMP2-SCS methods with the resoln. of identity technique is effective and reliable an alternative to the semi-empirical d. functional theory methods and Green's function-based calcns. of ionization potentials for large mol. systems such as silicon-based or org.-based solar cells, for which the IP-EOM-CCSD calcns. are too expensive for routine calcns.

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**1**Yeh, N.; Yeh, P. Organic solar cells: Their developments and potentials.*Renew. Sust. Energy Rev.*2013,*21*, 421, DOI: 10.1016/j.rser.2012.12.0461https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXksFags78%253D&md5=5b1dad002a9c806cbc670b98ce8a8ccfOrganic solar cells: Their developments and potentialsYeh, Naichia; Yeh, PulinRenewable & Sustainable Energy Reviews (2013), 21 (), 421-431CODEN: RSERFH; ISSN:1364-0321. (Elsevier Ltd.)This paper aims to review the developments and the potentials of org. photovoltaic, which has caught the attention of researchers of optoelectronics. The paper briefly introduces the physics underlying org. photovoltaic devices of donor-acceptor interfaces. It examines a variety of materials and architectures that benefit the performance of the org. photovoltaic cell; along with the important photogeneration factors including the exciton diffusion length as well as charge transport, sepn., and collection. The authors have reviewed the recent understanding of the mechanisms that govern these photocurrent generation steps and sketched out the search for alternative materials and device architectures. The review also discusses areas where active researches should be directed for cell efficiency improvement and outlines the issues to be resolved in order to speed up the commercialization.**2**Poelking, C.; Benduhn, J.; Spoltore, D.; Schwarze, M.; Roland, S.; Piersimoni, F.; Neher, D.; Leo, K.; Vandewal, K.; Andrienko, D. Open-circuit voltage of organic solar cells: interfacial roughness makes the difference.*Commun. Phys.*2022,*5*, 307, DOI: 10.1038/s42005-022-01084-x2https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XjtVWhtrzJ&md5=25621b124d3ad3c461ba6f824ddd9d05Open-circuit voltage of organic solar cells: interfacial roughness makes the differencePoelking, Carl; Benduhn, Johannes; Spoltore, Donato; Schwarze, Martin; Roland, Steffen; Piersimoni, Fortunato; Neher, Dieter; Leo, Karl; Vandewal, Koen; Andrienko, DenisCommunications Physics (2022), 5 (1), 307CODEN: CPOHDJ; ISSN:2399-3650. (Nature Portfolio)Org. photovoltaics (PV) is an energy-harvesting technol. that offers many advantages, such as flexibility, low wt. and cost, as well as environmentally benign materials and manufg. techniques. Despite growth of power conversion efficiencies to around 19% in the last years, org. PVs still lag behind inorg. PV technologies, mainly due to high losses in open-circuit voltage. Understanding and improving open circuit voltage in org. solar cells is challenging, as it is controlled by the properties of a donor-acceptor interface where the optical excitations are sepd. into charge carriers. Here, we provide an electrostatic model of a rough donor-acceptor interface and test it exptl. on small mol. PV materials systems. The model provides concise relationships between the open-circuit voltage, photovoltaic gap, charge-transfer state energy, and interfacial morphol. In particular, we show that the electrostatic bias generated across the interface reduces the photovoltaic gap. This neg. influence on open-circuit voltage can, however, be circumvented by adjusting the morphol. of the donor-acceptor interface.**3**Delgado, J. L.; Bouit, P.-A.; Filippone, S.; Herranz, M. A. Á.; Martín, N. Organic photovoltaics: a chemical approach.*Chem. Commun.*2010,*46*, 4853, DOI: 10.1039/c003088k3https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXotFensL8%253D&md5=429cc61efe0178bd3a61bdc0fb03eb63Organic photovoltaics: a chemical approachDelgado, Juan Luis; Bouit, Pierre-Antoine; Filippone, Salvatore; Herranz, Ma. Angeles; Martin, NazarioChemical Communications (Cambridge, United Kingdom) (2010), 46 (27), 4853-4865CODEN: CHCOFS; ISSN:1359-7345. (Royal Society of Chemistry)A review. The fundamental contribution of chem. to the multidisciplinary field of org. photovoltaics is presented in a systematic way through the wide variety of org. compds. synthesized to be successfully used in photovoltaic devices. Basic processes in org. solar cells, bulk heterojunction plastic solar cells based on semiconducting π-conjugated polymers and fullerenes, mol. bulk heterojunction solar cells based on π-conjugated oligomers or dyes and fullerenes, and org. solar cells with innovative carbon nanostructures are discussed.**4**Alizadeh, E.; Sanche, L. Precursors of Solvated Electrons in Radiobiological Physics and Chemistry.*Chem. Rev.*2012,*112*, 5578, DOI: 10.1021/cr300063r4https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XptFSqsr8%253D&md5=b1b53c2bd231e74451c88ce810dc8712Precursors of Solvated Electrons in Radiobiological Physics and ChemistryAlizadeh, Elahe; Sanche, LeonChemical Reviews (Washington, DC, United States) (2012), 112 (11), 5578-5602CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)A review. The following topics are discussed: Secondary electrons in water; low-energy electrons (LEEs) and damage to cellular constituents (DNA, protein subunits, amino acids, peptides); LEEs in Radiobiol. and radiation therapy; Prehydrated electrons interacting with biomols.**5**Alizadeh, E.; Orlando, T. M.; Sanche, L. Biomolecular Damage Induced by Ionizing Radiation: The Direct and Indirect Effects of Low-Energy Electrons on DNA.*Annu. Rev. Phys. Chem.*2015,*66*, 379, DOI: 10.1146/annurev-physchem-040513-1036055https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXnvFartr0%253D&md5=d8456b410ea528c73665154ecd28fb72Biomolecular damage induced by ionizing radiation: the direct and indirect effects of low-energy electrons on DNAAlizadeh, Elahe; Orlando, Thomas M.; Sanche, LeonAnnual Review of Physical Chemistry (2015), 66 (), 379-398CODEN: ARPLAP; ISSN:0066-426X. (Annual Reviews)Many exptl. and theor. advances have recently allowed the study of direct and indirect effects of low-energy electrons (LEEs) on DNA damage. In an effort to explain how LEEs damage the human genome, researchers have focused efforts on LEE interactions with bacterial plasmids, DNA bases, sugar analogs, phosphate groups, and longer DNA moieties. Here, we summarize the current understanding of the fundamental mechanisms involved in LEE-induced damage of DNA and complex biomol. films. Results obtained by several labs. on films prepd. and analyzed by different methods and irradiated with different electron-beam current densities and fluencies are presented. Despite varied conditions (e.g., film thicknesses and morphologies, intrinsic water content, substrate interactions, and extrinsic atm. compns.), comparisons show a striking resemblance in the types of damage produced and their yield functions. The potential of controlling this damage using mol. and nanoparticle targets with high LEE yields in targeted radiation-based cancer therapies is also discussed.**6**Kumar, A.; Becker, D.; Adhikary, A.; Sevilla, M. D. Reaction of Electrons with DNA: Radiation Damage to Radiosensitization.*Int. J. Mol. Sci.*2019,*20*, 3998, DOI: 10.3390/ijms201639986https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhsVCksb8%253D&md5=bfe7563709da52fb29a011eda5ff0dbdReaction of electrons with DNA: radiation damage to radiosensitizationKumar, Anil; Becker, David; Adhikary, Amitava; Sevilla, Michael D.International Journal of Molecular Sciences (2019), 20 (16), 3998CODEN: IJMCFK; ISSN:1422-0067. (MDPI AG)This review article provides a concise overview of electron involvement in DNA radiation damage. The review begins with the various states of radiation-produced electrons: Secondary electrons (SE), low energy electrons (LEE), electrons at near zero kinetic energy in water (quasi-free electrons, (e-qf)) electrons in the process of solvation in water (presolvated electrons, e-pre), and fully solvated electrons (e-aq). a current summary of the structure of e-aq, and its reactions with DNA-model systems is presented. Theor. works on redn. potentials of DNA-bases were found to be in agreement with expts. This review points out the proposed role of LEE-induced frank DNA-strand breaks in ion-beam irradiated DNA. The final section presents radiation-produced electron-mediated site-specific formation of oxidative neutral aminyl radicals from azidonucleosides and the evidence of radiosensitization provided by these aminyl radicals in azidonucleoside-incorporated breast cancer cells.**7**Zheng, Y.; Cloutier, P.; Hunting, D. J.; Sanche, L.; Wagner, J. R. Chemical Basis of DNA Sugar–Phosphate Cleavage by Low-Energy Electrons.*J. Am. Chem. Soc.*2005,*127*, 16592, DOI: 10.1021/ja054129q7https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXhtFynurjI&md5=86b374ae7e0fc8790818421c0624bf41Chemical basis of DNA sugar-phosphate cleavage by low-energy electronsZheng, Yi; Cloutier, Pierre; Hunting, Darel J.; Sanche, Leon; Wagner, J. RichardJournal of the American Chemical Society (2005), 127 (47), 16592-16598CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)DNA damage by low-energy electrons (LEE) was examd. using a novel system in which thin solid films of oligonucleotide tetramers (CGTA and GCAT) were irradiated with monoenergetic electrons (10 eV) under ultrahigh vacuum. The products of irradn. were examd. by HPLC. These analyses permitted the quantitation of 16 nonmodified nucleobase, nucleoside, and nucleotide fragments of each tetramer resulting from the cleavage of phosphodiester and N-glycosidic bonds. The distribution of nonmodified products suggests a mechanism of damage involving initial electron attachment to nucleobase moieties, followed by electron transfer to the sugar-phosphate backbone, and subsequent dissocn. of the phosphodiester bond. Moreover, virtually all the nonmodified fragments contained a terminal phosphate group at the site of cleavage. These results demonstrate that the phosphodiester bond breaks by a distinct pathway in which the neg. charge localizes on the phosphodiester bond giving rise to nonmodified fragments with an intact phosphate group. Conversely, the radical must localize on the sugar moiety to give as yet unidentified modifications. In summary, the reaction of LEE with simple tetramers involved dissociative electron attachment leading to phosphodiester bond cleavage and the formation of nonmodified fragments.**8**Dong, Y.; Gao, Y.; Liu, W.; Gao, T.; Zheng, Y.; Sanche, L. Clustered DNA Damage Induced by 2–20 eV Electrons and Transient Anions: General Mechanism and Correlation to Cell Death.*J. Phys. Chem. Lett.*2019,*10*, 2985, DOI: 10.1021/acs.jpclett.9b010638https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXpvVCgs7Y%253D&md5=d13c08acf2926347d423c1527cc61649Clustered DNA Damage Induced by 2-20 eV Electrons and Transient Anions: General Mechanism and Correlation to Cell DeathDong, Yanfang; Gao, Yingxia; Liu, Wenhui; Gao, Ting; Zheng, Yi; Sanche, LeonJournal of Physical Chemistry Letters (2019), 10 (11), 2985-2990CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)The mechanisms of action of low-energy electrons (LEEs) generated in large quantities by ionizing radiation constitute an essential element of our understanding of early events in radiolysis and radiobiol. We present the 2-20 eV electron energy dependence of the yields of base damage (BD), BD-related crosslinks (CLs), and non-double-strand break (NDSB) clustered damage induced in DNA. These new yield functions are generated by the impact of LEEs on plasmid DNA films. The damage is analyzed by gel electrophoresis with and without enzyme treatment. Maxima at 5 and 10 eV in BDs and BD-related CLs yield functions, and two others, at 6 and 10 eV, in those of NDSB clustered damage are ascribed to core-excited transient anions that decay into bond-breaking channels. The mechanism causing all types of DNA damages can be attributed to the capture of a single electron by a base followed by multiple different electron transfer pathways.**9**Mukherjee, M.; Tripathi, D.; Brehm, M.; Riplinger, C.; Dutta, A. K. Efficient EOM-CC-based Protocol for the Calculation of Electron Affinity of Solvated Nucleobases: Uracil as a Case Study.*J. Chem. Theory Comput.*2021,*17*, 105, DOI: 10.1021/acs.jctc.0c006559https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXislWhsbrK&md5=8e321848942809c54c1df1acb5aa313dEfficient EOM-CC-based Protocol for the Calculation of Electron Affinity of Solvated Nucleobases: Uracil as a Case StudyMukherjee, Madhubani; Tripathi, Divya; Brehm, Martin; Riplinger, Christoph; Dutta, Achintya KumarJournal of Chemical Theory and Computation (2021), 17 (1), 105-116CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present an explicit solvation protocol for the calcn. of electron affinity values of the solvated nucleobases. The protocol uses a quantum mechanics/mol. mechanics (QM/MM) approach based on the newly implemented domain-based pair natural orbital EOM-CCSD (equation-of-motion coupled-cluster single-double) method. The stability of the solvated nucleobase anion is sensitive to the local distribution of the water mols. around the nucleobase, and the calcd. electron affinity values converge slowly with respect to the no. of snapshots and the size of the water box. The use of nonpolarizable water mols. leads to an overestimation of the electron affinity and makes the result sensitive to the size of the QM region in the QM/MM calcn. The electron affinity values, although sensitive to the size of the basis set, lead to an almost const. blue shift of the electron affinity upon the increase in the basis set. The present protocol allows for a controllable description of the various parameters affecting the electron affinity value, and the calcd. adiabatic electron affinity values are in excellent agreement with exptl. results.**10**Verma, P.; Ghosh, D.; Dutta, A. K. Electron Attachment to Cytosine: The Role of Water.*J. Phys. Chem. A*2021,*125*, 4683, DOI: 10.1021/acs.jpca.0c1019910https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXht1Srt7nJ&md5=c95143e520d9d09017588b3c5b87190cElectron Attachment to Cytosine: The Role of WaterVerma, Pooja; Ghosh, Debashree; Dutta, Achintya KumarJournal of Physical Chemistry A (2021), 125 (22), 4683-4694CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)We present an EOM-CCSD-based quantum mech./mol. mech. (QM/MM) study on the electron attachment process to solvated cytosine. The electron attachment in the bulk solvated cytosine occurs through a doorway mechanism, where the initial electron is localized on water. The electron is subsequently transferred to cytosine by the mixing of electronic and nuclear degrees of freedom, which occurs on an ultrafast time scale. The bulk water environment stabilizes the cytosine-bound anion by an extensive hydrogen-bond network and drastically enhances the electron transfer rate from that obsd. in the gas phase. Microhydration studies cannot reproduce the effect of the bulk water environment on the electron attachment process, and one needs to include a large no. of water mols. in the calcn. to obtain converged results. The predicted adiabatic electron affinity and electron transfer rate obtained from our QM/MM calcns. are consistent with the available exptl. results.**11**Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects.*Phys. Rev.*1965,*140*, A1133, DOI: 10.1103/PhysRev.140.A1133There is no corresponding record for this reference.**12**Zhan, C.-G.; Nichols, J. A.; Dixon, D. A. Ionization Potential, Electron Affinity, Electronegativity, Hardness, and Electron Excitation Energy: Molecular Properties from Density Functional Theory Orbital Energies.*J. Phys. Chem. A*2003,*107*, 4184, DOI: 10.1021/jp022577412https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXjt1Shtbk%253D&md5=71feb2642a03fc6ada0177fd5e1f153dIonization Potential, Electron Affinity, Electronegativity, Hardness, and Electron Excitation Energy: Molecular Properties from Density Functional Theory Orbital EnergiesZhan, Chang-Guo; Nichols, Jeffrey A.; Dixon, David A.Journal of Physical Chemistry A (2003), 107 (20), 4184-4195CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)Representative at. and mol. systems, including various inorg. and org. mols. with covalent and ionic bonds, have been studied by using d. functional theory. The calcns. were done with the commonly used exchange-correlation functional B3LYP followed by a comprehensive anal. of the calcd. highest-occupied and lowest-unoccupied Kohn-Sham orbital (HOMO and LUMO) energies. The basis set dependence of the DFT results shows that the economical 6-31+G* basis set is generally sufficient for calcg. the HOMO and LUMO energies (if the calcd. LUMO energies are neg.) for use in correlating with mol. properties. The directly calcd. ionization potential (IP), electron affinity (EA), electronegativity (χ), hardness (η), and first electron excitation energy (τ) are all in good agreement with the available exptl. data. A generally applicable linear correlation relationship exists between the calcd. HOMO energies and the exptl./calcd. IPs. We have also found satisfactory linear correlation relationships between the calcd. LUMO energies and exptl./calcd. EAs (for the bound anionic states), between the calcd. av. HOMO/LUMO energies and χ values, between the calcd. HOMO-LUMO energy gaps and η values, and between the calcd. HOMO-LUMO energy gaps and exptl./calcd. first excitation energies. By using these linear correlation relationships, the calcd. HOMO and LUMO energies can be employed to semiquant. est. ionization potential, electron affinity, electronegativity, hardness, and first excitation energy.**13**Perdew, J. P.; Levy, M. Comment on “Significance of the highest occupied Kohn–Sham eigenvalue.*Phys. Rev. B*1997,*56*, 16021, DOI: 10.1103/PhysRevB.56.1602113https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXotFWntbg%253D&md5=174bb39db5f10f6c0862d5837d672ba0Comment on "Significance of the highest occupied Kohn-Sham eigenvalue"Perdew, John P.; Levy, MelPhysical Review B: Condensed Matter (1997), 56 (24), 16021-16028CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)With more explanation than usual and without appeal to Janak's theorem, we discuss the statement and proof of the ionization potential theorems for the exact Kohn-Sham d.-functional theory of a many-electron system: (1) For any av. electron no. N between the integers Z - 1 and Z, and thus for N → Z from below, the highest occupied or partly occupied Kohn-Sham orbital energy is minus the ionization energy of the Z-electron system. (2) For Z - 1 < N < Z, the exact Kohn-Sham effective potential vs(r) tends to zero as |r| → ∞. We then argue that an objection to these theorems [L. Kleinman, Phys. Rev. B 56, 12042 (1997)] overlooks a crucial step in the proof of theorem (2): The asymptotic exponential decay of the exact electron d. of the Z-electron system is controlled by the exact ionization energy, but the decay of an approx. d. is not controlled by the approx. ionization energy. We discuss relevant evidence from the numerical construction of the exact Kohn-Sham potential. In particular, we point out a model two-electron problem for which the ionization potential theorems are exactly confirmed. Finally, we comment on related issues: the self-interaction correction, the discontinuity of the exact Kohn-Sham potential as N passes through the integer Z, and the generalized sum rule on the exchange-correlation hole.**14**Levy, M.; Perdew, J. P.; Sahni, V. Exact differential equation for the density and ionization energy of a many-particle system.*Phys. Rev. A*1984,*30*, 2745, DOI: 10.1103/PhysRevA.30.2745There is no corresponding record for this reference.**15**Zhang, G.; Musgrave, C. B. Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations.*J. Phys. Chem. A*2007,*111*, 1554, DOI: 10.1021/jp061633o15https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXht1yrs7w%253D&md5=707bb4d5e592c5592f93045e5cef67ddComparison of DFT Methods for Molecular Orbital Eigenvalue CalculationsZhang, Gang; Musgrave, Charles B.Journal of Physical Chemistry A (2007), 111 (8), 1554-1561CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)We report how closely the Kohn-Sham HOMO and LUMO eigenvalues of 11 d. functional theory (DFT) functionals, resp., correspond to the neg. ionization potentials (-IPs) and electron affinities (EAs) of a test set of mols. We also report how accurately the HOMO-LUMO gaps of these methods predict the lowest excitation energies using both time-independent and time-dependent DFT (TD-DFT). The 11 DFT functionals include the local spin d. approxn. (LSDA), five generalized gradient approxn. (GGA) functionals, three hybrid GGA functionals, one hybrid functional, and one hybrid meta GGA functional. We find that the HOMO eigenvalues predicted by KMLYP, BH and HLYP, B3LYP, PW91, PBE, and BLYP predict the -IPs with av. abs. errors of 0.73, 1.48, 3.10, 4.27, 4.33, and 4.41 eV, resp. The LUMOs of all functionals fail to accurately predict the EAs. Although the GGA functionals inaccurately predict both the HOMO and LUMO eigenvalues, they predict the HOMO-LUMO gap relatively accurately (∼0.73 eV). On the other hand, the LUMO eigenvalues of the hybrid functionals fail to predict the EA to the extent that they include HF exchange, although increasing HF exchange improves the correspondence between the HOMO eigenvalue and -IP so that the HOMO-LUMO gaps are inaccurately predicted by hybrid DFT functionals. We find that TD-DFT with all functionals accurately predicts the HOMO-LUMO gaps. A linear correlation between the calcd. HOMO eigenvalue and the exptl. -IP and calcd. HOMO-LUMO gap and exptl. lowest excitation energy enables us to derive a simple correction formula.**16**Grimme, S. Semiempirical hybrid density functional with perturbative second-order correlation.*J. Chem. Phys.*2006,*124*, 034108, DOI: 10.1063/1.214895416https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XptVGnuw%253D%253D&md5=e0e89576e15f6a7c9fb40756b601dc66Semiempirical hybrid density functional with perturbative second-order correlationGrimme, StefanJournal of Chemical Physics (2006), 124 (3), 034108/1-034108/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A new hybrid d. functional for general chem. applications is proposed. It is based on a mixing of std. generalized gradient approxns. (GGAs) for exchange by Becke (B) and for correlation by Lee, Yang, and Parr (LYP) with Hartree-Fock (HF) exchange and a perturbative second-order correlation part (PT2) that is obtained from the Kohn-Sham (GGA) orbitals and eigenvalues. This virtual orbital-dependent functional contains only two global parameters that describe the mixt. of HF and GGA exchange (ax) and of the PT2 and GGA correlation (c), resp. The parameters are obtained in a least-squares-fit procedure to the G2/97 set of heat of formations. Opposed to conventional hybrid functionals, the optimum ax is found to be quite large (53% with c = 27%) which at least in part explains the success for many problematic mol. systems compared to conventional approaches. The performance of the new functional termed B2-PLYP is assessed by the G2/97 std. benchmark set, a second test suite of atoms, mols., and reactions that are considered as electronically very difficult (including transition-metal compds., weakly bonded complexes, and reaction barriers) and comparisons with other hybrid functionals of GGA and meta-GGA types. According to many realistic tests, B2-PLYP can be regarded as the best general purpose d. functional for mols. (e.g., a mean abs. deviation for the two test sets of only 1.8 and 3.2 kcal/mol compared to about 3 and 5 kcal/mol, resp., for the best other d. functionals). Very importantly, also the max. and minium errors (outliers) are strongly reduced (by about 10-20 kcal/mol). Furthermore, very good results are obtained for transition state barriers but unlike previous attempts at such a good description, this definitely comes not at the expense of equil. properties. Preliminary calcns. of the equil. bond lengths and harmonic vibrational frequencies for diat. mols. and transition-metal complexes also show very promising results. The uniformity with which B2-PLYP improves for a wide range of chem. systems emphasizes the need of (virtual) orbital-dependent terms that describe nonlocal electron correlation in accurate exchange-correlation functionals. From a practical point of view, the new functional seems to be very robust and it is thus suggested as an efficient quantum chem. method of general purpose.**17**Goerigk, L.; Grimme, S. Double-hybrid density functionals.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2014,*4*, 576, DOI: 10.1002/wcms.119317https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhvVelu7jI&md5=c2c8a4d2d17cea5bc4a9c559d42742c8Double-hybrid density functionalsGoerigk, Lars; Grimme, StefanWiley Interdisciplinary Reviews: Computational Molecular Science (2014), 4 (6), 576-600CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)Double-hybrid d. functionals (DHDFs) are reviewed in this study. In DHDFs parts of conventional d. functional theory (DFT) exchange and correlation are replaced by contributions from nonlocal Fock-exchange and second-order perturbative correlation. The latter portion is based on the well-known MP2 wave-function approach in which, however, Kohn-Sham orbitals are used to calc. its contribution. First, related methods preceding this idea are reviewed, followed by a thorough discussion of the first modern double-hybrid B2-PLYP. Parallels and differences between B2-PLYP and its various successors are then outlined. This discussion is rounded off with representative thermochem. examples demonstrating that DHDFs belong to the most robust and accurate DFT approaches currently available. This anal. also presents hitherto unpublished results for recently developed DHDFs. Finally, how double-hybrids can be combined with linear-response time-dependent DFT is also outlined and the value of this approach for electronically excited states is shown. WIREs Comput Mol Sci 2014, 4:576-600. doi: 10.1002/wcms.1193 For further resources related to this article, please visit the . Conflict of interest: The authors have declared no conflicts of interest for this article.**18**Brémond, É.; Ciofini, I.; Sancho-García, J. C.; Adamo, C. Nonempirical Double-Hybrid Functionals: An Effective Tool for Chemists.*Acc. Chem. Res.*2016,*49*, 1503, DOI: 10.1021/acs.accounts.6b0023218https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xht1ynsb7O&md5=f2722f267174ead8e7ab3e5daf92a76bNonempirical Double-Hybrid Functionals: An Effective Tool for ChemistsBremond, Eric; Ciofini, Ilaria; Sancho-Garcia, Juan Carlos; Adamo, CarloAccounts of Chemical Research (2016), 49 (8), 1503-1513CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)A review. D. functional theory (DFT) emerged in the last two decades as the most reliable tool for the description and prediction of properties of mol. systems and extended materials, coupling in an unprecedented way high accuracy and reasonable computational cost. This success rests also on the development of more and more performing d. functional approxns. (DFAs). Indeed, the Achilles' heel of DFT is represented by the exchange-correlation contribution to the total energy, which, being unknown, must be approximated. Since the beginning of the 1990s, global hybrids (GH) functionals, where an explicit dependence of the exchange-correlation energy on occupied Kohn-Sham orbitals is introduced thanks to a fraction of Hartree-Fock-like exchange, imposed themselves as the most reliable DFAs for chem. applications. However, if these functionals normally provide results of sufficient accuracy for most of the cases analyzed, some properties, such as thermochem. or dispersive interactions, can still be significantly improved. A possible way out is represented by the inclusion, into the exchange-correlation functional, of an explicit dependence on virtual Kohn-Sham orbitals via perturbation theory. This leads to a new class of functionals, called double-hybrids (DHs). In this Account, we describe our nonempirical approach to DHs, which, following the line traced by the Perdew-Burke-Ernzerhof approach, allows for the definition of a GH (PBE0) and a DH (QIDH) model. In such a way, a whole family of nonempirical functionals, spanning on the highest rungs of the Perdew's quality scale, is now available and competitive with other-more empirical-DFAs. Discussion of selected cases, ranging from thermochem. and reactions to weak interactions and excitation energies, not only show the large range of applicability of nonempirical DFAs, but also underline how increasing the no. of theor. constraints parallels with an improvement of the DFA's numerical performances. This fact further consolidates the strong theor. framework of nonempirical DFAs.Finally, even if nonempirical DH approaches are still computationally expensive, relying on the fact that they can benefit of all tech. enhancements developed for speeding up post-Hartree-Fock methods, there is substantial hope for their near future routine application to the description and prediction of complex chem. systems and reactions.**19**Martin, J. M. L.; Santra, G. Empirical Double-Hybrid Density Functional Theory: A ‘Third Way’ in Between WFT and DFT.*Isr. J. Chem.*2020,*60*, 787, DOI: 10.1002/ijch.20190011419https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXitlyhsrnM&md5=182610d9f5560d261abf36f80a6d9d2eEmpirical Double-Hybrid Density Functional Theory: A 'Third Way' in Between WFT and DFTMartin, Jan M. L.; Santra, GolokeshIsrael Journal of Chemistry (2020), 60 (8-9), 787-804CODEN: ISJCAT; ISSN:0021-2148. (Wiley-VCH Verlag GmbH & Co. KGaA)A review. Double hybrid d. functional theory arguably sits on the seamline between wavefunction methods and DFT: it represents a special case of Rung 5 on the "Jacob's Ladder" of John P. Perdew. For large and chem. diverse benchmarks such as GMTKN55, empirical double hybrid functionals with dispersion corrections can achieve accuracies approaching wavefunction methods at a cost not greatly dissimilar to hybrid DFT approaches, provided RI-MP2 and/or another MP2 acceleration techniques are available in the electronic structure code. Only a half-dozen or fewer empirical parameters are required. For vibrational frequencies, accuracies intermediate between CCSD and CCSD(T) can be achieved, and performance for other properties is encouraging as well. Organometallic reactions can likewise be treated well, provided static correlation is not too strong. Further prospects are discussed, including range-sepd. and RPA-based approaches.**20**Su, N. Q.; Xu, X. The XYG3 Type of Doubly Hybrid Density Functionals.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2016,*6*, 721, DOI: 10.1002/wcms.127420https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhslKgsbfP&md5=be9356f372be073ff2a1794c47a9fc11The XYG3 type of doubly hybrid density functionalsSu, Neil Qiang; Xu, XinWiley Interdisciplinary Reviews: Computational Molecular Science (2016), 6 (6), 721-747CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)Doubly hybrid (DH) functionals have emerged as a new class of d. functional approxns. (DFAs), which not only have a nonlocal orbital-dependent component in the exchange part, but also incorporate the information of unoccupied orbitals in the correlation part, being at the top rung of Perdew's view of Jacob's ladder in DFAs. This review article focuses on the XYG3 type of DH (xDH) functionals, which use a low rung functional to perform the self-consistent-field calcn. to generate orbitals and densities, with which a top rung DH functional is used for final energy evaluation. We will discuss the theor. background of the xDH functionals, briefly reviewing the adiabatic connection formalism, coordinate scaling relations, and Goerling-Levy perturbation theory. General performance of the xDH functionals will be presented for both energies and structures. In particular, we will present the fractional charge behaviors of the xDH functionals, examg. the self-interaction errors, the delocalization errors and the deviation from the linearity condition, as well as their effects on the predicted ionization potentials, electron affinities and fundamental gaps. This provides a theor. rationale for the obsd. good performance of the xDH functionals. WIREs Comput Mol Sci 2016, 6:721-747. doi: 10.1002/wcms.1274 For further resources related to this article, please visit the .**21**Sancho-García, J. C.; Adamo, C. Double-hybrid density functionals: Merging wavefunction and density approaches to get the best of both worlds.*Phys. Chem. Chem. Phys.*2013,*15*, 14581, DOI: 10.1039/c3cp50907a21https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXht1KhtbvF&md5=4216f41fe053cdc2348840cbd2567f0cDouble-hybrid density functionals: merging wavefunction and density approaches to get the best of both worldsSancho-Garcia, J. C.; Adamo, C.Physical Chemistry Chemical Physics (2013), 15 (35), 14581-14594CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)A review. We review why and how double-hybrid d. functionals have become new leading actors in the field of computational chem., thanks to the combination of an unprecedented accuracy together with large robustness and reliability. Similar to their predecessors, the widely employed hybrid d. functionals, they are rooted in the Adiabatic Connection Method from which they emerge in a natural way. We present recent achievements concerning applications to chem. systems of the most interest, and current extensions to deal with challenging issues such as non-covalent interactions and excitation energies. These promising methods, despite a slightly higher computational cost than other typical d.-based models, are called to play a key role in the near future and can thus pave the way towards new discoveries or advances.**22**Geertsen, J.; Rittby, M.; Bartlett, R. J. The equation-of-motion coupled-cluster method: Excitation energies of Be and CO.*Chem. Phys. Lett.*1989,*164*, 57, DOI: 10.1016/0009-2614(89)85202-922https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3cXkt12nsbw%253D&md5=00bf6668a536c3c45dc538396deb5ba7The equation-of-motion coupled-cluster method: excitation energies of beryllium and carbon monoxideGeertsen, Jan; Rittby, Magnus; Bartlett, Rodney J.Chemical Physics Letters (1989), 164 (1), 57-62CODEN: CHPLBC; ISSN:0009-2614.The equation-of-motion coupled-cluster (EOM-CC) method for the calcn. of excitation energies is presented. The procedure is based upon representing an excited state as an excitation from a coupled-cluster ground state and the excitation energies are obtained by solving a non-Hermitian eigenvalue problem. Numerical applications are reported for Be and CO, and the compared to full CI, Fock space multi-ref. coupled-cluster, multi-ref. MBPT, and propagator results.**23**Stanton, J. F.; Bartlett, R. J. The equation of motion coupled-cluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited state properties.*J. Chem. Phys.*1993,*98*, 7029, DOI: 10.1063/1.46474623https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXksFKgu78%253D&md5=bb8b7c7ea2e69d1272a8e98ee83d9be7The equation-of-motion, coupled-cluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited-state propertiesStanton, John F.; Bartlett, Rodney J.Journal of Chemical Physics (1993), 98 (9), 7029-39CODEN: JCPSA6; ISSN:0021-9606.A comprehensive overview of the equation of motion coupled-cluster (EOM-CC) method and its application to mol. systems is presented. By exploiting the biorthogonal nature of the theory, it is shown that excited-state properties and transition strengths can be evaluated via a generalized expectation-value approach that incorporates both the bra and ket state wave functions. Reduced d. matrixes defined by this procedure are given by closed form expressions. For the root of the EOM-CC effective Hamiltonian that corresponds to the ground state, the resulting equations are equiv. to the usual expressions for normal single-ref. CC d. matrixes. Thus, the method described in this paper provides a universal definition of coupled-cluster d. matrixes, providing a link between EOM-CC and traditional ground state CC theory. Excitation energy, oscillator strength, and property calcns. are illustrated by means of several numerical examples, including comparisons with full CI calcns. and a detailed study of the 10 lowest electronically excited states of the cyclic isomer of C4.**24**Watts, J. D.; Bartlett, R. J. The inclusion of connected triple excitations in the equation-of-motion coupled-cluster method.*J. Chem. Phys.*1994,*101*, 3073, DOI: 10.1063/1.46762024https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXlvVOmu7c%253D&md5=850291febfd7c61d08ef73f2ed7ba367The inclusion of connected triple excitations in the equation-of-motion coupled-cluster methodWatts, John D.; Bartlett, RodneyJournal of Chemical Physics (1994), 101 (4), 3073-8CODEN: JCPSA6; ISSN:0021-9606.The implementation of connected triple excitations in the equation-of-motion (EOM) coupled-cluster (CC) method for excitation energies is reported for the first time. The ref. state is described by the complete CC singles, doubles, and triples (CCSDT) method. Excited states are generated from the ref. state wave function by the action of a linear excitation operator including single, double, and triple excitations. The excited state wave functions and energies are obtained by diagonalizing the effective Hamiltonian e-THeT, where T is the cluster operator for the ref. state, in the space of singly, doubly, and triply excited determinants. Comparison is made with full CI excitation energies for several examples (CH+, Be, SiH2, and CH2). These show that EOM-CCSDT is able to describe states which are doubly excited relative to the ref. state, as well as singly excited states. Calcns. of several excitation energies of BH using an extended basis set are also reported, and show good agreement with expt.**25**Bartlett, R. J. Coupled-cluster theory and its equation-of-motion extensions.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2012,*2*, 126, DOI: 10.1002/wcms.7625https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhvFGls7c%253D&md5=b75a016dfe3ed5488b83de78ceabd0c7Coupled-cluster theory and its equation-of-motion extensionsBartlett, Rodney J.Wiley Interdisciplinary Reviews: Computational Molecular Science (2012), 2 (1), 126-138CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)A review. Coupled-cluster theory offers today's ref. quantum chem. method for most of the problems encountered in electronic structure theory. It has been instrumental in establishing the now well-known paradigm of converging, many-body methods, MBPT2 < CCD < CCSD < MBPT4 < CCSD(T) < CCSDT-n < CCSDT < CCSDT(Q) < CCSDTQ-n < CCSDTQ < fullCI. Many-body perturbation theory (MBPT) for second, MBPT2, and fourth-order MBPT4; and coupled-cluster (CC) theory for different categories of excitations, singles, doubles, triples, quadruples (SDTQ). Although built on the same basic concept as CI (CI), many-body methods fundamentally improve upon CI approxns. by introducing the property of size extensivity, meaning that contrary to any truncated CI all terms properly scale with the no. of electrons in the problem. This fundamental aspect of many-electron methods leads to the exceptional performance of CC theory and its finite-order MBPT approxns. plus its equation-of-motion extensions for excited, ionized, and electron attached states. This brief overview will describe formal aspects of the theory which should be understood by perspective users of CC methods. We will also comment on some current developments that are improving the theory's accuracy or applicability.**26**Sneskov, K.; Christiansen, O. Excited state coupled cluster methods.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2012,*2*, 566, DOI: 10.1002/wcms.9926https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhtlWiu7fN&md5=96241519af990fde76ce3473a0b87231Excited state coupled cluster methodsSneskov, Kristian; Christiansen, OveWiley Interdisciplinary Reviews: Computational Molecular Science (2012), 2 (4), 566-584CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)We review coupled cluster (CC) theory for electronically excited states. We outline the basics of a CC response theory framework that allows the transfer of the attractive accuracy and convergence properties assocd. with CC methods over to the calcn. of electronic excitation energies and properties. Key factors affecting the accuracy of CC excitation energy calcns. are discussed as are some of the key CC models in this field. To aid both the practitioner as well as the developer of CC excited state methods, we also briefly discuss the key computational steps in a working CC response implementation. Approaches aimed at extending the application range of CC excited state methods either in terms of mol. size and phenomena or in terms of environment (soln. and proteins) are also discussed.**27**Nooijen, M.; Bartlett, R. J. Similarity transformed equation-of-motion coupled-cluster theory: Details, examples, and comparisons.*J. Chem. Phys.*1997,*107*, 6812, DOI: 10.1063/1.47492227https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXmsFOgsrk%253D&md5=0358e456ea5cd5607529aa3d2c874ce1Similarity transformed equation-of-motion coupled-cluster theory: Details, examples, and comparisonsNooijen, Marcel; Bartlett, Rodney J.Journal of Chemical Physics (1997), 107 (17), 6812-6830CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The similarity transformed equation-of-motion coupled-cluster (STEOM-CC) method is presented in full detail. Comparisons are made with the Fock space coupled-cluster (FSCC) method and the equation-of-motion coupled-cluster (EOM-CC) scheme. The role of implicit triple excitations and, relatedly, charge transfer separability in STEOM is discussed. The dependence on the choice of active space in STEOM is addressed and criteria for the selection of the active space are given. The evaluation of properties within STEOM is outlined and a large no. of illustrative examples of STEOM is presented.**28**Stanton, J. F.; Gauss, J. Perturbative treatment of the similarity transformed Hamiltonian in equation-of-motion coupled-cluster approximations.*J. Chem. Phys.*1995,*103*, 1064, DOI: 10.1063/1.46981728https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXmvFCktrk%253D&md5=7a402877a01649f980a13981f512dad3Perturbative treatment of the similarity transformed Hamiltonian in equation-of-motion coupled-cluster approximationsStanton, John F.; Gauss, JuergenJournal of Chemical Physics (1995), 103 (3), 1064-76CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A series of size-consistent approxns. to the equation-of-motion coupled cluster method in the singles and doubles approxn. (EOM-CCSD) are developed by subjecting the similarity transformed Hamiltonian ‾H = exp(-T)H exp(T) to a perturbation expansion. Attention is directed to N and N - 1 electron final state realizations of the method defined by truncation of ‾H at second order. Explicit spin-orbital equations for the energy and its first deriv. are documented for both approaches [EOMEE-CCSD(2) and EOMIP-CCSD(2), resp.], and have been implemented in a large-scale quantum chem. program. Vertical ionization potentials calcd. by EOMIP-CCSD(2) are shown to be equiv. to those of an approach presented recently by Nooijen and Snijders [J. Chem. Phys. 102, 1681,(1995)]. Applications of both EOMIP-CCSD(2) and EOMEE-CCSD(2) provide results for final state properties that compare favorably with those obtained in full EOM-CCSD calcns. Anal. of the computational aspects of the approx. and full EOM-CCSD methods shows that the cost of EOMIP-CCSD(2) energy and gradient calcns. scales in proportion to the fifth power of the basis set size, a significant savings over the sixth power dependence of EOMIP-CCSD. This feature is of great practical importance, as it shows that this N - 1 electron final state approach has a large domain of applicability and is therefore likely to become a valuable tool for application calcns. On the other hand, the same cannot be said for EOMEE-CCSD(2) since its asymptotic computational dependence and storage requirements are the same as the full EOMEE-CCSD method.**29**Stanton, J. F.; Gauss, J. Analytic energy derivatives for ionized states described by the equation-of-motion coupled cluster method.*J. Chem. Phys.*1994,*101*, 8938, DOI: 10.1063/1.46802229https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXjt1yitL8%253D&md5=86ccf94994d88b83e0eedafd6ca3ae97Analytical energy derivatives for ionized states described by the equation-of-motion coupled cluster methodStanton, John F.; Gauss, JuergenJournal of Chemical Physics (1994), 101 (10), 8938-44CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The theory for analytic energy derivs. of excited electronic states described by the equation-of-motion coupled cluster (EOM-CC) method has been generalized to treat cases in which ref. and final states differ in the no. of electrons. While this work specializes to the sector of Fock space that corresponds to ionization of the ref., the approach can be trivially modified for electron attached final states. Unlike traditional coupled cluster methods that are based on single determinant ref. functions, several electronic configurations are treated in a balanced way by EOM-CC. Therefore, this quantum chem. approach is appropriate for problems that involve important nondynamic electron correlation effects. Furthermore, a fully spin adapted treatment of doublet electronic states is guaranteed when a spin restricted closed shell ref. state is used-a desirable feature that is not easily achieved in std. coupled cluster approaches. The efficient implementation of analytic gradients reported here allows this variant of EOM-CC theory to be routinely applied to multidimensional potential energy surfaces for the first time. Use of the method is illustrated by an investigation of the formyloxyl radical (HCOO), which suffers from notorious symmetry breaking effects.**30**Pieniazek, P. A.; Bradforth, S. E.; Krylov, A. I. Charge localization and Jahn–Teller distortions in the benzene dimer cation.*J. Chem. Phys.*2008,*129*, 074104, DOI: 10.1063/1.296910730https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXhtVamtL3N&md5=e905d21ec9bbe491d0b3909566a139beCharge localization and Jahn-Teller distortions in the benzene dimer cationPieniazek, Piotr A.; Bradforth, Stephen E.; Krylov, Anna I.Journal of Chemical Physics (2008), 129 (7), 074104/1-074104/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Jahn-Teller (JT) distortions and charge localization in the benzene dimer cation are analyzed using the equation-of-motion coupled cluster with single and double substitutions for ionization potential (EOM-IP-CCSD) method. Ionization of the dimer changes the bonding from noncovalent to covalent and induces significant geometrical distortions, e.g., shorter interfragment distance and JT displacements. Relaxation along interfragment coordinates lowers the energy of the t-shaped and displaced sandwich isomers by 0.07 and 0.23 eV, resp., whereas JT displacements result in addnl. 0.18 and 0.23 eV. Energetically, the effect of JT distortion on the dimer is similar to the monomer where JT relaxation lowers the energy by 0.18 eV. While the change in the interfragment distance has dramatic spectroscopic consequences, the JT distortion causes only a small perturbation in the electronic spectra. The two geometrical relaxations in the t-shaped isomer lead to opposing effects on hole localization. Intermol. relaxation leads to an increased delocalization, whereas JT ring distortion localizes the charge. In the sandwich isomers, breaking the symmetry by ring rotation does not induce considerable charge localization. The optimization and property calcns. were performed using a new implementation of EOM-IP-CCSD energies and gradients in the Q-CHEM electronic structure package. (c) 2008 American Institute of Physics.**31**Musiał, M.; Kucharski, S. A.; Bartlett, R. J. Equation-of-motion coupled cluster method with full inclusion of the connected triple excitations for ionized states: IP-EOM-CCSDT.*J. Chem. Phys.*2003,*118*, 1128, DOI: 10.1063/1.152701331https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXhvFWgug%253D%253D&md5=9697f8157100ea77f8d6f614750c83d9Equation-of-motion coupled cluster method with full inclusion of the connected triple excitations for ionized states: IP-EOM-CCSDTMusial, Monika; Kucharski, Stanislaw A.; Bartlett, Rodney J.Journal of Chemical Physics (2003), 118 (3), 1128-1136CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The equation-of-motion (EOM) coupled cluster (CC) method with full inclusion of the connected triple excitations for ionization energies has been formulated and implemented. Using proper factorization of the three- and four-body parts of the effective Hamiltonian, an efficient computational procedure has been proposed for IP-EOM-CCSDT which at the EOM level requires no-higher-than nocc3nvir4 scaling. The method is calibrated by the evaluation of the valence vertical ionization potentials for CO, N2, and F2 mols. for several basis sets up to 160 basis functions. At the basis set limit, errors vary from 0.0 to 0.2 eV, compared to "exptl." vertical ionization potentials.**32**Nooijen, M.; Bartlett, R. J. Equation of motion coupled cluster method for electron attachment.*J. Chem. Phys.*1995,*102*, 3629, DOI: 10.1063/1.46859232https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXjvFGqu74%253D&md5=8ad0ce69f40ddcaa984fe7c304af5e28Equation of motion coupled cluster method for electron attachmentNooijen, Marcel; Bartlett, Rodney J.Journal of Chemical Physics (1995), 102 (9), 3629-47CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The electron attachment equation of motion coupled cluster (EA-EOMCC) method is derived which enables detn. of the various bound states of an (N+1)-electron system and the corresponding energy eigenvalues relative to the energy of an N-electron CCSD ref. state. Detailed working equations for the EA-EOMCC method are derived using diagrammatic techniques for both closed-shell and open-shell CCSD ref. states based upon a single determinant. The EA-EOMCC method is applied to a variety of different problems, the main purpose being to establish its prospects and limitations. The results from EA-EOMCC calcns. are compared to other EOMCC approaches, starting from different ref. states, as well as other theor. methods and exptl. values, where available. We have investigated electron affinities for a wide selection of both closed-shell and open-shell systems. Excitation spectra of atoms and mols. with an odd no. of electrons are obtained, taking the closed-shell ground state of the ion as a ref. in the EA-EOMCC calcn. Finally we consider excitation spectra of some closed-shell systems, and find in particular that the electron attachment approach is capable of yielding accurate triplet excitation energies in an efficient way.**33**Musiał, M.; Bartlett, R. J. Equation-of-motion coupled cluster method with full inclusion of connected triple excitations for electron-attached states: EA-EOM-CCSDT.*J. Chem. Phys.*2003,*119*, 1901, DOI: 10.1063/1.158465733https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXltlyqsrw%253D&md5=5c6a045a2bf6bf07f5d3674930f03e8cEquation-of-motion coupled cluster method with full inclusion of connected triple excitations for electron-attached states: EA-EOM-CCSDTMusial, Monika; Bartlett, Rodney J.Journal of Chemical Physics (2003), 119 (4), 1901-1908CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We extend the full triples equation-of-motion (EOM) coupled cluster (CC) method to electron attached states. Proper factorization of the three- and four-body parts of the effective Hamiltonian makes it possible to achieve for the EA-EOM part a scaling no higher than nocc2nvir5. The method is calibrated by the evaluation of the valence vertical electron affinities for the C2 and O3 mols. for several basis sets up to 160 basis functions. For C2, EA-EOM-CCSDT gives 3.24 eV at the extrapolated basis limit, while the exptl. adiabatic EA is equal to 3.27 ± 0.008 eV. For O3 the agreement is ∼1.9 eV compared to an adiabatic value of 2.1 eV.**34**Schirmer, J. Beyond the random-phase approximation: A new approximation scheme for the polarization propagator.*Phys. Rev. A*1982,*26*, 2395, DOI: 10.1103/PhysRevA.26.239534https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL38Xmt1eisbk%253D&md5=491a01718b7aeaff6b8eec59b5a201d9Beyond the random-phase approximation: A new approximation scheme for the polarization propagatorSchirmer, JochenPhysical Review A: Atomic, Molecular, and Optical Physics (1982), 26 (5), 2395-416CODEN: PLRAAN; ISSN:0556-2791.Within the framework of the many-body Green's-function method, a new approach is given to the polarization propagator for finite Fermi systems. This approach makes explicit use of the diagrammatic perturbation expansion for the polarization propagator, and reformulates the exact summation in terms of a simple algebraic scheme, referred to as the algebraic diagrammatic construction (ADC). The ADC defines in a natural way a set of approxn. schemes (nth-order ADC schemes) which represent infinite partial summations exact up to nth order of perturbation theory. In contrast to the random-phase-approxn. (RPA)-like schemes, the corresponding math. procedures are essentially Hermitian eigenvalue problems in limited configuration spaces of unperturbed excited configurations. Explicit equations for the 1st- and 2nd-order ADC schemes are derived. These schemes are thoroughly discussed and compared with the Tamm-Dancoff approxn. and RPA schemes.**35**Schirmer, J.; Cederbaum, L. S.; Walter, O. New approach to the one-particle Green’s function for finite Fermi systems.*Phys. Rev. A*1983,*28*, 1237, DOI: 10.1103/PhysRevA.28.123735https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3sXlt1ymsrk%253D&md5=98a05eac7efa66da70a417ae38e94f7fNew approach to the one-particle Green's function for finite Fermi systemsSchirmer, J.; Cederbaum, L. S.; Walter, O.Physical Review A: Atomic, Molecular, and Optical Physics (1983), 28 (3), 1237-59CODEN: PLRAAN; ISSN:0556-2791.A new approach to the one-particle Green's functions G for finite electronic systems is presented. This approach is based on the diagrammatic perturbation expansions of the Green's function and of the dynamic self-energy part M related to G via the Dyson equation. The exact summation of the latter expansion is reformulated in terms of a simple algebraic form referred to as algebraic diagrammatic construction (ADC). The ADC defines in a systematic way a set of approxn. schemes (nth-order ADC schemes) that represent infinite partial summations for M and (via the Dyson equation) for G being complete through nth order of perturbation theory. The corresponding math. procedures are essentially Hermitian eigenvalue problems in restricted configuration spaces of unperturbed ionic configurations. Explicit equations for the second-, third-, and fourth-order ADC schemes are derived and analyzed.**36**Schirmer, J.; Trofimov, A. B.; Stelter, G. A non-Dyson third-order approximation scheme for the electron propagator.*J. Chem. Phys.*1998,*109*, 4734, DOI: 10.1063/1.47708536https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXlslyrs70%253D&md5=84afab4f57ce9e4a20c40730b51f0b51A non-Dyson third-order approximation scheme for the electron propagatorSchirmer, J.; Trofimov, A. B.; Stelter, G.Journal of Chemical Physics (1998), 109 (12), 4734-4744CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An efficient third-order propagator method to compute ionization potentials and electron affinities of atoms and mols. is presented. The development is based on the algebraic diagrammatic construction (ADC) representing a specific reformulation of the diagrammatic perturbation series of the electron propagator G(ω). In contrast with previous approxn. schemes, relying on the Dyson equation and approxns. for the self-energy part, the ADC procedure here is applied directly to the (N.-+.1)-electron parts G-(ω) and G+(ω), resp., of the electron propagator. This leads to decoupled secular equations for the ionization energies ((N-1)-electron part) and electron affinities ((N+1)-electron part), resp. In comparison with the Dyson-type approach, there is a substantial redn. of the secular matrix dimension opposed by a small addnl. expense in computing some second- and third-order contributions to the secular matrix elements. The relationship of the non-Dyson ADC(3) method to coupled cluster methods is outlined.**37**Trofimov, A. B.; Schirmer, J. Molecular ionization energies and ground- and ionic-state properties using a non-Dyson electron propagator approach.*J. Chem. Phys.*2005,*123*, 144115, DOI: 10.1063/1.204755037https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXhtFCmsrrM&md5=cb9ab355f330f71830544839568f2a82Molecular ionization energies and ground- and ionic-state properties using a non-Dyson electron propagator approachTrofimov, A. B.; Schirmer, J.Journal of Chemical Physics (2005), 123 (14), 144115/1-144115/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An earlier proposed propagator method for the treatment of mol. ionization is tested in first applications. The method referred to as the non-Dyson third-order algebraic-diagrammatic construction [nD-ADC(3)] approxn. for the electron propagator represents a computationally promising alternative to the existing Dyson ADC(3) method. The advantage of the nD-ADC(3) scheme is that the (N±1)-electronic parts of the one-particle Green's function are decoupled from each other and the corresponding equations can be solved sep. For a test of the method the nD-ADC(3) results for the vertical ionization transitions in C2H4, CO, CS, F2, H2CO, H2O, HF, N2, and Ne are compared with available exptl. and theor. data including results of full CI (FCI) and coupled cluster computations. The mean error of the nD-ADC(3) ionization energies relative to the exptl. and FCI results is about 0.2 eV. The nD-ADC(3) method, scaling as n5 with the no. of orbitals, requires the soln. of a relatively simple Hermitian eigenvalue problem. The method renders access to ground-state properties such as dipole moments. Moreover, also one-electron properties of (N±1) electron states can now be studied as a consequence of a specific intermediate-state representation (ISR) formulation of the nD-ADC approach. Corresponding second-order ISR equations are presented.**38**Dempwolff, A. L.; Schneider, M.; Hodecker, M.; Dreuw, A. Efficient implementation of the non-Dyson third-order algebraic diagrammatic construction approximation for the electron propagator for closed- and open-shell molecules.*J. Chem. Phys.*2019,*150*, 064108, DOI: 10.1063/1.508167438https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXjtFGnsrc%253D&md5=8cfceddce18399114cc4ec211aa488beEfficient implementation of the non-Dyson third-order algebraic diagrammatic construction approximation for the electron propagator for closed- and open-shell moleculesDempwolff, Adrian L.; Schneider, Matthias; Hodecker, Manuel; Dreuw, AndreasJournal of Chemical Physics (2019), 150 (6), 064108/1-064108/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A novel efficient implementation of the non-Dyson algebraic diagrammatic construction (ADC) scheme of the (N - 1)-part of the electron propagator up to third order of perturbation theory is presented. Due to the underlying spin-orbital formulation, for the first time, the computation of ionization potentials of open-shell radicals is thus possible via non-Dyson ADC schemes. Thorough evaluation of the accuracy, applicability, and capabilities of the new method reveals a mean error of 0.15 eV for closed- as well as open-shell atoms and mols. (c) 2019 American Institute of Physics.**39**Banerjee, S.; Sokolov, A. Y. Third-order algebraic diagrammatic construction theory for electron attachment and ionization energies: Conventional and Green’s function implementation.*J. Chem. Phys.*2019,*151*, 224112, DOI: 10.1063/1.513177139https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXitlyhs7vO&md5=4634701de50853ee43f0f8bf4d137e94Third-order algebraic diagrammatic construction theory for electron attachment and ionization energies: Conventional and Green's function implementationBanerjee, Samragni; Sokolov, Alexander Yu.Journal of Chemical Physics (2019), 151 (22), 224112/1-224112/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present implementation of second- and third-order algebraic diagrammatic construction (ADC) theory for efficient and accurate computations of mol. electron affinities (EA), ionization potentials (IP), and densities of states [EA-/IP-ADC(n), n = 2, 3]. Our work utilizes the non-Dyson formulation of ADC for the single-particle propagator and reports working equations and benchmark results for the EA-ADC(2) and EA-ADC(3) approxns. We describe two algorithms for solving EA-/IP-ADC equations: (i) conventional algorithm that uses iterative diagonalization techniques to compute low-energy EA, IP, and d. of states and (ii) Green's function algorithm (GF-ADC) that solves a system of linear equations to compute d. of states directly for a specified spectral region. To assess the accuracy of EA-ADC(2) and EA-ADC(3), we benchmark their performance for a set of atoms, small mols., and five DNA/RNA nucleobases. As our next step, we demonstrate the efficiency of our GF-ADC implementation by computing core-level K-, L-, and M-shell ionization energies of a zinc atom without introducing the core-valence sepn. approxn. Finally, we use EA- and IP-ADC methods to compute the bandgaps of equally spaced hydrogen chains Hn with n up to 150, providing their ests. near thermodn. limit. Our results demonstrate that EA-/IP-ADC(n) (n = 2, 3) methods are efficient and accurate alternatives to widely used electronic structure methods for simulations of electron attachment and ionization properties. (c) 2019 American Institute of Physics.**40**Hodecker, M.; Dempwolff, A. L.; Schirmer, J.; Dreuw, A. Theoretical analysis and comparison of unitary coupled-cluster and algebraic-diagrammatic construction methods for ionization.*J. Chem. Phys.*2022,*156*, 074104, DOI: 10.1063/5.007096740https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XktVGgtb0%253D&md5=41cafc86bcb38668ee0419f245f6a837Theoretical analysis and comparison of unitary coupled-cluster and algebraic-diagrammatic construction methods for ionizationHodecker, Manuel; Dempwolff, Adrian L.; Schirmer, Jochen; Dreuw, AndreasJournal of Chemical Physics (2022), 156 (7), 074104CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)This article describes a novel approach for the calcn. of ionization potentials (IPs), or, more generally, electron-detachment energies, based on a unitary coupled-cluster (UCC) parameterization of the ground-state wave function. Explicit working equations for a scheme referred to as IP-UCC3 are given, providing electron-detachment energies and spectroscopic amplitudes of electron-detached states dominated by one-hole excitations correct through third order. In the derivation, an expansion of the UCC transformed Hamiltonian involving Bernoulli nos. as expansion coeffs. is employed. Both the secular matrix and the effective transition moments are shown to be essentially equiv. to the strict third-order algebraic-diagrammatic construction scheme for the electron propagator (IP-ADC). Interestingly, due to the Bernoulli expansion, neglecting triple substitutions in the UCC expansion manifold does not affect the third-order consistency of the IP-UCC effective transition moments. Finally, the equivalence between ADC and UCC excited-state schemes is shown to not hold in fourth or higher order due to a different treatment of the correlated excited-state basis. (c) 2022 American Institute of Physics.**41**Dempwolff, A. L.; Hodecker, M.; Dreuw, A. Vertical ionization potential benchmark for unitary coupled-cluster and algebraic-diagrammatic construction methods.*J. Chem. Phys.*2022,*156*, 054114, DOI: 10.1063/5.007904741https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XivV2isbY%253D&md5=10d48d54a39c35dc5a41c4dcf9edc85bVertical ionization potential benchmark for unitary coupled-cluster and algebraic-diagrammatic construction methodsDempwolff, Adrian L.; Hodecker, Manuel; Dreuw, AndreasJournal of Chemical Physics (2022), 156 (5), 054114CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The performance of several methods for the calcn. of vertical ionization potentials (IPs) or, more generally, electron-detachment energies based on unitary coupled-cluster (UCC) theory and the algebraic-diagrammatic construction (ADC) scheme is evaluated with respect to benchmark data computed at the level of equation-of-motion coupled-cluster theory, including single, double, and triple excitations (IP-EOM-CCSDT). Based on a statistical evaluation of about 200 electron-detached states of 41 mols., the second-order methods IP-ADC(2) and IP-UCC2 show modest accuracies with IP-EOM-CCSDT as ref., exposing a mean signed error and a std. deviation of the error of -0.54 ± 0.50 and -0.49 ± 0.54 eV, resp., accompanied by a mean abs. error (MAE) of 0.61 and 0.58 eV, resp. The strict third-order IP-ADC method demonstrates an accuracy of 0.26 ± 0.35 eV (MAE = 0.35 eV), while the IP-UCC3 method is slightly more accurate with 0.24 ± 0.26 eV (MAE = 0.29 eV). Employing the static self-energy computed using the Dyson expansion method (DEM) improves the IP-ADC(3) performance to 0.27 ± 0.28 eV, with the mean abs. error of this method being 0.32 eV. However, employing the simpler improved fourth-order scheme Σ(4+) for the static self-energy provides almost identical results as the DEM. Based on the quality of the present benchmark results, it therefore appears not necessary to use the computationally more demanding DEM. (c) 2022 American Institute of Physics.**42**Dempwolff, A. L.; Paul, A. C.; Belogolova, A. M.; Trofimov, A. B.; Dreuw, A. Intermediate state representation approach to physical properties of molecular electron-detached states. I. Theory and implementation.*J. Chem. Phys.*2020,*152*, 024113, DOI: 10.1063/1.513779242https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXpt1KntQ%253D%253D&md5=5643d2d8b6776767051d1e2952f4d973Intermediate state representation approach to physical properties of molecular electron-detached states. I. Theory and implementationDempwolff, Adrian L.; Paul, Alexander C.; Belogolova, Alexandra M.; Trofimov, Alexander B.; Dreuw, AndreasJournal of Chemical Physics (2020), 152 (2), 024113CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The third-order non-Dyson algebraic-diagrammatic construction approach to the electron propagator [IP-ADC(3)] is extended using the intermediate state representation (ISR) formalism, allowing the wave functions and properties of mol. states with detached electron to be studied. The second-order ISR equations [ISR(2)] for the one-particle (transition) d. matrix have been derived and implemented in the Q-CHEM program. The approach is completely general and enables evaluation of arbitrary one-particle operators and interpretation of electron detachment processes in terms of d.-based quantities. The IP-ADC(3)/ISR(2) equations were implemented for Ŝz-adapted intermediate states, allowing open-shell mols. to be studied using UHF refs. As a first test for computations of ground state properties, dipole moments of various closed- and open-shell mols. have been computed by means of electron detachment from the corresponding anions. The results are in good agreement with exptl. data. The potential of IP-ADC(3)/ISR(2) for the interpretation of photoelectron spectra is demonstrated for the galvinoxyl free radical. (c) 2020 American Institute of Physics.**43**Dempwolff, A. L.; Paul, A. C.; Belogolova, A. M.; Trofimov, A. B.; Dreuw, A. Intermediate state representation approach to physical properties of molecular electron-detached states. II. Benchmarking.*J. Chem. Phys.*2020,*152*, 024125, DOI: 10.1063/1.513779443https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXht1Gis74%253D&md5=4efd24e4bf7a51c5b5e77a48abe7e70fIntermediate state representation approach to physical properties of molecular electron-detached states. II. BenchmarkingJournal of Chemical Physics (2020), 152 (2), 024125CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The third-order algebraic-diagrammatic construction method for studies of electron detachment processes within the electron propagator framework [IP-ADC(3)] was extended to treat the properties of mol. states with a detached electron using the intermediate state representation (ISR) formalism. The second-order ISR(2) equations for the one-particle (transition) d. matrix have been derived and implemented as an extension of the IP-(U)ADC(3) method available in the Q-CHEM program. As a first systematic test of the present IP-(U)ADC(3)/ISR(2) method, the dipole moments of various electronic states of closed- and open-shell mols. have been computed and compared to full CI (FCI) results. The present study employing FCI benchmarks also provides the first rigorous ests. for the accuracy of electron detachment energies obtained using the IP-ADC(3) method. (c) 2020 American Institute of Physics.**44**Dempwolff, A. L.; Belogolova, A. M.; Trofimov, A. B.; Dreuw, A. Intermediate state representation approach to physical properties of molecular electron-attached states: Theory, implementation, and benchmarking.*J. Chem. Phys.*2021,*154*, 104117, DOI: 10.1063/5.004333744https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXmtlGjtLo%253D&md5=66242d27d374e13ea6608cfc2c66eee8Intermediate state representation approach to physical properties of molecular electron-attached states: Theory, implementation, and benchmarkingDempwolff, Adrian L.; Belogolova, Alexandra M.; Trofimov, Alexander B.; Dreuw, AndreasJournal of Chemical Physics (2021), 154 (10), 104117CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Computational schemes for comprehensive studies of mol. electron-attached states and the calcn. of electron affinities (EAs) are formulated and implemented employing the intermediate state representation (ISR) formalism and the algebraic-diagrammatic construction approxn. for the electron propagator (EA-ADC). These EA-ADC(n)/ISR(m) schemes allow for a consistent treatment of not only electron affinities and pole strengths up to third-order of perturbation theory (n = 3) but also one-electron properties of electron-attached states up to second order (m = 2). The EA-ADC/ISR equations were implemented in the Q-CHEM program for Ŝz-adapted intermediate states, allowing also open-shell systems to be studied using UHF refs. For benchmarking of the EA-(U)ADC/ISR schemes, EAs and dipole moments of various electron-attached states of small closed- and open-shell mols. were computed and compared to full CI data. As an illustrative example, EA-ADC(3)/ISR(2) has been applied to the thymine-thymine (6-4) DNA photolesion. (c) 2021 American Institute of Physics.**45**Ortiz, J. V. Electron propagator theory: an approach to prediction and interpretation in quantum chemistry.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2013,*3*, 123, DOI: 10.1002/wcms.111645https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXmtFaiurY%253D&md5=de0c9211fddba230feff0336718bd7a1Electron propagator theory: an approach to prediction and interpretation in quantum chemistryOrtiz, Joseph VincentWiley Interdisciplinary Reviews: Computational Molecular Science (2013), 3 (2), 123-142CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)A review. Electron propagator theory provides a practical means of calcg. electron binding energies, Dyson orbitals, and ground-state properties from first principles. This approach to ab initio electronic structure theory also facilitates the interpretation of its quant. predictions in terms of concepts that closely resemble those of one-electron theories. An explanation of the phys. meaning of the electron propagator's poles and residues is followed by a discussion of its couplings to more complicated propagators. These relationships are exploited in superoperator theory and lead to a compact form of the electron propagator that is derived by matrix partitioning. Expressions for ref.-state properties, relationships to the extended Koopmans's theorem technique for evaluating electron binding energies, and connections between Dyson orbitals and transition probabilities follow from this discussion. The inverse form of the Dyson equation for the electron propagator leads to a strategy for obtaining electron binding energies and Dyson orbitals that generalizes the Hartree-Fock equations through the introduction of the self-energy operator. All relaxation and correlation effects reside in this operator, which has an energy-dependent, nonlocal form that is systematically improvable. Perturbative arguments produce several, convenient (e.g. partial third order, outer valence Green's function, and second-order, transition-operator) approxns. for the evaluation of valence ionization energies, electron affinities, and core ionization energies. Renormalized approaches based on Hartree-Fock or approx. Brueckner orbitals are employed when correlation effects become qual. important. Ref.-state total energies based on contour integrals in the complex plane and gradients of electron binding energies enable exploration of final-state potential energy surfaces.**46**Ortiz, J. V. Partial third-order quasiparticle theory: Comparisons for closed-shell ionization energies and an application to the borazine photoelectron spectrum.*J. Chem. Phys.*1996,*104*, 7599, DOI: 10.1063/1.47146846https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28XivFGrt7k%253D&md5=bdaf740c62b87ccd783585b50ae231fbPartial third-order quasiparticle theory: comparisons for closed-shell ionization energies and an application to the Borazine photoelectron spectrumOrtiz, J. V.Journal of Chemical Physics (1996), 104 (19), 7599-7605CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Valence ionization energies of a set closed-shell mols. calcd. in a partial 3rd-order (P3) quasiparticle approxn. of the electron propagator have an av. abs. error of 0.19 eV. Diagonal elements of the self-energy matrix include all 2nd-order and some 3rd-order self-energy diagrams. Because of its 5th power dependence on basis set size and its independence from electron repulsion integrals with four virtual indexes, this method has considerable potential for large mols. Formal and computational comparisons with other electron propagator techniques illustrate the advantages of the P3 procedure. Addnl. applications to benzene and borazine display the efficacy of the P3 propagator in assigning photoelectron spectra. In the borazine spectrum, 2E' and 2A2' final states are responsible for an obsd. feature at 14.76 eV. Another peak at 17.47 eV is assigned to a 2E' final state.**47**Corzo, H. H.; Galano, A.; Dolgounitcheva, O.; Zakrzewski, V. G.; Ortiz, J. V. NR2 and P3+: Accurate, Efficient Electron-Propagator Methods for Calculating Valence, Vertical Ionization Energies of Closed-Shell Molecules.*J. Phys. Chem. A*2015,*119*, 8813, DOI: 10.1021/acs.jpca.5b0094247https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXht1KmsbvJ&md5=a47a13b49be29fb616c6b71fdddd42c1NR2 and P3+: Accurate, Efficient Electron-Propagator Methods for Calculating Valence, Vertical Ionization Energies of Closed-Shell MoleculesCorzo, H. H.; Galano, Annia; Dolgounitcheva, O.; Zakrzewski, V. G.; Ortiz, J. V.Journal of Physical Chemistry A (2015), 119 (33), 8813-8821CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)Two accurate and computationally efficient electron-propagator (EP) methods for calcg. the valence, vertical ionization energies (VIEs) of closed-shell mols. have been identified through comparisons with related approxns. VIEs of a representative set of closed-shell mols. were calcd. with EP methods using 10 basis sets. The most easily executed method, the diagonal, second-order (D2) EP approxn., produces results that steadily rise as basis sets are improved toward values based on extrapolated coupled-cluster singles and doubles plus perturbative triples calcns., but its mean errors remain unacceptably large. The outer valence Green function, partial third-order and renormalized partial third-order methods (P3+), which employ the diagonal self-energy approxn., produce markedly better results but have a greater tendency to overestimate VIEs with larger basis sets. The best combination of accuracy and efficiency with a diagonal self-energy matrix is the P3+ approxn., which exhibits the best trends with respect to basis-set satn. Several renormalized methods with more flexible nondiagonal self-energies also have been examd.: the two-particle, one-hole Tamm-Dancoff approxn. (2ph-TDA), the third-order algebraic diagrammatic construction or ADC(3), the renormalized third-order (3+) method, and the nondiagonal second-order renormalized (NR2) approxn. Like D2, 2ph-TDA produces steady improvements with basis set augmentation, but its av. errors are too large. Errors obtained with 3+ and ADC(3) are smaller on av. than those of 2ph-TDA. These methods also have a greater tendency to overestimate VIEs with larger basis sets. The smallest av. errors occur for the NR2 approxn.; these errors decrease steadily with basis augmentations. As basis sets approach satn., NR2 becomes the most accurate and efficient method with a nondiagonal self-energy.**48**Ortiz, J. V. An efficient, renormalized self-energy for calculating the electron binding energies of closed-shell molecules and anions.*Int. J. Quantum Chem.*2005,*105*, 803, DOI: 10.1002/qua.2066448https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXht1SiurzI&md5=a6ef2e2bf7f9dee84c7b36ffec7b1002An efficient, renormalized self-energy for calculating the electron binding energies of closed-shell molecules and anionsOrtiz, J. V.International Journal of Quantum Chemistry (2005), 105 (6), 803-808CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)The energy-dependent, nonlocal correlation potential known as the self-energy that appears in the Dyson equation has a pole and residue structure that enables renormalizations of its low-order, perturbative contributions to be estd. The partial third-order (P3) approxn. has been extensively applied to the ionization energies of closed-shell, org. mols. and is the most successful example of a low-order, self-energy method. A renormalization based on the P3 self-energy ests. higher-order contributions by scaling low-order terms that chiefly describe final-state relaxation. The resulting P3 + self-energy retains the accuracy and efficiency of the P3 approxn., but also improves the latter method's performance with respect to the calcn. of anion electron detachment energies without the introduction of adjustable parameters. An application to an anion that previously has yielded only to more intricate treatments of electron correlation demonstrates the power of this simple, new approxn.**49**Ortiz, J. V. A nondiagonal, renormalized extension of partial third-order quasiparticle theory: Comparisons for closed-shell ionization energies.*J. Chem. Phys.*1998,*108*, 1008, DOI: 10.1063/1.47546349https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXitVSgsA%253D%253D&md5=5bffa8187f2aceebad008ded498bfc74A nondiagonal, renormalized extension of partial third-order quasiparticle theory: comparisons for closed-shell ionization energiesOrtiz, J. V.Journal of Chemical Physics (1998), 108 (3), 1008-1014CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Valence ionization energies of a set closed-shell mols. calcd. in a nondiagonal, renormalized approxn. of the electron propagator have an av. abs. error of 0.17 eV. This procedure extends the partial third order, quasiparticle approxn. of J. Chem. Phys. 104, 7599 (1996) that has proven successful in many applications. Elements of the self-energy matrix include all second-order and many higher-order terms. Because of its fifth power dependence on basis set size and its independence from electron repulsion integrals with four virtual orbital indexes, this method has considerable promise for large mols. Formal and computational comparisons with renormalized electron propagator techniques that are complete through third-order illustrate the advantages of this procedure.**50**Opoku, E.; Pawłowski, F.; Ortiz, J. V. A new generation of diagonal self-energies for the calculation of electron removal energies.*J. Chem. Phys.*2021,*155*, 204107, DOI: 10.1063/5.007084950https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXis1KisLrE&md5=6395803c7d665e06864cc840e89c5452A new generation of diagonal self-energies for the calculation of electron removal energiesOpoku, Ernest; Pawlowski, Filip; Ortiz, J. V.Journal of Chemical Physics (2021), 155 (20), 204107CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A new generation of diagonal self-energy approxns. in ab initio electron propagator theory for the calcn. of electron removal energies of mols. and mol. ions has been derived from an intermediately normalized, Hermitized super-operator metric. These methods and widely used antecedents such as the outer valence Green's function and the approx. renormalized partial third order method are tested with respect to a dataset of vertical ionization energies generated with a valence, triple-ζ, correlation-consistent basis set and a converged series of many-body calcns. whose accuracy approaches that of full CI. Several modifications of the diagonal second-order self-energy, a version of G0W0 theory based on Tamm-Dancoff excitations and several non-diagonal self-energies are also included in the tests. All new methods employ canonical Hartree-Fock orbitals. No adjustable or empirical parameters appear. A hierarchy of methods with optimal accuracy for a given level of computational efficiency is established. Several widely used diagonal self-energy methods are rendered obsolete by the new hierarchy whose members, in order of increasing accuracy, are (1) the opposite-spin non-Dyson diagonal second-order or os-nD-D2, (2) the approx. renormalized third-order quasiparticle or Q3+ , (3) the renormalized third-order quasiparticle or RQ3, (4) the approx. renormalized linear third-order or L3+ , and (5) the renormalized linear third-order or RL3 self-energies. (c) 2021 American Institute of Physics.**51**Opoku, E.; Pawłowski, F.; Ortiz, J. V. Electron Propagator Theory of Vertical Electron Detachment Energies of Anions: Benchmarks and Applications to Nucleotides.*J. Phys. Chem. A*2023,*127*, 1085, DOI: 10.1021/acs.jpca.2c0837251https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3sXhtlGgu78%253D&md5=2529454b60cc520fbd4334e1529b5e7dElectron Propagator Theory of Vertical Electron Detachment Energies of Anions: Benchmarks and Applications to NucleotidesOpoku, Ernest; Pawlowski, Filip; Ortiz, J. V.Journal of Physical Chemistry A (2023), 127 (4), 1085-1101CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)A new generation of ab initio electron-propagator self-energy approxns. that is free of adjustable parameters is tested on a benchmark set of 55 vertical electron detachment energies of closed-shell anions. Comparisons with older self-energy approxns. indicate that several new methods that make the diagonal self-energy approxn. in the canonical Hartree-Fock orbital basis provide superior accuracy and computational efficiency. These methods, their acronyms, mean abs. errors (in eV) and arithmetic bottlenecks expressed in terms of occupied (O) and virtual (V) orbitals are the opposite-spin, non-Dyson, diagonal second-order method (os-nD-D2, 0.2, OV2), the approx. renormalized quasiparticle third-order method (Q3+ , 0.15, O2V3) and the approx. renormalized, non-Dyson, linear, third-order method (nD-L3+ , 0.1, OV4). The BD-T1 (Brueckner Doubles with Triple field operators) nondiagonal electron-propagator method provides such close agreement with coupled-cluster single, double and perturbative triple replacement total energy differences that it may be used as an alternative means of obtaining std. data. The new methods with diagonal self-energy matrixes are the foundation of a composite procedure for estg. basis-set effects. This model produces accurate predictions and clear interpretations based on Dyson orbitals for the photoelectron spectra of the nucleotides found in DNA.**52**Gilbert, A. T. B.; Besley, N. A.; Gill, P. M. W. Self-Consistent Field Calculations of Excited States Using the Maximum Overlap Method (MOM).*J. Phys. Chem. A*2008,*112*, 13164, DOI: 10.1021/jp801738f52https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXhtValurbL&md5=3baaf7b15c1c6fcd86bc3c071deacfadSelf-Consistent Field Calculations of Excited States Using the Maximum Overlap Method (MOM)Gilbert, Andrew T. B.; Besley, Nicholas A.; Gill, Peter M. W.Journal of Physical Chemistry A (2008), 112 (50), 13164-13171CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)We present a simple algorithm, which we call the max. overlap method (MOM), for finding excited-state solns. to SCF equations. Instead of using the aufbau principle, the algorithm maximizes the overlap between the occupied orbitals on successive SCF iterations. This prevents variational collapse to the ground state and guides the SCF process toward the nearest, rather than the lowest energy, soln. The resulting excited-state solns. can be treated in the same way as the ground-state soln. and, in particular, derivs. of excited-state energies can be computed using ground-state code. We assess the performance of our method by applying it to a variety of excited-state problems including the calcn. of excitation energies, charge-transfer states, and excited-state properties.**53**Bagus, P. S. Self-Consistent-Field Wave Functions for Hole States of Some Ne-Like and Ar-Like Ions.*Phys. Rev.*1965,*139*, A619, DOI: 10.1103/PhysRev.139.A61953https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaF2MXktlGgu7w%253D&md5=c09be4ef8d7fb5554f0bb12eba07b08aSelf-consistent-field wave functions for hole states of some Ne-like and Ar-like ionsBagus, P. S.Physical Review (1965), 139 (3A), 619-34CODEN: PHRVAO; ISSN:0031-899X.Analytic self-consistent-field (SCF) wave functions were computed for the ground states of the closed-shell at. systems F-, Ne, and Na+ and Cl-, Ar, and K+; and for those ground and excited states of the open-shell systems which are obtained by removing a single electron from any one of the occupied shells of these closed-shell systems. Details of the calcn. of the functions are presented, with emphasis on a justification of the procedures used for the calcns. for excited states. A high accuracy is obtained; the calcns. for the closed-shell systems give the most accurate analytic SCF wave functions which have yet been reported. Ionization potentials are compared with exptl. values. Computed ionization potentials for the removal of a 2s electron from Cl-, Ar, and K+, for which no direct exptl. data are available, are estd. to be accurate to 1%. The removal of an electron from the outermost s shell increases the correlation energy, in contradiction to the predictions of a recently proposed semi- empirical scheme for estg. the correlation energy. For example, the magnitude of the correlation energy of the lowest 2S state of Ar+ is ∼4 ev. greater than the magnitude of the correlation energy of neutral Ar. The effect of the nonzero off-diagonal Lagrangian multipliers is important for the inner shell hole states.**54**Triguero, L.; Pettersson, L. G. M.; Ågren, H. Calculations of near-edge x-ray-absorption spectra of gas-phase and chemisorbed molecules by means of density-functional and transition-potential theory.*Phys. Rev. B*1998,*58*, 8097, DOI: 10.1103/PhysRevB.58.809754https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXmtVCnt7k%253D&md5=9b70092828315cc859abab4d13173b5bCalculations of near-edge x-ray-absorption spectra of gas-phase and chemisorbed molecules by means of density-functional and transition-potential theoryTriguero, L.; Pettersson, L. G. M.; Agren, H.Physical Review B: Condensed Matter and Materials Physics (1998), 58 (12), 8097-8110CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)The authors explore the utility of d.-functional theory (DFT) in conjunction with the transition-potential (TP) method to simulate x-ray-absorption spectra. Calcns. on a set of small C-contg. mols. and chemisorbed species show that this provides a viable option for obtaining excitation energies and oscillator strengths close to the exptl. accuracy of core-valence transitions. Systematic variations in energy positions and intensities of the different spectra in the test series were studied, and comparison is made with respect to the static exchange-, SCF, and explicit electron-correlation methods. The choice between std. exchange-correlation functionals is of little consequence for the valence resonant, here π*, parts of the x-ray-absorption spectra, while the long-range behavior of presently available functionals is found not to be completely satisfactory for Rydberg-like transitions. Implementing a basis set augmentation technique, DFT methods still account well for most of the salient features in the near-edge x-ray-absorption spectra, save for the multielectron transitions in the near continuum, and for some loss of Rydberg structure. For clusters modeling surface adsorbates, the DFT transition potential method reproduces well the spectral compression and intensity redn. for the valence level absorption compared to the free phase, provided fairly large clusters are taken into account. While for near-edge x-ray-absorption fine-structure (NEXAFS) spectra of free mols. the DFT-TP and Hartree-Fock/static exchange methods have complementary advantages, the DFT-TP method is clearly to be preferred when using clusters to simulate NEXAFS spectra of surface adsorbates.**55**Lee, J.; Small, D. W.; Head-Gordon, M. Excited states via coupled cluster theory without equation-of-motion methods: Seeking higher roots with application to doubly excited states and double core hole states.*J. Chem. Phys.*2019,*151*, 214103, DOI: 10.1063/1.512879555https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXit12iurvL&md5=df69beffae8f965202030d76eba217a0Excited states via coupled cluster theory without equation-of-motion methods: Seeking higher roots with application to doubly excited states and double core hole statesLee, Joonho; Small, David W.; Head-Gordon, MartinJournal of Chemical Physics (2019), 151 (21), 214103/1-214103/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In this work, we revisited the idea of using the coupled-cluster (CC) ground state formalism to target excited states. Our main focus was targeting doubly excited states and double core hole states. Typical equation-of-motion (EOM) approaches for obtaining these states struggle without higher-order excitations than doubles. We showed that by using a non-Aufbau determinant optimized via the max. overlap method, the CC ground state solver can target higher energy states. Furthermore, just with singles and doubles (i.e., CCSD), we demonstrated that the accuracy of ΔCCSD and ΔCCSD(T) (triples) far surpasses that of EOM-CCSD for doubly excited states. The accuracy of ΔCCSD(T) is nearly exact for doubly excited states considered in this work. For double core hole states, we used an improved ansatz for greater numerical stability by freezing core hole orbitals. The improved methods, core valence sepn. (CVS)-ΔCCSD and CVS-ΔCCSD(T), were applied to the calcn. of the double ionization potential of small mols. Even without relativistic corrections, we obsd. qual. accurate results with CVS-ΔCCSD and CVS-ΔCCSD(T). Remaining challenges in ΔCC include the description of open-shell singlet excited states with the single-ref. CC ground state formalism as well as excited states with genuine multireference character. The tools and intuition developed in this work may serve as a stepping stone toward directly targeting arbitrary excited states using ground state CC methods. (c) 2019 American Institute of Physics.**56**Meissner, L.; Balková, A.; Bartlett, R. J. Multiple solutions of the single-reference coupled-cluster method.*Chem. Phys. Lett.*1993,*212*, 177, DOI: 10.1016/0009-2614(93)87127-O56https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXmtFSqurc%253D&md5=0546e409176ee2281807fbdcbaad2b83Multiple solutions of the single-reference coupled-cluster methodMeissner, Leszek; Balkova, Anna; Bartlett, Rodney J.Chemical Physics Letters (1993), 212 (1-2), 177-84CODEN: CHPLBC; ISSN:0009-2614.The nonlinear coupled-cluster (CC) equations possess several solns. They describe various excited states as long as they contain a contribution from the ref. function. The authors study the addnl. solns. of the single-ref. CC equations and demonstrate that unlike the ground state, the approx. soln. for excited states in general does not satisfy the cluster condition nor the std. proof of extensivity. For such states that makes the CC approxn. poorer than that for the analogous CI.**57**Zheng, X.; Cheng, L. Performance of Delta-Coupled-Cluster Methods for Calculations of Core-Ionization Energies of First-Row Elements.*J. Chem. Theory Comput.*2019,*15*, 4945, DOI: 10.1021/acs.jctc.9b0056857https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhsV2jsbzO&md5=9552d88cc44e3b3aff173462e03ea4edPerformance of Delta-Coupled-Cluster Methods for Calculations of Core-Ionization Energies of First-Row ElementsZheng, Xuechen; Cheng, LanJournal of Chemical Theory and Computation (2019), 15 (9), 4945-4955CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A thorough study of the performance of delta-coupled-cluster (ΔCC) methods for calcns. of core-ionization energies for elements of the first long row of the periodic table is reported. Inspired by the core-valence sepn. (CVS) scheme in response theories, a simple CVS scheme of excluding the vacant core orbital from the CC treatment has been adopted to solve the convergence problem of the CC equations for core-ionized states. Dynamic correlation effects have been shown to make important contributions to the computed core-ionization energies, esp. to chem. shifts of these quantities. The max. abs. error (MaxAE) and std. deviation (SD) of delta-Hartree-Fock results for chem. shifts of core-ionization energies with respect to the corresponding exptl. values amt. to more than 1.7 and 0.6 eV, resp. In contrast, the inclusion of electron correlation in ΔCC singles and doubles augmented with a noniterative triples correction [ΔCCSD(T)] method significantly reduces the corresponding deviations to around 0.3 and 0.1 eV. With the consideration of basis set effects and the corrections to the CVS approxn., ΔCCSD(T) has been shown to provide highly accurate results for abs. values of core-ionization energies, with a MaxAE of 0.22 eV and SD of 0.13 eV. To further demonstrate the usefulness of ΔCCSD(T), calcns. of carbon K-edge ionization energies of Et trifluoroacetate, a mol. of significant interest to the study of X-ray spectroscopy and dynamics, are reported.**58**Hirata, S.; Hermes, M. R.; Simons, J.; Ortiz, J. V. General-Order Many-Body Green’s Function Method.*J. Chem. Theory Comput.*2015,*11*, 1595, DOI: 10.1021/acs.jctc.5b0000558https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXjtF2lsr0%253D&md5=220dfeed02c0268f49d7941bd2ad0e73General-Order Many-Body Green's Function MethodHirata, So; Hermes, Matthew R.; Simons, Jack; Ortiz, J. V.Journal of Chemical Theory and Computation (2015), 11 (4), 1595-1606CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Electron binding energies are evaluated as differences in total energy between the N- and (N ± 1)-electron systems calcd. by the nth-order Moller-Plesset perturbation (MPn) theory using the same set of orbitals. The MPn energies up to n = 30 are, in turn, obtained by the determinant-based method of Knowles et al. (Chem. Phys. Lett.1985, 113, 8-12). The zeroth- through third-order electron binding energies thus detd. agree with those obtained by solving the Dyson equation in the diagonal and frequency-independent approxns. of the self-energy. However, as n → ∞, they converge at the exact basis-set solns. from the Dyson equation with the exact self-energy, which is nondiagonal and frequency-dependent. This suggests that the MPn energy differences define an alternative diagrammatic expansion of Koopmans-like electron binding energies, which takes into account the perturbation corrections from the off-diagonal elements and frequency dependence of the irreducible self-energy. Our anal. shows that these corrections are included as semireducible and linked-disconnected diagrams, resp., which are also found in a perturbation expansion of the electron binding energies of the equation-of-motion coupled-cluster methods. The rate of convergence of the electron binding energies with respect to n and its acceleration by Pade approximants are also discussed.**59**Hirata, S.; Doran, A. E.; Knowles, P. J.; Ortiz, J. V. One-particle many-body Green’s function theory: Algebraic recursive definitions, linked-diagram theorem, irreducible-diagram theorem, and general-order algorithms.*J. Chem. Phys.*2017,*147*, 044108, DOI: 10.1063/1.499483759https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXht1GjtbfO&md5=a3f66c1ad67ee4656e9010e7089656e8One-particle many-body Green's function theory: Algebraic recursive definitions, linked-diagram theorem, irreducible-diagram theorem, and general-order algorithmsHirata, So; Doran, Alexander E.; Knowles, Peter J.; Ortiz, J. V.Journal of Chemical Physics (2017), 147 (4), 044108/1-044108/31CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A thorough anal. and numerical characterization of the whole perturbation series of one-particle many-body Green's function (MBGF) theory is presented in a pedagogical manner. Three distinct but equiv. algebraic (first-quantized) recursive definitions of the perturbation series of the Green's function are derived, which can be combined with the well-known recursion for the self-energy. Six general-order algorithms of MBGF are developed, each implementing one of the three recursions, the ΔMPn method (where n is the perturbation order) [S. Hirata et al., J. Chem. Theory Comput. 11, 1595 (2015)], the automatic generation and interpretation of diagrams, or the numerical differentiation of the exact Green's function with a perturbation-scaled Hamiltonian. They all display the identical, nondivergent perturbation series except ΔMPn, which agrees with MBGF in the diagonal and frequency-independent approxns. at 1 ≤ n ≤ 3 but converges at the full-configuration-interaction (FCI) limit at n = ∞ (unless it diverges). Numerical data of the perturbation series are presented for Koopmans and non-Koopmans states to quantify the rate of convergence towards the FCI limit and the impact of the diagonal, frequency-independent, or ΔMPn approxn. The diagrammatic linkedness and thus size-consistency of the one-particle Green's function and self-energy are demonstrated at any perturbation order on the basis of the algebraic recursions in an entirely time-independent (frequency-domain) framework. The trimming of external lines in a one-particle Green's function to expose a self-energy diagram and the removal of reducible diagrams are also justified math. using the factorization theorem of Frantz and Mills. Equivalence of ΔMPn and MBGF in the diagonal and frequency-independent approxns. at 1 ≤ n ≤ 3 is algebraically proven, also ascribing the differences at n = 4 to the so-called semi-reducible and linked-disconnected diagrams. (c) 2017 American Institute of Physics.**60**Goerigk, L.; Hansen, A.; Bauer, C.; Ehrlich, S.; Najibi, A.; Grimme, S. A look at the density functional theory zoo with the advanced GMTKN55 database for general main group thermochemistry, kinetics and noncovalent interactions.*Phys. Chem. Chem. Phys.*2017,*19*, 32184, DOI: 10.1039/C7CP04913G60https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhslajtLnF&md5=f9393c9e3907336c4da053743797f8dfA look at the density functional theory zoo with the advanced GMTKN55 database for general main group thermochemistry, kinetics and noncovalent interactionsGoerigk, Lars; Hansen, Andreas; Bauer, Christoph; Ehrlich, Stephan; Najibi, Asim; Grimme, StefanPhysical Chemistry Chemical Physics (2017), 19 (48), 32184-32215CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)We present the GMTKN55 benchmark database for general main group thermochem., kinetics and noncovalent interactions. Compared to its popular predecessor GMTKN30, it allows assessment across a larger variety of chem. problems - with 13 new benchmark sets being presented for the first time - and it also provides ref. values of significantly higher quality for most sets. GMTKN55 comprises 1505 relative energies based on 2462 single-point calcns. and it is accessible to the user community via a dedicated website. Herein, we demonstrate the importance of better ref. values, and we re-emphasize the need for London-dispersion corrections in d. functional theory (DFT) treatments of thermochem. problems, including Minnesota methods. We assessed 217 variations of dispersion-cor. and -uncorrected d. functional approxns., and carried out a detailed anal. of 83 of them to identify robust and reliable approaches. Double-hybrid functionals are the most reliable approaches for thermochem. and noncovalent interactions, and they should be used whenever tech. feasible. These are, in particular, DSD-BLYP-D3(BJ), DSD-PBEP86-D3(BJ), and B2GPPLYP-D3(BJ). The best hybrids are ωB97X-V, M052X-D3(0), and ωB97X-D3, but we also recommend PW6B95-D3(BJ) as the best conventional global hybrid. At the meta-generalized-gradient (meta-GGA) level, the SCAN-D3(BJ) method can be recommended. Other meta-GGAs are outperformed by the GGA functionals revPBE-D3(BJ), B97-D3(BJ), and OLYP-D3(BJ). We note that many popular methods, such as B3LYP, are not part of our recommendations. In fact, with our results we hope to inspire a change in the user community's perception of common DFT methods. We also encourage method developers to use GMTKN55 for cross-validation studies of new methodologies.**61**Goerigk, L.; Grimme, S. A general database for main group thermochemistry, kinetics, and noncovalent interactions – Assessment of common and reparameterized (meta-)GGA density functionals.*J. Chem. Theory Comput.*2010,*6*, 107, DOI: 10.1021/ct900489g61https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhsVClurvM&md5=6452b32bf508de27fb37c221b8fdfdd4A General Database for Main Group Thermochemistry, Kinetics, and Noncovalent Interactions - Assessment of Common and Reparameterized (meta-)GGA Density FunctionalsGoerigk, Lars; Grimme, StefanJournal of Chemical Theory and Computation (2010), 6 (1), 107-126CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a quantum chem. benchmark database for general main group thermochem., kinetics, and noncovalent interactions (GMTKN24). It is an unprecedented compilation of 24 different, chem. relevant subsets that either are taken from already existing databases or are presented here for the first time. The complete set involves a total of 1.049 at. and mol. single point calcns. and comprises 731 data points (relative chem. energies) based on accurate theor. or exptl. ref. values. The usefulness of the GMTKN24 database is shown by applying common d. functionals on the (meta-)generalized gradient approxn. (GGA), hybrid-GGA, and double-hybrid-GGA levels to it, including an empirical London dispersion correction. Furthermore, we refitted the functional parameters of four (meta-)GGA functionals based on a fit set contg. 143 systems, comprising seven chem. different problems. Validation against the GMTKN24 and the mol. structure (bond lengths) databases shows that the reparameterization does not change bond lengths much, whereas the description of energetic properties is more prone to the parameters' values. The empirical dispersion correction also often improves for conventional thermodn. problems and makes a functional's performance more uniform over the entire database. The refitted functionals typically have a lower mean abs. deviation for the majority of subsets in the proposed GMTKN24 set. This, however, is also often accompanied at the expense of poor performance for a few other important subsets. Thus, creating a broadly applicable (and overall better) functional by just reparameterizing existing ones seems to be difficult. Nevertheless, this benchmark study reveals that a reoptimized (i.e., empirical) version of the TPSS-D functional (oTPSS-D) performs well for a variety of problems and may meet the stds. of an improved functional. We propose validation against this new compilation of benchmark sets as a definitive way to evaluate a new quantum chem. method's true performance.**62**Curtiss, L. A.; Raghavachari, K.; Trucks, G. W.; Pople, J. A. Gaussian-2 theory for molecular energies of first-and second-row compounds.*J. Chem. Phys.*1991,*94*, 7221, DOI: 10.1063/1.46020562https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3MXksFOlsr4%253D&md5=2de9eaceadf097004bde9659ee043f42Gaussian-2 theory for molecular energies of first- and second-row compoundsCurtiss, Larry A.; Raghavachari, Krishnan; Trucks, Gary W.; Pople, John A.Journal of Chemical Physics (1991), 94 (11), 7221-30CODEN: JCPSA6; ISSN:0021-9606.The Gaussian-2 theor. procedure (G2 theory), based on ab-initio MO theory, for calcn. of mol. energies (atomization energies, ionization potentials, electron affinities, and proton affinities) of compds. contg. first- (Li-F) and second-row atoms (Na-Cl) is presented. This new theor. procedure adds three features to G1 theory (P., et al., 1989; C., et al., 1990), including a correction for nonadditivity of diffuse-sp and 2df basis-set extensions, a basis-set extension contg. a third d-function on nonhydrogen atoms and a second p-function on hydrogen atoms, and a modification of the higher level correction. G2 theory is a significant improvement over G1 theory, because it eliminates a no. of deficiencies present in G1 theory. Of particular importance is the improvement in atomization energies of ionic mols. such as LiF and hydrides such as C2H6,NH3, N2H4, H2O2, and CH3SH. The av. abs. deviation from expt. of atomization energies of 39 first-row compds. is reduced from 1.42 to 0.92 kcal/mol. In addn., G2 theory gives improved performance for hypervalent species and electron affinities of second-row species (the av. deviation from expt. of electron affinities of second-row species is reduced from 1.94 to 1.08 kcal/mol). Finally, G2 atomization energies for another 43 mols., not previously studied with G1 theory, many of which have uncertain exptl. data, are presented and differences with expt. are assessed.**63**Śmiga, S.; Franck, O.; Mussard, B.; Buksztel, A.; Grabowski, I.; Luppi, E.; Toulouse, J. Self-consistent double-hybrid density-functional theory using the optimized-effective-potential method.*J. Chem. Phys.*2016,*145*, 144102, DOI: 10.1063/1.496431963https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xhs1yjurzJ&md5=39d1fb62cea025accba2744dcb59e1f9Self-consistent double-hybrid density-functional theory using the optimized-effective-potential methodSmiga, Szymon; Franck, Odile; Mussard, Bastien; Buksztel, Adam; Grabowski, Ireneusz; Luppi, Eleonora; Toulouse, JulienJournal of Chemical Physics (2016), 145 (14), 144102/1-144102/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We introduce an orbital-optimized double-hybrid (DH) scheme using the optimized-effective-potential (OEP) method. The orbitals are optimized using a local potential corresponding to the complete exchange-correlation energy expression including the second-order Moller-Plesset correlation contribution. We have implemented a one-parameter version of this OEP-based self-consistent DH scheme using the BLYP d.-functional approxn. and compared it to the corresponding non-self-consistent DH scheme for calcns. on a few closed-shell atoms and mols. While the OEP-based self-consistency does not provide any improvement for the calcns. of ground-state total energies and ionization potentials, it does improve the accuracy of electron affinities and restores the meaning of the LUMO orbital energy as being connected to a neutral excitation energy. Moreover, the OEP-based self-consistent DH scheme provides reasonably accurate exchange-correlation potentials and correlated densities. (c) 2016 American Institute of Physics.**64**Śmiga, S.; Grabowski, I.; Witkowski, M.; Mussard, B.; Toulouse, J. Self-Consistent Range-Separated Density-Functional Theory with Second-Order Perturbative Correction via the Optimized-Effective-Potential Method.*J. Chem. Theory Comput.*2020,*16*, 211, DOI: 10.1021/acs.jctc.9b0080764https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BB3MfotVCrsA%253D%253D&md5=e2e43924a3c8e3a5668b8e023cafe47bSelf-Consistent Range-Separated Density-Functional Theory with Second-Order Perturbative Correction via the Optimized-Effective-Potential MethodSmiga Szymon; Grabowski Ireneusz; Witkowski Mateusz; Mussard Bastien; Toulouse JulienJournal of chemical theory and computation (2020), 16 (1), 211-223 ISSN:.We extend the range-separated double-hybrid RSH+MP2 method (Angyan, J. G.; et al. Phys. Rev. A2005, 72, 012510), combining long-range HF exchange and MP2 correlation with a short-range density functional to a fully self-consistent version using the optimized-effective-potential technique in which the orbitals are obtained from a local potential including the long-range HF and MP2 contributions. We test this approach, that we name RS-OEP2, on a set of small closed-shell atoms and molecules. For the commonly used value of the range-separation parameter μ = 0.5 bohr(-1), we find that self-consistency does not seem to bring any improvement for total energies, ionization potentials, and electronic affinities. However, contrary to the non-self-consistent RSH+MP2 method, the present RS-OEP2 method gives a LUMO energy which physically corresponds to a neutral excitation energy and gives local exchange-correlation potentials which are reasonably good approximations to the corresponding Kohn-Sham quantities. At a finer scale, we find that RS-OEP2 gives largely inaccurate correlation potentials and correlated densities, which points to the need of further improvement of this type of range-separated double hybrids.**65**Cohen, A. J.; Mori-Sánchez, P.; Yang, W. Second-Order Perturbation Theory with Fractional Charges and Fractional Spins.*J. Chem. Theory Comput.*2009,*5*, 786, DOI: 10.1021/ct800541965https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXjtlOhtLw%253D&md5=cd0e821eee7816ae162f2bb3d14ba2c6Second-Order Perturbation Theory with Fractional Charges and Fractional SpinsCohen, Aron J.; Mori-Sanchez, Paula; Yang, WeitaoJournal of Chemical Theory and Computation (2009), 5 (4), 786-792CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The behavior of MP2 for fractional occupations is investigated. The consideration of fractional charge behavior gives a simple derivation of an expression for the chem. potential (or the deriv. of energy with respect to the no. of electrons) of MP2. A generalized optimized effective potential formalism (OEP) has been developed in which the OEP is a nonlocal potential, which can be applied to explicit functionals of the orbitals and eigenvalues and also facilitates the evaluation of the chem. potential. The MP2 deriv. improves upon the corresponding Koopmans' theorem in Hartree-Fock theory for the ionization energy and also gives a good est. of the electron affinity. In strongly correlated systems with degeneracies and fractional spins, MP2 diverges, and another cor. second-order perturbative method ameliorates this failure for the energy but still does not recapture the correct behavior for the energy derivs. that yield the gap. Overall we present a view of wave function based methods and their behavior for fractional charges and spins that offers insight into the application of these methods to challenging chem. problems.**66**Su, N. Q.; Yang, W.; Mori-Sánchez, P.; Xu, X. Fractional Charge Behavior and Band Gap Predictions with the XYG3 Type of Doubly Hybrid Density Functionals.*J. Phys. Chem. A*2014,*118*, 9201, DOI: 10.1021/jp502999266https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXotlWjur4%253D&md5=7b62ae2403f6c31d8b4e8c620e128097Fractional Charge Behavior and Band Gap Predictions with the XYG3 Type of Doubly Hybrid Density FunctionalsSu, Neil Qiang; Yang, Weitao; Mori-Sanchez, Paula; Xu, XinJournal of Physical Chemistry A (2014), 118 (39), 9201-9211CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)In this work, we examine the fractional charge behaviors of doubly hybrid (DH) functionals. By plotting the ground-state energies E and energy derivs. for atoms and mols. with fractional electron nos. N, we directly quantify the delocalization errors of some representative DH functionals such as B2PLYP, XYG3, and XYGJ-OS. Numerical assessments on ionization potentials (IPs), electron affinities (EAs), and fundamental gaps, from either integer no. calcns. or energy deriv. calcns., are provided. It is shown that the XYG3 type of DH functionals gives good agreement between their energy derivs. and the exptl. IPs, EAs, and gaps, as expected from their nearly straight line fractional charge behaviors.**67**Mussard, B.; Toulouse, J. Fractional-charge and fractional-spin errors in range-separated density-functional theory.*Mol. Phys.*2017,*115*, 161, DOI: 10.1080/00268976.2016.121391067https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xht1yqsbbO&md5=7d6e54d5e0aa61f9d751af2846eac407Fractional-charge and fractional-spin errors in range-separated density-functional theoryMussard, Bastien; Toulouse, JulienMolecular Physics (2017), 115 (1-2), 161-173CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)We investigate fractional-charge and fractional-spin errors in range-sepd. d.-functional theory (DFT). Specifically, we consider the range-sepd. hybrid (RSH) method which combines long-range Hartree-Fock (HF) exchange with a short-range semilocal exchange-correlation d. functional, and the RSH+MP2 method which adds long-range, second-order Moller-Plesset (MP2) correlation. Results on atoms and mols. show that the fractional-charge errors obtained in RSH are much smaller than in the std. Kohn-Sham (KS) scheme applied with semilocal or hybrid approxns., and also generally smaller than in the std. HF method. The RSH+MP2 method tends to have smaller fractional-charge errors than std. MP2 for the most diffuse systems, but larger fractional-charge errors for the more compact systems. Even though the individual contributions to the fractional-spin errors in the H atom coming from the short-range exchange and correlation d.-functional approxns. are smaller than the corresponding contributions for the full-range exchange and correlation d.-functional approxns., RSH gives fractional-spin errors that are larger than in the std. KS scheme and only slightly smaller than in std. HF. Adding long-range MP2 correlation only leads to infinite fractional-spin errors. This work clarifies the successes and limitations of range-sepd. DFT approaches for eliminating self-interaction and static-correlation errors.**68**Su, N. Q.; Xu, X. Insights into Direct Methods for Predictions of Ionization Potential and Electron Affinity in Density Functional Theory.*J. Phys. Chem. Lett.*2019,*10*, 2692, DOI: 10.1021/acs.jpclett.9b0105268https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXovVKnsbc%253D&md5=7556f302c3c7856511cefeb2290a20f8Insights into Direct Methods for Predictions of Ionization Potential and Electron Affinity in Density Functional TheorySu, Neil Qiang; Xu, XinJournal of Physical Chemistry Letters (2019), 10 (11), 2692-2699CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)Vertical ionization potential (IP) and electron affinity (EA) are fundamental mol. properties, while the Δ method and the direct method are the widely used approaches to compute these properties. The Δ method is calcd. by taking the total energy difference of the initial and final states, whose reliability is seriously affected by the issue assocd. with the imbalanced treatment of these two states. The direct method based on the derivs. involving only one single-state calcn. can yield a quasiparticle spectrum whose accuracy, on the other hand, is mostly affected by the levels of approx. mol. structure theories. Because of the aforementioned issues, EA prediction can be particularly problematic. Here we present, for the first time, an analytic theory on the derivation and realization of generalized Kohn-Sham (KS) eigenvalues of doubly hybrid (DH) functionals that depend on both occupied and unoccupied orbitals. The method based on the KS eigenvalues of neutral systems, termed the NKS method, is found to suffer little from the imbalance issue, while it is only the NKS method that can offer accurate EA prediction from a good functional approxn., such as the XYG3 type of DH functionals. Being less sensitive to the size of basis sets, the NKS method is of great significance for its application to large systems. The insights gained in this work are useful for the calcn. of properties assocd. with small energy differences while emphasizing the importance of the development of generalized functionals that rely on both occupied and unoccupied orbitals.**69**Beste, A.; Vázquez-Mayagoitia, Á.; Ortiz, J. V. Direct ΔMBPT(2) method for ionization potentials, electron affinities, and excitation energies using fractional occupation numbers.*J. Chem. Phys.*2013,*138*, 074101, DOI: 10.1063/1.479062669https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXis1Wrtbo%253D&md5=621540912c7873be7720dd6baa5aef35Direct ΔMBPT(2) method for ionization potentials, electron affinities, and excitation energies using fractional occupation numbersBeste, Ariana; Vazquez-Mayagoitia, Alvaro; Ortiz, J. V.Journal of Chemical Physics (2013), 138 (7), 074101/1-074101/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A direct method (D-ΔMBPT(2)) to calc. 2nd-order ionization potentials (IPs), electron affinities (EAs), and excitation energies is developed. The ΔMBPT(2) method is defined as the correlated extension of the ΔHF method. Energy differences are obtained by integrating the energy deriv. with respect to occupation nos. over the appropriate parameter range. This is made possible by writing the 2nd-order energy as a function of the occupation nos. Relaxation effects are fully included at the SCF level. This is in contrast to linear response theory, which makes the D-ΔMBPT(2) applicable not only to single excited but also higher excited states. We show the relationship of the D-ΔMBPT(2) method for IPs and EAs to a 2nd-order approxn. of the effective Fock-space coupled-cluster Hamiltonian and a 2nd-order electron propagator method. We also discuss the connection between the D-ΔMBPT(2) method for excitation energies and the CIS-MP2 method. Finally, as a proof of principle, we apply our method to calc. ionization potentials and excitation energies of some small mols. For IPs, the ΔMBPT(2) results compare well to the 2nd-order soln. of the Dyson equation. For excitation energies, the deviation from equation of motion coupled cluster singles and doubles increases when correlation becomes more important. When using the numerical integration technique, we encounter difficulties that prevented us from reaching the ΔMBPT(2) values. Most importantly, relaxation beyond the Hartree-Fock level is significant and needs to be included in future research. (c) 2013 American Institute of Physics.**70**Gu, Y.; Xu, X. Extended Koopmans’ theorem in the adiabatic connection formalism: Applied to doubly hybrid density functionals.*J. Chem. Phys.*2020,*153*, 044109, DOI: 10.1063/5.001074370https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhsVOqtLrK&md5=613aa454c3423f076a2da0dd1ba9c2ceExtended Koopmans' theorem in the adiabatic connection formalism: Applied to doubly hybrid density functionalsGu, Yonghao; Xu, XinJournal of Chemical Physics (2020), 153 (4), 044109CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A rigorous framework that combines the extended Koopmans' theorem (EKT) with the adiabatic connection (AC) formalism of d. functional theory is developed here, namely, EKT-AC, to calc. the vertical ionization potentials (IPs) of mol. systems. When applied to the doubly hybrid d. functional approxns. (DH-DFAs), the EKT-DH approach is established for the B2PLYP-type DHs with one-parameter and two-parameters, as well as the XYG3-type DHs. Based on EKT-DH, an approxn. of the KT type is introduced, leading to the KT-DH approach. The IP-condition that the calcd. vertical IPs with EKT-DH or KT-DH are to reproduce the exptl. IPs closely is applied to investigate the commonly used DH-DFAs for such a purpose and is utilized as a principle for DH-DFA developments. Considering the systematic improvements, as well as its numeric stability, we recommend the KT-B2GPPLYP approach as a pragmatic way for vertical IP calcns. (c) 2020 American Institute of Physics.**71**Grimme, S.; Neese, F. Double-hybrid density functional theory for excited electronic states of molecules.*J. Chem. Phys.*2007,*127*, 154116, DOI: 10.1063/1.277285471https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXht1ejs7fP&md5=99b6be2c14771032a2aa44c9827d64afDouble-hybrid density functional theory for excited electronic states of moleculesGrimme, Stefan; Neese, FrankJournal of Chemical Physics (2007), 127 (15), 154116/1-154116/18CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Double-hybrid d. functionals are based on a mixing of std. generalized gradient approxns. (GGAs) for exchange and correlation with Hartree-Fock (HF) exchange and a perturbative second-order correlation part (PT2) that is obtained from the Kohn-Sham (GGA) orbitals and eigenvalues. This virtual orbital-dependent functional (dubbed B2PLYP) contains only two empirical parameters that describe the mixt. of HF and GGA exchange (ax) and of the PT2 and GGA correlation (ac), resp. Extensive testing has recently demonstrated the outstanding accuracy of this approach for various ground state problems in general chem. applications. The method is extended here without any further empirical adjustments to electronically excited states in the framework of time-dependent d. functional theory (TD-DFT) or the closely related Tamm-Dancoff approxn. (TDA-DFT). In complete analogy to the ground state treatment, a scaled second-order perturbation correction to CI with singles (CIS(D)) wave functions developed some years ago by Head-Gordon et al. is computed on the basis of d. functional data and added to the TD(A)-DFT/GGA excitation energy. The method is implemented by applying the resoln. of the identity approxn. and the efficiency of the code is discussed. Extensive tests for a wide variety of mols. and excited states (of singlet, triplet, and doublet multiplicities) including electronic spectra are presented. In general, rather accurate excitation energies (deviations from ref. data typically <0.2 eV) are obtained that are mostly better than those from std. functionals. Still, systematic errors are obtained for Rydberg (too low on av. by about 0.3 eV) and charge-transfer transitions but due to the relatively large ax parameter (0.53), B2PLYP outperforms most other functionals in this respect. Compared to conventional HF-based CIS(D), the method is more robust in electronically complex situations due to the implicit account of static correlation effects by the GGA parts. The (D) correction often works in the right direction and compensates for the overestimation of the transition energy at the TD level due to the elevated fraction of HF exchange in the hybrid GGA part. Finally, the limitations of the method are discussed for challenging systems such as transition metal complexes, cyanine dyes, and multireference cases.**72**Hirata, S.; Head-Gordon, M. Time-dependent density functional theory within the Tamm–Dancoff approximation.*Chem. Phys. Lett.*1999,*314*, 291, DOI: 10.1016/S0009-2614(99)01149-572https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXnsl2rsr0%253D&md5=1b62f410de6c2a2193f1011d42f389c5Time-dependent density functional theory within the Tamm-Dancoff approximationHirata, S.; Head-Gordon, M.Chemical Physics Letters (1999), 314 (3,4), 291-299CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)A computationally simple method for mol. excited states, namely, the Tamm-Dancoff approxn. to time-dependent d. functional theory, is proposed and implemented. This method yields excitation energies for several closed- and open-shell mols. that are essentially of the same quality as those obtained from time-dependent d. functional theory itself, when the same exchange-correlation functional is used.**73**Foresman, J. B.; Head-Gordon, M.; Pople, J. A.; Frisch, M. J. Toward a systematic molecular orbital theory for excited states.*J. Phys. Chem.*1992,*96*, 135, DOI: 10.1021/j100180a03073https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK38Xks1GhsA%253D%253D&md5=fc67a2af6aef6358d605cefe654b991aToward a systematic molecular orbital theory for excited statesForesman, James B.; Head-Gordon, Martin; Pople, John A.; Frisch, Michael J.Journal of Physical Chemistry (1992), 96 (1), 135-49CODEN: JPCHAX; ISSN:0022-3654.The methodol. and computational considerations necessary for the detn. of the ab initio energy, wave function, and gradient of a mol. in an electronically excited state using MO theory are discussed. A fundamental level of theory is reexamd. which was employed several years ago for the interpretation of the electronic spectra of simple org. mols.: CI (CI) among all singly substituted determinants using a Hartree-Fock ref. state. Several new enhancements to this general theory are given. First, it is shown how the "CI-singles" wave function can be used to compute efficiently the analytic first deriv. of the energy in order to obtain accurate properties and optimized geometries for a wide range of mols. in their excited states. Secondly, a computer program is described which allows these computations to be done in a "direct" fashion, with no disk storage required for the two-electron repulsion integrals. This allows investigations of systems with large nos. of atoms (or large nos. of basis functions). Thirdly it is shown how the CI-singles approxn. can be cor. via second-order Moeller-Plesset perturbation theory to produce a level of theory for excited states which further includes some effects of electronic correlation. The relative success of the model as a function of basis set indicates that a judicious choice of basis set is needed in order to evaluate its performance adequately. Application of the method to the excited states of formaldehyde, ethylene, pyridine, and porphin demonstrates the utility of CI-singles theory.**74**Head-Gordon, M.; Rico, R. J.; Oumi, M.; Lee, T. J. A doubles correction to electronic excited states from configuration interaction in the space of single substitutions.*Chem. Phys. Lett.*1994,*219*, 21, DOI: 10.1016/0009-2614(94)00070-074A doubles correction to electronic excited states from configuration interaction in the space of single substitutionsHead-Gordon, Martin; Rico, Rudolph J.; Oumi, Manabu; Lee, Timothy J.Chemical Physics Letters (1994), 219 (1-2), 21-9CODEN: CHPLBC; ISSN:0009-2614.A perturbative correction to the method of CI with single substitutions (CIS) is presented. This CIS(D) correction approx. introduces the effect of double substitutions which are absent in CIS excited states. CIS(D) is a second-order perturbation expansion of the coupled-cluster excited state method, restricted to single and double substitutions, in a series in which CIS is zeroth order, and the first-order correction vanishes. CIS(D) excitation energies are size consistent and the calculational complexity scales with the fifth power of mol. size, akin to second-order Moeller-Plesset theory for the ground state. Calcns. on singlet excited states of ethylene,