**Cite This:**

*J. Chem. Theory Comput.*2021, 17, 1, 139-150

# Energy-Based Molecular Orbital Localization in a Specific Spatial RegionClick to copy article linkArticle link copied!

- Tommaso Giovannini
*****Tommaso GiovanniniDepartment of Chemistry, Norwegian University of Science and Technology, 7491 Trondheim, Norway*****Email: [email protected]More by Tommaso Giovannini - Henrik Koch
*****Henrik KochScuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy*****Email: [email protected]More by Henrik Koch

## Abstract

We present a novel energy-based localization procedure able to localize molecular orbitals into predefined spatial regions. The method is defined in a multiscale framework based on the multilevel Hartree–Fock approach. In particular, the system is partitioned into active and inactive fragments. The localized molecular orbitals are obtained maximizing the repulsion between the two fragments. The method is applied to several cases including both conjugated and non-conjugated systems. Our multiscale approach is compared with reference values for both ground-state properties, such as dipole moments, and local excitation energies. The proposed approach is useful to extend the application range of high-level electron correlation methods. In fact, the reduced number of molecular orbitals can lead to a large reduction in the computational cost of correlated calculations.

## 1. Introduction

## 2. Theory

**D**=

**D**

^{A}+

**D**

^{B}). The total HF energy can be written as

**h**and

**G**are the usual one- and two-electron matrices and

*h*

_{nuc}is the nuclear repulsion. The

**G**(

**D**)

^{X}with

*X*= {

*A*,

*B*} matrix is defined as

**D**corresponds to the full HF density. However, in order to reduce the computational cost, in MLHF the density of fragment

*A*is minimized in the field generated by the density

*B*, which is kept fixed. This procedure is performed by minimizing the energy (see eq 1) in the MO basis of the active part, reducing the dimensionality of the problem. The MO basis is defined in terms of the full AO basis set, without performing any AO truncation, except linear dependencies. The quality of the MLHF results will therefore depend on both the accuracy of the decomposed density and the decomposition algorithm. In MLHF, the Fock matrix in AO basis is expressed by differentiating eq 1 with respect to

**D**

^{A}

*G*

_{μν}(

**D**

^{B}) is a one-electron contribution because the

**D**

^{B}density is kept frozen during the SCF procedure.

**D**is the converged HF density for the entire system. However, eq 1 does not have an apparent physical interpretation because the different energy terms are not assigned to the individual fragments. Such a physical insight can be achieved by dividing the one-electron term into the kinetic (

**T**) and the electron-nuclear attraction operators for the two parts (

**V**

^{A}and

**V**

^{B}). Thus, eq 1 can be written as

*h*

_{nuc}

^{A},

*h*

_{nuc}

^{B}, and

*h*

_{nuc}

^{AB}are nuclear repulsion terms;

*E*

_{A}and

*E*

_{B}are the energies of the two fragments, whereas

*E*

_{AB}is the interaction energy. The latter term is composed of the electron–nuclear attraction between

*A*and

*B*and vice versa and the coulomb and exchange interactions between the two fragments.

**D**that is decomposed into two densities belonging to two fragments.

*E*

_{A}(see eq 4) can be minimized (denoted MLHF-A). In such a case, the Fock matrix reads

*E*

_{A}and

*E*

_{B}is minimized. From the computational point of view,

*E*

_{A}+

*E*

_{B}can be rewritten by considering that the total density

**D**=

**D**

^{A}+

**D**

^{B}remains constant during occupied–occupied rotations. This means that

**D**

^{B}can be expressed in terms of it as

**D**

^{B}=

**D**−

**D**

^{A}. Therefore, the sum of

*A*and

*B*energies reads

**D**

^{A}

**G**(

**D**) is instead the two-electron interaction between the active and the constant total density

**D**. The Fock matrix of the active part can be written as

*V*

_{μν}

^{A}–

*V*

_{μν}

^{B}), a two-electron contribution on the active density 2

*G*

_{μν}(

**D**

^{A}), and a constant contribution due to the total density

*G*

_{μν}(

**D**).

*A*and

*B*parts in MLHF-AB (see eq 6) is equivalent to maximizing the interaction energy

*E*

_{AB}. Physically, this means that the repulsion between the two parts is maximized, and the occupied orbitals obtained by this scheme are those maximally located in the two fragments. For this reason, MLHF-AB can be viewed as an extension to fragment-based methods of the Edmiston–Ruedenberg MO localization procedure; (3) although our approach conceptually differs from the latter because it is energy-guided. Finally, notice that when

**D**is the HF converged density of the entire system, the MLHF-AB localization will provide MOs which are localized in the predefined fragments. In this case, the localization only depends on the number of electrons which are assigned to each fragment, but it is independent of the chosen decomposition algorithm.

## 3. Computational Procedure

1. | Construction of the initial density by means of superposition of atomic densities, (72) followed by diagonalization of the initial Fock matrix. | ||||

2. | Partitioning of the resulting density into (8) I and J are the diagonal elements which are decomposed, D̃ is the submatrix of D containing the selected diagonal elements, and L_{αI} are the Cholesky orbitals. In particular, the number of D diagonal elements which are selected corresponds to the correct number of occupied orbitals (n_{o}) of the active fragment, that is the largest n_{o} diagonals. As a result of the decomposition, the active Cholesky MOs are obtained and the active density matrix D^{A} is trivially constructed (see eq 8). As stated above, the active virtual orbitals are constructed by means of PAOs obtained from the AOs centered in the active fragment. The active and inactive occupied orbitals are projected out from them. The obtained PAOs are defined in terms of the full AO basis set and are orthonormalized through the Lowdin procedure. The threshold for removing the linear dependencies is set to 10^{–6}. The inactive density D^{B} is instead obtained as a difference between the total density D and the active one D^{A}. | ||||

3. | The energy defined in eq 1 is minimized in the MO basis of the active part. In this way, the dimensionality of the problem is reduced, because, although the active MO coefficients are defined in the whole AO basis set, their number corresponds to the selected active orbitals only. Therefore, all AO matrices defined in eq 1 can be transformed in the active MO basis using the active MO coefficients. | ||||

4. | The total density | ||||

5. | The energy of | ||||

6. | From MLHF-A/AB occupied MO coefficients, active and inactive densities are constructed and a new MLHF calculation is restarted from point 3 until convergence is reached. For all results reported in this paper, three macrocycles MLHF–MLHF-A/AB are sufficient to reach full convergence of the energy. It is worth noticing that since the MLHF calculation is restarted from the MO coefficients obtained at the 5th step, the total computational cost of MLHF-A/AB is only twice that of a standard MLHF calculation. |

## 4. Numerical Applications

*S*)-nicotine (in its most stable conformer (76)), and [2,2]paracyclophane (PCP) (see Figure 2 for the molecular structures). Molecular geometries of ANS and nicotine are optimized at the B3LYP/aug-cc-pVDZ by using Gaussian 16 package. (84) The graphene sheet is constructed by setting the C–C distance to 1.42 Å and the C–H distance to 1.07 Å. (85) The PCP geometry is taken from ref (82). Graphene and ANS are chosen because they are conjugated systems. The conjugation is broken by our definition of the active regions as depicted in Figure 2a,b (in both cases, the bonding electrons are assigned to the inactive part). In the case of nicotine and PCP, single covalent bonds are cut and the bonding electrons are assigned to the active fragment (see Figure 2c,d). Hereby, we demonstrate the generality of our procedure, which can be applied to different cases (single/double bond cutting) and to different definitions of the active region.

_{2}

^{p}of an MO φ

_{p}is defined as (67)

_{p}is defined as the square root of μ

_{2}

^{p}. We also defined ξ as the average value of σ

_{p}, that is, ξ is a measure of the mean locality of the considered set of MOs. (1) In this paper, MLHF-A/AB MOs are compared with canonical MLHF ones (named Cholesky because they are obtained through a Cholesky decomposition of the initial density matrix) that are also localized with the Boys procedure (Cholesky–Boys). (67) Notice that in Boys localization, the sum over p of μ

_{2}

^{p}in eq 9 is minimized, (67) and the obtained MOs can therefore be used as a reference for both MLHF-A and MLHF-AB approaches.

### 4.1. MLHF-A/AB Localized MOs

_{p}are also reported (see Sections S1.2 and S1.3 in the Supporting Information for the spreads of all occupied valence orbitals). First, we notice that in both ANS and graphene, Cholesky orbitals have the largest spread on average (ξ) and the largest maximum MO spread (max{σ

_{p}}). As expected, both parameters are reduced by Cholesky–Boys. The MOs calculated by both methods are delocalized over the whole molecule (for both ANS and graphene).

*σ*

_{p}} are reduced compared to the corresponding Cholesky counterparts. It is also worth noticing significant differences between MLHF-A and MLHF-AB, in particular for ANS. In fact, the most diffuse MLHF-A MO has a tail connecting active and inactive fragments, which should be absent since bonding electrons are assigned to the inactive part. Such a tail is completely absent in the case of MLHF-AB. From a physical point of view, this is not surprising. In fact, in the MLHF-AB procedure (see eq 4), the occupied orbitals of the active and inactive fragments are rotated in order to minimize the sum of the two energies. As stated above, such a rotation corresponds to maximizing the interaction energy between the two parts, that is, to maximizing the repulsion between them. As a consequence, the active occupied orbitals calculated by MLHF-AB are more localized on the active part.

_{p}} for ANS computed by using cc-pVTZ and aug-cc-pVDZ give very similar results, thus showing the consistency of our approach when diffuse functions are included (see Table S2 in the Supporting Information). The observations for ANS and graphene also apply to nicotine and PCP, whose MOs and corresponding spreads are reported in Sections S1.4 and S1.5 in the Supporting Information. For the latter systems, the ξ for Cholesky–Boys are lower than the corresponding MLHF-A/AB counterparts, but the MOs also spread in the inactive region. To illustrate the robustness of our approach, a different definition of active/inactive parts of ANS is also investigated (see Section S1.2.2 in the Supporting Information). The calculated results confirm the findings discussed here.

### 4.2. Ground-State Dipole Moments

### 4.3. MLHF-AB Basis Set Dependence

*X*

^{–3}approach, to extrapolate the asymptotic convergence of the correlation energies. (88,89) The results for the three studied systems are reported in Figure 7. For all three moieties, the CCSD correlation energies show an asymptotic convergence, thus demonstrating that our approach indeed has a basis set limit. This is not surprising and directly follows from the fact that the basis set is not truncated in the definition of the MO basis in the MLHF-AB approach.

### 4.4. MLHF-AB Versus Projection-Based Approaches

*n*

_{o}=

*n*

_{el}/2) similarly to MLHF-AB calculations. The MOs belonging to the active fragment have to be selected on some mathematical criterion. Here, we calculate the percentage (

*p*

_{i}

^{A}) of the

*i*-th MO in the active part

*A*as

*C*

_{iμ}is the MO coefficient of the

*i*-th MO in AO basis {μ}. The

*n*

_{o}active MOs in projected-HF calculations are then selected as those having the highest percentage in the active atoms. It is worth pointing out that the active MOs in projected-HF models can also be selected as those having a percentage ≥50%, instead of fixing the number of active MOs to

*n*

_{o}. However, when applied to PES studies, such a choice leads to unavoidable PES discontinuities because a different number of active MOs may be selected depending on the active–inactive distance. Also, different methods to calculate the MO percentage in A can be arbitrarily chosen; thus the results are not unique. For these reasons, we prefer to keep the number of active MOs fixed to

*n*

_{o}. We notice that such arbitrariness is almost absent in MLHF-AB calculations, which only depend on the active–inactive partitioning of the electrons in the studied system.

*D*

_{μν}

^{A}= ∑

_{ij}

*C*

_{μi}

*C*

_{jν}), and the active energy is calculated as the

*E*

^{A}term in eq 4.

*n*

_{o}is fixed to 21 because, as stated above, the bonding electrons are assigned to the active fragment.

### 4.5. Absorption Energies

### 4.6. Summary and Conclusions

## Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.0c00737.

Cartesian coordinates of the studied molecules; parameters of the calculations; and data related to Figures 3–5 and 9 (PDF)

Molecular geometry of acetone (XYZ)

Molecular geometry of graphene (XYZ)

Molecular geometry of ANS (XYZ)

Molecular geometry of nicotine (XYZ)

Molecular geometry of benzene (XYZ)

Molecular geometry of PCP (XYZ)

## 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

We acknowledge Sarai Dery Folkestad and Ida-Marie Høyvik (NTNU) for discussions on technical aspects of the implementation. We acknowledge Chiara Cappelli (SNS) for computer resources. We acknowledge funding from the Marie Sklodowska-Curie European Training Network “COSINE—Computational Spectroscopy in Natural Sciences and Engineering”, grant agreement no. 765739, and the Research Council of Norway through FRINATEK projects 263110 and 275506.

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(CTA) enables sepn. of the forward and backward charge transfer components for each pair of mols. in the system. The key feature of ALMO CTA is that all charge transfer terms have corresponding well defined energetic effects that measure the contribution of the given term to the overall energetic stabilization of the system. To simplify anal. of charge transfer effects, the concept of chem. significant complementary occupied-virtual orbital pairs (COVPs) is introduced. COVPs provide a simple description of intermol. electron transfer effects in terms of just a few localized orbitals. ALMO CTA is applied to understand fundamental aspects of donor-acceptor interactions in borane adducts, synergic bonding in classical and nonclassical metal carbonyls, and multiple intermol. hydrogen bonds in a complex of isocyanuric acid and melamine. These examples show that the ALMO CTA results are generally consistent with the existing conceptual description of intermol. bonding. 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The first is a fully self-consistent treatment of the energy lowering due to polarization, which is evaluated by a SCF calcn. in which the MO coeffs. are constrained to be block-diagonal (absolutely localized) in the interacting mols. to prohibit charge transfer. The second new feature is the ability to sep. forward and back-donation in the charge-transfer energy term using a perturbative approxn. starting from the optimized block-diagonal ref. The newly proposed EDA method is used to understand the fundamental aspects of intermol. interactions such as the degree of covalency in the hydrogen bonding in water and the contributions of forward and back-donation in synergic bonding in metal complexes. Addnl., it is demonstrated that this method can be used to identify the factors controlling the interaction of the mol. hydrogen with open metal centers in potential hydrogen storage materials and the interaction of methane with rhenium complexes.**7**Aquilante, F.; Bondo Pedersen, T.; Sánchez de Merás, A.; Koch, H. Fast noniterative orbital localization for large molecules.*J. Chem. Phys.*2006,*125*, 174101, DOI: 10.1063/1.2360264Google Scholar7https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XhtF2htr7F&md5=b0a20bc9f75b8c1a05088911eb5979b5Fast noniterative orbital localization for large moleculesAquilante, Francesco; Bondo Pedersen, Thomas; Sanchez de Meras, Alfredo; Koch, HenrikJournal of Chemical Physics (2006), 125 (17), 174101/1-174101/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We use Cholesky decompn. of the d. matrix in AO basis to define a new set of occupied MO coeffs. Anal. of the resulting orbitals ("Cholesky MOs") demonstrates their localized character inherited from the sparsity of the d. matrix. Comparison with the results of traditional iterative localization schemes shows minor differences with respect to a no. of suitable measures of locality, particularly the scaling with system size of orbital pair domains used in local correlation methods. The Cholesky procedure for generating orthonormal localized orbitals is noniterative and may be made linear scaling. Although our present implementation scales cubically, the algorithm is significantly faster than any of the conventional localization schemes. In addn., since this approach does not require starting orbitals, it will be useful in local correlation treatments on top of diagonalization-free Hartree-Fock optimization algorithms.**8**Høyvik, I.-M.; Jansik, B.; Jørgensen, P. Orbital localization using fourth central moment minimization.*J. Chem. 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We propose that the std. orbital spread (the square root of the second central moment) and fourth moment orbital spread (the fourth root of the fourth central moment) are used as complementary measures to characterize the locality of an orbital, irresp. of localization scheme. (c) 2012 American Institute of Physics.**9**Jansík, B.; Høst, S.; Kristensen, K.; Jørgensen, P. Local orbitals by minimizing powers of the orbital variance.*J. Chem. Phys.*2011,*134*, 194104, DOI: 10.1063/1.3590361Google Scholar9https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXmtl2rtbs%253D&md5=a74f01980b9c752491296842255513c8Local orbitals by minimizing powers of the orbital varianceJansik, Branislav; Host, Stinne; Kristensen, Kasper; Jorgensen, PoulJournal of Chemical Physics (2011), 134 (19), 194104/1-194104/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)It is demonstrated that a set of local orthonormal Hartree-Fock (HF) MOs can be obtained for both the occupied and virtual orbital spaces by minimizing powers of the orbital variance using the trust-region algorithm. For a power exponent equal to one, the Boys localization function is obtained. For increasing power exponents, the penalty for delocalized orbitals is increased and smaller max. orbital spreads are encountered. Calcns. on superbenzene, C60, and a fragment of the titin protein show that for a power exponent equal to one, delocalized outlier orbitals may be encountered. These disappear when the exponent is larger than one. For a small penalty, the occupied orbitals are more local than the virtual ones. When the penalty is increased, the locality of the occupied and virtual orbitals becomes similar. 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For an exponential parametrization of the localization function only small steps are allowed, and the standard trust radius update therefore has been replaced by a scheme where the direction of the step is determined using a conservative estimate of the trust radius and the length of the step is determined from a line search along the obtained direction. Numerical results for large molecular systems have shown that large steps can then safely be taken, and a robust and nearly monotonic convergence is obtained.**12**Ziółkowski, M.; Jansik, B.; Jørgensen, P.; Olsen, J. Maximum locality in occupied and virtual orbital spaces using a least-change strategy.*J. Chem. Phys.*2009,*131*, 124112, DOI: 10.1063/1.3230604Google Scholar12https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhtFymsr%252FJ&md5=598c8d47345f5b2e573b4f158aaa6e99Maximum locality in occupied and virtual orbital spaces using a least-change strategyZiolkowski, Marcin; Jansik, Branislav; Joergensen, Poul; Olsen, JeppeJournal of Chemical Physics (2009), 131 (12), 124112/1-124112/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A new strategy is introduced for obtaining localized orthonormal Hartree-Fock (HF) orbitals where the underlying principle is to minimize the size of the transformation matrix from the AO basis to the HF optimized orbital basis. The new strategy gives both localized occupied and localized virtual orbital spaces. The locality of the occupied orbital space is similar to one obtained using std. localization schemes. For the virtual space, std. localization schemes fail to give local orbitals while the new strategy gives a virtual space which has a locality similar to the one of a Loewdin orthonormalization of the AO basis. Since Loewdin orthonormalization gives the most local orthonormal basis functions in the sense that they have the largest similarity with the local at. basis functions, the new strategy thus allows the orthonormal basis to become optimized without introducing significant delocalization. (c) 2009 American Institute of Physics.**13**Gianinetti, E.; Raimondi, M.; Tornaghi, E. Modification of the Roothaan equations to exclude BSSE from molecular interaction calculations.*Int. J. Quantum Chem.*1996,*60*, 157– 166, DOI: 10.1002/(sici)1097-461x(1996)60:1<157::aid-qua17>3.0.co;2-cGoogle Scholar13https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28Xltlejtro%253D&md5=3ff2d4a5aa9db639ee511c3e0f5e8b61Modification of the Roothaan equations to exclude BSSE from molecular interactions calculationsGianinetti, E.; Raimondi, M.; Tornaghi, E.International Journal of Quantum Chemistry (1996), 60 (1), 157-166CODEN: IJQCB2; ISSN:0020-7608. (Wiley)The Roothaan equations have been modified to compute mol. interactions between weakly bonded systems at the SCF level of theory without the basis set superposition error (BSSE). The increase in complication with respect to the usual SCF algorithm is negligible. Calcn. of the SCF energy on large systems, such as nucleic acid pair, does not pose any computational problem. At the same time, it is shown that a modest change in basis-set quality from 3-21G to 6-31G changes the binding energy by about 50% when computed according to std. SCF "supermol." techniques, while remaining practically const. when computed without introducing BSSE. Bader anal. shows that the amt. of charge transferred between the interacting units is of the same order of magnitude when performed on std. SCF wave functions and those computed using the new method. The large difference between the corresponding computed energies is thus ascribed to the BSSE.**14**Stoll, H.; Wagenblast, G.; Preuβ, H. On the use of local basis sets for localized molecular orbitals.*Theor. Chim. Acta*1980,*57*, 169– 178, DOI: 10.1007/bf00574903Google Scholar14https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3cXmtFSmtLk%253D&md5=d43b312f617f80e6e8e957f2f47113ceOn the use of local basis sets for localized molecular orbitalsStoll, Hermann; Wagenblast, Gerhard; Preuss, HeinzwernerTheoretica Chimica Acta (1980), 57 (2), 169-78CODEN: TCHAAM; ISSN:0040-5744.Two procedures are discussed for the direct variational optimization of localized MO's which are expanded in local subsets of the mol. basis set. A Newton-Raphson approach is more efficient than an iterative diagonalization scheme. The effect of the basis-set truncation on the quality of ab initio SCF results is investigated for Be, Li2, HF, H2O, NH3, CH4 and C2H6.**15**Li, W.; Ni, Z.; Li, S. Cluster-in-molecule local correlation method for post-Hartree–Fock calculations of large systems.*Mol. Phys.*2016,*114*, 1447– 1460, DOI: 10.1080/00268976.2016.1139755Google Scholar15https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhvFyhsrs%253D&md5=f4425783d8e50aac3d940428c1727f95Cluster-in-molecule local correlation method for post-Hartree-Fock calculations of large systemsLi, Wei; Ni, Zhigang; Li, ShuhuaMolecular Physics (2016), 114 (9), 1447-1460CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)Our recent developments on cluster-in-mol. (CIM) local correlation method are reviewed in this paper. In the CIM method, the correlation energy of a large system can be approx. obtained from electron correlation calcns. on a series of clusters, each of which contains a subset of occupied and virtual localised MOs in a certain region. The CIM method is a linear scaling method and its inherent parallelisation allows electron correlation calcns. of very large systems to be feasible at ordinary workstations. In the illustrative applications, this approach is applied to investigate the conformational energy differences, reaction barriers, and binding energies of large systems at the levels of Moller-Plesset perturbation theory and coupled-cluster theory.**16**Zhang, X.; Carter, E. A. Subspace Density Matrix Functional Embedding Theory: Theory, Implementation, and Applications to Molecular Systems.*J. Chem. Theory Comput.*2018,*15*, 949– 960, DOI: 10.1021/acs.jctc.8b00990Google ScholarThere is no corresponding record for this reference.**17**Govind, N.; Wang, Y. A.; Carter, E. A. Electronic-structure calculations by first-principles density-based embedding of explicitly correlated systems.*J. Chem. Phys.*1999,*110*, 7677– 7688, DOI: 10.1063/1.478679Google Scholar17https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXit1yhs7c%253D&md5=19399936ef64e98248dc5ba77fc7bb9dElectronic-structure calculations by first-principles density-based embedding of explicitly correlated systemsGovind, Niranjan; Wang, Yan Alexander; Carter, Emily A.Journal of Chemical Physics (1999), 110 (16), 7677-7688CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A first-principles embedding theory that combines the salient features of d. functional theory (DFT) and traditional quantum chem. methods is presented. The method involves constructing a DFT-based embedding potential and then using it as a one-electron operator within a very accurate ab initio calcn. We demonstrate how DFT calcns. can be systematically improved via this procedure. The scheme is tested using two closed shell systems, a toy model Li2Mg2, and the exptl. well characterized CO/Cu(111) system. Our results are in good agreement with near full CI calcns. in the former case and exptl. adsorbate binding energies in the latter. This method provides the means to systematically include electron correlation in a local region of a condensed phase.**18**Yu, K.; Carter, E. A. Extending density functional embedding theory for covalently bonded systems.*Proc. Natl. Acad. Sci. U.S.A.*2017,*114*, E10861– E10870, DOI: 10.1073/pnas.1712611114Google Scholar18https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhvFWgsrnK&md5=bc56a71c5f634ea5ab305ee6bfd17415Extending density functional embedding theory for covalently bonded systemsYu, Kuang; Carter, Emily A.Proceedings of the National Academy of Sciences of the United States of America (2017), 114 (51), E10861-E10870CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)Quantum embedding theory aims to provide an efficient soln. to obtain accurate electronic energies for systems too large for full-scale, high-level quantum calcns. It adopts a hierarchical approach that divides the total system into a small embedded region and a larger environment, using different levels of theory to describe each part. Previously, we developed a d.-based quantum embedding theory called d. functional embedding theory (DFET), which achieved considerable success in metals and semiconductors. In this work, we extend DFET into a d.-matrix-based nonlocal form, enabling DFET to study the stronger quantum couplings between covalently bonded subsystems. We name this theory d.-matrix functional embedding theory (DMFET), and we demonstrate its performance in several test examples that resemble various real applications in both chem. and biochem. DMFET gives excellent results in all cases tested thus far, including predicting isomerization energies, proton transfer energies, and HOMO-LUMO gaps for local chromophores. Here, we show that DMFET systematically improves the quality of the results compared with the widely used state-of-the-art methods, such as the simple capped cluster model or the widely used ONIOM method.**19**Huang, C.; Pavone, M.; Carter, E. A. Quantum mechanical embedding theory based on a unique embedding potential.*J. Chem. Phys.*2011,*134*, 154110, DOI: 10.1063/1.3577516Google Scholar19https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXkvFGgu7Y%253D&md5=93874a27b3413cf28c74fed290d71250Quantum mechanical embedding theory based on a unique embedding potentialHuang, Chen; Pavone, Michele; Carter, Emily A.Journal of Chemical Physics (2011), 134 (15), 154110/1-154110/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We remove the nonuniqueness of the embedding potential that exists in most previous quantum mech. embedding schemes by letting the environment and embedded region share a common embedding (interaction) potential. To efficiently solve for the embedding potential, an optimized effective potential method is derived. This embedding potential, which eschews use of approx. kinetic energy d. functionals, is then used to describe the environment while a correlated wavefunction (CW) treatment of the embedded region is employed. We first demonstrate the accuracy of this new embedded CW (ECW) method by calcg. the van der Waals binding energy curve between a hydrogen mol. and a hydrogen chain. We then examine the prototypical adsorption of CO on a metal surface, here the Cu(111) surface. In addn. to obtaining proper site ordering (top site most stable) and binding energies within this theory, the ECW exhibits dramatic changes in the p-character of the CO 4σ and 5σ orbitals upon adsorption that agree very well with x-ray emission spectra, providing further validation of the theory. Finally, we generalize our embedding theory to spin-polarized quantum systems and discuss the connection between our theory and partition d. functional theory. (c) 2011 American Institute of Physics.**20**Libisch, F.; Huang, C.; Carter, E. A. Embedded correlated wavefunction schemes: Theory and applications.*Acc. Chem. Res.*2014,*47*, 2768– 2775, DOI: 10.1021/ar500086hGoogle Scholar20https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXptVahu7c%253D&md5=c38754b2218f3584b29849f2d7c4bb1dEmbedded Correlated Wavefunction Schemes: Theory and ApplicationsLibisch, Florian; Huang, Chen; Carter, Emily A.Accounts of Chemical Research (2014), 47 (9), 2768-2775CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)A review. Ab initio modeling of matter has become a pillar of chem. research: with ever-increasing computational power, simulations can be used to accurately predict, for example, chem. reaction rates, electronic and mech. properties of materials, and dynamical properties of liqs. Many competing quantum mech. methods have been developed over the years that vary in computational cost, accuracy, and scalability: d. functional theory (DFT), the workhorse of solid-state electronic structure calcns., features a good compromise between accuracy and speed. However, approx. exchange-correlation functionals limit DFT's ability to treat certain phenomena or states of matter, such as charge-transfer processes or strongly correlated materials. Furthermore, conventional DFT is purely a ground-state theory: electronic excitations are beyond its scope. Excitations in mols. are routinely calcd. using time-dependent DFT linear response; however applications to condensed matter are still limited. By contrast, many-electron wavefunction methods aim for a very accurate treatment of electronic exchange and correlation. Unfortunately, the assocd. computational cost renders treatment of more than a handful of heavy atoms challenging. On the other side of the accuracy spectrum, parametrized approaches like tight-binding can treat millions of atoms. In view of the different (dis-)advantages of each method, the simulation of complex systems seems to force a compromise: one is limited to the most accurate method that can still handle the problem size. For many interesting problems, however, compromise proves insufficient. A possible soln. is to break up the system into manageable subsystems that may be treated by different computational methods. The interaction between subsystems may be handled by an embedding formalism. In this Account, we review embedded correlated wavefunction (CW) approaches and some applications. We first discuss our d. functional embedding theory, which is formally exact. We show how to det. the embedding potential, which replaces the interaction between subsystems, at the DFT level. CW calcns. are performed using a fixed embedding potential, i.e., a non-self-consistent embedding scheme. We demonstrate this embedding theory for two challenging electron transfer phenomena: (1) initial oxidn. of an aluminum surface and (2) hot-electron-mediated dissocn. of hydrogen mols. on a gold surface. In both cases, the interaction between gas mols. and metal surfaces were treated by sophisticated CW techniques, with the remainder of the extended metal surface being treated by DFT. Our embedding approach overcomes the limitations of conventional Kohn-Sham DFT in describing charge transfer, multiconfigurational character, and excited states. From these embedding simulations, we gained important insights into fundamental processes that are crucial aspects of fuel cell catalysis (i.e., O2 redn. at metal surfaces) and plasmon-mediated photocatalysis by metal nanoparticles. Moreover, our findings agree very well with exptl. observations, while offering new views into the chem. We finally discuss our recently formulated potential-functional embedding theory that provides a seamless, first-principles way to include back-action onto the environment from the embedded region.**21**Knizia, G.; Chan, G. K.-L. Density matrix embedding: A simple alternative to dynamical mean-field theory.*Phys. Rev. Lett.*2012,*109*, 186404, DOI: 10.1103/physrevlett.109.186404Google Scholar21https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhslClsL7J&md5=91a195c35bc9aca0f09f19ce1f6ac795Density matrix embedding: a simple alternative to dynamical mean-field theoryKnizia, Gerald; Chan, Garnet Kin-LicPhysical Review Letters (2012), 109 (18), 186404/1-186404/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)We introduce d. matrix embedding theory (DMET), a quantum embedding theory for computing frequency-independent quantities, such as ground-state properties, of infinite systems. Like dynamical mean-field theory, DMET maps the bulk interacting system to a simpler impurity model and is exact in the noninteracting and at. limits. Unlike dynamical mean-field theory, DMET is formulated in terms of the frequency-independent local d. matrix, rather than the local Green's function. In addn., it features a finite, algebraically constructible bath of only one bath site per impurity site, with no bath discretization error. Frequency independence and the minimal bath make DMET a computationally simple and efficient method. We test the theory in the one-dimensional and two-dimensional Hubbard models at and away from half filling, and we find that compared to benchmark data, total energies, correlation functions, and metal-insulator transitions are well reproduced, at a tiny computational cost.**22**Knizia, G.; Chan, G. K.-L. Density matrix embedding: A strong-coupling quantum embedding theory.*J. Chem. Theory Comput.*2013,*9*, 1428– 1432, DOI: 10.1021/ct301044eGoogle Scholar22https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXivVyitr8%253D&md5=8b6d527b333cf4e4f06fb18181fe30f3Density Matrix Embedding: A Strong-Coupling Quantum Embedding TheoryKnizia, Gerald; Chan, Garnet Kin-LicJournal of Chemical Theory and Computation (2013), 9 (3), 1428-1432CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We extend our d. matrix embedding theory (DMET) [Phys. Rev. Lett.2012, 109, 186404] from lattice models to the full chem. Hamiltonian. DMET allows the many-body embedding of arbitrary fragments of a quantum system, even when such fragments are open systems and strongly coupled to their environment (e.g., by covalent bonds). In DMET, empirical approaches to strong coupling, such as link atoms or boundary regions, are replaced by a small, rigorous quantum bath designed to reproduce the entanglement between a fragment and its environment. We describe the theory and demonstrate its feasibility in strongly correlated hydrogen ring and grid models; these are not only beyond the scope of traditional embeddings but even challenge conventional quantum chem. methods themselves. We find that DMET correctly describes the notoriously difficult sym. dissocn. of a 4 × 3 hydrogen atom grid, even when the treated fragments are as small as single hydrogen atoms. We expect that DMET will open up new ways of treating complex strongly coupled, strongly correlated systems in terms of their individual fragments.**23**Sayfutyarova, E. R.; Sun, Q.; Chan, G. K.-L.; Knizia, G. Automated construction of molecular active spaces from atomic valence orbitals.*J. Chem. Theory Comput.*2017,*13*, 4063– 4078, DOI: 10.1021/acs.jctc.7b00128Google Scholar23https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXht1WmtL7L&md5=2c8d3c8062fa13f4f4e68c6432bb65b1Automated Construction of Molecular Active Spaces from Atomic Valence OrbitalsSayfutyarova, Elvira R.; Sun, Qiming; Chan, Garnet Kin-Lic; Knizia, GeraldJournal of Chemical Theory and Computation (2017), 13 (9), 4063-4078CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We introduce the at. valence active space (AVAS), a simple and well-defined automated technique for constructing active orbital spaces for use in multi-configuration and multi-ref. (MR) electronic structure calcns. Concretely, the technique constructs active MOs capable of describing all relevant electronic configurations emerging from a targeted set of at. valence orbitals (e.g., the metal d orbitals in a redcoordination complex). This is achieved via a linear transformation of the occupied and unoccupied orbital spaces from an easily obtainable single-ref. wavefunction (such as from a Hartree-Fock or Kohn-Sham calcns.) based on projectors to targeted at. valence orbitals. We discuss the premises, theory, and implementation of the idea, and several of its variations are tested. To investigate the performance and accuracy, we calc. the excitation energies for various transition metal complexes in typical application scenarios. Addnl., we follow the homolytic bond breaking process of a Fenton reaction along its reaction coordinate. While the described AVAS technique is not an universal soln. to the active space problem, its premises are fulfilled in many application scenarios of transition metal chem. and bond dissocn. processes. In these cases the technique makes MR calcns. easier to execute, easier to reproduce by any user, and simplifies the detn. of the appropriate size of the active space required for accurate results.**24**Azarias, C.; Russo, R.; Cupellini, L.; Mennucci, B.; Jacquemin, D. Modeling excitation energy transfer in multi-BODIPY architectures.*Phys. Chem. Chem. Phys.*2017,*19*, 6443– 6453, DOI: 10.1039/c7cp00427cGoogle Scholar24https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhvF2gsr8%253D&md5=605a3d7a2d97ba93b7f20b0fc2337a19Modeling excitation energy transfer in multi-BODIPY architecturesAzarias, Cloe; Russo, Roberto; Cupellini, Lorenzo; Mennucci, Benedetta; Jacquemin, DenisPhysical Chemistry Chemical Physics (2017), 19 (9), 6443-6453CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)The excitation energy transfer (EET) allowing the concn. of the energy has been investigated in several multi-BODIPY architectures with the help of an approach coupling time dependent d. functional theory to an implicit solvation scheme, the polarizable continuum simulation, physicochem. We have first considered several strategies to compute the electronic coupling in a dyad varying the size of the donor/acceptor units, the bridge, the geometries and conformations. We have next studied the electronic coupling in three different architectures for which the EET rate consts. have been exptl. measured both from luminescence and transient absorption data and from Forster intramol. energy transfer :: ditto.**25**Mennucci, B.; Corni, S. Multiscale modelling of photoinduced processes in composite systems.*Nat. Rev. Chem.*2019,*3*, 315– 330, DOI: 10.1038/s41570-019-0092-4Google ScholarThere is no corresponding record for this reference.**26**Warshel, A.; Levitt, M. Theoretical studies of enzymic reactions: dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme.*J. Mol. Biol.*1976,*103*, 227– 249, DOI: 10.1016/0022-2836(76)90311-9Google Scholar26https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE28XktFKhtr0%253D&md5=f34df33b5971b6b02bd03be95dcd7ba5Theoretical studies of enzymic reactions: dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozymeWarshel, A.; Levitt, M.Journal of Molecular Biology (1976), 103 (2), 227-49CODEN: JMOBAK; ISSN:0022-2836.A general method for detailed study of enzymic reactions is presented. The method considers the complete enzyme-substrate complex together with the surrounding solvent and evaluates all the different quantum mech. and classical energy factors that can affect the reaction pathway. These factors include the quantum mech. energies assocd. with bond cleavage and charge redistribution of the substrate and the classical energies of steric and electrostatic interactions between the substrate and the enzyme. The electrostatic polarization of the enzyme atoms and the orientation of the dipoles of the surrounding H2O mols. is simulated by a microscopic dielec. model. The solvation energy resulting from this polarization is considerable and must be included in any realistic calcn. of chem. reactions involving anything more than an isolated mol. in vacuo. Without it, acidic groups can never become ionized and the charge distribution on the substrate will not be reasonable. The same dielec. model can also be used to study the reaction of the substrate in soln. In this way the reaction in soln. can be compared with the enzymic reaction. The stability of the carbonium ion intermediate formed in the cleavage of a glycosidic bond by lysozyme was studied. Electrostatic stabilization is an important factor in increasing the rate of the reaction step that leads to the formation of the carbonium ion intermediate. Steric factors, such as the strain of the substrate on binding to lysozyme, do not seem to contribute significantly.**27**Giovannini, T.; Egidi, F.; Cappelli, C. Molecular spectroscopy of aqueous solutions: a theoretical perspective.*Chem. Soc. Rev.*2020,*49*, 5664– 5677, DOI: 10.1039/c9cs00464eGoogle Scholar27https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhsFSlsrzJ&md5=b37e9d2ce60827c120a1090f07213c09Molecular spectroscopy of aqueous solutions: a theoretical perspectiveGiovannini, Tommaso; Egidi, Franco; Cappelli, ChiaraChemical Society Reviews (2020), 49 (16), 5664-5677CODEN: CSRVBR; ISSN:0306-0012. (Royal Society of Chemistry)Computational spectroscopy is an invaluable tool to both accurately reproduce the spectra of mol. systems and provide a rationalization for the underlying physics. However, the inherent difficulty to accurately model systems in aq. solns., owing to water's high polarity and ability to form hydrogen bonds, has severely hampered the development of the field. In this tutorial review we present a technique developed and tested in recent years based on a fully atomistic and polarizable classical modeling of water coupled with a quantum mech. description of the solute. Thanks to its unparalleled accuracy and versatility, this method can change the perspective of computational and exptl. chemists alike.**28**Giovannini, T.; Egidi, F.; Cappelli, C. Theory and algorithms for chiroptical properties and spectroscopies of aqueous systems.*Phys. Chem. Chem. Phys.*2020,*22*, 22864– 22879, DOI: 10.1039/d0cp04027dGoogle Scholar28https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhvFensL%252FM&md5=dfe27b9b9b9a3ada60788b6b69da8ba5Theory and algorithms for chiroptical properties and spectroscopies of aqueous systemsGiovannini, Tommaso; Egidi, Franco; Cappelli, ChiaraPhysical Chemistry Chemical Physics (2020), 22 (40), 22864-22879CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)Chiroptical properties and spectroscopies are valuable tools to study chiral mols. and assign abs. configurations. The spectra that result from chiroptical measurements may be very rich and complex, and hide much of their information content. For this reason, the interplay between expts. and calcns. is esp. useful, provided that all relevant physico-chem. interactions that are present in the exptl. sample are accurately modelled. The inherent difficulty assocd. to the calcn. of chiral signals of systems in aq. solns. requires the development of specific tools, able to account for the peculiarities of water-solute interactions, and esp. its ability to form hydrogen bonds. In this perspective we discuss a multiscale approach, which we have developed and challenged to model the most used chiroptical techniques.**29**Mennucci, B. Polarizable Continuum Model.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2012,*2*, 386– 404, DOI: 10.1002/wcms.1086Google Scholar29https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XovVyrsro%253D&md5=dffcb7dcec69845b8e8bbc40692fd1abPolarizable continuum modelMennucci, BenedettaWiley Interdisciplinary Reviews: Computational Molecular Science (2012), 2 (3), 386-404CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)A review. The polarizable continuum model (PCM) is a computational method originally formulated 30 years ago but still today it represents one of the most successful examples among continuum solvation models. Such a success is mainly because of the continuous improvements, both in terms of computational efficiency and generality, made by all the people involved in the PCM project. The result of these efforts is that nowadays, PCM, with all its different variants, is the default choice in many computational codes to couple a quantum-mech. (QM) description of a mol. system with a continuum description of the environment. In this review, a brief presentation of the main methodol. and computational aspects of the method will be given together with an anal. of strengths and crit. issues of its coupling with different QM methods. Finally, some examples of applications will be presented and discussed to show the potentialities of PCM in describing the effects of environments of increasing complexity.**30**Giovannini, T.; Puglisi, A.; Ambrosetti, M.; Cappelli, C. Polarizable QM/MM approach with fluctuating charges and fluctuating dipoles: the QM/FQFμ model.*J. Chem. Theory Comput.*2019,*15*, 2233– 2245, DOI: 10.1021/acs.jctc.8b01149Google Scholar30https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXltFCjsb4%253D&md5=5b19d342033f29a11dd0cbcc703f2b2aPolarizable QM/MM Approach with Fluctuating Charges and Fluctuating Dipoles: The QM/FQFμ ModelGiovannini, Tommaso; Puglisi, Alessandra; Ambrosetti, Matteo; Cappelli, ChiaraJournal of Chemical Theory and Computation (2019), 15 (4), 2233-2245CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The novel polarizable FQFμ force field is proposed and coupled to a quantum mech. (QM) SCF Hamiltonian. The peculiarity of the resulting QM/FQFμ approach stands in the fact the polarization effects are modeled in terms of both fluctuating charges and dipoles, which vary as a response to the external elec. field/potential. Remarkably, QM/FQFμ is defined in terms of three parameters: electronegativity and chem. hardness, which are well-defined in d. functional theory, and polarizability, which is phys. observable. Such parameters are numerically adjusted to reproduce full QM ref. electrostatic energy values. The model is challenged against test mol. systems in aq. soln., showing remarkable accuracy and thus highlighting its potentialities for future extensive applications.**31**Gordon, M. S.; Fedorov, D. G.; Pruitt, S. R.; Slipchenko, L. V. Fragmentation methods: A route to accurate calculations on large systems.*Chem. Rev.*2012,*112*, 632– 672, DOI: 10.1021/cr200093jGoogle Scholar31https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhtVyhurjJ&md5=7fa407f4c831f6c15c23d76fde206ba0Fragmentation Methods: A Route to Accurate Calculations on Large SystemsGordon, Mark S.; Fedorov, Dmitri G.; Pruitt, Spencer R.; Slipchenko, Lyudmila V.Chemical Reviews (Washington, DC, United States) (2012), 112 (1), 632-672CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)A review including the following topics: methodologies, software and parallel computing, applications, and conclusions and prognosis.**32**Collins, M. A.; Bettens, R. P. A. Energy-based molecular fragmentation methods.*Chem. Rev.*2015,*115*, 5607– 5642, DOI: 10.1021/cr500455bGoogle Scholar32https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXmtVajsbY%253D&md5=a83ac50604af530b6d89c94f1a6b6df6Energy-Based Molecular Fragmentation MethodsCollins, Michael A.; Bettens, Ryan P. A.Chemical Reviews (Washington, DC, United States) (2015), 115 (12), 5607-5642CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)A review including the following topics: methods and principles, applications and examples, and speculations and future developments etc.**33**Pruitt, S. R.; Bertoni, C.; Brorsen, K. R.; Gordon, M. S. Efficient and accurate fragmentation methods.*Acc. Chem. Res.*2014,*47*, 2786– 2794, DOI: 10.1021/ar500097mGoogle Scholar33https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXns1Cmu7k%253D&md5=e6b7179e37a2fb45d912348377ccaa4fEfficient and Accurate Fragmentation MethodsPruitt, Spencer R.; Bertoni, Colleen; Brorsen, Kurt R.; Gordon, Mark S.Accounts of Chemical Research (2014), 47 (9), 2786-2794CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)A review. Three novel fragmentation methods that are available in the electronic structure program GAMESS (general at. and mol. electronic structure system) are discussed in this Account. The fragment MO (FMO) method can be combined with any electronic structure method to perform accurate calcns. on large mol. species with no reliance on capping atoms or empirical parameters. The FMO method is highly scalable and can take advantage of massively parallel computer systems. For example, the method has been shown to scale nearly linearly on up to 131 000 processor cores for calcns. on large water clusters. There have been many applications of the FMO method to large mol. clusters, to biomols. (e.g., proteins), and to materials that are used as heterogeneous catalysts. The effective fragment potential (EFP) method is a model potential approach that is fully derived from first principles and has no empirically fitted parameters. Consequently, an EFP can be generated for any mol. by a simple preparatory GAMESS calcn. The EFP method provides accurate descriptions of all types of intermol. interactions, including Coulombic interactions, polarization/induction, exchange repulsion, dispersion, and charge transfer. The EFP method has been applied successfully to the study of liq. water, π-stacking in substituted benzenes and in DNA base pairs, solvent effects on pos. and neg. ions, electronic spectra and dynamics, non-adiabatic phenomena in electronic excited states, and nonlinear excited state properties. The effective fragment MO (EFMO) method is a merger of the FMO and EFP methods, in which interfragment interactions are described by the EFP potential, rather than the less accurate electrostatic potential. The use of EFP in this manner facilitates the use of a smaller value for the distance cut-off (Rcut). Rcut dets. the distance at which EFP interactions replace fully quantum mech. calcns. on fragment-fragment (dimer) interactions. The EFMO method is both more accurate and more computationally efficient than the most commonly used FMO implementation (FMO2), in which all dimers are explicitly included in the calcn. While the FMO2 method itself does not incorporate three-body interactions, such interactions are included in the EFMO method via the EFP self-consistent induction term. Several applications (ranging from clusters to proteins) of the three methods are discussed to demonstrate their efficacy. The EFMO method will be esp. exciting once the analytic gradients have been completed, because this will allow geometry optimizations, the prediction of vibrational spectra, reaction path following, and mol. dynamics simulations using the method.**34**Collins, M. A.; Cvitkovic, M. W.; Bettens, R. P. A. The combined fragmentation and systematic molecular fragmentation methods.*Acc. Chem. Res.*2014,*47*, 2776– 2785, DOI: 10.1021/ar500088dGoogle Scholar34https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhtVGksLnK&md5=3d4318bae1bf7fdc64592326da80d823The Combined Fragmentation and Systematic Molecular Fragmentation MethodsCollins, Michael A.; Cvitkovic, Milan W.; Bettens, Ryan P. A.Accounts of Chemical Research (2014), 47 (9), 2776-2785CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)A review. Chem., particularly org. chem., is mostly concerned with functional groups: amines, amides, alcs., ketones, and so forth. This is because the reactivity of mols. can be categorized in terms of the reactions of these functional groups, and by the influence of other adjacent groups in the mol. These simple truths ought to be reflected in the electronic structure and electronic energy of mols., as reactivity is detd. by electronic structure. However, sophisticated ab initio quantum calcns. of the mol. electronic energy usually do not make these truths apparent. In recent years, several computational chem. groups have discovered methods for estg. the electronic energy as a sum of the energies of small mol. fragments, or small sets of groups. By decompg. mols. into such fragments of adjacent functional groups, researchers can est. the electronic energy to chem. accuracy; not just qual. trends, but accurate enough to understand reactivity. In addn., this has the benefit of cutting down on both computational time and cost, as the necessary calcn. time increases rapidly with an increasing no. of electrons. Even with steady advances in computer technol., progress in the study of large mols. is slow. In this Account, we describe two related "fragmentation" methods for treating mols., the combined fragmentation method (CFM) and systematic mol. fragmentation (SMF). In addn., we show how we can use the SMF approach to est. the energy and properties of nonconducting crystals, by fragmenting the periodic crystal structure into relatively small pieces. A large part of this Account is devoted to simple overviews of how the methods work. We also discuss the application of these approaches to calcg. reactivity and other useful properties, such as the NMR and vibrational spectra of mols. and crystals. These applications rely on the ability of these fragmentation methods to accurately est. derivs. of the mol. and crystal energies. Finally, to provide some common applications of CFM and SMF, we present some specific examples of energy calcns. for moderately large mols. For computational chemists, this fragmentation approach represents an important practical advance. It reduces the computer time required to est. the energies of mols. so dramatically, that accurate calcns. of the energies and reactivity of very large org. and biol. mols. become feasible.**35**Pruitt, S. R.; Addicoat, M. A.; Collins, M. A.; Gordon, M. S. The fragment molecular orbital and systematic molecular fragmentation methods applied to water clusters.*Phys. Chem. Chem. Phys.*2012,*14*, 7752– 7764, DOI: 10.1039/c2cp00027jGoogle Scholar35https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XmvVykurg%253D&md5=f817e5de29f6069d8a560043ba808eeaThe fragment molecular orbital and systematic molecular fragmentation methods applied to water clustersPruitt, Spencer R.; Addicoat, Matthew A.; Collins, Michael A.; Gordon, Mark S.Physical Chemistry Chemical Physics (2012), 14 (21), 7752-7764CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)Two electronic structure methods, the fragment MO (FMO) and systematic mol. fragmentation (SMF) methods, that are based on fragmenting a large mol. system into smaller, more computationally tractable components (fragments), are presented and compared with fully ab initio results for the predicted binding energies of water clusters. It is demonstrated that, even when explicit three-body effects are included (esp. necessary for water clusters due to their complex hydrogen-bonded networks) both methods present viable, computationally efficient alternatives to fully ab initio quantum chem.**36**Khaliullin, R. Z.; Head-Gordon, M.; Bell, A. T. An efficient self-consistent field method for large systems of weakly interacting components.*J. Chem. Phys.*2006,*124*, 204105, DOI: 10.1063/1.2191500Google Scholar36https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28Xls1amsrg%253D&md5=c9b37664a7993434200f464d93ee752bAn efficient self-consistent field method for large systems of weakly interacting componentsKhaliullin, Rustam Z.; Head-Gordon, Martin; Bell, Alexis T.Journal of Chemical Physics (2006), 124 (20), 204105/1-204105/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An efficient method for removing the SCF diagonalization bottleneck is proposed for systems of weakly interacting components. The method is based on the equations of the locally projected SCF for mol. interactions (SCF MI) which utilize absolutely localized nonorthogonal MOs expanded in local subsets of the at. basis set. A generalization of direct inversion in the iterative subspace for nonorthogonal MOs is formulated to increase the rate of convergence of the SCF MI equations. Single Roothaan step perturbative corrections are developed to improve the accuracy of the SCF MI energies. The resulting energies closely reproduce the conventional SCF energy. Extensive test calcns. are performed on water clusters up to several hundred mols. Compared to conventional SCF, speedups of the order of (N/O)2 have been achieved for the diagonalization step, where N is the size of the AO basis, and O is the no. of occupied MOs.**37**Ding, F.; Manby, F. R.; Miller, T. F., III Embedded mean-field theory with block-orthogonalized partitioning.*J. Chem. Theory Comput.*2017,*13*, 1605– 1615, DOI: 10.1021/acs.jctc.6b01065Google Scholar37https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXjsVCgtr8%253D&md5=cb3d6244a2abae04cfcc771dd1abb0c9Embedded Mean-Field Theory with Block-Orthogonalized PartitioningDing, Feizhi; Manby, Frederick R.; Miller, Thomas F.Journal of Chemical Theory and Computation (2017), 13 (4), 1605-1615CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Embedded mean-field theory (EMFT) provides a simple, flexible framework for describing subsystems at different levels of mean-field theory. Subsystems are defined by partitioning a one-particle basis set, with a natural choice being the AO basis. Although generally well behaved, EMFT with AO partitioning can exhibit unphys. collapse of the self-consistent soln. To avoid this issue, we introduce subsystem partitioning of a block-orthogonalized (BO) basis set; this eliminates the unphys. collapse without significantly increasing computational cost. We also investigate a non-self-consistent implementation of EMFT, in which the d. matrix is obtained using BO partitioning and the final energy evaluated using AO partitioning; this d.-cor. EMFT approach is found to yield more accurate energies than BO partitioning while also avoiding issues of the unphys. collapse. Using these refined implementations of EMFT, previously proposed descriptions of the exact-exchange coupling between subsystems are compared: although the EX1 coupling scheme is slightly more accurate than EX0, the small improvement does not merit its substantially greater computational cost.**38**Wen, X.; Graham, D. S.; Chulhai, D. V.; Goodpaster, J. D. Absolutely Localized Projection-Based Embedding for Excited States.*J. Chem. Theory Comput.*2020,*16*, 385– 398, DOI: 10.1021/acs.jctc.9b00959Google Scholar38https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXit1Crs7vL&md5=1fcf757ad15dc04d3f07fcdbfaa0c495Absolutely Localized Projection-Based Embedding for Excited StatesWen, Xuelan; Graham, Daniel S.; Chulhai, Dhabih V.; Goodpaster, Jason D.Journal of Chemical Theory and Computation (2020), 16 (1), 385-398CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a quantum embedding method that allows for the calcn. of local excited states embedded in a Kohn-Sham d. functional theory (DFT) environment. Projection-based quantum embedding methodologies provide a rigorous framework for performing DFT-in-DFT and wave function in DFT (WF-in-DFT) calcns. The use of abs. localization, where the d. of each subsystem is expanded in only the basis functions assocd. with the atoms of that subsystem, provide improved computationally efficiency for WF-in-DFT calcns. by reducing the no. of orbitals in the WF calcn. In this work, we extend absolutely localized projection-based quantum embedding to study localized excited states using EOM-CCSD-in-DFT and TDDFT-in-DFT. The embedding results are highly accurate compared to the corresponding canonical EOM-CCSD and TDDFT results on the full system, with TDDFT-in-DFT frequently more accurate than canonical TDDFT. The abs. localization method is shown to eliminate the spurious low-lying excitation energies for charge transfer states and prevent over delocalization of excited states. Addnl., we attempt to recover the environment response caused by the electronic excitations in the high-level subsystem using different schemes and compare their accuracy. Finally, we apply this method to the calcn. of the excited state energy of green fluorescent protein and show that we systematically converge to the full system results. Here we demonstrate how this method can be useful in understanding excited states, specifically which chem. moieties polarize to the excitation. This work shows absolutely localized projection-based quantum embedding can treat local electronic excitations accurately, and make computationally expensive WF methods applicable to systems beyond current computational limits.**39**Bennie, S. J.; Curchod, B. F. E.; Manby, F. R.; Glowacki, D. R. Pushing the limits of EOM-CCSD with projector-based embedding for excitation energies.*J. Phys. Chem. Lett.*2017,*8*, 5559– 5565, DOI: 10.1021/acs.jpclett.7b02500Google Scholar39https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhslahu73E&md5=2d435b51099d12860b8decee7d0b1a08Pushing the Limits of EOM-CCSD with Projector-Based Embedding for Excitation EnergiesBennie, Simon J.; Curchod, Basile F. E.; Manby, Frederick R.; Glowacki, David R.Journal of Physical Chemistry Letters (2017), 8 (22), 5559-5565CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)The calcn. of accurate excitation energies using ab initio electronic structure methods such as std. equation of motion coupled cluster singles and doubles (EOM-CCSD) has been cost prohibitive for large systems. In this work, we use a simple projector-based embedding scheme to calc. the EOM-CCSD excitation energies of acrolein solvated in water mols. modeled using d. functional theory (DFT). We demonstrate the accuracy of this approach gives excitation energies within 0.01 eV of full EOM-CCSD, but with significantly reduced computational cost. This approach is also shown to be relatively invariant to the choice of functional used in the environment and allows for the description of systems with large nos. of basis functions ( > 1000) to be treated using state-of-the-art wave function methods. The flexibility of embedding to select orbitals to add to the excited-state method provides insights into the origins of the excitations and can reduce artifacts that could arise in traditional linear response time-dependent DFT (LR-TDDFT).**40**Chen, X.; Gao, J. Fragment Exchange Potential for Realizing Pauli Deformation of Inter-Fragment Interactions.*J. Phys. Chem. Lett.*2020,*11*, 4008, DOI: 10.1021/acs.jpclett.0c00933Google Scholar40https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXnsVCgtLc%253D&md5=bd62e3595e459e781d1e3e585ee8cb34Fragment Exchange Potential for Realizing Pauli Deformation of Interfragment InteractionsChen, Xin; Gao, JialiJournal of Physical Chemistry Letters (2020), 11 (10), 4008-4016CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)In fragment-based methods, the lack of explicit short-range exchange interactions between monomers can result in unphys. deformation in charge d. In this study, we describe a fragment exchange potential (XFP) to explicitly account for interfragmental Pauli deformation. In our implementation, a Kohn-Sham exchange potential is adopted along with the Yukawa potential. The method has been validated by comparison of the computed exchange energies using the XFP potential with results obtained from antisymmetrized fragmental orbitals on the S66x8 data set contg. 528 bimol. interactions of equil. and arbitrary geometries. It was also found that it is only necessary to deploy numerical grids on atoms within their van der Waals contacts, significantly reducing the small, albeit extra, computational cost. We anticipate that the XFP presented here may be applied to mol. dynamics simulations of macromols. using a fragment-based quantum mech. potential with improved SCF convergence and computational accuracy.**41**Fertitta, E.; Booth, G. H. Energy-weighted density matrix embedding of open correlated chemical fragments.*J. Chem. Phys.*2019,*151*, 014115, DOI: 10.1063/1.5100290Google Scholar41https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhtlahs7fI&md5=ea84185b8268d86273c4e725802c4ab7Energy-weighted density matrix embedding of open correlated chemical fragmentsFertitta, Edoardo; Booth, George H.Journal of Chemical Physics (2019), 151 (1), 014115/1-014115/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a multiscale approach to efficiently embed an ab initio correlated chem. fragment described by its energy-weighted d. matrixes and entangled with a wider mean-field many-electron system. This approach, first presented by Fertitta and Booth [Phys. Rev. B 98, 235132 (2018)], is here extended to account for realistic long-range interactions and broken symmetry states. The scheme allows for a systematically improvable description in the range of correlated fluctuations out of the fragment into the system, via a self-consistent optimization of a coupled auxiliary mean-field system. It is discussed that the method has rigorous limits equiv. to the existing quantum embedding approaches of both dynamical mean-field theory and d. matrix embedding theory, to which this method is compared, and the importance of these correlated fluctuations is demonstrated. We derive a self-consistent local energy functional within the scheme and demonstrate the approach for hydrogen rings, where quant. accuracy is achieved despite only a single atom being explicitly treated. (c) 2019 American Institute of Physics.**42**Manby, F. R.; Stella, M.; Goodpaster, J. D.; Miller, T. F., III A simple, exact density-functional-theory embedding scheme.*J. Chem. Theory Comput.*2012,*8*, 2564– 2568, DOI: 10.1021/ct300544eGoogle Scholar42https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhtVCmtLrP&md5=07b2276726a58eb3df5b733e0c75580bA Simple, Exact Density-Functional-Theory Embedding SchemeManby, Frederick R.; Stella, Martina; Goodpaster, Jason D.; Miller, Thomas F.Journal of Chemical Theory and Computation (2012), 8 (8), 2564-2568CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)D. functional theory (DFT) provides a formally exact framework for quantum embedding. The appearance of nonadditive kinetic energy contributions in this context poses significant challenges, but using optimized effective potential (OEP) methods, various groups have devised DFT-in-DFT methods that are equiv. to Kohn-Sham (KS) theory on the whole system. This being the case, we note that a very considerable simplification arises from doing KS theory instead. We then describe embedding schemes that enforce Pauli exclusion via a projection technique, completely avoiding numerically demanding OEP calcns. Illustrative applications are presented using DFT-in-DFT, wave-function-in-DFT, and wave-function-in-Hartree-Fock embedding, and using an embedded many-body expansion.**43**Fornace, M. E.; Lee, J.; Miyamoto, K.; Manby, F. R.; Miller, T. F., III Embedded mean-field theory.*J. Chem. Theory Comput.*2015,*11*, 568– 580, DOI: 10.1021/ct5011032Google Scholar43https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXmvFSjsA%253D%253D&md5=e8a2a875dbcb5a16e6f7eacbeb93108aEmbedded Mean-Field TheoryFornace, Mark E.; Lee, Joonho; Miyamoto, Kaito; Manby, Frederick R.; Miller, Thomas F.Journal of Chemical Theory and Computation (2015), 11 (2), 568-580CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We introduce embedded mean-field theory (EMFT), an approach that flexibly allows for the embedding of one mean-field theory in another without the need to specify or fix the no. of particles in each subsystem. EMFT is simple, is well-defined without recourse to parameters, and inherits the simple gradient theory of the parent mean-field theories. In this paper, we report extensive benchmarking of EMFT for the case where the subsystems are treated using different levels of Kohn-Sham theory, using PBE or B3LYP/6-31G* in the high-level subsystem and LDA/STO-3G in the low-level subsystem; we also investigate different levels of d. fitting in the two subsystems. Over a wide range of chem. problems, we find EMFT to perform accurately and stably, smoothly converging to the high-level of theory as the active subsystem becomes larger. In most cases, the performance is at least as good as that of ONIOM, but the advantages of EMFT are highlighted by examples that involve partitions across multiple bonds or through arom. systems and by examples that involve more complicated electronic structure. EMFT is simple and parameter free, and based on the tests provided here, it offers an appealing new approach to a multiscale electronic structure.**44**Gordon, M. S.; Smith, Q. A.; Xu, P.; Slipchenko, L. V. Accurate first principles model potentials for intermolecular interactions.*Annu. Rev. Phys. Chem.*2013,*64*, 553– 578, DOI: 10.1146/annurev-physchem-040412-110031Google Scholar44https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXntVCrs7g%253D&md5=883c719d2d6d4a4afd3333a45e7cd5abAccurate first principles model potentials for intermolecular interactionsGordon, Mark S.; Smith, Quentin A.; Xu, Peng; Slipchenko, Lyudmila V.Annual Review of Physical Chemistry (2013), 64 (), 553-578CODEN: ARPLAP; ISSN:0066-426X. (Annual Reviews Inc.)A review. The general effective fragment potential (EFP) method provides model potentials for any mol. that is derived from first principles, with no empirically fitted parameters. The EFP method has been interfaced with most currently used ab initio single-ref. and multireference quantum mechanics (QM) methods, ranging from Hartree-Fock and coupled cluster theory to multireference perturbation theory. The most recent innovations in the EFP model have been to make the computationally expensive charge transfer term much more efficient and to interface the general EFP dispersion and exchange repulsion interactions with QM methods. Following a summary of the method and its implementation in generally available computer programs, these most recent new developments are discussed.**45**Gordon, M. S.; Slipchenko, L.; Li, H.; Jensen, J. H. The effective fragment potential: a general method for predicting intermolecular interactions.*Annu. Rep. Comput. Chem.*2007,*3*, 177– 193, DOI: 10.1016/s1574-1400(07)03010-1Google Scholar45https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXkt1Wnsrk%253D&md5=4456c9613581b41f675131118db393b2The effective fragment potential: a general method for predicting intermolecular interactionsGordon, Mark S.; Slipchenko, Lyudmilla; Li, Hui; Jensen, Jan H.Annual Reports in Computational Chemistry (2007), 3 (), 177-193CODEN: ARCCC3; ISSN:1574-1400. (Elsevier B.V.)A review.**46**Sun, Q.; Chan, G. K.-L. Quantum embedding theories.*Acc. Chem. Res.*2016,*49*, 2705– 2712, DOI: 10.1021/acs.accounts.6b00356Google Scholar46https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhslyntrbM&md5=5908130580b088e4ea49c788bda516d8Quantum Embedding TheoriesSun, Qiming; Chan, Garnet Kin-LicAccounts of Chemical Research (2016), 49 (12), 2705-2712CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)A review. In complex systems, it is often the case that the region of interest forms only one part of a much larger system. The idea of joining two different quantum simulations - a high level calcn. on the active region of interest, and a low level calcn. on its environment - formally defines a quantum embedding. While any combination of techniques constitutes an embedding, several rigorous formalisms have emerged that provide for exact feedback between the embedded system and its environment. These three formulations: d. functional embedding, Green's function embedding, and d. matrix embedding, resp., use the single-particle d., single-particle Green's function, and single-particle d. matrix as the quantum variables of interest. Many excellent reviews exist covering these methods individually. However, a unified presentation of the different formalisms is so far lacking. Indeed, the various languages commonly used, functional equations for d. functional embedding, diagrammatics for Green's function embedding, and entanglement arguments for d. matrix embedding, make the three formulations appear vastly different. In this Account, we introduce the basic equations of all three formulations in such a way as to highlight their many common intellectual strands. While we focus primarily on a straightforward theor. perspective, we also give a brief overview of recent applications and possible future developments. The first section starts with d. functional embedding, where we introduce the key embedding potential via the Euler equation. We then discuss recent work concerning the treatment of the nonadditive kinetic potential, before describing mean-field d. functional embedding and wave function in d. functional embedding. We finish the section with extensions to time-dependence and excited states. The second section is devoted to Green's function embedding. Here, we use the Dyson equation to obtain equations that parallel as closely as possible the d. functional embedding equations, with the hybridization playing the role of the embedding potential. Embedding a high-level self-energy within a low-level self-energy is treated analogously to wave function in d. functional embedding. The numerical computation of the high-level self-energy allows us to briefly introduce the bath representation in the quantum impurity problem. We then consider translationally invariant systems to bring in the important dynamical mean-field theory. Recent developments to incorporate screening and long-range interactions are discussed.The third section concerns d. matrix embedding. Here, we first highlight some math. complications assocd. with a simple Euler equation derivation, arising from the open nature of fragments. This motivates the d. matrix embedding theory, where we use the Schmidt decompn. to represent the entanglement through bath orbitals. The resulting impurity plus bath formulation resembles that of dynamical mean-field theory. We discuss the numerical self-consistency assocd. with using a high-level correlated wave function with a mean-field low-level treatment, and connect the resulting numerical inversion to that used in d. functional embedding. We finish with perspectives on the future of all three methods.**47**Chulhai, D. V.; Goodpaster, J. D. Projection-based correlated wave function in density functional theory embedding for periodic systems.*J. Chem. Theory Comput.*2018,*14*, 1928– 1942, DOI: 10.1021/acs.jctc.7b01154Google Scholar47https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXjs1Wjurc%253D&md5=19bfdc5a3700990c68f3fec4bddc35fbProjection-Based Correlated Wave Function in Density Functional Theory Embedding for Periodic SystemsChulhai, Dhabih V.; Goodpaster, Jason D.Journal of Chemical Theory and Computation (2018), 14 (4), 1928-1942CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a level shift projection operator-based embedding method for systems with periodic boundary conditions - where the "active" subsystem can be described using either d. functional theory (DFT) or correlated wave function (WF) methods, and the "environment" is described using DFT. Our method allows for k-point sampling, is shown to be exactly equal to the canonical DFT soln. of the full system under the limit that we use the full system basis to describe each subsystem, and can treat the active subsystem either with periodic boundary conditions - in what we term "periodic-in-periodic" embedding - or as a mol. cluster - in "cluster-in-periodic" embedding. We explore each of these methods, and show that cluster WF-in-periodic DFT embedding can accurately calc. the absorption energy of CO on to a Si(100)-2x1 surface.**48**Chulhai, D. V.; Goodpaster, J. D. Improved accuracy and efficiency in quantum embedding through absolute localization.*J. Chem. Theory Comput.*2017,*13*, 1503– 1508, DOI: 10.1021/acs.jctc.7b00034Google Scholar48https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXjvVWitr4%253D&md5=21d0bbf40f2e5d7efb40dd1d8358e584Improved Accuracy and Efficiency in Quantum Embedding through Absolute LocalizationChulhai, Dhabih V.; Goodpaster, Jason D.Journal of Chemical Theory and Computation (2017), 13 (4), 1503-1508CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Projection-based quantum embedding methodologies provide a framework for performing wave function-in-d. functional theory (WF-in-DFT) calcns. The total WF-in-DFT energy is dependent on the partitioning of the total system and requires similar partitioning in each system for accurate energy differences. To achieve this, we enforce an abs. localization of the WF orbitals to basis functions only assocd. with the WF subsystem. This abs. localization, followed by iterative optimization of the subsystems' orbitals, provides improved energy differences for WF-in-DFT while simultaneously improving the computational efficiency.**49**Goodpaster, J. D.; Barnes, T. A.; Manby, F. R.; Miller, T. F., III Density functional theory embedding for correlated wavefunctions: Improved methods for open-shell systems and transition metal complexes.*J. Chem. Phys.*2012,*137*, 224113, DOI: 10.1063/1.4770226Google Scholar49https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhvVeqsrrP&md5=17536904dd5e5e360ccfa9144590782dDensity functional theory embedding for correlated wavefunctions: Improved methods for open-shell systems and transition metal complexesGoodpaster, Jason D.; Barnes, Taylor A.; Manby, Frederick R.; Miller, Thomas F., IIIJournal of Chemical Physics (2012), 137 (22), 224113/1-224113/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)D. functional theory (DFT) embedding provides a formally exact framework for interfacing correlated wave-function theory (WFT) methods with lower-level descriptions of electronic structure. Here, we report techniques to improve the accuracy and stability of WFT-in-DFT embedding calcns. In particular, we develop spin-dependent embedding potentials in both restricted and unrestricted orbital formulations to enable WFT-in-DFT embedding for open-shell systems, and develop an orbital-occupation-freezing technique to improve the convergence of optimized effective potential calcns. that arise in the evaluation of the embedding potential. The new techniques are demonstrated in applications to the van-der-Waals-bound ethylene-propylene dimer and to the hexaaquairon(II) transition-metal cation. Calcn. of the dissocn. curve for the ethylene-propylene dimer reveals that WFT-in-DFT embedding reproduces full CCSD(T) energies to within 0.1 kcal/mol at all distances, eliminating errors in the dispersion interactions due to conventional exchange-correlation (XC) functionals while simultaneously avoiding errors due to subsystem partitioning across covalent bonds. Application of WFT-in-DFT embedding to the calcn. of the low-spin/high-spin splitting energy in the hexaaquairon(II) cation reveals that the majority of the dependence on the DFT XC functional can be eliminated by treating only the single transition-metal atom at the WFT level; furthermore, these calcns. demonstrate the substantial effects of open-shell contributions to the embedding potential, and they suggest that restricted open-shell WFT-in-DFT embedding provides better accuracy than unrestricted open-shell WFT-in-DFT embedding due to the removal of spin contamination. (c) 2012 American Institute of Physics.**50**Goodpaster, J. D.; Barnes, T. A.; Manby, F. R.; Miller, T. F., III Accurate and systematically improvable density functional theory embedding for correlated wavefunctions.*J. Chem. Phys.*2014,*140*, 18A507, DOI: 10.1063/1.4864040Google ScholarThere is no corresponding record for this reference.**51**Goodpaster, J. D.; Ananth, N.; Manby, F. R.; Miller, T. F., III Exact nonadditive kinetic potentials for embedded density functional theory.*J. Chem. Phys.*2010,*133*, 084103, DOI: 10.1063/1.3474575Google Scholar51https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXhtVKhtrzK&md5=4b907560a876e6f16ae576a3a68013f2Exact nonadditive kinetic potentials for embedded density functional theoryGoodpaster, Jason D.; Ananth, Nandini; Manby, Frederick R.; Miller, Thomas F., IIIJournal of Chemical Physics (2010), 133 (8), 084103/1-084103/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We describe an embedded d. functional theory (DFT) protocol in which the nonadditive kinetic energy component of the embedding potential is treated exactly. At each iteration of the Kohn-Sham equations for constrained electron d., the Zhao-Morrison-Parr constrained search method for constructing Kohn-Sham orbitals is combined with the King-Handy expression for the exact kinetic potential. We use this formally exact embedding protocol to calc. ionization energies for a series of three- and four-electron at. systems, and the results are compared to embedded DFT calcns. that utilize the Thomas-Fermi (TF) and the Thomas-Fermi-von Weisacker approxns. to the kinetic energy functional. These calcns. illustrate the expected breakdown due to the TF approxn. for the nonadditive kinetic potential, with errors of 30%-80% in the calcd. ionization energies; by contrast, the exact protocol is found to be accurate and stable. To significantly improve the convergence of the new protocol, we introduce a d.-based switching function to map between the exact nonadditive kinetic potential and the TF approxn. in the region of the nuclear cusp, and we demonstrate that this approxn. has little effect on the accuracy of the calcd. ionization energies. Finally, we describe possible extensions of the exact protocol to perform accurate embedded DFT calcns. in large systems with strongly overlapping subsystem densities. (c) 2010 American Institute of Physics.**52**Zhang, K.; Ren, S.; Caricato, M. Multi-state QM/QM Extrapolation of UV/Vis Absorption Spectra with Point Charge Embedding.*J. Chem. Theory Comput.*2020,*16*, 4361– 4372, DOI: 10.1021/acs.jctc.0c00339Google Scholar52https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhtVaqsrvJ&md5=ee6a5afce1a5a569cc85154682f31257Multistate QM/QM Extrapolation of UV/Vis Absorption Spectra with Point Charge EmbeddingZhang, Kaihua; Ren, Sijin; Caricato, MarcoJournal of Chemical Theory and Computation (2020), 16 (7), 4361-4372CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The simulation of UV/vis absorption spectra of large chromophores is prohibitively expensive with accurate quantum mech. (QM) methods. Thus, hybrid methods, which treat the core chromophoric region at a high level of theory while the substituent effects are treated with a more computationally efficient method, may provide the best compromise between cost and accuracy. The ONIOM (Our own N-layered Integrated MO mol. Mechanics) method has proved successful at describing ground-state processes. However, for excited states, it suffers from difficulties in matching the correct excited states among the different levels of theory. We devised an approach, based on the ONIOM extrapolation formula, to combine two QM levels of theory to extrapolate entire excitation bands rather than individual states. In this contribution, we extend the same QM/QM hybrid scheme to include polarization effects on the core region through point charge embedding. The charges are computed to reproduce the electrostatic potential of the entire chromophore at the low level of theory, with proper constraints to avoid overpolarization issues at the boundary between layers. We test this approach on a variety of model compds. that show how the multistate QM/QM-embedding scheme is able to accurately reproduce the spectrum of the entire system at the high level of theory better than (i) the bare QM/QM hybrid scheme, (ii) the low-level calcn. on the entire system, and (iii) the high-level calcn. on the core region.**53**Ramos, P.; Papadakis, M.; Pavanello, M. Performance of frozen density embedding for modeling hole transfer reactions.*J. Phys. Chem. B*2015,*119*, 7541– 7557, DOI: 10.1021/jp511275eGoogle Scholar53https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXlvV2hsLo%253D&md5=b1d39bae8c0024b8a1c3f5d5abf18554Performance of Frozen Density Embedding for Modeling Hole Transfer ReactionsRamos, Pablo; Papadakis, Markos; Pavanello, MicheleJournal of Physical Chemistry B (2015), 119 (24), 7541-7557CODEN: JPCBFK; ISSN:1520-5207. (American Chemical Society)We have carried out a thorough benchmark of the frozen d.-embedding (FDE) method for calcg. hole transfer couplings. We have considered 10 exchange-correlation functionals, 3 nonadditive kinetic energy functionals, and 3 basis sets. Overall, we conclude that with a 7% mean relative unsigned error, the PBE and PW91 functionals coupled with the PW91k nonadditive kinetic energy functional and a TZP basis set constitute the most stable and accurate levels of theory for hole transfer coupling calcns. The FDE-ET method is found to be an excellent tool for computing diabatic couplings for hole transfer reactions.**54**Pavanello, M.; Neugebauer, J. Modelling charge transfer reactions with the frozen density embedding formalism.*J. Chem. Phys.*2011,*135*, 234103, DOI: 10.1063/1.3666005Google Scholar54https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhs1eltbbP&md5=4d693d900e148be551dccab8478efe76Modelling charge transfer reactions with the frozen density embedding formalismPavanello, Michele; Neugebauer, JohannesJournal of Chemical Physics (2011), 135 (23), 234103/1-234103/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The frozen d. embedding (FDE) subsystem formulation of d.-functional theory is a useful tool for studying charge transfer reactions. In this work charge-localized, diabatic states are generated directly with FDE and used to calc. electronic couplings of hole transfer reactions in two π-stacked nucleobase dimers of B-DNA: 5'-GG-3' and 5'-GT-3'. The calcns. rely on two assumptions: the two-state model, and a small differential overlap between donor and acceptor subsystem densities. The resulting electronic couplings agree well with benchmark values for those exchange-correlation functionals that contain a high percentage of exact exchange. Instead, when semilocal GGA functionals are used the electronic couplings are grossly overestimated. (c) 2011 American Institute of Physics.**55**Wesolowski, T. A.; Shedge, S.; Zhou, X. Frozen-density embedding strategy for multilevel simulations of electronic structure.*Chem. Rev.*2015,*115*, 5891– 5928, DOI: 10.1021/cr500502vGoogle Scholar55https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXnsVSqu7s%253D&md5=ff2f073b8c897f887f137326c46ba06dFrozen-Density Embedding Strategy for Multilevel Simulations of Electronic StructureWesolowski, Tomasz A.; Shedge, Sapana; Zhou, XiuwenChemical Reviews (Washington, DC, United States) (2015), 115 (12), 5891-5928CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)A review. The following topics are discussed: Frozen-D. Embedding Theory (FDET); Extensions and Formalisms Related to FDET; Approxns. in FDET for Multilevel Simulations; Numerical Simulations Using Approximated FDET Embedding Potentials.**56**Jacob, C. R.; Neugebauer, J.; Visscher, L. A flexible implementation of frozen-density embedding for use in multilevel simulations.*J. Comput. Chem.*2008,*29*, 1011– 1018, DOI: 10.1002/jcc.20861Google Scholar56https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXks1altLc%253D&md5=675c0696a25bcca330c15a85902a9115Software news and update a flexible implementation of frozen-density embedding for use in multilevel simulationsJacob, Christoph R.; Neugebauer, Johannes; Visscher, LucasJournal of Computational Chemistry (2008), 29 (6), 1011-1018CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)A new implementation of frozen-d. embedding (FDE) in the Amsterdam D. Functional (ADF) program package is presented. FDE is based on a subsystem formulation of d.-functional theory (DFT), in which a large system is assembled from an arbitrary no. of subsystems, which are coupled by an effective embedding potential. The new implementation allows both an optimization of all subsystems as a linear-scaling alternative to a conventional DFT treatment, the calcn. of one active fragment in the presence of a frozen environment, and intermediate setups, in which individual subsystems are fully optimized, partially optimized, or completely frozen. It is shown how this flexible setup can facilitate the application of FDE in multilevel simulations.**57**Wesolowski, T. A.; Warshel, A. Frozen density functional approach for ab initio calculations of solvated molecules.*J. Phys. Chem.*1993,*97*, 8050– 8053, DOI: 10.1021/j100132a040Google Scholar57https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXkvVSlu7o%253D&md5=e671a5774b1afd337d42c7ec460a36e7Frozen density functional approach for ab initio calculations of solvated moleculesWesolowski, Tomasz Adam; Warshel, AriehJournal of Physical Chemistry (1993), 97 (30), 8050-3CODEN: JPCHAX; ISSN:0022-3654.A new d. functional method for treatment of chem. processes in soln. is presented. The method is based on freezing the electron d. of the solvent mols., while solving the eigenvalue problem for the solute Hamiltonian, which includes the effective potential of the solvent mols. The method is developed and examd. in the simple case of one solvent and one solute mol. The results are encouraging and the deviation between the unfrozen and frozen treatments can be attributed to the kinetic energy functional used. The method can be implemented in ab initio calcns. of solvation free energies, following a recent pseudopotential approach [Vaidehi et al., 1992].**58**Sæther, S.; Kjærgaard, T.; Koch, H.; Høyvik, I.-M. Density-Based Multilevel Hartree–Fock Model.*J. Chem. Theory Comput.*2017,*13*, 5282– 5290, DOI: 10.1021/acs.jctc.7b00689Google Scholar58https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhsFGhs7zN&md5=9c3da3ee31d5baf4e6fb2680e53636bcDensity-Based Multilevel Hartree-Fock ModelSaether, Sandra; Kjaergaard, Thomas; Koch, Henrik; Hoeyvik, Ida-MarieJournal of Chemical Theory and Computation (2017), 13 (11), 5282-5290CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We introduce a d.-based multilevel Hartree-Fock (HF) method where the electronic d. is optimized in a given region of the mol. (the active region). Active MOs are generated by a decompn. of a starting guess AO d., whereas the inactive MOs (which constitute the remainder of the d.) are never generated or referenced. The MO formulation allows for a significant dimension redn. by transforming from the AO basis to the active MO basis. All interactions between the inactive and active regions of the mol. are retained, and an exponential parametrization of orbital rotations ensures that the active and inactive d. matrixes sep., and in sum, satisfy the symmetry, trace, and idempotency requirements. Thus, the orbital spaces stay orthogonal, and furthermore, the total d. matrix represents a single Slater determinant. In each iteration, the (level-shifted) Newton equations in the active MO basis are solved to obtain the orbital transformation matrix. The approach is equiv. to variationally optimizing only a subset of the MOs of the total system. In this orbital space partitioning, no bonds are broken and no a priori orbital assignments are carried out. In the limit of including all orbitals in the active space, we obtain an MO d.-based formulation of full HF.**59**Høyvik, I.-M. Convergence acceleration for the multilevel Hartree–Fock model.*Mol. Phys.*2020,*118*, 1626929, DOI: 10.1080/00268976.2019.1626929Google ScholarThere is no corresponding record for this reference.**60**Aquilante, F.; Boman, L.; Boström, J.; Koch, H.; Lindh, R.; de Merás, A. S.; Pedersen, T. B.*Linear-Scaling Techniques in Computational Chemistry and Physics*; Springer, 2011; pp 301– 343.Google ScholarThere is no corresponding record for this reference.**61**Sánchez de Merás, A. M. J.; Koch, H.; Cuesta, I. G.; Boman, L. Cholesky decomposition-based definition of atomic subsystems in electronic structure calculations.*J. Chem. Phys.*2010,*132*, 204105, DOI: 10.1063/1.3431622Google Scholar61https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXmvVajtbg%253D&md5=5306234a00b3081733cd735911c4e4e9Cholesky decomposition-based definition of atomic subsystems in electronic structure calculationsSanchez de Meras, Alfredo M. J.; Koch, Henrik; Cuesta, Inmaculada Garcia; Boman, LinusJournal of Chemical Physics (2010), 132 (20), 204105/1-204105/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Decompg. the Hartree-Fock one-electron d. matrix and a virtual pseudodensity matrix, we obtain an orthogonal set of normalized MOs with local character to be used in post-Hartree-Fock calcns. The applicability of the procedure is illustrated by calcg. CCSD(T) energies and CCSD mol. properties in reduced active spaces. (c) 2010 American Institute of Physics.**62**Pulay, P. Second and third derivatives of variational energy expressions: Application to multiconfigurational self-consistent field wave functions.*J. Chem. Phys.*1983,*78*, 5043– 5051, DOI: 10.1063/1.445372Google Scholar62https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3sXhvVGrtrg%253D&md5=9508b76b43bcd8a98623e0bd0c6dab64Second and third derivatives of variational energy expressions: application to multiconfigurational self-consistent field wave functionsPulay, PeterJournal of Chemical Physics (1983), 78 (8), 5043-51CODEN: JCPSA6; ISSN:0021-9606.General anal. expressions are given for the second and third derivs. of constrained variational energy expressions. Variational energy expressions and odd-order derivs. have a distinct advantage over nonvariational (e.g., perturbative) energy expressions and even-order derivs. In particular, the first-order wave function suffices to det. the derivs. of the variational energy up to third order. The coupled-perturbed MC-SCF equations, obtained from the general results, are equiv., with minor corrections, to the ones very recently presented by Y. Osamura, et al., (1982). Explicit expressions are given for the second and third derivs. of the MC-SCF energy. Computational implementation is briefly discussed.**63**Saebo, S.; Pulay, P. Local treatment of electron correlation.*Annu. Rev. Phys. Chem.*1993,*44*, 213– 236, DOI: 10.1146/annurev.pc.44.100193.001241Google Scholar63https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXhsFSqtg%253D%253D&md5=6f98c42474702e87f679e786681a84a5Local treatment of electron correlationSaebo, Svein; Pulay, PeterAnnual Review of Physical Chemistry (1993), 44 (), 213-36CODEN: ARPLAP; ISSN:0066-426X.A review with 88 refs. The topics include: general strategies, localized MOs, local correlation methods, and problems and future prospects.**64**Culpitt, T.; Brorsen, K. R.; Hammes-Schiffer, S. Communication: Density functional theory embedding with the orthogonality constrained basis set expansion procedure.*J. Chem. Phys.*2017,*146*, 211101, DOI: 10.1063/1.4984777Google Scholar64https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXpt1Kqs7w%253D&md5=f9788d5b309079a0dae883a4ae2e784bCommunication: Density functional theory embedding with the orthogonality constrained basis set expansion procedureCulpitt, Tanner; Brorsen, Kurt R.; Hammes-Schiffer, SharonJournal of Chemical Physics (2017), 146 (21), 211101/1-211101/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)D. functional theory (DFT) embedding approaches have generated considerable interest in the field of computational chem. because they enable calcns. on larger systems by treating subsystems at different levels of theory. To circumvent the calcn. of the non-additive kinetic potential, various projector methods have been developed to ensure the orthogonality of MOs between subsystems. Herein the orthogonality constrained basis set expansion (OCBSE) procedure is implemented to enforce this subsystem orbital orthogonality without requiring a level shifting parameter. This scheme is a simple alternative to existing parameter-free projector-based schemes, such as the Huzinaga equation. The main advantage of the OCBSE procedure is that excellent convergence behavior is attained for DFT-in-DFT embedding without freezing any of the subsystem densities. For the three chem. systems studied, the level of accuracy is comparable to or higher than that obtained with the Huzinaga scheme with frozen subsystem densities. Allowing both the high-level and low-level DFT densities to respond to each other during DFT-in-DFT embedding calcns. provides more flexibility and renders this approach more generally applicable to chem. systems. It could also be useful for future extensions to embedding approaches combining wavefunction theories and DFT. (c) 2017 American Institute of Physics.**65**Hégely, B.; Nagy, P. R.; Ferenczy, G. G.; Kállay, M. Exact density functional and wave function embedding schemes based on orbital localization.*J. Chem. Phys.*2016,*145*, 064107, DOI: 10.1063/1.4960177Google Scholar65https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xhtlalu7%252FP&md5=4e1dbf60af141a2d3e9e0d10a5ced940Exact density functional and wave function embedding schemes based on orbital localizationHegely, Bence; Nagy, Peter R.; Ferenczy, Gyorgy G.; Kallay, MihalyJournal of Chemical Physics (2016), 145 (6), 064107/1-064107/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Exact schemes for the embedding of d. functional theory (DFT) and wave function theory (WFT) methods into lower-level DFT or WFT approaches are introduced utilizing orbital localization. First, a simple modification of the projector-based embedding scheme of Manby and co-workers [J. Chem. Phys. 140, 18A507 (2014)] is proposed. We also use localized orbitals to partition the system, but instead of augmenting the Fock operator with a somewhat arbitrary level-shift projector we solve the Huzinaga-equation, which strictly enforces the Pauli exclusion principle. Second, the embedding of WFT methods in local correlation approaches is studied. Since the latter methods split up the system into local domains, very simple embedding theories can be defined if the domains of the active subsystem and the environment are treated at a different level. The considered embedding schemes are benchmarked for reaction energies and compared to quantum mechanics (QM)/mol. mechanics (MM) and vacuum embedding. We conclude that for DFT-in-DFT embedding, the Huzinaga-equation-based scheme is more efficient than the other approaches, but QM/MM or even simple vacuum embedding is still competitive in particular cases. Concerning the embedding of wave function methods, the clear winner is the embedding of WFT into low-level local correlation approaches, and WFT-in-DFT embedding can only be more advantageous if a non-hybrid d. functional is employed. (c) 2016 American Institute of Physics.**66**Huzinaga, S.; Cantu, A. A. Theory of separability of many-electron systems.*J. Chem. Phys.*1971,*55*, 5543– 5549, DOI: 10.1063/1.1675720Google Scholar66https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE38Xhsl2ruw%253D%253D&md5=57073e274c58fc9dd1649b90d089272cTheory of separability of many-electron systemsHuzinaga, S.; Cantu, A. A.Journal of Chemical Physics (1971), 55 (12), 5543-9CODEN: JCPSA6; ISSN:0021-9606.At. and mol. systems are often intuitively sepd. into almost independent subsystems as, for example, the core and valence parts of an atom. Consequently, if this sepn. provides a good approxn., one can obtain the states of the system from the states of the subsystems which best represent the entire system. In the light of the work of McWeeny, in which one assumes strong orthogonality among subsystem wavefunctions, an effective Hamiltonian is detd. for a given subsystem which should properly describe the states of that subsystem. Previous work has dealt with an improper effective Hamiltonian.**67**Boys, S. F. Construction of some molecular orbitals to be approximately invariant for changes from one molecule to another.*Rev. Mod. Phys.*1960,*32*, 296, DOI: 10.1103/revmodphys.32.296Google Scholar67https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaF3MXht1amuro%253D&md5=dd797e8d0da21499495500b2f3e49496Construction of some molecular orbitals to be approximately invariant for changes from one molecule to anotherBoys, S. F.Reviews of Modern Physics (1960), 32 (), 296-9CODEN: RMPHAT; ISSN:0034-6861.The concept of the invariant orbital is introduced and defined as an orbital for part of a system which remains invariant under chem. changes occurring at some distance. Methods for computing invariant orbitals are outlined.**68**Christiansen, O.; Koch, H.; Jørgensen, P. The second-order approximate coupled cluster singles and doubles model CC2.*Chem. Phys. Lett.*1995,*243*, 409– 418, DOI: 10.1016/0009-2614(95)00841-qGoogle Scholar68https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXotFCmtL4%253D&md5=1978d2062ade66a8a9a04af8378fa0ddThe second-order approximate coupled cluster singles and doubles model CC2Christiansen, Ove; Koch, Henrik; Jorgensen, PoulChemical Physics Letters (1995), 243 (5,6), 409-18CODEN: CHPLBC; ISSN:0009-2614. (Elsevier)An approx. coupled cluster singles and doubles model is presented, denoted CC2. The CC2 total energy is of second-order Moeller-Plesset perturbation theory (MP2) quality. The CC2 linear response function is derived. Unlike MP2, excitation energies and transition moments can be obtained in CC2. A hierarchy of coupled cluster models, CCS, CC2, CCSD, CC3, CCSDT, etc., is presented where CC2 and CC3 are approx. coupled cluster models defined by similar approxns. Higher levels give increased accuracy at increased computational effort. The scaling of CCS, CC2, CCSD, CC3, and CCSDT is N4, N5, N6, N7, and N8, resp., where N is the no. of orbitals. Calcns. of excitation energies for Be, N2, and C2H4 are performed, and results compared with those obtained with the second-order polarization propagator approach SOPPA.**69**Myhre, R. H.; Koch, H. The multilevel CC3 coupled cluster model.*J. Chem. Phys.*2016,*145*, 044111, DOI: 10.1063/1.4959373Google Scholar69https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xht1Kmur7E&md5=6f62a1c36755e3209a5cbbf86d3502c8The multilevel CC3 coupled cluster modelMyhre, Rolf H.; Koch, HenrikJournal of Chemical Physics (2016), 145 (4), 044111/1-044111/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present an efficient implementation of the closed shell multilevel coupled cluster method where coupled cluster singles and doubles (CCSD) is used for the inactive orbital space and CCSD with perturbative triples (CC3) is employed for the smaller active orbital space. Using Cholesky orbitals, the active space can be spatially localized and the computational cost is greatly reduced compared to full CC3 while retaining the accuracy of CC3 excitation energies. For the small org. mols. considered we achieve up to two orders of magnitude redn. in the computational requirements. (c) 2016 American Institute of Physics.**70**Folkestad, S. D.; Koch, H. Multilevel CC2 and CCSD Methods with Correlated Natural Transition Orbitals.*J. Chem. Theory Comput.*2020,*16*, 179, DOI: 10.1021/acs.jctc.9b00701Google Scholar70https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BB3MfgslGluw%253D%253D&md5=e42d0da9ebf9a3346e33474271046907Multilevel CC2 and CCSD Methods with Correlated Natural Transition OrbitalsFolkestad Sarai Dery; Koch Henrik; Koch HenrikJournal of chemical theory and computation (2020), 16 (1), 179-189 ISSN:.In the multilevel coupled cluster approach, an active orbital space is treated at a higher level of coupled cluster theory than the remaining inactive orbitals. We introduce the multilevel CC2 method where CC2 is used for the active orbital space. Furthermore, we present a simplified formulation of the multilevel CCSD method where CCSD is used for the active space. The simplification lies in the evaluation of the CC2 amplitudes in the inactive space; these CC2 amplitudes have previously been determined iteratively. We use correlated natural transition orbitals to determine the active orbital spaces. The convergence of the multilevel CC2 and multilevel CCSD valence excitation energies is established with proof-of-concept calculations. The methods are also applied to two larger systems: p-nitroaniline in water and amoxicillin. The calculations on the p-nitroaniline-water system illustrate the usefulness of multilevel coupled cluster methods for molecules in solution and for charge transfer excitations.**71**Folkestad, S. D.; Kjønstad, E. F.; Myhre, R. H.; Andersen, J. H.; Balbi, A.; Coriani, S.; Giovannini, T.; Goletto, L.; Haugland, T. S.; Hutcheson, A.; Høyvik, I.-M.; Moitra, T.; Paul, A. C.; Scavino, M.; Skeidsvoll, A. S.; Tveten, Å. H.; Koch, H. eT 1.0: An open source electronic structure program with emphasis on coupled cluster and multilevel methods.*J. Chem. Phys.*2020,*152*, 184103, DOI: 10.1063/5.0004713Google Scholar71https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXptleitbc%253D&md5=07f57f25437d1255896e90628884dba1eT 1.0: An open source electronic structure program with emphasis on coupled cluster and multilevel methodsFolkestad, Sarai D.; Kjoenstad, Eirik F.; Myhre, Rolf H.; Andersen, Josefine H.; Balbi, Alice; Coriani, Sonia; Giovannini, Tommaso; Goletto, Linda; Haugland, Tor S.; Hutcheson, Anders; Hoeyvik, Ida-Marie; Moitra, Torsha; Paul, Alexander C.; Scavino, Marco; Skeidsvoll, Andreas S.; Tveten, Aasmund H.; Koch, HenrikJournal of Chemical Physics (2020), 152 (18), 184103CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The eT program is an open source electronic structure package with emphasis on coupled cluster and multilevel methods. It includes efficient spin adapted implementations of ground and excited singlet states, as well as equation of motion oscillator strengths, for CCS, CC2, CCSD, and CC3. Furthermore, eT provides unique capabilities such as multilevel Hartree-Fock and multilevel CC2, real-time propagation for CCS and CCSD, and efficient CC3 oscillator strengths. With a coupled cluster code based on an efficient Cholesky decompn. algorithm for the electronic repulsion integrals, eT has similar advantages as codes using d. fitting, but with strict error control. Here, we present the main features of the program and demonstrate its performance through example calcns. Because of its availability, performance, and unique capabilities, we expect eT to become a valuable resource to the electronic structure community. (c) 2020 American Institute of Physics.**72**Lehtola, S. Assessment of initial guesses for self-consistent field calculations. Superposition of atomic potentials: Simple yet efficient.*J. Chem. Theory Comput.*2019,*15*, 1593– 1604, DOI: 10.1021/acs.jctc.8b01089Google Scholar72https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhsFSqt7g%253D&md5=0f04ec8103ff5427d261bcf89b07630bAssessment of Initial Guesses for Self-Consistent Field Calculations. Superposition of Atomic Potentials: Simple yet EfficientLehtola, SusiJournal of Chemical Theory and Computation (2019), 15 (3), 1593-1604CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Electronic structure calcns., such as in the Hartree-Fock or Kohn-Sham d. functional approach, require an initial guess for the MOs. The quality of the initial guess has a significant impact on the speed of convergence of the SCF procedure. Popular choices for the initial guess include the one-electron guess from the core Hamiltonian, the extended Hueckel method, and the superposition of at. densities (SAD). Here, we discuss alternative guesses obtained from the superposition of at. potentials (SAP), which is easily implementable even in real-space calcns. We also discuss a variant of SAD which produces guess orbitals by purifn. of the d. matrix that could also be used in real-space calcns., as well as a parameter-free variant of the extended Hueckel method, which resembles the SAP method and is easy to implement on top of existing SAD infrastructure. The performance of the core Hamiltonian, the SAD and the SAP guesses as well as the extended Hueckel variant is assessed in non-relativistic calcns. on a dataset of 259 mols. ranging from the first to the fourth periods by projecting the guess orbitals onto precomputed, converged SCF solns. in single- to triple-ζ basis sets. It is shown that the proposed SAP guess is the best guess on av. The extended Hueckel guess offers a good alternative, with less scatter in accuracy.**73**Koch, H.; Sánchez de Merás, A.; Pedersen, T. B. Reduced scaling in electronic structure calculations using Cholesky decompositions.*J. Chem. Phys.*2003,*118*, 9481– 9484, DOI: 10.1063/1.1578621Google Scholar73https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXjvVGrt74%253D&md5=f01a7505fbdba8e5e85fbd34cff8fdfbReduced scaling in electronic structure calculations using Cholesky decompositionsKoch, Henrik; Sanchez de Meras, Alfredo; Pedersen, Thomas BondoJournal of Chemical Physics (2003), 118 (21), 9481-9484CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We demonstrate that substantial computational savings are attainable in electronic structure calcns. using a Cholesky decompn. of the two-electron integral matrix. In most cases, the computational effort involved calcg. the Cholesky decompn. is less than the construction of one Fock matrix using a direct O(N2) procedure.**74**Christiansen, O.; Manninen, P.; Jørgensen, P.; Olsen, J. Coupled-cluster theory in a projected atomic orbital basis.*J. Chem. Phys.*2006,*124*, 084103, DOI: 10.1063/1.2173249Google Scholar74https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28Xis1ags7Y%253D&md5=b5de844b127d028c7c0841ce1ec16771Coupled-cluster theory in a projected atomic orbital basisChristiansen, Ove; Manninen, Pekka; Joergensen, Poul; Olsen, JeppeJournal of Chemical Physics (2006), 124 (8), 084103/1-084103/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a biorthogonal formulation of coupled-cluster (CC) theory using a redundant projected AO (PAO) basis. The biorthogonal formulation provides simple equations, where the projectors involved in the definition of the PAO basis are absorbed in the integrals. Explicit expressions for the coupled-cluster singles and doubles equations are derived in the PAO basis. The PAO CC equations can be written in a form identical to the std. MO CC equations, only with integrals that are related to the AO integrals through different transformation matrixes. The dependence of cluster amplitudes, integrals, and correlation energy contributions on the distance between the participating at. centers and on the no. of involved at. centers is illustrated in numerical case studies. It is also discussed how the present reformulation of the CC equations opens new possibilities for reducing the no. of involved parameters and thereby the computational cost.**75**Myhre, R. H.; Sánchez de Merás, A. M. J.; Koch, H. Multi-level coupled cluster theory.*J. Chem. Phys.*2014,*141*, 224105, DOI: 10.1063/1.4903195Google Scholar75https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXitV2mur%252FJ&md5=3c46b9a81bb4e87079c833660f5ab1b0Multi-level coupled cluster theoryMyhre, Rolf H.; Sanchez de Meras, Alfredo M. J.; Koch, HenrikJournal of Chemical Physics (2014), 141 (22), 224105/1-224105/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a general formalism where different levels of coupled cluster theory can be applied to different parts of the mol. system. The system is partitioned into subsystems by Cholesky decompn. of the one-electron Hartree-Fock d. matrix. In this way the system can be divided across chem. bonds without discontinuities arising. The coupled cluster wave function is defined in terms of cluster operators for each part and these are detd. from a set of coupled equations. The total wave function fulfills the Pauli-principle across all borders and levels of electron correlation. We develop the assocd. response theory for this multi-level coupled cluster theory and present proof of principle applications. The formalism is an essential tool in order to obtain size-intensive complexity in the calcn. of local mol. properties. (c) 2014 American Institute of Physics.**76**Egidi, F.; Segado, M.; Koch, H.; Cappelli, C.; Barone, V. A benchmark study of electronic excitation energies, transition moments, and excited-state energy gradients on the nicotine molecule.*J. Chem. Phys.*2014,*141*, 224114, DOI: 10.1063/1.4903307Google Scholar76https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXitFahsrvK&md5=6501af06f537c1b5ff920630ecc77615A benchmark study of electronic excitation energies, transition moments, and excited-state energy gradients on the nicotine moleculeEgidi, Franco; Segado, Mireia; Koch, Henrik; Cappelli, Chiara; Barone, VincenzoJournal of Chemical Physics (2014), 141 (22), 224114/1-224114/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In this work, we report a comparative study of computed excitation energies, oscillator strengths, and excited-state energy gradients of (S)-nicotine, chosen as a test case, using multireference methods, coupled cluster singles and doubles, and methods based on time-dependent d. functional theory. This system was chosen because its apparent simplicity hides a complex electronic structure, as several different types of valence excitations are possible, including n-π*, π-π*, and charge-transfer states, and in order to simulate its spectrum it is necessary to describe all of them consistently well by the chosen method. (c) 2014 American Institute of Physics.**77**Marder, S. R.; Beratan, D. N.; Cheng, L.-T. Approaches for optimizing the first electronic hyperpolarizability of conjugated organic molecules.*Science*1991,*252*, 103– 106, DOI: 10.1126/science.252.5002.103Google Scholar77https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3MXitlylt7g%253D&md5=4da3b8c6cf261058f7d419755fb28fd2Approaches for optimizing the first electronic hyperpolarizability of conjugated organic moleculesMarder, S. R.; Beratan, D. N.; Cheng, L. T.Science (Washington, DC, United States) (1991), 252 (5002), 103-6CODEN: SCIEAS; ISSN:0036-8075.A two-state, four-orbital, independent electron anal. of the first optical mol. hyperpolarizability, β, leads to the prediction that |β| maximizes at a combination of donor and acceptor strengths for a given conjugated bridge. Mol. design strategies that focus on the energetic manipulations of the bridge states are proposed for the optimization of β. The limitations of mol. classes based on common bridge structures are highlighted, and more promising candidates are described. Exptl. results supporting the validity of this approach are presented.**78**Grigorenko, A. N.; Polini, M.; Novoselov, K. S. Graphene plasmonics.*Nat. Photonics*2012,*6*, 749, DOI: 10.1038/nphoton.2012.262Google Scholar78https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38Xhs1ajtbzJ&md5=041a227bbcc24796abb7c8cc2a301f84Graphene plasmonicsGrigorenko, A. N.; Polini, M.; Novoselov, K. S.Nature Photonics (2012), 6 (11), 749-758CODEN: NPAHBY; ISSN:1749-4885. (Nature Publishing Group)A review. Two rich and vibrant fields of investigation-graphene physics and plasmonics-strongly overlap. Not only does graphene possess intrinsic plasmons that are tunable and adjustable, but a combination of graphene with noble-metal nanostructures promises a variety of exciting applications for conventional plasmonics. The versatility of graphene means that graphene-based plasmonics may enable the manuf. of novel optical devices working in different frequency ranges-from terahertz to the visible-with extremely high speed, low driving voltage, low power consumption and compact sizes. Here we review the field emerging at the intersection of graphene physics and plasmonics.**79**Gibson, S. E.; Knight, J. D. [2.2] Paracyclophane derivatives in asymmetric catalysis.*Org. Biomol. Chem.*2003,*1*, 1256– 1269, DOI: 10.1039/b300717kGoogle Scholar79https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXkt1Krsrs%253D&md5=311d3a4932478b32153aaca1b357892a[2.2]paracyclophane derivatives in asymmetric catalysisGibson, Susan E.; Knight, Jamie D.Organic & Biomolecular Chemistry (2003), 1 (8), 1256-1269CODEN: OBCRAK; ISSN:1477-0520. (Royal Society of Chemistry)A review. The growing importance of [2.2]paracyclophane derivs. as planar chiral ligands was highlighted. Comprehensive coverage of the applications of mono- and disubstituted [2.2]paracyclophane derivs. in asym. catalysis was provided. Each section of the review was supplemented with a description of typical approaches used to access classes of cyclophanes under discussion. A review.**80**Gleiter, R.; Hopf, H.*Modern Cyclophane Chemistry*; John Wiley & Sons, 2006.Google ScholarThere is no corresponding record for this reference.**81**Grimme, S. On the Importance of Electron Correlation Effects for the π-π Interactions in Cyclophanes.*Chem.—Eur. J.*2004,*10*, 3423– 3429, DOI: 10.1002/chem.200400091Google Scholar81https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXmtFKrtLk%253D&md5=ddd58fd3cc59dd5705bf54731a79f51fOn the importance of electron correlation effects for the π-π interactions in cyclophanesGrimme, StefanChemistry - A European Journal (2004), 10 (14), 3423-3429CODEN: CEUJED; ISSN:0947-6539. (Wiley-VCH Verlag GmbH & Co. KGaA)Correlated ab initio quantum chem. methods based on second-order perturbation theory and d. functional theory (DFT) together with large AO basis sets are used to calc. the structures of four cyclophanes with two arom. rings and one sulfur-contg. phane with one arom. ring. The calcd. geometrical data for [2.2]paracyclophane, cyclophane (superphane), 8,16-dimethyl[2.2]metacyclophane, 16-methyl[2.2]metaparacyclophane, and 2,6,15-trithia[34,10][7]metacyclophane are compared to exptl. data from x-ray crystal structure detns. In all cases, very accurate theor. predictions are obtained from the recently developed spin-component-scaled MP2 (SCS-MP2) method, in which the deviations are within the exptl. accuracy and expected crystal-packing or vibrational effects. Esp. the interring distances, which are detd. by a detailed balance between attractive van der Waals (dispersive) and repulsive (Pauli) contributions, are very sensitive to the level of theory employed. While std. MP2 theory in general overestimates the dispersive interactions (π-π correlations) between the two arom. rings leading to too short distances (between 3 and 8 pm), the opposite is obsd. for DFT methods (errors up to 15 pm). An explicit account of dispersive-type electron correlation effects between the clamped arom. units is essential for a quant. description of cyclophane structures. To distinguish these effects from normal van der Waals interactions, the term overlap-dispersive interaction may be employed. The popular B3LYP hybrid d. functional offers no advantage over the pure PBE functional that at least qual. accounts for some of the dispersive effects. The use of properly polarized AO basis sets of at least valence-triple-ζ quality is strongly recommended to obtain quant. predictions with traditional wave function methods.**82**Demissie, T. B.; Dodziuk, H.; Waluk, J.; Ruud, K.; Pietrzak, M.; Vetokhina, V.; Szymański, S.; Jaźwiński, J.; Hopf, H. Structure, NMR and Electronic Spectra of [m.n]Paracyclophanes with Varying Bridges Lengths (m, n = 2–4).*J. Phys. Chem. A*2016,*120*, 724– 736, DOI: 10.1021/acs.jpca.5b12168Google Scholar82https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XpsFahsw%253D%253D&md5=f914778424b4b40f59da3c494fbcac52Structure, NMR and Electronic Spectra of [m.n]Paracyclophanes with Varying Bridges Lengths (m, n = 2-4)Demissie, Taye B.; Dodziuk, Helena; Waluk, Jacek; Ruud, Kenneth; Pietrzak, Mariusz; Vetokhina, Volha; Szymanski, Slawomir; Jazwinski, Jaroslaw; Hopf, HenningJournal of Physical Chemistry A (2016), 120 (5), 724-736CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)Extending our earlier studies on cyclophanes, we here report the structure, chem. shifts, spin-spin coupling consts., absorption and emission properties of [m.n]paracyclophanes, m, n = 2-4, obtained using a combination of exptl. and computational techniques. Accurate values of proton chem. shifts as well as of JHH for the bridges are detd. The exptl. chem. shifts, coupling consts., absorption and emission wavelengths are satisfactorily reproduced using d. functional theory calcns., using both the B3LYP and ωB97X-D functionals. The geometries predicted using a functional that includes dispersion corrections (ωB97X-D) are in a better agreement with available exptl. values than those obtained using the B3LYP method. Up to 8 UV-vis absorption/emission bands have been obsd. (or anticipated in the region below 200 nm) and assigned on the basis of quantum-chem. calcns. Optimized excited-state geometries showed that the distances between the arom. bridgehead carbon atoms of all the [m.n]paracyclophanes in the excited state decrease compared to the ground-state geometries by ca. 0.2-0.9 Å, the largest being for [4.4]paracyclophane, though the rather large differences in the calcd. emission wavelength compared to expt. cast some doubts on the accuracy of the excited-state geometries.**83**Bachrach, S. M. DFT Study of [2.2]-, [3.3]-, and [4.4]Paracyclophanes: Strain Energy, Conformations, and Rotational Barriers.*J. Phys. Chem. A*2011,*115*, 2396– 2401, DOI: 10.1021/jp111523uGoogle Scholar83https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXisFWntL4%253D&md5=ad0fafbb4e607229085988310f3f8fb0DFT Study of [2.2]-, [3.3]-, and [4.4]Paracyclophanes: Strain Energy, Conformations, and Rotational BarriersBachrach, Steven M.Journal of Physical Chemistry A (2011), 115 (11), 2396-2401CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)The three smallest sym. paracyclophanes, having tethers with two, three, or four methylene groups, have been examd. with four d. functional methods (B3LYP, M06-2x, B97-D, ωB97X-D). The geometries predicted by functionals accounting for medium-range correlation or long-range exchange and/or dispersion are in close agreement with expt. In addn., these methods provide similar ests. of the strain energy of the paracylcophanes, which decrease with increasing tether length. [4.4]Paracyclophane is nearly strain-free, reflecting the small out-of-plane distortion of its Ph rings. Lastly, the barrier for interconversion of the conformers of [3.3]paracylcophane is computed in close agreement with expt., and an est. for Ph rotation in [4.4]paracyclophane of about 19 kcal mol-1 is predicted by the DFT methods employed.**84**Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Petersson, G. A.; Nakatsuji, H.; Li, X.; Caricato, M.; Marenich, A. V.; Bloino, J.; Janesko, B. G.; Gomperts, R.; Mennucci, B.; Hratchian, H. P.; Ortiz, J. V.; Izmaylov, A. F.; Sonnenberg, J. L.; Williams-Young, D.; Ding, F.; Lipparini, F.; Egidi, F.; Goings, J.; Peng, B.; Petrone, A.; Henderson, T.; Ranasinghe, D.; Zakrzewski, V. G.; Gao, J.; Rega, N.; Zheng, G.; Liang, W.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Throssell, K.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.; Bearpark, M. J.; Heyd, J. J.; Brothers, E. N.; Kudin, K. N.; Staroverov, V. N.; Keith, T. A.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A. P.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Millam, J. M.; Klene, M.; Adamo, C.; Cammi, R.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Farkas, O.; Foresman, J. B.; Fox, D. J.*Gaussian 16*, Revision A.03; Gaussian Inc., Wallingford CT, 2016.Google ScholarThere is no corresponding record for this reference.**85**Castro Neto, A. H.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K. The electronic properties of graphene.*Rev. Mod. Phys.*2009,*81*, 109, DOI: 10.1103/revmodphys.81.109Google Scholar85https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXksVamsLY%253D&md5=d4b07bf6507d26df9b0447a25131bf18The electronic properties of grapheneCastro Neto, A. H.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K.Reviews of Modern Physics (2009), 81 (1), 109-162CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)A review. This article reviews the basic theor. aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external elec. and magnetic fields, or by altering sample geometry and/or topol. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and no. of layers. Edge (surface) states in graphene depend on the edge termination (zigzag or armchair) and affect the phys. properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.**86**Zhang, D. W.; Zhang, J. Z. H. Molecular fractionation with conjugate caps for full quantum mechanical calculation of protein–molecule interaction energy.*J. Chem. Phys.*2003,*119*, 3599– 3605, DOI: 10.1063/1.1591727Google Scholar86https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXlvFCqsLo%253D&md5=4952ee44ff52f306b22fb230f7919ff6Molecular fractionation with conjugate caps for full quantum mechanical calculation of protein-molecule interaction energyZhang, Da W.; Zhang, J. Z. H.Journal of Chemical Physics (2003), 119 (7), 3599-3605CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A scheme to calc. fully quantum mech. (ab initio) interaction energy involving a macromol. like protein is presented. In this scheme, the protein is decompd. into individual amino acid-based fragments that are treated with proper mol. caps. The interaction energy between any mol. and the given protein is given by the summation of interactions between the mol. and individually capped protein fragments. This scheme, termed mol. fractionation with conjugate caps (MFCC), makes it possible and practical to carry out full quantum mech. (ab initio) calcn. of intermol. interaction energies involving proteins or other similar biol. mols. Numerical tests performed on the interaction energies between a water mol. and three small peptides demonstrate that the MFCC method can give excellent ab initio interaction energies compared to the exact treatment in which the whole peptides are included in the calcn. The current scheme scales linearly with the at. size of the protein and can be directly applied to calcg. real protein-mol. interaction energies by using fully quantum (ab initio) methods that are otherwise impossible. The success of the current method is expected to have a powerful impact in our prediction of protein interaction energies including, e.g., protein-drug interactions.**87**Gauss, J.; Stanton, J. F. The equilibrium structure of benzene.*J. Phys. Chem. A*2000,*104*, 2865– 2868, DOI: 10.1021/jp994408yGoogle Scholar87https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXhs1Omurg%253D&md5=867664cf8026794f7dd76c3c0bea68caThe Equilibrium Structure of BenzeneGauss, Juergen; Stanton, John F.Journal of Physical Chemistry A (2000), 104 (13), 2865-2868CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)The re structure of benzene is revised on the basis of high-level quantum chem. calcns. at the CCSD(T)/cc-pVQZ level as well a reanal. of the exptl. rotational consts. using computed vibrational corrections. A least-squares fit to empirically detd. Be consts. yields re(CC) = 1.3914 ± 0.0010 Å and re(CH) = 1.0802 ± 0.0020 Å; the latter distance is significantly shorter than the best previous est. based on exptl. data. Comparison of computed rg and rz distances with expt. as well as considerations of bond lengthening due to anharmonicity are consistent with the estd. re distance, indicating that the recommended structural parameters are very accurate.**88**Helgaker, T.; Klopper, W.; Koch, H.; Noga, J. Basis-set convergence of correlated calculations on water.*J. Chem. Phys.*1997,*106*, 9639– 9646, DOI: 10.1063/1.473863Google Scholar88https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXjvVCgu78%253D&md5=f4689c1b38fe30eb721e9cd7d607bdf7Basis-set convergence of correlated calculations on waterHelgaker, Trygve; Klopper, Wim; Koch, Henrik; Noga, JozefJournal of Chemical Physics (1997), 106 (23), 9639-9646CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The basis-set convergence of the electronic correlation energy in the water mol. is investigated at the second-order Moller-Plesset level and at the coupled-cluster singles-and-doubles level with and without perturbative triples corrections applied. The basis-set limits of the correlation energy are established to within 2mEh by means of (1) extrapolations from sequences of calcns. using correlation-consistent basis sets and (2) from explicitly correlated calcns. employing terms linear in the inter-electronic distances rij. For the extrapolations to the basis-set limit of the correlation energies, fits of the form a + bX-3 (where X is two for double-zeta sets, three for triple-zeta sets, etc.) are found to be useful. CCSD(T) calcns. involving as many as 492 AOs are reported.**89**Halkier, A.; Helgaker, T.; Jørgensen, P.; Klopper, W.; Koch, H.; Olsen, J.; Wilson, A. K. Basis-set convergence in correlated calculations on Ne, N2, and H2O.*Chem. Phys. Lett.*1998,*286*, 243– 252, DOI: 10.1016/s0009-2614(98)00111-0Google Scholar89https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXitVGqsLo%253D&md5=04274821d9c7fa664e9588855ed9a061Basis-set convergence in correlated calculations on Ne, N2, and H2OHalkier, Asger; Helgaker, Trygve; Jorgensen, Poul; Klopper, Wim; Koch, Henrik; Olsen, Jeppe; Wilson, Angela K.Chemical Physics Letters (1998), 286 (3,4), 243-252CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)Valence and all-electron correlation energies of Ne, N2, and H2O at fixed exptl. geometries are computed at the levels of second-order perturbation theory (MP2) and coupled cluster theory with singles and doubles excitations (CCSD), and singles and doubles excitations with a perturbative triples correction (CCSD(T)). Correlation-consistent polarized valence and core-valence basis sets up to sextuple zeta quality are employed. Guided by basis-set limits established by rij-dependent methods, a no. of extrapolation schemes for use with the correlation-consistent basis sets are investigated. Among the schemes considered here, a linear least-squares procedure applied to the quintuple and sextuple zeta results yields the most accurate extrapolations.**90**Russ, N. J.; Crawford, T. D. Potential energy surface discontinuities in local correlation methods.*J. Chem. Phys.*2004,*121*, 691– 696, DOI: 10.1063/1.1759322Google Scholar90https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXlt1Srs74%253D&md5=75329cc988b46c6dd08086372d0b2627Potential energy surface discontinuities in local correlation methodsRuss, Nicholas J.; Crawford, T. DanielJournal of Chemical Physics (2004), 121 (2), 691-696CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We have examd. the occurrence of discontinuities in bond-breaking potential energy surfaces given by local correlation methods based on the Pulay-Saebo orbital domain approach. Our anal. focuses on three prototypical dissocg. systems: the C-F bond in fluoromethane, the C-C bond in singlet, ketene, and the central C-C bond in propadienone. We find that such discontinuities do not occur in cases of homolytic bond cleavage due to the inability of the Pipek-Mezey orbital localization method to sep. singlet-coupled charges on distant fragments. However, for heterolytic bond cleavage, such as that obsd. in singlet ketene and propadienone, discontinuities occur both at stretched geometries and near equil. These discontinuities are usually small, but may be of the same order of magnitude as the localization error in some cases.**91**Giovannini, T.; Lafiosca, P.; Cappelli, C. A General Route to Include Pauli Repulsion and Quantum Dispersion Effects in QM/MM Approaches.*J. Chem. Theory Comput.*2017,*13*, 4854– 4870, DOI: 10.1021/acs.jctc.7b00776Google Scholar91https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhsVyis7nP&md5=2d275dbd740d8d79d4e8b349c5e0f932A General Route to Include Pauli Repulsion and Quantum Dispersion Effects in QM/MM ApproachesGiovannini, Tommaso; Lafiosca, Piero; Cappelli, ChiaraJournal of Chemical Theory and Computation (2017), 13 (10), 4854-4870CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A methodol. to account for non-electrostatic interactions in Quantum Mech. (QM)/Mol. Mechanics (MM) approaches is developed. Formulations for Pauli repulsion and dispersion energy, explicitly depending on the QM d., are derived. Such expressions are based on the definition of an auxiliary d. on the MM portion and the Tkatchenko-Scheffler (TS) approach, resp. The developed method is general enough to be applied to any QM/MM method and partition, provided an accurate tuning of a small no. of parameters is obtained. The coupling of the method with both nonpolarizable and fully polarizable QM/fluctuating charge (FQ) approaches is reported and applied. A suitable parametrization for the aq. soln., so that its most representative features are well reproduced, is outlined. Then, the obtained parametrization and method are applied to calc. the non-electrostatic (repulsion and dispersion) interaction energy of nicotine in aq. soln.**92**Giovannini, T.; Lafiosca, P.; Chandramouli, B.; Barone, V.; Cappelli, C. Effective yet Reliable Computation of Hyperfine Coupling Constants in Solution by a QM/MM Approach: Interplay Between Electrostatics and Non-electrostatic Effects.*J. Chem. Phys.*2019,*150*, 124102, DOI: 10.1063/1.5080810Google Scholar92https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXmtVClt7k%253D&md5=35f2df3b7518c3ba66567633728a9a44Effective yet reliable computation of hyperfine coupling constants in solution by a QM/MM approach: Interplay between electrostatics and non-electrostatic effectsGiovannini, Tommaso; Lafiosca, Piero; Chandramouli, Balasubramanian; Barone, Vincenzo; Cappelli, ChiaraJournal of Chemical Physics (2019), 150 (12), 124102/1-124102/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In this paper, we have extended to the calcn. of hyperfine coupling consts., the model recently proposed by some of the present authors [Giovannini et al., J. Chem. Theory Comput. 13, 4854-4870 (2017)] to include Pauli repulsion and dispersion effects in Quantum Mech./Mol. Mechanics (QM/MM) approaches. The peculiarity of the proposed approach stands in the fact that repulsion/dispersion contributions are explicitly introduced in the QM Hamiltonian. Therefore, such terms not only enter the evaluation of energetic properties but also propagate to mol. properties and spectra. A novel parametrization of the electrostatic fluctuating charge force field has been developed, thus allowing a quant. reprodn. of ref. QM interaction energies. Such a parametrization has been then tested against the prediction of EPR parameters of prototypical nitroxide radicals in aq. solns. (c) 2019 American Institute of Physics.**93**Høyvik, I.-M.; Myhre, R. H.; Koch, H. Correlated natural transition orbitals for core excitation energies in multilevel coupled cluster models.*J. Chem. Phys.*2017,*146*, 144109, DOI: 10.1063/1.4979908Google Scholar93https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC1cvotVeguw%253D%253D&md5=11542a30b8d30f0b20b1b9def6012e69Correlated natural transition orbitals for core excitation energies in multilevel coupled cluster modelsHoyvik Ida-Marie; Myhre Rolf Heilemann; Koch HenrikThe Journal of chemical physics (2017), 146 (14), 144109 ISSN:.In this article, we present a black-box approach for the selection of orbital spaces when computing core excitation energies in the multilevel coupled cluster (MLCC) framework. Information available from the lower level of theory is used to generate correlated natural transition orbitals (CNTOs) for the high-level calculation by including both singles and doubles information in the construction of the transition orbitals. The inclusion of the doubles excitation information is essential to obtain a set of orbitals that all contain physical information, in contrast to the natural transition orbitals where only a small subset of the virtual orbitals contains physical information. The CNTOs may be included in an active space based on a cutoff threshold for the eigenvalues corresponding to the orbitals. We present MLCC results for core excitation energies calculated using coupled cluster singles and doubles (CCSD) in the inactive space and CCSD with perturbative triples (CC3) in the active space. The use of CNTOs results in small errors compared to full CC3.**94**Giovannini, T.; Ambrosetti, M.; Cappelli, C. Quantum Confinement Effects on Solvatochromic Shifts of Molecular Solutes.*J. Phys. Chem. Lett.*2019,*10*, 5823– 5829, DOI: 10.1021/acs.jpclett.9b02318Google Scholar94https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhslOqu7zN&md5=8b800363ebc75b80483b7334d58e6107Quantum Confinement Effects on Solvatochromic Shifts of Molecular SolutesGiovannini, Tommaso; Ambrosetti, Matteo; Cappelli, ChiaraJournal of Physical Chemistry Letters (2019), 10 (19), 5823-5829CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)We demonstrate the pivotal role of quantum mechanics d. confinement effects on solvatochromic shifts. In particular, by resorting to a quantum mechanics/mol. mechanics (QM/MM) approach capable of accounting for confinement effects we successfully reproduce vacuo-to-water solvatochromic shifts for dark n → π* and bright π → π* transitions of acrolein and dark n → π* transitions of pyridine and pyrimidine without the need of including explicit water mols. in the QM portion. Remarkably, our approach is also able to dissect the effects of the single forces acting on the solute-solvent couple and allows for a rationalization of the exptl. findings in terms of physicochem. quantities.**95**Su, P.; Li, H. Energy decomposition analysis of covalent bonds and intermolecular interactions.*J. Chem. Phys.*2009,*131*, 014102, DOI: 10.1063/1.3159673Google Scholar95https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXosFGhsbY%253D&md5=d7ee8efa3cbafd0c33b37f42f14b1a9bEnergy decomposition analysis of covalent bonds and intermolecular interactionsSu, Peifeng; Li, HuiJournal of Chemical Physics (2009), 131 (1), 014102/1-014102/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An energy decompn. anal. method is implemented for the anal. of both covalent bonds and intermol. interactions on the basis of single-determinant Hartree-Fock (HF) (restricted closed shell HF, restricted open shell HF, and unrestricted open shell HF) wavefunctions and their d. functional theory analogs. For HF methods, the total interaction energy from a supermol. calcn. is decompd. into electrostatic, exchange, repulsion, and polarization terms. Dispersion energy is obtained from second-order Moller-Plesset perturbation theory and coupled-cluster methods such as CCSD and CCSD(T). Similar to the HF methods, Kohn-Sham d. functional interaction energy is decompd. into electrostatic, exchange, repulsion, polarization, and dispersion terms. Tests on various systems show that this algorithm is simple and robust. insights are provided by the energy decompn. anal. into H2, methane C-H, and ethane C-C covalent bond formation, CH3CH3 internal rotation barrier, water, ammonia, ammonium, and hydrogen fluoride hydrogen bonding, van der Waals interaction, DNA base pair formation, NH3NH3 and NH3CO coordinate bond formation, Cu-ligand interactions, as well as LiF, LiCl, NaF, and NaCl ionic interactions. (c) 2009 American Institute of Physics.**96**Boulanger, E.; Thiel, W. Toward QM/MM simulation of enzymatic reactions with the drude oscillator polarizable force field.*J. Chem. Theory Comput.*2014,*10*, 1795– 1809, DOI: 10.1021/ct401095kGoogle Scholar96https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXktlensbs%253D&md5=aefd8893a08a886db738d0a5d3d5361bToward QM/MM Simulation of Enzymatic Reactions with the Drude Oscillator Polarizable Force FieldBoulanger, Eliot; Thiel, WalterJournal of Chemical Theory and Computation (2014), 10 (4), 1795-1809CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The polarization of the environment can influence the results from hybrid quantum mech./mol. mech. (QM/MM) simulations of enzymic reactions. In this article, we address several tech. aspects in the development of polarizable QM/MM embedding using the Drude Oscillator (DO) force field. We propose a stable and converging update of the DO polarization state for geometry optimizations and a suitable treatment of the QM/MM-DO boundary when the QM and MM regions are sepd. by cutting through a covalent bond. We assess the performance of our approach by computing binding energies and geometries of three selected complexes relevant to biomol. modeling, namely the water trimer, the N-methylacetamide dimer, and the cationic bis(benzene)sodium sandwich complex. Using a recently published MM-DO force field for proteins, we evaluate the effect of MM polarization on the QM/MM energy profiles of the enzymic reactions catalyzed by chorismate mutase and by p-hydroxybenzoate hydroxylase. We find that inclusion of MM polarization affects the computed barriers by about 10%.**97**Senn, H. M.; Thiel, W. QM/MM methods for biomolecular systems.*Angew. Chem., Int. Ed.*2009,*48*, 1198– 1229, DOI: 10.1002/anie.200802019Google Scholar97https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXitFOqs7g%253D&md5=c51da58b0525651c71f9c393a79023beQM/MM methods for biomolecular systemsSenn, Hans Martin; Thiel, WalterAngewandte Chemie, International Edition (2009), 48 (7), 1198-1229CODEN: ACIEF5; ISSN:1433-7851. (Wiley-VCH Verlag GmbH & Co. KGaA)A review. Combined quantum-mechanics/mol.-mechanics (QM/MM) approaches have become the method of choice for modeling reactions in biomol. systems. Quantum-mech. (QM) methods are required for describing chem. reactions and other electronic processes, such as charge transfer or electronic excitation. However, QM methods are restricted to systems of up to a few hundred atoms. However, the size and conformational complexity of biopolymers calls for methods capable of treating up to several 100,000 atoms and allowing for simulations over time scales of tens of nanoseconds. This is achieved by highly efficient, force-field-based mol. mechanics (MM) methods. Thus to model large biomols. the logical approach is to combine the two techniques and, to use a QM method for the chem. active region (e.g., substrates and co-factors in an enzymic reaction) and an MM treatment for the surroundings (e.g., protein and solvent). The resulting schemes are commonly referred to as combined or hybrid QM/MM methods. They enable the modeling of reactive biomol. systems at a reasonable computational effort while providing the necessary accuracy.

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**1**Høyvik, I.-M.; Jørgensen, P. Characterization and generation of local occupied and virtual Hartree–Fock orbitals.*Chem. Rev.*2016,*116*, 3306– 3327, DOI: 10.1021/acs.chemrev.5b004921https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC28nosFaksw%253D%253D&md5=f2c43fdd7bbc4ce8a64fa67a3653b8d5Characterization and Generation of Local Occupied and Virtual Hartree-Fock OrbitalsHoyvik Ida-Marie; Jorgensen PoulChemical reviews (2016), 116 (5), 3306-27 ISSN:.The scope of this review article is to discuss the locality of occupied and virtual orthogonal Hartree-Fock orbitals generated by localization function optimization. Locality is discussed from the stand that an orbital is local if it is confined to a small region in space. Focusing on locality measures that reflects the spatial extent of the bulk of an orbital and the thickness of orbital tails, we discuss, with numerical illustrations, how the locality may be reported for individual orbitals as well as for sets of orbitals. Traditional and more recent orbital localization functions are reviewed, and the locality measures are used to compare the locality of the orbitals generated by the different localization functions, both for occupied and virtual orbitals. Numerical illustrations are given also for large molecular systems and for cases where diffuse functions are included in the atomic orbital basis. In addition, we have included a discussion on the physical and mathematical limitations on orbital locality.**2**Ma, Q.; Werner, H.-J. Explicitly correlated local coupled-cluster methods using pair natural orbitals.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2018,*8*, e1371, DOI: 10.1002/wcms.1371There is no corresponding record for this reference.**3**Edmiston, C.; Ruedenberg, K. Localized atomic and molecular orbitals.*Rev. Mod. Phys.*1963,*35*, 457, DOI: 10.1103/revmodphys.35.4573https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaF2cXjsFKqtg%253D%253D&md5=8a2f5fbe3d160c8371c8e258cec2972aLocalized atomic and molecular orbitalsEdmiston, Clyde; Ruedenberg, KlausReviews of Modern Physics (1963), 35 (3), 457-65CODEN: RMPHAT; ISSN:0034-6861.An exact method is described for finding those M.O.'s which maximize the sum of the orbital self-repulsion energies. These orbitals, called localized M.O.'s, are analyzed, and an application to the construction of localized orbitals from the 1s and the 2s Slater orbitals in O is given.**4**Boughton, J. W.; Pulay, P. Comparison of the boys and Pipek–Mezey localizations in the local correlation approach and automatic virtual basis selection.*J. Comput. Chem.*1993,*14*, 736– 740, DOI: 10.1002/jcc.5401406154https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXlt1Gntbc%253D&md5=7a43c82108697943ba41bb8cd6363d9cComparisons of the Boys and Pipek-Mezey localizations in the local correlation approach and automatic virtual basis selectionBoughton, James W.; Pulay, PeterJournal of Computational Chemistry (1993), 14 (6), 736-40CODEN: JCCHDD; ISSN:0192-8651.The authors' implementation of Pipek-Mezey electron population localization is described. It is compared with other localization schemes, and its use in the framework of the local-correlation method is discussed. For such use, this localization is shown to be clearly superior to the Boys localization method in the case of phys. well-localized systems. The authors' current algorithm for selection of local virtual spaces is also described.**5**Khaliullin, R. Z.; Bell, A. T.; Head-Gordon, M. Analysis of charge transfer effects in molecular complexes based on absolutely localized molecular orbitals.*J. Chem. Phys.*2008,*128*, 184112, DOI: 10.1063/1.29120415https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXmtVCrt74%253D&md5=bb29c1ef7bd4aadcaba0827a04d4f138Analysis of charge transfer effects in molecular complexes based on absolutely localized molecular orbitalsKhaliullin, Rustam Z.; Bell, Alexis T.; Head-Gordon, MartinJournal of Chemical Physics (2008), 128 (18), 184112/1-184112/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A new method based on absolutely localized MOs (ALMOs) is proposed to measure the degree of intermol. electron d. delocalization (charge transfer) in mol. complexes. ALMO charge transfer anal. (CTA) enables sepn. of the forward and backward charge transfer components for each pair of mols. in the system. The key feature of ALMO CTA is that all charge transfer terms have corresponding well defined energetic effects that measure the contribution of the given term to the overall energetic stabilization of the system. To simplify anal. of charge transfer effects, the concept of chem. significant complementary occupied-virtual orbital pairs (COVPs) is introduced. COVPs provide a simple description of intermol. electron transfer effects in terms of just a few localized orbitals. ALMO CTA is applied to understand fundamental aspects of donor-acceptor interactions in borane adducts, synergic bonding in classical and nonclassical metal carbonyls, and multiple intermol. hydrogen bonds in a complex of isocyanuric acid and melamine. These examples show that the ALMO CTA results are generally consistent with the existing conceptual description of intermol. bonding. The results also show that charge transfer and the energy lowering due to charge transfer are not proportional to each other, and some interesting differences emerge which are discussed. Addnl., according to ALMO CTA, the amt. of electron d. transferred between mols. is significantly smaller than charge transfer estd. from various population anal. methods. (c) 2008 American Institute of Physics.**6**Khaliullin, R. Z.; Cobar, E. A.; Lochan, R. C.; Bell, A. T.; Head-Gordon, M. Unravelling the origin of intermolecular interactions using absolutely localized molecular orbitals.*J. Phys. Chem. A*2007,*111*, 8753– 8765, DOI: 10.1021/jp073685z6https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXotFGnt7s%253D&md5=d470a6f50fd08e7c6d81e6214a613fb1Unravelling the Origin of Intermolecular Interactions Using Absolutely Localized Molecular OrbitalsKhaliullin, Rustam Z.; Cobar, Erika A.; Lochan, Rohini C.; Bell, Alexis T.; Head-Gordon, MartinJournal of Physical Chemistry A (2007), 111 (36), 8753-8765CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)An energy decompn. anal. (EDA) method is proposed to isolate phys. relevant components of the total intermol. interaction energies such as the contribution from interacting frozen monomer densities, the energy lowering due to polarization of the densities, and the further energy lowering due to charge-transfer effects. This method is conceptually similar to existing EDA methods such as Morokuma anal. but includes several important new features. The first is a fully self-consistent treatment of the energy lowering due to polarization, which is evaluated by a SCF calcn. in which the MO coeffs. are constrained to be block-diagonal (absolutely localized) in the interacting mols. to prohibit charge transfer. The second new feature is the ability to sep. forward and back-donation in the charge-transfer energy term using a perturbative approxn. starting from the optimized block-diagonal ref. The newly proposed EDA method is used to understand the fundamental aspects of intermol. interactions such as the degree of covalency in the hydrogen bonding in water and the contributions of forward and back-donation in synergic bonding in metal complexes. Addnl., it is demonstrated that this method can be used to identify the factors controlling the interaction of the mol. hydrogen with open metal centers in potential hydrogen storage materials and the interaction of methane with rhenium complexes.**7**Aquilante, F.; Bondo Pedersen, T.; Sánchez de Merás, A.; Koch, H. Fast noniterative orbital localization for large molecules.*J. Chem. Phys.*2006,*125*, 174101, DOI: 10.1063/1.23602647https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XhtF2htr7F&md5=b0a20bc9f75b8c1a05088911eb5979b5Fast noniterative orbital localization for large moleculesAquilante, Francesco; Bondo Pedersen, Thomas; Sanchez de Meras, Alfredo; Koch, HenrikJournal of Chemical Physics (2006), 125 (17), 174101/1-174101/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We use Cholesky decompn. of the d. matrix in AO basis to define a new set of occupied MO coeffs. Anal. of the resulting orbitals ("Cholesky MOs") demonstrates their localized character inherited from the sparsity of the d. matrix. Comparison with the results of traditional iterative localization schemes shows minor differences with respect to a no. of suitable measures of locality, particularly the scaling with system size of orbital pair domains used in local correlation methods. The Cholesky procedure for generating orthonormal localized orbitals is noniterative and may be made linear scaling. Although our present implementation scales cubically, the algorithm is significantly faster than any of the conventional localization schemes. In addn., since this approach does not require starting orbitals, it will be useful in local correlation treatments on top of diagonalization-free Hartree-Fock optimization algorithms.**8**Høyvik, I.-M.; Jansik, B.; Jørgensen, P. Orbital localization using fourth central moment minimization.*J. Chem. Phys.*2012,*137*, 224114, DOI: 10.1063/1.47698668https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhvVeqs73O&md5=e92a99f3dbecf4d4248dec889b6cad63Orbital localization using fourth central moment minimizationHoyvik, Ida-Marie; Jansik, Branislav; Jorgensen, PoulJournal of Chemical Physics (2012), 137 (22), 224114/1-224114/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a new orbital localization function based on the sum of the fourth central moments of the orbitals. To improve the locality, we impose a power on the fourth central moment to act as a penalty on the least local orbitals. With power two, the occupied and virtual Hartree-Fock orbitals exhibit a more rapid tail decay than orbitals from other localization schemes, making them suitable for use in local correlation methods. We propose that the std. orbital spread (the square root of the second central moment) and fourth moment orbital spread (the fourth root of the fourth central moment) are used as complementary measures to characterize the locality of an orbital, irresp. of localization scheme. (c) 2012 American Institute of Physics.**9**Jansík, B.; Høst, S.; Kristensen, K.; Jørgensen, P. Local orbitals by minimizing powers of the orbital variance.*J. Chem. Phys.*2011,*134*, 194104, DOI: 10.1063/1.35903619https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXmtl2rtbs%253D&md5=a74f01980b9c752491296842255513c8Local orbitals by minimizing powers of the orbital varianceJansik, Branislav; Host, Stinne; Kristensen, Kasper; Jorgensen, PoulJournal of Chemical Physics (2011), 134 (19), 194104/1-194104/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)It is demonstrated that a set of local orthonormal Hartree-Fock (HF) MOs can be obtained for both the occupied and virtual orbital spaces by minimizing powers of the orbital variance using the trust-region algorithm. For a power exponent equal to one, the Boys localization function is obtained. For increasing power exponents, the penalty for delocalized orbitals is increased and smaller max. orbital spreads are encountered. Calcns. on superbenzene, C60, and a fragment of the titin protein show that for a power exponent equal to one, delocalized outlier orbitals may be encountered. These disappear when the exponent is larger than one. For a small penalty, the occupied orbitals are more local than the virtual ones. When the penalty is increased, the locality of the occupied and virtual orbitals becomes similar. In fact, when increasing the cardinal no. for Dunning's correlation consistent basis sets, it is seen that for larger penalties, the virtual orbitals become more local than the occupied ones. We also show that the local virtual HF orbitals are significantly more local than the redundant projected AOs, which often have been used to span the virtual orbital space in local correlated wave function calcns. Our local MOs thus appear to be a good candidate for local correlation methods. (c) 2011 American Institute of Physics.**10**Høyvik, I.-M.; Kristensen, K.; Kjærgaard, T.; Jørgensen, P.*Thom H. Dunning, Jr.*; Springer, 2015; pp 287– 296.There is no corresponding record for this reference.**11**Høyvik, I.-M.; Jansik, B.; Jørgensen, P. Trust region minimization of orbital localization functions.*J. Chem. Theory Comput.*2012,*8*, 3137– 3146, DOI: 10.1021/ct300473g11https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC28rgs1Kmuw%253D%253D&md5=772fa2fd6fa818647a51c36c05a7241dTrust Region Minimization of Orbital Localization FunctionsHoyvik Ida-Marie; Jansik Branislav; Jorgensen PoulJournal of chemical theory and computation (2012), 8 (9), 3137-46 ISSN:1549-9618.The trust region method has been applied to the minimization of localization functions, and it is shown that both local occupied and local virtual Hartree-Fock (HF) orbitals can be obtained. Because step sizes are size extensive in the trust region method, large steps may be required when the method is applied to large molecular systems. For an exponential parametrization of the localization function only small steps are allowed, and the standard trust radius update therefore has been replaced by a scheme where the direction of the step is determined using a conservative estimate of the trust radius and the length of the step is determined from a line search along the obtained direction. Numerical results for large molecular systems have shown that large steps can then safely be taken, and a robust and nearly monotonic convergence is obtained.**12**Ziółkowski, M.; Jansik, B.; Jørgensen, P.; Olsen, J. Maximum locality in occupied and virtual orbital spaces using a least-change strategy.*J. Chem. Phys.*2009,*131*, 124112, DOI: 10.1063/1.323060412https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhtFymsr%252FJ&md5=598c8d47345f5b2e573b4f158aaa6e99Maximum locality in occupied and virtual orbital spaces using a least-change strategyZiolkowski, Marcin; Jansik, Branislav; Joergensen, Poul; Olsen, JeppeJournal of Chemical Physics (2009), 131 (12), 124112/1-124112/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A new strategy is introduced for obtaining localized orthonormal Hartree-Fock (HF) orbitals where the underlying principle is to minimize the size of the transformation matrix from the AO basis to the HF optimized orbital basis. The new strategy gives both localized occupied and localized virtual orbital spaces. The locality of the occupied orbital space is similar to one obtained using std. localization schemes. For the virtual space, std. localization schemes fail to give local orbitals while the new strategy gives a virtual space which has a locality similar to the one of a Loewdin orthonormalization of the AO basis. Since Loewdin orthonormalization gives the most local orthonormal basis functions in the sense that they have the largest similarity with the local at. basis functions, the new strategy thus allows the orthonormal basis to become optimized without introducing significant delocalization. (c) 2009 American Institute of Physics.**13**Gianinetti, E.; Raimondi, M.; Tornaghi, E. Modification of the Roothaan equations to exclude BSSE from molecular interaction calculations.*Int. J. Quantum Chem.*1996,*60*, 157– 166, DOI: 10.1002/(sici)1097-461x(1996)60:1<157::aid-qua17>3.0.co;2-c13https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28Xltlejtro%253D&md5=3ff2d4a5aa9db639ee511c3e0f5e8b61Modification of the Roothaan equations to exclude BSSE from molecular interactions calculationsGianinetti, E.; Raimondi, M.; Tornaghi, E.International Journal of Quantum Chemistry (1996), 60 (1), 157-166CODEN: IJQCB2; ISSN:0020-7608. (Wiley)The Roothaan equations have been modified to compute mol. interactions between weakly bonded systems at the SCF level of theory without the basis set superposition error (BSSE). The increase in complication with respect to the usual SCF algorithm is negligible. Calcn. of the SCF energy on large systems, such as nucleic acid pair, does not pose any computational problem. At the same time, it is shown that a modest change in basis-set quality from 3-21G to 6-31G changes the binding energy by about 50% when computed according to std. SCF "supermol." techniques, while remaining practically const. when computed without introducing BSSE. Bader anal. shows that the amt. of charge transferred between the interacting units is of the same order of magnitude when performed on std. SCF wave functions and those computed using the new method. The large difference between the corresponding computed energies is thus ascribed to the BSSE.**14**Stoll, H.; Wagenblast, G.; Preuβ, H. On the use of local basis sets for localized molecular orbitals.*Theor. Chim. Acta*1980,*57*, 169– 178, DOI: 10.1007/bf0057490314https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3cXmtFSmtLk%253D&md5=d43b312f617f80e6e8e957f2f47113ceOn the use of local basis sets for localized molecular orbitalsStoll, Hermann; Wagenblast, Gerhard; Preuss, HeinzwernerTheoretica Chimica Acta (1980), 57 (2), 169-78CODEN: TCHAAM; ISSN:0040-5744.Two procedures are discussed for the direct variational optimization of localized MO's which are expanded in local subsets of the mol. basis set. A Newton-Raphson approach is more efficient than an iterative diagonalization scheme. The effect of the basis-set truncation on the quality of ab initio SCF results is investigated for Be, Li2, HF, H2O, NH3, CH4 and C2H6.**15**Li, W.; Ni, Z.; Li, S. Cluster-in-molecule local correlation method for post-Hartree–Fock calculations of large systems.*Mol. Phys.*2016,*114*, 1447– 1460, DOI: 10.1080/00268976.2016.113975515https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhvFyhsrs%253D&md5=f4425783d8e50aac3d940428c1727f95Cluster-in-molecule local correlation method for post-Hartree-Fock calculations of large systemsLi, Wei; Ni, Zhigang; Li, ShuhuaMolecular Physics (2016), 114 (9), 1447-1460CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)Our recent developments on cluster-in-mol. (CIM) local correlation method are reviewed in this paper. In the CIM method, the correlation energy of a large system can be approx. obtained from electron correlation calcns. on a series of clusters, each of which contains a subset of occupied and virtual localised MOs in a certain region. The CIM method is a linear scaling method and its inherent parallelisation allows electron correlation calcns. of very large systems to be feasible at ordinary workstations. In the illustrative applications, this approach is applied to investigate the conformational energy differences, reaction barriers, and binding energies of large systems at the levels of Moller-Plesset perturbation theory and coupled-cluster theory.**16**Zhang, X.; Carter, E. A. Subspace Density Matrix Functional Embedding Theory: Theory, Implementation, and Applications to Molecular Systems.*J. Chem. Theory Comput.*2018,*15*, 949– 960, DOI: 10.1021/acs.jctc.8b00990There is no corresponding record for this reference.**17**Govind, N.; Wang, Y. A.; Carter, E. A. Electronic-structure calculations by first-principles density-based embedding of explicitly correlated systems.*J. Chem. Phys.*1999,*110*, 7677– 7688, DOI: 10.1063/1.47867917https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXit1yhs7c%253D&md5=19399936ef64e98248dc5ba77fc7bb9dElectronic-structure calculations by first-principles density-based embedding of explicitly correlated systemsGovind, Niranjan; Wang, Yan Alexander; Carter, Emily A.Journal of Chemical Physics (1999), 110 (16), 7677-7688CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A first-principles embedding theory that combines the salient features of d. functional theory (DFT) and traditional quantum chem. methods is presented. The method involves constructing a DFT-based embedding potential and then using it as a one-electron operator within a very accurate ab initio calcn. We demonstrate how DFT calcns. can be systematically improved via this procedure. The scheme is tested using two closed shell systems, a toy model Li2Mg2, and the exptl. well characterized CO/Cu(111) system. Our results are in good agreement with near full CI calcns. in the former case and exptl. adsorbate binding energies in the latter. This method provides the means to systematically include electron correlation in a local region of a condensed phase.**18**Yu, K.; Carter, E. A. Extending density functional embedding theory for covalently bonded systems.*Proc. Natl. Acad. Sci. U.S.A.*2017,*114*, E10861– E10870, DOI: 10.1073/pnas.171261111418https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhvFWgsrnK&md5=bc56a71c5f634ea5ab305ee6bfd17415Extending density functional embedding theory for covalently bonded systemsYu, Kuang; Carter, Emily A.Proceedings of the National Academy of Sciences of the United States of America (2017), 114 (51), E10861-E10870CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)Quantum embedding theory aims to provide an efficient soln. to obtain accurate electronic energies for systems too large for full-scale, high-level quantum calcns. It adopts a hierarchical approach that divides the total system into a small embedded region and a larger environment, using different levels of theory to describe each part. Previously, we developed a d.-based quantum embedding theory called d. functional embedding theory (DFET), which achieved considerable success in metals and semiconductors. In this work, we extend DFET into a d.-matrix-based nonlocal form, enabling DFET to study the stronger quantum couplings between covalently bonded subsystems. We name this theory d.-matrix functional embedding theory (DMFET), and we demonstrate its performance in several test examples that resemble various real applications in both chem. and biochem. DMFET gives excellent results in all cases tested thus far, including predicting isomerization energies, proton transfer energies, and HOMO-LUMO gaps for local chromophores. Here, we show that DMFET systematically improves the quality of the results compared with the widely used state-of-the-art methods, such as the simple capped cluster model or the widely used ONIOM method.**19**Huang, C.; Pavone, M.; Carter, E. A. Quantum mechanical embedding theory based on a unique embedding potential.*J. Chem. Phys.*2011,*134*, 154110, DOI: 10.1063/1.357751619https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXkvFGgu7Y%253D&md5=93874a27b3413cf28c74fed290d71250Quantum mechanical embedding theory based on a unique embedding potentialHuang, Chen; Pavone, Michele; Carter, Emily A.Journal of Chemical Physics (2011), 134 (15), 154110/1-154110/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We remove the nonuniqueness of the embedding potential that exists in most previous quantum mech. embedding schemes by letting the environment and embedded region share a common embedding (interaction) potential. To efficiently solve for the embedding potential, an optimized effective potential method is derived. This embedding potential, which eschews use of approx. kinetic energy d. functionals, is then used to describe the environment while a correlated wavefunction (CW) treatment of the embedded region is employed. We first demonstrate the accuracy of this new embedded CW (ECW) method by calcg. the van der Waals binding energy curve between a hydrogen mol. and a hydrogen chain. We then examine the prototypical adsorption of CO on a metal surface, here the Cu(111) surface. In addn. to obtaining proper site ordering (top site most stable) and binding energies within this theory, the ECW exhibits dramatic changes in the p-character of the CO 4σ and 5σ orbitals upon adsorption that agree very well with x-ray emission spectra, providing further validation of the theory. Finally, we generalize our embedding theory to spin-polarized quantum systems and discuss the connection between our theory and partition d. functional theory. (c) 2011 American Institute of Physics.**20**Libisch, F.; Huang, C.; Carter, E. A. Embedded correlated wavefunction schemes: Theory and applications.*Acc. Chem. Res.*2014,*47*, 2768– 2775, DOI: 10.1021/ar500086h20https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXptVahu7c%253D&md5=c38754b2218f3584b29849f2d7c4bb1dEmbedded Correlated Wavefunction Schemes: Theory and ApplicationsLibisch, Florian; Huang, Chen; Carter, Emily A.Accounts of Chemical Research (2014), 47 (9), 2768-2775CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)A review. Ab initio modeling of matter has become a pillar of chem. research: with ever-increasing computational power, simulations can be used to accurately predict, for example, chem. reaction rates, electronic and mech. properties of materials, and dynamical properties of liqs. Many competing quantum mech. methods have been developed over the years that vary in computational cost, accuracy, and scalability: d. functional theory (DFT), the workhorse of solid-state electronic structure calcns., features a good compromise between accuracy and speed. However, approx. exchange-correlation functionals limit DFT's ability to treat certain phenomena or states of matter, such as charge-transfer processes or strongly correlated materials. Furthermore, conventional DFT is purely a ground-state theory: electronic excitations are beyond its scope. Excitations in mols. are routinely calcd. using time-dependent DFT linear response; however applications to condensed matter are still limited. By contrast, many-electron wavefunction methods aim for a very accurate treatment of electronic exchange and correlation. Unfortunately, the assocd. computational cost renders treatment of more than a handful of heavy atoms challenging. On the other side of the accuracy spectrum, parametrized approaches like tight-binding can treat millions of atoms. In view of the different (dis-)advantages of each method, the simulation of complex systems seems to force a compromise: one is limited to the most accurate method that can still handle the problem size. For many interesting problems, however, compromise proves insufficient. A possible soln. is to break up the system into manageable subsystems that may be treated by different computational methods. The interaction between subsystems may be handled by an embedding formalism. In this Account, we review embedded correlated wavefunction (CW) approaches and some applications. We first discuss our d. functional embedding theory, which is formally exact. We show how to det. the embedding potential, which replaces the interaction between subsystems, at the DFT level. CW calcns. are performed using a fixed embedding potential, i.e., a non-self-consistent embedding scheme. We demonstrate this embedding theory for two challenging electron transfer phenomena: (1) initial oxidn. of an aluminum surface and (2) hot-electron-mediated dissocn. of hydrogen mols. on a gold surface. In both cases, the interaction between gas mols. and metal surfaces were treated by sophisticated CW techniques, with the remainder of the extended metal surface being treated by DFT. Our embedding approach overcomes the limitations of conventional Kohn-Sham DFT in describing charge transfer, multiconfigurational character, and excited states. From these embedding simulations, we gained important insights into fundamental processes that are crucial aspects of fuel cell catalysis (i.e., O2 redn. at metal surfaces) and plasmon-mediated photocatalysis by metal nanoparticles. Moreover, our findings agree very well with exptl. observations, while offering new views into the chem. We finally discuss our recently formulated potential-functional embedding theory that provides a seamless, first-principles way to include back-action onto the environment from the embedded region.**21**Knizia, G.; Chan, G. K.-L. Density matrix embedding: A simple alternative to dynamical mean-field theory.*Phys. Rev. Lett.*2012,*109*, 186404, DOI: 10.1103/physrevlett.109.18640421https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhslClsL7J&md5=91a195c35bc9aca0f09f19ce1f6ac795Density matrix embedding: a simple alternative to dynamical mean-field theoryKnizia, Gerald; Chan, Garnet Kin-LicPhysical Review Letters (2012), 109 (18), 186404/1-186404/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)We introduce d. matrix embedding theory (DMET), a quantum embedding theory for computing frequency-independent quantities, such as ground-state properties, of infinite systems. Like dynamical mean-field theory, DMET maps the bulk interacting system to a simpler impurity model and is exact in the noninteracting and at. limits. Unlike dynamical mean-field theory, DMET is formulated in terms of the frequency-independent local d. matrix, rather than the local Green's function. In addn., it features a finite, algebraically constructible bath of only one bath site per impurity site, with no bath discretization error. Frequency independence and the minimal bath make DMET a computationally simple and efficient method. We test the theory in the one-dimensional and two-dimensional Hubbard models at and away from half filling, and we find that compared to benchmark data, total energies, correlation functions, and metal-insulator transitions are well reproduced, at a tiny computational cost.**22**Knizia, G.; Chan, G. K.-L. Density matrix embedding: A strong-coupling quantum embedding theory.*J. Chem. Theory Comput.*2013,*9*, 1428– 1432, DOI: 10.1021/ct301044e22https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXivVyitr8%253D&md5=8b6d527b333cf4e4f06fb18181fe30f3Density Matrix Embedding: A Strong-Coupling Quantum Embedding TheoryKnizia, Gerald; Chan, Garnet Kin-LicJournal of Chemical Theory and Computation (2013), 9 (3), 1428-1432CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We extend our d. matrix embedding theory (DMET) [Phys. Rev. Lett.2012, 109, 186404] from lattice models to the full chem. Hamiltonian. DMET allows the many-body embedding of arbitrary fragments of a quantum system, even when such fragments are open systems and strongly coupled to their environment (e.g., by covalent bonds). In DMET, empirical approaches to strong coupling, such as link atoms or boundary regions, are replaced by a small, rigorous quantum bath designed to reproduce the entanglement between a fragment and its environment. We describe the theory and demonstrate its feasibility in strongly correlated hydrogen ring and grid models; these are not only beyond the scope of traditional embeddings but even challenge conventional quantum chem. methods themselves. We find that DMET correctly describes the notoriously difficult sym. dissocn. of a 4 × 3 hydrogen atom grid, even when the treated fragments are as small as single hydrogen atoms. We expect that DMET will open up new ways of treating complex strongly coupled, strongly correlated systems in terms of their individual fragments.**23**Sayfutyarova, E. R.; Sun, Q.; Chan, G. K.-L.; Knizia, G. Automated construction of molecular active spaces from atomic valence orbitals.*J. Chem. Theory Comput.*2017,*13*, 4063– 4078, DOI: 10.1021/acs.jctc.7b0012823https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXht1WmtL7L&md5=2c8d3c8062fa13f4f4e68c6432bb65b1Automated Construction of Molecular Active Spaces from Atomic Valence OrbitalsSayfutyarova, Elvira R.; Sun, Qiming; Chan, Garnet Kin-Lic; Knizia, GeraldJournal of Chemical Theory and Computation (2017), 13 (9), 4063-4078CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We introduce the at. valence active space (AVAS), a simple and well-defined automated technique for constructing active orbital spaces for use in multi-configuration and multi-ref. (MR) electronic structure calcns. Concretely, the technique constructs active MOs capable of describing all relevant electronic configurations emerging from a targeted set of at. valence orbitals (e.g., the metal d orbitals in a redcoordination complex). This is achieved via a linear transformation of the occupied and unoccupied orbital spaces from an easily obtainable single-ref. wavefunction (such as from a Hartree-Fock or Kohn-Sham calcns.) based on projectors to targeted at. valence orbitals. We discuss the premises, theory, and implementation of the idea, and several of its variations are tested. To investigate the performance and accuracy, we calc. the excitation energies for various transition metal complexes in typical application scenarios. Addnl., we follow the homolytic bond breaking process of a Fenton reaction along its reaction coordinate. While the described AVAS technique is not an universal soln. to the active space problem, its premises are fulfilled in many application scenarios of transition metal chem. and bond dissocn. processes. In these cases the technique makes MR calcns. easier to execute, easier to reproduce by any user, and simplifies the detn. of the appropriate size of the active space required for accurate results.**24**Azarias, C.; Russo, R.; Cupellini, L.; Mennucci, B.; Jacquemin, D. Modeling excitation energy transfer in multi-BODIPY architectures.*Phys. Chem. Chem. Phys.*2017,*19*, 6443– 6453, DOI: 10.1039/c7cp00427c24https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhvF2gsr8%253D&md5=605a3d7a2d97ba93b7f20b0fc2337a19Modeling excitation energy transfer in multi-BODIPY architecturesAzarias, Cloe; Russo, Roberto; Cupellini, Lorenzo; Mennucci, Benedetta; Jacquemin, DenisPhysical Chemistry Chemical Physics (2017), 19 (9), 6443-6453CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)The excitation energy transfer (EET) allowing the concn. of the energy has been investigated in several multi-BODIPY architectures with the help of an approach coupling time dependent d. functional theory to an implicit solvation scheme, the polarizable continuum simulation, physicochem. We have first considered several strategies to compute the electronic coupling in a dyad varying the size of the donor/acceptor units, the bridge, the geometries and conformations. We have next studied the electronic coupling in three different architectures for which the EET rate consts. have been exptl. measured both from luminescence and transient absorption data and from Forster intramol. energy transfer :: ditto.**25**Mennucci, B.; Corni, S. Multiscale modelling of photoinduced processes in composite systems.*Nat. Rev. Chem.*2019,*3*, 315– 330, DOI: 10.1038/s41570-019-0092-4There is no corresponding record for this reference.**26**Warshel, A.; Levitt, M. Theoretical studies of enzymic reactions: dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme.*J. Mol. Biol.*1976,*103*, 227– 249, DOI: 10.1016/0022-2836(76)90311-926https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE28XktFKhtr0%253D&md5=f34df33b5971b6b02bd03be95dcd7ba5Theoretical studies of enzymic reactions: dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozymeWarshel, A.; Levitt, M.Journal of Molecular Biology (1976), 103 (2), 227-49CODEN: JMOBAK; ISSN:0022-2836.A general method for detailed study of enzymic reactions is presented. The method considers the complete enzyme-substrate complex together with the surrounding solvent and evaluates all the different quantum mech. and classical energy factors that can affect the reaction pathway. These factors include the quantum mech. energies assocd. with bond cleavage and charge redistribution of the substrate and the classical energies of steric and electrostatic interactions between the substrate and the enzyme. The electrostatic polarization of the enzyme atoms and the orientation of the dipoles of the surrounding H2O mols. is simulated by a microscopic dielec. model. The solvation energy resulting from this polarization is considerable and must be included in any realistic calcn. of chem. reactions involving anything more than an isolated mol. in vacuo. Without it, acidic groups can never become ionized and the charge distribution on the substrate will not be reasonable. The same dielec. model can also be used to study the reaction of the substrate in soln. In this way the reaction in soln. can be compared with the enzymic reaction. The stability of the carbonium ion intermediate formed in the cleavage of a glycosidic bond by lysozyme was studied. Electrostatic stabilization is an important factor in increasing the rate of the reaction step that leads to the formation of the carbonium ion intermediate. Steric factors, such as the strain of the substrate on binding to lysozyme, do not seem to contribute significantly.**27**Giovannini, T.; Egidi, F.; Cappelli, C. Molecular spectroscopy of aqueous solutions: a theoretical perspective.*Chem. Soc. Rev.*2020,*49*, 5664– 5677, DOI: 10.1039/c9cs00464e27https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhsFSlsrzJ&md5=b37e9d2ce60827c120a1090f07213c09Molecular spectroscopy of aqueous solutions: a theoretical perspectiveGiovannini, Tommaso; Egidi, Franco; Cappelli, ChiaraChemical Society Reviews (2020), 49 (16), 5664-5677CODEN: CSRVBR; ISSN:0306-0012. (Royal Society of Chemistry)Computational spectroscopy is an invaluable tool to both accurately reproduce the spectra of mol. systems and provide a rationalization for the underlying physics. However, the inherent difficulty to accurately model systems in aq. solns., owing to water's high polarity and ability to form hydrogen bonds, has severely hampered the development of the field. In this tutorial review we present a technique developed and tested in recent years based on a fully atomistic and polarizable classical modeling of water coupled with a quantum mech. description of the solute. Thanks to its unparalleled accuracy and versatility, this method can change the perspective of computational and exptl. chemists alike.**28**Giovannini, T.; Egidi, F.; Cappelli, C. Theory and algorithms for chiroptical properties and spectroscopies of aqueous systems.*Phys. Chem. Chem. Phys.*2020,*22*, 22864– 22879, DOI: 10.1039/d0cp04027d28https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhvFensL%252FM&md5=dfe27b9b9b9a3ada60788b6b69da8ba5Theory and algorithms for chiroptical properties and spectroscopies of aqueous systemsGiovannini, Tommaso; Egidi, Franco; Cappelli, ChiaraPhysical Chemistry Chemical Physics (2020), 22 (40), 22864-22879CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)Chiroptical properties and spectroscopies are valuable tools to study chiral mols. and assign abs. configurations. The spectra that result from chiroptical measurements may be very rich and complex, and hide much of their information content. For this reason, the interplay between expts. and calcns. is esp. useful, provided that all relevant physico-chem. interactions that are present in the exptl. sample are accurately modelled. The inherent difficulty assocd. to the calcn. of chiral signals of systems in aq. solns. requires the development of specific tools, able to account for the peculiarities of water-solute interactions, and esp. its ability to form hydrogen bonds. In this perspective we discuss a multiscale approach, which we have developed and challenged to model the most used chiroptical techniques.**29**Mennucci, B. Polarizable Continuum Model.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2012,*2*, 386– 404, DOI: 10.1002/wcms.108629https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XovVyrsro%253D&md5=dffcb7dcec69845b8e8bbc40692fd1abPolarizable continuum modelMennucci, BenedettaWiley Interdisciplinary Reviews: Computational Molecular Science (2012), 2 (3), 386-404CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)A review. The polarizable continuum model (PCM) is a computational method originally formulated 30 years ago but still today it represents one of the most successful examples among continuum solvation models. Such a success is mainly because of the continuous improvements, both in terms of computational efficiency and generality, made by all the people involved in the PCM project. The result of these efforts is that nowadays, PCM, with all its different variants, is the default choice in many computational codes to couple a quantum-mech. (QM) description of a mol. system with a continuum description of the environment. In this review, a brief presentation of the main methodol. and computational aspects of the method will be given together with an anal. of strengths and crit. issues of its coupling with different QM methods. Finally, some examples of applications will be presented and discussed to show the potentialities of PCM in describing the effects of environments of increasing complexity.**30**Giovannini, T.; Puglisi, A.; Ambrosetti, M.; Cappelli, C. Polarizable QM/MM approach with fluctuating charges and fluctuating dipoles: the QM/FQFμ model.*J. Chem. Theory Comput.*2019,*15*, 2233– 2245, DOI: 10.1021/acs.jctc.8b0114930https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXltFCjsb4%253D&md5=5b19d342033f29a11dd0cbcc703f2b2aPolarizable QM/MM Approach with Fluctuating Charges and Fluctuating Dipoles: The QM/FQFμ ModelGiovannini, Tommaso; Puglisi, Alessandra; Ambrosetti, Matteo; Cappelli, ChiaraJournal of Chemical Theory and Computation (2019), 15 (4), 2233-2245CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The novel polarizable FQFμ force field is proposed and coupled to a quantum mech. (QM) SCF Hamiltonian. The peculiarity of the resulting QM/FQFμ approach stands in the fact the polarization effects are modeled in terms of both fluctuating charges and dipoles, which vary as a response to the external elec. field/potential. Remarkably, QM/FQFμ is defined in terms of three parameters: electronegativity and chem. hardness, which are well-defined in d. functional theory, and polarizability, which is phys. observable. Such parameters are numerically adjusted to reproduce full QM ref. electrostatic energy values. The model is challenged against test mol. systems in aq. soln., showing remarkable accuracy and thus highlighting its potentialities for future extensive applications.**31**Gordon, M. S.; Fedorov, D. G.; Pruitt, S. R.; Slipchenko, L. V. Fragmentation methods: A route to accurate calculations on large systems.*Chem. Rev.*2012,*112*, 632– 672, DOI: 10.1021/cr200093j31https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhtVyhurjJ&md5=7fa407f4c831f6c15c23d76fde206ba0Fragmentation Methods: A Route to Accurate Calculations on Large SystemsGordon, Mark S.; Fedorov, Dmitri G.; Pruitt, Spencer R.; Slipchenko, Lyudmila V.Chemical Reviews (Washington, DC, United States) (2012), 112 (1), 632-672CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)A review including the following topics: methodologies, software and parallel computing, applications, and conclusions and prognosis.**32**Collins, M. A.; Bettens, R. P. A. Energy-based molecular fragmentation methods.*Chem. Rev.*2015,*115*, 5607– 5642, DOI: 10.1021/cr500455b32https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXmtVajsbY%253D&md5=a83ac50604af530b6d89c94f1a6b6df6Energy-Based Molecular Fragmentation MethodsCollins, Michael A.; Bettens, Ryan P. A.Chemical Reviews (Washington, DC, United States) (2015), 115 (12), 5607-5642CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)A review including the following topics: methods and principles, applications and examples, and speculations and future developments etc.**33**Pruitt, S. R.; Bertoni, C.; Brorsen, K. R.; Gordon, M. S. Efficient and accurate fragmentation methods.*Acc. Chem. Res.*2014,*47*, 2786– 2794, DOI: 10.1021/ar500097m33https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXns1Cmu7k%253D&md5=e6b7179e37a2fb45d912348377ccaa4fEfficient and Accurate Fragmentation MethodsPruitt, Spencer R.; Bertoni, Colleen; Brorsen, Kurt R.; Gordon, Mark S.Accounts of Chemical Research (2014), 47 (9), 2786-2794CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)A review. Three novel fragmentation methods that are available in the electronic structure program GAMESS (general at. and mol. electronic structure system) are discussed in this Account. The fragment MO (FMO) method can be combined with any electronic structure method to perform accurate calcns. on large mol. species with no reliance on capping atoms or empirical parameters. The FMO method is highly scalable and can take advantage of massively parallel computer systems. For example, the method has been shown to scale nearly linearly on up to 131 000 processor cores for calcns. on large water clusters. There have been many applications of the FMO method to large mol. clusters, to biomols. (e.g., proteins), and to materials that are used as heterogeneous catalysts. The effective fragment potential (EFP) method is a model potential approach that is fully derived from first principles and has no empirically fitted parameters. Consequently, an EFP can be generated for any mol. by a simple preparatory GAMESS calcn. The EFP method provides accurate descriptions of all types of intermol. interactions, including Coulombic interactions, polarization/induction, exchange repulsion, dispersion, and charge transfer. The EFP method has been applied successfully to the study of liq. water, π-stacking in substituted benzenes and in DNA base pairs, solvent effects on pos. and neg. ions, electronic spectra and dynamics, non-adiabatic phenomena in electronic excited states, and nonlinear excited state properties. The effective fragment MO (EFMO) method is a merger of the FMO and EFP methods, in which interfragment interactions are described by the EFP potential, rather than the less accurate electrostatic potential. The use of EFP in this manner facilitates the use of a smaller value for the distance cut-off (Rcut). Rcut dets. the distance at which EFP interactions replace fully quantum mech. calcns. on fragment-fragment (dimer) interactions. The EFMO method is both more accurate and more computationally efficient than the most commonly used FMO implementation (FMO2), in which all dimers are explicitly included in the calcn. While the FMO2 method itself does not incorporate three-body interactions, such interactions are included in the EFMO method via the EFP self-consistent induction term. Several applications (ranging from clusters to proteins) of the three methods are discussed to demonstrate their efficacy. The EFMO method will be esp. exciting once the analytic gradients have been completed, because this will allow geometry optimizations, the prediction of vibrational spectra, reaction path following, and mol. dynamics simulations using the method.**34**Collins, M. A.; Cvitkovic, M. W.; Bettens, R. P. A. The combined fragmentation and systematic molecular fragmentation methods.*Acc. Chem. Res.*2014,*47*, 2776– 2785, DOI: 10.1021/ar500088d34https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhtVGksLnK&md5=3d4318bae1bf7fdc64592326da80d823The Combined Fragmentation and Systematic Molecular Fragmentation MethodsCollins, Michael A.; Cvitkovic, Milan W.; Bettens, Ryan P. A.Accounts of Chemical Research (2014), 47 (9), 2776-2785CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)A review. Chem., particularly org. chem., is mostly concerned with functional groups: amines, amides, alcs., ketones, and so forth. This is because the reactivity of mols. can be categorized in terms of the reactions of these functional groups, and by the influence of other adjacent groups in the mol. These simple truths ought to be reflected in the electronic structure and electronic energy of mols., as reactivity is detd. by electronic structure. However, sophisticated ab initio quantum calcns. of the mol. electronic energy usually do not make these truths apparent. In recent years, several computational chem. groups have discovered methods for estg. the electronic energy as a sum of the energies of small mol. fragments, or small sets of groups. By decompg. mols. into such fragments of adjacent functional groups, researchers can est. the electronic energy to chem. accuracy; not just qual. trends, but accurate enough to understand reactivity. In addn., this has the benefit of cutting down on both computational time and cost, as the necessary calcn. time increases rapidly with an increasing no. of electrons. Even with steady advances in computer technol., progress in the study of large mols. is slow. In this Account, we describe two related "fragmentation" methods for treating mols., the combined fragmentation method (CFM) and systematic mol. fragmentation (SMF). In addn., we show how we can use the SMF approach to est. the energy and properties of nonconducting crystals, by fragmenting the periodic crystal structure into relatively small pieces. A large part of this Account is devoted to simple overviews of how the methods work. We also discuss the application of these approaches to calcg. reactivity and other useful properties, such as the NMR and vibrational spectra of mols. and crystals. These applications rely on the ability of these fragmentation methods to accurately est. derivs. of the mol. and crystal energies. Finally, to provide some common applications of CFM and SMF, we present some specific examples of energy calcns. for moderately large mols. For computational chemists, this fragmentation approach represents an important practical advance. It reduces the computer time required to est. the energies of mols. so dramatically, that accurate calcns. of the energies and reactivity of very large org. and biol. mols. become feasible.**35**Pruitt, S. R.; Addicoat, M. A.; Collins, M. A.; Gordon, M. S. The fragment molecular orbital and systematic molecular fragmentation methods applied to water clusters.*Phys. Chem. Chem. Phys.*2012,*14*, 7752– 7764, DOI: 10.1039/c2cp00027j35https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XmvVykurg%253D&md5=f817e5de29f6069d8a560043ba808eeaThe fragment molecular orbital and systematic molecular fragmentation methods applied to water clustersPruitt, Spencer R.; Addicoat, Matthew A.; Collins, Michael A.; Gordon, Mark S.Physical Chemistry Chemical Physics (2012), 14 (21), 7752-7764CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)Two electronic structure methods, the fragment MO (FMO) and systematic mol. fragmentation (SMF) methods, that are based on fragmenting a large mol. system into smaller, more computationally tractable components (fragments), are presented and compared with fully ab initio results for the predicted binding energies of water clusters. It is demonstrated that, even when explicit three-body effects are included (esp. necessary for water clusters due to their complex hydrogen-bonded networks) both methods present viable, computationally efficient alternatives to fully ab initio quantum chem.**36**Khaliullin, R. Z.; Head-Gordon, M.; Bell, A. T. An efficient self-consistent field method for large systems of weakly interacting components.*J. Chem. Phys.*2006,*124*, 204105, DOI: 10.1063/1.219150036https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28Xls1amsrg%253D&md5=c9b37664a7993434200f464d93ee752bAn efficient self-consistent field method for large systems of weakly interacting componentsKhaliullin, Rustam Z.; Head-Gordon, Martin; Bell, Alexis T.Journal of Chemical Physics (2006), 124 (20), 204105/1-204105/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An efficient method for removing the SCF diagonalization bottleneck is proposed for systems of weakly interacting components. The method is based on the equations of the locally projected SCF for mol. interactions (SCF MI) which utilize absolutely localized nonorthogonal MOs expanded in local subsets of the at. basis set. A generalization of direct inversion in the iterative subspace for nonorthogonal MOs is formulated to increase the rate of convergence of the SCF MI equations. Single Roothaan step perturbative corrections are developed to improve the accuracy of the SCF MI energies. The resulting energies closely reproduce the conventional SCF energy. Extensive test calcns. are performed on water clusters up to several hundred mols. Compared to conventional SCF, speedups of the order of (N/O)2 have been achieved for the diagonalization step, where N is the size of the AO basis, and O is the no. of occupied MOs.**37**Ding, F.; Manby, F. R.; Miller, T. F., III Embedded mean-field theory with block-orthogonalized partitioning.*J. Chem. Theory Comput.*2017,*13*, 1605– 1615, DOI: 10.1021/acs.jctc.6b0106537https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXjsVCgtr8%253D&md5=cb3d6244a2abae04cfcc771dd1abb0c9Embedded Mean-Field Theory with Block-Orthogonalized PartitioningDing, Feizhi; Manby, Frederick R.; Miller, Thomas F.Journal of Chemical Theory and Computation (2017), 13 (4), 1605-1615CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Embedded mean-field theory (EMFT) provides a simple, flexible framework for describing subsystems at different levels of mean-field theory. Subsystems are defined by partitioning a one-particle basis set, with a natural choice being the AO basis. Although generally well behaved, EMFT with AO partitioning can exhibit unphys. collapse of the self-consistent soln. To avoid this issue, we introduce subsystem partitioning of a block-orthogonalized (BO) basis set; this eliminates the unphys. collapse without significantly increasing computational cost. We also investigate a non-self-consistent implementation of EMFT, in which the d. matrix is obtained using BO partitioning and the final energy evaluated using AO partitioning; this d.-cor. EMFT approach is found to yield more accurate energies than BO partitioning while also avoiding issues of the unphys. collapse. Using these refined implementations of EMFT, previously proposed descriptions of the exact-exchange coupling between subsystems are compared: although the EX1 coupling scheme is slightly more accurate than EX0, the small improvement does not merit its substantially greater computational cost.**38**Wen, X.; Graham, D. S.; Chulhai, D. V.; Goodpaster, J. D. Absolutely Localized Projection-Based Embedding for Excited States.*J. Chem. Theory Comput.*2020,*16*, 385– 398, DOI: 10.1021/acs.jctc.9b0095938https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXit1Crs7vL&md5=1fcf757ad15dc04d3f07fcdbfaa0c495Absolutely Localized Projection-Based Embedding for Excited StatesWen, Xuelan; Graham, Daniel S.; Chulhai, Dhabih V.; Goodpaster, Jason D.Journal of Chemical Theory and Computation (2020), 16 (1), 385-398CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a quantum embedding method that allows for the calcn. of local excited states embedded in a Kohn-Sham d. functional theory (DFT) environment. Projection-based quantum embedding methodologies provide a rigorous framework for performing DFT-in-DFT and wave function in DFT (WF-in-DFT) calcns. The use of abs. localization, where the d. of each subsystem is expanded in only the basis functions assocd. with the atoms of that subsystem, provide improved computationally efficiency for WF-in-DFT calcns. by reducing the no. of orbitals in the WF calcn. In this work, we extend absolutely localized projection-based quantum embedding to study localized excited states using EOM-CCSD-in-DFT and TDDFT-in-DFT. The embedding results are highly accurate compared to the corresponding canonical EOM-CCSD and TDDFT results on the full system, with TDDFT-in-DFT frequently more accurate than canonical TDDFT. The abs. localization method is shown to eliminate the spurious low-lying excitation energies for charge transfer states and prevent over delocalization of excited states. Addnl., we attempt to recover the environment response caused by the electronic excitations in the high-level subsystem using different schemes and compare their accuracy. Finally, we apply this method to the calcn. of the excited state energy of green fluorescent protein and show that we systematically converge to the full system results. Here we demonstrate how this method can be useful in understanding excited states, specifically which chem. moieties polarize to the excitation. This work shows absolutely localized projection-based quantum embedding can treat local electronic excitations accurately, and make computationally expensive WF methods applicable to systems beyond current computational limits.**39**Bennie, S. J.; Curchod, B. F. E.; Manby, F. R.; Glowacki, D. R. Pushing the limits of EOM-CCSD with projector-based embedding for excitation energies.*J. Phys. Chem. Lett.*2017,*8*, 5559– 5565, DOI: 10.1021/acs.jpclett.7b0250039https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhslahu73E&md5=2d435b51099d12860b8decee7d0b1a08Pushing the Limits of EOM-CCSD with Projector-Based Embedding for Excitation EnergiesBennie, Simon J.; Curchod, Basile F. E.; Manby, Frederick R.; Glowacki, David R.Journal of Physical Chemistry Letters (2017), 8 (22), 5559-5565CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)The calcn. of accurate excitation energies using ab initio electronic structure methods such as std. equation of motion coupled cluster singles and doubles (EOM-CCSD) has been cost prohibitive for large systems. In this work, we use a simple projector-based embedding scheme to calc. the EOM-CCSD excitation energies of acrolein solvated in water mols. modeled using d. functional theory (DFT). We demonstrate the accuracy of this approach gives excitation energies within 0.01 eV of full EOM-CCSD, but with significantly reduced computational cost. This approach is also shown to be relatively invariant to the choice of functional used in the environment and allows for the description of systems with large nos. of basis functions ( > 1000) to be treated using state-of-the-art wave function methods. The flexibility of embedding to select orbitals to add to the excited-state method provides insights into the origins of the excitations and can reduce artifacts that could arise in traditional linear response time-dependent DFT (LR-TDDFT).**40**Chen, X.; Gao, J. Fragment Exchange Potential for Realizing Pauli Deformation of Inter-Fragment Interactions.*J. Phys. Chem. Lett.*2020,*11*, 4008, DOI: 10.1021/acs.jpclett.0c0093340https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXnsVCgtLc%253D&md5=bd62e3595e459e781d1e3e585ee8cb34Fragment Exchange Potential for Realizing Pauli Deformation of Interfragment InteractionsChen, Xin; Gao, JialiJournal of Physical Chemistry Letters (2020), 11 (10), 4008-4016CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)In fragment-based methods, the lack of explicit short-range exchange interactions between monomers can result in unphys. deformation in charge d. In this study, we describe a fragment exchange potential (XFP) to explicitly account for interfragmental Pauli deformation. In our implementation, a Kohn-Sham exchange potential is adopted along with the Yukawa potential. The method has been validated by comparison of the computed exchange energies using the XFP potential with results obtained from antisymmetrized fragmental orbitals on the S66x8 data set contg. 528 bimol. interactions of equil. and arbitrary geometries. It was also found that it is only necessary to deploy numerical grids on atoms within their van der Waals contacts, significantly reducing the small, albeit extra, computational cost. We anticipate that the XFP presented here may be applied to mol. dynamics simulations of macromols. using a fragment-based quantum mech. potential with improved SCF convergence and computational accuracy.**41**Fertitta, E.; Booth, G. H. Energy-weighted density matrix embedding of open correlated chemical fragments.*J. Chem. Phys.*2019,*151*, 014115, DOI: 10.1063/1.510029041https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhtlahs7fI&md5=ea84185b8268d86273c4e725802c4ab7Energy-weighted density matrix embedding of open correlated chemical fragmentsFertitta, Edoardo; Booth, George H.Journal of Chemical Physics (2019), 151 (1), 014115/1-014115/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a multiscale approach to efficiently embed an ab initio correlated chem. fragment described by its energy-weighted d. matrixes and entangled with a wider mean-field many-electron system. This approach, first presented by Fertitta and Booth [Phys. Rev. B 98, 235132 (2018)], is here extended to account for realistic long-range interactions and broken symmetry states. The scheme allows for a systematically improvable description in the range of correlated fluctuations out of the fragment into the system, via a self-consistent optimization of a coupled auxiliary mean-field system. It is discussed that the method has rigorous limits equiv. to the existing quantum embedding approaches of both dynamical mean-field theory and d. matrix embedding theory, to which this method is compared, and the importance of these correlated fluctuations is demonstrated. We derive a self-consistent local energy functional within the scheme and demonstrate the approach for hydrogen rings, where quant. accuracy is achieved despite only a single atom being explicitly treated. (c) 2019 American Institute of Physics.**42**Manby, F. R.; Stella, M.; Goodpaster, J. D.; Miller, T. F., III A simple, exact density-functional-theory embedding scheme.*J. Chem. Theory Comput.*2012,*8*, 2564– 2568, DOI: 10.1021/ct300544e42https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhtVCmtLrP&md5=07b2276726a58eb3df5b733e0c75580bA Simple, Exact Density-Functional-Theory Embedding SchemeManby, Frederick R.; Stella, Martina; Goodpaster, Jason D.; Miller, Thomas F.Journal of Chemical Theory and Computation (2012), 8 (8), 2564-2568CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)D. functional theory (DFT) provides a formally exact framework for quantum embedding. The appearance of nonadditive kinetic energy contributions in this context poses significant challenges, but using optimized effective potential (OEP) methods, various groups have devised DFT-in-DFT methods that are equiv. to Kohn-Sham (KS) theory on the whole system. This being the case, we note that a very considerable simplification arises from doing KS theory instead. We then describe embedding schemes that enforce Pauli exclusion via a projection technique, completely avoiding numerically demanding OEP calcns. Illustrative applications are presented using DFT-in-DFT, wave-function-in-DFT, and wave-function-in-Hartree-Fock embedding, and using an embedded many-body expansion.**43**Fornace, M. E.; Lee, J.; Miyamoto, K.; Manby, F. R.; Miller, T. F., III Embedded mean-field theory.*J. Chem. Theory Comput.*2015,*11*, 568– 580, DOI: 10.1021/ct501103243https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXmvFSjsA%253D%253D&md5=e8a2a875dbcb5a16e6f7eacbeb93108aEmbedded Mean-Field TheoryFornace, Mark E.; Lee, Joonho; Miyamoto, Kaito; Manby, Frederick R.; Miller, Thomas F.Journal of Chemical Theory and Computation (2015), 11 (2), 568-580CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We introduce embedded mean-field theory (EMFT), an approach that flexibly allows for the embedding of one mean-field theory in another without the need to specify or fix the no. of particles in each subsystem. EMFT is simple, is well-defined without recourse to parameters, and inherits the simple gradient theory of the parent mean-field theories. In this paper, we report extensive benchmarking of EMFT for the case where the subsystems are treated using different levels of Kohn-Sham theory, using PBE or B3LYP/6-31G* in the high-level subsystem and LDA/STO-3G in the low-level subsystem; we also investigate different levels of d. fitting in the two subsystems. Over a wide range of chem. problems, we find EMFT to perform accurately and stably, smoothly converging to the high-level of theory as the active subsystem becomes larger. In most cases, the performance is at least as good as that of ONIOM, but the advantages of EMFT are highlighted by examples that involve partitions across multiple bonds or through arom. systems and by examples that involve more complicated electronic structure. EMFT is simple and parameter free, and based on the tests provided here, it offers an appealing new approach to a multiscale electronic structure.**44**Gordon, M. S.; Smith, Q. A.; Xu, P.; Slipchenko, L. V. Accurate first principles model potentials for intermolecular interactions.*Annu. Rev. Phys. Chem.*2013,*64*, 553– 578, DOI: 10.1146/annurev-physchem-040412-11003144https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXntVCrs7g%253D&md5=883c719d2d6d4a4afd3333a45e7cd5abAccurate first principles model potentials for intermolecular interactionsGordon, Mark S.; Smith, Quentin A.; Xu, Peng; Slipchenko, Lyudmila V.Annual Review of Physical Chemistry (2013), 64 (), 553-578CODEN: ARPLAP; ISSN:0066-426X. (Annual Reviews Inc.)A review. The general effective fragment potential (EFP) method provides model potentials for any mol. that is derived from first principles, with no empirically fitted parameters. The EFP method has been interfaced with most currently used ab initio single-ref. and multireference quantum mechanics (QM) methods, ranging from Hartree-Fock and coupled cluster theory to multireference perturbation theory. The most recent innovations in the EFP model have been to make the computationally expensive charge transfer term much more efficient and to interface the general EFP dispersion and exchange repulsion interactions with QM methods. Following a summary of the method and its implementation in generally available computer programs, these most recent new developments are discussed.**45**Gordon, M. S.; Slipchenko, L.; Li, H.; Jensen, J. H. The effective fragment potential: a general method for predicting intermolecular interactions.*Annu. Rep. Comput. Chem.*2007,*3*, 177– 193, DOI: 10.1016/s1574-1400(07)03010-145https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXkt1Wnsrk%253D&md5=4456c9613581b41f675131118db393b2The effective fragment potential: a general method for predicting intermolecular interactionsGordon, Mark S.; Slipchenko, Lyudmilla; Li, Hui; Jensen, Jan H.Annual Reports in Computational Chemistry (2007), 3 (), 177-193CODEN: ARCCC3; ISSN:1574-1400. (Elsevier B.V.)A review.**46**Sun, Q.; Chan, G. K.-L. Quantum embedding theories.*Acc. Chem. Res.*2016,*49*, 2705– 2712, DOI: 10.1021/acs.accounts.6b0035646https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhslyntrbM&md5=5908130580b088e4ea49c788bda516d8Quantum Embedding TheoriesSun, Qiming; Chan, Garnet Kin-LicAccounts of Chemical Research (2016), 49 (12), 2705-2712CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)A review. In complex systems, it is often the case that the region of interest forms only one part of a much larger system. The idea of joining two different quantum simulations - a high level calcn. on the active region of interest, and a low level calcn. on its environment - formally defines a quantum embedding. While any combination of techniques constitutes an embedding, several rigorous formalisms have emerged that provide for exact feedback between the embedded system and its environment. These three formulations: d. functional embedding, Green's function embedding, and d. matrix embedding, resp., use the single-particle d., single-particle Green's function, and single-particle d. matrix as the quantum variables of interest. Many excellent reviews exist covering these methods individually. However, a unified presentation of the different formalisms is so far lacking. Indeed, the various languages commonly used, functional equations for d. functional embedding, diagrammatics for Green's function embedding, and entanglement arguments for d. matrix embedding, make the three formulations appear vastly different. In this Account, we introduce the basic equations of all three formulations in such a way as to highlight their many common intellectual strands. While we focus primarily on a straightforward theor. perspective, we also give a brief overview of recent applications and possible future developments. The first section starts with d. functional embedding, where we introduce the key embedding potential via the Euler equation. We then discuss recent work concerning the treatment of the nonadditive kinetic potential, before describing mean-field d. functional embedding and wave function in d. functional embedding. We finish the section with extensions to time-dependence and excited states. The second section is devoted to Green's function embedding. Here, we use the Dyson equation to obtain equations that parallel as closely as possible the d. functional embedding equations, with the hybridization playing the role of the embedding potential. Embedding a high-level self-energy within a low-level self-energy is treated analogously to wave function in d. functional embedding. The numerical computation of the high-level self-energy allows us to briefly introduce the bath representation in the quantum impurity problem. We then consider translationally invariant systems to bring in the important dynamical mean-field theory. Recent developments to incorporate screening and long-range interactions are discussed.The third section concerns d. matrix embedding. Here, we first highlight some math. complications assocd. with a simple Euler equation derivation, arising from the open nature of fragments. This motivates the d. matrix embedding theory, where we use the Schmidt decompn. to represent the entanglement through bath orbitals. The resulting impurity plus bath formulation resembles that of dynamical mean-field theory. We discuss the numerical self-consistency assocd. with using a high-level correlated wave function with a mean-field low-level treatment, and connect the resulting numerical inversion to that used in d. functional embedding. We finish with perspectives on the future of all three methods.**47**Chulhai, D. V.; Goodpaster, J. D. Projection-based correlated wave function in density functional theory embedding for periodic systems.*J. Chem. Theory Comput.*2018,*14*, 1928– 1942, DOI: 10.1021/acs.jctc.7b0115447https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXjs1Wjurc%253D&md5=19bfdc5a3700990c68f3fec4bddc35fbProjection-Based Correlated Wave Function in Density Functional Theory Embedding for Periodic SystemsChulhai, Dhabih V.; Goodpaster, Jason D.Journal of Chemical Theory and Computation (2018), 14 (4), 1928-1942CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a level shift projection operator-based embedding method for systems with periodic boundary conditions - where the "active" subsystem can be described using either d. functional theory (DFT) or correlated wave function (WF) methods, and the "environment" is described using DFT. Our method allows for k-point sampling, is shown to be exactly equal to the canonical DFT soln. of the full system under the limit that we use the full system basis to describe each subsystem, and can treat the active subsystem either with periodic boundary conditions - in what we term "periodic-in-periodic" embedding - or as a mol. cluster - in "cluster-in-periodic" embedding. We explore each of these methods, and show that cluster WF-in-periodic DFT embedding can accurately calc. the absorption energy of CO on to a Si(100)-2x1 surface.**48**Chulhai, D. V.; Goodpaster, J. D. Improved accuracy and efficiency in quantum embedding through absolute localization.*J. Chem. Theory Comput.*2017,*13*, 1503– 1508, DOI: 10.1021/acs.jctc.7b0003448https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXjvVWitr4%253D&md5=21d0bbf40f2e5d7efb40dd1d8358e584Improved Accuracy and Efficiency in Quantum Embedding through Absolute LocalizationChulhai, Dhabih V.; Goodpaster, Jason D.Journal of Chemical Theory and Computation (2017), 13 (4), 1503-1508CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Projection-based quantum embedding methodologies provide a framework for performing wave function-in-d. functional theory (WF-in-DFT) calcns. The total WF-in-DFT energy is dependent on the partitioning of the total system and requires similar partitioning in each system for accurate energy differences. To achieve this, we enforce an abs. localization of the WF orbitals to basis functions only assocd. with the WF subsystem. This abs. localization, followed by iterative optimization of the subsystems' orbitals, provides improved energy differences for WF-in-DFT while simultaneously improving the computational efficiency.**49**Goodpaster, J. D.; Barnes, T. A.; Manby, F. R.; Miller, T. F., III Density functional theory embedding for correlated wavefunctions: Improved methods for open-shell systems and transition metal complexes.*J. Chem. Phys.*2012,*137*, 224113, DOI: 10.1063/1.477022649https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhvVeqsrrP&md5=17536904dd5e5e360ccfa9144590782dDensity functional theory embedding for correlated wavefunctions: Improved methods for open-shell systems and transition metal complexesGoodpaster, Jason D.; Barnes, Taylor A.; Manby, Frederick R.; Miller, Thomas F., IIIJournal of Chemical Physics (2012), 137 (22), 224113/1-224113/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)D. functional theory (DFT) embedding provides a formally exact framework for interfacing correlated wave-function theory (WFT) methods with lower-level descriptions of electronic structure. Here, we report techniques to improve the accuracy and stability of WFT-in-DFT embedding calcns. In particular, we develop spin-dependent embedding potentials in both restricted and unrestricted orbital formulations to enable WFT-in-DFT embedding for open-shell systems, and develop an orbital-occupation-freezing technique to improve the convergence of optimized effective potential calcns. that arise in the evaluation of the embedding potential. The new techniques are demonstrated in applications to the van-der-Waals-bound ethylene-propylene dimer and to the hexaaquairon(II) transition-metal cation. Calcn. of the dissocn. curve for the ethylene-propylene dimer reveals that WFT-in-DFT embedding reproduces full CCSD(T) energies to within 0.1 kcal/mol at all distances, eliminating errors in the dispersion interactions due to conventional exchange-correlation (XC) functionals while simultaneously avoiding errors due to subsystem partitioning across covalent bonds. Application of WFT-in-DFT embedding to the calcn. of the low-spin/high-spin splitting energy in the hexaaquairon(II) cation reveals that the majority of the dependence on the DFT XC functional can be eliminated by treating only the single transition-metal atom at the WFT level; furthermore, these calcns. demonstrate the substantial effects of open-shell contributions to the embedding potential, and they suggest that restricted open-shell WFT-in-DFT embedding provides better accuracy than unrestricted open-shell WFT-in-DFT embedding due to the removal of spin contamination. (c) 2012 American Institute of Physics.**50**Goodpaster, J. D.; Barnes, T. A.; Manby, F. R.; Miller, T. F., III Accurate and systematically improvable density functional theory embedding for correlated wavefunctions.*J. Chem. Phys.*2014,*140*, 18A507, DOI: 10.1063/1.4864040There is no corresponding record for this reference.**51**Goodpaster, J. D.; Ananth, N.; Manby, F. R.; Miller, T. F., III Exact nonadditive kinetic potentials for embedded density functional theory.*J. Chem. Phys.*2010,*133*, 084103, DOI: 10.1063/1.347457551https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXhtVKhtrzK&md5=4b907560a876e6f16ae576a3a68013f2Exact nonadditive kinetic potentials for embedded density functional theoryGoodpaster, Jason D.; Ananth, Nandini; Manby, Frederick R.; Miller, Thomas F., IIIJournal of Chemical Physics (2010), 133 (8), 084103/1-084103/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We describe an embedded d. functional theory (DFT) protocol in which the nonadditive kinetic energy component of the embedding potential is treated exactly. At each iteration of the Kohn-Sham equations for constrained electron d., the Zhao-Morrison-Parr constrained search method for constructing Kohn-Sham orbitals is combined with the King-Handy expression for the exact kinetic potential. We use this formally exact embedding protocol to calc. ionization energies for a series of three- and four-electron at. systems, and the results are compared to embedded DFT calcns. that utilize the Thomas-Fermi (TF) and the Thomas-Fermi-von Weisacker approxns. to the kinetic energy functional. These calcns. illustrate the expected breakdown due to the TF approxn. for the nonadditive kinetic potential, with errors of 30%-80% in the calcd. ionization energies; by contrast, the exact protocol is found to be accurate and stable. To significantly improve the convergence of the new protocol, we introduce a d.-based switching function to map between the exact nonadditive kinetic potential and the TF approxn. in the region of the nuclear cusp, and we demonstrate that this approxn. has little effect on the accuracy of the calcd. ionization energies. Finally, we describe possible extensions of the exact protocol to perform accurate embedded DFT calcns. in large systems with strongly overlapping subsystem densities. (c) 2010 American Institute of Physics.**52**Zhang, K.; Ren, S.; Caricato, M. Multi-state QM/QM Extrapolation of UV/Vis Absorption Spectra with Point Charge Embedding.*J. Chem. Theory Comput.*2020,*16*, 4361– 4372, DOI: 10.1021/acs.jctc.0c0033952https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhtVaqsrvJ&md5=ee6a5afce1a5a569cc85154682f31257Multistate QM/QM Extrapolation of UV/Vis Absorption Spectra with Point Charge EmbeddingZhang, Kaihua; Ren, Sijin; Caricato, MarcoJournal of Chemical Theory and Computation (2020), 16 (7), 4361-4372CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The simulation of UV/vis absorption spectra of large chromophores is prohibitively expensive with accurate quantum mech. (QM) methods. Thus, hybrid methods, which treat the core chromophoric region at a high level of theory while the substituent effects are treated with a more computationally efficient method, may provide the best compromise between cost and accuracy. The ONIOM (Our own N-layered Integrated MO mol. Mechanics) method has proved successful at describing ground-state processes. However, for excited states, it suffers from difficulties in matching the correct excited states among the different levels of theory. We devised an approach, based on the ONIOM extrapolation formula, to combine two QM levels of theory to extrapolate entire excitation bands rather than individual states. In this contribution, we extend the same QM/QM hybrid scheme to include polarization effects on the core region through point charge embedding. The charges are computed to reproduce the electrostatic potential of the entire chromophore at the low level of theory, with proper constraints to avoid overpolarization issues at the boundary between layers. We test this approach on a variety of model compds. that show how the multistate QM/QM-embedding scheme is able to accurately reproduce the spectrum of the entire system at the high level of theory better than (i) the bare QM/QM hybrid scheme, (ii) the low-level calcn. on the entire system, and (iii) the high-level calcn. on the core region.**53**Ramos, P.; Papadakis, M.; Pavanello, M. Performance of frozen density embedding for modeling hole transfer reactions.*J. Phys. Chem. B*2015,*119*, 7541– 7557, DOI: 10.1021/jp511275e53https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXlvV2hsLo%253D&md5=b1d39bae8c0024b8a1c3f5d5abf18554Performance of Frozen Density Embedding for Modeling Hole Transfer ReactionsRamos, Pablo; Papadakis, Markos; Pavanello, MicheleJournal of Physical Chemistry B (2015), 119 (24), 7541-7557CODEN: JPCBFK; ISSN:1520-5207. (American Chemical Society)We have carried out a thorough benchmark of the frozen d.-embedding (FDE) method for calcg. hole transfer couplings. We have considered 10 exchange-correlation functionals, 3 nonadditive kinetic energy functionals, and 3 basis sets. Overall, we conclude that with a 7% mean relative unsigned error, the PBE and PW91 functionals coupled with the PW91k nonadditive kinetic energy functional and a TZP basis set constitute the most stable and accurate levels of theory for hole transfer coupling calcns. The FDE-ET method is found to be an excellent tool for computing diabatic couplings for hole transfer reactions.**54**Pavanello, M.; Neugebauer, J. Modelling charge transfer reactions with the frozen density embedding formalism.*J. Chem. Phys.*2011,*135*, 234103, DOI: 10.1063/1.366600554https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhs1eltbbP&md5=4d693d900e148be551dccab8478efe76Modelling charge transfer reactions with the frozen density embedding formalismPavanello, Michele; Neugebauer, JohannesJournal of Chemical Physics (2011), 135 (23), 234103/1-234103/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The frozen d. embedding (FDE) subsystem formulation of d.-functional theory is a useful tool for studying charge transfer reactions. In this work charge-localized, diabatic states are generated directly with FDE and used to calc. electronic couplings of hole transfer reactions in two π-stacked nucleobase dimers of B-DNA: 5'-GG-3' and 5'-GT-3'. The calcns. rely on two assumptions: the two-state model, and a small differential overlap between donor and acceptor subsystem densities. The resulting electronic couplings agree well with benchmark values for those exchange-correlation functionals that contain a high percentage of exact exchange. Instead, when semilocal GGA functionals are used the electronic couplings are grossly overestimated. (c) 2011 American Institute of Physics.**55**Wesolowski, T. A.; Shedge, S.; Zhou, X. Frozen-density embedding strategy for multilevel simulations of electronic structure.*Chem. Rev.*2015,*115*, 5891– 5928, DOI: 10.1021/cr500502v55https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXnsVSqu7s%253D&md5=ff2f073b8c897f887f137326c46ba06dFrozen-Density Embedding Strategy for Multilevel Simulations of Electronic StructureWesolowski, Tomasz A.; Shedge, Sapana; Zhou, XiuwenChemical Reviews (Washington, DC, United States) (2015), 115 (12), 5891-5928CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)A review. The following topics are discussed: Frozen-D. Embedding Theory (FDET); Extensions and Formalisms Related to FDET; Approxns. in FDET for Multilevel Simulations; Numerical Simulations Using Approximated FDET Embedding Potentials.**56**Jacob, C. R.; Neugebauer, J.; Visscher, L. A flexible implementation of frozen-density embedding for use in multilevel simulations.*J. Comput. Chem.*2008,*29*, 1011– 1018, DOI: 10.1002/jcc.2086156https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXks1altLc%253D&md5=675c0696a25bcca330c15a85902a9115Software news and update a flexible implementation of frozen-density embedding for use in multilevel simulationsJacob, Christoph R.; Neugebauer, Johannes; Visscher, LucasJournal of Computational Chemistry (2008), 29 (6), 1011-1018CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)A new implementation of frozen-d. embedding (FDE) in the Amsterdam D. Functional (ADF) program package is presented. FDE is based on a subsystem formulation of d.-functional theory (DFT), in which a large system is assembled from an arbitrary no. of subsystems, which are coupled by an effective embedding potential. The new implementation allows both an optimization of all subsystems as a linear-scaling alternative to a conventional DFT treatment, the calcn. of one active fragment in the presence of a frozen environment, and intermediate setups, in which individual subsystems are fully optimized, partially optimized, or completely frozen. It is shown how this flexible setup can facilitate the application of FDE in multilevel simulations.**57**Wesolowski, T. A.; Warshel, A. Frozen density functional approach for ab initio calculations of solvated molecules.*J. Phys. Chem.*1993,*97*, 8050– 8053, DOI: 10.1021/j100132a04057https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXkvVSlu7o%253D&md5=e671a5774b1afd337d42c7ec460a36e7Frozen density functional approach for ab initio calculations of solvated moleculesWesolowski, Tomasz Adam; Warshel, AriehJournal of Physical Chemistry (1993), 97 (30), 8050-3CODEN: JPCHAX; ISSN:0022-3654.A new d. functional method for treatment of chem. processes in soln. is presented. The method is based on freezing the electron d. of the solvent mols., while solving the eigenvalue problem for the solute Hamiltonian, which includes the effective potential of the solvent mols. The method is developed and examd. in the simple case of one solvent and one solute mol. The results are encouraging and the deviation between the unfrozen and frozen treatments can be attributed to the kinetic energy functional used. The method can be implemented in ab initio calcns. of solvation free energies, following a recent pseudopotential approach [Vaidehi et al., 1992].**58**Sæther, S.; Kjærgaard, T.; Koch, H.; Høyvik, I.-M. Density-Based Multilevel Hartree–Fock Model.*J. Chem. Theory Comput.*2017,*13*, 5282– 5290, DOI: 10.1021/acs.jctc.7b0068958https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhsFGhs7zN&md5=9c3da3ee31d5baf4e6fb2680e53636bcDensity-Based Multilevel Hartree-Fock ModelSaether, Sandra; Kjaergaard, Thomas; Koch, Henrik; Hoeyvik, Ida-MarieJournal of Chemical Theory and Computation (2017), 13 (11), 5282-5290CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We introduce a d.-based multilevel Hartree-Fock (HF) method where the electronic d. is optimized in a given region of the mol. (the active region). Active MOs are generated by a decompn. of a starting guess AO d., whereas the inactive MOs (which constitute the remainder of the d.) are never generated or referenced. The MO formulation allows for a significant dimension redn. by transforming from the AO basis to the active MO basis. All interactions between the inactive and active regions of the mol. are retained, and an exponential parametrization of orbital rotations ensures that the active and inactive d. matrixes sep., and in sum, satisfy the symmetry, trace, and idempotency requirements. Thus, the orbital spaces stay orthogonal, and furthermore, the total d. matrix represents a single Slater determinant. In each iteration, the (level-shifted) Newton equations in the active MO basis are solved to obtain the orbital transformation matrix. The approach is equiv. to variationally optimizing only a subset of the MOs of the total system. In this orbital space partitioning, no bonds are broken and no a priori orbital assignments are carried out. In the limit of including all orbitals in the active space, we obtain an MO d.-based formulation of full HF.**59**Høyvik, I.-M. Convergence acceleration for the multilevel Hartree–Fock model.*Mol. Phys.*2020,*118*, 1626929, DOI: 10.1080/00268976.2019.1626929There is no corresponding record for this reference.**60**Aquilante, F.; Boman, L.; Boström, J.; Koch, H.; Lindh, R.; de Merás, A. S.; Pedersen, T. B.*Linear-Scaling Techniques in Computational Chemistry and Physics*; Springer, 2011; pp 301– 343.There is no corresponding record for this reference.**61**Sánchez de Merás, A. M. J.; Koch, H.; Cuesta, I. G.; Boman, L. Cholesky decomposition-based definition of atomic subsystems in electronic structure calculations.*J. Chem. Phys.*2010,*132*, 204105, DOI: 10.1063/1.343162261https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXmvVajtbg%253D&md5=5306234a00b3081733cd735911c4e4e9Cholesky decomposition-based definition of atomic subsystems in electronic structure calculationsSanchez de Meras, Alfredo M. J.; Koch, Henrik; Cuesta, Inmaculada Garcia; Boman, LinusJournal of Chemical Physics (2010), 132 (20), 204105/1-204105/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Decompg. the Hartree-Fock one-electron d. matrix and a virtual pseudodensity matrix, we obtain an orthogonal set of normalized MOs with local character to be used in post-Hartree-Fock calcns. The applicability of the procedure is illustrated by calcg. CCSD(T) energies and CCSD mol. properties in reduced active spaces. (c) 2010 American Institute of Physics.**62**Pulay, P. Second and third derivatives of variational energy expressions: Application to multiconfigurational self-consistent field wave functions.*J. Chem. Phys.*1983,*78*, 5043– 5051, DOI: 10.1063/1.44537262https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3sXhvVGrtrg%253D&md5=9508b76b43bcd8a98623e0bd0c6dab64Second and third derivatives of variational energy expressions: application to multiconfigurational self-consistent field wave functionsPulay, PeterJournal of Chemical Physics (1983), 78 (8), 5043-51CODEN: JCPSA6; ISSN:0021-9606.General anal. expressions are given for the second and third derivs. of constrained variational energy expressions. Variational energy expressions and odd-order derivs. have a distinct advantage over nonvariational (e.g., perturbative) energy expressions and even-order derivs. In particular, the first-order wave function suffices to det. the derivs. of the variational energy up to third order. The coupled-perturbed MC-SCF equations, obtained from the general results, are equiv., with minor corrections, to the ones very recently presented by Y. Osamura, et al., (1982). Explicit expressions are given for the second and third derivs. of the MC-SCF energy. Computational implementation is briefly discussed.**63**Saebo, S.; Pulay, P. Local treatment of electron correlation.*Annu. Rev. Phys. Chem.*1993,*44*, 213– 236, DOI: 10.1146/annurev.pc.44.100193.00124163https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXhsFSqtg%253D%253D&md5=6f98c42474702e87f679e786681a84a5Local treatment of electron correlationSaebo, Svein; Pulay, PeterAnnual Review of Physical Chemistry (1993), 44 (), 213-36CODEN: ARPLAP; ISSN:0066-426X.A review with 88 refs. The topics include: general strategies, localized MOs, local correlation methods, and problems and future prospects.**64**Culpitt, T.; Brorsen, K. R.; Hammes-Schiffer, S. Communication: Density functional theory embedding with the orthogonality constrained basis set expansion procedure.*J. Chem. Phys.*2017,*146*, 211101, DOI: 10.1063/1.498477764https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXpt1Kqs7w%253D&md5=f9788d5b309079a0dae883a4ae2e784bCommunication: Density functional theory embedding with the orthogonality constrained basis set expansion procedureCulpitt, Tanner; Brorsen, Kurt R.; Hammes-Schiffer, SharonJournal of Chemical Physics (2017), 146 (21), 211101/1-211101/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)D. functional theory (DFT) embedding approaches have generated considerable interest in the field of computational chem. because they enable calcns. on larger systems by treating subsystems at different levels of theory. To circumvent the calcn. of the non-additive kinetic potential, various projector methods have been developed to ensure the orthogonality of MOs between subsystems. Herein the orthogonality constrained basis set expansion (OCBSE) procedure is implemented to enforce this subsystem orbital orthogonality without requiring a level shifting parameter. This scheme is a simple alternative to existing parameter-free projector-based schemes, such as the Huzinaga equation. The main advantage of the OCBSE procedure is that excellent convergence behavior is attained for DFT-in-DFT embedding without freezing any of the subsystem densities. For the three chem. systems studied, the level of accuracy is comparable to or higher than that obtained with the Huzinaga scheme with frozen subsystem densities. Allowing both the high-level and low-level DFT densities to respond to each other during DFT-in-DFT embedding calcns. provides more flexibility and renders this approach more generally applicable to chem. systems. It could also be useful for future extensions to embedding approaches combining wavefunction theories and DFT. (c) 2017 American Institute of Physics.**65**Hégely, B.; Nagy, P. R.; Ferenczy, G. G.; Kállay, M. Exact density functional and wave function embedding schemes based on orbital localization.*J. Chem. Phys.*2016,*145*, 064107, DOI: 10.1063/1.496017765https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xhtlalu7%252FP&md5=4e1dbf60af141a2d3e9e0d10a5ced940Exact density functional and wave function embedding schemes based on orbital localizationHegely, Bence; Nagy, Peter R.; Ferenczy, Gyorgy G.; Kallay, MihalyJournal of Chemical Physics (2016), 145 (6), 064107/1-064107/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Exact schemes for the embedding of d. functional theory (DFT) and wave function theory (WFT) methods into lower-level DFT or WFT approaches are introduced utilizing orbital localization. First, a simple modification of the projector-based embedding scheme of Manby and co-workers [J. Chem. Phys. 140, 18A507 (2014)] is proposed. We also use localized orbitals to partition the system, but instead of augmenting the Fock operator with a somewhat arbitrary level-shift projector we solve the Huzinaga-equation, which strictly enforces the Pauli exclusion principle. Second, the embedding of WFT methods in local correlation approaches is studied. Since the latter methods split up the system into local domains, very simple embedding theories can be defined if the domains of the active subsystem and the environment are treated at a different level. The considered embedding schemes are benchmarked for reaction energies and compared to quantum mechanics (QM)/mol. mechanics (MM) and vacuum embedding. We conclude that for DFT-in-DFT embedding, the Huzinaga-equation-based scheme is more efficient than the other approaches, but QM/MM or even simple vacuum embedding is still competitive in particular cases. Concerning the embedding of wave function methods, the clear winner is the embedding of WFT into low-level local correlation approaches, and WFT-in-DFT embedding can only be more advantageous if a non-hybrid d. functional is employed. (c) 2016 American Institute of Physics.**66**Huzinaga, S.; Cantu, A. A. Theory of separability of many-electron systems.*J. Chem. Phys.*1971,*55*, 5543– 5549, DOI: 10.1063/1.167572066https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE38Xhsl2ruw%253D%253D&md5=57073e274c58fc9dd1649b90d089272cTheory of separability of many-electron systemsHuzinaga, S.; Cantu, A. A.Journal of Chemical Physics (1971), 55 (12), 5543-9CODEN: JCPSA6; ISSN:0021-9606.At. and mol. systems are often intuitively sepd. into almost independent subsystems as, for example, the core and valence parts of an atom. Consequently, if this sepn. provides a good approxn., one can obtain the states of the system from the states of the subsystems which best represent the entire system. In the light of the work of McWeeny, in which one assumes strong orthogonality among subsystem wavefunctions, an effective Hamiltonian is detd. for a given subsystem which should properly describe the states of that subsystem. Previous work has dealt with an improper effective Hamiltonian.**67**Boys, S. F. Construction of some molecular orbitals to be approximately invariant for changes from one molecule to another.*Rev. Mod. Phys.*1960,*32*, 296, DOI: 10.1103/revmodphys.32.29667https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaF3MXht1amuro%253D&md5=dd797e8d0da21499495500b2f3e49496Construction of some molecular orbitals to be approximately invariant for changes from one molecule to anotherBoys, S. F.Reviews of Modern Physics (1960), 32 (), 296-9CODEN: RMPHAT; ISSN:0034-6861.The concept of the invariant orbital is introduced and defined as an orbital for part of a system which remains invariant under chem. changes occurring at some distance. Methods for computing invariant orbitals are outlined.**68**Christiansen, O.; Koch, H.; Jørgensen, P. The second-order approximate coupled cluster singles and doubles model CC2.*Chem. Phys. Lett.*1995,*243*, 409– 418, DOI: 10.1016/0009-2614(95)00841-q68https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXotFCmtL4%253D&md5=1978d2062ade66a8a9a04af8378fa0ddThe second-order approximate coupled cluster singles and doubles model CC2Christiansen, Ove; Koch, Henrik; Jorgensen, PoulChemical Physics Letters (1995), 243 (5,6), 409-18CODEN: CHPLBC; ISSN:0009-2614. (Elsevier)An approx. coupled cluster singles and doubles model is presented, denoted CC2. The CC2 total energy is of second-order Moeller-Plesset perturbation theory (MP2) quality. The CC2 linear response function is derived. Unlike MP2, excitation energies and transition moments can be obtained in CC2. A hierarchy of coupled cluster models, CCS, CC2, CCSD, CC3, CCSDT, etc., is presented where CC2 and CC3 are approx. coupled cluster models defined by similar approxns. Higher levels give increased accuracy at increased computational effort. The scaling of CCS, CC2, CCSD, CC3, and CCSDT is N4, N5, N6, N7, and N8, resp., where N is the no. of orbitals. Calcns. of excitation energies for Be, N2, and C2H4 are performed, and results compared with those obtained with the second-order polarization propagator approach SOPPA.**69**Myhre, R. H.; Koch, H. The multilevel CC3 coupled cluster model.*J. Chem. Phys.*2016,*145*, 044111, DOI: 10.1063/1.495937369https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xht1Kmur7E&md5=6f62a1c36755e3209a5cbbf86d3502c8The multilevel CC3 coupled cluster modelMyhre, Rolf H.; Koch, HenrikJournal of Chemical Physics (2016), 145 (4), 044111/1-044111/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present an efficient implementation of the closed shell multilevel coupled cluster method where coupled cluster singles and doubles (CCSD) is used for the inactive orbital space and CCSD with perturbative triples (CC3) is employed for the smaller active orbital space. Using Cholesky orbitals, the active space can be spatially localized and the computational cost is greatly reduced compared to full CC3 while retaining the accuracy of CC3 excitation energies. For the small org. mols. considered we achieve up to two orders of magnitude redn. in the computational requirements. (c) 2016 American Institute of Physics.**70**Folkestad, S. D.; Koch, H. Multilevel CC2 and CCSD Methods with Correlated Natural Transition Orbitals.*J. Chem. Theory Comput.*2020,*16*, 179, DOI: 10.1021/acs.jctc.9b0070170https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BB3MfgslGluw%253D%253D&md5=e42d0da9ebf9a3346e33474271046907Multilevel CC2 and CCSD Methods with Correlated Natural Transition OrbitalsFolkestad Sarai Dery; Koch Henrik; Koch HenrikJournal of chemical theory and computation (2020), 16 (1), 179-189 ISSN:.In the multilevel coupled cluster approach, an active orbital space is treated at a higher level of coupled cluster theory than the remaining inactive orbitals. We introduce the multilevel CC2 method where CC2 is used for the active orbital space. Furthermore, we present a simplified formulation of the multilevel CCSD method where CCSD is used for the active space. The simplification lies in the evaluation of the CC2 amplitudes in the inactive space; these CC2 amplitudes have previously been determined iteratively. We use correlated natural transition orbitals to determine the active orbital spaces. The convergence of the multilevel CC2 and multilevel CCSD valence excitation energies is established with proof-of-concept calculations. The methods are also applied to two larger systems: p-nitroaniline in water and amoxicillin. The calculations on the p-nitroaniline-water system illustrate the usefulness of multilevel coupled cluster methods for molecules in solution and for charge transfer excitations.**71**Folkestad, S. D.; Kjønstad, E. F.; Myhre, R. H.; Andersen, J. H.; Balbi, A.; Coriani, S.; Giovannini, T.; Goletto, L.; Haugland, T. S.; Hutcheson, A.; Høyvik, I.-M.; Moitra, T.; Paul, A. C.; Scavino, M.; Skeidsvoll, A. S.; Tveten, Å. H.; Koch, H. eT 1.0: An open source electronic structure program with emphasis on coupled cluster and multilevel methods.*J. Chem. Phys.*2020,*152*, 184103, DOI: 10.1063/5.000471371https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXptleitbc%253D&md5=07f57f25437d1255896e90628884dba1eT 1.0: An open source electronic structure program with emphasis on coupled cluster and multilevel methodsFolkestad, Sarai D.; Kjoenstad, Eirik F.; Myhre, Rolf H.; Andersen, Josefine H.; Balbi, Alice; Coriani, Sonia; Giovannini, Tommaso; Goletto, Linda; Haugland, Tor S.; Hutcheson, Anders; Hoeyvik, Ida-Marie; Moitra, Torsha; Paul, Alexander C.; Scavino, Marco; Skeidsvoll, Andreas S.; Tveten, Aasmund H.; Koch, HenrikJournal of Chemical Physics (2020), 152 (18), 184103CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The eT program is an open source electronic structure package with emphasis on coupled cluster and multilevel methods. It includes efficient spin adapted implementations of ground and excited singlet states, as well as equation of motion oscillator strengths, for CCS, CC2, CCSD, and CC3. Furthermore, eT provides unique capabilities such as multilevel Hartree-Fock and multilevel CC2, real-time propagation for CCS and CCSD, and efficient CC3 oscillator strengths. With a coupled cluster code based on an efficient Cholesky decompn. algorithm for the electronic repulsion integrals, eT has similar advantages as codes using d. fitting, but with strict error control. Here, we present the main features of the program and demonstrate its performance through example calcns. Because of its availability, performance, and unique capabilities, we expect eT to become a valuable resource to the electronic structure community. (c) 2020 American Institute of Physics.**72**Lehtola, S. Assessment of initial guesses for self-consistent field calculations. Superposition of atomic potentials: Simple yet efficient.*J. Chem. Theory Comput.*2019,*15*, 1593– 1604, DOI: 10.1021/acs.jctc.8b0108972https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhsFSqt7g%253D&md5=0f04ec8103ff5427d261bcf89b07630bAssessment of Initial Guesses for Self-Consistent Field Calculations. Superposition of Atomic Potentials: Simple yet EfficientLehtola, SusiJournal of Chemical Theory and Computation (2019), 15 (3), 1593-1604CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Electronic structure calcns., such as in the Hartree-Fock or Kohn-Sham d. functional approach, require an initial guess for the MOs. The quality of the initial guess has a significant impact on the speed of convergence of the SCF procedure. Popular choices for the initial guess include the one-electron guess from the core Hamiltonian, the extended Hueckel method, and the superposition of at. densities (SAD). Here, we discuss alternative guesses obtained from the superposition of at. potentials (SAP), which is easily implementable even in real-space calcns. We also discuss a variant of SAD which produces guess orbitals by purifn. of the d. matrix that could also be used in real-space calcns., as well as a parameter-free variant of the extended Hueckel method, which resembles the SAP method and is easy to implement on top of existing SAD infrastructure. The performance of the core Hamiltonian, the SAD and the SAP guesses as well as the extended Hueckel variant is assessed in non-relativistic calcns. on a dataset of 259 mols. ranging from the first to the fourth periods by projecting the guess orbitals onto precomputed, converged SCF solns. in single- to triple-ζ basis sets. It is shown that the proposed SAP guess is the best guess on av. The extended Hueckel guess offers a good alternative, with less scatter in accuracy.**73**Koch, H.; Sánchez de Merás, A.; Pedersen, T. B. Reduced scaling in electronic structure calculations using Cholesky decompositions.*J. Chem. Phys.*2003,*118*, 9481– 9484, DOI: 10.1063/1.157862173https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXjvVGrt74%253D&md5=f01a7505fbdba8e5e85fbd34cff8fdfbReduced scaling in electronic structure calculations using Cholesky decompositionsKoch, Henrik; Sanchez de Meras, Alfredo; Pedersen, Thomas BondoJournal of Chemical Physics (2003), 118 (21), 9481-9484CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We demonstrate that substantial computational savings are attainable in electronic structure calcns. using a Cholesky decompn. of the two-electron integral matrix. In most cases, the computational effort involved calcg. the Cholesky decompn. is less than the construction of one Fock matrix using a direct O(N2) procedure.**74**Christiansen, O.; Manninen, P.; Jørgensen, P.; Olsen, J. Coupled-cluster theory in a projected atomic orbital basis.*J. Chem. Phys.*2006,*124*, 084103, DOI: 10.1063/1.217324974https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28Xis1ags7Y%253D&md5=b5de844b127d028c7c0841ce1ec16771Coupled-cluster theory in a projected atomic orbital basisChristiansen, Ove; Manninen, Pekka; Joergensen, Poul; Olsen, JeppeJournal of Chemical Physics (2006), 124 (8), 084103/1-084103/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a biorthogonal formulation of coupled-cluster (CC) theory using a redundant projected AO (PAO) basis. The biorthogonal formulation provides simple equations, where the projectors involved in the definition of the PAO basis are absorbed in the integrals. Explicit expressions for the coupled-cluster singles and doubles equations are derived in the PAO basis. The PAO CC equations can be written in a form identical to the std. MO CC equations, only with integrals that are related to the AO integrals through different transformation matrixes. The dependence of cluster amplitudes, integrals, and correlation energy contributions on the distance between the participating at. centers and on the no. of involved at. centers is illustrated in numerical case studies. It is also discussed how the present reformulation of the CC equations opens new possibilities for reducing the no. of involved parameters and thereby the computational cost.**75**Myhre, R. H.; Sánchez de Merás, A. M. J.; Koch, H. Multi-level coupled cluster theory.*J. Chem. Phys.*2014,*141*, 224105, DOI: 10.1063/1.490319575https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXitV2mur%252FJ&md5=3c46b9a81bb4e87079c833660f5ab1b0Multi-level coupled cluster theoryMyhre, Rolf H.; Sanchez de Meras, Alfredo M. J.; Koch, HenrikJournal of Chemical Physics (2014), 141 (22), 224105/1-224105/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a general formalism where different levels of coupled cluster theory can be applied to different parts of the mol. system. The system is partitioned into subsystems by Cholesky decompn. of the one-electron Hartree-Fock d. matrix. In this way the system can be divided across chem. bonds without discontinuities arising. The coupled cluster wave function is defined in terms of cluster operators for each part and these are detd. from a set of coupled equations. The total wave function fulfills the Pauli-principle across all borders and levels of electron correlation. We develop the assocd. response theory for this multi-level coupled cluster theory and present proof of principle applications. The formalism is an essential tool in order to obtain size-intensive complexity in the calcn. of local mol. properties. (c) 2014 American Institute of Physics.**76**Egidi, F.; Segado, M.; Koch, H.; Cappelli, C.; Barone, V. A benchmark study of electronic excitation energies, transition moments, and excited-state energy gradients on the nicotine molecule.*J. Chem. Phys.*2014,*141*, 224114, DOI: 10.1063/1.490330776https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXitFahsrvK&md5=6501af06f537c1b5ff920630ecc77615A benchmark study of electronic excitation energies, transition moments, and excited-state energy gradients on the nicotine moleculeEgidi, Franco; Segado, Mireia; Koch, Henrik; Cappelli, Chiara; Barone, VincenzoJournal of Chemical Physics (2014), 141 (22), 224114/1-224114/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In this work, we report a comparative study of computed excitation energies, oscillator strengths, and excited-state energy gradients of (S)-nicotine, chosen as a test case, using multireference methods, coupled cluster singles and doubles, and methods based on time-dependent d. functional theory. This system was chosen because its apparent simplicity hides a complex electronic structure, as several different types of valence excitations are possible, including n-π*, π-π*, and charge-transfer states, and in order to simulate its spectrum it is necessary to describe all of them consistently well by the chosen method. (c) 2014 American Institute of Physics.**77**Marder, S. R.; Beratan, D. N.; Cheng, L.-T. Approaches for optimizing the first electronic hyperpolarizability of conjugated organic molecules.*Science*1991,*252*, 103– 106, DOI: 10.1126/science.252.5002.10377https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3MXitlylt7g%253D&md5=4da3b8c6cf261058f7d419755fb28fd2Approaches for optimizing the first electronic hyperpolarizability of conjugated organic moleculesMarder, S. R.; Beratan, D. N.; Cheng, L. T.Science (Washington, DC, United States) (1991), 252 (5002), 103-6CODEN: SCIEAS; ISSN:0036-8075.A two-state, four-orbital, independent electron anal. of the first optical mol. hyperpolarizability, β, leads to the prediction that |β| maximizes at a combination of donor and acceptor strengths for a given conjugated bridge. Mol. design strategies that focus on the energetic manipulations of the bridge states are proposed for the optimization of β. The limitations of mol. classes based on common bridge structures are highlighted, and more promising candidates are described. Exptl. results supporting the validity of this approach are presented.**78**Grigorenko, A. N.; Polini, M.; Novoselov, K. S. Graphene plasmonics.*Nat. Photonics*2012,*6*, 749, DOI: 10.1038/nphoton.2012.26278https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38Xhs1ajtbzJ&md5=041a227bbcc24796abb7c8cc2a301f84Graphene plasmonicsGrigorenko, A. N.; Polini, M.; Novoselov, K. S.Nature Photonics (2012), 6 (11), 749-758CODEN: NPAHBY; ISSN:1749-4885. (Nature Publishing Group)A review. Two rich and vibrant fields of investigation-graphene physics and plasmonics-strongly overlap. Not only does graphene possess intrinsic plasmons that are tunable and adjustable, but a combination of graphene with noble-metal nanostructures promises a variety of exciting applications for conventional plasmonics. The versatility of graphene means that graphene-based plasmonics may enable the manuf. of novel optical devices working in different frequency ranges-from terahertz to the visible-with extremely high speed, low driving voltage, low power consumption and compact sizes. Here we review the field emerging at the intersection of graphene physics and plasmonics.**79**Gibson, S. E.; Knight, J. D. [2.2] Paracyclophane derivatives in asymmetric catalysis.*Org. Biomol. Chem.*2003,*1*, 1256– 1269, DOI: 10.1039/b300717k79https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXkt1Krsrs%253D&md5=311d3a4932478b32153aaca1b357892a[2.2]paracyclophane derivatives in asymmetric catalysisGibson, Susan E.; Knight, Jamie D.Organic & Biomolecular Chemistry (2003), 1 (8), 1256-1269CODEN: OBCRAK; ISSN:1477-0520. (Royal Society of Chemistry)A review. The growing importance of [2.2]paracyclophane derivs. as planar chiral ligands was highlighted. Comprehensive coverage of the applications of mono- and disubstituted [2.2]paracyclophane derivs. in asym. catalysis was provided. Each section of the review was supplemented with a description of typical approaches used to access classes of cyclophanes under discussion. A review.**80**Gleiter, R.; Hopf, H.*Modern Cyclophane Chemistry*; John Wiley & Sons, 2006.There is no corresponding record for this reference.**81**Grimme, S. On the Importance of Electron Correlation Effects for the π-π Interactions in Cyclophanes.*Chem.—Eur. J.*2004,*10*, 3423– 3429, DOI: 10.1002/chem.20040009181https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXmtFKrtLk%253D&md5=ddd58fd3cc59dd5705bf54731a79f51fOn the importance of electron correlation effects for the π-π interactions in cyclophanesGrimme, StefanChemistry - A European Journal (2004), 10 (14), 3423-3429CODEN: CEUJED; ISSN:0947-6539. (Wiley-VCH Verlag GmbH & Co. KGaA)Correlated ab initio quantum chem. methods based on second-order perturbation theory and d. functional theory (DFT) together with large AO basis sets are used to calc. the structures of four cyclophanes with two arom. rings and one sulfur-contg. phane with one arom. ring. The calcd. geometrical data for [2.2]paracyclophane, cyclophane (superphane), 8,16-dimethyl[2.2]metacyclophane, 16-methyl[2.2]metaparacyclophane, and 2,6,15-trithia[34,10][7]metacyclophane are compared to exptl. data from x-ray crystal structure detns. In all cases, very accurate theor. predictions are obtained from the recently developed spin-component-scaled MP2 (SCS-MP2) method, in which the deviations are within the exptl. accuracy and expected crystal-packing or vibrational effects. Esp. the interring distances, which are detd. by a detailed balance between attractive van der Waals (dispersive) and repulsive (Pauli) contributions, are very sensitive to the level of theory employed. While std. MP2 theory in general overestimates the dispersive interactions (π-π correlations) between the two arom. rings leading to too short distances (between 3 and 8 pm), the opposite is obsd. for DFT methods (errors up to 15 pm). An explicit account of dispersive-type electron correlation effects between the clamped arom. units is essential for a quant. description of cyclophane structures. To distinguish these effects from normal van der Waals interactions, the term overlap-dispersive interaction may be employed. The popular B3LYP hybrid d. functional offers no advantage over the pure PBE functional that at least qual. accounts for some of the dispersive effects. The use of properly polarized AO basis sets of at least valence-triple-ζ quality is strongly recommended to obtain quant. predictions with traditional wave function methods.**82**Demissie, T. B.; Dodziuk, H.; Waluk, J.; Ruud, K.; Pietrzak, M.; Vetokhina, V.; Szymański, S.; Jaźwiński, J.; Hopf, H. Structure, NMR and Electronic Spectra of [m.n]Paracyclophanes with Varying Bridges Lengths (m, n = 2–4).*J. Phys. Chem. A*2016,*120*, 724– 736, DOI: 10.1021/acs.jpca.5b1216882https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XpsFahsw%253D%253D&md5=f914778424b4b40f59da3c494fbcac52Structure, NMR and Electronic Spectra of [m.n]Paracyclophanes with Varying Bridges Lengths (m, n = 2-4)Demissie, Taye B.; Dodziuk, Helena; Waluk, Jacek; Ruud, Kenneth; Pietrzak, Mariusz; Vetokhina, Volha; Szymanski, Slawomir; Jazwinski, Jaroslaw; Hopf, HenningJournal of Physical Chemistry A (2016), 120 (5), 724-736CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)Extending our earlier studies on cyclophanes, we here report the structure, chem. shifts, spin-spin coupling consts., absorption and emission properties of [m.n]paracyclophanes, m, n = 2-4, obtained using a combination of exptl. and computational techniques. Accurate values of proton chem. shifts as well as of JHH for the bridges are detd. The exptl. chem. shifts, coupling consts., absorption and emission wavelengths are satisfactorily reproduced using d. functional theory calcns., using both the B3LYP and ωB97X-D functionals. The geometries predicted using a functional that includes dispersion corrections (ωB97X-D) are in a better agreement with available exptl. values than those obtained using the B3LYP method. Up to 8 UV-vis absorption/emission bands have been obsd. (or anticipated in the region below 200 nm) and assigned on the basis of quantum-chem. calcns. Optimized excited-state geometries showed that the distances between the arom. bridgehead carbon atoms of all the [m.n]paracyclophanes in the excited state decrease compared to the ground-state geometries by ca. 0.2-0.9 Å, the largest being for [4.4]paracyclophane, though the rather large differences in the calcd. emission wavelength compared to expt. cast some doubts on the accuracy of the excited-state geometries.**83**Bachrach, S. M. DFT Study of [2.2]-, [3.3]-, and [4.4]Paracyclophanes: Strain Energy, Conformations, and Rotational Barriers.*J. Phys. Chem. A*2011,*115*, 2396– 2401, DOI: 10.1021/jp111523u83https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXisFWntL4%253D&md5=ad0fafbb4e607229085988310f3f8fb0DFT Study of [2.2]-, [3.3]-, and [4.4]Paracyclophanes: Strain Energy, Conformations, and Rotational BarriersBachrach, Steven M.Journal of Physical Chemistry A (2011), 115 (11), 2396-2401CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)The three smallest sym. paracyclophanes, having tethers with two, three, or four methylene groups, have been examd. with four d. functional methods (B3LYP, M06-2x, B97-D, ωB97X-D). The geometries predicted by functionals accounting for medium-range correlation or long-range exchange and/or dispersion are in close agreement with expt. In addn., these methods provide similar ests. of the strain energy of the paracylcophanes, which decrease with increasing tether length. [4.4]Paracyclophane is nearly strain-free, reflecting the small out-of-plane distortion of its Ph rings. Lastly, the barrier for interconversion of the conformers of [3.3]paracylcophane is computed in close agreement with expt., and an est. for Ph rotation in [4.4]paracyclophane of about 19 kcal mol-1 is predicted by the DFT methods employed.**84**Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Petersson, G. A.; Nakatsuji, H.; Li, X.; Caricato, M.; Marenich, A. V.; Bloino, J.; Janesko, B. G.; Gomperts, R.; Mennucci, B.; Hratchian, H. P.; Ortiz, J. V.; Izmaylov, A. F.; Sonnenberg, J. L.; Williams-Young, D.; Ding, F.; Lipparini, F.; Egidi, F.; Goings, J.; Peng, B.; Petrone, A.; Henderson, T.; Ranasinghe, D.; Zakrzewski, V. G.; Gao, J.; Rega, N.; Zheng, G.; Liang, W.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Throssell, K.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.; Bearpark, M. J.; Heyd, J. J.; Brothers, E. N.; Kudin, K. N.; Staroverov, V. N.; Keith, T. A.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A. P.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Millam, J. M.; Klene, M.; Adamo, C.; Cammi, R.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Farkas, O.; Foresman, J. B.; Fox, D. J.*Gaussian 16*, Revision A.03; Gaussian Inc., Wallingford CT, 2016.There is no corresponding record for this reference.**85**Castro Neto, A. H.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K. The electronic properties of graphene.*Rev. Mod. Phys.*2009,*81*, 109, DOI: 10.1103/revmodphys.81.10985https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXksVamsLY%253D&md5=d4b07bf6507d26df9b0447a25131bf18The electronic properties of grapheneCastro Neto, A. H.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K.Reviews of Modern Physics (2009), 81 (1), 109-162CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)A review. This article reviews the basic theor. aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external elec. and magnetic fields, or by altering sample geometry and/or topol. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and no. of layers. Edge (surface) states in graphene depend on the edge termination (zigzag or armchair) and affect the phys. properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.**86**Zhang, D. W.; Zhang, J. Z. H. Molecular fractionation with conjugate caps for full quantum mechanical calculation of protein–molecule interaction energy.*J. Chem. Phys.*2003,*119*, 3599– 3605, DOI: 10.1063/1.159172786https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXlvFCqsLo%253D&md5=4952ee44ff52f306b22fb230f7919ff6Molecular fractionation with conjugate caps for full quantum mechanical calculation of protein-molecule interaction energyZhang, Da W.; Zhang, J. Z. H.Journal of Chemical Physics (2003), 119 (7), 3599-3605CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A scheme to calc. fully quantum mech. (ab initio) interaction energy involving a macromol. like protein is presented. In this scheme, the protein is decompd. into individual amino acid-based fragments that are treated with proper mol. caps. The interaction energy between any mol. and the given protein is given by the summation of interactions between the mol. and individually capped protein fragments. This scheme, termed mol. fractionation with conjugate caps (MFCC), makes it possible and practical to carry out full quantum mech. (ab initio) calcn. of intermol. interaction energies involving proteins or other similar biol. mols. Numerical tests performed on the interaction energies between a water mol. and three small peptides demonstrate that the MFCC method can give excellent ab initio interaction energies compared to the exact treatment in which the whole peptides are included in the calcn. The current scheme scales linearly with the at. size of the protein and can be directly applied to calcg. real protein-mol. interaction energies by using fully quantum (ab initio) methods that are otherwise impossible. The success of the current method is expected to have a powerful impact in our prediction of protein interaction energies including, e.g., protein-drug interactions.**87**Gauss, J.; Stanton, J. F. The equilibrium structure of benzene.*J. Phys. Chem. A*2000,*104*, 2865– 2868, DOI: 10.1021/jp994408y87https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXhs1Omurg%253D&md5=867664cf8026794f7dd76c3c0bea68caThe Equilibrium Structure of BenzeneGauss, Juergen; Stanton, John F.Journal of Physical Chemistry A (2000), 104 (13), 2865-2868CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)The re structure of benzene is revised on the basis of high-level quantum chem. calcns. at the CCSD(T)/cc-pVQZ level as well a reanal. of the exptl. rotational consts. using computed vibrational corrections. A least-squares fit to empirically detd. Be consts. yields re(CC) = 1.3914 ± 0.0010 Å and re(CH) = 1.0802 ± 0.0020 Å; the latter distance is significantly shorter than the best previous est. based on exptl. data. Comparison of computed rg and rz distances with expt. as well as considerations of bond lengthening due to anharmonicity are consistent with the estd. re distance, indicating that the recommended structural parameters are very accurate.**88**Helgaker, T.; Klopper, W.; Koch, H.; Noga, J. Basis-set convergence of correlated calculations on water.*J. Chem. Phys.*1997,*106*, 9639– 9646, DOI: 10.1063/1.47386388https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXjvVCgu78%253D&md5=f4689c1b38fe30eb721e9cd7d607bdf7Basis-set convergence of correlated calculations on waterHelgaker, Trygve; Klopper, Wim; Koch, Henrik; Noga, JozefJournal of Chemical Physics (1997), 106 (23), 9639-9646CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The basis-set convergence of the electronic correlation energy in the water mol. is investigated at the second-order Moller-Plesset level and at the coupled-cluster singles-and-doubles level with and without perturbative triples corrections applied. The basis-set limits of the correlation energy are established to within 2mEh by means of (1) extrapolations from sequences of calcns. using correlation-consistent basis sets and (2) from explicitly correlated calcns. employing terms linear in the inter-electronic distances rij. For the extrapolations to the basis-set limit of the correlation energies, fits of the form a + bX-3 (where X is two for double-zeta sets, three for triple-zeta sets, etc.) are found to be useful. CCSD(T) calcns. involving as many as 492 AOs are reported.**89**Halkier, A.; Helgaker, T.; Jørgensen, P.; Klopper, W.; Koch, H.; Olsen, J.; Wilson, A. K. Basis-set convergence in correlated calculations on Ne, N2, and H2O.*Chem. Phys. Lett.*1998,*286*, 243– 252, DOI: 10.1016/s0009-2614(98)00111-089https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXitVGqsLo%253D&md5=04274821d9c7fa664e9588855ed9a061Basis-set convergence in correlated calculations on Ne, N2, and H2OHalkier, Asger; Helgaker, Trygve; Jorgensen, Poul; Klopper, Wim; Koch, Henrik; Olsen, Jeppe; Wilson, Angela K.Chemical Physics Letters (1998), 286 (3,4), 243-252CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)Valence and all-electron correlation energies of Ne, N2, and H2O at fixed exptl. geometries are computed at the levels of second-order perturbation theory (MP2) and coupled cluster theory with singles and doubles excitations (CCSD), and singles and doubles excitations with a perturbative triples correction (CCSD(T)). Correlation-consistent polarized valence and core-valence basis sets up to sextuple zeta quality are employed. Guided by basis-set limits established by rij-dependent methods, a no. of extrapolation schemes for use with the correlation-consistent basis sets are investigated. Among the schemes considered here, a linear least-squares procedure applied to the quintuple and sextuple zeta results yields the most accurate extrapolations.**90**Russ, N. J.; Crawford, T. D. Potential energy surface discontinuities in local correlation methods.*J. Chem. Phys.*2004,*121*, 691– 696, DOI: 10.1063/1.175932290https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXlt1Srs74%253D&md5=75329cc988b46c6dd08086372d0b2627Potential energy surface discontinuities in local correlation methodsRuss, Nicholas J.; Crawford, T. DanielJournal of Chemical Physics (2004), 121 (2), 691-696CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We have examd. the occurrence of discontinuities in bond-breaking potential energy surfaces given by local correlation methods based on the Pulay-Saebo orbital domain approach. Our anal. focuses on three prototypical dissocg. systems: the C-F bond in fluoromethane, the C-C bond in singlet, ketene, and the central C-C bond in propadienone. We find that such discontinuities do not occur in cases of homolytic bond cleavage due to the inability of the Pipek-Mezey orbital localization method to sep. singlet-coupled charges on distant fragments. However, for heterolytic bond cleavage, such as that obsd. in singlet ketene and propadienone, discontinuities occur both at stretched geometries and near equil. These discontinuities are usually small, but may be of the same order of magnitude as the localization error in some cases.**91**Giovannini, T.; Lafiosca, P.; Cappelli, C. A General Route to Include Pauli Repulsion and Quantum Dispersion Effects in QM/MM Approaches.*J. Chem. Theory Comput.*2017,*13*, 4854– 4870, DOI: 10.1021/acs.jctc.7b0077691https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhsVyis7nP&md5=2d275dbd740d8d79d4e8b349c5e0f932A General Route to Include Pauli Repulsion and Quantum Dispersion Effects in QM/MM ApproachesGiovannini, Tommaso; Lafiosca, Piero; Cappelli, ChiaraJournal of Chemical Theory and Computation (2017), 13 (10), 4854-4870CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A methodol. to account for non-electrostatic interactions in Quantum Mech. (QM)/Mol. Mechanics (MM) approaches is developed. Formulations for Pauli repulsion and dispersion energy, explicitly depending on the QM d., are derived. Such expressions are based on the definition of an auxiliary d. on the MM portion and the Tkatchenko-Scheffler (TS) approach, resp. The developed method is general enough to be applied to any QM/MM method and partition, provided an accurate tuning of a small no. of parameters is obtained. The coupling of the method with both nonpolarizable and fully polarizable QM/fluctuating charge (FQ) approaches is reported and applied. A suitable parametrization for the aq. soln., so that its most representative features are well reproduced, is outlined. Then, the obtained parametrization and method are applied to calc. the non-electrostatic (repulsion and dispersion) interaction energy of nicotine in aq. soln.**92**Giovannini, T.; Lafiosca, P.; Chandramouli, B.; Barone, V.; Cappelli, C. Effective yet Reliable Computation of Hyperfine Coupling Constants in Solution by a QM/MM Approach: Interplay Between Electrostatics and Non-electrostatic Effects.*J. Chem. Phys.*2019,*150*, 124102, DOI: 10.1063/1.508081092https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXmtVClt7k%253D&md5=35f2df3b7518c3ba66567633728a9a44Effective yet reliable computation of hyperfine coupling constants in solution by a QM/MM approach: Interplay between electrostatics and non-electrostatic effectsGiovannini, Tommaso; Lafiosca, Piero; Chandramouli, Balasubramanian; Barone, Vincenzo; Cappelli, ChiaraJournal of Chemical Physics (2019), 150 (12), 124102/1-124102/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In this paper, we have extended to the calcn. of hyperfine coupling consts., the model recently proposed by some of the present authors [Giovannini et al., J. Chem. Theory Comput. 13, 4854-4870 (2017)] to include Pauli repulsion and dispersion effects in Quantum Mech./Mol. Mechanics (QM/MM) approaches. The peculiarity of the proposed approach stands in the fact that repulsion/dispersion contributions are explicitly introduced in the QM Hamiltonian. Therefore, such terms not only enter the evaluation of energetic properties but also propagate to mol. properties and spectra. A novel parametrization of the electrostatic fluctuating charge force field has been developed, thus allowing a quant. reprodn. of ref. QM interaction energies. Such a parametrization has been then tested against the prediction of EPR parameters of prototypical nitroxide radicals in aq. solns. (c) 2019 American Institute of Physics.**93**Høyvik, I.-M.; Myhre, R. H.; Koch, H. Correlated natural transition orbitals for core excitation energies in multilevel coupled cluster models.*J. Chem. Phys.*2017,*146*, 144109, DOI: 10.1063/1.497990893https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC1cvotVeguw%253D%253D&md5=11542a30b8d30f0b20b1b9def6012e69Correlated natural transition orbitals for core excitation energies in multilevel coupled cluster modelsHoyvik Ida-Marie; Myhre Rolf Heilemann; Koch HenrikThe Journal of chemical physics (2017), 146 (14), 144109 ISSN:.In this article, we present a black-box approach for the selection of orbital spaces when computing core excitation energies in the multilevel coupled cluster (MLCC) framework. Information available from the lower level of theory is used to generate correlated natural transition orbitals (CNTOs) for the high-level calculation by including both singles and doubles information in the construction of the transition orbitals. The inclusion of the doubles excitation information is essential to obtain a set of orbitals that all contain physical information, in contrast to the natural transition orbitals where only a small subset of the virtual orbitals contains physical information. The CNTOs may be included in an active space based on a cutoff threshold for the eigenvalues corresponding to the orbitals. We present MLCC results for core excitation energies calculated using coupled cluster singles and doubles (CCSD) in the inactive space and CCSD with perturbative triples (CC3) in the active space. The use of CNTOs results in small errors compared to full CC3.**94**Giovannini, T.; Ambrosetti, M.; Cappelli, C. Quantum Confinement Effects on Solvatochromic Shifts of Molecular Solutes.*J. Phys. Chem. Lett.*2019,*10*, 5823– 5829, DOI: 10.1021/acs.jpclett.9b0231894https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhslOqu7zN&md5=8b800363ebc75b80483b7334d58e6107Quantum Confinement Effects on Solvatochromic Shifts of Molecular SolutesGiovannini, Tommaso; Ambrosetti, Matteo; Cappelli, ChiaraJournal of Physical Chemistry Letters (2019), 10 (19), 5823-5829CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)We demonstrate the pivotal role of quantum mechanics d. confinement effects on solvatochromic shifts. In particular, by resorting to a quantum mechanics/mol. mechanics (QM/MM) approach capable of accounting for confinement effects we successfully reproduce vacuo-to-water solvatochromic shifts for dark n → π* and bright π → π* transitions of acrolein and dark n → π* transitions of pyridine and pyrimidine without the need of including explicit water mols. in the QM portion. Remarkably, our approach is also able to dissect the effects of the single forces acting on the solute-solvent couple and allows for a rationalization of the exptl. findings in terms of physicochem. quantities.**95**Su, P.; Li, H. Energy decomposition analysis of covalent bonds and intermolecular interactions.*J. Chem. Phys.*2009,*131*, 014102, DOI: 10.1063/1.315967395https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXosFGhsbY%253D&md5=d7ee8efa3cbafd0c33b37f42f14b1a9bEnergy decomposition analysis of covalent bonds and intermolecular interactionsSu, Peifeng; Li, HuiJournal of Chemical Physics (2009), 131 (1), 014102/1-014102/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An energy decompn. anal. method is implemented for the anal. of both covalent bonds and intermol. interactions on the basis of single-determinant Hartree-Fock (HF) (restricted closed shell HF, restricted open shell HF, and unrestricted open shell HF) wavefunctions and their d. functional theory analogs. For HF methods, the total interaction energy from a supermol. calcn. is decompd. into electrostatic, exchange, repulsion, and polarization terms. Dispersion energy is obtained from second-order Moller-Plesset perturbation theory and coupled-cluster methods such as CCSD and CCSD(T). Similar to the HF methods, Kohn-Sham d. functional interaction energy is decompd. into electrostatic, exchange, repulsion, polarization, and dispersion terms. Tests on various systems show that this algorithm is simple and robust. insights are provided by the energy decompn. anal. into H2, methane C-H, and ethane C-C covalent bond formation, CH3CH3 internal rotation barrier, water, ammonia, ammonium, and hydrogen fluoride hydrogen bonding, van der Waals interaction, DNA base pair formation, NH3NH3 and NH3CO coordinate bond formation, Cu-ligand interactions, as well as LiF, LiCl, NaF, and NaCl ionic interactions. (c) 2009 American Institute of Physics.**96**Boulanger, E.; Thiel, W. Toward QM/MM simulation of enzymatic reactions with the drude oscillator polarizable force field.*J. Chem. Theory Comput.*2014,*10*, 1795– 1809, DOI: 10.1021/ct401095k96https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXktlensbs%253D&md5=aefd8893a08a886db738d0a5d3d5361bToward QM/MM Simulation of Enzymatic Reactions with the Drude Oscillator Polarizable Force FieldBoulanger, Eliot; Thiel, WalterJournal of Chemical Theory and Computation (2014), 10 (4), 1795-1809CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The polarization of the environment can influence the results from hybrid quantum mech./mol. mech. (QM/MM) simulations of enzymic reactions. In this article, we address several tech. aspects in the development of polarizable QM/MM embedding using the Drude Oscillator (DO) force field. We propose a stable and converging update of the DO polarization state for geometry optimizations and a suitable treatment of the QM/MM-DO boundary when the QM and MM regions are sepd. by cutting through a covalent bond. We assess the performance of our approach by computing binding energies and geometries of three selected complexes relevant to biomol. modeling, namely the water trimer, the N-methylacetamide dimer, and the cationic bis(benzene)sodium sandwich complex. Using a recently published MM-DO force field for proteins, we evaluate the effect of MM polarization on the QM/MM energy profiles of the enzymic reactions catalyzed by chorismate mutase and by p-hydroxybenzoate hydroxylase. We find that inclusion of MM polarization affects the computed barriers by about 10%.**97**Senn, H. M.; Thiel, W. QM/MM methods for biomolecular systems.*Angew. Chem., Int. Ed.*2009,*48*, 1198– 1229, DOI: 10.1002/anie.20080201997https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXitFOqs7g%253D&md5=c51da58b0525651c71f9c393a79023beQM/MM methods for biomolecular systemsSenn, Hans Martin; Thiel, WalterAngewandte Chemie, International Edition (2009), 48 (7), 1198-1229CODEN: ACIEF5; ISSN:1433-7851. (Wiley-VCH Verlag GmbH & Co. KGaA)A review. Combined quantum-mechanics/mol.-mechanics (QM/MM) approaches have become the method of choice for modeling reactions in biomol. systems. Quantum-mech. (QM) methods are required for describing chem. reactions and other electronic processes, such as charge transfer or electronic excitation. However, QM methods are restricted to systems of up to a few hundred atoms. However, the size and conformational complexity of biopolymers calls for methods capable of treating up to several 100,000 atoms and allowing for simulations over time scales of tens of nanoseconds. This is achieved by highly efficient, force-field-based mol. mechanics (MM) methods. Thus to model large biomols. the logical approach is to combine the two techniques and, to use a QM method for the chem. active region (e.g., substrates and co-factors in an enzymic reaction) and an MM treatment for the surroundings (e.g., protein and solvent). The resulting schemes are commonly referred to as combined or hybrid QM/MM methods. They enable the modeling of reactive biomol. systems at a reasonable computational effort while providing the necessary accuracy.

## Supporting Information

## Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.0c00737.

Cartesian coordinates of the studied molecules; parameters of the calculations; and data related to Figures 3–5 and 9 (PDF)

Molecular geometry of acetone (XYZ)

Molecular geometry of graphene (XYZ)

Molecular geometry of ANS (XYZ)

Molecular geometry of nicotine (XYZ)

Molecular geometry of benzene (XYZ)

Molecular geometry of PCP (XYZ)

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