**Cite This:**

*J. Chem. Theory Comput.*2023, 19, 20, 6859-6890

# TURBOMOLE: Today and Tomorrow

- Yannick J. FranzkeYannick J. FranzkeFachbereich Chemie, Philipps-Universität Marburg, Hans-Meerwein-Str. 4, 35032 Marburg, GermanyMore by Yannick J. Franzke
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- Christof HolzerChristof HolzerInstitute of Theoretical Solid State Physics, Karlsruhe Institute of Technology (KIT), Wolfgang-Gaede-Str. 1, 76131 Karlsruhe, GermanyMore by Christof Holzer
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- Josefine H. AndersenJosefine H. AndersenDTU Chemistry, Department of Chemistry, Technical University of Denmark, Kemitorvet Building 207, DK-2800 Kongens Lyngby, DenmarkMore by Josefine H. Andersen
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- Sonia CorianiSonia CorianiDTU Chemistry, Department of Chemistry, Technical University of Denmark, Kemitorvet Building 207, DK-2800 Kongens Lyngby, DenmarkMore by Sonia Coriani
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- Fabio Della SalaFabio Della SalaInstitute for Microelectronics and Microsystems (CNR-IMM), Via Monteroni, Campus Unisalento, 73100 Lecce, ItalyCenter for Biomolecular Nanotechnologies @UNILE, Istituto Italiano di Tecnologia, Via Barsanti, 73010 Arnesano, ItalyMore by Fabio Della Sala
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- Eduardo FabianoEduardo FabianoInstitute for Microelectronics and Microsystems (CNR-IMM), Via Monteroni, Campus Unisalento, 73100 Lecce, ItalyCenter for Biomolecular Nanotechnologies @UNILE, Istituto Italiano di Tecnologia, Via Barsanti, 73010 Arnesano, ItalyMore by Eduardo Fabiano
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- Daniil A. FedotovDaniil A. FedotovDTU Chemistry, Department of Chemistry, Technical University of Denmark, Kemitorvet Building 207, DK-2800 Kongens Lyngby, DenmarkInstitute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, IsraelMore by Daniil A. Fedotov
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- Susanne FürstSusanne FürstInstitut für Chemie, Theoretische Chemie/Quantenchemie, Sekr. C7, Technische Universität Berlin, Straße des 17 Juni 135, 10623, Berlin, GermanyMore by Susanne Fürst
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- Sebastian GillhuberSebastian GillhuberInstitute of Inorganic Chemistry, Karlsruhe Institute of Technology (KIT), Engesserstr. 15, 76131 Karlsruhe, GermanyMore by Sebastian Gillhuber
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- Robin GrotjahnRobin GrotjahnDepartment of Chemistry, University of California, Irvine, 1102 Natural Sciences II, Irvine, California 92697-2025, United StatesMore by Robin Grotjahn
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- Martin KauppMartin KauppInstitut für Chemie, Theoretische Chemie/Quantenchemie, Sekr. C7, Technische Universität Berlin, Straße des 17 Juni 135, 10623, Berlin, GermanyMore by Martin Kaupp
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- Max KehryMax KehryInstitute of Physical Chemistry, Karlsruhe Institute of Technology (KIT), Fritz-Haber-Weg 2, 76131 Karlsruhe, GermanyMore by Max Kehry
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- Marjan KrstićMarjan KrstićInstitute of Theoretical Solid State Physics, Karlsruhe Institute of Technology (KIT), Wolfgang-Gaede-Str. 1, 76131 Karlsruhe, GermanyMore by Marjan Krstić
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- Fabian MackFabian MackInstitute of Physical Chemistry, Karlsruhe Institute of Technology (KIT), Fritz-Haber-Weg 2, 76131 Karlsruhe, GermanyMore by Fabian Mack
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- Sourav MajumdarSourav MajumdarDepartment of Chemistry, University of California, Irvine, 1102 Natural Sciences II, Irvine, California 92697-2025, United StatesMore by Sourav Majumdar
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- Brian D. NguyenBrian D. NguyenDepartment of Chemistry, University of California, Irvine, 1102 Natural Sciences II, Irvine, California 92697-2025, United StatesMore by Brian D. Nguyen
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- Shane M. ParkerShane M. ParkerDepartment of Chemistry, Case Western Reserve University, 10900 Euclid Ave, Cleveland, Ohio 44106 United StatesMore by Shane M. Parker
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- Fabian PaulyFabian PaulyInstitute of Physics, University of Augsburg, Universitätsstr. 1, 86159 Augsburg, GermanyMore by Fabian Pauly
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- Eva PerltEva PerltOtto-Schott-Institut für Materialforschung, Friedrich-Schiller-Universität Jena, Löbdergraben 32, 07743 Jena, GermanyMore by Eva Perlt
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- Tim SchraderTim SchraderOtto-Schott-Institut für Materialforschung, Friedrich-Schiller-Universität Jena, Löbdergraben 32, 07743 Jena, GermanyMore by Tim Schrader
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- Enrico TapaviczaEnrico TapaviczaDepartment of Chemistry and Biochemistry, California State University, Long Beach, 1250 Bellflower Boulevard, Long Beach, California 90840-9507, United StatesMore by Enrico Tapavicza
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- Robert S. TreßRobert S. TreßLehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, 44801 Bochum, GermanyMore by Robert S. Treß
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- Vamsee VooraVamsee VooraDepartment of Chemical Sciences, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, IndiaMore by Vamsee Voora
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- Artur WodyńskiArtur WodyńskiInstitut für Chemie, Theoretische Chemie/Quantenchemie, Sekr. C7, Technische Universität Berlin, Straße des 17 Juni 135, 10623, Berlin, GermanyMore by Artur Wodyński
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- Jason M. YuJason M. YuMore by Jason M. Yu
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- Benedikt ZerullaBenedikt ZerullaInstitute of Nanotechnology, Karlsruhe Institute of Technology (KIT), Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen GermanyMore by Benedikt Zerulla
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*****David P. TewPhysical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QZ, United Kingdom*****E-mail: [email protected]More by David P. Tew - , and
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*****Florian WeigendFachbereich Chemie, Philipps-Universität Marburg, Hans-Meerwein-Str. 4, 35032 Marburg, Germany*****E-mail: [email protected]More by Florian Weigend

## Abstract

TURBOMOLE is a highly optimized software suite for large-scale quantum-chemical and materials science simulations of molecules, clusters, extended systems, and periodic solids. TURBOMOLE uses Gaussian basis sets and has been designed with robust and fast quantum-chemical applications in mind, ranging from homogeneous and heterogeneous catalysis to inorganic and organic chemistry and various types of spectroscopy, light–matter interactions, and biochemistry. This Perspective briefly surveys TURBOMOLE’s functionality and highlights recent developments that have taken place between 2020 and 2023, comprising new electronic structure methods for molecules and solids, previously unavailable molecular properties, embedding, and molecular dynamics approaches. Select features under development are reviewed to illustrate the continuous growth of the program suite, including nuclear electronic orbital methods, Hartree–Fock-based adiabatic connection models, simplified time-dependent density functional theory, relativistic effects and magnetic properties, and multiscale modeling of optical properties.

This publication is licensed under

### License Summary*

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*Disclaimer

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### License Summary*

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

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

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

*Disclaimer

This summary highlights only some of the key features and terms of the actual license. It is not a license and has no legal value. Carefully review the actual license before using these materials.

### License Summary*

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

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

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

*Disclaimer

This summary highlights only some of the key features and terms of the actual license. It is not a license and has no legal value. Carefully review the actual license before using these materials.

#### SPECIAL ISSUE

This article is part of the

## 1. Introduction

*GW*-Bethe–Salpeter equation (BSE) methods, second-order Møller–Plesset (MP2) theory, and coupled-cluster (CC) methods. The code is based on Gaussian basis sets and has been pivotal for the development of many fast and low-scaling algorithms in the past three decades, such as integral-direct methods, the resolution-of-the-identity (RI) approximation, fast multipole methods, imaginary frequency integration, Laplace transform, and pair natural orbital methods.

*J*(40,41)) and exchange contributions, (RI-

*K*(42,43)), as well as its generalization to post-HF theories such as MP2 and CC2 (44−46) and its multipole-accelerated version for extended systems (MARI-

*J*), were other critical innovations. (47) The RI methods are still cornerstones in many modern implementations and outstanding features of the program suite. (23,31,48−68) More recent additions include explicitly correlated wave function methods up to CCSD(T) and BCCD(T), (68,69) efficient pair natural orbital (PNO) approaches, (70−73) solvation models and embedding, (74−83) two-component relativistic methods, (84−86) GW-BSE type methods (52,87,88) real-time (RT) TDDFT, (89) and nonadiabatic molecular dynamics. (90)

## 2. Brief Feature Overview

module | functionality | Fork-SMP | OpenMP | MPI | OpenMP/MPI | GPU |
---|---|---|---|---|---|---|

dscf | HF/DFT energy | √ | √ | √ | √ | √ |

grad | HF/DFT gradient | √ | √ | √ | √ | √ |

ridft | RI-HF/RI-DFT energy | √ | √ | √ | X | √ |

rdgrad | RI-HF/RI-DFT gradient | √ | √ | √ | X | √ |

aoforce | HF/DFT Hessian | √ | √ | √ | √ | √ |

escf | HF/DFT/GW-BSE excitation energies | √ | √ | √ | √ | √ |

egrad | HF/DFT excited-state gradient | √ | √ | X | X | √ |

mpshift | NMR/EPR parameters (HF/DFT/MP2) | X | √ | X | X | √ |

evib | electron transport (HF/DFT) | √ | √ | √ | √ | X |

odft | orbital-dependent DFT energies | X | √ | X | X | X |

mpgrad | MP2 energy, gradient | √ | X | √ | X | X |

ricc2 | RI-MP2, ADC(2), CC2 energies, gradients, spectra | X | √ | √ | √ | X |

pnoccsd | PNO-MPPT and PNO–CC energies with F12 | X | √ | √ | √ | X |

ccsdf12 | CCSD, CCSD(T) energies with F12 | X | √ | X | X | X |

rirpa | RPA energy, gradient | X | √ | X | X | X |

riper | periodic HF/DFT energy, gradient | X | √ | X | X | X |

^{a}

Fork-SMP (94) and the OpenMP version (46,62,95−97) are restricted to calculations on a single node. MPI (5,98−101) and OpenMP/MPI hybrid (102) implementations allow for the use of multiple nodes. The availability of first- and second-order derivatives as well as excitation energies is also indicated.

## 3. Recent Developments

### 3.1. Local Hybrid Functionals for Strong Correlation and Range-Separated Local Hybrid Functionals

*GW*. (119) From the variety of possible applications and evaluations, many of which have been touched upon in the 2020 overview of TURBOMOLE (92) and in a 2019 comprehensive review of local hybrids, (110) we highlight in particular the outstanding performance of LHs for mixed-valence systems (120−122) and for phosphorescence spectra. (123−125) We also point to further recent LH publications and to reviews by other authors. (111,117,119,126−128)

*q*

_{AC}factor. (134) Most recently, the more advanced scLH22t functional has been constructed. (135) It is based on the more recent and overall more accurate LH20t functional. (120) Using a damping factor for smaller NDC contributions, an almost complete decoupling between the underlying LH20t and the added

*q*

_{AC}factor could be obtained. That is, the optimized parameters of LH20t, as well as its excellent performance for weakly correlated situations (e.g., for GMTKN55 main-group energetics), are retained, but FSEs and the related spin-restricted dissociation curves of covalent bonds are dramatically improved. (135) Notably, the

*q*

_{AC}factor forms part of a new LMF (Figure 1).

### 3.2. Inclusion of the Current Density in DFT

**r**) is a commonly used ingredient in many functionals to detect iso-orbital regions or describe the inhomogeneity of the electron density ρ(

**r**). For its extension τ(

**r**,

*t*), used in the time-dependent Kohn–Sham formalism (TDKS), it has been shown (17) that this quantity is not invariant under a gauge transformation in the external potential. Substitution of τ by its generalization (141−143) $\hat{\tau}(\mathbf{r},t)=\tau (\mathbf{r},t)-\frac{|{\mathbf{j}}_{\mathrm{P}}(\mathbf{r},t){|}^{2}}{2\rho (\mathbf{r},t)}$, where

**j**

_{P}is the paramagnetic current density, restores gauge invariance. This leads to additional terms in the magnetic orbital rotation Hessian in linear-response TDDFT calculations, accounting for the response of

**j**

_{P}. While the original implementation of these terms in TURBOMOLE dates back to 2012 (release V6.4), we highlight four recent important updates.

*n*→ π* excitations are significantly more affected by restoring gauge invariance than most π → π* excitations with the exception of π → π

_{⊥}

^{*}excitations, where the dominantly contributing molecular orbitals (MOs) are perpendicular (⊥) to each other. (145) These findings suggest that a reassessment of previously reported TDDFT results obtained with τ-dependent functionals is warranted, particularly for cases that are potentially more sensitive due to the choice of the functional, the type of excitation, or both. Gauge invariance is restored by default with τ-dependent functionals at moderate (nonhybrid functionals) or negligible (hybrid functionals) additional computational cost. (17) Recently, excited-state gradients and quadratic response properties for τ̂-dependent meta-generalized gradient approximations (mGGAs) have been implemented, enabling gauge invariant computations of excited-state equilibrium structures, relaxed dipole moments, (dynamic) hyperpolarizabilities, and two-photon absorption cross sections. (147) These developments will be available in a future TURBOMOLE release.

**j**

_{P}is required for gauge invariance of magnetic properties and implemented for magnetizabilities, (119) NMR coupling constants, (119) and NMR shifts of closed-shell (119,148) and open-shell systems, (118) as well as EPR hyperfine coupling constants (149) and g-tensors. (118) Recent findings indeed hint at the inclusion of the current density response also being crucial for NMR and EPR properties. (118,119,148) Especially for open-shell systems, neglecting the current density leads to large deviations, (118) as shown in Figure 3.

**j**

_{P}-dependent terms, which have a profound impact on many properties. (151,152) Current density functional theory (CDFT) for τ-based functionals is available for ground-state energies (module ridft) and gradients (module rdgrad) and linear response properties (modules escf and mpshift). (151) Given the profound impact of

**j**

_{P}, we therefore strongly recommend the exclusive use of the current-dependent forms.

### 3.3. Methods for Finite Magnetic Fields

*GW*/BSE, (177,178) RPA, (179) and CC2. (178) Through the calculation of electronic ground states, molecular gradients, and electronic excitations, a wide variety of applications for molecular spectroscopy in magnetic fields are now accessible.

### 3.4. EPR Properties and Single-Molecule Magnets

*in silico*study of f-element chemistry. To provide accurate descriptions of compounds containing these heavy elements, robust DFT routines are leveraged to deliver a balanced treatment of dynamic correlation, static correlation, solvation, and relativistic effects. More recently, these developments have enabled the discovery of new species with novel chemistry, subsequently necessitating new and improved computational methods capable of describing them.

_{3}]

^{−}, [Lu(NR

_{2})

_{3}]

^{−}, and [Lu(OAr*)

_{3}]

^{−}(OAr* = 2,6-Ad

_{2}-4-

*t*-Bu-C

_{6}H

_{2}O, Ad = adamantyl,

*t*-Bu=

*tert*-butyl, R = SiMe

_{3}with Me = methyl), were found to exhibit exotic EPR properties, with [Lu(OAr*)

_{3}]

^{−}producing hyperfine coupling (HFC) constants of unforeseen magnitude and furthermore demonstrating extended magnetic coherence facilitated by a hyperfine clock transition. A primary investigation of these species with nonrelativistic HFC operators attributed the large hyperfine coupling constants to a sizable Fermi contact contribution from the highest occupied MO (HOMO) of each system. However, the predictions of hyperfine coupling constants and g-tensor values themselves produced errors of roughly one order of magnitude, strongly suggesting the need for more rigorous methods. (181)

*g*-tensor calculations is ensured by the gauge including atomic orbitals, (36,39,149) which are crucial for systems with a spatially distributed spin density. These methods were implemented, and their performance is shown here for the aforementioned SMM [Lu(OAr*)

_{3}]

^{−}shown in Figure 5. The all-electron relativistic methods lead to good agreement with the experiment, as shown in Table 2. The HFC constant of [Lu(OAr*)

_{3}]

^{−}is dominated by the scalar-relativistic contribution due to the localization of the spin-density in the 6s/d HOMO producing a large Fermi contact interaction. The importance of the paramagnetic spin–orbit contribution increases with the number of unpaired electrons and the scalar formulation, as well as the spin–orbit perturbation theory (SOPT) break down for systems such as [TbPc

_{2}]

^{−}with six unpaired electrons, Pc = bis(phthalocyaninato). (38,39,117,151) The self-consistent 2c methods are thus pivotal.

[Lu(OAr*)_{3}]^{−} | [TbPc_{2}]^{−} | ||||
---|---|---|---|---|---|

method | A_{xx} | A_{yy} | A_{zz} | A | |

TPSS | SR | 3192 | 3192 | 3148 | 190 |

cTPSS | SO | 3190 | 3190 | 3153 | 375 |

ωB97X-D | SR | 3489 | 3489 | 3456 | 136 |

ωB97X-D | SO | 3464 | 3464 | 3448 | 488 |

TMHF | SR | 3225 | 3225 | 3171 | 293 |

cTMHF | SO | 3208 | 3208 | 3161 | 526 |

Expt. | 3500 | 3500 | 3400 | 519 |

^{a}

HFC is given in MHz. SR and SO denote scalar and spin–orbit relativistic results (x2c-TZVPall-2c/x2c-SVPall-2c basis), respectively.

*g*-tensor, and electric-field gradient as well as the nuclear quadrupole interaction tensor will further become available for users. (39)

### 3.5. NMR Coupling Constants Across the Periodic Table of Elements

*i.e.*, methods based on the Dirac equation are introduced. For such methods, the FC, SD, and PSO terms are coupled due to spin–orbit interaction, and they come with drastically increased computational demands. Nevertheless, when using a local X2C ansatz, (84−86) large-scale calculations are possible, as illustrated in Figure 6 for the Karplus relationship of Me

_{3}Sn–CH

_{2}–CHR–SnMe

_{3}, where R is different substituents (Me = CH

_{3}). The relativistic DFT approach reproduces the experimental findings with fairly good agreement. Improvements are possible with the correlation kernel augmented BSE (cBSE) based on the Green’s function

*GW*ansatz. Here, the DFT response equations are replaced with their BSE counter parts. (61,113)

_{2}Sn] (SIDipp = 1,3-bis(2,6-di-isopropylphenyl)-imidazolidin-2-ylidene) with 137 atoms (187) takes about 44 min (PBE) and 55 h (PBE0) using 12 OpenMP threads of an Intel Xeon Gold 6212U CPU (2.40 GHz). (86) Notably, using a single NVIDIA A100 GPU, the PBE0 timing can be reduced to 3.5 h.

### 3.6. Paramagnetic NMR Shieldings and Shifts

**σ**

_{I}

^{tot}for a nucleus

*I*reads

*S*denoting the spin, μ

_{e}denoting the Bohr magneton, γ

_{I}denoting the gyromagnetic ratio of nucleus

*I*,

*k*

_{B}denoting the Boltzmann constant, and

*T*denoting the temperature. Here, the orbital contribution

**σ**

^{orb}is the straightforward open-shell generalization of the closed-shell limit. (63,118) Additionally, a temperature-dependent contribution arises, which includes the HFC tensor

*A*

_{I}of nucleus

*I*and the

*g*-tensor

*g*already discussed in section 3.4. Both the HFC and the

*g*-tensor depend on spin–orbit coupling. For the calculation of

^{1}H/

^{13}C pNMR spectra of large molecules, a perturbative treatment of spin–orbit coupling is preferred over the 2c ansätze due to lower computational costs. (149) The viability of the pertubative ansatz in the X2C framework is demonstrated for two negatively charged Ru(III) complexes in Figure 7, which depicts the good agreement between calculated results and the experimentally measured (189) pNMR

^{1}H and

^{13}C shifts of the two compounds.

### 3.7. Ring Currents of Heavy-Metal Clusters

^{1}H NMR spectra. This shift is a consequence of the cyclic electron delocalization associated with the π-orbitals, which deshield the nuclei due to an induced ring current. (195) This magnetically induced current density may be calculated indirectly with the nucleus-independent chemical shift (196) (NICS) or directly using TURBOMOLE’s interface to the GIMIC program, which was reworked for release V7.7 and now supports open-shell calculations. (197−200) The latter approach is more flexible and also applicable to complicated multicyclic systems. (201−204)

_{12}]

^{4–}cluster features a nonlocalizable π-orbital around the {Bi

_{12}} torus, which leads to a ring current. (205) Figure 9 shows a streamlined representation of this ring current, whose strength amounts to about 25 nA/T. (119,205)

_{12}]

^{3–}shows a strong ring current, (63) as the same π-orbital is occupied by two electrons. (205,206) Moreover, prismatic {Bi

_{6}}-based clusters such as [(CpRu)

_{3}Bi

_{6}]

^{−}show ring currents of more than 25 nA/T. (207) Therefore, all-metal clusters help to push the frontiers of aromaticity.

### 3.8. Characterization of Novel Electronic Configurations in f-Block Elements by DFT and RPA Methods

^{2+}complexes possessing the 4f

^{n}5d

^{1}configuration in the [Ln(C

_{5}H

_{4}SiMe

_{3})

_{3}]

^{−}(Ln = Ce–Nd, Gd–Er), (214−217) [Ln{N(SiMe

_{3})

_{2}}

_{3}]

^{−}(Ln = La, Gd), (218,219) and [Ln(Cp

^{iPr5})

_{2}] (Ln = Tb, Dy) series. (220) The preference for the 4f

^{n}5d

^{1}configuration relative to the traditional 4f

^{n+1}occupation of Ln

^{2+}results from the stabilization of the Ln 5d

_{z2}orbital by the trigonal ligand environment or extremely bulky ligands. Nontraditional Ln

^{2+}complexes show a characteristic intense absorption band in the visible range due to excitations from the occupied Ln 5d orbital. The prediction of UV–vis spectra of low-valent lanthanide complexes, in particular those with a nontraditional configuration, is improved by including diffuse augmentation in lanthanide basis sets. (221) DFT and RPA methods were employed to examine the strong ferromagnetic coupling between the Ln

^{3+}centers in [(C

_{5}Me

_{5})

_{2}Ln(μ-S)

_{2}Mo(μ-S)

_{2}Ln(C

_{5}Me

_{5})

_{2}]

^{−}(Ln = Y, Gd, Tb, Dy) and the Mo→Ln electronic excitations in the near-infrared spectral region. (222) Excited-state studies of [Ln(C

_{5}Me

_{5})

_{2}(C

_{5}Me

_{4}H)] and [Ln(C

_{5}Me

_{5})

_{2}(η

^{3}-C

_{3}H

_{4})] complexes (Ln = Y, Dy, Lu) using TDDFT elucidated their unexpected photochemical activation, which was used to reduce dinitrogen and sulfur and to polymerize isoprene. (223,224)

^{iPr5})

_{2}] (An = Th, U, Pu, Am, Bk, No, Lr) (pentaisopropylcyclopentadienyl = Cp

^{iPr5}) using DFT predicted ground states with a linear ligand arrangement of

*S*

_{10}symmetry and significant An 6d orbital occupation for An = Th, U, Lr. (225) The calculations were carried out with the TPSS exchange–correlation functional, (226) Stuttgart–Cologne scalar-relativistic small-core effective core potentials (ECPs), (227) and the corresponding basis sets. (228,229) Mixed 5f/6d occupation was predicted in the corresponding Pu complex, while the An = Am, Bk, No complexes were found to have 5f

^{n+1}configurations. The Pu and Am complexes showed a slight deviation from the perfectly symmetric structure, with Cp–M–Cp bending angles of 11° and 12°, respectively. The simulated absorption spectra showed intense peaks in the UV–vis range due to the metal–ligand charge transfer excitations from the An 6d orbital shown in Figure 10. Comparisons with the previously experimentally known Ln analogs (Ln = Dy, Tb) (220) suggested that the synthesis of the predicted actinocene complexes was thermodynamically feasible. The computational predictions received experimental confirmation for An = U while the results were still under review. Layfield and co-workers reported the synthesis of the linear

*S*

_{10}-symmetric “second-generation” uranocene [U(Cp

^{iPr5})

_{2}]. (230) The U–Cp centroid distance was determined from crystallographic studies as 2.504 Å, in good agreement with the computational result of 2.483 Å. The measured UV–vis spectra showed broad and intense absorption, as predicted by TDDFT calculations.

### 3.9. Damped Response Theory

*e.g.*, the resonant inelastic X-ray scattering (RIXS) transition amplitudes, (237) as well as properties at imaginary frequencies, like the C

_{6}dispersion coefficient. (232,238−240)

#### 3.9.1. Damped Response for Multiscale Modeling

*GW*-BSE method, being especially useful for core excitations. (113,255)

_{ex}= ω +

*iγ*, where ω and γ are the real and imaginary parts of the external field, respectively, the coupled perturbed equation (254,256)

**A**and

**B**are defined as

_{p}marks orbital or quasiparticle energies,

*v*

_{pq,rs}is a Coulomb integral,

*f*

_{pq,rs}is the exchange-correlation kernel, if present, and

*K*

_{pq,rs}is a exchange integral. The precise kinds of

*f*

_{pq,rs}and

*K*

_{pq,rs}depend on the chosen method. (254)

**p**

^{ν}describes the external perturbation,

*e.g.*, an electric or magnetic field. (254,256) The polarizability can then be calculated as

#### 3.9.2. Damped Response Theory for One-Photon Absorption and CD Spectra with RI-CC2

**A**

_{SS},

**A**

_{SD}, and

**A**

_{DS}are, respectively, the singles–singles, singles-doubles, and doubles-singles blocks of the CC2 Jacobi matrix. Δ

^{ij}

_{ab}= ε

_{a}– ε

_{i}+ ε

_{b}– ε

_{j}– ω are frequency-shifted orbital energy differences, and ω and γ are again the real and imaginary parts of the frequency of the external field, respectively. The effective right-hand sides are

_{S}

^{x}and ξ

_{D}

^{x}are the single and doubles parts of the right-hand sides in the nonpartitioned form, respectively. (232,253) The partitioned formulation that avoids the need to store double excitation amplitudes and four-index integrals allows applications to system sizes otherwise not accessible at the CC2 level.

### 3.10. Nonadiabatic Molecular Dynamics Simulation for Spectroscopic Observables

_{1}undergo ultrafast decay to the S

_{1}state and thus are not directly involved in fluorescence or photoactivated reactions. However, exceptions to Kasha’s rule are well-known in molecules in the gas phase, such as azulene and pyrene. (268)

_{2}state, in agreement with experiment. (271) BCDP obeys Kasha’s rule and emits only from S

_{1}. Previous studies assigned the high-energy shoulder in the pyrene fluorescence spectrum to non-Kasha emission as a result of the reverse S

_{1}→ S

_{2}internal conversion. (268,272−275) Multistate NAMD simulations describe the non-Kasha behavior as a combination of S

_{1}→ S

_{2}transitions and the change in the diabatic character of the S

_{1}state from dark (L

_{b}) to bright (L

_{a}) at points of near degeneracy between the S

_{1}and S

_{2}states. The high-energy shoulder in the fluorescence spectrum of pyrene can be understood as originating from excited states with diabatic bright (L

_{a}) character. The S

_{2}lifetime in pyrene was computed by an exponential fit of the decay of the state population as 63 fs, in agreement with the experiment value of 85 fs in methanol. (276) The S

_{1}lifetime of azulene was computed to be 2.2 ps in comparison to the experimental result of 1.4 ps in cyclohexane. (277) The computed lifetime of the S

_{1}state of BCDP was found to be 0.8 ps.

_{2}→ S

_{n}transient absorption that decays rapidly and a S

_{1}→ S

_{n}transition (

*n*> 4), respectively. (276,278) NAMD simulations predict S

_{2}→ S

_{4}and S

_{1}→ S

_{4}bands at 1500 and 800 nm, respectively. The time evolution of the S

_{1}and S

_{2}states is in good agreement with experiment, while the absorption maxima (λ(S

_{2}→ S

_{4}) = 1500 nm and λ(S

_{1}→ S

_{4}) = 800 nm) are red-shifted due to truncation of the electronic excitation space.

### 3.11. Generating Function Methods for Vibrationally Resolved Electronic Spectroscopy

*i.e.*, vibronic) spectra has been advanced by switching from the time-independent approach, (289) which requires a tedious sum-overstates evaluation of Franck–Condon overlaps, to the time-dependent formalism (generating function approach), where the spectrum is given by the Fourier transform of an appropriate time-correlation function. (288,290−292) The radless module (288,293,294) makes use of the generating function method to compute vibrationally resolved absorption and emission spectra, as well as photoelectron ionization spectra. Spectra can be computed within the global harmonic approximation, which only requires equilibrium geometries for initial and final structures as well as vibrational spectra of both structures. The method accounts for the full Duschinsky rotation, (295) taking into account differences in initial and final state structures and vibrational modes. Due to its efficiency, the method can be applied to large molecules, such as polyaromatic hydrocarbons. (123,124,293,296,297) An extension of the module further allows the computation of emission and absorption spectra arising from singly occupied vibrationally excited initial states, allowing the simulation of single vibronic level (SVL) fluorescence (293) and vibrationally promoted electronic resonance (VIPER) spectra. (298)

*ab initio*TGA (300−303) the Hessian of the potential energy is updated over time, in the single-Hessian version, (304−306) implemented in TURBOMOLE as part of the radless module, (294) the Hessian is kept constant throughout the dynamics. Therefore, the overall additional cost compared to the conventional harmonic approximation is that of a single

*ab initio*trajectory in the final electronic state, which is simulated using the frog module. Since the trajectory experiences the true anharmonic PES, the method can account for anharmonicity at least approximately, although it cannot describe more subtle quantum dynamics, such as wavepacket splitting or tunneling. The TGA approach has proven to be especially useful in systems with a large displacement between the ground- and excited-state minima and in systems with a double-well-shaped PES along a low-frequency mode in the final electronic state. In such molecules, the harmonic approach typically fails because the global harmonic PES constructed around one of the wells is not adequate. Moreover, in contrast to the global harmonic methods, the TGA results often depend weakly on the choice of the Hessian, as illustrated in Figure 14. Overall, the implementation in TURBOMOLE combines these advantages of TGA with accurate and efficient excited-state electronic structures, such as ADC(2) and CC2 methods.

### 3.12. Molecular Properties from Self-Consistent GKS-spRPA

#### 3.12.1. Applications to Nonvalence Anionic States and X-ray Emission Spectroscopy

*ΔE*, computed by taking the difference between core and valence IPs and the oscillator strengths,

*f*

_{osc.}, are evaluated within a frozen orbital approximation

_{c}and ϕ

_{v}are the core orbital and valence GKS-spRPA orbitals involved in the XE process, respectively. The AC GKS-spRPA approach was used in conjunction with the scalar-relativistic (SR) X2C approach and uncontracted basis sets to estimate highly accurate XE spectra for molecules containing second- and third-period elements, for example, see Figure 16. Using uncontracted basis sets, the XE energies were found to have MAEs of 0.7 eV for both second and third period-based XE. The X2C-based AC GKS-spRPA approach thus enables the simulation of nonresonant X-ray emission within a simple one-particle picture while avoiding the use of empirical shifts or core-hole reference states. The latter is an appealing aspect, since issues related to variational instability, which are present in core–hole reference based methods, are avoided.

#### 3.12.2. Orbital Ordering in Quinacridone

*G*

_{0}

*W*

_{0}IPs. (320,321) The canonical GKS-spRPA orbital ordering qualitatively agrees with the one OT-RSH one down to HOMO(−3), see Figure 17. Furthermore, the negative GKS-spRPA HOMO energy of 7.07 eV is close to the experimental IP of 7.23 eV, (322) whereas the negative PBE HOMO energy of 4.92 eV is significantly too small.

### 3.13. Subquadratic-Scaling PNO-CCSD(T) and PNO-CCSD(T)-F12 Methods

^{–7}) PNO truncation threshold was used in all cases. The def2-TZVPP basis and def2-TZVP were used for C

_{n}H

_{2n+2}and Na

_{n}Cl

_{n}, respectively, and calculations were run on a 48 core Intel processor with 200 Gb of memory. For the linear systems, the observed scaling is subquadratic, with C

_{128}H

_{256}taking 15 h to complete. For the globular systems, the observed scaling is subcubic, with Na

_{50}Cl

_{50}taking 45 h to complete. The F12 calculations are 2–3× more costly than non-F12 calculations but provide energies close to the basis set limit without requiring basis sets with a large number of AOs per atom.

### 3.14. Real-Time TDDFT for Molecules

### 3.15. Developments of the DFT-Based Embedding Implementations

#### 3.15.1. Frozen Density Embedding Implementation

update scheme | successive | simultaneous |
---|---|---|

T(40), min | 122.8 | 71.4 |

T(1), min | 369.6 | 586.2 |

speedup | 3.0 | 8.2 |

n_{FaT} | 6 | 10 |

^{a}

*T*(*X*) represents the computational wall time obtained using *X* CPU cores. *n*_{FaT} gives the number of FaT iterations where it is necessary to reach convergence.

#### 3.15.2. FDE and Projection-Based Embedding for Molecules and Solids

_{2}O)

_{20}, acetone + (H

_{2}O)

_{35}, and acetone + (H

_{2}O)

_{48}clusters at only a fraction of the computational cost. While beneficial, it is also important to acknowledge the limitations of the implementation as well, such as being restricted to only two closed-shell subsystems and using only LDA/GGA functionals for the embedding potential.

CC2 | CC2-in-DFT (periodic) | |||
---|---|---|---|---|

Ac + (H_{2}O)_{20} | Ac + (H_{2}O)_{35} | Ac + (H_{2}O)_{48} | Ac in 3D H_{2}O | |

ΔE, eV | 0.19 | 0.19 | 0.21 | 0.20 |

N_{bf} | 640 | 1000 | 1312 | 2872 |

T, h | 0.88 | 8.36 | 37.33 | 4.10 |

^{a}

*T* denotes the wall time of the CC2 solvatochromic shift calculations.

### 3.16. Periodic Hartree–Fock Exchange

*k*-mesh determines in turn the size of the Born–von Kármán supercell. We have demonstrated (328) that a minimum image convention (338) removes the divergence for discrete

*k*-meshes. While calculations with periodic HF exchange may be unstable for small supercells, stable SCF calculations and convergence of total energies are typically achieved for sufficiently large sizes of the supercells. The size of the supercell or

*k*-mesh that is required for a reliable energy depends on the locality of the density matrix and hence both the electronic structure of the studied material and the chosen basis set. For selected insulators and semiconductors, we have demonstrated that HF total energies converge exponentially with the number of

*k*-points, (328) see Figure 22.

## 4. Select Features under Development

### 4.1. Nuclear Electronic Orbital Method

*trans*-Zundel isomer H

_{9}O

_{4}

^{+}. (349) The nuclear orbital energies, as calculated by the NEO-HF method, give an estimate of the binding energies of the protons. While the four outer ones have energies from −450 to −436 kcal/mol, the central one is the least stable with an energy of −417 kcal/mol. If the outer two water molecules are removed, the energies change to between −400.5 and −399.6 kcal/mol for the four outer protons and −371 kcal/mol for the central one. This hints at the ionic cluster being stabilized by the outer water molecules.

*ab initio*occupation of nuclear orbitals at the respective sites instead of placing classical nuclei according to intuition.

### 4.2. Hartree–Fock-Based Adiabatic Connection Models

*e.g.*, it overestimates the correlation energy in large systems (355) and diverges for systems with a vanishing gap. (356) For this reason, several regularized and/or scaled MP2 methods have been developed. (356−358)

*E*

_{c}is given as a nonlinear function of

*E*

_{MP2},

*E*

_{x}

^{HF}(the HF exchange energy), and two semilocal functionals of the HF density (

*W*

_{c}=

*W*

_{c}[ρ

^{HF}] and

*W*

_{c}

^{′}=

*W*

_{c}

^{′}[ρ

^{HF}]). The latter are derived from the strong-correlation regime, (360,361) and we have

*de facto*includes an infinite-order resummation of the MP correlation series thanks to the interpolation with the strong-correlation limit, as in the more conventional AC based on DFT. (362) The nonlinear function

*F*can be approximated by modeling the HFAC curve at various coupling strengths (359) using known exact asymptotic conditions. (361,363) Consequently,

*F*satisfies two important limits:

*G*is a nonlinear function whose form depends on the choice of

*F*. For well-behaving approximations of

*F*, the condition in eq 11b yields a finite energy whenever

*E*

_{MP2}→ – ∞, thus removing one main limitation of MP2 and DH functionals for systems with a vanishing gap. The condition (eq 11a) is an exact condition, (359) which is violated in all the regularized MP2 methods. (356,357) Thus, the HFAC method allows us to overcome the main drawbacks of the MP2 approach within a well-defined theoretical framework at the small extra cost of a post-HF semilocal DFT calculation.

*F*have been proposed (

*e.g.*, ISI, (362,364) RevISI, (365) and MPACF1 (363)), and they have been implemented in TURBOMOLE together with the currently available DFT approximations for

*W*

_{c}and

*W*

_{c}

^{′}. (360,366) Note that eq 10 is not size-consistent for systems composed of different species of fragments (as

*F*is a nonlinear function). However, a size-consistent correction (SCC) (367) can be readily computed with TURBOMOLE at no additional costs, allowing the calculation of dissociation curves.

_{2}dissociation in a restricted formalism, a prototype of a strongly correlated system. (129−131,357,364) All methods work well close to the equilibrium geometry. However, for a larger separation where the energy-gap closes, MP2 and DH functionals rapidly diverge. HFAC methods, however, remain well behaved, yielding a finite interaction energy, see eq 11b. The exact result is not reproduced, as the available HFAC functionals are approximated and do not take into account the recent theory developments. (361) Further development and testing are thus required, and the HFAC implementation in TURBOMOLE represents an efficient platform to this end.

### 4.3. Approximate TDDFT Approaches

*J*technique. (374−377) Another efficient, though approximated, path is to perform a semiempirical tight-binding linear-response (TBLR) approximation, (378−380) using first-principles KS orbitals and eigenvalues. TBLR methods speed up the TDDFT calculation by about two orders of magnitude. TBLR is accompanied by a loss in accuracy of about 0.1–0.2 eV, which is comparable to the overall TDDFT accuracy. (381,382) More recently, it has been shown that, for semilocal XC functionals, the TBLR approaches can be considered as an approximation of the RI-TDDFT scheme with only one s-type Gaussian basis function per atom in the RI auxiliary basis set (TDDFT-as) (383) and with the three index RI integrals replaced with a Löwdin approximation. (378−380) Instead, in the TDDFT-as method, the latter approximation is not employed and, moreover, the calculation of the semilocal XC kernel contribution on the grid is not required, (383) as it can be modeled/approximated by the same exponent α of the s-type Gaussian auxiliary basis function. However, the exponent α needs to be optimized for each atom type separately.

_{120}, with

*T*

_{d}symmetry calculated using the def-SVP (41) basis set. Second, a fullerene with

*C*

_{1}symmetry calculated using the def2-TZVP (270) basis set. Both systems contain only a single type of atom. We optimized α by minimizing the root-mean square (RMS) averaged excitation energy error

*E*

_{avg}, avoiding state flipping, (383) and considering 400

*t*

_{2}(600

*a*) excited states for the silver nanoparticle (fullerene). The optimization yields α

_{Ag}= 0.036 with

*E*

_{avg}= 5 meV and α

_{C}= 0.18 with

*E*

_{avg}= 12 meV for Ag

_{120}and C

_{60}, respectively, as reported in Table 5. The amazing accuracy is also retained when the maximum error on all excitation energies

*E*

_{max}, including many optically dark states in

*C*

_{60}, is considered. This is demonstrated in Table 5 and Figure 25, where we report the absorption spectra of the two systems considered. The TDDFT-as absorption spectra in Figure 25, reported on a log scale to also highlight states with low oscillator strengths, can be hardly distinguished from the reference TDDFT results in a wide energy range for both systems investigated. Compared to TBLR approaches, (383) the accuracy of TDDFT-as is therefore increased by an order of magnitude, showing that it is a highly competitive approach.

Ag_{120} | C_{60} | |
---|---|---|

α | 0.036 | 0.18 |

E_{avg}, meV | 5 | 12 |

E_{max}, meV | 11 | 39 |

TDDFT RI-J, s | 5719 | 7148 |

TDDFT-as RI-J, s | 116 | 104 |

TDDFT XC, s | 22363 | 29845 |

TDDFT-as XC, s | 0 | 0 |

^{a}

In the TDDFT(-as) calculations, the 1s core orbital and the 4s4p orbitals were kept frozen in C_{60} and Ag_{120}, respectively.

*E*

_{avg}= 60 meV (384) compared to an average error of

*E*

_{avg}= 240 meV for sTDDFT. (378,379) Thus, the TDDFT-as and TDDFT-ris methods, both fully available in the next release, are efficient and accurate approximations of standard TDDFT, providing a significantly less empirical alternative to TBLR approaches. Thanks to the flexible and efficient implementation, accurate simulations of the absorption spectra of large nanoparticles and organic molecules are available at a fraction of the computational cost of standard TDDFT.

### 4.4. Multiscale Modeling Extensions for the Nonlinear Optical Response of Molecular Materials

*ab initio*-based T-matrix approach available to a broader scientific community interested in a bottom-up approach of simulating complex artificial molecular materials and photonic devices. Future work will be dedicated to nonlinear optical properties, as this topic is currently seeing increasing interest in the scientific community. The change in the molecular dipole moment

*Δμ*

_{i}(polarization) upon exposure to the oscillating external electric field

*E*

_{i}at the excitation frequencies is often expressed as a power series of the incident field

*E*.

_{ij}denotes the polarizability, β

_{ijk}denotes the first hyperpolarizability, γ

_{ijkl}denotes the second hyperpolarizability and so on. Currently, optical multiscale studies are limited to linear response, taking into account α

_{ij}. (254,261) To take into account nonlinear effects,

*i.e.*, β

_{ijk}and/or γ

_{ijkl}, the additionally arising quadratic (and/or cubic) response terms of eq 12 need to be taken into account. While TURBOMOLE already allows calculation of the first hyperpolarizabilities β

_{ijk}for real frequencies, ongoing work is dedicated to expanding this toward general complex frequencies. Ultimately, this will allow not only studies of the nonlinear light–matter interactions on the individual molecular level but also the construction of “hyper-T-matrices”. The latter can be used to investigate for example second-harmonic generation (SHG) efficiencies, macroscopic second-order susceptibilities, and two-photon absorption of photonic devices made from molecular materials.

### 4.5. Relativistic Effects and Magnetic Properties of Periodic Systems

*i.e.*, ferromagnetic, antiferromagnetic, and noncollinear spin textures.

method | L–L | Γ–Γ | X–X | L–X |
---|---|---|---|---|

NR | 3.89 | 3.42 | 3.71 | 1.48 |

1c ECP | 3.49 | 2.16 | 2.98 | 0.65 |

2c ECP | 3.25 | 1.82 | 2.69 | 0.41 |

4c DKS | 3.25 | 1.88 | 2.74 | 0.49 |

^{a}

Nonrelativistic calculations (NR) are performed with the TZVPalls2/TZVPall basis set, (387) whereas the ECP-based 1c and 2c calculations use the dhf-SVP(-2c) bases. (388) Results taken from ref (385). Four-component Dirac–Kohn–Sham (DKS) reference values are taken from ref (389), employing the uncontracted Dyall-VDZ basis. (390,391).

## 5. Outlook

*modus operandi*for large scientific coding projects, the need to secure original authorship and demonstrate scientific independence often conflicts with sharing plans and code, taking collective responsibility, and avoiding “technical debt”. As a result, TURBOMOLE has historically not been particularly easy to use, contribute to, or interface with other codes. TURBOMOLE GmbH was founded precisely to address these issues and has provided a framework to advance common goals and improve code quality. Nevertheless, incentives to collaborate and adopt sustainable coding practices remain few and far between. The future of the TURBOMOLE project will vitally depend on whether the conditions set by the environment,

*i.e.*, academic institutions, funding agencies, reviewers, the developers, and not least the users, foster a thriving and collaborative community, which incentivizes continued investment in the code base.

## Acknowledgments

All past and present developers’ contributions to the TURBOMOLE project are gratefully acknowledged. A list of TURBOMOLE contributors is available on the TURBOMOLE website. (91) Y. J. Franzke was supported by fellowships from Fonds der Chemischen Industrie (FCI, German Chemical Industry Fonds), Deutscher Akademischer Austauschdienst (DAAD, German Academic Exchange Service), and TURBOMOLE GmbH. C. Holzer and M. Krstić gratefully acknowledge funding by Volkswagen Stiftung. T. Begušić and E. Tapavicza acknowledge scientific support from J. Vaníček in the development and application of the TGA method. F. Della Sala acknowledges the financial support from ICSC–Centro Nazionale di Ricerca in High Performance Computing, Big Data and Quantum Computing, funded by European Union–NextGenerationEU–PNRR. D. A. Fedotov and S. Coriani acknowledge support from the European Unions Horizon 2020 research and innovation program under the Marie Skłodowska-Curie European Training Network COSINE (grant agreement no. 765739). J. H. Andersen and S. Coriani acknowledge financial support from the Independent Research Fund Denmark-DFF-FNU RP2 (grant no. 7014-00258B). S. Gillhuber is supported by a fellowship from Fonds der Chemischen Industrie (FCI no. 110160). R. Grotjahn acknowledges support via a Walter-Benjamin postdoctoral fellowship funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), no. 501114520. C. Hättig acknowledges support by the Deutsche Forschungsgemeneinschaft (DFG) via project Ha 2588/10-1. The Kaupp group has been supported by the Deutsche Forschungsgemeinschaft (DFG) via projects KA1187/14-1 and KA1187/14-2. M. Kehry acknowledges financial support by the DFG through the Transregional CRC 88 “Cooperative Effects in Homo- and Heterometallic Complexes” (project C1). F. Mack acknowledges support from TURBOMOLE GmbH and from the DFG through the CRC 1176 (Project Q5). The material in section 3.8 and parts of the material in section 3.12 is based upon work supported by the US National Science Foundation under CHE-2102568. The material in section 3.2 and in section 3.10 is based upon work supported by the US Department of Energy, Office of Basic Energy Sciences, under award number DE-SC0018352. A. Pausch was supported by a fellowship from Fonds der Chemischen Industrie and Studienstiftung des deutschen Volkes (German Academic Scholarship Foundation). E. Perlt and T. Schrader acknowledge support from the Carl Zeiss Foundation within the CZS Breakthroughs Program. M. Sierka and M. Sharma gratefully acknowledge financial support from Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within the CRC 1375 NOA, project A4, and from the Carl Zeiss Foundation within the CZS Breakthroughs Program. B. Samal and V. K. Voora were supported by the Department of Atomic Energy, Government of India, under project no. RTI2001. F. Weigend acknowledges support from the DFG through the Collaborative Research Centre (CRC) 1573 (Project Q). J.M. Yu acknowledges support from the US National Science Foundation under Grant DGE-1839285. B. Zerulla acknowledges support by the KIT through the “Virtual Materials Design” (VIRTMAT) project.

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(American Institute of Physics)A scheme that fully exploits mol. point group symmetry in direct 2nd-order many-body perturbation theory chem. shift calcns. based on gauge-including AOs is presented and implemented. Representative calcns. for the mols. B4(tBu)4 (Td symmetry) and [AlCp]4 (D2d symmetry) involving >600 basis functions demonstrate the applicability of the developed program to large sym. mols. (which could be otherwise not treated) and indicate the importance of electron correlation effects for the reliable prediction of NMR chem. shifts in larger mols.**8**Weiss, H.; Ahlrichs, R.; Häser, M. A direct algorithm for self-consistent-field linear-response theory and application to C_{60}-excitation-energies, oscillator-strengths, and frequency-dependent polarizabilities.*J. Chem. Phys.*1993,*99*, 1262– 1270, DOI: 10.1063/1.465370Google Scholar8https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXlvFSrur0%253D&md5=fff31402931eeaafe17933a6a2c6c472A direct algorithm for self-consistent-field linear response theory and application to C60 fullerene: excitation energies, oscillator strengths, and frequency-dependent polarizabilitiesWeiss, Horst; Ahlrichs, Reinhart; Haeser, MarcoJournal of Chemical Physics (1993), 99 (2), 1262-70CODEN: JCPSA6; ISSN:0021-9606.The authors present a direct self-consistent-field (SCF)-type algorithm and its implementation for the computation of linear response properties: excitation energies, oscillator strengths, and frequency-dependent polarizabilities within the time-dependent SCF or RPA. The treatment of singles CI for electronic excitations and Hartree-Fock instability criteria are covered as special cases. The algorithm is based on proven direct SCF methodol. This, together with full exploitation of mol. symmetry, opens the way to the treatment of large mols. Applications to C60 strongly support the assignment of the lowest-lying dipole allowed transition to the strong band at 3.8 eV.**9**Treutler, O.; Ahlrichs, R. Efficient molecular numerical integration schemes.*J. Chem. Phys.*1995,*102*, 346– 354, DOI: 10.1063/1.469408Google Scholar9https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXivVCkt7s%253D&md5=fd650c3774a1a95ad0cd95054392cbbaEfficient molecular numerical integration schemesTreutler, Oliver; Ahlrichs, ReinhartJournal of Chemical Physics (1995), 102 (1), 346-54CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)New grids for three-dimensional numerical integration are introduced. They include a new mapping for radial integration of the Gauss-Chebyshev type which seems to surpass in accuracy the existing integration schemes as proposed by Becke [J. Chem. Phys. 88, 2547 (1988)], Murray et al. [Mol. Phys., 78, 997 (1993)], or Gill et al. [Chem. Phys. Lett. 209, 506 (1993)]. Lebedev grids are employed for spherical integration. Open ended quadrature schemes are presented using the efficient Lobatto formula for the θ integration. These grids are employed for self-consistent d. functional calcns. using local approxn. and nonlocal corrections and are implemented into the program package TURBOMOLE. The results of grid tests and demonstrative applications of energy and esp. anal. gradient calcns. are given.**10**Bauernschmitt, R.; Ahlrichs, R. Treatment of electronic excitations within the adiabatic approximation of time dependent density functional theory.*Chem. Phys. Lett.*1996,*256*, 454– 464, DOI: 10.1016/0009-2614(96)00440-XGoogle Scholar10https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28XksFWltrs%253D&md5=484b6c8f0fa99d77d9aa02693ee6103aTreatment of electronic excitations within the adiabatic approximation of time dependent density functional theoryBauernschmitt, Ruedger; Ahlrichs, ReinhartChemical Physics Letters (1996), 256 (4,5), 454-464CODEN: CHPLBC; ISSN:0009-2614. (Elsevier)Time dependent d. functional methods are applied in the adiabatic approxn. to compute low-lying electronic excitations of N2, ethylene, formaldehyde, pyridine and porphin. Out of various local, gradient-cor. and hybrid (including exact exchange) functionals, the best results are obtained for the three-parameter Lee-Yang-Parr (B3LYP) functional proposed by Becke. B3LYP yields excitation energies about 0.4 eV too low but typically gives the correct ordering of states and constitutes a considerable improvement over HF-based approaches requiring comparable numerical work.**11**Furche, F.; Perdew, J. P. The performance of semi-local and hybrid density functionals in 3*d*transition metal chemistry.*J. Chem. Phys.*2006,*124*, 044103, DOI: 10.1063/1.2162161Google Scholar11https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XhtFertL8%253D&md5=0197e9427d342f56b90ff5629dcb39e3The performance of semilocal and hybrid density functionals in 3d transition-metal chemistryFurche, Filipp; Perdew, John P.Journal of Chemical Physics (2006), 124 (4), 044103/1-044103/27CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We investigate the performance of contemporary semilocal and hybrid d. functionals for bond energetics, structures, dipole moments, and harmonic frequencies of 3d transition-metal (TM) compds. by comparison with gas-phase expts. Special attention is given to the nonempirical metageneralized gradient approxn. (meta-GGA) of Tao, Perdew, Staroverov, and Scuseria (TPSS) [Phys. Rev. Lett. 91, 146401 (2003)], which has been implemented in TURBOMOLE for the present work. Trends and error patterns for classes of homologous compds. are analyzed, including dimers, monohydrides, mononitrides, monoxides, monofluorides, polyat. oxides and halogenides, carbonyls, and complexes with org. π ligands such as benzene and cyclopentadienyl. Weakly bound systems such as Ca2, Mn2, and Zn2 are discussed. We propose a ref. set of reaction energies for benchmark purposes. Our all-electron results with quadruple zeta valence basis sets validate semilocal d.-functional theory as the workhorse of computational TM chem. Typical errors in bond energies are substantially larger than in (org.) main group chem., however. The Becke-Perdew'86 [Phys. Rev. A 38, 3098 (1988); Phys. Rev. B 33, 8822 (1986)] GGA and the TPSS meta-GGA have the best price/performance ratio, while the TPSS hybrid functional achieves a slightly lower mean abs. error in bond energies. The popular Becke three-parameter hybrid B3LYP underbinds significantly and tends to overestimate bond distances; we give a possible explanation for this. We further show that hybrid mixing does not reduce the width of the error distribution on our ref. set. The error of a functional for the s-d transfer energy of a TM atom does not predict its error for TM bond energies and bond lengths. For semilocal functionals, self-interaction error in one- and three-electron bonds appears to be a major source of error in TM reaction energies. Nevertheless, TPSS predicts the correct ground-state symmetry in the vast majority of cases and rarely fails qual. This further confirms TPSS as a general purpose functional that works throughout the periodic table. We also give workstation timing comparisons for the 645-atom protein crambin.also published in Virtual J. Biol. Phys. Res. 2006, 11.

**12**Furche, F. Molecular tests of the random phase approximation to the exchange-correlation energy functional.*Phys. Rev. B*2001,*64*, 195120, DOI: 10.1103/PhysRevB.64.195120Google Scholar12https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3MXot1WjtLc%253D&md5=9002931f39f5832bf1c0551d273839eaMolecular tests of the random phase approximation to the exchange-correlation energy functionalFurche, FilippPhysical Review B: Condensed Matter and Materials Physics (2001), 64 (19), 195120/1-195120/8CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)The exchange-correlation energy functional within the RPA (RPA) is recast into an explicitly orbital-dependent form. A method to evaluate the functional in finite basis sets is introduced. The basis set dependence of the RPA correlation energy is analyzed. Extrapolation using large, correlation-consistent basis sets is essential for accurate ests. of RPA correlation energies. The potential energy curve of N2 is discussed. The RPA is found to recover most of the strong static correlation at large bond distance. Atomization energies of main-group mols. are rather uniformly underestimated by the RPA. The method performs better than generalized-gradient-type approxns. (GGA's) only for some electron-rich systems. However, the RPA functional is free of error cancellation between exchange and correlation, and behaves qual. correct in the high-d. limit, as is demonstrated by the coupling strength decompn. of the atomization energy of F2. The GGA short-range correlation correction to the RPA by Yan, Perdew, and Kurth [Phys. Rev. B 61,16,430 (2000)] does not seem to improve atomization energies consistently.**13**Bates, J. E.; Furche, F. Communication: Random phase approximation renormalized many-body perturbation theory.*J. Chem. Phys.*2013,*139*, 171103, DOI: 10.1063/1.4827254Google Scholar13https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXhsleiu73P&md5=bbfd81d02296e4acbc349e365c8b9b70Communication: Random phase approximation renormalized many-body perturbation theoryBates, Jefferson E.; Furche, FilippJournal of Chemical Physics (2013), 139 (17), 171103/1-171103/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We derive a renormalized many-body perturbation theory (MBPT) starting from the RPA. This RPA-renormalized perturbation theory extends the scope of single-ref. MBPT methods to small-gap systems without significantly increasing the computational cost. The leading correction to RPA, termed the approx. exchange kernel (AXK), substantially improves upon RPA atomization energies and ionization potentials without affecting other properties such as barrier heights where RPA is already accurate. Thus, AXK is more balanced than second-order screened exchange, which tends to overcorrect RPA for systems with stronger static correlation. Similarly, AXK avoids the divergence of second-order Moller-Plesset (MP2) theory for small gap systems and delivers a much more consistent performance than MP2 across the periodic table at comparable cost. RPA+AXK thus is an accurate, non-empirical, and robust tool to assess and improve semi-local d. functional theory for a wide range of systems previously inaccessible to first-principles electronic structure calcns. (c) 2013 American Institute of Physics.**14**Chen, G. P.; Voora, V. K.; Agee, M. M.; Balasubramani, S. G.; Furche, F. Random-phase approximation methods.*Annu. Rev. Phys. Chem.*2017,*68*, 421– 445, DOI: 10.1146/annurev-physchem-040215-112308Google Scholar14https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXksVCqtrw%253D&md5=770e90bfd4f1d98f0dfacd23add15003Random-Phase Approximation MethodsChen, Guo P.; Voora, Vamsee K.; Agee, Matthew M.; Balasubramani, Sree Ganesh; Furche, FilippAnnual Review of Physical Chemistry (2017), 68 (), 421-445CODEN: ARPLAP; ISSN:0066-426X. (Annual Reviews)RPA methods are rapidly emerging as cost-effective validation tools for semilocal d. functional computations. We present the theor. background of RPA in an intuitive rather than formal fashion, focusing on the phys. picture of screening and simple diagrammatic anal. A new decompn. of the RPA correlation energy into plasmonic modes leads to an appealing visualization of electron correlation in terms of charge d. fluctuations. Recent developments in the areas of beyond-RPA methods, RPA correlation potentials, and efficient algorithms for RPA energy and property calcns. are reviewed. The ability of RPA to approx. capture static correlation in mols. is quantified by an anal. of RPA natural occupation nos. We illustrate the use of RPA methods in applications to small-gap systems such as open-shell d- and f-element compds., radicals, and weakly bound complexes, where semilocal d. functional results exhibit strong functional dependence.**15**Chen, G. P.; Agee, M. M.; Furche, F. Performance and scope of perturbative corrections to random-phase approximation energies.*J. Chem. Theory Comput.*2018,*14*, 5701– 5714, DOI: 10.1021/acs.jctc.8b00777Google Scholar15https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhslOlsrfM&md5=15027735d3a819dacfdeceef8ba62b99Performance and Scope of Perturbative Corrections to Random-Phase Approximation EnergiesChen, Guo P.; Agee, Matthew M.; Furche, FilippJournal of Chemical Theory and Computation (2018), 14 (11), 5701-5714CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)It has been suspected since the early days of the RPA that corrections to RPA correlation energies result mostly from short-range correlation effects and are thus amenable to perturbation theory. Here we test this hypothesis by analyzing formal and numerical results for the most common beyond-RPA perturbative corrections, including the bare second-order exchange (SOX), second-order screened exchange (SOSEX), and approx. exchange kernel (AXK) methods. Our anal. is facilitated by efficient and robust algorithms based on the resoln.-of-the-identity (RI) approxn. and numerical frequency integration, which enable benchmark beyond-RPA calcns. on medium- and large-size mols. with size-independent accuracy. The AXK method systematically improves upon RPA, SOX, and SOSEX for reaction barrier heights, reaction energies, and noncovalent interaction energies of main-group compds. The improved accuracy of AXK compared with SOX and SOSEX is attributed to stronger screening of bare SOX in AXK. For reactions involving transition-metal compds., particularly 3d transition-metal dimers, the AXK correction is too small and can even have the wrong sign. These observations are rationalized by a measure ‾α of the effective coupling strength for beyond-RPA correlation. When the effective coupling strength increases beyond a crit. ‾α value of approx. 0.5, the RPA errors increase rapidly and perturbative corrections become unreliable. Thus, perturbation theory can systematically correct RPA but only for systems and properties qual. well captured by RPA, as indicated by small ‾α values.**16**Voora, V. K.; Balasubramani, S. G.; Furche, F. Variational generalized Kohn-Sham approach combining the random-phase-approximation and Green’s-function methods.*Phys. Rev. A*2019,*99*, 012518, DOI: 10.1103/PhysRevA.99.012518Google Scholar16https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhtFGrt7rK&md5=6a32a536f1b15fb37a635cbbf6928959Variational generalized Kohn-Sham approach combining the random-phase-approximation and Green's-function methodsVoora, Vamsee K.; Balasubramani, Sree Ganesh; Furche, FilippPhysical Review A (2019), 99 (1), 012518CODEN: PRAHC3; ISSN:2469-9934. (American Physical Society)A generalized Kohn-Sham (GKS) scheme which variationally minimizes the RPA ground-state energy with respect to the GKS one-particle d. matrix is put forward. We introduce the notion of functional self-consistent schemes, which vary the one-particle Kohn-Sham (KS) potential entering an explicitly potential-dependent exchange-correlation (XC) energy functional for a given d., and distinguish them from orbital-self-consistent (OSC) schemes, which vary the d., or the orbitals, d. matrix, or KS potential generating the d. It is shown that, for explicitly potential-dependent XC functionals, existing OSC schemes such as the optimized effective potential method violate the Hellmann-Feynman theorem for the d., producing a spurious discrepancy between the KS d. and the correct Hellmann-Feynman d. for approx. functionals. A functional self-consistency condition is derived which resolves this discrepancy by requiring the XC energy to be stationary with respect to the KS potential at fixed d. We approx. impose functional self-consistency by semicanonical projection (sp) of the Perdew, Burke, and Ernzerhof KS Hamiltonian. Variational OSC minimization of the resulting GKS-spRPA energy functional leads to a nonlocal correlation potential whose off-diagonal blocks correspond to orbital rotation gradients, while its diagonal blocks are related to the RPA self-energy at a real frequency. Quasiparticle GW energies are a first-order perturbative limit of the GKS-spRPA orbital energies; the lowest-order change of the total energy captures the renormalized singles excitation correction to RPA. GKS-spRPA orbital energies are found to approx. ionization potentials and fundamental gaps of atoms and mols. more accurately than semilocal d. functional approxns. (SL DFAs) or G0W0 and correct the spurious behavior of SL DFAs for neg. ions. GKS-spRPA energy differences are uniformly more accurate than the SL-RPA ones; improvements are modest for covalent bonds but substantial for weakly bound systems. GKS-spRPA energy minimization also removes the spurious max. in the SL-RPA potential energy curve of Be2 and produces a single Coulson-Fischer point at ∼2.7 times the equil. bond length in H2. GKS-spRPA thus corrects most d.-driven errors of SL-RPA, enhances the accuracy of RPA energy differences for electron-pair-conserving processes, and provides an intuitive one-electron GKS picture yielding ionization potentials energies and gaps of GW quality.**17**Bates, J. E.; Furche, F. Harnessing the meta-generalized gradient approximation for time-dependent density functional theory.*J. Chem. Phys.*2012,*137*, 164105, DOI: 10.1063/1.4759080Google Scholar17https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhsFGksLrF&md5=d2e6a673d974d619dfe23292efc905b1Harnessing the meta-generalized gradient approximation for time-dependent density functional theoryBates, Jefferson E.; Furche, FilippJournal of Chemical Physics (2012), 137 (16), 164105/1-164105/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)D. functionals within the meta-generalized gradient approxn. (MGGA) are widely used for ground-state electronic structure calcns. However, the gauge variance of the kinetic energy d. τ confounds applications of MGGAs to time-dependent systems, excited states, magnetic properties, and states with strong spin-orbit coupling. Becke and Tao used the paramagnetic c.d. to construct a gauge invariant generalized kinetic energy d. ̂τ. We show that τW ≤ ̂τ, where τW is the von Weizsaecker kinetic energy d. of a one-electron system. Thus, replacing τ by ̂τ leads to current-dependent MGGAs (cMGGAs) that are not only gauge invariant but also restore the accuracy of MGGAs in iso-orbital regions for time-dependent and current-carrying states. The current dependence of cMGGAs produces a vector exchange-correlation (XC) potential in the time-dependent adiabatic Kohn-Sham (KS) equations. While MGGA response properties of current-free ground states become manifestly gauge-variant to second order, linear response properties are affected by a new XC kernel appearing in the cMGGA magnetic orbital rotation Hessian. This kernel reflects the first-order coupling of KS orbitals due to changes in the paramagnetic c.d. and has apparently been ignored in previous MGGA response implementations. Inclusion of the current dependence increases total computation times by less than 50%. Benchmark applications to 109 adiabatic excitation energies using the Tao-Perdew-Staroverov-Scuseria (TPSS) MGGA and its hybrid version TPSSh show that cMGGA excitation energies are slightly lower than the MGGA ones on av., but exhibit fewer outliers. Similarly, the optical rotations of 13 small org. mols. show a small but systematic improvement upon inclusion of the magnetic XC kernel. We conclude that cMGGAs should replace MGGAs in all applications involving time-dependent or current-carrying states. (c) 2012 American Institute of Physics.**18**Plessow, P.; Weigend, F. Seminumerical calculation of the Hartree-Fock exchange matrix: Application to two-component procedures and efficient evaluation of local hybrid density functionals.*J. Comput. Chem.*2012,*33*, 810– 816, DOI: 10.1002/jcc.22901Google Scholar18https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38Xkt1GqtA%253D%253D&md5=12ab51bba04721dec5b05ebb1571c871Seminumerical calculation of the Hartree-Fock exchange matrix: Application to two-component procedures and efficient evaluation of local hybrid density functionalsPlessow, Philipp; Weigend, FlorianJournal of Computational Chemistry (2012), 33 (7), 810-816CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)A two-component extension of the seminumerical procedure for the calcn. of the Hartree-Fock (HF) exchange matrix recently presented by Neese et al. (Chem Phys 2009, 356, 98) was implemented into the program system TURBOMOLE. This allows for efficient self-consistent treatment of spin-orbit coupling at HF and hybrid d. functional theory level. One-component HF calcns. were performed to study the accuracy of integration grids and the exploitation of the mol. point group symmetry. The efficiency was tested, and for one-component calcns., compared to the implementation realized by Neese. Local hybrid d. functionals can be evaluated with this technique. The "prototype" of this class of functionals, Lh-BLYP, was applied to an org. mol. with more than 150 atoms. © 2012 Wiley Periodicals, Inc.; J. Comput. Chem., 2012.**19**Maier, T. M.; Bahmann, H.; Kaupp, M. Efficient Semi-numerical Implementation of Global and Local Hybrid Functionals for Time-Dependent Density Functional Theory.*J. Chem. Theory Comput.*2015,*11*, 4226– 4237, DOI: 10.1021/acs.jctc.5b00624Google Scholar19https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhtlemu73F&md5=9584d7e20b543b5fad82018d9360e5aaEfficient Semi-numerical Implementation of Global and Local Hybrid Functionals for Time-Dependent Density Functional TheoryMaier, Toni M.; Bahmann, Hilke; Kaupp, MartinJournal of Chemical Theory and Computation (2015), 11 (9), 4226-4237CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Local hybrid functionals with position-dependent exact-exchange admixt. offer increased flexibility compared to global hybrids. For sufficiently advanced functionals of this type, this is expected to hold also for a wide range of electronic excitations within time-dependent d. functional theory (TDDFT). Following a recent semi-numerical implementation of local hybrid functionals for ground-state self-consistent-field calcns., the first linear-response TDDFT implementation of local hybrids is reported, using a semi-numerical integration technique. The timings and accuracy of the semi-numerical implementation are evaluated by comparison with anal. schemes for time-dependent Hartree-Fock (TDHF) and for the TPSSh global hybrid. In combination with the RI approxn. to the Coulomb part of the kernel, the semi-numerical implementation is faster than the existing anal. TDDFT/TDHF implementation of global hybrid functionals in the TURBOMOLE code, even for small systems and moderate basis sets. Moreover, timings for global and local hybrids are practically equal for the semi-numerical scheme. The way to TDDFT calcns. with local hybrid functionals for large systems is thus now open, and more sophisticated parametrizations of local hybrids may be evaluated.**20**Furche, F.; Ahlrichs, R.; Wachsmann, C.; Weber, E.; Sobanski, A.; Vögtle, F.; Grimme, S. Circular dichroism of helicenes investigated by time-dependent density functional theory.*J. Am. Chem. Soc.*2000,*122*, 1717– 1724, DOI: 10.1021/ja991960sGoogle Scholar20https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXhtVCrsbc%253D&md5=cac069718b58a340de6c97e693379411Circular Dichroism of Helicenes Investigated by Time-Dependent Density Functional TheoryFurche, Filipp; Ahlrichs, Reinhart; Wachsmann, Claudia; Weber, Edwin; Sobanski, Adam; Voegtle, Fritz; Grimme, StefanJournal of the American Chemical Society (2000), 122 (8), 1717-1724CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)It is shown that mol. electronic CD can systematically be investigated by means of adiabatic time-dependent d. functional theory (TDDFT). We briefly summarize the theory and outline its extension for the calcn. of rotatory strengths. A new, efficient algorithm has been implemented in the TURBOMOLE program package for the present work, making large-scale applications feasible. The study of CD in helicenes has played a crucial role in the understanding of mol. optical activity. We present the first ab initio simulation of electronic CD spectra of [n]helicenes, n = 4-7, 12. Substituent effects are considered for the 2,15-dicyano and 2,15-dimethoxy derivates of hexahelicene; exptl. CD spectra of these compds. were newly recorded for the present work. The calcns. correctly reproduce the most important spectral features and greatly facilitate interpretation. We propose assignments of the low-energy bands in terms of individual excited states. Changes in the obsd. spectra depending on the no. of rings and substitution patterns are worked out and rationalized. Merits and limitations of TDDFT in chem. applications are discussed.**21**Furche, F.; Rappoport, D. Density functional methods for excited states: equilibrium structure and electronic spectra In*Computational Photochemistry*; Olivucci, M., Ed.; Computational and Theoretical Chemistry, Vol. 16; Elsevier: Amsterdam, The Netherlands, 2005; pp 93– 128.Google ScholarThere is no corresponding record for this reference.**22**Kattannek, M.*Entwicklung und Implementierung optimierter Algorithmen für molekulare Hartree-Fock- und Dichtefunktional-Rechnungen*. Ph.D. Thesis, Universität Karlsruhe (TH), Karlsruhe, Germany, 2006.Google ScholarThere is no corresponding record for this reference.**23**Eshuis, H.; Yarkony, J.; Furche, F. Fast computation of molecular random phase approximation correlation energies using resolution of the identity and imaginary frequency integration.*J. Chem. Phys.*2010,*132*, 234114, DOI: 10.1063/1.3442749Google Scholar23https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXnslSlsLw%253D&md5=90e538090088f1ded996c402ffc7b342Fast computation of molecular random phase approximation correlation energies using resolution of the identity and imaginary frequency integrationEshuis, Henk; Yarkony, Julian; Furche, FilippJournal of Chemical Physics (2010), 132 (23), 234114/1-234114/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The RPA is an increasingly popular post-Kohn-Sham correlation method, but its high computational cost has limited mol. applications to systems with few atoms. Here we present an efficient implementation of RPA correlation energies based on a combination of resoln. of the identity (RI) and imaginary frequency integration techniques. We show that the RI approxn. to four-index electron repulsion integrals leads to a variational upper bound to the exact RPA correlation energy if the Coulomb metric is used. Auxiliary basis sets optimized for second-order Moller-Plesset (MP2) calcns. are well suitable for RPA, as is demonstrated for the HEAT and MOLEKEL benchmark sets. Using imaginary frequency integration rather than diagonalization to compute the matrix square root necessary for RPA, evaluation of the RPA correlation energy requires O(N4logN) operations and O(N3) storage only; the price for this dramatic improvement over existing algorithms is a numerical quadrature. We propose a numerical integration scheme that is exact in the two-orbital case and converges exponentially with the no. of grid points. For most systems, 30-40 grid points yield μH accuracy in triple zeta basis sets, but much larger grids are necessary for small gap systems. The lowest-order approxn. to the present method is a post-Kohn-Sham frequency-domain version of opposite-spin Laplace-transform RI-MP2. Timings for polyacenes with up to 30 atoms show speed-ups of two orders of magnitude over previous implementations. The present approach makes it possible to routinely compute RPA correlation energies of systems well beyond 100 atoms, as is demonstrated for the octapeptide angiotensin II. (c) 2010 American Institute of Physics.**24**Furche, F.; Krull, B. T.; Nguyen, B. D.; Kwon, J. Accelerating molecular property calculations with nonorthonormal Krylov space methods.*J. Chem. Phys.*2016,*144*, 174105, DOI: 10.1063/1.4947245Google Scholar24https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xntlygsb0%253D&md5=db5f81c09ea89acd83838ae2f6e4ed69Accelerating molecular property calculations with nonorthonormal Krylov space methodsFurche, Filipp; Krull, Brandon T.; Nguyen, Brian D.; Kwon, JakeJournal of Chemical Physics (2016), 144 (17), 174105/1-174105/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We formulate Krylov space methods for large eigenvalue problems and linear equation systems that take advantage of decreasing residual norms to reduce the cost of matrix-vector multiplication. The residuals are used as subspace basis without prior orthonormalization, which leads to generalized eigenvalue problems or linear equation systems on the Krylov space. These nonorthonormal Krylov space (nKs) algorithms are favorable for large matrixes with irregular sparsity patterns whose elements are computed on the fly, because fewer operations are necessary as the residual norm decreases as compared to the conventional method, while errors in the desired eigenpairs and soln. vectors remain small. We consider real sym. and symplectic eigenvalue problems as well as linear equation systems and Sylvester equations as they appear in CI and response theory. The nKs method can be implemented in existing electronic structure codes with minor modifications and yields speed-ups of 1.2-1.8 in typical time-dependent Hartree-Fock and d. functional applications without accuracy loss. The algorithm can compute entire linear subspaces simultaneously which benefits electronic spectra and force const. calcns. requiring many eigenpairs or soln. vectors. The nKs approach is related to difference d. methods in electronic ground state calcns. and particularly efficient for integral direct computations of exchange-type contractions. By combination with resoln.-of-the-identity methods for Coulomb contractions, three- to fivefold speed-ups of hybrid time-dependent d. functional excited state and response calcns. are achieved. (c) 2016 American Institute of Physics.**25**Deglmann, P.; Furche, F.; Ahlrichs, R. An efficient implementation of second analytical derivatives for density functional methods.*Chem. Phys. Lett.*2002,*362*, 511– 518, DOI: 10.1016/S0009-2614(02)01084-9Google Scholar25https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XntVWku7k%253D&md5=e989b0048565e1e5f5c620e8d235fdf4An efficient implementation of second analytical derivatives for density functional methodsDeglmann, Peter; Furche, Filipp; Ahlrichs, ReinhartChemical Physics Letters (2002), 362 (5,6), 511-518CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)An implementation is presented of anal. 2nd-order geometric derivs. for d. functional methods using Gaussian basis sets. Key features include a stable and efficient numerical quadrature, the direct iterative soln. of the coupled perturbed Kohn-Sham equations, integral prescreening based on rigorous ests., and exploitation of point group symmetry for all finite groups. Benchmark results indicate a moderate cubic growth of CPU and storage requirements with system size; low symmetry mols. with ≤100 heavy atoms can be treated on personal computers. The performance of gradient cor. functionals in predicting IR spectra of larger mols. is exemplified for transition metal carbonyl complexes.**26**Grimme, S.; Furche, F.; Ahlrichs, R. An improved method for density functional calculations of the frequency-dependent optical rotation.*Chem. Phys. Lett.*2002,*361*, 321– 328, DOI: 10.1016/S0009-2614(02)00975-2Google Scholar26https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38Xls1Wnu74%253D&md5=c8dee9031fb2b82d73d409ccd28e8a1aAn improved method for density functional calculations of the frequency-dependent optical rotationGrimme, Stefan; Furche, Filipp; Ahlrichs, ReinhartChemical Physics Letters (2002), 361 (3,4), 321-328CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)D. functional theory calcns. of the frequency-dependent optical rotation (OR) provide a simple and effective way to det. the abs. configuration of chiral mols. The authors report two major improvements of the method here. Firstly, the dipole velocity form of the OR is introduced. In contrast to the commonly employed dipole length form, the dipole velocity form is gauge-origin invariant in any basis set. The authors present a basis set convergence study which shows that basis sets of augmented double zeta quality yield sufficiently accurate results in most cases. The deviation between the length and velocity forms provides a convenient measure of the basis set error. Secondly, the authors extend the resoln. of the identity (RI) approxn. to frequency-dependent OR calcns. Using the Coulomb norm and auxiliary basis sets optimized for ground-state calcns., the authors obtain RI errors that are negligible compared to errors due to the basis set and the d. functional. Total computation times are reduced by a up to a factor of 10, as is demonstrated for a benchmark set of eight different mols.**27**Deglmann, P.; May, K.; Furche, F.; Ahlrichs, R. Nuclear second analytical derivative calculations using auxiliary basis set expansions.*Chem. Phys. Lett.*2004,*384*, 103– 107, DOI: 10.1016/j.cplett.2003.11.080Google Scholar27https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXisFyksw%253D%253D&md5=f9f90b97e779f11013e25c471a1edd06Nuclear second analytical derivative calculations using auxiliary basis set expansionsDeglmann, Peter; May, Klaus; Furche, Filipp; Ahlrichs, ReinhartChemical Physics Letters (2004), 384 (1-3), 103-107CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)We present an efficient implementation of second order anal. derivs. (force consts.) within d. functional theory which applies to non-hybrid functionals. The dominant step in force const. treatments concerns the soln. of the coupled perturbed Kohn-Sham equations (CPKS). We employ an auxiliary basis expansion, the RI-J technique (resoln. of the identity for the interelectronic Coulomb energy J), to accelerate the soln. of the CPKS equations. This results in pronounced redns. of computational work and implies insignificant losses in accuracy as is shown by extensive tests. A force const. calcn. of cyanocobalamin is reported as a demonstrative application.**28**Rappoport, D.; Furche, F. Analytical time-dependent density functional derivative methods within the RI-J approximation, an approach to excited states of large molecules.*J. Chem. Phys.*2005,*122*, 064105, DOI: 10.1063/1.1844492Google Scholar28https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXhtlWrtbs%253D&md5=1f02a196cd9cf27df85d89b6bc671803Analytical time-dependent density functional derivative methods within the RI-J approximation, an approach to excited states of large moleculesRappoport, Dmitrij; Furche, FilippJournal of Chemical Physics (2005), 122 (6), 064105/1-064105/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Time-dependent d. functional theory (TDDFT) is now well established as an efficient method for mol. excited state treatments. In this work, we introduce the resoln. of the identity approxn. for the Coulomb energy (RI-J) to excited state gradient calcns. In combination with nonhybrid functionals, the RI-J approxn. leads to speed ups in total timings of an order of magnitude compared to the conventional method; this is demonstrated for oligothiophenes with up to 40 monomeric units and adamantane clusters. We assess the accuracy of the computed adiabatic excitation energies, excited state structures, and vibrational frequencies on a set of 36 excited states. The error introduced by the RI-J approxn. is found to be negligible compared to deficiencies of std. basis sets and functionals. Auxiliary basis sets optimized for ground states are suitable for excited state calcns. with small modifications. In conclusion, the RI-J approxn. significantly extends the scope of applications of anal. TDDFT deriv. methods in photophysics and photochem.**29**Rappoport, D.; Furche, F. Lagrangian approach to molecular vibrational Raman intensities using time-dependent hybrid density functional theory.*J. Chem. Phys.*2007,*126*, 201104, DOI: 10.1063/1.2744026Google Scholar29https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXmtlCgsr4%253D&md5=6f1cc04320089e743cfd14e3d35c7006Lagrangian approach to molecular vibrational Raman intensities using time-dependent hybrid density functional theoryRappoport, Dmitrij; Furche, FilippJournal of Chemical Physics (2007), 126 (20), 201104/1-201104/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The authors propose a new route to vibrational Raman intensities based on anal. derivs. of a fully variational polarizability Lagrangian. The Lagrangian is constructed to recover the neg. frequency-dependent polarizability of time-dependent Hartree-Fock or adiabatic (hybrid) d. functional theory at its stationary point. By virtue of the variational principle, first-order polarizability derivs. can be computed without using deriv. MO coeffs. As a result, the intensities of all Raman-active modes within the double harmonic approxn. are obtained at approx. the same cost as the frequency-dependent polarizability itself. This corresponds to a redn. of the scaling of computational expense by one power of the system size compared to a force const. calcn. and to previous implementations. Since the Raman intensity calcn. is independent of the harmonic force const. calcn. more, computationally demanding d. functionals or basis sets may be used to compute the polarizability gradient without much affecting the total time required to compute a Raman spectrum. As illustrated for fullerene C60, the present approach considerably extends the domain of mol. vibrational Raman calcns. at the (hybrid) d. functional level. The accuracy of abs. and relative Raman intensities of benzene obtained using the PBE0 hybrid functional is assessed by comparison with expt.**30**Send, R.; Furche, F. First-order nonadiabatic couplings from time-dependent hybrid density functional response theory: Consistent formalism, implementation, and performance.*J. Chem. Phys.*2010,*132*, 044107, DOI: 10.1063/1.3292571Google Scholar30https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXhtlCjurw%253D&md5=bde1d95c7ca694b0ed7b96b9efc75c05First-order nonadiabatic couplings from time-dependent hybrid density functional response theory: Consistent formalism, implementation, and performanceSend, Robert; Furche, FilippJournal of Chemical Physics (2010), 132 (4), 044107/1-044107/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)First-order nonadiabatic coupling matrix elements (NACMEs) are key for phenomena such as nonradiative transitions and excited-state decay, yet a consistent and practical first principles treatment has been elusive for mols. with more than a few heavy atoms. Here we present theory, implementation using Gaussian basis sets, and benchmarks of first-order NACMEs between ground and excited states in the framework of time-dependent hybrid d. functional theory (TDDFT). A time-dependent response approach to NACMEs which avoids explicit computation of excited-state wave functions is outlined. In contrast to previous approaches, the present treatment produces exact anal. deriv. couplings between time-dependent Kohn-Sham (TDKS) determinants in a finite atom-centered basis set. As in anal. gradient theory, deriv. MO coeffs. can be eliminated, making the computational cost independent of the no. of nuclear degrees of freedom. Our expression reduces to the exact Chernyak-Mukamel formula for first-order NACMEs in the complete basis-set limit, but greatly improves basis-set convergence in finite atom-centered basis sets due to addnl. Pulay type terms. The Chernyak-Mukamel formula is shown to be equiv. to the Hellmann-Feynman contribution in anal. gradient theory. Our formalism may be implemented in TDDFT anal. excited-state gradient codes with minor modifications. Tests for systems with up to 147 atoms show that evaluation of first-order NACMEs causes total computation times to increase by an insignificant 10% on av. The resoln.-of-the-identity approxn. for the Coulomb energy (RI-J) reduces the computational cost by an order of magnitude for nonhybrid functionals, while errors are insignificant with std. auxiliary basis sets. We compare the computed NACMEs to full CI (FCI) in benchmark results for diat. mols.; hybrid TDDFT and FCI are found to be in agreement for regions of the potential energy curve where the Kohn-Sham ground-state ref. is stable and the character of the excitation is properly captured by the present functionals. With these developments, nonadiabatic mol. dynamics simulations of mol. systems in the 100 atoms regime are within reach. (c) 2010 American Institute of Physics.**31**Burow, A. M.; Bates, J. E.; Furche, F.; Eshuis, H. Analytical first-order molecular properties and forces within the adiabatic connection random phase approximation.*J. Chem. Theory Comput.*2014,*10*, 180– 194, DOI: 10.1021/ct4008553Google Scholar31https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXhsl2ntLvP&md5=40b45f75169f713f462399a8b89406feAnalytical First-Order Molecular Properties and Forces within the Adiabatic Connection Random Phase ApproximationBurow, Asbjorn M.; Bates, Jefferson E.; Furche, Filipp; Eshuis, HenkJournal of Chemical Theory and Computation (2014), 10 (1), 180-194CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The RPA is an increasingly popular method for computing mol. ground-state correlation energies within the adiabatic connection fluctuation-dissipation theorem framework of d. functional theory. We present an efficient anal. implementation of first-order RPA mol. properties and nuclear forces using the resoln.-of-the-identity (RI) approxn. and imaginary frequency integration. The centerpiece of our approach is a variational RPA energy Lagrangian invariant under unitary transformations of occupied and virtual ref. orbitals, resp. Its construction requires the soln. of a single coupled-perturbed Kohn-Sham equation independent of the no. of perturbations. Energy gradients with respect to nuclear displacements and other first-order properties such as one-particle densities or dipole moments are obtained from partial derivs. of the Lagrangian. Our RPA energy gradient implementation exhibits the same O(N4logN) scaling with system size N as a single-point RPA energy calcn. In typical applications, the cost for computing the entire gradient vector with respect to nuclear displacements is ∼5 times that of a single-point RPA energy calcn. Derivs. of the quadrature nodes and wts. used for frequency integration are essential for RPA gradients with an accuracy consistent with RPA energies and can be included in our approach. The quality of RPA equil. structures is assessed by comparison to accurate theor. and exptl. data for covalent main group compds., weakly bonded dimers, and transition metal complexes. RPA outperforms semilocal functionals as well as second-order Moller-Plesset (MP2) theory, which fails badly for the transition metal compds. Dipole moments of polarizable mols. and weakly bound dimers show a similar trend. RPA harmonic vibrational frequencies are nearly of coupled cluster singles, doubles, and perturbative triples quality for a set of main group compds. Compared to the ring-coupled cluster based implementation of Rekkedal et al., our method scales better by two powers of N and supports a semilocal Kohn-Sham ref. The latter is essential for the good performance of RPA in small-gap systems.**32**Ou, Q.; Bellchambers, G. D.; Furche, F.; Subotnik, J. E. First-order derivative couplings between excited states from adiabatic TDDFT response theory.*J. Chem. Phys.*2015,*142*, 064114, DOI: 10.1063/1.4906941Google Scholar32https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXis12jsLg%253D&md5=9d1e4cbc511a1af4fbe78b3c852ed160First-order derivative couplings between excited states from adiabatic TDDFT response theoryOu, Qi; Bellchambers, Gregory D.; Furche, Filipp; Subotnik, Joseph E.Journal of Chemical Physics (2015), 142 (6), 064114/1-064114/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a complete derivation of deriv. couplings between excited states in the framework of adiabatic time-dependent d. functional response theory. Explicit working equations are given and the resulting deriv. couplings are compared with deriv. couplings from a pseudo-wavefunction ansatz. For degenerate excited states, i.e., close to a conical intersection (CI), the two approaches are identical apart from an antisym. overlap term. However, if the difference between two excitation energies equals another excitation energy, the couplings from response theory exhibit an unphys. divergence. This spurious behavior is a result of the adiabatic or static kernel approxn. of time-dependent d. functional theory leading to an incorrect anal. structure of the quadratic response function. Numerical examples for couplings close to a CI and for well-sepd. electronic states are given. (c) 2015 American Institute of Physics.**33**Klawohn, S.; Bahmann, H.; Kaupp, M. Implementation of Molecular Gradients for Local Hybrid Density Functionals Using Seminumerical Integration Techniques.*J. Chem. Theory Comput.*2016,*12*, 4254– 4262, DOI: 10.1021/acs.jctc.6b00486Google Scholar33https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhtFyitL3L&md5=58d4778ffdc0df9ad20dac50b2d3f30aImplementation of Molecular Gradients for Local Hybrid Density Functionals Using Seminumerical Integration TechniquesKlawohn, Sascha; Bahmann, Hilke; Kaupp, MartinJournal of Chemical Theory and Computation (2016), 12 (9), 4254-4262CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present the first implementation of the deriv. of the local hybrid exchange-correlation energy with respect to the displacement of nuclei in a Gaussian-type at. basis set. This extends a recent efficient implementation of local hybrid functionals for self-consistent Kohn-Sham and linear-response TDDFT calcns. into the TURBOMOLE program package. In contrast to seminumerical schemes for global exact-exchange admixts. and to the related SCF and TDDFT implementations of local hybrid functionals, addnl. anal. integrals have to be evaluated at each grid point in the case of mol. gradients. The overall efficiency of the present scheme is improved through prescreening with the d. matrix (P-junctions), as well as with spherical overlap ests. (S-junctions). Comparative timings for structure optimizations with local vs global hybrid functionals are discussed while gauging the accuracy for S- and P-junctions using varying thresholds. Local hybrids are furthermore assessed for structure optimization and harmonic vibrational frequency calcns. (using numerical second derivs.) of a selection of test systems, comparing with exptl. data and some widely used d. functionals.**34**Parker, S. M.; Rappoport, D.; Furche, F. Quadratic Response Properties from TDDFT: Trials and Tribulations.*J. Chem. Theory Comput.*2018,*14*, 807– 819, DOI: 10.1021/acs.jctc.7b01008Google Scholar34https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhvFOiur%252FP&md5=56465efec8e59421f7160aee3ad2be98Quadratic Response Properties from TDDFT: Trials and TribulationsParker, Shane M.; Rappoport, Dmitrij; Furche, FilippJournal of Chemical Theory and Computation (2018), 14 (2), 807-819CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The authors report on the efficient TURBOMOLE implementation of quadratic response properties within the time-dependent d. functional theory (TDDFT) context that includes the static and dynamic dipole hyperpolarizability, ground-to-excited-state 2-photon absorption amplitudes (through a single residue) and state-to-state 1-photon absorption amplitudes (through a double residue). The implementation makes full use of arbitrary (including non-Abelian) point-group symmetry as well as permutational symmetry and enables the calcn. of nonlinear properties with hybrid d. functionals for mols. with hundreds of atoms and thousands of basis functions at a cost that is a fixed multiple of the cost of the corresponding linear properties. Using the PBE0 hybrid d. functional, excited-state absorption spectra computed within the pseudowavefunction approach contain the qual. features of transient absorption spectra tracking excimer formation in perylene diimide dimers, 2-photon absorption cross sections for highly twisted fused porphyrin chains are semiquant. reproduced, and the computed dynamic hyperpolarizability of several calix[4]arene stereoisomers yield simulated hyper-Raleigh scattering signals consistent with expt. The incorrect pole structure of adiabatic TDDFT properties can cause incorrect excited-state absorption spectra and overly resonant hyperpolarizabilities, and discuss possible remedies.**35**Grotjahn, R.; Furche, F.; Kaupp, M. Development and Implementation of Excited-State Gradients for Local Hybrid Functionals.*J. Chem. Theory Comput.*2019,*15*, 5508– 5522, DOI: 10.1021/acs.jctc.9b00659Google Scholar35https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhs12jsbbE&md5=5943c43f7445f4cf7546917346d65f36Development and Implementation of Excited-State Gradients for Local Hybrid FunctionalsGrotjahn, Robin; Furche, Filipp; Kaupp, MartinJournal of Chemical Theory and Computation (2019), 15 (10), 5508-5522CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Local hybrid functionals are a relatively recent class of exchange-correlation functionals that use a real-space dependent admixt. of exact exchange. Here, we present the first implementation of time-dependent d. functional theory excited-state gradients for these functionals. Based on the ansatz of a fully variational auxiliary Lagrangian of the excitation energy, the working equations for the case of a local hybrid functional are deduced. For the implementation, we derive the third-order functional derivs. used in the hyper-kernel and kernel-gradients following a seminumerical integration scheme. The first assessment for a test set of small mols. reveals competitive performance for excited-state structural parameters with typical mean abs. errors (MAEs) of 1.2 pm (PBE0: 1.4 pm) for bond lengths as well as for vibrational frequencies with typical MAEs of 81 cm-1 (PBE0: 76 cm-1). Excellent performance was found for adiabatic triplet excitation energies with typical MAEs of 0.08 eV (PBE0: 0.32 eV). In a detailed case anal. of the first singlet and triplet excited states of formaldehyde the conceptional (dis-)advantages of the local hybrid scheme for excited-state gradients are exposed.**36**Franzke, Y. J.; Weigend, F. NMR Shielding Tensors and Chemical Shifts in Scalar-Relativistic Local Exact Two-Component Theory.*J. Chem. Theory Comput.*2019,*15*, 1028– 1043, DOI: 10.1021/acs.jctc.8b01084Google Scholar36https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXms1CksA%253D%253D&md5=ec986b998897a388332ce39db9b55d00NMR Shielding Tensors and Chemical Shifts in Scalar-Relativistic Local Exact Two-Component TheoryFranzke, Yannick J.; Weigend, FlorianJournal of Chemical Theory and Computation (2019), 15 (2), 1028-1043CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)An efficient formulation of scalar-relativistic NMR shielding tensors based on (one-electron) spin-free exact two-component theory (X2C) is presented. It utilizes the diagonal local approxn. to the unitary decoupling matrix (DLU), which we recently applied to anal. derivs. [J. Chem. Phys. 2018, 148, 104110]. This allows for routine calcns. of large mols. with heavy atoms. Here, the computation times of the NMR shielding tensors of all nuclei formally scale cubically with the size of the system, while memory demands increase quadratically. Efficiency is demonstrated for heavy-element clusters and organometallic complexes with more than 120 atoms and 3200 contracted basis functions. The accuracy of the DLU scheme is evaluated based on 13C, 17O, 29Si, 73Ge, 119Sn, 129Xe, 183W, and 207Pb NMR shielding consts. and chem. shifts using different basis sets. The finite nucleus model is available throughout.**37**Wodyński, A.; Kaupp, M. Density Functional Calculations of EPR g- and Hyperfine-Coupling Tensors Using the Exact Two-Component (X2C) Transformation and Efficient Approximations to the Two-Electron Spin-Orbit Terms.*J. Phys. Chem. A*2019,*123*, 5660– 5672, DOI: 10.1021/acs.jpca.9b03979Google Scholar37https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhtFaqsbbN&md5=3ee94bbab3a180c002df064fb62a40fdDensity Functional Calculations of Electron Paramagnetic Resonance g- and Hyperfine-Coupling Tensors Using the Exact Two-Component (X2C) Transformation and Efficient Approximations to the Two-Electron Spin-Orbit TermsWodynski, Artur; Kaupp, MartinJournal of Physical Chemistry A (2019), 123 (26), 5660-5672CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)A two-component quasirelativistic d. functional theory implementation of the computation of hyperfine and g-tensors at exact two-component (X2C) and Douglas-Kroll-Hess method (DKH) levels in the Turbomole code is reported and tested for a series of smaller 3d1, 4d1, and 5d1 complexes, as well as for some larger 5d7 Ir and Pt systems in comparison with earlier four-component matrix-Dirac-Kohn-Sham results. A main emphasis is placed on efficient approxns. to the two-electron spin-orbit contributions, comparing an existing implementation of two variants of Boettger's "scaled nuclear spin-orbit" (SNSO) approxn. in the code with a newly implemented at. mean-field spin-orbit (AMFSO) approxn. The different variants perform overall comparably well with the four-component data. The AMFSO approxn. has the added advantage of being able to include the spin-other-orbit contributions arising from the Gaunt term of relativistic electron-electron interactions. These are of comparably larger importance for the 3d complexes than for their heavier homologues. The excellent agreement between X2C and four-component ESR parameter results provides the opportunity to treat large systems efficiently and accurately with the computationally more expedient two-component quasirelativistic methodol.**38**Franzke, Y. J.; Yu, J. M. Hyperfine Coupling Constants in Local Exact Two-Component Theory.*J. Chem. Theory Comput.*2022,*18*, 323– 343, DOI: 10.1021/acs.jctc.1c01027Google Scholar38https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXislOmtrrL&md5=7e26b714f9da0928afd72eacf3772e2cHyperfine Coupling Constants in Local Exact Two-Component TheoryFranzke, Yannick J.; Yu, Jason M.Journal of Chemical Theory and Computation (2022), 18 (1), 323-343CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a highly efficient implementation of the electron-nucleus hyperfine coupling matrix within the one-electron exact two-component (X2C) theory. The complete deriv. of the X2C Hamiltonian is formed, i.e., the derivs. of the unitary decoupling transformation are considered. This requires the soln. of the response and Sylvester equations, consequently increasing the computational costs. Therefore, we apply the diagonal local approxn. to the unitary decoupling transformation (DLU). The finite nucleus model is employed for both the scalar potential and the vector potential. Two-electron picture-change effects are modeled with the (modified) screened nuclear spin-orbit approach. Our implementation is fully integral direct and OpenMP-parallelized. An extensive benchmark study regarding the Hamiltonian, the basis set, and the d. functional approxn. is carried out for a set of 12-17 transition-metal compds. The error introduced by DLU is negligible, and the DLU-X2C Hamiltonian accurately reproduces its four-component "fully" relativistic parent results. Functionals with a large amt. of Hartree-Fock exchange such as CAM-QTP-02 and ωB97X-D are generally favorable. The pure d. functional r2SCAN performs remarkably and even outperforms the common hybrid functionals TPSSh and CAM-B3LYP. Fully uncontracted basis sets or contracted quadruple-ζ bases are required for accurate results. The capability of our implementation is demonstrated for [Pt(C6Cl5)4]- with more than 4700 primitive basis functions and four rare-earth single-mol. magnets: [La(OAr*)3]-, [Lu(NR2)3]-, [Lu(OAr*)3]-, and [TbPc2]-. Here, the results with the spin-orbit DLU-X2C Hamiltonian are in an excellent agreement with the exptl. findings of all Pt, La, Lu, and Tb mols.**39**Franzke, Y. J.; Yu, J. M. Quasi-Relativistic Calculation of EPR*g*Tensors with Derivatives of the Decoupling Transformation, Gauge-Including Atomic Orbitals, and Magnetic Balance.*J. Chem. Theory Comput.*2022,*18*, 2246– 2266, DOI: 10.1021/acs.jctc.1c01175Google Scholar39https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XotlSntLs%253D&md5=7ce3f95f49504ba52bcdb96e68294306Quasi-Relativistic Calculation of EPR g Tensors with Derivatives of the Decoupling Transformation, Gauge-Including Atomic Orbitals, and Magnetic BalanceFranzke, Yannick J.; Yu, Jason M.Journal of Chemical Theory and Computation (2022), 18 (4), 2246-2266CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present an exact two-component (X2C) ansatz for the EPR g tensor using gauge-including AOs (GIAOs) and a magnetically balanced basis set expansion. In contrast to previous X2C and four-component relativistic ans.ovrddot.atze for the g tensor, this implementation results in a gauge-origin-invariant formalism. Furthermore, the derivs. of the relativistic decoupling matrix are incorporated to form the complete anal. deriv. of the X2C Hamiltonian. To reduce the assocd. computational costs, we apply the diagonal local approxn. to the unitary decoupling transformation (DLU). The quasi-relativistic X2C and DLU-X2C Hamiltonians accurately reproduce the results of the parent four-component relativistic theory when accounting for two-electron picture-change effects with the modified screened nuclear spin-orbit approxn. in the resp. one-electron integrals and integral derivs. According to our benchmark studies, the uncontracted Dyall and segmented-contracted Karlsruhe x2c-type basis sets perform well when compared to large even-tempered basis sets. Moreover, (range-sepd.) hybrid d. functional approxns. such as LC-ωPBE and ωB97X-D are needed to match the exptl. findings. The impact of the GIAOs depends on the distribution of the spin d., and their use may change the Δg shifts by 10-50% as shown for [(C5Me5)2Y(μ-S)2Mo(μ-S)2Y(C5Me5)2]-. Routine calcns. of large mols. are possible with widely available and comparably low-cost hardware as demonstrated for [Pt(C6Cl5)4]- with 3003 basis functions and three spin-(1/2) La(II) and Lu(II) compds., for which we observe good agreement with the exptl. findings.**40**Eichkorn, K.; Treutler, O.; Öhm, H.; Häser, M.; Ahlrichs, R. Auxiliary Basis Sets to Approximate Coulomb Potentials.*Chem. Phys. Lett.*1995,*240*, 283– 290, DOI: 10.1016/0009-2614(95)00621-AGoogle Scholar40https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXms1ylu74%253D&md5=2ccc5127a1afa8c7626fe8dad7fa71daAuxiliary basis sets to approximate Coulomb potentialsEichkorn, Karin; Treutler, Oliver; Oehm, Holger; Haeser, Marco; Ahlrichs, ReinhartChemical Physics Letters (1995), 240 (4), 283-90CODEN: CHPLBC; ISSN:0009-2614. (Elsevier)We demonstrate accuracy and computational efficiency resulting from an approx. treatment of Coulomb operators which is based on the expansion of mol. electron densities in atom-centered auxiliary basis sets. This is of special importance in d. functional methods which sep. the treatment of Coulomb and exchange-correlation terms. Auxiliary basis sets are optimized as much as possible for isolated atoms and then augmented for use in mol. electronic structure calcns. For mols. involving atoms up to Br this typically affects energies by only 10-4 au per atom, and computed structure consts. by less than 0.1 pm in bond distances and 0.1° in bond angles.**41**Eichkorn, K.; Weigend, F.; Treutler, O.; Ahlrichs, R. Auxiliary basis sets for main row atoms and transition metals and their use to approximate Coulomb potentials.*Theor. Chem. Acc.*1997,*97*, 119– 124, DOI: 10.1007/s002140050244Google Scholar41https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXmvFCisbg%253D&md5=d6af66738d1910b99d1f86310d315e46Auxiliary basis sets for main row atoms and transition metals and their use to approximate Coulomb potentialsEichkorn, Karin; Weigend, Florian; Treutler, Oliver; Ahlrichs, ReinhartTheoretical Chemistry Accounts (1997), 97 (1-4), 119-124CODEN: TCACFW; ISSN:1432-881X. (Springer)We present auxiliary basis sets for the atoms H to At - excluding the lanthanides - optimized for an efficient treatment of mol. electronic Coulomb interactions. For atoms beyond Kr our approach is based on effective core potentials to describe core electrons. The approx. representation of the electron d. in terms of the auxiliary basis has virtually no effect on computed structures and affects the energy by <10-4 a.u. per atom. Efficiency is demonstrated in applications for mols. with ≤300 atoms and 2500 basis functions.**42**Weigend, F. A fully direct RI-HF algorithm: Implementation, optimized auxiliary basis sets, demonstration of accuracy and efficiency.*Phys. Chem. Chem. Phys.*2002,*4*, 4285– 4291, DOI: 10.1039/b204199pGoogle Scholar42https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XmslKltb4%253D&md5=313d2de2c5b5a8182b102cfd6f6e10a2A fully direct RI-HF algorithm: Implementation, optimised auxiliary basis sets, demonstration of accuracy and efficiencyWeigend, FlorianPhysical Chemistry Chemical Physics (2002), 4 (18), 4285-4291CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)A direct HF algorithm using the resoln. of identity for Coulomb and exchange integrals (RI-HF) was implemented within the program system TURBOMOLE. A variational procedure for the optimization of auxiliary functions is presented as well as optimized auxiliary basis sets for large basis sets up to Br. The accuracy of RI-HF energies and of MP2 energies based on RI-HF wave functions is demonstrated for a large test set of mols. Accuracy of first order properties is documented for selected cases. The size dependency of the RI errors and the efficiency of the method are investigated for closo-boranes [BnHn]2- (n = 4-12).**43**Weigend, F. Hartree–Fock exchange fitting basis sets for H to Rn.*J. Comput. Chem.*2008,*29*, 167– 175, DOI: 10.1002/jcc.20702Google Scholar43https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BD2sjktVOjtw%253D%253D&md5=e16d82b5b49d9c043fc719723cfd69aeHartree-Fock exchange fitting basis sets for H to RnWeigend FlorianJournal of computational chemistry (2008), 29 (2), 167-75 ISSN:0192-8651.For elements H to Rn (except Lanthanides), a series of auxiliary basis sets fitting exchange and also Coulomb potentials in Hartree-Fock treatments (RI-JK-HF) is presented. A large set of small molecules representing nearly each element in all its common oxidation states was used to assess the quality of these auxiliary bases. For orbital basis sets of triple zeta valence and quadruple zeta valence quality, errors in total energies arising from the RI-JK approximation are below approximately 1 meV per atom in molecular compounds. Accuracy of RI-JK-approximated HF wave functions is sufficient for being used for post-HF treatments like Moller-Plesset perturbation theory, MP2. Compared to nonapproximated treatments, RI-JK-HF leads to large computational savings for quadruple zeta valence orbital bases and, in case of small to midsize systems, to significant savings for triple zeta valence bases.**44**Weigend, F.; Häser, M.; Patzelt, H.; Ahlrichs, R. RI-MP2: optimized auxiliary basis sets and demonstration of efficiency.*Chem. Phys. Lett.*1998,*294*, 143– 152, DOI: 10.1016/S0009-2614(98)00862-8Google Scholar44https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXlvVGjtrc%253D&md5=305af6c7dc6ec9a83cb2d4760adb9f3bRI-MP2: optimized auxiliary basis sets and demonstration of efficiencyWeigend, Florian; Haser, Marco; Patzelt, Holger; Ahlrichs, ReinhartChemical Physics Letters (1998), 294 (1,2,3), 143-152CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)Applications of the RI-MP2 method require high-quality auxiliary basis sets employed to approx. charge distributions. A variational procedure is proposed and applied to optimize auxiliary bases for main group and transition metal atoms which are tested for more than 350 mols. The RI approxn. affects mol. MP2 energies by less than 60 μEh per atom and equil. distances by less than 0.2 pm. We further comment on the extension from RHF to UHF and the exploitation of mol. symmetry. Applications to (Cu2S)n clusters and hydrocarbons CnH2n+2 document a significant redn. of computation times which allows for calcns. with up to 1000 basis functions in C1 symmetry.**45**Weigend, F.; Häser, M. RI-MP2: first derivatives and global consistency.*Theor. Chem. Acc.*1997,*97*, 331– 340, DOI: 10.1007/s002140050269Google Scholar45https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2sXmvFCitL8%253D&md5=14ae7a8718188931367e3a192be50351RI-MP2. First derivatives and global consistencyWeigend, Florian; Haser, MarcoTheoretical Chemistry Accounts (1997), 97 (1-4), 331-340CODEN: TCACFW; ISSN:1432-881X. (Springer)The evaluation of RI-MP2 first derivs. with respect to nuclear coordinates or with respect to an external elec. field is described. The prefix RI indicates the use of an approx. resoln. of identity in the Hilbert space of interacting charge distributions (Coulomb metric), i.e., the use of an auxiliary basis set to approx. charge distributions. The RI technique is applied to first derivs. of the MP2 correlation energy expression while the (restricted) Hartree-Fock ref. is treated in the usual way. Computational savings by a factor of 10 over conventional approaches are demonstrated in an application to porphyrin. The RI approxn. to MP2 derivs. does not entail any significant loss in accuracy. Finally, the relative energetic stabilities of a representative sample of closed-shell mols. built from first and second row elements were investigated by the RI-MP2 approach, and thus it is tested whether such properties that refer to potential energy hypersurfaces in a more global way can be described with similar consistency to the more locally defined derivs.**46**Hättig, C.; Weigend, F. CC2 excitation energy calculations on large molecules using the resolution of the identity approximation.*J. Chem. Phys.*2000,*113*, 5154– 5161, DOI: 10.1063/1.1290013Google Scholar46https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXmslWns7w%253D&md5=a4be1da94494fc32bd3e330b5718874aCC2 excitation energy calculations on large molecules using the resolution of the identity approximationHattig, Christof; Weigend, FlorianJournal of Chemical Physics (2000), 113 (13), 5154-5161CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A new implementation of the approx. coupled cluster singles and doubles method CC2 is reported, which is suitable for large scale integral-direct calcns. It employs the resoln. of the identity (RI) approxn. for two-electron integrals to reduce the CPU time needed for calcn. and I/O of these integrals. We use a partitioned form of the CC2 equations which eliminates the need to store double excitation cluster amplitudes. In combination with the RI approxn. this formulation of the CC2 equations leads to a reduced scaling of memory and disk space requirements with the no. of correlated electrons (n) and basis functions (N) to, resp., O(N2) and O(nN2), compared to O(n2N2) in previous implementations. The reduced CPU, memory and disk space requirements make it possible to perform CC2 calcns. with accurate basis sets on large mols., which would not be accessible with conventional implementations of the CC2 method. We present an application to vertical excitation energies of alkenes C2nH2n+2, for n=1-12, and report results for the lowest lying dipole-allowed transitions for the TZVPP basis sets, which for n=12 contain 1108 basis functions. Comparison with conventional CC2 results for the smaller alkenes show that for CC2 ground state energies and for excitation energies of valence states, the error due to the RI approxn. is negligible compared to the usual basis set error, if auxiliary basis sets are used, which have been optimized for MP2 energy calcns.**47**Sierka, M.; Hogekamp, A.; Ahlrichs, R. Fast evaluation of the Coulomb potential for electron densities using multipole accelerated resolution of identity approximation.*J. Chem. Phys.*2003,*118*, 9136– 9148, DOI: 10.1063/1.1567253Google Scholar47https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXjs1Gkurw%253D&md5=6230812b9a569f78764d98e156fda7d6Fast evaluation of the Coulomb potential for electron densities using multipole accelerated resolution of identity approximationSierka, Marek; Hogekamp, Annika; Ahlrichs, ReinhartJournal of Chemical Physics (2003), 118 (20), 9136-9148CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A new computational approach is presented that allows for an accurate and efficient treatment of the electronic Coulomb term in d. functional methods. This multipole accelerated resoln. of identity for J (MARI-J) method partitions the Coulomb interactions into the near- and far-field parts. The calcn. of the far-field part is performed by a straightforward application of the multipole expansions and the near-field part is evaluated employing expansion of mol. electron densities in atom-centered auxiliary basis sets (RI-J approxn.). Compared to full RI-J calcns., up to 6.5-fold CPU time savings are reported for systems with about 1000 atoms without any significant loss of accuracy. Other multipole-based methods are compared with regard to redn. of the CPU times vs. the conventional treatment of the Coulomb term. The MARI-J approach compares favorably and offers speedups approaching two orders of magnitude for mols. with about 400 atoms and more than 5000 basis functions. Our new method shows scalings as favorable as N1.5, where N is the no. of basis functions, for a variety of systems including dense three-dimensional mols. Calcns. on mols. with up to 1000 atoms and 7000 to 14 000 basis functions, depending on symmetry, can now be easily performed on single processor work stations. Details of the method implementation in the quantum chem. program TURBOMOLE are discussed.**48**Armbruster, M. K.; Weigend, F.; van Wüllen, C.; Klopper, W. Self-consistent treatment of spin-orbit interactions with efficient Hartree-Fock and density functional methods.*Phys. Chem. Chem. Phys.*2008,*10*, 1748– 1756, DOI: 10.1039/b717719dGoogle Scholar48https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXjtl2htbw%253D&md5=f7e0c0a3476a90f504d7b14c130f5018Self-consistent treatment of spin-orbit interactions with efficient Hartree-Fock and density functional methodsArmbruster, Markus K.; Weigend, Florian; van Wuellen, Christoph; Klopper, WimPhysical Chemistry Chemical Physics (2008), 10 (13), 1748-1756CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)Efficient SCF schemes including both scalar relativistic effects and spin-orbit (SO) interactions at Hartree-Fock (HF) and d. functional (DFT) levels are presented. SO interactions require the extension of std. procedures to two-component formalisms. Efficiency is achieved by using effective core potentials (ECPs) and by employing the resoln.-of-the-identity approxn. for the Coulomb part (RI-J) in pure DFT calcns. as well as also for the HF-exchange part (RI-JK) in the case of HF or hybrid-DFT treatments. The procedures were implemented in the program system TURBOMOLE; efficiency is demonstrated for comparably large systems, such as Pb54. Relevance of SO effects for electronic structure and stability is illustrated by treatments of small Pb and Po clusters with and without accounting for SO effects.**49**Baldes, A.; Weigend, F. Efficient two-component self-consistent field procedures and gradients: implementation in TURBOMOLE and application to Au_{20}^{–}.*Mol. Phys.*2013,*111*, 2617– 2624, DOI: 10.1080/00268976.2013.802037Google Scholar49https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXntVGntLk%253D&md5=1872aed4ded61a9030fba7b2f93c39b7Efficient two-component self-consistent field procedures and gradients: implementation in TURBOMOLE and application toBaldes, Alexander; Weigend, FlorianMolecular Physics (2013), 111 (16-17), 2617-2624CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)The implementation of gradients of two-component d. functional (DFT) and hybrid-DFT energies is presented as well as several improvements with respect to our previous version in the SCF (SCF) procedure: Calcn. of the Hartree-Fock (HF) exchange with four-center integrals, use of difference d. formalism and partial exploitation of symmetry. Efficiency is demonstrated by comparison to recently reported data and by calcn. of energy and gradient of Au147. As an example of use, structure parameters of Au-20 were optimized and subsequently vibration frequencies were calcd. numerically from the anal. gradients at two-component level. For this example, spin-orbit coupling leads to higher degeneracy of vibrational levels by lifting degeneracy of electronic levels, as the latter prevents Jahn-Teller distortion.**50**Kühn, M. Correlation Energies from the Two-Component Random Phase Approximation.*J. Chem. Theory Comput.*2014,*10*, 623– 633, DOI: 10.1021/ct400994xGoogle Scholar50https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXnsVKhtQ%253D%253D&md5=e24ec20ede799c4ab5b260e508389f26Correlation Energies from the Two-Component Random Phase ApproximationKuehn, MichaelJournal of Chemical Theory and Computation (2014), 10 (2), 623-633CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The correlation energy within the two-component RPA accounting for spin-orbit effects is derived. The resulting plasmon equation is rewritten - analogously to the scalar relativistic case - in terms of the trace of two Hermitian matrixes for (Kramers-restricted) closed-shell systems and then represented as an integral over imaginary frequency using the resoln. of the identity approxn. The final expression is implemented in the TURBOMOLE program suite. The code is applied to the computation of equil. distances and vibrational frequencies of heavy diat. mols. The efficiency is demonstrated by calcn. of the relative energies of the Oh-, D4h-, and C5v-sym. isomers of Pb6. Results within the RPA are obtained based on two-component Kohn-Sham ref.-state calcns., using effective-core potentials. These values are finally compared to other two-component and scalar relativistic methods, as well as exptl. data.**51**Krause, K.; Klopper, W. Implementation of the Bethe-Salpeter equation in the TURBOMOLE program.*J. Comput. Chem.*2017,*38*, 383– 388, DOI: 10.1002/jcc.24688Google Scholar51https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XitVWls7nE&md5=8ca152d28459b2f348adbe83e1bb5d46Implementation of the Bethe-Salpeter equation in the TURBOMOLE programKrause, Katharina; Klopper, WimJournal of Computational Chemistry (2017), 38 (6), 383-388CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)A software update solving the Bethe-Salpeter equation (BSE) is reported for the ESCF module of the TURBOMOLE program for the theor. description of electronically excited states of atoms and mols. A resoln.-of-the-identity (RI) approxn. is used for all two-electron electron-repulsion integrals that are required for solving the equation. Symmetry is utilized for the point group D2h and its subgroups, and the BSE approach can be applied in either a spin-restricted or a spin-unrestricted Kohn-Sham formalism. Triplet as well as singlet excited states of closed-shell atoms and mols. can be treated in the spin-restricted formalism. As a side product, the present software update also allows for the application of the RI approxn. to the Hartree-Fock exchange contribution that occurs when a hybrid functional is used in time-dependent d.-functional theory. © 2016 Wiley Periodicals, Inc.**52**Gui, X.; Holzer, C.; Klopper, W. Accuracy Assessment of GW Starting Points for Calculating Molecular Excitation Energies Using the Bethe-Salpeter Formalism.*J. Chem. Theory Comput.*2018,*14*, 2127– 2136, DOI: 10.1021/acs.jctc.8b00014Google Scholar52https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXjslKlurY%253D&md5=1dc9e328e6d018f74ec48d97d07534fcAccuracy Assessment of GW Starting Points for Calculating Molecular Excitation Energies Using the Bethe-Salpeter FormalismGui, Xin; Holzer, Christof; Klopper, WimJournal of Chemical Theory and Computation (2018), 14 (4), 2127-2136CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The performance of the Bethe-Salpeter equation (BSE) approach for the first-principles computation of singlet and triplet excitation energies of small org., closed-shell mols. has been assessed with respect to the quasiparticle energies used on input, obtained at various levels of GW theory. In the corresponding GW computations, quasiparticle energies have been computed for all orbital levels by means of using full spectral functions. The assessment reveals that, for valence excited states, quasiparticle energies obtained at the levels of eigenvalue-only self-consistent (evGW) or quasiparticle self-consistent theory (qsGW) are required to obtain results of comparable accuracy as in time-dependent d.-functional theory (TDDFT) using a hybrid functional such as PBE0. In contrast to TDDFT, however, the BSE approach performs well not only for valence excited states but also for excited states with Rydberg or charge-transfer character. To demonstrate the applicability of the BSE approach, computation times are reported for a set of arom. hydrocarbons. Furthermore, examples of computations of ordinary photoabsorption and electronic CD spectra are presented for (C60)2 and C84, resp.**53**Holzer, C.; Klopper, W. Ionized, electron-attached, and excited states of molecular systems with spin–orbit coupling: Two-component GW and Bethe–Salpeter implementations.*J. Chem. Phys.*2019,*150*, 204116, DOI: 10.1063/1.5094244Google Scholar53https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXkslOmsbk%253D&md5=955e52f1ba3fccfb6c8096e66e71d5dbIonized, electron-attached, and excited states of molecular systems with spin-orbit coupling: Two-component GW and Bethe-Salpeter implementationsHolzer, Christof; Klopper, WimJournal of Chemical Physics (2019), 150 (20), 204116/1-204116/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We have implemented and applied the GW method and the static screened Bethe-Salpeter equation (BSE) for calcg. linear-response properties for quasirelativistic mol. systems. Our ansatz is based on a two-component (2c) scheme that includes spin-orbit coupling as well as scalar relativistic effects. Efficient, state-of-the-art approaches including the analytic continuation (employing Pade approximants, scaling as N4 with system size N) and contour deformation schemes are presented to obtain the required 2c quasirelativistic GW quasiparticle energies. Screened exchange contributions are computed within the resoln.-of-the-identity approxn., and working equations for the 2c GW/BSE method are given. The performance of the 2c GW/BSE method is assessed, and results are compared to other methods and exptl. data. A robust iterative scheme for solving the eigenvalue problems occurring in the 2c GW/BSE and hybrid time-dependent d. functional theories is presented. (c) 2019 American Institute of Physics.**54**Kühn, M.; Weigend, F. Two-component hybrid time-dependent density functional theory within the Tamm-Dancoff approximation.*J. Chem. Phys.*2015,*142*, 034116, DOI: 10.1063/1.4905829Google Scholar54https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhtF2mu7g%253D&md5=3ffde89db3663342bb9defe38cc56a0eTwo-component hybrid time-dependent density functional theory within the Tamm-Dancoff approximationKuehn, Michael; Weigend, FlorianJournal of Chemical Physics (2015), 142 (3), 034116/1-034116/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We report the implementation of a two-component variant of time-dependent d. functional theory (TDDFT) for hybrid functionals that accounts for spin-orbit effects within the Tamm-Dancoff approxn. (TDA) for closed-shell systems. The influence of the admixt. of Hartree-Fock exchange on excitation energies is investigated for several atoms and diat. mols. by comparison to nos. for pure d. functionals obtained previously [M. Kuhn and F. Weigend, J. Chem. Theory Comput. 9, 5341 (2013)]. It is further related to changes upon switching to the local d. approxn. or using the full TDDFT formalism instead of TDA. Efficiency is demonstrated for a comparably large system, Ir(ppy)3 (61 atoms, 1501 basis functions, lowest 10 excited states), which is a prototype mol. for org. light-emitting diodes, due to its "spin-forbidden" triplet-singlet transition. (c) 2015 American Institute of Physics.**55**Burow, A. M.; Sierka, M.; Mohamed, F. Resolution of identity approximation for the Coulomb term in molecular and periodic systems.*J. Chem. Phys.*2009,*131*, 214101– 1–214101–6, DOI: 10.1063/1.3267858Google Scholar55https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhsFagu7zL&md5=2f511b951c7c63ac93846e07c3a54f52Resolution of identity approximation for the Coulomb term in molecular and periodic systemsBurow, Asbjoern M.; Sierka, Marek; Mohamed, FawziJournal of Chemical Physics (2009), 131 (21), 214101/1-214101/6CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A new formulation of resoln. of identity approxn. for the Coulomb term is presented, which uses atom-centered basis and auxiliary basis functions and treats mol. and periodic systems of any dimensionality on an equal footing. It relies on the decompn. of an auxiliary charge d. into charged and chargeless components. Applying the Coulomb metric under periodic boundary conditions constrains the explicit form of the charged part. The chargeless component is detd. variationally and converged Coulomb lattice sums needed for its detn. are obtained using chargeless linear combinations of auxiliary basis functions. The lattice sums are partitioned in near- and far-field portions which are treated through an anal. integration scheme employing two- and three-center electron repulsion integrals and multipole expansions, resp., operating exclusively in real space. Our preliminary implementation within the TURBOMOLE program package demonstrates consistent accuracy of the method across mol. and periodic systems. Using common auxiliary basis sets the errors of the approxn. are small, in av. about 20 μhartree per atom, for both mol. and periodic systems. (c) 2009 American Institute of Physics.**56**Łazarski, R.; Burow, A. M.; Sierka, M. Density functional theory for molecular and periodic systems using density fitting and continuous fast multipole methods.*J. Chem. Theory Comput.*2015,*11*, 3029– 3041, DOI: 10.1021/acs.jctc.5b00252Google Scholar56https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXptF2mtrg%253D&md5=3524e6338aea31618a21453c8fd325c3Density Functional Theory for Molecular and Periodic Systems Using Density Fitting and Continuous Fast Multipole MethodsLazarski, Roman; Burow, Asbjoern M.; Sierka, MarekJournal of Chemical Theory and Computation (2015), 11 (7), 3029-3041CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)An implementation of Kohn-Sham d. functional theory within the TURBOMOLE program package with Gaussian-type orbitals (GTO) as basis functions is reported that treats mol. and periodic systems of any dimensionality on an equal footing. Its key component is a combination of d. fitting/resoln. of identity (DF) approxn. and continuous fast multipole method (CFMM) applied for the electronic Coulomb term. This DF-CFMM scheme operates entirely in the direct space and partitions Coulomb interactions into far-field part evaluated using multipole expansions and near-field contribution calcd. employing d. fitting. Computational efficiency and favorable scaling behavior of our implementation approaching O(N) for the formation of Kohn-Sham matrix is demonstrated for various mol. and periodic systems including three-dimensional models with unit cells contg. up to 640 atoms and 19072 GTO basis functions.**57**Łazarski, R.; Burow, A. M.; Grajciar, L.; Sierka, M. Density functional theory for molecular and periodic systems using density fitting and continuous fast multipole method: Analytical gradients.*J. Comput. Chem.*2016,*37*, 2518– 2526, DOI: 10.1002/jcc.24477Google Scholar57https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xhtlykur7M&md5=c7faf4d9c2b7aae8c4065f56d3840a38Density functional theory for molecular and periodic systems using density fitting and continuous fast multipole method: Analytical gradientsLazarski, Roman; Burow, Asbjoern Manfred; Grajciar, Lukas; Sierka, MarekJournal of Computational Chemistry (2016), 37 (28), 2518-2526CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)A full implementation of anal. energy gradients for mol. and periodic systems is reported in the TURBOMOLE program package within the framework of Kohn-Sham d. functional theory using Gaussian-type orbitals as basis functions. Its key component is a combination of d. fitting (DF) approxn. and continuous fast multipole method (CFMM) that allows for an efficient calcn. of the Coulomb energy gradient. For exchange-correlation part the hierarchical numerical integration scheme (Burow and Sierka, Journal of Chem. Theory and Computation 2011, 7, 3097) is extended to energy gradients. Computational efficiency and asymptotic O(N) scaling behavior of the implementation is demonstrated for various mol. and periodic model systems, with the largest unit cell of hematite contg. 640 atoms and 19,072 basis functions. The overall computational effort of energy gradient is comparable to that of the Kohn-Sham matrix formation. © 2016 Wiley Periodicals, Inc.**58**Grajciar, L. Low-memory Iterative Density Fitting.*J. Comput. Chem.*2015,*36*, 1521– 1535, DOI: 10.1002/jcc.23961Google Scholar58https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXps12qs74%253D&md5=f6cc1b6f766d0f245fb4109ff0d30bfbLow-memory iterative density fittingGrajciar, LukasJournal of Computational Chemistry (2015), 36 (20), 1521-1535CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)A new low-memory modification of the d. fitting approxn. based on a combination of a continuous fast multipole method (CFMM) and a preconditioned conjugate gradient solver is presented. Iterative conjugate gradient solver uses preconditioners formed from blocks of the Coulomb metric matrix that decrease the no. of iterations needed for convergence by up to one order of magnitude. The matrix-vector products needed within the iterative algorithm are calcd. using CFMM, which evaluates them with the linear scaling memory requirements only. Compared with the std. d. fitting implementation, up to 15-fold redn. of the memory requirements is achieved for the most efficient preconditioner at a cost of only 25% increase in computational time. The potential of the method is demonstrated by performing d. functional theory calcns. for zeolite fragment with 2592 atoms and 121,248 auxiliary basis functions on a single 12-core CPU workstation. © 2015 Wiley Periodicals, Inc.**59**Becker, M.; Sierka, M. Density functional theory for molecular and periodic systems using density fitting and continuous fast multipole method: Stress tensor.*J. Comput. Chem.*2019,*40*, 2563– 2570, DOI: 10.1002/jcc.26033Google Scholar59https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhsVWhtLnE&md5=2b11e88191ae65e51cd8431dc07d9907Density functional theory for molecular and periodic systems using density fitting and continuous fast multipole method: Stress tensorBecker, Martin; Sierka, MarekJournal of Computational Chemistry (2019), 40 (29), 2563-2570CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)A full implementation of the anal. stress tensor for periodic systems is reported in the TURBOMOLE program package within the framework of Kohn-Sham d. functional theory using Gaussian-type orbitals as basis functions. It is the extension of the implementation of anal. energy gradients (Lazarski et al., Journal of Computational Chem. 2016, 37, 2518-2526) to the stress tensor for the purpose of optimization of lattice vectors. Its key component is the efficient calcn. of the Coulomb contribution by combining d. fitting approxn. and continuous fast multipole method. For the exchange-correlation (XC) part the hierarchical numerical integration scheme (Burow and Sierka, Journal of Chem. Theory and Computation 2011, 7, 3097-3104) is extended to XC wt. derivs. and stress tensor. The computational efficiency and favorable scaling behavior of the stress tensor implementation are demonstrated for various model systems. The overall computational effort for energy gradient and stress tensor for the largest systems investigated is shown to be at most two and a half times the computational effort for the Kohn-Sham matrix formation. © 2019 Wiley Periodicals, Inc.**60**Mack, F.; Schattenberg, C. J.; Kaupp, M.; Weigend, F. Nuclear Spin–Spin Couplings: Efficient Evaluation of Exact Exchange and Extension to Local Hybrid Functionals.*J. Phys. Chem. A*2020,*124*, 8529– 8539, DOI: 10.1021/acs.jpca.0c06897Google Scholar60https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhvVygtbjP&md5=53ae0ff734254141af28618a25627942Nuclear Spin-Spin Couplings: Efficient Evaluation of Exact Exchange and Extension to Local Hybrid FunctionalsMack, Fabian; Schattenberg, Caspar J.; Kaupp, Martin; Weigend, FlorianJournal of Physical Chemistry A (2020), 124 (41), 8529-8539CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)We present an efficient implementation for the computation of nuclear spin-spin coupling tensors within d. functional theory into the TURBOMOLE software suite. Emphasis is put on methods to efficiently evaluate the Hartree-Fock exchange needed for hybrid functionals: resoln. of the identity and seminumerical evaluation on a grid. Our algorithm allows for the selection of specific nuclei for the redn. of calcn. times. Further, the accuracy of locally dense basis sets in the d. functional theory framework is investigated. These features allow for the routine computation of coupling consts. in systems comprising about 100 carbon atoms within less than one day on a single CPU and within a few hours when using the OpenMP variant. Based on seminumerical integration, the first implementation of local hybrid functionals for spin-spin couplings is reported. This has allowed a preliminary evaluation of position-dependent exact-exchange admixt. in three local hybrid functionals for a set of 80 isotropic spin-spin couplings in 23 small main-group mols. against CC3 and MCSCF ref. data. Two of the local hybrids (LH14t-calPBE and LH07t-SVWN) are the top performers in the overall statistical evaluation compared to several std. functionals (TPSS, TPSSh, B3LYP, PBE0, and BHLYP), in particular, as they do not exhibit notable outliers for specific coupling types.**61**Franzke, Y. J.; Holzer, C.; Mack, F. NMR Coupling Constants Based on the Bethe–Salpeter Equation in the*GW*Approximation.*J. Chem. Theory Comput.*2022,*18*, 1030– 1045, DOI: 10.1021/acs.jctc.1c00999Google Scholar61https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XisFKnuw%253D%253D&md5=1c841a98a5c18fc81f76c04e9b9fc91cNMR Coupling Constants Based on the Bethe-Salpeter Equation in the GW ApproximationFranzke, Yannick J.; Holzer, Christof; Mack, FabianJournal of Chemical Theory and Computation (2022), 18 (2), 1030-1045CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present the first steps to extend the Green's function GW method and the Bethe-Salpeter equation (BSE) to mol. response properties such as NMR indirect spin-spin coupling consts. We discuss both a nonrelativistic one-component and a quasi-relativistic two-component formalism. The latter describes scalar-relativistic and spin-orbit effects and allows us to study heavy-element systems with reasonable accuracy. Efficiency is maintained by the application of the resoln. of the identity approxn. throughout. The performance is demonstrated using conventional central processing units (CPUs) and modern graphics processing units (GPUs) for mols. involving several thousand basis functions. Our results show that a large amt. of Hartree-Fock exchange is vital to provide a sufficient Kohn-Sham starting point to compute the GW quasi-particle energies. As the GW-BSE approach is generally less accurate for triplet excitations or related properties such as the Fermi-contact interaction, the admixt. of the Kohn-Sham correlation kernel through the contracted BSE (cBSE) method improves the results for NMR coupling consts. This leads to remarkable results when combined with the eigenvalue-only self-consistent variant (evGW) and Becke's half and half functional (BH&HLYP) or the CAM-QTP family. The developed methodol. is used to calc. the Karplus curve of tin mols., illustrating its applicability to extended chem. relevant mols. Here, the GW-cBSE method improves upon the chosen BH&HLYP Kohn-Sham starting points.**62**Reiter, K.; Mack, F.; Weigend, F. Calculation of Magnetic Shielding Constants with meta-GGA Functionals Employing the Multipole-Accelerated Resolution of the Identity: Implementation and Assessment of Accuracy and Efficiency.*J. Chem. Theory Comput.*2018,*14*, 191– 197, DOI: 10.1021/acs.jctc.7b01115Google Scholar62https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhvFOitbrL&md5=455aa59e6c1d9772d490ae812e7540c5Calculation of Magnetic Shielding Constants with meta-GGA Functionals Employing the Multipole-Accelerated Resolution of the Identity: Implementation and Assessment of Accuracy and EfficiencyReiter, Kevin; Mack, Fabian; Weigend, FlorianJournal of Chemical Theory and Computation (2018), 14 (1), 191-197CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a highly efficient implementation for d. functional calcns. of chem. shielding consts. It employs the multipole-accelerated resoln. of the identity for the calcn. of the Coulomb part, which complements the usage of low order scaling routines for the evaluation of the exchange-correlation part and the Hartree-Fock exchange part. Introduced errors for shifts of chem. shielding consts. of H, C, F and P are evaluated for resp. test sets of mols. and are related to the accuracy of shifts obtained with hybrid and non-hybrid functionals of the generalized gradient approxn. type as well as for meta-GGA functionals themselves, also considering solvation effects. Efficiency is demonstrated for α-D-glucose chains with more than 2500 atoms on a single CPU as well as with an OpenMP parallelized version.**63**Gillhuber, S.; Franzke, Y. J.; Weigend, F. Paramagnetic NMR Shielding Tensors and Ring Currents: Efficient Implementation and Application to Heavy Element Compounds.*J. Phys. Chem. A*2021,*125*, 9707– 9723, DOI: 10.1021/acs.jpca.1c07793Google Scholar63https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXitlGktrrE&md5=a838b21d8fd0400cc3f2cf1d9c936b52Paramagnetic NMR Shielding Tensors and Ring Currents: Efficient Implementation and Application to Heavy Element CompoundsGillhuber, Sebastian; Franzke, Yannick J.; Weigend, FlorianJournal of Physical Chemistry A (2021), 125 (44), 9707-9723CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)A review on efficient implementation of paramagnetic NMR shielding tensors and shifts in a nonrelativistic and scalar-relativistic d. functional theory framework. For the latter, we make use of the scalar exact two-component Hamiltonian in its local approxn., generally we apply the well established (multipole-accelerated) resoln. of the identity approxn. and the seminumerical exchange approxn. The perturbed d. matrix of a paramagnetic NMR shielding calcn. is further used to study the magnetically induced c.d. and ring currents of open-shell systems as illustrated for [U@Bi12]3-. [U@Bi12]3- features delocalized highest MOs and sustains a net diatropic ring current of ca. 18 nA/T through the Bi12 torus similar to the all-metal arom. heavy-element cluster [Th@Bi12]4-.**64**Friese, D. H.; Hättig, C.; Ruud, K. Calculation of two-photon absorption strengths with the approximate coupled cluster singles and doubles model CC2 using the resolution-of-identity approximation.*Phys. Chem. Chem. Phys.*2012,*14*, 1175– 1184, DOI: 10.1039/C1CP23045JGoogle Scholar64https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhs1egs73I&md5=0000c693109afd10246dc1941421a6d1Calculation of two-photon absorption strengths with the approximate coupled cluster singles and doubles model CC2 using the resolution-of-identity approximationFriese, Daniel H.; Haettig, Christof; Ruud, KennethPhysical Chemistry Chemical Physics (2012), 14 (3), 1175-1184CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)An implementation of two-photon absorption matrix elements using the approx. second-order coupled-cluster singles and doubles model CC2 is presented. In this implementation we use the resoln.-of-the-identity approxn. for the two-electron repulsion integrals to reduce the computational cost. To avoid storage of large arrays we introduce in addn. a numerical Laplace transformation of orbital energy denominators for the response of the doubles amplitudes. The error due to the numerical Laplace transformation is found to be negligible. Using this new implementation, we performed a series of benchmark calcns. on substituted benzene and azobenzene derivs. to get ref. values for TD-DFT results. We show that results obtained with the Coulomb-attenuated B3LYP functional are in reasonable agreement with the coupled-cluster results, whereas other d. functionals which do not have a long-range correction give considerably less accurate results. Applications to the AF240 dye mol. and a weakly bound mol. tweezer complex demonstrate that this new RI-CC2 implementation allows for the first time to compute two-photon absorption cross sections with a correlated wave function method for mols. with more than 70 atoms and to apply this method for benchmarking TD-DFT calcns. on mols. which are of particular relevance for exptl. studies of two-photon absorption.**65**Friese, D. H.; Hättig, C.; Rizzo, A. Origin-independent two-photon circular dichroism calculations in coupled cluster theory.*Phys. Chem. Chem. Phys.*2016,*18*, 13683– 13692, DOI: 10.1039/C6CP01653GGoogle Scholar65https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XntVyhtrk%253D&md5=4089784f6c1938766861722f0de81787Origin-independent two-photon circular dichroism calculations in coupled cluster theoryFriese, Daniel H.; Hattig, Christof; Rizzo, AntonioPhysical Chemistry Chemical Physics (2016), 18 (19), 13683-13692CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)We present the first origin-independent approach for the treatment of two-photon CD (TPCD) using coupled cluster methods. The approach is assessed concerning its behavior on the choice of the basis set and different coupled cluster methods. We also provide a comparison of results from CC2 with those from d. functional theory using the CAM-B3LYP functional. Concerning the basis set we note that in most cases an augmented triple zeta basis or a doubly augmented double zeta basis is needed for reasonably converged results. In the comparison of different coupled cluster methods results from CCSD, CC3 and CC2 have been found to be quite similar in most cases, while CCS results differ remarkably from the results at the higher levels. However, this proof-of-principle study also shows that further benchmarking of DFT and CC2 against accurate coupled cluster ref. values (e.g. CCSD or CC3) is needed.**66**Winter, N. O. C.; Hättig, C. Scaled opposite-spin CC2 for ground and excited states with fourth order scaling computational costs.*J. Chem. Phys.*2011,*134*, 184101, DOI: 10.1063/1.3584177Google Scholar66https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXmtVajtb4%253D&md5=7291417f6c3bce6fec1f54a489511024Scaled opposite-spin CC2 for ground and excited states with fourth order scaling computational costsWinter, Nina O. C.; Haettig, ChristofJournal of Chemical Physics (2011), 134 (18), 184101/1-184101/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An implementation of scaled opposite-spin CC2 (SOS-CC2) for ground and excited state energies is presented that requires only fourth order scaling computational costs. The SOS-CC2 method yields results with an accuracy comparable to the unscaled method. Furthermore the time-detg. fifth order scaling steps in the algorithm can be replaced by only fourth order scaling computational costs using a "resoln. of the identity" approxn. for the electron repulsion integrals and a Laplace transformation of the orbital energy denominators. This leads to a significant redn. of computational costs esp. for large systems. Timings for ground and excited state calcns. are shown and the error of the Laplace transformation is investigated. An application to a chlorophyll mol. with 134 atoms results in a speed-up by a factor of five and demonstrates how the new implementation extends the applicability of the method. A SOS variant of the algebraic diagrammatic construction through second order ADC(2), which arises from a simplification of the SOS-CC2 model, is also presented. The SOS-ADC(2) model is a cost-efficient alternative in particular for future extensions to spectral intensities and excited state structure optimizations. (c) 2011 American Institute of Physics.**67**Winter, N. O. C.; Hättig, C. Quartic scaling analytical gradients of scaled opposite-spin CC2.*Chem. Phys.*2012,*401*, 217– 227, DOI: 10.1016/j.chemphys.2011.10.002Google Scholar67https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XnvFCru7w%253D&md5=cd912e3b65951e6e1029ceed1cb56a03Quartic scaling analytical gradients of scaled opposite-spin CC2Winter, Nina O. C.; Haettig, ChristofChemical Physics (2012), 401 (), 217-227CODEN: CMPHC2; ISSN:0301-0104. (Elsevier B.V.)An implementation of anal. gradients, transition moments and excited state properties for scaled opposite-spin (SOS) CC2 and a SOS variant of the algebraic diagrammatic construction through second order ADC(2) is presented. The time-detg. fifth order scaling steps in the algorithms are replaced by schemes with only fourth order scaling computational costs using a "resoln. of the identity" approxn. for the electron repulsion integrals and a numerical Laplace transformation of the orbital energy denominators. This leads to a significant redn. of the computational costs for geometry optimizations of large systems. This work is an extension to the recently presented quartic scaling algorithm for SOS-CC2 energies for ground and excited states. The Laplace error for adiabatic electronic excitation energies and excited state structures is found to be very small. SOS-ADC(2) provides for adiabatic electronic excitation energies, excited state structure parameters, harmonic vibrational frequencies as well as dipole moments and transition strengths an accuracy similar to SOS-CC2. Timings for ground and excited state calcns. are reported and applications to a chlorophyll mol. and a ditopic 2,2':6',2''-bis(terpyridine) with 160 atoms demonstrate how the new implementation extends the applicability of these methods for large systems.**68**Hättig, C.; Tew, D. P.; Köhn, A. Communications: Accurate and efficient approximations to explicitly correlated coupled-cluster singles and doubles, CCSD-F12.*J. Chem. Phys.*2010,*132*, 231102, DOI: 10.1063/1.3442368Google Scholar68https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC3crnsl2hug%253D%253D&md5=7cd1250870f1ab2259fdcb7bff9438dcCommunications: Accurate and efficient approximations to explicitly correlated coupled-cluster singles and doubles, CCSD-F12Hattig Christof; Tew David P; Kohn AndreasThe Journal of chemical physics (2010), 132 (23), 231102 ISSN:.We propose a novel explicitly correlated coupled-cluster singles and doubles method CCSD(F12(*)), which retains the accuracy of CCSD-F12 while the computational costs are only insignificantly larger than those for a conventional CCSD calculation.**69**Tew, D. P. Explicitly correlated coupled-cluster theory with Brueckner orbitals.*J. Chem. Phys.*2016,*145*, 074103, DOI: 10.1063/1.4960655Google Scholar69https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhtleqtbrM&md5=4d9fe818ce0d520e35e4336716321471Explicitly correlated coupled-cluster theory with Brueckner orbitalsTew, David P.Journal of Chemical Physics (2016), 145 (7), 074103/1-074103/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Brueckner orbitals are the optimal orbitals for use in F12 explicitly correlated coupled-cluster (CC) treatments. A novel approach, Brueckner coupled-cluster doubles with perturbative triples BCCD(T)(F12*) is presented that avoids the expensive re-evaluation of F12 integrals throughout the orbital optimization and includes a newly derived basis set correction to the Brueckner ref. energy. The generalization of F12 theory to arbitrary non-Hartree-Fock refs. and to Fock operators that include scalar relativistic effects is also presented. The performance of the new Brueckner F12 method is assessed for a test set of 50 open- and closed-shell reactions and for the ionization potentials and electron affinities (EAs) of the first-row transition metal atoms. Benchmark basis set limit coupled-cluster singles, doubles and perturbative triples (CCSD(T)) and BCCD(T) values are reported for all energies in the test sets. BCCD(T)(F12*) performs systematically better than CCSD(T)(F12*) for electron affinities where orbital relaxation effects are significant. (c) 2016 American Institute of Physics.**70**Schmitz, G.; Hättig, C.; Tew, D. P. Explicitly Correlated PNO-MP2 and PNO-CCSD and their Application to the S66 Set and Large Molecular Systems.*Phys. Chem. Chem. Phys.*2014,*16*, 22167– 22178, DOI: 10.1039/C4CP03502JGoogle Scholar70https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhsVOhu7vK&md5=0166f76942f0475b69484a3b8fc1e155Explicitly correlated PNO-MP2 and PNO-CCSD and their application to the S66 set and large molecular systemsSchmitz, Gunnar; Haettig, Christof; Tew, David P.Physical Chemistry Chemical Physics (2014), 16 (40), 22167-22178CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)We present our current progress on the combination of explicit electron correlation with the pair natural orbital (PNO) representation. In particular we show cubic scaling PNO-MP2-F12, PNO-CCSD(2)‾F‾1‾2 and PNO-CCSD[F12] implementations. The PNOs are constructed using a hybrid scheme, where the PNOs are generated in a truncated doubles space, spanned by orbital specific virtuals obtained using an iterative eigenvector algorithm. We demonstrate the performance of our implementation through calcns. on a series of glycine chains. The accuracy of the local approxns. is assessed using the S66 benchmark set, and we report for the first time explicitly correlated CCSD results for the whole set and improved ests. for the CCSD/CBS limits. For several dimers the PNO-CCSD[F12] calcns. are more accurate than the current ref. values. Addnl., we present pilot applications of our PNO-CCSD[F12] code to host-guest interactions in a cluster model for zeolite H-ZSM-5 and in a calix[4]arene-water complex.**71**Schmitz, G.; Hättig, C. Perturbative triples correction for local pair natural orbital based explicitly correlated CCSD(F12*) using Laplace transformation techniques.*J. Chem. Phys.*2016,*145*, 234107, DOI: 10.1063/1.4972001Google Scholar71https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XitFamur%252FM&md5=40354d9b70054019928647f19f455b8fPerturbative triples correction for local pair natural orbital based explicitly correlated CCSD(F12*) using Laplace transformation techniquesSchmitz, Gunnar; Haettig, ChristofJournal of Chemical Physics (2016), 145 (23), 234107/1-234107/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present an implementation of pair natural orbital coupled cluster singles and doubles with perturbative triples, PNO-CCSD(T), which avoids the quasi-canonical triples approxn. (T0) where couplings due to off-diagonal Fock matrix elements are neglected. A numerical Laplace transformation of the canonical expression for the perturbative (T) triples correction is used to avoid an I/O and storage bottleneck for the triples amplitudes. Results for a test set of reaction energies show that only very few Laplace grid points are needed to obtain converged energy differences and that PNO-CCSD(T) is a more robust approxn. than PNO-CCSD(T0) with a reduced mean abs. deviation from canonical CCSD(T) results. We combine the PNO-based (T) triples correction with the explicitly correlated PNO-CCSD(F12*) method and investigate the use of specialized F12-PNOs in the conventional triples correction. We find that no significant addnl. errors are introduced and that PNO-CCSD(F12*)(T) can be applied in a black box manner. (c) 2016 American Institute of Physics.**72**Frank, M. S.; Schmitz, G.; Hättig, C. The PNO–MP2 gradient and its application to molecular geometry optimizations.*Mol. Phys.*2017,*115*, 343– 356, DOI: 10.1080/00268976.2016.1263762Google Scholar72https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XitFers77K&md5=26c91ecac95545dfb4b3c7dc0b832990The PNO-MP2 gradient and its application to molecular geometry optimisationsFrank, Marius S.; Schmitz, Gunnar; Haettig, ChristofMolecular Physics (2017), 115 (3), 343-356CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)A preliminary implementation of a gradient for MP2 based on the pair natural orbital approxn. (PNO-MP2) is presented. The accuracy of the PNO approxn. for geometries is investigated by comparing bond lengths and bond angles of the PNO-MP2 structures with results from canonical MP2. Special emphasis is put on the optimization of weakly bound complexes, such as differently sized water clusters. It is found that the error introduced by the PNO approxn. is already for moderate PNO thresholds negligible compared to the inherent error of MP2. However, too loose PNO thresholds lead to convergence difficulties in geometry optimisations. This is obsd. in particular for floppy groups as, for example, Me groups with low rotational barriers. Compared to optimisations with canonical MP2, the convergence thresholds have to be mainly energy and less coordinate based to comply with the slight roughness of the potential energy surface which is introduced by the PNO selection.**73**Schmitz, G.; Hättig, C. Accuracy of Explicitly Correlated Local PNO-CCSD(T).*J. Chem. Theory Comput.*2017,*13*, 2623– 2633, DOI: 10.1021/acs.jctc.7b00180Google Scholar73https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXnvFCntrs%253D&md5=8a62fe64e1d9d316806ac4e7700c833fAccuracy of Explicitly Correlated Local PNO-CCSD(T)Schmitz, Gunnar; Hattig, ChristofJournal of Chemical Theory and Computation (2017), 13 (6), 2623-2633CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)In recent years PNO-based local correlation methods have gained popularity since they allow Coupled Cluster (CC) calcns. with reduced computational costs, yet only a few systematic studies concerning their accuracy are available, in particular for the explicitly correlated versions. In this work we take a deeper look at the explicitly correlated local PNO-CCSD(F12*)(T0) and PNO-CCSD(F12*)(T) methods. The first variant uses the so-called semicanonical triples correction (T0) which neglects off-diagonal elements in the occupied block of the Fock matrix. In PNO-CCSD(F12*)(T) this approxn. is avoided by means of Laplace transformation techniques and convergence to the canonical results in the limit of no PNO truncation is restored. We assess the accuracy of both methods using well established benchmark sets for reaction energies and weak mol. interactions and take a look at a system with strong cooperative many-body effects. For reaction energies a close agreement with canonical methods is obsd., and chem. accuracy can be reached. Also for weak intermol. interactions the accuracy is easily controlled, and the methods even allow for improving existing benchmark data.**74**Laricchia, S.; Fabiano, E.; Della Sala, F. Frozen density embedding with hybrid functionals.*J. Chem. Phys.*2010,*133*, 164111, DOI: 10.1063/1.3494537Google Scholar74https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXhtlCgt7jK&md5=424396a749f4d1062a32d518d47a8f6fFrozen density embedding with hybrid functionalsLaricchia, S.; Fabiano, E.; Della Sala, F.Journal of Chemical Physics (2010), 133 (16), 164111/1-164111/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The Kohn-Sham equations with constrained electron d. are extended to hybrid exchange-correlation (XC) functionals. We derive the frozen d. embedding generalized Kohn-Sham (FDE-GKS) scheme which allows to treat the nonlocal exact-exchange in the subsystems. For practical calcns. we propose an approximated version of the FDE-GKS in which the nonadditive exchange potential is computed at a semilocal level. The proposed method is applied to compute the ground-state electronic properties of small test systems and selected DNA base pairs. The results of calcns. employing the hierarchy of XC functionals BLYP/B3LYP/BHLYP and PBE/PBE0 are presented, in order to analyze the effect of nonlocal exchange contributions, and compared with ref. coupled-cluster singles and doubles results. We find that the use of hybrid functionals leads to a significant improvement in the description of ground-state electronic properties of the investigated systems. The semilocal version of the FDE-GKS correctly reproduces the dipole and the electron d. distribution of the exact GKS supramol. system, with errors smaller than the ones obtained using conventional semilocal XC functionals. (c) 2010 American Institute of Physics.**75**Bachorz, R. A.; Bischoff, F. A.; Glöß, A.; Hättig, C.; Höfener, S.; Klopper, W.; Tew, D. P. The MP2-F12 method in the TURBOMOLE program package.*J. Comput. Chem.*2011,*32*, 2492– 2513, DOI: 10.1002/jcc.21825Google Scholar75https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXntlOqsbs%253D&md5=88445b01ce7627f43d7f7e20b81f9aa3The MP2-F12 method in the TURBOMOLE program packageBachorz, Rafal A.; Bischoff, Florian A.; Gloess, Andreas; Haettig, Christof; Hoefener, Sebastian; Klopper, Wim; Tew, David P.Journal of Computational Chemistry (2011), 32 (11), 2492-2513CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)A detailed description of the explicitly correlated second-order Moller-Plesset perturbation theory (MP2-F12) method, as implemented in the TURBOMOLE program package, is presented. The TURBOMOLE implementation makes use of d. fitting, which greatly reduces the prefactor for integral evaluation. Methods are available for the treatment of ground states of open- and closed-shell species, using unrestricted as well as restricted (open-shell) Hartree-Fock ref. determinants. Various methodol. choices and approxns. are discussed. The performance of the TURBOMOLE implementation is illustrated by example calcns. of the mols. leflunomide, prednisone, methotrexate, ethylenedioxytetrathiafulvalene, and a cluster model for the adsorption of methanol on the zeolite H-ZSM-5. Various basis sets are used, including the correlation-consistent basis sets specially optimized for explicitly correlated calcns. (cc-pVXZ-F12). © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011.**76**Schwabe, T.; Sneskov, K.; Haugaard Olsen, J. M.; Kongsted, J.; Christiansen, O.; Hättig, C. PERI-CC2: A Polarizable Embedded RI-CC2 Method.*J. Chem. Theory Comput.*2012,*8*, 3274– 3283, DOI: 10.1021/ct3003749Google Scholar76https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhtV2gsLnO&md5=39cd602c5d1fcf3e17d3985d5b43be7ePERI-CC2: A Polarizable Embedded RI-CC2 MethodSchwabe, Tobias; Sneskov, Kristian; Haugaard-Olsen, Jogvan Magnus; Kongsted, Jacob; Christiansen, Ove; Haettig, ChristofJournal of Chemical Theory and Computation (2012), 8 (9), 3274-3283CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a combination of the polarizable embedding (PE) method with the resoln.-of-the-identity implementation of the approx. coupled-cluster singles and doubles method CC2. The new approach, termed PERI-CC2, allows one to study excited state phenomena of large solvated mol. systems with an accurate correlated wave function method. Central to the PE approach is the advanced description of the environmental electrostatic potential and inclusion of polarization, and the quintessence of RI-CC2 is efficient access to excited state properties while retaining the accuracy assocd. with CC theory. To maintain efficiency, an approx. truncated CC2 d. is introduced to calc. the PE contributions. Explicitly, we derive the central equations and outline an implementation of polarizable embedding for the RI-CC2 approach. The new method is tested against previous PE-CC2 and PE-CCSD results for solvatochromic shifts, demonstrating how the important effects of polarization are incorporated well with PERI-CC2 but with a dramatically reduced overall computational cost. A follow-up investigation of the solvatochromic shift of uracil in aq. soln. further illustrates the potential of PERI-CC2. We discuss the need to explicitly incorporate several water mols. into the region treated by quantum mechanics in order to obtain a reliable and accurate description of the phys. effects when specific solute/solvent interactions as, e.g., hydrogen-bonds are involved.**77**Höfener, S. Coupled-cluster frozen-density embedding using resolution of the identity methods.*J. Comput. Chem.*2014,*35*, 1716– 1724, DOI: 10.1002/jcc.23679Google Scholar77https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC2cbktFSgtA%253D%253D&md5=df4fb2c06c7606f54d0871fa5cc1c937Coupled-cluster frozen-density embedding using resolution of the identity methodsHofener SebastianJournal of computational chemistry (2014), 35 (23), 1716-24 ISSN:.Frozen-density embedding (FDE) is combined with resolution of the identity (RI) Hartree-Fock and a RI-variant of a second-order approximate coupled-cluster singles and doubles (RI-CC2) to determine solvatochromic shifts for the lowest excitation energy of acetone and pyridazine, respectively, each solvated in different environments with total system sizes of about 2.5 nm diameter. The combination of FDE and RI-CC2 increases efficiency and enables the calculation of numerous snapshots with 100 to 300 molecules, also allowing for larger basis sets as well as diffuse functions needed for an accurate treatment of properties. The maximum errors in the solvatochromic shifts amount up to 0.2 eV, which are similar to other approximated studies in the literature.**78**Klamt, A.; Diedenhofen, M. Calculation of Solvation Free Energies with DCOSMO-RS.*J. Phys. Chem. A*2015,*119*, 5439– 5445, DOI: 10.1021/jp511158yGoogle Scholar78https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhvVOhsrc%253D&md5=30a6555a7c94dd502e32e6d8023b3986Calculation of Solvation Free Energies with DCOSMO-RSKlamt, Andreas; Diedenhofen, MichaelJournal of Physical Chemistry A (2015), 119 (21), 5439-5445CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)The concept of dielec. continuum models has turned out to be fruitful for the qual. description of solvation effects in quantum chem. calcns., although from a theor. perspective its basis is questionable, at least if applied to polar solvents, because the electrostatic nearest neighbor interactions in polar solvents are much too strong to be described by macroscopic dielec. continuum theory. On the basis of this insight, the Conductorlike Screening Model for Realistic Solvation (COSMO-RS) had been developed, which gives a thermodynamically consistent, quant. description of solvation effects in polar and nonpolar solvents, even in mixts. and at variable temp., starting from quantum chem. calcns. of solute and solvent mols. embedded in a virtual conductor (COSMO). Though COSMO-RS usually only requires quantum chem. calcns. in the conductor and thus does not allow for studying of the concrete solvent influence on the solute electron d., the direct COSMO-RS (DCOSMO-RS) was introduced, which uses the σ-potential, i.e., a solvent specific response function provided by COSMO-RS, as a replacement of the conductor or dielec. response employed in continuum solvation models. In this article we describe the current status of DCOSMO-RS and demonstrate the performance of the DCOSMO-RS approach for the prediction of free energies of solvation.**79**Hršak, D.; Marefat Khah, A.; Christiansen, O.; Hättig, C. Polarizable Embedded RI-CC2 Method for Two-Photon Absorption Calculations.*J. Chem. Theory Comput.*2015,*11*, 3669– 3678, DOI: 10.1021/acs.jctc.5b00496Google Scholar79https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhtFKjtr3J&md5=863d22fad001a083153824e23b16ead1Polarizable Embedded RI-CC2 Method for Two-Photon Absorption CalculationsHrsak, Dalibor; Marefat Khah, Alireza; Christiansen, Ove; Hattig, ChristofJournal of Chemical Theory and Computation (2015), 11 (8), 3669-3678CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The authors present a novel polarizable embedded resoln.-of-identity coupled cluster singles and approx. doubles (PERI-CC2) method for calcn. of 2-photon absorption (TPA) spectra of large mol. systems. The method was benchmarked for 3 types of systems: a H2O-solvated mol. of formamide, a uracil mol. in aq. soln., and a set of mutants of the channelrhodopsin (ChR) protein. The 1st test case shows that the PERI-CC2 method is in agreement with the PE-CC2 and PE-CCSD methods. The uracil test case indicates that the effects of H bonding on the TPA of a chromophore with the nearest environment is well-described with the PERI-CC2 method. The ChR calcn. shows that the PERI-CC2 method is suited and efficient for calcns. on proteins with medium-sized chromophores.**80**Reinholdt, P.; Nørby, M. S.; Kongsted, J. Modeling of Magnetic Circular Dichroism and UV/Vis Absorption Spectra Using Fluctuating Charges or Polarizable Embedding within a Resonant-Convergent Response Theory Formalism.*J. Chem. Theory Comput.*2018,*14*, 6391– 6404, DOI: 10.1021/acs.jctc.8b00660Google Scholar80https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXitFeisrvI&md5=f343771ed954b7392e59e02111279ef6Modeling of Magnetic Circular Dichroism and UV/Vis Absorption Spectra Using Fluctuating Charges or Polarizable Embedding within a Resonant-Convergent Response Theory FormalismReinholdt, Peter; Noerby, Morten S.; Kongsted, JacobJournal of Chemical Theory and Computation (2018), 14 (12), 6391-6404CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)In recent years computational methods based on embedding have become of increasing popularity when the aim is to introduce environmental effects into quantum chem. calcns. of mol. properties. This is due in particular to the efficiency of such methods while still retaining a high degree of accuracy compared to full quantum chem. treatments. The authors compare 2 popular embedding methods-fluctuating charges (FQ) and polarizable embedding (PE)-highlighting their similar math. structure. Based on a unified formulation of the 2 embedding methods, theory and implementation of these embedding methods within resonant-convergent response theory up to the level of quadratic response are presented. A numerical comparison between FQ and PE is presented for a set of solute-solvent systems based on calcns. of UV/visible and MCD spectra. In the cases considered the FQ and PE models to perform rather similarly, esp. upon introducing effects of explicit conformational sampling into the theor. predictions were found.**81**Marefat Khah, A.; Karbalaei Khani, S.; Hättig, C. Analytic Excited State Gradients for the QM/MM Polarizable Embedded Second-Order Algebraic Diagrammatic Construction for the Polarization Propagator PE-ADC(2).*J. Chem. Theory Comput.*2018,*14*, 4640– 4650, DOI: 10.1021/acs.jctc.8b00396Google Scholar81https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhtlyktLbK&md5=4b18351608dfa213492f857ea6bfb201Analytic Excited State Gradients for the QM/MM Polarizable Embedded Second-Order Algebraic Diagrammatic Construction for the Polarization Propagator PE-ADC(2)Marefat Khah, Alireza; Karbalaei Khani, Sarah; Haettig, ChristofJournal of Chemical Theory and Computation (2018), 14 (9), 4640-4650CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)An implementation of a QM/MM embedding in a polarizable environment is presented for second-order Moller-Plesset perturbation theory, MP2, for ground state energies and mol. gradients and for the second-order Algebraic Diagrammatic Construction, ADC(2), for excitation energies and excited state mol. gradients. In this implementation of PE-MP2 and PE-ADC(2), the polarizable embedded Hartree-Fock wave function is used as uncorrelated ref. state. The polarization-correlation cross terms for the ground and excited states are included in this model via an approx. coupling d. A Lagrangian formulation is used to derive the relaxed electron densities and mol. gradients. The resoln.-of-the-identity approxn. speeds up the calcn. of four-index electron repulsion integrals in the MO basis. As a first application, the method is used to study the photophys. properties of host-guest complexes where the accuracy and weaknesses of the model are also critically examd. It is demonstrated that the ground state geometries of the full quantum mech. calcn. for the supermol. can be well reproduced. For excited state geometries, the deviations from the supermol. calcn. are slightly larger, but still the environment effects are captured qual. correctly, and energy gaps between the ground and excited states are obtained with sufficient accuracy.**82**Sharma, M.; Sierka, M. Efficient Implementation of Density Functional Theory Based Embedding for Molecular and Periodic Systems Using Gaussian Basis Functions.*J. Chem. Theory Comput.*2022,*18*, 6892– 6904, DOI: 10.1021/acs.jctc.2c00380Google Scholar82https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XisF2kurzN&md5=c68685cf6a3a41d6ca52d92092d3425dEfficient Implementation of Density Functional Theory Based Embedding for Molecular and Periodic Systems Using Gaussian Basis FunctionsSharma, Manas; Sierka, MarekJournal of Chemical Theory and Computation (2022), 18 (11), 6892-6904CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A practical and effective implementation of d. functional theory based embedding is reported, which allows us to treat both periodic and aperiodic systems on an equal footing. Its essence is the expansion of orbitals and electron d. of the periodic system using Gaussian basis functions, rather than plane-waves, which provides a unique all-electron direct-space representation, thus avoiding the need for pseudopotentials. This makes the construction of embedding potential for a mol. active subsystem due to a periodic environment quite convenient, as transformation between representations is far from trivial. The three flavors of embedding, mol.-in-mol., mol.-in-periodic, and periodic-in-periodic embedding, are implemented using embedding potentials based on non-additive kinetic energy d. functionals (approx.) and level-shift projection operator (exact). The embedding scheme is coupled with a variety of correlated wave function theory (WFT) methods, thereby providing an efficient way to study the ground and excited state properties of low-dimensional systems using high-level methods for the region of interest. Finally, an implementation of real time-time-dependent d. functional embedding theory (RT-TDDFET) is presented that uses a projection operator-based embedding potential and provides accurate results compared to full RT-TDDFT for systems with uncoupled excitations. The embedding potential is calcd. efficiently using a combination of d. fitting and continuous fast multipole method for the Coulomb term. The applicability of (i) WFT-in-DFT embedding, in predicting the adsorption and excitation energies, and (ii) RT-TDDFET, in predicting the absorption spectra, is explored for various test systems.**83**Treß, R. S.; Hättig, C.; Höfener, S. Employing Pseudopotentials to Tackle Excited-State Electron Spill-Out in Frozen Density Embedding Calculations.*J. Chem. Theory Comput.*2022,*18*, 1737– 1747, DOI: 10.1021/acs.jctc.1c00732Google Scholar83https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XisF2jsrk%253D&md5=30214c4d7f86a2db10726e6d97218f0eEmploying Pseudopotentials to Tackle Excited-State Electron Spill-Out in Frozen Density Embedding CalculationsTress, Robert S.; Haettig, Christof; Hoefener, SebastianJournal of Chemical Theory and Computation (2022), 18 (3), 1737-1747CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)In frozen d. embedding (FDE), the properties of a target mol. are computed in the presence of an effective embedding potential, which accounts for the attractive and repulsive contributions of the environment. The formally exact embedding potential, however, is in practice calcd. using explicit kinetic-energy functionals for which the resulting potentials are in many cases not repulsive enough to account fully for Pauli repulsion by the electrons of the environment and to compensate thereby the strong electron-nuclear attraction. For the excited states on the target mol., this leads to charge spill-out when diffuse basis functions are included, which allow that valence electrons are excited to those regions of the environment where the strong nuclear attraction is not sufficiently compensated by repulsive contributions. To reduce this insufficiency, we propose in the present work the inclusion of at. all-electron pseudopotentials for all environment atoms on top of the conventional embedding potential. In the current work, the pseudopotentials are applied for computing vertical excitation energies of local excited states in complex systems employing the second-order algebraic diagrammatic construction (ADC(2)) scheme. The proposed approach leads to significantly reduced charge spill-out and an improved agreement of FDE and supermol. calcns. in the frozen solvent approxn. In particular, when diffuse functions are employed, the mean abs. deviation (MAD) is reduced from 0.27 to 0.05 eV for the investigated cases.**84**Peng, D.; Middendorf, N.; Weigend, F.; Reiher, M. An efficient implementation of two-component relativistic exact-decoupling methods for large molecules.*J. Chem. Phys.*2013,*138*, 184105, DOI: 10.1063/1.4803693Google Scholar84https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXnsVymt7k%253D&md5=f4614a020113ff004fa93d68072de1beAn efficient implementation of two-component relativistic exact-decoupling methods for large moleculesPeng, Daoling; Middendorf, Nils; Weigend, Florian; Reiher, MarkusJournal of Chemical Physics (2013), 138 (18), 184105/1-184105/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present an efficient algorithm for one- and two-component relativistic exact-decoupling calcns. Spin-orbit coupling is thus taken into account for the evaluation of relativistically transformed (one-electron) Hamiltonian. As the relativistic decoupling transformation has to be evaluated with primitive functions, the construction of the relativistic one-electron Hamiltonian becomes the bottleneck of the whole calcn. for large mols. For the established exact-decoupling protocols, a minimal matrix operation count is established and discussed in detail. We apply our recently developed local DLU scheme to accelerate this step. With our new implementation two-component relativistic d. functional calcns. can be performed invoking the resoln.-of-identity d.-fitting approxn. and (Abelian as well as non-Abelian) point group symmetry to accelerate both the exact-decoupling and the two-electron part. The capability of our implementation is illustrated at the example of silver clusters with up to 309 atoms, for which the cohesive energy is calcd. and extrapolated to the bulk. (c) 2013 American Institute of Physics.**85**Franzke, Y. J.; Middendorf, N.; Weigend, F. Efficient implementation of one- and two-component analytical energy gradients in exact two-component theory.*J. Chem. Phys.*2018,*148*, 104110, DOI: 10.1063/1.5022153Google Scholar85https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXktlGrtbw%253D&md5=a96b520e12b654f2b735d95dc42c450aEfficient implementation of one- and two-component analytical energy gradients in exact two-component theoryFranzke, Yannick J.; Middendorf, Nils; Weigend, FlorianJournal of Chemical Physics (2018), 148 (10), 104110/1-104110/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present an efficient algorithm for one- and two-component anal. energy gradients with respect to nuclear displacements in the exact two-component decoupling approach to the one-electron Dirac equation (X2C). Our approach is a generalization of the spin-free ansatz by Cheng and Gauss [J. Chem. Phys. 135, 084114 (2011)], where the perturbed one-electron Hamiltonian is calcd. by solving a first-order response equation. Computational costs are drastically reduced by applying the diagonal local approxn. to the unitary decoupling transformation (DLU) [D. Peng and M. Reiher, J. Chem. Phys. 136, 244108 (2012)] to the X2C Hamiltonian. The introduced error is found to be almost negligible as the mean abs. error of the optimized structures amts. to only 0.01 pm. Our implementation in TURBOMOLE is also available within the finite nucleus model based on a Gaussian charge distribution. For a X2C/DLU gradient calcn., computational effort scales cubically with the mol. size, while storage increases quadratically. The efficiency is demonstrated in calcns. of large silver clusters and organometallic iridium complexes. (c) 2018 American Institute of Physics.**86**Franzke, Y. J.; Mack, F.; Weigend, F. NMR Indirect Spin-Spin Coupling Constants in a Modern Quasirelativistic Density Functional Framework.*J. Chem. Theory Comput.*2021,*17*, 3974– 3994, DOI: 10.1021/acs.jctc.1c00167Google Scholar86https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXhtlCms7zJ&md5=996a3579680d29b955c1f16a2d8a4c18NMR Indirect Spin-Spin Coupling Constants in a Modern Quasi-Relativistic Density Functional FrameworkFranzke, Yannick J.; Mack, Fabian; Weigend, FlorianJournal of Chemical Theory and Computation (2021), 17 (7), 3974-3994CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A quasi-relativistic implementation of NMR indirect spin-spin coupling consts. is presented. The exact two-component (X2C) Hamiltonian and its diagonal local approxn. to the unitary decoupling transformation (DLU) are utilized together with the (modified) screened nuclear spin-orbit approach. In a restricted kinetic balance, the finite nucleus model is available for both the scalar and vector potentials. The implementation supports d. functionals up to the fourth rung of Jacob's ladder, i.e., (range-sepd.) hybrid and local hybrid functionals based on a seminumerical ansatz. We assess the quality of our quasi-relativistic X2C approach by comparison with "fully" relativistic four-component results for small main-group mols. and alkynyl compds. The mean abs. error introduced by the DLU scheme is less than 0.05 x 1019 T J-2 of the reduced coupling const. for the small main-group mols. and 0.5 Hz for the alkynyl compds. Thus, the error is significantly smaller than finite nucleus size effects for heavy elements. The basis set convergence and the impact of different d. functional approxns. are further studied. We propose a simple scheme to develop segmented-contracted relativistic all-electron basis sets for NMR spin-spin couplings. Our implementation allows us to perform calcns. of extended mols. with reasonable computational effort, which is illustrated for the 1J(119Sn, 31P) coupling const. of a low-valent tin phosphinidenide complex. The corresponding results are in good agreement with the exptl. findings.**87**van Setten, M. J.; Weigend, F.; Evers, F. The GW-Method for Quantum Chemistry Applications: Theory and Implementation.*J. Chem. Theory Comput.*2013,*9*, 232– 246, DOI: 10.1021/ct300648tGoogle Scholar87https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38Xhs1Wlsb3O&md5=05a4203f8be671fd7248bff230d1eeceThe GW-Method for Quantum Chemistry Applications: Theory and Implementationvan Setten, M. J.; Weigend, F.; Evers, F.Journal of Chemical Theory and Computation (2013), 9 (1), 232-246CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The GW-technol. corrects the Kohn-Sham (KS) single particle energies and single particle states for artifacts of the exchange-correlation (XC) functional of the underlying d. functional theory (DFT) calcn. We present the formalism and implementation of GW adapted for std. quantum chem. packages. Our implementation is tested using a typical set of mols. We find that already after the first iteration of the self-consistency cycle, G0W0, the deviations of quasi-particle energies from exptl. ionization potentials and electron affinities can be reduced by an order of magnitude against those of KS-DFT using GGA or hybrid functionals. Also, we confirm that even on this level of approxn. there is a considerably diminished dependency of the G0W0-results on the XC-functional of the underlying DFT.**88**Kaplan, F.; Harding, M. E.; Seiler, C.; Weigend, F.; Evers, F.; van Setten, M. J. Quasi-Particle Self-Consistent GW for Molecules.*J. Chem. Theory Comput.*2016,*12*, 2528– 2541, DOI: 10.1021/acs.jctc.5b01238Google Scholar88https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XnslCrsL4%253D&md5=56c82d6d964f0442879250973d6ed2d7Quasi-Particle Self-Consistent GW for MoleculesKaplan, F.; Harding, M. E.; Seiler, C.; Weigend, F.; Evers, F.; van Setten, M. J.Journal of Chemical Theory and Computation (2016), 12 (6), 2528-2541CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present the formalism and implementation of quasi-particle self-consistent GW (qsGW) and eigenvalue only quasi-particle self-consistent GW (evGW) adapted to std. quantum chem. packages. Our implementation is benchmarked against high-level quantum chem. computations (coupled-cluster theory) and exptl. results using a representative set of mols. Furthermore, we compare the qsGW approach for five mols. relevant for org. photovoltaics to self-consistent GW results (scGW) and analyze the effects of the self-consistency on the ground state d. by comparing calcd. dipole moments to their exptl. values. We show that qsGW makes a significant improvement over conventional G0W0 and that partially self-consistent flavors (in particular evGW) can be excellent alternatives.**89**Müller, C.; Sharma, M.; Sierka, M. Real-time time-dependent density functional theory using density fitting and the continuous fast multipole method.*J. Comput. Chem.*2020,*41*, 2573– 2582, DOI: 10.1002/jcc.26412Google Scholar89https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhvVSlsbbJ&md5=dcaa74b3bad92cbd952b3b639c28423cReal-time time-dependent density functional theory using density fitting and the continuous fast multipole methodMueller, Carolin; Sharma, Manas; Sierka, MarekJournal of Computational Chemistry (2020), 41 (30), 2573-2582CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)An implementation of real-time time-dependent d. functional theory (RT-TDDFT) within the TURBOMOLE program package is reported using Gaussian-type orbitals as basis functions, second and fourth order Magnus propagator, and the SCF as well as the predictor-corrector time integration schemes. The Coulomb contribution to the Kohn-Sham matrix is calcd. combining d. fitting approxn. and the continuous fast multipole method. Performance of the implementation is benchmarked for mol. systems with different sizes and dimensionalities. For linear alkane chains, the wall time for d. matrix time propagation step is comparable to the Kohn-Sham (KS) matrix construction. However, for larger two- and three-dimensional mols., with up to about 5,000 basis functions, the computational effort of RT-TDDFT calcns. is dominated by the KS matrix evaluation. In addn., the max. time step is evaluated using a set of small mols. of different polarities. The photoabsorption spectra of several mol. systems calcd. using RT-TDDFT are compared to those obtained using linear response time-dependent d. functional theory and coupled cluster methods.**90**Tapavicza, E.; Bellchambers, G.; Vincent, J. C.; Furche, F.*Ab initio*non-adiabatic dynamics.*Phys. Chem. Chem. Phys.*2013,*15*, 18336– 18348, DOI: 10.1039/c3cp51514aGoogle Scholar90https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXhsF2gtr3N&md5=20d83c5904d57818388bdf28f3e34bb8Ab initio non-adiabatic molecular dynamicsTapavicza, Enrico; Bellchambers, Gregory D.; Vincent, Jordan C.; Furche, FilippPhysical Chemistry Chemical Physics (2013), 15 (42), 18336-18348CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)A review. Adiabatic nuclear potential energy surfaces (PESs) defined via the Born-Oppenheimer (BO) approxn. are a fundamental concept underlying chem. reactivity theory. For a wide range of excited-state phenomena such as radiationless decay, energy and charge transfer, and photochem. reactions, the BO approxn. breaks down due to strong couplings between two or more BO PESs. Non-adiabatic mol. dynamics (NAMD) is the method of choice to model these processes. We review new developments in quantum-classical dynamics, anal. deriv. methods, and time-dependent d. functional theory (TDDFT) which have lead to a dramatic expansion of the scope of ab initio NAMD simulations for mol. systems in recent years. We focus on atom-centered Gaussian basis sets allowing highly efficient simulations for mols. and clusters, esp. in conjunction with hybrid d. functionals. Using anal. deriv. techniques, forces and deriv. couplings can be obtained with machine precision in a given basis set, which is crucial for accurate and stable dynamics. We illustrate the performance of surface-hopping TDDFT for photochem. reactions of the lowest singlet excited states of cyclohexadiene, several vitamin D derivs., and a bicyclic cyclobutene. With few exceptions, the calcd. quantum yields and excited state lifetimes agree qual. with expt. For systems with ∼50 atoms, the present Turbomole implementation allows NAMD simulations with 0.2-0.4 ns total simulation time using hybrid d. functionals and polarized double zeta valence basis sets on medium-size compute clusters. We close by discussing open problems and future directions.**91***TURBOMOLE*. https://www.turbomole.org (accessed 2022-12-01).Google ScholarThere is no corresponding record for this reference.**92**Balasubramani, S. G.; Chen, G. P.; Coriani, S.; Diedenhofen, M.; Frank, M. S.; Franzke, Y. J.; Furche, F.; Grotjahn, R.; Harding, M. E.; Hättig, C.; Hellweg, A.; Helmich-Paris, B.; Holzer, C.; Huniar, U.; Kaupp, M.; Marefat Khah, A.; Karbalaei Khani, S.; Müller, T.; Mack, F.; Nguyen, B. D.; Parker, S. M.; Perlt, E.; Rappoport, D.; Reiter, K.; Roy, S.; Rückert, M.; Schmitz, G.; Sierka, M.; Tapavicza, E.; Tew, D. P.; van Wüllen, C.; Voora, V. K.; Weigend, F.; Wodyński, A.; Yu, J. M. TURBOMOLE: Modular program suite for*ab initio*quantum-chemical and condensed-matter simulations.*J. Chem. Phys.*2020,*152*, 184107, DOI: 10.1063/5.0004635Google Scholar92https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXps1Ogs7s%253D&md5=836d350d0cdc56aeea644742a098b23bTURBOMOLE: Modular program suite for ab initio quantum-chemical and condensed-matter simulationsBalasubramani, Sree Ganesh; Chen, Guo P.; Coriani, Sonia; Diedenhofen, Michael; Frank, Marius S.; Franzke, Yannick J.; Furche, Filipp; Grotjahn, Robin; Harding, Michael E.; Hattig, Christof; Hellweg, Arnim; Helmich-Paris, Benjamin; Holzer, Christof; Huniar, Uwe; Kaupp, Martin; Marefat Khah, Alireza; Karbalaei Khani, Sarah; Muller, Thomas; Mack, Fabian; Nguyen, Brian D.; Parker, Shane M.; Perlt, Eva; Rappoport, Dmitrij; Reiter, Kevin; Roy, Saswata; Ruckert, Matthias; Schmitz, Gunnar; Sierka, Marek; Tapavicza, Enrico; Tew, David P.; van Wullen, Christoph; Voora, Vamsee K.; Weigend, Florian; Wodynski, Artur; Yu, Jason M.Journal of Chemical Physics (2020), 152 (18), 184107CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A review. TURBOMOLE is a collaborative, multi-national software development project aiming to provide highly efficient and stable computational tools for quantum chem. simulations of mols., clusters, periodic systems, and solns. The TURBOMOLE software suite is optimized for widely available, inexpensive, and resource-efficient hardware such as multi-core workstations and small computer clusters. TURBOMOLE specializes in electronic structure methods with outstanding accuracy-cost ratio, such as d. functional theory including local hybrids and the RPA, GW-Bethe-Salpeter methods, second-order Moller-Plesset theory, and explicitly correlated coupled-cluster methods. TURBOMOLE is based on Gaussian basis sets and has been pivotal for the development of many fast and low-scaling algorithms in the past three decades, such as integral-direct methods, fast multipole methods, the resoln.-of-the-identity approxn., imaginary frequency integration, Laplace transform, and pair natural orbital methods. This review focuses on recent addns. to TURBOMOLE's functionality, including excited-state methods, RPA and Green's function methods, relativistic approaches, high-order mol. properties, solvation effects, and periodic systems. A variety of illustrative applications along with accuracy and timing data are discussed. Moreover, available interfaces to users as well as other software are summarized. TURBOMOLE's current licensing, distribution, and support model are discussed, and an overview of TURBOMOLE's development workflow is provided. Challenges such as communication and outreach, software infrastructure, and funding are highlighted. (c) 2020 American Institute of Physics.**93**Furche, F.; Ahlrichs, R.; Hättig, C.; Klopper, W.; Sierka, M.; Weigend, F. Turbomole.*WIREs Comput. Mol. Sci.*2014,*4*, 91– 100, DOI: 10.1002/wcms.1162Google Scholar93https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXisFSjtr8%253D&md5=ba18d31f334d4e5ed83459cfb4aeb2d3TurbomoleFurche, Filipp; Ahlrichs, Reinhart; Haettig, Christof; Klopper, Wim; Sierka, Marek; Weigend, FlorianWiley Interdisciplinary Reviews: Computational Molecular Science (2014), 4 (2), 91-100CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)Turbomole is a highly optimized software package for large-scale quantum chem. simulations of mols., clusters, and periodic solids. Turbomole uses Gaussian basis sets and specializes on predictive electronic structure methods with excellent cost to performance characteristics, such as (time-dependent) d. functional theory (TDDFT), second-order Moller-Plesset theory, and explicitly correlated coupled cluster (CC) methods. These methods are combined with ultraefficient and numerically stable algorithms such as integral-direct and Laplace transform methods, resoln.-of-the-identity, pair natural orbitals, fast multipole, and low-order scaling techniques. Apart from energies and structures, a variety of optical, elec., and magnetic properties are accessible from anal. energy derivs. for electronic ground and excited states. Recent addns. include post-Kohn-Sham calcns. within the RPA, periodic calcns., spin-orbit couplings, explicitly correlated CC singles doubles and perturbative triples methods, CC singles doubles excitation energies, and nonadiabatic mol. dynamics simulations using TDDFT. A dedicated graphical user interface and a user support network are also available.**94**van Wüllen, C. Shared-memory parallelization of the TURBOMOLE programs AOFORCE, ESCF, and EGRAD: How to quickly parallelize legacy code.*J. Comput. Chem.*2011,*32*, 1195– 1201, DOI: 10.1002/jcc.21692Google Scholar94https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXivVGjs7c%253D&md5=7f4f3313dbd10b4a48c9cf04d0ca01d0Shared-memory parallelization of the TURBOMOLE programs AOFORCE, ESCF, and EGRAD: How to quickly parallelize legacy codevan Wuellen, ChristophJournal of Computational Chemistry (2011), 32 (6), 1195-1201CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)The programs ESCF, EGRAD, and AOFORCE are parts of the TURBOMOLE program package and compute excited-state properties and ground-state geometric hessians, resp., for Hartree-Fock and d. functional methods. The range of applicability of these programs has been extended by allowing them to use all CPU cores on a given node in parallel. The parallelization strategy is not new and duplicates what is std. today in the calcn. of ground-state energies and gradients. The focus is on how this can be achieved without needing extensive modifications of the existing serial code. The key ingredient is to fork off worker processes with sepd. address spaces as they are needed. Test calcns. on a mol. with about 80 atoms and 1000 basis functions show good parallel speedup up to 32 CPU cores. © 2010 Wiley Periodicals, Inc.; J. Comput. Chem., 2011.**95**Bachorz, R. A.; Bischoff, F. A.; Glöß, A.; Hättig, C.; Höfener, S.; Klopper, W.; Tew, D. P. The MP2-F12 Method in the TURBOMOLE Program Package.*J. Comput. Chem.*2011,*32*, 2492– 2513, DOI: 10.1002/jcc.21825Google Scholar95https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXntlOqsbs%253D&md5=88445b01ce7627f43d7f7e20b81f9aa3The MP2-F12 method in the TURBOMOLE program packageBachorz, Rafal A.; Bischoff, Florian A.; Gloess, Andreas; Haettig, Christof; Hoefener, Sebastian; Klopper, Wim; Tew, David P.Journal of Computational Chemistry (2011), 32 (11), 2492-2513CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)A detailed description of the explicitly correlated second-order Moller-Plesset perturbation theory (MP2-F12) method, as implemented in the TURBOMOLE program package, is presented. The TURBOMOLE implementation makes use of d. fitting, which greatly reduces the prefactor for integral evaluation. Methods are available for the treatment of ground states of open- and closed-shell species, using unrestricted as well as restricted (open-shell) Hartree-Fock ref. determinants. Various methodol. choices and approxns. are discussed. The performance of the TURBOMOLE implementation is illustrated by example calcns. of the mols. leflunomide, prednisone, methotrexate, ethylenedioxytetrathiafulvalene, and a cluster model for the adsorption of methanol on the zeolite H-ZSM-5. Various basis sets are used, including the correlation-consistent basis sets specially optimized for explicitly correlated calcns. (cc-pVXZ-F12). © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011.**96**Almaraz, E. P.; Furche, F. Initial OpenMP version of aoforce and escf, released with TURBOMOLE V6.3; TURBOMOLE GmbH: Karlsruhe, Germany, 2011.Google ScholarThere is no corresponding record for this reference.**97**Holzer, C.; Franzke, Y. J. OpenMP version of ridft, rdgrad, and egrad with contributions to mpshift, dscf, and grad; improved OpenMP version of aoforce and escf, released with TURBOMOLE V7.4 and further improved in TURBOMOLE V7.5; TURBOMOLE GmbH: Karlsruhe, Germany, 2020.Google ScholarThere is no corresponding record for this reference.**98**Hättig, C.; Hellweg, A.; Köhn, A. Distributed memory parallel implementation of energies and gradients for second-order Møller-Plesset perturbation theory with the resolution-of-the-identity approximation.*Phys. Chem. Chem. Phys.*2006,*8*, 1159– 1169, DOI: 10.1039/b515355gGoogle Scholar98https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BD283isFKisg%253D%253D&md5=a0a7b241faf2ffa621d484dc8862651fDistributed memory parallel implementation of energies and gradients for second-order Moller-Plesset perturbation theory with the resolution-of-the-identity approximationHattig Christof; Hellweg Arnim; Kohn AndreasPhysical chemistry chemical physics : PCCP (2006), 8 (10), 1159-69 ISSN:1463-9076.We present a parallel implementation of second-order Moller-Plesset perturbation theory with the resolution-of-the-identity approximation (RI-MP2). The implementation is based on a recent improved sequential implementation of RI-MP2 within the Turbomole program package and employs the message passing interface (MPI) standard for communication between distributed memory nodes. The parallel implementation extends the applicability of canonical MP2 to considerably larger systems. Examples are presented for full geometry optimizations with up to 60 atoms and 3300 basis functions and MP2 energy calculations with more than 200 atoms and 7000 basis functions.**99**Müller, T. Global Array toolkit based distributed shared memory version of ridft and rdgrad, released with TURBOMOLE V6.0; TURBOMOLE GmbH: Karlsruhe, Germany, 2011.Google ScholarThere is no corresponding record for this reference.**100**Müller, T. Parallel DFT in Turbomole, Linear Algebra. In*High Performance Computing in Chemistry*; Grotendorst, J., Ed.; NIC Series, Vol. 25; John von Neumann Institute for Computing: Jülich, Germany, 2005; pp 83– 107.Google ScholarThere is no corresponding record for this reference.**101**Müller, T. MPI based version of ridft and rdgrad with native distributed shared memory support, released with TURBOMOLE V7.2; TURBOMOLE GmbH: Karlsruhe, Germany, 2017.Google ScholarThere is no corresponding record for this reference.**102**van Wüllen, C. Hybrid OpenMP/MPI parallelization of dscf, grad, and aoforce, released with TURBOMOLE V7.2; TURBOMOLE GmbH: Karlsruhe, Germany, 2017.Google ScholarThere is no corresponding record for this reference.**103**Plessow, P. Reaction Path Optimization without NEB Springs or Interpolation Algorithms.*J. Chem. Theory Comput.*2013,*9*, 1305– 1310, DOI: 10.1021/ct300951jGoogle Scholar103https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXjtVCis7Y%253D&md5=95db3547abb2c77b8309804cec44c1a6Reaction Path Optimization without NEB Springs or Interpolation AlgorithmsPlessow, P.Journal of Chemical Theory and Computation (2013), 9 (3), 1305-1310CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)This letter describes a chain-of-states method that optimizes reaction paths under the sole constraint of equally spaced structures. In contrast to NEB (Nudged Elastic Band) and string methods, it requires no spring forces, interpolation algorithms, or other heuristics to control structure distribution. Rigorous use of a quadratic PES allows calcn. of an optimization step with a predefined distribution in Cartesian space. The method is a formal extension of single-structure quasi-Newton methods. An initial guess can be evolved, as in the growing string method.**104**Reiter, K.; Kühn, M.; Weigend, F. Vibrational circular dichroism spectra for large molecules and molecules with heavy elements.*J. Chem. Phys.*2017,*146*, 054102, DOI: 10.1063/1.4974897Google Scholar104https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXitVKht7Y%253D&md5=9af9378c74df28492fe2648ab17b8378Vibrational circular dichroism spectra for large molecules and molecules with heavy elementsReiter, Kevin; Kuehn, Michael; Weigend, FlorianJournal of Chemical Physics (2017), 146 (5), 054102/1-054102/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The authors present an implementation of vibrational CD (VCD) spectra in TURBOMOLE. The authors mainly followed the route proposed by Cheeseman [Chem. Phys. Lett. 252, 211(1996)] and extended the modules for calcg. the magnetic response and vibrational frequencies accordingly. The implementation allows for gauge origin invariant employment of effective core potentials, as demonstrated for Co(ppy)3, ppy = 2-Phenylpyridine. In this way, scalar relativistic effects are covered and heavy elements can be treated. Further, with the present implementation mol. symmetry may be efficiently exploited, which makes the calcn. of large (sym.) systems feasible. The calcn. of the VCD spectrum of icosahedral C622+0 is shown as an illustrative application. (c) 2017 American Institute of Physics.**105**Sierka, M.; Döbler, J.; Sauer, J.; Santambrogio, G.; Brümmer, M.; Wöste, L.; Janssens, E.; Meijer, G.; Asmis, K. Unexpected Structures of Aluminum Oxide Clusters in the Gas Phase.*Angew. Chem., Int. Ed.*2007,*46*, 3372– 3375, DOI: 10.1002/anie.200604823Google Scholar105https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXlsVKgt7w%253D&md5=25cd44b3fb8dc7e91c4591e9ccc7634bUnexpected structures of aluminum oxide clusters in the gas phaseSierka, Marek; Doebler, Jens; Sauer, Joachim; Santambrogio, Gabriele; Bruemmer, Mathias; Woeste, Ludger; Janssens, Ewald; Meijer, Gerard; Asmis, Knut R.Angewandte Chemie, International Edition (2007), 46 (18), 3372-3375CODEN: ACIEF5; ISSN:1433-7851. (Wiley-VCH Verlag GmbH & Co. KGaA)IR photodissocn. expts. and DFT optimizations implementing a genetic algorithm suggest that small aluminum oxide clusters have unusual structures in the gas phase. For example, the structure proposed for [(Al2O3)4]+ has no features in common with bulk alumina phases.**106**Steffen, C.; Thomas, K.; Huniar, U.; Hellweg, A.; Rubner, O.; Schroer, A. TmoleX – A graphical user interface for TURBOMOLE.*J. Comput. Chem.*2010,*31*, 2967– 2970, DOI: 10.1002/jcc.21576Google Scholar106https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXhtlyqtb7K&md5=a1396048b211ee0c29f818e18b2b7b24TmoleX-A graphical user interface for TURBOMOLESteffen, Claudia; Thomas, Klaus; Huniar, Uwe; Hellweg, Arnim; Rubner, Oliver; Schroer, AlexanderJournal of Computational Chemistry (2010), 31 (16), 2967-2970CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)We herein present the graphical user interface (GUI) TmoleX for the quantum chem. program package TURBOMOLE. TmoleX allows users to execute the complete work-flow of a quantum chem. investigation from the initial building a structure to the visualization of the results in a user friendly graphical front end. The purpose of TmoleX is to make TURBOMOLE easy to use and to provide a high degree of flexibility. Hence, it should be a valuable tool for most users from beginners to experts. The program is developed in Java and runs on Linux, Windows, and Mac platforms. It can be used to run calcns. on local desktops as well as on remote computers.**107***OpenMP*. https://www.openmp.org (accessed 2022-12-05).Google ScholarThere is no corresponding record for this reference.**108**MPI Forum. https://www.mpi-forum.org (accessed 2022-12-05).Google ScholarThere is no corresponding record for this reference.**109**Holzer, C. An improved seminumerical Coulomb and exchange algorithm for properties and excited states in modern density functional theory.*J. Chem. Phys.*2020,*153*, 184115, DOI: 10.1063/5.0022755Google Scholar109https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXitlelur7N&md5=cad44a4ffbef63e25cf320bbcd316688An improved seminumerical Coulomb and exchange algorithm for properties and excited states in modern density functional theoryHolzer, ChristofJournal of Chemical Physics (2020), 153 (18), 184115CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A seminumerical algorithm capable of performing large-scale (time-dependent) d. functional theory (TD-DFT) calcns. to ext. excitation energies and other ground-state and excited-state properties is outlined. The algorithm uses seminumerical integral techniques for evaluating Coulomb and exchange parts for a set of d. matrixes as occurring in std. TD-DFT or similar methods for the evaluation of vibrational frequencies. A suitable optimized de-aliasing procedure is introduced. The latter does not depend on further auxiliary quantities and retains the symmetry of a given d. matrix. The algorithm is self-contained and applicable to any orbital basis set available without the need for further auxiliary basis sets or optimized de-aliasing grids. Relativistic two-component excited-state TD-DFT calcns. are reported for the first time using the developed seminumerical algorithm for std. and local hybrid d. functional approxns. Errors are compared with the widely used "resoln. of the identity" (RI) approxns. for Coulomb (RI-J) and exchange integrals (RI-K). The fully seminumerical algorithm does not exhibit an enlarged error for std. DFT functionals compared to the RI approxn. For the more involved local hybrid functionals and within strong external fields, accuracy is even considerably improved. (c) 2020 American Institute of Physics.**110**Maier, T. M.; Arbuznikov, A. V.; Kaupp, M. Local hybrid functionals: Theory, implementation, and performance of an emerging new tool in quantum chemistry and beyond.*WIREs Comput. Mol. Sci.*2019,*9*, e1378, DOI: 10.1002/wcms.1378Google ScholarThere is no corresponding record for this reference.**111**Janesko, B. G. Replacing hybrid density functional theory: motivation and recent advances.*Chem. Soc. Rev.*2021,*50*, 8470– 8495, DOI: 10.1039/D0CS01074JGoogle Scholar111https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXhtF2qsL7L&md5=a4f6b67de8732090fc4437a4af48a674Replacing hybrid density functional theory: motivation and recent advancesJanesko, Benjamin G.Chemical Society Reviews (2021), 50 (15), 8470-8495CODEN: CSRVBR; ISSN:0306-0012. (Royal Society of Chemistry)A review. D. functional theory (DFT) is the most widely-used electronic structure approxn. across chem., physics, and materials science. Every year, thousands of papers report hybrid DFT simulations of chem. structures, mechanisms, and spectra. Unfortunately, hybrid DFT's accuracy is ultimately limited by tradeoffs between over-delocalization and under-binding. This review summarizes these tradeoffs, and introduces six modern attempts to go beyond them while maintaining hybrid DFT's relatively low computational cost: DFT+U, self-interaction corrections, localized orbital scaling corrections, local hybrid functionals, real-space nondynamical correlation, and our rung-3.5 approach. The review concludes with practical suggestions for DFT users to identify and mitigate these tradeoffs' impact on their simulations.**112**Bahmann, H.; Kaupp, M. Efficient Self-Consistent Implementation of Local Hybrid Functionals.*J. Chem. Theory Comput.*2015,*11*, 1540– 1548, DOI: 10.1021/ct501137xGoogle Scholar112https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXkt1OitbY%253D&md5=05843fda01f57cee5c624a471dba1f5dEfficient Self-Consistent Implementation of Local Hybrid FunctionalsBahmann, Hilke; Kaupp, MartinJournal of Chemical Theory and Computation (2015), 11 (4), 1540-1548CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Local hybrid d. functionals, with position-dependent exact-exchange admixt., are an important extension to the popular global hybrid functionals, promising improved accuracy for many properties. An efficient implementation is crucial to make local hybrids available for widespread application. The resoln.-of-the-identity approach used in previous implementations to compute nonstandard two-electron integrals has been found to require large uncontracted basis sets, rendering the cost of local hybrid calcns. impractical for large-scale systems. On the basis of recently promoted seminumerical implementations of exact exchange in global hybrid functionals, we present an efficient, self-consistent implementation of local hybrid functionals within the generalized Kohn-Sham scheme. The final cost of a local hybrid calcn. is equal to that of a meta-GGA global hybrid using the seminumerical algorithm. Since seminumerical schemes exhibit superior scaling with respect to system and basis set size over anal. exact exchange, and this advantage is not affected by a position-dependent admixt. of exact exchange, local hybrid calcns. for large systems are now possible.**113**Kehry, M.; Franzke, Y. J.; Holzer, C.; Klopper, W. Quasirelativistic two-component core excitations and polarisabilities from a damped-response formulation of the Bethe-Salpeter equation.*Mol. Phys.*2020,*118*, e1755064, DOI: 10.1080/00268976.2020.1755064Google Scholar113https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhsVGlsL%252FK&md5=5fa39e68d33845e275ab743d1a1dec3eQuasirelativistic two-component core excitations and polarisabilities from a damped-response formulation of the Bethe-Salpeter equationKehry, Max; Franzke, Yannick J.; Holzer, Christof; Klopper, WimMolecular Physics (2020), 118 (21-22), e1755064/1-e1755064/16CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)A damped-response formalism in the GW approxn. to the Bethe-Salpeter equation (BSE) is presented and implemented. It is based on a quasirelativistic two-component (2c) approach that includes scalar-relativistic and spin-orbit effects derived from the one-electron Dirac equation. A generalised solver, which also allows to calc. static, dynamic and damped-response polarisabilites, is discussed in detail. Throughout our implementation, the resoln.-of-the-identity approxn. is employed to reduce the computational effort. The performance of 2c GW-BSE is benchmarked against exptl. and high-level ab initio data and compared to std. d.-functional theory approaches, including modern local hybrid functionals for which a proper non-collinear kernel for Kramers-restricted systems is reported.**114**Grotjahn, R.; Lauter, G. J.; Haasler, M.; Kaupp, M. Evaluation of Local Hybrid Functionals for Electric Properties: Dipole Moments and Static and Dynamic Polarizabilities.*J. Phys. Chem. A*2020,*124*, 8346– 8358, DOI: 10.1021/acs.jpca.0c06939Google Scholar114https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhslKqur3K&md5=418ee23fa42144d81cd0ab6480aaf358Evaluation of Local Hybrid Functionals for Electric Properties: Dipole Moments and Static and Dynamic PolarizabilitiesGrotjahn, Robin; Lauter, Gregor J.; Haasler, Matthias; Kaupp, MartinJournal of Physical Chemistry A (2020), 124 (40), 8346-8358CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)Local hybrid functionals are a class of exchange-correlation functionals that feature a real-space dependent admixt. of exact (Hartree-Fock like) exchange governed by a local mixing function. Recently we reported the LH20t functional with wide chem. applicability and excellent performance for the GMTKN55 main-group energetics test suite (M. Haasler et al., J. Chem. Theory Comput.2020, 16, 5645-5657). Here, we present a systematic evaluation of earlier and recent local hybrid functionals for large test sets of dipole moments and static polarizabilities and for a smaller set of dynamic polarizabilities for heterocycles. Comparisons with coupled-cluster benchmark data show robust performance of all investigated local hybrids for dipole moments and polarizabilities. The two best local hybrids are the new LH20t and LH14t-calPBE. LH20t gives a percentage-relative mean-square deviation of 5.87% for the dipole moment test set and one of 4.30% for static polarizabilities. This is only slightly inferior to the currently best performances among rung 4 functionals. Most notably, no large outliers are obsd. in contrast to some other hybrid functionals. This shows that the current most highly parametrized (nine-parameter) LH20t functional clearly produces not only good energetics but also accurate electron densities and elec.-field responses. The influences of various aspects of local hybrids are examd. to aid in the further development of this class of functionals.**115**Schattenberg, C. J.; Reiter, K.; Weigend, F.; Kaupp, M. An Efficient Coupled-Perturbed Kohn-Sham Implementation of NMR Chemical Shift Computations with Local Hybrid Functionals and Gauge-Including Atomic Orbitals.*J. Chem. Theory Comput.*2020,*16*, 931– 943, DOI: 10.1021/acs.jctc.9b00944Google Scholar115https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXjt1ehtQ%253D%253D&md5=d29d9c33f5a8e9c8b1222cc252f1d66eAn Efficient Coupled-Perturbed Kohn-Sham Implementation of NMR Chemical Shift Computations with Local Hybrid Functionals and Gauge-Including Atomic OrbitalsSchattenberg, Caspar Jonas; Reiter, Kevin; Weigend, Florian; Kaupp, MartinJournal of Chemical Theory and Computation (2020), 16 (2), 931-943CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Nuclear shielding calcns. for local hybrid (LH) functionals with position-dependent exact-exchange admixts. within a coupled-perturbed Kohn-Sham (CPKS) framework were implemented into the TURBOMOLE code using efficient seminumerical integration techniques to deal with 2-electron integrals. When using gauge-including AOs, LHs generate addnl. terms within the pre-loop section of the CPKS scheme compared to global hybrid (GH) functionals, related to perturbed electron-repulsion integrals. These terms were implemented and tested in detail, together with dependencies on grid sizes and integral screening procedures. Even with relatively small grids, a seminumerical treatment of GHs reproduces anal. GH results with high accuracy while improving scaling with system and basis-set sizes significantly. The extra terms generated by LHs in the pre-loop part increase the scaling of that contribution slightly, but the advantages compared to the anal. scheme are largely retained, in particular for the typically large basis sets used in NMR shift calcns., allowing for a very efficient computational scheme. An initial comparison of 4 1st-generation LHs based on LDA exchange for a shielding test set of 15 small main-group mols. against high-level CCSD(T) benchmark data indicates a substantial redn. of the systematically underestimated shieldings compared to semilocal functionals or GHs for nonhydrogen nuclei when a so-called t-LMF is used to control the position dependence of the exact-exchange admixt. But proton shieldings are underestimated with this LMF, while an LH with a so-called s-LMF performs much better. These results are discussed in the context of experience for other properties, and they suggest directions for further improvements of LHs regarding nuclear shieldings.**116**Wodyński, A.; Kaupp, M. Noncollinear Relativistic Two-Component X2C Calculations of Hyperfine Couplings Using Local Hybrid Functionals. Importance of the High-Density Coordinate Scaling Limit.*J. Chem. Theory Comput.*2020,*16*, 314– 325, DOI: 10.1021/acs.jctc.9b00911Google Scholar116https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BB3MfpvVyrsQ%253D%253D&md5=b23843963f87de9c9be5f02a4b1e7c50Noncollinear Relativistic Two-Component X2C Calculations of Hyperfine Couplings Using Local Hybrid Functionals. Importance of the High-Density Coordinate Scaling LimitWodynski Artur; Kaupp MartinJournal of chemical theory and computation (2020), 16 (1), 314-325 ISSN:.Local hybrid functionals with position-dependent exact-exchange admixture have been implemented in the noncollinear spin form into a two-component X2C code and are evaluated for the hyperfine coupling tensors of a series of 3d, 4d, and 5d transition-metal complexes. One aim is to see if the potential of local hybrid functionals toward an improved balance between core-shell and valence-shell spin polarization, recently identified in nonrelativistic computations on 3d complexes (Schattenberg, C.; Maier, T. M.; Kaupp, M. J. Chem. Theory Comput.2018,14, 5653-5672), can be extended to the hyperfine couplings of heavier metal centers. The correctness of the two-component implementation is first established by comparison to previous computations for 3d systems with or without notable spin-orbit contributions to their hyperfine tensors, and the good performance of a standard "t-LMF" local mixing function is confirmed. However, when moving to 4d and 5d metal centers, the performance of such local mixing functions deteriorates. This is likely due to their violation of the homogeneous coordinate scaling condition in the high-density limit, which is particularly important for the core shells of heavier atoms. A local mixing function that respects this high-density limit performs notably better for heavier metal centers. However, it brings in much too high exact-exchange admixtures for the 3d systems and is too inflexible to simultaneously provide reasonable chemical accuracy in other areas. These results point to the ongoing need to develop improved local mixing functions and local hybrid functionals that exhibit favorable properties in different areas of space defined by very high and much lower electron densities.**117**Holzer, C.; Franzke, Y. J. A Local Hybrid Exchange Functional Approximation from First Principles.*J. Chem. Phys.*2022,*157*, 034108, DOI: 10.1063/5.0100439Google Scholar117https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XhvFWnsrrL&md5=7159a93d64bd3fb99e73344d525b20d8A local hybrid exchange functional approximation from first principlesHolzer, Christof; Franzke, Yannick J.Journal of Chemical Physics (2022), 157 (3), 034108CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Local hybrid functionals are a more flexible class of d. functional approxns., allowing for a position-dependent admixt. of exact exchange. This addnl. flexibility, however, comes with a more involved math. form and a more complicated design. A common denominator for previously constructed local hybrid functionals is the usage of thermochem. benchmark data to construct these functionals. Herein, we design a local hybrid functional without relying on benchmark data. Instead, we construct it in a more ab initio manner, following the principles of modern meta-generalized gradient approxns. and considering theor. constraints. To achieve this, we make use of the d. matrix expansion and a local mixing function based on an approx. correlation length. The accuracy of the developed d. functional approxn. is assessed for thermochem., excitation energies, polarizabilities, magnetizabilities, NMR (NMR) spin-spin coupling consts., NMR shieldings, and shifts, as well as EPR g-tensors and hyperfine coupling consts. Here, the new exchange functional shows a robust performance and is esp. well suited for atomization energies, barrier heights, excitation energies, NMR coupling consts., and EPR properties, whereas it loses some ground for the NMR shifts. Therefore, the designed functional is a major step forward for functionals that have been designed from first principles. (c) 2022 American Institute of Physics.**118**Franzke, Y. J.; Holzer, C. Impact of the current density on paramagnetic NMR properties.*J. Chem. Phys.*2022,*157*, 031102, DOI: 10.1063/5.0103898Google Scholar118https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XhvFWgsrbO&md5=efea0164ad2b17842712a77d3f42f491Impact of the current density on paramagnetic NMR propertiesFranzke, Yannick J.; Holzer, ChristofJournal of Chemical Physics (2022), 157 (3), 031102CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Meta-generalized gradient approxns. (meta-GGAs) and local hybrid functionals generally depend on the kinetic energy d. τ. For magnetic properties, this necessitates generalizations to ensure gauge invariance. In most implementations, τ is generalized by incorporating the external magnetic field. However, this introduces artifacts in the response of the d. matrix and does not satisfy the iso-orbital constraint. Here, we extend previous approaches based on the c.d. to paramagnetic NMR (NMR) shieldings and ESR (EPR) g-tensors. The impact is assessed for main-group compds. and transition-metal complexes considering 25 d. functional approxns. It is shown that the c.d. leads to substantial improvements-esp. for the popular Minnesota and strongly constrained and appropriately normed (SCAN) functional families. Thus, we strongly recommend to use the c.d. generalized τ in paramagnetic NMR and EPR calcns. with meta-GGAs. (c) 2022 American Institute of Physics.**119**Holzer, C.; Franzke, Y. J.; Kehry, M. Assessing the Accuracy of Local Hybrid Density Functional Approximations for Molecular Response Properties.*J. Chem. Theory Comput.*2021,*17*, 2928– 2947, DOI: 10.1021/acs.jctc.1c00203Google Scholar119https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXpslejsrY%253D&md5=9a7b4bbcbd5a5b3dd35fce208f203513Assessing the Accuracy of Local Hybrid Density Functional Approximations for Molecular Response PropertiesHolzer, Christof; Franzke, Yannick J.; Kehry, MaxJournal of Chemical Theory and Computation (2021), 17 (5), 2928-2947CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A comprehensive overview of the performance of local hybrid functionals for mol. properties like excited states, ionization potentials within the GW framework, polarizabilities, magnetizabilities, NMR chem. shifts, and NMR spin-spin coupling consts. is presented. We apply the generalization of the kinetic energy, τ, with the paramagnetic c.d. to all magnetic properties and the excitation energies from time-dependent d. functional theory. This restores gauge invariance for these properties. Different ansatze for local mixing functions such as the iso-orbital indicator, the correlation length, the Gorling-Levy second-order limit, and the spin polarization are compared. For the latter, we propose a modified version of the corresponding hyper-generalized gradient approxn. functional of Perdew, Staroverov, Tao, and Scuseria (PSTS) to allow for a numerically stable evaluation of the exchange-correlation kernel and hyperkernel. The PSTS functional leads to a very consistent improvement compared to the related TPSSh functional. It is further shown that the "best" choice of the local mixing function depends on the studied property and mol. class. While functionals based on the iso-orbital indicator lead to rather accurate excitation energies and ionization energies, the results are less impressive for NMR properties, for which a considerable dependence on the considered mol. test set and nuclei is obsd. Johnson's local hybrid functional based on the correlation length yields remarkable results for NMR shifts of compds. featuring heavy elements and also for the excitation energies of org. compds.**120**Haasler, M.; Maier, T. M.; Grotjahn, R.; Gückel, S.; Arbuznikov, A. V.; Kaupp, M. A Local Hybrid Functional with Wide Applicability and Good Balance between (De)Localization and Left–Right Correlation.*J. Chem. Theory Comput.*2020,*16*, 5645– 5657, DOI: 10.1021/acs.jctc.0c00498Google Scholar120https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhsVaqtrvJ&md5=c49b85ac199c8b4656c96e45e18fd29fA Local Hybrid Functional with Wide Applicability and Good Balance between (De)Localization and Left-Right CorrelationHaasler, Matthias; Maier, Toni M.; Grotjahn, Robin; Gueckel, Simon; Arbuznikov, Alexei V.; Kaupp, MartinJournal of Chemical Theory and Computation (2020), 16 (9), 5645-5657CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A new local hybrid functional, LH20t, with position-dependent exact-exchange admixt. governed by a simple local mixing function (g(r) = b·τW(r)/τ(r)), combined with gradient-cor. (PBE) exchange and meta-GGA (B95) correlation, as well as a second-order GGA-based pig2 calibration function to address the ambiguity of exchange-energy densities, has been constructed. The adjustable parameters of LH20t have been optimized in a multi-step procedure based on thermochem. kinetics data and measures of spurious nondynamical correlation. LH20t has subsequently been evaluated for the full GMTKN55 main-group energetics test suite, with and without an added DFT-D4 dispersion correction. Performance of the new functional in the GMTKN55 tests is excellent, better than any global hybrid so far, approaching the best results for any rung 4 functional, without any noticeable artifacts due to the gauge ambiguity. The robust performance across the board is combined with enhanced exact-exchange admixts. of > 70% near the nuclei and asymptotically (but low admixt. in bonds). This helps to provide excellent performance for a wide variety of excitation classes (core, valence singlet and triplet, Rydberg, short-range inter-valence charge-transfer) in TDDFT evaluations. Notably, LH20t is the first functional that provides simultaneously the correct description for the most extreme localized and delocalized cases of the MVO-10 test set of gas-phase mixed-valence systems. This outstanding performance for mixed-valence systems, which signals a very fine balance between reduced delocalization errors and a reasonable description of left-right correlation, is corroborated by tests on ground- and excited-state properties for org. and organometallic mixed-valence systems in soln.**121**Harrison, D. P.; Grotjahn, R.; Naher, M.; Ghazvini, S. M. B. H.; Mazzucato, D. M.; Korb, M.; Moggach, S. A.; Lambert, C.; Kaupp, M.; Low, P. J. Quantum Interference in Mixed-Valence Complexes: Tuning Electronic Coupling Through Substituent Effects.*Angew. Chem., Int. Ed.*2022,*61*, e202211000, DOI: 10.1002/anie.202211000Google Scholar121https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38Xis1ahsLrE&md5=76e5beb6999267ec787f6db1d1a0b081Quantum Interference in Mixed-Valence Complexes: Tuning Electronic Coupling Through Substituent EffectsHarrison, Daniel P.; Grotjahn, Robin; Naher, Masnun; Ghazvini, Seyed M. B. H.; Mazzucato, Daniel M.; Korb, Marcus; Moggach, Stephen A.; Lambert, Colin; Kaupp, Martin; Low, Paul J.Angewandte Chemie, International Edition (2022), 61 (45), e202211000CODEN: ACIEF5; ISSN:1433-7851. (Wiley-VCH Verlag GmbH & Co. KGaA)While 2- or 5-OMe groups on the bridging phenylene ring in [{Cp*(dppe)RuC≡C}2(μ-1,3-C6H4)]+ have little influence on the electronic structure of this weakly coupled mixed-valence complex, a 4-OMe substituent enhances ground state electron delocalization, and increases the intensity of the IVCT transition. Vibrational frequency and TDDFT calcns. (LH20t-D3(BJ), def2-SVP, COSMO (CH2Cl2)) on [{Cp*(dppe)RuC≡C}2(μ-1,3-C6H3-n-OMe)]+ (n = 2, 4, 5) models are in excellent agreement with the exptl. results. The stronger ground state coupling is attributed to the change in compn. of the β-HOSO brought about by the 4-OMe group, which is ortho or para to each of the metal fragments. The intensity of the IVCT transition increases with the greater overlap of the β-HOSO and β-LUSO, while the relative phases of the β-HOSO and β-LUSO in the 4-OMe substituted complex are consistent with predictions of constructive quantum interference from mol. circuit rules.**122**Gückel, S.; Safari, P.; Bagher Hosseini Ghazvini, S. M.; Hall, M. R.; Gluyas, J. B. G.; Kaupp, M.; Low, P. J. Iron Versus Ruthenium: Evidence for the Distinct Differences in the Electronic Structures of Hexa-1,3,5-triyn-1,6-diyl-bridged Complexes [Cp* (dppe){M}{μ-(C≡C)_{3}}{M(dppe)Cp*}]^{+}(M = Fe, Ru).*Organometallics*2021,*40*, 346– 357, DOI: 10.1021/acs.organomet.0c00681Google Scholar122https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXhs1elurY%253D&md5=3a31612c288171251124ffe4d0fb9d47Iron Versus Ruthenium: Evidence for the Distinct Differences in the Electronic Structures of Hexa-1,3,5-triyn-1,6-diyl-bridged Complexes [{Cp*(dppe)M}{μ-(C≡C)3}{M(dppe)Cp*}]+ (M = Fe, Ru)Gueckel, Simon; Safari, Parvin; Bagher Hosseini Ghazvini, Seyed Mohammad; Hall, Michael R.; Gluyas, Josef B. G.; Kaupp, Martin; Low, Paul J.Organometallics (2021), 40 (3), 346-357CODEN: ORGND7; ISSN:0276-7333. (American Chemical Society)The spectroelectrochem. generated IR and near-IR spectra of the homo-bimetallic, hexa-1,3,5-triyn-1,6-diyl-bridged complex cations [{Cp*(dppe)M}(μ-C≡CC≡CC≡C){M(dppe)Cp*}]+ (M = Fe, [1]+; Ru, [2]+) were analyzed using d. functional theory calcns. based on global (BLYP35) and local (LH20t) hybrid functionals. The differences in the no. of IR active ν(C≡CC≡CC≡C) modes in these complexes are attributed to the distinct electronic localization of the Fe(II)-Fe(III) mixed-valence cation [1]+ on the IR timescale, as opposed to the delocalized electronic character of [2]+.**123**Grotjahn, R.; Kaupp, M. Reliable TDDFT Protocol Based on a Local Hybrid Functional for the Prediction of Vibronic Phosphorescence Spectra Applied to Tris(2,2′-bipyridine)-Metal Complexes.*J. Phys. Chem. A*2021,*125*, 7099– 7110, DOI: 10.1021/acs.jpca.1c05101Google Scholar123https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXhslahs7fI&md5=46be0e277609db70f9cc5a861c24585aReliable TDDFT Protocol Based on a Local Hybrid Functional for the Prediction of Vibronic Phosphorescence Spectra Applied to Tris(2,2'-bipyridine)-Metal ComplexesGrotjahn, Robin; Kaupp, MartinJournal of Physical Chemistry A (2021), 125 (32), 7099-7110CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)An efficient computational protocol for the prediction of vibrationally resolved phosphorescence spectra is developed and validated for five tris(2,2'-bipyridine)-metal complexes ([M(bpy)3]n+, where M = Zn, Ru, Rh, Os, Ir). The outstanding feature of this protocol is the use of full linear-response time-dependent d. functional theory (TDDFT) for the excited-state triplet calcn., i.e., the commonly seen strategies employing the Tamm-Dancoff approxn. (TDA) or unrestricted d. functional theory (DFT) calcns. for the T1 state are not needed. This is achieved by the use of a local hybrid functional (LH12ct-SsirPW92) that features a real-space dependent admixt. of exact exchange governed by a local mixing function. The excellent performance of this LH for triplet excitation energies known from previous studies transfers to a remarkable mean abs. error of 0.06 eV for the phosphorescence 0-0 energies investigated herein, while the popular B3PW91 functional gives an error of 0.27 eV in TDDFT and 0.09 eV in unrestricted DFT calcns., resp. The advantages of the local hybrid are particularly apparent for excited states with a mixed-valence character. The influence of spin-orbit coupling was found to be significant for [Os(bpy)3]2+ red-shifting the 0-0 energy for phosphorescence by 0.17 eV, while the effect is negligible for the other complexes (<0.03 eV). The influence of the basis-set and integration-grid sizes is evaluated, and a computationally lighter protocol is validated that leads to drastic savings in computation time with negligible loss in accuracy.**124**Grotjahn, R.; Kaupp, M. Validation of Local Hybrid Functionals for Excited States: Structures, Fluorescence, Phosphorescence, and Vibronic Spectra.*J. Chem. Theory Comput.*2020,*16*, 5821– 5834, DOI: 10.1021/acs.jctc.0c00520Google Scholar124https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXhsVaqtrvN&md5=c56d8a90696f17e7a3171e72f49def8dValidation of Local Hybrid Functionals for Excited States: Structures, Fluorescence, Phosphorescence, and Vibronic SpectraGrotjahn, Robin; Kaupp, MartinJournal of Chemical Theory and Computation (2020), 16 (9), 5821-5834CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Local hybrid functionals are evaluated in linear-response TDDFT computations for a broad range of excited-state properties including excited-state structures, fluorescence, and phosphorescence energies and the vibronic shape of absorption and phosphorescence spectra. Computation of such properties requires the optimization of excited states, which is facilitated by the recent implementation of excited-state gradients for local hybrid functionals in the TURBOMOLE program (Grothjahn, R. et al., J. Chem. Theory Comput., 2019, 15, 5508). Comparison with coupled-cluster ref. values reveals competitive performance of local hybrids for excited-state bond lengths with particular advantages for carbon-halogen, carbon-carbon, and carbon-nitrogen bonds. As with most global and range-sepd. hybrid functionals, carbonyl and thionyl bonds in n → π* excited states are found to be too compact. For the emission energies, results depend on the multiplicity of the excited state. While the local hybrid functionals tested perform moderately well, comparable to global hybrids, for singlet states (fluorescence energies), they provide outstanding accuracy for triplet states (phosphorescence energies), only matched by those from the highly empirical M06-2X hybrid functional. The assessment of the shape of vibronic spectra reveals rather small differences between local hybrid functionals and conventional hybrid functionals with comparable exact-exchange admixt. The advantages for phosphorescence energies and the robust performance for the shape of vibronic spectra are combined to showcase the potential of local hybrid functionals for the prediction of phosphorescence spectra.**125**Grotjahn, R.; Kaupp, M. Assessment of hybrid functionals for singlet and triplet excitations: Why do some local hybrid functionals perform so well for triplet excitation energies?.*J. Chem. Phys.*2021,*155*, 124108, DOI: 10.1063/5.0063751Google Scholar125https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXitFaqt7zK&md5=f837614712fc49704b06b2774a6c7a9cAssessment of hybrid functionals for singlet and triplet excitations: Why do some local hybrid functionals perform so well for triplet excitation energies?Grotjahn, Robin; Kaupp, MartinJournal of Chemical Physics (2021), 155 (12), 124108CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The performance of various hybrid d. functionals is assessed for 105 singlet and 105 corresponding triplet vertical excitation energies from the QUEST database. The overall lowest mean abs. error is obtained with the local hybrid (LH) functional LH12ct-SsirPW92 with individual errors of 0.11 eV (0.11 eV) for singlet (triplet) n → π* excitations and 0.29 eV (0.17 eV) for π → π* excitations. This is slightly better than with the overall best performing global hybrid M06-2X [n → π*: 0.13 eV (0.17 eV), π → π*: 0.30 eV (0.20 eV)], while most other global and range-sepd. hybrids and some LHs suffer from the "triplet problem" of time-dependent d. functional theory. This is exemplified by correlating the errors for singlet and triplet excitations on a state-by-state basis. The excellent performance of LHs based on a common local mixing function, i.e., an LMF constructed from the spin-summed rather than the spin-resolved semilocal quantities, is systematically investigated by the introduction of a spin-channel interpolation scheme that allows us to continuously modulate the fraction of opposite-spin terms used in the LMF. The correlation of triplet and singlet errors is systematically improved for the n → π* excitations when larger fractions of the opposite-spin-channel are used in the LMF, whereas this effect is limited for the π → π* excitations. This strongly supports a previously made hypothesis that attributes the excellent performance of LHs based on a common LMF to cross-spin-channel nondynamical correlation terms. (c) 2021 American Institute of Physics.**126**Grotjahn, R.; Kaupp, M. A Look at Real-World Transition-Metal Thermochemistry and Kinetics with Local Hybrid Functionals.*Isr. J. Chem.*2022, e202200021, DOI: 10.1002/ijch.202200021Google ScholarThere is no corresponding record for this reference.**127**Schattenberg, C. J.; Kaupp, M. Extended Benchmark Set of Main-Group Nuclear Shielding Constants and NMR Chemical Shifts and Its Use to Evaluate Modern DFT Methods.*J. Chem. Theory Comput.*2021,*17*, 7602– 7621, DOI: 10.1021/acs.jctc.1c00919Google Scholar127https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXisFWgtLbJ&md5=c46bad9bcbadb805fbb4e7b1fb6ea6b5Extended Benchmark Set of Main-Group Nuclear Shielding Constants and NMR Chemical Shifts and Its Use to Evaluate Modern DFT MethodsSchattenberg, Caspar Jonas; Kaupp, MartinJournal of Chemical Theory and Computation (2021), 17 (12), 7602-7621CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)An extended theor. benchmark set, NS372, for light main-group nuclear shieldings and NMR shifts has been constructed based on high-level GIAO-CCSD(T)/pcSseg-3//CCSD(T)/cc-pVQZ ref. data. After removal of the large static-correlation cases O3, F3-, and BH from the statistical evaluations for the 17O, 19F, and 11B subsets, the benchmark comprises overall 372 shielding values in 117 mols. with a wide range of electronic-structure situations, contg. 124 1H, 14 11B, 93 13C, 43 15N, 31 17O, 47 19F, 14 31P, and 6 33S shielding consts. The CCSD(T)/pcSseg-3 data are shown to be close to the basis-set and method limit and thus provide an excellent benchmark to evaluate more approx. methods, such as d. functional approaches. This dataset has been used to evaluate Hartree-Fock (HF) and MP2, and a wide range of exchange-correlation functionals from local d. approxn. (LDA) to generalized gradient approxns. (GGAs) and meta-GGAs (focusing on their current-d. functional implementations), as well as global hybrid, range-sepd. hybrid, local hybrid, and double-hybrid functionals. Starting with abs. shielding consts., the DSD-PBEP86 double hybrid is confirmed to provide the highest accuracy, with an aggregate relative mean abs. error (rel.MAE) of only 0.9%, followed by MP2 (1.1%). MP2 and double hybrids only show larger errors for a few systems with the largest static-correlation effects. The double-hybrid B2GP-PLYP, the two local hybrids cLH12ct-SsirPW92 and cLH12ct-SsifPW92, and the current-d. functional meta-GGA cB97M-V follow closely behind (all 1.5%), as do some further functionals, cLH20t and cMN15-L (both 1.6%), as well as B2PLYP and KT3 (both 2.0%). Functionals on the lower rungs of the usual ladder offer the advantage of lower computational cost and access to larger mols. Closer examn. also reveals the best-performing methods for individual nuclei in the test set. Different ways of treating τ-dependent functionals are evaluated. When moving from abs. shielding consts. to chem. shifts, some of the methods can benefit from systematic error compensation, and the overall error range somewhat narrows. Further methods now achieve the 2% threshold of relative MAEs, including functionals based on TPSS (TPSSh, cmPSTS).**128**Schattenberg, C. J.; Lehmann, M.; Bühl, M.; Kaupp, M. Systematic Evaluation of Modern Density Functional Methods for the Computation of NMR Shifts of 3d Transition-Metal Nuclei.*J. Chem. Theory Comput.*2022,*18*, 273– 292, DOI: 10.1021/acs.jctc.1c00964Google Scholar128https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXivVGjt7bF&md5=22b07439179ce76cae423d88205dade5Systematic Evaluation of Modern Density Functional Methods for the Computation of NMR Shifts of 3d Transition-Metal NucleiSchattenberg, Caspar Jonas; Lehmann, Morten; Buehl, Michael; Kaupp, MartinJournal of Chemical Theory and Computation (2022), 18 (1), 273-292CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A wide range of d. functionals from all rungs of Jacob's ladder has been evaluated systematically for a set of exptl. 3d transition-metal NMR shifts of 70 complexes encompassing 12 x 49Ti, 10 x 51V, 10 x 53Cr, 11 x 55Mn, 9 x 57Fe, 9 x 59Co, and 9 x 61Ni shift values, as well as a diverse range of electronic-structure characteristics. The overall 39 functionals evaluated include one LDA, 8 GGAs, 7 meta-GGAs (including their current-d.-functional - CDFT - versions), 9 global hybrids, 4 range-sepd. hybrids, 8 local hybrids, and 2 double hybrids, and we also include Hartree-Fock and MP2 calcns. While recent evaluations of the same functionals for a very large coupled-cluster-based benchmark of main-group shieldings and shifts achieved in some cases aggregate percentage mean abs. errors clearly below 2%, the best results for the present 3d-nuclei set are in the range between 4 and 5%. Strikingly, the overall best-performing functionals are the recently implemented CDFT versions of two meta-GGAs, namely cM06-L (4.0%) and cVSXC (4.3%), followed by cLH14t-calPBE (4.9%), B3LYP (5.0%), and cLH07t-SVWN (5.1%), i.e. the previously best-performing global hybrid and two local hybrids. A no. of further functionals achieve aggregate deviations in the range 5-6%. Range-sepd. hybrids offer no particular advantage over global hybrids. Due to the overall poor performance of Hartree-Fock theory for all systems except the titanium complexes, MP2 and double-hybrid functionals are unsuitable for these 3d-nucleus shifts and provide large errors. Global hybrid functionals with larger EXX admixts., such as BHLYP or M06-2X, also perform poorly, and some other highly parametrized global hybrids also are unsuitable. For many functionals depending on local kinetic energy τ, their CDFT variants perform much better than their "non-CDFT" versions. This holds notably also for the above-mentioned M06-L and VSXC, while the effect is small for τ-dependent local hybrids and can even be somewhat detrimental to the agreement with expt. for a few other cases. The sepn. between well-performing and more poorly performing functionals is mainly detd. by their results for the most crit. nuclei 55Mn, 57Fe, and 59Co. Here either moderate exact-exchange admixts. or CDFT versions of meta-GGAs are beneficial for the accuracy. The overall deviations of the better-performing global or local hybrids are then typically dominated by the 53Cr shifts, where triplet instabilities appear to disfavor exact-exchange admixt. Further detailed analyses help to pinpoint specific nuclei and specific types of complexes that are challenges for a given functional.**129**Mori-Sánchez, P.; Cohen, A. J. The derivative discontinuity of the exchange–correlation functional.*Phys. Chem. Chem. Phys.*2014,*16*, 14378– 14387, DOI: 10.1039/C4CP01170HGoogle Scholar129https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhtVKgs7rN&md5=f6bc8c6b2fc2d935256eb70a1e760db0The derivative discontinuity of the exchange-correlation functionalMori-Sanchez, Paula; Cohen, Aron J.Physical Chemistry Chemical Physics (2014), 16 (28), 14378-14387CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)The deriv. discontinuity is a key concept in electronic structure theory in general and d. functional theory in particular. The electronic energy of a quantum system exhibits deriv. discontinuities with respect to different degrees of freedom that are a consequence of the integer nature of electrons. The classical understanding refers to the deriv. discontinuity of the total energy as a function of the total no. of electrons (N), but it can also manifest at const. N. Examples are shown in models including several hydrogen systems with varying nos. of electrons or nuclear charge (Z), as well as the 1-dimensional Hubbard model (1DHM). Two sides of the problem are investigated: first, the failure of currently used approx. exchange-correlation functionals in DFT and, second, the importance of the deriv. discontinuity in the exact electronic structure of mols., as revealed by full CI (FCI). Currently, all approx. functionals, including hybrids, miss the deriv. discontinuity, leading to basic errors that can be seen in many ways: from the complete failure to give the total energy of H2 and H2+, to the missing gap in Mott insulators such as stretched H2 and the thermodn. limit of the 1DHM, or a qual. incorrect d. in the HZ mol. with two electrons and incorrect electron transfer processes. Description of the exact particle behavior of electrons is emphasized, which is key to many important phys. processes in real systems, esp. those involving electron transfer, and offers a challenge for the development of new exchange-correlation functionals.**130**Janesko, B. G.; Proynov, E.; Kong, J.; Scalmani, G.; Frisch, M. J. Practical Density Functionals beyond the Overdelocalization–Underbinding Zero-Sum Game.*J. Phys. Chem. Lett.*2017,*8*, 4314– 4318, DOI: 10.1021/acs.jpclett.7b02023Google Scholar130https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhtlyhsLbF&md5=353871cb2039fb0410484284e249469bPractical Density Functionals beyond the Overdelocalization-Underbinding Zero-Sum GameJanesko, Benjamin G.; Proynov, Emil; Kong, Jing; Scalmani, Giovanni; Frisch, Michael J.Journal of Physical Chemistry Letters (2017), 8 (17), 4314-4318CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)D. functional theory (DFT) uses a d. functional approxn. (DFA) to add electron correlation to mean-field electronic structure calcns. Std. strategies (generalized gradient approxns. GGAs, meta-GGAs, hybrids, etc.) for building DFAs, no matter whether based on exact constraints or empirical parametrization, all face a zero-sum game between overdelocalization (fractional charge error, FC) and underestimation of covalent bonding (fractional spin error, FS). This work presents an alternative strategy. Practical "Rung 3.5" ingredients are used to implement insights from hyper-GGA DFAs that reduce both FS and FC errors. Prototypes of this strategy qual. improve FS and FC error over 40 years of std. DFAs while maintaining low cost and practical evaluation of properties. Numerical results ranging from transition metal thermochem. to absorbance peaks and excited-state geometry optimizations highlight this strategy's promise and indicate areas requiring further development.**131**Mori-Sánchez, P.; Cohen, A. J.; Yang, W. Discontinuous Nature of the Exchange-Correlation Functional in Strongly Correlated Systems.*Phys. Rev. Lett.*2009,*102*, 066403, DOI: 10.1103/PhysRevLett.102.066403Google Scholar131https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhvFaru7c%253D&md5=a85a142169d61c212421b483bfb68259Discontinuous Nature of the Exchange-Correlation Functional in Strongly Correlated SystemsMori-Sanchez, Paula; Cohen, Aron J.; Yang, WeitaoPhysical Review Letters (2009), 102 (6), 066403/1-066403/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)Std. approxns. for the exchange-correlation functional have been found to give big errors for the linearity condition of fractional charges, leading to delocalization error, and the constancy condition of fractional spins, leading to static correlation error. These two conditions are now unified and extended to states with both fractional charge and fractional spin to give a much more stringent condition: the exact energy functional is a plane, linear along the fractional charge coordinate and const. along the fractional spin coordinate with a line of discontinuity at the integer. Violation of this condition underlies the failure of all known approx. functionals to describe the gaps in strongly correlated systems. It is shown that explicitly discontinuous functionals of the d. or orbitals that go beyond these currently used smooth approxns. is the key for the application of d. functional theory to strongly correlated systems.**132**Becke, A. D. Density functionals for static, dynamical, and strong correlation.*J. Chem. Phys.*2013,*138*, 074109, DOI: 10.1063/1.4790598Google Scholar132https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXivVWgtbk%253D&md5=dfbd1d9c4f903da09863bb246eee34cbDensity functionals for static, dynamical, and strong correlationBecke, Axel D.Journal of Chemical Physics (2013), 138 (7), 074109/1-074109/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In this work, our exact-exchange-based static + dynamical correlation d. functional is generalized to include "strong" correlation, i.e., accurate computations on dissocg. chem. systems without breaking space or spin symmetries and without using multi-determinantal ref. states. Also, we introduce a strong-correlation benchmark set composed of space- and spin-symmetrized open-shell atoms on which the generalized functional is tested. Initial results are very promising. (c) 2013 American Institute of Physics.**133**Kong, J.; Proynov, E. Density Functional Model for Nondynamic and Strong Correlation.*J. Chem. Theory Comput.*2016,*12*, 133– 143, DOI: 10.1021/acs.jctc.5b00801Google Scholar133https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhvFWrsbfK&md5=4f9a95f191b9b6dab21926d4dedffb23Density Functional Model for Nondynamic and Strong CorrelationKong, Jing; Proynov, EmilJournal of Chemical Theory and Computation (2016), 12 (1), 133-143CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)A single-term d. functional model for the left-right nondynamic/strong electron correlation is presented based on single-determinant Kohn-Sham d. functional theory. It is derived from modeling the adiabatic connection for kinetic correlation energy based on phys. arguments, with the correlation potential energy based on the Becke'13 model. This functional satisfies some known scaling relationships for correlation functionals. The fractional spin error is further reduced substantially with a new d.-functional correction. Preliminary tests with self-consistent-field implementation show that the model, with only three empirical parameters, recovers the majority of left-right nondynamic/strong correlation upon bond dissocn. and performs reasonably well for atomization energies and singlet-triplet energy splittings. This study also demonstrates the feasibility of developing DFT functionals for nondynamic and strong correlation within the single-determinant KS scheme.**134**Wodyński, A.; Arbuznikov, A. V.; Kaupp, M. Local hybrid functionals augmented by a strong-correlation model.*J. Chem. Phys.*2021,*155*, 144101, DOI: 10.1063/5.0058917Google Scholar134https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXit1GlsrbP&md5=6a54022cf0d25bca0bf403adb30cb6adLocal hybrid functionals augmented by a strong-correlation modelWodynski, Artur; Arbuznikov, Alexei V.; Kaupp, MartinJournal of Chemical Physics (2021), 155 (14), 144101CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The strong-correlation factor of the recent KP16/B13 exchange-correlation functional has been adapted and applied to the framework of local hybrid (LH) functionals. The expression identifiable as nondynamical (NDC) and dynamical (DC) correlations in LHs is modified by inserting a position-dependent KP16/B13-style strong-correlation factor qAC(r) based on a local version of the adiabatic connection. Different ways of deriving this factor are evaluated for a simple one-parameter LH based on the local d. approxn. While the direct derivation from the LH NDC term fails due to known deficiencies, hybrid approaches, where the factor is detd. from the B13 NDC term as in KP16/B13 itself, provide remarkable improvements. In particular, a modified B13 NDC expression using Patra's exchange-hole curvature showed promising results. When applied to the simple LH as a first attempt, it reduces at. fractional-spin errors and deficiencies of spin-restricted bond dissocn. curves to a similar extent as the KP16/B13 functional itself while maintaining the good accuracy of the underlying LH for atomization energies and reaction barriers in weakly correlated situations. The performance of different NDC expressions in deriving strong-correlation corrections is analyzed, and areas for further improvements of strong-correlation cor. LHs and related approaches are identified. All the approaches evaluated in this work have been implemented self-consistently into a developers' version of the Turbomole program. (c) 2021 American Institute of Physics.**135**Wodyński, A.; Kaupp, M. Local Hybrid Functional Applicable to Weakly and Strongly Correlated Systems.*J. Chem. Theory Comput.*2022,*18*, 6111– 6123, DOI: 10.1021/acs.jctc.2c00795Google Scholar135https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XisV2jsb3I&md5=46d8a3f9c306961aa48bd7b016b28fa5Local Hybrid Functional Applicable to Weakly and Strongly Correlated SystemsWodynski, Artur; Kaupp, MartinJournal of Chemical Theory and Computation (2022), 18 (10), 6111-6123CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The recent idea (Wodynski, A.; Arbuznikov, A. V.; Kaupp M., J. Chem. Phys.2021,155, 144101) to augment local hybrid functionals by a strong-correlation (s.c.) factor obtained from the adiabatic connection in the spirit of the KP16 model has been extended and applied to generate the accurate s.c.-cor. local hybrid functional scLH22t. By damping small values of the ratio between nondynamical and dynamical correlation entering the correction factor, it has become possible to avoid double counting of nondynamical correlation for weakly correlated situations and thereby preserve the excellent accuracy of the underlying LH20t local hybrid for such cases almost perfectly. On the other hand, scLH22t improves substantially over LH20t in reducing fractional-spin errors (FSEs), in providing improved spin-restricted bond dissocn. curves, and in treating some typical systems with multireference character. The obtained FSEs are similar to those of the KP16/B13 model and slightly larger than for B13, but performance for weakly correlated systems is better than for these two related methods, which are also difficult to use self-consistently. The recent DM21 functional based on the training of a deep neural network still performs somewhat better than scLH22t but allows no phys. insights into the origins of reduced FSEs. Examn. of local mixing functions (LMFs) for the cor. scLH22t and uncorrected LH20t functionals provides further insights: in weakly correlated situations, the LMF remains essentially unchanged. Strong-correlation effects manifest in a redn. of the LMF values in certain regions of space, even to the extent of producing neg. LMF values. It is suggested that this is the mechanism by which also DM21, which may be viewed as a range-sepd. local hybrid, is able to reduce FSEs.**136**Haunschild, R.; Scuseria, G. E. Range-separated local hybrids.*J. Chem. Phys.*2010,*132*, 224106, DOI: 10.1063/1.3451078Google Scholar136https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXntlGntLs%253D&md5=2a0dc90756817a867a179a8dc3c899e2Range-separated local hybridsHaunschild, Robin; Scuseria, Gustavo E.Journal of Chemical Physics (2010), 132 (22), 224106/1-224106/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present two range-sepd. (screened and long-range-cor.) extensions to our recent work on local hybrids of generalized gradient approxn. exchange. Our screened local hybrid improves over HSE06 for thermochem. and barrier heights. The long-range-cor. (LC) local hybrid improves over LC-ωPBE for heats of formation and nonhydrogen transfer reaction barriers but does not reach the accuracy of LC-ωPBE for hydrogen transfer barrier heights. (c) 2010 American Institute of Physics.**137**Janesko, B. G.; Krukau, A. V.; Scuseria, G. E. Self-consistent generalized Kohn-Sham local hybrid functionals of screened exchange: Combining local and range-separated hybridization.*J. Chem. Phys.*2008,*129*, 124110, DOI: 10.1063/1.2980056Google Scholar137https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXht1WnsLvI&md5=7ed36b36d5edd1e85f6ecfe39ff9b553Self-consistent generalized Kohn-Sham local hybrid functionals of screened exchange: Combining local and range-separated hybridizationJanesko, Benjamin G.; Krukau, Aliaksandr V.; Scuseria, Gustavo E.Journal of Chemical Physics (2008), 129 (12), 124110/1-124110/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present local hybrid functionals that incorporate a position-dependent admixt. of short-range (screened) nonlocal exact [Hartree-Fock-type (HF)] exchange. We test two limiting cases: screened local hybrids with no long-range HF exchange and long-range-cor. local hybrids with 100% long-range HF exchange. Long-range-cor. local hybrids provide the exact asymptotic exchange-correlation potential in finite systems, while screened local hybrids avoid the problems inherent to long-range HF exchange in metals and small-bandgap systems. We treat these functionals self-consistently using the nonlocal exchange potential constructed from Kohn-Sham orbital derivs. Generalized Kohn-Sham calcns. with screened and long-range-cor. local hybrids can provide accurate mol. thermochem. and kinetics, comparable to existing local hybrids of full-range exchange. Generalized Kohn-Sham calcns. with existing full-range local hybrids provide results consistent with previous non-self-consistent and "localized local hybrid" calcns. These new functionals appear to provide a promising extension of existing local and range-sepd. hybrids. (c) 2008 American Institute of Physics.**138**Klawohn, S.; Bahmann, H. Self-Consistent Implementation of Hybrid Functionals with Local Range Separation.*J. Chem. Theory Comput.*2020,*16*, 953– 963, DOI: 10.1021/acs.jctc.9b01015Google Scholar138https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3cXjt1Cruw%253D%253D&md5=3a99a068321c22580969de8fc39c2554Self-Consistent Implementation of Hybrid Functionals with Local Range SeparationKlawohn, Sascha; Bahmann, HilkeJournal of Chemical Theory and Computation (2020), 16 (2), 953-963CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Building on the previously introduced concept of local range-sepn. (LRS) [Krukau et al. J. Chem. Phys. 129, 124103 (2008)], we report the first self-consistent implementation of hybrid exchange functionals with a position-dependent range-sepn. parameter. The two-electron integrals emerging from long-range exact exchange are calcd. semi-numerically. This avoids formerly suggested approxns. to the exact exchange part and paves the way for applications to larger and chem. relevant systems. Addnl., we investigate the role of short-range exchange in this LRS scheme by comparing the local d. approxn. and PBE-type functionals. We propose a semiempirical approach for the range-sepn. function based on common ingredients of semilocal exchange-correlation functionals. Four parameters are optimized to a small training set of atomization energies and barrier heights. In comparison with established hybrid and semilocal functionals, the LRS functional performs well for basic chem. properties. In addn., our best functional yields outer-valence spectra comparable to optimally-tuned approaches.**139**Fürst, S.; Haasler, M.; Grotjahn, R.; Kaupp, M. Full Implementation, Optimization, and Evaluation of a Range-Separated Local Hybrid Functional with Wide Accuracy for Ground and Excited States.*J. Chem. Theory Comput.*2023,*19*, 488– 502, DOI: 10.1021/acs.jctc.2c00782Google Scholar139https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3sXmsFGqtA%253D%253D&md5=37b74f5ab453e1fc912104ca392e7dd8Full Implementation, Optimization, and Evaluation of a Range-Separated Local Hybrid Functional with Wide Accuracy for Ground and Excited StatesFuerst, Susanne; Haasler, Matthias; Grotjahn, Robin; Kaupp, MartinJournal of Chemical Theory and Computation (2023), 19 (2), 488-502CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We report the first full and efficient implementation of range-sepd. local hybrid functionals (RSLHs) into the TURBOMOLE program package. This enables the computation of ground-state energies and nuclear gradients, as well as excitation energies. Regarding the computational effort, RSLHs scale like regular local hybrids (LHs) with system or basis-set size, and increase timings by a factor of 2-3 in total. An advanced RSLH, ωLH22t, has been optimized for atomization energies and reaction barriers. It is an extension of the recent LH20t local hybrid and is based on short-range PBE and long-range HF exchange-energy densities, a pig2 calibration function to deal with the gauge ambiguity of exchange-energy densities, as well as reoptimized B95c correlation. ωLH22t has been evaluated for a wide range of ground-state and excited-state quantities. It further improves upon the already successful LH20t for the GMTKN55 main-group energetics test suite, and it outperforms any global hybrid while performing close to the top rung-4 functional, ωB97M-V, for these evaluations when augmented by D4 dispersion corrections. ωLH22t performs excellently for transition-metal reactivity and provides good balance between delocalization errors and left-right correlation for mixed-valence systems, with a somewhat larger bias toward localized states compared to LH20t. It approaches the accuracy of the best local hybrids so far for core, valence singlet and triplet, as well as Rydberg excitation energies while improving strikingly on intra- and intermol. charge-transfer excitations, comparable to the most successful range-sepd. hybrids available.**140**Fürst, S.; Kaupp, M. Accurate Ionization Potentials, Electron Affinities, and Band Gaps from the ωLH22t Range-Separated Local Hybrid Functional: No Tuning Required.*J. Chem. Theory Comput.*2023,*19*(11), 3146– 3158, DOI: 10.1021/acs.jctc.3c00173Google Scholar140https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3sXhtValtbrI&md5=6c8dd4ae553eabdef3d8dc3b0dec04b5Accurate Ionization Potentials, Electron Affinities, and Band Gaps from the ωLH22t Range-Separated Local Hybrid Functional: No Tuning RequiredFuerst, Susanne; Kaupp, MartinJournal of Chemical Theory and Computation (2023), 19 (11), 3146-3158CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The optimal tuning (OT) of range-sepd. hybrid (RSH) functionals has been proposed as the currently most accurate DFT-based way to compute the relevant quantities required for charge-transfer processes in org. chromophores used in org. photovoltaics and related fields. The main drawback of OT-RSHs is that the system-specific tuning of the range-sepn. parameter is not size-consistent. It therefore also lacks transferability, e.g., when considering processes involving orbitals not involved in the tuning or for reactions between different chromophores. Here we show that the recently reported ωLH22t range-sepd. local hybrid functional provides ionization energies, electron affinities, and fundamental gaps on par with OT-RSH treatments, approaching the quality of GW results, without any need for system-specific tuning. This holds from relevant org. chromophores of varying sizes all the way to at. electron affinities. ωLH22t also gives excellent outer-valence quasiparticle spectra and is a generally accurate functional for both main-group and transition-metal energetics, as well as for a variety of excitation types. Range-sepd. local hybrid functionals are suggested as promising new quantum-chem. tools in mol. electronics.**141**Dobson, J. F. Alternative expressions for the Fermi hole curvature.*J. Chem. Phys.*1993,*98*, 8870– 8872, DOI: 10.1063/1.464444Google Scholar141https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXkvVSmt74%253D&md5=0623ae49360b47df1b9560a86baffe77Alternative expressions for the Fermi hole curvatureDobson, John F.Journal of Chemical Physics (1993), 98 (11), 8870-2CODEN: JCPSA6; ISSN:0021-9606.The Fermi hole curvature C(r,s) is defined as the Laplacian of the parallel-spin pair distribution, evaluated at zero sepn. r' = r for a pair of fermions in a many-fermion system. It has been used by a no. of authors to discuss electron localization, properties of the exchange and correlation hole, and exchange and correlation energies of inhomogeneous electron gases. Here, the discussion of this quantity is extended in two directions. First, for the special case of a single-determinant many-electron state, a previously derived macroscopic expression for C can be generalized in a simple fashion to apply to current-carrying states. Second, a recently given interpretation of C(r,s), in terms of relative kinetic energy of pairs, is valid for a general many-fermion state and is not limited to the single-determinant case investigated previously.**142**Becke, A. D. Current density in exchange-correlation functionals: Application to atomic states.*J. Chem. Phys.*2002,*117*, 6935– 6938, DOI: 10.1063/1.1503772Google Scholar142https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XnsF2itro%253D&md5=37fd49dbccbeadc7d61935a97183f1cfCurrent density in exchange-correlation functionals: Application to atomic statesBecke, Axel D.Journal of Chemical Physics (2002), 117 (15), 6935-6938CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An old and yet unsolved problem in d.-functional theory is the strong dependence of degenerate open-shell at. energies on the occupancy of the AOs. This arises from the fact that degenerate AOs of different ml do not have equiv. densities. Approx. d. functionals therefore give energies depending strongly on which orbitals are occupied. This problem is solved in the present work by incorporating c.d. into the calcns. using a current-d. dependent functional previously published by the author.**143**Tao, J. Explicit inclusion of paramagnetic current density in the exchange-correlation functionals of current-density functional theory.*Phys. Rev. B*2005,*71*, 205107, DOI: 10.1103/PhysRevB.71.205107Google Scholar143https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXlvVyit7o%253D&md5=3c44947594da7443bc13f38d2676305fExplicit inclusion of paramagnetic current density in the exchange-correlation functionals of current-density functional theoryTao, JianminPhysical Review B: Condensed Matter and Materials Physics (2005), 71 (20), 205107/1-205107/5CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)Current-d. functional theory has been formulated in terms of the paramagnetic (or canonical) current densities jpσ(r), where σ=↑,↓. Vignale and Rasolt argued that jpσ enter the exchange-correlation (xc) functional Exc[n↑,n↓,jp↑,jp↓] (abbreviated as Exc[nσ,jpσ]) only through the vorticities νσ(r)=.del.×[jpσ(r)/nσ(r)], i.e., Exc[nσ,jpσ]=Exc[nσ,νσ], where nσ(r) are the spin electron densities. This has been generally accepted. Alternatively this theory was also formulated in terms of the (gauge-invariant) phys. or full current densities jσ(r). We show that in both formulations the paramagnetic current densities jpσ can enter the exchange-correlation functional explicitly. While this discovery favors the Vignale-Rasolt formulation of current-d. functional theory, it admits of the explicit jpσ dependence, leading to a family of metageneralized gradient current-d. functionals and their hybrid versions.**144**Bates, J. E.; Heiche, M. C.; Liang, J.; Furche, F. Erratum: “Harnessing the meta-generalized gradient approximation for time-dependent density functional theory” [J. Chem. Phys. 137, 164105 (2012)].*J. Chem. Phys.*2022,*156*, 159902, DOI: 10.1063/5.0088123Google Scholar144https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XhtVGktbvO&md5=41725148e1cbc44f91a4b63e11861d3eErratum: "Harnessing the meta-generalized gradient approximation for time-dependent density functional theory" [J. Chem. Phys. 137, 164105 (2012)]Bates, Jefferson E.; Heiche, Maximillian C.; Liang, Jiashu; Furche, FilippJournal of Chemical Physics (2022), 156 (15), 159902CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)There is no expanded citation for this reference.**145**Grotjahn, R.; Furche, F.; Kaupp, M. Importance of imposing gauge invariance in time-dependent density functional theory calculations with meta-generalized gradient approximations.*J. Chem. Phys.*2022,*157*, 111102, DOI: 10.1063/5.0113083Google Scholar145https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XisVWnsb7K&md5=a7569b81324b08f987c6a0c78f759254Importance of imposing gauge invariance in time-dependent density functional theory calculations with meta-generalized gradient approximationsGrotjahn, Robin; Furche, Filipp; Kaupp, MartinJournal of Chemical Physics (2022), 157 (11), 111102CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)It has been known for more than a decade that the gauge variance of the kinetic energy d. τ leads to addnl. terms in the magnetic orbital rotation Hessian used in linear-response time-dependent d. functional theory (TDDFT), affecting excitation energies obtained with τ-dependent exchange-correlation functionals. While previous investigations found that a correction scheme based on the paramagnetic c.d. has a small effect on benchmark results, we report more pronounced effects here, in particular, for the popular M06-2X functional and for some other meta-generalized gradient approxns. (mGGAs). In the first part of this communication, this is shown by a reassessment of a set of five Ni(II) complexes for which a previous benchmark study that did not impose gauge invariance has found surprisingly large errors for excitation energies obtained with M06-2X. These errors are more than halved by restoring gauge invariance. The variable importance of imposing gauge invariance for different mGGA-based functionals can be rationalized by the deriv. of the mGGA exchange energy integrand with respect to τ. In the second part, a large set of valence excitations in small main-group mols. is analyzed. For M06-2X, several selected n → π* and π→π*.perp. excitations are heavily gauge-dependent with av. changes of -0.17 and -0.28 eV, resp., while π→π*‖ excitations are marginally affected (-0.04 eV). Similar patterns, but of the opposite signs, are found for SCAN0. The results suggest that reevaluation of previous gauge variant TDDFT results based on M06-2X and other mGGA functionals is warranted. (c) 2022 American Institute of Physics.**146**Sciortino, G.; Lihi, N.; Czine, T.; Maréchal, J.-D.; Lledós, A.; Garribba, E. Accurate prediction of vertical electronic transitions of Ni(II) coordination compounds via time dependent density functional theory.*Int. J. Quantum Chem.*2018,*118*, e25655, DOI: 10.1002/qua.25655Google ScholarThere is no corresponding record for this reference.**147**Grotjahn, R.; Furche, F. Gauge-Invariant Excited-State Linear and Quadratic Response Properties within the Meta-Generalized Gradient Approximation.*J. Chem. Theory Comput.*2023, DOI: 10.1021/acs.jctc.3c00259Google ScholarThere is no corresponding record for this reference.**148**Schattenberg, C. J.; Kaupp, M. Effect of the current dependence of tau-dependent exchange-correlation functionals on nuclear shielding calculations.*J. Chem. Theory Comput.*2021,*17*, 1469– 1479, DOI: 10.1021/acs.jctc.0c01223Google Scholar148https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXkvFGhu7k%253D&md5=a525b9ccc88060676bdb34e67c68c050Effect of the Current Dependence of Tau-Dependent Exchange-Correlation Functionals on Nuclear Shielding CalculationsSchattenberg, Caspar Jonas; Kaupp, MartinJournal of Chemical Theory and Computation (2021), 17 (3), 1469-1479CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Exchange-correlation functionals that depend on the local kinetic energy τ are widely used in many fields. This includes meta-generalized gradient approxn. (GGA) functionals and their global hybrid versions as well as local hybrid functionals with τ-dependent local mixing functions to det. position-dependent exact-exchange admixt. Under the influence of an external magnetic field, τ becomes dependent on the gauge of the magnetic vector potential and should thus be extended to a gauge-invariant formulation. The currently most widely used extension for nuclear shielding calcns. is that suggested by Maximoff and Scuseria. Using the recent first implementation of local hybrids in this framework, we have found unphys. paramagnetic contributions, which are most clearly identified for atoms but are also present in mols. These τMS artifacts are small for the TPSS or TPSSh functionals, significantly deshielding in the case of nonhydrogen nuclei for the M06-L and M06 functionals and significantly shielding in those cases for the first-generation τ-dependent local hybrids LH07t-SVWN and LH12ct-SsifPW92. We have therefore implemented an extension of a linear-response nuclear shielding code to the proper current-d. functional version of τ suggested by Dobson. Using τD eliminates the gauge dependence as well as the unphys. contributions introduced by τMS. A first evaluation of this implementation for a set of main-group nuclear shieldings against CCSD(T) benchmark data indicates rather small effects for the TPSS and TPSSh functionals but significant changes for the other functionals studied: the previously obsd. remarkable performance of the highly parameterized meta-GGA functional M06-L is found to be the result of error compensation with the lack of an explicit current dependence. Results for the M06 functional are improved somewhat but are still overall of relatively low accuracy. In contrast, too low proton shieldings found recently for τ-dependent local hybrids are improved significantly within the τD current-d. functional framework while preserving the outstanding performance of these functionals for other nuclei.**149**Bruder, F.; Franzke, Y. J.; Weigend, F. Paramagnetic NMR Shielding Tensors Based on Scalar Exact Two-Component and Spin-Orbit Perturbation Theory.*J. Phys. Chem. A*2022,*126*, 5050– 5069, DOI: 10.1021/acs.jpca.2c03579Google Scholar149https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XhvVGjurjF&md5=80d1f295029ba3b687ba1631d252bddeParamagnetic NMR Shielding Tensors Based on Scalar Exact Two-Component and Spin-Orbit Perturbation TheoryBruder, Florian; Franzke, Yannick J.; Weigend, FlorianJournal of Physical Chemistry A (2022), 126 (30), 5050-5069CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)The temp.-dependent Fermi-contact and pseudocontact terms are important contributions to the paramagnetic NMR shielding tensor. Herein, we augment the scalar-relativistic (local) exact two-component (X2C) framework with spin-orbit perturbation theory including the screened nuclear spin-orbit correction for the EPR hyperfine coupling and g tensor to compute these temp.-dependent terms. The accuracy of this perturbative ansatz is assessed with the self-consistent spin-orbit two-component and four-component treatments serving as ref. This shows that the Fermi-contact and pseudocontact interaction is sufficiently described for paramagnetic NMR shifts; however, larger deviations are found for the EPR spectra and the principle components of the EPR properties of heavy elements. The impact of the perturbative treatment is further compared to that of the d. functional approxn. and the basis set. Large-scale calcns. are routinely possible with the multipole-accelerated resoln. of the identity approxn. and the seminumerical exchange approxn., as shown for [CeTi6O3(OiPr)9(salicylate)6].**150**Maximoff, S. N.; Scuseria, G. E. Nuclear magnetic resonance shielding tensors calculated with kinetic energy density-dependent exchange-correlation functionals.*Chem. Phys. Lett.*2004,*390*, 408– 412, DOI: 10.1016/j.cplett.2004.04.049Google Scholar150https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXktVaisbc%253D&md5=8ee2b28c0522f39cb7c0b41250e5b1a3Nuclear magnetic resonance shielding tensors calculated with kinetic energy density-dependent exchange-correlation functionalsMaximoff, Sergey N.; Scuseria, Gustavo E.Chemical Physics Letters (2004), 390 (4-6), 408-412CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)Exchange-correlation functionals that depend on the noninteracting kinetic energy d. (τ) break gauge invariance in the presence of a magnetic field. This yields incorrect results for mol. magnetic properties. The authors propose a simple generalization of the kinetic energy d. that resolves this problem. The authors' modification is validated by computing NMR abs. isotropic shielding consts. and shielding anisotropies with τ-dependent functionals for a representative set of mols. The accuracy of τ-dependent approxns. is found to surpass that of common generalized gradient approxns. (GGA) and hybrid functionals for strongly deshielded nuclei.**151**Holzer, C.; Franzke, Y. J.; Pausch, A. Current density functional framework for spin–orbit coupling.*J. Chem. Phys.*2022,*157*, 204102, DOI: 10.1063/5.0122394Google Scholar151https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XivFOrs7vJ&md5=b4a643a3046e817b011ac86b842f0318Current density functional framework for spin-orbit couplingHolzer, Christof; Franzke, Yannick J.; Pausch, AnsgarJournal of Chemical Physics (2022), 157 (20), 204102CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Relativistic two-component d. functional calcns. are carried out in a non-collinear formalism to describe spin-orbit interactions, where the exchange-correlation functional is constructed as a generalization of the non-relativistic d. functional approxn. Contrary to non-relativistic d. functional theory (DFT), spin-orbit coupling, however, leads to a non-vanishing paramagnetic c.d. D. functionals depending on the kinetic energy d., such as meta-generalized gradient approxns., should therefore be constructed in the framework of current DFT (CDFT). The latter has previously exclusively been used in the regime of strong magnetic fields. Herein, we present a consistent CDFT approach for relativistic DFT, including spin-orbit coupling. Furthermore, we assess the importance of the c.d. terms for ground-state energies, excitation energies, NMR shielding, and spin-spin coupling consts., as well as hyperfine coupling consts., Δg-shifts, and the nuclear quadrupole interaction tensor in ESR (EPR) spectroscopy. The most notable changes are found for EPR properties. The impact of the current-dependent terms rises with the no. of unpaired electrons, and consequently, the EPR properties are more sensitive toward CDFT. Considerable changes are obsd. for the strongly constrained and appropriately normed functionals, as well as the B97M family and TASK. The c.d. terms are less important when exact exchange is incorporated. At the same time, the current-dependent kernel ensures the stability of response calcns. in all cases. We, therefore, strongly recommend to use the framework of CDFT for self-consistent spin-orbit calcns. (c) 2022 American Institute of Physics.**152**Pausch, A.; Holzer, C. Linear Response of Current-Dependent Density Functional Approximations in Magnetic Fields. J. Phys.*Chem. Lett.*2022,*13*, 4335– 4341, DOI: 10.1021/acs.jpclett.2c01082Google Scholar152https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38Xht1ejtr%252FM&md5=6a91e2ac0829341c3e06d9d482b25afcLinear Response of Current-Dependent Density Functional Approximations in Magnetic FieldsPausch, Ansgar; Holzer, ChristofJournal of Physical Chemistry Letters (2022), 13 (19), 4335-4341CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)This Letter outlines the steps and derivations that are necessary to apply d. functional approxns. that depend on the current and kinetic energy d. rigorously within the framework of linear-response methods, including adiabatic time-dependent c.d. functional theory. This includes systems with a non-zero c.d. in the ground state. The necessary exchange-correlation kernel for these d. functional approxns. is derived, and the matrix elements are given explicitly. Due to the gauge variance of the kinetic energy d. in an external magnetic field, having access to the proper current-dependent exchange-correlation kernel is necessary to recover gauge invariance for excited states. As a proof of principle application, the excited states of two small mols. in strong external magnetic fields are calcd. using linear-response time-dependent c.d. functional theory. Finally, the implications of the derived c.d.-dependent exchange-correlation kernel for systems with strong spin-orbit coupling are discussed.**153**Helgaker, T.; Jaszuński, M.; Ruud, K. Ab initio methods for the calculation of NMR shielding and indirect spin-spin coupling constants.*Chem. Rev.*1999,*99*, 293– 352, DOI: 10.1021/cr960017tGoogle Scholar153https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXotFajurY%253D&md5=fb3af66e68153aaee6b6b1f9c8639a4aAb initio methods for the calculation of NMR shielding and indirect spin-spin coupling constantsHelgaker, Trygve; Jaszunski, Michal; Ruud, KennethChemical Reviews (Washington, D. C.) (1999), 99 (1), 293-352CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)A review with 377 refs. on NMR parameters and ab initio methods for their calcn. After a discussion of the effective NMR spin-Hamiltonian and Ramsey's theory for its evaluation, the most important ab initio methods for the calcn. of NMR shieldings and spin-spin coupling consts. were presented: (i) Hartree-Fock approxn., (ii) FCI approxn., (iii) MCSCF approxn., (iv) coupled-cluster approxn., and (v) Moller-Plesset theory. The ab initio methods were compared to each other and their relative merits and deficiencies were analyzed. The highly accurate results for H2 and other diat. mols. and 2nd-row hydrides were presented and implications for larger mols. were discussed. The calcn. of spin-rotation consts. and nuclear quadrupole-coupling consts. was also described. The rovibrational effects detg. the temp. dependence of the NMR parameters and their calcn. from the geometry dependence of the NMR parameters were discussed. Finally, the role of intermol. forces and the influence of solvents on the measured NMR parameters were outlined.**154**Vaara, J. Theory and computation of nuclear magnetic resonance parameters.*Phys. Chem. Chem. Phys.*2007,*9*, 5399– 5148, DOI: 10.1039/b706135hGoogle Scholar154https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXhtFCgsr7L&md5=8270c55fe054cd1f6be67bb2bbc0d7dfTheory and computation of nuclear magnetic resonance parametersVaara, JuhaPhysical Chemistry Chemical Physics (2007), 9 (40), 5399-5418CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)A review. The art of quantum chem. electronic structure calcn. has over the last 15 years reached a point where systematic computational studies of magnetic response properties have become a routine procedure for mol. systems. One of their most prominent areas of application are the spectral parameters of NMR spectroscopy, due to the immense importance of this exptl. method in many scientific disciplines. This article attempts to give an overview on the theory and state-of-the-art of the practical computations in the field, in terms of the size of systems that can be treated, the accuracy that can be expected, and the various factors that would influence the agreement of even the most accurate imaginable electronic structure calcn. with expt. These factors include relativistic effects, thermal effects, as well as solvation/environmental influences, where my group was active. The dependence of the NMR spectra on external magnetic and optical fields is also briefly touched on.**155**Kjærgaard, T.; Coriani, S.; Ruud, K. Ab initio calculation of magnetic circular dichroism.*WIREs Comput. Mol. Sci.*2012,*2*, 443– 455, DOI: 10.1002/wcms.1091Google ScholarThere is no corresponding record for this reference.**156**Tellgren, E. I.; Soncini, A.; Helgaker, T. Nonperturbative ab initio calculations in strong magnetic fields using London orbitals.*J. Chem. Phys.*2008,*129*, 154114, DOI: 10.1063/1.2996525Google Scholar156https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXhtlSmtL%252FM&md5=f962bd8417bc0beb0479ab5299a125a1Nonperturbative ab initio calculations in strong magnetic fields using London orbitalsTellgren, Erik I.; Soncini, Alessandro; Helgaker, TrygveJournal of Chemical Physics (2008), 129 (15), 154114/1-154114/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A SCF (SCF) London-orbital computational scheme to perform gauge-origin independent nonperturbative calcns. for mols. in strong magnetic fields is presented. The crucial difference in the proposed approach with respect to common-origin finite-field SCF implementations consists in the evaluation of mol. integrals over the field-dependent mol. basis functions, which is tantamount to computing mol. integrals in a hybrid Gaussian and plane-wave basis set. The implementation of a McMurchie-Davidson scheme for the calcn. of the mol. integrals over London orbitals is discussed, and preliminary applications of the newly developed code to the calcn. of fourth-rank hypermagnetizabilities for a set of small mols., benzene, and cyclobutadiene are presented. The nonperturbative approach is particularly useful for studying the highly nonlinear response of paramagnetic closed-shell systems such as boron monohydride, or the π-electron response of cyclobutadiene. (c) 2008 American Institute of Physics.**157**Stopkowicz, S.; Gauss, J.; Lange, K. K.; Tellgren, E. I.; Helgaker, T. Coupled-cluster theory for atoms and molecules in strong magnetic fields.*J. Chem. Phys.*2015,*143*, 074110, DOI: 10.1063/1.4928056Google Scholar157https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhtlOgsrrN&md5=803b9a5e5228c809f7ec5967770cb29cCoupled-cluster theory for atoms and molecules in strong magnetic fieldsStopkowicz, Stella; Gauss, Jurgen; Lange, Kai K.; Tellgren, Erik I.; Helgaker, TrygveJournal of Chemical Physics (2015), 143 (7), 074110/1-074110/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An implementation of coupled-cluster (CC) theory to treat atoms and mols. in finite magnetic fields is presented. The main challenges for the implementation stem from the magnetic-field dependence in the Hamiltonian, or, more precisely, the appearance of the angular momentum operator, due to which the wave function becomes complex and which introduces a gauge-origin dependence. For this reason, an implementation of a complex CC code is required together with the use of gauge-including AOs to ensure gauge-origin independence. Results of coupled-cluster singles-doubles-perturbative-triples (CCSD(T)) calcns. are presented for atoms and mols. with a focus on the dependence of correlation and binding energies on the magnetic field. (c) 2015 American Institute of Physics.**158**Irons, T. J. P.; Zemen, J.; Teale, A. M. Efficient Calculation of Molecular Integrals over London Atomic Orbitals.*J. Chem. Theory Comput.*2017,*13*, 3636– 3649, DOI: 10.1021/acs.jctc.7b00540Google Scholar158https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhtFChsb7L&md5=dbec737cdb32d62c96bf2c20f35ef389Efficient Calculation of Molecular Integrals over London Atomic OrbitalsIrons, Tom J. P.; Zemen, Jan; Teale, Andrew M.Journal of Chemical Theory and Computation (2017), 13 (8), 3636-3649CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The use of London AOs (LAOs) in a nonperturbative manner enables the detn. of gauge-origin invariant energies and properties for mol. species in arbitrarily strong magnetic fields. Central to the efficient implementation of such calcns. for mol. systems is the evaluation of mol. integrals, particularly the electron repulsion integrals (ERIs). We present an implementation of several different algorithms for the evaluation of ERIs over Gaussian-type LAOs at arbitrary magnetic field strengths. The efficiencies of generalized McMurchie-Davidson (MD), Head-Gordon-Pople (HGP), and Rys quadrature schemes are compared. For the Rys quadrature implementation, we avoid the use of high precision arithmetic and interpolation schemes in the computation of the quadrature roots and wts., enabling the application of this algorithm seamlessly to a wide range of magnetic fields. The efficiency of each generalized algorithm is compared by numerical application, classifying the ERIs according to their total angular momenta and evaluating their performance for primitive and contracted basis sets. In common with zero-field integral evaluation, no single algorithm is optimal for all angular momenta; thus, a simple mixed scheme is put forward that selects the most efficient approach to calc. the ERIs for each shell quartet. The mixed approach is significantly more efficient than the exclusive use of any individual algorithm.**159**Sun, S.; Williams-Young, D. B.; Stetina, T. F.; Li, X. Generalized Hartree-Fock with Nonperturbative Treatment of Strong Magnetic Fields: Application to Molecular Spin Phase Transitions.*J. Chem. Theory Comput.*2019,*15*, 348– 356, DOI: 10.1021/acs.jctc.8b01140Google Scholar159https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXitlKmsbvJ&md5=573099c8f2198fc79ab35e36861e8e99Generalized Hartree-Fock with Nonperturbative Treatment of Strong Magnetic Fields: Application to Molecular Spin Phase TransitionsSun, Shichao; Williams-Young, David B.; Stetina, Torin F.; Li, XiaosongJournal of Chemical Theory and Computation (2019), 15 (1), 348-356CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a framework of an ab initio variational approach to effectively explore electronic spin phase transitions in mol. systems inside of a homogeneous magnetic field. In order to capture this phenomenon, the complex generalized Hartree-Fock (C-GHF) method is used in the spinor formalism with London orbitals. Recursive algorithms for computing the one- and two-electron integrals of London orbitals are also provided. A Pauli matrix representation of the C-GHF method is introduced to sep. spin contributions from the scalar part of the Fock matrix. Next, spin phase transitions in two different mol. systems are investigated in the presence of a strong magnetic field. Noncollinear spin configurations are obsd. during the spin phase transitions in H2 and a dichromium complex, [(H3N)4Cr(OH)2Cr(NH3)4]4+, with an increase in magnetic field strength. The competing driving forces of exchange coupling and the spin Zeeman effect have been shown to govern the spin phase transition and its transition rate. Addnl., the energetic contributions of the spin Zeeman, orbital Zeeman, and diamagnetic terms to the potential energy surface are also analyzed.**160**Pausch, A.; Gebele, M.; Klopper, W. Molecular point groups and symmetry in external magnetic fields.*J. Chem. Phys.*2021,*155*, 201101, DOI: 10.1063/5.0069859Google Scholar160https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXis1Wju7fO&md5=9b92e4df04e4beb6f13626f67b2d17bdMolecular point groups and symmetry in external magnetic fieldsPausch, Ansgar; Gebele, Melanie; Klopper, WimJournal of Chemical Physics (2021), 155 (20), 201101CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)As quantum-chem. calcns. of mols. in static external magnetic fields are becoming increasingly popular, the description of mol. symmetry under such conditions is also becoming more and more relevant. Using group theory, a general scheme of identifying the mol. point group in an external magnetic field is constructed. For both point groups that are non-existent in the absence of a field (C∞ and C∞h) and their double groups, the character tables are presented. General properties of all possible point groups are discussed, and it is math. proven that they are all Abelian. (c) 2021 American Institute of Physics.**161**David, G.; Irons, T. J. P.; Fouda, A. E. A.; Furness, J. W.; Teale, A. M. Self-Consistent Field Methods for Excited States in Strong Magnetic Fields: a Comparison between Energy- and Variance-Based Approaches.*J. Chem. Theory Comput.*2021,*17*, 5492– 5508, DOI: 10.1021/acs.jctc.1c00236Google Scholar161https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB3MXhslyhs73F&md5=a3f52e55a75af81796f82da9692c707fSelf-Consistent Field Methods for Excited States in Strong Magnetic Fields: a Comparison between Energy- and Variance-Based ApproachesDavid, Gregoire; Irons, Tom J. P.; Fouda, Adam E. A.; Furness, James W.; Teale, Andrew M.Journal of Chemical Theory and Computation (2021), 17 (9), 5492-5508CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)SCF methods for excited states offer an attractive low-cost route to study not only excitation energies but also properties of excited states. Here, we present the generalization of two SCF methods, the max. overlap method (MOM) and the σ-SCF method, to calc. excited states in strong magnetic fields and investigate their stability and accuracy in this context. These methods use different strategies to overcome the well-known variational collapse of energy-based optimizations to the lowest soln. of a given symmetry. The MOM tackles this problem in the definition of the orbital occupations to constrain the SCF procedure to converge on excited states, while the σ-SCF method is based on the minimization of the variance instead of the energy. To overcome the high computational cost of the variance minimization, we present a new implementation of the σ-SCF method with the resoln. of identity approxn., allowing the use of large basis sets, which is an important requirement for calcns. in strong magnetic fields. The accuracy of these methods is assessed by comparison with the benchmark literature data for He, H2, and CH+. The results reveal severe limitations of the variance-based scheme, which become more acute in large basis sets. In particular, many states are not accessible using variance optimization. Detailed anal. shows that this is a general feature of variance optimization approaches due to the masking of local min. in the optimization. In contrast, the MOM shows promising performance for computing excited states under these conditions, yielding results consistent with available benchmark data for a diverse range of electronic states.**162**Stetina, T. F.; Sun, S.; Williams-Young, D. B.; Li, X. Modeling Magneto-Photoabsorption Using Time-Dependent Complex Generalized Hartree-Fock.*ChemPhotoChem.*2019,*3*, 739– 746, DOI: 10.1002/cptc.201900161Google Scholar162https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhsV2jurfF&md5=c0405372479d963794209f28e22ebe99Modeling Magneto-Photoabsorption Using Time-Dependent Complex Generalized Hartree-FockStetina, Torin F.; Sun, Shichao; Williams-Young, David B.; Li, XiaosongChemPhotoChem (2019), 3 (9), 739-746CODEN: CHEMYH ISSN:. (Wiley-VCH Verlag GmbH & Co. KGaA)Magneto-optical spectroscopy provides a glimpse into the hidden electronic structure of mols. and materials that normally cannot be resolved with regular optical absorption spectroscopy. The magneto-optical signal is difficult to simulate computationally due to the necessity of modeling perturbations from both uniform magnetic and oscillating elec. fields. In this work, we introduce a time-dependent complex generalized Hartree-Fock framework for calcg. electronic magneto-optical spectra of at. and mol. systems. Complex-valued gauge including AOs are used in a non-perturbative treatment of the uniform magnetic field. The magneto-photoabsorption spectra of at. alkalis and arom. systems are investigated with a focus on understanding how spectroscopic signals change in the presence of a uniform magnetic field.**163**Monzel, L.; Pausch, A.; Peters, L. D. M.; Tellgren, E. I.; Helgaker, T.; Klopper, W. Molecular Dynamics of Linear Molecules in Strong Magnetic Fields.*J. Chem. Phys.*2022,*157*, 054106, DOI: 10.1063/5.0097800Google Scholar163https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XitVGgtb7P&md5=383b5104410d57b66ca3ed0514112bf2Molecular dynamics of linear molecules in strong magnetic fieldsMonzel, Laurenz; Pausch, Ansgar; Peters, Laurens D. M.; Tellgren, Erik I.; Helgaker, Trygve; Klopper, WimJournal of Chemical Physics (2022), 157 (5), 054106CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Mol. rotations and vibrations were extensively studied by chemists for decades, both exptl. using spectroscopic methods and theor. with the help of quantum chem. However, the theor. study of mol. rotations and vibrations in strong magnetic fields requires computationally more demanding tools. As such, proper calcns. of rotational and vibrational spectra were not feasible up until very recently. Rotational and vibrational spectra are presented for 2 small linear mols., H2 and LiH, in strong magnetic fields. By treating the nuclei as classical particles, trajectories for rotations and vibrations are simulated from ab initio mol. dynamics. Born-Oppenheimer potential energy surfaces are calcd. at the Hartree-Fock and MP2 levels of theory using London AOs to ensure gauge origin invariance. For the calcn. of nuclear trajectories, a highly efficient Tajima propagator is introduced, incorporating the Berry curvature tensor accounting for the screening of nuclear charges. (c) 2022 American Institute of Physics.**164**Ferrario, L.; de Martino, D.; Gänsicke, B. T. Magnetic white dwarfs.*Space Sci. Rev.*2015,*191*, 111– 169, DOI: 10.1007/s11214-015-0152-0Google ScholarThere is no corresponding record for this reference.**165**Mereghetti, S.; Pons, J. A.; Melatos, A.