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Quantum Electronic StructureJune 27, 2022

Excited-State Properties for Extended Systems: Efficient Hybrid Density Functional Methods
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Anna-Sophia Hehn*
Anna-Sophia Hehn
Department of Chemistry, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
*A.-S. Hehn. Email: [email protected]
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https://orcid.org/0000-0002-0498-740X
Beliz Sertcan
Beliz Sertcan
Department of Chemistry, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
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Fabian Belleflamme
Fabian Belleflamme
Department of Chemistry, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
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Sergey K. Chulkov
Sergey K. Chulkov
School of Mathematics and Physics, University of Lincoln, Brayford Pool, Lincoln LN67TS, United Kingdom
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Matthew B. Watkins
Matthew B. Watkins
School of Mathematics and Physics, University of Lincoln, Brayford Pool, Lincoln LN67TS, United Kingdom
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https://orcid.org/0000-0003-4215-684X
Jürg Hutter
Jürg Hutter
Department of Chemistry, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
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Journal of Chemical Theory and Computation

Cite this: J. Chem. Theory Comput. 2022, 18, 7, 4186–4202

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https://doi.org/10.1021/acs.jctc.2c00144

Published June 27, 2022

CC-BY-NC-ND 4.0 .

Abstract

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Time-dependent density functional theory has become state-of-the-art for describing photophysical and photochemical processes in extended materials because of its affordable cost. The inclusion of exact exchange was shown to be essential for the correct description of the long-range asymptotics of electronic interactions and thus a well-balanced description of valence, Rydberg, and charge-transfer excitations. Several approaches for an efficient treatment of exact exchange have been established for the ground state, while implementations for excited-state properties are rare. Furthermore, the high computational costs required for excited-state properties in comparison to ground-state computations often hinder large-scale applications on periodic systems with hybrid functional accuracy. We therefore propose two approximate schemes for improving computational efficiency for the treatment of exact exchange. Within the auxiliary density matrix method (ADMM), exact exchange is estimated using a relatively small auxiliary basis and the introduced basis set incompleteness error is compensated by an exchange density functional correction term. Benchmark results for a test set of 35 molecules demonstrate that the mean absolute error introduced by ADMM is smaller than 0.3 pm for excited-state bond lengths and in the range of 0.02–0.04 eV for vertical excitation, adiabatic excitation, and fluorescence energies. Computational timings for a series of covalent-organic frameworks demonstrate that a speed-up of at least 1 order of magnitude can be achieved for excited-state geometry optimizations in comparison to conventional hybrid functionals. The second method is to use a semiempirical tight binding approximation for both Coulomb and exchange contributions to the excited-state kernel. This simplified Tamm–Dancoff approximation (sTDA) achieves an accuracy comparable to approximated hybrid density functional theory when referring to highly accurate coupled-cluster reference data. We find that excited-state bond lengths deviate by 1.1 pm on average and mean absolute errors in vertical excitation, adiabatic excitation, and fluorescence energies are in the range of 0.2–0.5 eV. In comparison to ADMM-approximated hybrid functional theory, sTDA accelerates the computation of broad-band excitation spectra by 1 order of magnitude, suggesting its potential use for large-scale screening purposes.

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1. Introduction

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The description of excited states in large extended systems using quantum-mechanical approaches is still a challenge for theoretical spectroscopy. (1) In contrast to the electronic ground state, the quest of finding an appropriate model with a well-balanced accuracy–cost ratio for the excited state is complicated by various electronic states of different natures having to be described simultaneously. For instance, this renders the parametrization of transferable classical force fields for excited states an almost impractical task. The cost of highly accurate wave function methods scales prohibitively with system size, making routine large-scale applications impossible. When a robust and efficient tool set for the treatment of extended periodic systems is the aim, density functional theory and semiempirical tight binding approaches therefore represent a suitable compromise. While time-dependent density functional theory (TDDFT) has been established as efficient and broadly applicable for the treatment of excited states, (2,3) there has also been a recent revival of semiempirical approaches to extend treatable system sizes from hundreds to thousands of atoms. (4) Both the re-emergence of semiempirical methods as well as the continuing improvement of density functional approximations are among the most important future developments in computational chemistry. (5)

The correct description of exact exchange has proven to be crucial for an adequate treatment of excited states in TDDFT. It has a much greater influence on the geometrical displacement upon excitation, and thus on transition energies and spectra, than the exchange-correlation functional. (6) Benchmarks on molecular systems, including radicals and ions, suggest including 30–40% exact exchange to obtain reliable spectroscopic data. (6) Straight-forward evaluation of the two-electron exact exchange integrals leads to a formal scaling of N⁴ with system size N for localized basis sets, emphasizing the need for more cost-efficient approximation schemes. An overview of recent developments and the various existing algorithms can, e.g., be found in ref (7). One approach is the auxiliary density matrix method (ADMM), (8) where the gain in efficiency is achieved by evaluating a model exact exchange energy within a relatively small auxiliary basis augmented by a correction term based on a local exchange functional. The assumption that the basis set incompleteness error can be corrected in terms of a cost-efficient density functional correction term was shown to be well-founded, (7) and benchmarks including liquids, solids as well as proteins in solution with up to 3000 atoms revealed that memory requirements and computational efficiency are improved by at least 1 order of magnitude. (8) Kumar et al. showed that ADMM achieves good accuracy for ground- and excited-state energies, even though rather large errors were found for polarizabilities and hyperpolarizabilities. (9) The least complex variant of ADMM is based on a simple least-squares fitting projection of molecular orbitals onto the smaller auxiliary basis. More sophisticated ADMM variants have been proposed including density matrix purification and projections enforcing orthogonality or charge constraints. (7) In a comparison among the developed ADMM variants, additional constraints as well as an adequate choice of the exchange functional for the correction term were shown to further improve accuracy for total energies while retaining a comparable efficiency. With respect to other sophisticated exact exchange algorithms (10) including the pair-atomic resolution-of-the-identity method (PARI-K) (11,12) and the chain-of-spheres algorithm (COSX), (13−16) total as well as orbital, reaction, and atomization energies were found to be 1 to 3 orders of magnitude less accurate. However, ADMM achieves a computational efficiency comparable to the cost required for density fitting Coulomb algorithms. ADMM was thus classified as a promising model with an “impressive speedup”, encouraging further development to improve accuracy and calling for careful optimization of the required auxiliary basis sets. (10)

For even larger system sizes of several thousands of atoms, it is necessary to go from ab initio excited-state calculations toward semiempirical tight binding approximations. In recent years, a class of tight binding approaches was suggested by Grimme et al., (4) on the basis of the idea of reducing the computational effort of electron repulsion integrals while retaining an adequate description of the physics of electronic interactions. More specifically, Coulomb and exchange contributions are approximated using an electron repulsion operator that captures the correct long-range 1/R asymptotics by construction (17−19) but allows for a short-range empirical parametrization. It was shown that this choice provides a balanced description of valence, charge-transfer as well as Rydberg states. The developed methods were dubbed simplified TDDFT (sTDDFT) (20) or simplified Tamm–Dancoff approximation (sTDA) (21,22) corresponding to the related ab initio electronic structure method and were extended to treat, e.g., spin-flip excitations (23−26) or to be combined with the idea of range separation. (27) In contrast to other common tight binding approaches like density functional-based tight binding (DFTB), (28) sTDA and sTDDFT involve only a limited number of global parameters, which enables their straightforward application over the whole periodic table. (29) Benchmark results demonstrated that the simplified approaches achieve computational savings of at least 2 orders of magnitude while the loss in accuracy is minor with an average deviation of 0.2–0.3 eV for excitation energies when compared to conventional TDDFT or experiments. Using a semiempirical ground-state reference, sTDA errors were found to be slightly larger with mean absolute deviations in the range of 0.3–0.5 eV. Most importantly, accuracy was found to be consistent for both the low-energy valence as well as the high-energy Rydberg transitions. (21,22) Simplified approaches were also suggested for the Bethe–Salpeter equation (BSE) and the GW approximation. (30) These methods provided sGW ionization potentials within the GW100 test set differing only by 0.2 eV while improving the cubic scaling with system size. Deviations in sBSE excitation energies amount up to 0.5 eV and are thus in an error range comparable to those of sTDA and sTDDFT. (30) A closely related TDDFT+TB approach has been suggested by Visscher et al. that relies on the same monopole approximation for the electron repulsion integrals but is, in contrast to sTDA and sTDDFT, not designed for hybrid but rather pure density functionals. (31,32)

The mentioned advantages of ADMM and sTDA regarding their physically correct description of exact exchange qualify both approaches as promising for an efficient calculation of excited-state properties. Excited-state gradients within the framework of linear response TDDFT were pioneered for molecular systems by Van Caillie and Amos (33,34) as well as Furche and Ahlrichs. (35,36) This work was extended to periodic systems and plane wave basis sets. (37) Furthermore, TDDFT excited-state properties for exact exchange have been extended to global and local hybrid functionals (38,39) as well as range separation. (40) Implementations are also available for tight binding approaches including long-range corrections. (41) However, algorithms are most often restricted to molecular systems. We implemented sTDA- and ADMM-approximated hybrid density functional excited-state gradients based on the Gaussian and plane wave (GPW) framework within the CP2K program package. (42−44) The GPW methods allow for a natural extension of algorithms to periodic boundary conditions and thus enable the treatment of extended systems. The implementation is based on earlier works of Iannuzzi et al. (45) and Strand et al. (46) already featuring the calculation of ADMM excitation energies for model systems of up to 1000 atoms. (47) As outlined in section 2.1, the CP2K implementation is based on the Sternheimer equations, (48−50) thus depending solely on the occupied molecular and atomic orbital space. Such a formulation requires, e.g., adjustments in the formulation of the ADMM equations and the sTDA eigenvalue problem (see sections 2.2 and 2.3). Going from molecular to periodic systems also requires Coulomb interactions to be treated using Ewald summation techniques and the minimum image approximation to capture exact exchange (see section 2.4). We tested our excited-state gradient implementations using a molecular benchmark set of Jacquemin et al., (51) which is one of the state-of-the-art test sets for excited-state properties. ADMM-approximated hybrid functional and semiempirical sTDA kernels are compared to conventional hybrid functional TDDFT exploiting the therein provided EOM-CCSD geometries and excited-state (ES) data. Finally, in section 3.2, the computational efficiency of the proposed algorithms is demonstrated by applications on porous covalent-organic framework (COF) materials taken from the CURATED COFs database. (52,53)

2. Theoretical Background

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2.1. The Tamm–Dancoff Approximation

Within the Tamm–Dancoff approximation, (54) the excitation energy Ω and corresponding excited-state eigenvectors X for each excited state are defined in terms of the variational Lagrangian G

G [X, C, Ω, {\bar{W}}^{X}] = \sum_{κ k σ} X_{κ k σ}^{T} \sum_{λ l} [F_{κ λ σ} δ_{k l} - F_{k l σ} S_{κ λ}] X_{λ l σ} + \sum_{κ λ σ} D_{κ λ σ}^{X} K_{κ λ σ} [D^{X}]

(1a)

- \sum_{κ λ k l σ} Ω (X_{κ k σ}^{T} S_{κ λ} X_{λ l σ} - δ_{k l})

(1b)

- \sum_{k l σ} {({\bar{W}}_{k l σ}^{X})}^{T} \sum_{κ λ} \frac{1}{2} (C_{κ k σ}^{T} S_{κ λ} X_{λ l σ} + X_{κ k σ}^{T} S_{κ λ} C_{λ l σ})

(1c)

implying the stationary conditions

\frac{\partial G}{\partial X} = 0, \to, \sum_{κ k} [F_{μ κ σ} δ_{i k} - F_{i k σ} S_{μ κ}] X_{κ k σ} + \sum_{κ λ} Q_{μ κ}^{T} K_{κ λ σ} [D^{X}] C_{λ i σ} = \sum_{κ} Ω S_{μ κ} X_{κ i σ}

(2)

\frac{\partial G}{\partial Ω} = 0, \to, \sum_{κ λ σ} X_{κ i σ}^{T} S_{κ λ} X_{λ j σ} = δ_{i j}

(3)

\frac{\partial G}{\partial {\bar{W}}^{X}} = 0, \to, \sum_{κ λ k l σ} (C_{κ k σ}^{T} S_{κ λ} X_{λ l σ} + X_{κ k σ}^{T} S_{κ λ} C_{λ l σ}) = 0

(4)

Equation 2 represents a hermitian eigenvalue problem that is for molecular orbital (MO)-based formulations solved under the constraint that the excited-state eigenvectors remain orthonormalized (eq 3). (55) The Lagrange multiplier introducing the normalization constraint of eq 3 is thereby chosen to be equal to the excitation energy Ω according to the canonical gauge (eq 1b). In an atomic orbital (AO)-based formalism relying solely on occupied MOs, {i, j, k, l, ... }, and AOs, {μ, ν, κ, λ, ... }, it furthermore has to be ensured that the excited-state eigenvector X is orthogonal to the ground-state (GS) MO coefficients C (eq 4), a constraint introduced in the Lagrangian G via the Lagrange multiplier

{\bar{W}}^{X}

(eq 1c)

X_{i j σ} = \sum_{κ λ} C_{κ i σ}^{T} S_{κ λ} X_{λ j σ} = 0

(5)

By taking the derivative of G with respect to the excited-state eigenvectors X and projecting onto the occupied MO coefficients C, it can be shown that this second normalization constraint represents a projection of the kernel contributions onto the virtual space, already inserted in the eigenvalue problem of eq 2 in terms of the projection operator Q

Q_{μ ν σ} = δ_{μ ν} - \sum_{κ k} C_{μ k σ} C_{κ k σ}^{T} S_{κ ν}

(6)

Q_{μ ν σ}^{T} = δ_{ν μ} - \sum_{κ k} S_{ν κ} C_{κ k σ} C_{μ k σ}^{T}

(7)

Thus, the contribution of the

{\bar{W}}^{X}

constraint to eq 2 is taken into account by inserting Q, a reformulation that is given in detail in section 1 of the Supporting Information. Both Ω and

{\bar{W}}^{X}

constraints are ensured in the TDDFT module of CP2K by explicitly orthonormalizing the ES eigenvectors as well as by orthogonalizing the ES eigenvectors with respect to the MO coefficients at each step of the Davidson algorithm. Equation 5 implies the transformation rules from AO to MO basis and vice versa, which are given for vectors V and matrices M in general as

V_{μ i σ}^{AO} = \sum_{k} C_{μ k σ} V_{k i σ}^{MO}

(8)

V_{i j σ}^{MO} = \sum_{κ λ} C_{κ i σ}^{T} S_{κ λ} V_{λ j σ}^{AO}

(9)

M_{μ ν σ}^{AO} = \sum_{κ λ k l} S_{μ κ} C_{κ k σ}^{T} M_{k l σ}^{MO} C_{l λ σ} S_{λ ν}

(10)

M_{i j σ}^{MO} = \sum_{κ λ} C_{κ i σ}^{T} M_{κ λ σ}^{AO} C_{λ j σ}

(11)

with the overlap matrix S being defined in terms of the AOs φ_μ(r)

S_{μ ν} = \int φ_{μ} (r) φ_{ν} (r) d r

(12)

Depending on the applied density functional, the eigenvalue problem of eq 2 comprises one-electron h, Coulomb J, and exact exchange K^EX contributions as well as contributions because of the exchange-correlation (XC) potential

V_{σ}^{XC} (r)

or kernel

f_{σ σ'}^{XC} (r, r^{'})

, with the Kohn–Sham matrix F and the kernel matrix K being defined as

\begin{array}{l} F_{μ ν σ} [D] & = h_{μ ν} + J_{μ ν σ} [D] - a_{EX} K_{μ ν σ}^{EX} [D] + V_{μ ν σ}^{XC} \\ = h_{μ ν} + \sum_{κ λ σ^{'}} D_{κ λ σ'} [(μ ν | κ λ) - a_{EX} δ_{σ σ'} (μ κ | ν λ)] + V_{μ ν σ}^{XC} \end{array}

(13)

\begin{array}{l} K_{μ ν σ} [D^{X}] & = J_{μ ν σ} [D^{X}] - a_{EX} K_{μ ν σ}^{EX} [D^{X}] + \sum_{κ λ σ^{'}} f_{μ ν σ, κ λ σ'}^{XC} D_{κ λ σ'}^{X} \\ = \sum_{κ λ σ^{'}} D_{κ λ σ'}^{X} [(μ ν | κ λ) - a_{EX} δ_{σ σ'} (μ κ | ν λ) + f_{μ ν σ, κ λ σ'}^{XC}] \end{array}

(14)

where a_EX is a global parameter to scale the amount of exact exchange and the two-electron repulsion integrals are defined as

(μ ν | κ λ) = \int d r \int d r^{'} φ_{μ} (r) φ_{ν} (r) \frac{1}{| r - r^{'} |} φ_{κ} (r^{'}) φ_{λ} (r^{'})

(15)

The corresponding density matrices D and D^X are defined based on the MO coefficients C implying symmetrization

D_{μ ν σ} = \sum_{k} C_{μ k σ} C_{ν k σ}^{T}

(16)

D_{μ ν σ}^{X} = \frac{1}{2} \sum_{k} (X_{μ k σ} C_{ν k σ}^{T} + C_{μ k σ} X_{ν k σ}^{T})

(17)

Symmetrization is necessary to ensure that the linear response density is real. Furthermore, within the implementation for periodic systems using a Γ-point only description, we can assume real wave functions. The basis functions and MOs are periodically replicated, integrals are over the computational unit cell, and all Coulomb terms evaluated using Ewald sums.

2.2. Exact Exchange Using the Auxiliary Density Matrix Method (ADMM)

The basic idea of ADMM (7,56) is to introduce a small and rapidly decaying auxiliary density matrix

\overset{ˇ}{D}

to speed up the calculation of the exact Hartree–Fock exchange matrix, K^EX, with the latter being evaluated within a smaller auxiliary basis,

{\overset{ˇ}{μ}, \overset{ˇ}{ν}, \overset{ˇ}{κ}, \overset{ˇ}{λ}, ...}

. The total exchange energy contribution to the ES energy is thus approximated by a model term,

E^{EX} [\overset{ˇ}{D}]

, and to compensate the so-introduced basis set incompleteness error, a correction term based on a local density functional E^GGA is added

E^{EX} [D] \approx E^{EX} [\overset{ˇ}{D}] + [E^{GGA} [D] - E^{GGA} [\overset{ˇ}{D}]]

(18)

Different approaches have been developed to obtain the auxiliary density matrix

\overset{ˇ}{D}

and corresponding MO coefficients

\overset{ˇ}{C}

, e.g., by minimizing solely the square difference between the occupied orbital and auxiliary basis functions or by adding an orthonormality constraint to the fitting procedure. We will restrict the discussion to the ADMM2 variant, which is often also dubbed nonpurified wave function fitting and which can be expressed in terms of the projection matrix

\overset{ˇ}{U}

. For the sake of convenience, we will nevertheless refer to ADMM2-approximated results in the following by the general acronym ADMM. Auxiliary basis functions and corresponding auxiliary matrices are indicated as

\overset{ˇ}{μ}

and

\overset{ˇ}{M}

. The projection matrix

\overset{ˇ}{U}

is obtained according to

\overset{ˇ}{U} = {\overset{ˇ}{S}}^{- 1} \overset{ˇ}{V}

(19)

\overset{ˇ}{D} = \overset{ˇ}{C} {\overset{ˇ}{C}}^{T} = \overset{ˇ}{U} D {\overset{ˇ}{U}}^{T}

(20)

\overset{ˇ}{C} = \overset{ˇ}{U} C

(21)

based on the overlap matrices

\overset{ˇ}{S}

and

\overset{ˇ}{V}

of the auxiliary and mixed auxiliary-primary basis

{\overset{ˇ}{S}}_{\overset{ˇ}{μ} \overset{ˇ}{ν}} = \int {\overset{ˇ}{φ}}_{\overset{ˇ}{μ}} (r) {\overset{ˇ}{φ}}_{\overset{ˇ}{ν}} (r) d r

(22)

{\overset{ˇ}{V}}_{\overset{ˇ}{μ} ν} = \int {\overset{ˇ}{φ}}_{\overset{ˇ}{μ}} (r) φ_{ν} (r) d r

(23)

The exact exchange matrix K^EX (eq 14) is thus approximated within ADMM as

K_{μ ν σ}^{EX} [D^{X}] \approx K_{μ ν σ}^{EX, ADMM} [D^{X}] = \sum_{\overset{ˇ}{μ} \overset{ˇ}{ν}} {\overset{ˇ}{U}}_{\overset{ˇ}{μ} μ}^{T} {\overset{ˇ}{K}}_{\overset{ˇ}{μ} \overset{ˇ}{ν} σ}^{EX} [{\overset{ˇ}{D}}^{X}] {\overset{ˇ}{U}}_{\overset{ˇ}{ν} ν} + K_{μ ν σ}^{GGA, ES}

(24)

with the local GGA exchange density functional correction term

K_{μ ν σ}^{GGA, ES} = \sum_{κ λ σ^{'}} f_{μ ν σ, κ λ σ'}^{GGA} [D] D_{κ λ σ'}^{X} - \sum_{\overset{ˇ}{μ} \overset{ˇ}{ν}} \sum_{\overset{ˇ}{κ} \overset{ˇ}{λ} σ^{'}} {\overset{ˇ}{U}}_{\overset{ˇ}{μ} μ}^{T} f_{\overset{ˇ}{μ} \overset{ˇ}{ν} σ, \overset{ˇ}{κ} \overset{ˇ}{λ} σ'}^{GGA} [\overset{ˇ}{D}] \times {\overset{ˇ}{U}}_{\overset{ˇ}{ν} ν} \sum_{κ λ} {\overset{ˇ}{U}}_{\overset{ˇ}{κ} κ} D_{κ λ σ'}^{X} {\overset{ˇ}{U}}_{\overset{ˇ}{λ} λ}^{T}

(25)

If relying also on an ADMM-approximated GS reference, exchange contributions to the KS matrix F of eq 13 imply an analogous correction with the second term of eq 24 now depending on the XC potential

K_{μ ν σ}^{GGA, GS} = V_{μ ν σ}^{GGA} [D] - \sum_{\overset{ˇ}{μ} \overset{ˇ}{ν}} {\overset{ˇ}{U}}_{\overset{ˇ}{μ} μ}^{T} V_{\overset{ˇ}{μ} \overset{ˇ}{ν} σ}^{GGA} [\overset{ˇ}{D}] {\overset{ˇ}{U}}_{\overset{ˇ}{ν} ν}

(26)

2.3. Semiempirical Coulomb and Exchange Contributions within the Simplified Tamm–Dancoff Approximation (sTDA)

In contrast to conventional TDA, sTDA (21) neglects all contributions because of the exchange-correlation kernel and approximates the remaining two-electron repulsion integrals based on the semiempirical Mataga–Nishimoto–Ohno–Klopman operator γ(A, B). (17−19) The simplified kernel contribution to eq 2 is given as

\sum_{λ} K_{μ λ σ}^{sTDA} [D^{X}] C_{λ i σ} = \sum_{A B} (1 - s) γ^{J} (A, B) {\overset{\approx}{C}}_{μ i σ}^{B} \times \sum_{λ_{A} l σ^{'}} {\tilde{C}}_{λ_{A} l σ'} {\tilde{X}}_{λ_{A} l σ'} - \sum_{A B} γ^{EX} (A, B) \sum_{l} {\overset{\approx}{X}}_{μ l σ}^{B} q_{i l σ}^{A}

(27)

with γ(A, B) describing either Coulomb (J) or exchange (EX) interactions depending on the interatomic distance R_AB of atoms A and B

γ^{J} (A, B) = {(\frac{1}{{(R_{A B})}^{α} + η^{- α}})}^{1 / α}

(28)

γ^{EX} (A, B) = {(\frac{1}{{(R_{A B})}^{β} + {(a_{EX} η)}^{- β}})}^{1 / β}

(29)

Note that the nomenclature classifying eqs 28 and 29 as exchange- or Coulomb-like interaction operators differs from the original paper (21) to match with the definitions for ADMM. The parameter s is equal to −1 for singlet closed-shell wave functions. For triplet closed-shell, it is set to s = 1 and for open-shell wave functions to s = 0. Four different global parameters are included. The chemical hardness η, which is specified for each element according to ref (57), powers of α and β allowing to modify the distance dependence of γ independently for either Coulomb or exchange interactions, and, analogously to conventional TDA, a Fock-exchange mixing parameter a_EX. The latter can be chosen freely, it was adjusted for molecular systems and global hybrids (21) and in this case shows the best performance for a_EX = 0.5. (22) Furthermore, we chose to set γ^EX(A, B) to zero if a_EX = 0. The transition density charge q^A is defined as

q_{i j σ}^{A} = \sum_{κ_{A}} {\tilde{C}}_{κ_{A} i σ}^{T} {\tilde{C}}_{κ_{A} j σ}

(30)

with the sum over κ_A running over all atomic orbitals located at atom A.

\tilde{C}

\tilde{X}

are Löwdin transformed MO coefficients or excitation vectors and

\overset{\approx}{C}

and

\overset{\approx}{X}

related doubly contracted intermediates

{\tilde{C}}_{μ i σ} = \sum_{η} S_{μ η}^{1 / 2} C_{η i σ}

(31)

{\overset{\approx}{C}}_{μ i σ}^{A} = \sum_{η_{A}} S_{μ η_{A}}^{1 / 2} {\tilde{C}}_{η_{A} i σ}

(32)

2.4. Periodic Boundary Conditions

When the methods are generalized to account for periodic boundary conditions (PBC), an adequate description of the long-range Coulomb forces using Ewald summation techniques is required to ensure convergence of the slowly decaying potential at large distances. (58) In the case of sTDA, the Coulomb operator is therefore split into a semiempirical short-range and an exact long-range contribution

γ^{J} (A, B) \overset{PBC}{\to} γ_{PBC}^{J} (A, B) + \frac{1}{R_{A B}}

(33)

implying that the semiempirical electron repulsion operator

γ_{PBC}^{J}

for the short-range contribution is cut off at an atom-specific radius R_cut

\begin{array}{l} γ_{PBC}^{J} (A, B) = 0, if R_{A B} > R_{cut} \\ γ_{PBC}^{J} (A, B) = η, if R_{A B} < 10^{- 6} a.u. \\ γ_{PBC}^{J} (A, B) = γ^{J} (A, B), if 10^{- 6} a.u. \leq R_{A B} < (R_{cut} - R_{smooth}) \\ γ_{PBC}^{J} (A, B) = f (\bar{R}) γ^{J} (A, B) - \frac{f (\bar{R})}{R_{A B}}, if (R_{cut} - R_{smooth}) \leq R_{A B} \leq R_{cut} \end{array}

(34)

R_cut is defined according to the cutoff radius of the atomic basis functions and the function

f (\bar{R})

smoothes the potential around R_cut and is chosen such that both the first and second derivative of γ^J vanish at the cutoff borders

f (\bar{R}) = - 6 {\bar{R}}^{5} + 15 {\bar{R}}^{4} - 10 {\bar{R}}^{3} + 1

(35)

\bar{R} = R_{A B} - (R_{cut} - R_{smooth}) / R_{smooth}

(36)

Thus, the polynomial

f (\bar{R})

is constructed to include those terms of the Taylor expansion of 1/R, which ensure that

f (\bar{R} = 0) = \frac{d f (\bar{R} = 0)}{d R} = \frac{d^{2} f (\bar{R} = 0)}{d R^{2}} = 1

and

f (\bar{R} = 1) = \frac{d f (\bar{R} = 1)}{d R} = \frac{d^{2} f (\bar{R} = 1)}{d R^{2}} = 0

with integer expansion coefficients chosen accordingly, so that

γ_{PBC}^{J} (A, B) \to γ^{J} (A, B)

for the limit of R_AB → R_cut – R_smooth and

γ_{PBC}^{J} (A, B) \to 1 / R_{A B}

for R_AB → R_cut. R_smooth defines the width of the smoothing function and is set to R_smooth = 1 a.u. The long-range contribution 1/R_AB is calculated using the smooth particle mesh Ewald method as implemented for tight binding approaches in CP2K, (59) scaling as N logN with system size N. Exchange interactions based on γ^EX are treated by implying the minimum image convention, thus restricting the sum over AB to neighbors within the unit cell.

2.5. The Excited-State Lagrangian in the Tamm–Dancoff Approximation

Nuclear gradients for the excited state are state-of-the-art in many program codes, so that the general procedure will only be outlined in short to discuss modifications that are necessary for ADMM and sTDA kernels. To avoid the computational cost of calculating the first-order response of the MO coefficients and to take into account the geometry dependence of the AOs, the Lagrangian of eqs 1a, 1b, and 1c needs to be extended by means of two additional constraints and thus is given as

L [X, C, Ω, {\bar{W}}^{X}, \bar{Z}, {\bar{W}}^{C}] = G [X, C, Ω, {\bar{W}}^{X}]

(37a)

+ \sum_{κ k σ} {({\bar{Z}}_{κ k σ})}^{T} \sum_{λ} (F_{κ λ σ} C_{λ k σ} - S_{κ λ} C_{λ k σ} ϵ_{k σ})

(37b)

- \sum_{k l σ} {({\bar{W}}_{k l σ}^{C})}^{T} (S_{k l σ} - δ_{k l})

(37c)

The first additional constraint (eq 37b) ensures the stationarity of the GS Kohn–Sham (KS) equations, the Brillouin condition, which can be rearranged using the definition of Q to emphasize the equivalence with MO-based formulations

\sum_{i a σ} {\bar{Z}}_{i a σ} F_{a i σ} = \sum_{i κ λ σ} {\bar{Z}}_{i λ σ} Q_{κ λ σ}^{T} F_{κ i σ} = \sum_{i κ λ σ} {\bar{Z}}_{i λ σ} (F_{λ κ σ} C_{κ i σ} - S_{λ κ} C_{κ i σ} ϵ_{i σ})

(38)

Note that only the virtual-occupied part of the Z vector

\bar{Z}

is taken into account and that the occupied–occupied part is implied to be zero

{\bar{Z}}_{i j σ} = \sum_{κ λ} {\bar{Z}}_{i λ σ} S_{λ κ} C_{κ j σ} = 0

(39)

The second additional constraint (eq 37c) ensures the orthogonality of the occupied MOs and, in analogy to

{\bar{W}}^{X}

and

\bar{Z}

, the hereby introduced Lagrange multiplier

{\bar{W}}^{C}

is assumed to be symmetric.

{\bar{W}}^{C}

and

\bar{Z}

are determined by taking the derivative of L with respect to the MO coefficients C and projecting onto either the virtual or occupied orbital space

\frac{\partial L}{\partial C} C = 0 \to {\bar{W}}^{C};, \frac{\partial L}{\partial C} Q = 0 \to \bar{Z}

(40)

The final equations for the

{\bar{W}}^{C}

multiplier are given as

{\bar{W}}_{i j σ}^{C} = \frac{1}{2} H_{i j σ} [P] + \sum_{κ λ} X_{κ i σ}^{T} (Ω S_{κ λ} - F_{κ λ σ}) X_{λ j σ} + \sum_{κ λ σ^{'}} \sum_{η τ σ^{″}} D_{κ λ σ'}^{X} D_{η τ σ ″}^{X} g_{κ λ σ', η τ σ ″, i j σ}^{XC}

(41)

with the relaxed and unrelaxed difference density matrices P and T being defined as

P = T + D^{Z}

(42)

D_{μ ν σ}^{Z} = \frac{1}{2} \sum_{k} ({\bar{Z}}_{μ k σ} C_{ν k σ}^{T} + C_{μ k σ} {\bar{Z}}_{ν k σ}^{T})

(43)

T_{μ ν σ} = \sum_{k} X_{μ k σ} X_{ν k σ}^{T} - \frac{1}{2} \sum_{κ λ k l} (C_{μ k σ} X_{κ k σ}^{T} S_{κ λ} X_{λ l σ} C_{ν l σ}^{T} + C_{μ l σ} X_{κ l σ}^{T} S_{κ λ} X_{λ k σ} C_{ν k σ}^{T})

(44)

Note that the unrelaxed difference density matrix T in the AO basis as defined in eq 44 corresponds to the sum of the virtual–virtual and occupied–occupied blocks of the corresponding MO matrix T^MO with the mixed virtual–occupied blocks being equal to zero. Regarding the comparison to MO formulations, (35,36) the matrix

{\bar{W}}^{C}

as defined in eqs 37c, and 41 only comprises occupied–occupied contributions while analogous MO formulations treat the combined occupied–virtual space. Contributions referring to the virtual space of refs (35and36) are included in our formalism via Q projections, explicitly taken into account because of the reformulated Brillouin condition (eq 38) and the

{\bar{W}}^{X}

constraint. Furthermore, the intermediate H stems from the KS matrix contributions of eq 14 and thus the explicit formula depends on the chosen GS reference, given here for hybrid functionals with or without ADMM

H_{i j σ} [M] = \sum_{κ λ σ^{'}} M_{κ λ σ'} [2 (κ λ | i j σ) - a_{EX} δ_{σ σ'} [(κ i σ | λ j σ) + (κ j σ | λ i σ)] + 2 f_{κ λ σ', i j σ}^{XC}]

(45)

\begin{array}{l} H_{i j σ}^{ADMM} [M] & = \sum_{κ λ σ^{'}} M_{κ λ σ'} [2 (κ λ | i j σ) + 2 f_{κ λ σ', i j σ}^{XC}] \\ - a_{EX} \sum_{\overset{ˇ}{κ} \overset{ˇ}{λ} σ^{'}} δ_{σ σ'} {(\overset{ˇ}{U} M {\overset{ˇ}{U}}^{T})}_{\overset{ˇ}{κ} \overset{ˇ}{λ} σ'} [\sum_{η \overset{ˇ}{η} τ \overset{ˇ}{τ}} C_{η i σ}^{T} {\overset{ˇ}{U}}_{\overset{ˇ}{η} η}^{T} [(\overset{ˇ}{κ} \overset{ˇ}{η} | \overset{ˇ}{λ} \overset{ˇ}{τ}) + (\overset{ˇ}{κ} \overset{ˇ}{τ} | \overset{ˇ}{λ} \overset{ˇ}{η})] {\overset{ˇ}{U}}_{\overset{ˇ}{τ} τ} C_{τ j σ}] \\ - 2 a_{EX} [\sum_{κ λ σ^{'}} δ_{σ σ'} M_{κ λ σ'} f_{κ λ σ', i j σ}^{GGA} + \sum_{\overset{ˇ}{κ} \overset{ˇ}{λ} σ^{'}} δ_{σ σ'} {(\overset{ˇ}{U} M {\overset{ˇ}{U}}^{T})}_{\overset{ˇ}{κ} \overset{ˇ}{λ} σ'} [\sum_{η \overset{ˇ}{η} τ \overset{ˇ}{τ}} C_{η i σ}^{T} {\overset{ˇ}{U}}_{\overset{ˇ}{η} η}^{T} f_{\overset{ˇ}{κ} \overset{ˇ}{λ} σ', \overset{ˇ}{η} \overset{ˇ}{τ} σ}^{GGA} {\overset{ˇ}{U}}_{\overset{ˇ}{τ} τ} C_{τ j σ}]] \end{array}

(46)

The linear

\bar{Z}

vector equation

A \bar{Z} = - R

(47)

\sum_{κ} {\bar{Z}}_{κ i σ}^{T} [F_{κ μ σ} - S_{κ μ} ϵ_{i σ}] + H_{i μ σ} [D^{Z}] = - R_{i μ σ}

(48)

contains contributions on the left-hand side, which stem from the KS matrix (eq 13) and are independent of the chosen excited-state kernel K (eq 14). Such a dependence is solely included on the right-hand side R, summarizing a first term stemming from the KS matrix as well as kernel contributions that have to be adjusted for ADMM and sTDA

R_{i μ σ} = H_{i μ σ} [T] + 2 \sum_{κ} X_{κ i σ}^{T} K_{κ μ σ} [D^{X}] - 2 \sum_{κ k} X_{κ k σ}^{T} S_{κ μ} K_{i k σ} [D^{X}] + 2 \sum_{κ λ σ^{'}} \sum_{η τ σ^{″}} D_{κ λ σ'}^{X} D_{η τ σ ″}^{X} g_{κ λ σ', η τ σ ″, μ i σ}^{XC}

(49)

Finally, the gradient L^ζ with respect to the nuclear coordinate ζ can be written in terms of the effective difference density matrix Γ analogously to the formulations of refs (35and36)

L^{ζ} = \sum_{μ ν σ} [h_{μ ν}^{ζ} + V_{μ ν σ}^{XC (ζ)}] P_{μ ν σ} - \sum_{μ ν σ} S_{μ ν}^{ζ} Λ_{μ ν σ} + \sum_{μ ν κ λ σ σ^{'}} [{(μ ν | κ λ)}^{ζ} Γ_{μ ν σ κ λ σ'} + f_{μ ν σ κ λ σ'}^{XC (ζ)} D_{μ ν σ}^{X} D_{κ λ σ'}^{X}]

(50)

with the intermediates

Λ_{μ ν σ} = \sum_{k l} C_{μ k σ} {\bar{W}}_{k l σ}^{C} C_{ν l σ}^{T} + \frac{1}{2} \sum_{k} ϵ_{k σ} ({\bar{Z}}_{μ k σ} C_{ν k σ}^{T} + C_{μ k σ} {\bar{Z}}_{ν k σ}^{T}) + \sum_{k l} [C_{μ k σ}^{T} X_{ν l σ} + X_{μ k σ}^{T} C_{ν l σ}] K_{k l σ} [D^{X}] + \sum_{k l} (Ω + F_{k l σ}) δ_{k l} X_{μ k σ}^{T} X_{ν l σ}

(51)

Γ_{μ ν σ κ λ σ'} = P_{μ ν σ} D_{κ λ σ'} + D_{μ ν σ}^{X} D_{κ λ σ'}^{X} - a_{EX} δ_{σ σ'} [P_{μ κ σ} D_{ν λ σ'} + D_{μ κ σ}^{X} D_{ν λ σ'}^{X}]

(52)

Note that the total nuclear forces sum contributions because of the ES energy functional and the additional constraints, as listed in eq 50, as well as the GS energy contributions

E_{GS}^{ζ}

Regarding the discussed kernel options, additional terms for the gradient have to be considered for the transformed ADMM matrix of eq 24, with all contributions implying the chain rule

\begin{array}{l} \frac{\partial K_{μ ν σ}^{EX, ADMM} [D^{X}]}{\partial ζ} & \leftarrow {({({\overset{ˇ}{U}}^{T})}^{ζ} {\overset{ˇ}{K}}^{EX} [\overset{ˇ}{U} D^{X} {\overset{ˇ}{U}}^{T}] \overset{ˇ}{U})}_{μ ν σ} \\ + {({\overset{ˇ}{U}}^{T} {\overset{ˇ}{K}}^{EX} [\overset{ˇ}{U} D^{X} {\overset{ˇ}{U}}^{T}] {\overset{ˇ}{U}}^{ζ})}_{μ ν σ} \\ + {({\overset{ˇ}{U}}^{T} {\overset{ˇ}{K}}^{EX} [{\overset{ˇ}{U}}^{ζ} D^{X} {\overset{ˇ}{U}}^{T} + \overset{ˇ}{U} D^{X} {({\overset{ˇ}{U}}^{T})}^{ζ}] \overset{ˇ}{U})}_{μ ν σ} \\ + \sum_{\overset{ˇ}{μ} \overset{ˇ}{ν} \overset{ˇ}{κ} \overset{ˇ}{λ}} {\overset{ˇ}{U}}_{\overset{ˇ}{μ} μ}^{T} {(\overset{ˇ}{μ} \overset{ˇ}{κ} | \overset{ˇ}{ν} \overset{ˇ}{λ})}^{ζ} {\overset{ˇ}{U}}_{\overset{ˇ}{ν} ν} \sum_{κ λ} {\overset{ˇ}{U}}_{\overset{ˇ}{κ} κ} D_{κ λ σ}^{X} {\overset{ˇ}{U}}_{\overset{ˇ}{λ} λ}^{T} \end{array}

(53)

encompassing the gradient for the ADMM projection matrix

{\overset{ˇ}{U}}^{ζ}

{\overset{ˇ}{U}}^{ζ} = {\overset{ˇ}{S}}^{- 1} [{\overset{ˇ}{V}}^{ζ} - {\overset{ˇ}{S}}^{ζ} \overset{ˇ}{U}]

(54)

Analogous contributions have to be considered for the correction term of eq 25. sTDA kernels bear the advantage that they do not require third-order derivatives of the XC kernel, but additional derivatives of the overlap matrix S^1/2 (60)

{(\frac{\partial S^{1 / 2}}{\partial ζ})}_{μ ν} = \sum_{κ λ η τ} U_{μ κ}^{S} U_{η κ}^{S} {(\frac{\partial S}{\partial ζ})}_{η τ} U_{τ λ}^{S} {(\sqrt{s_{λ}} + \sqrt{s_{κ}})}^{- 1} U_{ν λ}^{S}

(55)

with the eigenvalues s and eigenvectors U^S of the overlap matrix. On the basis of the outlined Lagrange formalism, nuclear gradients were implemented for both ADMM-approximated hybrid functional theory and semiempirical sTDA in the CP2K program package, relying for the Z vector equation on already available linear response solvers. (45,46) Details on the applied Block–Davidson algorithm (61) and the preconditioners can, e.g., be found in refs (46and62). The current implementation of ES nuclear gradients paves the way for adiabatic and nonadiabatic nuclear dynamics, enabling, e.g., the combination with trajectory surface hopping methods assuming time derivative or empirical coupling vectors. (63)

3. Results and Discussion

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3.1. Tests for Accuracy: Benchmark Results for 35 Main-Group Molecules

To assess the accuracy of ADMM and sTDA excited-state properties, we performed a benchmark on a molecular test set by Budzak et al. containing 35 small molecules. The test set contains main-group atoms of the first and second row as well as sulfur, selenium, chlorine, and bromine. Reported EOM-CCSD geometries were obtained using Gaussian16, (64) correlating all electrons (including core electrons) and choosing a def2-TZVPP basis. (65) The nature of the GS and ES reference structures was furthermore checked by performing frequency calculations and computing T₁ diagnostics. We selected these structures as the highest-accuracy level reference and performed, for comparison, all-electron PBE0 calculations with the Turbomole program package, (66) first using the def2-TZVPP basis to unravel the effect of the underlying electronic structure method and then increasing the basis set to def2-QZVPP (67) quality to get an estimate for the basis set error. Deviations of PBE0 computations with CP2K to the mentioned EOM-CCSD and PBE0 reference data originate then mainly from core–shells being described in terms of pseudopotentials and correspondingly adapted basis sets. More specifically, we used pseudopotentials and basis sets that were optimized for the PBE0 functional with the numerical atom code of CP2K and that are available within the database and distribution of CP2K. (42,68) To check consistency of the chosen CP2K basis sets and pseudopotentials, we investigated a hierarchy of MOLOPT as well as correlation consistent ccGRB-X basis sets. Benchmark data on vertical singlet excitation energies, geometries of the first excited state, adiabatic excitation and fluorescence energies (for PBE0, ADMM-PBE0, and sTDA kernels based on PBE0 and ADMM-PBE0 references) are given for the two basis set families in comparison to EOM-CCSD/def2-TZVPP/Gaussian, PBE0/def2-TZVPP/Turbomole, and PBE0/def2-QZVPP/Turbomole references in the Supporting Information, demonstrating that MOLOPT and ccGRB-X basis sets yield comparable accuracy and that converged results are in general obtained for triple-ζ basis sets. For the sake of convenience, we therefore restrict the following discussion to subsets of MOLOPT type basis sets, concentrating on the assessment of the mentioned excited-state properties regarding (a) the accuracy of ADMM in comparison to conventional hybrid functional TDDFT as well as (b) the comparison of semiempirical sTDA and ADMM-approximated hybrid functional kernels in comparison to highly accurate EOM-CCSD reference data.

Regarding the performed benchmark, a coupling parameter with a default value of a_EX = 0.2 was chosen for the exchange contribution for sTDA computations throughout the following discussion. As noted in the Supporting Information, adjustments only had to be made in the case of formyl chloride to avoid dissociation upon ES geometry optimization. A default value for a_EX might not correspond to an optimal choice, but a thorough and careful optimization of a_EX for the investigated benchmark sets and the assessed extended materials is beyond the scope of this work. The general idea of ADMM is to reduce computational costs of the exact exchange contribution by trading in basis set incompleteness with a GGA correction term. The introduced error is then, however, dependent on both the auxiliary basis set size as well as the chosen exchange density functional. To discriminate between the two error sources, the discussed ADMM calculations for the molecular benchmark set are performed without correction. An optimally chosen GGA exchange functional contribution could improve the discussed results, with the latter representing the lowest limit for the ADMM accuracy. It was, e.g., shown in ref (7) that exchange functionals like KT3X and OPTX improve over PBEX for ground-state properties, but analogous investigations for the excited state are left for future work. For the subsequently discussed assessment of computational timings for periodic systems, we, however, included a PBEX exchange functional correction to account for the additionally required resources.

3.1.1. Excited-State Geometries: Impact of the Auxiliary Basis Set Size for ADMM Kernels

Relaxed excited-state geometries for PBE0 and ADMM-PBE0 kernels using MOLOPT and ccGRB-X basis sets as well as sTDA kernels based on PBE0 and ADMM-PBE0 references are reported in Tables S3 and S4. The data includes a corresponding statistical analysis on the error in the geometrical data with respect to EOM-CCSD/def2-TZVPP, PBE0/def2-TZVPP, and PBE0/def2-QZVPP references (Tables S6–S10). In contrast to the study of ref (51), we excluded first excited states of the eclipsed structures of nitrosomethane and trifluoronitrosomethane as these singlet states represent transition-state structures with an imaginary frequency. Because of the large number of geometries that had to be optimized considering different kernels and basis sets, we calculated vibrational frequencies only for those structures that differed significantly from the reference geometries, ensuring that they represent local minima. In the earlier benchmarks of ref (69), MAEs for bond lengths of GS structures are predicted to be in the range from 0.5 to 1.0 pm for hybrid functionals, with the largest errors found for CO, CN, CS, and CSe bonds. In general, it is concluded that deviations are largest for polarized bonds. Furthermore, increased errors were found when going from the GS to the ES rising up to a MAE range from 0.9 to 2.8 pm for bond lengths. In Figure 1, the error introduced by ADMM in bond lengths of GS or first singlet ES geometries is depicted for selected bonds of the 35 molecules of the benchmark set, with “ADMM error” being defined as the deviation that results from comparing the conventional PBE0 computations with approximated ADMM-PBE0 ones choosing the same primary basis set. The assessment is thus reflecting the accuracy of the auxiliary basis set size. Corresponding statistical data can be found in Tables S11, S12, and S13. The statistics are based only on the selected geometrical data listed explicitly in the Supporting Information following the selection of ref (51), omitting less polarized C–H bonds for larger molecules, but including all bonds that undergo significant changes to ensure meaningful MEs and MAEs. For both GS and ES, results relying on the smallest dzp basis, colored in green or blue depending on the chosen primary basis, deviate significantly from computations using larger auxiliary basis sets, depicted in orange and red. Predicted bond lengths are too long, with dzp errors amounting up to 3.8/3.5 pm for GS/ES geometries and being thus much larger than the corresponding error range for the tzp and tz2p auxiliary basis sets; both tzp and tz2p bonds combined with a TZVP or TZV2P primary basis deviate by at most 1.0 pm. Comparing GS and ES distributions, the error spread for the ES is as broad as for the GS. Both tzp distributions suggest a classification in two or more groups depending on the type of atoms involved and resulting in two maxima in the histograms. Bonds including H, S, Se, and C–C bonds can be associated with relatively small errors corresponding to the first maximum and Gaussian distribution around 0.0 pm. Larger deviations in the range from 0.3 to 0.6 pm can be found for oxygen or halogen bonds contributing to the second maximum in both GS and ES histograms. Outliers corresponding to the maximum errors of 1.0 pm can be traced back to nitrogen bonds. An analogous classification and assignment for the dzp auxiliary basis is less conclusive because of the broader distributions, but it still holds true with the exception that C–C bonds fall into the error range of the second maximum around 1.5 pm. For comparison, the ADMM error in selected angles of the optimized ES geometries is depicted in Figure 2, demonstrating that the impact of the chosen auxiliary basis on the angles is less dependent on the basis set size than it is for the bond lengths, even though the width of the Gaussian distribution is reduced when going from double- to triple-ζ basis sets. An analogous plot for the GS, given in Figure S1, looks nearly identical.

Figure 1. Error (in pm) for selected bond lengths for (a) optimized ground- and (b) first singlet excited-state geometries comparing conventional PBE0 and ADMM-PBE0 computations for different auxiliary (ABS) and primary basis sets (PBS) indicated as ABS/PBS.

Figure 2. Error (in deg) for selected angles for optimized first singlet excited-state geometries comparing conventional PBE0 and ADMM-PBE0 computations.

A similar and transferable result is found for dihedral angles investigated for a subset of 13 molecules for which excitation leads to a significantly bent excited-state structure. While dihedral angles of corresponding GS geometries are consistently predicted to describe a planar geometry for all auxiliary and primary basis set sizes, the bending angle of the excited state is more accurately described when increasing the cardinal number of the auxiliary basis from double- to triple-ζ size. Again, the effect of additional polarization functions as included within the largest tz2p basis is negligible. A plot for the ADMM error in dihedral angles analogous to Figure 2 is given in Figure S2.

A summarized look at the corresponding statistical data of Table 1, listing mean errors (MEs), mean absolute errors (MAEs), and standard deviations (STDs) for bond lengths, angles, and dihedral angles of ES geometries optimized within the various primary and auxiliary basis set combinations, emphasizes again the impact of the chosen auxiliary basis: while MAEs of the dzp basis amount up to 1.4 pm for bond lengths and up to 0.8° or 2.1° for bond and dihedral angles, tzp results are converged with remaining deviations of 0.3 pm/0.2°/0.3°. The ADMM error is negligibly small in comparison to the error of PBE0 or hybrid functionals in general, which was found to be for MAEs in the range from 0.9 to 2.8 pm for bond lengths in comparison to highly accurate coupled-cluster results. (69) ADMM error distributions are relatively broad for the dzp auxiliary basis and nearly equally narrow for tzp and tz2p basis sets. MEs and MAEs are of the same magnitude, indicating that the dzp auxiliary basis predicts bonds that are consistently too long and angles too narrow. This trend is in agreement with the finding that increasing the amount of exact exchange in hybrid functionals results in shorter bond lengths and larger bond angles. (39,70,71) Going to larger ADMM auxiliary basis set sizes achieves a more accurate description of the fraction of exact exchange that is included in the underlying hybrid functional. It thus increases the Hartree–Fock nature of the bond with the one-determinant Hartree–Fock wave function representing the maximum amount of bonding character and corresponding to the shortest bonds. ADMM ES bond lengths are therefore shortened, and bond angles broadened when going from double- to triple-ζ auxiliary basis sets. Studies on the effect of exact exchange on dihedral angles furthermore reveal that hybrid functionals improve on the overestimation of GGA functionals, a result that is thus in line with our findings following the same rationale. (72,73)

Table 1. Statistics Including Mean Errors (MEs), Mean Absolute Errors (MAEs), and Standard Deviations (STDs) for Different Auxiliary and Primary Basis Sets (ABS/PBS) Visualizing that the ADMM Error in ES Bond Lengths (pm), Angles (deg), and Dihedral Angles (deg) as well as in the First Adiabatic Excitation (E_ad), the First 10 Vertical Excitation (E_vert), and the First Fluorescence (E_fl) Energy (eV) Is Converged for Auxiliary Basis Sets of Triple-ζ Size

	Bonds (pm)			Angles (deg)			Dihedrals (deg)
ABS/PBS	ME	MAE	STD	ME	MAE	STD	ME	MAE	STD
dzp/DZVP	1.31	1.31	0.55	–0.74	0.85	1.19	2.07	2.07	1.20
dzp/TZVP	1.37	1.37	0.57	–0.75	0.85	1.13	1.94	1.94	1.11
dzp/TZV2P	1.38	1.38	0.56	–0.73	0.83	1.18	2.05	2.05	1.23
tzp/TZVP	0.21	0.23	0.25	–0.11	0.21	0.35	0.07	0.34	0.54
tzp/TZV2P	0.25	0.26	0.24	–0.13	0.20	0.32	0.30	0.36	0.49
tz2p/TZV2P	0.21	0.22	0.26	–0.10	0.18	0.28	0.02	0.13	0.16

	E_vert (eV)			E_ad (eV)			E_fl (eV)
ABS/PBS	ME	MAE	STD	ME	MAE	STD	ME	MAE	STD
dzp/DZVP	–0.064	0.095	0.117	–0.023	0.048	0.054	–0.055	0.080	0.074
dzp/TZVP	–0.058	0.091	0.109	–0.025	0.055	0.061	–0.057	0.083	0.075
dzp/TZV2P	–0.062	0.093	0.109	–0.028	0.054	0.058	–0.058	0.081	0.073
tzp/TZVP	–0.012	0.036	0.057	0.004	0.022	0.031	–0.002	0.022	0.030
tzp/TZV2P	–0.015	0.035	0.048	0.004	0.022	0.032	0.003	0.022	0.029
tz2p/TZV2P	–0.021	0.033	0.044	–0.004	0.016	0.022	–0.005	0.019	0.025

3.1.2. Excited-State Geometries: ADMM and sTDA Kernels in Comparison to EOM-CCSD References

To analyze the performance of ADMM-PBE0 and sTDA kernels in comparison to highly accurate EOM-CCSD reference data, we restricted investigations to basis sets of triple-ζ size. In Figure 3 the error in the geometrical data is given as normal distributions based on the corresponding MEs and STDs; more detailed statistical analysis and explicit errors listed for each molecule can be found in sections 3.4 and 3.5 of the Supporting Information. To give not only an upper limit for a desired high-accuracy reference but also a lower reference point of reachable performance, conventional hybrid functional PBE0 results are depicted in black, deviating by MEs of −0.15 pm/0.32°/0.80° from the EOM-CCSD reference. As for the preceding assessment on the ADMM auxiliary basis size, PBE0 results represent the basis set limit for ADMM computations, a fact that is reflected in the similarity of the PBE0 error distributions with the ones obtained for analogous ADMM computations, colored in orange and red for a tzp auxiliary and a TZVP or TZV2P primary basis, respectively. PBE0 and ADMM-PBE0 results are thus in agreement with the conclusions of ref (69), with the latter stating that global hybrids yield slightly too short bonds and that functionals with a rather small amount of exact exchange show negative signed MEs. MAEs are also in agreement, with our result of 0.9 pm for the triple-ζ computations being at the lower boundary of the reported MAE range from 0.9 to 2.8 pm for CC3 or CCSDR(3) references. The relatively good performance can probably be argued with our chosen EOM-CCSD reference, which was shown to yield underestimated TDDFT errors for strongly polarized bonds. (69) All in all, we thus confirm the tendency of PBE0 to provide too compact distances and that errors are most pronounced for polarized carbonyl bonds. In comparison, error distributions for sTDA kernels are increasingly broadened when going from PBE0/TZVP to ADMM-PBE0/TZVP+tzp references. The deviation of the semiempirical results is most pronounced for dihedral angles, with maximum errors amounting up to −13.29° and −14.1° for PBE0/TZVP and ADMM-PBE0/TZVP+tzp references. However, it should be noted that the error distribution for the dihedrals is still nearly equally broad as the corresponding PBE0 one with STDs of 3.2–3.4° for PBE0 and 4.0–4.1° for sTDA. The apparently large mean error of sTDA in dihedral angles is biased because of the fact that sTDA consistently overestimates the dihedral angle while PBE0 gives an equally broad distribution of both negative and positive signed errors. Because the computations were performed using, e.g., a default value of a_EX = 0.2 for the exact exchange scaling parameter, it remains to be investigated if the overestimation represents the general difficulty of recovering a weak stabilizing force or if it could be cured by optimizing the amount of exact exchange in line with refs (72and73).

Figure 3. Normal distributions based on the mean errors (ME) and standard deviations (STD) wrt EOM-CCSD reference geometries in (pm)/(deg)/(deg) for selected (a) bond lengths/(b) angles/(c) dihedral angles for optimized first singlet excited-state geometries comparing conventional PBE0, ADMM-PBE0, and sTDA computations using triple-ζ basis sets.

3.1.3. Vertical Excitation Energies

Benchmarking excited-state methods with the focus on vertical excitation energies has been established as a common assessment tool (see, e.g., ref (74) and references therein). Here one assumes that excitations occur without a change in geometry. Corresponding studies showed that TDDFT excitation energies, which are classified as Rydberg states or associated with a significant amount of charge transfer, are underestimated leading to errors on the order of several eV. (75) A common remedy is to include exact exchange, suggesting that the investigated ADMM-approximated hybrid kernel as well as the sTDA kernel with its motivation to capture the correct physics and asymptotics of electronic interactions could be well-suited compromises to retain sufficient accuracy while reaching high efficiency for a broad range of applications. Analyzing the nature of a transition is beyond the scope of the current work and, for the sake of convenience, we restricted the analysis of vertical excitation energies to a direct comparison of the first 10 excitation energies with states being assigned solely by the corresponding ES energy. Such a simplified comparison might not be justified and lead to wrong assignments when comparing different basis set sizes but also should give a valid assessment of the accuracy of the different kernels for basis sets of the same type. Figure 4 displays normal distributions based on the MEs and STDs for the first 10 vertical excitation energies for optimized geometries, comparing ADMM-PBE0 computations with conventional PBE0 results as well as assessing the performance of PBE0, ADMM-PBE0, and sTDA kernels with respect to a PBE0/TZV2P/CP2K and a PBE0/def2-QZVPP/TURBOMOLE reference. Corresponding statistical data is furthermore summarized in Tables 1 and 2 as well as in section 4 of the Supporting Information. As shown by the upper plot on the left, the auxiliary basis set size reduces, in analogy to our findings for the geometrical data, the ADMM error in the vertical excitation energies when increasing from double- to triple-ζ size and brings only a little further improvement when polarization functions are added. Taking the PBE0 computations within the largest TZV2P basis as a reference exposes the impact of the increasingly large primary basis set as visualized in the upper plot on the right: PBE0/DZVP as well as ADMM-PBE0/DZVP+dzp computations show an equally broad error distribution with MEs and STDs rising up to 0.42 and 0.72 eV, respectively. The broad distribution emphasizes the insufficiency of double-ζ basis sets and that discrepancies between DZVP and TZV2P basis sets are, as already commented on at the beginning of the section, presumably too large to allow for a straightforward comparison of excitation energies. On the other end of the accuracy spectrum, ADMM-PBE0/TZV2P+tz2p results have a relatively sharp distribution with a ME and STD of −0.02 and 0.04 eV. TZVP errors for the PBE0, ADMM-PBE0, and sTDA kernels lie between those extreme values, highlighting that ADMM has only a negligible impact on excitation energies and that the MEs of sTDA are below 0.28 or 0.19 eV depending on the chosen GS reference. This result is in agreement with the error range of 0.2–0.5 eV of the original molecular benchmark studies. (21,22)

Table 2. Statistics Including Mean Errors (MEs), Mean Absolute Errors (MAEs) and Standard Deviations (STDs) Visualizing the Error for Both ADMM-PBE0 and sTDA Kernels for Excited-State Geometries Comprising ES Bond Lengths (pm), Angles (deg), and Dihedral Angles (deg) as well as First Adiabatic Excitation (E_ad), Vertical Excitation (E_vert), and Fluorescence Energies (E_fl) (eV)^a

		Bonds (pm)			Angles (deg)			Dihedrals (deg)
Kernel	PBS+ABS	ME	MAE	STD	ME	MAE	STD	ME	MAE	STD
PBE0	TZVP	–0.15	0.93	1.18	0.32	1.24	2.20	0.80	2.34	3.21
ADMM-PBE0	TZVP+tzp	0.06	0.92	1.18	0.20	1.18	2.09	0.86	2.31	3.46
sTDA@PBE0	TZVP	0.10	1.06	1.50	–0.03	1.60	2.57	3.98	4.06	3.95
sTDA@ADMM-PBE0	TZVP+tzp	0.28	1.03	1.35	–0.05	1.51	2.34	4.03	4.14	4.06

		E_vert (eV)			E_ad (eV)			E_fl (eV)
		ME	MAE	STD	ME	MAE	STD	ME	MAE	STD
PBE0	DZVP	–0.02	0.05	0.06	–0.16	0.30	0.42	–0.05	0.06	0.05
PBE0	TZVP	–0.01	0.03	0.03	–0.15	0.29	0.41	–0.04	0.05	0.04
PBE0	TZV2P	0.00	0.02	0.03	–0.13	0.28	0.40	–0.02	0.02	0.02
ADMM-PBE0	TZVP+tzp	–0.01	0.03	0.04	–0.14	0.29	0.41	–0.05	0.07	0.06
sTDA@PBE0	TZVP	0.45	0.45	0.18	0.23	0.34	0.39	0.27	0.30	0.21
sTDA@ADMM-PBE0	TZVP+tzp	0.34	0.34	0.17	0.10	0.33	0.44	0.14	0.18	0.16

^{^a}

Errors in the geometrical data and E_ad are calculated with respect to the EOM-CCSD reference data of ref (51); errors in E_vert and E_fl are referring to PBE0/def2-QZVPP/TURBOMOLE reference computations.

Figure 4. Normal distributions based on the MEs and STDs for vertical excitation energies (in eV) depicting (a) the error introduced by ADMM-PBE0 in comparison to conventional PBE0 computations (upper left plot), (b) the performance of PBE0, ADMM-PBE0, and sTDA kernel computations using double- and triple-ζ basis sets in comparison to a PBE0/TZV2P/CP2K reference (upper right plot) as well as (c) an analogous assessment of the kernels with respect to a PBE0/def2-QZVPP/Turbomole reference (lower plot).

3.1.4. Adiabatic Excitation Energies and Fluorescence

Adiabatic excitation energies, defined as the energy difference between lowest vibrational levels of the relaxed ground- and excited-state energies, are often benchmarked to assess ES methods. In contrast to vertical excitation energies, adiabatic excitations have the advantage of being experimentally observable and less structure sensitive. They show not a linear but rather a quadratic dependence on nuclear displacements. (76) The difference between calculated vertical and adiabatic excitation energies is the ES relaxation energy, and fluorescence energies are defined as vertical de-excitation energies. The errors in adiabatic excitation and fluorescence energies for the investigated 35 molecules are depicted in Figures 5 and 6, the former comparing the ADMM errors for different auxiliary basis set sizes with respect to conventional PBE0 and the latter assessing the performance of ADMM and sTDA kernels for adiabatic excitation energies with respect to EOM-CCSD reference data and for fluorescence energies with respect to the performed PBE0/def2-QZVPP/Turbomole computations, respectively.

Figure 5. ADMM error distribution in (a) adiabatic excitation and (b) fluorescence energies comparing approximated ADMM-PBE0 and conventional PBE0 results for different auxiliary basis set sizes (in eV).

Figure 6. Comparison of (a) adiabatic excitation energies with respect to EOM-CCSD reference data and (b) fluorescence energies with respect to PBE0/def2-QZVPP/Turbomole reference data for PBE0, ADMM-PBE0, and sTDA kernels (in eV).

Explicit data and a corresponding statistical analysis is given in sections 5 and 6 of the Supporting Information. Despite the varying references, ADMM errors are comparable for both the excitation and de-excitation processes, ranging from −0.3 to −0.1 eV for the smallest dzp auxiliary basis. The tzp and tz2p auxiliary basis sets reduce the maximum error to +0.1 or −0.06 eV, respectively. Corresponding MEs are of meV magnitude and MAEs smaller than 0.02 eV. Analyzing the total error of the different kernels in the scatter plots of Figure 6 highlights that sTDA errors are consistently shifted to larger and more positive errors. The correlation between the relative shift of sTDA errors with respect to both PBE0 and EOM-CCSD results and the considered amount of exact exchange was not investigated any further, but will be part of future work. However, even when a nonoptimized fraction of 0.2 is chosen for the coupling parameter a_EX of eq 29, sTDA curves are close to the hybrid functional results when compared to EOM-CCSD reference data achieving statistical measures of the same order of magnitude, with MEs of 0.2/–0.1/–0.1 eV, MAEs of 0.3/0.3/0.3 eV, and maximum errors of −1.3/1.0/1.0 eV for sTDA/ADMM-PBE0/PBE0 adiabatic excitation energies. Switching to a PBE0/TURBOMOLE reference for fluorescence energies favors PBE0 and ADMM-PBE0 results over the semiempirical ones, with the former now only accounting for deviations in the basis sets and the ADMM error with MEs of −0.05/–0.05 eV, MAEs of 0.07/0.05 eV, and maximum errors of −0.2/0.1 eV for ADMM-PBE0/PBE0. The sTDA errors are, however, consistent, yielding an equivalent ME, MAE, and maximum error of 0.3 eV.

3.2. Increasing Computational Efficiency: Treating Extended Systems Using ADMM-PBE0 and sTDA Kernels

To analyze computational timings of ADMM-PBE0 and sTDA excited-state properties and to demonstrate the suitability of the two approaches for large-scale periodic systems, we investigated a series of porous COF materials taken from the CURATED COFs database. (52,53) The chosen subset includes the prototypes of the first ever synthesized COFs (77) as well as pairs of COFs that were demonstrated to show more or less bright fluorescence depending on either accordingly tuned linker molecules or the surrounding solvent. (78,79) The CURATED COFs database provides cleaned-up DFT-optimized GS structures at a GGA level of accuracy, obtained with the PBE functional, D3(BJ) correction, and DZVP-MOLOPT-SR basis sets. We reoptimized these geometries using a corresponding ADMM-PBE0 GS reference and triple-ζ ccGRB-T primary and tzp auxiliary basis sets to ensure a consistent accuracy of both structural and electronic GS and ES properties. Structural details are provided in the Supporting Information. In all calculations a truncated Coulomb operator with a radius of 4 Å has been used. In Table 3, detailed timings for the first 3 ES geometry optimization steps are given for COF 05000N2 for ADMM-approximated and sTDA kernels in comparison to conventional PBE0 results, with the latter relying on an exact analytical evaluation of the two-electron-four-center exchange integrals. The formal scaling of both excited-state energy and gradient implementations is cubic with respect to the number of atomic orbital basis functions and increases linearly with the number of computed excited states. Memory requirements and execution times for the main computational steps of the algorithm are analogous to corresponding density functional ground-state implementations, as analyzed in detail in refs (56) and (43). As outlined in ref (46), the computation of electron repulsion integrals (ERIs) represents the main bottleneck for hybrid functional computations, with the number of integrals and thus the required memory scaling quadratically with the number of atomic and occupied molecular orbitals, N × N_occ, when considering sparsity and applying Cauchy–Schwarz screening. At best linear scaling with a still significant prefactor can be achieved when relying on additional density matrix screening and when using truncated exchange operators.

Table 3. Explicit Computational Timings (s) for COF 05000N2 Comprising 84 Atoms/276 Electrons in the Unit Cell for PBE0, ADMM-PBE0, and sTDA@ADMM-PBE0 Kernels^a

	PBE0	ADMM-PBE0	sTDA@ADMM-PBE0
GS SCF	301	137	134
ES energy and gradient	1919	403	148
ERI for ES energy	687	86	33
DERI for ES gradient	1265	11	7
Total computation time	2220	540	282

^{^a}

Computations were performed using a ccGRB-T primary basis (2076 basis functions) and a tzp auxiliary basis (1164 basis functions). Timings are reported for a Intel Xeon E5-2670 processor using in total 4608 cores, analysing the cost for converging the GS SCF computation, for calculating the ES energy and gradient as well as for the computation of the ERIs needed for the ES energy calculation and the DERIs needed for the ES gradient.

The measured total computation time of

\approx 100

hours for a ccGRB-T basis with 2076 basis functions reflects that an analytical computation of exchange integrals within a large primary orbital basis cannot provide optimal efficiency for large periodic systems. The fact that 31% and 57% of the PBE0 computation times are solely attributed to the computation of ERIs and derivative electron repulsion integrals (DERIs) emphasizes the need for approximations. Applying ADMM with a triple-ζ auxiliary basis and thus reducing the basis set size by a factor of 2 to 1164 functions accelerates the computation by a factor of

\approx 4

with a remaining total time of 9.0 min. The small auxiliary basis reduces the number of ERIs and, in combination with increased sparsity and thus more efficient Schwarz screening, the reduction of the basis size results in timings for the integral evaluation that are reduced by a factor of

\approx 8

. More drastic is even the speed-up for the computation of DERIs because of the here applied density-matrix screening. In comparison to PBE0, the computation of DERIs is 115 times faster for ADMM-PBE0 and thus represents not 57% but solely 2% of the total costs. Timings comparable to ADMM-PBE0 can also be achieved with the semiempirical tight binding setup: sTDA reduces the total costs to 4.7 min, thus by a factor of

\approx 8

in comparison to conventional PBE0 and by a factor of

\approx 2

in comparison to ADMM. The speed-up with respect to ADMM is because of the faster ERI evaluation relying on the semiempirical description of both Coulomb and exchange contributions when solving the ES eigenvalue problem. sTDA and ADMM timings are on the same order of magnitude, but further acceleration could be expected when switching to a complete semiempirical setup for both GS and ES. We are also aware that absolute timings for the sTDA ES gradient computations could still be improved by further code optimization.

To give an estimate on the required relative costs of ADMM and sTDA for the computation of broad-band absorption spectra and for the geometry optimization of low-lying excited states, we report timings for all six COFs in Table 4. Computations were performed for the lowest 500 excited states and the geometry optimization of the first ES when a manifold of 8 singlet states was selected. The investigated system sizes range from unit cells of 84 to 300 atoms and include basis set sizes of 2076 to 7440 basis functions, respectively. Comparing the computational timings for the calculation of the lowest 500 excitation energies clearly shows the advantage of the semiempirical setup. While computations using the ADMM kernel take ≈4 to 10 h, sTDA is 1 order of magnitude faster with a maximum computation time of less than 44 min. With the current setup, the tight binding approach is, however, less favorable for ES geometry optimization. First, the number of required optimization cycles is increased for five out of six COFs. Second, the average computation time per optimization cycle is accelerated only by a factor of 1.4 to 5, so that ADMM-PBE0 and sTDA computation times are of the same order of magnitude. For both ADMM-PBE0 and sTDA computations, timings of 4–30 min per optimization cycle and total timings of 2.5–40 h for an ES geometry optimization can thus be expected for periodic systems in the size range of hundreds of atoms and using the indicated computational resources.

Table 4. Computational Timings (s) for ADMM-PBE0 and sTDA@ADMM-PBE0 Kernels for a Series of Fluorescent COFs as Required for the Computation of Broad-Band Absorption Spectra over the Lowest 500 Excited States as well as for the ES Geometry Optimization of the First Excited State when Selecting a Manifold of the Lowest 8 Singlet States^a

			Absorption spectra		ES optimization
			ADMM	sTDA	ADMM		sTDA
	# Atoms	# Basis	Time	Time	Time	Time/Step	Time	Time/Step
05000N2	84	2076	15758	2157	9590	461	8937	329
05001N2	192	4848	33678	2472	77332	1556	33837	637
15100N2	300	7440	36907	2623	155696	1729	37804	433
15101N2	192	5028	30931	1932	67962	1680	29258	417
20610N2	240	5700	30014	2119	59603	1086	16627	266
20611N2	262	6188	27965	1993	136122	1818	35955	362

^{^a}

Timings are reported for a Intel Xeon E5-2670 processor using in total 2304 cores.

To compare the relative accuracy of ADMM-PBE0 and sTDA for periodic systems, the computed broad-band spectra are depicted in Figure 7. Corresponding vertical excitation, adiabatic excitation, and fluorescence energies, as well as Stokes’ shifts are listed in Table 5. As highlighted by the six plots of Figure 7, sTDA and ADMM provide absorption spectra with qualitatively matching oscillator strengths. sTDA excitations are, however, in all cases shifted to smaller energies, with MAEs for the first and the first ten vertical excitation energies amounting both up to 0.31 eV. Because the exchange mixing parameter a_EX was set to the same value as for the molecular systems (a_EX = 0.2), it remains to be investigated if this apparently systematic shift could be improved by adjusting the amount of exchange. However, even with this default setting, the MAE in adiabatic excitation and fluorescence energies is within the error range found for molecular systems. ADMM-PBE0 and sTDA energies differ by 0.42 or 0.40 eV, respectively. Corresponding Stokes’ shifts are thus also predicted with a relatively small error of 0.17 eV.

Table 5. Vertical Excitation (E_vert), Adiabatic Excitation (E_ad), and Fluorescence Energies (E_fl) as well as Corresponding Stokes’ Shifts (λ) (eV) for ADMM-PBE0 and sTDA Kernels for the First Excited Singlet State for a Series of Fluorescent COFs

	E_vert		E_ad		E_fl		λ
	ADMM	sTDA	ADMM	sTDA	ADMM	sTDA	ADMM	sTDA
05000N2	4.71	4.95	4.40	3.87	4.49	4.16	0.22	0.79
05001N2	3.77	3.44	2.60	2.14	2.97	2.56	0.80	0.88
15100N2	3.54	3.08	2.27	1.85	2.43	1.98	1.11	1.10
15101N2	3.10	2.67	2.04	1.46	2.18	1.68	0.92	0.98
20610N2	3.97	3.67	2.82	2.45	3.08	2.74	0.89	0.93
20611N2	3.21	3.10	2.47	1.96	2.65	2.25	0.56	0.85
MAE	0.31		0.42		0.40		0.17

Figure 7. Comparison of ADMM-PBE0 and sTDA absorption spectra for a series of fluorescent COFs.

4. Conclusions

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Approaches based on TDDFT are well established to calculate excited-state properties. However, when dealing with periodic systems, the routine usage of hybrid functionals is often hindered because of the increased computational costs in comparison to GGA functionals. We presented two approximate schemes based on the ADMM and sTDA approaches with the goal to reach hybrid functional accuracy at a reduced cost. Benchmark results for molecular and periodic systems show that the two methods can indeed reach the required accuracy for excited-state properties and allow for an efficient calculation of many excited states. First, the implemented ADMM-approximated hybrid functional kernel leads to a speed-up of at least 1 order of magnitude in comparison to conventional hybrid functional TDDFT and thus enables practical calculations on large systems of hundreds of atoms at a moderate cost. Timings could be further improved by accelerating the evaluation of the exact exchange integrals using either standard resolution-of-the-identity approaches or seminumerical integration. In comparison to inherent methodological errors of TDDFT because of the chosen density functional, the loss in accuracy for ADMM-approximated kernels is minor when auxiliary basis sets of at least triple-ζ size are selected. ADMM/TZVP+tzp MAEs in excited-state bond lengths are in the range of 0.3 pm and corresponding vertical and adiabatic excitation and fluorescence energies are off by 0.02–0.04 eV. In contrast, conventional PBE0/TZVP bond lengths and adiabatic excitation energies show MAEs of 0.9 pm and 0.3 eV with respect to EOM-CCSD/def2-TZVPP reference data. The given ADMM accuracy estimate represents a lower limit that was found to be reliable when the simplest ADMM variant was chosen based on basis projection in combination with a triple-ζ auxiliary basis. It could be further improved by either adding an exchange density functional correction to compensate the incompleteness error or, regarding the perpendicular methodological error, by switching to more sophisticated ADMM schemes with the latter, however, increasing the complexity of implementation. The semiempirical sTDA method is particularly useful when aiming for broad-band absorption or emission spectra and efficient prescreening of large-scale test sets. An order of magnitude speedup was achieved with sTDA in comparison to ADMM when spectra over the lowest 500 excitation energies were computed, with resulting timings being in the range of minutes rather than hours. Because of the reduced amount of ERIs, the sTDA kernel also accelerates timings by a factor of 2 for each ES geometry optimization cycle. However, an increased number of optimization steps can lead to total timings comparable to ADMM. We expect that further savings could be achieved when optimizing the current sTDA ES gradient implementation as well as when choosing a semiempirical GS reference. In comparison to hybrid functional kernels, sTDA MAEs in excited-state bond lengths with respect to EOM-CCSD reference results are only slightly increased to 1.1 pm. According to the molecular benchmarks, the error estimate for vertical and adiabatic excitation and fluorescence energies can be assumed to be below 0.5 eV, a deviation that is slightly increased but still comparable to the MAE of 0.3 eV of conventional PBE0 with respect to EOM-CCSD references. Similar relative error ranges are found for the investigated COFs, and a comparison of ADMM and sTDA oscillator strengths suggests qualitative accuracy of the tight binding method. The most critical parameter in the sTDA method is the exact exchange scaling. This parameter should be adjusted with respect to the selected hybrid functional ground-state reference and a suboptimal value can lead to large geometrical errors and even negative excitation energies. In the case of formyl chloride, the scaling parameter had to be adjusted to avoid dissociation when the first excited singlet state was optimized. Further studies are required to investigate the optimal fraction of exact exchange for periodic systems because the choice might differ from the values that were originally suggested for molecular systems. Further empirical shifts might also be required when the sTDA excited-state method is combined with corresponding semiempirical ground-state calculations. A consistent setup remains to be investigated, but we expect it to pave the way to treat even larger systems. It also remains to be investigated in future work if efficiency and accuracy of ADMM-approximated hybrid functional TDDFT and sTDA are beneficial for excited-state molecular dynamic simulations.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.2c00144.

Treatment of the virtual space, structural information for molecular benchmark set, geometrical data on optimized molecular geometries, statistical analysis for ADMM and sTDA vertical excitation energies, adiabatic excitation energies and statistical analysis, fluorescence energies and statistical analysis, and structural information for covalent organic frameworks (PDF)

ct2c00144_si_001.pdf (1.09 MB)

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Author Information

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Corresponding Author
- Anna-Sophia Hehn - Department of Chemistry, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland; https://orcid.org/0000-0002-0498-740X; Email: [email protected]
Authors
- Beliz Sertcan - Department of Chemistry, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
- Fabian Belleflamme - Department of Chemistry, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
- Sergey K. Chulkov - School of Mathematics and Physics, University of Lincoln, Brayford Pool, Lincoln LN67TS, United Kingdom
- Matthew B. Watkins - School of Mathematics and Physics, University of Lincoln, Brayford Pool, Lincoln LN67TS, United Kingdom; https://orcid.org/0000-0003-4215-684X
- Jürg Hutter - Department of Chemistry, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
Notes
The authors declare no competing financial interest.

Acknowledgments

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We thank Dr. Augustin Bussy (University of Zurich) for his kind support and the valuable discussions. This work was supported in parts by the MARVEL National Centre for Competence in Research funded by the Swiss National Science Foundation (grant agreement ID 51NF40-182892). We thank the Swiss National Supercomputing Center (CSCS) for providing computational resources (Grant No. S1109). Furthermore, this work has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 798196. M.W. and S.C. were assisted by membership of the UK’s HEC Materials Chemistry Consortium, which is funded by EPSRC (EP/R029431).

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De wergifosse, Marc; Grimme, Stefan
Journal of Physical Chemistry A (2021), 125 (18), 3841-3851CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
A review. We review recent developments in the framework of simplified quantum chem. for excited state and optical response properties (sTD-DFT) and present future challenges for new method developments to improve accuracy and extend the range of application. In recent years, the scope of sTD-DFT was extended to mol. response calcns. of the polarizability, optical rotation, first hyperpolarizability, two-photon absorption (2PA), and excited-state absorption for large systems with hundreds to thousands of atoms. The recently introduced spin-flip simplified time-dependent d. functional theory (SF-sTD-DFT) variant enables an ultrafast treatment for diradicals and related strongly correlated systems. A few drawbacks were also identified, specifically for the computation of 2PA cross sections. We propose solns. to this problem and how to generally improve the accuracy of simplified schemes. New possible simplified schemes are also introduced for strongly correlated systems, e.g., with a second-order perturbative correlation correction. Interpretation tools that can ext. chem. structure-property relationships from excited state or response calcns. are also discussed. In particular, the recently introduced method-agnostic RespA approach based on natural response orbitals (NROs) as the key concept is employed.
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Grimme, S.; Schreiner, P. R. Computational Chemistry: The Fate of Current Methods and Future Challenges. Angew. Chem., Int. Ed. 2018, 57, 4170– 4176, DOI: 10.1002/anie.201709943
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Computational Chemistry: The Fate of Current Methods and Future Challenges
Grimme, Stefan; Schreiner, Peter R.
Angewandte Chemie, International Edition (2018), 57 (16), 4170-4176CODEN: ACIEF5; ISSN:1433-7851. (Wiley-VCH Verlag GmbH & Co. KGaA)
In this essay, we attempt to make predictions about the fate and development of the computational mol. sciences. Of course, it is not the first time that the future challenges for computational org. chem. and biochem. are considered; these were outlined in complementary contexts recently. In this article, the authors take a somewhat different perspective and emphasize the changes expected for chem. that are triggered by the rapid developments and increasingly stronger influences from theory, algorithms, and data-driven technologies. The authors of this essay are about the same age and have a general overview of a period of about 25 years during which they have actively contributed to the field of computational chem. Hence, it appears sensible to make predictions extending 25 years into the future, roughly to the year 2043 (when both authors will long be retired).
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Dierksen, M.; Grimme, S. The Vibronic Structure of Electronic Absorption Spectra of Large Molecules: A Time-Dependent Density Functional Study on the Influence of Exact Hartree-Fock Exchange. J. Phys. Chem. A 2004, 108, 10225– 10237, DOI: 10.1021/jp047289h
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The Vibronic Structure of Electronic Absorption Spectra of Large Molecules: A Time-Dependent Density Functional Study on the Influence of "Exact" Hartree-Fock Exchange
Dierksen, Marc; Grimme, Stefan
Journal of Physical Chemistry A (2004), 108 (46), 10225-10237CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
The functional dependence of excited-state geometries and normal modes calcd. with time-dependent d. functional theory (TDDFT) is studied from vibronic structure calcns. of the absorption spectra of large mols. For a set of mols. covering a wide range of different structures including org. dyes, biol. chromophores, and mols. of importance in material science, quantum mech. simulations of the vibronic structure are performed. In total over 40 singlet-singlet transitions of neutral closed-shell compds. and doublet-doublet transitions of neutral radicals, radical cations, and anions are considered. Calcns. with different std. d. functionals show that the predicted vibronic structure critically depends on the fraction of the exact Hartree-Fock exchange (EEX) included in hybrid functionals. The effect can been traced back to a large influence of EEX on the geometrical displacement upon excitation. On the contrary, the dependence of the results on the choice of the local exchange-correlation functional is rather small. From detailed comparisons with exptl. spectra conclusions are drawn concerning the optimum amt. of EEX mixing for a proper description of the excited-state properties. The relation of the quality of the simulated spectra with the errors for 0-0 transition energies is discussed. For the studied singlet-singlet π → π* transitions and the 1st strongly dipole-allowed transitions of PAH radical cations some rules of thumb concerning the optimum portion of EEX are derived. However, in general no universal amt. of EEX seems to exist that gives a uniformly good description for all systems and states. Nevertheless an inclusion of ∼30-40% of EEX in the functional is found empirically to yield in most cases simulated spectra that compare very well with those from expt. and thus seems to be necessary for an accurate description of the excited-state geometry. Pure d. functionals that are computationally more efficient provide less accurate spectra in most cases and their application is recommended solely for comparison purposes to obtain ests. for the reliability of the theor. predictions.
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Merlot, P.; Izsák, R.; Borgoo, A.; Kjærgaard, T.; Helgaker, T.; Reine, S. Charge-constrained auxiliary-density-matrix methods for the Hartree-Fock exchange contribution. J. Chem. Phys. 2014, 141, 094104, DOI: 10.1063/1.4894267
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Charge-constrained auxiliary-density-matrix methods for the Hartree-Fock exchange contribution
Merlot, Patrick; Izsak, Robert; Borgoo, Alex; Kjaergaard, Thomas; Helgaker, Trygve; Reine, Simen
Journal of Chemical Physics (2014), 141 (9), 094104/1-094104/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Three new variants of the auxiliary-d.-matrix method (ADMM) of Guidon, Hutter, and VandeVondele [J. Chem. Theory Comput. 6, 2348 (2010)] are presented with the common feature that they have a simplified constraint compared with the full orthonormality requirement of the earlier ADMM1 method. All ADMM variants are tested for accuracy and performance in all-electron B3LYP calcns. with several commonly used basis sets. The effect of the choice of the exchange functional for the ADMM exchange-correction term is also investigated. (c) 2014 American Institute of Physics.
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Guidon, M.; Hutter, J.; VandeVondele, J. Auxiliary Density Matrix Methods for Hartree-Fock Exchange Calculations. J. Chem. Theory Comput. 2010, 6, 2348– 2364, DOI: 10.1021/ct1002225
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Auxiliary Density Matrix Methods for Hartree-Fock Exchange Calculations
Guidon, Manuel; Hutter, Jurg; Vande Vondele, Joost
Journal of Chemical Theory and Computation (2010), 6 (8), 2348-2364CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
The calcn. of Hartree-Fock exchange (HFX) is computationally demanding for large systems described with high-quality basis sets. In this work, we show that excellent performance and good accuracy can nevertheless be obtained if an auxiliary d. matrix is employed for the HFX calcn. Several schemes to derive an auxiliary d. matrix from a high-quality d. matrix are discussed. Key to the accuracy of the auxiliary d. matrix methods (ADMM) is the use of a correction based on std. generalized gradient approxns. for HFX. ADMM integrates seamlessly in existing HFX codes and, in particular, can be employed in linear scaling implementations. Demonstrating the performance of the method, the effect of HFX on the structure of liq. water is investigated in detail using Born-Oppenheimer mol. dynamics simulations (300 ps) of a system of 64 mols. Representative for large systems are calcns. on a solvated protein (Rubredoxin), for which ADMM outperforms the corresponding std. HFX implementation by approx. a factor 20.
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Kumar, C.; Fliegl, H.; Jensen, F.; Teale, A. M.; Reine, S.; Kjærgaard, T. Accelerating Kohn-Sham response theory using density fitting and the auxiliary-density-matrix method. Int. J. Quantum Chem. 2018, 118, e25639 DOI: 10.1002/qua.25639
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Rebolini, E.; Izsák, R.; Reine, S. S.; Helgaker, T.; Pedersen, T. B. Comparison of Three Efficient Approximate Exact-Exchange Algorithms: The Chain-of-Spheres Algorithm, Pair-Atomic Resolution-of-the-Identity Method, and Auxiliary Density Matrix Method. J. Chem. Theory Comput. 2016, 12, 3514– 3522, DOI: 10.1021/acs.jctc.6b00074
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Comparison of Three Efficient Approximate Exact-Exchange Algorithms: The Chain-of-Spheres Algorithm, Pair-Atomic Resolution-of-the-Identity Method, and Auxiliary Density Matrix Method
Rebolini, Elisa; Izsak, Robert; Reine, Simen Sommerfelt; Helgaker, Trygve; Pedersen, Thomas Bondo
Journal of Chemical Theory and Computation (2016), 12 (8), 3514-3522CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
We compare the performance of three approx. methods for speeding up evaluation of the exchange contribution in Hartree-Fock and hybrid Kohn-Sham calcns.: the chain-of-spheres algorithm, and the auxiliary d. matrix method. Both the efficiency relative to that of a conventional linear-scaling algorithm and the accuracy of total, atomization, and orbital energies are compared for a subset contg. 25 of the 200 mols. in the Rx200 set using double-, triple-, and quadruple-ζ basis sets. The accuracy of relative energies is further compared for small alkane conformers (ACONF test set) and Diels-Alder reactions (DARC test set). Overall, we find that the COSX method provides good accuracy for orbital energies as well as total and relative energies, and the method delivers a satisfactory speedup. The PARI-K and in particular ADMM algorithms require further development and optimization to fully exploit their indisputable potential.
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Merlot, P.; Kjærgaard, T.; Helgaker, T.; Lindh, R.; Aquilante, F.; Reine, S.; Pedersen, T. B. Attractive electron–electron interactions within robust local fitting approximations. J. Comput. Chem. 2013, 34, 1486– 1496, DOI: 10.1002/jcc.23284
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Attractive electron-electron interactions within robust local fitting approximations
Merlot, Patrick; Kjaergaard, Thomas; Helgaker, Trygve; Lindh, Roland; Aquilante, Francesco; Reine, Simen; Pedersen, Thomas Bondo
Journal of Computational Chemistry (2013), 34 (17), 1486-1496CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
An anal. of Dunlap's robust fitting approach reveals that the resulting two-electron integral matrix is not manifestly pos. semidefinite when local fitting domains or non-Coulomb fitting metrics are used. We present a highly local approx. method for evaluating four-center two-electron integrals based on the resoln.-of-the-identity (RI) approxn. and apply it to the construction of the Coulomb and exchange contributions to the Fock matrix. In this pair-at. resoln.-of-the-identity (PARI) approach, at.-orbital (AO) products are expanded in auxiliary functions centered on the two atoms assocd. with each product. Numerical tests indicate that in 1% or less of all Hartree-Fock and Kohn-Sham calcns., the indefinite integral matrix causes nonconvergence in the self-consistent-field iterations. In these cases, the two-electron contribution to the total energy becomes neg., meaning that the electronic interaction is effectively attractive, and the total energy is dramatically lower than that obtained with exact integrals. In the vast majority of our test cases, however, the indefiniteness does not interfere with convergence. The total energy accuracy is comparable to that of the std. Coulomb-metric RI method. The speed-up compared with conventional algorithms is similar to the RI method for Coulomb contributions; exchange contributions are accelerated by a factor of up to eight with a triple-zeta quality basis set. A pos. semidefinite integral matrix is recovered within PARI by introducing local auxiliary basis functions spanning the full AO product space, as may be achieved by using Cholesky-decompn. techniques. Local completion, however, slows down the algorithm to a level comparable with or below conventional calcns. © 2013 Wiley Periodicals, Inc.
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Manzer, S. F.; Epifanovsky, E.; Head-Gordon, M. Efficient Implementation of the Pair Atomic Resolution of the Identity Approximation for Exact Exchange for Hybrid and Range-Separated Density Functionals. J. Chem. Theory Comput. 2015, 11, 518– 527, DOI: 10.1021/ct5008586
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Efficient Implementation of the Pair Atomic Resolution of the Identity Approximation for Exact Exchange for Hybrid and Range-Separated Density Functionals
Manzer, Samuel F.; Epifanovsky, Evgeny; Head-Gordon, Martin
Journal of Chemical Theory and Computation (2015), 11 (2), 518-527CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
An efficient new MO basis algorithm is reported implementing the pair at. resoln. of the identity approxn. (PARI) to evaluate the exact exchange contribution (K) to SCF methods, such as hybrid and range-sepd. hybrid d. functionals. The PARI approxn., in which AO basis function pairs are expanded using auxiliary basis functions centered only on their two resp. atoms, was recently investigated by Merlot et al. Our algorithm is significantly faster than quartic scaling RI-K, with an asymptotic exchange speedup for hybrid functionals of (1 + X/N), where N and X are the AO and auxiliary basis dimensions. The asymptotic speedup is 2 + 2X/N for range sepd. hybrids such as CAM-B3LYP, ωB97X-D, and ωB97X-V which include short- and long-range exact exchange. The obsd. speedup for exchange in ωB97X-V for a C68 graphene fragment in the cc-pVTZ basis is 3.4 relative to RI-K. Like conventional RI-K, our method greatly outperforms conventional integral evaluation in large basis sets; a speedup of 19 is obtained in the cc-pVQZ basis on a C54 graphene fragment. Negligible loss of accuracy relative to exact integral evaluation is demonstrated on databases of bonded and nonbonded interactions. We also demonstrate both anal. and numerically that the PARI-K approxn. is variationally stable.
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Neese, F.; Wennmohs, F.; Hansen, A.; Becker, U. Efficient, approximate and parallel Hartree–Fock and hybrid DFT calculations. A ‘chain-of-spheres’ algorithm for the Hartree–Fock exchange. Chem. Phys. 2009, 356, 98– 109, DOI: 10.1016/j.chemphys.2008.10.036
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Efficient, approximate and parallel Hartree-Fock and hybrid DFT calculations. A 'chain-of-spheres' algorithm for the Hartree-Fock exchange
Neese, Frank; Wennmohs, Frank; Hansen, Andreas; Becker, Ute
Chemical Physics (2009), 356 (1-3), 98-109CODEN: CMPHC2; ISSN:0301-0104. (Elsevier B.V.)
In this paper, the possibility is explored to speed up Hartree-Fock and hybrid d. functional calcns. by forming the Coulomb and exchange parts of the Fock matrix by different approxns. For the Coulomb part the previously introduced Split-RI-J variant of the well-known d. fitting' approxn. is used. The exchange part is formed by semi-numerical integration techniques that are closely related to Friesner's pioneering pseudo-spectral approach. Our potentially linear scaling realization of this algorithm is called the 'chain-of-spheres exchange' (COSX). A combination of semi-numerical integration and d. fitting is also proposed. Both Split-RI-J and COSX scale very well with the highest angular momentum in the basis sets. It is shown that for extended basis sets speed-ups of up to two orders of magnitude compared to traditional implementations can be obtained in this way. Total energies are reproduced with an av. error of <0.3 kcal/mol as detd. from extended test calcns. with various basis sets on a set of 26 mols. with 20-200 atoms and up to 2000 basis functions. Reaction energies agree to within 0.2 kcal/mol (Hartree-Fock) or 0.05 kcal/mol (hybrid DFT) with the canonical values. The COSX algorithm parallelizes with a speedup of 8.6 obsd. for 10 processes. Min. energy geometries differ by less than 0.3 pm in the bond distances and 0.5° in the bond angles from their canonical values. These developments enable highly efficient and accurate SCF calcns. including nonlocal Hartree-Fock exchange for large mols. In combination with the RI-MP2 method and large basis sets, second-order many body perturbation energies can be obtained for medium sized mols. with unprecedented efficiency. The algorithms are implemented into the ORCA electronic structure system.
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Izsák, R.; Neese, F. An overlap fitted chain of spheres exchange method. J. Chem. Phys. 2011, 135, 144105, DOI: 10.1063/1.3646921
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An overlap fitted chain of spheres exchange method
Izsak, Robert; Neese, Frank
Journal of Chemical Physics (2011), 135 (14), 144105/1-144105/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The "chain of spheres" (COS) algorithm, as part of the RIJCOSX SCF procedure, approximates the exchange term by performing analytic integration with respect to the coordinates of only one of the two electrons, whereas for the remaining coordinates, integration is carried out numerically. In the present work, we attempt to enhance the efficiency of the method by minimizing numerical errors in the COS procedure. The main idea is based on the work of Friesner and consists of finding a fitting matrix, Q, which leads the numerical and anal. evaluated overlap matrixes to coincide. Using Q, the evaluation of exchange integrals can indeed be improved. Improved results and timings are obtained with the present default grid setup for both single point calcns. and geometry optimizations. The fitting procedure results in a redn. of grid sizes necessary for achieving chem. accuracy. We demonstrate this by testing a no. of grids and comparing results to the fully analytic and the earlier COS approxns. This turns out to be favorable for total and reaction energies, for which chem. accuracy can now be reached with a corresponding ∼30% speedup over the original RIJCOSX procedure for single point energies. Results are slightly less favorable for the accuracy of geometry optimizations, but the procedure is still shown to yield geometries with errors well below the method inherent errors of the employed theor. framework. (c) 2011 American Institute of Physics.
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Laqua, H.; Thompson, T. H.; Kussmann, J.; Ochsenfeld, C. Highly Efficient, Linear-Scaling Seminumerical Exact-Exchange Method for Graphic Processing Units. J. Chem. Theory Comput. 2020, 16, 1456– 1468, DOI: 10.1021/acs.jctc.9b00860
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Highly Efficient, Linear-Scaling Seminumerical Exact-Exchange Method for Graphic Processing Units
Laqua, Henryk; Thompson, Travis H.; Kussmann, Joerg; Ochsenfeld, Christian
Journal of Chemical Theory and Computation (2020), 16 (3), 1456-1468CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
We present a highly efficient and asymptotically linear-scaling graphic processing unit accelerated seminumerical exact-exchange method (sn-LinK). We go beyond our previous central processing unit-based method (H. Laqua et al., 2018) by employing our recently developed integral bounds (T.H. Thomson and C. Ochsenfeld, 2019) and high-accuracy numerical integration grid (H. Laqua et al., 2018). The accuracy is assessed for several established test sets, providing errors significantly below 1mEh for the smallest grid. Moreover, a comprehensive performance anal. for large mols. between 62 and 1347 atoms is provided, revealing the outstanding performance of our method, in particular, for large basis sets such as the polarized quadruple-zeta level with diffuse functions.
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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: 10.1063/5.0022755
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An improved seminumerical Coulomb and exchange algorithm for properties and excited states in modern density functional theory
Holzer, Christof
Journal 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.
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Ohno, K. Some remarks on the Pariser-Parr-Pople method. Theor. Chim. Acta 1964, 2, 219– 227, DOI: 10.1007/BF00528281
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The Pariser-Parr-Pople method
Ohno, Kimio
Theoretica Chimica Acta (1964), 2 (3), 219-27CODEN: TCHAAM; ISSN:0040-5744.
Basic assumptions which characterize the Pariser-Parr (CA 49, 10725f)-Pople (CA 48, 8639g) method of computing mol. electronic wave functions are examd. crit. By restricted variational calcn. of the valence state of C and N atoms and ions, it is demonstrated that the usual methods of evaluation of center Coulomb integrals and at. core energies are rather good. A semitheoretical means of estg. the core resonance integral is proposed and shown to give fair agreement with the empirical values for C-C, O-O, C-N, and C-O bonds.
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Klopman, G. A Semiempirical Treatment of molecular Structures. II. Molecular Terms and Application to diatomic Molecules. J. Am. Chem. Soc. 1964, 86, 4550– 4557, DOI: 10.1021/ja01075a008
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A semiempirical treatment of molecular structures. II. Molecular terms and application to diatomic molecules
Klopman, G.
Journal of the American Chemical Society (1964), 86 (21), 4550-7CODEN: JACSAT; ISSN:0002-7863.
cf. CA 60, 12679e. A self-consistent semiempirical method which is designed for the calcn. of heats of formation and charge distribution of nonconjugated mols. is outlined. The method is based on an antisymmetrized product of mol. orbitals, simplified in such a way as to make the values of all involved integrals directly available from at. spectra (loc. cit.) and mol. bond distances. In a preliminary study, this method was used to calc. satisfactory values of bond energies and reasonable values of charge distributions in 80 diat. mols. (σ-bonded).
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Nishimoto, K.; Mataga, N. Z. Electronic Structure and Spectra of Some Nitrogen Heterocycles. Z. Phys. Chem. 1957, 12, 335, DOI: 10.1524/zpch.1957.12.5_6.335
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Electronic structure and spectra of some nitrogen heretocycles
Nishimoto, Kitisuke; Mataga, Noboru
Zeitschrift fuer Physikalische Chemie (Muenchen, Germany) (1957), 12 (), 335-8CODEN: ZPCFAX; ISSN:0044-3336.
The energies of lower excited states of pyridine, pyrazine, and sym-triazine and the oscillator strengths of the transitions to those excited states have been calcd. The results are listed and discussed.
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Bannwarth, C.; Grimme, S. A simplified time-dependent density functional theory approach for electronic ultraviolet and circular dichroism spectra of very large molecules. Comput. Theor. Chem. 2014, 1040–1041, 45– 53, DOI: 10.1016/j.comptc.2014.02.023
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A simplified time-dependent density functional theory approach for electronic ultraviolet and circular dichroism spectra of very large molecules
Bannwarth, Christoph; Grimme, Stefan
Computational & Theoretical Chemistry (2014), 1040-1041 (), 45-53CODEN: CTCOA5; ISSN:2210-271X. (Elsevier B.V.)
We present a simplified time-dependent d. functional theory approach (sTD-DFT) that allows fast computation of electronic UV or CD spectra of mols. with 500-1000 atoms. The matrix elements are treated in the same way as in the recently proposed simplified Tamm-Dancoff approach but instead of applying the Tamm-Dancoff approxn., the std. linear-response d. functional theory problem is solved. Compared to sTDA, the method leads to an increase in computation time (typically a factor of 2-5 compared to the corresponding sTDA) which is justified since the resulting transition dipole moments are in general of higher quality. This becomes important if spectral intensities (e.g. single-photon oscillator and rotatory transition strengths) are of interest. Comparison of UV and CD spectra obtained from sTD-DFT and sTDA for some typical systems employing std. hybrid functionals shows that both yield very similar excitation energies but the advantage of using the former approach for transition moments. In order to show the applicability of sTD-DFT to systems which are far beyond the scope of conventional TD-DFT, we present the CD spectrum of a substituted, chiral fullerene over a range of almost 1200 excited states. We propose this method as a more reliable alternative for the prediction esp. of the more challenging CD spectra.
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Grimme, S. A simplified Tamm-Dancoff density functional approach for the electronic excitation spectra of very large molecules. J. Chem. Phys. 2013, 138, 244104, DOI: 10.1063/1.4811331
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A simplified Tamm-Dancoff density functional approach for the electronic excitation spectra of very large molecules
Grimme, Stefan
Journal of Chemical Physics (2013), 138 (24), 244104/1-244104/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Two approxns. in the Tamm-Dancoff d. functional theory approach (TDA-DFT) to electronically excited states are proposed which allow routine computations for electronic UV- or CD spectra of mols. with 500-1000 atoms. Speed-ups compared to conventional time-dependent DFT (TD-DFT) treatments of about two to three orders of magnitude in the excited state part at only minor loss of accuracy are obtained. The method termed sTDA ("s" for simplified) employs atom-centered Loewdin-monopole based two-electron repulsion integrals with the asymptotically correct 1/R behavior and perturbative single excitation configuration selection. It is formulated generally for any std. global hybrid d. functional with given Fock-exchange mixing parameter ax. The method performs well for two std. benchmark sets of vertical singlet-singlet excitations for values of ax in the range 0.2-0.6. The mean abs. deviations from ref. data are only 0.2-0.3 eV and similar to those from std. TD-DFT. In three cases (two dyes and one polypeptide), good mutual agreement between the electronic spectra (up to 10-11 eV excitation energy) from the sTDA method and those from TD(A)-DFT is obtained. The computed UV- and CD-spectra of a few typical systems (e.g., C60, two transition metal complexes, [7]helicene, polyalanine, a supramol. aggregate with 483 atoms and about 7000 basis functions) compare well with corresponding exptl. data. The method is proposed together with medium-sized double- or triple-zeta type at.-orbital basis sets as a quantum chem. tool to investigate the spectra of huge mol. systems at a reliable DFT level. (c) 2013 American Institute of Physics.
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Grimme, S.; Bannwarth, C. Ultra-fast computation of electronic spectra for large systems by tight-binding based simplified Tamm-Dancoff approximation (sTDA-xTB). J. Chem. Phys. 2016, 145, 054103, DOI: 10.1063/1.4959605
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Ultra-fast computation of electronic spectra for large systems by tight-binding based simplified Tamm-Dancoff approximation (sTDA-xTB)
Grimme, Stefan; Bannwarth, Christoph
Journal of Chemical Physics (2016), 145 (5), 054103/1-054103/20CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The computational bottleneck of the extremely fast simplified Tamm-Dancoff approximated (sTDA) time-dependent d. functional theory procedure [S. Grimme, J. Chem. Phys. 138, 244104 (2013)] for the computation of electronic spectra for large systems is the detn. of the ground state Kohn-Sham orbitals and eigenvalues. This limits such treatments to single structures with a few hundred atoms and hence, e.g., sampling along mol. dynamics trajectories for flexible systems or the calcn. of chromophore aggregates is often not possible. The aim of this work is to solve this problem by a specifically designed semiempirical tight binding (TB) procedure similar to the well established self-consistent-charge d. functional TB scheme. The new special purpose method provides orbitals and orbital energies of hybrid d. functional character for a subsequent and basically unmodified sTDA procedure. Compared to many previous semiempirical excited state methods, an advantage of the ansatz is that a general eigenvalue problem in a non-orthogonal, extended AO basis is solved and therefore correct occupied/virtual orbital energy splittings as well as Rydberg levels are obtained. A key idea for the success of the new model is that the detn. of at. charges (describing an effective electron-electron interaction) and the one-particle spectrum is decoupled and treated by two differently parametrized Hamiltonians/basis sets. The three-diagonalization-step composite procedure can routinely compute broad range electronic spectra (0-8 eV) within minutes of computation time for systems composed of 500-1000 atoms with an accuracy typical of std. time-dependent d. functional theory (0.3-0.5 eV av. error). An easily extendable parametrization based on coupled-cluster and d. functional computed ref. data for the elements H-Zn including transition metals is described. The accuracy of the method termed sTDA-xTB is first benchmarked for vertical excitation energies of open- and closed-shell systems in comparison to other semiempirical methods and applied to exemplary problems in electronic spectroscopy. As side products of the development, a robust and efficient valence electron TB method for the accurate detn. of at. charges as well as a more accurate calcn. scheme of dipole rotatory strengths within the Tamm-Dancoff approxn. is proposed. (c) 2016 American Institute of Physics.
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de Wergifosse, M.; Bannwarth, C.; Grimme, S. A Simplified Spin-Flip Time-Dependent Density Functional Theory Approach for the Electronic Excitation Spectra of Very Large Diradicals. J. Phys. Chem. A 2019, 123, 5815– 5825, DOI: 10.1021/acs.jpca.9b03176
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A Simplified Spin-Flip Time-Dependent Density Functional Theory Approach for the Electronic Excitation Spectra of Very Large Diradicals
de Wergifosse, Marc; Bannwarth, Christoph; Grimme, Stefan
Journal of Physical Chemistry A (2019), 123 (27), 5815-5825CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
Experimentalists working with diradicals are often facing the question of what kind of species among singlet or triplet diradicals or closed-shell mols. are obsd. To treat large diradicals with a high d. of electronic states, we propose a simplified version of the spin-flip time-dependent d. functional theory (SF-TD-DFT) method for a fast computation of their state energies and absorption spectra with an accuracy similar to the nonsimplified scheme. An ultrafast tight-binding variant called SF-sTD-DFT-xTB is also developed to treat even larger systems. For a benchmark set of nine diradicals, good agreement between simplified and conventional SF excitation energies for std. functionals is found. This shows that the proposed parameterization is robust for a wide range of Fock exchange mixing values. With the asymptotically correct response integrals used in SF-sTD-DFT and a correction factor of √(2) for the transition moments, the SF-sTD-DFT/B5050LYP/cc-pVDZ method even outperforms the nonsimplified scheme at drastically reduced computational effort when comparing to the exptl. absorption spectra for this set of diradicals. To showcase the actual performance of the method, absorption spectra of two μ-hydroxo-bridged dimers of corrole tape Ga(III) complex derivs. were computed and compared to the expt., providing good qual. agreement. Finally, a comparison with the high-spin triplet spectrum of a perylene bisimide biradical and the one detd. at the SF-sTD-DFT level showed that at room temp., mostly triplet diradicals are present in soln.
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De Wergifosse, M.; Grimme, S. Nonlinear-response properties in a simplified time-dependent density functional theory (sTD-DFT) framework: Evaluation of excited-state absorption spectra. J. Chem. Phys. 2019, 150, 094112, DOI: 10.1063/1.5080199
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Nonlinear-response properties in a simplified time-dependent density functional theory (sTD-DFT) framework: Evaluation of excited-state absorption spectra
de Wergifosse, Marc; Grimme, Stefan
Journal of Chemical Physics (2019), 150 (9), 094112/1-094112/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The energy conversion efficiency of org. solar cells seems crucial for a clean future. The design of new light-harvesting devices needs an in-depth understanding of their optical properties, including the excited-state absorption (ESA). In biol., the optical characterization of photochem./phys. processes happening in photosynthetic pigments and proteins can be difficult to interpret due to their structural complexities. Exptl., an ultrafast transient absorption expt. can probe the excited state interaction with light. Quantum chem. could play an important role to model the transient absorption spectrum of excited states. However, systems that need to be investigated can be way too large for existent software implementations. In this contribution, we present the first sTDA/sTD-DFT (simplified time-dependent d. functional theory with and without Tamm Dancoff approxn.) implementation to evaluate the ESA of mols. The ultrafast ESA evaluation presents a negligible extra cost with respect to sTDA/sTD-DFT original schemes for std. ground state absorption. The sTD-DFT method shows ability to assign ESA spectra to the correct excited state. We showed that in the literature, wrong assignments were proposed as for the L34/L44 mixt. and N-methylfulleropyrrolidine. In addn., sTDA/sTD-DFT-xTB tight-binding variants are also available, allowing the evaluation of ESA for systems of a few thousands of atoms, e.g., the spectrum of the photoactive yellow protein composed of 1931 atoms. (c) 2019 American Institute of Physics.
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De Wergifosse, M.; Grimme, S. Nonlinear-response properties in a simplified time-dependent density functional theory (sTD-DFT) framework: Evaluation of the first hyperpolarizability. J. Chem. Phys. 2018, 149, 024108, DOI: 10.1063/1.5037665
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Nonlinear-response properties in a simplified time-dependent density functional theory (sTD-DFT) framework: Evaluation of the first hyperpolarizability
de Wergifosse, Marc; Grimme, Stefan
Journal of Chemical Physics (2018), 149 (2), 024108/1-024108/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Recent developments in nonlinear imaging microscopy show the need to implement new theor. tools, which are able to characterize nonlinear optical properties in an efficient way. For second-harmonic imaging microscopy (SHIM), quantum chem. could play an important role to design new exogenous dyes with enhanced first hyperpolarizabilities or to characterize the response origin in large endogenous biol. systems. Such methods should be able to screen a large no. of compds. while reproducing their trends and to treat large systems in reasonable computation times. To fulfill these requirements, we present a new simplified time-dependent d. functional theory (sTD-DFT) implementation to evaluate the first hyperpolarizability where the Coulomb and exchange integrals are approximated by short-range damped Coulomb interactions of transition d. monopoles. For an ultra-fast computation of the first hyperpolarizability, a tight-binding version (sTD-DFT-xTB) is also proposed. In our implementation, a sTD-DFT calcn. is more than 600 time faster with respect to a regular TD-DFT treatment, while the xTB version speeds up the entire calcn. further by at least two orders of magnitude. We challenge our implementation on three test cases: typical push-pull π-conjugated compds., fluorescent proteins, and a collagen model, which were selected to model requirements for SHIM applications. (c) 2018 American Institute of Physics.
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Bannwarth, C.; Seibert, J.; Grimme, S. Electronic Circular Dichroism of [16]Helicene With Simplified TD-DFT: Beyond the Single Structure Approach. Chirality 2016, 28, 365– 369, DOI: 10.1002/chir.22594
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Electronic Circular Dichroism of [16]Helicene With Simplified TD-DFT: Beyond the Single Structure Approach
Bannwarth, Christoph; Seibert, Jakob; Grimme, Stefan
Chirality (2016), 28 (5), 365-369CODEN: CHRLEP; ISSN:0899-0042. (Wiley-Liss, Inc.)
The electronic CD (ECD) spectrum of the recently synthesized [16]helicene and a deriv. comprising two triisopropylsilyloxy protection groups was computed by means of the very efficient simplified time-dependent d. functional theory (sTD-DFT) approach. Different from many previous ECD studies of helicenes, nonequil. structure effects were accounted for by computing ECD spectra on "snapshots" obtained from a mol. dynamics (MD) simulation including solvent mols. The trajectories are based on a mol. specific classical potential as obtained from the recently developed quantum chem. derived force field (QMDFF) scheme. The reduced computational cost in the MD simulation due to the use of the QMDFF (compared to ab-initio MD) as well as the sTD-DFT approach make realistic spectral simulations feasible for these compds. that comprise more than 100 atoms. While the ECD spectra of [16]helicene and its deriv. computed vertically on the resp. gas phase, equil. geometries show noticeable differences, these are "washed" out when nonequil. structures are taken into account. The computed spectra with two recommended d. functionals (ωB97X and BHLYP) and extended basis sets compare very well with the exptl. one. In addn. we provide an est. for the missing abs. intensities of the latter. The approach presented here could also be used in future studies to capture nonequil. effects, but also to systematically av. ECD spectra over different conformations in more flexible mols. Chirality 00:000-000, 2016. 2016 Wiley Periodicals, Inc.
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Risthaus, T.; Hansen, A.; Grimme, S. Excited states using the simplified Tamm-Dancoff-Approach for range-separated hybrid density functionals: development and application. Phys. Chem. Chem. Phys. 2014, 16, 14408– 14419, DOI: 10.1039/C3CP54517B
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Excited states using the simplified Tamm-Dancoff-Approach for range-separated hybrid density functionals: development and application
Risthaus, Tobias; Hansen, Andreas; Grimme, Stefan
Physical Chemistry Chemical Physics (2014), 16 (28), 14408-14419CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
The recently introduced sTDA methodol. [S. Grimme, J. Chem. Phys., 2013, 138, 244104] to compute excitation spectra of huge mol. systems is extended to range-sepd. hybrid (RSH) d. functionals. The three empirical parameters of the method which describe a screened two-electron interaction are obtained for some common RSH functionals (ωB97 family, CAM-B3LYP, LC-BLYP) from a fit to theor. SCS-CC2 ref. vertical excitation energies for a set of small to medium-sized chromophores. The method is cross-validated on a set of inter- and intramol. charge transfer states and a set composed of typical valence transitions. Overall small deviations from ref. data of only about 0.2-0.4 eV are found with best performance for CAM-B3LYP and ωB97X-D3. To demonstrate versatility and robustness of the new methodol., applications (the UV/Vis spectrum of the pyridine polymer and the ECD spectrum of (P)-[11]helicene) and frequently used charge transfer examples are discussed. In one case, 11 000 + excited electronic states of a system contg. 330 atoms were calcd. We show that the asymptotically correct sTDA-RSH combination yields results often superior to those based on global hybrids and that it opens up new possibilities for the computation of excited states in materials science and bio-mol. systems.
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Elstner, M.; Porezag, D.; Jungnickel, G.; Elsner, J.; Haugk, M.; Frauenheim, T.; Suhai, S.; Seifert, G. Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties. Phys. Rev. B 1998, 58, 7260– 7268, DOI: 10.1103/PhysRevB.58.7260
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Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties
Elstner, M.; Porezag, D.; Jungnickel, G.; Elsner, J.; Haugk, M.; Frauenheim, Th.; Suhai, S.; Seifert, G.
Physical Review B: Condensed Matter and Materials Physics (1998), 58 (11), 7260-7268CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
We outline details about an extension of the tight-binding (TB) approach to improve total energies, forces, and transferability. The method is based on a second-order expansion of the Kohn-Sham total energy in d.-functional theory (DFT) with respect to charge-d. fluctuations. The zeroth-order approach is equiv. to a common std. non-self-consistent (TB) scheme, while at second-order a transparent, parameter-free, and readily calculable expression for generalized Hamiltonian matrix elements may be derived. These are modified by a self-consistent redistribution of Mulliken charges (SCC). Besides the usual "band structure" and short-range repulsive terms the final approx. Kohn-Sham energy addnl. includes a Coulomb interaction between charge fluctuations. At large distances this accounts for long-range electrostatic forces between two point charges and approx. includes self-interaction contributions of a given atom if the charges are located at one and the same atom. We apply the new SCC scheme to problems where deficiencies within the non-SCC std. TB approach become obvious. We thus considerably improve transferability.
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Seibert, J.; Pisarek, J.; Schmitz, S.; Bannwarth, C.; Grimme, S. Extension of the element parameter set for ultra-fast excitation spectra calculation (sTDA-xTB). Mol. Phys. 2019, 117, 1104– 1116, DOI: 10.1080/00268976.2018.1510141
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Extension of the element parameter set for ultra-fast excitation spectra calculation (sTDA-xTB)
Seibert, Jakob; Pisarek, Jana; Schmitz, Sarah; Bannwarth, Christoph; Grimme, Stefan
Molecular Physics (2019), 117 (9-12), 1104-1116CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
The extension of the parameter set for an ultra-fast electronic excitation spectra calcn. is presented. The semiempirical theory based on a tight-binding approach, called extended tight-binding (xTB) in combination with the simplified Tamm-Dancoff approxn. (sTDA) shows remarkable accuracy at very low computational cost for the calcn. of vertical excitation energies of mols. It enables the possibility for computing even large systems up to thousands of atoms or sampling along mol. dynamic (MD) trajectories. The original publication of the sTDA-xTB method included parameters for the most important elements (H-Zn,Br,I). In this work, element parameters for 4d and 5d metals, and the missing ones in 4p, 5p and 6p element blocks are presented and analyzed for their quality. Comparisons to theory and expt. show that sTDA-xTB provides similar good results as for the elements in the original publication with an av. deviation of excitation energies of 0.3-0.5 eV.
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Cho, Y.; Bintrim, S. J.; Berkelbach, T. C. A simplified GW/BSE approach for charged and neutral excitation energies of large molecules and nanomaterials ; 2021, arXiv:2109.04421. DOI: 10.48550/arXiv.2109.04421 .
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Rüger, R.; van Lenthe, E.; Heine, T.; Visscher, L. Tight-binding approximations to time-dependent density functional theory ─ A fast approach for the calculation of electronically excited states. J. Chem. Phys. 2016, 144, 184103, DOI: 10.1063/1.4948647
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Tight-binding approximations to time-dependent density functional theory - A fast approach for the calculation of electronically excited states
Rueger, Robert; van Lenthe, Erik; Heine, Thomas; Visscher, Lucas
Journal of Chemical Physics (2016), 144 (18), 184103/1-184103/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We propose a new method of calcg. electronically excited states that combines a d. functional theory based ground state calcn. with a linear response treatment that employs approxns. used in the time-dependent d. functional based tight binding (TD-DFTB) approach. The new method termed time-dependent d. functional theory TD-DFT+TB does not rely on the DFTB parametrization and is therefore applicable to systems involving all combinations of elements. We show that the new method yields UV/Vis absorption spectra that are in excellent agreement with computationally much more expensive TD-DFT calcns. Errors in vertical excitation energies are reduced by a factor of two compared to TD-DFTB. (c) 2016 American Institute of Physics.
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Asadi-Aghbolaghi, N.; Pototschnig, J.; Jamshidi, Z.; Visscher, L. Effects of ligands on (de-)enhancement of plasmonic excitations of silver, gold and bimetallic nanoclusters: TD-DFT+TB calculations. Phys. Chem. Chem. Phys. 2021, 23, 17929– 17938, DOI: 10.1039/D1CP03220H
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Effects of ligands on (de-)enhancement of plasmonic excitations of silver, gold and bimetallic nanoclusters: TD-DFT+TB calculations
Asadi-Aghbolaghi, Narges; Pototschnig, Johann; Jamshidi, Zahra; Visscher, Lucas
Physical Chemistry Chemical Physics (2021), 23 (33), 17929-17938CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
Metal nanoclusters can be synthesized in various sizes and shapes and are typically protected with ligands to stabilize them. These ligands can also be used to tune the plasmonic properties of the clusters as the absorption spectrum of a protected cluster can be significantly altered compared to the bare cluster. In this paper, we computationally investigate the influence of thiolate ligands on the plasmonic intensity for silver, gold and alloy clusters. Using time-dependent d. functional theory with tight-binding approxns., TD-DFT+TB, we show that this level of theory can reproduce the broad exptl. spectra of Au144(SR)60 and Ag53Au91(SR)60 (R = CH3) compds. with satisfactory agreement. As TD-DFT+TB does not depend on atom-type parameters we were able to apply this approach on large ligand-protected clusters with various compns. With these calcns. we predict that the effect of ligands on the absorption can be a quenching as well as an enhancement. We furthermore show that it is possible to unambiguously identify the plasmonic peaks by the scaled Coulomb kernel technique and explain the influence of ligands on the intensity (de-)enhancement by analyzing the plasmonic excitations in terms of the dominant orbital contributions.
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Van Caillie, C.; Amos, R. D. Geometric derivatives of excitation energies using SCF and DFT. Chem. Phys. Lett. 1999, 308, 249– 255, DOI: 10.1016/S0009-2614(99)00646-6
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Geometric derivatives of excitation energies using SCF and DFT
Van Caillie, Carole; Amos, Roger D.
Chemical Physics Letters (1999), 308 (3,4), 249-255CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)
There is increasing interest in using the methods of time-dependent d. functional theory to calc. electronic excitation energies. We have implemented an analytic gradient method to find the geometric derivs. of the excitation energies. When added to the gradient for the ground state, this yields the excited-state energy derivs. This enables the efficient generation and searching of excited-state potential energy surfaces to obtain excited-state geometries and other properties. The initial implementation is for SCF methods and for the local d. approxn. Some examples of excited-state geometry optimizations are given.
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Van Caillie, C.; Amos, R. D. Geometric derivatives of density functional theory excitation energies using gradient-corrected functionals. Chem. Phys. Lett. 2000, 317, 159– 164, DOI: 10.1016/S0009-2614(99)01346-9
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Geometric derivatives of density functional theory excitation energies using gradient-corrected functionals
Van Caillie, C.; Amos, R. D.
Chemical Physics Letters (2000), 317 (1,2), 159-164CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)
D. functional theory (DFT) is having increasing success in predicting excitation energies using the methods of time-dependent DFT. As a result, it should be possible to generate potential energy surfaces for excited states by adding the excitation energy, as a function of geometry, to the ground-state energy. It is easier to find stationary points such as min. and transition states if the gradient of the energy is known. The present Letter extends earlier work on the gradients on excited-state surfaces using SCF and LDA (local d. approxn.) methods, to use gradient-cor. and hybrid functionals. Some examples of geometry optimizations are given.
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Furche, F.; Ahlrichs, R. Adiabatic time-dependent density functional methods for excited state properties. J. Chem. Phys. 2002, 117, 7433– 7447, DOI: 10.1063/1.1508368
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Adiabatic time-dependent density functional methods for excited state properties
Furche, Filipp; Ahlrichs, Reinhart
Journal of Chemical Physics (2002), 117 (16), 7433-7447CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
This work presents theory, implementation, and validation of excited state properties obtained from time-dependent d. functional theory (TDDFT). Based on a fully variational expression for the excited state energy, a compact derivation of first order properties is given. We report an implementation of analytic excited state gradients and charge moments for local, gradient cor., and hybrid functionals, as well as for the CI singles (CIS) and time-dependent Hartree-Fock (TDHF) methods. By exploiting analogies to ground state energy and gradient calcns., efficient techniques can be transferred to excited state methods. Benchmark results demonstrate that, for low-lying excited states, geometry optimizations are not substantially more expensive than for the ground state, independent of the mol. size. We assess the quality of calcd. adiabatic excitation energies, structures, dipole moments, and vibrational frequencies by comparison with accurate exptl. data for a variety of excited states and mols. Similar trends are obsd. for adiabatic excitation energies as for vertical ones. TDDFT is more robust than CIS and TDHF, in particular, for geometries differing significantly from the ground state min. The TDDFT excited state structures, dipole moments, and vibrational frequencies are of a remarkably high quality, which is comparable to that obtained in ground state d. functional calcns. Thus, yielding considerably more accurate results at similar computational cost, TDDFT rivals CIS as a std. method for calcg. excited state properties in larger mols.
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Furche, F.; Ahlrichs, R. Erratum: ”Time-dependent density functional methods for excited state properties” [J. Chem. Phys. 117, 7433 (2002)]. J. Chem. Phys. 2004, 121, 12772– 12773, DOI: 10.1063/1.1824903
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Adiabatic time-dependent density functional methods for excited state properties. [Erratum to document cited in CA138:044999]
Furche, Filipp; Ahlrichs, Reinhart
Journal of Chemical Physics (2004), 121 (24), 12772-12773CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
A factor 1/2 is missing in the expression for the diagonal elements of the energy weighted difference d. matrix W throughout the paper. The cor. versions of Equations (24), (A4), (A10), and (A13) and Tables V, VI, and VIII are given. The implementation and results presented in the paper were based on the correct expressions. Some of the calcd. spectroscopic consts. of the 1 1.sum.u- state of N2, of the 1 1.sum.u+ state of Mg2, and of the 2 1.sum.+ state of CuH in Tables V, VI, and VII are incorrect and should be replaced by the values in the cor. equations. The conclusions are not affected by these changes.
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Hutter, J. Excited state nuclear forces from the Tamm-Dancoff approximation to time-dependent density functional theory within the plane wave basis set framework. J. Chem. Phys. 2003, 118, 3928– 3934, DOI: 10.1063/1.1540109
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Excited state nuclear forces from the Tamm-Dancoff approximation to time-dependent density functional theory within the plane wave basis set framework
Hutter, Jurg
Journal of Chemical Physics (2003), 118 (9), 3928-3934CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
An efficient formulation of time-dependent linear response d. functional theory for the use within the plane wave basis set framework is presented. The method avoids the transformation of the Kohn-Sham matrix into the canonical basis and refs. virtual orbitals only through a projection operator. Using a Lagrangian formulation nuclear derivs. of excited state energies within the Tamm-Dancoff approxn. are derived. The algorithms were implemented into a pseudo potential/plane wave code and applied to the calcn. of adiabatic excitation energies, optimized geometries and vibrational frequencies of three low lying states of formaldehyde. An overall good agreement with other time-dependent d. functional calcns., multireference CI calcns. and exptl. data was found.
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Petrenko, T.; Kossmann, S.; Neese, F. Efficient time-dependent density functional theory approximations for hybrid density functionals: Analytical gradients and parallelization. J. Chem. Phys. 2011, 134, 054116, DOI: 10.1063/1.3533441
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Efficient time-dependent density functional theory approximations for hybrid density functionals: Analytical gradients and parallelization
Petrenko, Taras; Kossmann, Simone; Neese, Frank
Journal of Chemical Physics (2011), 134 (5), 054116/1-054116/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We present the implementation of efficient approxns. to time-dependent d. functional theory (TDDFT) within the Tamm-Dancoff approxn. (TDA) for hybrid d. functionals. For the calcn. of the TDDFT/TDA excitation energies and anal. gradients, we combine the resoln. of identity (RI-J) algorithm for the computation of the Coulomb terms and the recently introduced "chain of spheres exchange" (COSX) algorithm for the calcn. of the exchange terms. It is shown that for extended basis sets, the RIJCOSX approxn. leads to speedups of up to 2 orders of magnitude compared to traditional methods, as demonstrated for hydrocarbon chains. The accuracy of the adiabatic transition energies, excited state structures, and vibrational frequencies is assessed on a set of 27 excited states for 25 mols. with the CI singles and hybrid TDDFT/TDA methods using various basis sets. Compared to the canonical values, the typical error in transition energies is of the order of 0.01 eV. Similar to the ground-state results, excited state equil. geometries differ by less than 0.3 pm in the bond distances and 0.5° in the bond angles from the canonical values. The typical error in the calcd. excited state normal coordinate displacements is of the order of 0.01, and relative error in the calcd. excited state vibrational frequencies is less than 1%. The errors introduced by the RIJCOSX approxn. are, thus, insignificant compared to the errors related to the approx. nature of the TDDFT methods and basis set truncation. For TDDFT/TDA energy and gradient calcns. on Ag-TB2-helicate (156 atoms, 2732 basis functions), it is demonstrated that the COSX algorithm parallelizes almost perfectly (speedup ∼26-29 for 30 processors). The exchange-correlation terms also parallelize well (speedup ∼27-29 for 30 processors). The soln. of the Z-vector equations shows a speedup of ∼24 on 30 processors. The parallelization efficiency for the Coulomb terms can be somewhat smaller (speedup ∼15-25 for 30 processors), but their contribution to the total calcn. time is small. Thus, the parallel program completes a Becke3-Lee-Yang-Parr energy and gradient calcn. on the Ag-TB2-helicate in less than 4 h on 30 processors. We also present the necessary extension of the Lagrangian formalism, which enables the calcn. of the TDDFT excited state properties in the frozen-core approxn. The algorithms described in this work are implemented into the ORCA electronic structure system. (c) 2011 American Institute of Physics.
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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.9b00659
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Development and Implementation of Excited-State Gradients for Local Hybrid Functionals
Grotjahn, Robin; Furche, Filipp; Kaupp, Martin
Journal 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.
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Kretz, B.; Egger, D. A. Accurate Molecular Geometries in Complex Excited-State Potential Energy Surfaces from Time-Dependent Density Functional Theory. J. Chem. Theory Comput. 2021, 17, 357– 366, DOI: 10.1021/acs.jctc.0c00858
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Accurate Molecular Geometries in Complex Excited-State Potential Energy Surfaces from Time-Dependent Density Functional Theory
Kretz, Bernhard; Egger, David A.
Journal of Chemical Theory and Computation (2021), 17 (1), 357-366CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
The interplay of electronic excitations and structural changes in mols. impacts nonradiative decay and charge transfer in the excited state, thus influencing excited-state lifetimes and photocatalytic reaction rates in optoelectronic and energy devices. To capture such effects requires computational methods providing an accurate description of excited-state potential energy surfaces and geometries. We suggest time-dependent d. functional theory using optimally tuned range-sepd. hybrid (OT-RSH) functionals as an accurate approach to obtain excited-state mol. geometries. We show that OT-RSH provides accurate mol. geometries in excited-state potential energy surfaces that are complex and involve an interplay of local and charge-transfer excitations, for which conventional semilocal and hybrid functionals fail. At the same time, the nonempirical OT-RSH approach maintains the high accuracy of parametrized functionals (e.g., B3LYP) for predicting excited-state geometries of small org. mols. showing valence excited states.
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Sokolov, M.; Bold, B. M.; Kranz, J. J.; Höfener, S.; Niehaus, T. A.; Elstner, M. Analytical Time-Dependent Long-Range Corrected Density Functional Tight Binding (TD-LC-DFTB) Gradients in DFTB+: Implementation and Benchmark for Excited-State Geometries and Transition Energies. J. Chem. Theory Comput. 2021, 17, 2266– 2282, DOI: 10.1021/acs.jctc.1c00095
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Analytical Time-Dependent Long-Range Corrected Density Functional Tight Binding (TD-LC-DFTB) Gradients in DFTB+: Implementation and Benchmark for Excited-State Geometries and Transition Energies
Sokolov, Monja; Bold, Beatrix M.; Kranz, Julian J.; Hoefener, Sebastian; Niehaus, Thomas A.; Elstner, Marcus
Journal of Chemical Theory and Computation (2021), 17 (4), 2266-2282CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
The absorption and emission of light is a ubiquitous process in chem. and biol. processes, making a theor. description inevitable for understanding and predicting such properties. Although ab initio and DFT methods are capable of describing excited states with good accuracy in many cases, the investigation of dynamical processes and the need to sample the phase space in complex systems often requires methods with reduced computational costs but still sufficient accuracy. In the present work, we report the derivation and implementation of anal. nuclear gradients for time-dependent long-range cor. d. functional tight binding (TD-LC-DFTB) in the DFTB+ program. The accuracy of the TD-LC-DFTB potential-energy surfaces is benchmarked for excited-state geometries and adiabatic as well as vertical transition energies. The benchmark set consists of more than 100 org. mols. taken as subsets from available benchmark sets. The reported method yields a mean deviation of 0.31 eV for adiabatic excitation energies with respect to CC2. In order to study more subtle effects, seminumerical second derivs. based on the anal. gradients are employed to simulate vibrationally resolved UV/vis spectra. This extensive test exhibits few problematic cases, which can be traced back to the parametrization of the repulsive potential.
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Kühne, T. D.; Iannuzzi, M.; Del Ben, M.; Rybkin, V. V.; Seewald, P.; Stein, F.; Laino, T.; Khaliullin, R. Z.; Schütt, O.; Schiffmann, F.; Golze, D.; Wilhelm, J.; Chulkov, S.; Bani-Hashemian, M. H.; Weber, V.; Borštnik, U.; Taillefumier, M.; Jakobovits, A. S.; Lazzaro, A.; Pabst, H.; Müller, T.; Schade, R.; Guidon, M.; Andermatt, S.; Holmberg, N.; Schenter, G. K.; Hehn, A.; Bussy, A.; Belleflamme, F.; Tabacchi, G.; Glöß, A.; Lass, M.; Bethune, I.; Mundy, C. J.; Plessl, C.; Watkins, M.; VandeVondele, J.; Krack, M.; Hutter, J. CP2K: An electronic structure and molecular dynamics software package - Quickstep: Efficient and accurate electronic structure calculations. J. Chem. Phys. 2020, 152, 194103, DOI: 10.1063/5.0007045
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CP2K: An electronic structure and molecular dynamics software package - Quickstep: Efficient and accurate electronic structure calculations
Kuehne, Thomas D.; Iannuzzi, Marcella; Del Ben, Mauro; Rybkin, Vladimir V.; Seewald, Patrick; Stein, Frederick; Laino, Teodoro; Khaliullin, Rustam Z.; Schuett, Ole; Schiffmann, Florian; Golze, Dorothea; Wilhelm, Jan; Chulkov, Sergey; Bani-Hashemian, Mohammad Hossein; Weber, Valery; Borstnik, Urban; Taillefumier, Mathieu; Jakobovits, Alice Shoshana; Lazzaro, Alfio; Pabst, Hans; Mueller, Tiziano; Schade, Robert; Guidon, Manuel; Andermatt, Samuel; Holmberg, Nico; Schenter, Gregory K.; Hehn, Anna; Bussy, Augustin; Belleflamme, Fabian; Tabacchi, Gloria; Gloess, Andreas; Lass, Michael; Bethune, Iain; Mundy, Christopher J.; Plessl, Christian; Watkins, Matt; VandeVondele, Joost; Krack, Matthias; Hutter, Juerg
Journal of Chemical Physics (2020), 152 (19), 194103CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
A review. CP2K is an open source electronic structure and mol. dynamics software package to perform atomistic simulations of solid-state, liq., mol., and biol. systems. It is esp. aimed at massively parallel and linear-scaling electronic structure methods and state-of-the-art ab initio mol. dynamics simulations. Excellent performance for electronic structure calcns. is achieved using novel algorithms implemented for modern high-performance computing systems. This review revisits the main capabilities of CP2K to perform efficient and accurate electronic structure simulations. The emphasis is put on d. functional theory and multiple post-Hartree-Fock methods using the Gaussian and plane wave approach and its augmented all-electron extension. (c) 2020 American Institute of Physics.
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VandeVondele, J.; Krack, M.; Mohamed, F.; Parrinello, M.; Chassaing, T.; Hutter, J. Quickstep: Fast and accurate density functional calculations using a mixed Gaussian and plane waves approach. Comput. Phys. Commun. 2005, 167, 103– 128, DOI: 10.1016/j.cpc.2004.12.014
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QUICKSTEP: fast and accurate density functional calculations using a mixed Gaussian and plane waves approach
VandeVondele, Joost; Krack, Matthias; Mohamed, Fawzi; Parrinello, Michele; Chassaing, Thomas; Hutter, Juerg
Computer Physics Communications (2005), 167 (2), 103-128CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)
We present the Gaussian and plane waves (GPW) method and its implementation in which is part of the freely available program package CP2K. The GPW method allows for accurate d. functional calcns. in gas and condensed phases and can be effectively used for mol. dynamics simulations. We show how derivs. of the GPW energy functional, namely ionic forces and the Kohn-Sham matrix, can be computed in a consistent way. The computational cost of computing the total energy and the Kohn-Sham matrix is scaling linearly with the system size, even for condensed phase systems of just a few tens of atoms. The efficiency of the method allows for the use of large Gaussian basis sets for systems up to 3000 atoms, and we illustrate the accuracy of the method for various basis sets in gas and condensed phases. Agreement with basis set free calcns. for single mols. and plane wave based calcns. in the condensed phase is excellent. Wave function optimization with the orbital transformation technique leads to good parallel performance, and outperforms traditional diagonalisation methods. Energy conserving Born-Oppenheimer dynamics can be performed, and a highly efficient scheme is obtained using an extrapolation of the d. matrix. We illustrate these findings with calcns. using commodity PCs as well as supercomputers.
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Lippert, G.; Hutter, J.; Parrinello, M. A hybrid Gaussian and plane wave density functional scheme. Mol. Phys. 1997, 92, 477– 488, DOI: 10.1080/00268979709482119
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A hybrid Gaussian and plane wave density functional scheme
Lippert, Gerald; Hutter, Juerg; Parrinello, Michele
Molecular Physics (1997), 92 (3), 477-487CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis)
A d.-functional theory-based algorithm for periodic and nonperiodic ab initio calcns. is presented. This scheme uses pseudopotentials in order to integrate out the core electrons from the problem. The valence pseudo wave functions are expanded in Gaussian-type orbitals and the d. is represented in a plane wave auxiliary basis. The Gaussian basis functions make it possible to use the efficient anal. integration schemes and screening algorithms of quantum chem. Novel recursion relations are developed for the calcn. of the matrix elements of the d.-dependent Kohn-Sham self-consistent potential. At the same time the use of a plane wave basis for the electron d. permits efficient calcn. of the Hartree energy using fast Fourier transforms, thus circumventing one of the major bottlenecks of std. Gaussian based calcns. Furthermore, this algorithm avoids the fitting procedures that go along with intermediate basis sets for the charge d. The performance and accuracy of this new scheme are discussed and selected examples are given.
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Iannuzzi, M.; Chassaing, T.; Wallman, T.; Hutter, J. Ground and Excited State Density Functional Calculations with the Gaussian and Augmented-Plane-Wave Method. CHIMIA Int. J. Chem. 2005, 59, 499– 503, DOI: 10.2533/000942905777676164
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Ground and excited state density functional calculations with the Gaussian and Augmented-Plane-Wave method
Iannuzzi, Marcella; Chassaing, Thomas; Wallman, Thomas; Hutter, Jurg
Chimia (2005), 59 (7-8), 499-503CODEN: CHIMAD; ISSN:0009-4293. (Swiss Chemical Society)
The calcn. of the electronic structure of large systems by methods based on d. functional theory has recently gained a central role in mol. simulations. However, the extensive study of quantities like excited states and related properties is still out of reach due to high computational costs. We present a new implementation of a hybrid method, the Gaussian and APW (GAPW) method, where the electronic d. is partitioned in hard and soft contributions. The former are local terms naturally expanded in a Gaussian basis, whereas the soft contributions are expanded in plane-waves by using a low energy cutoff, without loss in accuracy, even for all-electron calcns. For the calcn. of excitation energies a recently developed, time-dependent d. functional response theory (TD-DFRT) technique is joined with the GAPW procedure. We demonstrate the accuracy of the method by comparison with std. quantum chem. calcns. for a set of small mols. To highlight the performance and efficiency of GAPW we show calcns. on systems with several thousands of basis functions.
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Strand, J.; Chulkov, S. K.; Watkins, M. B.; Shluger, A. L. First principles calculations of optical properties for oxygen vacancies in binary metal oxides. J. Chem. Phys. 2019, 150, 044702, DOI: 10.1063/1.5078682
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First principles calculations of optical properties for oxygen vacancies in binary metal oxides
Strand, Jack; Chulkov, Sergey K.; Watkins, Matthew B.; Shluger, Alexander L.
Journal of Chemical Physics (2019), 150 (4), 044702/1-044702/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Using an advanced computational methodol. implemented in CP2K, a non-local PBE0-TC-LRC d. functional and the recently implemented linear response formulation of the Time-dependent D. Functional Theory equations, we test the interpretation of the optical absorption and photoluminescence signatures attributed by previous exptl. and theor. studies to O-vacancies in two widely used oxides-cubic MgO and monoclinic (m)-HfO2. The results obtained in large periodic cells including up to 1000 atoms emphasize the importance of accurate predictions of defect-induced lattice distortions. They confirm that optical transitions of O-vacancies in 0, +1, and +2 charge states in MgO all have energies close to 5 eV. We test the models of photoluminescence of O-vacancies proposed in the literature. The photoluminescence of V+2O centers in m-HfO2 is predicted to peak at 3.7 eV and originate from radiative tunneling transition between a V+1O center and a self-trapped hole created by the 5.2 eV excitation. (c) 2019 American Institute of Physics.
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Poli, E.; Elliott, J. D.; Chulkov, S. K.; Watkins, M. B.; Teobaldi, G. The Role of Cation-Vacancies for the Electronic and Optical Properties of Aluminosilicate Imogolite Nanotubes: A Non-local, Linear-Response TDDFT Study. Front. Chem. 2019, 7, 210, DOI: 10.3389/fchem.2019.00210
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Cation-vacancies for the electronic and optical properties of aluminosilicate imogolite nanotubes and a non-local, linear-response TDDFT study
Poli, Emiliano; Elliott, Joshua D.; Chulkov, Sergey K.; Watkins, Matthew B.; Teobaldi, Gilberto
Frontiers in Chemistry (Lausanne, Switzerland) (2019), 7 (), 210CODEN: FCLSAA; ISSN:2296-2646. (Frontiers Media S.A.)
We report a combined non-local (PBE-TC-LRC) D. Functional Theory (DFT) and linear-response time-dependent DFT (LR-TDDFT) study of the structural, electronic, and optical properties of the cation-vacancy based defects in aluminosilicate (AlSi) imogolite nanotubes (Imo-NTs) that have been recently proposed on the basis of NMR (NMR) expts. Following numerical detn. of the smallest AlSi Imo-NT model capable of accommodating the defect-induced relaxation with negligible finite-size errors, we analyze the defect-induced structural deformations in the NTs and ensuing changes in the NTs' electronic structure. The NMR-derived defects are found to introduce both shallow and deep occupied states in the pristine NTs' band gap (BG). These BG states are found to be highly localized at the defect site. No empty defect-state is modeled for any of the considered systems. LR-TDDFT simulation of the defects reveal increased low-energy optical absorbance for all but one defects, with the appearance of optically active excitations at energies lower than for the defect-free NT. These results enable interpretation of the low-energy tail in the exptl. UV-vis spectra for AlSi NTs as being due to the defects. Finally, the PBE-TC-LRC-approximated exciton binding energy for the defects' optical transitions is found to be substantially lower (up to 0.8 eV) than for the pristine defect-free NT's excitations (1.1 eV).
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Sternheimer, R. On Nuclear Quadrupole Moments. Phys. Rev. 1951, 84, 244– 253, DOI: 10.1103/PhysRev.84.244
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Nuclear quadrupole moments
Sternheimer, R.
Physical Review (1951), 84 (), 244-53CODEN: PHRVAO; ISSN:0031-899X.
cf. C.A. 45, 443i. The correction to nuclear quadrupole moments on account of the quadrupole moment induced in the electron shells was obtained by solving the Schroedinger equation for the perturbed core wave functions for Li, Al, and Cl. The correction factor by which the average 〈l/r3〉 over the valence electron function in the equation for the quadrupole coupling should be multiplied to take account of the induced effect is 1.11, 0.83, and 0.68, resp. The previously described Thomas-Fermi calcn. of this effect was carried out for F, Cl, Cu, Br, Y, Ag, I, La, Lu, Pt, Tl, At, and U.
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Baroni, S.; de Gironcoli, S.; Dal Corso, A.; Giannozzi, P. Phonons and related crystal properties from density-functional perturbation theory. Rev. Mod. Phys. 2001, 73, 515– 562, DOI: 10.1103/RevModPhys.73.515
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Phonons and related crystal properties from density-functional perturbation theory
Baroni, Stefano; De Gironcoli, Stefano; Dal Corso, Andrea; Giannozzi, Paolo
Reviews of Modern Physics (2001), 73 (2), 515-562CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)
This article reviews with many refs. the current status of lattice-dynamical calcns. in crystals, using d.-functional perturbation theory, with emphasis on the plane-wave pseudopotential method. Several specialized topics are treated, including the implementation for metals, the calcn. of the response to macroscopic elec. fields and their relevance to long-wavelength vibrations in polar materials, the response to strain deformations, and higher-order responses. The success of this methodol. is demonstrated with a no. of applications existing in the literature.
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Sternheimer, R. M. Electronic Polarizabilities of Ions from the Hartree-Fock Wave Functions. Phys. Rev. 1954, 96, 951– 968, DOI: 10.1103/PhysRev.96.951
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Electron polarizabilities of ions from the Hartree-Fock wave functions
Sternheimer, R. M.
Physical Review (1954), 96 (), 951-68CODEN: PHRVAO; ISSN:0031-899X.
cf. ibid. 95, 655. The electronic polarizability α was calcd. for several ions by obtaining the perturbation of the wave functions by an external field from a numerical solution of the differential equation satisfied by the perturbation. For the He-like ions an analytic solution was obtained by using the wave functions of L.ovrddot.owdin (C.A. 47, 9135c). The calcd. values of α are, in general, 1 to 1.5 times the observed values. For several ions values were calcd. for the quadrupole polarizability, which measures the quadrupole moment induced in the ion by an external charge. The effect of the dipole moment induced in the ion on the elec. field at the nucleus is discussed.
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Budzak, S.; Scalmani, G.; Jacquemin, D. Accurate Excited-State Geometries: A CASPT2 and Coupled-Cluster Reference Database for Small Molecules. J. Chem. Theory Comput. 2017, 13, 6237– 6252, DOI: 10.1021/acs.jctc.7b00921
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Accurate Excited-State Geometries: A CASPT2 and Coupled-Cluster Reference Database for Small Molecules
Budzak, Simon; Scalmani, Giovanni; Jacquemin, Denis
Journal of Chemical Theory and Computation (2017), 13 (12), 6237-6252CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
We present an investigation of the excited-state structural parameters detd. for a large set of small compds. with the dual goals of defining ref. values for further works and assessing the quality of the geometries obtained with relatively cheap computational approaches. In the first stage, we compare the excited-state geometries obtained with ADC(2), CC2, CCSD, CCSDR(3), CC3, and CASPT2 and large at. basis sets. It is found that CASPT2 and CC3 results are generally in very good agreement with one another (typical differences of ca. 3 × 10-3 Å) when all electrons are correlated and when the aug-cc-pVTZ at. basis set is employed with both methods. In a second stage, a statistical anal. reveals that, on the one hand, the excited-state (ES) bond lengths are much more sensitive to the selected level of theory than their ground-state (GS) counterparts and, on the other hand, that CCSDR(3) is probably the most cost-effective method delivering accurate structures. Indeed, CCSD tends to provide too compact multiple bond lengths on an almost systematic basis, whereas both CC2 and ADC(2) tend to exaggerate these bond distances, with more erratic error patterns, esp. for the latter method. The deviations are particularly marked for the polarized CO and CN bonds, as well as for the puckering angle in formaldehyde homologues. In the last part of this contribution, we provide a series of CCSDR(3) GS and ES geometries of medium-sized mols. to be used as refs. in further investigations.
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Ongari, D.; Yakutovich, A. V.; Talirz, L.; Smit, B. Building a Consistent and Reproducible Database for Adsorption Evaluation in Covalent–Organic Frameworks. ACS Cent. Sci. 2019, 5, 1663– 1675, DOI: 10.1021/acscentsci.9b00619
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Building a Consistent and Reproducible Database for Adsorption Evaluation in Covalent-Organic Frameworks
Ongari, Daniele; Yakutovich, Aliaksandr V.; Talirz, Leopold; Smit, Berend
ACS Central Science (2019), 5 (10), 1663-1675CODEN: ACSCII; ISSN:2374-7951. (American Chemical Society)
We present a workflow that traces the path from the bulk structure of a cryst. material to assessing its performance in carbon capture from coal's postcombustion flue gases. This workflow is applied to a database of 324 covalent-org. frameworks (COFs) reported in the literature, to characterize their CO2 adsorption properties using the following steps: (1) optimization of the crystal structure (at. positions and unit cell) using d. functional theory, (2) fitting at. point charges based on the electron d., (3) characterizing the pore geometry of the structures before and after optimization, (4) computing carbon dioxide and nitrogen isotherms using grand canonical Monte Carlo simulations with an empirical interaction potential, and finally, (5) assessing the CO2 parasitic energy via process modeling. The full workflow has been encoded in the Automated Interactive Infrastructure and Database for Computational Science (AiiDA). Both the workflow and the automatically generated provenance graph of our calcns. are made available on the Materials Cloud, allowing peers to inspect every input parameter and result along the workflow, download structures and files at intermediate stages, and start their research right from where this work has left off. In particular, our set of CURATED (Clean, Uniform, and Refined with Automatic Tracking from Exptl. Database) COFs, having optimized geometry and high-quality DFT-derived point charges, are available for further investigations of gas adsorption properties. We plan to update the database as new COFs are being reported. An automated and reproducible computational workflow is proposed, to systematically optimize the geometry of covalent-org. frameworks and evaluate their performances for carbon capture and storage.
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Ongari, D.; Talirz, L.; Smit, B. Too Many Materials and Too Many Applications: An Experimental Problem Waiting for a Computational Solution. ACS Cent. Sci. 2020, 6, 1890– 1900, DOI: 10.1021/acscentsci.0c00988
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Too Many Materials and Too Many Applications: An Experimental Problem Waiting for a Computational Solution
Ongari, Daniele; Talirz, Leopold; Smit, Berend
ACS Central Science (2020), 6 (11), 1890-1900CODEN: ACSCII; ISSN:2374-7951. (American Chemical Society)
A review. Finding the best material for a specific application is the ultimate goal of materials discovery. However, there is also the reverse problem: when exptl. groups discover a new material, they would like to know all the possible applications this material would be promising for. Computational modeling can aim to fulfill this expectation, thanks to the sustained growth of computing power and the collective engagement of the scientific community in developing more efficient and accurate workflows for predicting materials' performances. We discuss the impact that reproducibility and automation of the modeling protocols have on the field of gas adsorption in nanoporous crystals. We envision a platform that combines these tools and enables effective matching between promising materials and industrial applications. We identify the opportunity for a computational platform for matching nanoporous materials and gas-related applications, motivating the development of automated and reproducible computational workflows.
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Hirata, S.; Head-Gordon, M. Time-dependent density functional theory within the Tamm-Dancoff approximation. Chem. Phys. Lett. 1999, 314, 291– 299, DOI: 10.1016/S0009-2614(99)01149-5
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Time-dependent density functional theory within the Tamm-Dancoff approximation
Hirata, S.; Head-Gordon, M.
Chemical Physics Letters (1999), 314 (3,4), 291-299CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)
A computationally simple method for mol. excited states, namely, the Tamm-Dancoff approxn. to time-dependent d. functional theory, is proposed and implemented. This method yields excitation energies for several closed- and open-shell mols. that are essentially of the same quality as those obtained from time-dependent d. functional theory itself, when the same exchange-correlation functional is used.
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Casida, M. E. In Recent Advances in Density Functional Methods, Part I; Chong, D. P., Ed.; World Scientific: Singapore, 1995; p 155.
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Guidon, M.; Hutter, J.; VandeVondele, J. Auxiliary Density Matrix Methods for Hartree-Fock Exchange Calculations. J. Chem. Theory Comput. 2010, 6, 2348– 2364, DOI: 10.1021/ct1002225
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Auxiliary Density Matrix Methods for Hartree-Fock Exchange Calculations
Guidon, Manuel; Hutter, Jurg; Vande Vondele, Joost
Journal of Chemical Theory and Computation (2010), 6 (8), 2348-2364CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
The calcn. of Hartree-Fock exchange (HFX) is computationally demanding for large systems described with high-quality basis sets. In this work, we show that excellent performance and good accuracy can nevertheless be obtained if an auxiliary d. matrix is employed for the HFX calcn. Several schemes to derive an auxiliary d. matrix from a high-quality d. matrix are discussed. Key to the accuracy of the auxiliary d. matrix methods (ADMM) is the use of a correction based on std. generalized gradient approxns. for HFX. ADMM integrates seamlessly in existing HFX codes and, in particular, can be employed in linear scaling implementations. Demonstrating the performance of the method, the effect of HFX on the structure of liq. water is investigated in detail using Born-Oppenheimer mol. dynamics simulations (300 ps) of a system of 64 mols. Representative for large systems are calcns. on a solvated protein (Rubredoxin), for which ADMM outperforms the corresponding std. HFX implementation by approx. a factor 20.
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Ghosh, D. C.; Islam, N. Whether electronegativity and hardness are manifest two different descriptors of the one and the same fundamental property of atoms? @ TA quest. Int. J. Quantum Chem. 2011, 111, 40– 51, DOI: 10.1002/qua.22415
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Whether electronegativity and hardness are manifest two different descriptors of the one and the same fundamental property of atoms-A quest
Ghosh, Dulal C.; Islam, Nazmul
International Journal of Quantum Chemistry (2011), 111 (1), 40-51CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
In this report, we have tried to reveal that there is much conceptual commonality between the two fundamental theor. descriptors of chem. and physics-the electronegativity and the hardness. The phys. hardness was introduced and theorized by condensed matter physicists. The chem. hardness was introduced by chemists to generalize and rationalize the HSAB principle. We have tried to establish that the phys. hardness and the chem. hardness with evolution of time have converged to one and the same general principle-the hardness. We have also tried to understand the phys. basis and operational significance of another very important descriptor arising out of theor. constructs of chem.-the electronegativity. We have relied upon the fact that, since these descriptors are not observables, there is no possibility of their quantum mech. evaluation. These descriptors, therefore, should be and must be reified before suggesting ansatz for their evaluation. We have dwelt at length upon the effort of d. functional definition and evaluation of electronegativity and hardness and discovered the inherent inner contradiction of the theory and measurement. We have also noted that a good no. of scientists hold the opinion that the d. functional formula of electronegativity is χ = I and that of hardness is η = I, where I is the ionization potential of the chem. system. This study concludes that the two fundamental descriptors-the hardness and the electronegativity originate from the same source-the electron attracting power of the screened nucleus upon valence electrons and discovers the surprising result that if one measures hardness, the electronegativity is simultaneously measured and vice versa. We have also explored the ansatz of semiempirical evaluation of electronegativity and hardness of at. systems involving the radius of the atoms. We have noted that the ansatz for electronegativity and hardness is the same. To justify our statement that if one measures hardness, the electronegativity is simultaneously measured and vice versa, we have used the evaluated set of hardness for 103 elements of the periodic table as a scale of electronegativity and found that such set of hardness data satisfies the sine qua non of a satisfactory scale of electronegativity. The electronegativity and the hardness are two different appearances of the one and the same fundamental property of atoms. They are different and also nondifferent. They are different in their fields of application. They are nondifferent when we discuss the basic philosophical structures of their origin and development. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011.
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Marx, D.; Hutter, J. In Modern Methods and Algorithms of Quantum Chemistry; Grotendorst, J., Ed.; John von Neumann Institute for Computing, Forschungszentrum Jülich: Jülich, Germany, 2000; pp 301– 449 (first edition, paperback, ISBN 3-00-005618-1) or pp 329–477 (second edition, hardcover, ISBN 3-00-005834-6), see http://www.theochem.rub.de/go/cprev.html.
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Essmann, U.; Perera, L.; Berkowitz, M. L.; Darden, T.; Lee, H.; Pedersen, L. G. A smooth particle mesh Ewald method. J. Chem. Phys. 1995, 103, 8577– 8593, DOI: 10.1063/1.470117
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A smooth particle mesh Ewald method
Essmann, Ulrich; Perera, Lalith; Berkowitz, Max L.; Darden, Tom; Lee, Hsing; Pedersen, Lee G.
Journal of Chemical Physics (1995), 103 (19), 8577-93CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The previously developed particle mesh Ewald method is reformulated in terms of efficient B-spline interpolation of the structure factors. This reformulation allows a natural extension of the method to potentials of the form 1/rp with p ≥ 1. Furthermore, efficient calcn. of the virial tensor follows. Use of B-splines in the place of Lagrange interpolation leads to analytic gradients as well as a significant improvement in the accuracy. The authors demonstrate that arbitrary accuracy can be achieved, independent of system size N, at a cost that scales as N log(N). For biomol. systems with many thousands of atoms and this method permits the use of Ewald summation at a computational cost comparable to that of a simple truncation method of 10 Å or less.
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Mayhall, N. J.; Raghavachari, K.; Hratchian, H. P. ONIOM-based QM:QM electronic embedding method using Löwdin atomic charges: Energies and analytic gradients. J. Chem. Phys. 2010, 132, 114107, DOI: 10.1063/1.3315417
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ONIOM-based QM:QM electronic embedding method using Lowdin atomic charges: Energies and analytic gradients
Mayhall, Nicholas J.; Raghavachari, Krishnan; Hratchian, Hrant P.
Journal of Chemical Physics (2010), 132 (11), 114107/1-114107/6CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We report a new quantum mech.:quantum mech. (QM:QM) method which provides explicit electronic polarization of the high-level region by using the Lowdin at. charges from the low-level region. This provides an embedding potential which naturally evolves with changes in nuclear geometry. However, this coupling of the high-level and low-level regions introduces complications in the energy gradient evaluation. Following previous work, we derive and implement efficient gradients where a single set of SCF response equations is solved. We provide results for the calcn. of deprotonation energies of a hydroxylated spherosiloxane cluster (Si8O12H7OH) and the dissocn. energy of a water mol. from a [ZnIm3(H2O)]2+ complex. The Lowdin charge embedding model provides results which are not only an improvement over mech. embedding (no electronic embedding) but which are also resistant to large overpolarization effects which occur more often with Mulliken charge embedding. Finally, a scaled-Loewdin charge embedding method is also presented which provides a method for fine tuning the extent of electronic polarization. (c) 2010 American Institute of Physics.
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Davidson, E. R. The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. J. Comput. Phys. 1975, 17, 87– 94, DOI: 10.1016/0021-9991(75)90065-0
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Chassaing, T. Excitation Energy Calculations with TD-DFT. PhD thesis, University of Zurich, Zurich, Switzerland, 2005.
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Ryabinkin, I. G.; Nagesh, J.; Izmaylov, A. F. Fast Numerical Evaluation of Time-Derivative Nonadiabatic Couplings for Mixed Quantum–Classical Methods. J. Phys. Chem. Lett. 2015, 6, 4200– 4203, DOI: 10.1021/acs.jpclett.5b02062
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Fast Numerical Evaluation of Time-Derivative Nonadiabatic Couplings for Mixed Quantum-Classical Methods
Ryabinkin, Ilya G.; Nagesh, Jayashree; Izmaylov, Artur F.
Journal of Physical Chemistry Letters (2015), 6 (21), 4200-4203CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
We have developed a numerical differentiation scheme that eliminates evaluation of overlap determinants in calcg. the time-deriv. nonadiabatic couplings (TDNACs). Evaluation of these determinants was the bottleneck in previous implementations of mixed quantum-classical methods using numerical differentiation of electronic wave functions in the Slater determinant representation. The central idea of our approach is, first, to reduce the analytic time derivs. of Slater determinants to time derivs. of MOs and then to apply a finite-difference formula. Benchmark calcns. prove the efficiency of the proposed scheme showing impressive several-order-of-magnitude speedups of the TDNAC calcn. step for midsize mols.
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Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Petersson, G. A.; Nakatsuji, H.; Li, X.; Caricato, M.; Marenich, A. V.; Bloino, J.; Janesko, B. G.; Gomperts, R.; Mennucci, B.; Hratchian, H. P.; Ortiz, J. V.; Izmaylov, A. F.; Sonnenberg, J. L.; Williams-Young, D.; Ding, F.; Lipparini, F.; Egidi, F.; Goings, J.; Peng, B.; Petrone, A.; Henderson, T.; Ranasinghe, D.; Zakrzewski, V. G.; Gao, J.; Rega, N.; Zheng, G.; Liang, W.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Throssell, K.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.; Bearpark, M. J.; Heyd, J. J.; Brothers, E. N.; Kudin, K. N.; Staroverov, V. N.; Keith, T. A.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A. P.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Millam, J. M.; Klene, M.; Adamo, C.; Cammi, R.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Farkas, O.; Foresman, J. B.; Fox, D. J. Gaussian 16, Revision C.01; Gaussian Inc.: Wallingford, CT, 2016.
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Weigend, F.; Ahlrichs, R. Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy. Phys. Chem. Chem. Phys. 2005, 7, 3297, DOI: 10.1039/b508541a
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Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy
Weigend, Florian; Ahlrichs, Reinhart
Physical Chemistry Chemical Physics (2005), 7 (18), 3297-3305CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
Gaussian basis sets of quadruple zeta valence quality for Rb-Rn are presented, as well as bases of split valence and triple zeta valence quality for H-Rn. The latter were obtained by (partly) modifying bases developed previously. A large set of more than 300 mols. representing (nearly) all elements-except lanthanides-in their common oxidn. states was used to assess the quality of the bases all across the periodic table. Quantities investigated were atomization energies, dipole moments and structure parameters for Hartree-Fock, d. functional theory and correlated methods, for which we had chosen Moller-Plesset perturbation theory as an example. Finally recommendations are given which type of basis set is used best for a certain level of theory and a desired quality of results.
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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, h.; 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: 10.1063/5.0004635
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TURBOMOLE: Modular program suite for ab initio quantum-chemical and condensed-matter simulations
Balasubramani, 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.
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Weigend, F.; Furche, F.; Ahlrichs, R. Gaussian basis sets of quadruple zeta valence quality for atoms H-Kr. J. Chem. Phys. 2003, 119, 12753– 12762, DOI: 10.1063/1.1627293
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Gaussian basis sets of quadruple zeta valence quality for atoms H-Kr
Weigend, Florian; Furche, Filipp; Ahlrichs, Reinhart
Journal of Chemical Physics (2003), 119 (24), 12753-12762CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We present Gaussian basis sets of quadruple zeta valence quality with a segmented contraction scheme for atoms H to Kr. This extends earlier work on segmented contracted split valence (SV) and triple zeta valence (TZV) basis sets. Contraction coeffs. and orbital exponents are fully optimized in at. Hartree-Fock (HF) calcns. As opposed to other quadruple zeta basis sets, the basis set errors in at. ground-state HF energies are less than 1 mEh and increase smoothly across the Periodic Table, while the no. of primitives is comparably small. Polarization functions are taken partly from previous work, partly optimized in at. MP2 calcns., and for a few cases detd. at the HF level for excited at. states nearly degenerate with the ground state. This leads to basis sets denoted QZVP for HF and d. functional theory (DFT) calcns., and for some atoms to a larger basis recommended for correlated treatments, QZVPP. We assess the performance of the basis sets in mol. HF, DFT, and MP2 calcns. for a sample of diat. and small polyat. mols. by a comparison of energies, bond lengths, and dipole moments with results obtained numerically or using very large basis sets. It is shown that basis sets of quadruple zeta quality are necessary to achieve an accuracy of 1 kcal/mol per bond in HF and DFT atomization energies. For compds. contg. third row as well as alk. and earth alk. metals it is demonstrated that the inclusion of high-lying core orbitals in the active space can be necessary for accurate correlated treatments. The QZVPP basis sets provide sufficient flexibility to polarize the core in those cases. All test calcns. indicate that the new basis sets lead to consistent accuracies in HF, DFT, or correlated treatments even in crit. cases where other basis sets may show deficiencies.
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VandeVondele, J.; Hutter, J. Gaussian basis sets for accurate calculations on molecular systems in gas and condensed phases. J. Chem. Phys. 2007, 127, 114105, DOI: 10.1063/1.2770708
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Gaussian basis sets for accurate calculations on molecular systems in gas and condensed phases
VandeVondele, Joost; Hutter, Jurg
Journal of Chemical Physics (2007), 127 (11), 114105/1-114105/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We present a library of Gaussian basis sets that has been specifically optimized to perform accurate mol. calcns. based on d. functional theory. It targets a wide range of chem. environments, including the gas phase, interfaces, and the condensed phase. These generally contracted basis sets, which include diffuse primitives, are obtained minimizing a linear combination of the total energy and the condition no. of the overlap matrix for a set of mols. with respect to the exponents and contraction coeffs. of the full basis. Typically, for a given accuracy in the total energy, significantly fewer basis functions are needed in this scheme than in the usual split valence scheme, leading to a speedup for systems where the computational cost is dominated by diagonalization. More importantly, binding energies of hydrogen bonded complexes are of similar quality as the ones obtained with augmented basis sets, i.e., have a small (down to 0.2 kcal/mol) basis set superposition error, and the monomers have dipoles within 0.1 D of the basis set limit. However, contrary to typical augmented basis sets, there are no near linear dependencies in the basis, so that the overlap matrix is always well conditioned, also, in the condensed phase. The basis can therefore be used in first principles mol. dynamics simulations and is well suited for linear scaling calcns.
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Brémond, E.; Savarese, M.; Adamo, C.; Jacquemin, D. Accuracy of TD-DFT Geometries: A Fresh Look. J. Chem. Theory Comput. 2018, 14, 3715– 3727, DOI: 10.1021/acs.jctc.8b00311
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Accuracy of TD-DFT Geometries: A Fresh Look
Bremond, Eric; Savarese, Marika; Adamo, Carlo; Jacquemin, Denis
Journal of Chemical Theory and Computation (2018), 14 (7), 3715-3727CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
We benchmark a panel of 48 DFT exchange-correlation functionals in the framework of TD-DFT optimizations of the geometry of valence singlet excited states. To this end, we use a set of 41 small- and medium-sized org. mols. for which ref. geometries were obtained at high level of theory, typically, CC3 or CCSDR(3), with the aug-cc-pVTZ at. basis set. For the ground-state parameters, the tested functionals provide av. deviations that are small (0.010 Å and 0.5° for bond lengths and valence angles) and not very sensitive to the selected (hybrid) functional, but the errors are larger for the most polarized bonds (CO, CN, and so on). Nevertheless, DFT has a tendency to provide too compact distances, a trend slightly enhanced for functionals including a large amt. of exact exchange. The av. errors largely increase when going to the excited-state for most bond types, i.e., TD-DFT delivers less accurate excited-state distances than DFT for ground state. In particular TD-DFT combined with hybrid functionals provides significantly too short CO and CS/CSe bonds with resp. av. errors in the -0.026/-0.052 Å and -0.015/-0.082 Å ranges, depending on the selected hybrid functional. For the carbonyl bonds, the sizes of the TD-DFT deviations obtained when selecting std. hybrid functionals are of the same order of magnitude as the EOM-CCSD ones.
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Chan, G. K.-L.; Handy, N. C. C8H8: a density functional theory study of molecular geometries introducing the localised bond density. J. Chem. Soc., Faraday Trans. 1996, 92, 3015– 3021, DOI: 10.1039/ft9969203015
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C8H8: A density functional theory study of molecular geometries introducing the localized bond density
Chan, Garnet K-L.; Handy, Nicholas C.
Journal of the Chemical Society, Faraday Transactions (1996), 92 (17), 3015-3021CODEN: JCFTEV; ISSN:0956-5000. (Royal Society of Chemistry)
D. functional theory was used with all the common exchange-correlation functionals to investigate the structures of three isomers of C8H8 found in F.A. Cotton's text, barrelene, cyclooctatetraene, tetramethylenecyclobutane and also ethane and ethene. All calcns. were performed with TZ2P basis sets and large quadrature. The results are compared with expt. and those obtained with Hartree-Fock theory. Delocalization in the three mols. is discussed. A localized bond d. is introduced to explain the transferability of the trends in the predictions of the functionals between different mols. Three-parameter adiabatic connection functionals are examd. and their usefulness in geometry prediction questioned. Finally a phys. picture of the correlation as modeled by d. functional theory is presented and used to explain trends in the overestimation or underestimation of bond lengths.
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Neumann, R.; Nobes, R. H.; Handy, N. C. Exchange functionals and potentials. Mol. Phys. 1996, 87, 1– 36, DOI: 10.1080/00268979600100011
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Exchange functionals and potentials
Neumann, Ralf; Nobes, Ross H.; Handy, Nicholas C.
Molecular Physics (1996), 87 (1), 1-36CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis)
The commonly used exchange-correlation functionals of d. functional theory and their potentials are examd. numerically following the first such investigation of Perdew. They are also investigated for Ne and Kr. Their behavior for large gradients of the d. and for large distances is not satisfactory. In particular, the correct asymptotic r-1 behavior is difficult to achieve. Following van Leeuwen and Baerends, this is linked to the energy εmax of the highest occupied orbital arising from the Kohn-Sham equations. This deficiency is linked also with the poor prediction of mol. polarizabilities. The Becke-Roussel (BR) exchange functional is examd., which is derived by assuming a hydrogen-like exchange hole at all spatial points, and it has the attraction of being dependent on both the kinetic energy d. and the Laplacian of the d. and has no adjustable parameters. Becke has presented encouraging results using this functional in a hybrid manner. Fully self-consistent Kohn-sham calcns. are performed using it in combination with Perdew's 1986 correlation functional. The results are very encouraging indeed, so much so that this exchange functional is the best generalized gradient approxn. (GGA) yet discovered. In particular, bond lengths of many mol. show a substantial improvement over results from other GGAs. For example, many CH bonds are now within exptl. accuracy, instead of being typically 0.02 Å too long. Our ab initio understanding of non-dynamic correlation and dynamic correlation is then linked with d. functional theory. It is argued that correlation functionals should pick up the local dynamic correlation, whereas exchange functionals should include non-dynamic correlation effects. For these reasons it is considered that exchange functionals are best modeled on a system for which there is effectively no non-dynamic correlation, for which the optimum example is the Ne atom. Thus, again following Becke and Roussel, the spherically averaged Hartree-Fock exchange hole for Ne is examd., compared with the BR model functional hole. An excellent overlap is found, and thus the above good results are explained. As a final contribution, the dissocn. of the H2 mol. is re-examd., looking at it in terms of the exchange hole. For a ref. electron near one proton A, the RHF model has half an exchange electron near it, and half the exchange electron near the other proton B, whereas the BR functional has one electron near the other proton b, whereas the Br functional has one electron near A, which is the correct picture. For this reason the (restricted) BR functional gives a greatly improved dissocn. curve for H2 when compared with the Hartree-Fock curve. In summary, the Becke-Roussel functional is found to be a most attractive exchange functional.
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Sancho-García, J. C.; Cornil, J. Assessment of recently developed exchange-correlation functionals for the description of torsion potentials in π-conjugated molecules. J. Chem. Phys. 2004, 121, 3096– 3101, DOI: 10.1063/1.1774976
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Assessment of recently developed exchange-correlation functionals for the description of torsion potentials in π-conjugated molecules
Sancho-Garcia, Juan Carlos; Cornil, Jerome
Journal of Chemical Physics (2004), 121 (7), 3096-3101CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Newly developed exchange-correlation functionals in d. functional theory (DFT) have been applied to describe conjugation effects in org. mols. The performance of the various approaches is assessed through the calcn. of torsion energy profiles and their crit. comparison with available exptl. data. Our results indicate that the OPTX-B95 exchange-correlation functional as well as its corresponding hybrid versions perform better than the well-established BLYP or B3LYP schemes when dealing with π-conjugated mols. In contrast, the recently introduced VSXC functional is not as reliable as other DFT methods for the systems examd. here.
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Vikramaditya, T.; Lin, S.-T. Limitations of Global Hybrids in Predicting the Geometries and Torsional Energy Barriers of Dimeric Systems and the Role of Hartree Fock and DFT Exchange. J. Comput. Chem. 2019, 40, 2810– 2818, DOI: 10.1002/jcc.26056
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Limitations of Global Hybrids in Predicting the Geometries and Torsional Energy Barriers of Dimeric Systems and the Role of Hartree Fock and DFT Exchange
Vikramaditya, Talapunur; Lin, Shiang-Tai
Journal of Computational Chemistry (2019), 40 (32), 2810-2818CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
Prediction of accurate geometries is a prerequisite for accurate prediction of mol. properties. Impact of Hartree Fock (HF) exchange (a0) on geometry in the framework of DFT is investigated by monitoring dihedral angles, bond length alternations, and torsional energy barriers of 10 dimeric systems against CCSD (ADZ/ATZ) benchmarks. A strong correlation is obsd. between the fraction of HF exchange, equil. dihedral angles, and the potential energy barriers in global hybrids. Full HF exchange is crit. to accurately predict the nonplanarity. Lower fractions of (a0)/larger DFT exchange (1-a0) results in overestimation of torsional energy barriers at 900 and underestimation at 00. Large contributions of (1-a0) in global hybrid functionals tend to overestimate torsional energy barriers (900) and are biased toward planar geometries. However, inclusion of larger fractions of (a0)/lower (1-a0) also overestimate the torsional energy barriers in syn-conformations due to the localization errors assocd. with HF exchange in global hybrids. Hence, irresp. of the fraction of HF/DFT exchange incorporated, global hybrids fail to accurately predict torsional energy barriers at 00 and 900 simultaneously. Long-range cor. (LC) functionals, which employ full HF exchange at longer regions, outperform global hybrid functionals in predicting geometries and torsional energy barriers of the dimeric mols. The distance dependence of (a0) thus provides a balanced fraction of HF exchange as the dihedral torsion varies. Impact of range sepn. parameter on geometries is marginal in altering the planarity/nonplanarity. However, range sepn. parameter within 0.20-0.40 bohr-1 predicts more reliable torsional energies and geometries. © 2019 Wiley Periodicals, Inc.
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Fang, C.; Oruganti, B.; Durbeej, B. How Method-Dependent Are Calculated Differences between Vertical, Adiabatic, and 0─0 Excitation Energies?. J. Phys. Chem. A 2014, 118, 4157– 4171, DOI: 10.1021/jp501974p
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How Method-Dependent Are Calculated Differences between Vertical, Adiabatic, and 0-0 Excitation Energies?
Fang, Changfeng; Oruganti, Baswanth; Durbeej, Bo
Journal of Physical Chemistry A (2014), 118 (23), 4157-4171CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
Through a large no. of benchmark studies, the performance of different quantum chem. methods in calcg. vertical excitation energies is today quite well established. These efforts have in recent years been complemented by a few benchmarks focusing instead on adiabatic excitation energies. However, it is much less well established how calcd. differences between vertical, adiabatic and 0-0 excitation energies vary between methods, which may be due to the cost of evaluating zero-point vibrational energy corrections for excited states. To fill this gap, we have calcd. vertical, adiabatic, and 0-0 excitation energies for a benchmark set of mols. covering both org. and inorg. systems. Considering in total 96 excited states and using both TD-DFT with a variety of exchange-correlation functionals and the ab initio CIS and CC2 methods, it is found that while the vertical excitation energies obtained with the various methods show an av. (over the 96 states) std. deviation of 0.39 eV, the corresponding std. deviations for the differences between vertical, adiabatic, and 0-0 excitation energies are much smaller: 0.10 (difference between adiabatic and vertical) and 0.02 eV (difference between 0-0 and adiabatic). These results provide a quant. measure showing that the calcn. of such quantities in photochem. modeling is well amenable to low-level methods. In addn., we also report on how these energy differences vary between chem. systems and assess the performance of TD-DFT, CIS, and CC2 in reproducing exptl. 0-0 excitation energies.
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Peach, M. J. G.; Benfield, P.; Helgaker, T.; Tozer, D. J. Excitation energies in density functional theory: An evaluation and a diagnostic test. J. Chem. Phys. 2008, 128, 044118, DOI: 10.1063/1.2831900
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Excitation energies in density functional theory: An evaluation and a diagnostic test
Peach, Michael J. G.; Benfield, Peter; Helgaker, Trygve; Tozer, David J.
Journal of Chemical Physics (2008), 128 (4), 044118/1-044118/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Electronic excitation energies are detd. using the CAM-B3LYP Coulomb-attenuated functional, together with a std. generalized gradient approxn. (GGA) and hybrid functional. The degree of spatial overlap between the occupied and virtual orbitals involved in an excitation is measured using a quantity Λ, and the extent to which excitation energy errors correlate with Λ is quantified. For a set of 59 excitations of local, Rydberg, and intramol. charge-transfer character in 18 theor. challenging main-group mols., CAM-B3LYP provides by far the best overall performance; no correlation is obsd. between excitation energy errors and Λ, reflecting the good quality, balanced description of all three categories of excitation. By contrast, a clear correlation is obsd. for the GGA and, to a lesser extent, the hybrid functional, allowing a simple diagnostic test to be proposed for judging the reliability of a general excitation from these functionals-when Λ falls below a prescribed threshold, excitations are likely to be in very significant error. The study highlights the ambiguous nature of the term "charge transfer," providing insight into the observation that while many charge-transfer excitations are poorly described by GGA and hybrid functionals, others are accurately reproduced. (c) 2008 American Institute of Physics.
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Send, R.; Kühn, M.; Furche, F. Assessing Excited State Methods by Adiabatic Excitation Energies. J. Chem. Theory Comput. 2011, 7, 2376– 2386, DOI: 10.1021/ct200272b
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Assessing Excited State Methods by Adiabatic Excitation Energies
Send, Robert; Kuhn, Michael; Furche, Filipp
Journal of Chemical Theory and Computation (2011), 7 (8), 2376-2386CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
We compile a 109-membered benchmark set of adiabatic excitation energies (AEEs) from high-resoln. gas-phase expts. Our data set includes a variety of org. chromophores with up to 46 atoms, radicals, and inorg. transition metal compds. Many of the 91 mols. in our set are relevant to atm. chem., photovoltaics, photochem., and biol. The set samples valence, Rydberg, and ionic states of various spin multiplicities. As opposed to vertical excitation energies, AEEs are rigorously defined by energy differences of vibronic states, directly observable, and insensitive to errors in equil. structures. We supply optimized ground state and excited state structures, which allows fast and convenient evaluation of AEEs with two single-point energy calcns. per system. We apply our benchmark set to assess the performance of time-dependent d. functional theory using common semilocal functionals and the CI singles method. Hybrid functionals such as B3LYP and PBE0 yield the best results, with mean abs. errors around 0.3 eV. We also investigate basis set convergence and correlations between different methods and between the magnitude of the excited state relaxation energy and the AEE error. A smaller, 15-membered subset of AEEs is introduced and used to assess the correlated wave function methods CC2 and ADC(2). These methods improve upon hybrid TDDFT for systems with single-ref. ground states but perform less well for radicals and small-gap transition metal compds. None of the investigated methods reaches "chem. accuracy" of 0.05 eV in AEEs.
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Côté, A. P.; Benin, A. I.; Ockwig, N. W.; O’Keeffe, M.; Matzger, A. J.; Yaghi, O. M. Porous, Crystalline, Covalent Organic Frameworks. Science 2005, 310, 1166– 1170, DOI: 10.1126/science.1120411
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Porous, Crystalline, Covalent Organic Frameworks
Cote, Adrien P.; Benin, Annabelle I.; Ockwig, Nathan W.; O'Keeffe, Michael; Matzger, Adam J.; Yaghi, Omar M.
Science (Washington, DC, United States) (2005), 310 (5751), 1166-1170CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)
Covalent org. frameworks (COFs) have been designed and successfully synthesized by condensation reactions of Ph diboronic acid {C6H4[B(OH)2]2} and hexahydroxytriphenylene [C18H6(OH)6]. Powder x-ray diffraction studies of the highly cryst. products (C3H2BO)6•(C9H12)1 (COF-1) and C9H4BO2 (COF-5) revealed expanded porous graphitic layers that are either staggered (COF-1, P63/mmc) or eclipsed (COF-5, P6/mmm). Their crystal structures are entirely held by strong bonds between B, C, and O atoms to form rigid porous architectures with pore sizes ranging from 7 to 27 angstroms. COF-1 and COF-5 exhibit high thermal stability (to temps. up to 500° to 600°C), permanent porosity, and high surface areas (711 and 1590 square meters per g, resp.).
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Das, G.; Biswal, B. P.; Kandambeth, S.; Venkatesh, V.; Kaur, G.; Addicoat, M.; Heine, T.; Verma, S.; Banerjee, R. Chemical sensing in two dimensional porous covalent organic nanosheets. Chem. Sci. 2015, 6, 3931– 3939, DOI: 10.1039/C5SC00512D
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Chemical sensing in two dimensional porous covalent organic nanosheets
Das, Gobinda; Biswal, Bishnu P.; Kandambeth, Sharath; Venkatesh, V.; Kaur, Gagandeep; Addicoat, Matthew; Heine, Thomas; Verma, Sandeep; Banerjee, Rahul
Chemical Science (2015), 6 (7), 3931-3939CODEN: CSHCCN; ISSN:2041-6520. (Royal Society of Chemistry)
Two new imide-based cryst., porous, and chem. stable covalent org. frameworks (COFs) (TpBDH and TfpBDH) have been successfully synthesized employing solvothermal crystn. route. Furthermore, thin layered covalent org. nanosheets (CONs) were derived from these bulk COFs by the simple liq. phase exfoliation method. These 2D CONs showcase increased luminescence intensity compared to their bulk counterparts (COFs). Notably, TfpBDH-CONs showcase good selectivity and prominent, direct visual detection towards different nitroarom. analytes over TpBDH-CONs. Quite interestingly, TfpBDH-CONs exhibit a superior "turn-on" detection capability for 2,4,6-trinitrophenol (TNP) in the solid state, but conversely, they also show a "turn-off" detection in the dispersion state. These findings describe a new approach towards developing an efficient, promising fluorescence chemosensor material for both visual and spectroscopic detection of nitroarom. compds. with very low [10-5 (M)] analyte concns.
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Das, P.; Chakraborty, G.; Tyagi, S.; Mandal, S. K. Design of Fluorescent and Robust Covalent Organic Framework Host Matrices for Illuminating Mechanistic Insight into Solvatochromic Decoding. ACS Appl. Mater. Interfaces 2020, 12, 52527– 52537, DOI: 10.1021/acsami.0c12785
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Design of Fluorescent and Robust Covalent Organic Framework Host Matrices for Illuminating Mechanistic Insight into Solvatochromic Decoding
Das, Prasenjit; Chakraborty, Gouri; Tyagi, Sparsh; Mandal, Sanjay K.
ACS Applied Materials & Interfaces (2020), 12 (47), 52527-52537CODEN: AAMICK; ISSN:1944-8244. (American Chemical Society)
Two functional covalent org. frameworks (COFs), constructed from 3-connected triazine-based amine or hydrazine with linear dialdehyde, are decorated with mol. docking sites to showcase their solvatochromic decoding behavior toward volatile solvent mols. (VSMs). These luminescent and cryst. COFs, namely, COF-N and COF-NN, are characterized by numerous anal. techniques. After accommodation of different VSMs as guests, the inclusion compds. of COF-N and COF-NN display solvatochromism. More fascinatingly, the singlet energy, band gaps, and lifetime of these VSM-accommodated COF-N and COF-NN are linearly correlated with the properties of VSMs. D. functional theory (DFT) and Monte Carlo simulation studies further support the interaction of VSMs with COF-N and COF-NN. The presence of extra amine functionality in COF-NN leads to the better interaction with VSMs and, therefore, results in different modes of interaction and correlation. Considering their inestimable chem. diversity, this study introduces a new path for finely tuned solvatochromic properties by COFs.
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Abstract
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Figure 1
Figure 1. Error (in pm) for selected bond lengths for (a) optimized ground- and (b) first singlet excited-state geometries comparing conventional PBE0 and ADMM-PBE0 computations for different auxiliary (ABS) and primary basis sets (PBS) indicated as ABS/PBS.
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Figure 2
Figure 2. Error (in deg) for selected angles for optimized first singlet excited-state geometries comparing conventional PBE0 and ADMM-PBE0 computations.
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Figure 3
Figure 3. Normal distributions based on the mean errors (ME) and standard deviations (STD) wrt EOM-CCSD reference geometries in (pm)/(deg)/(deg) for selected (a) bond lengths/(b) angles/(c) dihedral angles for optimized first singlet excited-state geometries comparing conventional PBE0, ADMM-PBE0, and sTDA computations using triple-ζ basis sets.
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Figure 4
Figure 4. Normal distributions based on the MEs and STDs for vertical excitation energies (in eV) depicting (a) the error introduced by ADMM-PBE0 in comparison to conventional PBE0 computations (upper left plot), (b) the performance of PBE0, ADMM-PBE0, and sTDA kernel computations using double- and triple-ζ basis sets in comparison to a PBE0/TZV2P/CP2K reference (upper right plot) as well as (c) an analogous assessment of the kernels with respect to a PBE0/def2-QZVPP/Turbomole reference (lower plot).
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Figure 5
Figure 5. ADMM error distribution in (a) adiabatic excitation and (b) fluorescence energies comparing approximated ADMM-PBE0 and conventional PBE0 results for different auxiliary basis set sizes (in eV).
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Figure 6
Figure 6. Comparison of (a) adiabatic excitation energies with respect to EOM-CCSD reference data and (b) fluorescence energies with respect to PBE0/def2-QZVPP/Turbomole reference data for PBE0, ADMM-PBE0, and sTDA kernels (in eV).
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Figure 7
Figure 7. Comparison of ADMM-PBE0 and sTDA absorption spectra for a series of fluorescent COFs.
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References
This article references 79 other publications.
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  A review. A detailed understanding of the properties of electronic excited states and the reaction mechanisms that mols. undergo after light irradn. is a fundamental ingredient for following light-driven natural processes and for designing novel photonic materials. The aim of this review is to present an overview of the ab initio quantum chem. and time-dependent d. functional theory methods that can be used to model spectroscopy and photochem. in mol. systems. The applicability and limitations of the different methods as well as the main frontiers are discussed. To illustrate the progress achieved by excited-state chem. in the recent years as well as the main challenges facing computational chem., three main applications that reflect the authors' experience are addressed: the UV/Vis spectroscopy of org. mols., the assignment of absorption and emission bands of organometallic complexes, and finally, the obtainment of non-adiabatic photoinduced pathways mediated by conical intersections. In the latter case, special emphasis is put on the photochem. of DNA. These applications show that the description of electronically excited states is a rewarding but challenging area of research.
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  A review of single-ref. ab initio excited state methods, which are applicable to large mols. and do not explicitly include correlation through the ground-state wave function. CIS, TDHF, and TDDFT are rigorously introduced by outlining their derivations and theor. footing, where special emphasis is put on their relations to each other. Different methods for the anal. of complicated electronically excited states are reviewed and comparatively discussed. The applicability of the presented methods is outlined, their limitations are show, and out their strengths and weaknesses are pointed out.
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  TD-DFT benchmarks: A review
  Laurent, Adele D.; Jacquemin, Denis
  International Journal of Quantum Chemistry (2013), 113 (17), 2019-2039CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
  A review. Time-Dependent D. Functional Theory (TD-DFT) has become the most widely-used theor. approach to simulate the optical properties of both org. and inorg. mols. In this contribution, we review TD-DFT benchmarks that have been performed during the last decade. The aim is often to pinpoint the most accurate or adequate exchange-correlation functional(s). We present both the different strategies used to assess the functionals and the main results obtained in terms of accuracy. In particular, we discuss both vertical and adiabatic benchmarks and comparisons with both exptl. and theor. ref. transition energies. More specific benchmarks (oscillator strengths, excited-state geometries, dipole moments, vibronic shapes, etc.) are summarized as well. © 2013 Wiley Periodicals, Inc.
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4. 4
  de Wergifosse, M.; Grimme, S. Perspective on Simplified Quantum Chemistry Methods for Excited States and Response Properties. J. Phys. Chem. A 2021, 125, 3841– 3851, DOI: 10.1021/acs.jpca.1c02362
  4
  Perspective on Simplified Quantum Chemistry Methods for Excited States and Response Properties
  De wergifosse, Marc; Grimme, Stefan
  Journal of Physical Chemistry A (2021), 125 (18), 3841-3851CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
  A review. We review recent developments in the framework of simplified quantum chem. for excited state and optical response properties (sTD-DFT) and present future challenges for new method developments to improve accuracy and extend the range of application. In recent years, the scope of sTD-DFT was extended to mol. response calcns. of the polarizability, optical rotation, first hyperpolarizability, two-photon absorption (2PA), and excited-state absorption for large systems with hundreds to thousands of atoms. The recently introduced spin-flip simplified time-dependent d. functional theory (SF-sTD-DFT) variant enables an ultrafast treatment for diradicals and related strongly correlated systems. A few drawbacks were also identified, specifically for the computation of 2PA cross sections. We propose solns. to this problem and how to generally improve the accuracy of simplified schemes. New possible simplified schemes are also introduced for strongly correlated systems, e.g., with a second-order perturbative correlation correction. Interpretation tools that can ext. chem. structure-property relationships from excited state or response calcns. are also discussed. In particular, the recently introduced method-agnostic RespA approach based on natural response orbitals (NROs) as the key concept is employed.
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5. 5
  Grimme, S.; Schreiner, P. R. Computational Chemistry: The Fate of Current Methods and Future Challenges. Angew. Chem., Int. Ed. 2018, 57, 4170– 4176, DOI: 10.1002/anie.201709943
  5
  Computational Chemistry: The Fate of Current Methods and Future Challenges
  Grimme, Stefan; Schreiner, Peter R.
  Angewandte Chemie, International Edition (2018), 57 (16), 4170-4176CODEN: ACIEF5; ISSN:1433-7851. (Wiley-VCH Verlag GmbH & Co. KGaA)
  In this essay, we attempt to make predictions about the fate and development of the computational mol. sciences. Of course, it is not the first time that the future challenges for computational org. chem. and biochem. are considered; these were outlined in complementary contexts recently. In this article, the authors take a somewhat different perspective and emphasize the changes expected for chem. that are triggered by the rapid developments and increasingly stronger influences from theory, algorithms, and data-driven technologies. The authors of this essay are about the same age and have a general overview of a period of about 25 years during which they have actively contributed to the field of computational chem. Hence, it appears sensible to make predictions extending 25 years into the future, roughly to the year 2043 (when both authors will long be retired).
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6. 6
  Dierksen, M.; Grimme, S. The Vibronic Structure of Electronic Absorption Spectra of Large Molecules: A Time-Dependent Density Functional Study on the Influence of Exact Hartree-Fock Exchange. J. Phys. Chem. A 2004, 108, 10225– 10237, DOI: 10.1021/jp047289h
  6
  The Vibronic Structure of Electronic Absorption Spectra of Large Molecules: A Time-Dependent Density Functional Study on the Influence of "Exact" Hartree-Fock Exchange
  Dierksen, Marc; Grimme, Stefan
  Journal of Physical Chemistry A (2004), 108 (46), 10225-10237CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
  The functional dependence of excited-state geometries and normal modes calcd. with time-dependent d. functional theory (TDDFT) is studied from vibronic structure calcns. of the absorption spectra of large mols. For a set of mols. covering a wide range of different structures including org. dyes, biol. chromophores, and mols. of importance in material science, quantum mech. simulations of the vibronic structure are performed. In total over 40 singlet-singlet transitions of neutral closed-shell compds. and doublet-doublet transitions of neutral radicals, radical cations, and anions are considered. Calcns. with different std. d. functionals show that the predicted vibronic structure critically depends on the fraction of the exact Hartree-Fock exchange (EEX) included in hybrid functionals. The effect can been traced back to a large influence of EEX on the geometrical displacement upon excitation. On the contrary, the dependence of the results on the choice of the local exchange-correlation functional is rather small. From detailed comparisons with exptl. spectra conclusions are drawn concerning the optimum amt. of EEX mixing for a proper description of the excited-state properties. The relation of the quality of the simulated spectra with the errors for 0-0 transition energies is discussed. For the studied singlet-singlet π → π* transitions and the 1st strongly dipole-allowed transitions of PAH radical cations some rules of thumb concerning the optimum portion of EEX are derived. However, in general no universal amt. of EEX seems to exist that gives a uniformly good description for all systems and states. Nevertheless an inclusion of ∼30-40% of EEX in the functional is found empirically to yield in most cases simulated spectra that compare very well with those from expt. and thus seems to be necessary for an accurate description of the excited-state geometry. Pure d. functionals that are computationally more efficient provide less accurate spectra in most cases and their application is recommended solely for comparison purposes to obtain ests. for the reliability of the theor. predictions.
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7. 7
  Merlot, P.; Izsák, R.; Borgoo, A.; Kjærgaard, T.; Helgaker, T.; Reine, S. Charge-constrained auxiliary-density-matrix methods for the Hartree-Fock exchange contribution. J. Chem. Phys. 2014, 141, 094104, DOI: 10.1063/1.4894267
  7
  Charge-constrained auxiliary-density-matrix methods for the Hartree-Fock exchange contribution
  Merlot, Patrick; Izsak, Robert; Borgoo, Alex; Kjaergaard, Thomas; Helgaker, Trygve; Reine, Simen
  Journal of Chemical Physics (2014), 141 (9), 094104/1-094104/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Three new variants of the auxiliary-d.-matrix method (ADMM) of Guidon, Hutter, and VandeVondele [J. Chem. Theory Comput. 6, 2348 (2010)] are presented with the common feature that they have a simplified constraint compared with the full orthonormality requirement of the earlier ADMM1 method. All ADMM variants are tested for accuracy and performance in all-electron B3LYP calcns. with several commonly used basis sets. The effect of the choice of the exchange functional for the ADMM exchange-correction term is also investigated. (c) 2014 American Institute of Physics.
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8. 8
  Guidon, M.; Hutter, J.; VandeVondele, J. Auxiliary Density Matrix Methods for Hartree-Fock Exchange Calculations. J. Chem. Theory Comput. 2010, 6, 2348– 2364, DOI: 10.1021/ct1002225
  8
  Auxiliary Density Matrix Methods for Hartree-Fock Exchange Calculations
  Guidon, Manuel; Hutter, Jurg; Vande Vondele, Joost
  Journal of Chemical Theory and Computation (2010), 6 (8), 2348-2364CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  The calcn. of Hartree-Fock exchange (HFX) is computationally demanding for large systems described with high-quality basis sets. In this work, we show that excellent performance and good accuracy can nevertheless be obtained if an auxiliary d. matrix is employed for the HFX calcn. Several schemes to derive an auxiliary d. matrix from a high-quality d. matrix are discussed. Key to the accuracy of the auxiliary d. matrix methods (ADMM) is the use of a correction based on std. generalized gradient approxns. for HFX. ADMM integrates seamlessly in existing HFX codes and, in particular, can be employed in linear scaling implementations. Demonstrating the performance of the method, the effect of HFX on the structure of liq. water is investigated in detail using Born-Oppenheimer mol. dynamics simulations (300 ps) of a system of 64 mols. Representative for large systems are calcns. on a solvated protein (Rubredoxin), for which ADMM outperforms the corresponding std. HFX implementation by approx. a factor 20.
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9. 9
  Kumar, C.; Fliegl, H.; Jensen, F.; Teale, A. M.; Reine, S.; Kjærgaard, T. Accelerating Kohn-Sham response theory using density fitting and the auxiliary-density-matrix method. Int. J. Quantum Chem. 2018, 118, e25639 DOI: 10.1002/qua.25639
  There is no corresponding record for this reference.
10. 10
  Rebolini, E.; Izsák, R.; Reine, S. S.; Helgaker, T.; Pedersen, T. B. Comparison of Three Efficient Approximate Exact-Exchange Algorithms: The Chain-of-Spheres Algorithm, Pair-Atomic Resolution-of-the-Identity Method, and Auxiliary Density Matrix Method. J. Chem. Theory Comput. 2016, 12, 3514– 3522, DOI: 10.1021/acs.jctc.6b00074
  10
  Comparison of Three Efficient Approximate Exact-Exchange Algorithms: The Chain-of-Spheres Algorithm, Pair-Atomic Resolution-of-the-Identity Method, and Auxiliary Density Matrix Method
  Rebolini, Elisa; Izsak, Robert; Reine, Simen Sommerfelt; Helgaker, Trygve; Pedersen, Thomas Bondo
  Journal of Chemical Theory and Computation (2016), 12 (8), 3514-3522CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  We compare the performance of three approx. methods for speeding up evaluation of the exchange contribution in Hartree-Fock and hybrid Kohn-Sham calcns.: the chain-of-spheres algorithm, and the auxiliary d. matrix method. Both the efficiency relative to that of a conventional linear-scaling algorithm and the accuracy of total, atomization, and orbital energies are compared for a subset contg. 25 of the 200 mols. in the Rx200 set using double-, triple-, and quadruple-ζ basis sets. The accuracy of relative energies is further compared for small alkane conformers (ACONF test set) and Diels-Alder reactions (DARC test set). Overall, we find that the COSX method provides good accuracy for orbital energies as well as total and relative energies, and the method delivers a satisfactory speedup. The PARI-K and in particular ADMM algorithms require further development and optimization to fully exploit their indisputable potential.
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11. 11
  Merlot, P.; Kjærgaard, T.; Helgaker, T.; Lindh, R.; Aquilante, F.; Reine, S.; Pedersen, T. B. Attractive electron–electron interactions within robust local fitting approximations. J. Comput. Chem. 2013, 34, 1486– 1496, DOI: 10.1002/jcc.23284
  11
  Attractive electron-electron interactions within robust local fitting approximations
  Merlot, Patrick; Kjaergaard, Thomas; Helgaker, Trygve; Lindh, Roland; Aquilante, Francesco; Reine, Simen; Pedersen, Thomas Bondo
  Journal of Computational Chemistry (2013), 34 (17), 1486-1496CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
  An anal. of Dunlap's robust fitting approach reveals that the resulting two-electron integral matrix is not manifestly pos. semidefinite when local fitting domains or non-Coulomb fitting metrics are used. We present a highly local approx. method for evaluating four-center two-electron integrals based on the resoln.-of-the-identity (RI) approxn. and apply it to the construction of the Coulomb and exchange contributions to the Fock matrix. In this pair-at. resoln.-of-the-identity (PARI) approach, at.-orbital (AO) products are expanded in auxiliary functions centered on the two atoms assocd. with each product. Numerical tests indicate that in 1% or less of all Hartree-Fock and Kohn-Sham calcns., the indefinite integral matrix causes nonconvergence in the self-consistent-field iterations. In these cases, the two-electron contribution to the total energy becomes neg., meaning that the electronic interaction is effectively attractive, and the total energy is dramatically lower than that obtained with exact integrals. In the vast majority of our test cases, however, the indefiniteness does not interfere with convergence. The total energy accuracy is comparable to that of the std. Coulomb-metric RI method. The speed-up compared with conventional algorithms is similar to the RI method for Coulomb contributions; exchange contributions are accelerated by a factor of up to eight with a triple-zeta quality basis set. A pos. semidefinite integral matrix is recovered within PARI by introducing local auxiliary basis functions spanning the full AO product space, as may be achieved by using Cholesky-decompn. techniques. Local completion, however, slows down the algorithm to a level comparable with or below conventional calcns. © 2013 Wiley Periodicals, Inc.
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12. 12
  Manzer, S. F.; Epifanovsky, E.; Head-Gordon, M. Efficient Implementation of the Pair Atomic Resolution of the Identity Approximation for Exact Exchange for Hybrid and Range-Separated Density Functionals. J. Chem. Theory Comput. 2015, 11, 518– 527, DOI: 10.1021/ct5008586
  12
  Efficient Implementation of the Pair Atomic Resolution of the Identity Approximation for Exact Exchange for Hybrid and Range-Separated Density Functionals
  Manzer, Samuel F.; Epifanovsky, Evgeny; Head-Gordon, Martin
  Journal of Chemical Theory and Computation (2015), 11 (2), 518-527CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  An efficient new MO basis algorithm is reported implementing the pair at. resoln. of the identity approxn. (PARI) to evaluate the exact exchange contribution (K) to SCF methods, such as hybrid and range-sepd. hybrid d. functionals. The PARI approxn., in which AO basis function pairs are expanded using auxiliary basis functions centered only on their two resp. atoms, was recently investigated by Merlot et al. Our algorithm is significantly faster than quartic scaling RI-K, with an asymptotic exchange speedup for hybrid functionals of (1 + X/N), where N and X are the AO and auxiliary basis dimensions. The asymptotic speedup is 2 + 2X/N for range sepd. hybrids such as CAM-B3LYP, ωB97X-D, and ωB97X-V which include short- and long-range exact exchange. The obsd. speedup for exchange in ωB97X-V for a C68 graphene fragment in the cc-pVTZ basis is 3.4 relative to RI-K. Like conventional RI-K, our method greatly outperforms conventional integral evaluation in large basis sets; a speedup of 19 is obtained in the cc-pVQZ basis on a C54 graphene fragment. Negligible loss of accuracy relative to exact integral evaluation is demonstrated on databases of bonded and nonbonded interactions. We also demonstrate both anal. and numerically that the PARI-K approxn. is variationally stable.
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13. 13
  Neese, F.; Wennmohs, F.; Hansen, A.; Becker, U. Efficient, approximate and parallel Hartree–Fock and hybrid DFT calculations. A ‘chain-of-spheres’ algorithm for the Hartree–Fock exchange. Chem. Phys. 2009, 356, 98– 109, DOI: 10.1016/j.chemphys.2008.10.036
  13
  Efficient, approximate and parallel Hartree-Fock and hybrid DFT calculations. A 'chain-of-spheres' algorithm for the Hartree-Fock exchange
  Neese, Frank; Wennmohs, Frank; Hansen, Andreas; Becker, Ute
  Chemical Physics (2009), 356 (1-3), 98-109CODEN: CMPHC2; ISSN:0301-0104. (Elsevier B.V.)
  In this paper, the possibility is explored to speed up Hartree-Fock and hybrid d. functional calcns. by forming the Coulomb and exchange parts of the Fock matrix by different approxns. For the Coulomb part the previously introduced Split-RI-J variant of the well-known d. fitting' approxn. is used. The exchange part is formed by semi-numerical integration techniques that are closely related to Friesner's pioneering pseudo-spectral approach. Our potentially linear scaling realization of this algorithm is called the 'chain-of-spheres exchange' (COSX). A combination of semi-numerical integration and d. fitting is also proposed. Both Split-RI-J and COSX scale very well with the highest angular momentum in the basis sets. It is shown that for extended basis sets speed-ups of up to two orders of magnitude compared to traditional implementations can be obtained in this way. Total energies are reproduced with an av. error of <0.3 kcal/mol as detd. from extended test calcns. with various basis sets on a set of 26 mols. with 20-200 atoms and up to 2000 basis functions. Reaction energies agree to within 0.2 kcal/mol (Hartree-Fock) or 0.05 kcal/mol (hybrid DFT) with the canonical values. The COSX algorithm parallelizes with a speedup of 8.6 obsd. for 10 processes. Min. energy geometries differ by less than 0.3 pm in the bond distances and 0.5° in the bond angles from their canonical values. These developments enable highly efficient and accurate SCF calcns. including nonlocal Hartree-Fock exchange for large mols. In combination with the RI-MP2 method and large basis sets, second-order many body perturbation energies can be obtained for medium sized mols. with unprecedented efficiency. The algorithms are implemented into the ORCA electronic structure system.
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14. 14
  Izsák, R.; Neese, F. An overlap fitted chain of spheres exchange method. J. Chem. Phys. 2011, 135, 144105, DOI: 10.1063/1.3646921
  14
  An overlap fitted chain of spheres exchange method
  Izsak, Robert; Neese, Frank
  Journal of Chemical Physics (2011), 135 (14), 144105/1-144105/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The "chain of spheres" (COS) algorithm, as part of the RIJCOSX SCF procedure, approximates the exchange term by performing analytic integration with respect to the coordinates of only one of the two electrons, whereas for the remaining coordinates, integration is carried out numerically. In the present work, we attempt to enhance the efficiency of the method by minimizing numerical errors in the COS procedure. The main idea is based on the work of Friesner and consists of finding a fitting matrix, Q, which leads the numerical and anal. evaluated overlap matrixes to coincide. Using Q, the evaluation of exchange integrals can indeed be improved. Improved results and timings are obtained with the present default grid setup for both single point calcns. and geometry optimizations. The fitting procedure results in a redn. of grid sizes necessary for achieving chem. accuracy. We demonstrate this by testing a no. of grids and comparing results to the fully analytic and the earlier COS approxns. This turns out to be favorable for total and reaction energies, for which chem. accuracy can now be reached with a corresponding ∼30% speedup over the original RIJCOSX procedure for single point energies. Results are slightly less favorable for the accuracy of geometry optimizations, but the procedure is still shown to yield geometries with errors well below the method inherent errors of the employed theor. framework. (c) 2011 American Institute of Physics.
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15. 15
  Laqua, H.; Thompson, T. H.; Kussmann, J.; Ochsenfeld, C. Highly Efficient, Linear-Scaling Seminumerical Exact-Exchange Method for Graphic Processing Units. J. Chem. Theory Comput. 2020, 16, 1456– 1468, DOI: 10.1021/acs.jctc.9b00860
  15
  Highly Efficient, Linear-Scaling Seminumerical Exact-Exchange Method for Graphic Processing Units
  Laqua, Henryk; Thompson, Travis H.; Kussmann, Joerg; Ochsenfeld, Christian
  Journal of Chemical Theory and Computation (2020), 16 (3), 1456-1468CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  We present a highly efficient and asymptotically linear-scaling graphic processing unit accelerated seminumerical exact-exchange method (sn-LinK). We go beyond our previous central processing unit-based method (H. Laqua et al., 2018) by employing our recently developed integral bounds (T.H. Thomson and C. Ochsenfeld, 2019) and high-accuracy numerical integration grid (H. Laqua et al., 2018). The accuracy is assessed for several established test sets, providing errors significantly below 1mEh for the smallest grid. Moreover, a comprehensive performance anal. for large mols. between 62 and 1347 atoms is provided, revealing the outstanding performance of our method, in particular, for large basis sets such as the polarized quadruple-zeta level with diffuse functions.
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16. 16
  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: 10.1063/5.0022755
  16
  An improved seminumerical Coulomb and exchange algorithm for properties and excited states in modern density functional theory
  Holzer, Christof
  Journal 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.
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17. 17
  Ohno, K. Some remarks on the Pariser-Parr-Pople method. Theor. Chim. Acta 1964, 2, 219– 227, DOI: 10.1007/BF00528281
  17
  The Pariser-Parr-Pople method
  Ohno, Kimio
  Theoretica Chimica Acta (1964), 2 (3), 219-27CODEN: TCHAAM; ISSN:0040-5744.
  Basic assumptions which characterize the Pariser-Parr (CA 49, 10725f)-Pople (CA 48, 8639g) method of computing mol. electronic wave functions are examd. crit. By restricted variational calcn. of the valence state of C and N atoms and ions, it is demonstrated that the usual methods of evaluation of center Coulomb integrals and at. core energies are rather good. A semitheoretical means of estg. the core resonance integral is proposed and shown to give fair agreement with the empirical values for C-C, O-O, C-N, and C-O bonds.
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18. 18
  Klopman, G. A Semiempirical Treatment of molecular Structures. II. Molecular Terms and Application to diatomic Molecules. J. Am. Chem. Soc. 1964, 86, 4550– 4557, DOI: 10.1021/ja01075a008
  18
  A semiempirical treatment of molecular structures. II. Molecular terms and application to diatomic molecules
  Klopman, G.
  Journal of the American Chemical Society (1964), 86 (21), 4550-7CODEN: JACSAT; ISSN:0002-7863.
  cf. CA 60, 12679e. A self-consistent semiempirical method which is designed for the calcn. of heats of formation and charge distribution of nonconjugated mols. is outlined. The method is based on an antisymmetrized product of mol. orbitals, simplified in such a way as to make the values of all involved integrals directly available from at. spectra (loc. cit.) and mol. bond distances. In a preliminary study, this method was used to calc. satisfactory values of bond energies and reasonable values of charge distributions in 80 diat. mols. (σ-bonded).
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19. 19
  Nishimoto, K.; Mataga, N. Z. Electronic Structure and Spectra of Some Nitrogen Heterocycles. Z. Phys. Chem. 1957, 12, 335, DOI: 10.1524/zpch.1957.12.5_6.335
  19
  Electronic structure and spectra of some nitrogen heretocycles
  Nishimoto, Kitisuke; Mataga, Noboru
  Zeitschrift fuer Physikalische Chemie (Muenchen, Germany) (1957), 12 (), 335-8CODEN: ZPCFAX; ISSN:0044-3336.
  The energies of lower excited states of pyridine, pyrazine, and sym-triazine and the oscillator strengths of the transitions to those excited states have been calcd. The results are listed and discussed.
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20. 20
  Bannwarth, C.; Grimme, S. A simplified time-dependent density functional theory approach for electronic ultraviolet and circular dichroism spectra of very large molecules. Comput. Theor. Chem. 2014, 1040–1041, 45– 53, DOI: 10.1016/j.comptc.2014.02.023
  20
  A simplified time-dependent density functional theory approach for electronic ultraviolet and circular dichroism spectra of very large molecules
  Bannwarth, Christoph; Grimme, Stefan
  Computational & Theoretical Chemistry (2014), 1040-1041 (), 45-53CODEN: CTCOA5; ISSN:2210-271X. (Elsevier B.V.)
  We present a simplified time-dependent d. functional theory approach (sTD-DFT) that allows fast computation of electronic UV or CD spectra of mols. with 500-1000 atoms. The matrix elements are treated in the same way as in the recently proposed simplified Tamm-Dancoff approach but instead of applying the Tamm-Dancoff approxn., the std. linear-response d. functional theory problem is solved. Compared to sTDA, the method leads to an increase in computation time (typically a factor of 2-5 compared to the corresponding sTDA) which is justified since the resulting transition dipole moments are in general of higher quality. This becomes important if spectral intensities (e.g. single-photon oscillator and rotatory transition strengths) are of interest. Comparison of UV and CD spectra obtained from sTD-DFT and sTDA for some typical systems employing std. hybrid functionals shows that both yield very similar excitation energies but the advantage of using the former approach for transition moments. In order to show the applicability of sTD-DFT to systems which are far beyond the scope of conventional TD-DFT, we present the CD spectrum of a substituted, chiral fullerene over a range of almost 1200 excited states. We propose this method as a more reliable alternative for the prediction esp. of the more challenging CD spectra.
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21. 21
  Grimme, S. A simplified Tamm-Dancoff density functional approach for the electronic excitation spectra of very large molecules. J. Chem. Phys. 2013, 138, 244104, DOI: 10.1063/1.4811331
  21
  A simplified Tamm-Dancoff density functional approach for the electronic excitation spectra of very large molecules
  Grimme, Stefan
  Journal of Chemical Physics (2013), 138 (24), 244104/1-244104/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Two approxns. in the Tamm-Dancoff d. functional theory approach (TDA-DFT) to electronically excited states are proposed which allow routine computations for electronic UV- or CD spectra of mols. with 500-1000 atoms. Speed-ups compared to conventional time-dependent DFT (TD-DFT) treatments of about two to three orders of magnitude in the excited state part at only minor loss of accuracy are obtained. The method termed sTDA ("s" for simplified) employs atom-centered Loewdin-monopole based two-electron repulsion integrals with the asymptotically correct 1/R behavior and perturbative single excitation configuration selection. It is formulated generally for any std. global hybrid d. functional with given Fock-exchange mixing parameter ax. The method performs well for two std. benchmark sets of vertical singlet-singlet excitations for values of ax in the range 0.2-0.6. The mean abs. deviations from ref. data are only 0.2-0.3 eV and similar to those from std. TD-DFT. In three cases (two dyes and one polypeptide), good mutual agreement between the electronic spectra (up to 10-11 eV excitation energy) from the sTDA method and those from TD(A)-DFT is obtained. The computed UV- and CD-spectra of a few typical systems (e.g., C60, two transition metal complexes, [7]helicene, polyalanine, a supramol. aggregate with 483 atoms and about 7000 basis functions) compare well with corresponding exptl. data. The method is proposed together with medium-sized double- or triple-zeta type at.-orbital basis sets as a quantum chem. tool to investigate the spectra of huge mol. systems at a reliable DFT level. (c) 2013 American Institute of Physics.
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22. 22
  Grimme, S.; Bannwarth, C. Ultra-fast computation of electronic spectra for large systems by tight-binding based simplified Tamm-Dancoff approximation (sTDA-xTB). J. Chem. Phys. 2016, 145, 054103, DOI: 10.1063/1.4959605
  22
  Ultra-fast computation of electronic spectra for large systems by tight-binding based simplified Tamm-Dancoff approximation (sTDA-xTB)
  Grimme, Stefan; Bannwarth, Christoph
  Journal of Chemical Physics (2016), 145 (5), 054103/1-054103/20CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The computational bottleneck of the extremely fast simplified Tamm-Dancoff approximated (sTDA) time-dependent d. functional theory procedure [S. Grimme, J. Chem. Phys. 138, 244104 (2013)] for the computation of electronic spectra for large systems is the detn. of the ground state Kohn-Sham orbitals and eigenvalues. This limits such treatments to single structures with a few hundred atoms and hence, e.g., sampling along mol. dynamics trajectories for flexible systems or the calcn. of chromophore aggregates is often not possible. The aim of this work is to solve this problem by a specifically designed semiempirical tight binding (TB) procedure similar to the well established self-consistent-charge d. functional TB scheme. The new special purpose method provides orbitals and orbital energies of hybrid d. functional character for a subsequent and basically unmodified sTDA procedure. Compared to many previous semiempirical excited state methods, an advantage of the ansatz is that a general eigenvalue problem in a non-orthogonal, extended AO basis is solved and therefore correct occupied/virtual orbital energy splittings as well as Rydberg levels are obtained. A key idea for the success of the new model is that the detn. of at. charges (describing an effective electron-electron interaction) and the one-particle spectrum is decoupled and treated by two differently parametrized Hamiltonians/basis sets. The three-diagonalization-step composite procedure can routinely compute broad range electronic spectra (0-8 eV) within minutes of computation time for systems composed of 500-1000 atoms with an accuracy typical of std. time-dependent d. functional theory (0.3-0.5 eV av. error). An easily extendable parametrization based on coupled-cluster and d. functional computed ref. data for the elements H-Zn including transition metals is described. The accuracy of the method termed sTDA-xTB is first benchmarked for vertical excitation energies of open- and closed-shell systems in comparison to other semiempirical methods and applied to exemplary problems in electronic spectroscopy. As side products of the development, a robust and efficient valence electron TB method for the accurate detn. of at. charges as well as a more accurate calcn. scheme of dipole rotatory strengths within the Tamm-Dancoff approxn. is proposed. (c) 2016 American Institute of Physics.
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23. 23
  de Wergifosse, M.; Bannwarth, C.; Grimme, S. A Simplified Spin-Flip Time-Dependent Density Functional Theory Approach for the Electronic Excitation Spectra of Very Large Diradicals. J. Phys. Chem. A 2019, 123, 5815– 5825, DOI: 10.1021/acs.jpca.9b03176
  23
  A Simplified Spin-Flip Time-Dependent Density Functional Theory Approach for the Electronic Excitation Spectra of Very Large Diradicals
  de Wergifosse, Marc; Bannwarth, Christoph; Grimme, Stefan
  Journal of Physical Chemistry A (2019), 123 (27), 5815-5825CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
  Experimentalists working with diradicals are often facing the question of what kind of species among singlet or triplet diradicals or closed-shell mols. are obsd. To treat large diradicals with a high d. of electronic states, we propose a simplified version of the spin-flip time-dependent d. functional theory (SF-TD-DFT) method for a fast computation of their state energies and absorption spectra with an accuracy similar to the nonsimplified scheme. An ultrafast tight-binding variant called SF-sTD-DFT-xTB is also developed to treat even larger systems. For a benchmark set of nine diradicals, good agreement between simplified and conventional SF excitation energies for std. functionals is found. This shows that the proposed parameterization is robust for a wide range of Fock exchange mixing values. With the asymptotically correct response integrals used in SF-sTD-DFT and a correction factor of √(2) for the transition moments, the SF-sTD-DFT/B5050LYP/cc-pVDZ method even outperforms the nonsimplified scheme at drastically reduced computational effort when comparing to the exptl. absorption spectra for this set of diradicals. To showcase the actual performance of the method, absorption spectra of two μ-hydroxo-bridged dimers of corrole tape Ga(III) complex derivs. were computed and compared to the expt., providing good qual. agreement. Finally, a comparison with the high-spin triplet spectrum of a perylene bisimide biradical and the one detd. at the SF-sTD-DFT level showed that at room temp., mostly triplet diradicals are present in soln.
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24. 24
  De Wergifosse, M.; Grimme, S. Nonlinear-response properties in a simplified time-dependent density functional theory (sTD-DFT) framework: Evaluation of excited-state absorption spectra. J. Chem. Phys. 2019, 150, 094112, DOI: 10.1063/1.5080199
  24
  Nonlinear-response properties in a simplified time-dependent density functional theory (sTD-DFT) framework: Evaluation of excited-state absorption spectra
  de Wergifosse, Marc; Grimme, Stefan
  Journal of Chemical Physics (2019), 150 (9), 094112/1-094112/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The energy conversion efficiency of org. solar cells seems crucial for a clean future. The design of new light-harvesting devices needs an in-depth understanding of their optical properties, including the excited-state absorption (ESA). In biol., the optical characterization of photochem./phys. processes happening in photosynthetic pigments and proteins can be difficult to interpret due to their structural complexities. Exptl., an ultrafast transient absorption expt. can probe the excited state interaction with light. Quantum chem. could play an important role to model the transient absorption spectrum of excited states. However, systems that need to be investigated can be way too large for existent software implementations. In this contribution, we present the first sTDA/sTD-DFT (simplified time-dependent d. functional theory with and without Tamm Dancoff approxn.) implementation to evaluate the ESA of mols. The ultrafast ESA evaluation presents a negligible extra cost with respect to sTDA/sTD-DFT original schemes for std. ground state absorption. The sTD-DFT method shows ability to assign ESA spectra to the correct excited state. We showed that in the literature, wrong assignments were proposed as for the L34/L44 mixt. and N-methylfulleropyrrolidine. In addn., sTDA/sTD-DFT-xTB tight-binding variants are also available, allowing the evaluation of ESA for systems of a few thousands of atoms, e.g., the spectrum of the photoactive yellow protein composed of 1931 atoms. (c) 2019 American Institute of Physics.
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25. 25
  De Wergifosse, M.; Grimme, S. Nonlinear-response properties in a simplified time-dependent density functional theory (sTD-DFT) framework: Evaluation of the first hyperpolarizability. J. Chem. Phys. 2018, 149, 024108, DOI: 10.1063/1.5037665
  25
  Nonlinear-response properties in a simplified time-dependent density functional theory (sTD-DFT) framework: Evaluation of the first hyperpolarizability
  de Wergifosse, Marc; Grimme, Stefan
  Journal of Chemical Physics (2018), 149 (2), 024108/1-024108/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Recent developments in nonlinear imaging microscopy show the need to implement new theor. tools, which are able to characterize nonlinear optical properties in an efficient way. For second-harmonic imaging microscopy (SHIM), quantum chem. could play an important role to design new exogenous dyes with enhanced first hyperpolarizabilities or to characterize the response origin in large endogenous biol. systems. Such methods should be able to screen a large no. of compds. while reproducing their trends and to treat large systems in reasonable computation times. To fulfill these requirements, we present a new simplified time-dependent d. functional theory (sTD-DFT) implementation to evaluate the first hyperpolarizability where the Coulomb and exchange integrals are approximated by short-range damped Coulomb interactions of transition d. monopoles. For an ultra-fast computation of the first hyperpolarizability, a tight-binding version (sTD-DFT-xTB) is also proposed. In our implementation, a sTD-DFT calcn. is more than 600 time faster with respect to a regular TD-DFT treatment, while the xTB version speeds up the entire calcn. further by at least two orders of magnitude. We challenge our implementation on three test cases: typical push-pull π-conjugated compds., fluorescent proteins, and a collagen model, which were selected to model requirements for SHIM applications. (c) 2018 American Institute of Physics.
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26. 26
  Bannwarth, C.; Seibert, J.; Grimme, S. Electronic Circular Dichroism of [16]Helicene With Simplified TD-DFT: Beyond the Single Structure Approach. Chirality 2016, 28, 365– 369, DOI: 10.1002/chir.22594
  26
  Electronic Circular Dichroism of [16]Helicene With Simplified TD-DFT: Beyond the Single Structure Approach
  Bannwarth, Christoph; Seibert, Jakob; Grimme, Stefan
  Chirality (2016), 28 (5), 365-369CODEN: CHRLEP; ISSN:0899-0042. (Wiley-Liss, Inc.)
  The electronic CD (ECD) spectrum of the recently synthesized [16]helicene and a deriv. comprising two triisopropylsilyloxy protection groups was computed by means of the very efficient simplified time-dependent d. functional theory (sTD-DFT) approach. Different from many previous ECD studies of helicenes, nonequil. structure effects were accounted for by computing ECD spectra on "snapshots" obtained from a mol. dynamics (MD) simulation including solvent mols. The trajectories are based on a mol. specific classical potential as obtained from the recently developed quantum chem. derived force field (QMDFF) scheme. The reduced computational cost in the MD simulation due to the use of the QMDFF (compared to ab-initio MD) as well as the sTD-DFT approach make realistic spectral simulations feasible for these compds. that comprise more than 100 atoms. While the ECD spectra of [16]helicene and its deriv. computed vertically on the resp. gas phase, equil. geometries show noticeable differences, these are "washed" out when nonequil. structures are taken into account. The computed spectra with two recommended d. functionals (ωB97X and BHLYP) and extended basis sets compare very well with the exptl. one. In addn. we provide an est. for the missing abs. intensities of the latter. The approach presented here could also be used in future studies to capture nonequil. effects, but also to systematically av. ECD spectra over different conformations in more flexible mols. Chirality 00:000-000, 2016. 2016 Wiley Periodicals, Inc.
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27. 27
  Risthaus, T.; Hansen, A.; Grimme, S. Excited states using the simplified Tamm-Dancoff-Approach for range-separated hybrid density functionals: development and application. Phys. Chem. Chem. Phys. 2014, 16, 14408– 14419, DOI: 10.1039/C3CP54517B
  27
  Excited states using the simplified Tamm-Dancoff-Approach for range-separated hybrid density functionals: development and application
  Risthaus, Tobias; Hansen, Andreas; Grimme, Stefan
  Physical Chemistry Chemical Physics (2014), 16 (28), 14408-14419CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
  The recently introduced sTDA methodol. [S. Grimme, J. Chem. Phys., 2013, 138, 244104] to compute excitation spectra of huge mol. systems is extended to range-sepd. hybrid (RSH) d. functionals. The three empirical parameters of the method which describe a screened two-electron interaction are obtained for some common RSH functionals (ωB97 family, CAM-B3LYP, LC-BLYP) from a fit to theor. SCS-CC2 ref. vertical excitation energies for a set of small to medium-sized chromophores. The method is cross-validated on a set of inter- and intramol. charge transfer states and a set composed of typical valence transitions. Overall small deviations from ref. data of only about 0.2-0.4 eV are found with best performance for CAM-B3LYP and ωB97X-D3. To demonstrate versatility and robustness of the new methodol., applications (the UV/Vis spectrum of the pyridine polymer and the ECD spectrum of (P)-[11]helicene) and frequently used charge transfer examples are discussed. In one case, 11 000 + excited electronic states of a system contg. 330 atoms were calcd. We show that the asymptotically correct sTDA-RSH combination yields results often superior to those based on global hybrids and that it opens up new possibilities for the computation of excited states in materials science and bio-mol. systems.
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28. 28
  Elstner, M.; Porezag, D.; Jungnickel, G.; Elsner, J.; Haugk, M.; Frauenheim, T.; Suhai, S.; Seifert, G. Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties. Phys. Rev. B 1998, 58, 7260– 7268, DOI: 10.1103/PhysRevB.58.7260
  28
  Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties
  Elstner, M.; Porezag, D.; Jungnickel, G.; Elsner, J.; Haugk, M.; Frauenheim, Th.; Suhai, S.; Seifert, G.
  Physical Review B: Condensed Matter and Materials Physics (1998), 58 (11), 7260-7268CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
  We outline details about an extension of the tight-binding (TB) approach to improve total energies, forces, and transferability. The method is based on a second-order expansion of the Kohn-Sham total energy in d.-functional theory (DFT) with respect to charge-d. fluctuations. The zeroth-order approach is equiv. to a common std. non-self-consistent (TB) scheme, while at second-order a transparent, parameter-free, and readily calculable expression for generalized Hamiltonian matrix elements may be derived. These are modified by a self-consistent redistribution of Mulliken charges (SCC). Besides the usual "band structure" and short-range repulsive terms the final approx. Kohn-Sham energy addnl. includes a Coulomb interaction between charge fluctuations. At large distances this accounts for long-range electrostatic forces between two point charges and approx. includes self-interaction contributions of a given atom if the charges are located at one and the same atom. We apply the new SCC scheme to problems where deficiencies within the non-SCC std. TB approach become obvious. We thus considerably improve transferability.
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29. 29
  Seibert, J.; Pisarek, J.; Schmitz, S.; Bannwarth, C.; Grimme, S. Extension of the element parameter set for ultra-fast excitation spectra calculation (sTDA-xTB). Mol. Phys. 2019, 117, 1104– 1116, DOI: 10.1080/00268976.2018.1510141
  29
  Extension of the element parameter set for ultra-fast excitation spectra calculation (sTDA-xTB)
  Seibert, Jakob; Pisarek, Jana; Schmitz, Sarah; Bannwarth, Christoph; Grimme, Stefan
  Molecular Physics (2019), 117 (9-12), 1104-1116CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
  The extension of the parameter set for an ultra-fast electronic excitation spectra calcn. is presented. The semiempirical theory based on a tight-binding approach, called extended tight-binding (xTB) in combination with the simplified Tamm-Dancoff approxn. (sTDA) shows remarkable accuracy at very low computational cost for the calcn. of vertical excitation energies of mols. It enables the possibility for computing even large systems up to thousands of atoms or sampling along mol. dynamic (MD) trajectories. The original publication of the sTDA-xTB method included parameters for the most important elements (H-Zn,Br,I). In this work, element parameters for 4d and 5d metals, and the missing ones in 4p, 5p and 6p element blocks are presented and analyzed for their quality. Comparisons to theory and expt. show that sTDA-xTB provides similar good results as for the elements in the original publication with an av. deviation of excitation energies of 0.3-0.5 eV.
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30. 30
  Cho, Y.; Bintrim, S. J.; Berkelbach, T. C. A simplified GW/BSE approach for charged and neutral excitation energies of large molecules and nanomaterials ; 2021, arXiv:2109.04421. DOI: 10.48550/arXiv.2109.04421 .
  There is no corresponding record for this reference.
31. 31
  Rüger, R.; van Lenthe, E.; Heine, T.; Visscher, L. Tight-binding approximations to time-dependent density functional theory ─ A fast approach for the calculation of electronically excited states. J. Chem. Phys. 2016, 144, 184103, DOI: 10.1063/1.4948647
  31
  Tight-binding approximations to time-dependent density functional theory - A fast approach for the calculation of electronically excited states
  Rueger, Robert; van Lenthe, Erik; Heine, Thomas; Visscher, Lucas
  Journal of Chemical Physics (2016), 144 (18), 184103/1-184103/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We propose a new method of calcg. electronically excited states that combines a d. functional theory based ground state calcn. with a linear response treatment that employs approxns. used in the time-dependent d. functional based tight binding (TD-DFTB) approach. The new method termed time-dependent d. functional theory TD-DFT+TB does not rely on the DFTB parametrization and is therefore applicable to systems involving all combinations of elements. We show that the new method yields UV/Vis absorption spectra that are in excellent agreement with computationally much more expensive TD-DFT calcns. Errors in vertical excitation energies are reduced by a factor of two compared to TD-DFTB. (c) 2016 American Institute of Physics.
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32. 32
  Asadi-Aghbolaghi, N.; Pototschnig, J.; Jamshidi, Z.; Visscher, L. Effects of ligands on (de-)enhancement of plasmonic excitations of silver, gold and bimetallic nanoclusters: TD-DFT+TB calculations. Phys. Chem. Chem. Phys. 2021, 23, 17929– 17938, DOI: 10.1039/D1CP03220H
  32
  Effects of ligands on (de-)enhancement of plasmonic excitations of silver, gold and bimetallic nanoclusters: TD-DFT+TB calculations
  Asadi-Aghbolaghi, Narges; Pototschnig, Johann; Jamshidi, Zahra; Visscher, Lucas
  Physical Chemistry Chemical Physics (2021), 23 (33), 17929-17938CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
  Metal nanoclusters can be synthesized in various sizes and shapes and are typically protected with ligands to stabilize them. These ligands can also be used to tune the plasmonic properties of the clusters as the absorption spectrum of a protected cluster can be significantly altered compared to the bare cluster. In this paper, we computationally investigate the influence of thiolate ligands on the plasmonic intensity for silver, gold and alloy clusters. Using time-dependent d. functional theory with tight-binding approxns., TD-DFT+TB, we show that this level of theory can reproduce the broad exptl. spectra of Au144(SR)60 and Ag53Au91(SR)60 (R = CH3) compds. with satisfactory agreement. As TD-DFT+TB does not depend on atom-type parameters we were able to apply this approach on large ligand-protected clusters with various compns. With these calcns. we predict that the effect of ligands on the absorption can be a quenching as well as an enhancement. We furthermore show that it is possible to unambiguously identify the plasmonic peaks by the scaled Coulomb kernel technique and explain the influence of ligands on the intensity (de-)enhancement by analyzing the plasmonic excitations in terms of the dominant orbital contributions.
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33. 33
  Van Caillie, C.; Amos, R. D. Geometric derivatives of excitation energies using SCF and DFT. Chem. Phys. Lett. 1999, 308, 249– 255, DOI: 10.1016/S0009-2614(99)00646-6
  33
  Geometric derivatives of excitation energies using SCF and DFT
  Van Caillie, Carole; Amos, Roger D.
  Chemical Physics Letters (1999), 308 (3,4), 249-255CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)
  There is increasing interest in using the methods of time-dependent d. functional theory to calc. electronic excitation energies. We have implemented an analytic gradient method to find the geometric derivs. of the excitation energies. When added to the gradient for the ground state, this yields the excited-state energy derivs. This enables the efficient generation and searching of excited-state potential energy surfaces to obtain excited-state geometries and other properties. The initial implementation is for SCF methods and for the local d. approxn. Some examples of excited-state geometry optimizations are given.
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34. 34
  Van Caillie, C.; Amos, R. D. Geometric derivatives of density functional theory excitation energies using gradient-corrected functionals. Chem. Phys. Lett. 2000, 317, 159– 164, DOI: 10.1016/S0009-2614(99)01346-9
  34
  Geometric derivatives of density functional theory excitation energies using gradient-corrected functionals
  Van Caillie, C.; Amos, R. D.
  Chemical Physics Letters (2000), 317 (1,2), 159-164CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)
  D. functional theory (DFT) is having increasing success in predicting excitation energies using the methods of time-dependent DFT. As a result, it should be possible to generate potential energy surfaces for excited states by adding the excitation energy, as a function of geometry, to the ground-state energy. It is easier to find stationary points such as min. and transition states if the gradient of the energy is known. The present Letter extends earlier work on the gradients on excited-state surfaces using SCF and LDA (local d. approxn.) methods, to use gradient-cor. and hybrid functionals. Some examples of geometry optimizations are given.
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35. 35
  Furche, F.; Ahlrichs, R. Adiabatic time-dependent density functional methods for excited state properties. J. Chem. Phys. 2002, 117, 7433– 7447, DOI: 10.1063/1.1508368
  35
  Adiabatic time-dependent density functional methods for excited state properties
  Furche, Filipp; Ahlrichs, Reinhart
  Journal of Chemical Physics (2002), 117 (16), 7433-7447CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  This work presents theory, implementation, and validation of excited state properties obtained from time-dependent d. functional theory (TDDFT). Based on a fully variational expression for the excited state energy, a compact derivation of first order properties is given. We report an implementation of analytic excited state gradients and charge moments for local, gradient cor., and hybrid functionals, as well as for the CI singles (CIS) and time-dependent Hartree-Fock (TDHF) methods. By exploiting analogies to ground state energy and gradient calcns., efficient techniques can be transferred to excited state methods. Benchmark results demonstrate that, for low-lying excited states, geometry optimizations are not substantially more expensive than for the ground state, independent of the mol. size. We assess the quality of calcd. adiabatic excitation energies, structures, dipole moments, and vibrational frequencies by comparison with accurate exptl. data for a variety of excited states and mols. Similar trends are obsd. for adiabatic excitation energies as for vertical ones. TDDFT is more robust than CIS and TDHF, in particular, for geometries differing significantly from the ground state min. The TDDFT excited state structures, dipole moments, and vibrational frequencies are of a remarkably high quality, which is comparable to that obtained in ground state d. functional calcns. Thus, yielding considerably more accurate results at similar computational cost, TDDFT rivals CIS as a std. method for calcg. excited state properties in larger mols.
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36. 36
  Furche, F.; Ahlrichs, R. Erratum: ”Time-dependent density functional methods for excited state properties” [J. Chem. Phys. 117, 7433 (2002)]. J. Chem. Phys. 2004, 121, 12772– 12773, DOI: 10.1063/1.1824903
  36
  Adiabatic time-dependent density functional methods for excited state properties. [Erratum to document cited in CA138:044999]
  Furche, Filipp; Ahlrichs, Reinhart
  Journal of Chemical Physics (2004), 121 (24), 12772-12773CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  A factor 1/2 is missing in the expression for the diagonal elements of the energy weighted difference d. matrix W throughout the paper. The cor. versions of Equations (24), (A4), (A10), and (A13) and Tables V, VI, and VIII are given. The implementation and results presented in the paper were based on the correct expressions. Some of the calcd. spectroscopic consts. of the 1 1.sum.u- state of N2, of the 1 1.sum.u+ state of Mg2, and of the 2 1.sum.+ state of CuH in Tables V, VI, and VII are incorrect and should be replaced by the values in the cor. equations. The conclusions are not affected by these changes.
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37. 37
  Hutter, J. Excited state nuclear forces from the Tamm-Dancoff approximation to time-dependent density functional theory within the plane wave basis set framework. J. Chem. Phys. 2003, 118, 3928– 3934, DOI: 10.1063/1.1540109
  37
  Excited state nuclear forces from the Tamm-Dancoff approximation to time-dependent density functional theory within the plane wave basis set framework
  Hutter, Jurg
  Journal of Chemical Physics (2003), 118 (9), 3928-3934CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  An efficient formulation of time-dependent linear response d. functional theory for the use within the plane wave basis set framework is presented. The method avoids the transformation of the Kohn-Sham matrix into the canonical basis and refs. virtual orbitals only through a projection operator. Using a Lagrangian formulation nuclear derivs. of excited state energies within the Tamm-Dancoff approxn. are derived. The algorithms were implemented into a pseudo potential/plane wave code and applied to the calcn. of adiabatic excitation energies, optimized geometries and vibrational frequencies of three low lying states of formaldehyde. An overall good agreement with other time-dependent d. functional calcns., multireference CI calcns. and exptl. data was found.
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38. 38
  Petrenko, T.; Kossmann, S.; Neese, F. Efficient time-dependent density functional theory approximations for hybrid density functionals: Analytical gradients and parallelization. J. Chem. Phys. 2011, 134, 054116, DOI: 10.1063/1.3533441
  38
  Efficient time-dependent density functional theory approximations for hybrid density functionals: Analytical gradients and parallelization
  Petrenko, Taras; Kossmann, Simone; Neese, Frank
  Journal of Chemical Physics (2011), 134 (5), 054116/1-054116/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We present the implementation of efficient approxns. to time-dependent d. functional theory (TDDFT) within the Tamm-Dancoff approxn. (TDA) for hybrid d. functionals. For the calcn. of the TDDFT/TDA excitation energies and anal. gradients, we combine the resoln. of identity (RI-J) algorithm for the computation of the Coulomb terms and the recently introduced "chain of spheres exchange" (COSX) algorithm for the calcn. of the exchange terms. It is shown that for extended basis sets, the RIJCOSX approxn. leads to speedups of up to 2 orders of magnitude compared to traditional methods, as demonstrated for hydrocarbon chains. The accuracy of the adiabatic transition energies, excited state structures, and vibrational frequencies is assessed on a set of 27 excited states for 25 mols. with the CI singles and hybrid TDDFT/TDA methods using various basis sets. Compared to the canonical values, the typical error in transition energies is of the order of 0.01 eV. Similar to the ground-state results, excited state equil. geometries differ by less than 0.3 pm in the bond distances and 0.5° in the bond angles from the canonical values. The typical error in the calcd. excited state normal coordinate displacements is of the order of 0.01, and relative error in the calcd. excited state vibrational frequencies is less than 1%. The errors introduced by the RIJCOSX approxn. are, thus, insignificant compared to the errors related to the approx. nature of the TDDFT methods and basis set truncation. For TDDFT/TDA energy and gradient calcns. on Ag-TB2-helicate (156 atoms, 2732 basis functions), it is demonstrated that the COSX algorithm parallelizes almost perfectly (speedup ∼26-29 for 30 processors). The exchange-correlation terms also parallelize well (speedup ∼27-29 for 30 processors). The soln. of the Z-vector equations shows a speedup of ∼24 on 30 processors. The parallelization efficiency for the Coulomb terms can be somewhat smaller (speedup ∼15-25 for 30 processors), but their contribution to the total calcn. time is small. Thus, the parallel program completes a Becke3-Lee-Yang-Parr energy and gradient calcn. on the Ag-TB2-helicate in less than 4 h on 30 processors. We also present the necessary extension of the Lagrangian formalism, which enables the calcn. of the TDDFT excited state properties in the frozen-core approxn. The algorithms described in this work are implemented into the ORCA electronic structure system. (c) 2011 American Institute of Physics.
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39. 39
  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.9b00659
  39
  Development and Implementation of Excited-State Gradients for Local Hybrid Functionals
  Grotjahn, Robin; Furche, Filipp; Kaupp, Martin
  Journal 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.
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40. 40
  Kretz, B.; Egger, D. A. Accurate Molecular Geometries in Complex Excited-State Potential Energy Surfaces from Time-Dependent Density Functional Theory. J. Chem. Theory Comput. 2021, 17, 357– 366, DOI: 10.1021/acs.jctc.0c00858
  40
  Accurate Molecular Geometries in Complex Excited-State Potential Energy Surfaces from Time-Dependent Density Functional Theory
  Kretz, Bernhard; Egger, David A.
  Journal of Chemical Theory and Computation (2021), 17 (1), 357-366CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  The interplay of electronic excitations and structural changes in mols. impacts nonradiative decay and charge transfer in the excited state, thus influencing excited-state lifetimes and photocatalytic reaction rates in optoelectronic and energy devices. To capture such effects requires computational methods providing an accurate description of excited-state potential energy surfaces and geometries. We suggest time-dependent d. functional theory using optimally tuned range-sepd. hybrid (OT-RSH) functionals as an accurate approach to obtain excited-state mol. geometries. We show that OT-RSH provides accurate mol. geometries in excited-state potential energy surfaces that are complex and involve an interplay of local and charge-transfer excitations, for which conventional semilocal and hybrid functionals fail. At the same time, the nonempirical OT-RSH approach maintains the high accuracy of parametrized functionals (e.g., B3LYP) for predicting excited-state geometries of small org. mols. showing valence excited states.
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41. 41
  Sokolov, M.; Bold, B. M.; Kranz, J. J.; Höfener, S.; Niehaus, T. A.; Elstner, M. Analytical Time-Dependent Long-Range Corrected Density Functional Tight Binding (TD-LC-DFTB) Gradients in DFTB+: Implementation and Benchmark for Excited-State Geometries and Transition Energies. J. Chem. Theory Comput. 2021, 17, 2266– 2282, DOI: 10.1021/acs.jctc.1c00095
  41
  Analytical Time-Dependent Long-Range Corrected Density Functional Tight Binding (TD-LC-DFTB) Gradients in DFTB+: Implementation and Benchmark for Excited-State Geometries and Transition Energies
  Sokolov, Monja; Bold, Beatrix M.; Kranz, Julian J.; Hoefener, Sebastian; Niehaus, Thomas A.; Elstner, Marcus
  Journal of Chemical Theory and Computation (2021), 17 (4), 2266-2282CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  The absorption and emission of light is a ubiquitous process in chem. and biol. processes, making a theor. description inevitable for understanding and predicting such properties. Although ab initio and DFT methods are capable of describing excited states with good accuracy in many cases, the investigation of dynamical processes and the need to sample the phase space in complex systems often requires methods with reduced computational costs but still sufficient accuracy. In the present work, we report the derivation and implementation of anal. nuclear gradients for time-dependent long-range cor. d. functional tight binding (TD-LC-DFTB) in the DFTB+ program. The accuracy of the TD-LC-DFTB potential-energy surfaces is benchmarked for excited-state geometries and adiabatic as well as vertical transition energies. The benchmark set consists of more than 100 org. mols. taken as subsets from available benchmark sets. The reported method yields a mean deviation of 0.31 eV for adiabatic excitation energies with respect to CC2. In order to study more subtle effects, seminumerical second derivs. based on the anal. gradients are employed to simulate vibrationally resolved UV/vis spectra. This extensive test exhibits few problematic cases, which can be traced back to the parametrization of the repulsive potential.
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42. 42
  Kühne, T. D.; Iannuzzi, M.; Del Ben, M.; Rybkin, V. V.; Seewald, P.; Stein, F.; Laino, T.; Khaliullin, R. Z.; Schütt, O.; Schiffmann, F.; Golze, D.; Wilhelm, J.; Chulkov, S.; Bani-Hashemian, M. H.; Weber, V.; Borštnik, U.; Taillefumier, M.; Jakobovits, A. S.; Lazzaro, A.; Pabst, H.; Müller, T.; Schade, R.; Guidon, M.; Andermatt, S.; Holmberg, N.; Schenter, G. K.; Hehn, A.; Bussy, A.; Belleflamme, F.; Tabacchi, G.; Glöß, A.; Lass, M.; Bethune, I.; Mundy, C. J.; Plessl, C.; Watkins, M.; VandeVondele, J.; Krack, M.; Hutter, J. CP2K: An electronic structure and molecular dynamics software package - Quickstep: Efficient and accurate electronic structure calculations. J. Chem. Phys. 2020, 152, 194103, DOI: 10.1063/5.0007045
  42
  CP2K: An electronic structure and molecular dynamics software package - Quickstep: Efficient and accurate electronic structure calculations
  Kuehne, Thomas D.; Iannuzzi, Marcella; Del Ben, Mauro; Rybkin, Vladimir V.; Seewald, Patrick; Stein, Frederick; Laino, Teodoro; Khaliullin, Rustam Z.; Schuett, Ole; Schiffmann, Florian; Golze, Dorothea; Wilhelm, Jan; Chulkov, Sergey; Bani-Hashemian, Mohammad Hossein; Weber, Valery; Borstnik, Urban; Taillefumier, Mathieu; Jakobovits, Alice Shoshana; Lazzaro, Alfio; Pabst, Hans; Mueller, Tiziano; Schade, Robert; Guidon, Manuel; Andermatt, Samuel; Holmberg, Nico; Schenter, Gregory K.; Hehn, Anna; Bussy, Augustin; Belleflamme, Fabian; Tabacchi, Gloria; Gloess, Andreas; Lass, Michael; Bethune, Iain; Mundy, Christopher J.; Plessl, Christian; Watkins, Matt; VandeVondele, Joost; Krack, Matthias; Hutter, Juerg
  Journal of Chemical Physics (2020), 152 (19), 194103CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  A review. CP2K is an open source electronic structure and mol. dynamics software package to perform atomistic simulations of solid-state, liq., mol., and biol. systems. It is esp. aimed at massively parallel and linear-scaling electronic structure methods and state-of-the-art ab initio mol. dynamics simulations. Excellent performance for electronic structure calcns. is achieved using novel algorithms implemented for modern high-performance computing systems. This review revisits the main capabilities of CP2K to perform efficient and accurate electronic structure simulations. The emphasis is put on d. functional theory and multiple post-Hartree-Fock methods using the Gaussian and plane wave approach and its augmented all-electron extension. (c) 2020 American Institute of Physics.
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43. 43
  VandeVondele, J.; Krack, M.; Mohamed, F.; Parrinello, M.; Chassaing, T.; Hutter, J. Quickstep: Fast and accurate density functional calculations using a mixed Gaussian and plane waves approach. Comput. Phys. Commun. 2005, 167, 103– 128, DOI: 10.1016/j.cpc.2004.12.014
  43
  QUICKSTEP: fast and accurate density functional calculations using a mixed Gaussian and plane waves approach
  VandeVondele, Joost; Krack, Matthias; Mohamed, Fawzi; Parrinello, Michele; Chassaing, Thomas; Hutter, Juerg
  Computer Physics Communications (2005), 167 (2), 103-128CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)
  We present the Gaussian and plane waves (GPW) method and its implementation in which is part of the freely available program package CP2K. The GPW method allows for accurate d. functional calcns. in gas and condensed phases and can be effectively used for mol. dynamics simulations. We show how derivs. of the GPW energy functional, namely ionic forces and the Kohn-Sham matrix, can be computed in a consistent way. The computational cost of computing the total energy and the Kohn-Sham matrix is scaling linearly with the system size, even for condensed phase systems of just a few tens of atoms. The efficiency of the method allows for the use of large Gaussian basis sets for systems up to 3000 atoms, and we illustrate the accuracy of the method for various basis sets in gas and condensed phases. Agreement with basis set free calcns. for single mols. and plane wave based calcns. in the condensed phase is excellent. Wave function optimization with the orbital transformation technique leads to good parallel performance, and outperforms traditional diagonalisation methods. Energy conserving Born-Oppenheimer dynamics can be performed, and a highly efficient scheme is obtained using an extrapolation of the d. matrix. We illustrate these findings with calcns. using commodity PCs as well as supercomputers.
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44. 44
  Lippert, G.; Hutter, J.; Parrinello, M. A hybrid Gaussian and plane wave density functional scheme. Mol. Phys. 1997, 92, 477– 488, DOI: 10.1080/00268979709482119
  44
  A hybrid Gaussian and plane wave density functional scheme
  Lippert, Gerald; Hutter, Juerg; Parrinello, Michele
  Molecular Physics (1997), 92 (3), 477-487CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis)
  A d.-functional theory-based algorithm for periodic and nonperiodic ab initio calcns. is presented. This scheme uses pseudopotentials in order to integrate out the core electrons from the problem. The valence pseudo wave functions are expanded in Gaussian-type orbitals and the d. is represented in a plane wave auxiliary basis. The Gaussian basis functions make it possible to use the efficient anal. integration schemes and screening algorithms of quantum chem. Novel recursion relations are developed for the calcn. of the matrix elements of the d.-dependent Kohn-Sham self-consistent potential. At the same time the use of a plane wave basis for the electron d. permits efficient calcn. of the Hartree energy using fast Fourier transforms, thus circumventing one of the major bottlenecks of std. Gaussian based calcns. Furthermore, this algorithm avoids the fitting procedures that go along with intermediate basis sets for the charge d. The performance and accuracy of this new scheme are discussed and selected examples are given.
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45. 45
  Iannuzzi, M.; Chassaing, T.; Wallman, T.; Hutter, J. Ground and Excited State Density Functional Calculations with the Gaussian and Augmented-Plane-Wave Method. CHIMIA Int. J. Chem. 2005, 59, 499– 503, DOI: 10.2533/000942905777676164
  45
  Ground and excited state density functional calculations with the Gaussian and Augmented-Plane-Wave method
  Iannuzzi, Marcella; Chassaing, Thomas; Wallman, Thomas; Hutter, Jurg
  Chimia (2005), 59 (7-8), 499-503CODEN: CHIMAD; ISSN:0009-4293. (Swiss Chemical Society)
  The calcn. of the electronic structure of large systems by methods based on d. functional theory has recently gained a central role in mol. simulations. However, the extensive study of quantities like excited states and related properties is still out of reach due to high computational costs. We present a new implementation of a hybrid method, the Gaussian and APW (GAPW) method, where the electronic d. is partitioned in hard and soft contributions. The former are local terms naturally expanded in a Gaussian basis, whereas the soft contributions are expanded in plane-waves by using a low energy cutoff, without loss in accuracy, even for all-electron calcns. For the calcn. of excitation energies a recently developed, time-dependent d. functional response theory (TD-DFRT) technique is joined with the GAPW procedure. We demonstrate the accuracy of the method by comparison with std. quantum chem. calcns. for a set of small mols. To highlight the performance and efficiency of GAPW we show calcns. on systems with several thousands of basis functions.
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46. 46
  Strand, J.; Chulkov, S. K.; Watkins, M. B.; Shluger, A. L. First principles calculations of optical properties for oxygen vacancies in binary metal oxides. J. Chem. Phys. 2019, 150, 044702, DOI: 10.1063/1.5078682
  46
  First principles calculations of optical properties for oxygen vacancies in binary metal oxides
  Strand, Jack; Chulkov, Sergey K.; Watkins, Matthew B.; Shluger, Alexander L.
  Journal of Chemical Physics (2019), 150 (4), 044702/1-044702/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Using an advanced computational methodol. implemented in CP2K, a non-local PBE0-TC-LRC d. functional and the recently implemented linear response formulation of the Time-dependent D. Functional Theory equations, we test the interpretation of the optical absorption and photoluminescence signatures attributed by previous exptl. and theor. studies to O-vacancies in two widely used oxides-cubic MgO and monoclinic (m)-HfO2. The results obtained in large periodic cells including up to 1000 atoms emphasize the importance of accurate predictions of defect-induced lattice distortions. They confirm that optical transitions of O-vacancies in 0, +1, and +2 charge states in MgO all have energies close to 5 eV. We test the models of photoluminescence of O-vacancies proposed in the literature. The photoluminescence of V+2O centers in m-HfO2 is predicted to peak at 3.7 eV and originate from radiative tunneling transition between a V+1O center and a self-trapped hole created by the 5.2 eV excitation. (c) 2019 American Institute of Physics.
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47. 47
  Poli, E.; Elliott, J. D.; Chulkov, S. K.; Watkins, M. B.; Teobaldi, G. The Role of Cation-Vacancies for the Electronic and Optical Properties of Aluminosilicate Imogolite Nanotubes: A Non-local, Linear-Response TDDFT Study. Front. Chem. 2019, 7, 210, DOI: 10.3389/fchem.2019.00210
  47
  Cation-vacancies for the electronic and optical properties of aluminosilicate imogolite nanotubes and a non-local, linear-response TDDFT study
  Poli, Emiliano; Elliott, Joshua D.; Chulkov, Sergey K.; Watkins, Matthew B.; Teobaldi, Gilberto
  Frontiers in Chemistry (Lausanne, Switzerland) (2019), 7 (), 210CODEN: FCLSAA; ISSN:2296-2646. (Frontiers Media S.A.)
  We report a combined non-local (PBE-TC-LRC) D. Functional Theory (DFT) and linear-response time-dependent DFT (LR-TDDFT) study of the structural, electronic, and optical properties of the cation-vacancy based defects in aluminosilicate (AlSi) imogolite nanotubes (Imo-NTs) that have been recently proposed on the basis of NMR (NMR) expts. Following numerical detn. of the smallest AlSi Imo-NT model capable of accommodating the defect-induced relaxation with negligible finite-size errors, we analyze the defect-induced structural deformations in the NTs and ensuing changes in the NTs' electronic structure. The NMR-derived defects are found to introduce both shallow and deep occupied states in the pristine NTs' band gap (BG). These BG states are found to be highly localized at the defect site. No empty defect-state is modeled for any of the considered systems. LR-TDDFT simulation of the defects reveal increased low-energy optical absorbance for all but one defects, with the appearance of optically active excitations at energies lower than for the defect-free NT. These results enable interpretation of the low-energy tail in the exptl. UV-vis spectra for AlSi NTs as being due to the defects. Finally, the PBE-TC-LRC-approximated exciton binding energy for the defects' optical transitions is found to be substantially lower (up to 0.8 eV) than for the pristine defect-free NT's excitations (1.1 eV).
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48. 48
  Sternheimer, R. On Nuclear Quadrupole Moments. Phys. Rev. 1951, 84, 244– 253, DOI: 10.1103/PhysRev.84.244
  48
  Nuclear quadrupole moments
  Sternheimer, R.
  Physical Review (1951), 84 (), 244-53CODEN: PHRVAO; ISSN:0031-899X.
  cf. C.A. 45, 443i. The correction to nuclear quadrupole moments on account of the quadrupole moment induced in the electron shells was obtained by solving the Schroedinger equation for the perturbed core wave functions for Li, Al, and Cl. The correction factor by which the average 〈l/r3〉 over the valence electron function in the equation for the quadrupole coupling should be multiplied to take account of the induced effect is 1.11, 0.83, and 0.68, resp. The previously described Thomas-Fermi calcn. of this effect was carried out for F, Cl, Cu, Br, Y, Ag, I, La, Lu, Pt, Tl, At, and U.
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49. 49
  Baroni, S.; de Gironcoli, S.; Dal Corso, A.; Giannozzi, P. Phonons and related crystal properties from density-functional perturbation theory. Rev. Mod. Phys. 2001, 73, 515– 562, DOI: 10.1103/RevModPhys.73.515
  49
  Phonons and related crystal properties from density-functional perturbation theory
  Baroni, Stefano; De Gironcoli, Stefano; Dal Corso, Andrea; Giannozzi, Paolo
  Reviews of Modern Physics (2001), 73 (2), 515-562CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)
  This article reviews with many refs. the current status of lattice-dynamical calcns. in crystals, using d.-functional perturbation theory, with emphasis on the plane-wave pseudopotential method. Several specialized topics are treated, including the implementation for metals, the calcn. of the response to macroscopic elec. fields and their relevance to long-wavelength vibrations in polar materials, the response to strain deformations, and higher-order responses. The success of this methodol. is demonstrated with a no. of applications existing in the literature.
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50. 50
  Sternheimer, R. M. Electronic Polarizabilities of Ions from the Hartree-Fock Wave Functions. Phys. Rev. 1954, 96, 951– 968, DOI: 10.1103/PhysRev.96.951
  50
  Electron polarizabilities of ions from the Hartree-Fock wave functions
  Sternheimer, R. M.
  Physical Review (1954), 96 (), 951-68CODEN: PHRVAO; ISSN:0031-899X.
  cf. ibid. 95, 655. The electronic polarizability α was calcd. for several ions by obtaining the perturbation of the wave functions by an external field from a numerical solution of the differential equation satisfied by the perturbation. For the He-like ions an analytic solution was obtained by using the wave functions of L.ovrddot.owdin (C.A. 47, 9135c). The calcd. values of α are, in general, 1 to 1.5 times the observed values. For several ions values were calcd. for the quadrupole polarizability, which measures the quadrupole moment induced in the ion by an external charge. The effect of the dipole moment induced in the ion on the elec. field at the nucleus is discussed.
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51. 51
  Budzak, S.; Scalmani, G.; Jacquemin, D. Accurate Excited-State Geometries: A CASPT2 and Coupled-Cluster Reference Database for Small Molecules. J. Chem. Theory Comput. 2017, 13, 6237– 6252, DOI: 10.1021/acs.jctc.7b00921
  51
  Accurate Excited-State Geometries: A CASPT2 and Coupled-Cluster Reference Database for Small Molecules
  Budzak, Simon; Scalmani, Giovanni; Jacquemin, Denis
  Journal of Chemical Theory and Computation (2017), 13 (12), 6237-6252CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  We present an investigation of the excited-state structural parameters detd. for a large set of small compds. with the dual goals of defining ref. values for further works and assessing the quality of the geometries obtained with relatively cheap computational approaches. In the first stage, we compare the excited-state geometries obtained with ADC(2), CC2, CCSD, CCSDR(3), CC3, and CASPT2 and large at. basis sets. It is found that CASPT2 and CC3 results are generally in very good agreement with one another (typical differences of ca. 3 × 10-3 Å) when all electrons are correlated and when the aug-cc-pVTZ at. basis set is employed with both methods. In a second stage, a statistical anal. reveals that, on the one hand, the excited-state (ES) bond lengths are much more sensitive to the selected level of theory than their ground-state (GS) counterparts and, on the other hand, that CCSDR(3) is probably the most cost-effective method delivering accurate structures. Indeed, CCSD tends to provide too compact multiple bond lengths on an almost systematic basis, whereas both CC2 and ADC(2) tend to exaggerate these bond distances, with more erratic error patterns, esp. for the latter method. The deviations are particularly marked for the polarized CO and CN bonds, as well as for the puckering angle in formaldehyde homologues. In the last part of this contribution, we provide a series of CCSDR(3) GS and ES geometries of medium-sized mols. to be used as refs. in further investigations.
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52. 52
  Ongari, D.; Yakutovich, A. V.; Talirz, L.; Smit, B. Building a Consistent and Reproducible Database for Adsorption Evaluation in Covalent–Organic Frameworks. ACS Cent. Sci. 2019, 5, 1663– 1675, DOI: 10.1021/acscentsci.9b00619
  52
  Building a Consistent and Reproducible Database for Adsorption Evaluation in Covalent-Organic Frameworks
  Ongari, Daniele; Yakutovich, Aliaksandr V.; Talirz, Leopold; Smit, Berend
  ACS Central Science (2019), 5 (10), 1663-1675CODEN: ACSCII; ISSN:2374-7951. (American Chemical Society)
  We present a workflow that traces the path from the bulk structure of a cryst. material to assessing its performance in carbon capture from coal's postcombustion flue gases. This workflow is applied to a database of 324 covalent-org. frameworks (COFs) reported in the literature, to characterize their CO2 adsorption properties using the following steps: (1) optimization of the crystal structure (at. positions and unit cell) using d. functional theory, (2) fitting at. point charges based on the electron d., (3) characterizing the pore geometry of the structures before and after optimization, (4) computing carbon dioxide and nitrogen isotherms using grand canonical Monte Carlo simulations with an empirical interaction potential, and finally, (5) assessing the CO2 parasitic energy via process modeling. The full workflow has been encoded in the Automated Interactive Infrastructure and Database for Computational Science (AiiDA). Both the workflow and the automatically generated provenance graph of our calcns. are made available on the Materials Cloud, allowing peers to inspect every input parameter and result along the workflow, download structures and files at intermediate stages, and start their research right from where this work has left off. In particular, our set of CURATED (Clean, Uniform, and Refined with Automatic Tracking from Exptl. Database) COFs, having optimized geometry and high-quality DFT-derived point charges, are available for further investigations of gas adsorption properties. We plan to update the database as new COFs are being reported. An automated and reproducible computational workflow is proposed, to systematically optimize the geometry of covalent-org. frameworks and evaluate their performances for carbon capture and storage.
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53. 53
  Ongari, D.; Talirz, L.; Smit, B. Too Many Materials and Too Many Applications: An Experimental Problem Waiting for a Computational Solution. ACS Cent. Sci. 2020, 6, 1890– 1900, DOI: 10.1021/acscentsci.0c00988
  53
  Too Many Materials and Too Many Applications: An Experimental Problem Waiting for a Computational Solution
  Ongari, Daniele; Talirz, Leopold; Smit, Berend
  ACS Central Science (2020), 6 (11), 1890-1900CODEN: ACSCII; ISSN:2374-7951. (American Chemical Society)
  A review. Finding the best material for a specific application is the ultimate goal of materials discovery. However, there is also the reverse problem: when exptl. groups discover a new material, they would like to know all the possible applications this material would be promising for. Computational modeling can aim to fulfill this expectation, thanks to the sustained growth of computing power and the collective engagement of the scientific community in developing more efficient and accurate workflows for predicting materials' performances. We discuss the impact that reproducibility and automation of the modeling protocols have on the field of gas adsorption in nanoporous crystals. We envision a platform that combines these tools and enables effective matching between promising materials and industrial applications. We identify the opportunity for a computational platform for matching nanoporous materials and gas-related applications, motivating the development of automated and reproducible computational workflows.
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54. 54
  Hirata, S.; Head-Gordon, M. Time-dependent density functional theory within the Tamm-Dancoff approximation. Chem. Phys. Lett. 1999, 314, 291– 299, DOI: 10.1016/S0009-2614(99)01149-5
  54
  Time-dependent density functional theory within the Tamm-Dancoff approximation
  Hirata, S.; Head-Gordon, M.
  Chemical Physics Letters (1999), 314 (3,4), 291-299CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)
  A computationally simple method for mol. excited states, namely, the Tamm-Dancoff approxn. to time-dependent d. functional theory, is proposed and implemented. This method yields excitation energies for several closed- and open-shell mols. that are essentially of the same quality as those obtained from time-dependent d. functional theory itself, when the same exchange-correlation functional is used.
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55. 55
  Casida, M. E. In Recent Advances in Density Functional Methods, Part I; Chong, D. P., Ed.; World Scientific: Singapore, 1995; p 155.
  There is no corresponding record for this reference.
56. 56
  Guidon, M.; Hutter, J.; VandeVondele, J. Auxiliary Density Matrix Methods for Hartree-Fock Exchange Calculations. J. Chem. Theory Comput. 2010, 6, 2348– 2364, DOI: 10.1021/ct1002225
  56
  Auxiliary Density Matrix Methods for Hartree-Fock Exchange Calculations
  Guidon, Manuel; Hutter, Jurg; Vande Vondele, Joost
  Journal of Chemical Theory and Computation (2010), 6 (8), 2348-2364CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  The calcn. of Hartree-Fock exchange (HFX) is computationally demanding for large systems described with high-quality basis sets. In this work, we show that excellent performance and good accuracy can nevertheless be obtained if an auxiliary d. matrix is employed for the HFX calcn. Several schemes to derive an auxiliary d. matrix from a high-quality d. matrix are discussed. Key to the accuracy of the auxiliary d. matrix methods (ADMM) is the use of a correction based on std. generalized gradient approxns. for HFX. ADMM integrates seamlessly in existing HFX codes and, in particular, can be employed in linear scaling implementations. Demonstrating the performance of the method, the effect of HFX on the structure of liq. water is investigated in detail using Born-Oppenheimer mol. dynamics simulations (300 ps) of a system of 64 mols. Representative for large systems are calcns. on a solvated protein (Rubredoxin), for which ADMM outperforms the corresponding std. HFX implementation by approx. a factor 20.
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57. 57
  Ghosh, D. C.; Islam, N. Whether electronegativity and hardness are manifest two different descriptors of the one and the same fundamental property of atoms? @ TA quest. Int. J. Quantum Chem. 2011, 111, 40– 51, DOI: 10.1002/qua.22415
  57
  Whether electronegativity and hardness are manifest two different descriptors of the one and the same fundamental property of atoms-A quest
  Ghosh, Dulal C.; Islam, Nazmul
  International Journal of Quantum Chemistry (2011), 111 (1), 40-51CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
  In this report, we have tried to reveal that there is much conceptual commonality between the two fundamental theor. descriptors of chem. and physics-the electronegativity and the hardness. The phys. hardness was introduced and theorized by condensed matter physicists. The chem. hardness was introduced by chemists to generalize and rationalize the HSAB principle. We have tried to establish that the phys. hardness and the chem. hardness with evolution of time have converged to one and the same general principle-the hardness. We have also tried to understand the phys. basis and operational significance of another very important descriptor arising out of theor. constructs of chem.-the electronegativity. We have relied upon the fact that, since these descriptors are not observables, there is no possibility of their quantum mech. evaluation. These descriptors, therefore, should be and must be reified before suggesting ansatz for their evaluation. We have dwelt at length upon the effort of d. functional definition and evaluation of electronegativity and hardness and discovered the inherent inner contradiction of the theory and measurement. We have also noted that a good no. of scientists hold the opinion that the d. functional formula of electronegativity is χ = I and that of hardness is η = I, where I is the ionization potential of the chem. system. This study concludes that the two fundamental descriptors-the hardness and the electronegativity originate from the same source-the electron attracting power of the screened nucleus upon valence electrons and discovers the surprising result that if one measures hardness, the electronegativity is simultaneously measured and vice versa. We have also explored the ansatz of semiempirical evaluation of electronegativity and hardness of at. systems involving the radius of the atoms. We have noted that the ansatz for electronegativity and hardness is the same. To justify our statement that if one measures hardness, the electronegativity is simultaneously measured and vice versa, we have used the evaluated set of hardness for 103 elements of the periodic table as a scale of electronegativity and found that such set of hardness data satisfies the sine qua non of a satisfactory scale of electronegativity. The electronegativity and the hardness are two different appearances of the one and the same fundamental property of atoms. They are different and also nondifferent. They are different in their fields of application. They are nondifferent when we discuss the basic philosophical structures of their origin and development. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011.
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58. 58
  Marx, D.; Hutter, J. In Modern Methods and Algorithms of Quantum Chemistry; Grotendorst, J., Ed.; John von Neumann Institute for Computing, Forschungszentrum Jülich: Jülich, Germany, 2000; pp 301– 449 (first edition, paperback, ISBN 3-00-005618-1) or pp 329–477 (second edition, hardcover, ISBN 3-00-005834-6), see http://www.theochem.rub.de/go/cprev.html.
  There is no corresponding record for this reference.
59. 59
  Essmann, U.; Perera, L.; Berkowitz, M. L.; Darden, T.; Lee, H.; Pedersen, L. G. A smooth particle mesh Ewald method. J. Chem. Phys. 1995, 103, 8577– 8593, DOI: 10.1063/1.470117
  59
  A smooth particle mesh Ewald method
  Essmann, Ulrich; Perera, Lalith; Berkowitz, Max L.; Darden, Tom; Lee, Hsing; Pedersen, Lee G.
  Journal of Chemical Physics (1995), 103 (19), 8577-93CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The previously developed particle mesh Ewald method is reformulated in terms of efficient B-spline interpolation of the structure factors. This reformulation allows a natural extension of the method to potentials of the form 1/rp with p ≥ 1. Furthermore, efficient calcn. of the virial tensor follows. Use of B-splines in the place of Lagrange interpolation leads to analytic gradients as well as a significant improvement in the accuracy. The authors demonstrate that arbitrary accuracy can be achieved, independent of system size N, at a cost that scales as N log(N). For biomol. systems with many thousands of atoms and this method permits the use of Ewald summation at a computational cost comparable to that of a simple truncation method of 10 Å or less.
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60. 60
  Mayhall, N. J.; Raghavachari, K.; Hratchian, H. P. ONIOM-based QM:QM electronic embedding method using Löwdin atomic charges: Energies and analytic gradients. J. Chem. Phys. 2010, 132, 114107, DOI: 10.1063/1.3315417
  60
  ONIOM-based QM:QM electronic embedding method using Lowdin atomic charges: Energies and analytic gradients
  Mayhall, Nicholas J.; Raghavachari, Krishnan; Hratchian, Hrant P.
  Journal of Chemical Physics (2010), 132 (11), 114107/1-114107/6CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We report a new quantum mech.:quantum mech. (QM:QM) method which provides explicit electronic polarization of the high-level region by using the Lowdin at. charges from the low-level region. This provides an embedding potential which naturally evolves with changes in nuclear geometry. However, this coupling of the high-level and low-level regions introduces complications in the energy gradient evaluation. Following previous work, we derive and implement efficient gradients where a single set of SCF response equations is solved. We provide results for the calcn. of deprotonation energies of a hydroxylated spherosiloxane cluster (Si8O12H7OH) and the dissocn. energy of a water mol. from a [ZnIm3(H2O)]2+ complex. The Lowdin charge embedding model provides results which are not only an improvement over mech. embedding (no electronic embedding) but which are also resistant to large overpolarization effects which occur more often with Mulliken charge embedding. Finally, a scaled-Loewdin charge embedding method is also presented which provides a method for fine tuning the extent of electronic polarization. (c) 2010 American Institute of Physics.
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61. 61
  Davidson, E. R. The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. J. Comput. Phys. 1975, 17, 87– 94, DOI: 10.1016/0021-9991(75)90065-0
  There is no corresponding record for this reference.
62. 62
  Chassaing, T. Excitation Energy Calculations with TD-DFT. PhD thesis, University of Zurich, Zurich, Switzerland, 2005.
  There is no corresponding record for this reference.
63. 63
  Ryabinkin, I. G.; Nagesh, J.; Izmaylov, A. F. Fast Numerical Evaluation of Time-Derivative Nonadiabatic Couplings for Mixed Quantum–Classical Methods. J. Phys. Chem. Lett. 2015, 6, 4200– 4203, DOI: 10.1021/acs.jpclett.5b02062
  63
  Fast Numerical Evaluation of Time-Derivative Nonadiabatic Couplings for Mixed Quantum-Classical Methods
  Ryabinkin, Ilya G.; Nagesh, Jayashree; Izmaylov, Artur F.
  Journal of Physical Chemistry Letters (2015), 6 (21), 4200-4203CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
  We have developed a numerical differentiation scheme that eliminates evaluation of overlap determinants in calcg. the time-deriv. nonadiabatic couplings (TDNACs). Evaluation of these determinants was the bottleneck in previous implementations of mixed quantum-classical methods using numerical differentiation of electronic wave functions in the Slater determinant representation. The central idea of our approach is, first, to reduce the analytic time derivs. of Slater determinants to time derivs. of MOs and then to apply a finite-difference formula. Benchmark calcns. prove the efficiency of the proposed scheme showing impressive several-order-of-magnitude speedups of the TDNAC calcn. step for midsize mols.
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64. 64
  Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Petersson, G. A.; Nakatsuji, H.; Li, X.; Caricato, M.; Marenich, A. V.; Bloino, J.; Janesko, B. G.; Gomperts, R.; Mennucci, B.; Hratchian, H. P.; Ortiz, J. V.; Izmaylov, A. F.; Sonnenberg, J. L.; Williams-Young, D.; Ding, F.; Lipparini, F.; Egidi, F.; Goings, J.; Peng, B.; Petrone, A.; Henderson, T.; Ranasinghe, D.; Zakrzewski, V. G.; Gao, J.; Rega, N.; Zheng, G.; Liang, W.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Throssell, K.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.; Bearpark, M. J.; Heyd, J. J.; Brothers, E. N.; Kudin, K. N.; Staroverov, V. N.; Keith, T. A.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A. P.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Millam, J. M.; Klene, M.; Adamo, C.; Cammi, R.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Farkas, O.; Foresman, J. B.; Fox, D. J. Gaussian 16, Revision C.01; Gaussian Inc.: Wallingford, CT, 2016.
  There is no corresponding record for this reference.
65. 65
  Weigend, F.; Ahlrichs, R. Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy. Phys. Chem. Chem. Phys. 2005, 7, 3297, DOI: 10.1039/b508541a
  65
  Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy
  Weigend, Florian; Ahlrichs, Reinhart
  Physical Chemistry Chemical Physics (2005), 7 (18), 3297-3305CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
  Gaussian basis sets of quadruple zeta valence quality for Rb-Rn are presented, as well as bases of split valence and triple zeta valence quality for H-Rn. The latter were obtained by (partly) modifying bases developed previously. A large set of more than 300 mols. representing (nearly) all elements-except lanthanides-in their common oxidn. states was used to assess the quality of the bases all across the periodic table. Quantities investigated were atomization energies, dipole moments and structure parameters for Hartree-Fock, d. functional theory and correlated methods, for which we had chosen Moller-Plesset perturbation theory as an example. Finally recommendations are given which type of basis set is used best for a certain level of theory and a desired quality of results.
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66. 66
  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, h.; 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: 10.1063/5.0004635
  66
  TURBOMOLE: Modular program suite for ab initio quantum-chemical and condensed-matter simulations
  Balasubramani, 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.
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67. 67
  Weigend, F.; Furche, F.; Ahlrichs, R. Gaussian basis sets of quadruple zeta valence quality for atoms H-Kr. J. Chem. Phys. 2003, 119, 12753– 12762, DOI: 10.1063/1.1627293
  67
  Gaussian basis sets of quadruple zeta valence quality for atoms H-Kr
  Weigend, Florian; Furche, Filipp; Ahlrichs, Reinhart
  Journal of Chemical Physics (2003), 119 (24), 12753-12762CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We present Gaussian basis sets of quadruple zeta valence quality with a segmented contraction scheme for atoms H to Kr. This extends earlier work on segmented contracted split valence (SV) and triple zeta valence (TZV) basis sets. Contraction coeffs. and orbital exponents are fully optimized in at. Hartree-Fock (HF) calcns. As opposed to other quadruple zeta basis sets, the basis set errors in at. ground-state HF energies are less than 1 mEh and increase smoothly across the Periodic Table, while the no. of primitives is comparably small. Polarization functions are taken partly from previous work, partly optimized in at. MP2 calcns., and for a few cases detd. at the HF level for excited at. states nearly degenerate with the ground state. This leads to basis sets denoted QZVP for HF and d. functional theory (DFT) calcns., and for some atoms to a larger basis recommended for correlated treatments, QZVPP. We assess the performance of the basis sets in mol. HF, DFT, and MP2 calcns. for a sample of diat. and small polyat. mols. by a comparison of energies, bond lengths, and dipole moments with results obtained numerically or using very large basis sets. It is shown that basis sets of quadruple zeta quality are necessary to achieve an accuracy of 1 kcal/mol per bond in HF and DFT atomization energies. For compds. contg. third row as well as alk. and earth alk. metals it is demonstrated that the inclusion of high-lying core orbitals in the active space can be necessary for accurate correlated treatments. The QZVPP basis sets provide sufficient flexibility to polarize the core in those cases. All test calcns. indicate that the new basis sets lead to consistent accuracies in HF, DFT, or correlated treatments even in crit. cases where other basis sets may show deficiencies.
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68. 68
  VandeVondele, J.; Hutter, J. Gaussian basis sets for accurate calculations on molecular systems in gas and condensed phases. J. Chem. Phys. 2007, 127, 114105, DOI: 10.1063/1.2770708
  68
  Gaussian basis sets for accurate calculations on molecular systems in gas and condensed phases
  VandeVondele, Joost; Hutter, Jurg
  Journal of Chemical Physics (2007), 127 (11), 114105/1-114105/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We present a library of Gaussian basis sets that has been specifically optimized to perform accurate mol. calcns. based on d. functional theory. It targets a wide range of chem. environments, including the gas phase, interfaces, and the condensed phase. These generally contracted basis sets, which include diffuse primitives, are obtained minimizing a linear combination of the total energy and the condition no. of the overlap matrix for a set of mols. with respect to the exponents and contraction coeffs. of the full basis. Typically, for a given accuracy in the total energy, significantly fewer basis functions are needed in this scheme than in the usual split valence scheme, leading to a speedup for systems where the computational cost is dominated by diagonalization. More importantly, binding energies of hydrogen bonded complexes are of similar quality as the ones obtained with augmented basis sets, i.e., have a small (down to 0.2 kcal/mol) basis set superposition error, and the monomers have dipoles within 0.1 D of the basis set limit. However, contrary to typical augmented basis sets, there are no near linear dependencies in the basis, so that the overlap matrix is always well conditioned, also, in the condensed phase. The basis can therefore be used in first principles mol. dynamics simulations and is well suited for linear scaling calcns.
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69. 69
  Brémond, E.; Savarese, M.; Adamo, C.; Jacquemin, D. Accuracy of TD-DFT Geometries: A Fresh Look. J. Chem. Theory Comput. 2018, 14, 3715– 3727, DOI: 10.1021/acs.jctc.8b00311
  69
  Accuracy of TD-DFT Geometries: A Fresh Look
  Bremond, Eric; Savarese, Marika; Adamo, Carlo; Jacquemin, Denis
  Journal of Chemical Theory and Computation (2018), 14 (7), 3715-3727CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  We benchmark a panel of 48 DFT exchange-correlation functionals in the framework of TD-DFT optimizations of the geometry of valence singlet excited states. To this end, we use a set of 41 small- and medium-sized org. mols. for which ref. geometries were obtained at high level of theory, typically, CC3 or CCSDR(3), with the aug-cc-pVTZ at. basis set. For the ground-state parameters, the tested functionals provide av. deviations that are small (0.010 Å and 0.5° for bond lengths and valence angles) and not very sensitive to the selected (hybrid) functional, but the errors are larger for the most polarized bonds (CO, CN, and so on). Nevertheless, DFT has a tendency to provide too compact distances, a trend slightly enhanced for functionals including a large amt. of exact exchange. The av. errors largely increase when going to the excited-state for most bond types, i.e., TD-DFT delivers less accurate excited-state distances than DFT for ground state. In particular TD-DFT combined with hybrid functionals provides significantly too short CO and CS/CSe bonds with resp. av. errors in the -0.026/-0.052 Å and -0.015/-0.082 Å ranges, depending on the selected hybrid functional. For the carbonyl bonds, the sizes of the TD-DFT deviations obtained when selecting std. hybrid functionals are of the same order of magnitude as the EOM-CCSD ones.
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70. 70
  Chan, G. K.-L.; Handy, N. C. C8H8: a density functional theory study of molecular geometries introducing the localised bond density. J. Chem. Soc., Faraday Trans. 1996, 92, 3015– 3021, DOI: 10.1039/ft9969203015
  70
  C8H8: A density functional theory study of molecular geometries introducing the localized bond density
  Chan, Garnet K-L.; Handy, Nicholas C.
  Journal of the Chemical Society, Faraday Transactions (1996), 92 (17), 3015-3021CODEN: JCFTEV; ISSN:0956-5000. (Royal Society of Chemistry)
  D. functional theory was used with all the common exchange-correlation functionals to investigate the structures of three isomers of C8H8 found in F.A. Cotton's text, barrelene, cyclooctatetraene, tetramethylenecyclobutane and also ethane and ethene. All calcns. were performed with TZ2P basis sets and large quadrature. The results are compared with expt. and those obtained with Hartree-Fock theory. Delocalization in the three mols. is discussed. A localized bond d. is introduced to explain the transferability of the trends in the predictions of the functionals between different mols. Three-parameter adiabatic connection functionals are examd. and their usefulness in geometry prediction questioned. Finally a phys. picture of the correlation as modeled by d. functional theory is presented and used to explain trends in the overestimation or underestimation of bond lengths.
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71. 71
  Neumann, R.; Nobes, R. H.; Handy, N. C. Exchange functionals and potentials. Mol. Phys. 1996, 87, 1– 36, DOI: 10.1080/00268979600100011
  71
  Exchange functionals and potentials
  Neumann, Ralf; Nobes, Ross H.; Handy, Nicholas C.
  Molecular Physics (1996), 87 (1), 1-36CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis)
  The commonly used exchange-correlation functionals of d. functional theory and their potentials are examd. numerically following the first such investigation of Perdew. They are also investigated for Ne and Kr. Their behavior for large gradients of the d. and for large distances is not satisfactory. In particular, the correct asymptotic r-1 behavior is difficult to achieve. Following van Leeuwen and Baerends, this is linked to the energy εmax of the highest occupied orbital arising from the Kohn-Sham equations. This deficiency is linked also with the poor prediction of mol. polarizabilities. The Becke-Roussel (BR) exchange functional is examd., which is derived by assuming a hydrogen-like exchange hole at all spatial points, and it has the attraction of being dependent on both the kinetic energy d. and the Laplacian of the d. and has no adjustable parameters. Becke has presented encouraging results using this functional in a hybrid manner. Fully self-consistent Kohn-sham calcns. are performed using it in combination with Perdew's 1986 correlation functional. The results are very encouraging indeed, so much so that this exchange functional is the best generalized gradient approxn. (GGA) yet discovered. In particular, bond lengths of many mol. show a substantial improvement over results from other GGAs. For example, many CH bonds are now within exptl. accuracy, instead of being typically 0.02 Å too long. Our ab initio understanding of non-dynamic correlation and dynamic correlation is then linked with d. functional theory. It is argued that correlation functionals should pick up the local dynamic correlation, whereas exchange functionals should include non-dynamic correlation effects. For these reasons it is considered that exchange functionals are best modeled on a system for which there is effectively no non-dynamic correlation, for which the optimum example is the Ne atom. Thus, again following Becke and Roussel, the spherically averaged Hartree-Fock exchange hole for Ne is examd., compared with the BR model functional hole. An excellent overlap is found, and thus the above good results are explained. As a final contribution, the dissocn. of the H2 mol. is re-examd., looking at it in terms of the exchange hole. For a ref. electron near one proton A, the RHF model has half an exchange electron near it, and half the exchange electron near the other proton B, whereas the BR functional has one electron near the other proton b, whereas the Br functional has one electron near A, which is the correct picture. For this reason the (restricted) BR functional gives a greatly improved dissocn. curve for H2 when compared with the Hartree-Fock curve. In summary, the Becke-Roussel functional is found to be a most attractive exchange functional.
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72. 72
  Sancho-García, J. C.; Cornil, J. Assessment of recently developed exchange-correlation functionals for the description of torsion potentials in π-conjugated molecules. J. Chem. Phys. 2004, 121, 3096– 3101, DOI: 10.1063/1.1774976
  72
  Assessment of recently developed exchange-correlation functionals for the description of torsion potentials in π-conjugated molecules
  Sancho-Garcia, Juan Carlos; Cornil, Jerome
  Journal of Chemical Physics (2004), 121 (7), 3096-3101CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Newly developed exchange-correlation functionals in d. functional theory (DFT) have been applied to describe conjugation effects in org. mols. The performance of the various approaches is assessed through the calcn. of torsion energy profiles and their crit. comparison with available exptl. data. Our results indicate that the OPTX-B95 exchange-correlation functional as well as its corresponding hybrid versions perform better than the well-established BLYP or B3LYP schemes when dealing with π-conjugated mols. In contrast, the recently introduced VSXC functional is not as reliable as other DFT methods for the systems examd. here.
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73. 73
  Vikramaditya, T.; Lin, S.-T. Limitations of Global Hybrids in Predicting the Geometries and Torsional Energy Barriers of Dimeric Systems and the Role of Hartree Fock and DFT Exchange. J. Comput. Chem. 2019, 40, 2810– 2818, DOI: 10.1002/jcc.26056
  73
  Limitations of Global Hybrids in Predicting the Geometries and Torsional Energy Barriers of Dimeric Systems and the Role of Hartree Fock and DFT Exchange
  Vikramaditya, Talapunur; Lin, Shiang-Tai
  Journal of Computational Chemistry (2019), 40 (32), 2810-2818CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
  Prediction of accurate geometries is a prerequisite for accurate prediction of mol. properties. Impact of Hartree Fock (HF) exchange (a0) on geometry in the framework of DFT is investigated by monitoring dihedral angles, bond length alternations, and torsional energy barriers of 10 dimeric systems against CCSD (ADZ/ATZ) benchmarks. A strong correlation is obsd. between the fraction of HF exchange, equil. dihedral angles, and the potential energy barriers in global hybrids. Full HF exchange is crit. to accurately predict the nonplanarity. Lower fractions of (a0)/larger DFT exchange (1-a0) results in overestimation of torsional energy barriers at 900 and underestimation at 00. Large contributions of (1-a0) in global hybrid functionals tend to overestimate torsional energy barriers (900) and are biased toward planar geometries. However, inclusion of larger fractions of (a0)/lower (1-a0) also overestimate the torsional energy barriers in syn-conformations due to the localization errors assocd. with HF exchange in global hybrids. Hence, irresp. of the fraction of HF/DFT exchange incorporated, global hybrids fail to accurately predict torsional energy barriers at 00 and 900 simultaneously. Long-range cor. (LC) functionals, which employ full HF exchange at longer regions, outperform global hybrid functionals in predicting geometries and torsional energy barriers of the dimeric mols. The distance dependence of (a0) thus provides a balanced fraction of HF exchange as the dihedral torsion varies. Impact of range sepn. parameter on geometries is marginal in altering the planarity/nonplanarity. However, range sepn. parameter within 0.20-0.40 bohr-1 predicts more reliable torsional energies and geometries. © 2019 Wiley Periodicals, Inc.
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74. 74
  Fang, C.; Oruganti, B.; Durbeej, B. How Method-Dependent Are Calculated Differences between Vertical, Adiabatic, and 0─0 Excitation Energies?. J. Phys. Chem. A 2014, 118, 4157– 4171, DOI: 10.1021/jp501974p
  74
  How Method-Dependent Are Calculated Differences between Vertical, Adiabatic, and 0-0 Excitation Energies?
  Fang, Changfeng; Oruganti, Baswanth; Durbeej, Bo
  Journal of Physical Chemistry A (2014), 118 (23), 4157-4171CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
  Through a large no. of benchmark studies, the performance of different quantum chem. methods in calcg. vertical excitation energies is today quite well established. These efforts have in recent years been complemented by a few benchmarks focusing instead on adiabatic excitation energies. However, it is much less well established how calcd. differences between vertical, adiabatic and 0-0 excitation energies vary between methods, which may be due to the cost of evaluating zero-point vibrational energy corrections for excited states. To fill this gap, we have calcd. vertical, adiabatic, and 0-0 excitation energies for a benchmark set of mols. covering both org. and inorg. systems. Considering in total 96 excited states and using both TD-DFT with a variety of exchange-correlation functionals and the ab initio CIS and CC2 methods, it is found that while the vertical excitation energies obtained with the various methods show an av. (over the 96 states) std. deviation of 0.39 eV, the corresponding std. deviations for the differences between vertical, adiabatic, and 0-0 excitation energies are much smaller: 0.10 (difference between adiabatic and vertical) and 0.02 eV (difference between 0-0 and adiabatic). These results provide a quant. measure showing that the calcn. of such quantities in photochem. modeling is well amenable to low-level methods. In addn., we also report on how these energy differences vary between chem. systems and assess the performance of TD-DFT, CIS, and CC2 in reproducing exptl. 0-0 excitation energies.
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75. 75
  Peach, M. J. G.; Benfield, P.; Helgaker, T.; Tozer, D. J. Excitation energies in density functional theory: An evaluation and a diagnostic test. J. Chem. Phys. 2008, 128, 044118, DOI: 10.1063/1.2831900
  75
  Excitation energies in density functional theory: An evaluation and a diagnostic test
  Peach, Michael J. G.; Benfield, Peter; Helgaker, Trygve; Tozer, David J.
  Journal of Chemical Physics (2008), 128 (4), 044118/1-044118/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Electronic excitation energies are detd. using the CAM-B3LYP Coulomb-attenuated functional, together with a std. generalized gradient approxn. (GGA) and hybrid functional. The degree of spatial overlap between the occupied and virtual orbitals involved in an excitation is measured using a quantity Λ, and the extent to which excitation energy errors correlate with Λ is quantified. For a set of 59 excitations of local, Rydberg, and intramol. charge-transfer character in 18 theor. challenging main-group mols., CAM-B3LYP provides by far the best overall performance; no correlation is obsd. between excitation energy errors and Λ, reflecting the good quality, balanced description of all three categories of excitation. By contrast, a clear correlation is obsd. for the GGA and, to a lesser extent, the hybrid functional, allowing a simple diagnostic test to be proposed for judging the reliability of a general excitation from these functionals-when Λ falls below a prescribed threshold, excitations are likely to be in very significant error. The study highlights the ambiguous nature of the term "charge transfer," providing insight into the observation that while many charge-transfer excitations are poorly described by GGA and hybrid functionals, others are accurately reproduced. (c) 2008 American Institute of Physics.
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76. 76
  Send, R.; Kühn, M.; Furche, F. Assessing Excited State Methods by Adiabatic Excitation Energies. J. Chem. Theory Comput. 2011, 7, 2376– 2386, DOI: 10.1021/ct200272b
  76
  Assessing Excited State Methods by Adiabatic Excitation Energies
  Send, Robert; Kuhn, Michael; Furche, Filipp
  Journal of Chemical Theory and Computation (2011), 7 (8), 2376-2386CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  We compile a 109-membered benchmark set of adiabatic excitation energies (AEEs) from high-resoln. gas-phase expts. Our data set includes a variety of org. chromophores with up to 46 atoms, radicals, and inorg. transition metal compds. Many of the 91 mols. in our set are relevant to atm. chem., photovoltaics, photochem., and biol. The set samples valence, Rydberg, and ionic states of various spin multiplicities. As opposed to vertical excitation energies, AEEs are rigorously defined by energy differences of vibronic states, directly observable, and insensitive to errors in equil. structures. We supply optimized ground state and excited state structures, which allows fast and convenient evaluation of AEEs with two single-point energy calcns. per system. We apply our benchmark set to assess the performance of time-dependent d. functional theory using common semilocal functionals and the CI singles method. Hybrid functionals such as B3LYP and PBE0 yield the best results, with mean abs. errors around 0.3 eV. We also investigate basis set convergence and correlations between different methods and between the magnitude of the excited state relaxation energy and the AEE error. A smaller, 15-membered subset of AEEs is introduced and used to assess the correlated wave function methods CC2 and ADC(2). These methods improve upon hybrid TDDFT for systems with single-ref. ground states but perform less well for radicals and small-gap transition metal compds. None of the investigated methods reaches "chem. accuracy" of 0.05 eV in AEEs.
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77. 77
  Côté, A. P.; Benin, A. I.; Ockwig, N. W.; O’Keeffe, M.; Matzger, A. J.; Yaghi, O. M. Porous, Crystalline, Covalent Organic Frameworks. Science 2005, 310, 1166– 1170, DOI: 10.1126/science.1120411
  77
  Porous, Crystalline, Covalent Organic Frameworks
  Cote, Adrien P.; Benin, Annabelle I.; Ockwig, Nathan W.; O'Keeffe, Michael; Matzger, Adam J.; Yaghi, Omar M.
  Science (Washington, DC, United States) (2005), 310 (5751), 1166-1170CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)
  Covalent org. frameworks (COFs) have been designed and successfully synthesized by condensation reactions of Ph diboronic acid {C6H4[B(OH)2]2} and hexahydroxytriphenylene [C18H6(OH)6]. Powder x-ray diffraction studies of the highly cryst. products (C3H2BO)6•(C9H12)1 (COF-1) and C9H4BO2 (COF-5) revealed expanded porous graphitic layers that are either staggered (COF-1, P63/mmc) or eclipsed (COF-5, P6/mmm). Their crystal structures are entirely held by strong bonds between B, C, and O atoms to form rigid porous architectures with pore sizes ranging from 7 to 27 angstroms. COF-1 and COF-5 exhibit high thermal stability (to temps. up to 500° to 600°C), permanent porosity, and high surface areas (711 and 1590 square meters per g, resp.).
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78. 78
  Das, G.; Biswal, B. P.; Kandambeth, S.; Venkatesh, V.; Kaur, G.; Addicoat, M.; Heine, T.; Verma, S.; Banerjee, R. Chemical sensing in two dimensional porous covalent organic nanosheets. Chem. Sci. 2015, 6, 3931– 3939, DOI: 10.1039/C5SC00512D
  78
  Chemical sensing in two dimensional porous covalent organic nanosheets
  Das, Gobinda; Biswal, Bishnu P.; Kandambeth, Sharath; Venkatesh, V.; Kaur, Gagandeep; Addicoat, Matthew; Heine, Thomas; Verma, Sandeep; Banerjee, Rahul
  Chemical Science (2015), 6 (7), 3931-3939CODEN: CSHCCN; ISSN:2041-6520. (Royal Society of Chemistry)
  Two new imide-based cryst., porous, and chem. stable covalent org. frameworks (COFs) (TpBDH and TfpBDH) have been successfully synthesized employing solvothermal crystn. route. Furthermore, thin layered covalent org. nanosheets (CONs) were derived from these bulk COFs by the simple liq. phase exfoliation method. These 2D CONs showcase increased luminescence intensity compared to their bulk counterparts (COFs). Notably, TfpBDH-CONs showcase good selectivity and prominent, direct visual detection towards different nitroarom. analytes over TpBDH-CONs. Quite interestingly, TfpBDH-CONs exhibit a superior "turn-on" detection capability for 2,4,6-trinitrophenol (TNP) in the solid state, but conversely, they also show a "turn-off" detection in the dispersion state. These findings describe a new approach towards developing an efficient, promising fluorescence chemosensor material for both visual and spectroscopic detection of nitroarom. compds. with very low [10-5 (M)] analyte concns.
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79. 79
  Das, P.; Chakraborty, G.; Tyagi, S.; Mandal, S. K. Design of Fluorescent and Robust Covalent Organic Framework Host Matrices for Illuminating Mechanistic Insight into Solvatochromic Decoding. ACS Appl. Mater. Interfaces 2020, 12, 52527– 52537, DOI: 10.1021/acsami.0c12785
  79
  Design of Fluorescent and Robust Covalent Organic Framework Host Matrices for Illuminating Mechanistic Insight into Solvatochromic Decoding
  Das, Prasenjit; Chakraborty, Gouri; Tyagi, Sparsh; Mandal, Sanjay K.
  ACS Applied Materials & Interfaces (2020), 12 (47), 52527-52537CODEN: AAMICK; ISSN:1944-8244. (American Chemical Society)
  Two functional covalent org. frameworks (COFs), constructed from 3-connected triazine-based amine or hydrazine with linear dialdehyde, are decorated with mol. docking sites to showcase their solvatochromic decoding behavior toward volatile solvent mols. (VSMs). These luminescent and cryst. COFs, namely, COF-N and COF-NN, are characterized by numerous anal. techniques. After accommodation of different VSMs as guests, the inclusion compds. of COF-N and COF-NN display solvatochromism. More fascinatingly, the singlet energy, band gaps, and lifetime of these VSM-accommodated COF-N and COF-NN are linearly correlated with the properties of VSMs. D. functional theory (DFT) and Monte Carlo simulation studies further support the interaction of VSMs with COF-N and COF-NN. The presence of extra amine functionality in COF-NN leads to the better interaction with VSMs and, therefore, results in different modes of interaction and correlation. Considering their inestimable chem. diversity, this study introduces a new path for finely tuned solvatochromic properties by COFs.
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Supporting Information
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- Treatment of the virtual space, structural information for molecular benchmark set, geometrical data on optimized molecular geometries, statistical analysis for ADMM and sTDA vertical excitation energies, adiabatic excitation energies and statistical analysis, fluorescence energies and statistical analysis, and structural information for covalent organic frameworks (PDF)
- ct2c00144_si_001.pdf (1.09 MB)
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Excited-State Properties for Extended Systems: Efficient Hybrid Density Functional MethodsClick to copy article linkArticle link copied!

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1. Introduction

2. Theoretical Background

2.1. The Tamm–Dancoff Approximation

2.2. Exact Exchange Using the Auxiliary Density Matrix Method (ADMM)

2.3. Semiempirical Coulomb and Exchange Contributions within the Simplified Tamm–Dancoff Approximation (sTDA)

2.4. Periodic Boundary Conditions

2.5. The Excited-State Lagrangian in the Tamm–Dancoff Approximation

3. Results and Discussion

3.1. Tests for Accuracy: Benchmark Results for 35 Main-Group Molecules

3.1.1. Excited-State Geometries: Impact of the Auxiliary Basis Set Size for ADMM Kernels

Figure 1

Figure 2

3.1.2. Excited-State Geometries: ADMM and sTDA Kernels in Comparison to EOM-CCSD References

Figure 3

3.1.3. Vertical Excitation Energies

Figure 4

3.1.4. Adiabatic Excitation Energies and Fluorescence

Figure 5

Figure 6

3.2. Increasing Computational Efficiency: Treating Extended Systems Using ADMM-PBE0 and sTDA Kernels

Figure 7

4. Conclusions

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Excited-State Properties for Extended Systems: Efficient Hybrid Density Functional Methods
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