The method of dispersion correction as an add-on to standard Kohn-Sham density functional theory (DFT-D) has been refined regarding higher accuracy, broader range of applicability, and less empiricism. The main new ingredients are atom-pairwise specific dispersion coefficients and cutoff radii that are both computed from first principles. The coefficients for new eighth-order dispersion terms are computed using established recursion relations. System (geometry) dependent information is used for the first time in a DFT-D type approach by employing the new concept of fractional coordination numbers (CN). They are used to interpolate between dispersion coefficients of atoms in different chemical environments. The method only requires adjustment of two global parameters for each density functional, is asymptotically exact for a gas of weakly interacting neutral atoms, and easily allows the computation of atomic forces. Three-body nonadditivity terms are considered. The method has been assessed on standard benchmark sets for inter- and intramolecular noncovalent interactions with a particular emphasis on a consistent description of light and heavy element systems. The mean absolute deviations for the S22 benchmark set of noncovalent interactions for 11 standard density functionals decrease by 15%-40% compared to the previous (already accurate) DFT-D version. Spectacular improvements are found for a tripeptide-folding model and all tested metallic systems. The rectification of the long-range behavior and the use of more accurate C(6) coefficients also lead to a much better description of large (infinite) systems as shown for graphene sheets and the adsorption of benzene on an Ag(111) surface. For graphene it is found that the inclusion of three-body terms substantially (by about 10%) weakens the interlayer binding. We propose the revised DFT-D method as a general tool for the computation of the dispersion energy in molecules and solids of any kind with DFT and related (low-cost) electronic structure methods for large systems.

http://dns2.asia.edu.tw/~ysho/YSHO-English/1000%20WC/PDF/J%20Che%20Phy132,%20154104.pdf

A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu

: The unpolarized absorption and circular dichroism spectra of the fundamental vibrational transitions of the chiral molecule, 4-methyl-2-oxetanone, are calculated ab initio. Harmonic force fields are obtained using Density Functional Theory (DFT), MP2, and SCF methodologies and a 5S4P2D/3S2P (TZ2P) basis set. DFT calculations use the Local Spin Density Approximation (LSDA), BLYP, and Becke3LYP (B3LYP) density functionals. Mid-IR spectra predicted using LSDA, BLYP, and B3LYP force fields are of significantly different quality, the B3LYP force field yielding spectra in clearly superior, and overall excellent, agreement with experiment. The MP2 force field yields spectra in slightly worse agreement with experiment than the B3LYP force field. The SCF force field yields spectra in poor agreement with experiment.The basis set dependence of B3LYP force fields is also explored: the 6-31G* and TZ2P basis sets give very similar results while the 3-21G basis set yields spectra in substantially worse agreements with experiment. jg

Ab initio calculation of vibrational absorption and circular dichroism spectra using density functional force fields

In this tutorial review, we show how Time-Dependent Density Functional Theory (TD-DFT) has become a popular tool for computing the signatures of electronically excited states, and more specifically, the properties directly related to the optical (absorption and emission) spectra of molecules. We discuss the properties that can be obtained with widely available programs as well as how to account for the environmental effects (solvent and surfaces) and present recent applications in these fields. We next expose the transformation of the TD-DFT results into chemically intuitive parameters (colours as well as charge-transfer distances). Eventually, the non-specialised reader will find a series of advices and warnings necessary to perform her/his first TD-DFT calculations.

The calculations of excited-state properties with Time-Dependent Density Functional Theory

Extensive Time-Dependent Density Functional Theory (TD-DFT) calculations have been carried out in order to obtain a statistically meaningful analysis of the merits of a large number of functionals. To reach this goal, a very extended set of molecules (∼500 compounds, >700 excited states) covering a broad range of (bio)organic molecules and dyes have been investigated. Likewise, 29 functionals including LDA, GGA, meta-GGA, global hybrids, and long-range-corrected hybrids have been considered. Comparisons with both theoretical references and experimental measurements have been carried out. On average, the functionals providing the best match with reference data are, one the one hand, global hybrids containing between 22% and 25% of exact exchange (X3LYP, B98, PBE0, and mPW1PW91) and, on the other hand, a long-range-corrected hybrid with a less-rapidly increasing HF ratio, namely LC-ωPBE(20). Pure functionals tend to be less consistent, whereas functionals incorporating a larger fraction of exact exchange tend to underestimate significantly the transition energies. For most treated cases, the M05 and CAM-B3LYP schemes deliver fairly small deviations but do not outperform standard hybrids such as X3LYP or PBE0, at least within the vertical approximation. With the optimal functionals, one obtains mean absolute deviations smaller than 0.25 eV, though the errors significantly depend on the subset of molecules or states considered. As an illustration, PBE0 and LC-ωPBE(20) provide a mean absolute error of only 0.14 eV for the 228 states related to neutral organic dyes but are completely off target for cyanine-like derivatives. On the basis of comparisons with theoretical estimates, it also turned out that CC2 and TD-DFT errors are of the same order of magnitude, once the above-mentioned hybrids are selected.

Extensive TD-DFT Benchmark: Singlet-Excited States of Organic Molecules.

Time-Dependent Density Functional Theory (TD-DFT) has become the most widely-used theoretical approach to simulate the optical properties of both organic and inorganic molecules. 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 exchangecorrelation 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 experimental and theoretical reference transition energies. More specific benchmarks (oscillator strengths, excited-state geometries, dipole moments, vibronic shapes, etc.) are summarized as well. V C 2013 Wiley Periodicals, Inc.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/qua.24438

TD-DFT benchmarks: A review

The functional dependence of excited-state geometries and normal modes calculated with time-dependent density functional theory (TDDFT) is investigated on the basis of vibronic structure calculations of the absorption spectra of large molecules. For a set of molecules covering a wide range of different structures including organic dyes, biological chromophores, and molecules of importance in material science, quantum mechanical simulations of the vibronic structure are performed. In total over 40 singlet−singlet transitions of neutral closed-shell compounds and doublet−doublet transitions of neutral radicals, radical cations, and anions are considered. Calculations with different standard density 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 r...

The Vibronic Structure of Electronic Absorption Spectra of Large Molecules: A Time-Dependent Density Functional Study on the Influence of “Exact” Hartree−Fock Exchange

Calculations of the vibronic structure in electronic spectra of large organic molecules based on density functional methods are presented. The geometries of the excited states are obtained from time-dependent density functional (TDDFT) calculations employing the B3LYP hybrid functional. The vibrational functions and transition dipole moment derivatives are calculated within the harmonic approximation by finite difference of analytical gradients and the transition dipole moment, respectively. Normal mode mixing is taken into account by the Duschinsky transformation. The vibronic structure of strongly dipole-allowed transitions is calculated within the Franck-Condon approximation. Weakly dipole-allowed and dipole-forbidden transitions are treated within the Franck-Condon-Herzberg-Teller and Herzberg-Teller approximation, respectively. The absorption spectra of several organic pi systems (anthracene, pentacene, pyrene, octatetraene, styrene, azulene, phenoxyl) are calculated and compared with experimental data. For dipole-allowed transitions in general a very good agreement between theory and experiment is obtained. This indicates the good quality of the optimized geometries and harmonic force fields. Larger errors are found for the weakly dipole-allowed S0 --> S1 transition of pyrene which can tentatively be assigned to TDDFT errors for the relative energies of excited states close to the target state. The weak bands of azulene and phenoxyl are very well described within the Franck-Condon approximation which can be explained by the large energy gap (>1.2 eV) to higher-lying excited states leading to small vibronic couplings. Once corrections are made for the errors in the theoretical 0-0 transition energies, the TDDFT approach to calculate vibronic structure seems to outperform both widely used ab initio methods based on configuration interaction singles or complete active space self-consistent field wave functions and semiempirical treatments regarding accuracy, applicability, and computational effort. Together with the parallel computer implementations employed, the present approach appears to be a valuable tool for a quantitative description and detailed understanding of electronic excitation processes in large molecules.

Density functional calculations of the vibronic structure of electronic absorption spectra.

A general and efficient approach for the calculation of Franck–Condon integrals (FCIs) of large molecules is presented. In a first step, by exploiting the diagonally dominant and sparse structure of the Duschinsky matrix, a model system is constructed for which the Duschinsky matrix takes a block-diagonal form. For each of these blocks separately, the FCIs are calculated discarding all below a certain threshold. From those integrals retained the FCIs of the model system are obtained by simple multiplication. These serve as an estimate for the FCIs of the exact system which are calculated for those integrals which lie above a certain threshold. By systematically decreasing the threshold, the simulation can be reliably converged to the exact result with an arbitrary accuracy. Using this scheme, a considerable reduction of the number of FCIs which have to be calculated is achieved which leads to an improved scaling behavior of the computational effort with system size. The approach has been tested thoroughly for a set of molecules including difficult cases. For the larger systems a speedup of up to three orders of magnitude compared to an exact calculation is observed while the errors can be kept negligible. With this approach accurate calculations of FCIs are feasible also for large molecules encountered in “real-life” chemistry, especially biochemistry and material science.

An efficient approach for the calculation of Franck-Condon integrals of large molecules.

There has been a considerable interest in the chiroptical properties of molecules whose chirality is exclusively due to an isotopic substitution and numerous examples for the electronic circular dichroism (CD) spectra of isotopically chiral systems have been reported in literature. Four different explanations have been proposed for the mechanism as to how the isotopic substitution induces a chiral perturbation of the otherwise achiral electronic wave function; however, up to now no conclusive answer has been given about the dominating effect responsible for the experimental observations. In this study we will present, for the first time, fully quantum-mechanical calculations of the CD spectra of three different molecular systems with isotopically engendered chirality. As examples, we consider the spectra of organic molecules with ketone and α-diketone carbonyl and diene chromophores. The effect of vibronic couplings for the reorientation of the electric and magnetic transition dipole moments is taken into...

A theoretical study of the chiroptical properties of molecules with isotopically engendered chirality

It is shown that the general solution for the computation of the pseudorotation matrix for satisfying the Eckart axis conditions recently proposed by Dymarsky and Kudin [J. Chem. Phys. 122, 124103 (2005)] is mathematically equivalent to a solution described in the Appendix of the paper of Pickett and Strauss [J. Am. Chem. Soc. 92, 7281 (1970)].

Marc Dierksen

Papers

The Vibronic Structure of Electronic Absorption Spectra of Large Molecules: A Time-Dependent Density Functional Study on the Influence of “Exact” Hartree−Fock Exchange

Density functional calculations of the vibronic structure of electronic absorption spectra.

An efficient approach for the calculation of Franck-Condon integrals of large molecules.

A theoretical study of the chiroptical properties of molecules with isotopically engendered chirality

Comment on "Computation of the pseudorotation matrix to satisfy the Eckart axis conditions" [J. Chem. Phys. 122, 124103 (2005)].