Benchmarking the performance of time-dependent density functional methods

The performance of 24 density functionals, including 14 meta-generalized gradient approximation (mGGA) functionals, is assessed for the calculation of vertical excitation energies against an experimental benchmark set comprising 14 small- to medium-sized compounds with 101 total excited states. The experimental benchmark set consists of singlet, triplet, valence, and Rydberg excited states. The global-hybrid (GH) version of the Perdew-Burke-Ernzerhoff GGA density functional (PBE0) is found to offer the best overall performance with a mean absolute error (MAE) of 0.28 eV. The GH-mGGA Minnesota 2006 density functional with 54% Hartree-Fock exchange (M06-2X) gives a lower MAE of 0.26 eV, but this functional encounters some convergence problems in the ground state. The local density approximation functional consisting of the Slater exchange and Volk-Wilk-Nusair correlation functional (SVWN) outperformed all non-GH GGAs tested. The best pure density functional performance is obtained with the local version of the Minnesota 2006 mGGA density functional (M06-L) with an MAE of 0.41 eV.     

Singly and doubly excited states of butadiene, acrolein, and glyoxal: Geometries and electronic spectra

Excited-state geometries and electronic spectra of butadiene, acrolein, and glyoxal have been investigated by the symmetry adapted cluster configuration interaction (SAC-CI) method in their s-trans conformation. Valence and Rydberg states below the ionization threshold have been precisely calculated with sufficiently flexible basis sets. Vertical and adiabatic excitation energies were well reproduced and the detailed assignments were given taking account of the second moments. The deviations of the vertical excitation energies from the experiment were less than 0.3 eV for all cases. The SAC-CI geometry optimization has been applied to some valence and Rydberg excited states of these molecules in the planar structure. The optimized ground- and excited-state geometries agree well with the available experimental values; deviations lie within 0.03 A and 0.7 degrees for the bond lengths and angles, respectively. The force acting on the nuclei caused by the excitations has been discussed in detail by calculating the SAC-CI electron density difference between the ground and excited states; the geometry relaxation was well interpreted with the electrostatic force theory. In Rydberg excitations, geometry changes were also noticed. Doubly excited states (so-called 2 (1)A(g) states) were investigated by the SAC-CI general-R method considering up to quadruple excitations. The characteristic geometrical changes and large energetic relaxations were predicted for these states.     

Performance of the M11 and M11-L density functionals for calculations of electronic excitation energies by adiabatic time-dependent density functional theory

Adiabatic time-dependent density functional theory is a powerful method for calculating electronic excitation energies of complex systems, but the quality of the results depends on the choice of approximate density functional. In this article we test two promising new density functionals, M11 and M11-L, against databases of 214 diverse electronic excitation energies, and we compare the results to those for 16 other density functionals of various kinds and to time-dependent Hartree-Fock. Charge transfer excitations are well known to be the hardest challenge for TDDFT. M11 is a long-range-corrected hybrid meta-GGA, and it shows better performance for charge transfer excitations than any of the other functionals except M06-HF, which is a specialized functional that does not do well for valence excitations. Several other long-range-corrected hybrid functionals also do well, and we especially recommend M11, ωB97X, and M06-2X for general spectroscopic applications because they do exceptionally well on ground-state properties as well as excitation energies. Local functionals are preferred for many applications to extended systems because of their significant cost advantage for large systems. M11-L is a dual-range local functional and-unlike all previous local functionals-it has good performance for Rydberg states as well as for valence states. Thus it is highly recommended for excitation energy calculations on extended systems.     

Calculation of near-edge X-ray absorption fine structure with the CIS(D) method

We report an investigation into the calculation of near-edge X-ray absorption fine structure with the CIS(D) method. Core excitation energies computed with time-dependent density functional theory using standard exchange-correlation functionals are systematically underestimated. CIS(D) predicts core excitation energies that are closer to experiment. However, excitation energies for Rydberg states are too low with respect to valence states, and for some systems spectra that are qualitatively incorrect are obtained. A scaled opposite spin only approach is proposed that reduces the error in the computed core excitation energies, and results in spectra that are in good agreement with experiment.     

The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals

We present two new hybrid meta exchange- correlation functionals, called M06 and M06-2X. The M06 functional is parametrized including both transition metals and nonmetals, whereas the M06-2X functional is a high-nonlocality functional with double the amount of nonlocal exchange (2X), and it is parametrized only for nonmetals.The functionals, along with the previously published M06-L local functional and the M06-HF full-Hartree–Fock functionals, constitute the M06 suite of complementary functionals. We assess these four functionals by comparing their performance to that of 12 other functionals and Hartree–Fock theory for 403 energetic data in 29 diverse databases, including ten databases for thermochemistry, four databases for kinetics, eight databases for noncovalent interactions, three databases for transition metal bonding, one database for metal atom excitation energies, and three databases for molecular excitation energies. We also illustrate the performance of these 17 methods for three databases containing 40 bond lengths and for databases containing 38 vibrational frequencies and 15 vibrational zero point energies. We recommend the M06-2X functional for applications involving main-group thermochemistry, kinetics, noncovalent interactions, and electronic excitation energies to valence and Rydberg states. We recommend the M06 functional for application in organometallic and inorganometallic chemistry and for noncovalent interactions. Electronic Supplementary Material The online version of this article (doi:) contains supplementary material, which is available to authorized users. Contribution to the Mark S. Gordon 65th Birthday Festschrift Issue.     

Performance of time-dependent density functional and Green functions methods for calculations of excitation energies in radicals and for Rydberg electronic states

Time-dependent density functional (TD-DFT) and perturbation theory-based outer valence Green functions (OVGF) methods have been tested for calculations of excitation energies for a set of radicals, molecules, and model clusters simulating points defects in silica. The results show that the TD-DFT approach may give unreliable results not only for diffuse Rydberg states, but also for electronic states involving transitions between MOs localized in two remote from each other spatial regions, for example, for charge-transfer excitations. For the. O-SiX(3) clusters, where X is a single-valence group, TD-DFT predicts reasonable excitation energies but incorrect sequence of electronic transitions. For a number of cases where TD-DFT is shown to be unreliable, the OVGF approach can provide better estimates of excitation energies, but this method also is not expected to perform universally well. The OVGF performance is demonstrated to be satisfactory for excitations with predominantly single-determinant wave functions where the deviations of the calculated energies from experiment should not exceed 0.1-0.3 eV. However, for more complicated transitions involving multiple bonds or for excited states with multireference wave functions the OVGF approach is less reliable and error in the computed energies can reach 0.5-1 eV.     

Benchmark results for empirical post-GGA functionals: difficult exchange problems and independent tests

Many of the most promising new density functionals have improved the treatment of non-local exchange effects with the help of semi-empirical information and more sophisticated recipes for combining Hartree-Fock and local exchange approximations. In order to quantify recent advancements and identify directions for improvement, we have examined a broad spectrum of test problems. We evaluate the performance of several new hybrid density functionals (ωB97, ωB97X, ωB97X-D, LRC-ωPBEh, M06, M06-2X, and M06-HF) on a variety of chemical problems, some sensitive to the treatment of exact exchange (which we have hoped to systematically improve) and some which require a balanced treatment of correlation. Since all of the functionals under consideration are parameterized with ground-state thermochemical data, the benchmark aims to determine the applicability of the new density functionals to cases that have not been considered in the optimization of the semi-empirical parameters. The first class of benchmarks includes the excitation energies of 21 molecules (83 states) primarily from a recent benchmark conducted by Tozer and co-workers, with some additional references from data made available from the groups of Thiel and Truhlar. We briefly examine the conformational preferences of a small peptide and complete our study with two recently published sets of data that have shown large, systematic errors in simple alkane thermochemistry. While our results indicate that the more general hybrids currently under development perform well for problems outside of their parameterization and improve over the standard hybrid density functionals in an essentially systematic way, there is still a significant self-interaction error in the more difficult cases. Functionals based on a range-separation of exchange and functionals depending on the kinetic-energy density both perform comparably, and there is evidence for complementary strengths.     

Excited states and electronic spectra of annulated dinuclear free-base phthalocyanines: a theoretical study on near-infrared-absorbing dyes

The electronic excited states and electronic absorption spectra of annulated dinuclear free-base phthalocyanine (C(58)H(30)N(16)) are studied through quantum chemical calculations using the symmetry-adapted cluster-configuration interaction (SAC-CI) method. Three tautomers are possible with respect to the position of the pyrrole protons; therefore, the SAC-CI calculations for these tautomers were performed. The structures of the Q-band states are discussed based on the character of their molecular orbitals. The lower energy shift of the Q-bands because of dimerization is explained by the decrease in the HOMO-LUMO gaps resulting from the bonding and antibonding interactions between the monomer units. The electronic dipole moments of the nonsymmetric tautomer were calculated, and the possibility of charge-separated excited states is discussed. The relative energies of these tautomers are examined using density functional theory (DFT) calculations for several peripheral substituents. The relative energies of these tautomers significantly depend on the substituents, and therefore, the abundance ratios of the three tautomers were affected by the substituents. The absorption spectra were simulated from the SAC-CI results weighted by the Boltzmann factors obtained from the DFT calculations. The SAC-CI spectra reproduce the experimental findings well. The thermal-averaged SAC-CI spectra could explain the observed substituent effect on the structure of the Q-bands in terms of the relative stabilities and the abundance ratios of the tautomers. The SAC-CI and time-dependent density functional theory calculations are also compared. The CAM-B3LYP results agreed with the trends of the SAC-CI results; however, the CAM-B3LYP calculation overestimated the excitation energies in comparison with the SAC-CI and experimental results.     

Probing the valence character of O 1s-->Rydberg excited O2 by participator Auger decay measurements and partial ion yield spectroscopy following x-ray absorption

The valence character of O 1s-->Rydberg excited O2 is investigated by means of participator Auger decay spectroscopy, performed at selected photon energies across the K-shell resonance region, and by means of partial ion yield x-ray absorption spectroscopy. For several of the excitation energies studied, the authors find substantial sigma*(4Sigmau-, 2Sigmau-) valence character being mixed with nssigma and npsigma (4Sigmau-, 2Sigmau-) Rydberg states. An experimental indication of a coupling between the channels associated with quartet and doublet ion cores is considered and discussed. New spectroscopic constants are derived for the singly ionized X 2Pig state of O2 based on the observation of at least 20 vibrational sublevels.     

Full configuration interaction calculation of the low lying valence and Rydberg states of BeH

The all-electron full configuration interaction (FCI) vertical excitation energies for some low lying valence and Rydberg excited states of BeH are presented in this article. A basis set of valence atomic natural orbitals has been augmented with a series of Rydberg orbitals that have been generated as centered onto the Be atom. The resulting basis set can be described as 4s2p1d/2s1p (Be/H) + 4s4p3d. It allows to calculate Rydberg states up to n= {3,4,5} of the s, p, and d series of Rydberg states. The FCI vertical ionization potential for the same basis set and geometry amounts to 8.298 eV. Other properties such as FCI electric dipole and quadrupole moments and FCI transition dipole and quadrupole moments have also been calculated. The results provide a set of benchmark values for energies, wave functions, properties, and transition properties for the five electron BeH molecule. Most of the states have large multiconfigurational character in spite of their essentially single excited nature and a number of them present an important Rydberg-valence mixing that is achieved through the mixed nature of the particle MO of the single excitations.     

Electronic spectrum of F<Subscript>2</Subscript>CO: theoretical calculations of vertical excitation energies and intensities

In this work, the linear response formalism with a triples-corrected CCSD reference wave function, LR-CCSDR(3), is applied to the calculation of vertical excitation energies of singlet states of the F₂CO molecule. A basis set of atomic natural orbitals augmented with a series of Rydberg functions has been used in the calculations. A large number of electronically excited states were calculated, and the valence, Rydberg, or mixed character of the states were investigated. In addition, the molecular quantum defect orbital (MQDO) method has been used to determine transition intensities involving Rydberg states. Excitation energies and transition intensities for Rydberg states with n > 3 are reported for the first time.     

Density functional and spin-orbit ab initio study of CF3Br: molecular properties and electronic curve crossing

Quantum chemical calculations of CF(3)Br and the CF(3) radical are performed using density functional theory (DFT) and time-dependent DFT (TDDFT). Molecular structures, vibrational frequencies, dipole moment, bond dissociation energy, and vertical excitation energies of CF(3)Br are calculated and compared with available experimental results. The performance of six hybrid and five hybrid meta functionals in DFT and TDDFT calculations are evaluated. The ωB97X, B3PW91, and M05-2X functionals give very good results for molecular structures, vibrational frequencies, and vertical excitation energies, respectively. The ωB97X functional calculates well the dipole moment of CF(3)Br. B3LYP, one of the most widely used functionals, does not perform well for calculations of the C-Br bond length, bond dissociation energy, and vertical excitation energies. Potential energy curves of the low-lying excited states of CF(3)Br are obtained using the multiconfigurational spin-orbit ab initio method. The crossing point between 2A(1) and 3E states is located near the C-Br bond length of 2.45 ?. Comparison with CH(3)Br shows that fluorination does not alter the location of the crossing point. The relation between the calculated potential energy curves and recent experimental result is briefly discussed.     

Expanded electronic states of the fluorine molecule

Potential curves of electronically excited states of F₂ with an expanded outer orbital have been calculated using a modified frozen core technique: The ionic core has been described with a two-determinant wave function and for the excited states a mixing of configurations with different cores has been employed. An investigation of the valence shell states of F₂ is presented and potential curves for a singly excited as well as a doubly excited V-state of ¹Σ_u⁺ symmetry have been calculated. Further a low lying two-configuration state resulting from simultaneous excitation to a valence and a Rydberg orbital is predicted.     

Exponentially generated wave functions and excited states of benzene

Usefulness of the exponentially generated wave function approach is shown. We first give an overview of the SAC (symmetry adapted cluster) and SAC-CI study on the valence and Rydberg excitations and ionizations of benzene including both and spaces. The importance of the reorganization effect is found for the T₃(³B_2u), S₂(¹B_1u), and S₃(¹E_1u) states, so-called V states. A first systematic calculation is reported for the Rydberg excited states. Next, the idea of the exponentially generated wave function (EGWF) theory is explained. New exponential-type operators and new wave functions associated with them are defined. The mixed or multi use of these exponential operators is shown to be effective both physically and practically. We call the resultant wave functions MEG (multi-exponentially generated) wave functions. We then explain the algorithm of calculations and show some results on the potential energy curves of the ground, excited, and quasi-degenerate states of some diatomics and triatomics.     

Excited states and electronic spectra of extended tetraazaporphyrins

Electronic excited states, electronic absorption, and magnetic circular dichroism (MCD) spectra of free-base tetraazaporphyrin (TAP), phthalocyanine (Pc), naphthalocyanine (Nc), and anthracocyanine (Ac) were studied by quantum chemical calculations using the symmetry-adapted cluster-configuration interaction (SAC-CI) method. Not only optically allowed states including the Q- and B-bands but also optically forbidden states were calculated for transitions whose excitation energies were lower than 4.5 eV. The present SAC-CI calculations consistently assigned the absorption and MCD peaks as optically allowed π→π(?) excitations, although these calculations using double-zeta basis limit quantitative agreement and discussion. For Nc and Ac, excited states beyond the four-orbital model appeared in the low-energy region. The low-energy shifts of the Q-bands with the extension of molecular size were explained by the orbital energies. The splitting of the Q-bands decreases with extension of the molecular size. This feature was reproduced by the SAC-CI calculations but the configuration interaction with single excitations and time-dependent density functional theory calculations failed to reproduce this trend. Electron correlation in the excited states is important in reproducing this splitting of the Q-bands and in describing the energy difference between the B(2u) and B(3u) states of free-base porphyrins.