Takao Tsuneda

On Koopmans' theorem in density functional theory.

This paper clarifies why long-range corrected (LC) density functional theory gives orbital energies quantitatively. First, the highest occupied molecular orbital and the lowest unoccupied molecular orbital energies of typical molecules are compared with the minus vertical ionization potentials (IPs) and electron affinities (EAs), respectively. Consequently, only LC exchange functionals are found to give the orbital energies close to the minus IPs and EAs, while other functionals considerably underestimate them. The reproducibility of orbital energies is hardly affected by the difference in the short-range part of LC functionals. Fractional occupation calculations are then carried out to clarify the reason for the accurate orbital energies of LC functionals. As a result, only LC functionals are found to keep the orbital energies almost constant for fractional occupied orbitals. The direct orbital energy dependence on the fractional occupation is expressed by the exchange self-interaction (SI) energy through the potential derivative of the exchange functional plus the Coulomb SI energy. On the basis of this, the exchange SI energies through the potential derivatives are compared with the minus Coulomb SI energy. Consequently, these are revealed to be cancelled out only by LC functionals except for H, He, and Ne atoms.

Long‐range correction for density functional theory

Long‐range correction for exchange functionals in Kohn–Sham density functional theory and its applicability are reviewed. Long‐range correction simply supplements the long‐range exchange effect in exchange functionals by replacing the Hartree–Fock exchange integral with the long‐range part of exchange functionals. Nevertheless, this correction has solved many problems in Kohn–Sham calculations. Using this correction, valence occupied and unoccupied orbital energies are quantitatively reproduced in a comprehensive manner for the first time. Long‐range correction has also solved the underestimations of charge transfer excitation energies and oscillator strengths in time‐dependent Kohn–Sham calculations and has clearly improved poor optical response properties such as hyperpolarizability in coupled‐perturbed Kohn–Sham and finite‐field calculations. Moreover, this correction has drastically improved the reproducibility of van der Waals bonds by simply combining with conventional van der Waals calculation methods. We, therefore, believe that the long‐range correction clearly extends the applicability of the Kohn–Sham method in future quantum chemistry calculations.

Reaction energetics on long-range corrected density functional theory: Diels–Alder reactions

The possibility of quantitative reaction analysis on the orbital energies of long‐range corrected density functional theory (LC‐DFT) is presented. First, we calculated the Diels–Alder reaction enthalpies that have been poorly given by conventional functionals including B3LYP functional. As a result, it is found that the long‐range correction drastically improves the reaction enthalpies. The barrier height energies were also computed for these reactions. Consequently, we found that dispersion correlation correction is also crucial to give accurate barrier height energies. It is, therefore, concluded that both long‐range exchange interactions and dispersion correlations are essentially required in conventional functionals to investigate Diels–Alder reactions quantitatively. After confirming that LC‐DFT accurately reproduces the orbital energies of the reactant and product molecules of the Diels–Alder reactions, the global hardness responses, the halves of highest occupied molecular orbital (HOMO)‐lowest unoccupied molecular orbital (LUMO) energy gaps, along the intrinsic reaction coordinates of two Diels–Alder reactions were computed. We noticed that LC‐DFT results satisfy the maximum hardness rule for overall reaction paths while conventional functionals violate this rule on the reaction pathways. Furthermore, our results also show that the HOMO‐LUMO gap variations are close to the reaction enthalpies for these Diels–Alder reactions. Based on these results, we foresee quantitative reaction analysis on the orbital energies.

Self-interaction corrections in density functional theory.

Self-interaction corrections for Kohn-Sham density functional theory are reviewed for their physical meanings, formulations, and applications. The self-interaction corrections get rid of the self-interaction error, which is the sum of the Coulomb and exchange self-interactions that remains because of the use of an approximate exchange functional. The most frequently used self-interaction correction is the Perdew-Zunger correction. However, this correction leads to instabilities in the electronic state calculations of molecules. To avoid these instabilities, several self-interaction corrections have been developed on the basis of the characteristic behaviors of self-interacting electrons, which have no two-electron interactions. These include the von Weizsäcker kinetic energy and long-range (far-from-nucleus) asymptotic correction. Applications of self-interaction corrections have shown that the self-interaction error has a serious effect on the states of core electrons, but it has a smaller than expected effect on valence electrons. This finding is supported by the fact that the distribution of self-interacting electrons indicates that they are near atomic nuclei rather than in chemical bonds.

A theoretical investigation on photocatalytic oxidation on the TiO2 surface

The TiO(2) photocatalytic oxidation mechanism was theoretically investigated by using long-range corrected time-dependent density functional theory (LC-TDDFT) with a cluster model of the anatase TiO(2)(001) surface. We found that LC-TDDFT with the cluster model quantitatively reproduces the photoexcitations of the TiO(2) surface by calculating the electronic spectra of a clean TiO(2) surface and one with oxygen defects. We calculated the electronic spectra of a molecularly adsorbed TiO(2) surface for the adsorptions of phenol, methanol, and methane molecules as typical organic molecules. We obtained the surprising result that the main peak of the phenol-adsorbed TiO(2) surface, which overlaps with the main peak of the clean TiO(2) surface, corresponds to charge transfers from the phenol molecule to the TiO(2) surface. This indicates that the TiO(2) photocatalytic oxidation proceeds through direct charge transfer excitation from the substrate molecules to the TiO(2) surface. In contrast, we found slight and no charge transfer for methanol and methane adsorption, respectively, in agreement with the experimental findings for their reactivities. In light of these results, we propose a new mechanism for heterogeneous TiO(2) photocatalytic oxidations.

Density functional theory for comprehensive orbital energy calculations

This study reveals the reason core 1s orbital energies and the highest occupied molecular orbital (HOMO) energies of hydrogen and rare gas atoms are underestimated by long-range corrected (LC) density functional theory (DFT), which quantitatively reproduces the HOMO energies of other systems and the lowest unoccupied molecular orbital (LUMO) energies. Applying the pseudospectral regional (PR) self-interaction correction (SIC) drastically improved the underestimated orbital energies in LC-DFT calculations, while maintaining or improving the accuracies in the calculated valence HOMO and LUMO energies. This indicates that the self-interaction error in exchange functionals causes the underestimations of core 1s orbital energies and the HOMO energies of hydrogen and rare gas atoms in LC-DFT calculations. To clarify the reason for the improvement, the fractional occupation dependences of total electronic energies and orbital energies were examined. The calculated results clearly showed that the LC-PR functional gives almost linear dependences of total electronic energies for a slight decrease in the occupation number of core 1s orbitals, although this linear dependence disappears for significant decrease due to the shrinking of exchange self-interaction regions. It was also clarified that the PRSIC hardly affects the occupation number dependences of the total electronic energies and orbital energies for the fractional occupations of HOMOs and LUMOs. As a result, it was concluded that core orbital energies are obtained accurately by combining LC-DFT with PRSIC.

Modified regional self-interaction correction method based on the pseudospectral method.

A modification of the regional self-interaction correction (RSIC) scheme (Tsuneda et al., J. Comput. Chem. 2003, 24, 1592), pseudospectral RSIC (PSRSIC), is proposed to eliminate the self-interaction errors (SIEs) especially in core regions. PSRSIC reduces the SIEs by substituting the HF exchange energy density calculated with the use of the pseudospectral technique for the exchange energy in the SI-domain region. PSRSIC is combined with the long-range correction (LC) scheme. TDDFT calculations with LC-PSRSIC yield all of the core-, valence-, Rydberg-, and charge-transfer-excitation energies with reasonable accuracy. Core-ionization energies are also well-reproduced by LC-PSRSIC.

LONG-RANGE-CORRECTED TIME-DEPENDENT DENSITY FUNCTIONAL STUDY ON ELECTRONIC SPECTRA OF FIVE-MEMBERED RING COMPOUNDS AND FREE-BASE PORPHYRIN

Long-range-corrected time-dependent density functional theory (LC-TDDFT) was applied to five-membered ring compounds (cyclopentadiene, pyrrole, and furan molecules) and free-base porphyrin. The vertical π-π* and Rydberg excitation energies and corresponding oscillator strengths were calculated by LC-TDDFT. The LC scheme obviously improved the Rydberg excitation energies and oscillator strengths of these systems, which have been fairly underestimated by TDDFT with conventional pure and hybrid B3LYP functionals. On the whole, LC-TDDFT results were very close to the results of the ab initio symmetry-adapted cluster configuration interaction (SAC-CI) method for most excitations. However, LC-TDDFT is poor in describing doubly excited states such as the 1 1A1 state of five-membered ring compounds.

Reactivity index based on orbital energies.

This study shows that the chemical reactivities depend on the orbital energy gaps contributing to the reactions. In the process where a reaction only makes progress through charge transfer with the minimal structural transformation of the reactant, the orbital energy gap gradient (OEGG) between the electron‐donating and electron‐accepting orbitals is proven to be very low. Using this relation, a normalized reaction diagram is constructed by plotting the normalized orbital energy gap with respect to the normalized intrinsic reaction coordinate. Application of this reaction diagram to 43 fundamental reactions showed that the majority of the forward reactions provide small OEGGs in the initial stages, and therefore, the initial processes of the forward reactions are supposed to proceed only through charge transfer. Conversely, more than 60% of the backward reactions are found to give large OEGGs implying very slow reactions associated with considerable structural transformations. Focusing on the anti‐activation‐energy reactions, in which the forward reactions have higher barriers than those of the backward ones, most of these reactions are shown to give large OEGGs for the backward reactions. It is also found that the reactions providing large OEGGs in the forward directions inconsistent with the reaction rate constants are classified into SN2, symmetric, and methyl radical reactions. Interestingly, several large‐OEGG reactions are experimentally established to get around the optimum pathways. This indicates that the reactions can take significantly different pathways from the optimum ones provided no charge transfer proceeds spontaneously without the structural transformations of the reactants.

On low-lying excited states of extended nanographenes

Low‐lying excited states of planarly extended nanographenes are investigated using the long‐range corrected (LC) density functional theory (DFT) and the spin‐flip (SF) time‐dependent density functional theory (TDDFT) by exploring the long‐range exchange and double‐excitation correlation effects on the excitation energies, band gaps, and exciton binding energies. Optimizing the geometries of the nanographenes indicates that the long‐range exchange interaction significantly improves the CC bond lengths and amplify their bond length alternations with overall shortening the bond lengths. The calculated TDDFT excitation energies show that long‐range exchange interaction is crucial to provide accurate excitation energies of small nanographenes and dominate the exciton binding energies in the excited states of nanographenes. It is, however, also found that the present long‐range correction may cause the overestimation of the excitation energy for the infinitely wide graphene due to the discrepancy between the calculated band gaps and vertical ionization potential (IP) minus electron affinity (EA) values. Contrasting to the long‐range exchange effects, the SF‐TDDFT calculations show that the double‐excitation correlation effects are negligible in the low‐lying excitations of nanographenes, although this effect is large in the lowest excitation of benzene molecule. It is, therefore, concluded that long‐range exchange interactions should be incorporated in TDDFT calculations to quantitatively investigate the excited states of graphenes, although TDDFT using a present LC functional may provide a considerable excitation energy for the infinitely wide graphene mainly due to the discrepancy between the calculated band gaps and IP–EA values.