We propose a new long-range correction scheme that combines generalized-gradient-approximation (GGA) exchange functionals in density-functional theory (DFT) with the ab initio Hartree–Fock exchange integral by using the standard error function. To develop this scheme, we suggest a new technique that constructs an approximate first-order density matrix that corresponds to a GGA exchange functional. The calculated results of the long-range correction scheme are found to support a previous argument that the lack of the long-range interactions in conventional exchange functionals may be responsible for the underestimation of 4s−3d interconfigurational energies of the first-row transition metals and for the overestimation of the longitudinal polarizabilities of π-conjugated polyenes in DFT calculations.
We apply the long-range correction (LC) scheme for exchange functionals of density functional theory to time-dependent density functional theory (TDDFT) and examine its efficiency in dealing with the serious problems of TDDFT, i.e., the underestimations of Rydberg excitation energies, oscillator strengths, and charge-transfer excitation energies. By calculating vertical excitation energies of typical molecules, it was found that LC-TDDFT gives accurate excitation energies, within an error of 0.5 eV, and reasonable oscillator strengths, while TDDFT employing a pure functional provides 1.5 eV lower excitation energies and two orders of magnitude lower oscillator strengths for the Rydberg excitations. It was also found that LC-TDDFT clearly reproduces the correct asymptotic behavior of the charge-transfer excitation energy of ethylene-tetrafluoroethylene dimer for the long intramolecular distance, unlike a conventional far-nucleus asymptotic correction scheme. It is, therefore, presumed that poor TDDFT results for pure functionals may be due to their lack of a long-range orbital-orbital interaction.
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.
The symmetry-adapted-cluster (SAC) expansion of an exact wavefunction is given. It is constructed from the generators of the symmetry-adapted excited configurations having the symmetry under consideration, and includes their higher-order effect and self-consistency effect. It is different from the conventional cluster expansions in several important points, and is suitable for applications to open-shell systems as well as closed-shell systems. The variational equation for the SAC wavefunction has a form similar to the generalized Brillouin theorem in accordance with the inclusion of the higher-order effect and the self-consistency effect. We have expressed some existing open-shell orbital theories equivalently in the conventional cluster expansion formulas, and on this basis, we have given the pseudo-orbital theory which is an extension of open-shell orbital theory in the SAC expansion formula.
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