An approximate Kohn-Sham exchange-correlation potential xc SAOP is developed with the method of statistical averaging of ͑model͒ orbital potentials ͑SAOP͒ and is applied to the calculation of excitation energies as well as of static and frequency-dependent multipole polarizabilities and hyperpolarizabilities within time-dependent density functional theory ͑TDDFT͒. xc SAOP provides high quality results for all calculated response properties and a substantial improvement upon the local density approximation ͑LDA͒ and the van Leeuwen-Baerends ͑LB͒ potentials for the prototype molecules CO, N 2 , CH 2 O, and C 2 H 4 . For the first three molecules and the lower excitations of the C 2 H 4 the average error of the vertical excitation energies calculated with xc SAOP approaches the benchmark accuracy of 0.1 eV for the electronic spectra.

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Density functional calculations on the (non)linear optical properties of conjugated molecular chains using currently popular exchange-correlation (xc) potentials give overestimations of several orders of magnitude. By analyzing "exact" and Krieger-Li-Iafrate xc potentials, the error is traced back to an incorrect electric field dependence of the "response part" of the xc potential in local and gradientcorrected density approximations, which lack a linear term counteracting the applied electric field.PACS numbers: 31.15. Ew, 42.65.An, 71.15.Mb, 77.22.Ej The nonlinear optical (NLO) properties of molecules are of considerable current interest, both from the fundamental and technological points of view [1][2][3]. Prototype systems exhibiting large NLO responses like the polyacetylene (PA) chains have been studied intensively with conventional, ab initio Hartree-Fock (HF) based theoretical techniques [2]. Because density functional theory (DFT) [4,5] usually provides clearly improved accuracy with respect to HF at similar or lower computational cost, it seems tempting to apply DFT to the prediction of the NLO properties of large, conjugated molecular chains.Our calculations on static hyperpolarizabilities (determining the NLO response in the static limit) of such chains show, both here and in Ref. [6], that the local density approximation (LDA) and generalized gradient approximations (GGAs) in DFT provide very poor results. Whereas previously reported LDA errors are typically 10% for dielectric constants, we find overestimations of several orders of magnitude for the second hyperpolarizability g. In view of the respectable accuracy which is usually obtained in DFT calculations, this is highly surprising and deserves a detailed analysis, as this error may be among the largest in the history of DFT calculations. Our analysis allows us to pinpoint the weakness of the LDA. It will be shown below that, if an electric field, E, is applied, the so-called response part of the exact exchange-correlation (xc) potential develops a global behavior counteracting the applied field. Such behavior is not present in the LDA or GGA potentials. Our results substantiate and further elucidate the findings of Gonze, Ghosez, and Godby [7] and others [8][9][10][11][12][13], who pointed out the existence of such a counteracting linear potential in the exact y xc and provided the first physical interpretation for it [10,13].Description of the problem.-For this work, extensive calculations have been performed on PA and hydrogen chains. The LDA results described below have been obtained in the same manner as earlier calculations on the polarizability (a) and second hyperpolarizability (g) of C 60 [14]. Details of the computational procedure, the geometries, etc., have been given in another paper [6], where several aspects of the LDA problem (size of electron correlation correction, effect of bond length alternation) are described more fully.From the HF-based ab initio studies [15] on the NLO properties of PA chains, it is known that even sim...

Response calculations in the framework of time-dependent density-functional theory ͑TDDFT͒ have by now been shown to surpass time-dependent Hartree-Fock ͑TDHF͒ calculations in both accuracy and efficiency. This makes TDDFT an important tool for the calculation of frequency-dependent ͑hyper͒polarizabilities, excitation energies, and related properties of medium-sized and large molecules. Two separate approximations are made in the linear DFT response calculations. The first approximation concerns the exchange-correlation ͑xc͒ potential, which determines the form of the Kohn-Sham orbitals and their one-electron energies, while the second approximation involves the so-called xc kernel f xc , which determines the xc contribution to the frequencydependent screening. By performing calculations on small systems with accurate xc potentials, constructed from ab initio densities, we can test the relative importance of the two approximations for different properties and systems, thus showing what kind of improvement can be expected from future, more refined, approximations to these xc functionals. We find that in most, but not all, cases, improvements to v xc seem more desirable than improvements to f xc .

Kohn−Sham solutions are constructed from ab initio densities obtained with multireference configuration interaction (MRCI) calculations for the transition state (TS) and for the intermediate complex (IC) of the prototype symmetrical SN2 reaction F- + CH3F → FCH3 + F-. The calculated KS exchange and correlation energies, E x KS and E c KS, as well as the exchange and exchange-correlation (xc) energy densities ε x KS(r) and ε xc KS(r), are compared with the corresponding quantities of the standard generalized gradient approximation (GGA). GGA functionals substantially underestimate the repulsive exchange contribution to the central barrier of the SN2 reaction, thus producing a too low barrier. A similar problem arises in a number of other bonding situations, and a qualitative rule is put forward to predict success or failure of standard GGAs in molecular calculations, depending on the type of chemical bonding. For systems with two-center two-electron bonds (standard covalent bonds), two-center four-electron Pauli repulsion (interacting closed shells), and three-center three-electron bonds, current GGAs (or minor modifications) are expected to perform successfully. In these cases the GGA exchange functional represents exchange and (if it is present) nondynamical Coulomb correlation, while the GGA correlation functional represents dynamical Coulomb correlation. Contrary to this, for systems with three-center four-electron bonds (TS of the SN2 reaction), two-center three-electron bonds, and two-center one-electron bonds, for which the exchange hole is delocalized over all interacting fragments and efficient nondynamical correlation is hampered by the unfavorable electron count, the GGA exchange functionals still yield nondynamical correlation, which is in these cases spurious, the GGAs thus overestimating the relative stability of these systems.

The density functional definition of exchange and correlation differs from the traditional one. In order to calculate the density functional theory ͑DFT͒, quantities accurately, molecular Kohn-Sham ͑KS͒ solutions have been obtained from ab initio wave functions for the homonuclear diatomic molecules Li 2 , N 2 , F 2 . These afford the construction of the KS determinant ⌿ s and the calculation of its total electronic energy E KS and the kinetic, nuclear-attraction and Coulomb repulsion components T s , V, W H as well as the ͑DFT͒ exchange energy E x and correlation energy E c . Comparison of these DFT quantities has been made on one hand with the corresponding HartreeFock ͑HF͒ quantities and on the other hand with local density approximation ͑LDA͒ and generalized gradient approximation ͑GGA͒. Comparison with HF shows that the correlation errors in the components T, V, and W H of the total energy are much larger for HF than KS determinantal wave functions. However, the total energies E KS and E HF appear to be close to each other, as well as the exchange energies E x and E x HF and correlation energies E c and E c HF . The KS determinantal wave function and the KS orbitals therefore correspond to much improved kinetic and Coulombic energies, while having only a slightly larger total correlation energy. It is stressed that these properties of the Kohn-Sham orbitals make them very suitable for use in the molecular orbital theories of chemistry. Comparison of the accurate Kohn-Sham exchange and correlation energies with LDA and GGA shows that the GGA exchange energies are consistently too negative, while the GGA correlation energies are not negative enough. It is argued that the GGA exchange functionals represent effectively not only exchange, but also the molecular non-dynamical correlation, while the GGA correlation functionals represent dynamical correlation only. © 1997 American Institute of Physics. ͓S0021-9606͑97͒00337-1͔

The Kohn-Sham ͑KS͒ solution is constructed from an accurate CI density and the KS exchange and correlation energies E x and E c , as well as the corresponding exchange and exchange-correlation energy densities ⑀ x (r) and ⑀ xc (r), which are obtained for the hydrogen abstraction reaction HϩH 2 and the symmetrical four-center exchange reaction H 2 ϩH 2 . The KS quantities are compared with those of the standard GGAs. Comparison shows that the GGA exchange functional represents both exchange and molecular nondynamical left-right correlation, while the GGA correlation functional represents only the dynamical part of the correlation. This role of the GGA exchange functional is especially important for the transition states ͑TS͒ of the reactions where the left-right correlation is enhanced. Standard GGAs tend to underestimate the barrier height for the reaction HϩH 2 and to overestimate it for the reaction H 2 ϩH 2 . For H 2 ϩH 2 the Kohn-Sham orbital degeneracy in the square TS is represented with an equi-ensemble KS solution for both accurate KS/CI and GGA, while near the TS ensemble solutions with unequal occupations of the degenerate highest occupied orbitals are obtained. For the GGA ensemble solution a special ensemble formula for the GGA exchange functional is proposed. Application of this formula to the H 2 ϩH 2 reaction reduces appreciably the reaction barriers calculated with GGAs and leads to much better agreement with the accurate value. The too low GGA barriers for the HϩH 2 reaction are attributed to overestimation of the dynamical correlation in the TS by the GGA correlation functionals. In order to correct this error, it is recommended to modify the dependence of the approximate correlation functionals on the local polarization with the purpose of reducing the approximate correlation energy for intermediate values, which are expected to characterize the TS's of radical abstraction reactions.

A new method based on linear response theory is proposed for the determination of the KohnSham potential corresponding to a given electron density. The method is very precise and aords a comparison between Kohn-Sham potentials calculated from correlated reference densities expressed in Slater-(STO) and Gaussian-type orbitals (GTO). In the latter case the KS potential exhibits large oscillations that are not present in the exact potential. These oscillations are related to similar oscillations in the local error function d i r h À e i u i r when SCF orbitals (either KohnSham or Hartree-Fock) are expressed in terms of Gaussian basis functions. Even when using very large Gaussian basis sets, the oscillations are such that extreme care has to be exercised in order to distinguish genuine characteristics of the KS potential, such as intershell peaks in atoms, from the spurious oscillations. For a density expressed in GTOs, the Laplacian of the density will exhibit similar spurious oscillations. A previously proposed iterative local updating method for generating the Kohn-Sham potential is evaluated by comparison with the present accurate scheme. For a density expressed in GTOs, it is found to yield a smooth`a verage'' potential after a limited number of cycles. The oscillations that are peculiar to the GTO density are constructed in a slow process requiring very many cycles.