Improving virtual Kohn–Sham orbitals and eigenvalues: Application to excitation energies and static polarizabilities

Abstract

Conventional continuum exchange-correlation functionals (e.g., local density approximation, generalized gradient approximation) offer a poor description of many response properties, such as static polarizabilities and single photon vertical excitation energies to Rydberg states. These deficiencies are related to errors in the virtual Kohn–Sham orbitals and eigenvalues, which arise due to a fundamental deficiency in the potentials of conventional continuum functionals. Namely, although these potentials approximately average over the exact integer discontinuity in energetically important regions, they fail to do so asymptotically, because they vanish. Our recent functional HCTH [J. Chem. Phys. 109, 6264 (1998)] was designed with this deficiency in mind, although its potential still fails to exhibit the appropriate asymptotic form. In this paper, we present a new procedure that explicitly corrects this asymptotic deficiency for any continuum functional. Self-consistent Kohn–Sham calculations are performed using a corrected potential, which equals the conventional potential δEXC[ρα,ρβ]/δρσ(r) in energetically important regions, but which asymptotically behaves in the required average manner −(1/r)+Iσ+εHOMO,σ. The quantity −(1/r) is determined using a nonlocal expression; Iσ is an approximate σ spin ionization potential; and εHOMO,σ is the highest occupied σ spin eigenvalue. By applying this correction to the HCTH potential, we accurately reproduce the hydrogen atom eigenvalue spectrum, without significantly changing the total energy. We determine corrected orbitals and eigenvalues for a variety of molecules, and use them to compute excitation energies and static polarizabilities. We compare the results with those from a variety of other exchange-correlation functionals. Excitations to Rydberg states are determined as accurately as those to valence states; for CO, N2, H2CO, and C2H4, mean absolute errors are less than 0.35 eV. The static isotropic polarizabilities of 14 small molecules are of MP2 quality.

Excitation Energies for Transition Metal Compounds from Time-Dependent Density Functional Theory. Applications to MnO4-, Ni(CO)4, and Mn2(CO)10

“…13,14 Perhaps the most popular application of time-dependent density functional theory (TD-DFT) in the molecular regime has been the calculation of excitation energies, in which many groups have, by now, been involved. 5,[15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33] The equations from which the excitation energies are obtained are well-established. 18,20,25 They are formally quite similar to the time-dependent Hartree-Fock (TDHF) equations (TDHF is also known as random phase approximation (RPA)) and can be solved efficiently 25 by using iterative techniques, such as the Davidson algorithm.…”

Orbital-Free Embedding Effective Potential in Analytically Solvable Cases

“…The situation reminds of that of going from the system with N electrons to that of N +1 electrons: in order to satisfy both requirements, the potential will be essentially shifted by a "constant", (−1/2)−(−1/8)), thus shifting the eigenvalue [33][34][35][36]. This happens over all space, except for the asymptotic region, where the ϕ 2s 2 dominates.…”

Long-range excitations in time-dependent density functional theory

“…Typically the results are accurate to within a few tenths of an eV (although there are exceptions, see eg. [8,12,15,25]). When excitations are well-separated, the matrix may be simplified in a "single pole approximation (SPA)" [39,40,41] that gives the true frequency ω as a shift from a KS transition: for a closed-shell molecule,…”

“…Approximations enter at two stages in usual linear response calculations: first, in the xc contribution to the one-body ground-state Kohn-Sham (KS) potential out of which bare excitations are calculated. For example, its too rapid asymptotic decay in LDA/GGA causes problems for high-lying bound states [7,8,9,10]. Second, the dynamic xc kernel must be approximated; this corrects the bare KS transitions towards the true transitions.…”