We have shown analytically and numerically that the value of the total exchange-correlation charge, QXC, determines the rate of the asymptotic decay the corresponding exchange-
correlation potential via Eq. (1.66). The subtle aspect of applying this rule is that, to obtain the correct values of QXC it is necessary to analyze the behavior of the potential
at the nucleus (every nucleus in a molecule). If the potential has a Coulombic singularity at a nucleus (as GGAs do), the corresponding exchange-correlation charge will contain a non-zero point charge contribution which cannot be accounted for by numerical quadra- tures. This point charge contribution must be evaluated explicitly by Eq. (2.17).
As for the nonzero values of QXC reported by Liu et al. [2] for the LDA and GGA
exchange-correlation potentials, we believe that they were a combined result of numerical errors of the Zhao–Morrison–Parr procedure and residual contributions of the Coulombi- cally decaying Fermi–Amaldi potential which was used as a fixed reference to represent each vXC(r). This explanation is consistent with the fact that most values of QXC in
Table I of Ref. 2 are close to −1.
The relationship between the magnitude of QXC and the asymptotic behavior of
vXC(r) has important practical implications. For example, to construct a model exchange-
correlation potential with the correct −1/r asymptotic decay , it is sufficient to model a distribution qXC(r) that integrates to −1. This can be accomplished by taking an
exchange-correlation charge distribution that integrates to zero (e.g., LDAx) and reshap- ing it [10], or by adding a point exchange-correlation charge at the nucleus (in which case vXC(r) will be singular at r = 0). The overall effect of the latter approach should be
qualitatively similar to the effect of depopulating the highest-occupied molecular orbital of the system in a self-consistent Kohn–Sham calculation [23, 24]. At the same time, it is not possible to obtain a Coulombic potential by taking the LDA exchange-correlation
charge distribution and simply scaling it, because the total exchange-correlation charge will remain zero.
An approximate Kohn–Sham potential obtained from a model exchange-correlation charge distribution using Eq. (1.63) will generally not be a functional derivative of any energy expression. Although it is possible to assign reasonable energy values to non- integrable Kohn–Sham potentials in many different ways [25–27], a better strategy would be to find the constraints on qXC(r) which ensure that the corresponding vXC(r) is a func-
tional derivative. In Chapter 3, we derive these constraints in the form of an analytical test that a model function qXC(r) must pass in order for the corresponding vXC(r) to be
Bibliography
[1] C. Filippi, X. Gonze, and C. J. Umrigar, “Generalized gradient approximations to density functional theory: Comparison with exact results”, in Recent Developments and Applications of Modern Density Functional Theory, edited by J. M. Seminario (Elsevier, Amsterdam, 1996), pp. 295–326.
[2] S. Liu, P. W. Ayers, and R. G. Parr, “Alternative definition of exchange-correlation charge in density functional theory”, J. Chem. Phys. 111, 6197 (1999).
[3] Q. Zhao, R. C. Morrison, and R. G. Parr, “From electron densities to Kohn–Sham kinetic energies, orbital energies, exchange-correlation potentials, and exchange- correlation energies”, Phys. Rev. A 50, 2138 (1994).
[4] S. J. A. van Gisbergen, V. P. Osinga, O. V. Gritsenko, R. van Leeuwen, J. G. Sni- jders, and E. J. Baerends, “Improved density functional theory results for frequency- dependent polarizabilities, by the use of an exchange-correlation potential with cor- rect asymptotic behavior”, J. Chem. Phys. 105, 3142 (1996).
[5] M. E. Casida, C. Jamorski, K. C. Casida, and D. R. Salahub, “Molecular exci- tation energies to high-lying bound states from time-dependent density-functional response theory: Characterization and correction of the time-dependent local den- sity approximation ionization threshold”, J. Chem. Phys. 108, 4439 (1998).
[6] D. J. Tozer and N. C. Handy, “Improving virtual Kohn–Sham orbitals and eigenval- ues: Application to excitation energies and static polarizabilities”, J. Chem. Phys. 109, 10180 (1998).
[7] R. Baer, E. Livshits, and U. Salzner, “Tuned range-separated hybrids in density- functional theory”, Annu. Rev. Phys. Chem. 61, 85 (2010).
[8] C.-W. Tsai, Y.-C. Su, G.-D. Li, and J.-D. Chai, “Assessment of density functional methods with correct asymptotic behavior”, Phys. Chem. Chem. Phys. 15, 8352 (2013).
[9] W. Cencek and K. Szalewicz, “On asymptotic behavior of density functional the- ory”, J. Chem. Phys. 139, 024104 (2013).
[10] X. Andrade and A. Aspuru-Guzik, “Prediction of the derivative discontinuity in density functional theory from an electrostatic description of the exchange and correlation potential”, Phys. Rev. Lett. 107, 183002 (2011).
[11] N. I. Gidopoulos and N. N. Lathiotakis, “Constraining density functional approx- imations to yield self-interaction free potentials”, J. Chem. Phys. 136, 224109 (2012).
[12] J. P. Perdew and Y. Wang, “Accurate and simple analytic representation of the electron-gas correlation energy”, Phys. Rev. B 45, 13244 (1992).
[13] R. van Leeuwen and E. J. Baerends, “Exchange-correlation potential with correct asymptotic behavior”, Phys. Rev. A 49, 2421 (1994).
[14] J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple”, Phys. Rev. Lett. 77, 78, 1396(E) (1997), 3865 (1996).
[15] A. D. Becke, “Density-functional exchange-energy approximation with correct asymp- totic behavior”, Phys. Rev. A 38, 3098 (1988).
[16] A. P. Gaiduk and V. N. Staroverov, “Construction of integrable model Kohn–Sham potentials by analysis of the structure of functional derivatives”, Phys. Rev. A 83, 012509 (2011).
[17] A. P. Gaiduk and V. N. Staroverov, “How to tell when a model Kohn–Sham po- tential is not a functional derivative”, J. Chem. Phys. 131, 044107 (2009).
[18] C. F. Bunge, J. A. Barrientos, and A. V. Bunge, “Roothaan–Hartree–Fock ground- state atomic wave functions: Slater-type orbital expansions and expectation values for z = 2–54”, At. Data Nucl. Data Tables 53, 113 (1993).
[19] mathematica, Version 9.0, Wolfram Research, Inc., Champaign, IL, 2012. [20] E. Engel, J. A. Chevary, L. D. Macdonald, and S. H. Vosko, “Asymptotic proper-
ties of the exchange energy density and the exchange potential of finite systems: Relevance for generalized gradient approximations”, Z. Phys. D 23, 7 (1992). [21] C. J. Umrigar and X. Gonze, “Accurate exchange-correlation potentials and total-
energy components for the helium isoelectronic series”, Phys. Rev. A 50, 3827 (1994).
[23] A. P. Gaiduk, D. S. Firaha, and V. N. Staroverov, “Improved electronic excitation energies from shape-corrected semilocal Kohn–Sham potentials”, Phys. Rev. Lett. 108, 253005 (2012).
[24] A. P. Gaiduk, D. Mizzi, and V. N. Staroverov, “Self-interaction correction scheme for approximate Kohn–Sham potentials”, Phys. Rev. A 86, 052518 (2012).
[25] A. P. Gaiduk, S. K. Chulkov, and V. N. Staroverov, “Reconstruction of density functionals from Kohn–Sham potentials by integration along density scaling paths”, J. Chem. Theory Comput. 5, 699 (2009).
[26] A. P. Gaiduk and V. N. Staroverov, “A generalized gradient approximation for exchange derived from the model potential of van Leeuwen and Baerends”, J. Chem. Phys. 136, 064116 (2012).
[27] P. D. Elkind and V. N. Staroverov, “Energy expressions for Kohn–Sham potentials and their relation to the Slater–Janak theorem”, J. Chem. Phys. 136, 124115 (2012).
Chapter 3
Integrability conditions for model
potentials constructed using the
exchange-correlation charge
distribution
3.1
Introduction
Two methodological challenges of potential-driven KS DFT are: (i) how to assign an energy to a given model potential and (ii) how to identify and construct integrable model potentials. The first problem was solved by the line integral method [1] of van Leeuwen and Baerends [Eq. (1.59)]. The second problem was addressed by Ou-Yang and Levy [2] followed by van Leeuwen and Baerends [1] who derived the basic integrability condition for model potentials:
δvXC(r)
δρ(r0) =
δvXC(r0)
δρ(r) . (3.1)
The above equation can serve as the starting point for deriving more practical integrabil- ity conditions for potentials restricted to certain analytic forms. Consider, for example, an explicitly density-dependent potential, vXC ≡ vXC(ρ, ∇ρ, ∇2ρ). For such a potential,
evaluating both sides of Eq. (3.1), followed by their comparison, leads to the integrability condition of Eq. (1.62) [3].
In this chapter, we derive analytic integrability conditions for model potentials ex- pressed in the form of the electrostatic integral of Eq. (1.64). To this end, we first obtain
tribution. Derivations presented in this work require the knowledge of several properties of the Dirac delta function and its derivatives, which are summarized in AppendixA.