Paradox of Self-Interaction Correction

How Can Anything So Right Be So Wrong?

John P. Perdew*,†; Adrienn Ruzsinszky*; Jianwei Sun*; Mark R. Pederson‡,1    * Department of Physics, Temple University, Philadelphia, Pennsylvania, USA
† Department of Chemistry, Temple University, Philadelphia, Pennsylvania, USA
‡ Department of Chemistry, Johns Hopkins University, Baltimore, Maryland, USA
1 Corresponding author: email address: mpeder10@jhu.edu

1 Introduction

Kohn–Sham density functional theory (Kohn and Sham, 1965) is a formally exact construction of the ground-state energy and electron density for a system of electrons with mutual Coulomb repulsion in the presence of a multiplicative scalar external potential. The construction proceeds by solving self-consistent one-electron equations for the occupied Kohn–Sham orbitals, fictional objects used to build up the electron density, and the noninteracting part of the kinetic energy. The many-electron effects are incorporated via the exchange-correlation energy as a functional of the density, Exc[n↑,n↓], and its functional derivative or exchange-correlation potential vxcσ([n↑,n↓];r). In practice, the exchange-correlation energy has to be approximated. This approach is very widely used for the computation of atoms, molecules, and condensed matter, because of its useful balance between computational efficiency and accuracy.

The exact exchange-correlation energy is defined (Gunnarsson and Lundqvist, 1976; Langreth and Perdew, 1975, 1977) so that

[n]+Exc[n↑,n↓]=∫01dλ〈Ψλ|Vˆee|Ψλ〉.

Here,

[n]=12∫d3r∫d3r′n(r)n(r′)|r−r′|,

is the Hartree electrostatic self-repulsion energy of the total electron density n(r) = n↑(r) + n↓(r), the sum of up-spin and down-spin contributions. ˆee is the electron–electron Coulomb repulsion operator. And Ψλ is the ground-state wavefunction for electrons with interaction Vˆee and with density (r)=〈Ψλ|nˆ(r)|Ψλ〉 independent of coupling constant λ. The spin-dependent external scalar potential vσλ(r) varies between the Kohn–Sham effective potential at λ = 0 and the physical external potential at λ = 1. We can write Exc as the sum of exchange and correlation energies, where the exchange energy Ex is defined by

[n]+Ex[n↑,n↓]=〈Ψ0|Vˆee|Ψ0〉.

Typically Ψ0 is a single Slater determinant of Kohn–Sham orbitals, and Ex differs from Hartree–Fock exchange only via the small difference between Kohn–Sham and Hartree–Fock orbitals.

The exchange energy and the correlation energy are nonpositive. They arise because, as an electron moves through the density, it creates around itself exchange and correlation holes (Gunnarsson and Lundqvist, 1976) which reduce its repulsion energy with the other electrons. The exchange hole arises from self-interaction correction and wavefunction antisymmetry under particle exchange, and its density integrates to − 1, while the correlation hole arises from Coulomb repulsion, and its density integrates to 0. While the exchange-correlation energy can be a small fraction of the total energy, it is nature's glue (Kurth and Perdew, 2000) that creates most of the binding of one atom to another in a molecule or solid.

For any spin-up one-electron density n1(r), the Coulomb repulsion operator vanishes so

[n1]+Ex[n1,0]=0,

c[n1,0]=0.

The functional Exc of Eq. (1) is defined for ground-state spin-densities, but it has a natural continuation to all fully-spin-polarized one-electron densities, given by Eqs. (4) and (5), since the Coulomb repulsion operator vanishes for all such densities. This continuation is not only natural but also physical: It makes the solutions of the Kohn–Sham orbital equations exact for one-electron systems, not only in their ground states but also in their excited states and time-dependent states. It is also the choice made in the Hartree–Fock and self-interaction-corrected Hartree approximations. Approximate functionals that satisfy Eqs. (4) and (5) are said to be one-electron self-interaction-free (Perdew and Zunger, 1981). For other functionals, the numerical values of the right-hand sides are self-interaction errors (SIE) for exchange and correlation, respectively, and their sum is the total self-interaction error.

Semilocal approximations have single-integral form,

xcsl[n↑,n↓]=∫d3rnϵxcsl(n↑,n↓,∇n↑,∇n↓,τ↑,τ↓),

and are popular because of their computational efficiency. The original local spin-density approximation (Gunnarsson and Lundqvist, 1976; Kohn and Sham, 1965) uses only the spin-density arguments, the generalized gradient approximation (GGA) (Becke, 1988; Langreth and Mehl, 1983; Lee et al., 1988; Perdew and Wang, 1986; Perdew et al., 1996) adds the spin-density gradients, and the meta-GGA (Becke and Roussel, 1989; Perdew et al., 1999; Sun et al., 2012, 2013; Tao et al., 2003; Van Voorhis and Scuseria, 1998) adds the positive kinetic energy densities

σ(r)=Σαoccupied12|∇ψασ|2

of the Kohn–Sham orbitals ψασ. The exchange-correlation energy per particle xcsl may be constructed to satisfy exact constraints on Exc, and the addition of more arguments permits the satisfaction of more constraints with resulting greater accuracy. For some GGA s (e.g., PBE) and meta-GGA s (e.g., TPSS), this construction is nonempirical. But no semilocal functional can satisfy Eq. (4), because of the full nonlocality of U[n], and only the meta-GGA can satisfy Eq. (5). Hybrid functionals (Becke, 1993; Ernzerhof and Scuseria, 1999; Stephens et al., 1994) add an exact-exchange ingredient, e.g.,

xchybrid=(1−a)Exsl+aExexact+Ecsl.

Typically they achieve higher accuracy through empirical selection of the mixing parameter (e.g., 0.25) and not through...