Accurate effective potential for density amplitude and the corresponding Kohn–Sham exchange–correlation potential calculated from approximate wavefunctions

Abstract

Over the past few years it has been pointed out that direct inversion of accurate but approximate ground state densities leads to Kohn–Sham exchange–correlation (xc) potentials that can differ significantly from the exact xc potential of a given system. On the other hand, the corresponding wavefunction based construction of exchange-correlation potential as done by Baerends et al and Staroverov et al obviates such problems and leads to potentials that are very close to the true xc potential. In this paper, we provide an understanding of why the wavefunction based approach gives the exchange–correlation potential accurately. Our understanding is based on the work of Levy, Perdew and Sahni (LPS) who gave an equation for the square root of density (density amplitude) and the expression for the associated effective potential in the terms of the corresponding wavefunction. We show that even with the use of approximate wavefunctions the LPS expression gives accurate effective and exchange–correlation potentials. Based on this we also identify the source of difference between the potentials obtained from a wavefunction and those given by the inversion of the associated density. Finally, we suggest exploring the possibility of obtaining accurate ground-state density from an approximate wavefunction for a system by making use of the LPS effective potential.

Inverse Kohn-Sham Density Functional Theory: Progress and Challenges

“…will lead to the same density. This many-to-one problem is at the root of the relation between the exact external potential and 'effective' external potentials discussed by Gaiduk et al 56 and Kumar et al 44 . Second, the electron density cannot be represented exactly on finite Gaussian basis sets.…”

“…Its computational cost is thus out of reach for calculations on large systems. A detailed discussion explaining the success of wavefunction based methods compared to pure density inversions was provided recently by Kumar et al 44 . We use mRKS in this article as benchmark to compare with other purely density-based approaches.…”

On the Geometric Potential and the Relationship between the Exact Electron Factorization and Density Functional Theory

“…The separation of a wavefunction into a marginal and a conditional part was first considered for the electron-nuclear problem [29] and has subsequently been transferred to the many-electron problem [27], which lead to first studies of the properties of and the connections between v, v KS , and v P , for atoms [30][31][32][33] and diatomics [30,[34][35][36], and to further studies of the conditional wavefunction in the DFT literature [37][38][39][40][41][42][43][44]. Recently, the formalism of the wavefunction separation has been further developed for the electron-nuclear problem and been termed the exact factorization [45][46][47].…”

Inverse Kohn–Sham Density Functional Theory: Progress and Challenges

“…Its computational cost is thus out of reach for calculations on large systems. A detailed discussion explaining the success of wave function-based methods compared to pure density inversions (i.e., where only the densities are needed) was provided recently by Kumar et al We concentrate on pure Kohn–Sham inversions , referred to as iKS in this Perspective. Methods other than self-consistent calculations have also been designed, many of which feature constrained optimizations. ,,− …”

“…First, for a given finite OBS {ϕ μ ′ }, all the XC potentials that produce the same Fock matrices will lead to the same density. This many-to-one problem is at the root of the relation between the exact external potential and “effective” external potentials discussed by Gaiduk et al and Kumar et al Second, the electron density cannot be represented exactly on finite Gaussian basis sets . Small input errors will lead to large oscillations given the ill-posed nature of iKS. , …”