Reconstruction of Density Functionals from Kohn−Sham Potentials by Integration along Density Scaling Paths

Abstract

We demonstrate by specific examples that if a Kohn-Sham exchange-correlation potential is given explicitly in terms of the electron density and its derivatives, then one can easily reconstruct the parent density functional by evaluating analytically (or numerically with one-dimensional quadratures) the van Leeuwen-Baerends line integral (Phys. Rev. A 1995, 51, 170-178) along a path of (coordinate)-scaled densities. The choice of a density scaling path amounts to defining the gauge of the resultant exchange-correlation energy density. The well-known Levy-Perdew virial relation for exchange potentials can be viewed as an analytical line integral along the electron-number-conserving uniform density scaling path. Energies obtained from model exchange-correlation potentials should be interpreted with caution because the reconstructed density functional is unique (up to a gauge transformation) only if the model Kohn-Sham potential is a functional derivative.

A generalized gradient approximation for exchange derived from the model potential of van Leeuwen and Baerends

“…28 No such problems arise if v x ([ρ]; r) is a functional derivative because in that case the Levy-Perdew relation yields the same energy value as the (fully invariant) parent functional. 29 All this means that the Levy-Perdew formula is an acceptable way of constructing energy functionals only for exchange potentials that are functional derivatives. It is quite feasible, at least at the level of generalized gradient approximations, to develop model potentials that are by construction functional derivatives.…”

Communication: Explicit construction of functional derivatives in potential-driven density-functional theory

“…11 Their method involves integration of the potential along a path of suitably parametrized densities, which can be done using numerical quadratures or even analytically. 12 The second problem has not been tackled so far, but it cannot be ignored if one wants to use model potentials to calculate observable physical properties. This is because approximate KohnSham potentials that are not functional derivatives generate spurious forces that cause the energy to depend on the system's position, [13][14][15] give rise to unphysical self-excitations, 16 and inflict other unwelcome artifacts.…”

Hydrogen Molecule Dissociation Curve with Functionals Based on the Strictly Correlated Regime

“…61,62 In general, however, a given approximation for the xc potential is not necessarily a functional derivative, which means that the xc energy corresponding to it is not well-defined, although different solutions have been proposed. 61 rather than the usual xc potential that goes to zero asymptotically.…”

“…61,62 In general, however, a given approximation for the xc potential is not necessarily a functional derivative, which means that the xc energy corresponding to it is not well-defined, although different solutions have been proposed. 61 rather than the usual xc potential that goes to zero asymptotically. Although in general the model potential would not be a functional derivative, the corresponding physical energy could be obtained without the need of a line integral, 61,62 as it would be always given by the sum of the occupied KS eigenvalues.…”