A method for calculating the Kohn-Sham exchange-correlation potential, vXC(r), from a given electronic wavefunction is devised and implemented. It requires on input one-and two-electron density matrices and involves construction of the generalized Fock matrix. The method is free from numerical limitations and basis-set artifacts of conventional schemes for constructing vXC(r) in which the potential is recovered from a given electron density, and is simpler than various many-body techniques. The chief significance of this development is that it allows one to directly probe the functional derivative of the true exchange-correlation energy functional and to rigorously test and improve various density-functional approximations. Most existing methods for generating exact exchangecorrelation potentials fit the function v XC (r) to a given ground-state ρ(r) via the Kohn-Sham equations either by iterative updates [5][6][7][8] or through some constrained optimization [9][10][11]. The target densities are usually obtained from ab initio wavefunctions which are themselves discarded. Because small changes in ρ(r) can induce large changes in v XC (r) [12], potential-reconstruction methods that use only ρ(r) as input suffer from numerical instabilities. Moreover, electron densities generated using ubiquitous Gaussian basis sets correspond to exchangecorrelation potentials that wildly oscillate and diverge [13][14][15][16], a result that is formally correct but unwanted. KohnSham potentials can be also constructed by many-body methods [17][18][19][20][21], but these techniques are quite elaborate and often require solving an integral equation for v XC (r), which is a challenge by itself.Here, we propose a radically different method for computing the exchange-correlation potential of a given many-electron system, which avoids the above difficulties. In this method, the functional derivative of the exact E XC [ρ] is obtained directly from the system's electronic wavefunction. The approach represents a nontrivial generalization of our technique for constructing Kohn-Sham potentials corresponding to Hartree-Fock (HF) electron densities [22,23] and is conceptually related to the wavefunction-based analysis of Kohn-Sham potentials developed by Baerends and co-workers [24][25][26][27][28].The basic idea of our approach is to derive two expressions for the local electron energy balance, one of which originates from the Kohn-Sham equations, the other from the Schrödinger equation. When one expression is subtracted from the other under the assumption that the Kohn-Sham and wavefunction-based densities are equal, the system's electrostatic potentials cancel out and the difference gives an explicit formula for v XC (r). For simplicity, the treatment presented in this Letter is restricted to electronic singlet ground states described with closed-shell Kohn-Sham determinants, and assumes that all basis functions and orbitals are real (although the notation for complex conjugate is retained).Accomplishing the first part of this plan is easy....
Given a set of canonical Kohn-Sham orbitals, orbital energies, and an external potential for a many-electron system, one can invert the Kohn-Sham equations in a single step to obtain the corresponding exchange-correlation potential, vXC(r). For orbitals and orbital energies that are solutions of the Kohn-Sham equations with a multiplicative vXC(r) this procedure recovers vXC(r) (in the basis set limit), but for eigenfunctions of a non-multiplicative one-electron operator it produces an orbital-averaged potential. In particular, substitution of Hartree-Fock orbitals and eigenvalues into the Kohn-Sham inversion formula is a fast way to compute the Slater potential. In the same way, we efficiently construct orbital-averaged exchange and correlation potentials for hybrid and kinetic-energy-density-dependent functionals. We also show how the Kohn-Sham inversion approach can be used to compute functional derivatives of explicit density functionals and to approximate functional derivatives of orbital-dependent functionals.
We describe a method for constructing a hierarchy of model potentials approximating the functional derivative of a given orbital-dependent exchange-correlation functional with respect to electron density. Each model is derived by assuming a particular relationship between the self-consistent solutions of Kohn-Sham (KS) and generalized Kohn-Sham (GKS) equations for the same functional. In the KS scheme, the functional is differentiated with respect to density, in the GKS scheme--with respect to orbitals. The lowest-level approximation is the orbital-averaged effective potential (OAEP) built with the GKS orbitals. The second-level approximation, termed the orbital-consistent effective potential (OCEP), is based on the assumption that the KS and GKS orbitals are the same. It has the form of the OAEP plus a correction term. The highest-level approximation is the density-consistent effective potential (DCEP), derived under the assumption that the KS and GKS electron densities are equal. The analytic expression for a DCEP is the OCEP formula augmented with kinetic-energy-density-dependent terms. In the case of exact-exchange functional, the OAEP is the Slater potential, the OCEP is roughly equivalent to the localized Hartree-Fock approximation and related models, and the DCEP is practically indistinguishable from the true optimized effective potential for exact exchange. All three levels of the proposed hierarchy require solutions of the GKS equations as input and have the same affordable computational cost.
The exact exchange-correlation potential of a stretched heteronuclear diatomic molecule exhibits a localized upshift in the region around the more electronegative atom; by this device the Kohn-Sham scheme ensures that the molecule dissociates into neutral atoms. Baerends and co-workers showed earlier that the upshift originates in the response part of the exchange-correlation potential. We describe a reliable numerical method for constructing the response potential of a given many-electron system and report accurate plots of this quantity. We also demonstrate that the step feature itself can be obtained directly from the interacting wavefunction of the system by computing the so-called average local electron energy. These findings illustrate in previously unavailable detail the mechanism of the formation of the upshift and the role played by static correlation in this process.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.