Hellmann-Feynman, virial, and scaling requisites for the exact universal density functionals. Shape of the correlation potential and diamagnetic susceptibility for atoms

Mel Levy and John P. Perdew
Phys. Rev. A 32, 2010 – Published 1 October 1985
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Abstract

By the Hellmann-Feynman theorem, the density n(r) of many electrons in the presence of external potential v(r) obeys the relationships Fd3r n(r)∇v(r)=0 and Fd3r n(r)r×∇v(r)=0. By the virial theorem, the interacting kinetic and electron-electron repulsion expectation values obey 2T[n]+Vee[n]=-Fd3r n(r)r⋅∇[δT/δn(r)+δVee/δn(r)]. The exchange energy functional Ex[n] and potential vx([n];r)≡δEx/δn(r) must satisfy Ex[n]+Fd3r n(r)r⋅∇vx([n];r)=0, while the correlation energy and potential must satisfy Ec[n]+Fd3r n(r)r⋅∇vc([n];r)<0. Somewhat counterintuitively, it is not true that T[nγ]=γ2T[n] and Vee[nγ]=γVee[n], where nγ(r)≡γ3n(γr) is a scaled density with scale factor γ≠1. In fact, it is impossible to partition the exact Hohenberg-Kohn functional into a piece that scales as γ2 and a piece that scales as γ, even if complete freedom with the partitioning is allowed. Instead there are universal scaling inequalities.

For instance, T[nγ]+Vee[nγ]<γ2T[n]+ γVee[n] and T[nγ]+γVee[nγ]>γ2(T[n ]+Vee[n]), and consequent inequalities involving Ec[n]. All the above virial and scaling requisites are universal in that they are independent of external potential and they must hold for arbitrary proper n. In addition, for the ground-state energy (E) and n of any atom or molecule at its equilibrium nuclear configuration, there is the inequality E<-Ts[n], where Ts is the noninteracting kinetic energy. In the closed-shell tight-binding limit, the correlation potential obeys Fd3r n(r)r⋅∇vc([n];r)=0, and so cannot be a monotonic function of r for an atom in this limit.

Further, (∂/∂γ)Ec[nγ]γ=1=E c[n]+Tc[n]=-Fd3r n(r)r⋅∇vc([n];r), which implies that the exact Ec should be fairly insensitive to scaling. With the help of the ionization-potential theorem, it is argued that the exact vc([n];r) in an atom often has a positive part. Common approximations to the correlation potential are examined for their effects upon the highest occupied Kohn-Sham orbital energy and the density moment 〈r2〉, and these effects are found to be related. Further improvements needed in the approximate correlation potentials are relatively large, but not nearly so large as those recently suggested for the atoms Ne, Ar, Kr, and Xe: The discrepancy between theoretical values of 〈r2〉 from Hartree-Fock or configuration-interaction calculations, and experimental values from measured diamagnetic susceptibilities, is tentatively resolved in favor of theory.

  • Received 15 March 1985

DOI:https://doi.org/10.1103/PhysRevA.32.2010

©1985 American Physical Society

Authors & Affiliations

Mel Levy

  • Department of Chemistry and Quantum Theory Group, Tulane University, New Orleans, Louisiana 70118

John P. Perdew

  • Department of Physics and Quantum Theory Group, Tulane University, New Orleans, Louisiana 70118

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Vol. 32, Iss. 4 — October 1985

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