Article pubs.acs.org/JCTC
Cite This: J. Chem. Theory Comput. 2018, 14, 4246−4253
Construction of Fermi Potentials from Electronic Wave Functions Egor Ospadov and Viktor N. Staroverov*
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Department of Chemistry, The University of Western Ontario, London, Ontario N6A 5B7, Canada ABSTRACT: The Fermi potential, vF(r), is the nonclassical part of the multiplicative effective potential appearing in the one-particle Schrödinger-type equation for the square root of the electron density. The usual way of constructing vF(r) by inverting that equation produces unsatisfactory results when applied to electron densities expanded in Gaussian basis sets. We suggest a different method that is based on an explicit formula for vF(r) in terms of the interacting one- and two-electron reduced density matrices of the system. This method is exact in the basisset limit and yields accurate approximations to the basis-set-limit vF(r) when applied to reduced density matrices represented in terms of finite basis sets. Illustrative applications involve atomic and molecular wave functions generated at various levels of ab initio theory. It is also shown how to construct the Pauli and exchange-correlation potentials of any system starting with only vF(r).
1. INTRODUCTION Levy, Perdew, and Sahni1 considered a density functional scheme in which the total electronic energy functional of the groundstate electron density ρ(r) is partitioned as E[ρ] = TW[ρ] +
∫ ρ(r)v(r)dr + EH[ρ] + G[ρ]
is an unknown potential that is the subject of this paper. Given an approximation to G[ρ], eqs 1 and 4 allow one, in principle, to obtain the ground-state electron density and energy of the system without generating Kohn−Sham orbitals. There is no universally accepted terminology associated with eq 4. Levy et al.1 and Buijse et al.2 discussed a single effective potential veff that is equal to vH + vF in our notation. March and Holas3,4 used the term “effective potential for the boson problem” and symbol vB for the quantity which we would write here as v + vH + vF. Finzel and Ayers5 referred to δG/δρ as the “Fermi potential” and denoted it by the symbol vF, the name and notation which we adopt here. The Fermi potential is a fundamental quantity closely related to the exchange-correlation potential vXC(r) of the Kohn− Sham scheme, the Pauli potential3,4,6−10 (see below), and other concepts.11,12 It is therefore desirable to have a reliable general method for constructing vF(r) for a given ρ(r). At first glance, the problem seems trivial: from eq 4, one immediately has
(1)
where TW[ρ] =
1 8
2
∫ |∇ρρ((rr))|
dr
(2)
is the von Weizsäcker kinetic energy, v(r) is the external potential of the system, E H [ρ ] =
1 2
∫
ρ(r)ρ(r′) dr dr′ |r − r′|
(3)
is the Hartree (Coulomb) energy, and G[ρ] is a universal functional that includes the remainder of E[ρ]. Minimization of E[ρ] with respect to ρ(r), subject to the constraint ∫ ρ(r)dr = N, leads to an Euler−Lagrange equation which may be cast as the following one-particle Schrödinger-type equation:1 ÄÅ ÉÑ ÅÅ 1 2 Ñ ÅÅ− ∇ + v(r) + vH(r) + vF(r)ÑÑÑθ(r) = μθ(r) ÅÅÇ 2 ÑÑÖ (4)
vF(r) = μ − v(r) − vH(r) +
δE H[ρ] = δρ(r)
∫ |rρ−(r′r)′| dr′
vF(r) = μ − v(r) − vH(r) +
δG[ρ] δρ(r) © 2018 American Chemical Society
1 ∇2 ρ(r) 1 |∇ρ(r)|2 − 4 ρ(r) 8 ρ2 (r) (8)
(5)
This formula is undoubtedly correct in a complete (infinite) basis set, and it can be relied upon to construct Fermi (and Pauli) potentials from accurate densities on numerical grids.13−15 Unfortunately, it proves useless in practical calculations using finite
(6)
Received: May 21, 2018 Published: July 2, 2018
is the Hartree (Coulomb) potential, and vF(r) =
(7)
or equivalently
where θ(r) = ρ1/2(r), μ is the chemical potential, vH(r) =
1 ∇2 θ(r) 2 θ(r)
4246
DOI: 10.1021/acs.jctc.8b00490 J. Chem. Theory Comput. 2018, 14, 4246−4253
Article
Journal of Chemical Theory and Computation Gaussian basis sets because the term involving ∇2ρ(r) oscillates wildly when ρ(r) is expanded in Gaussian-type functions.16−18 On a more fundamental level, this has to do with the fact that if the domain of an operator  is infinite-dimensional, then the matrix representation of  −1 in a finite basis set is not the inverse of the matrix of  in the same finite basis. In any case, the severely limited utility of eq 8 is well-known, yet no definitive general-purpose method for constructing vF(r) in finite basis sets has been available. In this work, we suggest such a method for cases where the interacting wave function of the system or at least the two-electron reduced density matrix (2-RDM) is available.
G(x1, x1′) = h(̂ r1)γ(x1; x1′) + 2
(15)
where h(̂ r) = + v(r), γ(x1; x1′) is the interacting 1-RDM, and Γ(x1,x2;x1′,x2′) is the interacting 2-RDM. Subtracting eq 11 from eq 9, we obtain hole vF(r) = vXC (r) − ϵ̅ WF(r) +
(16)
ρ WF (r) =
τ WF(r) =
1 2
∑ nj|∇χj (r)|2 (18)
j
where χj(r) are the eigenfunctions of ρ (r;r′) and nj (0 ≤ nj ≤ 2) are the associated eigenvalues. Using eqs 17 and 18, applying the Lagrange identity28 to the product ρWFτWF, and dividing the result by ρWF, we obtain WF
(9)
1 [∇r ·∇r′ρ(r; r′)]r ′= r 2
τPWF(r) =
1 2ρ
WF
(r)
∑ ninj |χi (r)∇χj (r) − χj (r)∇χi (r)|2 i
