Density functional theory on phase space

Abstract

Forty-five years after the point de départ [Hohenberg and Kohn, Phys Rev, 1964, 136, B864] of density functional theory, its applications in chemistry and the study of electronic structures keep steadily growing. However, the precise form of the energy functional in terms of the electron density still eludes us-and possibly will do so forever [Schuch and Verstraete, Nat Phys, 2009, 5, 732]. In what follows we examine a formulation in the same spirit with phase space variables. The validity of Hohenberg-Kohn-Levy-type theorems on phase space is recalled. We study the representability problem for reduced Wigner functions, and proceed to analyze properties of the new functional. Along the way, new results on states in the phase space formalism of quantum mechanics are established. Natural Wigner orbital theory is developed in depth, with the final aim of constructing accurate correlation-exchange functionals on phase space. A new proof of the overbinding property of the Müller functional is given. This exact theory supplies its home at long last to that illustrious ancestor, the Thomas-Fermi model. 2 Even before the results in Ref.[4] the problem has been regarded as well nigh intractable. Nevertheless, there are the formal solution by Garrod and Percus [5], still an appropriate reference for N-representability; Ayers' reformulation of the latter [6]; and other recent remarkable progress both on it and on pending questions of the representability problem for 1-matrices [7][8][9][10]. We return to the Ayers method in the next section. 4 We still find most illuminating the treatment of the relations between phase space representations given by Cahill and Glauber [29] long ago.Also the form factors are easily expressed in terms of d 1 . As dequantization and reduction commute, we may also use formulas analogous to (3) for reduced density matrices. At each order, the reduced Wigner distributions contain the same information as the corresponding reduced density matrices. It is convenient to have "inversion formulas" to recover the latter matrices from the Wigner functions. It

Series expansions for an exact two-electron wave function in terms of Löwdin's renormalized natural orbitals

“…The application of the spectral form in Eq. (3) results [12] in the exact expectation values for the above energy terms. Now, we truncate (tr) the series expansion as (5) where S N ≡ ( N n=0 P n ) 1/2 in order to have a unit normalization.…”

“…In this equation the coupling-dependent P n numbers [the eigenvalues of the exact one-particle density matrix, derived from the exact ψ ex (x 1 ,x 2 )] and the φ n (x) functions (the eigenfunctions of the same matrix) are given [9,11,12] by the following expressions:…”

Exact time evolution of the pair distribution function for an entangled two-electron initial state

“…According to the above motivation, the present study rests on the exact ground-state solution of a frequently employed [13][14][15][16][17][18][19][20] two-electron model Hamiltonian…”

Lowest excited configuration of harmonium

“…Now, for the analysis of harmonium the phase space representation of quantum mechanics recommends itself. The deep reason for this is the metaplectic invariance of that formalism [22], hidden in the standard approach: This made it easy to solve the sign dilemma in the exact Löwdin-Shull-Kutzelnigg formula [7,8] for γ 2 in terms of γ 1 for two-electron systems [23,24]. We come to this at the end of the next section.…”

“…We come to this at the end of the next section. Such a phase-space description was taken up first by Dahl [25] and then developed, within the context of a phase-space density functional theory (WDFT), by Blanchard, Ebrahimi-Fard, and ourselves [23,24,[26][27][28].…”