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Physical Review B

Computational pair density functional theory: A proposal for the kinetic energy functional

Katsuhiko Higuchi, Masahiko Higuchi

Published Oct 22, 2010• 26 citations
Publisher

Abstract

We present a computational pair density (PD) functional theory using multiple Slater determinants in constructing the PD. Compared to the effective initial scheme [M. Higuchi and K. Higuchi, Phys. Rev. B 78, 125101 (2008)], the search region for the ground-state PD is substantially extended within the set of $N$-representable PDs. The merit of the present scheme is to guarantee the resultant PD to be $N$-representable and to describe correlation effects beyond the effective initial scheme, which enables us to tackle with the development of the approximate form of the kinetic energy functional. We derive two restrictive conditions on the kinetic energy functional. With the aid of these restrictive conditions, we also propose several types of approximate functionals of the kinetic energy. To check the abilities and limitations of the present scheme including such approximate functionals, actual calculations are performed for the neutral neon atom. The results show that the present scheme provides the $N$-representable and correlated PDs, though it would have some limitations. What we need to do first for improvement of the computational PD functional theory is that the development of the approximate functional of the kinetic energy is further advanced along the present scheme.

DOI
10.1103/physrevb.82.155135

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We present a computational pair density (PD) functional theory using multiple Slater determinants in constructing the PD. Compared to the effective initial scheme [M. Higuchi and K. Higuchi, Phys. Rev. B 78, 125101 (2008)], the search region for the ground-state PD is substantially extended within the set of NNN-representable PDs. The merit of the present scheme is to guarantee the resultant PD to be NNN-representable and to describe correlation effects beyond the effective initial scheme, which enables us to tackle with the development of the approximate form of the kinetic energy functional. We derive two restrictive conditions on the kinetic energy functional. With the aid of these restrictive conditions, we also propose several types of approximate functionals of the kinetic energy. To check the abilities and limitations of the present scheme including such approximate functionals, actual calculations are performed for the neutral neon atom. The results show that the present scheme provides the NNN-representable and correlated PDs, though it would have some limitations. What we need to do first for improvement of the computational PD functional theory is that the development of the approximate functional of the kinetic energy is further advanced along the present scheme.
We propose a practical scheme for calculating the ground-state pair density (PD) by utilizing the correlated wave function. As the correlated wave function, we adopt a linear combination of the single Slater determinants that are constructed from the solutions of the initial scheme[Higuchi M and Higuchi K 2007 Physica B 387 , 117]. The single-particle equation is derived by performing the variational principle within the set of PDs that are constructed from such correlated wave functions. Since the search region of the PD is substantially extended as compared with the initial scheme, it is expected that the present scheme can cover more correlation effects. The single-particle equation is practical, and may be easily applied to actual calculations.
We reconfirm the performance of the initial scheme for calculating the ground-state pair density (Higuchi and Higuchi 2007 Physica B 387 117, 2008 Phys. Rev. B 78 125101) by using the alternative approximation of the correlating kinetic energy functional. It is shown that about 20% of the correlation energy can be reproduced by the initial scheme, irrespective of the approximate form of the correlating kinetic energy functional. On the basis of the initial scheme, various kinds of schemes that go beyond the initial one can be developed. We illustrate two kinds of computational schemes.
To check the validity of the recently proposed kinetic energy (KE) functional of the pair density (PD) functional theory, we perform the numerical calculation for the neutral Ne atom. This is the first computational trial for the PD functional theory which includes the KE functional that is approximately derived on the basis of the rigorous expression with the coupling-constant integration. In this work we adopt the correlated Thomas-Fermi functional as the approximate KE functional. Although it is shown that non-negligible errors exist in the resulting PD, the present results will be useful as reference data in cases of developing further the approximate KE functional.
We present a density-functional scheme for calculating the pair density (PD) by means of the correlated wave function. The Jastrow wave function is adopted as the correlated wave function. By using the lowest-order approximation to the Jastrow wave function PDs, the search region for the ground-state PD is substantially extended as compared with our previous theory [Physica B 387, 117 (2007)]. The variational principle results in simultaneous equations that are practicable to calculate the ground-state PD.
We derive the coupling-constant expression for the kinetic energy (KE) functional of the pair-density-functional theory. Such the expression can be effectively used in developing the approximate KE functional. Indeed, on the basis of this expression, we present an approximate form of the KE functional by means of the perturbation theory. Furthermore, as the first step of an alternative way of obtaining the approximate KE functional, we derive several relations and bounds for the KE functional by investigating how the KE functional changes with the nonuniform scaling, rotation, and translation of electron coordinates. Such relations and bounds can be utilized as restrictive conditions on the approximate form. Thus, the coupling-constant expression, relations and bounds obtained here provide useful tools for devising various types of the approximate KE functional that is indispensable for advancing the pair-density-functional theory.
In pair density-functional theory, the only unknown piece of the energy is the kinetic energy TTT as a functional of the pair density P(x1,x2)P({x}_{1},{x}_{2})P(x1​,x2​). Although T[P]T\phantom{\rule{0.1em}{0ex}}[P]T[P] has a simpler structure than the Hohenberg-Kohn functional of conventional density-functional theory, computational requirements are still moderate. In the present work, a particularly convenient model system to represent many-electron pair densities is introduced. This ``boson pair model'' (BPM) approximately treats electron pairs as noninteracting bosons. The resulting explicit model for the kinetic energy T2[P]{T}_{2}[P]T2​[P] is shown to be exact for two-electron systems and a lower bound to T[P]T\phantom{\rule{0.1em}{0ex}}[P]T[P] for more than two electrons. The one- and two-particle density matrices obtained from the BPM yield upper bounds for the corresponding many-electron quantities. This suggests a partitioning T[P]=T2[P]+Teff[P]T\phantom{\rule{0.1em}{0ex}}[P]={T}_{2}[P]+{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]T[P]=T2​[P]+Teff​[P], where only the remainder ${T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]\ensuremath{\geqslant}0$ needs to be approximated. If the BPM is constrained to yield the exact ground-state pair density, a two-electron Schr\"odinger equation with an effective local two-particle potential results; the latter is identified as a sum of the bare Coulomb interaction and the functional derivative of Teff[P]{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]Teff​[P]. This self-consistent scheme to minimize the energy with respect to PPP is more efficient than previous procedures. Further information on the functional derivative of Teff[P]{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]Teff​[P] is derived from a contracted Schr\"odinger equation. Since Teff[P]{T}_{\mathrm{eff}}\phantom{\rule{0.1em}{0ex}}[P]Teff​[P] is explicitly known in the two-electron and noninteracting (Hartree-Fock) limits, the present method provides an alternative to density-matrix functional theories, which can be exact in the same limits and are similar in computational cost.