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arXiv:0801.0615

$N$-representability of the Jastrow wave function pair density of the lowest-order

Katsuhiko Higuchi, Masahiko Higuchi

Published Jan 4, 2008• 0 citations
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Abstract

We have recently proposed a density functional scheme for calculating the ground-state pair density (PD) within the Jastrow wave function PDs of the lowest-order (LO-Jastrow PDs) [M. Higuchi and K. Higuchi, Phys. Rev. A \textbf{75}, 042510 (2007)]. However, there remained an arguable problem on the $N$-representability of the LO-Jastrow PD. In this paper, the sufficient conditions for the $N$-representability of the LO-Jastrow PD are derived. These conditions are used as the constraints on the correlation function of the Jastrow wave function. A concrete procedure to search the suitable correlation function is also presented.

DOI
10.48550/arxiv.0801.0615

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NNN-representability of the Jastrow wave function pair density of the lowest-order
We have recently proposed a density functional scheme for calculating the ground-state pair density (PD) within the Jastrow wave function PDs of the lowest-order (LO-Jastrow PDs) [M. Higuchi and K. Higuchi, Phys. Rev. A \textbf{75}, 042510 (2007)]. However, there remained an arguable problem on the NNN-representability of the LO-Jastrow PD. In this paper, the sufficient conditions for the NNN-representability of the LO-Jastrow PD are derived. These conditions are used as the constraints on the correlation function of the Jastrow wave function. A concrete procedure to search the suitable correlation function is also presented.
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 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.
The ground-state wave function of an infinite system of fermions is approximated by the state-independent Jastrow ansatz. In order to optimize the pair correlation function, the Euler-Lagrange equations of the variational problem for the energy expectation value are derived. The short- and long-range behaviors of the optimum pair correlation function are discussed. Application of graphical techniques and use of rigorous results on the connection between the slope of the static structure function as $k\ensuremath{\rightarrow}0$+ and long-range Jastrow correlations allow one to prove that the optimum pair correlation function behaves like $1+O({r}^{\ensuremath{-}2})$ as $r\ensuremath{\rightarrow}\ensuremath{\infty}$. A connection is derived between the weight of the long-range correlations and the Landau parameters. As simple examples the limit of Bose statistics and the electron-gas problem are considered. The consequences of these investigations on numerical calculations and their relation to alternative expansion methods are investigated.