Coupling-constant expression and exact relations for the kinetic-energy functional in pair-density-functional theory

Abstract

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.

Approximate forms of the pair-density-functional kinetic energy on the basis of a rigorous expression with coupling-constant integration

“…One is how the variational principle with respect to the PD is performed with keeping the search region of PDs within the set of V-representable PDs [1][2][3][4][29][30][31][32][33][34][35][36][37][38][39][40]. Another problem is that the approximate form of the KE functional has to be developed in a form of the PD functional, as mentioned above [41][42][43][44][45]. These two problems should be solved simultaneously.…”

Scaling method for the pair-density-functional theory in combination with energy functionals satisfying the virial theorem: Checking the validity via atomic-structure calculations

“…(II) Another problem in developing the PD functional theory is that the approximate form of the kinetic energy (KE) is needed to be developed as a functional of the PD [35][36][37][38][39]. This is because the KE cannot be rigorously written by only the PD [35][36][37][38][39].…”

“…(b) The other is to develop the approximate form by imposing on it relations and bounds that are fulfilled by the exact KE functional as restrictive conditions [33,34,39]. Along each strategy, we have begun to propose the approximate forms of the KE functional [33,34,39].…”

“…This is because the KE cannot be rigorously written by only the PD [35][36][37][38][39]. Concerning this problem, we have taken two strategies for developing the approximate functional.…”

“…Concerning this problem, we have taken two strategies for developing the approximate functional. (a) One is to develop the approximate form on the basis of the coupling-constant expression for the KE functional [39]. (b) The other is to develop the approximate form by imposing on it relations and bounds that are fulfilled by the exact KE functional as restrictive conditions [33,34,39].…”