An alternative derivation of orbital-free density functional theory

Abstract

Polymer self-consistent field theory techniques are used to derive quantum density functional theory without the use of the theorems of density functional theory. Instead, a free energy is obtained from a partition function that is constructed directly from a Hamiltonian so that the results are, in principle, valid at finite temperatures. The main governing equations are found to be a set of modified diffusion equations, and the set of self-consistent equations are essentially identical to those of a ring polymer system. The equations are shown to be equivalent to Kohn-Sham density functional theory and to reduce to classical density functional theory, each under appropriate conditions. The obtained noninteracting kinetic energy functional is, in principle, exact but suffers from the usual orbital-free approximation of the Pauli exclusion principle in addition to the exchange-correlation approximation. The equations are solved using the spectral method of polymer self-consistent field theory, which allows the set of modified diffusion equations to be evaluated for the same computational cost as solving a single diffusion equation. A simple exchange-correlation functional is chosen, together with a shell-structure-based Pauli potential, in order to compare the ensemble average electron densities of several isolated atom systems to known literature results. The agreement is excellent, justifying the alternative formalism and numerical method. Some speculation is provided on considering the timelike parameter in the diffusion equations, which is related to temperature, as having dimensional significance, and thus picturing pointlike quantum particles instead as nonlocal, polymerlike, threads in a higher dimensional thermal-space. A consideration of the double-slit experiment from this point of view is speculated to provide results equivalent to the Copenhagen interpretation. Thus, the present formalism may be considered as a type of "pilot-wave," realist, perspective on density functional theory.

Atomic shell structure from an orbital-free-related density-functional-theory Pauli potential

“…It was found in Ref. [17] that the SCFT equations resulting from the quantum statistical mechanics derivation are identical in form to the SCFT equations of a system of ring polymers derived from classical statistical mechanics [25]. Ring polymers, which are macromolecules that formed closed rings without free ends, are mathematically parametrized in SCFT along their backbone by a contour variable embedded in three-dimensional (3D) space, whereas the contour parameter for quantum particles corresponds to an independent variable, specifically, an inverse thermal energy, β = 1/k B T , where k B is Boltzmann's constant and T is temperature, as shown in Fig.…”

“…Recently, an alternative derivation of OF-DFT was presented that did not use the theorems of DFT, but instead used quantum statistical mechanics to derive a temperature dependent free energy functional of the quantum particle density [17]. The derivation used techniques from polymer self-consistent field theory (SCFT) [18][19][20][21][22] and reduced to KS-DFT, assuming a strict enforcement of the Pauli exclusion principle.…”

“…In Ref. [17], correlations were ignored, and a simple shellstructure-based (SSB) Pauli potential together with a local density approximation exchange term (LDAX), as presented by Finzel [23], was adopted. The results showed correct qualitative shell-structure, essentially identical to that found by Finzel [23], but the energies of the structures were far from literature values [24].…”

“…As mentioned, Ref. [17] showed that SCFT reduces to DFT assuming a strict enforcement of the Pauli exclusion principle. This demonstrates that DFT fills the role in quantum systems that SCFT holds in polymer systems.…”

“…As mentioned, the derivation of DFT in Ref. [17] did not use the theorems of DFT. The DFT theorems show that the properties of a many-body wave function can be described by a single-particle electron density [7], but this one-to-one mapping can be reversed.…”