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Density-functional theory formulated in terms of functional integrals

Gérald Faussurier

Published Jan 1, 2025• 0 citations
Publisher

Abstract

In a previous study, the author formulated the density functional theory in terms of functional integrals. It was valid at zero and finite temperature. It was possible to derive the Hohenberg and Kohn formulation at zero temperature and the Mermin formulation at finite temperature of the density functional theory, which states that the energy or the grand potential are functionals of the true density of the system considered. In particular, the Kohn and Sham equations are proven to appear naturally by performing a saddle-point evaluation of a specific functional integral. This result is valid at zero or finite temperature. Unfortunately, the expression of the grand potential given in our previous work differs from the usual expression found in the literature. In this short paper, we derive the common expression of the grand potential in the framework of the density functional theory by starting from the expression given in this previous work. This completes the formulation of the density functional theory using functional integrals. This work could be of interest to people working in the field of quantum Monte Carlo methods at finite temperature.

DOI
10.1063/5.0230680

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In a previous study, the author formulated the density functional theory in terms of functional integrals. It was valid at zero and finite temperature. It was possible to derive the Hohenberg and Kohn formulation at zero temperature and the Mermin formulation at finite temperature of the density functional theory, which states that the energy or the grand potential are functionals of the true density of the system considered. In particular, the Kohn and Sham equations are proven to appear naturally by performing a saddle-point evaluation of a specific functional integral. This result is valid at zero or finite temperature. Unfortunately, the expression of the grand potential given in our previous work differs from the usual expression found in the literature. In this short paper, we derive the common expression of the grand potential in the framework of the density functional theory by starting from the expression given in this previous work. This completes the formulation of the density functional theory using functional integrals. This work could be of interest to people working in the field of quantum Monte Carlo methods at finite temperature.
A new approach for the calculation of ground states of many-electron systems is proposed via an integral formulation of the Hohenberg-Kohn-Sham density-functional theory. Only equations for the total electron density are involved; orbitals are not employed. Exchange and correlation effects are incorporated. In place of the set of single-electron equations, the total electron density is explicitly expressed in terms of the Kohn-Sham effective local potential through multidimensional integrations. The development is based on the first-order density matrix as obtained from the one-body Green's function in polygonal and Fourier path-integral representations. This might open up the possibility of ab initio calculations for molecules with very many electrons. It also provides explicit solutions to two long-standing problems: electron kinetic energy and momentum density as functionals of the total electron density. The formulation can also be used in calculations for general fermions.
A rigorous derivation of the density functional in the Hohenberg-Kohn theory is presented. With no assumption regarding the magnitude of the electric coupling constant e2e^2e2 (or correlation), this work provides a firm basis for first-principles calculations. Using the auxiliary field method, in which e2e^2e2 need not be small, we show that the bosonic loop expansion of the exchange-correlation functional can be reorganized so as to be expressed entirely in terms of the Kohn-Sham single-particle orbitals and energies. The excitations of the many-particle system can be obtained within the same formalism. We also explicitly demonstrate at zero-temperature the single-particle limit, the weak-coupling limit of the energy functional, and its application to homogeneous electron gas.
Following a recent work [Gál, Phys. Rev. A64, 062503 (2001)], a simple derivation of the density-functional correction of the Hartree–Fock equations, the Hartree–Fock–Kohn–Sham equations, is presented, completing an integrated view of quantum mechanical theories, in which the Kohn–Sham equations, the Hartree–Fock–Kohn–Sham equations and the ground-state Schrödinger equation formally stem from a common ground: density-functional theory, through its Euler equation for the ground-state density. Along similar lines, the Kohn–Sham formulation of the Hartree–Fock approach is also considered. Further, it is pointed out that the exchange energy of density-functional theory built from the Kohn–Sham orbitals can be given by degree-two homogeneous N-particle density functionals (N = 1, 2, …), forming a sequence of degree-two homogeneous exchange-energy density functionals, the first element of which is minus the classical Coulomb-repulsion energy functional.
In this work, the zero-temperature limit of the thermodynamic spin-density functional theory is investigated. The coarse-grained approach to the equilibrium density operator is used to describe the equilibrium state. The characteristic functions of a macrostate are introduced and their zero-temperature limits are investigated. A detailed discussion of the spin-grand-canonical ensemble in the entropy and energy representations is performed. The maps between the state function variables at 0K limit are rigorously studied for both representations. In the spin-canonical ensemble, the energy surface and the discontinuity pattern are investigated. Finally, based on the maps between the state function variables at 0K limit, the Hohenberg-Kohn theorem for the systems with non-integer electron and spin numbers at zero-temperature limit is formulated.
The formalism developed in the first paper of the series [arXiv:0901.1060] is applied to two thermodynamic systems: (i) of three global observables (the energy, the total electron number and the spin number), (ii) of one global observable (the internal electron energy) and two local (position-dependent) observables (the total electron density and the spin density). The two-component potential of the many-electron system of interest is constructed of a scalar external potential and a collinear magnetic field (coupled only with the spin operator). Various equilibrium characteristics of two systems are defined and investigated. Conditions for the equivalence between two systems (the same equilibrium density matrix demanded) are derived and thoroughly discussed. The applicability of the Hohenberg-Kohn theorem is extended to the thermodynamic spin-density functional theory. Obtained results provide a rigorous mathematical foundation for future derivation of the zero-temperature limit of this theory and determination of its chemical reactivity indices.
The Hohenberg-Kohn theorem and the Kohn-Sham equations, which are at the basis of the density-functional theory, are reformulated in terms of a particular many-body density, which is translational and Galilean invariant and therefore is relevant for self-bound systems. In a similar way that there is a unique relation between the one-body density and the external potential that gives rise to it, we demonstrate that there is a unique relation between that particular many-body density and a definite many-body potential. The energy is then a functional of this density, and its minimization leads to the ground-state energy of the system. As a proof of principle, the analogous of the Kohn-Sham equation is solved in the specific case of 4He^{4}\mathrm{He}4He atomic clusters, to put in evidence the advantages of this formulation in terms of physical insights.