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International Journal of Quantum Chemistry

Overcomplete density functional description

Jerome K. Percus

Published Jan 1, 1998• 5 citations
Publisher

Abstract

Although formally exact expressions can be written for thermodynamic potentials of many-particle systems in external fields, Legendre transformation to particle density as the controlling field has many virtues, insensitivity in the face of singular states being notable. It is also true that nonuniform systems, classical or quantum, which lead to simple closed-form solutions, tend to do so principally in the density functional form. However, as gleaned from the heavily investigated cases of classical lattice gas networks, this may be achieved only if additional auxiliary densities are appended, with respect to which the associated thermodynamic potential is stationary. Here, we successfully apply the overcompleteness strategy to the prototypical case of free fermion ground states in one-dimensional space, extrapolate to three-dimensional space, and apply a basic one-functional variational ansatz. The qualitative properties of the resulting free energy and potential-density profile relations are in accord with those that any valid formulation must satisfy. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 573–580, 1998

DOI
10.1002/(sici)1097-461x(1998)69:4<573::aid-qua15>3.0.co;2-0

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Although formally exact expressions can be written for thermodynamic potentials of many-particle systems in external fields, Legendre transformation to particle density as the controlling field has many virtues, insensitivity in the face of singular states being notable. It is also true that nonuniform systems, classical or quantum, which lead to simple closed-form solutions, tend to do so principally in the density functional form. However, as gleaned from the heavily investigated cases of classical lattice gas networks, this may be achieved only if additional auxiliary densities are appended, with respect to which the associated thermodynamic potential is stationary. Here, we successfully apply the overcompleteness strategy to the prototypical case of free fermion ground states in one-dimensional space, extrapolate to three-dimensional space, and apply a basic one-functional variational ansatz. The qualitative properties of the resulting free energy and potential-density profile relations are in accord with those that any valid formulation must satisfy. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 573–580, 1998
The kinetic energy, or thermodynamic potential, of a non-interacting Fermion system, as a functional of the non-uniform density pattern evoked by an external potential, serves as a crucial tool in the analysis of interacting system ground states. Examples from classical lattice gas and continuum fluid theory are cited to show that analytic tractability can be dramatically enhanced by the introduction of auxiliary density fields with respect to which a suitable thermodynamic potential is stationary. This strategy is developed for the prototype of one-dimensional spinless Fermions, resulting in an exact highly overcomplete representation. A semi-classical variational ansatz recovers the familiar Thomas-Fermi plus renormalized Weiszacker expression, as well as leading corrections, and a simple extension creates a simple modification of the former. The corresponding 3-dimensional format is set up, and evaluated by extrapolation from the 1-dimensional case.
The basic concern of density functional theory is the groundstate energy of an inhomogeneous interacting many-particle system as a function of the external potential acting on its density (and possibly as a function of the particle number, allowing for the consideration of ionization energies, affinities, and excitation gaps). This groundstate energy function is replaced by its Legendre transform as a function of the particle density, and the Legendre back transformation leads via insertion of an interaction-free reference system to an effective one-particle equation whose solution finds the ground-state energy and density to any given external potential. As was seen in Part I, this theory provides rather a reformulation than a solution of the many-body problem for the groundstate, because it does not explicitly provide the Legendre transform H [ n ] to E [ v, N ] (cf. (6.35, 6.31, and 6.36). The original many-body problem is now contained in H [ n ] which only can be guessed, although not without success. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
A density-functional theory (DFT) of lattice fermion models is presented, which uses the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ as basic variable. A simple, explicit approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ of the Hubbard model is derived from exact dimer results, scaling properties of $W[\ensuremath{\gamma}]$ and known limits. Systematic tests on the one-dimensional chain show a remarkable agreement with the Bethe-ansatz exact solution for all interaction regimes and band fillings. New results are obtained for the ground-state energy and charge-excitation gap in two dimensions. A successful description of strong electron correlations within DFT is achieved.
The relative performance of trial wave functions expressed as linear combination of correlated exponentials has been tested on a variety of systems. The results are compared against other correlated functions commonly used in the literature to assess the capabilities of the proposed ansatz. A possible departure from the simple exponential functional form used in previous works is discussed, along with its advantages and drawbacks. We also discuss how to implement an efficient optimization procedure for this correlated basis set. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 23–33, 1999
In this article we derive for an arbitrary, real, local (multiplicative), time-dependent (TD) external potential, the differential form of the virial theorem for the pure state in TD Schrödinger theory. We contrast this pure-state theorem with the many-body theory equation of motion for both equilibrium and nonequilibrium phenomena. We also derive the corresponding pure-state theorem for a model system of noninteracting fermions with equivalent TD density. These theorems are valid for both adiabatic and sudden switching on of the external potential. The theorems furthermore lead to a line-integral expression for the local effective potential of the noninteracting system that may be provided the physical interpretation, at each instant of time, as being the work done to move an electron in the force of a conservative field. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 341–347, 2000