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arXiv:cond-mat/0210129 · Physical Review Letters

Quantal Density Functional Theory of Degenerate States

Viraht Sahni, Xiao‐Yin Pan

Published Mar 27, 2003• 28 citations
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Abstract

The treatment of degenerate states within Kohn-Sham density functional theory is a problem of long-standing and current interest. We propose a solution to this mapping from the interacting degenerate system to that of the noninteracting fermion model whereby the equivalent density and energy are obtained via the unifying physical framework of quantal density functional theory. We describe the quantal theory of both ground and excited degenerate states, and for the cases of both pure state and ensemble v-representable densities. The quantal description further provides a rigorous physical interpretation of the corresponding Kohn-Sham energy functionals of the density, ensemble density, bidensity and ensemble bidensity, and of their respective functional derivatives. We conclude with examples of the mappings within the quantal theory.

DOI
10.1103/physrevlett.90.123001

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The treatment of degenerate states within Kohn-Sham density functional theory is a problem of long-standing and current interest. We propose a solution to this mapping from the interacting degenerate system to that of the noninteracting fermion model whereby the equivalent density and energy are obtained via the unifying physical framework of quantal density functional theory. We describe the quantal theory of both ground and excited degenerate states, and for the cases of both pure state and ensemble v-representable densities. The quantal description further provides a rigorous physical interpretation of the corresponding Kohn-Sham energy functionals of the density, ensemble density, bidensity and ensemble bidensity, and of their respective functional derivatives. We conclude with examples of the mappings within the quantal theory.
Abstract The quantal density functional theory (Q‐DFT) of excited states is the description of the physics of the mapping from any bound nondegenerate excited state of Schrödinger theory to that of the s‐system of noninteracting Fermions with equivalent density ρk(r), energy Ek, and ionization potential Ik. The s‐system may either be in an excited state with the same configuration as in Schrödinger theory or in a ground state with a consequently different configuration. The Q‐DFT description of the s‐system is in terms of a conservative field ℱk(r), whose electron‐interaction ℰee(r) and correlation‐kinetic 𝒵 (r) components are separately representative of electron correlations due to the Pauli exclusion principle and Coulomb repulsion, and correlation‐kinetic effects, respectively. The sources of these fields are expectations of Hermitian operators taken with respect to the system wavefunction. The local electron‐interaction potential vee(r) of the s‐system, representative of all the many‐body correlations, is the work done to move an electron in the force of the field ℱk(r). The electron interaction Eee and correlation‐kinetic Tc components of the total energy Ek may be expressed in integral virial form in terms of their respective fields. The difference between the s‐system in its ground or excited state representation is due entirely to correlation‐kinetic effects. The highest occupied eigenvalue of the s‐system differential equation in either case is minus the ionization potential Ik. In this work we demonstrate the transformation of an excited state of Schrödinger theory, as represented by the first excited singlet state of the exactly solvable Hooke's atom, to that of noninteracting Fermions in their ground state with equivalent excited state density, energy, and ionization potential. To further prove the Fermions are in a ground state, we solve the corresponding singlet s‐system differential equation numerically for the vee(r) determined, and obtain the excited state density from the zero node orbitals generated. The resulting total energy is also the same. In addition, the single eigenvalue determined corresponds to minus the ionization potential of the excited state. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001
A particle-density distribution is said to be v-representable if it is the ground-state density for some potential v(r). By developing the idea of proper order, proposed earlier [J. Chen and M. J. Stott, Phys. Rev. A 44, 2816 (1991)], to any number of energy levels which are involved in possible ground-state configurations, we show that any reasonable density with an arbitrary number of spinless fermions is either pure-state or ensemble v-representable. This result supports the general applicability of the Kohn-Sham method [Phys. Rev. 140, A1133 (1965)] in density-functional theory.
Based on the work of Görling and that of Levy and Nagy, a density-functional formalism for many fermionic excited states is explored through a careful and rigorous analysis of the excited-state density to external potential mapping. It is shown that knowledge of the ground-state density is a must to fix the mapping from an excited-state density to an external potential. This is the excited-state counterpart of the Hohenberg–Kohn theorem, where instead of the ground-state density the density of the excited state gives the true many-body wavefunctions of the system. Further, the excited-state Kohn–Sham system is defined by comparing its non-interacting kinetic energy with the true kinetic energy. The theory is demonstrated by studying a large number of atomic systems.
The State--Specific Kohn--Sham Density Functional Theory [arXiv:physics/0506037] is used to derive the Kohn-Sham exchange-correlation potential $\vxc$ and exchange-correlation energy $\Eco$ as explicit functionals of vsv_svs​ and ϕ1ϕ_1ϕ1​, where vsv_svs​ is the local, one-body potential from the Kohn--Sham equations, and ϕ1ϕ_1ϕ1​ is the spinless one-particle density matrix from the Kohn--Sham noninteracting state, say $|ϕ_1\ran$. In other words, $|ϕ_1\ran$ is the ground state eigenfunction of the noninteracting Schrödinger equation with the one-body potential vsv_svs​. For simplicity, we only consider noninteracting states that are closed-shell states and interacting states that are nondegenerate, singlet ground-states.
The formalism of Kohn and Sham uses a specific (model) hamiltonian which highly simplifies the many-electron problem to that of noninteracting fermions. The theorem of Hohenberg and Kohn tells us that, for a given ground state density, this hamiltonian is unique. In principle, this density can be chosen as that of the real, interacting system. To obtain the energy, or other properties of the real system, approximations are needed. Working with non interacting fermions is an important simplification, but it may be easier to produce approximations with different choices of the model hamiltonian. The feature that the exact density is (ideally) reproduced can be kept in the newly defined fictitious systems. Using model hamiltonians having the same form as the physical one, that is, being built of one- and two-body operators, allows to approach the physical hamiltonian arbitrarily close, and thus a systematic reduction of the approximations.
The theory of Kohn and Sham treats a system of interacting electrons by establishing a correspondence between this system and a noninteracting system. In this paper we develop a new formulation of this theory, which differs from the original by its rigor and by the more correct treatment of the kinetic energy term. This new formulation has the additional advantage that it bypasses the vvv representability requisite of the original Hohenberg-Kohn-Sham theory.