# Quantal Density Functional Theory of Excited States: Application to an Exactly Solvable Model

## Document Type

Peer-Reviewed Article

## Publication Date

8-2001

## 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 ρ*k*(*r*), energy *E**k*, and ionization potential *I**k*. 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 ℱ*k*(*r*), whose electron‐interaction ℰ*ee*(*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 *v**ee*(*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 ℱ*k*(*r*). The electron interaction *E**ee* and correlation‐kinetic *T**c* components of the total energy *E**k* 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 *I**k*. 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 *v**ee*(*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

## DOI

doi.org/10.1002/qua.1519

## Recommended Citation

Slamet, M., & Sahni, V. (2001). Quantal density functional theory of excited states: Application to an exactly solvable model. *International Journal of Quantum Chemistry*, *85*(4‐5), 436-448. Doi: 10.1002/qua.1519