Quantal Density Functional Theory: A Local Effective Potential Theory Complement to Schrödinger Theory

Abstract

Quantal density functional theory (Q-DFT) is a local effective potential description of the Schrödinger theory of electrons. In such a description, the Hamiltonian is written such that the electron correlations due to the Pauli principle and Coulomb repulsion are incorporated into a local (multiplicative) potential energy term. This potential also includes the contributions of these correlations to the kinetic energy—the Correlation-Kinetic component. The energy is obtained from the properties of the model system. Stationary-state Q-DFT is a description of the map** from a nondegenerate or degenerate ground or excited state of a system of electrons in a static electromagnetic field to one of noninteracting fermions experiencing the same external field and possessing the same electronic density $\rho (\mathbf {r})$ and physical current density $\mathbf {j}(\mathbf {r})$. The state of the model system is arbitrary. The map** is in terms of ‘classical’ fields: an Electron-Interaction field representative of correlations due to the Pauli principle and Coulomb repulsion (which may be decomposed into its Hartree, Pauli and Coulomb components), and one descriptive of Correlation-Kinetic effects. The sources of these fields are quantum-mechanical expectation values of Hermitian operators. The local potential in which all these many-body effects are ensconced has a rigorous physical interpretation: It is the work done in the force of a conservative effective field which is the sum of the Electron-Interaction and Correlation-Kinetic fields. Within Q-DFT, the separate contributions to the energy due to the Pauli principle and Coulomb repulsion, as well as the Correlation-Kinetic energy are expressed in integral virial form in terms of the field representative of each property. The highest occupied eigenvalue of the differential equation of the model system of fermions, irrespective of its state, is the negative of the Ionization Potential. As such, Q-DFT constitutes a complement to Schrödinger theory. Properties of the model fermionic system arrived at via Q-DFT are discussed: (a) The non-uniqueness of the local electron-interaction potential in which the many-body correlations are embedded; (b) The proof that this non-uniqueness is solely due to Correlation-Kinetic effects; (c) The non-uniqueness of the model system wave function; (d) That it is solely the Correlation-Kinetic effects which contribute to the discontinuity in the local electron-interaction potential as the electron number passes an integer value. To elucidate Q-DFT, the map** from two different 2-dimensional 2-electron semiconductor quantum dots one in a ground and the other in its first excited singlet $2^{1} S$ state, to one of noninteracting fermions in a ground state possessing the same corresponding $\{ \rho (\mathbf {r}), \mathbf {j}(\mathbf {r}) \}$ is described. A brief description of the Q-DFT of the density amplitude is provided. Time-dependent (TD) Q-DFT of a system of electrons in a TD electromagnetic field, mapped to one of noninteracting fermions experiencing the same external field and possessing the same TD density $\rho (\mathbf {y})$ and physical current density $\mathbf {j}(\mathbf {y})$; $\mathbf {y} = \mathbf {r} t$, is described. In a manner similar to the stationary-state case, the local electron-interaction potential is the work done, at each instant of time, in the force of a conservative effective field which is the sum of the TD Electron-Interaction and Correlation-Kinetic fields.