Local effective potential theory of electronic structure is the mapping from a system of electrons in an external field to one of noninteracting fermions or bosons with the same electronic density. The energy and ionization potential are also thereby determined. The mappings may be achieved via either quantal density-functional theory (QDFT) or Hohenberg-Kohn-Sham density-functional theory (HKS-DFT). The wave function for the model fermionic system is a Slater determinant of spin orbitals, whereas that for the model bosons is the density amplitude. In the QDFT mappings, the contributions of the electron correlations due to the Pauli exclusion principle, Coulomb repulsion and correlation-kinetic effects are separately delineated. It has been proved via QDFT that the contribution of Pauli and Coulomb correlations to these model systems is the same; the difference lies solely in their correlation-kinetic component. In this paper, we apply the QDFT of the density amplitude to study the mapping to the bosonic model. The application is to atoms and performed at the Hartree-Fock theory level of electron correlations. A principal conclusion is that correlation-kinetic effects play a significant role in the mapping to the bosonic model, whereas they are negligible in the mapping to the model fermions. For the bosonic model, this contribution increases with electron number, becoming nearly as significant as those due to the corresponding electron-interaction (the sum of the Hartree and Pauli) term. The significance of the correlation-kinetic effects will be further enhanced on the inclusion of Coulomb correlations and the corresponding correlation-kinetic contributions. The consequences of these conclusions for the HKS-DFT of the density amplitude are discussed, as are directions for future work.
Time-dependent quantal density-functional theory (Q-DFT) is a description of the s-system of noninteracting fermions with electronic density equivalent to that of Schr\"odinger theory, in terms of fields whose sources are quantum-mechanical expectations of Hermitian operators. The theory delineates and defines the contribution of each type of electron correlation to the local electron-interaction potential ${\ensuremath{\nu}}_{\mathrm{ee}}(\mathbf{r},t)$ of the s system. These correlations are due to the Pauli exclusion principle, Coulomb repulsion, correlation-kinetic, and correlation-current-density effects, the latter two resulting, respectively, from the difference in kinetic energy and current density between the interacting Schr\"odinger and noninteracting systems. We employ Q-DFT to prove the following sum rules and properties of the s system: (i) the components of the potential due to these correlations separately exert no net force on the system; (ii) the torque of the potential is finite and due solely to correlation-current-density effects; (iii) two sum rules involving the curl of the dynamic electron-interaction kernel defined as the functional derivative of ${\ensuremath{\nu}}_{\mathrm{ee}}(\mathbf{r},t)$ are derived and shown to depend on the frequency dependent correlation-current-density effect. Furthermore, via adiabatic coupling constant (\ensuremath{\lambda}) perturbation theory, we prove: (iv) the exchange potential ${\ensuremath{\nu}}_{x}(\mathbf{r},t)$ is the work done in a conservative field representative of Pauli correlations and lowest-order $O(\ensuremath{\lambda})$ correlation-kinetic and correlation-current-density effects; (v) the correlation potential ${\ensuremath{\nu}}_{c}(\mathbf{r},t)$ commences in $O({\ensuremath{\lambda}}^{2}),$ and, at each order, it is the work done in a conservative field representative of Coulomb correlations and correlation-kinetic and correlation-current-density effects; (vi) we derive the integral virial theorem relating ${\ensuremath{\nu}}_{\mathrm{ee}}(\mathbf{r},t)$ to the electron-interaction and correlation-kinetic energy for arbitrary coupling constant strength \ensuremath{\lambda}, and show there are no explicit correlation-current-density contributions to the energy. From this integral virial theorem we (vii) obtain the fully interacting $(\ensuremath{\lambda}=1)$ and exchange-only $(\ensuremath{\lambda}=0)$ integral virial theorems as special cases, the latter showing there is no explicit correlation-kinetic contribution to the exchange energy; and (viii) write expressions for the electron-interaction and correlation-kinetic actions for arbitrary coupling constant \ensuremath{\lambda} in terms of the corresponding fields.
Local effective potential theory, both stationary-state and time-dependent, constitutes the mapping from a system of electrons in an external field to one of the noninteracting fermions possessing the same basic variable such as the density, thereby enabling the determination of the energy and other properties of the electronic system. This paper is a description via Quantal Density Functional Theory (QDFT) of the electron correlations that must be accounted for in such a mapping. It is proved through QDFT that independent of the form of external field, (a) it is possible to map to a model system possessing all the basic variables; and that (b) with the requirement that the model fermions are subject to the same external fields, the only correlations that must be considered are those due to the Pauli exclusion principle, Coulomb repulsion, and Correlation–Kinetic effects. The cases of both a static and time-dependent electromagnetic field, for which the basic variables are the density and physical current density, are considered. The examples of solely an external electrostatic or time-dependent electric field constitute special cases. An efficacious unification in terms of electron correlations, independent of the type of external field, is thereby achieved. The mapping is explicated for the example of a quantum dot in a magnetostatic field, and for a quantum dot in a magnetostatic and time-dependent electric field.
Abstract The Schrödinger–Pauli (SP) theory of electrons in an electromagnetic field explicitly accounts for the electron spin moment. The many‐electron theory is complemented via a new descriptive perspective namely, that of the individual electron via its equation of motion or “Quantal Newtonian” first law. The law is in terms of ‘classical’ fields whose sources are quantum mechanical expectation values of Hermitian operators taken with respect to the system wave function. The law states that each electron experiences an external and an internal field, the sum of which vanish. The external field is the sum of the binding electrostatic and a Lorentz field. The internal field is a sum of fields representative respectively of electron correlations due to the Pauli principle and Coulomb repulsion; the electron density; kinetic effects; and an internal magnetic field. The energy can be expressed in integral virial form in terms of these fields. The law is elucidated by application to the 23S state of a quantum dot in a magnetic field. It is proved that the Hamiltonian is an exactly known and universal functional of the wave function. This generalizes the SP equation, and reveals that its eigenfunctions and eigenvalues may be determined self‐consistently. A quantal density functional theory (QDFT) of the SP system is developed whereby additional properties are determined. A physical interpretation of Spin‐DFT based on the QDFT mapping is provided. Further generalizations of the present work to the temporal case, and relativistic Dirac theory, are proposed.
The Wigner regime of a system of electrons in an external field is characterized by a low electron density and a high electron-interaction energy relative to the kinetic energy. The low-correlation regime is in turn described by a high electron density and an electron-interaction energy smaller than the kinetic energy. The Wigner regime of a nonuniform-electron-density system is investigated via quantal density functional theory (QDFT). Within QDFT, the contributions of electron correlations due to the Pauli exclusion principle, Coulomb repulsion, and correlation-kinetic effects are separately delineated and explicitly defined. The nonuniform-electron-density system studied is that of the Hooke's atom in the Wigner regime, for which the exact wave function is derived. As such, the results of the QDFT analysis are exact. It is observed that in comparison to the low-correlation case, not only is the electron-interaction energy greater than the kinetic energy as a fraction of the total energy, but so are its individual Hartree, Pauli, and Coulomb components. The ionization potential as a fraction of the total energy too is greater. But most significantly, in the Wigner regime, the correlation-kinetic energy as a fraction of both the electron-interaction and the total energy is substantially greater than in the low-correlation case. Hence, we propose that the Wigner regime now also be characterized by a high correlation-kinetic energy. The kinetic energy as a fraction of the total energy, however, is less than in the low-correlation case. This fact and the high correlation-kinetic energy value in the Wigner regime is explained by the concept of “quantal compression” of the kinetic energy density derived from QDFT. The corresponding results for the low-correlation case are in turn a consequence of a “quantal decompression” of the kinetic energy density. From the QDFT analysis, the exact values for the Kohn-Sham theory “exchange-correlation” and “correlation” energy functionals of the density and their respective functional derivatives are also obtained. These results ought to be of value in the construction and testing of approximate energy functionals valid for the Wigner regime.
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.