Viraht Sahni

Quantal Density Functional Theory

Quantal density functional theory (Q–DFT) is a physical local effective potential theory of electronic structure of both ground and excited states. It constitutes the mapping from any state of an interacting system of N electrons in a time-dependent external field as described by Schrodinger theory to one of noninteracting fermions in the same external field and possessing the same quantum-mechanical properties of the basic variables. Time-independent Q–DFT constitutes a special case. The Q–DFT mapping can be to any arbitrary state of the model system. Q–DFT is based on the ‘Quantal Newtonian’ second and first laws of both the interacting and noninteracting systems. As such it is a description in terms of ‘classical’ fields derived from quantal sources as experienced by each model fermion. The internal field components are separately representative of electron correlations due to the Pauli Exclusion Principle, Coulomb repulsion, kinetic effects and the density. Thus, as opposed to Schrodinger theory, within Q–DFT, the separate contributions to the total energy and local potential due to the Pauli principle, Coulomb repulsion, and the correlation contribution to the kinetic energy—the Correlation-Kinetic effects—are explicitly defined in terms of fields representative of these correlations. The local potential incorporating all the many-body effects is the work done in the force of a conservative effective field which is the sum of these fields. The many-body components of the energy are expressed in integral virial form in terms of the individual fields representative of the different electron correlations. Various sum rules for the model system such as the Integral Virial Theorem, Ehrenfest’s Theorem, the Zero Force and the Torque Sum Rule are derived. Q–DFT is explicated by application to both a ground and excited state of a model system in the low electron-correlation regime, and to a ground state in the Wigner high-electron correlation regime. A new characterization of the Wigner regime based on the newly discovered significance of Correlation-Kinetic effects is proposed. The multiplicity of potentials as obtained via Q–DFT which can generate the same basic variables, and the significance of Correlation-Kinetic effects in such mappings, is discussed. The Q–DFT of degenerate states is described, as is the Q–DFT of Hartree and Hartree-Fock theories.

Atomic shell structure in Hartree—Fock theory

Abstract The radial density distribution function computed from the spherically averaged numerical Hartree—Fock density is found to reveal distinct topological features characteristic of the shell structure of atoms beyond Ar for which hitherto reported studies based on the analytical Hartree—Fock density do not show the formal shell structure. Based on the numerical evidence, the location of the midpoint corresponding to the last two successive points of inflection in the distribution function is proposed to define the core—valence separation for heavier atoms ( Z ⩾19).

Quantum-mechanical interpretation of density functional theory

This article describes the rigorous quantum-mechanical interpretation of Hohenberg-Kohn-Sham density-functional theory based on the original ideas of Harbola and Sahni, and of their extension by Holas and March. The local electron-interaction potential v ee KS (r) of density-functional theory is defined mathematically as the functional derivative v ee KS (r)=δE ee KS [ρ]/δρ(r), where E ee KS [ρ] is the electron-interaction energy functional of the density ρ(r). This functional and its derivative incorporate the effects of Pauli and Coulomb correlations as well as those of the correlation contribution to the kinetic energy. The potential v ee KS (r) also has the following physical interpretation. It is the work done to move an electron in a field ℱ(r), which is the sum of two fields. The first, ℰee(r), is representative of Pauli and Coulomb correlations, and is determined by Coulombs law from its source charge which is the pair-correlation density. The second field, \(Z_{t_c } (r)\), represents the correlation contribution to the kinetic energy, and is proportional to the difference of fields derived from the kinetic-energy-density tensor for the interacting and non-interacting systems. The field ℱ(r) is conservative, and thus the work done in this field is path-independent. The quantum-mechanical electron-interaction energy component Eee[ρ] of E ee KS [ρ] is the energy of interaction between the electronic and pair-correlation densities. The correlation-kinetic-energy component Tc[ρ] can also be written in terms of its source through the field \(Z_{t_c } (r)\). Some results for finite atomic (both ground and excited states) and extended metal surface systems derived via the interpretation are presented. Certain consequences of the physical interpretation such as the understanding of Slater theory, and the implications with regard to electron correlations within approximate Kohn-Sham theory are also discussed.

Hartree-Fock theory of the inhomogeneous electron gas at metallic surfaces

Abstract In this paper we derive fundamental properties of the interacting inhomogeneous electron gas at jellium metal surfaces within the Hartree-Fock approximation. We also discuss the question of what constitutes quantum-mechanically the image charge at a metal surface, and provide a physical interpretation for the quantum-mechanical origin of the image potential. The self-consistent solution of the Hartree-Fock equations is in general formidable due to the integral exchange operator. We show that from a more fundamental physical viewpoint, the difficulty for the metal surface physics problem manifests itself in the complex structure of the orbital-dependent exchange charge densities that give rise to the orbital-dependent potentials. However, it is possible to solve the problem of the Pauli-correlated electron gas within the exchange-only formalism of density-functional theory. By employing the variational principle for the energy with the exchange energy component treated in its non-local form, and the variationally accurate “displaced-profile-change-in-self-consistent-field” expression for the work function, we derive rigorous upper bounds to the surface energy and accurate work functions. In order to understand what constitutes quantum-mechanically the image charge at a metal surface we study the structure of the average exchange charge density or Fermi hole as an electron is removed from within a metal to infinity outside. The study shows that the Fermi hole is localized to the surface region and is part of the image charge only for electron positions close to the surface. As the electron is removed further into the vacuum region, the width of the hole increases. In the asymptotic limit when the electron is removed to infinity, the hole is completely delocalized and spread throughout the crystal, its center of mass being singular. As a consequence it appears that it is the Coulomb hole charge distribution localized at the surface that is the image charge, but this has yet to be shown. Finally, we provide insights into the quantum-mechanical origin of the image potential. These ideas are based on our interpretation of the local exchange-correlation potential of density-functional theory as being the work done to remove an electron against the electric field of its Fermi-Coulomb hole charge density. Since the Coulomb hole charge is zero, the image potential in the asymptotic region far from the surface is the work done against the electric field of its Fermi hole. The results of preliminary calculations confirm this conclusion. Whether the Coulomb hole contributes to making the total effective potential the image potential for electron positions closer to the surface is yet unanswered. The above physical interpretation is based on the fact that the Fermi-Coulomb hole charge distribution is dynamic as a function of electron position. This explains why the exchange potential as calculated by the Slater method is incorrect. As such we also present the complete structure of the Slater potential at a metal surface and show that it leads to an erroneous value in the interior of the metal and that its asymptotic structure though image-potential-like has a coefficient approximately twice as large as that of the image potential.

Quantum-mechanical interpretation of time-dependent density-functional theory

Abstract In this paper we derive the differential virial theorem for both time-dependent (TD) Schrodinger theory and Kohn-Sham (KS) density-functional theory. As such we obtain an exact integral expression for the TD electron-interaction potential of KS theory that is independent of the choice of action. The expression, valid for general TD phenomena other than the ionization or scattering process, is afforded the physical interpretation at each instant of time as being the work done to move an electron in a conservative field. The field is a sum of four component fields representative of Pauli and Coulomb correlations, correlation-kinetic effects, and as discovered here, correlations due to the difference in current densities of the KS and Schrodinger systems. The interpretation further reduces to that for the corresponding electron-interaction potential of stationary-state KS theory for time-independent external potentials.

Integral coalescence conditions in D≥2, dimension space

We have derived the integral form of the cusp and node coalescence conditions satisfied by the wave function at the coalescence of two charged particles in D⩾2 dimension space. From it we have obtained the differential form of the coalescence conditions. These expressions reduce to the well-known integral and differential coalescence conditions in D=3 space. It follows from the results derived that the approximate Laughlin wave function for the fractional quantum Hall effect satisfies the node coalescence condition. It is further noted that the integral form makes evident that unlike the electron–nucleus coalescence condition, the differential form of the electron–electron coalescence condition cannot be expressed in terms of the electron density at the point of coalescence. From the integral form, the integral and differential coalescence conditions for the pair-correlation function in D⩾2 dimension space are also derived. The known differential form of the pair function cusp condition for the uniform el...

Time‐dependent differential virial theorems

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 Schrodinger 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.

Quantal density functional theory of degenerate states

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.

Local exchange-correlation potential from the force field of the Fermi-Coulomb hole charge for non-symmetric systems

Abstract It is proposed that for non-symmetrical electronic density systems for which the curl of the electric field due to the Fermi-Coulomb hole charge distribution may not vanish, the local effective many-body potential be obtained as the work done against the irrotational component of this field.

Analytical asymptotic structure of the exchange and correlation potentials at a metal surface

Abstract We derive the exact analytical asymptotic structure of the Kohn-Sham theory exchange potential and thereby of the correlation potential at a semi-infinite jellium metal surface. The exchange potential is image-like ( −A x ), where A depends on the Fermi energy and surface barrier height, and is precisely 1 4 for stable jellium.