We have extended Sham's work to derive the density-functional-theory (DFT) first-gradient-co... more We have extended Sham's work to derive the density-functional-theory (DFT) first-gradient-correction coefficient in the expansion for the screened-Coulomb exchange energy. For finite screening, this coefficient is equivalent to that derived within Hartree-Fock theory (HFT). It reduces to the bare-Coulomb interaction value in the limit of no screening, in which limit, as is well known, the HFT coefficient is singular. Due to a universal feature of the DFT coefficient derived (a feature not possessed by the HFT coefficient), it has been possible to demonstrate in a general manner conclusions on the convergence of this expansion arrived at in previous work of ours.
The two nondegenerate ground-state theorems of Hohenberg-Kohn (HK) are described with an emphasis... more The two nondegenerate ground-state theorems of Hohenberg-Kohn (HK) are described with an emphasis on new understandings of the first theorem (HK1) and of its proof. Via HK1, the concept of a basic variable of quantum mechanics, a gauge invariant property knowledge of which uniquely determines the Hamiltonian to within a constant, and hence the wave functions of the system, is developed. HK1 proves that the basic variable is the nondegenerate ground state density. HK1 is generalized via a density preserving unitary transformation to prove the wave function must be a functional of the density and a gauge function of the coordinates in order for the wave function written as a functional to be gauge variant. A corollary proves that degenerate Hamiltonians representing different physical systems but yet possessing the same density cannot be distinguished on the basis of HK1. (This does not constitute a violation of HK1 as the Hamiltonians differ by a constant.) The primacy of the electron number N in the proof of the HK theorems is stressed. The Percus-Levy-Lieb (PLL) constrained-search path from the density to the wave functions is described. It is noted that the HK path is more fundamental, as knowledge of the property that constitutes the basic variable, as gleaned from HK1, is essential for the constrained-search proof of PLL. The Gunnarsson-Lundqvist theorems, the extension of the HK theorems to the lowest excited state of symmetry different from that of the ground state are described. The Runge-Gross (RG) theorems for time-dependent theory, with an emphasis on the first theorem (RG1), are explained. RG1 proves the basic variables to be the density and the current density. A density preserving unitary transformation generalizes RG1 to prove the wave function must be a functional of the density and a gauge function of the coordinates and time. A hierarchy based on gauge functions thereby exists for the fundamental first theorems of density functional theory. A corollary to RG1 similar to that for the time-independent case is proved. Kohn-Sham theory, a ground state theory, which constitutes the mapping from the interacting system to one of noninteracting fermions of the same density, is formulated. As this mapping is based on the HK theorems, the description of the model system is mathematical in that the energy is in terms of functionals of the density, and the local potentials defined as the corresponding functional derivatives.
As time-independent ground state Quantal density functional theory (Q-DFT) is a description in te... more As time-independent ground state Quantal density functional theory (Q-DFT) is a description in terms of ‘classical’ fields and quantal sources of the mapping from the interacting system of electrons as described by Schrodinger theory to one of noninteracting fermions possessing the same nondegenrate ground state density, it provides a rigorous physical interpretation of the energy functionals and functional derivatives (potentials) of Kohn-Sham (KS) theory. The KS ‘exchange-correlation’ potential is the work done in a conservative effective field that is the sum of the Pauli-Coulomb and Correlation-Kinetic fields. The KS ‘exchange-correlation’ energy is the sum of the Pauli-Coulomb and the Correlation-Kinetic energies, these energies being defined in integral virial form in terms of the corresponding fields. Via adiabatic coupling constant perturbation theory applied to Q-DFT, it is shown that KS ‘exchange’ is representative of electron correlations due to the Pauli Exclusion Principle and lowest-order Correlation-Kinetic effects. KS ‘correlation’ in turn is representative of Coulomb correlations and second- and higher-order Correlation-Kinetic effects. The Optimized Potential Method (OPM) integro-differential equations are derived. As the OPM is equivalent to KS theory, Q-DFT thus also provides a physical interpretation of the OPM equations. It further provides the interpretation of the energy functionals and functional derivatives (potentials) of the KS Hartree and Hartree-Fock theories.
Schrodinger theory of the electronic structure of matter—N electrons in the presence of an extern... more Schrodinger theory of the electronic structure of matter—N electrons in the presence of an external time-dependent field—is described from the perspective of the individual electron. The corresponding equation of motion is expressed via the ‘Quantal Newtonian’ second law, the first law being a description of the stationary state case. This description of Schrodinger theory is ‘Newtonian’ in that it is in terms of ‘classical’ fields which pervade space, and whose sources are quantum-mechanical expectations of Hermitian operators taken with respect to the system wave function. In addition to the external field, each electron experiences an internal field, the components of which are representative of correlations due to the Pauli Exclusion Principle and Coulomb repulsion, the kinetic effects, and the density. The resulting motion of the electron is described by a response field. Ehrenfest’s theorem is derived by showing the internal field vanishes on summing over all the electrons. The ‘Newtonian’ perspective is then explicated for both a ground and excited state of an exactly solvable model. Various facets of quantum mechanics such as the Integral Virial Theorem, the Harmonic Potential Theorem, the quantum-mechanical ‘hydrodynamical’ equations in terms of fields, coalescence constraints, and the asymptotic structure of the wave function and density are derived. The equivalence of the ‘Quantal Newtonian’ second law and the Euler equation of Quantum Fluid Dynamics is proved.
Abstract The applicability of variational principles for single-particle expectation values to th... more Abstract The applicability of variational principles for single-particle expectation values to the inhomogeneous electron gas at surfaces is demonstrated. Jellium metal surface densities, correct to second order, are determined by application of the formalism to the operator W= ∑ i δ(x i −x) . The surface dipole barrier thus obtained is shown to lead to essentially exact results over a substantial range of the variational parameter employed. The procedure for inclusion of the discrete ionic lattice is also indicated.
We have extended Sham's work to derive the density-functional-theory (DFT) first-gradient-co... more We have extended Sham's work to derive the density-functional-theory (DFT) first-gradient-correction coefficient in the expansion for the screened-Coulomb exchange energy. For finite screening, this coefficient is equivalent to that derived within Hartree-Fock theory (HFT). It reduces to the bare-Coulomb interaction value in the limit of no screening, in which limit, as is well known, the HFT coefficient is singular. Due to a universal feature of the DFT coefficient derived (a feature not possessed by the HFT coefficient), it has been possible to demonstrate in a general manner conclusions on the convergence of this expansion arrived at in previous work of ours.
The two nondegenerate ground-state theorems of Hohenberg-Kohn (HK) are described with an emphasis... more The two nondegenerate ground-state theorems of Hohenberg-Kohn (HK) are described with an emphasis on new understandings of the first theorem (HK1) and of its proof. Via HK1, the concept of a basic variable of quantum mechanics, a gauge invariant property knowledge of which uniquely determines the Hamiltonian to within a constant, and hence the wave functions of the system, is developed. HK1 proves that the basic variable is the nondegenerate ground state density. HK1 is generalized via a density preserving unitary transformation to prove the wave function must be a functional of the density and a gauge function of the coordinates in order for the wave function written as a functional to be gauge variant. A corollary proves that degenerate Hamiltonians representing different physical systems but yet possessing the same density cannot be distinguished on the basis of HK1. (This does not constitute a violation of HK1 as the Hamiltonians differ by a constant.) The primacy of the electron number N in the proof of the HK theorems is stressed. The Percus-Levy-Lieb (PLL) constrained-search path from the density to the wave functions is described. It is noted that the HK path is more fundamental, as knowledge of the property that constitutes the basic variable, as gleaned from HK1, is essential for the constrained-search proof of PLL. The Gunnarsson-Lundqvist theorems, the extension of the HK theorems to the lowest excited state of symmetry different from that of the ground state are described. The Runge-Gross (RG) theorems for time-dependent theory, with an emphasis on the first theorem (RG1), are explained. RG1 proves the basic variables to be the density and the current density. A density preserving unitary transformation generalizes RG1 to prove the wave function must be a functional of the density and a gauge function of the coordinates and time. A hierarchy based on gauge functions thereby exists for the fundamental first theorems of density functional theory. A corollary to RG1 similar to that for the time-independent case is proved. Kohn-Sham theory, a ground state theory, which constitutes the mapping from the interacting system to one of noninteracting fermions of the same density, is formulated. As this mapping is based on the HK theorems, the description of the model system is mathematical in that the energy is in terms of functionals of the density, and the local potentials defined as the corresponding functional derivatives.
As time-independent ground state Quantal density functional theory (Q-DFT) is a description in te... more As time-independent ground state Quantal density functional theory (Q-DFT) is a description in terms of ‘classical’ fields and quantal sources of the mapping from the interacting system of electrons as described by Schrodinger theory to one of noninteracting fermions possessing the same nondegenrate ground state density, it provides a rigorous physical interpretation of the energy functionals and functional derivatives (potentials) of Kohn-Sham (KS) theory. The KS ‘exchange-correlation’ potential is the work done in a conservative effective field that is the sum of the Pauli-Coulomb and Correlation-Kinetic fields. The KS ‘exchange-correlation’ energy is the sum of the Pauli-Coulomb and the Correlation-Kinetic energies, these energies being defined in integral virial form in terms of the corresponding fields. Via adiabatic coupling constant perturbation theory applied to Q-DFT, it is shown that KS ‘exchange’ is representative of electron correlations due to the Pauli Exclusion Principle and lowest-order Correlation-Kinetic effects. KS ‘correlation’ in turn is representative of Coulomb correlations and second- and higher-order Correlation-Kinetic effects. The Optimized Potential Method (OPM) integro-differential equations are derived. As the OPM is equivalent to KS theory, Q-DFT thus also provides a physical interpretation of the OPM equations. It further provides the interpretation of the energy functionals and functional derivatives (potentials) of the KS Hartree and Hartree-Fock theories.
Schrodinger theory of the electronic structure of matter—N electrons in the presence of an extern... more Schrodinger theory of the electronic structure of matter—N electrons in the presence of an external time-dependent field—is described from the perspective of the individual electron. The corresponding equation of motion is expressed via the ‘Quantal Newtonian’ second law, the first law being a description of the stationary state case. This description of Schrodinger theory is ‘Newtonian’ in that it is in terms of ‘classical’ fields which pervade space, and whose sources are quantum-mechanical expectations of Hermitian operators taken with respect to the system wave function. In addition to the external field, each electron experiences an internal field, the components of which are representative of correlations due to the Pauli Exclusion Principle and Coulomb repulsion, the kinetic effects, and the density. The resulting motion of the electron is described by a response field. Ehrenfest’s theorem is derived by showing the internal field vanishes on summing over all the electrons. The ‘Newtonian’ perspective is then explicated for both a ground and excited state of an exactly solvable model. Various facets of quantum mechanics such as the Integral Virial Theorem, the Harmonic Potential Theorem, the quantum-mechanical ‘hydrodynamical’ equations in terms of fields, coalescence constraints, and the asymptotic structure of the wave function and density are derived. The equivalence of the ‘Quantal Newtonian’ second law and the Euler equation of Quantum Fluid Dynamics is proved.
Abstract The applicability of variational principles for single-particle expectation values to th... more Abstract The applicability of variational principles for single-particle expectation values to the inhomogeneous electron gas at surfaces is demonstrated. Jellium metal surface densities, correct to second order, are determined by application of the formalism to the operator W= ∑ i δ(x i −x) . The surface dipole barrier thus obtained is shown to lead to essentially exact results over a substantial range of the variational parameter employed. The procedure for inclusion of the discrete ionic lattice is also indicated.
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