Viraht Sahni | Graduate Center and Brooklyn College of the City University of New York (original) (raw)

Papers by Viraht Sahni

Physical review, Jan 15, 1979

Physical review, Dec 15, 1980

The inhomogeneous electron gas at a jellium metal surface is studied in the Haitree-Pock approxim... more The inhomogeneous electron gas at a jellium metal surface is studied in the Haitree-Pock approximation by' Kohn-Sham density functional theory. Rigorous upper bounds to the surface energy are derived by application of the Rayleigh-Ritz variational principle for the energy, the surface kinetic, electrostatic, and nonlocal exchange energy functionals being determined exactly for the accurate linear-potential model electronic wave functions. The densities obtained by the energy minimization constraint are then employed to determine work-function results via the variationally accurate "displaced-profile change-in-self-consistent-field" expression. The theoretical basis of this non-self-consistent procedure and its demonstrated accuracy for the fully correlated system (as treated within the local-density approximation for exchange and correlationj leads us to conclude these results for the surface energies and work functions to be essentially exact. Work-function values are also determined by the Koopmans'-theorem expression, both for these densities as well as for those obtained by satisfaction of the constraint set on the electrostatic potential by the Budd-Vannimenus theorem. The use of the Hartree-Fock results in the accurate estimation of correlation-effect contributions to these surface properties of the nonuniform electron gas is also indicated. In addition, the original work and approximations made by Bardeen in this attempt at a solution of the Hartree-Fock problem are briefly reviewed in order to contrast with the present work.

ChemPhysChem, Aug 17, 2022

Physical review, Mar 15, 1984

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.

Solid State Communications, Feb 1, 1977

A one-parameter model potential variational calculation of jelliummetal surface energies in the l... more A one-parameter model potential variational calculation of jelliummetal surface energies in the local density approximation is performed, the results very closely approximating those of the self-consistent calculations of Lang and Kohn. The use of this model potential for the determination of superior upper bounds for the surface energy and the study of various density gradient expansions is indicated.

Physical review, Feb 15, 1979

One-parameter variational calculations, employing different Kohn-Sham-type energy functionals of ... more One-parameter variational calculations, employing different Kohn-Sham-type energy functionals of the density, are performed to determine the surface energy, relaxation dipole barrier, and work function for the most densely packed crystal faces of 12 simple metals (Al, Pb, Zn, Mg, Li, Ca, Sr, Ba, Na K, Rb, and-Cs). The variational single-particle wave functions used are those generated from the linear-potential model, and the surface-energy functionals considered are those within the local-density approximation (LDA) for exchange and correlation, and the LDA with wave-vector analysis and gradient-expansion corrections. Both the first density-gradient correction coefficients due to Geldart-Rasolt, and the first and second gradient coefficients of Gupta-Singwi are employed in the determination of these properties. A comparative study of these gradient expansions and wave-vector-analysis corrections to the LDA exchange-correlation energy is also performed for jelljum metal surfaces. The ions of the crystal are included via the Ashcroft pseudopotential, and a general and exact expression for the work function, including band-structure effects, is derived for the case when the &onic lattice is represented by such local pseudopotentials. The results of the primarily analytic calculations indicate that the bounds obtained within the LDA are superior to the perturbative results of Lang and Kohn, and that it is necessary to include corrections to the LDA value of the exchange-correlation energy if results comparable to existing experimental values are to be obtained. For mediumand low-density metals, the energy functionals with the wave-vector and gradient-expansion corrections lead to essentially equivalent results and closely approximate the experimental values. Although all three energy functionals lead to accurate results for the high-density metals (with the exception of Pb), the gradient-expansion values are generally superior. The results for the surface dipole barrier demonstrate that the density profile at real-metal surfaces is substantially different from that at the surface of jellium metal. The corresponding work functions obtained are, however, similar to the jellium-model values and fairly insensitive to the choice of energy functional employed. These results for the work functions compare well with polycrystalline metal experimental values.

Chemical sciences journal, Jun 30, 2016

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.

Journal of Molecular Structure-theochem, Apr 1, 2000

In this paper we describe Schrödinger theory from the perspective of classical fields whose sourc... more In this paper we describe Schrödinger theory from the perspective of classical fields whose sources are quantal expectations of Hermitian operators. As such these fields may be considered as being intrinsic to and thereby descriptive of the quantum system. The perspective is valid for both ground and bound excited-states. The fields, whose existence is inferred via the differential virial theorem, are as follows: (i) an electron-interaction field E ee (r,t) derived via Coulomb's law from the paircorrelation density; (ii) a kinetic-field Z(r,t) derived as the derivative of the kinetic-energy-density tensor obtained from the spinless single-particle density matrix; (iii) a differential density field D(r,t) which is the gradient of the Laplacian of the electron density; and (iv) a current-density field J(r,t) derived as the time derivative of the current density whose source too is the spinless single-particle density matrix. The total energy and its components may be expressed in terms of these fields: the electron-interaction potential and kinetic energies via the fields E ee (r,t) and Z(r,t), respectively, and the external potential energy via a conservative field which is the sum of all the fields present. The field perspective is illustrated by application to the exactly solvable stationary ground-state of the Hooke's atom, its extension to the time-dependent case being surmised via the Harmonic Potential theorem. Finally, we note that both Schrödinger and Kohn-Sham density-functional theory are now describable in terms of classical fields derived from quantal sources.

Springer eBooks, 2016

The use of general descriptive names, registered names, trademarks, etc. in this publication does... more The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Physical review, Sep 1, 1975

A recently developed variational formalism for the determination of the reduced single-particle d... more A recently developed variational formalism for the determination of the reduced single-particle density matrix, correct to second order, is applied to the ground state of the helium isoelectronic sequence, For a Slater-determinant-type trial wave function the method requires the initial determination of either the charge density or equivalently its cosine Fourier transform for spherically symmetric systems. The trial wave function employed is a one-parameter Hartree product of hydrogenic functions and use is made of the highly accurate analytic expressions derived elsewhere for the Fourier transform of the charge density for the helium sequence. Analytic expressions for the single-particle density matrix are obtained and the internal self-consistency of the technique with regard to the Kato cusp condition is discussed. These ex pressions are then employed to obtain closed form analytic expressions for the momentum density valid for the entire isoelectronic, sequence. These results are subsequently employed to obtain expressions for the expectation values of the operators p", ps p~, and p~' and the Compton profile in the impulse approximation. Analytic Hartree-Pock calculations for these properties are also performed, and the results of the variational calculation are compared with these results and those of many-parameter correlated wave-function calculations wherever possible. It is observed that the results of the single-parameter variational calculation for helium are highly accurate and improve further for each heavier element of the isoelectronic sequence.

Physical review, Feb 15, 1977

Jellium metal surface properties including the dipole barrier, work function, and surface energy ... more Jellium metal surface properties including the dipole barrier, work function, and surface energy are obtained in the linear-potential approximation to the effective potential at the surface. The metal surface position and field strength are determined, respectively, by the requirement of overall charge neutrality and the constraint set on the electrostatic potential by the Budd-Vannimenus theorem. The surface energies are obtained both within the local density approximation and by application of a sum rule due to Vannimenus and Budd, and the two methods compared. The calculations are primarily analytic and all properties, with the exception of the exchange-correlation energy, are given in terms of universal functions of the field strength, The effects of correlation on the various properties are studied by employing three different approximations for the correlation energy per particle. The results obtained employing the Wigner expression for the correlation energy closely approximate those of Lang and Kohn. The use of different correlation functions, however, leads to only small differences in the results for the dipole barrier, work function, and the exchange-correlation contribution to the energy, but the results for the total surface energy are significantly different.

Physical review, Nov 15, 1977

The density-gradient expansion for the kinetic energy is studied by application of the expansion ... more The density-gradient expansion for the kinetic energy is studied by application of the expansion to an inhomogeneous system of noninteracting fermions, and its convergence demonstrated. The inhomogeneity in the density is created by assuming the electrons move in an effective potential which is linear in the positive half space and constant elsewhere. It is shown that the original von Weisacker coefficient of the first densitygradient correction is inappropriate for both rapidly and slowly varying densities. The coefficient reduced by 16 4249

Local effective potential theory of electronic structure is the mapping from a system of electron... more 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 densityfunctional theory ͑QDFT͒ or Hohenberg-Kohn-Sham density-functional theory ͑HKS-DFT͒. 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 Pauli͒ 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.

Springer eBooks, 2016

We generalize the quantal density-functional theory (QDFT) of electrons in the presence of an ext... more We generalize the quantal density-functional theory (QDFT) of electrons in the presence of an external electrostatic field E(r) = −∇v(r) to include an external magnetostatic field B(r) = ∇ × A(r), where {v(r),A(r)} are the respective scalar and vector potentials. The generalized QDFT, valid for nondegenerate ground and excited states, is the mapping from the interacting system of electrons to a model of noninteracting fermions with the same density ρ(r) and physical current density j(r), and from which the total energy can be obtained. The properties {ρ(r),j(r)} constitute the basic quantum-mechanical variables because, as proved previously, for a nondegenerate ground state they uniquely determine the potentials {v(r),A(r)}. The mapping to the noninteracting system is arbitrary in that the model fermions may be either in their ground or excited state. The theory is explicated by application to a ground state of the exactly solvable (two-dimensional) Hooke's atom in a magnetic field, with the mapping being to a model system also in its ground state. The majority of properties of the model are obtained in closed analytical or semianalytical form. A comparison with the corresponding mapping from a ground state of the (three-dimensional) Hooke's atom in the absence of a magnetic field is also made.

Solid State Communications, Sep 1, 1978

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.

Physical review, May 15, 1982

In this work we comment on the accuracy and sensitivity of metallic surface energies and work fun... more In this work we comment on the accuracy and sensitivity of metallic surface energies and work functions to the choice of lpcal and nonlocal exchange and correlation-energy functionals used to represent the interacting inhomogeneous electron gas. For a Paulicorrelated system we compare the results of these properties as obtained within the localdensity (LDA) and gradient-expansion (GEA) approximations (with the a priori gradient coefficient of Sham) with those determined by Sahni and Ma from the exact nonlocal Hartree-Fock energy functional. The proven accurate non-self-consistent procedure whereby the surface energies are determined by application of the variational principle for the energy, and the work functions by the variationally accurate "displaced-profile change-in-self-consistent-field expression, is applied in conjunction with linear-potential model densities to each energy functional. It is observed that although the LDA surface energies are substantially in error, the corresponding work functions are within one-tenth of an eV of the exact results. The most significant fact to emerge from these calculations, however, is the remarkable equivalence of the GEA and the exact Hartree-Fock energy functionals in terms of the densities, surface energies, and work functions to which they give rise. The present conclusions about the sensitivity of the surface energy and the contrasting insensitivity of the work function to the choice of energy functional are in agreement with the results of previous work on fully correlated nonuniform systems as approximated within both the LDA for exchange-correlation and by the nonlocal-wavevector-analysis formalism.

Physical review, Jan 15, 1979

Physical review, Dec 15, 1980

The inhomogeneous electron gas at a jellium metal surface is studied in the Haitree-Pock approxim... more The inhomogeneous electron gas at a jellium metal surface is studied in the Haitree-Pock approximation by' Kohn-Sham density functional theory. Rigorous upper bounds to the surface energy are derived by application of the Rayleigh-Ritz variational principle for the energy, the surface kinetic, electrostatic, and nonlocal exchange energy functionals being determined exactly for the accurate linear-potential model electronic wave functions. The densities obtained by the energy minimization constraint are then employed to determine work-function results via the variationally accurate "displaced-profile change-in-self-consistent-field" expression. The theoretical basis of this non-self-consistent procedure and its demonstrated accuracy for the fully correlated system (as treated within the local-density approximation for exchange and correlationj leads us to conclude these results for the surface energies and work functions to be essentially exact. Work-function values are also determined by the Koopmans'-theorem expression, both for these densities as well as for those obtained by satisfaction of the constraint set on the electrostatic potential by the Budd-Vannimenus theorem. The use of the Hartree-Fock results in the accurate estimation of correlation-effect contributions to these surface properties of the nonuniform electron gas is also indicated. In addition, the original work and approximations made by Bardeen in this attempt at a solution of the Hartree-Fock problem are briefly reviewed in order to contrast with the present work.

ChemPhysChem, Aug 17, 2022

Physical review, Mar 15, 1984

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.

Solid State Communications, Feb 1, 1977

A one-parameter model potential variational calculation of jelliummetal surface energies in the l... more A one-parameter model potential variational calculation of jelliummetal surface energies in the local density approximation is performed, the results very closely approximating those of the self-consistent calculations of Lang and Kohn. The use of this model potential for the determination of superior upper bounds for the surface energy and the study of various density gradient expansions is indicated.

Physical review, Feb 15, 1979

One-parameter variational calculations, employing different Kohn-Sham-type energy functionals of ... more One-parameter variational calculations, employing different Kohn-Sham-type energy functionals of the density, are performed to determine the surface energy, relaxation dipole barrier, and work function for the most densely packed crystal faces of 12 simple metals (Al, Pb, Zn, Mg, Li, Ca, Sr, Ba, Na K, Rb, and-Cs). The variational single-particle wave functions used are those generated from the linear-potential model, and the surface-energy functionals considered are those within the local-density approximation (LDA) for exchange and correlation, and the LDA with wave-vector analysis and gradient-expansion corrections. Both the first density-gradient correction coefficients due to Geldart-Rasolt, and the first and second gradient coefficients of Gupta-Singwi are employed in the determination of these properties. A comparative study of these gradient expansions and wave-vector-analysis corrections to the LDA exchange-correlation energy is also performed for jelljum metal surfaces. The ions of the crystal are included via the Ashcroft pseudopotential, and a general and exact expression for the work function, including band-structure effects, is derived for the case when the &onic lattice is represented by such local pseudopotentials. The results of the primarily analytic calculations indicate that the bounds obtained within the LDA are superior to the perturbative results of Lang and Kohn, and that it is necessary to include corrections to the LDA value of the exchange-correlation energy if results comparable to existing experimental values are to be obtained. For mediumand low-density metals, the energy functionals with the wave-vector and gradient-expansion corrections lead to essentially equivalent results and closely approximate the experimental values. Although all three energy functionals lead to accurate results for the high-density metals (with the exception of Pb), the gradient-expansion values are generally superior. The results for the surface dipole barrier demonstrate that the density profile at real-metal surfaces is substantially different from that at the surface of jellium metal. The corresponding work functions obtained are, however, similar to the jellium-model values and fairly insensitive to the choice of energy functional employed. These results for the work functions compare well with polycrystalline metal experimental values.

Chemical sciences journal, Jun 30, 2016

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.

Journal of Molecular Structure-theochem, Apr 1, 2000

In this paper we describe Schrödinger theory from the perspective of classical fields whose sourc... more In this paper we describe Schrödinger theory from the perspective of classical fields whose sources are quantal expectations of Hermitian operators. As such these fields may be considered as being intrinsic to and thereby descriptive of the quantum system. The perspective is valid for both ground and bound excited-states. The fields, whose existence is inferred via the differential virial theorem, are as follows: (i) an electron-interaction field E ee (r,t) derived via Coulomb's law from the paircorrelation density; (ii) a kinetic-field Z(r,t) derived as the derivative of the kinetic-energy-density tensor obtained from the spinless single-particle density matrix; (iii) a differential density field D(r,t) which is the gradient of the Laplacian of the electron density; and (iv) a current-density field J(r,t) derived as the time derivative of the current density whose source too is the spinless single-particle density matrix. The total energy and its components may be expressed in terms of these fields: the electron-interaction potential and kinetic energies via the fields E ee (r,t) and Z(r,t), respectively, and the external potential energy via a conservative field which is the sum of all the fields present. The field perspective is illustrated by application to the exactly solvable stationary ground-state of the Hooke's atom, its extension to the time-dependent case being surmised via the Harmonic Potential theorem. Finally, we note that both Schrödinger and Kohn-Sham density-functional theory are now describable in terms of classical fields derived from quantal sources.

Springer eBooks, 2016

The use of general descriptive names, registered names, trademarks, etc. in this publication does... more The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Physical review, Sep 1, 1975

A recently developed variational formalism for the determination of the reduced single-particle d... more A recently developed variational formalism for the determination of the reduced single-particle density matrix, correct to second order, is applied to the ground state of the helium isoelectronic sequence, For a Slater-determinant-type trial wave function the method requires the initial determination of either the charge density or equivalently its cosine Fourier transform for spherically symmetric systems. The trial wave function employed is a one-parameter Hartree product of hydrogenic functions and use is made of the highly accurate analytic expressions derived elsewhere for the Fourier transform of the charge density for the helium sequence. Analytic expressions for the single-particle density matrix are obtained and the internal self-consistency of the technique with regard to the Kato cusp condition is discussed. These ex pressions are then employed to obtain closed form analytic expressions for the momentum density valid for the entire isoelectronic, sequence. These results are subsequently employed to obtain expressions for the expectation values of the operators p", ps p~, and p~' and the Compton profile in the impulse approximation. Analytic Hartree-Pock calculations for these properties are also performed, and the results of the variational calculation are compared with these results and those of many-parameter correlated wave-function calculations wherever possible. It is observed that the results of the single-parameter variational calculation for helium are highly accurate and improve further for each heavier element of the isoelectronic sequence.

Physical review, Feb 15, 1977

Jellium metal surface properties including the dipole barrier, work function, and surface energy ... more Jellium metal surface properties including the dipole barrier, work function, and surface energy are obtained in the linear-potential approximation to the effective potential at the surface. The metal surface position and field strength are determined, respectively, by the requirement of overall charge neutrality and the constraint set on the electrostatic potential by the Budd-Vannimenus theorem. The surface energies are obtained both within the local density approximation and by application of a sum rule due to Vannimenus and Budd, and the two methods compared. The calculations are primarily analytic and all properties, with the exception of the exchange-correlation energy, are given in terms of universal functions of the field strength, The effects of correlation on the various properties are studied by employing three different approximations for the correlation energy per particle. The results obtained employing the Wigner expression for the correlation energy closely approximate those of Lang and Kohn. The use of different correlation functions, however, leads to only small differences in the results for the dipole barrier, work function, and the exchange-correlation contribution to the energy, but the results for the total surface energy are significantly different.

Physical review, Nov 15, 1977

The density-gradient expansion for the kinetic energy is studied by application of the expansion ... more The density-gradient expansion for the kinetic energy is studied by application of the expansion to an inhomogeneous system of noninteracting fermions, and its convergence demonstrated. The inhomogeneity in the density is created by assuming the electrons move in an effective potential which is linear in the positive half space and constant elsewhere. It is shown that the original von Weisacker coefficient of the first densitygradient correction is inappropriate for both rapidly and slowly varying densities. The coefficient reduced by 16 4249

Local effective potential theory of electronic structure is the mapping from a system of electron... more 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 densityfunctional theory ͑QDFT͒ or Hohenberg-Kohn-Sham density-functional theory ͑HKS-DFT͒. 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 Pauli͒ 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.

Springer eBooks, 2016

We generalize the quantal density-functional theory (QDFT) of electrons in the presence of an ext... more We generalize the quantal density-functional theory (QDFT) of electrons in the presence of an external electrostatic field E(r) = −∇v(r) to include an external magnetostatic field B(r) = ∇ × A(r), where {v(r),A(r)} are the respective scalar and vector potentials. The generalized QDFT, valid for nondegenerate ground and excited states, is the mapping from the interacting system of electrons to a model of noninteracting fermions with the same density ρ(r) and physical current density j(r), and from which the total energy can be obtained. The properties {ρ(r),j(r)} constitute the basic quantum-mechanical variables because, as proved previously, for a nondegenerate ground state they uniquely determine the potentials {v(r),A(r)}. The mapping to the noninteracting system is arbitrary in that the model fermions may be either in their ground or excited state. The theory is explicated by application to a ground state of the exactly solvable (two-dimensional) Hooke's atom in a magnetic field, with the mapping being to a model system also in its ground state. The majority of properties of the model are obtained in closed analytical or semianalytical form. A comparison with the corresponding mapping from a ground state of the (three-dimensional) Hooke's atom in the absence of a magnetic field is also made.

Solid State Communications, Sep 1, 1978

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.

Physical review, May 15, 1982

In this work we comment on the accuracy and sensitivity of metallic surface energies and work fun... more In this work we comment on the accuracy and sensitivity of metallic surface energies and work functions to the choice of lpcal and nonlocal exchange and correlation-energy functionals used to represent the interacting inhomogeneous electron gas. For a Paulicorrelated system we compare the results of these properties as obtained within the localdensity (LDA) and gradient-expansion (GEA) approximations (with the a priori gradient coefficient of Sham) with those determined by Sahni and Ma from the exact nonlocal Hartree-Fock energy functional. The proven accurate non-self-consistent procedure whereby the surface energies are determined by application of the variational principle for the energy, and the work functions by the variationally accurate "displaced-profile change-in-self-consistent-field expression, is applied in conjunction with linear-potential model densities to each energy functional. It is observed that although the LDA surface energies are substantially in error, the corresponding work functions are within one-tenth of an eV of the exact results. The most significant fact to emerge from these calculations, however, is the remarkable equivalence of the GEA and the exact Hartree-Fock energy functionals in terms of the densities, surface energies, and work functions to which they give rise. The present conclusions about the sensitivity of the surface energy and the contrasting insensitivity of the work function to the choice of energy functional are in agreement with the results of previous work on fully correlated nonuniform systems as approximated within both the LDA for exchange-correlation and by the nonlocal-wavevector-analysis formalism.