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Viraht Sahni

Publications and source records attributed to Viraht Sahni.

17 recordsLinked to original sources
arXiv2021

Wave Function Identity: A New Symmetry for 2-electron Systems in an Electromagnetic Field

Stationary-state Schr{ö}dinger-Pauli theory is a description of electrons with a spin moment in an external electromagnetic field. For 2-electron systems as described by the Schr{ö}dinger-Pauli theory Hamiltonian with a symmetrical binding potential, we report a new symmetry operation of the electronic coordinates. The symmetry operation is such that it leads to the equality of the transformed wave function to the wave function. This equality is referred to as the Wave Function Identity. The symmetry operation is a two-step process: an interchange of the spatial coordinates of the electrons whilst keeping their spin moments unchanged, followed by an inversion. The Identity is valid for arbitrary structure of the binding potential, arbitrary electron interaction of the form $w(|{\bf{r}} - {\bf{r}}'|)$, all bound electronic states, and arbitrary dimensionality. It is proved that the exact wave functions satisfy the Identity. On application of the permutation operation for fermions to the identity, it is shown that the parity of the singlet states is even and that of triplet states odd. As a consequence, it follows that at electron-electron coalescence, the singlet state wave functions satisfy the cusp coalescence constraint, and triplet state wave functions the node coalescence condition. Further, we show that the parity of the singlet state wave functions about all points of electron-electron coalescence is even, and that of the triplet state wave functions odd. The Wave Function Identity and the properties on parity, together with the Pauli principle, are then elucidated by application to the $2$-dimensional $2$-electron `artificial atoms' or semiconductor quantum dots in a magnetic field in their first excited singlet $2^{1}S$ and triplet $2^{3} S$ states.

physics.atom-ph↗
arXiv2019

New Perspectives on the Schr{ö}dinger-Pauli Theory of Electrons: Part I

Schr{ö}dinger-Pauli (SP) theory is a description of electrons in the presence of a static electromagnetic field in which the interaction of the magnetic field with both the orbital and spin moments is explicitly considered. The theory is described from a new perspective, viz. that of the individual electron via its equation of motion or `Quantal Newtonian' first law. The law leads to new physical and mathematical insights into the theory. 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 electrostatic and a Lorentz field. The internal field is a sum of fields: the electron-interaction, differential density, kinetic, and internal magnetic fields. These fields are respectively representative of a property of the system: electron correlations due to the Pauli exclusion principle and Coulomb repulsion, the electron density, kinetic effects, and the physical current density. The energy components can be expressed in integral virial form in terms of these fields. The law leads to the further understanding that the Hamiltonian is an exactly known and universal functional of the wave function. This allows for the generalization of the SP equation, which then proves it to be intrinsically self-consistent. A Quantal density functional (local effective potential) theory of the SP system is developed. Further generalizations of the present work to the temporal case, and relativistic Dirac theory are proposed.

quant-ph↗
arXiv2019

New Perspectives on the Schr{ö}dinger-Pauli Theory of Electrons: Part II: Application to the Triplet State of a Quantum Dot in a Magnetic Field

The Schrödinger-Pauli theory of electrons in the presence of a static electromagnetic field can be described from the perspective 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 wave function. The law states that the sum of the external and internal fields experienced by each electron vanishes. The external field is the sum of the binding electrostatic and Lorentz fields. The internal field is the sum of fields representative of properties of the system: electron correlations due to the Pauli exclusion principle and Coulomb repulsion; the electron density; kinetic effects; the current density. Thus, the internal field is a sum of the electron-interaction, differential density, kinetic, and internal magnetic fields. The energy can be expressed in integral virial form in terms of these fields. Via this perspective, the Schrödinger-Pauli equation can be written in a generalized form which then shows it to be intrinsically self-consistent. This new perspective is explicated by application to the triplet $2^{3}S$ state of a 2-D 2-electron quantum dot in a magnetic field. The quantal sources of the density; the paramagnetic, diamagnetic, and magnetization current densities; pair-correlation density; the Fermi-Coulomb hole charge; and the single-particle density matrix are obtained, and from them the corresponding fields determined. The fields are shown to satisfy the `Quantal Newtonian' first law. The components of the energy too are determined from these fields. Finally, the example is employed to demonstrate the intrinsic self-consistent nature of the Schrödinger-Pauli equation.

quant-ph↗
arXiv2016

Properties of the Schrödinger Theory of Electrons in Electromagnetic Fields

The Schrödinger theory of electrons in an external electromagnetic field can be described from the perspective of the individual electron via the `Quantal Newtonian' laws (or differential virial theorems). These laws are in terms of `classical' fields whose sources are quantal expectations of Hermitian operators taken with respect to the wave function. The laws reveal the following physics: (a) In addition to the external field, each electron experiences an internal field whose components are representative of a specific property of the system such as the correlations due to the Pauli exclusion principle and Coulomb repulsion, the electron density, kinetic effects, and an internal magnetic field component. (The response of the electron is described by the current density field.); (b) The scalar potential energy of an electron is the work done in a conservative field which is the sum of the internal and Lorentz fields. It is thus inherently related to the properties of the system. Its constituent property-related components are hence known. It is a known functional of the wave function; (c) As such the Hamiltonian is a functional of the wave function, thereby revealing the intrinsic self-consistent nature of the Schrödinger equation. This then provides a path for the determination of the exact wave function. (d) With the Schrödinger equation written in self-consistent form, the Hamiltonian now admits via the Lorentz field a new term that explicitly involves the external magnetic field. The new understandings are explicated for the stationary state case by application to a quantum dot in a magnetostatic field in both a ground and excited state. For the time-dependent case, the same states of the quantum dot in both a magnetostatic and a time-dependent electric field are considered.

quant-ph↗
arXiv2015

Hohenberg-Kohn Theorems in Electrostatic and Uniform Magnetostatic Fields

The Hohenberg-Kohn (HK) theorems of bijectivity between the external scalar potential and the gauge invariant nondegenerate ground state density, and the consequent Euler variational principle for the density, are proved for arbitrary electrostatic field and the constraint of fixed electron number. The HK theorems are generalized for spinless electrons to the added presence of an external uniform magnetostatic field by introducing the new constraint of fixed canonical orbital angular momentum. Thereby a bijective relationship between the external scalar and vector potentials, and the gauge invariant nondegenerate ground state density and physical current density, is proved. A corresponding Euler variational principle in terms of these densities is also developed. These theorems are further generalized to electrons with spin by imposing the added constraint of fixed canonical orbital and spin angular momentum. The proofs differ from the original HK proof, and explicitly account for the many-to-one relationship between the potentials and the nondegenerate ground state wave function.

cond-mat.str-el↗
arXiv2007

Asymptotic near nucleus structure of the electron-interaction potential in local effective potential theories

In local effective potential theories of electronic structure, the electron correlations due to the Pauli exclusion principle, Coulomb repulsion, and correlation-kinetic effects, are all incorporated in the local electron-interaction potential $v_{ee}({\bf r})$. In previous work, it has been shown that for spherically symmetric or sphericalized systems, the asymptotic near nucleus expansion of this potential is $v_{ee}(r) = v_{ee}(0) + βr + O(r^2)$, with $v_{ee}(0)$ being finite. By assuming that the Schrödinger and local effective potential theory wave functions are analytic near the nucleus of atoms, we prove the following via Quantal density functional theory (Q-DFT): (i) correlations due to the Pauli principle and Coulomb correlations do not contribute to the linear structure; (ii) these Pauli and Coulomb correlations contribute quadratically; (iii) the linear structure is {\em solely} due to correlation-kinetic effects, the contributions of these effects being determined analytically. We also derive by application of adiabatic coupling constant perturbation theory via Q-DFT (iv) the asymptotic near nucleus expansion of the Hohenberg-Kohn-Sham theory exchange $v_x({\bf r})$ and correlation $v_c({\bf r})$ potentials. These functions also approach the nucleus linearly with the linear term of $v_x({\bf r})$ being {\em solely} due to the lowest-order correlation kinetic effects, and the linear term of $v_c({\bf r})$ being due {\em solely} to the higher-order correlation kinetic contributions. The above conclusions are equally valid for systems of arbitrary symmetry, provided spherical averages of the properties are employed.

cond-mat.mtrl-sci↗
arXiv2005

Determination of Wave Function Functionals: The Constrained-Search--Variational Method

In a recent paper [Phys. Rev. Lett. \textbf{93}, 130401 (2004)], we proposed the idea of expanding the space of variations in variational calculations of the energy by considering the approximate wave function $ψ$ to be a functional of functions $ χ: ψ= ψ[χ]$ rather than a function. The space of variations is expanded because a search over the functions $χ$ can in principle lead to the true wave function. As the space of such variations is large, we proposed the constrained-search-- variational method whereby a constrained search is first performed over all functions $χ$ such that the wave function functional $ψ[χ]$ satisfies a physical constraint such as normalization or the Fermi-Coulomb hole sum rule, or leads to the known value of an observable such as the diamagnetic susceptibility, nuclear magnetic constant or Fermi contact term. A rigorous upper bound to the energy is then obtained by application of the variational principle. A key attribute of the method is that the wave function functional is accurate throughout space, in contrast to the standard variational method for which the wave function is accurate only in those regions of space contributing principally to the energy. In this paper we generalize the equations of the method to the determination of arbitrary Hermitian single-particle operators as applied to two-electron atomic and ionic systems. The description is general and applicable to both ground and excited states. A discussion on excited states in conjunction with the theorem of Theophilou is provided.

physics.chem-ph↗
arXiv2005

Determination of a Wave Function Functional

In this paper we propose the idea of expanding the space of variations in standard variational calculations for the energy by considering the wave function $ψ$ to be a functional of a set of functions $χ: ψ= ψ[χ]$, rather than a function. In this manner a greater flexibility to the structure of the wave function is achieved. A constrained search in a subspace over all functions $χ$ such that the wave function functional $ψ[χ]$ satisfies a constraint such as normalization or the Fermi-Coulomb hole charge sum rule, or the requirement that it lead to a physical observable such as the density, diamagnetic susceptibility, etc. is then performed. A rigorous upper bound to the energy is subsequently obtained by variational minimization with respect to the parameters in the approximate wave function functional. Hence, the terminology, the constrained-search variational method. The \emph{rigorous} construction of such a constrained-search--variational wave function functional is demonstrated by example of the ground state of the Helium atom.

physics.atom-ph↗
arXiv2005

Bijectivity of the Normalization and Fermi-Coulomb Hole Sum Rules for Approximate Wave Functions

We prove the bijectivity of the constraints of normalization and of the Fermi-Coulomb hole charge sum rule at each electron position for approximate wave functions. This bijectivity is surprising in light of the fact that normalization depends upon the probability of finding an electron at some position, whereas the Fermi-Coulomb hole sum rule depends on the probability of two electrons staying apart because of correlations due to the Pauli exclusion principle and Coulomb repulsion. We further demonstrate the bijectivity of these sum rules by example.

physics.chem-ph↗
arXiv2005

The constrained-search--variational method: application to the ground state of Helium atom

n a recent paper we proposed the expansion of the space of variations in energy calculations by considering the approximate wave function $ψ$ to be a functional of functions $χ: ψ= ψ[χ]$ rather than a function. For the determination of such a wave function functional, a constrained search is first performed over the subspace of all functions $χ$ such that $ψ[χ]$ satisfies a physical constraint or leads to the known value of an observable. A rigorous upper bound to the energy is then obtained by application of the variational principle. To demonstrate the advantages of the expansion of variational space, we apply the constrained-search--variational method to the ground state of the negative ion of atomic Hydrogen, the Helium atom, and its isoelectronic sequence. The method is equally applicable to excited states, and its extension to such states in conjunction with the theorem of Theophilou is also described.

physics.atom-ph↗
arXiv2004

On the Hylleraas Coordinates

The Hylleraas coordinates $s=r_{1}+r_{2}$, $t=r_{1}-r_{2}$, $u=|{\bf r}_{1}-{\bf r}_{2}|$ are the natural coordinates for the determination of properties of the Helium atom, the positive ions of its isoelectronic sequence, and the negative Hydrogen ion. In this paper, we derive a new expression for integrals representing properties such as the energy, normalization and expectation of arbitrary operators, as written in the $(s,t,u)$ coordinates. The expression derived is valid for both \emph{finite} and \emph{infinite} space. The integrals for the various properties are comprised in each case of two components $A$ and $B$. The contribution of these components to the volume of integration and the normalization of a wave function for finite space, and in variational calculations of the ground state energy of the Helium atom confined in a finite volume is demonstrated by example. We prove that when the integration space is \emph{infinite}, the expression for the energy and other properties employed by Hylleraas corresponds \emph{only} to that of integral $A$. We further prove that for the approximate variational wave functions employed by Hylleraas and other authors, the contribution of the term $B$ vanishes. This contribution also vanishes for the exact wave function. It is interesting to note that the component $B$ to the integral is not mentioned in the literature. A principle purpose of the paper, therefore, is to point out the existence of this term.

physics.atom-ph↗
arXiv2003

Coalescence constraints of many-body systems in one dimension

For one-dimensional many-body systems interacting via the \textit{Coulomb force} and with \textit{arbitrary} external potential energy, we derive (\textit{i}) the \textit{node coalescence condition} for the wave function. This condition rigorously proves the following: (\textit{ii}) that the particles satisfy \textit{only} a node coalescence condition; (\textit{iii}) that irrespective of their charge or statistics, the particles cannot coalesce; (\textit{iv}) that the particles cannot cross each other, and must be ordered; (\textit{v}) the particles are therefore distinguishable; (\textit{vi}) as such their statistics are not significant; (\textit{vii}) conclusions similar to those of the spin-statistics theorem of quantum field theory are arrived at via non-relativistic quantum mechanics; (\textit{viii}) the noninteracting system \textit{cannot} be employed as the lowest-order in a perturbation theory of the interacting system. (\textit{ix}) Finally, the coalescence condition for particles with the short-ranged delta-function interaction and \textit{arbitrary} external potential energy, is also derived. These particles can coalesce and cross each other.

cond-mat.mtrl-sci↗
arXiv2003

Integral Coalescence Conditions In $D \geq 2$ Dimension Space

We have derived the integral form of the cusp and node coalescence conditions satisfied by the wavefunction at the coalescence of two charged particles in $D \geq 2$ 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 wavefunction 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 $D\geq 2$ dimension space are also derived. The known differential form of the pair function cusp condition for the uniform electron gas in dimensions $D = 2,3$ constitute a special case of the result derived.

cond-mat.mtrl-sci↗
arXiv2003

Criticality of Electron-Nucleus Cusp Condition to Local Effective Potential Energy Theories

Local(multiplicative) effective potential energy theories of electronic structure comprise the transformation of the Schr{ö}dinger equation for interacting fermi systems to model noninteracting fermi or bose systems whereby the equivalent density and energy are obtained. By employing the integrated form of the Kato electron-nucleus cusp condition, we prove that the effective electron -interaction potential energy of these model fermions or bosons is finite at a nucleus. The proof is general and valid for arbitrary system whether it be atomic, molecular, or solid state, and for arbitrary state and symmetry. This then provides justification for all prior work in the literature based on the assumption of finiteness of this potential energy at a nucleus. We further demonstrate the criticality of the electron-nucleus cusp condition to such theories by example of the Hydrogen molecule. We show thereby that both model system effective electron-interaction potential energies, as determined from densities derived from accurate wave functions, will be singular at the nucleus unless the wave function satisfies the electron-nucleus cusp condition.

physics.atom-ph↗
arXiv2003

Quantal Density Functional Theory of Degenerate States

The treatment of degenerate states within Kohn-Sham density functional theory (KS-DFT) is a problem of longstanding 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 (Q-DFT). We describe the Q-DFT of \textit{both} ground and excited degenerate states, and for the cases of \textit{both} pure state and ensemble v-representable densities. This then further provides a rigorous physical interpretation of the density and bidensity energy functionals, and of their functional derivatives, of the corresponding KS-DFT. We conclude with examples of the mappings within Q-DFT.

cond-mat.mtrl-sci↗
arXiv2003

Corollary to the Hohenberg-Kohn Theorem

In this paper we construct such a set of `degenerate' Hamiltonians $\hat{H}$, which differ by an `intrinsic' constant but represent different physical systems yet possess the same ground state density. . Thus, although the proof of Hohenberg-Kohn (HK) theorem is independent of whether the constant $C$ is additive or intrinsic, its applicability is restricted to excluding the case of the latter. This constitutes the corollary to the theorem.The corollary has also been extended to the time-dependent version of the HK theorem.

cond-mat.mtrl-sci↗