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Xiao-Yin Pan

Publications and source records attributed to Xiao-Yin Pan.

29 records · Page 2Linked to original sources
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↗