Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:physics/0402066 · Physical Review Letters

Determination of a Wave Function Functional

Xiao‐Yin Pan, Viraht Sahni, Lou Massa

Published Sep 21, 2004• 20 citations
arXivHTMLPDF

Abstract

We propose expanding the space of variations in traditional variational calculations for the energy by considering the wave function $\ensuremath{\psi}$ to be a functional of a set of functions $\ensuremath{\chi}\ensuremath{\mathbin:}\ensuremath{\psi}=\ensuremath{\psi}[\ensuremath{\chi}]$, rather than a function. A constrained search in a subspace over all functions $\ensuremath{\chi}$ such that the functional $\ensuremath{\psi}[\ensuremath{\chi}]$ satisfies a sum rule or leads to a physical observable is then performed. An upper bound to the energy is subsequently obtained by variational minimization. The rigorous construction of such a constrained-search--variational wave function functional is demonstrated.

DOI
10.1103/physrevlett.93.130401

Related papers

Feynman Variational Method

Lukong Cornelius Fai

In most cases solving a quantum mechanical problem exactly is not always possible. In such a case, an analytic solution to the Schrödinger equation is appropriate. This results to numerical solutions that may not necessarily be the best approach. Interest sometimes may not necessarily be in the detailed eigenfunctions, but rather in the energy levels and the qualitative features of the eigenfunctions. So, the numerical solutions are usually less intuitively reasonable. It is possible to find the values of the energy levels to be slightly sensitive to the deviation of the wave function from its true form. So, the expectation value of the energy for an approximate wave function can be a very good estimate of the corresponding energy eigenvalue. Using an approximate wave function dependent on some set of variational parameters and minimizing its energy by these variational parameters yield such energy estimates. This technique is the so-called variational method due to this minimization procedure.

CRC Press eBooks•Feb 19, 2021•0 citations

Excited states of 4He droplets

R. Guardiola, Jesús Navarro, M. Portesi

We study low-lying excited states of ${}^{4}\mathrm{He}$ clusters up to a cluster size of 40 atoms in a variational framework. The ansatz wave function combines two- and three-body correlations, coming from a translationally invariant configuration interaction description, and Jastrow-type short-range correlation. We have previously used this scheme to determine the ground-state energies of ${}^{4}\mathrm{He}$ and ${}^{3}\mathrm{He}$ clusters. Here we present an extension of this ansatz wave function having a good quantum angular momentum L. The variational procedure is applied independently to the cases with $L=0,2,4,$ and upper bounds for the corresponding energies are thus obtained. Moreover, centroid energies for L excitations are calculated through the use of sum rules. A comparison with previous calculations is also made.

Physical review. B, Condensed matter•May 24, 2001•6 citations

Importance of parametrizing constraints in quantum-mechanical variational calculations

Kwong T. Chung, A. K. Bhatia

In variational calculations of quantum mechanics, constraints are sometimes imposed explicitly on the wave function. These constraints, which are deduced by physical arguments, are often not uniquely defined. In this work, we will point out the advantage of parametrizing constraints and letting the variational principle determine the best possible constraint for the problem. Examples are carried out to show the surprising effectiveness of the variational method if constraints are parametrized. It is also shown that misleading results may be obtained if a constraint is not parametrized.

Physical Review A•Feb 1, 1992•2 citations

Implicit density-functional theory

Bin Liu, Jerome K. Percus

A fermion ground state energy functional is set up in terms of particle density, relative pair density, and kinetic energy tensor density. It satisfies a minimum principle if constrained by a complete set of compatibility conditions. A partial set, which thereby results in a lower bound energy under minimization, is obtained from the solution of model systems, as well as a small number of exact sum rules. Prototypical application is made to several one-dimensional spinless noninteracting models. The effectiveness of ``atomic'' constraints on model ``molecules'' is observed, as well as the structure of systems with only finitely many bound states.

arXiv:physics/0604128•Physical Review A•Jul 24, 2006•13 citations

Nodal variational principle for excited states

Federico E. Zahariev, Mark S. Gordon, Mel Levy

It is proven that the exact excited-state wave function and energy may be obtained by minimizing the energy expectation value of trial wave functions that are constrained only to have the correct nodes of the state of interest. This excited-state nodal minimum principle has the advantage that it requires neither minimization with the constraint of wave-function orthogonality to all lower eigenstates nor the antisymmetry of the trial wave functions. It is also found that the minimization over the entire space can be partitioned into several interconnected minimizations within the individual nodal regions, and the exact excited-state energy may be obtained by a minimization in just one or several of these nodal regions. For the proofs of the theorem, it is observed that the many-electron eigenfunction (excited state as well as ground state), restricted to a nodal region, is equivalent to a ground-state wave function of one electron in a higher-dimensional space; and, alternatively, an explicit excited-state energy variational expression is utilized by generalizing the Jacobi method of multiplicative variation. In corollaries, error functions are constructed for cases for which the nodes are not necessarily exact. The exact nodes minimize the energy error functions with respect to nodal variations. Simple numerical illustrations of the error functions are presented.

arXiv:1609.02241•Physical Review A•Jul 31, 2018•10 citations

Variational orthogonalization

Farrokh Atai, Jens Hoppe, Mariusz Hynek +1 more

We introduce variational methods for finding approximate eigenfunctions and eigenvalues of quantum Hamiltonians by constructing a set of orthogonal wave functions which approximately solve the eigenvalue equation.

arXiv:1307.4010•Jul 15, 2013•1 citation
We propose expanding the space of variations in traditional variational calculations for the energy by considering the wave function \ensuremathψ\ensuremath{\psi}\ensuremathψ to be a functional of a set of functions \ensuremathχ\ensuremath:\ensuremathψ=\ensuremathψ[\ensuremathχ]\ensuremath{\chi}\ensuremath{\mathbin:}\ensuremath{\psi}=\ensuremath{\psi}[\ensuremath{\chi}]\ensuremathχ\ensuremath:\ensuremathψ=\ensuremathψ[\ensuremathχ], rather than a function. A constrained search in a subspace over all functions \ensuremathχ\ensuremath{\chi}\ensuremathχ such that the functional \ensuremathψ[\ensuremathχ]\ensuremath{\psi}[\ensuremath{\chi}]\ensuremathψ[\ensuremathχ] satisfies a sum rule or leads to a physical observable is then performed. An upper bound to the energy is subsequently obtained by variational minimization. The rigorous construction of such a constrained-search--variational wave function functional is demonstrated.
In most cases solving a quantum mechanical problem exactly is not always possible. In such a case, an analytic solution to the Schrödinger equation is appropriate. This results to numerical solutions that may not necessarily be the best approach. Interest sometimes may not necessarily be in the detailed eigenfunctions, but rather in the energy levels and the qualitative features of the eigenfunctions. So, the numerical solutions are usually less intuitively reasonable. It is possible to find the values of the energy levels to be slightly sensitive to the deviation of the wave function from its true form. So, the expectation value of the energy for an approximate wave function can be a very good estimate of the corresponding energy eigenvalue. Using an approximate wave function dependent on some set of variational parameters and minimizing its energy by these variational parameters yield such energy estimates. This technique is the so-called variational method due to this minimization procedure.
We study low-lying excited states of 4He{}^{4}\mathrm{He}4He clusters up to a cluster size of 40 atoms in a variational framework. The ansatz wave function combines two- and three-body correlations, coming from a translationally invariant configuration interaction description, and Jastrow-type short-range correlation. We have previously used this scheme to determine the ground-state energies of 4He{}^{4}\mathrm{He}4He and 3He{}^{3}\mathrm{He}3He clusters. Here we present an extension of this ansatz wave function having a good quantum angular momentum L. The variational procedure is applied independently to the cases with L=0,2,4,L=0,2,4,L=0,2,4, and upper bounds for the corresponding energies are thus obtained. Moreover, centroid energies for L excitations are calculated through the use of sum rules. A comparison with previous calculations is also made.
In variational calculations of quantum mechanics, constraints are sometimes imposed explicitly on the wave function. These constraints, which are deduced by physical arguments, are often not uniquely defined. In this work, we will point out the advantage of parametrizing constraints and letting the variational principle determine the best possible constraint for the problem. Examples are carried out to show the surprising effectiveness of the variational method if constraints are parametrized. It is also shown that misleading results may be obtained if a constraint is not parametrized.
A fermion ground state energy functional is set up in terms of particle density, relative pair density, and kinetic energy tensor density. It satisfies a minimum principle if constrained by a complete set of compatibility conditions. A partial set, which thereby results in a lower bound energy under minimization, is obtained from the solution of model systems, as well as a small number of exact sum rules. Prototypical application is made to several one-dimensional spinless noninteracting models. The effectiveness of ``atomic'' constraints on model ``molecules'' is observed, as well as the structure of systems with only finitely many bound states.
It is proven that the exact excited-state wave function and energy may be obtained by minimizing the energy expectation value of trial wave functions that are constrained only to have the correct nodes of the state of interest. This excited-state nodal minimum principle has the advantage that it requires neither minimization with the constraint of wave-function orthogonality to all lower eigenstates nor the antisymmetry of the trial wave functions. It is also found that the minimization over the entire space can be partitioned into several interconnected minimizations within the individual nodal regions, and the exact excited-state energy may be obtained by a minimization in just one or several of these nodal regions. For the proofs of the theorem, it is observed that the many-electron eigenfunction (excited state as well as ground state), restricted to a nodal region, is equivalent to a ground-state wave function of one electron in a higher-dimensional space; and, alternatively, an explicit excited-state energy variational expression is utilized by generalizing the Jacobi method of multiplicative variation. In corollaries, error functions are constructed for cases for which the nodes are not necessarily exact. The exact nodes minimize the energy error functions with respect to nodal variations. Simple numerical illustrations of the error functions are presented.