Hypervirial Theorems for Variational Wave Functions in Scattering Theory

Taking the ground state of helium as an example, evidence is presented in support of the theory that when an approximate wave function satisfies a hypervirial theorem, then the expectation values of certain operators calculated using that wave function will be quite accurate. Firstly, it is discovered which hypervirial theorems are satisfied by various simple approximate wave functions when their disposable parameters are chosen to optimize the energy. Appropriate expectation values are tested, and found to be reasonably accurate. Secondly, the optimum energy criterion is abandoned in certain cases, and parameters are chosen to satisfy alternative hypervirial theorems. This leads to improvement in expectation values related to the hypervirial operators. Some advantages and drawbacks of the hypervirial method of fixing parameters in approximate wave functions are briefly discussed.

The functionals of the Kohn and Sil variational principles of atomic collision theory contain second-order spatial derivatives of the wave-function. Ipso facto the "second-order" generalised Euler-Lagrange differential equation is a necessary condition for these functionals to yield a stationary value. To illustrate this we provide a classical prescription for the time-dependent and time-independent quantum potential scattering problems, generalising the latter to the three-body problem. As practical examples where the presence of second-order derivatives in the Lagrangian density makes a non-trivial contribution to the equations of motion, we consider the optimisation of translation factors in a trial wave-function using the Sil variational principle and the optimisation of a variable charge using the Kohn variational principle. As further examples of Lagrangian densities where second-order derivatives may occur we briefly mention electromagnetism and relativistic quantum mechanics.

Certain aspects of Rayleigh-Schr\"odinger (RS) perturbation theory and of the variational principle are combined in a new perturbationally oriented variational approach for obtaining approximate solutions to Schr\"odinger's time-independent equation. This perturbational-variational (PV) procedure is applicable for all perturbed Hamiltonian operators which can be expanded as $H(\ensuremath{\lambda})=\ensuremath{\Sigma}{H}_{t}{\ensuremath{\lambda}}^{t}$, where $\ensuremath{\lambda}$ is an external parameter governing the strength of the perturbing terms; in particular, the theory is developed here for the important special form, $H={H}_{0}+{H}_{1}\ensuremath{\lambda}$. It is shown how the application of the PV procedure to an arbitrary analytic approximate wave function yields the variational parameters embedded in the wave function, the wave function itself, the corresponding approximate energy, and other expectation values, all in the form of energetically optimized PV expansions in powers of $\ensuremath{\lambda}$ to any desired order. A variational wave-operator formalism is introduced to deal with the multivariant expansions required to form the PV expansions. The remainder theorem and the variational Hellmann-Feynman theorem are derived and their role in the PV methodology is discussed. The principal advantages of the new method are: (1) Unlike RS perturbation theory, there is no requirement that the exact solution of a simpler unperturbed system be known; (2) the explicit $\ensuremath{\lambda}$ dependence of the optimum quantities is analytically derived; (3) the numerical aspects of a given variational problem are simplified because the variational equations generated by the PV procedure are almost completely linearized in the variational parameters; and (4) the PV expansions form a natural framework for the systematic classification and evaluation of all perturbed variational wave functions in respect to their quality. The PV classification scheme, previously presented in bare outline, is rigorously derived. The insight obtained from PV theory is further illustrated by the derivation and application of the variational integral Hellmann-Feynman theorem, the virial theorem, and a theorem relating to the expectation value of the unperturbed potential. The a priori PV classification of perturbed variational wave functions is extended to the perturbed variational parameters embedded in the wave functions; it is shown that the anomalous PV expansions of some open-shell parameters and orbitals can be simply explained in terms of PV theory.

In this article, I consider the use of the Kohn variational principle in obtaining the reaction matrix, which determines the positive energy solution to the partial wave Schrodinger equation in coordinate space. The formulation considered is appropriate for electron scattering from an atom or molecule. In its usual form, the Kohn variational principle provides the relationship between the exact reaction matrix, independent of first-order errors in the trial wavefunction, and the trial reaction matrix. This relationship is usually called the Kato identity. It is obtained using the second form of Green's theorem for a sphere of infinite radius. I show how a Kohn-like variational principle, using Green's theorem for a sphere of arbitrary radius, provides an expression for the exact, coordinate space, integral equation amplitudes in terms of the trial amplitudes. The results are again exact to second order in the trial wavefunction. Applications of the present results to electron-molecule frame transformations and algebraic scattering theory are then suggested.

It is proved that if a scaling parameter alpha contained in suitably chosen trial functions F1T and F2T is optimized using Kohn's variational principle, then the virial theorem is satisfied by the optimized variational functions and the variationally determined scattering amplitude. A generalization of the theorem to arbitrary values of the scaling parameter is also established. In addition the virial theorem for the scattering amplitude is applied to the elastic scattering of electrons by atomic hydrogen.

APPLICATION OF THE CHANNEL COUPLING ARRAY THEORY TO SOME ATOMIC THREE-BODY SYSTEMS

The method of convolutions is used to form smoothed functions from approximate wave functions which are discontinuous or have discontinuous first derivatives. As a parameter ε in the smoothed function decreases, the smoothed function approaches the approximate wave function. The expectation values for physical properties corresponding to the approximate wave function are defined to be the limit as ε approaches zero of the expectation values corresponding to the smoothed function. It is found that if the approximate wave function is discontinuous, the corresponding expectation value for the kinetic energy is infinite. Therefore, it seems unlikely that discontinuous approximate wave functions can ever be useful. However, if the approximate wave function is continuous but has a discontinuity in its first derivative, then, as a result of the discontinuity, there is a contribution δT̄12 to the expectation value of the kinetic energy. For a one-dimensional problem δT̄12=−(ℏ2/2m)ψ*(0) [ψ2′(0)−ψ1′(0)].Here ψ(0) is the value of the approximate wave function at the point zero where the discontinuity in its first derivative occurs, and ψ1′(0) and ψ2′(0) are the first derivatives of ψ taken from the left and from the right, respectively, at this point. Similarly, for an N-dimensional problem having a surface S12=0 over which the first derivatives of the approximate wave function are discontinuous, δT̄12= ∫ surfaceψ*[(∂/∂n) (ψ2−ψ1)]dS12.Here the ∂/∂n is the normal derivative with the normal pointed from region 1 towards region 2.

Variational (Rayleigh-Ritz) methods are applied to local quantum field theory. For scalar theories the wave functional is parametrized in the form of a superposition of Gaussians and the expectation value of the Hamiltonian is expressed in a form that can be minimized numerically. A scheme of successive refinements of the superposition is proposed that may converge to the exact functional. As an illustration, a simple numerical approximation for the effective potential is worked out based on minimization with respect to five variational parameters. A variational principle is formulated for the fermion vacuum energy as a functional of the scalar fields to which the fermions are coupled. The discussion in this paper is given for scalar and fermion interactions in 1+1 dimensions. The extension to higher dimensions encounters a more involved structure of ultraviolet divergences and is deferred to future work.

We propose new basis sets for linear algebraic variational calculations of transition amplitudes for quantum-mechanical scattering problems. These basis sets are hybrids of those that yield the Kohn variational principle (KVP) and those that yield the generalized Newton variational principle (GNVP) when substituted in Schlessinger's stationary expression for the T operator. Trial calculations show that efficiencies almost as great as that of the GNVP and much greater than the KVP can be obtained even for basis sets with the majority of the members independent of energy.

A computational procedure for calculating correlated wave functions is proposed for three-particle systems interacting through Coulomb forces. Calculations are carried out for the muonic helium atom. Variational wave functions which explicitly contain interparticle coordinates are presented for the ground and excited states. General Hylleraas-type trial functions are used as the basis for the correlated wave functions. Excited-state energies of the muonic helium atom computed from 1- and 35-term wave functions are listed for four states.

Generally "exact" quantum Monte Carlo computations for the ground state of many bosons make use of importance sampling. The importance sampling is based either on a guiding function or on an initial variational wave function. Here we investigate the need of importance sampling in the case of path integral ground state (PIGS) Monte Carlo. PIGS is based on a discrete imaginary time evolution of an initial wave function with a nonzero overlap with the ground state, which gives rise to a discrete path which is sampled via a Metropolis-like algorithm. In principle the exact ground state is reached in the limit of an infinite imaginary time evolution, but actual computations are based on finite time evolutions and the question is whether such computations give unbiased exact results. We have studied bulk liquid and solid (4)He with PIGS by considering as initial wave function a constant, i.e., the ground state of an ideal Bose gas. This implies that the evolution toward the ground state is driven only by the imaginary time propagator, i.e., there is no importance sampling. For both phases we obtain results converging to those obtained by considering the best available variational wave function (the shadow wave function) as initial wave function. Moreover we obtain the same results even by considering wave functions with the wrong correlations, for instance, a wave function of a strongly localized Einstein crystal for the liquid phase. This convergence is true not only for diagonal properties such as the energy, the radial distribution function, and the static structure factor, but also for off-diagonal ones, such as the one-body density matrix. This robustness of PIGS can be traced back to the fact that the chosen initial wave function acts only at the beginning of the path without affecting the imaginary time propagator. From this analysis we conclude that zero temperature PIGS calculations can be as unbiased as those of finite temperature path integral Monte Carlo. On the other hand, a judicious choice of the initial wave function greatly improves the rate of convergence to the exact results.

The role of basis set expansions in the relative performance of the schwinger and the Newton variation principles

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTVariational Principle for a Particle in a BoxJuan I. Casaubon and Graham Doggett View Author Information Facultad de Ciencias Exactas y Naturales, Universidad de Belgrano y CONICET, Villanueva 1324, 1426 Buenos Aires, Argentina Chemistry Department, University of York, Heslington, York YO10 5DD, UKCite this: J. Chem. Educ. 2000, 77, 9, 1221Publication Date (Web):September 1, 2000Publication History Received3 August 2009Published online1 September 2000Published inissue 1 September 2000https://pubs.acs.org/doi/10.1021/ed077p1221https://doi.org/10.1021/ed077p1221research-articleACS PublicationsRequest reuse permissionsArticle Views1245Altmetric-Citations8LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InRedditEmail Other access optionsGet e-Alertsclose SUBJECTS:Chemical calculations,Energy,Excited states,Wave function Get e-Alerts

Approximate wave functions for the hydrogen molecule ground state previously available in the literature are analyzed quantitatively into approximate natural spin orbital functions with particular attention to the corresponding occupation numbers. The analysis demonstrates the very great similarity of all such trial wave functions, and especially the largely molecular orbital nature of the wave function. In addition it shows a close relationship between the molecular orbital and valence bond functions, and the importance for allowing for angular correlation of the electrons by including terms dependent upon the azimuthal coordinate. The analysis particularly demonstrates that approximate natural spin orbital occupation numbers are nearly invariant under a wide variety of choices of basis functions, and therefore are particularly suitable for comparison of different approximate functions and for discussion of their respective properties.

The conventional tensor-network states employ real-space product states as reference wave functions. Here, we propose a many-variable variational Monte Carlo (mVMC) method combined with tensor networks by taking advantages of both to study fermionic models. The variational wave function is composed of a pair product wave function operated by real space correlation factors and tensor networks. Moreover, we can apply quantum number projections, such as spin, momentum and lattice symmetry projections, to recover the symmetry of the wave function to further improve the accuracy. We benchmark our method for one- and two-dimensional Hubbard models, which show significant improvement over the results obtained individually either by mVMC or by tensor network. We have applied the present method to hole doped Hubbard model on the square lattice, which indicates the stripe charge/spin order coexisting with a weak $d$-wave superconducting order in the ground state for the doping concentration less than 0.3, where the stripe oscillation period gets longer with increasing hole concentration. The charge homogeneous and highly superconducting state also exists as a metastable excited state for the doping concentration less than 0.25.