Hypervirial Theorems Research Articles

The classical and quantum mechanical virial theorem

In Section 1 an extensive bibliographical review is given for the classical and quantum virial and for the hypervirial theorems, with some references to the special and general relativistic cases. Classical mechanics concepts are discussed from the point of view of “objectivism.” Some difficulties are examined concerning adiabatic and static approximations, partitioning, boundary conditions, constraints, and external interactions, and concepts used in analytical mechanics as related to the virial theorem. Connections of the quantum virial theorem to the Hellmann-Feynman theorem, force concept, partitioning and boundary conditions are mentioned briefly. In Section 2 the virial theorem is extended to periodic wave functions of the Bloch type. A corrective term, arising from the surface integral, takes account of the values of the wave function and its derivatives at the boundary of the integration space, values which are not null in the case of Bloch wave functions. The result is applied to solid state in the case of the monoelectronic approximation.

Hypervirial theorems for atomic bound and scattering states

The quantum-mechanical hypervirial (HV) theorems for atomic bound and scattering states are expressed as the expectation values of the commutator [W, H] between a virial operator W and the system Hamiltonian H. They provide various relationships between the average kinetic and potential energies, as well as the scattering phase shifts. Since the virial integral with an approximate wavefunction generally contains the information on the first-order error of the solution, we examine the applicability of the HV theorems not only for the purpose of testing approximate solutions of the Schrödinger equations, but also improving the solution by optimizing the parameters in the trial functions. To illustrate the approach, extensive numerical studies have been carried out for typical two-electron atomic systems in their bound and scattering states. The study is especially warranted for the scattering states, where simple criteria to test the quality of solutions are not readily available. We show that, with judicious choices of W, the hypervirial tests can provide useful checks on the accuracy of approximate solutions, and a way to determine the optimal values for the (nonlinear) parameters in the approximate solutions. Several points of caution in applications of the HV tests are explained.

Calculation of the electric hypershielding at the nuclei of molecules in a strong magnetic field

The third-rank electric hypershielding at the nuclei of 14 small molecules has been evaluated at the Hartree-Fock level of accuracy, by a pointwise procedure for the geometrical derivatives of magnetic susceptibilities and by a straightforward use of its definition within the Rayleigh-Schrodinger perturbation theory. The connection between these two quantities is provided by the Hellmann-Feynman theorem. The magnetically induced hypershielding at the nuclei accounts for distortion of molecular geometry caused by strong magnetic fields and for related changes of magnetic susceptibility. In homonuclear diatomics H(2), N(2), and F(2), a field along the bond direction squeezes the electron cloud toward the center, determining shorter but stronger bond. It is shown that constraints for rotational and translational invariances and hypervirial theorems provide a natural criterion for Hartree-Fock quality of computed nuclear electric hypershielding.

GENERALIZED HYPERVIRIAL THEOREM FOR THE D DIMENSIONAL SINGLE-PARTICLE SYSTEM AND ITS APPLICATIONS

By using the Hamiltonian identity, we present a generalized hypervirial theorem for the D dimensional single-particle system with arbitrary potential. It is shown that this generalized hypervirial theorem is powerful in deriving the Blanchard's and Kramers' recurrence relations among the matrix elements. We apply those recurrence relations to some physical systems, exactly solvable and unsolvable, such as the pseudoharmonic oscillator, the Morse, the modified Pöschl-Teller, the Lennard-Jones, the Buckingham and the Yukawa potentials. The Blanchard's and Kramers' recurrence relations in two dimensions are also briefly mentioned.

Two-tier formulation of multichannel scattering theory and hypervirial theorems

Complex dynamics of a multichannel scattering may be treated by a variational procedure. But the conventional variational principles are not readily applicable because of (i) the intrinsic difficulties of unstable fluctuations in the calculated amplitudes as functions of nonlinear parameters and (ii) lack of criteria to optimize the solutions. A two-tier theory is formulated in which the complex dynamical mixing and the asymptotic channel sector are treated separately, but as a coupled system, such that the instability problem (i) is resolved naturally in a mathematically consistent way, even when most of the weakly coupled open channels are neglected. The resulting solution is stable, but not necessarily optimal. Modified forms of hypervirial theorems are introduced to optimize the solutions, thus rectifying the shortcoming (ii). Thus, the reformulated theory for the scattering states, coupled with a properly chosen hypervirial theorem, can be applied effectively to many-body, multichannel scattering systems.

Quantum hypervirial theorems

The quantum mechanical hypervirial theorems (HVT) are given by the expectation values of a commutator between the virial operator W and the system Hamiltonian H. We propose application of the HVT to testing and improving approximate solutions of the Schroedinger equations. This is especially relevant for scattering states, where simple testing criteria are not readily available. The HVT, with judicious choices for W, can provide a criterion to test the accuracy of approximate solutions, both for the bound excited states and scattering states, and also ways to determine an optimal set of parameter values as the wave function improves.

Partial hypervirial theorems for atomic-molecular systems

A family of partial hypervirial theorems for physical properties of pairs of particles in three-and four-particle atomic-molecular systems is considered. The partial hypervirial theorems generalize the partial virial theorems proposed earlier. The sum rule is formulated, according to which the expectation values of the derivatives with respect to the interparticle distances are related to the products of the charges of pairs of the particles. This rule is used to check the accuracy of the calculation of variational wave functions for the positronium ion e − e − e +. It is shown that the sum rule is two orders of magnitude more sensitive to the inaccuracy of the calculation of the wave function than the partial virial theorems and is five or six orders of magnitude more sensitive to it than the conventional virial theorem.

Singular integrals and their application to a hypervirial theorem

Hypervirial theorems provide relationships exact electronic wave functions must satisfy, and the extent to which this is the case is a measure (additional to the energy) of wave function quality. The hypervirial relation known as the Vinti equation has been proposed for this purpose, but its application has been hampered by the absence of analytical formulas for the singular integrals occurring therein. The authors' methods for singular integrals arising in atomic computations [J. Chem. Phys. 121, 6323 (2004)] resolve this bottleneck; quality assessments based on the Vinti equation are provided here for a number of wave functions of varying complexity describing the He isoelectronic series (from ${\mathrm{H}}^{\ensuremath{-}}$ through ${\mathrm{Ne}}^{8+}$).

Temperatures: old, new and middle aged

This study explores the various ways in which a ‘temperature’ can be introduced and calculated in a molecular simulation. We focus on the relatively recent formula of Rugh, and a simplified version called the ‘configurational’ temperature, T config, which became popular after Rugh's work. We derive formulae for these various ‘temperatures’ for the special case of the soft–sphere fluid, with the pair interaction, , where ε and σ set the energy and length scales, respectively, and for different n values. We show with a number of simple test cases that the various prescriptions of temperature give the same result as the thermodynamic temperature at equilibrium. Although much of the current work uses the configurational temperature, Rugh's temperature, T R, has the advantage that it has the thermodynamic value in circumstances where the configurational temperature is not defined (such as in the limit of the ideal gas, where Rugh's temperature becomes the kinetic temperature). The configurational temperature is a particular example of the hypervirial theorem. Using the particle forces and their gradients, we calculate the first and second hypervirial temperatures, which we find to be equal, even though the force moments (i.e. ) and their derivatives themselves do not conform to a Gaussian distribution. Various moment ratios were shown to have a linear dependence on the reciprocal of the density, at not too high densities. We explored the behaviour of the kinetic and configurational temperatures for some non–equilibrium systems, where quite different trends were observed in these two quantities in any transient or non–stationary stage. The kinetic temperature was found to respond more rapidly to perturbations than the configurational temperature, as might be expected since usually energy is more rapidly transported than matter.

Configurational temperatures and interactions in charge-stabilized colloid

A system's temperature can be expressed in terms of its constituents' instantaneous positions rather than their momenta. Such configurational temperature definitions offer substantial benefits for experimental studies of soft condensed matter systems, most notably their applicability to overdamped systems whose instantaneous momenta may not be accessible. We demonstrate that the configurational temperature formalism can be derived from the classical hypervirial theorem, and introduce a hierarchy of hyperconfigurational temperature definitions, which are particularly well suited for experimental studies. We then use these analytical tools to probe the electrostatic interactions in monolayers of charge-stabilized colloidal spheres confined by parallel glass surfaces. The configurational and hyperconfigurational temperatures, together with a thermodynamic sum rule, provide previously lacking self-consistency tests for interaction measurements based on digital video microscopy, and thereby cast light on controversial reports of confinement-induced like-charge attractions. We further introduce a method to determine unknown parameters in a model potential by using consistency of the configurational and hyperconfigurational temperatures as a set of constraints. This approach, in principle, also should provide the basis for a model-free estimation of the pair potential.

The HVT technique and the uncertainty relation for central potentials

The quantum mechanical hypervirial theorems (HVT) technique is used to treat the so-called ‘uncertainty’ relation for quite a general class of central potential wells, including the (reduced) Poeschl–Teller and the Gaussian one. It is shown that this technique is quite suitable in deriving an approximate analytic expression in the form of a truncated power series expansion for .

Exponential representation in the Coulomb three-body problem

The exponential representation in the Coulomb three-body problem is considered. It is shown that the exponential variational expansion in relative coordinates r32, r31 and r21 has a number of advantages for the bound state calculations in Coulomb three-body systems. Moreover, it appears that the exponential (or Laplace–Fourier) representation of the Coulomb three-body problem is an optimal approach to analyse and solve various three-body problems. The optimization of nonlinear parameters in the trial wavefunctions is also considered. The developed methods are used to determine the highly accurate ground 11S(L = 0)-state energies and other bound state properties for a number of He-like two-electron ions (Li+, Be2+, B3+, C4+, N5+, O6+, F7+ and Ne8+). To represent the ground state energies of these He-like ions we apply the Z−1 expansion. The asymptotic form of the ground state wavefunctions at large electron–nuclear distances for the He-like ions is briefly discussed. Considered hypervirial theorems are of great interest for these ions, since they allow one to obtain some useful relations between different expectation values. The generalization of the exponential variational expansion in relative coordinates to the four-body non-relativistic systems is also considered.

Expectation values ofrqbetween Dirac and quasirelativistic wave functions in the quantum-defect approximation

A search is conducted for the determination of expectation values of r{sup q} between Dirac and quasirelativistic radial wave functions in the quantum-defect approximation. The phenomenological and supersymmetry-inspired quantum-defect models, which have proven so far to yield accurate results, are used. The recursive structure of formulas derived on the basis of the hypervirial theorem enables us to develop explicit relations for arbitrary values of q. Detailed numerical calculations concerning alkali-metal-like ions of the Li, Na, and Cu isoelectronic sequences confirm the superiority of supersymmetry-based quantum-defect theory over quantum-defect orbital and exact orbital quantum number approximations. It is also shown that relativistic rather than quasirelativistic treatment may be used for consistent inclusion of relativistic effects.

Study on Application of Hypervirial Theorem in the Variational Method**The project supported by National Natural Science Foundation of China

We discuss a methodology problem which is crucially important for solving the Schödinger equation in terms of the variational method. We present a complete analysis on the application of the hypervirial theorem for judging the quality of the trial wavefunction without invoking the precise solutions.

The HVT technique and the treatment of two basic inequalities

The quantum mechanical hypervirial theorems (HVT) technique is used in treating two basic inequalities relating the ground-state mean square radius of the orbit of a particle in a central potential and its kinetic energy, respectively, to the spacing of the two lowest energy levels ΔE. For quite a wide class of those potentials, the parameters of which also lead to a sufficiently small dimensionless quantity s, the difference between the two sides of each inequality is of order s3 and higher (while ΔE is of order s and higher), and thus it is expected in general to be quite small.

Simple algebraic recursion formulas for calculation of propagation characteristics of graded-index waveguides

Using the perturbation method with the aid of the hypervirial theorem and Hellmann–Feynman theorem in quantum mechanics, simple algebraic recursion formulas for calculating propagation constants of graded-index planar optical waveguides in successive orders of approximation are derived. Algebraic formulas for calculation of the modal field functions are also presented. Numerical results for typical graded-index waveguides (Gaussian, exponential, complementary error function and symmetric Epstein profiles) demonstrate that the present method can be applied to an accurate calculation of the propagation characteristics to an arbitrary order of perturbation approximation.

Fundamental properties of parametric functionals in quantum chemistry

A theoretical interpretation and justification of the basic approaches proposed in parametrical methods is presented in this paper. The idea of providing a fundamental understanding of these methods can be considered as a guide for further development in this area. The application of functional analysis concepts to understanding the nature of parametric functionals in terms of the Riesz and Weiestrass's theorems is discussed. In addition, applications of virial and hypervirial theorems to constrain the type of functional and the parameters used in these methods is addressed. A justification of the use of a single determinantal wavefunction and the nature and conditions of optimized orbital basis in terms of different electronic environments involved in different interactions was also established.

Interplay between shell effects and electron correlations in quantum dots

We use the Path Integral Monte Carlo method to investigate the interplay between shell effects and electron correlations in single quantum dots with up to 12 electrons. By use of an energy estimator based on the hypervirial theorem of Hirschfelder we study the energy contributions of different interaction terms in detail. We discuss under which conditions the total spin of the electrons is given by Hund's rule, and the temperature dependence of the crystallization effects.

1/Nexpansions for central potentials revisited in the light of hypervirial and Hellmann-Feynman theorems and the principle of minimal sensitivity

The hypervirial and Hellmann-Feynman theorems are used in the methods of 1/N expansion to construct Rayleigh-Schrodinger perturbation expansion for bound-state energy eigenvalues of spherical symmetric po- tentials. An iteration procedure of calculating correction terms of arbitrarily high orders is obtained for any kind of 1/N expansion. The recurrence formulas for three variants of the 1/N expansion are considered in this work, namely, the 1/n expansion and the shifted and unshifted 1/N expansions which are applied to the Gaussian and Patil potentials. As a result, their credibility could be reliably judged when account is taken of high-order terms of the eigenenergies. It is also found that there is a distinct advantage in using the shifted 1/ N expansion over the two other versions. However, the shifted 1/N expansion diverges for s states and in certain cases is not applicable as far as complicated potentials are concerned. In an effort to solve these problems we have incorporated the principle of minimal sensitivity in the shifted 1/ N expansion as a first step toward extending the scope of applicability of that technique, and then we have tested the obtained approach to some unfavorable cases of the Patil and Hellmann potentials. The agreement between our numerical calculations and reference data is quite satisfactory.

Mass number dependence in Λ–hypernuclei by means of the Hypervirial Theorems

The Hypervirial Theorems are applied to a wide class of single particle nuclear potentials, in order to study the mass number dependence of various quantities in Lambda hypernuclei. A very efficient model is assumed for the radius parameter of the potentials and the limitations of the whole method are discussed.