Integral Hellmann-Feynman theorem as a criterion of the quality of wave functions of atomic-molecular systems

It is proposed to use the integral Hellmann-Feynman theorem for the quality control and refinement of atomic and molecular wave functions. Its validity is verified for variational wave functions of the positronium ion e ⁺ e ^? e ^?, the negative ion of the hydrogen atom p _∞ ⁺ e ^? e ^? (H^?), and mesomolecules μ^?π⁺π⁺ and μ^? p ⁺ p ⁺. The relative violation of this theorem (10^?2) is six orders of magnitude larger than the error of the energy calculation (10^?8), which demonstrates its high sensitivity to the quality of wave functions. A way of refining wave functions on the basis of combination of the integral Hellmann-Feynman theorem for exactly solvable model and real atomic-molecular systems is proposed. A rule for verification of the mutual consistency of the wave functions of any three quantum-mechanical systems is formulated.     

Generalization of the Hellmann-Feynman theorem

The well-known Hellmann-Feynman theorem of quantum mechanics connected with the derivative of the eigenvalues with respect to a parameter upon which the Hamiltonian depends, is generalized to include cases in which the domain of definition of the Hamiltonian of the system also depends on that parameter.     

Calculation of the Peierls-Yoccoz translational mass for Hartree-Fock wave functions

The Peierls-Yoccoz projection method for translation is applied to Hartree-Fock wave functions constructed on a mesh for ${}^4\text{He}$ and ${}^{16}\text{O}$. As expected, the mass parameters differ from the nuclear masses, indicating a breaking of galilean invariance.     

Lithium crystal properties from high-quality Hartree-Fock wave functions

Properties derived from lithium b.c.c. crystal Hartree-Fock (HF) calculations, using LCAO-type Bloch functions and including 2s, 2p and 3d Slater-type basis orbitals, are presented. These include total energies, lattice parameters, band gaps, widths of the occupied part of valence band, bulk modulus and Fermi surface geometry. Energetic quantities are in qualitative agreement with our earlier work. Their disagreements with experiment again underscore some inadequacies of HF. The bulk modulus is overestimated by about 50%, which is typical for HF-based molecular force constants. The Fermi surface is calculated to have a maximum distortion (in the 〈110〉 direction) of about 3.5%. This is in agreement with the best experiments. The sensitivity of predicted distortions to Brillouin zone integration points is discussed.     

Can the SCA with relativistic Hartree-Fock wave functions describe the high energy ionisation probability

Recently experimental data of the L-shell ionisation probability of Au in the impact parameter region around the L-shell radius have been reported with α-particle energies in the range of 12–50 MeV. A very large discrepancy growing with increasing energy was found between the experimental and theoretical values. Refined calculations using the semiclassical approximation (SCA) with hyperbolic projectile trajectories and relativistic Hartree-Fock electron wave functions show, that this discrepancy can still be explained unsufficiently.     

Numerical calculation of localized (Wannier) functions from ab-initio Hartree-Fock wave functions

The Fourier transformation necessary to obtain localized (Wannier) functions from delocalized ab-initio crystal Hartree-Fock (Bloch) functions is carried out for the one-dimensional models of solids. Combinations of Wannier functions from different bands lead to even more localized functions in close analogy to localization in molecules.     

Quasi-free electron scattering from 12C using Hartree-Fock wave functions

The energy spectrum and the angular correlation distributions for the reaction ¹²C(e, e'p)¹¹B are calculated within the framework of DWBA using spherical Hartree-Fock wave functions. A comparison with experiment and some of the earlier calculations is made. It is observed that a large spatial extent of the wave functions significantly improves the agreement.     

Liapunov theorem for modular functions

We extend Liapunov Theorem to modular functions on complemented lattices.     

Statistical properties of the Hamiltonian generating phase state derived by using the generalized Hellmann-Feynman theorem

We study statistical properties of the Hamiltonian generating phase state. Using the generalized Hellmann-Feynman theorem for ensemble average, we derive its mean energy and find the ratio of the mean energies contributed from the term a^†a to that from . The relation on the entropy-variation with respect to the dynamic parameters ω and κ is also examined.     

Application of the Hellmann-Feynman theorem to the study of substitutional impurities in a linear chain of atoms

The Hellmann-Feynman theorem is applied to calculate the effects produced by substitutional impurities in a linear chain of atoms. In particular, within the tight binding Hamiltonian, we present exact results for the change in energy of the system caused by the impurity.     

Flow patterns of rotating time-dependent Hartree-Fock wave packets

A soluble model (Elliott's model in two dimensions) is used to study how well flow patterns and features of rotational motion are represented by the time-dependent Hartree-Fock approximation. Due to the spreading of the wave packet in the exact Schrödinger time-evolution, the agreement is good only for phenomena which probe the current during a short time interval.     

A theorem for the distribution functions of hard-sphere fluids

The thermal diffusion factor α_T for the gas mixtures N₂-CO₂, Ar-CO and Ar-NO has been measured as functions of temperature and composition by the two-bulb method. As at least one of the components of the mixture is diatomic or triatomic the thermal diffusion factor is likely to be influenced by inelastic collision. The α_T data have been analysed in terms of the available elastic and inelastic theories. The results show the comparatively small effect of inelastic collision on thermal diffusion. The necessity of using more realistic intermolecular potentials to interpret α_T data has been pointed out.     

Taylor's theorem for analytic functions of operators

We discuss analytic functions on a Banach algebra into itself. In particular expressions for derivatives are given as well as convergent Taylor expansions.     

A unified optical theorem for scalar and vectorial wave fields

The generalized optical theorem is an integral relation for the angle-dependent scattering amplitude of an inhomogeneous scattering object embedded in a homogeneous background. It has been derived separately for several scalar and vectorial wave phenomena. Here a unified optical theorem is derived that encompasses the separate versions for scalar and vectorial waves. Moreover, this unified theorem also holds for scattering by anisotropic elastic and piezoelectric scatterers as well as bianisotropic (non-reciprocal) EM scatterers.     

An Abelian and Tauberian theorem for the stieltjes transform of generalized functions

Using the technique of quasiasymptotics, we describe the asymptotic relations for the Stieltjes transform of generalized functions. The Tauberian condition given here is more general than the assumption by which the Stieltjes orginal is a non-negative measure. Examples are given.Presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.