Michele Battezzati

Modulated perturbation theory for molecular interactions. III. Variational calculations of the second‐order energy for the ground state of H2+

Hylleraas variational calculations of the second‐order energy for the ground state of H2+ have been performed in the modified form of Murrell–Shaw–Musher–Amos (MS–MA) perturbation theory including exchange described in Paper I. Simple first‐order trial functions consisting in a single optimized Slater function for each multipole are found to be effective in representing nonexpanded multipole components of the interaction energy. The modulated form of the theory includes a large part of the spherical component of the polarization function already in first order, leaving dipole polarization as the dominant second‐order contribution.

Modulated perturbation theory for molecular interactions

Modulated Murrell-Shaw-Musher-Amos perturbation theory is used in second order to calculate the interaction energy between two ground-state hydrogen atoms in the region of intermediate internuclear separations from 0 to 6 bohr. Using a small set of variationally optimized two-electron functions accounting for the majority of induction and for spherical and dipole contributions to dispersion, fairly accurate results are obtained for the interaction over the whole range of distances including the united atom. A marked reduction in the induction energy is observed in short range when second-order exchange is introduced.

The calculation of London coefficients by the variation-perturbation method

Abstract The London coefficients for the dispersion interaction between LiLi, BeBe and LiBe are calculated by the variation-perturbation method using the whole atomic hamiltonian as H 0 and the Hartree-Fock approximation for the upper turbed wavefunction ϕ 0 . A single excited valence configuration with optimized orbital exponents gives accurate results for Li, whereas comparable accuracy for Be is obtained when part of the valence correlation energy in the ground as well as in the excited state is accounted for through a limited configuration interaction.

A diffusion equation for Brownian motion with arbitrary frictional coefficient: Application to the turnover problem

After a brief re-exposition of the procedure devised by the author in order to reobtain a diffusion equation from the equations of the motion of a mechanical system driven by a random force, this method is applied to derive a third-order diffusion equation for an anharmonic oscillator undergoing Brownian motion. This equation is exact to first-order in the parameter of anharmonicity, and is valid for arbitrary values of the frictional coefficient. The confrontation of this equation with a similar equation obtained previously by asymptotic expansion in inverse powers of the frictional coefficient, shows that although the two equations are different, nevertheless they reduce to the same equation (within the limits of validity of each approximation scheme) when they are both reduced to second order. An asymptotic formula for the mean first-passage time (MFPT) for escaping over a barrier is then proved in the low-temperature limit, which is related to an eigenvalue of the diffusion operator, and to the soluti...

Second-order MS-MA calculations for the triplet state of H2 in the van der Waals region

Abstract Second-order MS-MA calculations by the variation-perturbation method for the triplet state of H 2 show that in the van der Waals region induction including exchange is practically negligible and the depth and position of the potential minimum are determined by the competition between repulsive first-order exchange and attractive Coulomb dispersion. Accurate interaction energies are obtained in second order only if all non-expanded multipoles up to the dipole-octupole term are properly accounted for.

Asymptotic evaluation of the Keesom integral

A simple physically transparent formula is obtained in the asymptotic evaluation of the Keesom integral K(a) for large values of the parameter a. Its derivation is described in some detail and the asymptotic formula is compared with the results obtained from direct three-dimensional numerical integration.

Analytical evaluation of the short-range interaction energy for ground state H+2

This work aims at the extension of the power series in R and ln R of interaction energy k and separation constant A for the system of two protons and one electron in the Born–Oppenheimer approximation. The wave equation is separated into confocal elliptic coordinates, in order to obtain two one-dimensional problems. The approach that we present here is based upon our previous calculations, where a relation between A and k for the (outer) ξ-equation was obtained in two different but substantially equivalent ways: either by an integral equation representation of the solution or by a logarithmic-perturbative expansion, in powers of the energy difference k, from the united atom state. At variance with our previous work, we make use of the (inner) η-equation of a remarkably simple determinantal equation proposed long ago by Hylleraas for which we develop a recursive method of calculation. By combining the two procedures we obtain the solution of the problem by solving for the coefficients of the R and ln R expansion. We calculate in this way coefficients up to O(R7) and show how higher order coefficients may be evaluated recursively.

Short-range interactions in ground state H2+ using fluctuation theory techniques

Abstract Fluctuation theory techniques are used to evaluate short-range interactions in ground state H 2 + by calculating in a direct way small energy differences taken with respect to the ground state of the united atom He + . Appropriate coordinate transformation allows to obtain inner and outer Hermitean eigenvalue equations, which are then solved perturbatively in positive powers of the internuclear distance R choosing He + as unperturbed problem. The separation constants are deduced from the regularity conditions on the functions. Next, the function endowed with the right symmetry is constructed by solving an integral equation in a perturbative way. A second-order approximation gives short-range interaction coefficients which are exact up to R 5 .

Exchange energy in a double-well potential profile from fluctuation theory

This article shows that the asymptotic quantum mechanical exchange energies and wave functions in a bistable potential can be obtained by computing the complex-valued first-passage time across the potential barrier from a transient state endowed with Smoluchowski boundary conditions, whose probability density is concentrated on one side of the barrier. This interpretation is validated by one of the authors’ previous work on diffusion in a random field, showing that this model yields a diffusion equation equivalent to quantum mechanical equations.

Nonexpanded multipoles for induction energy

Abstract A compact analytical formula giving all nonexpanded multipoles for the induction energy between neutral spherical atoms in S-states has been derived following a method previously proposed for the dispersion interaction and assuming a STO representation for the transition amplitudes. The screening of the nuclear charge by the electron cloud is taken into account from the beginning, thereby eliminating the poles in the Fourier-transform of the densities which are responsible for the long-range interactions.