H. H. Michels

Molecular Orbital Studies of the Ground and Low‐Lying Excited States of the HeH+ Molecular Ion

The molecular ion HeH+ has been examined using a method of molecular orbital calculation in which the total wavefunction is taken as a linear combination of configurations, each a properly antisymmetrized product of flexible, elliptic functions. We have examined the ground 1Σ state and 14 excited states of 1Σ, 3Σ, 1Π, and 3Π symmetry. Including the ground state, there are a total of seven bound molecular states for this system, all arising from atomic states with principal quantum number of 2 or less. All higher molecular states are indicated to be repulsive. Contrary to previous investigations, we find that the first excited 1Σ state is weakly bound and can support several vibrational levels. An extensive analysis of the final optimized wavefunctions has been made and, in addition to the potential curves, we have evaluated spectroscopic constants, values of the radiative transition moments, and certain other molecular quantities. An analysis of experimental studies which require a detailed knowledge of t...

Multicenter Integrals in Quantum Mechanics. I. Expansion of Slater‐Type Orbitals about a New Origin

Convenient and accurate formulas have been found which implement the spherical‐harmonic expansion of a Slater‐type orbital about a point displaced from the orbitals center. These formulas include, as a special case, the expansion of spherically symmetric orbitals which has been reported previously. The results are presented in algebraic form, suitable for machine computation, and have been tested for the transformation of any Slater‐type orbital with n≤7, l≤3, | m |≤l to an arbitrary space‐fixed point.By the use of this analysis, in conjunction with the properties of the rotation group, it is now possible to evaluate directly, the general multicenter electron‐repulsion integral expressed in terms of Slater‐type orbitals. The convenience of this direct approach suggests that previous analyses in terms of Gaussian orbitals or Gaussian approximations to Slater‐type orbitals, replete with their poor radial dependence, may no longer offer any computational advantages.

Molecular Orbital Studies of Excited States of HeH

The total energy of HeH was computed using a method of molecular orbital calculation in which the total wavefunction is taken as a linear combination of configurations, each a properly antisymmetrized product of flexible one‐electron functions in an elliptic coordinate system. We have examined the lowest 2Π state of this system and both of the 2Σ excited states which arise from the interaction of ground‐state He and a 2p or 2s H atom. The lowest 2Π state of HeH and the first excited 2Σ state are found to be bonding with calculated dissociation energies of 1.935 and 2.336 eV, respectively. A comparison of these results with those previously reported for the ground state of HeH and the molecular ion HeH+ indicates that the charge distribution about the helium atom is similar in all of these systems and it is predicted that, with the single exception of the ground state, all of the doublet states of HeH up to the ionization limit of the H atom are bonding and can be represented by very similar potential ener...

Variational Approach to Perturbation Theory. I. Application to the Calculation of Atomic Polarizabilities

A variational approach to perturbation theory is derived which establishes the relationship between variational and conventional approaches that utilize expansions in terms of atomic or molecular eigenfunctions. This approach is based upon the observation that the variational approach to perturbation theory is formally equivalent to a variational method for constructing a special set of excited‐state wavefunctions. A mathematical demonstration of this equivalence has been formulated. With the use of this new approach to perturbation theory, we have calculated both the static and dynamic polarizabilities of the two‐electron helium isoelectronic series, H−, He, and Li+ and a four‐electron system represented by the beryllium atom. Multiconfiguration open‐shell wavefunctions were employed. These flexible wavefunctions permit a quantitative analysis of the near‐degenerate s—p orbital mixing in the ground state of the beryllium atom. The inclusion of this orbital degeneracy effect results in a substantial reduc...

Multicenter Integrals in Quantum Mechanics. II. Evaluation of Electron‐Repulsion Integrals for Slater‐Type Orbitals

A unified method for the evaluation of electron‐repulsion integrals for Slater‐type orbitals is described. This method applies to every electron‐repulsion integral needed for polyatomic molecule calculations, and is sufficiently general to handle integrals over orbitals with higher quantum numbers, located at arbitrary spatial points with arbitrary relative orientations.Computation programs have been prepared to implement this analysis with emphasis on both the requirements of speed and accuracy in evaluating the large number of electron‐repulsion integrals required for a polyatomic calculation. Comparisons with other methods are given which indicate that this unified approach can result in at least an order‐of‐magnitude saving in computation time for large molecular systems.

Configuration interaction studies of the HeH+ molecular ion. I Singlet sigma states

The method of superposition of configurations is applied to the singlet sigma states of HeH+ which correlate to the separated atom states of principal quantum number less than or equal to 3. The calculations are carried out for internuclear separations in the ranges 0(.1)34.5, 35.0(.5)50.0 bohr. Energy eigenvalues and the expectation values of the electron coordinate along the internuclear line are discussed in this paper. Dipole transition matrix elements and radial coupling matrix elements are presented in a companion paper. The quality of the calculations is exhibited by comparisons of the length and velocity forms of the dipole matrix elements, by the Hellmann‐Feynman theorem, and by the approach to atomic properties at large internuclear separations. Supplementary calculations of higher quality wavefunctions for the lowest states are also reported. These are used to help estimate the extent to which the results of the main calculations have converged toward the exact values. Comparisons with other wo...

Quantum‐Mechanical Studies of Several Helium—Lithium Interaction Potentials

The total energies for a number of states of HeLi were computed using a method of molecular orbital calculation in which the total wavefunction is taken as a linear combination of spatial configurations, each a properly antisymmetrized product of flexible one‐electron functions in an elliptic coordinate system. Calculations were performed for the 2Σ ground state and a number of excited 2Σ, 2Π, 4Σ, and 4Π states of HeLi. The 1Σ ground state of the molecular ion HeLi+ was also examined to facilitate an evaluation of the neutral excited states. The lowest 2Π and 4Π states as well as two highly excited 2Σ states were found to be weakly bonding with dissociation energies of less than 1 eV. The importance of configuration interaction upon the accuracy of the calculations was examined and a method of obtaining approximate energies for highly excited states was developed. On the basis of these calculations, it is concluded that the weak bonding evidenced by some of the states of this system is due largely to the ...

Configuration interaction studies of HeH+ molecular ion. II Dipole and radial coupling matrix elements for the singlet sigma states

Configuration interaction calculations of the dipole, gradient, and radial coupling matrix elements among the lowest 10–12 1Σ states of HeH+ are reported for internuclear separations in the range 0≤R≤50 a.u. The forms 〈n|∂/∂R|m〉 and (Em − En)−1 〈n|∂V/∂R|m〉 of the radial coupling element are compared. The results are compared with those of other investigators. The accurate evaluation of these matrix elements in the present work demonstrates the feasibility of a quantitative ab initio treatment of low‐energy collisions between light atoms.

Elastic Scattering, Diffusion, and Excitation Transfer of Metastable Helium in Helium

The dynamics of the resonance collision of metastable and normal helium atoms has been examined and calculations have been performed for elastic scattering, diffusion, and excitation transfer. To facilitate this analysis, accurate ab initio calculations of the pertinent interaction potentials have been carried out. The use of these more accurate potential curves in the framework of the original dynamical formulation of the collision process by Buckingham and Dalgarno leads to serious discrepancies between calculations and experiment for the elastic scattering cross section and the coefficient of diffusion. The rate constant for the excitation transfer reaction, however, is in reasonable agreement with experiment, over a wide range of temperatures.

Configuration interaction studies of the HeH+ molecular ion. IV. The triplet sigma, pi, and delta states

The method of superposition of configurations was applied to the triplet sigma, pi, and delta states of HeH+ which correlate to the separated atom states of principal quantum number less than or equal to 3. The calculations were done for internuclear separations, 0⩽R⩽65.5 a.u., on a mesh adequate for interpolation. Similar calculations on the singlet states have already been reported. The present calculations complete the accurate evaluation of the potential energy curves for this system which are required for low‐ and intermediate‐energy collision studies. In addition to the energy eigenvalues and eigenfunctions, dipole, gradient, and radial coupling matrix elements were calculated for the sigma and pi states. Primarily, this paper presents information on the eigenvalues. The accuracy of the triplet‐state calculations is comparable to that obtained for the singlet states. The similarities and differences in the pattern of avoided crossings for the triplet and singlet states are exhibited. These are mainly determined by the differences in the triplet and singlet energy‐level schemes of the united and separated atoms.