Another criterion for gaussian approximations of slater-type orbitals

An alternative criterion, which enforces gaussian expansions of Slater-type atomic orbitals to have similar shortrange and long-range suitabilities, is used. Such an approximation is deduced from weighted expansions recently proposed by Ehrenson in this journal. These new expansions should clarify scaling in molecular calculations.     

Computer-generated formulas for overlap integrals of slater-type orbitals

Using a modified form of Sharma's method for the expansion of a Slater-type orbital in spherical harmonics about a displaced center, a general expression for the overlap integral between two orbitals is derived that is equivalent to that given by Sharma. By use of a simple kind of “computer algebra” this expression is developed here into a formula that is equivalent to earlier known ones and may have computational advantages when the generated formula coefficients are stored. This method is capable of extension to more complicated integrals.     

Calculating overlap intergals with slater-type orbitals in the molecular coordinate system

Kirov Azerbaidzhan University. Translated from Zhurnal Strukturnoi Khimii, Vol. 28, No. 5, pp. 148–149, September–October, 1987.     

Molecular orbital calculations based on linear combinations of fragment orbitals

A semiempirical MO method based on localized fragment orbitals has been developed, which is particularly suited for the construction of orbital correlation diagrams for the discussion of the electronic structure of complex molecules in terms of fragments and their interactions. The method allows for the inclusion of experimental ionization potentials and electron affinities of the fragments within the calculation of the Fock matrix elements and may thus form the basis of an interpretation of photoelectron spectra, comparable to the interpretation of UV spectra by means of the MIM method of Longuet-Higgins and Murrell. Several levels of approximation are discussed using the acrolein molecule as an example.     

Unified analytical evaluation of the two-center coulomb and hybrid integrals over slater-type orbitals

Two-center Coulomb and hybrid integrals are combined into a single expression by use of the formulas given by the author for the expansion of real and complex STO 's about a point displaced from the orbital center. The final results are expressed in terms of the overlap integrals between STO 's. Analytical formulas for the evaluation of these integrals have been established recently by the author.     

Evaluation of two-center overlap and coulomb integrals derived from slater-type orbitals

The strategy for the evaluation of two-center overlap and Coulomb integrals is illustrated by using computer-generated formulas produced from 1s orbitals. There is no loss in generality as these formulas have the same structure for higher quantum numbers. For small parameter values, singularities may be removed from the formulas by expanding exponential functions and collecting coefficients of like powers by machine. Suitable terminated expansions, along with exact formulas, permit high accuracy throughout the entire possible range of parameter values.     

New expressions for the overlap integral of two linear harmonic oscillator wavefunctions

Three new compact analytical expressions for the overlap integral of the wavefunctions of two distorted and displaced linear harmonic oscillators have been otained: unlike Ansbacher's formula, the do not lead to divergence when there is no distortion.     

Evaluation of fourier transform of two-center charge distribution for arbitrary slater-type orbitals

Fourier transform of two-center charge distributions corresponding to arbitrary Slater-type orbitals are evaluated by a Gaussian quadrature procedure without any preliminary series expansion of the integrand. Convergence and accuracy of the method are discussed and illustrated.     

Computation of derivatives for parameter optimization in least-squares fitting of linear combinations of Slater-type orbitals by Gaussians

The approximation of a linear combination of Slater-type orbitals in terms of Gaussian functions is a many-parameter optimization problem. Formulas for computation of the gradient of the overlap in parameter space are reported. An alternative method of computing the gradient is described, which is of general applicability. This technique permits the exact evaluation of a derivative, without derivation and programming of its analytic expression.     

Application of modified Neumann expansion for analytical evaluation of two-center,two-electron integrals over slater-type orbitals

A modified form of the Neumann expansion in terms of products of orthogonal polynomials for the inverse interelectronic distance r¹₁₂ is proposed. This expansion has been applied in order to derive a unified analytical formula for two-center and two-electron integrals over Slater-type orbitals. The results are equivalent to those given recently by Yasui and Saika, but the expansion itself can be used for building up a realistic algorithm for evaluation of three- and four-electron integrals determined by using correlated variational wave functions.     

On the dynamics of a linear and a nonlinear quantum oscillator with randomly changing harmonic frequency

Numerical experiments with a nonlinear (λχ⁴) oscillator which has its harmonic frequency changing randomly with time reveal certain interesting features of its dynamics of quantum evolution. When λ = 0, the level populations are seen to oscillate. But, as the nonlinear coupling is switched on (λ > 0), a threshold is reached at λ = λ_c when the evolution is seen to be characterized by an abrupt transition dominantly to the highest available state of the unperturbed (initial) oscillator. It is shown that this transition probability is maximized at a particular value of λ. The time threshold for this transition decreases with increasing nonlinear coupling strength. The numerically obtained structures of the underlying quantum-phase spaces of the linear and nonlinear random oscillators are examined. Possible use of these results in a problem of chemical origin is explored. © 1997 John Wiley & Sons, Inc.     

Surface reaction dynamics and the harmonic oscillator

One of the fundamental steps in chemical reaction dynamics involves breaking reactant bonds. This is facilitated by placement of energy into the vibrational degrees of freedom associated with the bond. Here we present a model for vibrational excitation in molecule-surface collisions in which the equilibrium geometry of the (diatomic) molecule varies with distance from the surface. The special feature of this model is that the potential energy surfaces for bound nuclear motion are constructed from quadratic potentials, thus enabling analytic solutions. Comparisons are made between exact results obtained from a purely classical trajectory model and various hybrid models in which the internal vibrational modes are treated quantum mechanically in the harmonic limit.     

The magnus expansion for the damped harmonic oscillator

It is shown that he time-evolution operator for the damped harmonic oscillator can be written as the exponential of an anti-Hermitean operator provided t is close enough to zero. The range of applicability of the Magnus expansion is discussed.     

General harmonic oscillator integrals by the operator method

The operator technique with a minimum of commutator algebra is employed to calculate matrix elements of any number of operators between distorted, displaced harmonic oscillator wavefunctions. The results are valid for multidimensional integrals, and regardless of the extent of the Duschinsky effect. General recursion relations useful in machine calculations are given. The formalism is illustrated for the well-known one-dimensional Franck–Condon integrals.