Herbert W. Jones

Convergence analysis of the addition theorem of Slater orbitals and its application to three-center nuclear attraction integrals

The mathematical foundation of the methods using addition theorems to evaluate multicenter integrals over Slater-type orbitals is actually well understood. However, many numerical aspects of such approaches still require further investigations. In the framework of these methods, multicenter integrals are generally represented by infinite series which under certain circumstances are very slowly convergent. Accordingly, the determination of the convergence type of such series is of great importance since it allows one to choose adequately the convergence accelerator to be used in the summation procedure. In this work, the convergence of the two-range addition theorem proposed by Barnett and Coulson [Philos. Trans. R. Soc. London, Ser. A 243, 221 (1951)] is analyzed. The results obtained from this study are then applied to study the convergence of three-center nuclear integrals, and most importantly, to discuss the choice of the convergence accelerator to be used in the summation procedure.

The löwdin α function and its application to the multi-center molecular integral problem over slater-type orbitals

Abstract In this paper we trace the evolution of the Lowdin α-function method in its application to multicenter molecular integrals over Slater-type orbitais (STOs). As is well-known, any STO displaced from the origin can be expanded in an infinite series of spherical harmonics; the functional coefficients have been designated as Lowdin α functions. These a functions can be represented as exponentials multiplied by polynomials in the displacement distance and the radial distance. The polynomials are used to construct a C matrix with integer elements. To avoid cancellation errors in some cases, the exponentials are expanded to obtain E matrices for interior regions and F matrices for exterior regions. We believe that this careful approach to molecular integrals will succeed in producing accurate and rapid evaluation of the integrals needed in STO basis-set methods or quantum chemistry.

Eigenvalues to arbitrary precision for one‐dimensional Schrödinger equations by the shooting method using integer arithmetic

It would seem that limiting computer computations to numbers with a fixed number of decimal digits would inhibit flexibility. Software programs such as Mathematica permit numerical algebra to be done exactly in terms of the ratio of integers. Hence, a single Taylor series representation of a function can span the entire range needed for a corresponding independent variable. We decided to find eigenvalues to 14 decimal digits by solving one-dimensional Schrodinger equations by the “shooting method” by employing a single Taylor series in each case. With more terms in the series, higher accuracy may be obtained by evaluations at larger asymptotic values. The problems solved were the 1s, 2s, and 2p hydrogen atom, the harmonic oscillator, the quartic potential, and the double-well potential. Noteworthy is the use of the asymptotic condition for the derivative of the eigenfunction as well as its value; this permits the determination of a lower and upper bound for the eigenvalues. The eigenfunctions determined are continuous rather than evaluated only over a grid, thus permitting easy and accurate evaluations of matrix elements by Gaussian quadrature. Also, theoretically accurate normalization constants are found for the eigenfunctions.

The Philosophy and Strategy for STO Integral Evaluation Using the C-Matrix Algebraic Single-Center Expansion Method

Sufficient work has now been done to establish the viability of the algebraic C-matrix single-center expansion method for the evaluation of integrals over Slater-type orbitals. Explicit formulas for two-center overlap1,2, Coulomb3, and hybrid4 integrals have been automatically generated by computer using this method. Accurate numerical evaluation5 of these formulas for all values of parameters can be achieved by machine manipulation of these formulas into appropriate Taylor series. The policy of carrying out all algebraic manipulations by computer and using different formulas for different ranges of parameters re-establishes the use of formulas for integral evaluation after the abandonment of this approach many years ago.

Orbit generation by use of Taylor's expansion

Feynman (I) presents a method of tracing the orbit of a particle in a field of force. He makes each step of the orbit by calculating an average velocity for the particle based on its initial velocity and the acceleration it is subjected to over a finite length of time, At, and calculating the new location of the particle, assuming it moves in a straight line. Since the acceleration of the particle is known as a function of position, it is possible to use Taylors expansion to get a higher order approximation. The question arises as to what is the most efficient way to plot an orbit over a cycle: should we use a few steps and a higher order of approximation, or many steps with the lowest order of approximation. It must be realized that the higher the order of approximation the more CPU time is required per step, but the steps may be larger to achieve the same degree of accuracy in the complete orbit; similarly, the more steps used, the more CPU is required. Several orders of approximations were used on Feynmans problem to decide which order leads to the least CPU time for equal accuracy in the determination of the orbits period. It was found that~the third order approximation had almost twice the efficiency of any order from