Harry F. King

Evaluation of molecular integrals over Gaussian basis functions

This paper is concerned with the efficient computation of the ubiquitous electron repulsion integral in molecular quantum mechanics. Differences and similarities in organization of existing Gaussian integral programs are discussed, and a new strategy is developed. An analysis based on the theory of orthogonal polynomials yields a general formula for basis functions of arbitrarily high angular momentum. (ηiηj∥ηkηl) = Σα=1,nIx(uα) Iy(uα) I*z(uα) By computing a large block of integrals concurrently, the same I factors may be used for many different integrals. This method is computationally simple and numerically well behaved. It has been incorporated into a new molecular SCF program HONDO. Preliminary tests indicate that it is competitive with existing methods especially for highly angularly dependent functions.

Molecular symmetry. II. Gradient of electronic energy with respect to nuclear coordinates

Symmetry methods employed in the HONDO ab initio polyatomic SCF program are extended to the analytic computation of energy gradients. Validity of the Hellmann–Feynman theorem is not assumed, i.e., all two‐electron contributions to the gradient are included explicitly. The method is geared to the efficient computation of entire blocks of two‐electron integrals. Just one of a set of symmetrically related blocks must be computed. The gradient contribution from each unique block is multiplied by q4, the number of equivalent blocks, and added into a ’’skeleton gradient vector,’’ all other blocks are simply omitted. After processing molecular integrals, the true gradient vector is generated by projecting the symmetric component out of the skeleton vector. The analysis is based on Eqs. (26) and (33) which are valid for many variational wavefunctions including restricted closed shell and unrestricted open shell self‐consistent field functions. We also extend the use of translational symmetry proposed previously b...

Numerical integration using rys polynomials

Abstract We define and discuss the properties of manifolds of polynomials J n ( t , x ) and R n ( t , x ), called Rys polynomials, which are orthonormal with respect to the weighting factor exp(− xt 2 ) on a finite interval of t . Numerical quadrature based on Rys polynomials provides an alternative approach to the computation of integrals commonly encountered in molecular quantum mechanics. This gives rise to a curve fitting problem for the roots and quadrature weights as a function of the x parameter. We have used Chebyshev approximation for small x and an asymptotic expansion for large x . A modified Christoffel-Darboux equation applicable to Rys polynomials is derived and used to obtain alternative formulas for Rys quadrature weight factors.

Theory of spin‐orbit coupling. Application to singlet–triplet interaction in the trimethylene biradical

Efficient methods are developed for the computation of spin‐orbit coupling constants in polyatomic molecules using complete active space multiconfiguration self‐consistent field wave functions. All electron–nuclear and electron–electron spin‐orbit interactions in the Breit–Pauli Hamiltonian are retained without storing or transforming spin‐orbit integrals. This technique is applied to the calculation of spin‐orbit coupling constants between singlet and triplet electronic states. Allowing nonorthogonality of the singlet and triplet molecular orbitals in the active space improves the quality of the wave functions and presents no serious computational difficulties. To test the method, spin‐orbit coupling constants are computed for the diatomic molecules NH, OH+, PH, and O2 and compared with similar calculations reported in the literature. Calculations are also carried out for the organic biradical trimethylene (ĊH2CH2ĊH2). The coupling constant is found to vary from 0 to 2.5 cm−1 depending upon geometry. It ...

Rotational States of a Tetrahedron in a Cubic Crystal Field

The Schrodinger equation is solved for the rotational states of a rigid cube whose center of mass is fixed at a point of symmetry in an external field. This is shown to be equivalent to the equation of motion for a regular tetrahedron in a field with symmetry Oh. The quantum states are classified under the direct product group Ō×O, where Ō is an octahedral group of rotations about body‐fixed axes and O is a similar but distinct group of rotations about space‐fixed axes. The wavefunctions are obtained by an expansion in a series of symmetry‐adapted linear combinations of spherical‐top functions. Projection operators are defined which generate the various linear combinations. Closed‐form expressions are obtained for the matrix elements of the Hamiltonian and of the projection operators. Upper and lower bounds are computed for representative energy levels. The results demonstrate that using basis sets of practical size, high accuracy can be obtained. This analysis forms the basis of a more general theory of ...

Electron Correlation in Closed Shell Systems. I. Perturbation Theory Using Gaussian‐Type Geminals

The calculation of electron correlation energy in closed‐shell atoms and molecules is approached using Rayleigh‐Schroedinger perturbation theory with the symmetric sum of Hartree‐Fock operators for H0. The alleged advantages of using a VN−1 potential are questioned. Variational equations for first‐order pair correlation functions are computed for He, Be, B+ and Ne by expansion in linear combinations of correlated Gaussian‐type geminal basis functions containing r122 in the exponent. Such functions form mathematically complete sets, have convenient symmetry properties, and are integrable in closed form. An extensive search for optimum exponential parameters yielded trial functions for each of the ss orbital pairs giving better than 99% of the limiting second‐order pair energy using only five basis functions per pair. Similar but less thorough studies of sp and pp pairs in neon are also reported. Careful attention is paid to computational accuracy. An upper bound of −0.3428 a.u. is established on the second...

Molecular symmetry. III. Second derivatives of electronic energy with respect to nuclear coordinates

Symmetry methods employed in the ab initio polyatomic program HONDO are extended to the analytic computation of the energy Hessian matrix. A ’’skeleton’’ Hessian matrix is calculated from the unique blocks of electron repulsion integrals. The true Hessian matrix is generated by projecting the symmetric component out of the skeleton Hessian. The analysis is valid for many wave functions, including closed‐ or open‐shell restricted and unrestricted Hartree–Fock wave functions, multiconfiguration Hartree–Fock wave functions, and configuration interaction wave functions. We also extend the use of translational invariance previously used for energy gradient calculations. To illustrate the method, we compare the computer time required for the two‐electron contribution to the Hessian matrix of eclipsed ethane, using Pople’s 6‐31G basis set and D3h symmetry and various subgroups of D3h. Computational times are roughly inversely proportional to the order of the point group.

Analytic computation of energy derivatives - Relationships among partial derivatives of a variationally determined function

This paper considers three functions of several variables, W(r,x), λ(r), and E(r), related by E(r)=W[r,λ(r)] and the condition that W(r,x) be stationary with respect to variations of x when x=λ. Formulas are presented which relate coefficients in the Taylor series expansions of these three functions. We call λ the response function. Partial derivatives of the response function are obtained by solution of a recursive system of linear equations. Solution through order n yields derivatives of E through order 2n+1. This analysis extends Pulay’s demonstration of the applicability of Wigner’s 2n+1 rule to partial derivatives in coupled perturbation theory. A four‐term second derivative formula is shown to be numerically more stable than the usual two‐term formula. We refute previous claims in the literature that energy derivatives are stationary properties of the wave function.

Implementation of a parallel direct SCF algorithm on distributed memory computers

A parallel direct self‐consistent field (SCF) algorithm for distributed memory computers is described. Key features of the algorithm are its ability to achieve a load balance dynamically, its modest memory requirements per processor, and its ability to utilize the full eightfold index permutation symmetry of the two‐electron integrals despite the fact that entire copies of the Fock and density matrices are not present in each processors local memory. The algorithm is scalable and, accordingly, has the potential to function efficiently on hundreds of processors. With the algorithm described here, a calculation employing several thousand basis functions can be carried out on a distributed memory machine with 100 or more processors each with just 4 MBytes of RAM and no disk. The Fock matrix build portion of the algorithm has been implemented on a 16‐node Intel iPSC/2. Results from benchmark calculations are encouraging. The algorithm shows excellent load balance when run on 4, 8, or 16 processors and displays almost ideal speed‐up in going from 4 to 16 processors. Preliminary benchmark calculations have also been carried out on an Intel Paragon.

Some Theorems Concerning Symmetry, Angular Momentum, and Completeness of Atomic Geminals with Explicit r12 Dependence

Several results are presented which have important implications for the calculation of many‐electron wavefunctions. Electron‐pair functions are constructed to be eigenfunctions of the two‐electron inversion, permutation, and angular‐momentum operators with quantum numbers L and M. For a given L and M there exist 2L+1 particular linear combinations of binary products of spherical harmonics called generators. These 2L+1 generators multiplied by members of a complete set of S‐type geminals constitute a complete set of geminal basis functions with that angular momentum. This approach establishes a connection between a formulation in terms of Eulerian angles developed by Wigner and others, and analyses based on configuration interaction expansions. One of several possible forms for the complete set of S‐type geminals employs only 1s‐type orbitals and a correlation factor, R(r12).Mixing of symmetry may occur when a strongly orthogonal geminal is projected out from a symmetry‐adapted one. This problem is easily ...