A Prototype Three Center Integral

A prototype three center exchange integral over Slater orbitals is reduced to an in nite series A closed expression is given for the general term The generalization to the orbitals of arbitrary quantum number is outlined The derivations and the production of this paper were mechanized Introduction The evaluation of the full range of molecular integrals over Slater orbitals ETOs remains an unsolved problem The so called function method showed promise in the early days of computa tional chemistry but convergence problems arose in particular circumstances Gaussian orbitals have dominated the eld since the s see for example Several attempts to solve the ETO problem were reported in With the advent of supercomputers the author restarted the exploration of the function method A way was found to sum the slowly convergent series and it was applied to a prototype Good results were obtained using numerical quadrature to compute the successive terms of the series Now the analytical reduction of these terms has been accomplished using symbolic computation and some general methods have been tested on a set of relatively simple one electron integrals Here we deal with a prototype three center two electron exchange integral as a preliminary to the unrestricted treatment of abab and abac integrals to be provided shortly In the following expression for the Slater orbital of electron i centered on nucleus c k is the screening constant and q designates the species characterized by the quantum numbers nq lq mq and q Nq k is the normalization factor c k rci Nq k rq ci e krciPq lq cos ci cosmq ci sinmq ci sq In the following expression for the general two electron integral r is the inter electron distance and V V denote the space of the two electrons X q q q c c c c k k k k Z Z q c k rc q c k rc r q c k rc q c k rc dV dV We use the function expansion to evaluate the following prototype exchange integral It contains four ctional s orbitals X s abac k k k k N s k k k k Z Z e k ra k rb k ra k rc ra rb r ra rc dV dV Expansion and angular integration We reduce this to an in nite series and provide a closed expression for the general term The generalization to exchange integrals containing orbitals of unrestricted quantum number is in hand It is discussed brie y in the nal section of this note All the auxiliary functions and almost all the steps in the analysis needed for this generalization are covered by the present note Work also is in hand on the development of asymptotic formulas for the terms of the in nite series that allow analytic summation of the tail to give power series in the molecular parameters that converge much more rapidly than the original series when direct summation is impractical Unrestricted precision arithmetic will overcome the loss of accuracy in the di erences of relatively large quantities that the expressions constructed here generate This work has been made feasible by the use of mechanized symbolic computation We used Mathematica Release supplemented by mathscape to facilitate the mechanized production of this note Expansion and angular integration We evaluate the integrals and in particular by expressing the entire integrand in terms of polar coordinates centered on nucleus A The factor r is expanded by means of the generating function de nition of Legendre functions p and written here as r X n n r r Pn cos where r r are the distances of the two electrons from a common origin is the angle subtended at this origin by the line that joins them and n u v u v n u v n The exponential factors containing rb and rc are expaned by the function theorem If ra and rb are the distances of a point P from centers A and B is the distance between these centers and a is the angle subtended at A by the line BP then r b e rb X n n p ra m n ra Pn cos a For m n ra In ra Kn ra where the In z and Kn z are modi ed spherical Bessel functions The functions for higher m values are constructed from those for m by recurrence formulas To facilitate the mechanized operations we write ghi for the angle d GHI Then in terms of coordinates based on A we write r X l l ra ra Pl cos a e k rb rb X i i p ra ab i k ra k ab Pi cos ba Expansion and angular integration e k rc rc X j j p ra ac j k ra k ac Pj cos ca In terms of polar coordinates a single space integral is Z s dV Z r Z Z s r sin dr d d Here we use coordinates ra ba ba and ra ca ca for the two electrons The aximuths ba ca are measured around AB and AC from the common plane ABC Following substitution of into the summation is brought outside the multiple integral To integrate over the angular coordinates of the two electrons we use the orthogonality properties of surface harmonics in the following form p Z Z Ps cos Yn sin d d n Yn if s n otherwise where Yn is a surface harmonic of degree n and is the angle between the lines from the origin to points and In the notation that we use for the angular coordinates Z