First and second derivatives of two electron integrals over Cartesian Gaussians using Rys polynomials

Formulas are developed for the first and second derivatives of two electron integrals over Cartesian Gaussians. Integrals and integral derivatives are evaluated by the Rys polynomial method. Higher angular momentum functions are not used to calculate the integral derivatives; instead the integral formulas are differentiated directly to produce compact and efficient expressions for the integral derivatives. The use of this algorithm in the ab initio molecular orbital programs gaussian 80 and gaussian 82 is discussed. Representative timings for some small molecules with several basis sets are presented. This method is compared with previously published algorithms and its computational merits are discussed.