On the implicit integral character of Roothaan's expansion

The molecular generator coordinate Hartree-Fock method is reviewed. The con nection between a quadrature solution of the generator coordinate Hartree-F ock equations and Roothaan's equations is stressed. The relation between li near expansion coefficients and generator coordinate weight functions is di scussed and a numerical and analytical example is provided for the 1s orbit al of the hydrogen atom represented as the integral transform of a Gaussian function. For the same example, the Gauss-Labatto quadrature is employed t o emphasize the implicit integral character of Roothaan's equations. As a m ajor conclusion, the interpretation that every LCAO calculation is actually performing integrations of the Griffin-Wheeler equations is advanced. Basi s sets are therefore abscissas of the implicit quadrature used in the integ ration, whereas the linear coefficients automatically incorporate the corre sponding weights. Subsequently, it is shown how to extract the generator co ordinate weight function from the LCAO coefficients which has the advantage of being a characteristic of the physical system under study and not of th e particular calculation being carried out. As such: basis set design becom es how to efficiently sample the weight function.