Formulation of molecular integrals over Gaussian functions treatable by both the Laplace and Fourier transforms of spatial operators by using derivative of Fourier-kernel multiplied Gaussians

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Abstract
General recurrence formulas for molecular integrals over Gaussian functions are derived by introducing the derivative of Fourier-kernel multiplied Gaussians (DFGs). The DFG allows us to formulate on the same ground molecular integrals over the Cartesian Gaussians, modified Hermite Gaussians, and the Gaussians multiplied by phase factors exp[ik⋅(r−R)] with spatial operators including any number of both the Laplace and Fourier transforms for one- and two-electron spatial operators. Thus the present formulation has a wider applicability than that given by Obara and Saika [J. Chem. Phys. 89, 1540(1988)], where the basis functions are the Cartesian Gaussians and the spatial operators are those in the Laplace transform with at most one kernel of the Fourier transform. Furthermore the present formulation inherits the characteristic features of the above one, such as being capable of dealing with (1) molecular integrals with both nonrelativistic and relativistic spatial operators, (2) any order of the derivative of these integrals with respect to the function centers, and (3) leading us to the efficient scheme of computing the integrals by virtue of the recursive nature of the formulation.