MolSSI Integral Reference Project
Gaussian Product Theorem


The Gaussian Product Theorem (GPT) states that the product of two gaussians G1,G2 can be represented as a third gaussian G3 with a center a line connecting G1 and G2. This theorem is used extensively in quantum chemistry to simplify integrals.

MIRP contains functionality to compute some of the terms from the GPT which will be used within integral kernels.

Given two gaussians $G_1(R) = e^{-\alpha_1 r_{AR}^2}$ and $G_2(R) = e^{-\alpha_2 r_{BR}^2}$ centered on $A$ and $B$, respectively, and with $r_{AR}$ and $r_{BR}$ representing the distance between the center of the gaussian and a given point $R$

\begin{eqnarray*} \gamma &= \alpha_1 + \alpha_2 \\ \overline{AB}^2 &= (A_x - B_x)^2 + (A_y - B_y)^2 + (A_z - B_z)^2 \\ P_x &= \frac{(\alpha_1 A_x + \alpha_2 B_x)}{\gamma} \\ P_y &= \frac{(\alpha_1 A_y + \alpha_2 B_y)}{\gamma} \\ P_z &= \frac{(\alpha_1 A_z + \alpha_2 B_z)}{\gamma} \\ \overline{PA}_x &= P_x - A_x \qquad\qquad \overline{PA}_y = P_y - A_y \qquad\qquad \overline{PA}_z = P_z - A_z \\ \overline{PB}_x &= P_x - B_x \qquad\qquad \overline{PB}_y = P_y - B_y \qquad\qquad \overline{PB}_z = P_z - B_z \end{eqnarray*}

Functions in MIRP

In MIRP, the parameters to the Gaussian Product Theorem can be calculated via the following functions: