# Kinetic Energy Evaluation Integral Evaluation Program

I'm reading Ostlund's Modern Quantum Chemistry. In Appendix A, the kinetic energy integral is evaluated using the Gaussian Basis functions to be

$$\left(A\left| -\frac{1}{2}\nabla^2 \right| B\right) = \alpha\beta/(\alpha + \beta)[3 - 2\alpha\beta/(\alpha + \beta) |\mathbf{R}_A - \mathbf{R}_B|^2][\pi/(\alpha + \beta)]^{3/2} \\ \times \exp [-\alpha\beta/(\alpha + \beta)|\mathbf{R}_A - \mathbf{R}_B|^2] \tag{A.11}\label{kin-en.int}$$

So, in the integral evaluation the Gaussian functions themselves are not used, but in the computer program he is evaluating the integral using

T11=T11+T(A1(I),A1(J),0.0D0)*D1(I)*D1(J)


The function T() is calculating the equation above \eqref{kin-en.int}.

I can't understand why he is multiplying by D1 and D2 which are the Gaussian functions themselves. $$g_\mathrm{1s}(\alpha) = (2\alpha/\pi)^{3/4}\mathrm{e}^{-\alpha\mathbf{r}^2}$$

• Just a comment, if you are looking to write your own integral code, I highly recommend taking a look at the following link joshuagoings.com/2017/04/28/integrals – Erik Kjellgren Feb 4 at 16:12
• Thank you Erik, it is very helpful – SssunnN Feb 8 at 7:55