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.
$$g_\mathrm{1s}(\alpha) = (2\alpha/\pi)^{3/4}\mathrm{e}^{-\alpha\mathbf{r}^2}$$
The equation for the Kinetic Energy matrix element that you quote is for two unnormalised 1s Gaussians. The d factors contain the normalisation factors and the contraction coefficients - look more carefully at the code, you have got it slightly wrong what the d's represent.

Talking of the code please, Please don't use that as a model for your own Fortran - that style is getting on for 50 years out of date!