Kinetic Energy Evaluation Integral Evaluation Program Ostlund I'm reading Ostlund's Modern Quantum Chemistry. In Appendix B, the kinetic energy integral is evaluated using the Gaussian Basis functions as

T11=T11+T(A1(I),A1(J),0.0D0)*D1(I)*D1(J)
T12=T12+T(A1(1),A2(J),R2).D1(I).D2(J) 
T22=T22+T(A2(1),A2(J).0.0D0).D2(1).D2(J) 


I understand T11 to be the kinetic energy integral using the first atom orbitals so it will give the kinetic energy of the electron of the first atom, and this is why it is passing zero as the bond length parameter. The same is T22 but for the second atom.

What is T12 used for, aren't T11 and T22 enough to calculate the total kinetic energy?
The off-diagonal elements $T_{\mu\nu}, (\mu \ne \nu)$ are just as necessary as the diagonal elements $T_{\mu\mu}$ to represent the kinetic energy operator in the chosen AO basis set. Substituting a few of Szabo/Ostlund's equations into each other (in my edition, 3.147-3.154), one finds
$$
F_{\mu\nu} = T_{\mu\nu} + V_{\mu\nu}^{\text{nucl}} + \sum_{\lambda\kappa} P_{\lambda\kappa}\left[ \left( \mu\nu \vert \kappa\lambda  \right) - \frac{1}{2} \left( \mu\lambda \vert \kappa \nu\right) \right]
$$
where $F$ is the Fock matrix, $V^{\text{nucl}}$ represents the nuclear attraction term, $P$ is a density matrix and $\left( \mu\nu \vert \kappa\lambda  \right)$ are two-electron repulsion integrals over the AO basis functions (denoted greek indices). Of course, your Fock matrix may be diagonal already, if you have a very special case and basis set, but in general and to perform the SCF calculation, the off-diagonal parts cannot be ignored.