Kinetic Energy Evaluation Integral Evaluation Program Ostlund
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.