Direct Exchange versus the indirect Exchange of the t-J Hamiltonian in HTSC Cuprates

The direct exchange integral and the transfer integral are calculated for the anti-bonding wave-function of the HTSC cuprates. The direct exchange integral yields a value three times larger than the indirect exchange of the t-J Hamiltonian. It also predicts positive exchange energy for neighboring parallel spins, contrary to the prediction of zero energy by the t-J Hamiltonian. A modified t-J Hamiltonian is proposed, and the implications of the results are discussed.