The Hamiltonian in Nonorthogonal Representations

In order to find the second quantized Hamiltonian in a nonorthogonal representation, we proceed in a similar manner as we did in finding the anticommutation rules of Eq. (13.9). That is, we introduce again an auxiliary orthonormal set / for which the result is known  

This form of the Hamiltonian is not very useful since the integral list refers to the auxiliary basis set. Let us transform it back to the original basis  

Several equivalent forms of this Hamiltonian can be put down by performing the summation over some indices of S A convenient form is obtained, for instance, when transforming all fern-indices of the integrals as well as all annihilation operators to the reciprocal space  

The Hamiltonian of Eq. (13.25) is formally very similar to the second quantized representation in an orthogonal basis, cf. Eq. (13.21). In fact, it can be manipulated in a similar manner since the anticommutation rules of Eq. (13.13) hold. An essential difference is that the integral list in Eq. (13.25) is not symmetric  

The total Hamiltonian is still Hermitian, of course. This is possible because Xp is not the adjoint of Xp- One has to be careful, however, when considering partitionings of Eq. (13.25), since individual terms of this Hamiltonian are not Hermitian. 

---