It sounds like you’re interested in quantum computation for quantum chemistry, but don’t have a strong background in quantum chemistry. Please read Sections IIIB2 and IIID of https://doi.org/10.1103/RevModPhys.92.015003, as well as Appendix A of https://doi.org/10.1021/acs.chemrev.8b00803, if you haven’t already.
Changing between 6-31G and CC-PVDZ is not a unitary transformation. Those give different sets of atomic orbitals for a given molecular geometry, which are not related by any linear transformation, unitary or otherwise. You cannot write one set of atomic orbitals as a linear combination of the others.
It sounds like your method is truncating the space of virtual orbitals based on occupation numbers?
If this is a quantum computing project, then the Hamiltonians are certainly not directly calculated from the 2RDM. However, they cannot possibly be computed from the overlap matrix either. On its own, the overlap matrix gives zero information about the Hamiltonian. You need to combine that with OEI and TEI integrals in a not-necessarily orthogonal basis set for that to be at all useful. Look over your code again and figure out what it is you need. “How do I use the Psi interface,” I can help with. “What exactly do I need for my research project/external software,” I cannot.