I don’t know what you mean by “know the Hamiltonian of my system for a given set of orbitals.” For any set of orthonormal orbitals, the molecular Hamiltonian is given as a linear combination of second quantized operators, with expansion coefficients given by one and two-electron integrals. Are you asking how you can get a program to compute the integrals for you? Are you trying to learn how to compute the integrals “from scratch”?
Psi is designed for molecular systems and does not support a Hubbard chain.
The “only s-orbital” approximation is not even qualitatively correct for most atoms. For hydrogen atoms, it’s qualitatively correct. Choose a STO-3G or STO-6G basis set, which is the bare minimum.
Forming the Hamiltonian matrix for a molecule is prohibitively expensive and is only ever done for pedagogical purposes. See here for an example.