Refactor get_spin_hamiltonian() to adhere to the single responsibility principle

[wsttiger]

get_spin_hamiltonian() is ~650 lines long. It violates the Single Responsibility Principle (each method should only do one thing).

Consider refactoring into smaller methods (e.g., _run_mean_field_calculation(), _run_mp2_calculation(), _compute_integrals_unrestricted(), etc.).

This would improve:

-Testability
-Readability
-Maintainability
-Reusability

Originally posted by @wsttiger in #309 (review)