Calculating first and second derivatives of a molecular Hamiltonian?

Question

I'm interested in computing first and second derivatives of molecular Hamiltonians with respect to nuclear coordinates. I've been using Psi4 and PySCF to perform Hartree-Fock calculations, and I was wondering if there is any way to use these packages (or similar Python packages) to compute first and second derivatives analytically, by solving the coupled-perturbed Hartree-Fock equations?

Answer 1

The derivatives of a molecular Hamiltonian with respect to nuclear coordinates can be performed analytically (as you correctly pointed out), and therefore does not need PySCF or Psi4, which are programs for doing numerical calculations for things that cannot be done analytically. This is how it's done:

\begin{align} H &= -\sum_{i}\frac{1}{2}\nabla_{i}^{2}-\sum_{i,A}\frac{Z_{A}}{\left|r_{i}-R_{A}\right|}+\sum_{A>B}\frac{Z_{A}Z_{B}}{\left|R_{A}-R_{B}\right|}+\sum_{i>j}\frac{1}{r_{ij}},\tag{1}\\ \frac{\partial H}{\partial R} &= \sum_{i,A}\frac{Z_{A}}{\left|r_{i}-R_{A}\right|^{2}}-\sum_{A>B}\frac{Z_{A}Z_{B}}{R^{2}}. \tag{2} \end{align}

More interesting is how to calculate derivatives of the energy or wavefunction. It can be done for the energy via the misappropriately named Hellmann-Feynman theorem, but often electronic structure programs prefer to do it in other ways. A lot of detail can be written about derivatives in electronic structure programs (for example, about linear response theory), depending on what your specific final goal is, so while the answer to "how to analytically calculate derivatives of a molecular Hamiltonian" has been given in Eq. 2 above, you might consider asking a new question with more details about the specific end goal in your project.