From: Bolla Marianna [email protected]
Date: May 13, 2025
Hello David,
Please, find attached a short writeup about Hamiltonian. I wrote it for a student, based on my discussions with a chemist (first author in Ref. [1]) and on the book [6] (the chemisst uses it as a course material). I can send you the pdf version of this book, but the chemist told that the Hückel theory, akin to the theory in Ref. [2] were out-dated. However, I am still interested how a simple or edge-weighted graph can be built based on molecular Hamiltonians. As those are sparse graphs, I use the non-backtracking spectra to find clusters of the nodes (electronic wave functions).
I have realized that I am not able to solve this problem myself, interdisciplinary collaboration is needed, akin to your suggested AI projects. My Finnish coauthor is a physicist, but he does not know too much about Hamiltonians, the chemist is better. I look forward to your possible contribution too.
Best regards,
Marianna
From: [email protected]
To: Bolla Marianna [email protected]
Date: May 13, 2025
Hi Marianna,
Ok, I am going through the writeup, it makes sense so far. I see it's constructed from the matrix formulation of a differential equation, as is usual, with the differential equation being the time-independent Schroedinger equation H * psi = E * psi, with psi as the wavefunction, E as the (scalar) energy eigenvalues (observed from spectroscopy) and H being the Hamiltonian (which is itself formed from a differential equation that minimizes the system's energy by setting its partial differentials to 0 to find the minimum of the system energy). So far, so good. To get it into matrix formulation we usually replace the coordinates with vectors, where the summation over differing indices will lead to a matrix (ex. H will be a matrix because it the kinetic term is p^2/m where p is momentum, momentum is a derivative of position (corresponds to velocity), and in matrix formulation P^2 is really p dot p*, or p_i * p_j so varying over i and j will give a matrix.
Now I need to look up modern best-practices for forming the Hamiltonian in molecular systems. There are probably going to be a number of simplifying assumptions resulting in an altered Hamiltonian H, and that will require knowing something about the molecules and which contributions contribute meaningfully to measured energy levels (ex. self-interactions are small effects - this is where the chemist comes in). Then I need to understand how this relates to a graph formulation.
Ok, let me read and research some and I will get back to you in a few days to see how well I understand the problem and expressing it properly in mathematical formulation.
Thanks! David
From: Bolla Marianna [email protected]