# On diagonal terms in the Coulomb matrix

I have seen many machine learning algorithms for prediction of quantum chemistry properties that use Coulomb matrix as their input. Coulomb matrix is defined as,

$$\boldsymbol{M}_{i j}^{\mathrm{Coulomb}}=\left\{\begin{array}{cc}{0.5 Z_{i}^{2.4}} & {\text { for } i=j} \\ {\frac{Z_{i} Z_{j}}{R_{i j}}} & {\text { for } i \neq j}\end{array}\right.,$$

where the diagonal terms:

... can be seen as the interaction of an atom with itself and are essentially a polynomial fit of the atomic energies to the nuclear charge.

I was trying to read more about these diagonal terms and how exactly this equation is fitted. Essentially, something that says: "... and from there we get $$0.5Z^{2.4}$$." If these energies can be extended to bonds as well, as the off-diagonal term in Coulomb matrix represents repulsion forces rather than potentials, any references would be appreciated as well.

• The model seems to stem from Rupp et al Phys. Rev. Lett.,108(5):058301, 2012, but I did not find much explanation for it there. – Buck Thorn Sep 28 '19 at 6:45
• @Blade could you clarify what you mean by extending these energies to bonds. There are methods (e.g. bag of bonds) related to Coulomb matrices which use bond energies, rather than atom energies , to predict an overall atomization energy. – Tyberius Sep 29 '19 at 22:19
• @Tyberius For instance, in molecular mechanics, we describe bond potentials using bond stretching interactions (either harmonic oscillator or Morse function). I was basically wondering how the empirical relation above was derived, so that I see if there are similar empirical relations for bond energies as well. I'm not sure if there is an answer to the second part of my question, just wondering if there is such a thing. – Blade Sep 29 '19 at 23:21

In terms of how they got the relation for the diagonal elements, I believe it is relatively straightforward. Given a list of atomic energies (energy of atom relative to separated nucleus and electrons), one can try to fit these values to a function of nuclear charge. If you plot a few values, it suggests trying an exponential fit (I don't know why all the papers say polynomial fit, as this could not lead to a noninteger exponent). So, you can try to fit the values to a function of the form $$aZ^b$$ using, for example, least-squares regression. You should wind up with something pretty close to the equation given in your question, with some variation because I don't know what atomic energies they used to do the fit. (I'm also not sure why the function isn't negative since the atomic energies are negative. My best guess is the sign winds up not being important as a descriptor in the network).