How to compute the electronic energy reported by Gaussian from the AlphaOrbitalEnergies?

I followed the instructions on [Wolfram Language & System Documentation Center — FCHK (.fchk)](https://reference.wolfram.com/language/ref/format/FCHK.html) to analyze the `*.fchk` file.


I ran an energy calculation with Gaussian and converted the results to `*.fchk` files. The `*.fchk` files were imported to Mathematica. I extracted the `AlphaOrbitalEnergies` from [`MetaInformation`](https://reference.wolfram.com/language/ref/MetaInformation.html) provided in the `*.fchk` files. Is there a way to compute the electronic energy reported by Gaussian from the `AlphaOrbitalEnergies`?


For my example calculation of methane with HF/3-21G, the electronic energy is $\pu{-39.976406 Ha}.$


I tried adding up all the energies from `AlphaOrbitalEnergies` and I got $\pu{-2.18778 Ha}.$
Next, I tried to only add the energies of occupied orbitals and I got the value of $\pu{-13.7344 Ha}.$


How to obtain the electronic energy from the alpha orbital energy?




This is a common question in quantum chemistry.
(It's common enough that one colleague specifically mentions this every time he teaches a quantum chemistry class.)


I'm going to assume that when you say AlphaOrbitalEnergies that you have a singlet species, so the alpha and beta orbital energies are identical.


For a Hartree Fock calculation, the total SCF energy is:


$$
E=\sum\_{i(\text { occupied })}\left\langle\phi\_{i}|h| \phi\_{i}\right\rangle+\sum\_{i>j(\text { occupied })}\left[J\_{i, j}-K\_{i, j}\right]
$$


If I add up the occupied orbital energies (i.e., two electrons in each orbital), I get:


$$
\sum\_{\mathrm{i}(\text { occupied })} \epsilon\_{i}=\sum\_{\mathrm{i}(\text { occupied })}\left\langle\phi\_{i}|h| \phi\_{i}\right\rangle+\sum\_{\mathrm{i}, \mathrm{j}(\text { occupied })}\left[J\_{i, j}-K\_{i, j}\right]
$$


Notice that the second term is different. Notice that if I'm adding up the occupied orbital energies I'll be double-counting the term with Coulomb - exchange energies.


In principal, you could get the total energy from the orbital energies, but you would have to do work:


$$
E=\frac{1}{2}\left[\sum\_{\mathrm{i}(\text { occupied })}\left\langle\phi\_{i}|h| \phi\_{i}\right\rangle+\sum\_{\mathrm{i}(\text { occupied })} \epsilon\_{i}\right]
$$


On the other hand, I believe the FCHK files include the total SCF energy already.


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