Definition of Electronic energy

Question

Using Born-Oppenheimer approximation, we separate the nuclear and electronic parts of the wavefunction, and treat them separately. We include the terms that have direct dependence on the electronic positions in the electronic Hamiltonian,

$$\hat{H}_{e} = \hat{T}_e + \hat{V}_{eN} + \hat{V}_{ee}$$

Thus, the total Hamiltonian is given by:

$$ \hat{H} = \hat{H}_{e} + \hat{V}_{NN} $$

So, what is the definition of electronic energy? According to the source mentioned below, electronic energy does not include $\hat{V}_{NN}$, which makes sense. But, all the electronic structure packages quote the Total energy as the electronic energy. The following is quoted from Gaussian website:

Zero-point correction=                  .023261 (Hartree/Particle)
Thermal correction to Energy=           .026094
Thermal correction to Enthalpy=         .027038
Thermal correction to Gibbs Free Energy= .052698
Sum of electronic and zero-point Energies=-527.492585
Sum of electronic and thermal Energies= -527.489751      
Sum of electronic and thermal Enthalpies=-527.488807
Sum of electronic and thermal Free Energies=-527.463147

Why is this discrepancy? What is the actual definition of electronic energy?

Source for my argument: http://vergil.chemistry.gatech.edu/notes/quantrev/node31.html

Answer 1

Electronic energy is indeed an eigenvalue of electronic Hamiltonian, and, as you correctly pointed out, it doesn't include the contribution from repulsions of the fixed nuclei (which has nothing to do with electrons). But sometimes the total energy for a particular nuclear configuration (one that includes the above mentioned interactions contribution) is still (incorrectly) termed "electronic energy" while just an eigenvalue of electronic Hamiltonian is called "pure electronic energy". That is essentially wrong, but it is sort of a custom.

And I would not say that "all the electronic structure packages quote the Total energy as the electronic energy". Did you check all of them? For instance, I quickly checked one of my DALTON calculations, and found this:

@    Final DFT energy:           -210.248305383291                 
@    Nuclear repulsion:           158.997839473485
@    Electronic energy:          -369.246144856776

So the first one is the (total) energy, then come its two components with the right names.

Same for ORCA:

Total Energy       :         -210.25104538 Eh           -5721.22181 eV

Components:
Nuclear Repulsion  :          158.99783872 Eh            4326.55115 eV
Electronic Energy  :         -369.24888411 Eh          -10047.77296 eV

GAMESS-US (here first two terms = electronic energy)

        ONE ELECTRON ENERGY =    -596.6619030866
        TWO ELECTRON ENERGY =     228.7817683952
   NUCLEAR REPULSION ENERGY =     158.9978510234
                              ------------------
               TOTAL ENERGY =    -208.8822836680

NWChem (middle three terms = electronic energy)

         Total DFT energy =     -210.248032406037
      One electron energy =     -597.377034042581
           Coulomb energy =      258.596296569204
    Exchange-Corr. energy =      -30.465145956066
 Nuclear repulsion energy =      158.997851023405

And I don't want to continue, since I'm pretty sure that most of the programs use the right terms.