Simplified Notation for the Hamiltonian

Now that we know the functional form for the wavefunction in Hartree-Fock theory, let's re-examine the Hamiltonian to make it look as simple as possible. In the process, we will bury some complexity that would have to be taken care of later (in the evaluation of integrals).

We will define a one-electron operator $h$ as follows

$$h(i) = - \frac{1}{2} \nabla^2_{i} - \sum_{A} \frac{Z_A}{r_{iA}},$$ (32)

and a two-electron operator $v(i,j)$ as

$$v(i,j) = \frac{1}{r_{ij}}.$$ (33)

Sometimes this is also called $g(i,j)$. Note that, in another simplification, we have begun writing $h(i)$ as shorthand for $h({\mathbf x}_i)$, and $v(i,j)$ as shorthand for $v({\mathbf x}_i, {\mathbf x}_j)$.

Now we can write the electronic Hamiltonian much more simply, as

$${\hat H}_{el} = \sum_i h(i) + \sum_{i<j} v(i,j) + V_{NN}.$$ (34)

Since $V_{NN}$ is just a constant for the fixed set of nuclear coordinates $\{ {\mathbf R} \}$, we will ignore it for now (it doesn't change the eigenfunctions, and only shifts the eigenvalues).