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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
as follows
![\begin{displaymath}
h(i) = - \frac{1}{2} \nabla^2_{i} - \sum_{A} \frac{Z_A}{r_{iA}},
\end{displaymath}](img116.png) |
(32) |
and a two-electron operator
as
![\begin{displaymath}
v(i,j) = \frac{1}{r_{ij}}.
\end{displaymath}](img118.png) |
(33) |
Sometimes this is also called
. Note that, in another
simplification, we have begun writing
as shorthand for
, and
as shorthand for
.
Now we can write the electronic Hamiltonian much more simply, as
![\begin{displaymath}
{\hat H}_{el} = \sum_i h(i) + \sum_{i<j} v(i,j) + V_{NN}.
\end{displaymath}](img123.png) |
(34) |
Since
is just a constant for the fixed set of nuclear coordinates
, we will ignore it for now (it doesn't change the
eigenfunctions, and only shifts the eigenvalues).
Next: Matrix Elements of the
Up: intro_estruc
Previous: Slater Determinants
David Sherrill
2003-08-07