# Hamiltonian operator

## Hamiltonian operator tutorial

The Hamiltonian contains one- and two-electron terms. The two-electron terms (summed over i and j) are just the repulsion potential energies between all pairs of electrons. Thus:

${\displaystyle {\hat {H}}=\sum _{i}{\hat {h}}_{i}+\sum _{i}\sum _{j}{\frac {1}{r_{ij}}}}$

${\displaystyle 1/r_{ij}\,\!}$ is the repulsion between a pair of electrons (distance ${\displaystyle r_{ij}\,\!}$ apart).

The one-electron terms (summed over i) are more varied. For each electron, there is a kinetic energy term and a sum of attractive potential energy terms for each nucleus, A, in the molecule.

${\displaystyle {\hat {h}}_{i}=-{\frac {1}{2}}\nabla _{i}^{2}-\sum _{A}{\frac {Z_{A}}{r_{Ai}}}\,\!}$

${\displaystyle -1/2\nabla _{i}^{2}\,\!}$ is the kinetic energy term with:

${\displaystyle \nabla _{i}^{2}={\frac {\partial ^{2}}{\partial x_{i}^{2}}}+{\frac {\partial ^{2}}{\partial y_{i}^{2}}}+{\frac {\partial ^{2}}{\partial z_{i}^{2}}}\,\!}$

${\displaystyle Z_{A}/r_{Ai}\,\!}$ is the coulombic attraction between electron i and nucleus A.

${\displaystyle Z_{A}\,\!}$ is the nuclear charge (atomic number) of atom A and ${\displaystyle r_{Ai}\,\!}$ is the distance between electron i and nucleus A.