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:

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

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

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

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

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

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

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

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