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.

${\hat{h}}_i=-\frac{1}{2}\underset{i}{\overset{2}{\mathrm{\nabla }}}-\sum _A\frac{Z_A}{r_{Ai}}$

$-1/2\underset{i}{\overset{2}{\mathrm{\nabla }}}$ is the kinetic energy term with:

$\underset{i}{\overset{2}{\mathrm{\nabla }}}=\frac{{\partial }^2}{\partial x_i^2}+\frac{{\partial }^2}{\partial y_i^2}+\frac{{\partial }^2}{\partial z_i^2}$

$Z_A/r_{Ai}$ 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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