Hamiltonian operator

From Wikiversity

Jump to: navigation, search

[edit] 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.

Return to Quantitative MO theory - Return to Empirical MO methods
Retrieved from "http://en.wikiversity.org/wiki/Hamiltonian_operator"