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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