Hamiltonian operator

## Hamiltonian operator tutorial[edit | edit source]

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:

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

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

h ^ i = − 1 2 ∇ i 2 − ∑ A Z A r A i {\displaystyle {\hat {h}}_{i}=-{\frac {1}{2}}\nabla _{i}^{2}-\sum _{A}{\frac {Z_{A}}{r_{Ai}}}\,\\!}

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

∇ i 2 = ∂ 2 ∂ x i 2 + ∂ 2 ∂ y i 2 + ∂ 2 ∂ z i 2 {\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}}}\,\\!}

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

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

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