Hartree-Fock

Hartree-Fock#

1.  Multi-electron \(\textbf{Schr\"odinger}\) Equation#

\[ \hat{H}=\sum_ih_i + \sum_i\sum_j\frac{1}{r_{ij}} \]

For a single electron integral \(\epsilon_i=\int\mathcal{X_i}^*(x_i)h_i\mathcal{X_i}(x_i)dx_i\)

For two electron integral: \(\epsilon_i=\int\mathcal{X_1}^*(x_1)\mathcal{X_2}^*(x_2)h_i\mathcal{X_1}(x_1)\mathcal{X_2}(x_2)dx_1dx_2\)

Energy contribution

  • Coulomb: \(J =\int\mathcal{X_1}^*(x_1)\mathcal{X_2}^*(x_2)\frac{1}{|r_{12}|}\mathcal{X_1}(x_1)\mathcal{X_2}(x_2)dx_1dx_2 = \int|\mathcal{X_1}(x_1)^2|\frac{1}{r_{12}}|\mathcal{X_2}(x_2)^2|dx_1dx_2\)

  • Exchange: \(K =\)

So, for exchange behavior for electron orbitals with inverse spin, the exchange term will be zero.

2.  Hartree-Fock equation#

3.  Linear combination of atomic orbitals (LCAO)#

\[\begin{split} \mathcal{X_i}=\sum C_{ui}\psi_u\\ \hat{F}(x_1)\mathcal{X_i} =\epsilon_i\mathcal{X_i}(x_1)\\ \Rightarrow \hat{F}(x_1)\sum C_{ui}\psi_u = \epsilon_i\sum C_{ui}\psi_u \end{split}\]

  • \(\psi_u\) is a basis function set, \(\psi\in \{\psi_{100},\psi_{200},\psi_{300}\}\)

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