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A mig redactar…
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$$ \psi (\mathbf {r} {1},s{1},...,\mathbf {r} {Z},s{Z})=\prod _{i}^{Z}\phi {n{i}}(\mathbf {r} {i},s{i}) $$
$$ \psi (\mathbf {r} {1},s{1},...,\mathbf {r} {Z},s{Z})={\frac {1}{\sqrt {Z!}}}\det {\begin{bmatrix}\phi {n{1}}(\mathbf {r} {1},s{1})&\phi {n{1}}(\mathbf {r} {2},s{2})&...&\phi {n{1}}(\mathbf {r} {Z},s{Z})\\\phi {n{2}}(\mathbf {r} {1},s{1})&\phi {n{2}}(\mathbf {r} {2},s{2})&...&\phi {n{2}}(\mathbf {r} {Z},s{Z})\\...&...&...&...\\\phi {n{Z}}(\mathbf {r} {1},s{1})&\phi {n{Z}}(\mathbf {r} {2},s{2})&...&\phi {n{Z}}(\mathbf {r} {Z},s{Z})\end{bmatrix}} $$
Tenen l’avantatge (respecte el simple producte de Hartre) que anti-simetritzen la funció d’ona, és a dir la fan compatible amb el principi d’exclusió de Pauli.
$$ \Psi_\text{HF}(\vec{r}_1,\vec{r}_2) = \frac{1}{\sqrt{2}}\left[\psi_a(\vec{r}_1)\psi_b(\vec{r}_2) - \psi_a(\vec{r}_2)\psi_b(\vec{r}_1)\right] $$
$$ \langle \Psi_\text{HF} | O_{12} | \Psi_\text{HF} \rangle = J_{ab} - K_{ab} $$
On
$$ \begin{aligned} J_{ab} &= \iint \psi_a^(\vec{r}_1)\psi_b^(\vec{r}2) O{12} \psi_a(\vec{r}_1)\psi_b(\vec{r}2) \, d^3 r_1 d^3 r_2 \quad \text{(Coulomb)} \\ K{ab} &= \iint \psi_a^(\vec{r}_1)\psi_b^(\vec{r}2) O{12} \psi_a(\vec{r}_2)\psi_b(\vec{r}_1) \, d^3 r_1 d^3 r_2 \quad \text{(Intercanvi)} \end{aligned} $$