2. Base \(1s_A,1s_B\) et problème généralisé
2.1 Combinaison linéaire
\[ \psi=c_A\phi_A+c_B\phi_B. \] \[ \mathbf S=\begin{pmatrix}1&S\\S&1\end{pmatrix},\qquad S=\langle\phi_A|\phi_B\rangle, \] \[ \mathbf H=\begin{pmatrix}H_{AA}&H_{AB}\\H_{AB}&H_{AA}\end{pmatrix}. \]2.2 Principe variationnel
\[ \mathcal E(\mathbf c)=\frac{\mathbf c^\dagger\mathbf H\mathbf c}{\mathbf c^\dagger\mathbf S\mathbf c}, \qquad \mathbf H\mathbf c=E_{\mathrm{el}}\mathbf S\mathbf c. \]2.3 États de symétrie
\[ \psi_g=\frac{\phi_A+\phi_B}{\sqrt{2(1+S)}},\qquad \psi_u=\frac{\phi_A-\phi_B}{\sqrt{2(1-S)}}. \]2.4 Éléments de matrice
\[ H_{AA}=\left\langle\phi_A\left|-\frac12\nabla^2-\frac1{r_A}-\frac1{r_B}\right|\phi_A\right\rangle, \] \[ H_{AB}=\left\langle\phi_A\left|-\frac12\nabla^2-\frac1{r_A}-\frac1{r_B}\right|\phi_B\right\rangle. \]En utilisant l’équation atomique :
\[ H_{AA}=-\frac12+J,\qquad J=\left\langle\phi_A\left|-\frac1{r_B}\right|\phi_A\right\rangle, \] \[ H_{AB}=-\frac12S+K,\qquad K=\left\langle\phi_A\left|-\frac1{r_A}\right|\phi_B\right\rangle. \]