Linear Combination of Atomic Orbitals

Uses Schrodinger’s Equation and Born-Oppenheimer Approximation to find molecular geometry.

To find internuclear distance between 2 $H$ atoms $a$ and $b$ in $H_2$, solve

$$\hat{H}\Psi (R)=E\Psi (R).$$

This is equivalent to combining wave function/probability distributions of 2 orbitals.

1. Fix $R_{ab}$
2. Simplify with Born-Oppenheimer

$$\left[-\frac{h^2}{2m}\frac{d^2}{d{R}^2}+V(R)\right]\Psi (R)=E\Psi (R)⟹V(R)\Psi (R)=E\Psi (R)$$

3. Solve for allowed energy levels, $E$, and $\Psi$ for electrons ($H_2$ has 2 electrons with 6 possible coordinates)
4. Repeat at new $R_{ab}$
5. Plot lowest energy as function of $R_{ab}$