Variational Methods

10.1 The Variational Principle

For any trial wavefunction $\psi_{\text{trial}}$ (normalised), the expectation value of the Hamiltonian is an upper bound on the true ground state energy:

$E_{\text{trial} = \langle\psi_{\text{trial}|\hat{H}|\psi_{\text{trial}\rangle \geq E_0}}}$

The equality holds if and only if $\psi_{\text{trial} = \psi_0}$.

10.2 The Hydrogen Molecule Ion $H_2^+$

The simplest molecule: one electron in the field of two protons separated by distance $R$. The Hamiltonian:

$\hat{H} = -\frac{\hbar^2}{2m_e}\nabla^2 - \frac{e^2}{4\pi\varepsilon_0 r_A} - \frac{e^2}{4\pi\varepsilon_0 r_B} + \frac{e^2}{4\pi\varepsilon_0 R}$

LCAO trial function: $\psi_\pm = N_\pm[\psi_{1s}(\mathbf{r}_A) \pm \psi_{1s}(\mathbf{r}_B)]$

The energies:

$E_\pm(R) = E_{1s} + \frac{e^2}{4\pi\varepsilon_0 R} + \frac{J \pm K}{1 \pm S}$

Where $S = \langle\psi_A|\psi_B\rangle$ is the overlap integral, $J$ is the Coulomb integral, and $K$ is the exchange integral.

• $E_-$ (bonding): has a minimum at $R \approx 2.5\,a_0$Giving a binding energy of $\sim 1.8$ eV (experiment: 2.8 eV).
• $E_+$ (antibonding): monotonically decreases, no bound state.

10.3 The Hydrogen Molecule $H_2$

With two electrons, the full Hamiltonian includes the electron-electron repulsion. Using the variational method with properly (anti)symmetrised spatial-spin wavefunctions:

Bonding (singlet): $E_{\text{singlet} = 2E_{1s} + \frac{e^2}{R} + \frac{2J + 2K}{1 + S^2}}$

Antibonding (triplet): $E_{\text{triplet} = 2E_{1s} + \frac{e^2}{R} + \frac{2J - 2K}{1 - S^2}}$

The equilibrium bond length is $R_e \approx 1.4\,a_0$ with binding energy $\sim 3.5$ eV (experiment: 4.75 eV).