Energy Levels in a Crystalline Solid — Question 1

Question 1

Coupling two identical atomic orbitals splits one level into two

Two orthonormal site orbitals have Hamiltonian $H=\begin{pmatrix}E_0&-t\\-t&E_0\end{pmatrix}$ with $t>0$. Spin does not affect the orbital energies.

1. Find the normalized orbital eigenstates and energies.

2. Place two noninteracting electrons in the lowest state and find their total energy relative to uncoupled atoms.

Question 1 - Solution

Coupling two identical atomic orbitals splits one level into two

(a) Method and calculation

$$\det(H-EI)=(E_0-E)^2-t^2=0\Rightarrow\boxed{E_\pm=E_0\pm t}.$$$$\boxed{|\psi_-\rangle=(|1\rangle+|2\rangle)/\sqrt2,\quad|\psi_+\rangle=(|1\rangle-|2\rangle)/\sqrt2}.$$

(b) Interpretation and further calculation

$$E_{2e}=2(E_0-t),\quad E_{\mathrm{uncoupled}}=2E_0.$$$$\boxed{\Delta E_{2e}=-2t}.$$

Physical check

Two opposite spins may occupy the lower spatial orbital. The model neglects electron repulsion and lattice energy, so this energy lowering alone is not a complete binding-energy calculation.