Past exam of the quantum condensed matter course of the University of Cambridge / 2022 / Section B / 5 / a

Choose one atomic orbital ${\varphi }_A$ and ${\varphi }_B$ on the two atoms and approximate a molecular orbital by the linear combination of atomic orbitalsProjecting the Schrödinger equation onto the two orbitals gives a generalized eigenvalue problem $H\mathbf{c}=ES\mathbf{c}$, where $H_{ij}=⟨{\varphi }_i|\hat{H}|{\varphi }_j⟩$ and $S_{ij}=⟨{\varphi }_i|{\varphi }_j⟩$. In an orthonormal basis the matrix isThe coupling mixes the atomic states and produces a lower bonding molecular orbital and upper antibonding molecular orbital. For equivalent real orbitals with overlap $s$ the normalized combinations are $({\varphi }_A\pm {\varphi }_B)/\sqrt{2(1\pm s)}$; the precise energy ordering follows from the Hamiltonian matrix elements. This restricted orbital basis is an approximation to the molecule's wavefunction, rather than a claim that exact isolated-atom orbitals remain unchanged in the molecule.