= Solution
Choose one <atomic orbital> $\phi_A$ and $\phi_B$ on the two atoms and approximate a <molecular orbital> by the <linear combination of atomic orbitals>
$$
\psi=c_A\phi_A+c_B\phi_B.
$$
Projecting the <Schrödinger equation> onto the two orbitals gives a <generalized eigenvalue problem> $H\mathbf c=ES\mathbf c$, where $H_{ij}=\langle\phi_i|\hat H|\phi_j\rangle$ and $S_{ij}=\langle\phi_i|\phi_j\rangle$. In an orthonormal basis the matrix is
$$
H=\begin{pmatrix}\epsilon_A&t\\t^*&\epsilon_B\end{pmatrix},
\qquad
E_\pm=\frac{\epsilon_A+\epsilon_B}{2}\pm\sqrt{\frac{(\epsilon_A-\epsilon_B)^2}{4}+|t|^2}.
$$
The 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 $(\phi_A\pm\phi_B)/\sqrt{2(1\pm s)}$; 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.
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