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In the nearly-free electron model, a weak periodic potential has Fourier components $V_{\mathbf{G}}$ that couple plane waves at $\mathbf{k}$ and $\mathbf{k}-\mathbf{G}$. Far from a degeneracy their admixture gives small perturbative energy shifts. At a Bragg plane, however,so degenerate perturbation theory is required. In one dimension the first such crossing is at $k=\pm G/2$, the boundary of the first Brillouin zone.

Within the degenerate pair the eigenvalues are ${ϵ}_{G/2}\pm |V_G|$. ThusThe eigenstates are standing-wave combinations of counterpropagating waves. Their charge densities sit differently relative to the ionic potential, giving different potential energies and removing the free-electron degeneracy. The resulting avoided crossing produces continuous allowed energy bands separated by a forbidden interval. A Fourier component or symmetry-allowed coupling is needed: geometry alone does not require every possible gap to be nonzero.

At nonzero band hybridization, the crossing becomes an avoided crossing. The separation at $k_c$ is $2\alpha$. If $\alpha \ll E_{\mathrm{fl}}$, appreciable mixing occurs only where $|A{k}^2-E_{\mathrm{fl}}|\lesssim \alpha$; away from that region each branch retains nearly its original orbital character. For $\alpha \sim E_{\mathrm{fl}}$, the avoided crossing is broad and both branches are substantially reshaped, as shown in the figure.

Use $E_{-}$ for the lower branch and $E_{+}$ for the upper one. For $\alpha >0$,The lower branch increases towards $E_{\mathrm{fl}}$ from below. Thus an indirect gap exists in the exactly flat-band model. In the unbounded-parabola limit its lower bound isA finite Brillouin zone gives a finite maximum of $E_{-}$ below $E_{\mathrm{fl}}$ and hence a slightly larger indirect gap. This hybridization gap differs from the local direct gap $2\alpha$.

The material can therefore be a band insulator when the lower band is full and the upper band empty. With two spin-degenerate orbitals per unit cell, the required filling isWith spin degeneracy removed, replace two by one for each independent spin sector. Other fillings are metallic in this noninteracting model.

Opposite parity of same-centre s orbitals and axial p orbitals makes their left and right nearest-neighbour hopping integrals opposite in sign. Their Bloch wave sum is consequently odd in wavevector and vanishes at the zone centre and boundary. Away from those points, orbital hybridization repels coupled levels and can produce an avoided crossing.

For orbital angular momentum $\ell =1$ and spin $s=1/2$, considerThe quantum number $m_j=m_{\ell }+m_s$ is conserved. The stretched states with $m_j=\pm 3/2$ have energies $K\pm 2a$. The remaining eigenvalues follow from two $2\times 2$ blocks with off-diagonal entries $\sqrt{2}K$:The subscripts denote $m_j=\pm 1/2$. For $K>0$, the zero-field multiplets are $j=3/2$ at $K$ and $j=1/2$ at $-2K$. Their weak-field Zeeman splitting is $g_{j}a{m}_j$, with Landé g-factors $4/3$ and $2/3$, respectively. When $a\gg |K|$, the Paschen-Back effect gives approximate product states $(m_{\ell },m_s)$ and energies $a(m_{\ell }+2m_s)+2K{m}_{\ell }{m}_s$. Mixing within a fixed $m_j$ causes an avoided crossing; different $m_j$ sectors can cross because the Hamiltonian does not couple them.