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A symmetry transformation acts on localized atomic orbitals both by moving their centres and by transforming their angular shapes. Only centres left in place can contribute to the trace of the corresponding group representation. At each fixed centre, sum the diagonal coefficients of the orbital transformation: a reversed p orbital contributes $-1$, whereas a fixed s orbital contributes $+1$. For Cartesian displacement vectors at a fixed atom, use the trace of the full polar-vector matrix, not merely the number of fixed components. This rule generates characters of a representation without writing its entire matrix.

The opposite spatial parity of the s orbital and axial p orbital gives opposite signs for left and right $s$–$p$ hopping. With real ${\beta }_{sp}$, one consistent Bloch-basis convention givesThe resulting energies areThus s-p mixing in a one-dimensional tight-binding chain lowers the lower branch and raises the upper branch wherever the coupling is nonzero. For separated atomic levels, the lower branch remains mainly $s$ and the upper mainly $p$, with increasing admixture where their uncoupled energies approach.

The upper-right panel of the figure in part (a)(ii) compares the mixed curves with dashed unmixed curves. The mixing vanishes at both $k=0$ and $\pi /a$, so those endpoints remain unchanged. If the unmixed curves approach or cross in the interior, the nonzero coupling produces an avoided crossing.

The drawn $x,y$ axes pass through opposite sites, rather than along the square edges. In the D4h molecular point group class order $(E,2C_4,C_2,2C_2^{\prime },2C_2^{\prime \prime },i,2S_4,{\sigma }_h,2{\sigma }_v,2{\sigma }_d)$, an s orbital contributes $+1$ whenever its centre is fixed and zero otherwise. ThusFor example, a transverse twofold rotation about the printed $x$ axis fixes sites 1 and 3, while its diagonal counterpart exchanges all four sites. Reflection in the square plane fixes every site. Taking character inner products, with class sizes $(1,2,1,2,2,1,2,1,2,2)$, givesThese have dimensions $1+1+2=4$. The $B_{1g}$ label is tied to the printed diagonal-axis convention; rotating the transverse coordinate labels by $45^{\circ }$ would interchange $B_{1g}$ and $B_{2g}$.

Use the trapezium's printed axes, with principal axis $z$ in its plane and $y$ perpendicular to the plane. For either site pair $(1,2)$ or $(3,4)$, the group representation has characters of a representation $(2,0,2,0)$ in the order $(E,C_2(z),\sigma (xz),\sigma (yz))$: identity and the molecular-plane mirror fix both s orbitals, while the other two operations exchange them. The character inner products give ${\Gamma }_{12}={\Gamma }_{34}=A_1\oplus B_1$ in the C2v molecular point group.

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.