= Solution
In the <Bloch wave> sum, $N$ is the number of sites, $R_j=ja$ the equilibrium position of site $j$, $\phi(r-R_j)$ a localized atomic orbital there, and $k$ the crystal wavevector. The phase factor gives the translation character of the <Bloch state>. The factor $1/\sqrt N$ normalizes it when the site orbitals are treated as orthonormal; appreciable intersite overlap requires an overlap-dependent normalization.
For a <tight-binding model>, choose localized orbitals and calculate the on-site and intersite Hamiltonian matrix elements. Form Bloch sums for each orbital type and solve $H(k)c=E(k)S(k)c$ at each $k$, where $S(k)$ is the orbital-overlap matrix. In the <orthogonal tight-binding approximation>, $S=I$. The eigenvalues give the energy bands and the eigenvectors their orbital composition; the <nonorthogonal nearest-neighbour tight-binding band> illustrates how retained overlap modifies this procedure.
For one orthonormal s orbital per site, keep on-site energy $\epsilon_s$ and real nearest-neighbor <hopping integral> $t=\langle\phi_j|H|\phi_{j+1}\rangle$. The two neighbors contribute phases $e^{\pm ika}$, giving $E_s(k)=\epsilon_s+2t\cos ka$. The band-curvature definition $1/m^*=E''(k)/\hbar^2$ gives
$$
\boxed{m^*(0)=-\frac{\hbar^2}{2ta^2},\qquad m^*(\pm\pi/a)=\frac{\hbar^2}{2ta^2},\qquad W=4|t|.}
$$
With the usual bonding convention $t<0$ for these s orbitals, the center is a minimum with positive electron effective mass and the edge a maximum with negative electron effective mass. Hole masses near a filled-band maximum have the opposite sign.
For p orbitals aligned along the chain, the <orbital phase and the sign of one-dimensional hopping> reverse the usual center-versus-edge ordering. Choose every $p_x$ orbital with its positive lobe on the right. Equal neighboring Bloch coefficients then bring opposite-sign lobes together between sites, giving an antibonding state at $k=0$. Alternating coefficients at $k=\pi/a$ make those facing lobes have the same sign, producing the lower bonding state. Thus a nearest-neighbor model has $E_p(k)=\epsilon_p+2t_\sigma\cos ka$ with the usual $t_\sigma>0$, maximum at zero and minima at the zone edges. This uses a fixed common orbital-phase convention at all sites.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2005-orbital-band-curvature.png]
{title=Opposite band curvature for bonding s orbitals and chain-aligned p orbitals}
{description=Original nearest-neighbor dispersions with independent zero on-site energies and unit hopping magnitudes. The lower diagrams show orbital signs for equal site coefficients at the zone center: s lobes face with the same sign, while neighboring $p_x$ lobes face with opposite signs. Energies and orbital sizes are schematic.}
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