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For $t(a)=-2E_1{e}^{-a/a_1}$ with $E_1,a_1>0$, a nearest-neighbour cubic tight-binding model has zone-centre band effective massIts logarithmic derivative is $1/a_1-2/a$. The unique minimum occurs at $a=2a_1$ and has value ${\hslash }^2\exp (2)/(16E_1{a}_1^2)$. The explicit $a^2$ in band curvature competes with the exponentially weakening hopping integral. Extrapolation to vanishing atomic separation lies outside the localized-orbital approximation.

The electronic bandwidth is the difference between the maximum and minimum of one energy band over the Brillouin zone. It has units of energy, unlike a frequency bandwidth in signal processing. A nearest-neighbour tight-binding model on a primitive cubic lattice with equal real hopping integrals $t$ has $E(\mathbf{k})=E_0+2t{\sum }_{i=1}^3\cos (k_{i}a)$ and $W=12|t|$.

A hopping integral is an off-diagonal Hamiltonian matrix element between localized orbitals on different lattice sites. It has units of energy and controls the dispersion in a tight-binding model. Its sign and phase depend on the orbital phase convention; observable band energies do not. The sign of a real nearest-neighbour $t$ in $E(k)=E_0+2t\cos (ka)$ determines the location of the minimum in that convention.

In the orthogonal tight-binding approximation, localized basis states satisfy $⟨i|j⟩={\delta }_{ij}$ while the Hamiltonian retains nonzero hopping integrals. These are independent matrix elements: orthogonal states need not have vanishing Hamiltonian coupling. One can construct localized orthonormal orbitals from overlapping atomic orbitals; truncating the resulting Hamiltonian to nearest neighbours is a further approximation. An overlap matrix must instead be retained for raw nonorthogonal orbitals.

The band effective mass is determined by the second derivative of the dispersion relation:With the stated sign convention and $t>0$, the band minimum is at $k=0$. Expanding the cosine, or differentiating twice, givesComparison with a free Electron's kinetic energy ${\hslash }^2{k}^2/(2m_e)$ therefore requiresThe hopping integral has units of energy. A negative $t$ would move the minimum to the edge of the Brillouin zone and give the same positive bottom-of-band effective mass with $|t|$ in this expression.

Both models respect Bloch's theorem and describe Electrons in a periodic lattice, but they start from different useful limits.

In the nearly-free electron model, start with extended plane waves and free-electron energies ${\epsilon }_{\mathbf{k}}^0={\hslash }^2{k}^2/(2m)$. A weak periodic potential has Fourier coefficients $V_{\mathbf{G}}$ at reciprocal lattice vectors and couples $\mathbf{k}$ to $\mathbf{k}-\mathbf{G}$. Away from degeneracies its corrections are small. Near a Bragg scattering, retain the two resonant plane waves and diagonalizeThe resulting eigenvalues areAt the crossing the gap is $\boxed{2|V_{\mathbf{G}}|}$. The two standing-wave combinations place electronic density differently relative to the periodic potential and therefore acquire different energies. Folding the many free parabolas into the first Brillouin zone produces the repeated band labels; the nonzero potential opens avoided crossings between coupled states. A Fourier component forbidden by symmetry need not open its corresponding first-order gap.

In the tight-binding model, instead start with localized orbitals on well-separated atoms or sites, with discrete on-site energies. Their overlap permits tunneling represented by hopping integrals. Bloch sums of these orbitals diagonalize the lattice translation symmetry. For one orthonormal orbital per site in a one-dimensional nearest-neighbor chain,The many initially degenerate copies of one atomic orbital broaden into a band. Other atomic orbitals or multiple sites per primitive cell produce additional bands; their hybridization creates bonding and antibonding combinations and may open gaps. For two orbitals per cell, one illustrative Hamiltonian iswith eigenvalues $(E_A+E_B)/2\pm \sqrt{(E_A-E_B{)}^2/4+|t_1+t_2{e}^{-ika}{|}^2}$. This shows explicitly how two bands arise from the two cell degrees of freedom, even when the isolated sites have equal on-site energies.

For each orbital degree of freedom per primitive cell there is one band per spin; band filling and the presence of a gap at the Fermi level determine metallic or insulating behavior. The nearly-free limit emphasizes weak Bragg scattering of extended states, while tight binding emphasizes hopping between localized states. They are complementary approximations to electronic band structure, rather than two incompatible explanations of why periodic solids have bands.

Use one orthonormalized localized atomic orbital $|j⟩$ at each position $ja$. In the nearest-neighbour tight-binding model, translation invariance gives the on-site energy and real hopping integralThe Bloch state isSince $H|j⟩=E_0|j⟩+t|j-1⟩+t|j+1⟩$, shifting the two summation indices givesThe hopping integral is an energy, and its sign sets whether $k=0$ is a minimum or maximum. $E_0$ is the diagonal matrix element in the crystal Hamiltonian, which can differ from an isolated-atom energy.

The orthogonal tight-binding approximation matters. If raw neighbouring orbitals retain a nonzero orbital overlap $s$ in their normalization, projection instead gives a generalized eigenvalue problem and the nonorthogonal nearest-neighbour tight-binding band $E(k)=[E_0+2h\cos (ka)]/[1+2s\cos (ka)]$. The stated cosine band follows after orthonormalization and nearest-neighbour truncation, or after neglecting that overlap metric while retaining the hopping it motivates.

A site in a primitive cubic lattice has six nearest neighbours, at $\pm a\hat{\mathbf{x}}$, $\pm a\hat{\mathbf{y}}$ and $\pm a\hat{\mathbf{z}}$. Repeating the Bloch state calculation with equal hopping integrals $t$ along all six bonds givesThe cosine sum ranges from $-3$ to $3$ within the first Brillouin zone. ThereforeFor $t<0$ the minimum is at the zone centre; for $t>0$ the maximum is there.