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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.

An energy band formed when localized impurity levels hybridize. For hydrogenic donor states, a typical donor separation $n^{-1/3}$ gives an exponentially small hopping integral and hence an exponentially narrow bandwidth when $n^{-1/3}\gg a_b$. An idealized half-filled band with two spin states per donor has spin-summed density of states of order $2n/W$ per volume; its Pauli paramagnetism is $\chi \sim {\mu }_0(g{\mu }_B/2{)}^2(2n/W)$. That expression requires $k_{B}T\ll W$ and does not by itself describe localization or strong Coulomb interactions.

In an orthogonal one-orbital nearest-neighbor tight-binding model, the hopping integral is $t=⟨{\varphi }_j|H|{\varphi }_{j+1}⟩$ and the dispersion is the displayed cosine. Using the same orbital-phase convention at every site, equal s-orbital coefficients usually make a bonding state at the zone center, so $t<0$ and $k=0$ is a band minimum. For $p_x$ orbitals along the chain, with every positive lobe pointing right, equal site coefficients bring opposite-sign lobes together between atoms. This is antibonding; alternating coefficients are bonding, giving the usual $t_{\sigma }>0$ and a band maximum at zero. Changing conventions requires transforming the Bloch basis consistently. The physical statement concerns the curvature and orbital character of a specified Bloch state, not an isolated hopping sign divorced from its basis.

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.

An overlap estimate must use nearest magnetic-ion separations. For the face-centered cubic lattice of Mn, $d_{\mathrm{Mn}}=a_{\mathrm{Mn}}/\sqrt{2}=3.147\mathring{\mathrm{A}}$; for the body-centered cubic lattice of Fe, $d_{\mathrm{Fe}}=\sqrt{3}{a}_{\mathrm{Fe}}/2=2.477\mathring{\mathrm{A}}$. The supplied radial functions have inverse decay lengths ${\kappa }_Z=Z/(3a_B)$. Their intersite overlap therefore contains approximately $e^{-{\kappa }_{Z}d}$, up to polynomial and angular factors. The decisive exponent difference isThe exponential orbital-tail estimate of exchange thus gives a hopping suppression approximately $e^{-8.99}\simeq 1.25\times 10^{-4}$ before polynomial corrections. Keeping the supplied $Z^{7/2}{r}^2$ radial prefactor in a midpoint overlap proxy givesThis proxy is not an evaluated hopping integral, but confirms that the polynomial factors do not undo the exponential loss. If comparable Coulomb denominators and nonexponential prefactors are assumed, a kinetic-exchange scale proportional to $t^2/U$ is suppressed by roughly $10^{-8}$–$10^{-7}$. Direct intersite Coulomb exchange likewise contains products of the small tails. Coordination-number differences cannot compensate for many orders of magnitude.

By contrast, the measured ordering-temperature ratio supplied in the question is $122/1043\simeq 0.117$, not an exponentially tiny number. Within this overlap model, direct Mn–Mn exchange cannot explain MnO's substantial magnetic ordering scale by extrapolating the Fe mechanism. Another exchange path is required.

This is an order-of-magnitude argument, not an exact relation between an itinerant Curie temperature and a localized-spin Néel temperature. The hydrogenic functions use the printed bare $Z$ values; actual solid-state orbitals are screened and hybridized. Their literal absolute exchange scales should not be treated as quantitative predictions for Fe or MnO.

In the orthogonal nearest-neighbour tight-binding model,The on-site integral ${\alpha }_s$ sets the band centre, while the hopping integral ${\beta }_s$ describes coupling between neighbouring orbitals. With a consistent phase convention, the bonding combination at the zone centre has ${\beta }_s<0$.

For p orbitals directed along the chain, useHere every positive $p$ lobe points in the same spatial direction. Neighbouring equal-sign coefficients then bring opposite-sign facing lobes together and make the zone-centre combination antibonding; the alternating zone-boundary combination is bonding. Thus the usual axial $p$ hopping has the opposite sign to $s$ hopping. Orbital phase and the sign of one-dimensional hopping must be treated consistently; replacing only ${\alpha }_s$ by ${\alpha }_p$ while silently keeping the same physical hopping sign would give the wrong band ordering.

A hydrogenic donor state has an exponentially decaying envelope $\psi (r)\propto e^{-r/a_b}$. Neighbouring donors at typical separation $r_d\sim n^{-1/3}$ therefore have exponentially small orbital overlap. The corresponding hopping integral is of order $E_b{e}^{-r_d/a_b}$, apart from algebraic factors and factors from the number of neighbours. Hybridizing many such localized states produces an impurity band with a bandwidth of the same scale:The order-one constant $\alpha$ absorbs details of the donor spacing and envelope. The exponential dependence expresses quantum tunnelling between weakly overlapping donor orbitals, rather than thermal excitation over the donor binding energy.

The tight-binding model uses Bloch sums of localized atomic-like or Wannier orbitals,Including ${\mathbf{\tau }}_{\alpha }$ in the phase is an equivalent basis convention. Overlap and hopping between these localized orbitals create dispersive bands. Narrow transition-metal $d$ bands and the bonding/antibonding bands of covalent solids are natural examples; graphene's $p_z$ bands are another. The model can be accurate beyond weak hopping when enough orbitals and hopping integrals are retained.

In the Bloch wave sum, $N$ is the number of sites, $R_j=ja$ the equilibrium position of site $j$, $\varphi (r-R_j)$ 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 ${ϵ}_s$ and real nearest-neighbor hopping integral $t=⟨{\varphi }_j|H|{\varphi }_{j+1}⟩$. The two neighbors contribute phases $e^{\pm ika}$, giving $E_s(k)={ϵ}_s+2t\cos ka$. The band-curvature definition $1/m^{\ast }=E^{\prime \prime }(k)/{\hslash }^2$ givesWith 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 $E_p(k)={ϵ}_p+2t_{\sigma }\cos ka$ 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.

Bloch's theorem states that an eigenfunction in a periodic potential can be written ${\psi }_{\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot \mathbf{r}}{u}_{\mathbf{k}}(\mathbf{r})$, with $u_{\mathbf{k}}$ lattice-periodic. In the orthogonal tight-binding approximation, choose localized orthonormal orbitals $|j⟩$ with on-site matrix element $E_0$ and real nearest-neighbour hopping integral $t$. The Bloch sum $|k⟩=N^{-1/2}{\sum }_j{e}^{ikja}|j⟩$ is an eigenstate because each neighbour shift contributes a phase $e^{\pm ika}$. ThusThe simple-cubic nearest-neighbour tight-binding band has six neighbours, giving $E(\mathbf{k})=E_0+2t[\cos (k_{x}a)+\cos (k_{y}a)+\cos (k_{z}a)]$. The cosine sum ranges from $-3$ to $3$, so the positive bandwidth is $W=12|t|$. The stated hopping formula is negative when $E_1>0$, so the printed $12t$ must be read as a bandwidth magnitude.

Near the zone centre, $E=E_0+6t-t{a}^2{k}^2+O(k^4)$. To leading order the constant-energy surfaces are spheres, and comparison with $E_{\Gamma }+{\hslash }^2{k}^2/(2m^{\ast })$ givesThis is the effective mass for overlap-weighted exponential hopping. With $s=a/a_1$, its logarithmic derivative is $d\ln m^{\ast }/ds=1-2/s-1/(1+s)$. The positive stationary point is $s=1+\sqrt{3}$, and it is a minimum. The formula grows at both very small and very large $s$; the small-separation divergence is a mathematical extrapolation into a regime where localized tight binding itself becomes unreliable.

For the bandwidth, $W=24E_1(1+s)e^{-s}$ and $d\ln W/d\ln a=-s^2/(1+s)$. At $s=1$, a fractional spacing change $-0.01$ therefore givesThe exact ratio $W(0.99a_1)/W(a_1)=(1.99/2)e^{0.01}=1.004999916\dots$ checks the first-order result. Compression strengthens the hopping and widens the band.

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.

If a tight-binding Hamiltonian is written with coefficient $-t_{ij}$ multiplying $c_i^{†}{c}_j$, projecting the one-electron Hamiltonian gives the displayed relation. An alternative convention defines the coefficient itself as the hopping integral and has no leading minus. Orthogonality removes constant scalar potentials from off-diagonal matrix elements, but does not remove a spatially varying potential. Orbital rephasing changes both matrix elements together and cannot repair an algebraic mismatch between these two conventions in a fixed basis.

For one orthonormal localized orbital per simple-cubic site and real equal nearest-neighbour hopping integral $t$, the six neighbours give the displayed energy band. Each cosine ranges from minus one to one, so the positive bandwidth is $12|t|$, independently of the sign convention for $t$. Near the zone centre, $E=E_0+6t-t{a}^2{k}^2+O(k^4)$ and the band effective mass is $-{\hslash }^2/(2t{a}^2)$. It is positive there for $t<0$; for $t>0$ the zone centre is a maximum instead.

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.

Let ${\mathbf{a}}_1,{\mathbf{a}}_2$ be linearly independent primitive translations of a two-dimensional Bravais lattice. Take one orthonormal orbital per site, on-site energy $E_0$, and equal real hopping integrals $-\lambda$ to the four neighbours at $\pm {\mathbf{a}}_1,\pm {\mathbf{a}}_2$. A Bloch state givesThe two phases can vary independently across a reciprocal primitive cell. Their cosine sum reaches both $2$ and $-2$, so the energy band ranges from $E_0-4|\lambda |$ to $E_0+4|\lambda |$, with electronic bandwidth $8|\lambda |$.

The independence assumption matters. If instead ${\mathbf{a}}_2=2{\mathbf{a}}_1$, the phases are locked. Writing $c=\cos (\mathbf{k}\cdot {\mathbf{a}}_1)$ gives $\cos \theta +\cos 2\theta =2(c+1/4{)}^2-9/8$, whose range is $[-9/8,2]$. For $\lambda >0$ that one-dimensional model has electronic bandwidth $25\lambda /4$, rather than $8\lambda$.