Tight-binding & Bloch's theorem

The foundational move of single-particle solid-state theory: a Hamiltonian commuting with the lattice-translation group is block-diagonalized by crystal momentum $\mathbf{k}$, and within each block the LCAO (tight-binding) ansatz turns the band problem into a finite generalized eigenproblem $H(\mathbf{k})c=E(\mathbf{k})S(\mathbf{k})c$. Every $\mathbf{q}$-resolved eigenproblem downstream — the dynamical matrix above all — is the same structure: Fourier transform real-space couplings, diagonalize at each point of the Brillouin zone.

Bloch’s theorem

Let $T_{\mathbf{R}}$ translate by a Bravais lattice vector $\mathbf{R}$. Since the crystal potential is lattice-periodic, $[H,T_{\mathbf{R}}]=0$ for all $\mathbf{R}$, and the $T_{\mathbf{R}}$ commute among themselves ($T_{\mathbf{R}}{T}_{{\mathbf{R}}^{\prime }}=T_{\mathbf{R}+{\mathbf{R}}^{\prime }}$). Simultaneous eigenstates of $H$ and the abelian translation group carry a one-dimensional irrep of that group, labelled by a wavevector $\mathbf{k}$ in the first Brillouin zone:

$$T_{\mathbf{R}}{\psi }_{n\mathbf{k}}=e^{-i\mathbf{k}\cdot \mathbf{R}}{\psi }_{n\mathbf{k}}.$$

Equivalently the eigenstates factor into a plane wave times a cell-periodic part,

$${\psi }_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot \mathbf{r}}{u}_{n\mathbf{k}}(\mathbf{r}),u_{n\mathbf{k}}(\mathbf{r}+\mathbf{R})=u_{n\mathbf{k}}(\mathbf{r}),$$

with $n$ the band index. The label $\mathbf{k}$ lives in ${\mathbb{T}}^d={\mathbb{R}}^d/2\pi b$ (the BZ as a torus): $\mathbf{k}$ and $\mathbf{k}+\mathbf{G}$ are physically identical for any reciprocal lattice vector $\mathbf{G}$, the gauge $e^{i\mathbf{k}\cdot \mathbf{r}}$ being the only thing that distinguishes them. Inserting the factorized form into $H\psi =E\psi$ gives the cell Hamiltonian

$$H(\mathbf{k})=e^{-i\mathbf{k}\cdot \mathbf{r}}H{e}^{i\mathbf{k}\cdot \mathbf{r}},H(\mathbf{k})u_{n\mathbf{k}}=E_n(\mathbf{k})u_{n\mathbf{k}},$$

a family of operators acting on the single primitive cell with periodic boundary conditions, one for each $\mathbf{k}$. The continuous problem is thereby reduced to one cell at the cost of a parameter $\mathbf{k}$. The two standard conventions for the phase — the “periodic” gauge $e^{i\mathbf{k}\cdot \mathbf{r}}$ used here versus the “atomic/TB” gauge $e^{i\mathbf{k}\cdot (\mathbf{r}-\mathbf{\tau })}$ that puts the orbital position $\mathbf{\tau }$ in the phase — differ by a $\mathbf{k}$-dependent unitary and matter the moment one differentiates in $\mathbf{k}$ (Berry connection, Wannier centers); flag the convention before computing any geometric quantity.

The LCAO / tight-binding eigenproblem

Expand in a basis of localized orbitals ${\varphi }_{\mu }(\mathbf{r}-{\mathbf{\tau }}_{\mu }-\mathbf{R})$ ($\mu$ indexing orbital and sublattice, ${\mathbf{\tau }}_{\mu }$ its position in the cell). Bloch sums build symmetry-adapted basis functions of definite $\mathbf{k}$:

$${\chi }_{\mu \mathbf{k}}(\mathbf{r})=\frac{1}{\sqrt{N}}\sum _{\mathbf{R}}{e}^{i\mathbf{k}\cdot (\mathbf{R}+{\mathbf{\tau }}_{\mu })}{\varphi }_{\mu }(\mathbf{r}-{\mathbf{\tau }}_{\mu }-\mathbf{R}).$$

Writing ${\psi }_{n\mathbf{k}}={\sum }_{\mu }{c}_{n\mu }(\mathbf{k}){\chi }_{\mu \mathbf{k}}$ and projecting onto the ${\chi }_{\nu \mathbf{k}}$ yields the secular equation

$$\boxed{H(\mathbf{k})c_n(\mathbf{k})=E_n(\mathbf{k})S(\mathbf{k})c_n(\mathbf{k})}$$

a generalized Hermitian eigenproblem of dimension = number of orbitals per cell, with

$$H_{\nu \mu }(\mathbf{k})=\sum _{\mathbf{R}}{e}^{i\mathbf{k}\cdot (\mathbf{R}+{\mathbf{\tau }}_{\mu }-{\mathbf{\tau }}_{\nu })}⟨{\varphi }_{\nu }(0)|H|{\varphi }_{\mu }(\mathbf{R})⟩,S_{\nu \mu }(\mathbf{k})=\sum _{\mathbf{R}}{e}^{i\mathbf{k}\cdot (\mathbf{R}+{\mathbf{\tau }}_{\mu }-{\mathbf{\tau }}_{\nu })}⟨{\varphi }_{\nu }(0)|{\varphi }_{\mu }(\mathbf{R})⟩.$$

The real-space matrix elements are the physical content:

• On-site energies ${\epsilon }_{\mu }=⟨{\varphi }_{\mu }(0)|H|{\varphi }_{\mu }(0)⟩$ (the $\mathbf{R}=0$, $\mu =\nu$ terms) — orbital level positions.
• Hopping (transfer) integrals $t_{\nu \mu }(\mathbf{R})=⟨{\varphi }_{\nu }(0)|H|{\varphi }_{\mu }(\mathbf{R})⟩$ — the amplitude to hop between orbitals separated by $\mathbf{R}+{\mathbf{\tau }}_{\mu }-{\mathbf{\tau }}_{\nu }$. Their angular dependence is captured compactly by the Slater–Koster two-center parameters ($V_{ss\sigma },V_{sp\sigma },V_{pp\sigma },V_{pp\pi },\dots$), with three-center and crystal-field corrections neglected in the two-center approximation.

The reality of $H(\mathbf{r})$ together with time-reversal gives $H(\mathbf{k}{)}^{\ast }=H(-\mathbf{k})$, hence $E_n(\mathbf{k})=E_n(-\mathbf{k})$; point-group operations of the crystal relate $H(\mathbf{k})$ at star-equivalent $\mathbf{k}$ and, via the little group, fix the degeneracies at high-symmetry points (see crystal symmetry & space groups).

Orthogonality and the overlap matrix

If the basis is orthonormal — Löwdin/symmetric-orthogonalized orbitals, or a Wannier basis — then $S(\mathbf{k})=1$ and one solves the ordinary eigenproblem $H(\mathbf{k})c=Ec$; this is the form used for model Hamiltonians and Wannier-interpolated bands. For genuine atomic orbitals (Gaussian or numerical AO bases, as in CRYSTAL or SIESTA) the orbitals on different sites overlap, $S(\mathbf{k})\ne 1$, and the generalized problem must be kept. $S(\mathbf{k})$ is Hermitian positive-definite (Gram matrix of linearly independent Bloch sums), so $E_n(\mathbf{k})$ are real and the eigenvectors can be $S$-orthonormalized, $c_n^{†}S(\mathbf{k})c_m={\delta }_{nm}$. Numerically one reduces to a standard problem via the Cholesky factor $S=L{L}^{†}$ and diagonalizes $L^{-1}H{L}^{-†}$; the same metric-aware reduction reappears, with an indefinite metric, in Colpa diagonalization.

The lattice-dynamics analogy

This is the unifying point worth stating explicitly. Lattice dynamics is the same theorem applied to the displacement field instead of the wavefunction. The harmonic equations of motion couple atomic displacements through real-space force constants ${\Phi }_{i\alpha ,j\beta }(\mathbf{R})$ — the exact analogue of hopping integrals — and Bloch’s theorem in the form of a plane-wave ansatz $u_{i\alpha }(\mathbf{R})={ϵ}_{i\alpha }(\mathbf{q})e^{i(\mathbf{q}\cdot \mathbf{R}-\omega t)}$ Fourier-transforms them into the dynamical matrix

$$D_{i\alpha ,j\beta }(\mathbf{q})=\frac{1}{\sqrt{M_i{M}_j}}\sum _{\mathbf{R}}{\Phi }_{i\alpha ,j\beta }(\mathbf{R})e^{-i\mathbf{q}\cdot \mathbf{R}},D(\mathbf{q})ϵ(\mathbf{q})={\omega }^2(\mathbf{q})ϵ(\mathbf{q}),$$

solved at each $\mathbf{q}$. The correspondence is term-for-term:

Electrons (tight binding)	Phonons (lattice dynamics)
crystal momentum $\mathbf{k}$	phonon wavevector $\mathbf{q}$
orbital/site index $\mu$	atom–Cartesian index $(i,\alpha )$
hopping $t_{\nu \mu }(\mathbf{R})$	force constant ${\Phi }_{i\alpha ,j\beta }(\mathbf{R})$
on-site energy ${\epsilon }_{\mu }$	self-term ${\Phi }_{i\alpha ,i\beta }(0)$
cell Hamiltonian $H(\mathbf{k})$	dynamical matrix $D(\mathbf{q})$
eigenvalue $E_n(\mathbf{k})$	${\omega }_{\nu }^2(\mathbf{q})$
(mass weighting → $S=1$)	mass weighting $1/\sqrt{M_i{M}_j}$

The genuine difference is the metric, not reality or symmetry: like $H(\mathbf{k})$, $D(\mathbf{q})$ is in general a complex Hermitian matrix — its off-diagonal blocks carry the $e^{-i\mathbf{q}\cdot \mathbf{R}}$ phases exactly as $H(\mathbf{k})$ does, with $D(\mathbf{q}{)}^{†}=D(\mathbf{q})$ from ${\Phi }_{i\alpha ,j\beta }(\mathbf{R})={\Phi }_{j\beta ,i\alpha }(-\mathbf{R})$, and it is real-symmetric only at $\Gamma$ and the other time-reversal-invariant momenta. The ${\omega }^2$ are real because $D(\mathbf{q})$ is Hermitian. What makes its eigenproblem ordinary rather than generalized is solely that mass-weighting already plays the role of the overlap, $S\to 1$, whereas the non-orthogonal AO basis keeps $S(\mathbf{k})\ne 1$. The exact constraint from translational invariance is the acoustic sum rule, ${\sum }_{j,\mathbf{R}}{\Phi }_{i\alpha ,j\beta }(\mathbf{R})=0$ — the sum runs over the partner atom $j$ as well as the cells $\mathbf{R}$ — which sends three branches to $\omega \to 0$ as $\mathbf{q}\to 0$, a universal constraint the electronic problem has no analogue of. Recognizing the shared skeleton is what makes the Green’s-function machinery, Wannier interpolation, and the finite-displacement phonon workflow read as variations on one theme.

Prerequisites

Linear algebra of periodic (lattice-translation–invariant) Hamiltonians · Fourier series on a Bravais lattice · generalized Hermitian eigenproblems

Builds toward: maximally-localized Wannier functions · the dynamical matrix · modern theory of polarization

Key references

• Bloch’s theorem & band theory — F. Bloch, Z. Phys. 52, 555 (1929); Ashcroft & Mermin, Solid State Physics (Holt, 1976), Ch. 8.
• Tight binding / LCAO — J. C. Slater & G. F. Koster, Phys. Rev. 94, 1498 (1954); W. A. Harrison, Electronic Structure and the Properties of Solids (Freeman, 1980); review: C. M. Goringe, D. R. Bowler & E. Hernández, Rep. Prog. Phys. 60, 1447 (1997).
• Non-orthogonal basis & Löwdin — P.-O. Löwdin, J. Chem. Phys. 18, 365 (1950); Cholesky reduction: Golub & Van Loan, Matrix Computations, 4th ed. (JHU Press, 2013).
• Lattice dynamics & the dynamical matrix — M. Born & K. Huang, Dynamical Theory of Crystal Lattices (Oxford, 1954); A. A. Maradudin, E. W. Montroll, G. H. Weiss & I. P. Ipatova, Theory of Lattice Dynamics in the Harmonic Approximation, 2nd ed. (Academic, 1971).
• Wannier connection — G. H. Wannier, Phys. Rev. 52, 191 (1937).