Tight-Binding Model

Learning Objectives

Section 4.2 started from plane waves and asked: what happens if I gently switch on the lattice? The lattice answered by opening tiny gaps. That works beautifully for sodium, aluminium, and most simple metals — materials where the valence electrons are almost free.

But take CdSe, MnO, NiO, or any d- or f-electron solid. Their valence electrons are not almost free. They sit close to the parent atomic orbitals; only a small fraction of their wavefunction leaks onto neighbouring sites. Try to expand them in plane waves and you need thousands — the basis is hopelessly inefficient. The physics begs for a different starting point: start from atomic orbitals and let the crystal weakly couple them. That is the tight-binding model, the second pillar of modern band theory and the language in which the Mn 3d-shell of our Mn:CdSe quantum dot will be most cleanly described.

By the end of this section you should be able to:

1. Construct a Bloch sum from a single atomic orbital and explain why it is the natural basis for a tight-binding Hamiltonian.
2. Identify the two physical parameters of the simplest TB model — the on-site energy $\varepsilon_0$ and the hopping integral $t$ — from their atomic-orbital integrals.
3. Derive the cosine dispersion $\varepsilon(k) = \varepsilon_0 - 2 t \cos(k a)$ for the 1D linear chain, and read the bandwidth, group velocity, and effective mass off it without doing any further algebra.
4. Generalise to two orbitals per cell (bonding/antibonding bands) and to a 2D square lattice $\varepsilon(\mathbf{k}) = -2 t [\cos(k_x a) + \cos(k_y a)]$.
5. Diagonalise a Bloch Hamiltonian in NumPy, recover the cosine band, and extract the bandwidth and effective mass numerically.
6. See how VASP exposes the same picture through LOBSTER / Wannier90 hopping integrals, LORBIT-projected DOS, and the COHP curves used to assess covalency.

The Other Extreme: Atoms First, Crystal Second

Imagine a row of hydrogen atoms placed very far apart — so far that their 1s orbitals do not overlap at all. Each electron sits on its own atom; the spectrum is the atomic 1s level, repeated N times. There are no bands. There is no transport. The material is a perfect insulator made of independent atoms.

Now bring the atoms closer. Their 1s tails begin to overlap. An electron on atom n can tunnel to atom n + 1 — quantum mechanics insists on it the moment the wavefunctions touch. The N-fold degeneracy of the isolated atoms splits into a band of width controlled by the overlap. Squeeze the atoms further and the band widens until, eventually, neighbouring bands touch and you reach the nearly-free regime of Section 4.2.

The Linear Chain of s-Orbitals

Take the simplest non-trivial system: a 1D chain of identical atoms, spacing $a$, one s-orbital $\phi(x)$ per atom, and only nearest-neighbour hopping. Label the atoms by integer $n$; the orbital on site $n$ is $\phi_n(x) = \phi(x - n a)$. We assume the $\{\phi_n\}$ are orthonormal: $\langle \phi_m | \phi_n \rangle = \delta_{mn}$ (this is exact for Wannier functions, an approximation for raw atomic orbitals — we deal with the correction in Section 4.8).

With only on-site and nearest-neighbour matrix elements, the Hamiltonian in this atomic-orbital basis is a giant tridiagonal matrix:

$H_{mn} = \varepsilon_0\,\delta_{mn} - t\,(\delta_{m,n+1} + \delta_{m,n-1})$

Two parameters set the entire model:

• The on-site energy $\varepsilon_0 = \langle \phi_n | H | \phi_n \rangle$ — the energy of an electron localised on a single atom.
• The hopping integral $t = -\langle \phi_n | H | \phi_{n+1} \rangle$ — (minus) the matrix element between neighbouring orbitals. The minus sign is convention; with it, $t > 0$ places the bonding state at the bottom of the band.

Bloch Sums: Atomic Orbitals That Respect Translation

Translation symmetry forbids us from using the localised $\phi_n$ directly: a translation by $a$ turns $\phi_n$ into $\phi_{n+1}$, so they are not eigenstates of the translation operator. To get eigenstates — which is what Bloch's theorem demands — we form a Bloch sum:

$\Phi_k(x) = \frac{1}{\sqrt{N}} \sum_{n} e^{i k a n}\,\phi_n(x)$

This is just a phase-weighted Fourier-style superposition of the atomic orbitals. Translating $x \to x + a$ relabels the sum ($n \to n - 1$) and pulls out a factor $e^{i k a}$ — exactly the Bloch phase. So $\Phi_k$ is a Bloch state with crystal momentum $k$, built from atomic Lego.

On-Site Energy and the Hopping Integral

Before computing the band, let's pin down the two parameters with their actual integral definitions, so you know what you are actually computing when VASP or Wannier90 reports them.

The on-site energy

$\varepsilon_0 = \int \phi^*_n(x)\,\hat{H}\,\phi_n(x)\,dx = \langle \phi_n | \hat{H} | \phi_n \rangle$

For an isolated atom this is just the atomic eigenvalue. In a crystal the surrounding ions shift it slightly — the so-called crystal-field correction — but the change is typically a fraction of $t$, so we lump it into the parameter and move on.

The hopping integral

$t = -\int \phi^*_n(x)\,\hat{H}\,\phi_{n+1}(x)\,dx$

The size of $t$ is governed almost entirely by the overlap of the two neighbouring orbitals: $t \propto e^{-\kappa a}$ for s-orbitals with decay constant $\kappa$. Doubling the bond length can shrink $t$ by an order of magnitude. That exponential sensitivity is why pressure, strain, and lattice distortions can change the bandwidth dramatically — and why metal-insulator transitions are so often driven by structural changes.

Order of magnitude estimates. For 3d transition-metal oxides, valence-band hopping is $t \sim 0.2$–$0.5$ eV. For Cd–Se covalent bonds in CdSe, $t \sim 1$–$2$ eV. For Mn 3d in Mn:CdSe, the d-d hopping is $t \sim 0.05$ eV — ten times smaller, hence the famously narrow Mn d-bands and the strong electron correlation that DFT alone cannot describe (Chapter 6).

Diagonalising the 1D Chain — The Cosine Band

Plug the Bloch sum $\Phi_k$ into the Schrödinger equation. Since the basis has a single orbital per cell, the Hamiltonian projected onto the Bloch sums is a single number:

$\varepsilon(k) = \langle \Phi_k | \hat{H} | \Phi_k \rangle$

Substitute the definition of $\Phi_k$, keep only on-site and nearest-neighbour matrix elements, and the sums collapse to:

$\varepsilon(k) = \varepsilon_0 - t\,e^{i k a} - t\,e^{-i k a} = \varepsilon_0 - 2 t \cos(k a)$

That is the entire band structure of a 1D s-band: a single cosine. Three immediate consequences:

1. The band sits between $\varepsilon_0 - 2t$ (at $k = 0$) and $\varepsilon_0 + 2t$ (at $k = \pm \pi/a$). The total bandwidth is $W = 4 t$.
2. The band bottom is the bonding Bloch sum — all atoms oscillating in phase. The band top is the antibonding sum — alternating signs.
3. Near $k = 0$, expanding the cosine gives $\varepsilon(k) \approx \varepsilon_0 - 2 t + t a^2 k^2$ — a parabola, just like a free electron, but with a different effective mass.

Interactive: Bloch Amplitude and the Cosine Dispersion

Below: a 9-atom slice of the chain (top), the $\varepsilon(k)$ band (bottom), and three sliders. Drag $k$ first — watch the cyan and orange bars on each atom flip from in-phase (bonding, band bottom) at $k = 0$ to alternating (antibonding, band top) at $k = \pi/a$. Then change $t$ and watch the entire band stretch like an accordion: the bandwidth bracket on the right shows you $W = 4t$ live.

Why the band has the shape it has. The cosine is not an accident of mathematics — it is the Fourier transform of a single “hop” on a 1D lattice. In any nearest-neighbour TB model on any lattice, the dispersion is $\varepsilon(\mathbf{k}) = \varepsilon_0 - t\,\sum_{\mathbf{\delta}} e^{i \mathbf{k}\cdot\mathbf{\delta}}$ where the sum is over neighbour vectors $\mathbf{\delta}$. Choose the neighbours, and the band shape follows.

Reading the Band: Bandwidth, Mass, and Group Velocity

Three quantities can be read straight off the cosine without any further calculation:

The negative effective mass at the band top is not a typo. The upper half of the band has its curvature flipped, so an electron accelerated by an electric field there moves opposite the field. This is exactly the “hole” behaviour of semiconductors — the empty states near the top of a filled valence band act like positive carriers.

Two Orbitals per Cell — Where Bonding/Antibonding Live

Most real materials have more than one orbital per cell. The smallest meaningful upgrade is two orbitals per cell, labelled $A$ and $B$ (think H₂ molecules arranged in a chain, or the alternating Cd / Se sublattices of CdSe). With on-site energies $\varepsilon_A$ and $\varepsilon_B$ and intra-cell hopping $t$, the Bloch Hamiltonian is now 2 × 2:

$H(k) = \begin{pmatrix} \varepsilon_A & -t f(k) \\ -t f^*(k) & \varepsilon_B \end{pmatrix},\quad f(k) = 1 + e^{i k a}$

Diagonalising gives two bands:

$\varepsilon_\pm(k) = \frac{\varepsilon_A + \varepsilon_B}{2} \pm \sqrt{\left(\frac{\varepsilon_A - \varepsilon_B}{2}\right)^2 + t^2 |f(k)|^2}$

When $\varepsilon_A = \varepsilon_B$ (a homonuclear chain) the two bands touch at $k = \pi/a$ — you have recovered the single cosine band by “unfolding”. When $\varepsilon_A \neq \varepsilon_B$ a gap of size $|\varepsilon_A - \varepsilon_B|$ opens at the BZ boundary — the same mechanism that makes ionic crystals (NaCl) and polar semiconductors (CdSe, GaAs) insulators even when the simple counting argument predicts a metal.

Up a Dimension: The 2D Square Lattice

Stack the chain into a 2D square lattice with one s-orbital per site, lattice constant $a$, and nearest-neighbour hopping $t$ in both $x$ and $y$ directions. The Bloch Hamiltonian is again 1 × 1 (one orbital per cell), and summing over the four nearest neighbours gives:

$\varepsilon(\mathbf{k}) = \varepsilon_0 - 2 t\bigl[\cos(k_x a) + \cos(k_y a)\bigr]$

Three special k-points organise the geometry of this band:

The saddle point at $X$ is responsible for the famous logarithmic van Hove singularity in the density of states — a feature that drives high-temperature superconductivity in cuprates and ferromagnetism in many transition-metal oxides.

Interactive: 2D Square-Lattice Band Surface

Left: heatmap of $\varepsilon(k_x, k_y)$ over the entire first Brillouin zone. The dotted contours mark the $\varepsilon = -2t,\,0,\,+2t$ iso-energy surfaces. Right: the band along the canonical path $\Gamma \to X \to M \to \Gamma$ you will see in every band-structure paper. Drag $(k_x, k_y)$ with the sliders — the purple dot follows on both panels simultaneously.

Code Walk-Through: Building H(k) and the Cosine Band

Time to compute. The script below builds the (1 × 1) Bloch Hamiltonian of the 1D chain, sweeps it across the first Brillouin zone, and verifies the bandwidth and effective-mass formulas analytically. Click any line on the right to see exactly which Python objects, NumPy calls, and physical numbers that line is producing.

[[-1.0000]]   (shape (1, 1), at k = 0)

1import numpy as np
2
3# === Tight-binding parameters (atomic units, hbar = m_e = 1) ===
4a = 1.0           # lattice constant
5eps0 = 0.0        # on-site energy of the s-orbital
6t = 0.5           # nearest-neighbour hopping integral
7
8# === Bloch Hamiltonian H(k) ===
9# For a 1D chain with one s-orbital per cell, the Bloch matrix is 1 x 1:
10#     H(k) = eps0 - 2 * t * cos(k * a)
11def H_bloch(k):
12    """Return the 1x1 Bloch Hamiltonian at wavevector k (atomic units)."""
13    return np.array([[eps0 - 2 * t * np.cos(k * a)]])
14
15# === Sample the band over the first Brillouin zone ===
16ks = np.linspace(-np.pi / a, np.pi / a, 9)
17print("    k          eps(k)")
18print("-" * 28)
19for k in ks:
20    eig = np.linalg.eigvalsh(H_bloch(k))[0]
21    print(f"{k:+.4f}     {eig:+.4f}")
22
23# === Bandwidth and effective mass at k = 0 ===
24W = 4 * t                              # full bandwidth W = 4 t
25m_star = 1.0 / (2 * t * a ** 2)        # m* = hbar^2 / (2 t a^2)
26print(f"\nBandwidth W = 4 t          = {W:.3f}")
27print(f"Effective mass m* (k = 0)  = {m_star:.3f}")

Run the script and you will see the following output:

1k          eps(k)
2----------------------------
3-3.1416     +1.0000
4-2.3562     +0.7071
5-1.5708     -0.0000
6-0.7854     -0.7071
7+0.0000     -1.0000
8+0.7854     -0.7071
9+1.5708     -0.0000
10+2.3562     +0.7071
11+3.1416     +1.0000
12
13Bandwidth W = 4 t          = 2.000
14Effective mass m* (k = 0)  = 1.000

Three sanity checks to perform before you trust this kind of code on a real material:

• The band is symmetric in $k$ (time-reversal symmetry: $\varepsilon(k) = \varepsilon(-k)$) ✓
• The endpoints $k = -\pi$ and $k = +\pi$ give identical energies (BZ periodicity) ✓
• The bandwidth is exactly 4t and the effective mass exactly $1/(2 t a^2)$ ✓

Multi-Orbital Crystals: Slater–Koster in One Picture

Real solids have several orbitals per atom (s, p, d …) and their hopping integrals depend on the relative orientation of the bond and the lobes. The systematic accounting is the Slater–Koster scheme (Slater & Koster, 1954). It expresses the matrix element between any two orbitals in a diatomic bond as a linear combination of just four irreducible parameters: $(ss\sigma)$, $(sp\sigma)$, $(pp\sigma)$, and $(pp\pi)$ — one per overlap symmetry.

Every full tight-binding calculation in this book reduces to: choose the orbital basis, look up (or fit) the Slater-Koster parameters, build a ~10 × 10 Bloch Hamiltonian, and diagonalise on a k-grid. The resulting bands match VASP/PAW DFT bands to a few percent in the valence region.

Tight Binding vs Nearly Free Electrons: Two Views, One Truth

Here is the same band, the same physics, expressed in the languages of the two opposite limits:

VASP's plane-wave DFT solves the full problem and is closer in spirit to the NFE side; once it's done, post-processing through Wannier functions translates the result into the TB language so chemists, theorists, and model-Hamiltonian practitioners can see what is going on.

VASP Connection: Wannier Functions, COHP, Projected DOS

VASP uses plane waves, but the post-processing pipeline gives you every tight-binding number you might want. Three workflows are worth knowing.

1. Projected DOS via LORBIT

Setting LORBIT = 11 in INCAR tells VASP to project the Kohn-Sham wavefunctions onto spherical harmonics centred on each ion. The output (PROCAR, DOSCAR) tells you which fraction of each band has s, p, or d character on each atom — the answer to “is this CdSe band Cd 5s or Se 4p?”.

1# INCAR — atom- and orbital-resolved DOS / PROCAR
2LORBIT = 11        # decompose by ion AND lm channel (s, py, pz, px, dxy, ...)
3NEDOS  = 3001      # smooth DOS grid
4ISMEAR = -5        # tetrahedron method — accurate DOS for insulators

2. Wannier90 — the actual TB Hamiltonian

With LWANNIER90 = .TRUE. VASP cooperates with the external Wannier90 program to construct maximally localised Wannier functions — the precise mathematical incarnation of the orbitals $\phi_n$ we have been treating symbolically. Wannier90 then writes the hopping integrals $t_{mn}(\mathbf{R})$ to a file (seedname_hr.dat) so you can build any $H(\mathbf{k})$ you want by Fourier transform — the same equation we wrote out by hand for the 1D chain, just with a few more orbitals and a richer $\mathbf{R}$-set.

1# wannier90.win (excerpt, for the Cd–Se 4 valence + 4 conduction bands)
2num_wann      = 8        # number of Wannier functions to construct
3num_bands     = 16       # bands to disentangle from
4projections
5 Cd:s; Cd:p
6 Se:s; Se:p
7end projections
8
9# Output of interest:
10#   wannier90_hr.dat   — H(R) hopping integrals (this IS the TB model)
11#   wannier90_band.dat — band structure built from H(k) = Σ_R H(R) e^{ikR}

3. LOBSTER + COHP — covalency in numbers

LOBSTER (the “Local Orbital Basis Suite Towards Electronic Structure Reconstruction”) reads VASP's plane-wave wavefunctions and rewrites them in a localised atomic-orbital basis. The output includes Crystal Orbital Hamilton Population (COHP) curves: integrated bonding character as a function of energy. A negative COHP at the Fermi level means the bond is anti-bonding there — a recipe for instability, often a signal of imminent magnetic or structural ordering.

1# lobsterin (run AFTER a converged VASP single-point with LORBIT = 11)
2COHPstartEnergy   -10
3COHPendEnergy       5
4basisSet pbeVaspFit2015
5includeOrbitals s p
6cohpBetween atom 1 atom 2
7cohpBetween atom 1 atom 3   # all symmetry-distinct neighbour pairs

Summary

1. The tight-binding model starts from atomic orbitals and lets the lattice couple them through a hopping integral $t$. It is the natural language whenever electrons are well-localised (d- and f-bands, 2D oxides, molecular crystals, the Mn d-shell of Mn:CdSe).
2. Bloch sums $\Phi_k = N^{-1/2}\sum_n e^{i k a n}\phi_n$ are the TB analogue of plane waves: translation eigenstates built from atomic Lego.
3. Two parameters — on-site energy $\varepsilon_0$ and hopping $t$ — determine the entire 1D s-band: $\varepsilon(k) = \varepsilon_0 - 2 t \cos(k a)$. Bandwidth is $W = 4t$; effective mass at the bottom is $m^* = \hbar^2/(2 t a^2)$.
4. Two orbitals per cell give two bands separated by an on-site energy gap $|\varepsilon_A - \varepsilon_B|$ — the same mechanism that opens the band gap of polar semiconductors and ionic insulators.
5. On the 2D square lattice the band is $\varepsilon(\mathbf{k}) = -2 t[\cos(k_x a) + \cos(k_y a)]$ — with a $-4t$ minimum at $\Gamma$, a saddle point at $X$, and a perfect-nesting Fermi surface at half-filling.
6. NFE and TB are two limits of the same problem. NFE wins for delocalised electrons (Na, Al); TB wins for localised ones (Mn d, oxides). The full DFT calculation done by VASP interpolates between them — and Wannier90, LOBSTER, and LORBIT-projected DOS expose the answer in TB language.

Next section we step back and ask the awkward question: every single-particle picture we have used so far ignores the fact that electrons repel each other. The free electron gas, the NFE avoided crossing, the tight-binding cosine — none of them contain a Coulomb interaction. The genuine many-body problem of ~10²³ interacting electrons is the next mountain to climb, and the path up it is called density functional theory.