Electronic Structure Methods Overview

An electronic-structure method supplies an approximation to a specified electronic problem. In a solid, the output may be an effective hopping Hamiltonian, a ground-state density and energy, an auxiliary band structure, or an interacting spectrum. These objects can resemble one another on an energy-versus-momentum plot while answering different physical questions.

Choose the requested observable before choosing the method. This page explains what the common solids methods produce, what information their outputs omit, and which additional calculation connects them to spectroscopy or material energetics. The formal derivations are linked beside each method; practical setup and convergence are developed in Band Structure Workflows.

Required background. Band Theory Overview supplies Bloch eigenvalues, filling, occupied subspaces and global gap definitions.

Helpful background. From Quantum Mechanics to Materials supplies physical reductions and parameter provenance. The Electronic Structure Methods Map compares general method families and their errors. Spectral Functions supplies poles, weights, continua and probe-dependent intensities.

Electronic bands are method-dependent objects

Fix the atomic structure, electron count, boundary conditions, magnetic state and relativistic treatment. Then name the output precisely. “The band gap of this material” is insufficient if one calculation returns a Kohn–Sham eigenvalue difference and the experiment measures the first bright neutral excitation.

Requested object	Suitable starting description	Additional question before a material claim
Symmetry, connectivity or a low-energy dispersion	Declared tight-binding or first-principles effective Hamiltonian	Does the retained subspace and its parameter range represent the system?
Ground-state structure or energy difference	A specified density functional, determinant method or correlated total-energy method	Are competing states compared with balanced physical models and numerical convergence?
Electron addition or removal peak	A one-particle Green function with a controlled pole estimate	Are residue, linewidth and incoherent weight consistent with a sharp peak?
Neutral optical excitation	A response calculation with appropriate electron–hole physics	Are matrix elements, screening, excitons and measurement geometry included?

These are starting descriptions, not an accuracy ranking. A more expensive calculation of the wrong object does not answer the original question. Band Gaps defines the distinctions among band, fundamental, optical and transport gaps.

Empirical and Wannier tight binding

A tight-binding matrix $H_{ab}(\mathbf k)$ acts in a chosen orbital subspace. Its eigenvalues are exact for that matrix, up to numerical error. Whether they describe a real crystal depends on the model construction. Parameters fitted to measured quasiparticle dispersions and parameters obtained by projecting Kohn–Sham bands have different meanings even if both are called hopping amplitudes.

A fit to energies does not uniquely fix eigenvectors, orbital embedding or response operators. For optical matrix elements, Berry curvature or coupling to a field, transform the relevant operators as well as the Hamiltonian. Eliminating remote bands may generate frequency dependence or additional couplings that a static short-range matrix discards.

Wannier interpolation changes the representation and sampling of a chosen band subspace. It does not by itself add missing electronic correlations. Marzari et al. (2012) describe this distinction between a localized representation and its applications. Use Wannierization Workflows for window selection, disentanglement, interpolation checks and transformed operators.

Hartree–Fock bands

Hartree–Fock optimizes a Slater determinant and includes its exchange exactly for the chosen interaction and representation. The variational energy statement concerns the determinant’s total energy for a fixed Hamiltonian, not a guarantee about every orbital eigenvalue. Frozen-orbital removal energies have a Koopmans interpretation; orbital relaxation and correlation change actual addition/removal energies.

In extended solids, unscreened exchange can behave poorly where electronic screening is essential. This is a physical approximation issue, not a failure that a denser momentum mesh necessarily cures. Hartree–Fock Approximation derives the variational construction and its limitations. A hybrid functional or a screened-exchange approximation must be identified separately from bare Hartree–Fock.

Kohn–Sham and generalized Kohn–Sham bands

The Kohn–Sham construction represents the ground-state density with auxiliary orbitals. Its total-energy functional contains kinetic, external, Hartree and exchange-correlation contributions. The occupied eigenvalue sum alone double counts some contributions and is not the total energy. Kohn and Sham (1965) provide the foundational self-consistent construction.

Practical calculations choose an approximate functional. A converged density or a plausible occupied-band dispersion does not turn every unoccupied eigenvalue into an electron-addition energy. Even exact multiplicative-potential Kohn–Sham theory generally needs the exchange-correlation derivative discontinuity to relate its gap to the fundamental gap; Band Gaps derives that relation. Hybrids usually use a generalized Kohn–Sham operator with nonlocal terms, so translating statements about the exact local-potential spectrum requires care.

The functional and any added Hubbard, hybrid or dispersion correction define the approximation being tested. Changing that choice is a model comparison, not a numerical convergence test. Approximate density-functional energies are not generally upper bounds to the exact many-electron energy. A self-consistent solution may also be only one metastable magnetic or orbital state.

Quasiparticle bands are a reduction of a spectrum

An interacting electron spectrum can have a narrow peak, broad continua and satellites at the same momentum. The spectral function contains their weights; a single curve usually displays only the selected peak position. The distinction between ground-state methods and excitation methods is developed by Onida, Reining and Rubio (2002).

For an isolated reference band, neglect off-diagonal mixing and use an energy variable $E$ in a common reference. Here $\Sigma^R$ denotes the exchange-correlation self-energy; the Hartree contribution is retained in the common reference Hamiltonian. Let $\varepsilon_{n\mathbf k}^{\mathrm{KS}}$ be a Kohn–Sham energy and define the diagonal correction

$$\Delta\Sigma_{n\mathbf k}(E) =\langle n\mathbf k|\Sigma^R(E)-v_{\mathrm{xc}}|n\mathbf k\rangle.$$

Subtracting the reference exchange-correlation potential avoids retaining it in addition to the replacement self-energy. The narrow-pole energy approximately satisfies

$$E_{n\mathbf k}^{\mathrm{QP}} =\varepsilon_{n\mathbf k}^{\mathrm{KS}} +\operatorname{Re}\Delta\Sigma_{n\mathbf k} \big(E_{n\mathbf k}^{\mathrm{QP}}\big).$$

Linearizing about the reference eigenvalue gives

$$\begin{aligned} E_{n\mathbf k}^{\mathrm{QP}} &\simeq\varepsilon_{n\mathbf k}^{\mathrm{KS}} +Z_{n\mathbf k}\operatorname{Re}\Delta\Sigma_{n\mathbf k} \big(\varepsilon_{n\mathbf k}^{\mathrm{KS}}\big),\\ Z_{n\mathbf k} &=\left[1-\left. \partial_E\operatorname{Re}\Delta\Sigma_{n\mathbf k}(E) \right|_{E=\varepsilon_{n\mathbf k}^{\mathrm{KS}}}\right]^{-1}. \end{aligned}$$

The factor evaluated at the reference energy is a linearization coefficient. For an isolated real pole, the actual spectral residue instead uses the derivative at the pole energy. They coincide only to the accuracy of the linearization. This scalar approximation needs a smooth self-energy and a recognizable pole. Near degenerate bands, treat the relevant matrix subspace. Near a satellite or a rapidly varying self-energy, several roots or a broad line can make a single quasiparticle energy misleading. The exact spectral definitions and residue conditions remain at Spectral Functions.

What GW adds and what it assumes

The GW approximation uses a one-particle propagator $G$ and a dynamically screened interaction $W$ to approximate the self-energy, with the vertex treated at its simplest level. Hedin (1965) develops the screened-interaction framework. In a one-shot $G_0W_0$ calculation, both input objects inherit a declared starting point; self-consistent variants update different parts of the problem and must be named.

Converge the basis, Brillouin-zone sampling, unoccupied-state representation, screening cutoff and frequency treatment relevant to the implementation. Agreement under these changes establishes numerical stability within the approximation. It does not prove that omitted vertex effects, a poor starting state or strong local correlations are negligible. Nor is increasing self-consistency a universal guarantee of improved agreement with experiment.

The imaginary self-energy and missing pole weight are physical information. Replacing the result by shifted real eigenvalues discards linewidths and satellites. Applying a constant “scissor” shift to conduction bands is an additional approximation: it assumes that the correction relevant to the intended use is sufficiently independent of momentum and band character.

Local correlation and beyond-static corrections

When local multiplets, fluctuating moments or Hubbard-band weight matter, a static band shift may miss the mechanism. A DFT+$U$ calculation adds a chosen static orbital-dependent correction; a material DMFT construction supplies a frequency-dependent local self-energy for a selected correlated subspace. Neither label removes the need to define orbitals, interactions and the correction for contributions already present in the starting description.

Single-site DMFT retains local dynamical correlations while approximating nonlocal self-energy structure. Its suitability depends on the material and question. Hubbard Physics in Materials explains the retained orbital space and interactions; DMFT for Quantum Materials develops the coupled impurity and lattice calculation. These methods are not automatic replacements for a controlled neutral-response calculation or for an assessment of nonlocal order.

Numerical convergence and approximation error are different

Convergence answers whether a numerical representation has stabilized a specified problem. It cannot establish that the problem contains the required physics. Distinguish the following changes:

Change	What it tests	What a stable result still does not establish
Increase basis size or real-space resolution	Basis completeness for the chosen Hamiltonian	Accuracy of a functional or frozen-core model
Refine the full-zone mesh and occupation treatment	Integration, filling, small pockets and extrema	Validity of the assumed magnetic state or structure
Tighten self-consistency and eigensolver tolerances	Residual numerical error at one solution	That this solution is the lowest-energy allowed state
Change functional, interaction model or retained subspace	Sensitivity to the chosen physics	A statistical confidence interval from the resulting spread
Change slab thickness, cell size or vacuum separation	Finite-size and image-interaction effects	That the chosen surface termination or defect ensemble is correct

Core treatment and relativistic approximation can affect different observables differently. A pseudopotential adequate for valence bonding does not automatically describe core spectroscopy. Spin–orbit coupling can be decisive for a small gap or inversion even when total energies appear almost unchanged. Compare like physical structures, electron counts and state constraints before assigning a difference to “the method.”

A high-symmetry band path is a visualization, not a full-zone extremum search or density-of-states integral. A small missed pocket can change the metal/insulator diagnosis without making the path plot look suspicious. For this reason, band interpolation accuracy and Brillouin-zone coverage are separate checks.

A comparison that separates the questions

Suppose increasing a plane-wave cutoff changes a global Kohn–Sham gap from $0.842$ to $0.846\,\mathrm{eV}$, and a denser mesh then gives $0.845\,\mathrm{eV}$. This shows small changes over the tested refinements; it does not bound all remaining error. A second functional giving $1.12\,\mathrm{eV}$ reveals a much larger approximation dependence. Neither calculation alone predicts a measured optical onset.

To interpret an onset at, say, $0.95\,\mathrm{eV}$, first establish the corresponding electron addition/removal gap, relevant direct transitions, selection rules, electron–hole interactions, phonon assistance and sample conditions. Numerical convergence, quasiparticle corrections and optical response answer successive questions; their numerical values should not be compared as interchangeable “gaps.” The numbers here are illustrative, not material data.

Match the computed object to the probe

Photoemission intensity is sensitive to the occupied spectral function, occupation factor and photoemission matrix elements, together with surface/final-state effects and instrumental resolution. A missing band in one geometry need not be absent from the electronic spectrum. Use Angle-Resolved Photoemission Spectroscopy for that inference.

Optical absorption is a neutral response. Even accurate addition/removal energies do not supply bound excitons or optical oscillator strengths automatically. An electron–hole response treatment, such as a suitable Bethe–Salpeter or time-dependent density-functional approximation, supplies a different object. Onida, Reining and Rubio (2002) explain this separation. Raman and Optical Spectroscopy explains how these responses appear in measurements.

For structural competition, use balanced total energies or free energies of the relevant states. A lower sum of occupied band eigenvalues is not a general phase-stability test. Temperature, phonons, configurational disorder and magnetic fluctuations can also change free-energy ordering beyond a clamped-nuclei zero-temperature electronic calculation.

For topology, an energy plot alone is incomplete. The occupied subspace, filling, symmetry and matrix structure must be correct over the required momentum domain. For transport, a Fermi velocity or density of states alone does not determine scattering, vertex corrections or contact effects. The calculated bands provide inputs to those further response calculations.

Exercises

1. Linearize a quasiparticle correction

At one isolated band and momentum, take $\varepsilon^{\mathrm{KS}}=0.50\,\mathrm{eV}$, $\operatorname{Re}\Delta\Sigma(\varepsilon^{\mathrm{KS}})=0.80\,\mathrm{eV}$ and $\partial_E\operatorname{Re}\Delta\Sigma=-0.25$. Compute the linearized $Z$ and quasiparticle energy. What information is still needed to justify a sharp quasiparticle interpretation?

Solution

$Z=1/(1+0.25)=0.80$, so $E^{\mathrm{QP}}\simeq0.50+0.80(0.80)=1.14\,\mathrm{eV}$. One still needs the imaginary self-energy or linewidth, the self-energy’s variation over the shift, nearby branches and possible satellite structure. A real root and a plausible slope alone do not verify the pole approximation.

2. Two poles from one reference level

In electronvolts, take a reference level $\varepsilon_0=0$ and a causal self-energy

$$\Sigma^R(E)=\frac{g^2}{E-E_s+i0^+},\qquad g=1\,\mathrm{eV},\quad E_s=3\,\mathrm{eV}.$$

Find both poles of $G^R(E)=[E+i0^+-\varepsilon_0-\Sigma^R(E)]^{-1}$ and their weights. Compare the pole near the reference level with its linearized estimate. This is a two-level spectral model, not a claim that a real satellite is always one sharp pole.

Solution

The pole equation is $E(E-3)-1=0$ in the stated energy units, giving

$$E_\pm=\frac{3\pm\sqrt{13}}2\,\mathrm{eV}.$$

Thus $E_-\simeq-0.3028\,\mathrm{eV}$ and $E_+\simeq3.3028\,\mathrm{eV}$. The residues are

$$Z_\pm=\left[1+\frac{g^2}{(E_\pm-E_s)^2}\right]^{-1}, \quad Z_-\simeq0.9160,\quad Z_+\simeq0.0840.$$

Their sum is one, as required for the projected spectral function of a normalized orbital. Keeping only the lower pole omits about $8.4\%$ of the weight. At $E=0$, $\Sigma=-1/3\,\mathrm{eV}$ and $\Sigma'=-1/9$, so $Z_{\mathrm{lin}}=0.9$ and $E_{\mathrm{lin}}=-0.3000\,\mathrm{eV}$. A good pole-energy estimate does not make its reference-point slope the exact spectral weight. Equivalently, diagonalizing the Hermitian matrix $\begin{pmatrix}0&1\\1&3\end{pmatrix}\,\mathrm{eV}$ gives these energies and the squared first-component weights; eliminating its second level produces the displayed self-energy.

3. Identify which conclusion a convergence check supports

A calculation converges a band gap to within $2\,\mathrm{meV}$ over its final mesh refinements, but two plausible magnetic states differ in total energy by $0.4\,\mathrm{meV}$ per cell and have different gaps. Does the gap test select the ground state? Does averaging the two gaps give a justified uncertainty estimate?

Solution

No. Gap convergence tests one output within an assumed state. The magnetic energy difference needs its own balanced convergence and approximation assessment; a tolerance in gap energy cannot be reused as an energy-per-cell error bar. Averaging outputs of competing phases has no general interpretation as the equilibrium gap or a statistical confidence interval. Report the state dependence and resolve the state-selection problem, or state it as an unresolved physical alternative.

4. Distinguish interpolation from a physical correction

A Wannier Hamiltonian reproduces the selected Kohn–Sham bands to $1\,\mathrm{meV}$ throughout a checked window. Its calculated optical onset disagrees with experiment. List three independent reasons that this excellent interpolation does not settle the discrepancy.

Solution

The source Kohn–Sham eigenvalues may differ from addition/removal energies; a neutral optical response may require electron–hole attraction and selection rules; and the projected velocity/dipole operators or excluded bands may not be adequate for the optical observable. Structure, occupation and experimental broadening are further possibilities. Interpolation accuracy verifies fidelity to a chosen source subspace, not every physical assumption feeding or following it.

References

• Hedin, Lars. “New Method for Calculating the One-Particle Green’s Function with Application to the Electron-Gas Problem.” Physical Review 139, A796–A823 (1965). doi:10.1103/PhysRev.139.A796.
• Kohn, Walter, and Lu J. Sham. “Self-Consistent Equations Including Exchange and Correlation Effects.” Physical Review 140, A1133–A1138 (1965). doi:10.1103/PhysRev.140.A1133.
• Marzari, Nicola, Arash A. Mostofi, Jonathan R. Yates, Ivo Souza, and David Vanderbilt. “Maximally localized Wannier functions: Theory and applications.” Reviews of Modern Physics 84, 1419–1475 (2012). doi:10.1103/RevModPhys.84.1419; arXiv:1112.5411.
• Onida, Giovanni, Lucia Reining, and Angel Rubio. “Electronic excitations: density-functional versus many-body Green’s-function approaches.” Reviews of Modern Physics 74, 601–659 (2002). doi:10.1103/RevModPhys.74.601.