Skip to content

Periodic Systems

Periodicity does not make an infinite electronic system finite by truncation. It makes the absolute cell label redundant. Translation symmetry reorganizes the infinite problem into independent one-cell problems labeled by the crystal momentum \(\boldsymbol k\). The cell contains the internal spatial structure, while \(\boldsymbol k\) retains the phase relation between translated cells.

This chapter makes that equivalence precise and then fixes the normalization and Fourier conventions used for periodic systems. No spatial basis is assumed. Hartree atomic units are used unless stated otherwise.

Lattice and Unit Cell

For a three-dimensional crystal, the Bravais lattice is

\[ \mathcal L = \left\{ \boldsymbol R = \sum_{\alpha=1}^3 n_\alpha\boldsymbol a_\alpha \;\middle|\; n_\alpha\in\mathbb Z \right\}, \]

where \(\boldsymbol a_\alpha\) are primitive lattice vectors. The primitive-cell volume is

\[ \Omega = \left| \boldsymbol a_1 \cdot \left( \boldsymbol a_2\times\boldsymbol a_3 \right) \right|. \]

The reciprocal primitive vectors satisfy

\[ \boldsymbol a_\alpha \cdot \boldsymbol b_\beta = 2\pi\delta_{\alpha\beta}, \]

and generate the reciprocal lattice

\[ \mathcal L^* = \left\{ \boldsymbol G = \sum_{\alpha=1}^3 m_\alpha\boldsymbol b_\alpha \;\middle|\; m_\alpha\in\mathbb Z \right\}. \]

An atom \(i\) in cell \(\boldsymbol R\) has position

\[ \boldsymbol r_{i\boldsymbol R} = \boldsymbol\tau_i+\boldsymbol R, \]

where \(\boldsymbol\tau_i\) is its position in a chosen unit cell. Changing the unit-cell representative of an atom does not change the infinite structure.

The formulas below use three-dimensional periodicity. Lower-dimensional periodicity follows by restricting lattice translations and reciprocal variables to the periodic subspace and specifying the electrostatic boundary conditions in the remaining directions.

Translation Symmetry

Let the translation operator act as

\[ \left( \hat T_{\boldsymbol R}\psi \right)(\boldsymbol r) = \psi(\boldsymbol r-\boldsymbol R). \]

A periodic one-electron Hamiltonian obeys

\[ \left[ \hat H, \hat T_{\boldsymbol R} \right] = 0, \qquad \boldsymbol R\in\mathcal L. \]

This statement includes local and nonlocal operators. If \(H_{ss'}(\boldsymbol r,\boldsymbol r')\) denotes the real-space kernel of \(\hat H\), translation symmetry is equivalently

\[ H_{ss'}( \boldsymbol r+\boldsymbol R, \boldsymbol r'+\boldsymbol R ) = H_{ss'}( \boldsymbol r, \boldsymbol r' ). \]

The Hamiltonian therefore depends on positions within the cell and on relative translations, not on an absolute cell origin. This redundancy is what permits the infinite cell index to be replaced by crystal momentum.

From the Infinite Crystal to Cell Problems

The lattice translations commute with one another and with \(\hat H\). Their simultaneous eigenvalues are phases. A state in the sector labeled by \(\boldsymbol k\) satisfies

\[ \hat T_{\boldsymbol R} \psi_{a\boldsymbol k} = \mathrm e^{ -\mathrm i\boldsymbol k\cdot\boldsymbol R } \psi_{a\boldsymbol k}. \]

Choose \(\boldsymbol x\) in the unit cell. Every point in the infinite crystal can be represented as \(\boldsymbol x+\boldsymbol R\), and the translation eigenvalue gives

\[ \psi_{a\boldsymbol k}( \boldsymbol x+\boldsymbol R ) = \mathrm e^{ +\mathrm i\boldsymbol k\cdot\boldsymbol R } \psi_{a\boldsymbol k}(\boldsymbol x). \]

This equation is a twisted boundary condition on a unit cell. For fixed \(\boldsymbol k\), the wavefunction in every translated cell is determined by its values in the unit cell and the phase \(\mathrm e^{\mathrm i\boldsymbol k\cdot\boldsymbol R}\). The absolute cell label is removed, but the coherence between cells is retained by \(\boldsymbol k\).

The infinite problem is therefore equivalent to the family

\[ \text{periodic infinite system} \quad\longleftrightarrow\quad \left\{ \text{one-cell problem with boundary twist }\boldsymbol k \right\}_{\boldsymbol k\in\mathrm{BZ}}. \]

One cell at one \(\boldsymbol k\) is only one translation sector. One cell together with all inequivalent \(\boldsymbol k\) sectors represents the complete infinite crystal.

Two crystal momenta differing by a reciprocal lattice vector define the same boundary condition because

\[ \mathrm e^{ \mathrm i(\boldsymbol k+\boldsymbol G)\cdot\boldsymbol R } = \mathrm e^{ \mathrm i\boldsymbol k\cdot\boldsymbol R }, \qquad \boldsymbol G\in\mathcal L^*. \]

Thus

\[ \boldsymbol k \sim \boldsymbol k+\boldsymbol G, \]

and an irredundant primitive cell of reciprocal space may be chosen as the first Brillouin zone.

Periodic Bloch Form

The boundary phase may be separated from the cell-internal function by writing

\[ \psi_{a\boldsymbol k}(\boldsymbol r) = \mathrm e^{ +\mathrm i\boldsymbol k\cdot\boldsymbol r } u_{a\boldsymbol k}(\boldsymbol r), \]

where

\[ u_{a\boldsymbol k}( \boldsymbol r+\boldsymbol R ) = u_{a\boldsymbol k}(\boldsymbol r). \]

This change of variables introduces no additional physical assumption. The Hamiltonian acting on periodic spinors is

\[ \hat H(\boldsymbol k) = \mathrm e^{ -\mathrm i\boldsymbol k\cdot\boldsymbol r } \hat H \mathrm e^{ +\mathrm i\boldsymbol k\cdot\boldsymbol r }, \]

and the eigenproblem becomes

\[ \hat H(\boldsymbol k) u_{a\boldsymbol k} = \varepsilon_{a\boldsymbol k} u_{a\boldsymbol k}. \]

Different \(\boldsymbol k\) sectors do not couple as long as lattice translation symmetry is exact. A basis or real-space discretization only determines how each cell problem is represented numerically; it does not create the \(\boldsymbol k\) decomposition.

Cell Normalization and Brillouin-Zone Averages

For two Bloch spinors at the same \(\boldsymbol k\), their pointwise inner product is lattice periodic. The inner product is therefore evaluated over the unit cell,

\[ \langle \psi_{a\boldsymbol k} \vert \psi_{b\boldsymbol k} \rangle_\Omega \equiv \int_\Omega \mathrm d\boldsymbol r\, \psi_{a\boldsymbol k}^\dagger(\boldsymbol r) \psi_{b\boldsymbol k}(\boldsymbol r) = \delta_{ab}. \]

This is the normalization used throughout. It applies at fixed \(\boldsymbol k\) and does not define overlaps between states belonging to different sectors.

The Brillouin-zone volume is

\[ \Omega_{\mathrm{BZ}} = \frac{(2\pi)^3}{\Omega}. \]

An intensive quantity per unit cell is obtained from a normalized Brillouin-zone average,

\[ \langle F\rangle_{\mathrm{BZ}} = \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\mathrm{BZ}} \mathrm d\boldsymbol k\, F(\boldsymbol k) = \frac{\Omega}{(2\pi)^3} \int_{\mathrm{BZ}} \mathrm d\boldsymbol k\, F(\boldsymbol k) \simeq \sum_{\boldsymbol k} w_{\boldsymbol k} F(\boldsymbol k), \qquad \sum_{\boldsymbol k}w_{\boldsymbol k}=1. \]

A finite \(\boldsymbol k\) mesh is a numerical quadrature for this continuous average, not a fundamental discretization of crystal momentum. The mesh may be complete or symmetry reduced. Its normalization is carried by \(w_{\boldsymbol k}\), not by the number of sampled points.

Point-group and antiunitary symmetries may reduce the integral to an irreducible part of the Brillouin zone. Multiplicity weights suffice for symmetry-invariant scalar integrands. Fields and matrices must additionally be transformed from each representative \(\boldsymbol k\) to its symmetry-related sectors.

The symbol \(N_{\boldsymbol k}\) is reserved for the cardinality of a complete, unreduced uniform mesh when a finite discrete Fourier pair is explicitly introduced below.

Density and Local Periodic Fields

For occupations \(f_{a\boldsymbol k}\), define the contribution from the states at one \(\boldsymbol k\) point to the density kernel by

\[ \rho_{ss'}^{\boldsymbol k}(\boldsymbol r,\boldsymbol r') = \sum_a f_{a\boldsymbol k} \psi_{a\boldsymbol k,s}(\boldsymbol r) \psi_{a\boldsymbol k,s'}^*(\boldsymbol r'). \]

The Bloch phases cancel under simultaneous translation:

\[ \rho_{ss'}^{\boldsymbol k}( \boldsymbol r+\boldsymbol R, \boldsymbol r'+\boldsymbol R ) = \rho_{ss'}^{\boldsymbol k}( \boldsymbol r, \boldsymbol r' ). \]

The real-space kernel of the periodic density operator is

\[ \rho_{ss'}(\boldsymbol r,\boldsymbol r') = \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\mathrm{BZ}} \mathrm d\boldsymbol k\, \rho_{ss'}^{\boldsymbol k}(\boldsymbol r,\boldsymbol r'). \]

Consequently, its diagonal is a periodic local field. The local spin-density matrix is

\[ n_{ss'}(\boldsymbol r) = \rho_{ss'}(\boldsymbol r,\boldsymbol r), \qquad \mathbf n(\boldsymbol r) = \left( n_{ss'}(\boldsymbol r) \right)_{ss'}. \]

The infinite crystal contains infinitely many electrons, so the electron number is quoted per unit cell:

\[ N_{\mathrm e} = \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\mathrm{BZ}} \mathrm d\boldsymbol k\, \sum_a f_{a\boldsymbol k} = \int_\Omega \mathrm d\boldsymbol r\, \operatorname{tr}_s \mathbf n(\boldsymbol r). \]

A finite quadrature uses the same weights in the density kernel and the electron count.

The scalar electron density, spin-polarization density, local potentials, and other translation-invariant local fields can therefore be stored on one cell. Their values still contain contributions from all occupied states and all \(\boldsymbol k\) sectors.

The folded auxiliary-basis expansion of a periodic density is defined in Auxiliary-Basis Expansion and Projection of Density.

The occupations refer to explicit spinor states. Any spin-degeneracy factor in a reduced spinless formulation is separate from the normalized Brillouin-zone average and hence from any numerical quadrature weights.

Fourier Conventions

The conceptual reduction above is basis independent. The following transform pairs fix how periodic fields and cell-shift-indexed matrix blocks are represented.

Periodic Fields

For a lattice-periodic field \(\mathbf x(\boldsymbol r)\),

\[ \mathbf x(\boldsymbol r) = \sum_{\boldsymbol G} \mathrm e^{ +\mathrm i\boldsymbol G\cdot\boldsymbol r } \mathbf x(\boldsymbol G), \]
\[ \mathbf x(\boldsymbol G) = \frac{1}{\Omega} \int_\Omega \mathrm d\boldsymbol r\, \mathrm e^{ -\mathrm i\boldsymbol G\cdot\boldsymbol r } \mathbf x(\boldsymbol r). \]

Cell-Shift-Indexed Matrix Blocks

When a translated family of localized states is used, define the Bloch sum

\[ \lvert\varphi_{\mu\boldsymbol k};s\rangle = \sum_{\boldsymbol R} \mathrm e^{ +\mathrm i\boldsymbol k\cdot\boldsymbol R } \lvert\varphi_{\mu\boldsymbol R};s\rangle. \]

The sum specifies the Bloch boundary condition and is evaluated through cell-normalized quantities. It is not a globally normalized finite-supercell state.

For a lattice-translation-invariant operator \(\hat X\), define its cell-shift-indexed blocks using the displacement of the right state relative to the left,

\[ X_{\mu s,\nu s'}(\boldsymbol R) \equiv \langle \varphi_{\mu\boldsymbol 0};s \lvert\hat X\rvert \varphi_{\nu\boldsymbol R};s' \rangle. \]

Let \(\mathbf X(\boldsymbol R)\) collect all orbital and spin components. The transform pair is

\[ \mathbf X(\boldsymbol k) = \sum_{\boldsymbol R} \mathrm e^{ +\mathrm i\boldsymbol k\cdot\boldsymbol R } \mathbf X(\boldsymbol R), \]
\[ \mathbf X(\boldsymbol R) = \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\mathrm{BZ}} \mathrm d\boldsymbol k\, \mathrm e^{ -\mathrm i\boldsymbol k\cdot\boldsymbol R } \mathbf X(\boldsymbol k). \]

If \(\hat X\) is Hermitian, then

\[ \mathbf X(\boldsymbol R)^\dagger = \mathbf X(-\boldsymbol R). \]

For fractional coordinates \(\boldsymbol k=\sum_\alpha k_\alpha\boldsymbol b_\alpha\) and integer lattice offsets \(\boldsymbol R=\sum_\alpha R_\alpha\boldsymbol a_\alpha\), the forward phase is

\[ \exp\left( +2\pi\mathrm i \sum_\alpha k_\alpha R_\alpha \right). \]

The positive sign follows jointly from the Bloch-sum phase and the convention that \(\boldsymbol R\) displaces the right state.

Finite Discrete Pair

Let \(\mathcal K\) be a complete, unreduced uniform mesh with \(N_{\boldsymbol k}\) points, and let \(\mathcal R_{\mathcal K}\) be a dual set of \(N_{\boldsymbol k}\) lattice translations. The discrete transform pair is

\[ \mathbf X_{\mathcal K}(\boldsymbol R) = \frac{1}{N_{\boldsymbol k}} \sum_{\boldsymbol k\in\mathcal K} \mathrm e^{ -\mathrm i\boldsymbol k\cdot\boldsymbol R } \mathbf X(\boldsymbol k), \qquad \boldsymbol R\in\mathcal R_{\mathcal K}, \]
\[ \mathbf X(\boldsymbol k) = \sum_{\boldsymbol R\in\mathcal R_{\mathcal K}} \mathrm e^{ +\mathrm i\boldsymbol k\cdot\boldsymbol R } \mathbf X_{\mathcal K}(\boldsymbol R), \qquad \boldsymbol k\in\mathcal K. \]

This pair is exact for the finite sampled data. As an approximation to the continuous infinite-crystal transform, it contains the usual sampling and real-space aliasing. A weighted or symmetry-reduced quadrature is not, by itself, an invertible discrete Fourier transform.

References