Second-order force constants#

Potential energy of phonon system is represented as functions of atomic positions:

\[V[\mathbf{r}(j_1 l_1),\ldots,\mathbf{r}(j_n l_N)],\]

where \(\mathbf{r}(jl)\) is the point of the \(j\)-th atom in the \(l\)-th unit cell and \(n\) and \(N\) are the number of atoms in a unit cell and the number of unit cells, respectively. A force and a second-order force constant \(\Phi_{\alpha \beta}\) are given by

\[F_\alpha(jl) = -\frac{\partial V }{\partial r_\alpha(jl)}\]


\[\Phi_{\alpha\beta}(jl, j'l') = \frac{\partial^2 V}{\partial r_\alpha(jl) \partial r_\beta(j'l')} = -\frac{\partial F_\beta(j'l')}{\partial r_\alpha(jl)},\]

respectively, where \(\alpha\), \(\beta\), …, are the Cartesian indices, \(j\), \(j'\), …, are the indices of atoms in a unit cell, and \(l\), \(l'\), …, are the indices of unit cells. In the finite displacement method, the equation for the force constants is approximated as

\[\Phi_{\alpha\beta}(jl, j'l') \simeq -\frac{ F_\beta(j'l';\Delta r_\alpha{(jl)}) - F_\beta(j'l')} {\Delta r_\alpha(jl)},\]

where \(F_\beta(j'l'; \Delta r_\alpha{(jl)})\) are the forces on atoms with a finite displacement \(\Delta r_\alpha{(jl)}\) and usually \(F_\beta(j'l') \equiv 0\).

Modified Parlinski-Li-Kawazoe method#

The following is a modified and simplified version of the Parlinski-Li-Kawazoe method, which is just a numerical fitting approach to obtain force constants from forces and displacements.

The last equation above is represented by matrices as

\[\mathbf{F} = - \mathbf{U} \mathbf{P},\]

where \(\mathbf{F}\), \(\mathbf{P}\), and \(\mathbf{U}\) for a pair of atoms, e.g. \(\{jl, j'l'\}\), are given by

\[\mathbf{F} = \begin{pmatrix} F_{x} & F_{y} & F_{z} \end{pmatrix},\]
\[\begin{split}\mathbf{P} = \begin{pmatrix} \Phi_{xx} & \Phi_{xy} & \Phi_{xz} \\ \Phi_{yx} & \Phi_{yy} & \Phi_{yz} \\ \Phi_{zx} & \Phi_{zy} & \Phi_{zz} \end{pmatrix},\end{split}\]
\[\begin{split}\mathbf{U} = \begin{pmatrix} \Delta r_{x} & \Delta r_{y} & \Delta r_{z} \\ \end{pmatrix}.\end{split}\]

The matrix equation is expanded for number of forces and displacements as follows:

\[\begin{split}\begin{pmatrix} \mathbf{F}_1 \\ \mathbf{F}_2 \\ \vdots \end{pmatrix} = - \begin{pmatrix} \mathbf{U}_1 \\ \mathbf{U}_2 \\ \vdots \end{pmatrix} \mathbf{P}.\end{split}\]

With sufficient number of atomic displacements, this may be solved by pseudo inverse such as

\[\begin{split}\mathbf{P} = - \begin{pmatrix} \mathbf{U}_1 \\ \mathbf{U}_2 \\ \vdots \end{pmatrix}^{+} \begin{pmatrix} \mathbf{F}_1 \\ \mathbf{F}_2 \\ \vdots \end{pmatrix}.\end{split}\]

Required number of atomic displacements to solve the simultaneous equations may be reduced using site-point symmetries. The matrix equation can be written using a symmetry operation as

\[\hat{R}(\mathbf{F}) = -\hat{R}(\mathbf{U})\mathbf{P},\]

where \(\hat{R}\) is the site symmetry operation centring at \(\mathbf{r}(jl)\). \(\hat{R}(\mathbf{F})\) and \(\hat{R}(\mathbf{U})\) are defined as \(\mathbf{RF}(\hat{R^{-1}}(j'l'))\) and \(\mathbf{RU}\), respectively, where \(\mathbf{R}\) is the matrix representation of the rotation operation. The combined simultaneous equations are built such as

\[\begin{split}\begin{pmatrix} \mathbf{F}^{(1)}_1 \\ \mathbf{F}^{(2)}_1 \\ \vdots \\ \mathbf{F}^{(1)}_2 \\ \mathbf{F}^{(2)}_2 \\ \vdots \end{pmatrix} = - \begin{pmatrix} \mathbf{U}^{(1)}_1 \\ \vdots \\ \mathbf{U}^{(2)}_1 \\ \mathbf{U}^{(1)}_2 \\ \mathbf{U}^{(2)}_2 \\ \vdots \end{pmatrix} \mathbf{P}.\end{split}\]

where the superscript with parenthesis gives the index of site-symmetry operations. This is solved by pseudo inverse.

Dynamical matrix#

In phonopy, a phase convention of dynamical matrix is used as follows:

(1)#\[D_{\alpha\beta}(jj',\mathbf{q}) = \frac{1}{\sqrt{m_j m_{j'}}} \sum_{l'} \Phi_{\alpha\beta}(j0, j'l') \exp(i\mathbf{q}\cdot[\mathbf{r}(j'l')-\mathbf{r}(j0)]),\]

where \(m\) is the atomic mass and \(\mathbf{q}\) is the wave vector. An equation of motion is written as

\[\sum_{j'\beta} D_{\alpha\beta}(jj',\mathbf{q}) e_\beta(j', \mathbf{q}\nu) = [ \omega(\mathbf{q}\nu) ]^2 e_\alpha(j, \mathbf{q}\nu).\]

where the eigenvector of the band index \(\nu\) at \(\mathbf{q}\) is obtained by the diagonalization of \(\mathbf{D}(\mathbf{q})\):

\[\sum_{j \alpha j' \beta}e_\alpha(j',\mathbf{q}\nu)^* D_{\alpha\beta}(jj',\mathbf{q}) e_\beta(j',\mathbf{q}\nu') = [\omega(\mathbf{q}\nu)]^2 \delta_{\nu\nu'}.\]

The atomic displacements \(\mathbf{u}\) are given as

\[u_\alpha(jl,t) = \left(\frac{\hbar}{2Nm_j}\right)^{\frac{1}{2}} \sum_{\mathbf{q},\nu}\left[\omega(\mathbf{q}\nu)\right]^{-\frac{1}{2}} \left[\hat{a}(\mathbf{q}\nu)\exp(-i\omega(\mathbf{q}\nu)t)+ \hat{a}^\dagger(\mathbf{-q}\nu)\exp({i\omega(\mathbf{q}\nu)}t)\right] \exp({i\mathbf{q}\cdot\mathbf{r}(jl)}) e_\alpha(j,\mathbf{q}\nu),\]

where \(\hat{a}^\dagger\) and \(\hat{a}\) are the creation and annihilation operators of phonon, \(\hbar\) is the reduced Planck constant, and \(t\) is the time.

Non-analytical term correction#

To treat long range interaction of macroscopic electric field induced by polarization of collective ionic motions near the \(\Gamma\)-point, non-analytical term is added to dynamical matrix (Non-analytical term correction). At \(\mathbf{q}\to\mathbf{0}\), the dynamical matrix with non-analytical term is given by,

\[D_{\alpha\beta}(jj',\mathbf{q}\to \mathbf{0}) = D_{\alpha\beta}(jj',\mathbf{q}=\mathbf{0}) + \frac{1}{\sqrt{m_j m_{j'}}} \frac{4\pi}{\Omega_0} \frac{\left[\sum_{\gamma}q_{\gamma}Z^{*}_{j,\gamma\alpha}\right] \left[\sum_{\gamma'}q_{\gamma'}Z^{*}_{j',\gamma'\beta}\right]} {\sum_{\alpha\beta}q_{\alpha}\epsilon_{\alpha\beta}^{\infty} q_{\beta}}.\]

Phonon frequencies at general q-points with long-range dipole-dipole interaction are calculated by the method of Gonze et al. (Correction by dipole-dipole interaction).

Thermodynamic properties#

Phonon number#

\[n = \frac{1}{\exp(\hbar\omega(\mathbf{q}\nu)/k_\mathrm{B} T)-1}\]

Harmonic phonon energy#

\[E = \sum_{\mathbf{q}\nu}\hbar\omega(\mathbf{q}\nu)\left[\frac{1}{2} + \frac{1}{\exp(\hbar\omega(\mathbf{q}\nu)/k_\mathrm{B} T)-1}\right]\]

Constant volume heat capacity#

\[\begin{split}C_V &= \left(\frac{\partial E}{\partial T} \right)_V \\ &= \sum_{\mathbf{q}\nu} k_\mathrm{B} \left(\frac{\hbar\omega(\mathbf{q}\nu)}{k_\mathrm{B} T} \right)^2 \frac{\exp(\hbar\omega(\mathbf{q}\nu)/k_\mathrm{B} T)}{[\exp(\hbar\omega(\mathbf{q}\nu)/k_\mathrm{B} T)-1]^2}\end{split}\]

Partition function#

\[Z = \exp(-\varphi/k_\mathrm{B} T) \prod_{\mathbf{q}\nu} \frac{\exp(-\hbar\omega(\mathbf{q}\nu)/2k_\mathrm{B} T)}{1-\exp(-\hbar\omega(\mathbf{q}\nu)/k_\mathrm{B} T)}\]

Helmholtz free energy#

\[\begin{split}F &= -k_\mathrm{B} T \ln Z \\ &= \varphi + \frac{1}{2} \sum_{\mathbf{q}\nu} \hbar\omega(\mathbf{q}\nu) + k_\mathrm{B} T \sum_{\mathbf{q}\nu} \ln \bigl[1 -\exp(-\hbar\omega(\mathbf{q}\nu)/k_\mathrm{B} T) \bigr]\end{split}\]


\[\begin{split}S &= -\frac{\partial F}{\partial T} \\ &= \frac{1}{2T} \sum_{\mathbf{q}\nu} \hbar\omega(\mathbf{q}\nu) \coth(\hbar\omega(\mathbf{q}\nu)/2k_\mathrm{B}T)-k_\mathrm{B} \sum_{\mathbf{q}\nu} \ln\left[2\sinh(\hbar\omega(\mathbf{q}\nu)/2k_\mathrm{B}T)\right]\end{split}\]

Thermal displacement#

Mean square displacement#

From Eq. (10.71) in the book “Thermodynamics of Crystal”, atomic displacement, u, is written by

\[u^\alpha(jl,t) = \left(\frac{\hbar}{2Nm_j}\right)^{\frac{1}{2}} \sum_{\mathbf{q},\nu}\left[\omega_\nu(\mathbf{q})\right]^{-\frac{1}{2}} \left[\hat{a}_\nu(\mathbf{q})\exp(-i\omega_\nu(\mathbf{q})t)+ \hat{a}^\dagger_\nu(\mathbf{-q})\exp({i\omega_\nu(\mathbf{q})}t)\right] \exp({i\mathbf{q}\cdot\mathbf{r}(jl)}) e^\alpha_\nu(j,\mathbf{q})\]

where j and l are the labels for the j-th atomic position in the l-th unit cell, t is the time, \(\alpha\) is an axis (a Cartesian axis in the default behavior of phonopy), m is the atomic mass, N is the number of the unit cells, \(\mathbf{q}\) is the wave vector, \(\nu\) is the index of phonon mode. e is the polarization vector of the atom jl and the band \(\nu\) at \(\mathbf{q}\). \(\mathbf{r}(jl)\) is the atomic position and \(\omega\) is the phonon frequency. \(\hat{a}^\dagger\) and \(\hat{a}\) are the creation and annihilation operators of phonon. The expectation value of the squared atomic displacement is calculated as,

\[\left\langle |u^\alpha(jl, t)|^2 \right\rangle = \frac{\hbar}{2Nm_j} \sum_{\mathbf{q},\nu}\omega_\nu(\mathbf{q})^{-1} (1+2n_\nu(\mathbf{q},T))|e^\alpha_\nu(j,\mathbf{q})|^2,\]

where \(n_\nu(\mathbf{q},T)\) is the phonon population, which is give by,

\[n_\nu(\mathbf{q},T) = \frac{1}{\exp(\hbar\omega_\nu(\mathbf{q})/\mathrm{k_B}T)-1},\]

where T is the temperature, and \(\mathrm{k_B}\) is the Boltzmann constant. The equation is calculated using the commutation relation of the creation and annihilation operators and the expectation values of the combination of the operations, e.g.,

\[\begin{split}[ \hat{a}_\nu(\mathbf{q}), \hat{a}^\dagger_{\nu'}(\mathbf{q'}) ] &= \delta(\mathbf{q}-\mathbf{q}')\delta_{\nu\nu'},\\ [ \hat{a}_\nu(\mathbf{q}), \hat{a}_{\nu'}(\mathbf{q'}) ] &= 0,\\ [ \hat{a}^\dagger_\nu(\mathbf{q}), \hat{a}^\dagger_{\nu'}(\mathbf{q'}) ] &= 0,\\ \langle|\hat{a}_\nu(\mathbf{q})\hat{a}_{\nu'}(\mathbf{q'})|\rangle &= 0,\\ \langle|\hat{a}^\dagger_\nu(\mathbf{q})\hat{a}^\dagger_{\nu'}(\mathbf{q'})|\rangle &= 0.\end{split}\]

Mean square displacement matrix#

Mean square displacement matrix is defined as follows:

\[\mathrm{U}_\text{cart}(j, T) = \frac{\hbar}{2Nm_j} \sum_{\mathbf{q},\nu}\omega_\nu(\mathbf{q})^{-1} (1+2n_\nu(\mathbf{q},T)) \mathbf{e}_\nu(j,\mathbf{q}) \otimes \mathbf{e}^*_\nu(j,\mathbf{q}).\]

This is a symmetry matrix and diagonal elements are same as mean square displacement calculated along Cartesian x, y, z directions.

Projection to an arbitrary axis#

In phonopy, eigenvectors are calculated in the Cartesian axes that are defined in the input structure file. Mean square displacement along an arbitrary axis is obtained projecting eigenvectors in the Cartesian axes as follows:

\[\left\langle |u(jl, t)|^2 \right\rangle = \frac{\hbar}{2Nm_j} \sum_{\mathbf{q},\nu}\omega_\nu(\mathbf{q})^{-1} (1+2n_\nu(\mathbf{q},T))| \hat{\mathbf{n}}\cdot\mathbf{e}_\nu(j,\mathbf{q})|^2\]

where \(\hat{\mathbf{n}}\) is an arbitrary unit direction.

Mean square displacement matrix in cif format#

According to the paper by Grosse-Kunstleve and Adams [J. Appl. Cryst., 35, 477-480 (2002)], mean square displacement matrix in the cif definition (aniso_U), \(\mathrm{U}_\text{cif}\), is obtained by

\[\mathrm{U}_\text{cif} = (\mathrm{AN})^{-1}\mathrm{U}_\text{cart} (\mathrm{AN})^{-\mathrm{T}},\]

where \(\mathrm{A}\) is the matrix to transform a point in fractional coordinates to the Cartesian coordinates and \(\mathrm{N}\) is the diagonal matrix made of reciprocal basis vector lengths as follows:

\[\begin{split}\mathrm{A} = \begin{pmatrix} a_x & b_x & c_x \\ a_y & b_y & c_y \\ a_z & b_z & c_z \end{pmatrix}\end{split}\]


\[\begin{split}\mathrm{N} = \begin{pmatrix} a^* & 0 & 0 \\ 0 & b^* & 0 \\ 0 & 0 & c^* \end{pmatrix}.\end{split}\]

\(a^*\), \(b^*\), \(c^*\) are defined without \(2\pi\).

Group velocity#


Phonopy calculates group velocity of phonon as follows:

\[\begin{split}\mathbf{v}_\mathrm{g}(\mathbf{q}\nu) = & \nabla_\mathbf{q} \omega(\mathbf{q}\nu) \\ =&\frac{\partial\omega(\mathbf{q}\nu)}{\partial \mathbf{q}} \\ =&\frac{1}{2\omega(\mathbf{q}\nu)}\frac{\partial[\omega(\mathbf{q}\nu)]^2}{\partial \mathbf{q}} \\ =&\frac{1}{2\omega(\mathbf{q}\nu)}\left<\mathbf{e}(\mathbf{q}\nu)\biggl| \frac{\partial D(\mathbf{q})} {\partial \mathbf{q}}\biggl|\mathbf{e}(\mathbf{q}\nu)\right>,\end{split}\]

where the meanings of the variables are found at Formulations.

Finite difference method#

In the previous versions, group velocity was calculated using finite difference method:

\[\mathbf{v}_\mathrm{g}(\mathbf{q}\nu) = \frac{1}{2\omega(\mathbf{q}\nu)}\left<\mathbf{e}(\mathbf{q}\nu)\biggl| \frac{\partial D(\mathbf{q})} {\partial \mathbf{q}}\biggl|\mathbf{e}(\mathbf{q}\nu)\right> \simeq \frac{1}{2\omega(\mathbf{q}\nu)} \left<\mathbf{e}(\mathbf{q}\nu)\biggl| \frac{\Delta D(\mathbf{q})} {\Delta \mathbf{q}}\biggl|\mathbf{e}(\mathbf{q}\nu)\right>.\]

Group velocity calculation with the finite difference method is still able to be activated using GV_DELTA_Q tag or --gv_delta_q option. \(\Delta\mathbf{q} = (\Delta q_x, \Delta q_y, \Delta q_z)\) is described in Cartesian coordinated in reciprocal space. In the implementation, central difference is employed, and \(+\Delta q_\alpha\) and \(-\Delta q_\alpha\) are taken to calculate group velocity, where \(\alpha\) is the Cartesian index in reciprocal space. \(\Delta q_\alpha\) is specified in the unit of reciprocal space distance by --gv_delta_q option or GV_DELTA_Q tag.

Physical unit conversion#

Phonopy calculates phonon frequencies based on input values from users. In the default case, the physical units of distance, atomic mass, force, and force constants are supposed to be \(\text{Angstrom}\), \(\text{AMU}\), \(\text{eV/Angstrom}\), and \(\text{eV/Angstrom}^2\), respectively, and the physical unit of the phonon frequency is converted to THz. This conversion is made as follows:

Internally phonon frequency has the physical unit of \(\sqrt{\text{eV/}(\text{Angstrom}^2\cdot \text{AMU})}\) in angular frequency. To convert this unit to THz (not angular frequency), the calculation of sqrt(EV/AMU)/Angstrom/(2*pi)/1e12 is made. EV, AMU, Angstrom are the values to convert them to those in the SI base unit, i.e., to Joule, kg, and metre, respectively. These values implemented in phonopy are found at a phonopy github page. This unit conversion factor can be manually specified. See FREQUENCY_CONVERSION_FACTOR.

The unit conversion factor in the BORN file is multiplied with the second term of the right hand side of the equation in Non-analytical term correction where this equation is written in atomic units (Gonze and Lee, 1997). The physical unit of the part of the equation corresponding to force constants:

\[\frac{4\pi}{\Omega_0} \frac{[\sum_{\gamma}q_{\gamma}Z^{*}_{j,\gamma\alpha}] [\sum_{\gamma'}q_{\gamma'}Z^{*}_{j',\gamma'\beta}]} {\sum_{\alpha\beta}q_{\alpha}\epsilon_{\alpha\beta}^{\infty} q_{\beta}}.\]

is \([\text{hartree}/\text{bohr}^2]\). In the default case for the VASP interface, internally \(\Omega_0\) is given in \(\text{Angstrom}^3\), while Born charges \(Z^{*}\) have physical unit of charge (\(e_{0}\)). Normally, physical units of Born charges obtained in many calculators are atomic ones. In atomic units Hartree energy is equal to \(1 \text{Ha} = e_{0}^{2}/a_{0}\) (\(a_{0}\) is Bohr radius), therefore the units of charge squared is \(e_{0}^{2}=[\text{hartree}\cdot \text{bohr}]\). In case for the VASP in order to convert the nonanalytical term, one has to convert units of energy (Ha to eV) and units of distance (Bohr to Angstrom). In total, the necessary unit conversion is \((\text{hartree} \rightarrow \text{eV}) \times (\text{bohr} \rightarrow \text{Angstrom})=14.4\). In the default case of the Wien2k interface, the conversion factor is \((\text{hartree} \rightarrow \text{mRy})=2000\). For the other interfaces, the conversion factors are similarly calculated following the unit systems employed in phonopy (Interfaces to calculators).

Crystal structure#

Coordinates in direct and reciprocal spaces#

As usual, in phonopy, the Born-von Karman boundary condition is assumed. Basis vectors of a primitive lattice are defined in three column vectors \(( \mathbf{a} \; \mathbf{b} \; \mathbf{c} )\). Coordinates of a point in the direct space \(\mathbf{r}\) is represented with respect to these basis vectors. The direct lattice points are given by \(i \mathbf{a} + j \mathbf{b} + k \mathbf{a}, \{i, j, k \in \mathbb{Z}\}\), and the points for atoms in a unit cell \(x \mathbf{a} + y \mathbf{b} + z \mathbf{a}, \{0 \le x, y, z < 1\}\). Basis vectors of the reciprocal lattice may be given by three row vectors, \(( \mathbf{a}^{*T} /\; \mathbf{b}^{*T} /\; \mathbf{c}^{*T} )\), but here they are defined as three column vectors as \(( \mathbf{a}^{*} \; \mathbf{b}^{*} \; \mathbf{c}^{*} )\) with

(2)#\[\begin{split}\mathbf{a}^{*} &= \frac{\mathbf{b} \times \mathbf{c}}{\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})}, \\ \mathbf{b}^{*} &= \frac{\mathbf{c} \times \mathbf{a}}{\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a})}, \\ \mathbf{c}^{*} &= \frac{\mathbf{a} \times \mathbf{b}}{\mathbf{c} \cdot (\mathbf{a} \times \mathbf{b})}.\end{split}\]

Coordinates of a point in the reciprocal space \(\mathbf{q}\) is represented with respect to these basis vectors, therefore \(q_x \mathbf{a}^{*} + q_y \mathbf{b}^{*} + q_z \mathbf{c}^{*}\). The reciprocal lattice points are given by \(G_x\mathbf{a}^{*} + G_y \mathbf{b}^{*} + G_z \mathbf{c}^{*}, \{G_x, G_y, G_z \in \mathbb{Z}\}\). Following these definition, phase factor should be represented as \(\exp(2\pi i\mathbf{q}\cdot\mathbf{r})\), however in phonopy documentation, \(2\pi\) is implicitly included and not shown, i.e., it is represented like \(\exp(i\mathbf{q}\cdot\mathbf{r})\) (e.g., see Eq. (1)). In the output of the reciprocal basis vectors, \(2\pi\) is not included, e.g., in band.yaml.

In phonopy, unless PRIMITIVE_AXES or PRIMITIVE_AXIS (or --pa option) is specified, basis vectors in direct space \(( \mathbf{a} \; \mathbf{b} \; \mathbf{c})\) are set from the input unit cell structure even if it is a supercell or a conventional unit cell having centring, therefore the basis vectors in the reciprocal space are given by Eq. (2). When using PRIMITIVE_AXES or PRIMITIVE_AXIS, \(( \mathbf{a} \; \mathbf{b} \; \mathbf{c})\) are set from those transformed by the transformation matrix \(M_\text{p}\) as written at PRIMITIVE_AXES or PRIMITIVE_AXIS, therefore \(( \mathbf{a}^{*} \; \mathbf{b}^{*} \; \mathbf{c}^{*} )\) are given by those calculated following Eq. (2) with this \(( \mathbf{a} \; \mathbf{b} \; \mathbf{c})\).

Commensurate points#

In phonopy, so-called commensurate points mean the q-points whose waves are confined in the supercell used in the phonon calculation.

To explain about the commensurate points, let basis vectors of a primitive cell in direct space cell be the column vectors \((\mathbf{a}_\mathrm{p} \; \mathbf{b}_\mathrm{p} \; \mathbf{c}_\mathrm{p})\) and those of the supercell be \((\mathbf{a}_\mathrm{s} \; \mathbf{b}_\mathrm{s} \; \mathbf{c}_\mathrm{s})\). The transformation of the basis vectors from the primitive cell to the supercell is written as

\[( \mathbf{a}_\mathrm{s} \; \mathbf{b}_\mathrm{s} \; \mathbf{c}_\mathrm{s} ) = ( \mathbf{a}_\mathrm{p} \; \mathbf{b}_\mathrm{p} \; \mathbf{c}_\mathrm{p} ) \boldsymbol{P}.\]

\(\boldsymbol{P}\) is given as a \(3\times 3\) matrix and its elements are all integers, which is a constraint we have. The resolution for q-points being the commensurate points is determined by \(\boldsymbol{P}\) since one period of a wave has to be bound by any of lattice points inside the supercell. Therefore the number of commensurate points becomes the same as the number of the primitive cell that can be contained in the supercell, i.e., \(\det(\boldsymbol{P})\).

Then let the basis vectors in reciprocal space be the column vectors \((\mathbf{a}^*_\mathrm{p} \; \mathbf{b}^*_\mathrm{p} \; \mathbf{c}^*_\mathrm{p})\). Note that often reciprocal vectors are deifned by row vectors, but column vectors are chosen here to formulate. Formally we see the set of besis vectors are \(3\times 3\) matrices, we have the following relation:

\[( \mathbf{a}^*_\mathrm{p} \; \mathbf{b}^*_\mathrm{p} \; \mathbf{c}^*_\mathrm{p} ) = ( \mathbf{a}_\mathrm{p} \; \mathbf{b}_\mathrm{p} \; \mathbf{c}_\mathrm{p} )^{-\mathbf{T}}.\]

Similarly for the supercell, we define a relation

\[( \mathbf{a}^*_\mathrm{s} \; \mathbf{b}^*_\mathrm{s} \; \mathbf{c}^*_\mathrm{s} ) = ( \mathbf{a}_\mathrm{s} \; \mathbf{b}_\mathrm{s} \; \mathbf{c}_\mathrm{s} )^{-\mathbf{T}}.\]


\[( \mathbf{a}^*_\mathrm{s} \; \mathbf{b}^*_\mathrm{s} \; \mathbf{c}^*_\mathrm{s} ) \boldsymbol{P}^{\mathrm{T}} = ( \mathbf{a}^*_\mathrm{p} \; \mathbf{b}^*_\mathrm{p} \; \mathbf{c}^*_\mathrm{p} ).\]

To multiply an arbitrary q-point \(\mathbf{q}\) on both sides

\[( \mathbf{a}^*_\mathrm{s} \; \mathbf{b}^*_\mathrm{s} \; \mathbf{c}^*_\mathrm{s} ) \boldsymbol{P}^{\mathrm{T}} \mathbf{q} = ( \mathbf{a}^*_\mathrm{p} \; \mathbf{b}^*_\mathrm{p} \; \mathbf{c}^*_\mathrm{p} ) \mathbf{q},\]

we find the constraint of a q-point being one of the commensurate points is the elements of \(\boldsymbol{P}^{\mathrm{T}} \mathbf{q}\) to be integers.