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

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

and

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

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 \(F_\beta(j'l') \equiv 0\).

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.

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

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

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.

To correct 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 are interpolated by the
method of Wang *et al.* (Interpolation scheme at general q-points with non-analytical term correction).

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

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

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

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

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

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 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.

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.

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:

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

and

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

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

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.

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 (\(\mathrm{\AA}^{-1}\) for the default case) by
`--gv_delta_q`

option or `GV_DELTA_Q`

tag.

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 \(\text{\AA}\), \(\text{AMU}\), \(\text{eV/\AA}\), and \(\text{eV/\AA}^2\), 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
\(\sqrt{\text{eV/}(\text{\AA}^2\cdot \text{AMU})}\) 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
with 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 \(\text{\AA}^3\). In total, the necessary unit conversion is \((\text{hartree} \rightarrow \text{eV}) \times (\text{bohr} \rightarrow \text{\AA})=14.4\). 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).

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_AXIS or PRIMITIVE_AXES (or `--pa`

option) is
specified, basis vectors in direct space \(( \mathbf{a} \;
\mathbf{b} \; \mathbf{c})\) are set from the input unit celll 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_AXIS or PRIMITIVE_AXES,
\(( \mathbf{a} \; \mathbf{b} \; \mathbf{c})\) are set from those
transformed by the transformation matrix \(M_\text{p}\) as written
at PRIMITIVE_AXIS or PRIMITIVE_AXES, therefore \(( \mathbf{a}^{*} \;
\mathbf{b}^{*} \; \mathbf{c}^{*} )\) are given by those calculated
following Eq. (2) with this \(( \mathbf{a}
\; \mathbf{b} \; \mathbf{c})\).

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

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

Then

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

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.