Acoustic Sum Rules and Invariance Conditions

The force constant matrix $\Phi_{i \alpha j \beta}(\mathbf{R})$ describes the interaction between atom $i$ and atom $j$ in a unit cell displaced by $\mathbf{R}$. It is defined as the second derivative of the potential energy with respect to atomic displacements. $$ \Phi_{i \alpha j \beta}(\mathbf{R}) = \Phi_{j \beta i \alpha }(-\mathbf{R}) $$

1. Acoustic Sum Rules

Below are Acoustic Sum Rules (ASR) obeyed by the force constant matrix. These rules are derived from the invariance of the total energy under global translations and rotations.

Translation Invariance:

This condition ensures that a rigid translation of the entire crystal does not change its potential energy.

\[ \sum_{\mathbf{R}, j} \Phi_{i \alpha j \beta}(\mathbf{R}) = 0, \]

where $i,j$ denote the atomic indices and $\alpha,\beta,\gamma,\delta$ denote the polarization directions. $\mathbf{R}$ is the position vector of the unit cell. The summation over $\mathbf{R}$ is in the Wigner-Seitz cell.

Rotation Invariance:

This condition ensures that a rigid rotation of the entire crystal does not change its potential energy.

\[ \sum_{\mathbf{R}, j} \left( \Phi_{ i \alpha j \beta} (\mathbf{R}) \tau^{\gamma}_{\mathbf{R} j} - \Phi_{ i \alpha j \gamma} (\mathbf{R}) \tau^{\beta}_{\mathbf{R} j} \right) = 0, \]

where $\boldsymbol{\tau}_{\mathbf{R} j} = \mathbf{R} + \boldsymbol{\tau}_j$ is the position of atom $j$ in the unit cell located at $\mathbf{R}$. $\boldsymbol{\tau}_j$ is the position of atom $j$ with respect to the unit cell.

Huang Conditions:

Huang conditions are related to the symmetry of the elastic constant tensor and are necessary for the consistency between the microscopic force constants and macroscopic elasticity.

\[ \sum_{\mathbf{R}, i, j} \Phi_{i \alpha j \beta} (\mathbf{R}) \tau^{\gamma}_{\mathbf{R} i j} \tau^{\delta}_{\mathbf{R} i j} = \sum_{\mathbf{R}, i, j} \Phi_{ i \gamma j \delta} (\mathbf{R}) \tau^{\alpha}_{\mathbf{R} ij} \tau^{\beta}_{\mathbf{R} ij} \]

where $\boldsymbol{\tau}_{\mathbf{R} i j} = \boldsymbol{\tau}_{i} - \boldsymbol{\tau}_{j} - \mathbf{R}$ is the vector between two atoms.

2. Enforcing Acoustic Sum Rules

It is often the case that the force constants obtained from standard ab initio calculations break acoustic sum rules due to the finite size of the grid or basis set, which leads to non-zero frequencies for the acoustic modes at the $\Gamma$ point.

In order to enforce ASR, the first step is to explicitly write the invariance conditions in the form of a matrix-vector product $A\Phi = 0$, where $A$ is the constraint matrix and $\Phi$ is the vectorized force constant matrix. Here, $N_A$ refers to the number of atoms in the unit cell.

1. Translation Operator

The translation operator tensor A is defined as:

\[ A^{i\alpha\beta}_{\mathbf{R}i'\alpha'j'\beta'} = \delta_{ii'} \delta_{\alpha\alpha'} \delta_{\beta\beta'} \]

Constraint:

\[ \sum_{\mathbf{R}, i', \alpha', j', \beta'} A^{i \alpha \beta}_{\mathbf{R} i' \alpha' j' \beta'} \Phi_{\mathbf{R} i' \alpha' j' \beta'} = 0 \quad \Rightarrow \quad 9 N_A \]

2. Rotation Operator

The rotation operator tensor A is:

\[ A^{i\alpha\beta\gamma}_{\mathbf{R}i'\alpha'j'\beta'} = \delta_{ii'} \delta_{\alpha\alpha'} \left( \delta_{\beta\beta'} \tau^{\gamma}_{\mathbf{R}j'} - \delta_{\gamma\beta'} \tau^{\beta}_{\mathbf{R}j'} \right) \]

Symmetry & Constraints:

\[ A^{i\alpha\beta\gamma} = - A^{i\alpha\gamma\beta} \quad \longrightarrow \quad 3 \cdot N_A \cdot (3) = 9 N_A \]

Resulting Equation ($A\Phi=0$):

\[ \sum_{\mathbf{R}, j'} \left( \Phi_{\mathbf{R} i \alpha j' \beta} \tau^{\gamma}_{\mathbf{R} j'} - \Phi_{\mathbf{R} i \alpha j' \gamma} \tau^{\beta}_{\mathbf{R} j'} \right) = 0 \]

3. Huang Operator

The Huang operator tensor A is:

\[ A^{\alpha\beta\gamma\delta}_{\mathbf{R}i'\alpha'j'\beta'} = \left[ \delta_{\alpha\alpha'} \delta_{\beta\beta'} \tau^{\gamma}_{\mathbf{R}i'j'} \tau^{\delta}_{\mathbf{R}i'j'} - \tau^{\alpha}_{\mathbf{R}i'j'} \tau^{\beta}_{\mathbf{R}i'j'} \delta_{\gamma\alpha'} \delta_{\delta\beta'} \right] \]

Symmetry & Constraints:

For $(\alpha, \beta) \longleftrightarrow (\gamma, \delta)$, $A^{\alpha\beta\gamma\delta}$ is anti-symmetric i.e,

\[ A^{\alpha\beta\gamma\delta} = - A^{\gamma\delta\alpha\beta} \]

For $(\alpha \leftrightarrow \beta)$ and $(\gamma \leftrightarrow \delta)$ exchanges, $A^{\alpha\beta\gamma\delta}$ is symmetric.

\[ \left. \begin{matrix} (\alpha \leftrightarrow \beta) \\ (\gamma \leftrightarrow \delta) \end{matrix} \right\} \longrightarrow \underline{\underline{15}} \]

Total constraints without reduction : 81 * Due to anti-symmetry property for $(\alpha, \beta) \leftrightarrow (\gamma, \delta)$, it reduces to 36 constraints. * Due to symmetry property for $(\alpha \leftrightarrow \beta)$ and $(\gamma \leftrightarrow \delta)$ further reduces to 15 constants.