Theory: use of symmetry

The usage of symmetry may substantially reduce the number of DFT runs required to obtain all elements of the FCM. The idea is the following:

For the given group $G$, all cell atoms can be split into ``orbits'' or ``stars'' of atoms, $\xi$. Atoms within an orbit can be obtained from each other by applying an appropriate symmetry operation $g\in G$; atoms from different orbits cannot be related to each other in this way. The idea is to find appropriate linear combinations of the atomic displacements so that the number of degrees of freedom be reduced.

This is done, first, by finding symmetry-adapted displacements for each orbit separately:

$$\eta_{\Gamma\xi f\alpha}=\sum_{i}U_{\xi f\alpha,i}^{(\Gamma)}u_{\xi i}$$ (2.5)

where $u_{\xi i}$ are atomic Cartesian displacements for all atoms belonging to the orbit $\xi$, $\Gamma$ is an irreducible representation, $f$ - its repetition and $\alpha$- dimension index (for 2D and 3D representations). Combining all symmetry-adapted displacements (for all orbits and all represnetations), we get the same number of degrees of freedom as originally, when Cartesian displacements $u_{i}$ were used. Therefore, one can write

$$\eta_{\nu}=\sum_{i}U_{\nu i}u_{i}$$ (2.6)

where the transformation matrix $U=\left\Vert U_{\nu i}\right\Vert$ is a square matrix. This matrix can be constructed using the projection operator method well known in the group theory (see, e.g. Ref. [L. Kantorovich and Livshits- Phys. Stat. Sol. (b) 174 (1992) 79]). It can also be constructed as a unitary and real matrix. The latter property is obtained by combining 1D complex conjugate irreducible representation into a single real 2D representation, see details in the same paper. Thus, one can relate atomic displacements back to the symmetry-adapted ones via

$$u_{i}=\sum_{\nu}U_{\nu i}\eta_{\nu}$$ (2.7)

or $\mathbf{u}=\mathbf{U}^{T}\eta$ in the matrix form. Inserting these expansions into the energy expression (2.2), one obtains an important result:

$$E-E_{0}=\sum_{\Gamma\xi f\alpha}F_{\Gamma\xi f\alpha}\eta_{\... ...ma\xi f\alpha}\eta_{\Gamma\xi^{\prime}f^{\prime}\alpha}+\ldots$$ (2.8)

Importantly, the sum

$$\sum_{i\in\xi}\sum_{j\in\xi^{\prime}}U_{\xi f\alpha,i}^{(\Ga... ...\alpha^{\prime}}\Phi_{\xi f,\xi^{\prime}f^{\prime}}^{(\Gamma)}$$ (2.9)

is diaginal in $\Gamma$ and does not depend on the dimension index $\alpha$. It is here where the simplification comes.

Therefore, the method works in the following way (again, for simplicity, the method of a single displacement is considered):

• firstly, a representation $\Gamma$ is chosen for the given orbit $\xi$
• then, atomic displacements are obtained for all atoms in the orbit using Eq. (2.7) (atoms of other orbits will not be displaced in this case)
• a DFT code is run for the new geometry and the Cartesian forces on all atoms of the cell are obtained
• the Cartesian forces $F_{i}$ behave similarly to atomic displacements; therefore, from all $F_{i}$ one can obtain their symmetry-adapted counterparts using
  
  $$F_{\Gamma\xi f\alpha}=\sum_{i}U_{\xi f\alpha,i}^{(\Gamma)}F_{i}$$
  
  

• using these forces, the symmetry adapted elements of the FCM, $\Phi_{\xi f,\xi^{\prime}f^{\prime}}^{(\Gamma)}$, can be obtained as explained in Section 2.6.8.1; note that the matrix does not depend on index $\alpha$; therefore, it is sufficient to use $\alpha=1$, which also reduces the number of DFT runs
• diagonalising the matrix $\Phi^{(\Gamma)}=\left\Vert \Phi_{\xi f,\xi^{\prime}f^{\prime}}^{(\Gamma)}\right\Vert$, one obtains squares of all phonon frequencies, $\omega_{\Gamma\lambda}^{2}$, for the given representation $\Gamma$, the index $\lambda$ numbering all solutions
• the eigenvectors, $\eta_{\Gamma\xi f\alpha}$, can be turned into actual displacements via Eq. (2.7); thus, normal coordinates for each mode $(\Gamma\lambda)$ can be obtained
• the procedure is repeated for another orbit $\xi$ and/or another $\Gamma$

Note that the FCM in Cartesian atomic displacements, $\Phi_{ij}$, can be restored from the matrices $\Phi^{(\Gamma)}$ using the inverse to the Eq. (2.9) transformation:

$$\Phi_{ij}=\sum_{\Gamma\alpha}\sum_{f\in\xi}\sum_{f^{\prime}\... ...rime}}^{(\Gamma)}U_{\xi^{\prime}f^{\prime}\alpha,j}^{(\Gamma)}$$ (2.10)

This transformation will prove useful if other atomic masses should be tried (isotops) and it will be used in option V3 of the main menu, Section 2.6.8.9.