Translational invariance

When all the atoms are moved by the same amount, i.e. the crystal is rigidly shifted, the force on each atom must be zero. This is a stronger constraint than the one in which it is the sum of the forces on each atom to be zero. The latter is expressed by:

$$\sum_{s,t,i} \Phi_{i s \alpha, 0 t \beta} = 0,$$ (9)

where $\Phi_{i s \alpha, 0 t \beta}$ is the force constant matrix, $s$ and $t$ run over the number of atoms $N$ in the primitive cell and $i$ over the $M$ lattice vectors included in the calculation. If this constraint is not satisfied, it is straightforward to impose it by subtracting from the calculated force on each atom the value ${\bf F}/(MN)$, where ${\bf F }= \sum_{s,i} {\bf F}_{si}$, and ${\bf F}_{si}$ is the force acting on atom $s$ in primitive cell $i$.

The former condition is:

$$\sum_{s,i} \Phi_{i s \alpha, 0 t \beta} = 0;  {\rm for  each}   t = 1, N.$$ (10)

Clearly, Eq. 10 implies Eq. 9, but the opposite is not true in general. However, it is Eq. 10 to imply that at ${\bf q}=(0,0,0)$ the three acoustic branches have identically zero frequencies.

The constraint in Eq. 10 has to be imposed in such a way that the force constant matrix remains symmetric: $\Phi_{\alpha,\beta} ({\bf R}_j + {\bf\tau}_t - {\bf R}_i - {\bf\tau}_s) = \Phi_{\beta,\alpha} (-[{\bf R}_j + {\bf\tau}_t - {\bf R}_i - {\bf \tau}_s ] )$. In the PHON code this is done iteratively, in a number of steps in which the symmetry is re-imposed at each step.

To impose translational invariance as described set the variable:

NTI = 20

Translational invariance is imposed iteratively, and $\sim 20$ iterations are usually enough.

The amount of output printed by the PHON is controlled by the variable IPRINT. IPRINT=0 will produce a minimal output, IPRINT=3 a verbose output, which also includes the dynamical matrix and its eigenvectors.