Theory: practical side

The precision of the calculation can be checked in the following way. For simplicity, we consider the first method. We start from $u_{1}$ and in a single energy calculation (without relaxation, of course!) all forces $F_{i}$ are determined. Thus, in a single calculation, the whole row, $\Phi_{1i}$ of the FCM is obtained. Then, we take $u_{2}$ and repeat the calculation. It gives the complete second row, $\Phi_{2i}$. This way, the whole matrix is obtained. However, the FCM should be symmetric. Due to the higher anharmonic terms that were dropped in deriving the working expression for the FCM, Eq. (2.3), this condition, however, will not be satisfied. Thus, by checking the relative difference between elements $\Phi_{ij}$ and $\Phi_{ji}$, one can check its precision.

There are basically two ways in improving precision. Firstly, one can reduce the trial displacements $u_{i}$. This is tricky as the forces on atoms are calculated numerically in any DFT code, so that forces may become unrealiable if $u_{i}$ are too small. Note that it is vital to have reliable forces in the DFT code you use. Normally, it is necessary to use as high precision in the code as possible, especially, if small displacements $u_{i}$ are to be used. The other option is to use more displacements, at least two (works up to the 4th order in the energy expression). Still, make sure that the forces are at least two orders of magnitude more accurate than in ordinary ground state relaxation calculations.