Forces with finite variance in periodic systems

The method present in section 1.1.1 can be easily generalized to periodic systems. It is sufficient use an auxiliary periodic function $\tilde \Psi$ with the same behavior of 1.12 close to nuclei. We have used the following form:

$$\tilde \Psi_{PBC} = Q_{PBC} \Psi_T$$

$$Q^\nu_{PBC} = Z_A \sum_{i=1}^{N_{elect}} \frac{ \frac{L}{2 \pi}\sin\left ( \frac{2 \pi}{L} ( x_\nu-R_A^\nu ) \right)}{r'_{iA}}$$

where $r'_{iA}$ is the periodic distance between the nucleus $A$ and the electron $i$ :

$$r'_{iA} = \frac{L}{\pi}\sqrt{\sin^2\left ( \frac{ \pi}{L} ( x_i^1... ... x_i^2-R^2_A ) \right) + \sin^2\left ( \frac{ \pi}{L} ( x_i^3-R^3_A ) \right)}.$$ (4.18)

Notice that $\Psi_{PBC}$ is a periodic function because we are using a periodic trial wave-function $\Psi_T$ (see section 4.1.2). This auxiliary function $\Psi_{PBC}$ removes the divergence in the bare force and it is consistent with the periodicity of the system. At variance of the case without periodic boundary conditions $\nabla^2 Q_{PBC}$ does not cancel exactly the term coming from the derivative of the ion-electron potential. Therefore we have to included both the Laplacian of the $Q_{PBC}$ and the derivatives of the ion-electron potential in the calculation of the forces. More precisely the expression we used for the force is:

$$\tilde F^\nu= F^\nu_{bare} - \frac{1}{2} \nabla^2 Q^\nu_{PBC} - \frac {\vec{\nabla}Q_{PBC}^\nu \vec{\nabla}\Psi _T }{\Psi_T}$$ (4.19)

where:

$$\nabla^\mu_i Q^\nu_{PBC} = Z_A \frac{2 \pi^2}{L^2} \frac{\sin(\frac {2 \pi x_i^\nu}{L}) \sin... ... \delta_{\mu,\nu} Z_A \frac{2 \pi }{L}\frac{\cos(\frac {2 \pi x_i^\nu}{L})}{r'}$$

$$\nabla_i^2 Q^\nu_{PBC} = Z_A \sum_{\mu =1,3} -\frac{\pi^2}{L^2} \frac{\sin(\frac{2 \pi x^\... ...t ) \cos \left (\frac{\pi x_i^\mu}{L} \right ) \right]^2 \frac{1}{r'^2} \right.$$

$$- \left. \cos^2 \left (\frac{\pi x_i^\mu}{L} \right )+\sin^2\left (\frac{\pi x_i^\mu}{L} \right) \right \} \frac{1}{r'^3}$$

$$- 2 Z_A \frac{\pi^2}{L^2} \left [-\sin \left (\frac{2 \pi x_i^\nu}{... ... \cos \left (\frac{2 \pi x_i^\nu}{L} \right) \frac{1}{r'^2}\right] \frac{1}{r'}$$