Periodic orbitals

In order to obtain periodic orbitals, starting from non periodic ones it is sufficient to replace the Cartesian coordinates $x_i$ with a simple periodic function $x'_i(x)$ that take into account the appropriate periodicity of the box. In this thesis we used:

$$x'_i = \frac{L}{ \pi } \sin\left ( \frac{ \pi x_i }{L }\right )$$ (4.3)

and the new distance is defined as

$$r'= \frac{L}{ \pi } \sqrt{\sum_{i=1}^3 \sin^2\left ( \frac{ \pi x_i }{L }\right )}$$ (4.4)

In doing so, we have only to compute gradients and Laplacian with the chain rule:

$$\frac{ \Phi(r')}{\partial x_i} = \frac{\partial \Phi(r') }{\partial r'} \frac{\partial r'}{ \partial x_i'} \frac{ \partial x_i'}{\partial x_i}$$

$$\frac{\partial^2 \Phi(r')}{\partial x_i^2} = \frac{\partial^2 \Phi(r')}{\partial r'^2} \left ( \frac{\partial ... ... )^2 + \frac{\partial r'}{x_i'} \frac{ \partial^2 x_i'}{\partial x_i^2}\right ]$$

where

$$\frac{ \partial x_i'}{\partial x_i} = \cos \left({\frac {\pi\,x_i}{L}} \right)$$

$$\frac{ \partial^2 x_i'}{\partial x_i^2} = - \frac{\pi}{L}\sin \left({\frac {\pi\,x_i}{L}} \right)$$

This transformation has been applied to all orbitals appearing in the wave-function and also to the one-body term and the two-body Jastrow.
We remark here that also the normalization constant of a given orbital has to be changed in a periodic system. Namely its integral over the simulation cell has to be equal to one.

$$\int^{L/2}_{-L/2} \int^{L/2}_{-L/2} \int^{L/2}_{-L/2} \phi^2(r_{iA})\, dx \, dy \, dz = 1$$ (4.5)

For instance a normal Gaussian in three-dimension:

$$\Phi(r) = \left ( \frac{2 k}{ \pi} \right )^{3/4} e^{-k r^2}$$ (4.6)

becomes after the substitution 4.3:

$$\Phi'(r') = \left ( L e^{-\frac{k L^2}{\pi^2}} I_0 \left [ \frac{k L^2 }{\pi^2} \right ] \right )^{-3/2} e^{ -k r'^2}$$ (4.7)

where $I_0$ is the modified Bessel function of the first kind and $L$ is the size of the simulation box and $r'$ is the periodic distance 4.4.