$\rho \rightarrow V_H$

From Hirose and Tsukada PRB. 51, 5278 (1995) and Hirose's thesis[4,5].

How to solve

$$\nabla^2 V_H(r) = -4 \pi \rho(r )$$ (191)

when $V_H(x,y,z=z_0)$ and $V_H(x,y,z=z_{l+1})$ are given.

change the coordinate $r=(x,y,z) \rightarrow (G_{\parallel },z)$, where $G_{\parallel }=(G _x,G_y)$.

$$V_H(r) = \sum_{G_{\parallel }} V_H(G_{\parallel },z) e^{i G_{\parallel } r_{\parallel }}$$ (192)

$$\rho(r) = \sum_{G_{\parallel }} \rho(G_{\parallel },z) e^{i G_{\parallel } r_{\parallel }}$$ (193)

$$\left(\frac{d^2}{dz^2} -G_\parallel ^2\right) V_H(G_{\parallel },z) = -4\pi \rho(G_{\parallel },z)$$ (194)

Introduce a Green function

$$\left(\frac{d^2}{dz^2} -G_\parallel ^2\right) G(G_\parallel ,z,z') = -\delta (z,z')$$ (195)

$$G(G_\parallel ,z) = \int dk G(G_\parallel ,k) e^{ikz}$$ (196)