Pair distribution

The pair distribution function (TBDM) in a homogeneous system is given by the formula (1.19)

$$g_2(\vert{\vec r_2}-{\vec r_1}\vert) = \frac{N(N-1)}{n^2}\frac{\i... ...R})\vert^2{d\vec r}_3...{d\vec r}_N}{\int\vert\psi^*({\bf R})\vert^2\,{\bf dR}}$$ (2.152)

Let us explain now how this formula is implemented in Monte Carlo calculation. We make summation over all pairs of particles:

$$g_2(r) =\frac{N(N-1)}{n^2 L} \frac{\int\delta({\vec r_1}-{\vec r_... ...\vert\psi({\bf R})\vert^2\,{\bf dR}}{\int\vert\psi^*({\bf R})\vert^2\,{\bf dR}}$$ (2.153)

If we do a discretization of the coordinate with spacing $h$ and introduce function $\vartheta_h(z)$ which is one if $z<h$ and zero otherwise, do summation over absolute value (the distribution is obviously symmetric) we obtain following expressions:

1)

In one dimensional system:

$$g^{1D}_2(r)=\left\langle \frac{2}{2hnN}\sum\limits_{i<j}\vartheta_h(\vert r_{ij}-r\vert)\vert \right\rangle$$ (2.154)

In a uncorrelated system $\vartheta_h(\vert z\vert) = 2h/L$ is constant and $g_2(z)=1-1/N$.

2)

In two-dimensional system distance $z$ enters explicitly in the expression of the pair distribution function leading to larger numerical variance at small distances

$$g^{2D}_2(z)= \left\langle \frac{2}{2\pi z hnN}\sum\limits_{i<j}\vartheta_h(\vert z_{ij}-z\vert)\vert\right\rangle$$ (2.155)

3)

In a three-dimensional system the corresponding expression is

$$g^{3D}_2(z)= \left\langle \frac{2}{4\pi z^2 hnN}\sum\limits_{i<j}\vartheta_h(\vert z_{ij}-z\vert)\vert\right\rangle$$ (2.156)