Static structure factor

It is natural to give the definition of the static structure factor $S(k)$ in the momentum space as the correlation function of the momentum distribution between elements $-\k$ and $\k$ (1.29):

$$N S(\k) = \langle\rho_{-\k}\rho_\k\rangle - \vert\langle\rho_\k\rangle\vert^2,$$ (2.133)

Using the properties of the Fourier component $\rho_{-\k} = (\rho_\k)^*$ it can be rewritten in a different way

$$N S(\k) = \langle\vert\rho_\k\vert^2\rangle - \vert\langle \rho_\k\rangle\vert^2$$ (2.134)

In the Diffusion Monte Carlo calculation the density distribution is approximated by the density of walkers (see (2.31))

$$n(\r) = \sum\limits_{i=1}^N \delta(\r-{\vec r}_i)$$ (2.135)

With the means of the Fourier transformation we express it in the momentum space

$$\rho_\k= \int e^{i\k\r} n(\r) d\r= \sum\limits_{i=1}^N e^{i\... ...=1}^N \cos\k{\vec r}_i +i \sum\limits_{i=1}^N \sin\k{\vec r}_i$$ (2.136)

and obtain a simple expression for the static structure factor

$$N S(\k) = \left< \left(\sum\limits_{i=1}^N \cos \k{\vec r}_i... ...left<\sum\limits_{i=1}^N \sin \k{\vec r}_i\right>\right\vert^2$$ (2.137)

In a trapped system there are no restrictions on the value of momentum $\k$, although, naturally, the momentum distribution vanishes for $k\ll 1/R$, where $R$ is the size of the system. Instead, if periodic boundary conditions are used, the value of momenta is quantized and is dependent on the size of the box

$$k_{n_{x,y,z}} = \frac{2\pi}{L} n_{x,y,z}$$ (2.138)

At the same in a homogeneous system the two last terms in (2.137) are vanishing.