Bijl-Jastrow wave function

The bosonic function must be symmetric with respect to exchange of two particles. The most natural way to construct the trial wave function of a system of Bosons is to consider a product of one-body and two-body terms (we neglect three-body and higher terms):

$$\Psi({{\vec r_1},...,{\vec r}_N}) =\prod\limits_{i=1}^N f_1({\vec r}_i)\prod\limits_{j<k}^N f_2(\vert{{\vec r}_j-{\vec r}_k}\vert)$$ (2.37)

This construction is called Bijl-Jastrow trial wave function. The one-body term $f_1(\r)$ accounts for the external potential and, commonly, has the same structure as the solution of an ideal system in the same external potential. The interaction between particles is accounted by the two-body Bijl-Jastrow term $f_2(r)$ which must go to a unity (uncorrelated value) at large distances. If the periodic boundary conditions are used the restriction on the two-body term is stronger, the function must go to the unity already at the half size of the simulation box. This condition ensures that the particles do not interact with their own images and no artificial correlations are introduced.

Once the exact type of the Bijl-Jastrow terms is chosen, one should also calculate the first and second derivatives in order to implement the QMC method. Actually the algorithm can be optimized by noticing (see Sec. 2.7) that the trial wave function always comes in one of the three combinations:

1) the logarithm of the Bijl-Jastrow term (is necessary for the Metropolis algorithm in the variational calculation and calculations of the non-local quantities, e.g. the one-body density matrix)

$$u(r) = \ln f(r)$$ (2.38)

2) the logarithmic derivative of the Bijl-Jastrow term (is needed for the calculation of the drift force (2.15))

$${\cal F}(r) = \frac{f'(r)}{f(r)}$$ (2.39)

3) the second derivative enters only in the calculation of the kinetic part of the local energy. The following combination is relevant:

$${\cal E}^{loc}(r) =-\frac{f''(r)}{f(r)}+\left(\frac{f'(r)}{f(r)}\right)^2 + \frac{mV_{int}(r)}{\hbar^2}-\frac{D-1}{r}\frac{f'(r)}{f(r)},$$ (2.40)

where $D$ is number of dimensions.