One-body Bijl-Jastrow term in an anisotropic trap

Let the external field be an anisotropic trap with the aspect ratio $\lambda$: $V_{ext}(\r)=\frac{1}{2}m\omega_\perp^2(x^2+y^2+\lambda^2z^2)$. We choose the one-body Bijl-Jastrow term (2.37) in form of a Gaussian with the widths $\alpha$ and $\beta$ being variational parameters:

$$f_1(\r) =\exp\{-\alpha(x^2+y^2)-\beta\lambda z^2\}$$ (2.41)

Then the one body contribution to the drift force is

$$\vec F_1({\vec r}_i) = -\left(2\alpha x, 2\alpha y, 2\beta\lambda z\right)$$ (2.42)

The local energy is

$$E^{loc}({\bf R}) = N(2\alpha+\beta\lambda) +\sum\limits_{j<k}^N{\... .....,{\vec r}_N})\vert^2 +\sum\limits_{i=1}^N\frac{x_i^2+y_i^2+\lambda^2z_i^2}{2}$$ (2.43)

where we used oscillator units: energy is measured in units of $\hbar\omega_\perp$ and the distances in units of the oscillator length $a_\perp$.