Local kinetic energy and the drift force

In this section we will find the expression of the local kinetic energy

$$T^{loc}({{\vec r_1},...,{\vec r}_N}) = -\frac{\hbar^2}{2m}\frac{\Delta\Psi({{\vec r_1},...,{\vec r}_N})}{\Psi({{\vec r_1},...,{\vec r}_N})}$$ (2.124)

Let us calculate the second derivative in two steps, as the first derivative is important for the calculation of the drift force. We consider the Bijl-Jastrow form (2.37) of the trial wave function and will express the final results in terms of one- and two- body Bijl-Jastrow terms $f_1$ and $f_2$. The gradient of the many-body trial wave function is given by

$$\vec\nabla_{{\vec r}_i}\Psi({{\vec r_1},...,{\vec r}_N})=\Psi({{\... ...\vert)}\frac{{{\vec r}_i-{\vec r}_k}}{\vert{{\vec r}_i-{\vec r}_k}\vert}\right)$$ (2.125)

The full expression for the Laplacian is

$$\Delta_{{\vec r}_i}\Psi({{\vec r_1},...,{\vec r}_N})\qquad=\qquad... ...t)}\frac{{{\vec r}_i-{\vec r}_k}}{\vert{{\vec r}_i-{\vec r}_k}\vert} \right)^2+$$

$$+\Psi({{\vec r_1},...,{\vec r}_N})\left( \frac{f_1''({\vec r}_i)}... ...ec r}_k}\vert)}{f_2(\vert{{\vec r}_i-{\vec r}_k}\vert)}\right)^2 \right]\right)$$ (2.126)

The kinetic energy can be written in a compact form

$$T^{loc}({{\vec r_1},...,{\vec r}_N}) = \frac{\hbar^2}{2m}\left\{\... ...-\sum\limits_{i=1}^N \vert\vec F_i({{\vec r_1},...,{\vec r}_N})\vert^2\right\},$$ (2.127)

where we introduced notation for the one- and two- body contribution to the local energy (see, also, (2.40))

$${\cal E}^{loc}_1(\vec r) = -\frac{f_1''(\vec r)}{f_1(\vec r)}+\left(\frac{f_1'(\vec r)}{f_1(\vec r)}\right)^2$$ (2.128)

$${\cal E}^{loc}_2(r) = -\frac{f_2''(r)}{f_2(r)}+\left(\frac{f_2'(r)}{f_2(r)}\right)^2$$ (2.129)

and introduced the drift force (see (2.39))

$$\vec F_i({{\vec r_1},...,{\vec r}_N}) = \frac{f_1'({\vec r}_i)}{f... ...c r}_k}\vert)}\frac{{{\vec r}_i-{\vec r}_k}}{\vert{{\vec r}_i-{\vec r}_k}\vert}$$ (2.130)