Exponentiation

It is convenient (see Eq.2.38) to do the exponentiation of the one- and two- body terms $u_1(\vec r) = \ln f_1(\vec r)$, $u_2(r) = \ln f_2(r)$. The point is that numerically a better precision is achieved by working with numbers of the same order. The formula for the kinetic energy becomes simpler

$$T^{loc}({{\vec r_1},...,{\vec r}_N}) = -\frac{\hbar^2}{2m}\left\{... ... +\sum\limits_{i=1}^N \vert\vec F_i({{\vec r_1},...,{\vec r}_N})\vert^2\right\}$$ (2.131)

with

$$\vec F_i({{\vec r_1},...,{\vec r}_N}) = u_1'({\vec r}_i)\frac{{\v... ...ec r}_k}\vert)\frac{{{\vec r}_i-{\vec r}_k}}{\vert{{\vec r}_i-{\vec r}_k}\vert}$$ (2.132)