Hard sphere trial wave function

The problem of scattering on a hard sphere potential (1.48) was studied in Sec. 1.3.2.2. In dilute systems for small interparticle distance $\r$ the two body Bijl Jastrow term $f_2(r)$ is well approximated by the solution $f(r)$ (1.51), i.e. by the wave function of a pair of particles in vacuum. At large distances the pair wave function asymptotically goes to a constant value, as the particles become uncorrelated.

Taking these facts into account we introduce the trial function in the following way[GBC99] (here we introduce dimensionless notation by measuring the distance $\r$ in units of the hard sphere radius $a_{3D}$ and energy $E$ in units of $\hbar^2/(ma^2_{3D})$)

$$f_2(r) = \left\{ {\begin{array}{ll} \displaystyle \frac{A\sin(\sq... ...eft\{-\frac{r}{\alpha}\,\right\},& \vert r\vert > R_{m}\\ \end{array}} \right.$$ (2.67)

The request the function be smooth at the matching point $R_{m}$, i.e.

1)

the function $f_2(r)$ itself must be continuous:

$$\frac{A\sin(\sqrt{2E}(R_{m}-1))}{R_{m}} = 1 - B \exp\left\{-\frac{R_{m}}{\alpha}\,\right\}$$ (2.68)

2)

derivative $f_2'(r)$ must be continuous

$$\frac{A\sqrt{2E}\cos(\sqrt{2E}(R_{m}-1))}{R_{m}} -\frac{A\sin(\sq... ...R_{m}-1))}{R_{m}^2} =\frac{B}{\alpha}\exp\left\{-\frac{R_{m}}{\alpha}\,\right\}$$ (2.69)

3)

the local energy $f_2(r)^{-1}(-\hbar^2\Delta_1/2m-\hbar^2\Delta_1/2m+ V_{int}({{\vec r}_i-{\vec r}_j}))f_2(r)$ must be continuous

$$2E = \frac{\left(\displaystyle\frac{1}{\alpha^2}-\frac{2}{R_{m}\a... ...}{\alpha}\,\right) }{1-B \exp\left(-\displaystyle\frac{R_{m}}{\alpha}\,\right)}$$ (2.70)

The solution of this system is

$$\left\{ {\begin{array}{l} A =\displaystyle\frac{R}{\sin(u(1-1/R))... ...\\ B =\displaystyle\frac{u^2 \exp(\xi)}{\xi^2-2\xi+ u^2}, \end{array}} \right.$$ (2.71)

where we used the notation $u = \sqrt{2E}R$ and $\xi = R/\alpha$. The value of $\xi$ is obtained from the equation

$$1-\frac{1}{R} = \frac{1}{u} \mathop{\rm arctg}\nolimits \frac{u(\xi-2)}{u^2+\xi-2}$$ (2.72)

There are three conditions for the determination of five unknown parameters, consequently two parameters are left free. The usual way to define them is minimize the variational energy in Variational Monte Carlo which yields an optimized trial wave function.