Matching to a constant

The wave function constructed in the previous section is long-ranged as it decays only as $1/r$ at large distances. This leads to overestimation of the correlations in deep BCS limit. For example at $a=-5R$ the variational energy is 8 times larger than the energy of a fermi gas. The fixed-node MC algorithm corrects this behavior and the resulting energy is close to the energy of a fermi gas.

$$f(r)=\left\{ \begin{array}{cc} \frac{A}{r}\sin \varkappa r, & r<R... ...c{\vert a\vert}{r}), & R<r<R_{m} \\ 1+C\exp (-k_{m}r), & R<r\end{array}\right.$$ (2.120)

The matching conditions fix values of the constants $A,B,C$:

$$\left\{ \begin{array}{lll} A&=&\frac{BR(1+\vert a\vert/R)}{\sin k... ... C&=&-\frac{B\vert a\vert\exp (-k_{m}R_{m})}{R_{m}^{2}k_{m}} \end{array}\right.$$ (2.121)