Optimum Jastrow Function for Few-Electron Ground States in a Quantum Dot: Reduction to a Three-Particle Problem

A new approach to calculating the optimum Jastrow wave function is presented for a system of $N$ electrons in a two-dimensional quantum dot. By introducing special derivative operators which act on differences of electron coordinates ${\mathbf{r}}_{\mathrm{ij}}={\mathbf{r}}_i-{\mathbf{r}}_j$ as if they were independent coordinates ${\mathbf{r}}_{\mathrm{ij}}\}i<j$, it is shown that the problem of finding the optimum $N$-particle Jastrow function reduces to a three-particle problem (for$N\ge 3$). This three-particle problem is then solved using a variational method to find the optimum pair function $\phi \left({\mathbf{r}}_{\mathrm{ij}}\right)$. A perpendiculat magnetic field may also be included in the problem.