Scaling of Coulomb and exchange-correlation effects with quantum dot size

The role of many-body effects in quantum dots is of both academic and practical interest. We study the electron-electron interaction within a simplified spherical quantum dot using the local spin-density approximation. We experiment with a variety of confining potentials (triangular, harmonic, square well, etc.) and with a varying number of electrons $(N=2\mathrm{}$ to 20). We carry out a detailed study of the scaling behavior of the “Hubbard $U\mathrm{”}$ potential, which is a measure of the capacitive energy, with quantum dot size R $\left(U\sim {1/R}^{\beta }\right).\mathrm{}$ We find that the scaling exponent $\beta$ is $\approx 1/2$ for harmonic confinement and equal to 1 for the square-well confinement. The dependence of the scaling exponents on the confining potential and the number of electrons N is elucidated. We also examine the relative importance of Coulomb, exchange, and correlation terms in the Hubbard U potential and find that correlation plays a relatively more important role at a larger size. We provide a partial explanation for the value of the exponent in the Appendix.