Crossover from bulk to few-electron limit in ultrasmall metallic grains

We study the properties of ultrasmall metallic grains with sizes in the range 20-400 electrons. Using a particle-hole version of the density-matrix renormalization-group (DMRG) method we compute condensation energies, spectroscopic gaps, pairing parameters, and particle-hole probabilities of the ground-state wave function. The results presented in this paper confirm that the bulk superconducting regime (large grains) and the fluctuation dominated regime (small grains) are qualitatively different, but show that the crossover between them is very smooth with no signs of critical level spacings separating them. We compare our DMRG results with the exact ones obtained with the Richardson solution finding complete agreement. We also propose a simplified version of the DMRG wave function, called the particle-hole BCS Ansatz, which agrees qualitatively with the DMRG solution and illustrates what is lacking in the projected BCS (PBCS) wave function in order to describe correctly the crossover. Finally we present a recursive method to compute norms and expectation values with the PBCS wave function.