Variational Principle Directly on the Coherent Pair Condensate. I. the BCS Case

We propose a scheme to perform the variational principle directly on the coherent pair condensate (VDPC). The result is equivalent to that of the so-called variation after particle-number projection, but now the particle number is always conserved and the time-consuming projection is avoided. This work considers VDPC+BCS. We derive analytical expressions for the average energy and its gradient in terms of the coherent pair structure. In addition, we give analytically the pair structure at the energy minimum, and discuss its asymptotic behavior away from the Fermi surface, which looks quite simple and allows easy physical interpretations. The new algorithm iterates these pair-structure expressions to minimize energy. We demonstrate the new algorithm in a semi-realistic example using the realistic $V_{low-k}$ interaction and large model spaces (up to 15 harmonic-oscillator major shells). The energy can be minimized to practically arbitrary precision. The result shows that the realistic $V_{low-k}$ interaction does not cause divergences in the pairing channel, although tiny occupation numbers (for example smaller than $10^{-5}$) contribute to the energy (by a few tens of keV). We also give analytical expressions for the gradient of energy with respect to changes of the canonical single-particle basis, which will be necessary for the next work in this series: VDPC+HFB.