Smooth Velocity-Dependent Potential and Nuclear Matter

A velocity-dependent potential of the form $V(\mathbf{r},\mathbf{p})=-V_1\mathrm{}\left(r\right)+p^2{e}^{-s^2{p}^2}{V}_2\mathrm{}\left(r\right)+V_2\mathrm{}\left(r\right)\mathrm{}{p}^2{e}^{-s^2{p}^2},$ where $V_1\mathrm{}\left(r\right)\mathrm{}$ and $V_2\mathrm{}\left(r\right)\mathrm{}$ are Gaussians, is used to fit the singlet (0-320-MeV) phase shifts for elastic nucleonnucleon scattering. This exponential velocity dependence replaces the hard core with a short-range repulsion which is much softer than the case of quadratic ($s=0\mathrm{}$) velocity dependence used by Green, Levinger, et al. The two-body scattering problem is solved in momentum space by numerical summation of the Born series; the ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ scattering length is calculated separately by a rapid matrix-inversion method. The applicability of ordinary many-nucleon perturbation theory for this interaction is tested by calculation of the first-order ($P_1$) and second-order ($P_2$) potential energy per particle of nuclear matter. A rapid singlet-even convergence rate of $\frac{P_2}{P_1}=4.3\%$ at $k_F=1.5\mathrm{}$ F follows as a result of the reduced off-energy-shell matrix elements of this two-body interaction. One also finds qualitative agreement with the singlet-state potential energy per particle obtained by Sprung et al. in their complete nuclear-matter calculation. The harmonic-oscillator matrix elements required for Hartree-Fock calculations of spherical nuclei are evaluated quite simply for this potential. Also, since this potential readily separates into relative $x$, $y$, and $z$ coordinates, it is well suited for Hartree-Fock calculations of deformed nuclei.