BORN-OPPENHEIMER AND HYPERSPHERICAL ADIABATIC APPROXIMATIONS IN THE NUCLEAR-MASS AND MOLECULAR-MASS LIMITS

Ground-state energies of a three-particle system composed of a distinct and two equal-mass particles interacting via local S-wave potentials is studied using Born-Oppenheimer and hyperspherical adiabatic approximations in the nuclear- and molecular-mass limits. The nuclear situation is simulated when the ratio m of equal to distinct particle masses is of the order of unity. The molecular situation is simulated for m >> 1, when the potential among the equal-mass particles is switched off. The ratio m was varied between the limits m is similar to 1 and m >> 1 and the depths and ranges of the potentials were varied to simulate intermediate situations. In all cases the extreme and the uncoupled hyperspherical adiabatic approximations, which are the lower and upper bounds for the exact ground-state energy, is found to define a very narrow band with the Born-Oppenheimer approximation lying very close to it. Thus the two types of approximations are always found to work surprisingly well.