A simple parameter-free wavefunction for the ground state of three-body systems

Full text: The study of the structure and stability of Coulombic three-body systems [m1m2m3], with arbitrary masses mi and charges zi (i = 1, 2, 3), has been the subject of many investigations. Recently, we have proposed a pedagogical, simple and parameter-free wavefunction for the ground state of two-electron atoms. The proposal was then generalized to more general atomic three-body systems in which one of the particles is positively charged (z3 > 0) and heavier than the other two which are negatively charged (z1 < 0, z2 < 0). Let νij = μijzizj where μij mimj/(mi + mj) (i ≠ j = 1, 2, 3) are the reduced masses. In terms of the interparticles coordinates r1 = r13, r2 = r23 and r12 (particle 3 is placed at the origin of the coordinates), the proposed wavefunction reads ψARGGEN = NARGGEN eν13r2(1 + ν12r12)[1 + c(r12 + r22], where NARGGEN is the normalization constant and c is replaced by an analytical expression in terms of (mi, zi) in order to minimize the mean energy of the ground state. The wavefunction ψARGGEN : has the same form for all systems; is parameter-free; is nodeless; satisfies, by construction, all two-particle cusp conditions; and yields reasonable ground state energies for several systems including the prediction of a bound state for H-, D-, T- and Mu-. A wavefunction with all these characteristics is presently not available in the literature. The simplicity of ψARGGEN is such that analytical expressions for the ground state energy can be derived. Hence, we have a useful predictive and simple analytical tool (which, to our knowledge, is not available in the literature) to estimate the energy, and therefore to study the stability, of exotic Coulombic three-body systems. In addition, our proposal is simple enough, but sufficiently accurate to be used as a starting point in calculations of collision cross sections. Of course due to its simplicity, energy values cannot compete with those obtained with advanced variational wavefunctions which involve large number of basis functions. However, the latter do not have a predictive character since they have to be optimized each time for a given three-body system, and in most cases, do not satisfy exactly Kato cusp conditions. For illustration, results will be shown for the several three-body systems. (author)