The quantum-mechanical Hamiltonian for tetraatomic systems in symmetric hyperspherical coordinates

Here, we obtain the explicit expression for the kinetic energy operator for the four-body problem in a system of symmetric or democratic hyperspherical coordinates which generalize those first introduced by F. T. Smith for the case of three particles. The derivation of the metric tensor also provides the corresponding classical kinetic energy functional. A novel feature is a parametrization of inertia tensor components by independent angular variables, which also span configurations of opposite chirality; it avoids the problem of interdependence of the range of variables by a proper analysis of the eigenvalues of a cubic secular equation, and can also be extended to cases of more than four bodies in a three-dimensional space. We also give formulae of the interatomic distances useful for the hyperspherical mapping of four-body potential-energy surfaces. Briefly discussed are also the kinetic energy and the potential-energy surface mapping in the asymmetric hyperspherical parametrization of Jacobi vectors and their Radau or orthogonal local variants.