The development of a high-accuracy, low-cost, numerical tensor method for the P states of three-particle systems

In this paper, we solve the non-relativistic many-body Schrödinger equation (SE) for the excited P states (L = 1) of three-particle systems with natural and unnatural parity. The exact separation of the rotational degrees of freedom from the internal degrees of freedom presented by Sadhukhan et al. [arXiv:2506.23962 (2025)] is exploited. The angular matrix elements involve the product of two bipolar harmonics and are calculated using an explicit analytical expression containing the Wigner 3j- and 6j-symbols. Using a symmetry and parity adapted wavefunction for P states in perimetric coordinates, the variational form of the SE in canonical tensor format is derived. This consists of a sum of triple tensor products of factor matrices and is an exact representation. From this canonical format, the SE can be solved in full tensor format or used to derive an exact representation as a second-order tensor train. Two formulations are presented. In the first, scaled perimetric coordinates are used; it is assumed that the like-charged particles have the same asymptotic behavior so that the orthonormality of the Laguerre functions can be exploited and the matrices are sparse. In the second, the like-charged particles are not constrained to have the same asymptotic behavior and an analytical integral is used which results in dense matrices. It is shown that by optimizing either one or two non-linear variational parameters, it is possible to obtain excited P state energies that are accurate to the picohartree, for atomic and molecular systems in less than 25 min on a standard desktop computer.