In this paper we present a unified treatment of three-body atoms and molecules using numerical tensor methods. The Schrödinger equation in perimetric coordinates is recast in a canonical tensor format. It is shown that the Schrödinger equation can be solved in this full-tensor format but that by using a low-rank tensor decomposition, in particular the tensor-train and quantized-tensor-train formats, energies accurate to at least the nanohartree can be obtained for the He atom (in which the mass of the uniquely charged particle is much greater than the other two particles), the positronium negative ion (${\mathrm{Ps}}^{-}$, in which all the masses are equal), and the non-Born-Oppenheimer ${\mathrm{H}}_2{}^{+}$ molecule (in which the mass of the uniquely charged particle is much smaller than the other two particles).

In this paper we derive a numerical tensor method to solve the many-body Schrödinger equation for three-particle systems. The Schrödinger equation is first represented in a canonical tensor format using a triple orthogonal Laguerre-based wave function in scaled perimetric coordinates. We demonstrate that the antisymmetry of the wave function can be explicitly included in the tensor method formulation by coupling the indices associated with the coordinates representing the identical fermions, e.g., electrons or muons. It is shown that energies accurate to the picohartree can be achieved in full tensor format in less than 3300 s for atomic and muonic molecular systems. Furthermore, this CPU time is reduced to less than 300 s, without compromising the accuracy of the energy or quality of the wave function, using a low-rank tensor-train decomposition.

In this paper, we investigate the maximum number of electrons that can be bound to a system of nuclei modelled by Hartree-Fock theory. We consider both the Restricted and Unrestricted Hartree-Fock models. We are taking a non-existence approach (necessary but not sufficient), in other words we are finding an upper bound on the maximum number of electrons. In giving a detailed account of the proof of Lieb’s bound [Theorem 1, Phys. Rev. A 29 (1984), 3018] for the Hartree-Fock models we establish several new auxiliary results, furthermore we propose a condition that, if satisfied, will give an improved upper bound on the maximum number of electrons within the Restricted Hartree-Fock model. For two-electron atoms we show that the latter condition holds.

The Complex Absorbing Potential (CAP) method is widely used to compute resonances in quantum chemistry, both for nonrelativistic and relativistic Hamiltonians. In the semiclassical limit ℏ → 0 we consider resonances near the real axis and we establish the CAP method rigorously for the perturbed Dirac operator by proving that individual resonances are perturbed eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa. The proofs are based on pseudodifferential operator theory and microlocal analysis.