Bound-state stability of Coulomb three-body systems using numerical tensor methods

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 (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msup><a:mrow><a:mi>Ps</a:mi></a:mrow><a:mo>−</a:mo></a:msup></a:math>, in which all the masses are equal), and the non-Born-Oppenheimer <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:msub><b:mi mathvariant="normal">H</b:mi><b:mn>2</b:mn></b:msub><b:msup><b:mrow/><b:mo>+</b:mo></b:msup></b:mrow></b:math> molecule (in which the mass of the uniquely charged particle is much smaller than the other two particles).