Many-body scattering theory methods as a means for solving bound-state problems: Applications of arrangement-channel quantum mechanics

We propose in this article that the non-Hermitian equations typical of some many-body scattering theories be used to help solve many-body bound-state problems. The basic idea is to exploit the channel nature of manybody bound states that must exist because bound states are obvious negative-energy extensions of scattering states. Since atomic, molecular, and nuclear systems all display multichannel effects for $E>\mathrm{}0\mathrm{}$, at least through Pauli-principle effects if not through mass-transfer reactions, this use of positive-energy methods for solving bound-state problems could have wide applicability. The development used here is based on the channel-component-state method of the channel-coupling-array theory, recently described in detail for the $E>\mathrm{}0\mathrm{}$ case, and various aspects of the formalism are discussed. Detailed calculations using simple approximations are discussed for ${\mathrm{H}}_2^{+}$, one of the simplest systems displaying channel structure. Comparison with the exact, Born-Oppenheimer results of Wind show that the non-Hermitian-equation, channel-component values of the equilibrium separation and total binding energy are accurate to within 2%, while the dissociation energy is accurate to 10%. The resulting wave function is identical to that arising from the simplest MO calculation, for which these numbers are less accurate than the preceding by at least a factor of 3. We also show that identical particle symmetry for the ${\mathrm{H}}_2^{+}$ case reduces the pair of coupled (two-channel) equations to a single equation with an exchange term. Similar reductions will occur for larger numbers of identical particles, thus suggesting application of the formalism to atomic structure problems. A detailed analysis of the present numerical results, their general implications, and possible applications is also given.