Extension of the correlated Gaussian hyperspherical method to more particles and dimensions

Abstract

The solution of the hyperangular Schrödinger equation for few-body systems using a basis of explicitly correlated Gaussians remains numerically challenging. This is in part due to the number of basis functions needed as the system size grows, but also due to the fact that the number of numerical integrations increases with the number of hyperangular degrees of freedom. This paper shows that the latter challenge is no more. Using a delta function to fix the hyperradius R, all matrix element calculations are reduced to a single numerical integration regardless of system size n or number of dimensions d. In the special case of d an even number, the matrix elements of the noninteracting system are fully analytical. We demonstrate the use of the new matrix elements for the 3-, 4-, and 5-body electron-positron systems with zero total angular momentum L, positive parity π, and varied spins S+ and S−.

Born-Oppenheimer study of two-component few-particle systems under one-dimensional confinement

“…A hyperspherical solution to the few-body problem in quasi-1D for finite-range interactions remains a significant challenge, although recent advances in the correlated Gaussian hyperspherical method [48] may make these calculations possible.…”

Universal few-body physics and cluster formation

“…Fig. 4 shows an example of the lowest potential energy curves obtained by (Daily and Greene, 2014) for a system of 3 electrons and 2 positrons. These potentials contain bound states of the different symmetries of this 5-body system, and they also describe the lowest energy scattering processes.…”

“…The horizontal solid lines ordered from lowest to highest indicate the asymptotic fragmentation threshold energies of Ps2+e − , Ps+Ps − , and 2Ps+e − . Adopted from (Daily and Greene, 2014).…”

Hyperspherical theory of the quantum Hall effect: The role of exceptional degeneracy

“…The literature in this field documents theoretical results that have been achieved in contexts as diverse as nuclear structure and reactivity [17][18][19][20][21], universal Efimov physics in cold atoms and molecules [22][23][24][25][26][27], few-electron atoms [28][29][30], and systems containing positrons and electrons [31][32][33][34]. Efimov's prediction [25] of a universal binding mechanism for three particles at very large scattering lengths can itself be viewed as an application of the adiabatic hyperspherical coordinate treatment in a problem where the method is exact, although Efimov did not himself express it in those terms.…”