Hyperspherical explicitly correlated Gaussian approach for few-body systems with finite angular momentum

Abstract

Within the hyperspherical framework, the solution of the time-independent Schrödinger equation for a nparticle system is divided into two steps: the solution of a Schrödinger-type equation in the hyperangular degrees of freedom and the solution of a set of coupled Schrödinger-type hyperradial equations. The solutions to the former provide effective potentials and coupling matrix elements that enter into the latter set of equations. This paper develops a theoretical framework to determine the effective potentials, as well as the associated coupling matrix elements, for few-body systems with finite angular momentum L = 1 and negative and positive parity . The hyperangular channel functions are expanded in terms of explicitly correlated Gaussian basis functions, and relatively compact expressions for the matrix elements are derived. The developed formalism is applicable to any n; however, for n 6, the computational demands are likely beyond present-day computational capabilities. A number of calculations relevant to cold-atom physics are presented, demonstrating that the developed approach provides a computationally efficient means to solving four-body bound and scattering problems with finite angular momentum on powerful desktop computers. Details regarding the implementation are discussed.

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

“…Earlier studies involved 3-and 4-body systems with vanishing angular momentum L and positive parity π [23,24], and recent developments have extended the method to systems of finite angular momentum and different parities [25]. The present study further extends the correlated Gaussian hyperspherical technique, developing a new approach for the calculation of matrix elements at fixed hyperradius.…”

“…The latter method involves a two-step process to diagonalize the full Hamiltonian. First, after expressing the Schrödinger equation using a single length, the hyperradius R, and the remaining degrees of freedom as hyperangles using hyperspherical coordinates [24][25][26][27][28][29][30][31], the hyperangular Schrödinger equation is solved parametrically in R. This leads to an infinite set of coupled "Born-Oppenheimer" potentials. Second, the set of onedimensional differential equations in R is solved.…”

Non-resonant Density of States Enhancement at Low Energies for Three or Four Neutrons

“…The second type of basis set implemented to solve the fixed-ρ Schrödinger equation is a linear combination of correlated Gaussian functions. [28][29][30][31] Following diagonalization of H ρ=const at each ρ, a Rayleigh-Ritz upper bound on the exact potential U ν (ρ) is obtained. The following theorem is important for our subsequent analysis below:…”

“…[35] N (LS)J π l eff C l Next consider the numerical computation of the adiabatic hyperspherical potential energy curves for the 3n and 4n systems. We use two different variational basis sets, an expansion into hyperspherical harmonics (extremely accurate at small and intermediate values of ρ) [36][37][38] and an expansion into correlated Gaussian basis functions (more accurate at large ρ) [28][29][30]. The HH basis produces well converged results for the quantities of interest, U ν (ρ) and W ν,ν (ρ), in a relatively large range of ρ values, 0−50fm and 0−30fm for 3n and 4n respectively.…”

Hyperspherical Asymptotics of a System of Four Charged Particles

“…Many few-body systems have been analyzed using the hyperspherical framework. For example, nuclear systems [3,4] utilizing an expansion in hyperspherical harmonics [5,6], systems of resonant short-range interactions [2,[7][8][9], and systems of three charges [1,[10][11][12][13][14][15][16][17][18][19] to name several. Hyperspherical methods have also been applied to molecular rearrangement collisions in physical chemistry, as discussed by Kuppermann for systems having up to five atoms in Ref.…”