Translationally-Invariant Coupled Cluster Theory Applied to the 4He Nucleus

Abstract

In this work we discuss a series of calculations for the 4 He nucleus which have been motivated by the coupled cluster method. For pedagogical reasons we restrict ourselves to the case of pure Wigner nucleon-nucleon interactions. All numerical work is done in the standard harmonic oscillator basis and with an exact treatment of the centre-of-mass motion. Particular emphasis is placed on elucidating the meaning of the coupled cluster wave function and its coordinate-space representation, as well as the relation with a variational-like use of the coupled cluster ansatz.

Variational Cluster Methods in Coordinate Space for Small Systems: Center of Mass Corrections Made Easy

“…We have only shown one set of data for the ICI-SUB(2) approximation as the two calculational schemes described above produce essentially the same ground-state energy in their respective converged limits. The results are very promising, with the energy relative to the Diffusion Monte Carlo results [1,4,5,6]missing at most 2 MeV for the MT-V and S3 potentials, whereas for the V7 potential we already have complete agreement by including only these one-and two-body TRI correlations. We need only between 4 and 7 terms in the expansion of Eq.…”

“…Nevertheless they are straightforward to evaluate. Their explicit forms will be presented elsewhere [9]. We solve this integro-differential equation (15) by discretization on a regular mesh, producing the generalized eigenvalue problem (16) (with summation convention implied) whose solution is straightforward.…”

“…In the preceding volume of this series [1] we presented work which started from an examination of the treatment of the CM motion in applications of the coupled cluster (CC) method to light systems. More specifically,in the original work by Zabolitzky on the 4 H e nucleus [2) with central forces (and whose ground-state spatial wavefunction is hence fully equivalent to that of the corresponding bosonic system), this problem was dealt with in a method due to Fink (3] by expressing the correlation operator e S in terms of the intrinsic coordinates, but imposing the statistics by the inclusion of a Slater determinant of harmonic oscillator (HO) wavefunctions in the original coordinates.…”

“…In Ref. [1), we presented an alternative approach which circumvented the problem altogether by restricting the correlation operator to produce only those excitations which explicitly leave the CM in the (Os) HO ground-state, thereby rendering the CM correction trivial. Unfortunately, this approach as we chose to implement it, suffered from the usual very slow convergence of the HO basis [1,4,5,6].…”

Exploring Many-Body Theories in Light Nuclei

“…(6) the theory is not translationally invariant (in finite systems), because the exponential factor spoils the basic factorizability property of the harmonic oscillator 'l'o wave function. For a system of bosons we have found [22] a simple modification of eq. (6) which preserves the translational invariance.…”