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Europhysics Letters (EPL)

Ritz variational treatment of the 4 He trimer

S. Geltman

Published Feb 1, 2009• 5 citations
Publisher

Abstract

The 4 He trimer is treated by the Ritz variational method using trial functions whose accuracy has been tested by a similar treatment of the dimer, for which an exact solution exists. These are nodeless functions containing four nonlinear parameters that are varied to give the minimum value for the ground-state energy. Agreement with measurement is obtained for the mean bond lengths of the trimer, but there is some disagreement with other non-variational calculations of the ground-state binding energy.

DOI
10.1209/0295-5075/85/33001

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The 4 He trimer is treated by the Ritz variational method using trial functions whose accuracy has been tested by a similar treatment of the dimer, for which an exact solution exists. These are nodeless functions containing four nonlinear parameters that are varied to give the minimum value for the ground-state energy. Agreement with measurement is obtained for the mean bond lengths of the trimer, but there is some disagreement with other non-variational calculations of the ground-state binding energy.
We calculated the 4{}^{4}4He trimer and tetramer ground and excited states with the LM2M2 potential using our Gaussian expansion method for ab initio variational calculations of few-body systems. The method has been extensively used for a variety of three-, four-, and five-body systems in nuclear physics and exotic atomic and molecular physics. The trimer (tetramer) wave function is expanded in terms of symmetric three- (four-) body Gaussian basis functions, ranging from very compact to very diffuse, without assumption of any pair correlation function. The calculated results for the trimer ground and excited states are in excellent agreement with values reported in the literature. The binding energies of the tetramer ground and excited states are obtained as 558.98 and 127.33 mK (0.93 mK below the trimer ground state), respectively. We found that precisely the same shape of the short-range correlation (${r}_{ij}\ensuremath{\lesssim}4$ \AA{}) in the dimer appears in the ground and excited states of the trimer and tetramer. The overlap function between the trimer excited state and the dimer ground state and that between the tetramer excited state and the trimer ground state are almost proportional to the dimer wave function in the asymptotic region (up to $\ensuremath{\sim}$1000 \AA{}). Also, the pair correlation functions of trimer and tetramer excited states are almost proportional to the squared dimer wave function. We then propose a model which predicts the binding energy of the first excited state of 4{}^{4}4HeN{}_{N}N​ ($N\ensuremath{\ge}3$) measured from the 4{}^{4}4He${}_{N\ensuremath{-}1}$ ground state to be nearly $\frac{N}{2(N\ensuremath{-}1)}{B}_{2}$ where B2{B}_{2}B2​ is the dimer binding energy.
The general form of the energy of the ground state of a many-fermion system is shown to be exactly of the form proposed by Brueckner and Bethe, without approximation. In a variational treatment, if the trial wave function is picked containing only pair correlations, together with all possible unlinked pairs, it is described by a two-body excitation matrix 〈m1m2∣A∣p1p2〉〈{m}_{1}{m}_{2}|A|{p}_{1}{p}_{2}〉〈m1​m2​∣A∣p1​p2​〉. Variation of this matrix in the Ritz-Rayleigh principle yields a set of integral equations of the scattering type for the matrix AAA. Hole-state energies are given self-consistently in terms of the matrix AAA, but particle-state energies are Hartree-Fock energies. This may be corrected for by widely enlarging the class of terms admitted into the wave function. If the approximation is then made of omitting a class of terms, defined as cross-linked clusters in $〈\ensuremath{\psi}|H|\ensuremath{\psi}〉$, the particle-state energies are easily renormalized. Variation then leads to an infinite hierarchy of integral equations.
We introduce a local measure of the quality of a trial wave function: the local variance. Using this tool we examine the pair function often adopted to construct wave functions for small helium clusters and apply it to a wave function for 4 He 2 using the TTY interaction potential. Our analysis shows that this commonly employed functional form should be improved in the short-range region. We introduce a new model based on the short-range behaviour of the exact ground-state solution of a Morse potential. The resulting compact trial wave function for 4 He 2 recovers 98% of the exact binding energy with only six variational parameters.