We calculated the ${}^4$He 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 ($r_{ij}\lesssim 4$ Å) 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 $\sim$1000 Å). 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 ${}^4$He${}_N$ ($N\ge 3$) measured from the ${}^4$He${}_{N-1}$ ground state to be nearly $\frac{N}{2(N-1)}{B}_2$ where $B_2$ is the dimer binding energy.

• Received 27 November 2011

DOI:https://doi.org/10.1103/PhysRevA.85.022502