We use the following approximation to the potential between two 4He atoms—V(r)=4420 exp(−r2/1.69)−1302 exp(−r2/2.44) (V in K, r in Å). We use the hyperspherical harmonics expansion, with Fabre’s optimal subset, to solve for the ground state of the 4He tetramer. The Schrödinger equation is transformed into an infinite set of second order coupled differential equations (CDE), which we truncate to M equations. We test our program for solving CDE in two appendices. We find that the energy for our model is less than −4.3 K.
The wave functions of the ground (Ψ 0 ) and the first excited (Ψ k ) states of He II in the second-order approximation, i.e., up to the first two corrections to the corresponding solutions for a weakly nonideal Bose gas, are determined by the collective variable method, which was proposed by Bogolyubov and Zubarev and developed in the studies by Yukhnovskii and Vakarchuk. The functions Ψ 0 and Ψ k = ψ k Ψ 0 are determined as the eigenfunctions of the N -particle Schrödinger equation from a system of coupled equations for Ψ 0 , Ψ k , and the quasiparticle spectrum E ( k ) of helium II. The results consist in the following: (1) these equations are solved numerically for a higher order approximation compared with those investigated earlier (the first-order approximation), and (2) Ψ 0 and ψ k are derived from a model potential of interaction between He 4 atoms (rather than from the structure factor as earlier) in which the potential barrier is joined with the attractive potential found from experiment. The height V 0 of the potential barrier is a free parameter. Except for V 0 , the model does not have any free parameters or functions. The calculated values of the structure factor, the ground-state energy E 0 , and the quasiparticle spectrum E ( k ) of He II are in agreement with the experimental values for V 0 ≈ 100 K. The second-order correction to the logarithm of Ψ 0 significantly affects the value of E 0 and provides the asymptotics E ( k → 0) = ck , while the second-order correction to ψ k slightly affects the E ( k ). The second-order corrections to Ψ 0 and ψ k have a smaller effect on the results compared with the first-order corrections, whereby the theory is in agreement with experiment; therefore, one may assume that the truncated Ψ 0 and ψ k well describe the microstructure of He II. Thus, the series for Ψ 0 and Ψ k can be truncated in spite of the fact that the expansion parameter is not very small (∼1/2).
We calculated the
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}$1000 \AA{}). Also, the pair correlation functions of trimer and tetramer excited states are almost proportional to the squared dimer wave function. We then propose a model which predicts the binding energy of the first excited state of
4He
N ($N\ensuremath{\ge}3$) measured from the
4He${}_{N\ensuremath{-}1}$ ground state to be nearly $\frac{N}{2(N\ensuremath{-}1)}{B}_{2}$ where
B2 is the dimer binding energy.
In a previous work [Phys. Rev. A 85, 022502 (2012)] we calculated, with the use of the Gaussian expansion method for few-body systems, the energy levels and spatial structures of the
4He trimer and tetramer ground and excited states using the LM2M2 potential, which has a very strong short-range repulsion. In this work, we calculate the same quantities using the currently most accurate
4He-
4He potential [Przybytek et al., Phys. Rev. Lett. 104, 183003 (2010)] that includes the adiabatic, relativistic, QED, and residual retardation corrections. Contributions of the corrections to the tetramer ground-(excited-) state energy $\ensuremath{-}573.90\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}132.70)$ mK are respectively $\ensuremath{-}4.13\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}1.52)$ mK,
+9.37(+3.48) mK, $\ensuremath{-}1.20\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}0.46)$ mK, and
+0.16(+0.07) mK. Further including other realistic
4He potentials, we calculated the binding energies of the trimer and tetramer ground and excited states,
B3(0),B3(1),B4(0), and
B4(1), respectively. We found that the four kinds of the binding energies for the different potentials exhibit perfect linear correlations between any two of them over the range of binding energies relevant for
4He atoms (namely, six types of the generalized Tjon lines are observed). The dimerlike-pair model for
4He clusters, proposed in the previous work, predicts a simple interaction-independent relation
B2B4(1)=B2B3(0)+32, which precisely explains the correlation between the tetramer excited-state energy and the trimer ground-state energy, with
B2 being the dimer binding energy.
The three-body Schrdinger equation is approximately solved in the hyperspherical coordinates and the binding energies of the three-body weakly bound systems are calculated with the purpose to find if He-He-Ba trimers could exist. Using the special feature of the B-spline function like the flexible and highly localized properties, hypersphercial potentials are obtained by modifying the knots distribution of the B-spline basis of different weakly bound three-atom systems. Employing the best empirical interaction potentials between each pair of particles, we obtain that in the ground state binding energies of the weakly bound typical three-atom systems, the bindings of the molecules, 4He-4He-138Ba, 4He-3He-138Ba and 3He-3He-138Ba are possible. The binding energies of these systems are shown in the order of 1 Kelvin, each system could support only one bound state. These weakly bound molecules can exist only in a very cold environment. To get insight into the geometry of the molecules, the features of the channel functions associated with the hyperspherical potential curves of each system are investigated.
The ground-state properties of
3He and
4He trimers and tetramers in infinite and restricted two-dimensional and quasi-two-dimensional (2D) space are studied. Using Monte Carlo calculation in two stages successively, simple variational Monte Carlo and diffusion Monte Carlo, we find that all trimers (except one of
3He with spin 3/2) and tetramers are bound. In infinite 2D space the binding energy of
4He trimer is $\ensuremath{-}183(1) \mathrm{mK}$ and of tetramer $\ensuremath{-}435(1) \mathrm{mK}.$ In the same environment for
3He, using Jastrow-Feenberg-Bruch wave functions trimer (spin 1/2) is bound with $\ensuremath{-}0.013(1) \mathrm{mK}$ and tetramer with $\ensuremath{-}0.021(1) \mathrm{mK}.$ Employing in the same procedure ``dimerlike'' wave functions for
3He molecules in infinite 2D space it is discovered that trimer prefers the structure of one dimer and one separate particle while a composition of two separate dimers is preferable for tetramer. In this case binding energies are $\ensuremath{-}0.020(1) \mathrm{mK}$ for trimer and $\ensuremath{-}0.040(1) \mathrm{mK}$ for tetramer. The binding energies of mixed molecules in infinite 2D space are $\ensuremath{-}74.3(4)$ mK ${(}^{3}\mathrm{He}{\ensuremath{-}}^{4}{\mathrm{He}}_{2});$ $\ensuremath{-}14(1)$ mK ${(}^{3}{\mathrm{He}}_{2}{\ensuremath{-}}^{4}\mathrm{He});$ $\ensuremath{-}254(4)$ mK ${(}^{3}\mathrm{He}{\ensuremath{-}}^{4}{\mathrm{He}}_{3});$ $\ensuremath{-}118(2)$ mK ${(}^{3}{\mathrm{He}}_{2}{\ensuremath{-}}^{4}{\mathrm{He}}_{2});$ $\ensuremath{-}11(1)$ mK ${(}^{3}{\mathrm{He}}_{3}{\ensuremath{-}}^{4}\mathrm{He}).$ In holding potentials with well-adjusted parameters there is a drastic increase of binding for trimer and tetramer in both sorts of helium atoms.