Weakly bound LiHe$_2$ molecules in the framework of three-dimensional Faddeev equations

TL;DR

This paper employs three-dimensional Faddeev equations to calculate binding energies of weakly bound LiHe$_2$ molecules, revealing two bound states supported by modern potential models and analyzing their proximity to two-body thresholds.

Contribution

It introduces a direct solution method for Faddeev equations applied to LiHe$_2$ molecules, providing new insights into their bound states and scattering properties.

Findings

Two bound states in LiHe$_2$ trimers supported by modern potentials

Excited state energy close to two-body threshold

Calculated scattering length for helium atom - HeLi dimer

Abstract

A method of direct solution of the Faddeev equations for the bound-state problem with zero total angular momentum is used to calculate the binding energies. The results for binding energies of He$_2$$^6$Li and He$_2$$^7$Li systems and helium atom - HeLi dimer scattering length are presented. The results show that modern potential models support two bound states in both trimers. In both cases the energy of the excited state is very close to the energy of the lowest two-body threshold.

Tables

Table 1. Table 1: Absolute values of dimer energies | ϵ d | subscript italic-ϵ 𝑑 |\epsilon_{d}| (in mK), the inverse wavenumber κ − 1 superscript 𝜅 1 \kappa^{-1} ( in Å) and the scattering length a 2 superscript 𝑎 2 a^{2} (in Å) for He-He and 4 He − - alkali-atoms calculated for the potentials used.

Dimer	$\left|{ϵ}_d\right|$ (mK)	${\kappa }^{-1}$ (Å)	$a^2$ (Å)	Dimer	$\left|{ϵ}_d\right|$ (mK)	${\kappa }^{-1}$ (Å)	$a^2$ (Å)
4He4He a	1.321	95.78	99.59	4He23Na	$28.97$	15.7	23.37
4He4He b	1.620	86.49	90.28	4He39K	$11.20$	24.4	33.32
4He6Li	1.515	81.63	89.42	4He85Rb	$10.27$	24.9	34.02
4He7Li	5.622	41.14	48.84	4He133Cs	$4.945$	35.5	45.32

Table 2. Table 2: Convergence for the 4 He 2 7 Li bound states energies with respect to the number of grid points for the Przybytek potential  PRZ2010 .

Ground state energy (mK)
$N_x$=$N_y$	$N_z=1$	$N_z=2$	$N_z=3$	$N_z=4$	$N_z=5$
20	-78.4124	-80.0524	-80.4523	-80.6422	-80.6622
30	-78.3750	-79.9701	-80.3877	-80.5630	-80.5937
40	-78.3849	-79.9695	-80.3853	-80.5690	-80.5977
50	-78.3877	-79.9727	-80.3869	-80.5712	-80.5990
Excited state energy (mK)
20	-5.6457	-5.6522	-5.6538	-5.6545	-5.6546
30	-5.6447	-5.6510	-5.6527	-5.6534	-5.6535
40	-5.6448	-5.6511	-5.6527	-5.6534	-5.6536
50	-5.6448	-5.6511	-5.6527	-5.6535	-5.6536

Table 3. Table 3: Bound-state energies for the He 2 Li systems and helium atom - HeLi dimer scattering length a 3 superscript 𝑎 3 a^{3} . The energies are given in mK and are relative to the three-body dissociation threshold. The scattering length is given in Å. The present results are compared to results given in references.

E (mK)/$a^3$ (Å)	present	present	Suno17	Wu	Suno13	Suno13	Suno10	Baccarelli	DiPaola	Stipanovic
He-He potential	TTY	Przybytek	LM2M2	LM2M2	Jeziorska	Jeziorska	LM2M2	LM2M2	TTY	HFDB
He-Li potental	KTTY	KTTY	KTTY	KTTY	KTTY	KTTY	Cvetko	Cvetko	KTTY	KTTY
$|$E${}_{{}_{}{}^{7}L{i}^{4}H{e}_2}{}^{}|$	79.35	80.60	79.36	78.73	76.32	81.29	64.26	73.3	80.0	81,03
$|$E${}_{{}_{}{}^{7}L{i}^{4}H{e}_2}{}^{\ast }|$	5.642	5.654	5.642	5.685	5.51	5.67	3.01	12.2
$a^3$	683	553
$|$E${}_{{}_{}{}^{6}L{i}^{4}H{e}_2}{}^{}|$	57.24	58.38	57.23			58.88		51.9		58,72
$|$E${}_{{}_{}{}^{6}L{i}^{4}H{e}_2}{}^{\ast }|$	1.940	2.049	1.937			2.09		7.9
$a^3$	191	144∗

Equations

$$-\frac{1}{r\Psi }\frac{\partial r\Psi }{\partial r}\underset{r\to 0}{\to }\frac{1}{a}.$$

$$\left(-\frac{d^2}{d{R}^2}-\frac{1}{R}\frac{d}{dR}+\frac{s_n^2}{R^2}\right)F_n(R)=E{F}_n(R),$$

$$-s_n\cos \left(s_n\frac{\pi }{2}\right)+\frac{8}{\sqrt{3}}\sin \left(s_n\frac{\pi }{6}\right)=0.$$

$$\underset{n\to \infty }{\lim }\frac{E_{n+1}}{E_n}=\exp (-2\pi /|s_0|)\equiv {\lambda }^{-2}\approx \frac{1}{515.035}\approx \frac{1}{22.7^2}.$$

$$\begin{array}[]{rcl}{\bf x}_{i}&=&\displaystyle\left(\frac{2m_{j}m_{k}}{m_{j}+m_{k}}\right)^{1/2}({\bf r}_{j}-{\bf r}_{k})\cr{\bf y}_{i}&=&\displaystyle\left(\frac{2m_{i}(m_{j}+m_{k})}{m_{i}+m_{j}+m_{k}}\right)^{1/2}\left({\bf r}_{i}-\displaystyle\frac{m_{j}{\bf r}_{j}+m_{k}{\bf r}_{k}}{m_{j}+m_{k}}\right)\end{array}$$

$${\mathbf{x}}_j={\mathsf{c}}_{ji}{\mathbf{x}}_i+{\mathsf{s}}_{ji}{\mathbf{y}}_i{\mathbf{y}}_j=-{\mathsf{s}}_{ji}{\mathbf{x}}_i+{\mathsf{c}}_{ji}{\mathbf{y}}_i.$$

$${\mathsf{c}}_{ji}$$

$$H=H_0+\sum _i{V}_i({\mathbf{x}}_i),$$

$$\Psi =\sum _i{\Phi }_i({\mathbf{x}}_i,{\mathbf{y}}_i),$$

$$(-{\Delta }_{{\mathbf{x}}_i}-{\Delta }_{{\mathbf{y}}_i}+V_i({\mathbf{x}}_i)-E){\Phi }_i({\mathbf{x}}_i,{\mathbf{y}}_i)=V_i({\mathbf{x}}_i)\sum _{k\ne i}{\Phi }_k({\mathbf{x}}_k,{\mathbf{y}}_k),$$

$$H_0=-\frac{{\partial }^2}{\partial x_i^2}-\frac{{\partial }^2}{\partial y_i^2}-(\frac{1}{x_i^2}+\frac{1}{y_i^2})\frac{\partial }{\partial z_i}(1-z_i^2)\frac{\partial }{\partial z_i},$$

$$x_i=|{\mathbf{x}}_i|,y_i=|{\mathbf{y}}_i|,z_i=\frac{({\mathbf{x}}_i,{\mathbf{y}}_i)}{x_i{y}_i},x_i,y_i\in [0,\infty ),z_i\in [-1,1].$$

$$x_j$$

$$(H_0+V_i(x_i)-E){\varphi }_i(x_i,y_i,z_i)=-V_i(x_i)\sum _{k\ne i}{\varphi }_k(x_k,y_k,z_k).$$

$$\begin{array}[]{rcl}\phi(x,y,z)=\psi_{d}(x)\exp({\rm i}\sqrt{E-\epsilon_{d}}\,y){\rm a}_{0}(z)+\displaystyle\frac{\exp({\rm i}\sqrt{E}\rho)}{\sqrt{\rho}}A(y/x,z),\end{array}$$

Full text

††journal: Few-Body Systems

1 Introduction

The Efimov effect is a remarkable phenomenon, which is an excellent illustration of the variety of possibilities arising when we transit from the two-body to the three-body problem. In 1970 V.Efimov Efimov1970_PLB ; Efimov1970_YAF proposed that three-body systems with short range interaction can have an infinite number of bound states when none of the two-particle subsystems has bound states but at least two of them have infinite scattering lengths. In such a case the scattering length $a$ is much lager than the range of the interaction $r_{0}$. The simplest situation described by Efimov Efimov1970_YAF corresponds to three identical neutral bosons interacting via short-range resonant interactions treated in the zero-range theory framework. In this theory it is assumed that the short-range region details of the interaction can be neglected and the wave function in the asymptotically free region can be parametrized by the scattering length. In order to reproduce correctly the two-body wave function in the region outside of the range of interaction $r_{0}$ one can use the Bethe-Peierls boundary conditions BethePeierls for the three-body wave function $\Psi$ when the two particles separated by $r$ come in contact

[TABLE]

The simplification which is used in the zero-range theory is to keep the same form of the wave function down to $r=0$, although this is unphysical at distances $r<r_{0}$. To describe the three-body system Efimov used the free Schrödinger equation written in hyperspherical coordinates Delves with Bethe-Peierls boundary conditions (1) which lead to the equation for a radial function

[TABLE]

where $R$is the hyperradius and $s_{n}$ is a solution of the following transcendental equation Efimov1970_YAF ; Danilov

[TABLE]

All the solutions of equation (3) are real, except one $s_{0}=1.0062378i$ which is purely imaginary, which results in an attractive effective potential in equation (2) for $n=0$. This attraction is the origin of the Efimov effect. In order to prevent the Thomas collapse Thomas an additional three-body boundary condition can be used to fix some of the three-body observables (the ground state energy or the particle-dimer scattering length). This boundary condition breaks the scale invariance under arbitrary scale transformations but still keeps the scale invariance under some discrete set of scale transformation with scaling factors being powers of $\lambda=\exp{(\pi/|s_{0}|)}$. Thus, in the limit $a\rightarrow\infty$ there is an infinite number of bound states, forming a geometric series of energies accumulated at the threshold. The following relationship holds for three identical bosons

[TABLE]

One of the best theoretically predicted three-body system with an excited state of the Efimov type is a naturally existing molecule of the helium trimer 4He3 (see, FBS2011 ; FBS2017 and refs. therein). The interaction between two helium atoms is quite weak and supports only one bound state with the energy about 1mK and a rather large scattering length about 100 Å. Only recently the long predicted weakly-bound excited state of the helium trimer was observed for the first time using a combination of Coulomb explosion imaging and cluster mass selection by matter wave diffraction Kunitski .

The first experimental evidence of the Efimov resonance was observed in an ulracold gas of 133Cs atoms in 2006 Kraemer . Experimentally, they observed a giant three-body recombination loss when the strength of two-body interaction was varied. More recently, the second Efimov resonance has been observed and the scaling factor for the Efimov period has been found to be $21.0$ Huang2014 ; Huang2015 , close to the universal ratio $\lambda=22.7$ in a homonuclear system Efimov1970_PLB . It was shown in Efimov1973 ; Esry2006 ; Wang2012 that the universal Efimov scaling is also valid for systems of non-identical particles. In particular, for a system composed of two heavy and one light atom the scaling factor $\lambda$ (it depends only on the masses of the particles) gets smaller as the mass imbalance increases. Experimentally, heteronuclear Efimov states have been searched for in 41K87Rb2 Barontini , 40K87Rb2 Bloom , 39K87Rb2 Wacker , 7Li87Rb2 Maier , 6Li133Cs2 Tung ; Ulmanis systems. Recent reviews on Efimov effect could be found in Naidon ; Greene .

There is a growing interest in the investigation of He2 - alkali-atom van-der-Waals systems, that are expected to be of Efimov nature. In addition to the Helium dimer, the He - alkali-atom interactions are even shallower and also support weakly bound states. In triatomic 4He2-alkali-atom systems presence of Efimov levels can be expected. Three-body recombination and atom-molecular collision in Helium-Helium-alkali-metal systems at ultracold temperatures have been studied using adiabatic hyperspherical representation in Ref. Suno09 ; Suno17 ; Wu . Here we use the Faddeev equations in total angular momentum representation to calculate the 4He26;7Li binding energies and a scattering length, which has not been studied before.

2 Method

The configuration space of three particles after elimination of the center of mass can be described in terms of three sets of Jacobi coordinates

[TABLE]

The set of coordinates $i$ describes a partitioning of the three particles into a pair $(jk)$ and a separate particle $i$. The Jacoby vectors with different indexes are related by an orthogonal transform

[TABLE]

The coefficients $c_{ji}$ and $s_{ji}$ are expressed through the masses of the particles

[TABLE]

and satisfy $c^{2}_{ji}+s^{2}_{ji}=1$.

The three-body system is described by the Hamiltonian

[TABLE]

where $H_{0}$ stands for the kinetic energy of the three particles and $V_{i}({\bf x}_{i})$ is the interaction potential acting in the pair $i$. Faddeev decomposition represents the wave function $\Psi$ in terms of the Faddeev components $\Phi_{i}$

[TABLE]

which satisfy the following set of equations MerkFadd

[TABLE]

where $E$ is the total energy of the system. In case of zero total angular momentum the angular degrees of freedom corresponding to collective rotation of the three-body system can be separated Kvits and the kinetic energy operator reduces to

[TABLE]

where $x_{i}$, $y_{i}$ and $z_{i}$ are intrinsic coordinates

[TABLE]

Due to (5) and (6) these coordinates are related by

[TABLE]

As a result we have a set of three-dimensional differential Faddeev equations

[TABLE]

When two particles of a system are identical, the Faddeev equations can be simplified. For example, for the He2Li atomic systems particles 1 and 2 corresponding to 4He atoms are identical and the Faddeev components $\phi_{1}(x_{1},y_{1},z_{1})$ and $\phi_{2}(x_{2},y_{2},z_{2})$ transform into each other under an appropriate rotation of the coordinate space. Therefore, it is sufficient to consider only two independent Faddeev components. In the case of three identical bosons all the Faddeev components take identical functional form, which makes it possible to reduce the system of three equations (12) to one equation.

Using the fact that the both dimers – 4He2 and 4HeLi – have a unique bound state, the asymptotic boundary condition for a bound state as $\rho=\sqrt{x^{2}+y^{2}}\rightarrow\infty$ and/or $y\rightarrow\infty$ reads as follows (see MerkFadd ; kea17 )

[TABLE]

where $\epsilon_{d}$ stands for the corresponding dimer energy while $\psi_{d}(x)$ denotes the dimer wave function. The coefficients ${\rm a}_{0}$ and $A(y/x,z)$ describe contributions into $\phi(x,y,z)$ from (2+1) and (1+1+1) channels respectively. The last term can be neglected for the states below the three-body threshold. For bound state calculations Dirichlet or Neumann boundary conditions can also be employed.

3 Results and Discussions

The Li-He interaction is described by the KTTY potential KTTY , theoretically derived by Kleinekathöfer, Tang, Toennies and Yiu with more accurate coefficients taken from Yan ; KLM-KTTY . Calculated values of the binding energy for 4He6Li is 1.512 mK and for 4He7Li is 5.622 mK. Such small values of binding energy give indication on possible existence of Efimov states in the corresponding He2Li triatomic systems.

In our calculations we use two different model potentials for He-He interaction - TTY TTY and Przybytek PRZ2010 . The purely theoretical TTY potential derived by Tang, Toennies and Yiu TTY is based on the perturbation theory and is described by a relatively simple analytical expression. The recent Przybytek PRZ2010 potential includes relativistic and quantum electrodynamics contributions as well as some accuracy improvements. Each of these potentials supports a single weakly bound state of the dimer. Calculated values of the binding energy $\epsilon_{d}$ for the corresponding dimers, the inverse wavenumber $\kappa^{-1}=(\hbar^{2}/2\mu\epsilon_{d})^{1/2}$ ($\mu$ is a reduced mass) and the atom-atom scattering length $a^{2}$ are presented in Table 1. Atomic masses for different isotopes are taken from Mills .

From Table 1 we can see that the inverse wave number is a good approximation for the scattering length, which indicates that the zero-range potential model (1) is applicable. The calculated binding energy of the helium dimer with Przybytek potential PRZ2010 is very close to its recent experimental value of $-151.9\pm 13.3$ neV $\approx-1.76\pm 0.15$ mK Zeller while the energy obtained with TTY potential TTY is closer to the previous experimental estimation of $-1.1^{+0.3}_{-0.2}$ mK $\approx-95^{+25}_{-15}$ neV Grisenti-exp-2000 . The choice of the He-He potential is especially important for 4He6Li, as switching between the two model potentials swaps the order of the two-body thresholds in the system. In the case of TTY potential the lowest two-body threshold corresponds to the 6LiHe system, while for the Przybytek potential it is the 4He2 dimer which is bound stronger.

All He-alkali-atom dimers are weakly bound, but the binding energies of HeLi and HeCs systems are of the same order as the binding energy of the Helium dimer. It suggests that in the corresponding He2-alkali-atom triatomic systems Efimov states might exist. Indeed, in calculations Suno09 ; Suno17 ; Wu ; kea17 an excited state located very close to the HeLi threshold has been found.

To calculate the binding energy of 4He26Li and 4He27Li trimers, we employed the equations (12), and the bound-state asymptotic boundary condition (13). The details of the numerical procedure are described in Sofianos ; Roudnev ; PAN2012 . The three-body interaction is expected to be small as in the case of helium trimer Cencek and we do not take it into account. Convergence tables for bound states of the 4He27Li trimer calculated with the Przybytek potential PRZ2010 are shown in Table 2. The bound state energies (with respect to the three-body break-up threshold) are presented for different numbers of grid points. The number of grid points in coordinates $x$ and $y$ are set equal, and the number of grid points in angular coordinate $z$ is varied independently. As it is seen from Table 2, the excited state is much less sensitive to the angular grid. Similar behavior had been observed for the helium trimer Sofianos ; Roudnev .

Our results for 4He27Li and 4He26Li trimers binding energies as well as the results obtained by other authors are summarized in Table 3. The results show that the both potential models support two bound states in the both trimers. The energy of the excited state is very close to the energy of the lowest two-body threshold. Different He-He potentials give $0.3$ mK difference in the helium dimer binding energy, which leads to the $\sim 1.2$mK difference in the binding energy of the ground state of He2Li trimers, although the energy of excited state of He27Li is practically unchanged (difference is $\sim 0.01$mK). As is mentioned above, for the He26Li system the lowest threshold is different for different potentials: for TTY it corresponds to the energy of HeLi bound state, while for Przybytek it is the bound state energy of the He2 dimer. So, for different potentials the absolute value of the excited state energy changes slightly, but the relative energy with respect to the two-body threshold remains practically the same.

Results of other authors in Table 3 are based on solving the Shrödinger equation. The adiabatic hyperspherical approach has been employed in Suno09 ; Suno17 ; Wu ; Suno13 ; Suno10 and variational calculations has been performed in Baccarelli ; DiPaola ; Stipanovic . The fourth column of Table 3 contains the results obtained by Suno in Suno17 using the adiabatic hyperspherical method. For the He-Li interaction he has used KTTY potential KTTY as in our calculations, but for the He-He interaction the LM2M2 potential LM2M2 has been used. However, LM2M2 and TTY potentials support 4He2 with the energies $\varepsilon_{d}=-1.31$ mK and $\varepsilon_{d}=-1.32$ mK, correspondingly Roudnev . So good agreement between our results and results from  Suno17 are not surprising (see columns 1 and 3 in Table 3).

The fifth column of Table 3 contains the results obtained by Wu et al. Wu using the mapping method within the adiabatic hyperspherical framework Kokoouline . The next two columns are the results of calculations by H. Suno, E. Hiyama and M. Kamimura Suno13 using the Gaussian expansion method and the adiabatic hyperspherical representation respectively, although with different He-He potentals. They employed the He-He potential suggested by Jeziorska et al. Jeziorska , which gives the helium dimer biding energy $-1.74$ mK which is lower than for Przybytek potential. The two methods give different results, but authors in Suno13 mentioned that the adiabatic hyperspherical representation was less accurate. The next column is the results of calculations by Suno and Esry Suno09 ; Suno10 by the adiabatic hyperspherical method. They also employed the He-He potential from Jeziorska , but for Li-He interaction Cvetko potential from Cvetko has been used. The potential proposed by Cvetko et al. Cvetko gives the HeLi smaller binding energy than the KTTY potential, namely $-2.8$ mK for 4He7Li and $-0.31$ mK in case of 4He6Li dimer. The ninth column contains the results obtained by Baccarelli et al. Baccarelli with the same potential as in Suno10 , but using a different computation method - variational calculations in terms of distributed Gaussian functions. The last two columns contain the results of Monte Carlo calculations by Di Paola et al. DiPaola and Stipanović et al. Stipanovic using TTY TTY and HFDB HFDB as He-He interactions.

We should also mention the first results obtained by Yuan and Lin Yuan using the adiabatic hyperspherical method which gives an upper bound to the ground state $-45.7$ mK for 4He27Li and $-31.4$ mK for 4He26Li and the prediction of the bound state energies made by Delfino et al. Delfino using the scaling ideas and zero-range model calculations. The preliminary Faddeev calculation using bipolar partial-wave expansion for searching Efimov states in 4He27Li system have been performed in kea17 . In these papers, however, the contribution of higher partial waves was underestimated because of the computational restrictions.

As it has been demonstrated in kea17 , the excited state of He27Li has a Efimov-type behavior similar to helium trimer system He3-Rev . To check for the Efimov-like state the original Li-He potential has been multiplied by a factor $\lambda$. An increase of the coupling constant $\lambda$ makes the potential more attractive and Efimov levels become weaker and disappear with further increase of $\lambda$. Indeed this situation is observed for the excited state energy of He27Li in contrast to the ground state energy whose absolute value increases continuously with increasing attraction.

The results for the He-atom – HeLi-dimer scattering length are presented in the last line of the Table 3 for each Li isotopes. The helium atom – helium-alkali-atom collisions at ultralow energies are studied in Ref. Suno14 by Suno and Esry using the adiabatic hyperspherical representation. In particular, they calculated the total cross section also for 4HeLi + 4He $\rightarrow$ 4HeLi + 4He elastic scattering. Our estimation of the cross section $\sigma=4\pi a^{2}$ at the threshold is $5.8\times 10^{-10}$ cm2 using TTY potential TTY and $3.8\times 10^{-10}$ cm2 using Przbytek potential PRZ2010 . These values agree with the value $\approx 3\times 10^{-10}$ cm2 obtained in Ref. Suno14 using SAPT potential Jeziorska at the energy $10^{-3}$ mK above the threshold.

4 Conclusion

We have used direct solution of the Faddeev equations for the bound-state and scattering problems with zero total angular momentum. The numerical algorithm is based on spline expansion of the Faddeev components combined with the tensor trick preconditioning and the Arnoldi algorithm for eigenanalysis. Calculations of the He26Li and He27Li ground and excited states show that the method is very efficient and allows one to obtain stable convergent results. Apparently, it performs better than the previously exploited method of the bipolar partial-wave expansion. Our results for 4He27Li and 4He26Li trimers binding energies show that different potential models support two bound states in both trimers. The energy of the excited state is very close to the energy of the lowest two-body threshold. In case of the He26Li system the lowest threshold is different for different potentials but the relative energy with respect to the lowest two-body threshold is practically the same.

Acknowledgements.

One of the author (EAK) would like to thank T. Frederico, A. Kievsky and P. Stipanović for stimulating discussions and also W. Sandhas and A.K. Motovilov for their constant interest to this work.