Full-range analytical potential for the a³ Σ⁺ state of LiNa: robust prediction of vibrational levels and scattering length

Abstract

Heteronuclear alkali diatomics in the lowest triplet state possess electric and magnetic dipoles, making them promising for quantum simulation of many-body physics. However, an analytical formula with physical transparency to describe the interaction potential of the heteronuclear alkali dimers involving Li atoms is still lacking. The present article shows that the Sheng–Tang–Toennies (STT) potential model with only two adjustable parameters describes the full-range potential energy curve (PEC) of the LiNa molecule in its lowest triplet electronic state (a³ Σ⁺) with high accuracy. Validation against high-accuracy ab initio data and experimental spectroscopy demonstrates exceptional agreement across all internuclear distances, with deviations <1% from ab initio calculations and Rydberg–Klein–Rees (RKR) potentials. The model achieves high-precision predictions for vibrational energy levels, which show a root-mean-square error (RMSE) of 0.1965 cm⁻¹ (11 observed states), and the s-wave scattering length was calculated as −71.28 a.u., consistent with experimental bounds (−76 ± 5 a.u.). This work demonstrates that the STT model is effective for modeling the PEC of heteronuclear alkali dimers involving a Li atom.

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I. Introduction

Ultracold gases of atoms and molecules have garnered significant attention due to their applications in probing quantum phenomena in physics and chemistry.¹ Heteronuclear alkali diatomic molecules in the lowest triplet electronic state possess both electric and magnetic dipole moments, making them promising candidates for quantum simulation of many-body physics.² It was found that those heteronuclear alkali diatomics with one Li atom have the largest permanent electric dipole moment and polarizabilities in both ground and excited electronic states, making them easy to manipulate using an external electric field.^3–6

The LiNa diatomic molecule, one of the simplest heteronuclear many-electron systems, exhibits an electric dipole moment (0.167 Debye) and a magnetic moment (2µ_B),^7,8 and has been extensively studied both theoretically and experimentally.^7,9–14 Recent advances include the creation and investigation of LiNa molecules in the ground rovibrational level of this triplet state. Notably, Son et al. demonstrated cooling of LiNa molecules to micro- and nanokelvin temperatures via collisions with ultracold Na atoms, with both species prepared in their stretched hyperfine spin states.¹⁵

Accurate potential energy curves (PECs) are crucial for understanding and controlling ultracold chemical reactions and quantum computing.^7,16,17 While ab initio methods, such as full configuration interaction (FCI), achieve unparalleled accuracy for few-electron systems, computational costs increase exponentially with electron count,¹⁸ making precise calculations for interactions between ultracold alkali-metal atoms challenging. Furthermore, ab initio interaction potentials often require fitting to analytical functions for practical application in property calculations. Historically, this has involved complex polynomial expansions with numerous parameters. For instance, Lesiuk et al. employed a 13-parameter formula to fit a Born–Oppenheimer potential derived from advanced composite methods.¹⁶ However, the predicted s-wave scattering length for using this PEC still underestimates experimental results. For LiNa in the a³ Σ⁺ triplet state, the most recent high-accuracy ab initio interaction potential energies were reported by Gronowski et al.⁷ and Ladjimi et al.¹³ Gronowski et al. reported the electronic and rovibrational structure of the a³ Σ⁺ triplet state of LiNa with spectroscopic accuracy (<0.5 cm⁻¹). Ladjimi and Tomza also reported the ab initio points for the interaction potential energy based on the coupled-cluster methods up to CCSDTQ. However, the analytical PEC was not provided in those papers. Consequently, both experimental and theoretical analyses necessitate accurate, faithful, analytical, and physically motivated PEC representations.

There are several potential models that were proposed to describe the interaction potentials of alkali dimers.^19–21 For example, Lau et al. proposed a model named LTT with three adjustable parameters describing the interaction potentials of the a³ Σ⁺_u triplet state of Na₂, K₂, Rb₂ and Cs₂.¹⁹ Bauer and Toennies proposed the BLTT model with four adjustable parameters to describe the interaction potentials of the a³ Σ⁺_u triplet state of Na₂, K₂, Rb₂ and Cs₂.²⁰ This model was shown to have a similar level of agreement with the spectroscopic term values as the LTT model. The advantage of BLTT over the LTT model is that the scattering length can be well reproduced by fitting the additional parameter compared to the LTT model. Schwarzer and Toennies proposed a semiempirical potential model with five adjustable parameters for the a³ Σ⁺ triplet state of NaCs, KCs and RbCs.²¹ However, all of those models are not able to describe the interaction potential of the alkali dimers involving the Li atom. It was shown that the interaction potentials of the a³ Σ⁺_u triplet state of Li₂ has different shape compared to the other alkali dimers.^19,21

Recently, the present group proposed a modified Tang–Toennies model named STT to describe the potential of the a³ Σ⁺_u triplet state of Li₂.¹⁷ This model just has two adjustable parameters which can be determined by the well depth D_e and its location R_e of the potential energy curve. The vibrational levels and the scattering length calculated from this model are in very good agreement with the experimental results. The STT model was proposed for the homonuclear dimers. Due to the good performance of this model for the a³ Σ⁺_u triplet state of Li₂, it is interesting to extend the applications of the STT model to LiNa.

In this paper, the STT potential model is applied to predict the interaction potential of the a³ Σ⁺ triplet state of LiNa. It is shown that the analytical potential energy curve for LiNa based on the STT model is in excellent agreement with the most recent ab initio and experimental results. With the obtained analytical potential energy curve, the vibrational energy levels and the scattering length are also predicted with high accuracy.

II. Analytical potential energy curve of the LiNa in a³ Σ⁺ state

The STT model builds upon the Tang–Toennies (TT) and Tang–Toennies–Yiu (TTY) models. Therefore, we first briefly introduce the TT and TTY models.

A. TT potential model

The TT model was proposed by Tang and Toennies in 1984. It is the sum of an attractive part and a repulsive part.²² The repulsive part is the Born–Mayer form A exp(−bR). The attractive part is the damped dispersion terms. It is usually written in the following forms.

A and b are determined by the conditions

V(R_e) = −D_e (2)

and

The depth of the potential well D_e and its location R_e can be converted into A and b by a well defined procedure.²³

B. TTY potential model

Considering that the short range repulsive potential is dominated by the exchange energy, Tang et al. modified the repulsive term in the TT model based on the surface integral method for exchange energy. The Born–Mayer term A exp(−bR) in the TT model is replaced with the term . Then the TTY model is given by the following form.²⁴Λ and λ are two adjustable parameters which can be determined from the transforms of A and b in the TT model.²⁵ The advantage of the TTY model over the TT model is that all the terms in the TTY model have a transparent physical meaning. All the parameters in the TTY model, in principle, can be calculated by the ab initio method.^24,26 It was shown that the TTY model indeed shows better performance than the TT model for the potential curves at the short range internuclear distances for the rare-gas dimers.²⁷

C. STT potential model

For the rare gas dimers, it has been shown that usually the Hartree–Fock Coulomb energy in the Hamiltonian ΔE^HF is a small fraction (10–35%) of the repulsive exchange energy.²⁸ Thus, a better approximation to the repulsive potential is to explicitly include the Hartree–Fock repulsive term in the exponential Born–Mayer term:

Then, the STT model was obtained and is given by¹⁷

and E_i is the well known ionization energy of the separate atoms. This is clear for the homonuclear dimers. For example, E_i is the ionization energy of the Li atom for Li₂.¹⁷ However, this is not the case for heteronuclear diatoms. There are two different atoms (Li and Na) with different ionization energies. Since the ionization energies of Li(E_Li = 0.1981418715 a.u.) and Na(E_Na = 0.188857595 a.u.) are very close to each other, it is reasonable to approximate the value of β to be the arithmetic mean or geometric mean .²⁹ This was also applied in the TT model for calculating the reduced parameters of the heteronuclear dimers.³⁰ For the present system, the arithmetic mean (0.62205 a.u.) and geometric mean (0.62201 a.u.) give very close numbers for β. The differences between these two numbers will not make observable changes to the potential. Here we simply take 0.62205 a.u. as the value of β.

B ₁ and B₂ values are determined by requiring that the potential energy and slope of in the potential of V_STT(R) to be the same as the Born–Mayer repulsive potential Ae^−bR in the TT model at the potential minimum, that is

To satisfy these conditions, B₁ and B₂ are required to be

where A and b are from the TT potential model. Thus, B₁ and B₂ are completely fixed without any fitting.

D. STT2 potential model

V _STT(R) does not approach the proper form as R → 0. This problem can be solved in a similar way as has been done for the V_TTY(R) model.²⁵ At small R, V_STT(R) will be gradually turned off and replaced with the proper expression obtained from the united atom perturbation theory.^31,32 This can be done without affecting D_e and R_e by defining

V_STT2(R) = V_short(R) + (1 − e^−αR)V_STT(R), (9)

where V_short(R) and α are expanded in the following form: thus

The 4 additional parameters a₁, a₂, a₃, and α can be determined by an appropriate choice of boundary conditions. We refer to those boundary conditions in ref. 25 and 33. With those boundary conditions considered, the additional 4 parameters can be calculated from the following formulas.

C = E_ua − E_A − E_B (13)

andwhere Z_A and Z_B are the nuclear charges and E_A and E_B are the energies of the separate atoms, and E_ua is the united atom energy with the same total mass, total nuclear charge, and electronic state of the separate atoms. Here, the ground state of the united triplet LiNa is the Si atom.

III. Comparisons of V_STT(R) with theoretical and experimental potential energy curves

The aforementioned STT model is employed to calculate the interaction potential of the a³ Σ⁺ state of the LiNa. The required values of the input parameters C₆ ∼ C₁₀, R_e and D_e, as well as the corresponding parameters derived using the STT model, are listed in Table 1. The higher order dispersion coefficients (C₁₂, C₁₄, C₁₆, C₁₈ and C₂₀) are computed via the following recurrence relation³⁴

Table 1 Parameters for LiNa, all in atomic units. R_e, D_e and the dispersion coefficients C₆–C₁₀ are taken from ref. 7 and 36, respectively. Those values for E are taken from ref. 37. The N_max = 10 in eqn (1), (4) and (6) and the higher order dispersion coefficients C₁₂ ∼ C₂₀ are calculated by the recurrence relation³⁴

Input parameters		Derived parameters
a Binding energies are determined by summing up all relevant ionization energies.^37
C 6	1.4729 × 10^3	A	2.04210
C 8	9.8851 × 10^4	b	0.78843
C 10	9.1880 × 10^6	B 1	14.42596
R e	8.924	B 2	0.004190
D e	−1.0475 × 10^−3	β	0.62205
E i (Li)	0.1981418715	Λ	0.010391
E i (Na)	0.188857595	λ	1.286222
E (Li)^a	−7.4779789	α	7.895049
E (Na)^a	−162.4307821	a 1	4.259003
E ua	−289.8982716	a 2	−0.992183
		a 3	0.056294

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This approximation was compared with more precise calculations for the case of two H atoms. It was found to be remarkably accurate, with the maximum error being less than 3.5%, and this error decreases as n increases.³⁵

The present calculated potential energy curve is shown in Fig. 1 (red solid line represents STT and red dashed line represents STT2). For comparison, the most recent two sets of ab initio data reported by Gronowski et al. and Ladjimi et al. (circles are taken from ref. 7, triangles are taken from ref. 13, and the RKR potential reported by Rvachov et al. (blue dashed-dotted line)¹¹ are shown in Fig. 1. The RKR potential was obtained to fit the vibrational binding energies from two-photon spectroscopy based on an X-representation potential model with 13 free parameters. In addition, the potential curve from Tang–Toennies (TT) (black dashed-dotted line) and Tang–Toennies–Yiu (TTY) (olive dotted line) models are also included.

Fig. 1 The potential energy curve for the a³ Σ⁺ state of LiNa calculated with the models of TT, TTY, STT, and STT2. The small circles and triangles are the raw data points of the ab initio results reported by Gronowski et al.⁷ and Ladjimi et al.¹³ The blue line is the point-wise RKR potential, which is taken from ref. 11.

Fig. 1 illustrates that the potential energy curves derived from the TT and STT models exhibit good consistency with both the ab initio results and the RKR data in the vicinity of the minimum internuclear distance R_e and at long-range distances. However, the interaction potential energy curve of the TTY model begins to deviate from the ab initio data points when the internuclear distance R is less than R_e. This deviation is due to the fact that the TTY model approaches zero as the internuclear distance approaches zero.³⁸ For distances R greater than R_e, the differences among the TT, TTY, and STT models are not readily observable with the scale used in Fig. 1. These differences can be seen in Fig. 2, which shows the repulsive potential energy curves of TT, TTY and STT models (based on the eqn (1), (4), (6), the differences among those models (TT, TTY and STT) are just from the repulsive potential). As can be seen from Fig. 2, the repulsive potential energy curve from the STT model (the red solid line in Fig. 2) lies between the two curves from the TT and TTY models for long – range distances (R > R_e). This is as expected because the repulsive potential in the STT model is a combination of the TT and TTY models. It is shown in Fig. 3 that these differences enable the STT model to be in excellent agreement with the ab initio and experimental results.

Fig. 2 Comparisons of the repulsive potential of the a³ Σ⁺ state of LiNa from TT (black dashed-dotted line), TTY (olive dotted-line) and the present STT (red solid line) models.

Fig. 3 Upper panel: The reduced potentials of the a³ Σ⁺ state of LiNa from the ab initio potential.⁷ Lower panel: The differences are shown between the corresponding reduced potentials and the reduced potential of the ab initio potential of Gronowski et al.⁷ The black solid line, the green solid line with crosses, the solid red line with triangle, and the blue solid line with circles are the differences between the TT, TTY, STT, and RKR potentials and the ab initio potential.

The STT2 model has the same potential energy curve as the STT model, except for the potential at internuclear distances R smaller than 1 a.u., as shown in Fig. 1. The advantage of the STT2 model over the STT model is that the STT2 model exhibits the correct physical behavior as the internuclear distance approaches zero. In the next section, the presently calculated potential energy will be used to derive the harmonic frequency, vibrational energy levels, and the scattering length, which depend only on the attractive potential. Therefore, the STT and STT2 models will yield the same results for these physical quantities.

Fig. 3 presents the differences in the reduced potentials of LiNa, where U(x) = V(xR_e)/|D_e|, from the TT, STT, and RKR models, corresponding to the ab initio data points. It is clearly shown that the present STT and RKR models are in excellent agreement with the ab initio results. The largest difference between the STT potential and the ab initio potential is less than 1% around 1.25R_e. The TT model generally overestimates the potential, and the largest difference between the TT potential and the ab initio potential exceeds 4% around 1.25R_e. For the TTY model, the potential is underestimated, and the largest difference between the TTY potential and the ab initio potential is larger than 3% at around 1.25R_e. As we have shown in Fig. 2, the repulsive potential energy curve of the STT model lies exactly between the two curves from the TT and TTY models. The combination of the repulsive potentials from the TT and TTY models compensates for the errors in each of these two models, which makes the STT model more suitable than the TT and TTY models for describing the potential energy curve (PEC) of the a³ Σ⁺ state of LiNa. Similar results were also found for the a³ Σ⁺_u state of Li₂.¹⁷

IV. Applications of the analytical potential energy curve of the LiNa in a³ Σ⁺ state

A. Harmonic frequency and vibrational energy levels

With the above obtained analytical potential energy curve of the LiNa in the a³ Σ⁺ state, the harmonic frequency and the vibrational energy levels can be calculated. The harmonic vibrational frequency is defined asin atomic units, where μ is the reduced mass of an isotopomer. Here, we consider isotopes of ²³Na⁶Li with atomic masses equal to

m(⁶Li) = 6.015123u,

m(²³Na) = 22.98977u.

The harmonic vibrational frequency of the lithium dimer based on the STT model is given by:

With the parameters listed in Table 1, the harmonic vibrational frequency ω_e of the LiNa is calculated and compared with previous reported results, as shown in the following table. The value of ω_e calculated from the present STT model is 44.47 cm⁻¹. It is clearly shown that the present result falls into the range of the previously reported ab initio results (39.9 cm⁻¹–46.0 cm⁻¹). It is also noticed that the predicted values of ω_e from TT and TTY are very close to the maximum and minimum values of the previous ab initio results, respectively (Table 2).

Table 2 The harmonic vibrational frequency ω_e of ⁶Li²³Na. Numbers are in cm⁻¹

Method	ω e	Reported
V TT	46.92	Present
V TTY	40.57	Present
V STT/STT2	44.47	Present
Ab initio	42.12	Ladjimi et al. (2024)^13
Ab initio	46.0	Mieszczanin et al. (2014)^39
Ab initio	40.56	Bellayouni et al. (2014)^40
Ab initio	39.9	Schmidt-Mink et al. (1984)^41

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Table 3 presents comparisons of the calculated vibrational energy levels obtained via various methods. Rvachov et al. reported that the triplet ground-state potential of LiNa possesses 11 vibrational states based on two-photon spectroscopy.⁹ The ab initio interaction potential for LiNa from Gronowski et al. also reproduces those vibrational levels with high accuracy.⁷ Here, the vibrational state binding energies reported by Rvachov et al.⁹ and Gronowski et al.⁷ are taken as the benchmark for comparison. Using the present STT potential model and the previous TT and TTY potential models, the vibrational energy levels of LiNa were derived by numerically solving the radial one-dimensional Schrödinger equation under the adiabatic approximation. The interaction potential energies at 1600 grid points, corresponding to internuclear separation ranging from 3.0 a.u. to 200 a.u. were utilized. The calculation was done using an in-house program, and has also been successfully verified using the standard program LEVEL16.⁴²

Table 3 Vibrational energy levels (J = 0) for the a³ Σ⁺ state of ⁶Li²³Na. The vibrational energies (E_v) are given in cm⁻¹. Experimental and ab initio data are taken from ref. 11 and 7, respectively. δ_rms represents the root-mean-square errors of the present results with respect to the experimental and ab initio data, respectively

v	Vibrational energy levels (J = 0)
	TT	TTY	STT/STT2	Ab initio	Exp.
a Experimental results are taken as the benchmark. b Ab initio results are taken as the benchmark.
0	207.028	210.094	208.129	208.2	208.0826
1	164.986	172.690	167.802	168.0	167.910
2	127.830	138.576	131.684	132.0	131.922
3	95.566	107.887	99.857	100.2	100.169
4	68.191	80.780	72.411	72.9	72.730
5	45.684	57.422	49.432	49.8	49.722
6	27.989	37.990	30.992	31.2	31.210
7	14.986	22.646	17.114	17.3	17.250
8	6.427	11.487	7.700	7.77	7.768
9	1.809	4.426	2.385	2.42	2.4137
10	0.169	0.977	0.304	0.314	0.3119
δ rms	3.0479	5.6265	0.1965	0.0778	0.0
	17.72%	71.76%	1.01%	0.25%	0.0
δ rms	3.1086	5.5683	0.2558	0.0	0.0778
	17.86%	71.12%	1.24%	0.0	0.25%

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It is clearly demonstrated in Table 3 that the 11 vibrational states observed in the experiment are well reproduced. Moreover, the calculated vibrational energy levels from the present STT model are in good agreement with the experiment and ab initio results. The root-mean-square errors (RMSEs) of the present results relative to the experimental and ab initio results are 0.1965 cm⁻¹ (1.01%) and 0.2558 cm⁻¹ (1.24%), respectively. Gronowski et al. have well illustrated that their reported rovibrational structure achieves spectroscopic accuracy (<0.5 cm⁻¹).⁷ When this error basis is taken into account, the vibrational energy levels calculated from the present STT model show excellent agreement with the ab initio results. The dissociation energy D₀ predicted by the present STT model is 208.129 cm⁻¹, which is also in excellent agreement with the experimental result of 208.0826 cm⁻¹.¹¹ Additionally, the calculated fundamental vibrational excitation energy is 40.327 cm⁻¹, in good agreement with the experimental value of 40.172 cm⁻¹.¹¹ It is also noted that the previous TT and TTY models are both unable to predict the vibrational energy levels with the high accuracy achieved by the present STT model. This further confirms that the TT or TTY model needs to be modified to describe the interaction potential of alkali dimers.^17,19–21

We also calculated the rotational constant B(v = 0), which is given by

B(v = 0) ≈ B_e − α_e/2 (17)

Here, B_e and α_e are Dunham coefficients derived from the force constants of the present potential.⁴³ The calculated rotational constant for the ground vibrational level is B(v = 0) = 4.617 GHz, which agrees excellently with the experimental value of 4.63 GHz¹¹ and the ab initio result of 4.616 GHz.⁷

B. Scattering length

The s-wave scattering length (a_s) fully characterizes scattering in the ultracold regime. As a crucial parameter for ultracold physics experiments, it is highly sensitive to the accuracy of the PEC, especially the position of the last weakly bound state.¹⁶ In recent years, a semiclassical formula developed by Gribakin and Flambaum⁴⁴ has been shown to predict the scattering length in good agreement with the fully quantum mechanical calculations.^17,45 This formula is written as follows:with
n = 6 and γ = (2μC₆)^1/2/ħ (ħ = 1 in the atomic unit). R₀ is defined by V(R₀) = 0. We have used this formula to compute the s-wave scattering length of ⁶Li²³Na with the present obtained potential. The scattering length is found to be −71.28 a.u., in excellent agreement with experimental and ab initio results. It is worth mentioning that the calculated scattering lengths based on the above formula are in good agreement with the fully quantum-mechanical method proposed by Meshkov et al.⁴⁶ The comparisons are shown in Table 4. It is interesting to see that the present value is in good agreement with the experimental value −76 ± 5 a.u., with the error bar considered.⁴⁷ For comparison, the results from the TT and TTY models are also given. It is clearly seen that the potential energy curves from the TT and TTY models are not accurate enough to give a proper estimation of the scattering length.

Table 4 The s-wave scattering length of ⁶Li²³Na. Numbers are in atomic units. Here, the fully quantum-mechanic method was proposed by Mechkov et al.⁴⁶

Potential	Method	Scattering length	Reported
V TT	Eqn (17)	5.38	Present
	Quantum-Mechanic	5.40
V TTY	Eqn (17)	47.74	Present
	Quantum-Mechanic	47.30
V STT/STT2	Eqn (17)	−71.28	Present
	Quantum-Mechanic	−71.34
V ^CCSD(T)BO	S-matrix formalism	−78	Gronowski et al. (2020)^7
V ^CCSD(T)BO + δVCorrection	S-matrix formalism	−84	Gronowski et al. (2020)^7
Experiment	Experiment	−74	Rvachov et al. (2018)^11
Experiment	Experiment	−76 ± 5	Schuster et al. (2012)^47

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V. Discussion

The repulsive potential in the present STT model incorporates the Born–Mayer term B₁e^−2βR from the TT model and also the exchange term from the TTY model. These two components decay differently as the internuclear distance R increases. Fig. 4 displays the repulsive potential curve along with the contributions from these two components. It can be found that the dominant contribution to the repulsive potential is the Born–Mayer term for short-range distances (R < 6 a.u.). However, the exchange term becomes the dominant contribution for long-range distances (R > 6 a.u.). This observation indicates that the repulsive potential of LiNa does not decay simply exponentially with R. This is the reason why both the TT and TTY models are not flexible enough to model the repulsive potential of LiNa. In addition, it is also understandable that the LTT¹⁹ and BLTT²⁰ models are unable to describe the interaction potential of alkali dimers involving the Li atom. The repulsive potentials in the LTT and BLTT models both decay exponentially with R.^19,20 Since short-range potentials are essential for processes like chemical reactions⁴⁸ and inelastic scattering,⁴⁹ the STT model makes it particularly well-suited for predicting the associated properties with high accuracy.

Fig. 4 Contributions of various components in the repulsive potential to the repulsive potential of the STT model for the a³ Σ⁺ state of LiNa. The dominant contribution to the repulsive potential varies as the internuclear distance changes from short to long ranges.

Given that the Sheng–Tang–Toennies (STT) model has accurately predicted the interaction potential of the LiNa molecule in its lowest triplet state, it would be interesting to explore the interaction potentials of other heteronuclear alkali dimers involving lithium atoms (LiK, LiRb and LiCs) based on the STT model. A preliminary verification (Fig. S1–S3) shows that the present STT model also predicts the interaction potentials of LiK, LiRb, and LiCs with high accuracy compared to ab initio results. This indicates that the model is reliable for describing the interaction potentials of heteronuclear alkali dimers containing lithium atoms. In addition, there is also a feasible approach for calculating the interaction potentials of LiK, LiRb and LiCs in their lowest triplet state. It has been shown that the reduced potentials of NaCs, KCs, RbCs, Na₂, K₂, Rb₂ and Cs₂ are highly similar.^19,21 Accordingly, it is reasonable to assume that LiNa, LiK, LiRb, LiCs and LiFr have the same reduced potential. By applying this assumption, the interaction potentials of LiK, LiRb, LiCs and LiFr can be derived using the reduced potential of LiNa obtained in this paper, along with its well depth D_e and the equilibrium distance R_e. We have previously shown that the potentials of rare gas dimers can be well predicted through this method.³³ The results for the LiNa, LiK, LiRb, LiCs and LiFr based on the present STT model will be reported in future work.

VI. Conclusions

By incorporating the explicit repulsive potential from TT and TTY models parameterized solely by the well depth D_e and equilibrium bond length R_e, the STT model achieves exceptional agreement with high-accuracy ab initio data, experimental Rydberg–Klein–Rees (RKR) potentials, and spectroscopic measurements for the LiNa molecule in its lowest triplet state. The deviations of the present STT model from benchmark ab initio PEC data remain below 1%. With the PEC from STT model applied, the harmonic frequency, vibrational energy levels, and scattering length are well predicted. The STT model successfully extends to heteronuclear alkali dimers involving lithium (LiNa), overcoming limitations of existing models (e.g., TT, TTY, LTT, and BLTT). In addition, its physical transparency, minimal parametrization, and robustness across short- to long-range interactions position it as a versatile tool for ultracold quantum simulations, reaction dynamics, and precision spectroscopy of dipole-rich molecules.

Author contributions

Dongying Li: data curation, formal analysis, and investigation. Xiaowei Sheng: methodology, writing, supervision, and funding acquisition.

Conflicts of interest

The authors declare no conflicts of interest.

Data availability

The data supporting the conclusions are quoted in the manuscript.

Supplementary information (SI) is available. See DOI: https://doi.org/10.1039/d5cp04966k.

Acknowledgements

The authors would like to express their gratitude to K. T. Tang and J. Peter Toennies for their insightful discussions that have contributed to this project over the past several years. This work is supported by the National Natural Science Foundation of China (No. 12574279 and No. 12174003), and the University Annual Scientific Research Plan of Anhui Province (No. 2022AH020019).

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