Neon dimers in momentum and configuration spaces | Scientific Reports

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This work investigates the rovibrational energy levels of the Neon dimer using the Lippmann-Schwinger equation in both configuration and momentum spaces. By analyzing the wave functions and interatomic separations, we provide a detailed characterization of the binding energies and spatial properties of the Neon dimer. Results indicate strong agreement between configuration and momentum space calculations, with deviations remaining below \(0.7\%\). Momentum space solutions exhibit enhanced precision for weakly bound states due to their localization at lower momenta. Comparisons with existing theoretical data validate the robustness of the approach, offering a reliable framework for studying other rare gas dimers and extending to larger clusters to deepen the understanding of intermolecular interactions in quantum systems.

The study of molecular interactions, particularly through the examination of intermolecular forces and bonds between atoms, provides profound insights into the physical behavior of matter at the microscopic level. Among the simplest quantum-mechanical molecular systems are rare gas dimers, which offer an ideal setting for investigating weakly bound states due to their lack of electronic complexity and chemical inertness. The neon dimer (\(\hbox {Ne}_2\)), a prototypical example, serves as an excellent model for probing the nuances of molecular bonding and quantum mechanical interactions. Neon gas clusters stand at a pivotal point in the rare gas family1, bridging the highly quantum nature of helium and the more classical regimes of heavier noble gases such as argon and krypton. Their weak van der Waals interactions and small binding energies render them exquisitely sensitive to subtle quantum effects, making them valuable test systems for theoretical models and computational methods2,3,4,5. Experimental investigations of neon dimers, trimers, and larger clusters-often through high-resolution spectroscopy6-provide critical benchmarks for refining potential energy surfaces, verifying quantum Monte Carlo simulations7, and sharpening machine learning approaches8. Recent research has pushed the boundaries beyond dimers to also explore Neon trimers and tetramers, thereby illuminating the interplay of three- and four-body forces in these weakly bound systems2,9,10,11,12. Developments in experimental techniques, such as high-resolution spectroscopy, and theoretical advances like Monte Carlo simulations, Faddeev-Yakubovsky formalisms, and variational methods, have further refined our understanding of these clusters. Moreover, modern computational approaches, including machine learning (ML) and deep neural networks, have been applied to rare gas systems, offering both data-driven predictions of intermolecular potentials and accelerated discovery of optimized interaction parameters8. Accurate determination of binding energies plays a pivotal role in understanding the equilibrium thermodynamic properties of rare gas clusters13. Precise binding energies guide the development of reliable potential energy surfaces and enable robust calculations of measurable quantities such as scattering lengths, virial coefficients, and bulk properties (e.g., compressibility and heat capacity) in both gaseous and condensed phases. This research focuses on the rovibrational energy calculations of neon dimers using the Lippmann-Schwinger (LS) equation in both momentum and configuration spaces, offering a dual perspective that enhances our understanding of these elusive quantum systems, while allowing a detailed analysis and comparison of predicted energy levels with those derived from other theoretical and empirical studies. The LS integral equation has been widely utilized across various areas of physics to investigate two-body bound and scattering states. Due to its non-perturbative nature, it is particularly effective in addressing weakly bound systems. Its applications range from quark-antiquark bound states in quantum chromodynamics14, to tetraquarks in a two-body diquark-antidiquark formulation15, Argon dimers in momentum space16, and the deuteron- proton-neutron bound state- using both relativistic potentials17,18and low-momentum interactions19. More recently, the LS approach has also been employed in condensed matter systems to explore excitons, the electron-hole bound states, in double-layer transition metal dichalcogenides (TMDs) with a dielectric spacer20. Despite the diversity of these applications, a unifying feature is the reliability of the LS method in accurately determining bound-state properties and yielding deep insights into the underlying interactions.

In the following sections, we outline the theoretical framework for solving the LS equations in momentum and configuration spaces, describe the computational methodology, and present detailed results for the Neon dimer’s binding energies and wavefunctions. We also discuss how these findings may be extended to larger Neon clusters and other rare gas dimers, potentially serving as a basis for further experimental and theoretical investigations in this field.

The nonrelativistic bound state of two particles with masses \(m_1\) and \(m_2\), relative momentum \({{\textbf{p}}}\), relative distance \({{\textbf{r}}}\), and interacting through an arbitrary interaction V, can be described by the homogeneous LS equation as

where \(\psi\) is the two-body (2B) wave function and \(G_0= (E-H_0)^{-1}\) is the free Green’s function which depends on 2B binding energy E. The representation of the LS equation (1) in configuration space leads to a three-dimensional integral equation as21

where v and j are the vibrational and rotational quantum numbers, respectively, \(k = \sqrt{2 \mu |E|}\), and \(\mu\) is the 2B reduced mass. By considering \(\psi _{vj}({{\textbf{r}}}) = P_j(x) R_{vj}(r)\), the radial part of the integral equation can be written as

where \(\rho = \sqrt{r^2+r'^2-2r r' x'}\). By having the 2B wave function, the root-mean-square distance between two particles can be obtained by

where the 2B wave function is normalized as follows

To ensure that the kernel of the LS integral equation (2) is of the Fredholm type and allows for stable numerical handling, the potential V(r) must decrease sufficiently rapidly at large distances and not be too singular at the origin.

While the LS equation can be solved in configuration space with high precision, it becomes problematic when applied to weakly bound systems. This is due to the wave function expansion over large distances, mandating large relative distance cutoffs and, leading to a significant increase in the required mesh points needed for discretizing the relative distance. This makes the problem computationally demanding. However, an alternative approach – numerically solving the LS equation in momentum space – can be beneficial, particularly for weakly bound systems where the wave function is predominantly concentrated at lower momenta. As a result, the computational accuracy can be controlled by applying a suitable mapping, reducing the computational demand while maintaining precision. With this approach, the representation of the LS equation (1) in momentum space takes the form of the following integral equation15,16,22

The input to the integral equation (6) is the partial-wave-projected two-body interaction \(V_j (p,p')\), derived from \(V({{\textbf{p}}},{{\textbf{p}}}') \equiv V(p,p',x)\) by

where p and \(p'\) are the magnitudes of initial and final 2B relative momentum vectors and \(x = \hat{{{\textbf{p}}}} \cdot \hat{{{\textbf{p}}}}'\). The matrix elements of the interaction in momentum space \(V(p,p',x)\) can be obtained from the Fourier transformation of 2B interaction in configuration space V(r) as

with the momentum transfer \(q=\sqrt{p^2+p'^2-2pp'x}\). The LS integral equations, as presented in Eqs. (3) and (6) for both configuration and momentum spaces, are applicable to any quantum two-body bound system and are not limited to the Neon dimer discussed in this paper. However, the momentum-space approach offers particular advantages for shallow bound states, such as \(\hbox {He}_2\). Since these weakly bound systems extend to large distances in configuration space, their representation in momentum space is more compact, leading to more accurate solutions with reduced computational complexity. The control over the number of mesh points required for discretizing continuous momentum variables at small momentum cutoffs is significantly more manageable than in configuration space, where large relative distances necessitate finer discretization.

In our calculations, we employ a well-established Neon-Neon interatomic potential of Ref23.

where \(R=\frac{r}{r_m}\), and \(f_{2n}(x)\) are the Tang-Toennies damping functions \(f_{2n}(x)=1-e^{-x}\,\sum \limits _{k=0}^{2n} \displaystyle \frac{x^k}{k!}\). The parameters of the potential, listed in Table 1, predict a minimum separation of 3.0895 Å with a depth of 42.153 K. In our calculations, we use the mass of Neon \(m_{^{20}Ne} = 20.1797\) amu, and take \(\frac{\hbar ^2}{m}=2.40388331361\) K \(\cdot\) Å\(^2\).

Ne–Ne interaction plotted as a function of the relative distance r (left panel), and as functions of the relative momenta p and \(p'\) for different rotational quantum numbers \(j = 0\text {--}8\) (right panel).

Figure 1 illustrates the Ne-Ne interaction in both configuration and momentum spaces. The left panel shows the potential as a function of the interatomic distance r. In the right panel, a few examples of the partial-wave-projected potential matrix elements \(V_j(p,p')\) are plotted for rotational quantum numbers \(j = 0\text {-}8\). These matrix elements, obtained from Eqs. (7) and (8), are shown as functions of the relative momenta p and \(p'\).

In this section, we present and analyze our numerical results for the neon dimer system. We focus primarily on the binding energies, the root-mean-square distances between neon atoms, and the wave functions obtained by solving the LS equation in both configuration and momentum spaces. A comparison with other theoretical and experimental data is also provided, highlighting the accuracy and consistency of our approach. The binding energies in both configuration, Eq. (3), and momentum, Eq. (6), spaces are calculated by solving eigenvalue equations. These equations can be schematically represented as

where the eigenvalue corresponding to the physical binding energy is equal to one. In Eq. (3), the energy dependency of the kernel is manifested in k, which is directly related to the energy E through the relationship \(k = \sqrt{2 \mu E}\). In contrast, Eq. (6) explicitly includes the energy dependency in the free propagator \(\frac{1}{E - \frac{p^2}{2\mu }}\). To calculate the binding energy, we employ the Newton-Raphson method to search in the 2B energy, beginning with two initial energy guesses and continuing searching to find a binding energy that yields an eigenvalue near one within a tolerance of \(10^{-6}\). For discretization of the continuous variables–relative momenta \(p, p'\), relative distances \(r, r'\), and angular variables \(x, x'\)–we employ Gauss-Legendre quadrature. The variables for momentum and distance are transformed through a hyperbolic-linear mapping from domain X: \([-1,+1]\) to Y: \([0,Y_1]+[Y_1,Y_2]+[Y_2, Y_3]\) using the formulas:

The domain \([0, Y_1] + [Y_1, Y_2]\) is handled using hyperbolic mapping, while \([Y_2, Y_3]\) is covered by linear mapping. This combined mapping allows precise control over the distribution of mesh points, ensuring that the wave functions’ structure is accurately captured at peaks and dips. For our analyses, we use 500 mesh points for relative momenta and require 2,000 mesh points for configuration space calculations to achieve similar accuracy. Table 2 presents the computed binding energies for neon dimers in both configuration (\(E_r\)) and momentum space (\(E_p\)) frameworks by solving the integral equations (3) and (6), correspondingly. The quantum numbers v and j denote vibrational and rotational levels, respectively. Overall, the relative percentage difference (\(\Delta \%\)) between the two methods is on the order of 0.02–\(0.70\%\), indicating a high degree of consistency. As expected, higher vibrational or rotational states exhibit slightly larger deviations, reflecting the increased sensitivity of excited, weakly bound states to the details of the potential. Nevertheless, the overall agreement between \(E_r\) and \(E_p\) remains excellent, underscoring the reliability of our numerical scheme in both domains. It is worth noting, however, that the neon dimer binding energies obtained from the LS equation in momentum space may be more accurate because weakly bound states are predominantly concentrated at lower momenta, requiring only a small momentum cutoff. In contrast, the wave functions in configuration space extend to large distances, demanding a larger spatial cutoff and thus a higher number of mesh points for discretizing the relative distance.

Table 3 reports the root-mean-square (RMS) distance \(\langle r^2 \rangle ^{1/2}\) for various rotational and vibrational states, computed within the configuration space approach using Eq. (4). As j or v increase, the nodal structure of the wave function changes and the dimer spends more time in the classically forbidden region, thus increasing the average distance. Notably, the RMS distance for \(v=2\) is significantly larger, underscoring the higher degree of spatial delocalization in the excited vibrational state.

Figure 2 shows the radial wave functions \(R_{vj}(r)\) and \(\psi _{vj}(p)\) for all eighteen neon ground and excited states, corresponding to different vibrational and rotational (v, j) quantum numbers. In the ground state \((v=0, j=0)\), the wave function in configuration space is sharply peaked around the potential minimum. As v or j increase, the wave function becomes more delocalized in real space, reflecting weaker binding and a higher likelihood of finding the atoms at larger separations. The number of nodes in the dimer wave function in configuration space corresponds directly to the vibrational quantum number v: for \(v=0\), there are no nodes; for \(v=1\), there is one node; and for \(v=2\), there are two nodes. A comparison of the wave function shapes in configuration and momentum space shows that, as v and j increase, the neon dimer wave functions become more compact at smaller momenta in momentum space, consistent with the characteristics of weakly bound excited states. For instance, while the wave function in configuration space for \(j=0\), \(v=2\) extends beyond 20 Å, the corresponding wave function in momentum space is confined within a much shorter range up to 2 Å\(^{-1}\). This confinement allows for more accurate computations with significantly fewer mesh points, almost four times fewer in momentum space than in configuration space, to achieve comparable accuracy in energy calculations.

Ne–Ne dimer wave functions \(R_{vj}(r)\) (upper panel) and \(\psi _{vj}(p)\) (lower panel), shown as functions of the relative distance r and momentum p, respectively.

Finally, Table 4 compares our computed energy differences relative to the ground state (\(E_{vj}-E_{00}\)) with those from other theoretical23,24,25and experimental works26,27in the literature. The overall agreement is excellent, with discrepancies typically remaining within the reported uncertainty ranges. Nonetheless, the consistency across these diverse datasets indicates that our LS-based approach, in both momentum and configuration spaces, effectively captures the essential physics of neon dimers. However, it should be noted that our calculations use the same Ne-Ne potential from Ref23., while Refs24,25. employ different Ne-Ne interactions, and Refs26,27. provide experimental data. Additionally, Ref23. uses the LEVEL computational toolkit28to solve the one-dimensional differential radial Schrödinger equation. In contrast, our approach involves solving the integral form of the LS equation in configuration space. Comparing our results with those from Ref23., we observe a relative percentage difference ranging from 0.03% to 0.86%. This difference primarily arises from the different methodologies used: the differential form of the Schrödinger equation in Ref23. versus the integral form of the LS equation in our study.

In summary, our results demonstrate that numerical solutions of the Lippmann-Schwinger equation in both momentum and configuration spaces can reliably describe neon dimer states, ranging from the ground state to excited rotational and vibrational levels. The small deviations between the two approaches validate the numerical algorithms and underscore the robustness of the LS formalism for weakly bound systems. However, we anticipate that momentum-space solutions are generally more accurate for higher excited states, since the corresponding wave functions are more localized at lower momenta, making the computational domain more manageable. In contrast, in configuration space, the wave function extends to larger distances for higher excited states, requiring larger spatial cutoffs and more mesh points to discretize the relative distance variable. By comparing our findings with established theoretical benchmarks, we confirm that our method provides a detailed and accurate picture of neon dimer binding properties, wave functions, and other observables pertinent to rare gas cluster physics. Moreover, the approach presented here can be readily adapted to other rare gas dimers. Extensions to larger clusters, such as trimers and tetramers, also appear feasible given the successful implementation of Faddeev-Yakubovsky methods for three- and four-body nuclear and atomic systems29,30,31,32,33,34,35,36, combined with the computational capabilities of modern supercomputer clusters. We are currently progressing in applying the Faddeev-Yakubovsky formalism in momentum space to calculate the binding energies and wave functions of He and Ne trimers and tetramers, with plans to extend it to other rare gas trimers and tetramers.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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M.R.H. acknowledges support from the National Science Foundation under Grant Nos. NSF-OIA-2430293 and NSF-PHY-2000029 at Central State University.

This study was funded by National Science Foundation (NSF-OIA-2430293 and NSF-PHY-2000029).

Department of Mechanical and Materials Engineering, Wright State University, Dayton, OH, 45435, USA

R. Ahnouch

College of Engineering, Science, Technology, and Agriculture, Central State University, Wilberforce, OH, 45384, USA

M. R. Hadizadeh

Department of Physics and Astronomy, Ohio University, Athens, OH, 45701, USA

M. R. Hadizadeh

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M.R.H. designed and directed the project, developed the theoretical formalism, and implemented the computational codes, while R.A. conducted the calculations for the Neon dimer binding energies. All authors discussed the results and contributed to the final manuscript.

Correspondence to M. R. Hadizadeh.

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Ahnouch, R., Hadizadeh, M.R. Neon dimers in momentum and configuration spaces. Sci Rep 15, 11928 (2025). https://doi.org/10.1038/s41598-025-97046-8

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