High-Temperature Optical Spectra of Diatomic Molecules: Influence of the Avoided Level Crossing

: In this study, we analyzed the light absorption by diatomic molecules or colliding atoms in a spectral region dominated by an avoided crossing of adiabatic state levels or crossing of the corresponding diabatic state levels. Our attention was focused on the low-resolution spectrum at a higher gas temperature under local thermodynamic equilibrium conditions. The absorption measurements of mixed vapors of potassium ( ≈ 80%) and cesium ( ≈ 20%) were made in the temperature range of 542–715 K and the infrared spectral range 900–1250 nm. In this area, the main spectral contributions were the broad bands of K 2 , Cs 2, and KCs molecules. There was a crossing of ( ) ( ) ( ) u A + + Σ and + Π state potential curves and the coupling of this state was due to the matrix element of the spin– orbit interaction. Using data for relevant electronic potential curves and transition dipole moments existing in the literature, the spectra of the molecular of , Cs 2, and KCs molecules were calculated. Full quantum mechanical and semi-quantum coupled channel calculations were done and compared with their non-coherent adiabatic or diabatic approximations. Through the comparison of our theoretical and experimental spectra, we identified all observed spectral features and determined the atoms’ number density and gas temperature.


Abstract:
In this study, we analyzed the light absorption by diatomic molecules or colliding atoms in a spectral region dominated by an avoided crossing of adiabatic state levels or crossing of the corresponding diabatic state levels. Our attention was focused on the low-resolution spectrum at a higher gas temperature under local thermodynamic equilibrium conditions. The absorption measurements of mixed vapors of potassium (≈80%) and cesium (≈20%) were made in the temperature range of 542-715 K and the infrared spectral range 900-1250 nm. In this area, the main spectral contributions were the broad

Introduction
Several approaches to the theoretical simulation of low-resolution, high-temperature optical spectra of diatomic molecules under local thermodynamic equilibrium conditions have been analyzed in detail by Beuc et al. [1]. Using relevant data for molecular potential curves and transition dipole moments, theoretical simulations can be powerful tools for identifying spectral features, gas temperatures, and atom number densities. Whereas in Beuc et al. [1], the optical transition between two well-isolated adiabatic states is analyzed, in this paper, the photon absorption from the lower adiabatic state into two excited non-radiative coupled electronic states is studied.
Many years ago, Devdariani et al. [2][3][4] and O'Callaghan et al. [5] studied this problem within semiclassical theory using a Landau-Zener approximation [6,7]. Although their analysis focused on the absorption and emission of light by colliding atoms, their conclusions can be qualitatively applied when interpreting quantum mechanical calculations of bound-bound molecular transitions.
The focus of this study was the spectral phenomenon that exists for all alkali metal homo-and hetero-nuclear molecular spectra. On the red side of the first resonant atomic transition doublet, the spectrum is dominated by molecular of the spin-orbit (SO) interaction, which has as a consequence the perturbation of the A X − spectrum. For several decades, low-resolution spectra of the A X − band at high temperatures have been widely experimentally and theoretically investigated, for example: Li2 [8][9][10][11], Na2 [1,8,11,12], K2 [11][12][13][14][15], Rb2 [16], and Cs2 [1,5,17,18]. The influence of   Π Ω state components whose angular moments Ω differ by 1 are included. The theoretical analysis in this study was focused on the low-resolution spectrum; therefore, this type of coupling can be neglected.
To our knowledge, there is no study of the high-temperature low-resolution A X − spectra of heteronuclear dimers in the literature. We measured the absorption spectrum of a potassium and cesium vapor mixture at temperatures of 542-715 K. In the infrared spectra from 900-1250 nm, the dominant contributions were the A X − bands of K2, Cs2, and KCs dimers. Using existing relevant potential curves and transition dipole moments obtained by quantum chemical calculations and the analysis of spectroscopic data, we calculated theoretical spectra in the measured wavelength range. We compared coupled channel quantum mechanical spectra calculations with coupled channel semiquantum calculations and showed that a time-efficient semi-quantum approach yielded results comparable to a full quantum approach. Using semi-classical theory [2][3][4][5], we defined the conditions under which coupled channel calculations can be approximated by non-coherent adiabatic or diabatic approximations. By comparing the experimental absorption coefficients with the theoretical simulations, we determined the temperature and potassium and cesium number densities in the gas mixture. The numerically time-efficient semi-quantum coupled channel calculus gave satisfactory good lowresolution spectra for K2, Cs2, and KCs dimers and can be used as an efficient alkali gas diagnostic tool.

Theoretical Background and Methods
The total Hamiltonian of a diatomic molecule is is the nuclei kinetic energy operator and µ is the molecular reduced mass.
which is represented as a sum of the product of the wavefunctions ( ) where In a spherical coordinate system, Xij can be written as wavefunction changes related to the interatomic distance R and A ij Χ depends on changes related to the rotation of the molecule (for details, see Janev et al. [34]). The functions ( ) , i φ r R belong to a complete set of orthogonal normalized functions that can be chosen in various ways.
The adiabatic representation is a uniquely defined base of wavefunctions that diagonalize the electronic Hamiltonian In the adiabatic representation, Equation (2) has the form: Potential curves ( ) i V R for adiabatic states of the same symmetry cannot intersect (Wigner-Neuman theorem) and they avoid crossing. In the region of avoided crossing, adiabatic states of the same symmetry are coupled by the matrix element R ij Χ (radial coupling). Note that the adiabatic or diabatic states whose electronic angular momentum Ω differ by 1 are coupled by the matrix element A ij Χ (angular coupling). In the region where one can neglect all matrix elements ij Χ (Born-Oppenheimer adiabatic approximation BOA), Equation (2) has the simple form: and the molecular wave function has the simple form Please note, in the case of molecules with large reduced masses, which is true for the dimers analyzed in this paper, the ( ) ii X R matrix elements are negligible.
The diabatic representation according to Lichten [35] is the base of electronic wavefunctions for which the matrix elements ij Χ are negligible ( 1 ij Χ << ) and the Equation (2) has the simple form The diabatic base for which = is called the radial diabatic base. Smith [36] has shown that the adiabatic and the radial diabatic base are connected through a unitary transformation. Unlike the uniquely defined adiabatic base, there is a class of radial diabatic bases that can be obtained using a unitary transformation (R independent) of one of them. In this study, we chose a diabatic base that diagonalized the electronic Hamiltonian at small interatomic distances.
The probability of an optical transition between the lower energy molecular state M ′′ and higher energy state M ′ is proportional to the square of the molecular dipole moment ( ) , r R D matrix element: where r R D r R is the electronic dipole moment for the transition between the adiabatic or diabatic states i and j.

One Excited Electronic State
If one assumes that the conditions for the Born-Oppenheimer approximation are satisfied, the wave function of the molecular state is given by one product of the nuclear and electronic wave function where ( ) ( ) ( ) is the Hönl-London factor. At thermodynamic equilibrium, the reduced absorption coefficient is [8,9,12]: is a statistical factor that depends on the symmetry of the electronic states and the temperature: where , A B S is the spin and   In each electronic state Λ , there is a finite number of bound and quasi-bound states with unitynormalized wavefunctions vJ ϕ Λ and an infinite continuum of free rovibrational states with energynormalized wavefunctions J ε φ Λ . Therefore, the sum over the rovibrational transitions in Equation (8) formally includes the integration over the bound-free, free-bound, and bound-bound transitions.
To calculate the energies and wavefunctions of rovibrational states, the FGH method (Fourier grid Hamiltonian) is used, where functions are represented on a finite number of grid points Ri (i = 1, …, N) [37]. On the grid of uniformly spaced points, where ( 1) In the above equation, the line profile and λ ∆ is equal to or larger than the line profile half-width and smaller than the instrumental profile half-width. Semi-quantum approximation (SQA) [1,15] give an even faster algorithm for calculating the lowresolution reduced absorption coefficient, which in the wavelength domain, has a form: where J is equal to the larger value of ′′ Λ and ′ Λ . This expression was formally obtained using a completely semi-classical procedure but was in a quantum-like form. An evaluation of the applicability, accuracy, and numerical efficiency of Equations (8), (11), (12), and (13) was extensively discussed in Beuc et al. [1]. One can describe the radial movement of atoms using a classical trajectory , where E is the energy of the molecule and ρ is the collision impact parameter. The wavefunction of the molecule in the electronic state ′ ′ Λ "dressed" [38] with photon frequency ν is The absorption cross-section of the optical transition between the lover ( Eρ ′′ Λ ) and the upper ( E ρ ′ Λ ) molecular state is: At thermodynamic equilibrium, the reduced absorption coefficient is given by the averaged absorption cross-section over parameters of the statistical ensemble E and ρ : The non-coherent quasi-static approximation is done using the first-order stationary phase approximation of the time-dependent integral in Equation (15), partial integration over ρ , and neglecting the rapidly oscillating terms [1,39]: , and the summation is over all real This approximation gives a good description of the spectra but diverges at the difference potential extremes, which are most often the consequence of the avoided crossing of the adiabatic electronic states' potential curves.
In the case where the difference potential has only one extreme point e R , a coherent uniform Airy approximation of the spectral profile is defined [1,39]. In the classically allowed region, the reduced absorption coefficient has the form: ∫ are integrals of the square of the Airy function and its first derivative, respectively. The uniform Airy approximation can be extended to the case where the transition difference potential has several extremes [1].

Two Coupled Excited Electronic States
In this paragraph, an optical transition from the ground electronic state ′′ Λ satisfying BOA to two coupled electronic states 1 ′ Λ and 2 ′ Λ is analyzed. It is assumed that the electronic wavefunction of the excited states belonging to the radial diabatic base diagonalizes the electronic Hamiltonian at small interatomic distances. The electronic wavefunction and potential for the ground state are and the corresponding transition dipole moments are 1, 2 The potential curves The corresponding adiabatic states' wavefunctions There is a simple relationship between the diabatic state potential difference ( ) d R ∆ and the adiabatic state potential difference Difference potentials and the corresponding transition dipole moments for the transition between the ground state and excited adiabatic states are: In the neighborhood of the diabatic state potential curves' crossing point, the potential curves The energies ( 1) , , , It is assumed here that both electronic states 1,2 ′ Λ have the same electronic angular moment ′ Λ . The coupling matrix element in the diabatic representation is ( ) , and in the adiabatic To solve the coupled channel Equation (21) numerically, it is suitable to use the diabatic representation and the FGH method [40]. Energies and radial wavefunctions can be determined via diagonalization of the 2N × 2N Hamiltonian matrix: (10) and The absorption cross-section for the transition from a rovibrational state ( , , ) v J ′′ ′′ ′′ Λ of the ground electronic state to the rovibrational states ( , , ) v J ′ ′ ′ Λ of the coupled excited electronic states is: where the transition energy is The reduced absorption coefficient for the transition from the lower electronic state ′′ Λ to the excited coupled electronic states in the adiabatic or diabatic representation can be written as: This is a full quantum mechanical coupled channel approach (QCC) in the adiabatic or diabatic representation. The first contribution represents an optical transition to the electronic state 1 ′ Λ , the second contribution represents the transition to the electronic state 2 ′ Λ , and third contribution represents their interference. ( 1) , , , where each contribution in Equation (26) can be calculated using Equations (8) or (12)

The Landau-Zener Model
To analyze the essential properties of the absorption spectrum for transitions in the neighborhood of the crossing of diabatic states' potential curves, simplifications inherent to the Landau-Zener model have been made. In the neighborhood of the crossing point c R , the potentials of the excited diabatic states can be approximated: , and for the potential of the ground electronic state, The transition difference potentials are Using the same approximation, the difference potentials for the transition between the ground and the excited adiabatic states and the corresponding transition dipole moments are: This is the non-coherent semi-classical diabatic approximation (SCDA) of the spectra, which is valid in both cases 1 2 0 α α > and 1 2 0 α α < .
In the case 1 2 0 α α > , a non-coherent sum of the semi-classical absorption coefficient for the transition to the excited adiabatic states' semi-classical adiabatic approximation (SCAA) has the form SCAA in the case 1 2 0 α α < can be obtained using the uniform Airy approximation (Equation (16)) [41]: The absorption coefficient is a function of the dimensionless reduced frequency (energy) ( ) Devdariani and coworkers [2][3][4] and O'Callaghan et al. [5] have studied the influence of nonadiabatic electronic states mixing on the shape of the spectrum within semi-classical atomic collision theory. The atomic collision can be divided into two half-collisions: the first refers to the motion from There are two scenarios involving a collision process: the initial conditions of the first halfcollision are 1 The cross-section for the transition between the lower molecular state ′′ Λ dressed with photon hν to the higher state ′ Λ for each half-collision is: Both groups of authors have calculated the spectral contribution for each half-collision and both scenarios of the initial conditions. After non-coherent summation of all contributions and averaging over the statistical ensemble at thermodynamic equilibrium, they obtained the relation for the absorption coefficient:  L ξ Ω but this is less important since it is very common that a scalar product 1 2 d d is small or equal to zero and the same holds for the second contribution in Equation (33). From Figure  1 it can be concluded that SCCC can be approximated for 0.1 ξ  using SCDA and for 0.2 ξ > using SCAA. We can generalize the conclusion to any case of diabatic electronic states' potential crossing, as long as this crossing is isolated from other crossings and the condition satisfied, and the Massey parameter is: The criteria of applicability for SCAA and SCDA obtained in semi-classical theory can be applied to both quantum mechanical and semi-quantum approaches.

Experiment
A T-type all-sapphire cell (ASC) of 4 cm in length and 1 cm in inner diameter was used for the transmission measurements in the K-Cs mixture vapor. The mixture in the sealed cell was 80% potassium and 20% cesium. The number density of atoms and molecules in the cell depends on the temperature at the tip of the side-arm finger (Tf) and the cell body (Tc). Tc was higher than Tf (at least 30 K) to prevent the condensation of the vapors on the inner side of the sapphire cell. The temperatures were controlled using two Chromel-Alumel Thermocouples.
In Figure 2a, we present the transmission intensities ( , ) for several cells' Tc and finger Tf temperatures in the infrared spectral region between 900 nm and 1250 nm.  Figure 2b. Assuming the local thermodynamic equilibrium at an effective temperature T, the linear absorption coefficient of the mixture of potassium and cesium vapor for a low-pressure binary approximation was:  Figure 2a,b, it was necessary to determine the reduced absorption coefficients of the K2, Cs2, and KCs molecules in the near-infrared part of the spectrum.

Near-Infrared Spectra of K2, KCs, and Cs2 Molecules
The dominant contribution to the near-infrared spectrum of alkali dimers is the A X − band. This molecular band is the result of optical transitions between all rovibrational states of the excited a + Π − Σ but its contribution in the spectral region of interest is negligible and will not be analyzed in this study.
Most quantum-chemical ab-initio calculations are usually non-relativistic calculations in which the electronic Hamiltonian does not include SO interactions. As a result of these calculations, the electronic wavefunctions in Hund`s case (a) are the representation and the corresponding electronic potentials, which are Using the diabatic states' wavefunctions and potentials, one can construct the corresponding adiabatic ( α and β ) state potentials and wavefunctions by diagonalizing matrix transition is dipole-forbidden, the adiabatic transition dipole moment X α − and X β − transitions have the simple form: To obtain the electronic potential curves and transition dipole moments required in our calculations, we used existing theoretical and experimental data from the literature. In the case of the K2 molecule, to construct potential curves, we combined ab-initio results [43], experimental data [19][20][21], and long-range region analytical results [44]. The spin-orbit function ( ) fs R δ is taken from Manaa et al. [21] and the transition dipole moment was from Yan and Meyer [43]. For the KCs molecule, all data were taken from the Supplementary Materials of Borsalino et al. [33]. In the case of the Cs2 molecule, for the diabatic state potentials, we used ab-initio results [45] (for states with 6S + 6P asymptote energies was shifted by +14 cm −1 ) and experimental data [24,25]. In the long-range region, the potential curves were smoothly matched with analytical curves [44]. The function ( ) fs R δ was taken from Bai et al. [25] and the transition dipole moment was from Allouche and Aubert-Frécon [46].
All relevant potential curves for K2, KCs, and Cs2 molecules are shown in Figure 3. It can be seen that the SO splitting function ( ) fs R δ in the K2 molecule was approximately ten times smaller than in the case of the KCs and Cs2 molecules. The diabatic states' potential curves ( ) V R had two well-separated crossing points for each molecule: K2 (9.0 Bohr, 46.3 Bohr), KCs (9.6 Bohr, 20.9 Bohr), and Cs2 (10.9 Bohr, 24.1 Bohr). In the same region, the potential curves of the adiabatic states avoided crossing. At the long-range crossing point, respectively. According to the discussion in Section 2.3., in the neighborhood of the diabatic state potentials' crossing point, the adiabatic state difference potentials were monotonic functions and the absorption spectra could be found using a non-coherent adiabatic approximation. Figure 4a-  At the short-range crossing point c R , the Massey parameter ξ at a typical experimental temperature 700 K is given in Table 1 for all dimers. The A b − coupling for the K2 dimer was much smaller than for Cs2 even for the KCs dimer. In Table 1, the probability of atomic motion in diabatic potential ( ) P T and the probability of atomic motion in adiabatic potential 1 ( ) P T − is given. It is evident that at the temperature of 700 K, in the K2 molecule, the atoms mainly moved in a diabatic potential, and for the KCs and especially Cs2 molecules, movement was in an adiabatic potential. The behavior of the rotational B ν constants also indicated the influence of the mixing of the two electronic states. The energy-dependent B ν constants for A , b , α , and β , as well as the coupled A and b states for all dimers, are shown in Figure 5a-c. For the K2 molecule, the coupled states B ν constants were grouped around the diabatic states constants, but in the case of KCs and especially Cs2 molecules, they were grouped around the adiabatic states B ν constants. These facts are consistent with the Massey parameter shown in Table 1. Using the full quantum (semi-quantum) approach, we calculated the reduced absorption coefficients for transitions between the X state and the coupled A and b states QCC (SQCC), noncoherent adiabatic approximation QAA (SQAA), and non-coherent diabatic approximation QDA (SQDA) of this transition. The rovibrational energies and radial wavefunctions were calculated using the FGH method via diagonalization of Hamiltonian matrices (Equations (10) and (22)). The number of grid points was N = 800 in the quantum mechanical calculation and N = 2000 in the semi-quantum calculation.
Using Equations (12) and (13), the rovibrational transition contributions were collected in bins of 0.4 nm and the spectra were smoothed using a Gaussian with a half-width of 0.9 nm for the quantum approach and 3.0 nm for the semi-quantum approach. Summation over the rotational quantum number J in the quantum approach was replaced by summation over the groups of n = 3 neighbor J values.
The numerical evaluation in this study was done using the Wolfram Mathematica 12.1 computing system. To calculate all the rovibrational contributions needed for the SQAA and SQDA spectra, 27 s of computer time was required, and for SQCC, 48 s was required. Using these data to calculate the absorption coefficient (Equation (13)) at a given temperature, 1 s of computer time was required for SQAA and SQCC, and 0.5 s was required for the SQDA spectra. For the full quantum mechanical approach, 340 s of computer time was required to calculate all the rovibrational contributions of the QAA and QDA spectra, and 830 s was required to calculate the QCC spectra. For the absorption coefficient evaluation at a given temperature, 70 s was required for the QCC and QAA spectra, and 38 s was required for the QDA spectra.
The reduced absorption coefficients at 700 K obtained using different approaches are shown in Figure 6. Each column in Figure 6 shows the different theoretical approaches for each of the dimers (K2, KCs, Cs2). The first row of Figure 6 shows the QCC and SQCC spectra, the second row shows the QAA and SQAA spectra, while the third row shows the QDA and SQDA spectra. It is noticeable that the non-coherent diabatic approximation (QDA, SQDA) gave an almost identical result as the coupled channel calculation (QCC, SQCC) in the case of K2. In contrast, the non-coherent adiabatic approximation (QAA, SQAA) gave very similar results to the coupled channel calculation (QCC, SQCC) in the case of KCs and an almost identical result in the case of the Cs2 molecule. Furthermore, excellent agreement of the semi quantum approximation (SQCC, SQAA, SQDA) with the full quantum calculation (QCC, QAA, QDA) was evident, especially in the case of the Cs2 molecule. are shown in the first (a,d,g), second (b,e,h), and third columns (c,f,i), respectively. The first row (Figure 6a,b,c) shows the QCC and SQCC spectra, the second row (d,e,f) shows the QAA and SQAA spectra, and the last row (g,h,i) shows the QDA and SQDA spectra.

The Comparison of the Experimental and Theoretical Absorption Coefficient
The absorption coefficient is temperature dependent, especially in the case of bound-bound transitions. Figure 7a-c shows the theoretical absorption coefficients of A X − transition calculated using the QCC approach at several experimental temperatures for K2, KCs, and Cs2 molecules. Using Equation (36), we aimed to obtain the best fit of the experimental spectrum and theoretical simulation by iteratively changing the following parameters: temperature T, potassium atom number density NK, and cesium atoms number density NCs. The iterative procedure started with the experimental temperature but the best agreement was obtained at the end of the iteration with a temperature that was about 20 K higher than the initial one. Theoretical simulations were done for two temperatures, as shown in Figure 7d In Figure 7d,e, the spectral contributions of K2, KCs, and Cs2 molecules calculated using the QCC approach are shown in magenta, green, and blue, respectively. By comparing the experimental and theoretical spectra, all important features in the experimental spectrum were identified. The peak at 1048.5 nm was related to the minimum of the K2 X α − transition difference potential at 1055 nm. The shoulder at 1068 nm was related to the minimum of the KCs X α − transition difference potential at 1072 nm. The broad oscillating structure around 1106 nm was related to the maximum of the KCs X β − transition difference potential at 1107 nm, and the peak at 1189 nm was related to the minimum at 1192 nm. The peak at 1208 nm was related to the minimum of the Cs2 X β − transition difference potential at 1205 nm.

Discussion and Conclusions
We studied the influence of the spin-orbit coupling of two excited diabatic electronic states of the same symmetry whose potential curves intersected on the high-temperature spectrum of the optical transition. In the neighborhood of the diabatic state potential curves' crossing point, the potential curves of the corresponding adiabatic states of the same symmetry avoided crossing and these states were mixed via the matrix element of radial non-adiabatic coupling. To obtain the correct spectrum of transitions from the isolated ground electronic state to the excited coupled diabatic or adiabatic states, the energies and wave functions of the excited states needed to be determined using a coupled channel calculus (QCC).
We distinguished two cases of the spectrum. In the neighborhood of an isolated diabatic state potential curves' crossing point, the transition difference potential curves of the excited adiabatic states were monotonic in the first case and had a minimum and a maximum in the second case. In the first case, based on the semi-classical analysis in Section 2.3, we concluded that a non-coherent adiabatic approach was a satisfactory approximation of the coupled channel calculation. Based on the discussion in Section 2.3, for the second case, we concluded that if the Massey parameter (Equation (35)) satisfied 0.1 ξ << , a non-coherent diabatic approach was a correct description of the spectrum, and if 0.2 ξ > , a non-coherent adiabatic approach was justified. The potentials of the adiabatic electronic states are often available in the literature but there is no data on the corresponding diabatic state potentials and the spin-orbit mixing of these states. Assuming that the Landau-Zener approximation is valid in the neighborhood where the adiabatic state potentials avoided crossing, the Massey parameter can be written: If only the adiabatic state potential curves are available, by using Equation (38), one can assess whether the non-coherent adiabatic approximation gives a satisfactory theoretical simulation of the spectrum. The semi-quantum approach is extensively studied in References [1,15,18], and in this paper, we analyzed its applicability in coupled channel calculations. As discussed in References [1,15], the semiquantum approximation describes the vibrational structure of the molecular band well but neglects the rotational structure. In the high-temperature, low-resolution spectra of molecules with a larger reduced mass, such as K2, KCs, and Cs2, the rotational structure was not resolved and the semiquantum coupled channel approach had a very good agreement with the quantum coupled channel calculation.
Due to the low computer time consumption, a semi-quantum approach was found to be an appropriate tool for gas diagnostics. In our experimental study of a potassium and cesium vapor mixture absorption spectrum, we used the semi-quantum method to determine the atom's number density and temperature. The alkali mixture investigated in this article was suitable for analyzing the influence of the diabatic state SO coupling on the A-X transition spectrum. In the case of the K2 molecule, due to the small SO interaction, the Massey parameter was small with 0.0051 ξ = , and the non-coherent diabatic approximation described the spectrum well. In contrast, in the case of the Cs2 molecule, the SO interaction and the Massey parameter were large with 0.75 ξ = , and the spectrum could be calculated well using a non-coherent adiabatic approximation. Low-resolution spectroscopy is applicable in the plasma diagnostics but can also be useful in the study of some fundamental phenomena and processes in the alkali gas. The experimental and theoretical methods presented here should also be applicable in the analysis of the SO interaction influence on the A-X transition spectra in the case of other heteronuclear alkali molecules, such as NaCs and RbCs.