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

^{*}

## Abstract

**:**

_{2}, Cs

_{2,}and KCs molecules. There was a crossing of ${A}^{1}{\Sigma}_{\left(u\right)}^{+}\left({0}_{\left(u\right)}^{+}\right)$ and ${b}^{3}{\Pi}_{\left(u\right)}\left({0}_{\left(u\right)}^{+}\right)$ state potential curves and the coupling of this state was due to the matrix element $\u27e8{A}^{1}{\Sigma}_{\left(u\right)}^{+}\left({0}_{\left(u\right)}^{+}\right)|{H}_{so}|{b}^{3}{\Pi}_{\left(u\right)}\left({0}_{\left(u\right)}^{+}\right)\u27e9$ of the spin–orbit interaction. Using data for relevant electronic potential curves and transition dipole moments existing in the literature, the spectra of the ${A}^{1}{\Sigma}_{\left(u\right)}^{+}\left({0}_{\left(u\right)}^{+}\right)-{X}^{1}{\Sigma}_{\left(g\right)}^{+}\left({0}_{\left(g\right)}^{+}\right)$ molecular bands of K

_{2}, 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.

## 1. Introduction

_{2}[8,9,10,11], Na

_{2}[1,8,11,12], K

_{2}[11,12,13,14,15], Rb

_{2}[16], and Cs

_{2}[1,5,17,18]. The influence of ${A}^{1}{\Sigma}_{\left(u\right)}^{+}\left({0}_{\left(u\right)}^{+}\right)-{b}^{3}{\Pi}_{\left(u\right)}\left({0}_{\left(u\right)}^{+}\right)$ coupling is directly observed only in the case of the heavy alkali dimers Rb

_{2}and Cs

_{2}due to their large SO interaction.

_{2}[19,20,21], Rb

_{2}[22,23], and Cs

_{2}[24,25], and heteronuclear dimers LiCs [26,27], NaCs [28], RbCs [29,30], and KCs [31,32,33]. In these theoretical analyses of spectroscopic data, the rotational couplings of ${b}^{3}{\Pi}_{\left(u\right)}\left({\Omega}_{\left(u\right)}^{(+,-)}\right)$ state components whose angular moments $\Omega $ 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.

_{2}, Cs

_{2}, 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 semi-quantum 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.

_{2}, Cs

_{2}, and KCs dimers and can be used as an efficient alkali gas diagnostic tool.

## 2. Theoretical Background and Methods

**r**. The eigenvector of the Hamiltonian $H\left(r,R\right)$ is the function:

_{ij}can be written as ${X}_{ij}={X}_{ij}^{R}+{X}_{ij}^{A}$, where ${X}_{ij}^{R}$ depends on electronic wavefunction changes related to the interatomic distance $R$ and ${X}_{ij}^{A}$ depends on changes related to the rotation of the molecule (for details, see Janev et al. [34]). The functions ${\varphi}_{i}\left(r,R\right)$ belong to a complete set of orthogonal normalized functions that can be chosen in various ways.

#### 2.1. One Excited Electronic State

_{i}(i = 1, …, N) [37]. On the grid of uniformly spaced points, where $\delta R={R}_{i+1}-{R}_{i}$ for the electronic state S with an electronic angular moment $L$ and electronic potential ${V}_{P}$, the N × N Hamiltonian matrix is defined as:

#### 2.2. Two Coupled Excited Electronic States

#### 2.3. The Landau–Zener Model

## 3. Results

#### 3.1. Experiment

_{f}) and the cell body (T

_{c}). T

_{c}was higher than T

_{f}(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.

_{c}and finger T

_{f}temperatures in the infrared spectral region between 900 nm and 1250 nm.

_{2}, Cs

_{2}, and KCs molecules, respectively, and ${N}_{K}(T)$ and ${N}_{Cs}(T)$ are the number densities of K and Cs atoms, respectively. To analyze the spectra shown in Figure 2a,b, it was necessary to determine the reduced absorption coefficients of the K

_{2}, Cs

_{2,}and KCs molecules in the near-infrared part of the spectrum.

#### 3.2. Near-Infrared Spectra of K_{2}, KCs, and Cs_{2} Molecules

^{−1}, cesium 554.0388 cm

^{−1}). In References [20,21,22,23,24,25,26,27,28,29,30,31,32,33], the functions ${\delta}_{3\Pi}(R)$ and ${\delta}_{fs}(R)$ are fit to the Morse potential form and differ slightly at small interatomic distances. For the sake of simplicity, in this study, we used the approximation ${\delta}_{3\Pi}(R)={\delta}_{fs}(R)$.

_{2}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 ${\delta}_{fs}(R)$ 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 Cs

_{2}molecule, for the diabatic state potentials, we used ab-initio results [45,46] (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 ${\delta}_{fs}(R)$ was taken from Bai et al. [25] and the transition dipole moment was from Allouche and Aubert-Frécon [47].

_{2}, KCs, and Cs

_{2}molecules are shown in Figure 3. It can be seen that the SO splitting function ${\delta}_{fs}(R)$ in the K

_{2}molecule was approximately ten times smaller than in the case of the KCs and Cs

_{2}molecules. The diabatic states’ potential curves ${V}_{A}(R)$ and ${V}_{b}(R)$ had two well-separated crossing points for each molecule: K

_{2}(9.0 Bohr, 46.3 Bohr), KCs (9.6 Bohr, 20.9 Bohr), and Cs

_{2}(10.9 Bohr, 24.1 Bohr). In the same region, the potential curves of the adiabatic states avoided crossing.

_{2}, KCs, and Cs

_{2}molecules, respectively. At the short-range crossing point, ${\scriptscriptstyle \frac{d}{dR}}{\Delta}_{A}\left(R\right){\scriptscriptstyle \frac{d}{dR}}{\Delta}_{b}{\left(R\right)}_{R={R}_{c}}<0$ and the difference potentials of the $\alpha -X$ and $\beta -X$ transitions had extrema in the neighborhood of the crossing point. As pointed out in Section 2.1 and Equation (16), the extremes of the difference potentials indicate the positions of the satellite rainbow in the spectrum. The difference potential of $\alpha -X$ had one minimum each at 985 nm, 1072 nm, and 1126 nm for the K

_{2}, KCs, and Cs

_{2}molecules, respectively. The $\beta -X$ transition difference potential had maxima and minima at (996 nm, 1055 nm), (1107 nm, 1192 nm), and (1197 nm, 1205 nm) for K

_{2}, KCs, and Cs

_{2}, respectively. The minima of the $X-\beta $ transition difference potential approximately coincided with the minima of $A-X$ transition difference potential. Figure 4d–f shows the relevant transitions dipole moments. In the neighborhood of the diabatic potentials’ crossing point, the dipole moments of the $\alpha -X$ and $\beta -X$ transitions changed significantly with the interatomic distance, especially in the case of the K

_{2}dimer.

_{2}dimer was much smaller than for Cs

_{2}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 K

_{2}molecule, the atoms mainly moved in a diabatic potential, and for the KCs and especially Cs

_{2}molecules, movement was in an adiabatic potential.

_{2}molecule, the coupled states ${B}_{\nu}$ constants were grouped around the diabatic states constants, but in the case of KCs and especially Cs

_{2}molecules, they were grouped around the adiabatic states ${B}_{\nu}$ constants. These facts are consistent with the Massey parameter shown in Table 1.

_{2}, KCs, Cs

_{2}). 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 K

_{2}. 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 Cs

_{2}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 Cs

_{2}molecule.

#### 3.3. The Comparison of the Experimental and Theoretical Absorption Coefficient

_{2}, KCs, and Cs

_{2}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 N

_{K}, and cesium atoms number density N

_{Cs}. 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,e. The sapphire cell was filled with potassium and cesium in an approximate ratio of ${N}_{K}/{N}_{Cs}=4.0$. Figure 7d,e show the results obtained for ${N}_{K}/{N}_{Cs}=4.0$ and ${N}_{K}/{N}_{Cs}=3.7$, respectively.

_{2,}KCs, and Cs

_{2}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 K

_{2}$\alpha -X$ transition difference potential at 1055 nm. The shoulder at 1068 nm was related to the minimum of the KCs $\alpha -X$ transition difference potential at 1072 nm. The broad oscillating structure around 1106 nm was related to the maximum of the KCs $\beta -X$ 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 Cs

_{2}$\beta -X$ transition difference potential at 1205 nm.

## 4. Discussion and Conclusions

_{2}, KCs, and Cs

_{2}, the rotational structure was not resolved and the semi-quantum coupled channel approach had a very good agreement with the quantum coupled channel calculation.

_{2}molecule, due to the small SO interaction, the Massey parameter was small with $\xi =0.0051$, and the non-coherent diabatic approximation described the spectrum well. In contrast, in the case of the Cs

_{2}molecule, the SO interaction and the Massey parameter were large with $\xi =0.75$, and the spectrum could be calculated well using a non-coherent adiabatic approximation.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Absorption spectra contributions ${L}_{d}(\xi ,\Omega )$ and ${L}_{\mathrm{int}}(\xi ,\Omega )$ as a function of the Massey parameter $\xi $ and reduced frequency $\Omega $ are shown in (

**a**,

**c**). The differences between these contributions and their adiabatic approximations ${L}_{d}^{a}(\xi ,\Omega )-{L}_{d}(\xi ,\Omega )$ and ${L}_{\mathrm{int}}^{a}(\xi ,\Omega )-{L}_{\mathrm{int}}(\xi ,\Omega )$ are shown in (

**b**,

**d**).

**Figure 2.**Relative intensity (in arbitrary units) of the light transmitted through a sapphire absorption cell at several temperatures ${T}_{c,f}$ is shown in (

**a**). The alkali mixture absorption coefficients for the corresponding temperatures are shown in (

**b**).

**Figure 3.**The electronic states’ potentials curves relevant for the analysis of the infrared spectra of K

_{2}, KCs, and Cs

_{2}molecules are shown in (

**a**–

**c**), respectively. The green curve shows the spin–orbit splitting function ${\delta}_{fs}(R)$ responsible for the coupling of the A and b states.

**Figure 4.**Difference potential curves of the A-X, b-X, α-X, and β-X transitions for K

_{2}, KCs, and Cs

_{2}molecules are shown in (

**a**–

**c**), respectively. The corresponding transition dipole moments are shown in (

**d**–

**f**). The labels on the left side of the upper panel denote energy, and the right-side denotes the wavelength of the transition.

**Figure 5.**Constants of A, b, $\alpha $, and $\beta $, as well as the coupled A and b electronic states, in the case of K

_{2}, KCs, and Cs

_{2}molecules are shown in (

**a**–

**c**), respectively.

**Figure 6.**Reduced absorption coefficients of K

_{2}, KCs, and Cs

_{2}molecules at temperature $T=700K$ are shown in the first (

**a**,

**d**,

**g**), second (

**b**,

**e**,

**h**), and third columns (

**c**,

**f**,

**i**), respectively. The first row (

**a**–

**c**) shows the QCC and SQCC spectra, the second row (

**d**–

**f**) shows the QAA and SQAA spectra, and the last row (

**g**–

**i**) shows the QDA and SQDA spectra.

**Figure 7.**The first column shows the reduced absorption coefficients of K

_{2}, KCs, and Cs

_{2}molecules (

**a**–

**c**, respectively) in the temperature range T = 550 K–750 K. The comparison of the experimental absorption coefficient (black curve) with the theoretical simulation (red curve) is in the second column (

**d**,

**e**). The theoretical simulations of the contributions of KCs, Cs

_{2}, and K

_{2}molecules are shown in green, blue, and magenta, respectively.

**Table 1.**The first row gives the crossing points ${R}_{c}$ of the potential curves of electronic states A and b for K

_{2}, KCs, and Cs

_{2}molecules. In the second row, the Massey parameters at the crossing points for the temperature T = 700 K are given. In the third and fourth rows, the averaged probabilities P of atoms moving in diabatic potentials and the probabilities for atoms moving in adiabatic electronic potentials are given, respectively.

K_{2} | KCs | Cs_{2} | |
---|---|---|---|

${R}_{c}$ | 9.0 | 9.57 | 10.87 |

$\xi (700\text{}K)$ | $0.0051$ | 0.21 | 0.75 |

$P(700\text{}K)$ | 0.87 | 0.12 | 0.0056 |

$1-P(700\text{}K)$ | 0.13 | 0.88 | 0.9944 |

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Beuc, R.; Pichler, G.
High-Temperature Optical Spectra of Diatomic Molecules: Influence of the Avoided Level Crossing. *Atoms* **2020**, *8*, 28.
https://doi.org/10.3390/atoms8020028

**AMA Style**

Beuc R, Pichler G.
High-Temperature Optical Spectra of Diatomic Molecules: Influence of the Avoided Level Crossing. *Atoms*. 2020; 8(2):28.
https://doi.org/10.3390/atoms8020028

**Chicago/Turabian Style**

Beuc, Robert, and Goran Pichler.
2020. "High-Temperature Optical Spectra of Diatomic Molecules: Influence of the Avoided Level Crossing" *Atoms* 8, no. 2: 28.
https://doi.org/10.3390/atoms8020028