Zhenlu
Hou
ab and
Linhua
Liu
*b
aState Nuclear Power Demonstration Plant Co. Ltd, Rongcheng City, Shandong Province 264312, People's Republic of China
bSchool of Energy and Power Engineering, Shandong University, Jinan, 250061, People's Republic of China. E-mail: liulinhua@sdu.edu.cn
First published on 6th December 2024
CH is one of the most spectroscopically studied diatomic molecules. The rovibronic spectra of the methylidyne radical (CH) in adiabatic and diabatic representations are obtained based on ab initio data, including 12 potential energy curves, 38 dipole moment curves, 79 spin–orbit coupling curves, and 18 electronic angular momentum coupling curves. We employed the internally contracted multireference configuration interaction method including the Davidson correction with the aug-cc-pV(5+d)Z basis set for the C atom and the aug-cc-pV5Z basis set for the H atom. The diabatic transformations are performed based on a property-based diabatisation method to remove the avoided crossings for the E 2Π–F 2Π and F 2Π–H 2Π pairs. The coupled nuclear motion Schrödinger equations are then solved using the Duo nuclear motion program to obtain the rovibronic spectra of CH for wavenumbers from 0 to 80000 cm−1 at 5000 K. An overall prediction of the rovibronic spectra of CH is provided in this work. Our results could be beneficial for future calculations of rovibronic spectra of CH and contribute to improving astronomical, chemical, and physical models.
Molecular spectroscopic data can be obtained by both experimental measurements and theoretical calculations. Previous experimental studies of CH are mainly concerned with transitions between the three low-lying electronic states. The spectra of A 2Δ–X 2Π, B 2Σ−–X 2Π, and C 2Σ+–X 2Π transitions were studied and analyzed by Gerö19 in 1941. Ubachs et al.20 measured the absorption spectra of the C 2Σ+–X 2Π transition of the CH molecule in 1986 by laser-induced fluorescence experiments. In 1996, Kepa et al.21 used conventional spectroscopy to record and analyze the spectral constants of the five (0-0, 0-1, 0-2, 1-0, and 1-1) bands of the B 2Σ−–X 2Π transition of the CH radical using conventional spectroscopy and obtained more precise molecular constants. Bembenek et al.22 obtained emission spectra of the C 2Σ+–X 2Π transition using a water-cooled Geissler-type tube in 1997. Medcraft et al.23 measured in 2019 in a supersonically expanding planar hydrocarbon plasma using a cavity-decay method for the C 2Σ+–X 2Π transition of CH rotationally resolved spectra of the electron leaps of CH and determined the equilibrium bond lengths of the C 2Σ+ state more precisely. However, due to factors such as experimental conditions, measurement environments, and experimental costs, it is difficult to study the high-lying electronic states of molecules in high-temperature environments via experimental measurements. Therefore, theoretical calculations have been widely used and the problems mentioned above can be alleviated using high-quality ab initio data.24 With the development of ab initio calculation methods, not only can the potential energy curves (PECs) of the high-lying electronic states and the coupling effects between different electronic states be obtained without the aid of any semiempirical and experimental parameters, but also the accuracy of the calculation results can be guaranteed. For example, Xiao et al.25,26 used the ab initio method to obtain the electronic structure parameters of the SH+ and PSe molecules and determined the spectroscopic constants of both of their bound states by taking into account the coupling effects between electronic states. Therefore, the ab initio data can be used as a baseline in the absence of spectroscopic experimental data.27
With the increasing demand for computational accuracy, more attention has been paid to the influence of non-adiabatic effects as well as coupling effects. The Born–Oppenheimer approximation for simplified calculations, which are basic approximations of the ab initio calculation method, has some impact on the accuracy of ab initio calculations. The Born–Oppenheimer approximation may lead to the occurrence of avoided crossings between electronic states which are spatially degenerate in energy at some internuclear distances. The phenomenon above may cause the subsequent theoretical calculations to differ significantly from the experimental results. Moreover, previous studies often ignore the coupling effects between electronic states, such as spin–orbit couplings (SOCs) and electronic angular momentum couplings (EAMCs), during the calculation of molecular spectroscopic data.6 For example, Brady et al.27 used an ab initio method to obtain the electronic structure parameters of the SO molecule and determined the absorption spectra generated by the bound-bound transitions, taking into account the coupling effects and nonadiabatic effects. The results show that the influence of both effects can make a significant difference in the obtained results. Semenov et al.28 obtained the rovibronic absorption spectra of PN based on ab initio data, including 9 PECs, 14 SOCs, 7 EAMCs, 9 permanent dipole moments (PDMs), and 8 transition dipole moments (TDMs).
Given the above consideration, this work aims to use the state-of-the-art ab initio method to calculate the electronic structures of CH, including the higher-lying electronic states. Duo29,30 and ExoCross31 are used to obtain the rovibronic spectra of CH with the consideration of the coupling effects between electronic states and the non-adiabatic effects. This work is organized as follows. Section 2 provides the theory and methods for computing the electronic structures of CH, as well as the diabatic transformation. Section 3 discusses the corresponding results. Finally, a conclusion is drawn in Section 4.
The transformation from the adiabatic to diabatic basis can be described by a unitary matrix U:
![]() | (1) |
![]() | (2) |
![]() | (3) |
The mixing angle β(R) should be equal to π/4 at the internuclear distance of the crossing point Rc and the path integral of non-adiabatic coupling (NAC) over each side of the crossing point should be π/4 (and the overall integral should be π/2). Therefore, the non-adiabatic coupling function is normalized as follows:
![]() | (4) |
The two-state electronic Hamiltonian can be expressed from the adiabatic potential energy curves Va1(R) and Va2(R):
![]() | (5) |
Based on the unitary transformation U(β(R)), the diabatic Hamiltonian can be obtained:
![]() | (6) |
The NACs can be obtained from the electronic wave functions using quantum chemical methods or modeled using the Lorentzian function forms, and the latter can reasonably describe the ab initio NACs in the vicinity of the crossing points,48–51 given as follows:
![]() | (7) |
The aforementioned method, called ‘property-based diabatisation’,27,39 is employed to construct the diabatic PECs. Minimizing the second derivatives of diabatic PECs in the vicinity of the crossing point Rc can obtain the smoothest PECs Vd1(R) and Vd2(R).
![]() | (8) |
![]() | ||
Fig. 1 PECs of CH calculated by the icMRCI method with the aug-cc-pV(5+d)Z basis set for the C atom and the aug-cc-pV5Z basis set for the H atom in the adiabatic representations. |
Dissociation limit | Molecular electronic states |
---|---|
C(1P)–H(2S) | H 2Π |
C(3P)–H(2S) | F 2Π |
C(5S)–H(2S) | c 4Σ−, d 6Σ− |
C(1S)–H(2S) | D 2Σ+ |
C(1D)–H(2S) | C 2Σ+, E 2Π, A 2Δ |
C(3P)–H(2S) | X 2Π, B 2Σ−, b 4Π, a 4Σ− |
In order to verify the reliability of our ab initio data, the spectroscopic constants computed in the present work are compared with those of the previous studies17,52–55 and with the experimental values56–62 in Table 2. Our adiabatic excitation energy (Te) and equilibrium internuclear distances (re) for X 2Π, B 2Σ−, a 4Σ−, C 2Σ+, E 2Π and A 2Δ states are close to the experimental values and the previous calculations. There are notable differences between our dissociation energy (De) for the B 2Σ− and C 2Σ+ states between the experimental values56,58–60 and the theoretical calculation results,17,52,54 which are close to those computed by Cui et al.53 The Te for the D 2Σ+ and c 4Σ− states as well as the re for H 2Π state are different from other results,17,55 which may result from the different levels of computational methods and the different basis sets. Overall, our ab initio PECs of CH are reliable and can be used to calculate the corresponding spectroscopic studies.
State | Ref. | T e (cm−1) | D e (eV) | r e (Å) |
---|---|---|---|---|
X 2Π | This work | 0.0 | 3.6275 | 1.10 |
Expt.56,57 | 0.0 | 3.472 | 1.120 | |
Ref. 17 | 0.0 | 3.447 | 1.120 | |
Ref. 52 | — | 3.413 | 1.123 | |
Ref. 53 | 0.0 | 3.669 | 1.1181 | |
B 2Σ− | This work | 26524.76 | 0.4764 | 1.20 |
Expt.58,59 | 26059.52 | 0.305 | 1.164 | |
Ref. 17 | 26140.17 | 0.244 | 1.175 | |
Ref. 52 | 26204.69 | 0.209 | 1.181 | |
Ref. 53 | 26007.57 | 0.438 | 1.1624 | |
a 4Σ− | This work | 5960.573 | — | 1.089 |
Expt.56 | 5844 | — | 1.085 | |
Ref. 17 | 6024.902 | 2.676 | 1.090 | |
Ref. 52 | 5847.462 | 2.680 | 1.093 | |
Ref. 53 | 6245.28 | 2.894 | 1.0872 | |
C 2Σ+ | This work | 31883.50 | 1.0950 | 1.116 |
Expt.56,60 | 31802.13 | 0.769 | 1.114 | |
Ref. 17 | 32205.40 | 0.738 | 1.116 | |
Ref. 52 | 32184.7 | 0.749 | 1.122 | |
Ref. 53 | 31961.97 | 0.930 | 1.1155 | |
Ref. 54 | 32184.7 | 0.749 | 1.118 | |
E 2Π | This work | 59337.22 | — | 1.150 |
Expt.62 | 60394.77 | — | 1.15 | |
Ref. 17 | 59346.60 | — | 1.1437 | |
Ref. 55 | 59243.06 | — | 1.1463 | |
A 2Δ | This work | 23168.85 | 2.0145 | 1.100 |
Expt.57,61 | 23147.88 | 1.836 | 1.103 | |
Ref. 17 | 23395.07 | 1.801 | 1.106 | |
Ref. 52 | 23510.83 | 1.760 | 1.109 | |
Ref. 53 | 23280.16 | 2.024 | 1.1032 | |
D 2Σ+ | This work | 52716.60 | 0.4643 | 1.6500 |
Ref. 17 | 47606.81 | 0.405 | 1.6635 | |
Ref. 55 | 47385.25 | — | 1.6611 | |
c 4Σ− | This work | 72162.93 | 0.9837 | 1.800 |
Ref. 55 | 55039.22 | — | 1.7797 | |
Ref. 17 | 55045.96 | 0.967 | 1.7866 | |
F 2Π | This work | 65395.45 | 3.3264 | 1.3500 |
Ref. 55 | 63312.55 | — | 1.3746 | |
Ref. 17 | 63503.92 | 3.285 | 1.3751 | |
H 2Π | This work | 65863.92 | — | 1.1270 |
Ref. 55 | 69895.35 | — | 1.3434 | |
Ref. 17 | 70350.04 | 2.647 | 1.3762 |
The dipole-allowed DMs for 12 electronic states of CH were calculated using the icMRCI method with the aug-cc-pV(5+d)Z basis set for the C atom and the aug-cc-pV5Z basis set for the H atom. The detailed PDMs and TDMs of CH molecule in the adiabatic representations are given in the ESI.†
Comparisons between our DMs of CH computed in this work and those computed by other researchers18,63–65 are made to verify the reliability of our ab initio data, which are shown in Fig. 3 and 4. As shown in Fig. 3, the TDMs of B 2Σ−–X 2Π and D 2Σ+–X 2Π transitions show reasonable agreement with those calculated by van Dishoeck.64 using ab initio self-consistent-field with configuration-interaction (CI) methods with the Gaussian atomic orbital (AO) basis set for carbon nucleus and the 5s primitive set of Huzinaga contracted to [3s] by Dunning for the H atom. There are notable differences between our TDMs of the B 2Σ−–X 2Π and C 2Σ+–X 2Π and those computed by Kanzler et al.63 in the short-range or long-range regions, who used the correlated, size-consistent, ab initio effective valence-shell dipole operator (μν) method with the basis set consisting of the Dunning triple-zeta Gaussian-type basis with 5s and 3p functions on carbon and 3s functions on hydrogen. The PDMs for the X 2Π–X 2Π transition are in excellent agreement with those calculated by Baluja et al.18 using the R-matrix method and are presented in Fig. 4. Other PDMs for the X 2Π–X 2Π, a 4Σ−–a 4Σ−, A 2Δ–A 2Δ, B 2Σ−–B 2Σ− and C 2Σ+–C 2Σ+ transitions are different from those calculated by Lie et al.65 based the “extended CI” wavefunctions66 and Kanzler et al.63 The above difference between our ab initio data and those computed by other researchers18,63–65 may result from the different levels of computational methods or different basis sets adopted. Overall, our ab initio data of CH are reliable and can be used to calculate the corresponding rovibronic spectra.
![]() | ||
Fig. 3 Comparison of the TDMs of CH obtained in this work with those calculated by Kanzler et al.63 and van Dishoeck et al.64 |
![]() | ||
Fig. 4 Comparison of PDMs for the electronic states of CH obtained in this work with those calculated by Lie et al.,65 Baluja et al.18 and Kanzler et al.63 |
All results of the ab initio EAMCs and SOCs for CH molecule are shown in the ESI.† SO and L represent spin–orbit and electronic angular moment matrix elements, respectively. The subscripts, X, Y, and Z, mean the component of the parameter on the corresponding axis. The values of these curves have been multiplied by (−i) for better exhibition, except for the Y-component. Table 3 provides values for the optimized NAC parameters α and Rc used to diabatise the energy degenerate pairs, which are obtained using the method mentioned in Section 2.2. The crossing points for F 2Π–E 2Π and H 2Π–F 2Π pairs are 1.37 Å and 1.74 Å, respectively. The Lorentzian parameters are 48.17 and 16.06 for the above pairs, respectively. In addition, the mixing angles increase from 0 to about 1.56 radians with a surge at Rc for both the F 2Π–E 2Π and H 2Π–F 2Π pairs, which are shown in Fig. 5(b). As shown in Fig. 5 (a), the NACs are symmetrical curves with a cusp in the vicinity of Rc, and the peaks of the NACs for the F 2Π–E 2Π and H 2Π–F 2Π pairs are about 26 Å−1 and 11 Å−1, respectively.
Electronic states | R c (Å) | α |
---|---|---|
E 2Π F 2Π | 1.37 | 48.17 |
F 2Π H 2Π | 1.74 | 16.06 |
A diabatic representation for PECs of the F 2Π, E 2Π and H 2Π states was obtained based on the parameters given above and the comparisons of PECs in the adiabatic and diabatic representations are shown in Fig. 6(a). The examples of comparisons of TDMs, SOCs, and EAMCs in the adiabatic and diabatic representations are presented in the ESI.† All examples suggest that the results are more reasonable after diabatisation due to the fact that coupling curves and DMs are smoothed out at the vicinity of avoided crossing points. Finally, all the TDMs, EAMCs, and SOCs in the diabatic representations are also shown in the ESI.†
![]() | ||
Fig. 6 Illustrations of the diabatic and adiabatic representations of PECs for E 2Π, F 2Π and H 2Π states. |
![]() | ||
Fig. 7 Comparisons of our adiabatic total absorption cross-sections with those calculated based on the states file (.states) and the transitions file (.trans) from Masseron et al.6 and Bernath67 at T = 5000 K (HWHM = 1 cm−1). |
For the CH molecule, the adiabatic absorption cross-sections of eight significant transitions that make the main contribution at temperatures of 5000 K are shown in Fig. 8. It can be seen that a 4Σ− → a 4Σ− and X 2Π → X 2Π transitions contribute more obviously in the wave number range of 0–6000 cm−1. Then, the contribution of the a 4Σ− → a 4Σ− transition decreases with the increasing wavenumbers, while the contribution of the X 2Π → X 2Π transition decreases firstly. The absorption cross-sections of X 2Π → A 2Δ, X 2Π → B 2Σ− and X 2Π → C 2Σ+ transitions tend to rise and then fall with the increasing wavenumbers. The absorption cross-sections of X 2Π → H 2Π and X 2Π → F 2Π transitions exist in a wider range of wave numbers and contribute significantly at wavenumbers above 60000 cm−1.
Fig. 9 shows the results for the eight transitions that are the main contributors to the total absorption cross-sections at 5000 K, which are obtained based on the diabatic electronic structure of the CH molecule. Compared with the diabatic results obtained, the adiabatic absorption cross-sections of X 2Π → X 2Π, X 2Π → A 2Δ, X 2Π → B 2Σ−, X 2Π → C 2Σ+, X 2Π → D 2Σ+ and a 4Σ− → a 4Σ− transitions show slight changes. The X 2Π → E′ 2Π transition contributes significantly to the total absorption cross-section in the wavenumbers ranging from 28000 to 35
000 cm−1 as well as in the high wavenumber region due to the potential well of E 2Π state becoming deeper after the diabatic transition. The absorption cross-section of the X 2Π → H′ 2Π transition also exhibits large variations in the range of wavenumbers less than 55
000 cm−1.
Fig. 10 shows the comparison of the adiabatic and diabatic absorption cross-sections of X 2Π → H 2Π and X 2Π → E 2Π transitions at 5000 K. Comparison of the adiabatic and diabatic PECs of E 2Π and H 2Π states is shown in Fig. 11. In the vibrational energy levels and dissociation limits of the diabatic H 2Π and E 2Π there appeared significant changes. Especially for the E 2Π state, the number of the vibrational energy levels and the dissociation energy increased. The absorption cross-sections of X 2Π → H 2Π and X 2Π → H′ 2Π transitions have huge differences at wavenumbers below about 58000 cm−1, since the vibrational energy levels and the dissociation energy of H 2Π state changed after the diabatisation. The absorption cross-sections of the X 2Π → E 2Π transition are mainly concentrated in the range 0–45
000 cm−1. For the X 2Π → E′ 2Π transition, the absorption cross-sections are distributed over the whole wavenumber range. To present the influence of the coupling effects and the non-adiabatic effects on the rovibronic spectra of CH, a comparison of the absorption cross-sections for X 2Π → E 2Π transition was made at 5000 K and is shown in Fig. 12. Differences of the absorption cross-sections for the X 2Π → E 2Π transition are visible at wavenumbers below about 44
000 cm−1, when only the coupling effect is considered. The absorption cross-sections differ widely over the entire range of wavenumbers considered in this work. Fig. 13 exhibits the comparison of the adiabatic absorption cross-sections for X 2Π → H 2Π and X 2Π → F 2Π transitions with/without the consideration of the coupling effects between electronic states at 5000 K. As shown in Fig. 13(a) and (b), the absorption cross-sections for the above two transitions in the adiabatic representations with/without the consideration of the coupling effects have similar trends over the whole wavenumber range. At the wavenumbers below about 40
000 cm−1, the absorption cross-sections with the consideration of the coupling effects are larger than those with no coupling effects, and there is an opposite pattern in the range 40
000–80
000 cm−1. The above differences between the absorption cross-sections certainly show that consideration of the non-adiabatic effects and the coupling effects is necessary to calculate the rovibronic spectra of CH.
![]() | ||
Fig. 10 Comparison of the adiabatic and diabatic absorption cross-sections for X 2Π → E 2Π and X 2Π → H 2Π transitions at 5000 K (HWHM = 1 cm−1). |
Footnote |
† Electronic supplementary information (ESI) available: The Duo input files consisting of the adiabatic data, and diabatic data with optimized PECs, respectively. The DMs, SOCs, and EAMCs of CH and the data of absorption cross-sections in Fig. 8 and 9. See DOI: https://doi.org/10.1039/d4cp03298e |
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