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Research Article | Open Access
New ab Initio Potential Energy Surfaces for the Renner-Teller Coupled and States of CH2
New ab initio potential energy surfaces (PESs) for the two lowest-lying singlet ′ and ′′ electronic states of CH2, coupled by the Renner-Teller (RT) effect and meant for the spectroscopic study, are presented. The surfaces are constructed using a dual-level strategy. The internally contracted multireference configuration interaction calculations with the Davidson correction, using the aug-cc-pVQZ basis set, are employed to obtain 3042 points at the lower level. The core and core-valence correlation effects are taken into account in the ab initio calculations with a modified optimized aug-cc-pCVQZ basis set for the higher-level points. The analytical representations of these PESs, with the inclusion of the nonadiabatic RT terms, are obtained by the nonlinear least-squares fit of the calculated points to three-body expansion. Quantum dynamical calculations are performed on these PESs, and the computed vibronic energy levels for the two singlet electronic states are in excellent agreement with experiment.
The CH2 biradical has been the subject of many theoretical and experimental studies, due to its distinct electronic characteristics and chemical and physical properties. It is the direct chemical precursor of the widely observed CH radical . As a crucial link in the photodissociation sequence of cometary methane, CH2 plays a significant role in the chemistry of hydrocarbon combustion and the astrophysics of interstellar medium [2–5].
Since Herzberg and Johns  carried out a detailed analysis of the high-resolution absorption spectrum of the singlet CH2 in the near ultraviolet region half a century ago, several groups [6–11] have observed the direct absorption spectra or subsequent monitoring of fluorescence via laser-induced fluorescence, stimulated emission pumping, dispersed fluorescence, and so forth. Experimental investigations of the spectroscopy provided abundant and precise rovibronic energy levels [12–16]. Recently, bands in the CH2- transition between 12500 and 13000 cm−1 were recorded at Doppler-limited resolution utilizing a transient frequency-modulation (FM) laser absorption spectrometer by Chang and coworkers . An unexpected and particularly complicated rovibronic structure was detected for and vibronic levels. Comparison of the calculated spectra on the available potential energy surfaces (PESs) [17–19] with the experimental spectra of Herzberg and Johns  indicates that the labeling of some vibrational levels is quite inconsistent, which can be ascribed to the local perturbations from the vibrational resonances and Renner-Teller (RT) effect.
The study of the RT effect on the vibronic levels for nonlinear three-atom molecules has been an active area for several decades [8, 17, 20–24]. The - transition of CH2, which is seen in the red and yellow parts of the spectrum, is one of the best examples in which the RT coupling should be observed . One of the consequences of the RT coupling is the inversion of the rotational structure in the lower component state () near the barrier to linearity. As explained by Jungen et al. , this reordering of the energy levels results from the transformation at high bending levels from bent to linear configurations . The two lowest-lying singlet electronic states of CH2, which become a degenerate pair at linearity, interact strongly with each other and are coupled dynamically [17, 24].
Theoretically, some researchers [3, 4, 26] examined a few electronic states of CH2 with state-of-the-art ab initio methods and provided significant clues to the understanding of the electronic structures. Bussery-Honvault and coworkers  computed an ab initio global PES for the first singlet state of CH2, where a mixed numerical and analytical method was employed in the PES construction. This surface shows no barrier for the C2V insertion, while a barrier of 4319 cm−1 (12.35 kcal/mol) is present for the collinear approach. Unfortunately, visible discrepancies were found between the theoretical calculations on this surface and the experimental results . Later, using the same ab initio methodology and similar fitting process, Bussery-Honvault and coworkers  constructed a PES for the second singlet state of CH2; it should be mentioned that their two lowest-lying singlet PESs are not degenerate at linearity. Joseph and Varandas  constructed a more accurate PES for the lowest singlet state of CH2 with the DMBE scaled-external-correlations method  and obtained very good agreement with the experimental rate constants, which recommends it for future dynamics studies [30, 32]. Furthermore, Dawes et al.  constructed the lowest singlet PES using the local interpolative moving least squares method and performed further spectroscopic calculations, which yielded vibrational frequencies with a root-mean-square error of a few wavenumbers relative to available measurements.
Liu and co-workers  revealed various PES intersection seams among the , , , and , states in the C()H2 reactive system systematically and determined the minimum energy crossing points (MECPs) accurately. The nonadiabatic interaction near MECPs may play an important role in spectroscopy and dynamics [35–40]. The lowest MECP  in C()H2, which is only 8797 cm−1 above the CH2 minimum and much lower in energy than all the other MECPs, is between the two low-lying and states, which become the degenerate pair at linearity and hence are strongly coupled by the RT effect. All the other MECPs  are above or somewhat below the C()+H2 asymptote in energy and thus are not expected to intervene most of the vibronic spectra of the two lowest-lying singlet electronic states. However, it is clear that the RT coupling must be taken into account in the vibronic energy level calculations of the two lowest-lying states.
A few PESs for the RT coupled and states have been developed to simulate the vibronic structure and electronic spectrum with the vibronic coupling included [17, 19, 24, 41]. However, the ab initio PESs of Green Jr. et al.  were empirically adjusted or shifted to fit the experimental data; other PESs [19, 24, 41] were constructed by fitting experimental data and a few ab initio points . However, there are therefore accurate and fully ab initio PESs for the two lowest-lying singlet electronic states and of CH2 with the RT terms required.
So far most of the ab initio work concerning CH2 has been based on the traditional correlated ab initio electronic structure calculations, that is, the so-called frozen core approximation, in which correlation effects involving the electrons in 1s core orbital of carbon are neglected. However, as noted by Peterson and Dunning , if the goals of a calculation are to obtain chemical accuracy of thermochemical properties, the effects of correlating the electrons in the core orbital generally must be addressed in the calculations. Of course, ab initio calculations including correlation effects of core electrons are very time-consuming. In this work, not only appropriate active space but also an optimized basis set with additional functions for describing core and core-valence correlation effects (called CV) is employed in our ab initio calculations, which can also guarantee that the two lowest-lying singlet PESs are degenerate at linearity. We further construct fully ab initio PESs for the two lowest-lying singlet electronic states and of CH2 with the inclusion of the nonadiabatic RT terms.
The organization of the present article is as follows. Section 2 describes the ab initio electronic structure calculations. The fitting of the ab initio energy points is presented in Section 3. The fitted PESs of CH2 (called MZB) and vibronic energy level calculations are discussed in Section 4. Finally, a summary is given in Section 5.
2. Electronic Structure and ab Initio Calculations
2.1. Electronic Structure
For computational convenience, the molecule is placed in the yz plane, and the electronic configuration of ground state is . The and molecular orbitals (MOs) mostly have carbon 1s and 2s characters, respectively. The , , and MOs mainly have the carbon 2p character lying along the twofold z axis, perpendicular and parallel with respect to the yz plane, respectively. In the Cs symmetry, the and orbitals become the orbitals, and the and orbitals become the orbitals.
The electronic configurations of CH2 can be represented as shown in Table 1 in the C2V or Cs symmetry.
The and states, which share the same spatial orbital configuration with two open-shell electrons parallelly or antiparallelly distributed in the outer and orbitals, correspond to the triplet and singlet configurations, respectively. While the and states correspond primarily to configurations with double occupation of the or orbital, they could be appropriately described by the two main configuration wave functions. The two CI coefficients and for have opposite signs with , while those for the state have the same sign with . Hence, multireference methods are required for an accurate ab initio description of the above excited states. The and states become the degenerate pair at linearity and hence are strongly mixed by the RT coupling, which is important to the vibronic calculations. The state correlates with the state at linearity. The next two states, and , become the degenerate pair at linearity. The spin-orbit coupling between the and singlet states is small in the well region and thus is neglected in this work.
2.2. ab Initio Calculations
Two levels of ab initio calculations were performed for the PES construction which involves a dual-level strategy . The electronic energies in the lower-level calculations are calculated with the state-averaged complete active space self-consistent field (SA-CASSCF) and internally contracted multireference configuration interaction (icMRCI) methods [44–47]. The active space consists of six electrons distributed among seven orbitals, which correspond to all valence electrons and valence orbitals and one additional 3s orbital of carbon which is of Rydberg character. The carbon 1s orbital is not correlated but optimized at the SA-CASSCF level. To get a good description of the two lowest-lying singlet states, especially in view of the importance of correct degeneracy between and at linear geometries and the influence of PES intersections, we simultaneously consider the five singlet states which correlate with the C()+H2 asymptote in the present ab initio calculations. Consequently, the SA-CASSCF calculations including three and two roots in the and symmetries are carried out to obtain the orbitals for further icMRCI calculations. The Davidson correction (denoted as +Q) is employed to include the correlation energy due to higher excitations. Dunning’s correlation-consistent polarized valence quadruple-zeta basis set augmented with diffuse functions (aug-cc-pVQZ) is used.
In the higher-level calculations, the methods and algorithm are the same as the lower-level, but the active space and basis set are different. The active space consists of all electrons distributed among eight orbitals, which include 1s, all valence, and 3s orbital of carbon. The carbon 1s orbital is correlated and also optimized at the SA-CASSCF level. To obtain the CV contribution to the energies, the aug-cc-pVQZ basis set for C atom is modified. The details of this scheme could be found elsewhere [42, 48–51], and only a brief outline will be given here. The seven inner 1s functions are contracted to two functions using the coefficients from the aug-cc-pVQZ basis set. The outer five functions are uncontracted as the six functions. Two tight and functions are added to the (3d, 2f, 1g) polarized set given by Peterson and Dunning  and Woon and Dunning . The additional functions are even tempered extensions of the valence sets, and the exponents of the added functions are 5.262 and 14.984 for the functions and 4.152 and 12.147 for the functions which are obtained by successively multiplying the corresponding tightest functions by the ratio of the first two compact functions [50, 51]. This core basis set, developed to treat both core and valence correlations, is of the form (12s6p5d4f1g)/[7s6p5d4f1g] and is designated as optACVQZ, which is much smaller than Dunning’s standard aug-cc-pCVQZ [42, 48, 49] and gives a good description of core electrons [50, 51].
For the ab initio calculation of nonadiabatic terms, the SA-CASSCF method is used. The active space is the same as that of the lower-level energy calculations, and the basis set employed here is the uncontracted aug-cc-pVQZ basis. The nonadiabatic terms, required for calculating for the RT coupling, are the matrix elements of electronic orbital angular momentum , and they are obtained as expectation values over the SA-CASSCF wave functions. The RT coupling takes effect when the molecule approaches linearity, that is, lies on the -axis, hence the matrix elements of and can be neglected. The details of the RT Hamiltonian have been discussed in another publication . So, three matrix elements of electronic angular momentum (designated as , , and ) are calculated, where , , .
All ab initio calculations reported in the present work were carried out using the MOLPRO2006.1 package of ab initio programs .
2.3. PES Grid
In order to cover the region of spectroscopic interest with two deep potential wells, ab initio calculations at icMRCI(6e, 7o)+Q/AVQZ and icMRCI(8e, 8o)+Q/optACVQZ levels were performed at 3042 and 273 symmetry unique geometries, respectively. These were chosen carefully to accurately represent the dynamically important regions, particularly the vicinities of the minimum and near linearity. As pointed out by Liu and co-workers , the pair ( and ) states conically cross not only the pair ( and ) states at around 3.4 bohr, but also the state () at 3.1 bohr in the stretching potential curves of linear H–C–H (with one of the C–H bond lengths fixed at 2.1 bohr).
We concentrate on the geometries with the CH bond length smaller than 3.0 bohrs in the process of the present PES construction. Dozens of geometries are also selected for the description of the higher energy regions. In the important regions, points were computed with small increments of 0.1–0.5 bohrs for bond length and 2.0–5.0° for bond angle, while in other regions coarser grids of 1.0-2.0 bohrs and 10.0–20.0° were used. Geometries with energies higher than 100 kcal/mol above the global minimum of were not totally neglected but assigned a very small weight. In the calculations of icMRCI(8e, 8o)/optACVQZ, the grids were chosen as the following: 1.4, 1.8, 2.0, 2.05, 2.2, 2.6, and 3.0 bohrs for CH distance; 80, 100, 120, 143, 155, 175, and 180° for ∠HCH. In the calculations for the nonadiabatic coupling terms, almost the same grids were selected as those used in the higher-level calculations.
3. Fit of the Potential Energy Surfaces
To construct the PESs for the and states of CH2, we choose three-body expansion functional forms for the analytical representation of the PESs with respect to the internal coordinates using a dual-level strategy, and nonadiabatic coupling terms are also taken into account. The idea of the dual-level strategy is to use two levels of ab initio calculations so as to reduce the number of higher-level points needed for fitting , and the basic scheme is as follows. First, a set of lower-level ab initio points without the CV effect, which are calculated at the icMRCI(6e, 7o)+Q/AVQZ level, is generated to construct a zeroth-order PES, (, b stands for the and states of CH2, resp.). Then, a set of higher-level points with the CV effect, calculated at the icMRCI(8e, 8o)+Q/optACVQZ level, is generated, and the data set of the energy differences of the two levels is used to construct a surface, . And it should be noted that the energies of the lower level are obtained from , instead of the ab initio calculations. In addition, the nonadiabatic coupling terms, , , and ( hereafter), are fitted into functions in the third stage for the future calculations. Each of these steps will now be described in more details as follows. The final adiabatic potential is expressed as
By applying the Levenberg-Marquardt technique for the nonlinear optimization, it was found that there are numerical problems for due to near linear dependence of the parameters, and thus quadruple precision arithmetic was needed to obtain convergence . To improve the fit, energy points below 60 kcal/mol relative to the PES minimum are weighted by a factor of 100, and energy points over 100 kcal/mol are weighted by 0.1. To accurately fit the intersection seam of the two lowest-lying singlet PESs at the linear configurations, all the points with HCH larger than 170° are weighted by a factor of 100.
3.1. Analytical Representation of
For the analytical representation of , the calculated energy points were fitted to a three-body expansion in curvilinear coordinates, which are believed to match the shape of the adiabatic PESs. For both electronic states, a Morse-type coordinate was used for the CH stretching mode, where () is the displacement from the equilibrium geometry of the corresponding electronic state. For the angular coordinate, several types of polynomials were tested, including , , , and . Finally, was employed to describe the HCH bending:
The parameters , , , and the are determined by performing unequally weighted least squares fit to the ab initio data.
3.2. Analytical Representation of
The core correlation surface was constructed in a similar way to the construction of , and the surfaces can be expressed as
3.3. Analytical Representation of the Nonadiabatic Coupling Terms
To fit the matrix elements of and into analytical representations, several types of polynomials have been tested, and the following one provides a good description of the nonadiabatic coupling terms, , , and : In the analytical representation above, the guarantees that and have a correct behavior at linearity. Several types of cos functions instead of were tested as angular coordinate, but very good results were not obtained. is found to be suitable for the description of the bending mode.
Many test calculations were performed with different polynomial orders , and the dependences are shown in Table S1 (see Table S1 in the Supplementary Material available online at doi: 10.1155/2012/236750). Although, the RMS errors in the fit become smaller as goes higher, we choose for the fit of due to numerical problems. The complete set of parameters amounts to a total of 286 linear coefficients and 3 nonlinear coefficients. The fit for has RMS errors of 31.88 and 142.88 cm−1 for the and states, respectively. Below 60 kcal/mol, the RMS errors are 6.64 and 5.94 cm−1, respectively. In the fit of , is taken as 6. The complete set of parameters amounts to a total of 85 linear coefficients. The fit for has RMS errors of 14.29 and 68.21 cm−1 for the and states, respectively. Below 60 kcal/mol, the RMS errors are 3.15 and 11.89 cm−1, respectively. The numerical values of all parameters to generate the surfaces and coupling terms reported in the present study are presented in Tables S2, S3, and S4.
4. Features of the Fitted Surfaces and Vibronic Energy Level Calculations
4.1. Adiabatic PESs
Figure 1 presents the contour plots for CH bonds stretching of our PESs keeping HCH fixed at 102.45° for the state, 144.36° for the state. We found the fitted potentials to be smooth and without any artificial oscillations. The degeneracy of the two lowest-lying singlet adiabatic potentials is illustrated in Figure 2. They are contour plots as functions of and HCH. Our fitted PESs for the two singlet states are degenerate at = () bohrs with HCH = 180°. The PES difference at linear geometries in higher energy regions (with energies higher than 60 kcal/mol above the global minimum of ) is due to the PES fitting error. To further improve the behavior of our PESs at linear geometries, we used an assumed angle dependence switching function to smoothly connect the two states above at linearity. Thus, this function is defined as follows:
Two adiabatic potentials for the and states of CH2, which are going to be degenerate at linearity, are represented as follows:
From the experimental side, a wide number of studies have led to the determination of accurate equilibrium geometries for the and states of CH2. Geometries and relative energies of minima obtained from our work along with the available experimental and other theoretical values are given in Table 2. We find the PES minimum of the state on our PESs to be located at bohrs and . Obviously, these results are in very good agreement with the experimental value CH2 available, = bohrs and . Despite the ground singlet state being widely studied, there have been limited research on the first excited singlet state . As can be seen, this state is the open-shell singlet analog of the state. As expected, its molecular orbitals and Mulliken atomic distributions are very similar to those of the ground state, which makes the ab initio calculation more difficult. The PES minimum for the state on our PESs is located at = 2.030 bohrs and HCH = 144.36°. In Figure 3, we have plotted the contours of the core corrections , as functions of and HCH of the state. The CV effects have a geometry dependence, and the shifts can be positive or negative. The core correlations varies from −400 to 300 cm−1. The inclusion of core and core-valence correlation decreases the bond lengths by 0.0051 and 0.0016 bohrs and increases the bond angle by 0.28 and 1.48° for the and states, respectively.
*Energies are relative to the C()+H2 asymptote.|
aOur ab initio values using icMRCI(6e, 7o)+Q/AVQZ.
bOur ab initio values using icMRCI(8e, 8o)+Q/optACVQZ.
cFrom our PESs.
dAb initio values using icMRCI(6e, 7o)/AVQZ from .
eAb initio values using icMR-ACPF(8e, 7o)/ACVQZ from .
fPES values from [27, 29].
gPES values from .
hExperimental values from .
iExperimental values from .
jExperimental values from .
The state of HCH at linear configuration, splitting into the two lowest-lying singlet electronic excited states and as the molecule is bent, is a prime example of the vibronic RT effect. In this case, the splitting will be proportional to , where and (= 1 for , and 2 for state) is the eigenvalue of , the axial component of electronic angular momentum at linearity. Although the other two higher excited states ( and ) calculated with icMRCI(6e, 7o)+Q/AVQZ are not involved in the present PESs, it is clearly seen that in this region the ab initio computed PECs show the behavior for the doubly degenerate pair ( and ) in Figure 4. The energy splittings between pair and pair states go quartically and quadratically, respectively, when approaching linearity.
While the energy changes with the CH bond stretched, the degeneracy of the and (or and ) of CH2 is not lifted so long as the molecule is linear. The barrier to linearity plays a very important role in quantum mechanical calculations of vibronic energy levels when the RT effect is considered . The height of the barrier to linearity in CH2 has been a long standing source of controversy. The range of reported barrier heights for linearity in the state of CH2 is quite large, varying from 8000 to 10000 cm−1 , which is summarized in Table 3. Herzberg and Johns originally gave a value of 8000 cm−1, estimated from the spacing of the bending vibrational levels . Duxbury and Jungen  obtained a barrier of 9800 cm−1 by fitting a bending potential function to the () levels. An earlier ab initio calculation gave a barrier height of 9600 cm−1 . Kalemos et al.  calculated this value to be 9217.7 cm−1. In 2009, the DMBE PES predicted 9644 cm−1 in agreement with the experimental determination of 9800 cm−1 .
aFrom Ref. . bFrom . cFrom . dFrom . eFrom . fFrom . gFrom . hFrom . iFrom . jFrom . kFrom . lFrom .mFrom .|
Green Jr. et al.  calculated an ab initio value of 9125 cm−1 and empirically adjusted it to about 8800 cm−1 according to the visible spectra around 15000 cm−1. And this value is in very good agreement with the derived value of 8600 ± 400 cm−1 from the experiment by Hartland et al. . The value of 8666 cm−1 was obtained from the PESs constructed by Gu et al.  via fitting experimental data and a few ab initio points. Liu et al. gave the value as 8797 cm−1 based upon the icMRCI(6e, 7o)+Q/AVQZ calculations with three reference states in the symmetry. Our ab initio calculation at the same level with five reference states gives the barrier to linearity as 8895.1 cm−1, but when the core correlation is taken into account, we obtain the ab initio calculated barrier to linearity as 8735.8 cm−1. The core correlations reduce this value by 160 cm−1. It may be due to the fact that the 1s electron is affected when the valence orbitals change from sp2 hybridization at the minimum area to sp hybridization at linearity. Our PESs predict a height of 8715.02 and 760.2 cm−1 for the barriers to linearity in the and states of CH2, respectively.
4.2. Renner-Teller Nonadiabatic Coupling Terms
The fit of the RT nonadiabatic coupling terms has an RMS error of 0.0060, 0.0225, and 0.0148 for , , and , respectively. The polynomial order is taken as 9. Note that is imaginary and its absolute value is thus used. Figures 5(a), 5(b), and 5(c) describe the stretching potential curves of the five singlet states (, , , and state) of CH2 as functions of bond length , with fixed at 2.09 bohrs and HCH at 180°, 175° and 160°, respectively; at linearity, and states become a degenerate pair. Due to the conical intersection between and at bohrs, , , and in Figure 5(d), which shows the dependence of the electronic matrix elements with stretching, are almost constants (2, 4, and 4) for bohrs due to quantization of the electronic angular momenta (, for the state); however, they change rapidly to 1 at 3.38 bohrs. Almost similar changing could be found in Figures 5(e) and 5(f) with HCH = 175 and 160° due to the PESs intersections. And much more interesting rapid changing can also be found in the other three states (, , and ), for example, , but this is beyond the subject of the present work.
In Figure 6, the contour plots for , , and of the and states of CH2 as function of CH stretching and HCH bending are presented. It is well noticed that along the HCH axis, the values of and but then as the molecule deviates from linearity the values of and decrease, and the values of increase at the short region but decrease at the long region. The values of and begin to drop rapidly and monotonically to 1 at bohrs due to the PES intersection at linearity. Figure 7 shows the variation of the electronic matrix elements for the two lowest-lying singlet electronic states as functions of the bending angle HCH, with the CH-distance optimized for the state. It is shown that, as the molecule bends, the curve of decreases monotonically from 2, that of rises from 4 basically in a monotonic way, but that of first goes down from 4 and then goes up.
4.3. Vibronic Energy Level Calculations
We have calculated the vibronic energy levels of the and states on our ab initio PESs employing the block improved relaxation scheme [61, 62] in the multiconfiguration time-dependent Hartree (MCTDH) method [63–65]. The energy results are listed in Table 4 and are compared with experiments and calculations by other groups. The variation of the electronic matrix elements with geometry is not considered, and other groups also adopted this treatment in previous calculations. For consistency, the energy levels in Table 4 are labeled by the bent molecular notation .
aFrom . bFrom .cFrom . dFrom . eFrom . fFrom . gFrom . hFrom . iFrom . jFrom . kFrom .|
Our calculated results are in excellent agreement with the experimental values, reflecting the accuracy of the constructed ab initio PESs. For the energy level of , our result 10828 cm−1 is closer to the experimental value 10827 cm−1 from Sears et al. [10, 12] than 10823 cm−1 from Herzberg and Johns . The experimental results from Sears et al. are more reliable [10, 12, 16]. Compared with the calculated results from Green Jr. et al., our results are closer to experiment . Generally speaking, our results are in slightly better agreement with the experimental values than the results of Gu et al. , and it should be noted that the semiempirical PESs used by Gu et al. were adjusted according to the experimental values while ours are fully ab initio ones.
In this work, we report fully ab initio PESs for the RT coupled and states of CH2 suitable for the spectroscopic study, based on the icMRCI+Q method using the AVQZ and a kind of optimized ACVQZ basis sets. The core and core-valence correlation effects are included, which are necessary for an accurate quantum chemical description of the CH2 electronic states. The analytical representations of the two lowest-lying singlet PESs, with the inclusion of the matrix elements of electronic angular momentum , , and , are obtained by fitting. The obtained PESs are smooth, and the two adiabatic potentials are exactly degenerate at linearity. The minimal energy structures and the barriers to linearity predicted by our PESs are in excellent agreement with available experimental data. Furthermore, the MCTDH quantum dynamical calculations are carried out on these new PESs, and the calculated vibronic energy levels are in excellent agreement with the experimental values. Further work on the construction of the global PESs for the and states, suitable for the C()+H2 reactive scattering studies, is in progress, and various PES intersections as revealed in our previous work , in particular conical intersections, will be included.
This work is supported by National Natural Science Foundation of China (nos. 20733005 and 21173232), Chinese Academy of Sciences, and Beijing National Laboratory for Molecular Sciences. The authors would like to thank Professor H. Partridge for useful discussions of the modified basis set optACVQZ for core-valence calculations.
Table S1 in the Supplementary Materials shows the dependence of RMS errors in the fit on the degree of the polynomials (M). The numerical values of all parameters to generate the surfaces and coupling terms reported in the present study are provided in Table S2, S3 and S4 of the Supplementary Materials.
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