L. H.
Coudert
*a,
Fan
Xie‡
b and
Melanie
Schnell
b
aUniversité Paris-Saclay, CNRS, Institut des Sciences Moléculaires d'Orsay, 91405 Orsay, France. E-mail: laurent.coudert@cnrs.fr
bDeutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany. E-mail: fan.xie@desy.de
First published on 15th April 2025
The microwave spectrum of the non-rigid trans-cyclohexanediamine (C6H10(NH2)2) is investigated. It displays a large amplitude interconversion motion during which both amino groups are rotated through 117° leading to tunneling splittings on the order of 21 MHz and line splittings on the order of 42 MHz for b- and c-type transitions. The tunneling is mediated by the quadrupole coupling hyperfine structure arising from both nitrogen atoms which leads to splittings on the same order of magnitude. The frequencies of the rotation-tunneling-hyperfine transitions are analyzed using a new theoretical model in which the large amplitude motion and the quadrupole coupling are treated simultaneously. Hyperfine matrix elements between (within) tunneling sublevels depend on the difference (sum) of the quadrupole coupling of the two nitrogen atoms. Using the theoretical formalism, 249 experimental frequencies are reproduced with an RMS value of 10 kHz, close to the experimental uncertainty. The spectroscopic parameters determined include usual rotational and distortion parameters; tunneling parameters describing the magnitude of the tunneling and its rotational dependence; and various components of the effective quadrupole coupling tensors.
Trans-1,2-cyclohexanediamine (C6H10(NH2)2), depicted in Fig. 1, is a non-rigid species displaying an internal hydrogen bond between its two amino groups and a prominent quadrupole coupling hyperfine structure arising from its two equivalent nitrogen atoms. The large amplitude interconversion motion displayed by this molecule is a concerted torsion of these two amino groups during which the donor group becomes the acceptor group and vice versa. Like in hydrazine, triply deuterated acetaldehyde, and dinitrogen pentoxide,5,7,8 the large amplitude motion exchanges the atoms giving rise to the hyperfine coupling. However, trans-1,2-cyclohexanediamine (CDA) is of greater theoretical interest because the tunneling splitting and the hyperfine coupling are of similar magnitude. The former is on the order of 20 MHz while the latter gives rise to splittings ranging from 0 to 5 MHz. For this reason, unlike in the aforementioned molecules,5,7,8 hyperfine coupling effects need to be evaluated within as well as between tunneling sublevels.
The microwave spectrum of CDA is experimentally and theoretically investigated in this paper. Its centimeter wave rotational-tunneling spectrum is recorded using a chirped-pulse Fourier transform microwave (CP-FTMW) spectrometer.10 The rotational-tunneling spectrum of the lowest lying conformer is analyzed with a new theoretical approach where the hyperfine coupling and the tunneling motion are treated simultaneously. It is found that non-diagonal hyperfine coupling matrix elements between different tunneling states are key to reproduce the experimental frequencies within experimental uncertainties.
ST | 1 | 2 | 3 | 4 | 5 | S1 | S2 | S3 |
---|---|---|---|---|---|---|---|---|
τ 1 | 310.1 | 193.0 | 283.9 | 190.3 | 74.0 | 298.1 | 245.4 | 156.0 |
τ 2 | 193.0 | 310.1 | 283.9 | 74.0 | 190.3 | 245.4 | 298.1 | 156.0 |
E | 0 | 253 | 67 | 439 | 1234 | |||
A | 2743 | 2739 | 2713 | 2735 | 2759 | |||
B | 1922 | 1940 | 1923 | 1930 | 1913 | |||
C | 1226 | 1223 | 1223 | 1222 | 1222 |
Table 1 emphasizes that the most feasible large amplitude motion is the interconversion motion connecting minima 1 and 2. This large amplitude motion is indeed characterized by a low barrier of only 439 cm−1. The corresponding tunneling path goes through saddle point S1, minimum 3, and saddle point S2, and is plotted in Fig. S1 of the ESI.† Another interconversion motion, connecting minima 4 and 5, goes through saddle point S3 and is characterized by a higher barrier of 1234 cm−1.
Ri = R + S−1(χ, θ, ϕ)·[ai(τ1, τ2) + di], i = 1 to 22, | (1) |
The three important configurations for the large amplitude interconversion motion are shown in Fig. 3. Configurations 1 and 2 correspond respectively to the minima 1 and 2 described in Section 3; the intermediate configuration, corresponding to minimum 3, displays a 2-fold axis of symmetry parallel to the molecule-fixed y axis. The equilibrium values of τ1 and τ2 describing Configuration 1 will be denoted τeq1 and τeq2 and are given in Table 1. The symmetry group to be used for the present tunneling problem is the permutation inversion symmetry group G2 isomorphic to the C2 point group.16 The character table of this group along with the effects of the symmetry operation on the large amplitude coordinates τ1 and τ2, and on the Eulerian angles can be found in Table III of Christen et al.16 where the permutation operation (01)(23)(45)(67)(89) should be changed into (12)(34)(56)(78) and the coordinates ζ1, ζ2 into τ1, τ2 for this work.
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Fig. 3 The two nonsuperimposable equilibrium configurations of CDA, numbered 1 and 2, which are connected by the large amplitude interconversion motion. The intermediate configuration for this tunneling motion is also drawn. The color coding of the amino groups' hydrogen atoms is the same as for Fig. 1(b). H3, H4, H5, and H6 are drawn in green, blue, yellow and red, respectively. The molecule-fixed xyz axis system is shown at the bottom center of the figure. |
In agreement with the IAM treatment,14,15 the Eulerian-type angles χ2, θ2, ϕ2 describing the rotational dependence of the tunneling splitting are determined solving eqn (33) of Hougen.14 The tunneling path retrieved in Section 3 and drawn in Fig. S1 of the ESI,† is used. It is parameterized with the coordinate η such that −1 ≤ η ≤ + 1, η = −1 at Configuration 1, η = 0 at the Intermediate Configuration, and η = +1 at Configuration 2. The parameterization of all internal coordinates in terms of η is given in Section S1 of the ESI.† Numerically solving eqn (33) of Hougen14 yields χ2 = 269.094°, θ2 = 0.577°, and ϕ2 = 89.094°. The 180° difference between χ2 and ϕ2 was theoretically substantiated in Christen et al.16 and arises because of the nature of the tunneling problem and because of the way the molecule-fixed xyz axis system is attached to the molecule.
![]() | (2) |
The Hamiltonian can be block-diagonalized using the symmetry-adapted ΨvJKα rovibrational wavefunctions defined in eqn (11) of Christen et al.16 which are characterized by the rotational quantum numbers JKα and the quantum number v which is 0 or 1. Their symmetry species is ΓS = ΓR ⊗ Γv, where Γv = A and B when v = 0 and 1, respectively. Hamiltonian matrix elements between two such wavefunctions, , are given in eqn (12) and (13) of Christen et al.16 for Δv = v′ − v′′ = 0 and ±1, respectively. In the limit of a slow rotational dependence of the tunneling splitting, when θ2 ≈ 0 and ϕ2 ≈ π/2, the Δv = ±1 matrix elements of the Hamiltonian vanish, v becomes the actual tunneling quantum number as emphasized by eqn (16) of Christen et al.,16 and the tunneling splitting is 2|h2|.
Rotation-tunneling energies are calculated considering a Hamiltonian matrix setup with symmetry-adapted basis set wavefunctions characterized by either ΓS = A or B. The resulting energy levels are assigned with the usual rotational quantum numbers JKaKc and the tunneling quantum number v; their eigenfunction can be obtained from eqn (15) of Christen et al.16 rewritten below for completeness:
![]() | (3) |
In accordance with symmetry considerations, the selection rules for a-type lines are Δv = 0, and for b- and c-type lines, Δv = ±1. This means that a-type lines consist of two nearly overlapped tunneling components while b- and c-type lines consist of two tunneling components separated by twice the tunneling splitting.
![]() | (4) |
Cy(π)·χ1(τ2, τ1)·Cy(π) = χ2(τ1, τ2), | (5) |
HQ = H+QO+R + H−QO−R, | (6) |
![]() | (7) |
The operators H±Q and O±R belong to the G2 symmetry species A and B for the upper and lower sign, respectively. The |(I1I2)IJF〉 hyperfine wavefunctions are used and correspond to the coupling scheme:
I1 + I2 = I ![]() ![]() | (8) |
|I, JKαv, FMF〉, | (9) |
![]() | (10) |
![]() | (11) |
![]() | (12) |
χ± = 〈Φ1|χ1 ± χ2|Φ1〉 ≈ χ1(τeq1, τeq2) ± χ2(τeq1, τeq2), | (13) |
![]() | (14) |
The matrix elements of the symmetry adapted hyperfine operators H±Q in eqn (10) should be computed using eqn (2)–(8) of Robinson and Cornwell.22 The ΔI = I′ − I′′ = 0, ±2 matrix elements of H+Q should be obtained from their eqn (6) and (8) where X+ should be set to 1; the ΔI = ±1 matrix elements of H−Q should be obtained from their eqn (7) where the product X−R should also be set to 1. In both cases, the result should be divided by (J + 1)(2J + 3)/2 because the Casimir's function used in the present paper is not defined in the same way as in Robinson and Cornwell.22
The dominant contribution in eqn (10) arises from the tensor χS± displaying diagonal K′ = K′′ and α′ = α′′ rotational matrix elements. Therefore, in eqn (11), the largest |Δv| = 0 and 1 hyperfine matrix elements involve χS+ and χS−, respectively. This means that matrix elements within (between) tunneling sublevels depend mainly on the sum (difference) of the quadrupole coupling of the two nitrogen atoms.
![]() | (15) |
The 2(2J + 1) rotational-tunneling levels arising for a given J-value are associated with hyperfine functions characterized by I = 0, 1, and 2 so that the basis set functions in this equation belong to the selected ΓRH symmetry species. With these basis set functions, the matrix of HRT is diagonal, with diagonal matrix elements equal to the rotation-tunneling energy, and the matrix of the hyperfine Hamiltonian HQ should be obtained from eqn (10). Diagonalization of the HRT + HQ matrix yields hyperfine energies that can be assigned with F and with I, although the latter is no longer a good quantum number, and with the rotation-tunneling quantum numbers JKaKc and v.
The results of the present approach are illustrated in the case of the hyperfine structure of the two tunneling components of the 101 ← 000 transition. Hyperfine levels are calculated using the rotational constants in Table 1, the values of θ2 and ϕ2 reported in Section 4.1, and the effective quadrupole coupling constants in Table 2. For the upper level, the hyperfine Hamiltonian matrices involve the six rotation-tunneling levels JKaKc,v with JKaKc = 101, 111, and 110; and v = 0 and 1. For the lower level, only two rotation-tunneling levels are involved with JKaKc = 000 and v = 0 and 1. The relative line intensities of hyperfine components were computed with the results of Cook and De Lucia.23 Several values of the tunneling parameter h2 ranging from 0 to −3 MHz were considered and the corresponding stick spectra are shown in Fig. 4. This figure emphasizes that the present approach leads to a fast variation of the hyperfine structure with the tunneling parameter h2. When h2 = 0, the molecule displays no tunneling and the hyperfine structure is that of a rigid molecule with two inequivalent nitrogen atoms. The hyperfine structure displays transitions with ΔI even and odd. For h2 = −3 MHz, the hyperfine structure is that of a molecule with two equivalent nitrogen atoms dominated by hyperfine components with ΔI even. The calculated frequencies of the two tunneling components are 3148.0008 and 3147.9992 MHz for v = 0 and 1, respectively, leading to a very small line splitting of only 1.6 kHz which cannot be seen with the scale of Fig. 4.
N 1 | N 2 | N 1 | N 2 | ||
---|---|---|---|---|---|
χ xx | −4.555 | 2.210 | χ xy | −1.183 | 0.967 |
χ yy | 2.334 | −1.876 | χ xz | 1.157 | 0.498 |
χ zz | 2.221 | −0.334 | χ yz | 0.680 | −3.742 |
In the final analysis, line frequencies were least-squares fitted to the rotation-tunneling and hyperfine parameters introduced in Section 4, computing the rotation-tunneling-hyperfine energies with the approach described in Section 4.4. Lines were given a weight equal to the inverse of their experimental uncertainty squared. Instead of the diagonal components of the tensors χ±, defined in eqn (14), the following hyperfine parameters were used:
χ± = (χ±xx − χ±yy)/2 and χ±z = 3/2χ±zz. | (16) |
All three diagonal components can be obtained from these equations remembering that the relation χ±xx + χ±yy + χ±zz = 0 holds.
The unitless standard deviation of the analysis was 1 and the root mean square deviation of the observed minus calculated residuals was 10 kHz. Table S1 of the ESI,† reports assignments, observed frequencies, and residuals. Inspection of this table shows that the largest residual is 33.5 kHz for the F = 1, I = 2 ← 1, 0 hyperfine component of the 212, 0 ← 101, 1 rotational-tunneling transition at 6386.7041 MHz.
Table 3 reports the fitted spectroscopic parameters. The rotation-tunneling parameter ϕ2 was constrained to the value computed in Section 4.1 and the hyperfine parameters χ+xz, χ+xy, χ+yz, χ−xz, χ−xy, and χ−yz, to those reported in Table 2. Table 3 shows that there is a very good agreement between fitted and calculated values for the rotational constants as the discrepancies are at most 9 MHz. A very satisfactory agreement also arises for the hyperfine parameters χ+ and χ+z for which the differences are less than 0.33 MHz. For the hyperfine parameters χ− and χ−z, the agreement is less satisfactory and this may be due to the fact that they are involved in non-diagonal matrix elements, as emphasized by eqn (10). For b- and c-type transitions, Fig. 5 of the paper and Fig. S2 and S3 of the ESI,† show comparisons between the observed spectrum and a spectrum computed from the results of the line frequency analysis. A fairly good agreement between observed and calculated spectra can be seen.
Parametera | Analysisb | Calculatedc |
---|---|---|
a Parameters are defined in Sections 4.2, 4.3 and 5. 249 hyperfine transitions were reproduced with a root mean square deviation of 10 kHz. b In MHz except for θ2 and ϕ2 which are in degrees. c Calculated value when available. d Constrained value (see text). | ||
h 2 | −10.573(1) | |
θ 2 | 1.311(27) | 0.577 |
ϕ 2 | 89.094d | |
A | 2734.155(1) | 2743 |
B | 1918.930(3) | 1922 |
C | 1224.707(1) | 1226 |
D K × 103 | 8.898(3300) | |
D KJ × 103 | −10.532(3800) | |
D J × 103 | 1.889(650) | |
d 2 × 103 | 0.876(320) | |
χ + | −1.409(4) | −1.401 |
χ + z | 2.492(8) | 2.830 |
χ + xz | 0.828d | |
χ + xy | −0.108d | |
χ + yz | −1.531d | |
χ − | 4.722(57) | 5.488 |
χ − z | −2.366(200) | −3.832 |
χ − xz | −0.329d | |
χ − xy | 1.075d | |
χ − yz | −2.211d |
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Fig. 5 Comparisons between the observed spectrum and a calculated stick spectrum for the 110, 1 ← 000, 0 and 110, 0 ← 000, 1 transitions. Both spectra are plotted as a function of the frequency in MHz. The lists of predicted hyperfine transitions are given in Table S2 of the ESI.† |
In this paper, the microwave spectrum of CDA has been recorded in the 2 to 12 GHz frequency range and analyzed using a new theoretical approach developed to account for both its large amplitude interconversion motion and its quadrupole coupling hyperfine structure. CDA is unusual from the spectroscopic point of view in that the effects of the large amplitude motion and those of the hyperfine quadrupole coupling are of the same order of magnitude. This leads to new effects, not observed in spectroscopic investigations of similar species,5,7,8 which in the new theoretical approach are described by matrix elements, given in eqn (10), of the quadrupole coupling Hamiltonian between rotational-tunneling levels split by the large amplitude motion.
The results of the line frequency analysis of the new microwave data are satisfactory as evidenced by an unitless standard deviation of 1. Fig. 5 of the paper and Fig. S2 and S3 of the ESI,† emphasize that there remain a few discrepancies between observed and calculated hyperfine patterns. These may be due to the neglect of the magnetic spin-rotation and spin–spin hyperfine couplings.29,30 Also, in a heavy molecule like CDA, low energy small amplitude vibrational modes may also prevent treating the ground vibrational state as an isolated state. The ab initio calculations carried out in this work yield a vibrational frequency of only 129 cm−1 for the lowest energy vibrational mode which, like in the parent species cyclohexane,31 is a skeletal mode involving mainly the carbon atoms.
An interesting outcome of the present investigation is shown in Fig. 6 where observed and calculated hyperfine patterns are shown for the 110, v ← 000, v, with v = 0 and 1, nearly overlapped rotation-tunneling transitions. Although both transitions are forbidden, the weak strength of their hyperfine components comes from the mixing between rotation-tunneling levels due to the hyperfine coupling Hamiltonian. In order to be compatible with the symmetry of the molecule, these hyperfine components are characterized by ΔI odd.
![]() | ||
Fig. 6 Comparisons between the observed spectrum and a calculated stick spectrum, plotted as a function of the frequency in MHz, for the hyperfine structure of the transitions 110, v ← 000, v, with v = 0 and 1. These transitions being forbidden, their hyperfine components gain strength through mixings due to the hyperfine coupling Hamiltonian. The lists of predicted hyperfine transitions are given in Table S2 of the ESI.† |
The matrix elements of the hyperfine Hamiltonian between rotational-tunneling levels split by the tunneling motion are described, in agreement with eqn (10), by the tensors χS− and χA− of eqn (14) which give rise to the hyperfine parameters χ−, χ−z, χ−xz, χ−xy, and χ−yz. Setting these five parameters to zero amounts to neglecting matrix elements of the hyperfine Hamiltonian between rotational-tunneling levels split by the tunneling motion. This is the large tunneling splitting limit which leads to a much less satisfactory RMS value of 34 kHz and does not allow us to predict the forbidden transitions shown in Fig. 6. In this limit, the fitting Hamiltonian can be shown to be equivalent to an SPFIT fitting32 with one set of rotational and distortion constants for each tunneling sublevels along with a tunneling splitting, a Coriolis coupling term in Fab, and quadrupole coupling constants equal to the average of those of the two nitrogen atoms.
Footnotes |
† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5cp00586h |
‡ Present address: Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui, 230026, China. Email: xiefan@ustc.edu.cn |
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