Low-temperature tunneling of CH₃ quantum rotor in van der Waals solids

Motional quantum effects of tunneling methyl radical isolated in solid gases as they appear on experimental electron paramagnetic resonance (EPR) spectra are examined. Obtained analytical expressions of the tunneling frequency for methyl rotor/torsional-oscillator utilizing localized Hermite polyn...

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Datum:	2019
Hauptverfasser:	Benetis, N.P., Zelenetckii, I.A., Dmitriev, Y.A.
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Sprache:	English
Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2019
Schriftenreihe:	Физика низких температур
Schlagworte:	Фізичні властивості кpіокpисталів
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spelling	nasplib_isofts_kiev_ua-123456789-1760952025-02-09T21:21:10Z Low-temperature tunneling of CH₃ quantum rotor in van der Waals solids Низькотемпературне тунелювання CH₃ квантового ротора в ван-дер-ваальсових твердих тілах Низкотемпературное туннелирование CH₃ квантового ротора в ван-дер-ваальсовых твердых телах Benetis, N.P. Zelenetckii, I.A. Dmitriev, Y.A. Фізичні властивості кpіокpисталів Motional quantum effects of tunneling methyl radical isolated in solid gases as they appear on experimental electron paramagnetic resonance (EPR) spectra are examined. Obtained analytical expressions of the tunneling frequency for methyl rotor/torsional-oscillator utilizing localized Hermite polynomials are compared to full numerical computations and tested against experimental EPR lineshape simulations. In particular, the X-band of methyl radical was displaying partial anisotropy averaging even at lowest temperatures. EPR lineshape simulations involving rotational dynamics were applied for the accurate determination of the potential barrier and the tunneling frequency. Tunneling frequency, as the splitting between the A and E torsional levels by the presence of a periodic C₃ model potential with periodic boundary conditions, was computed and related to the EPRlineshape alteration. The corresponding C₂ rotary tunneling about the in-plane axes of methyl was also studied while both the C₂ and C₃ rotations were compared with the rotation of deuteriated methyl radical. На основі виміряних експериментальних спектрів ЕПР представлено аналіз квантових ефектів, пов’язаних з тунелюванням метильних радикалів, захоплених в твердих газах. Отримано аналітичні вирази для частоти тунелювання метильного радикала навколо осей симетрії з використанням поліномів Ерміта. Ці результати порівнюються з чисельним розрахунком і з даними, отриманими моделюванням експериментальних спектрів ЕПР. Встановлено, зокрема, що спектри ЕПР Xдіапазону демонструють лише залишкову анізотропію, що означає усереднення анізотропії навіть при найнижчих температурах в експерименті. Моделювання спектрів ЕПР з урахуванням динаміки обертального руху використано для коректного отримання величин потенційних бар’єрів та частот тунелювання. Частоти тунелювання, які визначаються як величини розщеплення між A та E обертальними рівнями при наявності модельного C₃ потенціалу та періодичних граничних умов, розраховано та співвіднесено зі зміною форми спектра ЕПР. Також вивчено тунелювання радикала навколо осей C₂, що лежать в площині симетрії радикала. Представлено порівняння C₂ та C₃ обертань для протонованих й дейтерованих метильних радикалів. На основе измеренных экспериментальных спектров ЭПР представлен анализ квантовых эффектов, связанных с туннелированием метильных радикалов, захваченных в твердых газах. Получены аналитические выражения для частоты туннелирования метильного радикала вокруг осей симметрии с использованием полиномов Эрмита. Эти результаты сравниваются с численным расчетом и с данными, полученными моделированием экспериментальных спектров ЭПР. Установлено, в частности, что спектры ЭПР X-диапазона демонстрируют лишь остаточную анизотропию, что означает усреднение анизотропии даже при самых низких температурах в эксперименте. Моделирование спектров ЭПР с учетом динамики вращательного движения использовано для корректного получения величин потенциальных барьеров и частот туннелирования. Частоты туннелирования, определяемые как величины расщепления между A и E вращательными уровнями при наличии модельного C₃ потенциала и периодических граничных условий, были рассчитаны и соотнесены с изменением формы спектра ЭПР. Также изучено туннелирование радикала вокруг осей C₂, лежащих в плоскости симметрии радикала. Представлено сравнение C₂ и C₃ вращений для протонированных и дейтерированных метильных радикалов. N.P.B. is thankful to professor Nikolaos Kyratzis for assistance in certain numerical methods necessary for the best approach for the presentation of some parts in this work. Yu.A.D. and I.A.Z. acknowledge support of the Russian Foundation for Basic Research (RFBR), research project 16-02-00127a, for the experimental part of the study. 2019 Article Low-temperature tunneling of CH₃ quantum rotor in van der Waals solids / N.P. Benetis, I.A. Zelenetckii, Y.A. Dmitriev // Физика низких температур. — 2019. — Т. 45, № 4. — С. 495-510. — Бібліогр.: 38 назв. — англ. 0132-6414 https://nasplib.isofts.kiev.ua/handle/123456789/176095 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Фізичні властивості кpіокpисталів Фізичні властивості кpіокpисталів
spellingShingle	Фізичні властивості кpіокpисталів Фізичні властивості кpіокpисталів Benetis, N.P. Zelenetckii, I.A. Dmitriev, Y.A. Low-temperature tunneling of CH₃ quantum rotor in van der Waals solids Физика низких температур
description	Motional quantum effects of tunneling methyl radical isolated in solid gases as they appear on experimental electron paramagnetic resonance (EPR) spectra are examined. Obtained analytical expressions of the tunneling frequency for methyl rotor/torsional-oscillator utilizing localized Hermite polynomials are compared to full numerical computations and tested against experimental EPR lineshape simulations. In particular, the X-band of methyl radical was displaying partial anisotropy averaging even at lowest temperatures. EPR lineshape simulations involving rotational dynamics were applied for the accurate determination of the potential barrier and the tunneling frequency. Tunneling frequency, as the splitting between the A and E torsional levels by the presence of a periodic C₃ model potential with periodic boundary conditions, was computed and related to the EPRlineshape alteration. The corresponding C₂ rotary tunneling about the in-plane axes of methyl was also studied while both the C₂ and C₃ rotations were compared with the rotation of deuteriated methyl radical.
format	Article
author	Benetis, N.P. Zelenetckii, I.A. Dmitriev, Y.A.
author_facet	Benetis, N.P. Zelenetckii, I.A. Dmitriev, Y.A.
author_sort	Benetis, N.P.
title	Low-temperature tunneling of CH₃ quantum rotor in van der Waals solids
title_short	Low-temperature tunneling of CH₃ quantum rotor in van der Waals solids
title_full	Low-temperature tunneling of CH₃ quantum rotor in van der Waals solids
title_fullStr	Low-temperature tunneling of CH₃ quantum rotor in van der Waals solids
title_full_unstemmed	Low-temperature tunneling of CH₃ quantum rotor in van der Waals solids
title_sort	low-temperature tunneling of ch₃ quantum rotor in van der waals solids
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2019
topic_facet	Фізичні властивості кpіокpисталів
url	https://nasplib.isofts.kiev.ua/handle/123456789/176095
citation_txt	Low-temperature tunneling of CH₃ quantum rotor in van der Waals solids / N.P. Benetis, I.A. Zelenetckii, Y.A. Dmitriev // Физика низких температур. — 2019. — Т. 45, № 4. — С. 495-510. — Бібліогр.: 38 назв. — англ.
series	Физика низких температур
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first_indexed	2025-11-30T22:48:09Z
last_indexed	2025-11-30T22:48:09Z
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fulltext	Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 4, pp. 495–510 Low-temperature tunneling of CH3 quantum rotor in van der Waals solids Nikolas P. Benetis1, Ilia A. Zelenetckii2, and Yurij A. Dmitriev3 1Department of Environmental Engineering and Antipollution Control, Technological Educational Institute of Western Macedonia (TEI), Kila 50 100 Kozani, Greece 2Department of System Analysis and Control, Institute of Computer Science and Technology, Peter the Great St. Petersburg Polytechnic University, 29 Politekhnicheskaya ul., St. Petersburg 195251, Russia 3Ioffe Institute, 26 Politekhnicheskaya ul., St. Petersburg 194021, Russia E-mail: niben@eie.gr Received July 30, 2018, published online February 25, 2019 Motional quantum effects of tunneling methyl radical isolated in solid gases as they appear on experimental electron paramagnetic resonance (EPR) spectra are examined. Obtained analytical expressions of the tunneling frequency for methyl rotor/torsional-oscillator utilizing localized Hermite polynomials are compared to full nu- merical computations and tested against experimental EPR lineshape simulations. In particular, the X-band of methyl radical was displaying partial anisotropy averaging even at lowest temperatures. EPR lineshape simula- tions involving rotational dynamics were applied for the accurate determination of the potential barrier and the tunneling frequency. Tunneling frequency, as the splitting between the A and E torsional levels by the presence of a periodic C3 model potential with periodic boundary conditions, was computed and related to the EPR- lineshape alteration. The corresponding C2 rotary tunneling about the in-plane axes of methyl was also studied while both the C2 and C3 rotations were compared with the rotation of deuteriated methyl radical. Keywords: solid gases, methyl rotary tunneling, analytical tunneling frequency, tunnel frequency vs. hindering barrier. 1. Introduction A vast number of processes in solid state are realized through potential-barrier hindering dynamics of hosted neu- tral or charged particles/probes. For example, at low temper- atures, the only way for a particle to overcome a potential barrier is by quantum tunneling which becomes crucially enhanced for light atoms and molecules. Among those, me- thyl radical (CH3 and its isotope analogues) is considered as quantum rotator because of its small inertia. The radical may be stabilized in chemically inert matrices of which cryo- crystals are of particular interest. Monoatomic examples of host matrices are Ne, Ar, Kr, Xe, while molecular-gas host matrices of general cryogenic interest are hydrogen H2, ni- trogen N2, oxygen O2, carbon monoxide CO, carbon dioxide CO2, nitrous oxide N2O and methane CH4. The rotational behavior of methyl radical embedded in these matrices is very sensitive to the state and dynamics of the surrounding matrix molecules. Therefore, electron paramagnetic reso- nance (EPR) which provides information on such phenom- ena is a good tool to study the solid state behavior of such embedded radicals at very low temperature. The molecules of cryocrystals are held together by weak van der Waals forces and are among the simplest solids to test ab initio theoretical approaches. However, even in these solids, composed of most weakly bounded particles, the interactions contributing to the energy of the matrix particles may reach rather high values. For example, the central at- traction energy of a matrix molecule in solid CO2 is of the order of 10787 cal/mole [1] which is equivalent to 5428.3 K. The rotational tunneling and librations of the matrix isolated CH3 radical are, thus, the major mechanisms of the radical reorientation at the cryogenic temperatures. The reorienta- tional motion of CH3 in molecular cryocrystals was found to be complex in nature and included fast rotation around the molecular C3 axis, fast librations around the in-plane C2 axis, and slow tunneling tumbling about C2 axis [2–4]. The observed temperature dependence of methyl radical motion in these previous studies clearly evidences specific changes occurring in the orientation dynamics of the matrix mole- © Nikolas P. Benetis, Ilia A. Zelenetckii, and Yurij A. Dmitriev, 2019 Nikolas P. Benetis, Ilia A. Zelenetckii, and Yurij A. Dmitriev cules. The results are in line with a very recent study [5] by Krainyukova and Kuchta who reported new insight on the molecular structure and dynamics of solid CO2. These au- thors performed a high-energy electron diffraction study on solid carbon dioxide films in the temperature range 15–87 K and found hopping precession of molecules instead of sim- ple small-angle librations that should not exceed 5–6 de- grees. The relevant maximal angle deviations turned out to be as big as ~30 deg at the lowest temperatures (~15 K) and started to decrease with rise in temperature at ~45 K. This temperature point almost exactly coincides with the onset of the anisotropic C2 rotation we observed in our methyl radical EPR study in solid CO2 [4]. It is worth not- ing that, earlier, the rank-4 orientational order parameter η4 was found to decrease rapidly in pure solid CO2 at tem- peratures above 45 K [6]. This η4 parameter determines the correlative and rotational-unharmonic effects in the librational subsystem. In fact, EPR studies of CH3 stabi- lized in solid Kr films discovered formation of a highly disordered matrix structure for samples condensed from the gas phase at 4.2 K [7] as well as 16 K [8]. The structure accumulated the majority of trapped radicals which sur- vived annealing to 31 K [8]. The radicals in the disordered structure yielded very broad hyperfine (hf) lines compared to those at the centers in the symmetric substitutional sites of the regular Kr lattice. The broadening was found to orig- inate from the distribution of the radical g-factor due to the matrix effect. An attempt of assessing the disorder parame- ter was presented in Ref. 3. The present study is aimed at elucidating matrix effects in rotationally averaged methyl radical parameters and correlating tunneling rates to barriers of rotation for CH3 in van der Waals solids. The results are essential for convert- ing the radical to a probe to determine low-temperature structures and dynamics processes occurring in cryocrystals. An estimation of the tunneling particle mobility is possi- ble through the tunneling frequency [9] which can be used identically as the torsional splitting of the localized degen- erate vibrators’ ground level. The torsional splitting is thus per definition a temperature independent quantity as it re- quires an isolated quantum system. Experimentally, the tor- sional splitting is observable at lowest temperature while the definition of the tunneling frequency νt is given differ- ently by Stejskal and Gutowsky [10]: /1 e .iE RT t i i v Q −= ν∑ (1) The average energies iE of each originally two-fold de- generate localized torsional level, i, (C2 potential was as- sumed in that work), in the above Eq. (1) weights the cor- responding torsional splitting νi. The partition function Q, comprising the sum of the Boltzmann exponentials, is the normalization factor. The Boltzmann statistics used to ob- tain the populations of the torsional levels thus renders the above “tunneling frequency” νt as temperature-dependent function. There is, however, a fundamental drawback for the use- fulness of the definition in Eq. (1) in quantum systems; in particular, for large potentials of the order of 1000 K and higher. The difference in the energies of the torsional lev- els is too large to give any significant population to any oth- er levels than the ground level. In many other systems of tunneling methyl radical, simi- lar to our case, a simple estimation gives the observable ef- fect from the higher torsional levels for temperatures high- er than 250 K, while experimentally, the importance of the motional quantum effects become secondary for tempera- ture higher than 50 K. In this estimation the torsional ener- gies Eν = 3(BV3)1/2(ν + 1/2) were used, vide infra, for rota- tional energy B ca 7 K and potential barriers V3 = 1000 K. Notice also that barriers over 1200 K studied by Stejskal and Gutowsky [10] were exceedingly larger than the one used in the latter estimation and the ones of interest in the present work. Following the above discussion, we simplify our calcu- lations by not incorporating higher torsional levels than the ground level as also explained by Yamada et al. [11]. Fur- ther details of this approximation are discussed in the sec- tion about the theory of the tunneling-methyl radical. The general quantum mechanical problem of tunneling for hin- dering potentials of C3 and C2 symmetries are also treated easily in the present work within a numerical formalism. The applied method is also applicable for potentials of ar- bitrary complexity provided that its matrix elements in the simple Bloch-type basis set are available. The article is organized as follows: In Sec. 2, the exper- imental matrix shift of the EPR parameters and the anisot- ropy of methyl, CH3, radical isolated in four different ma- trices consisting of spherical molecules are analyzed in relation to the radical tunneling frequencies at cryogenic temperatures. The role of librations in the averaging of the parameter anisotropies is also considered for slightly high- er temperatures where quantum and classical motion coex- ist. The results are utilized in Secs. 3 and 4 to theoretically address the rotational rates and the libration amplitudes of the trapped methyl radicals even in matrices consisting of linear molecules. In particular, the tunneling frequencies of methyl radical against periodic C3 or C2 hindering potentials are computed both analytically and numerically exploiting the experimental EPR results. The EPR results correspond, respectively, to parallel or perpendicular rotary tunneling motions of methyl radical in different solid gas matrices, concerning exclusively very low-temperature quantum con- ditions. The main task is to show that a wealth of perceptible orientational motions is available for a 3D quantum rotator even at cryogenic temperatures by tunneling through barri- ers. Finally, after a short concluding Sec. 5, an Appendix demonstrating the mathematical details of the computations is included. 496 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 4 Low-temperature tunneling of CH3 quantum rotor in van der Waals solids 2. Hindered molecular rotation effect on the EPR parameters The EPR lineshape of CH3 trapped in van der Waals (vdW) solids at cryogenic temperatures demonstrates sig- nificant variability. It changes from nearly symmetric for methyl radical in solid p-H2 [12], to asymmetric spectrum with split anisotropic lines and additional satellite transi- tions of non-rotating methyls, in cold solids consisting of the linear CO2, N2O molecules. The above two sets of main and additional transitions are characterized by either isotropic or axial spin-Hamiltonian vs. rhombic spin- Hamiltonian, respectively [13,14], depending on the actual dispersion forces between matrix and CH3 radicals [15]. The experimental anisotropy variation of the EPR spectra of methyl radical in the present work was related to in- creasing rotational rate of the radical with temperature in- crease. The variation of the radical-matrix interaction in- fluences the matrix shifts of the EPR parameters and changes the radical dynamics governing the parameter av- eraging. Moreover, it is not possible to determine the exact static anisotropy of the g and A tensors experimentally, since methyl tunnels (rotates) even at the lowest experi- mental temperatures. Radical species isolated in a matrix cage are allowed to perform orientational and/or oscillatory motions. In case of a weak matrix coupling, the trapped molecules rotate almost freely in the host cage, yielding nearly isotropic EPR pa- rameters with small matrix shifts. Free rotation quantum effects at 4.2 K were observed early by Foner et al. [16] in the EPR experiments of NH2 radical. These effects were further explained by McConnel [17] who also foresaw the liquid He 1:1:1:1 methyl quartet. Extending the same basic considerations, high-resolution EPR spectra with obvious Pauli excluded EPR transitions of methyl hydrogen isotopo- mers were observed experimentally in cryogenic Ar matrix and interpreted accordingly [11]. Matrix effects of CH3 in solid gases of spherical particles, in particular, spectrum anisotropy, were studied semiquantitatively in [12], setting the following empirical coefficient, Arel, as its measure: 0 max 0 max rel 0 min 0 min 1 / 1 . A A A A A     = − +           (2) In Eq. (2), A0max and A0min are the amplitudes of the strong- est and weakest hyperfine (hf) components of the EPR quar- tet. It was shown that the anisotropy was governed mainly by short-range Pauli exclusion forces while the contribu- tion of the van der Waals attraction was negligible. Varying amplitudes and widths of the different hf components indi- cate residual magnetic anisotropy due to incomplete rota- tional averaging of the spectral parameters. Below the liquid helium temperature methyl radicals are in the ground rota- tion A-symmetry state yielding axially symmetric g and A hf tensors with parallel component coinciding with the highest C3 symmetry axis of the radical [13]. The spectrum averaging is achieved by additional fast rotation about the in-plane axes (perpendicular rotation). The rotation is char- acterized by tunneling frequency estimated as the reciprocal of the rotation correlation time, τcorr, considered as meas- ure of rotation hindrance. One of the aims of the present study is to verify the exclusive effect of the Pauli interaction on the spectrum anisotropy and obtain information about the CH3 rotation dynamics by measuring and correlating τcorr to the radical–matrix interaction. First, we address the case of methyl radical in solid Ar; at 4.2 K, the spectrum of solid Ar is composed of four hf lines with different amplitudes, A, and widths, ∆H [18]. The line intensities, estimated as the products A(∆H)2 are nearly equal yielding the well-known 1:1:1:1 low-tempera- ture intensity ratio. The different widths and amplitudes of the four components are due to the residual anisotropy af- ter partial rotational averaging. In Fig. 1(a), the reciprocal square roots of the hf amplitudes, 1/ A are proportional to the line width [12] that may be approximated by the following quadratic in mF formula [19]: 21 .F FH a b m c m A ∝ ∆ = + + (3) Here, mF is the projection of the coupled nuclear spin rep- resentation F = I1 + I2 + I3 of the three protons. The procedure of obtaining the CH3 EPR parameters and rotation correlation times was divided into two steps as follows: First, the experimental reciprocal square root am- plitudes of the hf components in Figs. 1(a)–(d) were fitted by adjusting the coefficients a, b and c of Eq. (3) assuming fast-motion regime conditions. The Origin 6.1 routine was used to determine the polynomial coefficients minimizing the standard deviation (SD) of the best curve fit through the experimental data. The EPR parameters were obtained from the above fit- ting procedure using reasonable estimates of the magnetic parameters as input in EPR spectral simulations by the EasySpin software. This procedure accounts for Lorentzian and/or Gaussian profiles allowing accurate comparison be- tween simulated and experimental line widths. It is a direct method, compared to using the standard expressions for the coefficients a, b, and c of Eq. (3), see Ref. 12 and refer- ences therein. 2.1. Determination of methyl radical tunneling frequency The effect of motional averaging of the EPR parameters anisotropy of methyl radical isolated in matrices of spheri- cally symmetrical particles is considered in this work. The perpendicular rotation of methyl in these matrices is not hindered compared to matrices of linear molecules where strong hindrance is the case [20]. For CH3 isolated in solid Ar, the approximate curve ob- tained in a least-squares fit of the experimental data is plot- Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 4 497 Nikolas P. Benetis, Ilia A. Zelenetckii, and Yurij A. Dmitriev ted in Fig. 1(a). To obtain the correlation time of the rota- tion, the simulation of the EPR spectrum with the EasySpin software [21] was performed in the fast-motion regime. The physical parameters of solid Ar and solid N2 are close as anticipated by the full mutual solubility of the two components. Therefore, matrix shifts of the EPR pa- rameters of methyl radicals trapped in Ar and N2 solids are expected to be close. Referring to the hf couplings, they are, indeed, nearly equal: −2.313(5) mT for CH3 in Ar com- pared to −2.317(5) mT for CH3 in N2 matrix [13]. The isotropic g-factor, however, shows a noticeable difference, 2.002322(56) and 2.00250(12) for CH3 in Ar and N2 solids, respectively. The g-factor matrix shift of methyl is due to admixture of the unpaired electron p-orbital of the radical with the frontier orbitals of the neighboring matrix particles. This effect is similar to the matrix isolated atomic hydrogen, whose g-factor matrix shift is obtained by admixture of its electron wave function with the p-orbitals of the nearest matrix particles after the required orthogonalization [22]. The principal quantum numbers of the orbitals of the outer (valence) Ar electrons are larger compared to the N2 molecule and would lead to an increased g-factor shift. To verify this possibility, we performed test simulations keep- ing the A-tensor anisotropy, \|\|A A A⊥∆ = − , equal to that of CH3 in solid N2, ∆A = 0.098 mT, and changing simultane- ously the g-tensor anisotropy, \|\|g g g⊥∆ = − , the individual line width, ∆H, and the τcorr to obtain the best possible curve fitting. The isotropic parts of the A- and g-tensors were set equal to those of CH3 in solid Ar: Aiso = −2.313 mT, giso = 2.002322. The individual line width was isotropic and the lineshape was Lorentzian. The best fit was obtained with ∆g = −1.73·10–4, ∆H = 0.0086 mT, τcorr = 24 ns. In Fig. 1(a), the results of the simulations fit well to the approximate curve. The peak-to-peak widths of the simulated hf components are 0.0248, 0.0163, 0.0141, 0.0172 mT counting from low to high fields and match well with experimental widths [18] 0.0229, 0.0183, 0.0136, 0.0204 mT. Fig. 1. (Color online) EasySpin simulations of the EPR spectra for matrix isolated CH3 at 4.2 K in solid gases: in Ar (a), Kr (b), Ne (c), and para-H2 (d). The reciprocal square root amplitudes of the hf components are plotted against the projection mF of the three coupled- proton spin representation F. The blue dashed line calculated using Eq. (3) fits the experimental data presented in black asterisks. The experimental data were taken from our earlier study [12]. The EasySpin simulation results indicated by red open circles are obtained with EPR parameters listed in the text. 498 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 4 Low-temperature tunneling of CH3 quantum rotor in van der Waals solids To make a complete consideration, we tried another ap- proach by fixing ∆g = −3.7·10–4 equal to that of the g- tensor anisotropy for CH3 in solid N2 [13] and changing the A-tensor anisotropy. Figure 2 summarizes the EasySpin obtained simulations. The upper spectrum is simulated using the EPR pa- rameters listed above and fits well with the CH3/Ar exper- imental data shown in Fig. 1(a). The next four simulated spectra show changes in the EPR lineshape with fixed ∆g and gradual increase of ∆A. The simulations begin to re- semble (however, poorly) the experimental results when a large value of the hf-tensor anisotropy, ∆A, is involved, not meeting the physics of the system. Indeed, the radical reor- ientation about the in-plane axes is considered as a combi- nation of very fast small-angle librations with the frequen- cy of the order of 1012 Hz and much slower full rotation with frequencies less than 109 Hz [20,24,25]. The aniso- tropies ∆g and ∆A, used in the spectral simulations, result from the rigid radical parameters, partially averaged by radi- cal librations. Previously, the averaged anisotropies were shown to gradually decrease with decreasing eccentricity of the matrix molecules in the order, CO2, N2O, CO, N2 [20]. Accordingly, ∆A for Ar matrix should be equal or somewhat smaller than N2 matrix. The case of Kr matrix is of particular interest. Indeed, the amplitude sequence of the EPR spectrum of CH3 in solid Kr matrix is a mirror image of the CH3 spectrum in practically all the other solid gas matrices, Ne, Ar, N2, CO, N2O, CO2 [13,18,19]. This “mirror” effect originates from the opposite sign of g-tensor anisotropy in Kr compared to Ar matrix [13,18], modifying the linear, cross relaxation term of Eq. (3). Another feature of the CH3–Kr system is the largest matrix shift of the g-tensor compared to CH3 in other matrices [13], including solid gases, zeolites, vycor glass, beryl, methane hydrate, and feldspar. On the contra- ry, the CH3–Kr hf coupling does not seem to demonstrate such striking matrix shift characteristics: the hf coupling anisotropy has the same sign as the CH3–Ar system [13] and the hf coupling matrix shift is of moderate value [20]. The EPR transitions recorded in the CH3–Kr experiments are somewhat broader [19] compared to those of the CH3– Ar system. The extra broadening is due to that natural Kr contains 11.5 % of the magnetically active 83Kr isotope that contributes to the superhyperfine broadening admixing in addition Gaussian profile to the EPR lineshape. On the basis of the hf component amplitude ratio [12] and the ex- perimental line widths [19], an excellent EasySpin simula- tion is presented in Fig. 1(b). The line widths of the simu- lated spectrum, 0.0275, 0.0234, 0.0247, 0.0313 mT, counting from the low- to the high-field component, match extremely well with the experimental data [19]: 0.0301, 0.0218, 0.0241, 0.0334 mT. The EPR parameters used in the simulations for CH3 in Ar, Kr, Ne, and para-H2 are listed in Table 1. The isotropic A- and g-tensor components were set equal to those of CH3 in solid gases from the pre- vious study [13]. It was found that both the hf anisotropy, ∆A = A\|\| − A⊥, and the g-tensor anisotropy, ∆g = g\|\| − g⊥, should be modified for best agreement of the simulations with the experiment. As a result, the following values were obtained: ∆A = 0.138 mT, ∆g = 1.13·10–4, τcorr = 17 ns. The opposite sign of ∆g compared to CH3 in Ar stems from the greater matrix shift of the g⊥ component compared to the g\|\| compo- nent reported earlier [13]. Comparing giso in the two noble gas matrices, it is concluded that Kr matrix affects the g- factor far more efficiently than Ar matrix. It is also worth to pay attention at the hyperfine tensor anisotropy yielded by the simulation. The value 0.138 mT lies between the aniso- tropies for CH3 in N2O and CO2 matrices, being among the largest hf anisotropies for matrix isolated CH3 in solid gases. The radical libration angles in the substitutional posi- tion of the regular solid Kr lattice are expected to be larger Fig. 2. EPR lineshape of CH3 in solid Ar simulated using differ- ent values of the g- and A-tensor anisotropies. The microwave resonance frequency was fres = 9.4003 GHz. The isotropic com- ponents of the tensors were set equal to those obtained experi- mentally: giso = 2.002322, Aiso = −2.313 mT. (a) ∆A ≡ A\|\| − A⊥ = = 0.098 mT, equal to the value measured for CH3 in solid N2, ∆g ≡ g\|\| − g⊥ = −1.73·10–4, τcorr = 24 ns; (b)–(e) ∆g = −3.7·10–4, equal to the value measured for CH3 in solid N2, and τcorr = 20 ns for all four simulations, whereas the A-tensor anisotropy ΔA in- creases gradually as 0.3, 0.5, 1.0, 1.2 mT, respectively. Figure 1 evidences that the simulation (a) is very close to the experimen- tally measured amplitude ratios of the hf components of the spec- trum recorded in the CH3–Ar experiments. Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 4 499 Nikolas P. Benetis, Ilia A. Zelenetckii, and Yurij A. Dmitriev than in Ar and N2 matrices because of the larger lattice con- stant of 6.13 Å in Kr, compared to 5.31 and 5.65 Å in Ar and N2, respectively. Large-angle librations should, how- ever, tend to decrease the hf anisotropy [20]. An explana- tion for this inconsistency could be the local rearrangement of the strained crystal lattice of quench-condensed Kr. The rearrangement of the cubic lattice to axially symmetric hex- agonal affects the axial anisotropy of the EPR parameters. This process was suggested previously when considering axially symmetrical EPR of H-atoms trapped in quench- condensed Kr [26] and saturation peculiarities of CH3 rad- icals in the same matrix [3]. Figure 1(c) shows the EPR lineshape simulation data of CH3 trapped in solid Ne. The best lineshape fit was ob- tained with g- and hf A-tensor anisotropies close to the ones used in the Ar matrix simulation as shown in Fig. 1(a) and Table 1. As a result, the correlation time τcorr = 28.5 ns is some- what greater compared to Ar matrix. Solid Ne is a tighter matrix compared to the solid Ar as seen from the lattice pa- rameters of 4.46 Å for Ne and 5.31 Å for Ar. As discussed above, a tighter matrix enables smaller radical libration an- gles, and less averaged anisotropies, yielding larger ∆A. The difference between the CH3 radical hf A-tensor anisotropies in Ne and Ar would be more pronounced if it were not for the semi-quantum nature of solid Ne, characterized by fair- ly large zero-point displacement amplitude of the Ne ma- trix atoms. A very small difference in the relative amplitudes of the CH3 hf components was recorded for methyl radical trapped in solid para-H2 [12]. In this case, even a small error in ex- perimental amplitudes may result in rather large uncertainty in 1/ A as a function of the projection of the total nuclear spin. This is evident from Fig. 1(d) where the experimental points deviate more from the theoretical fitting curve com- pared to the noble gas curves in Figs. 1(a)−(c). The devia- tion taken in the percentage of the overall spread of the hf- line amplitudes is addressed here. The EPR transitions recorded in the H2–CH3 experi- ment [12] were rather broad, with a peak-to-peak width of the third, mF = −1/2, component 0.022 mT compared to approximately 0.01 mT expected for radicals isolated in magnetically silent matrices. Indeed, the width of the third EPR hf component in solid Ar that has not magnetic iso- topes is 0.014 mT. The small, extra broadening in para-H2 is probably due to admixture of a small amount of ortho-H2 molecules generated during the para-hydrogen radiofre- quency discharge before sample condensation. The profiles of the simulated transitions were obtained as mixtures of Gaussian and Lorentzian components, applying 0.0045 mT Lorentzian width taken from the CH3–Kr experiment. On the basis of this value, the experimental width of the Gaussian contribution to the lineshape was calculated as 0.0196 mT. The closeness between the amplitudes of the hf components for CH3 in the para-H2 lineshape made it diffi- cult to select spectral parameters for best fitting. On the basis of results for the noble gas matrices, the estimated hf tensor anisotropy, ∆A, were of the order of 0.1 mT compared to 0.098 mT in Ar, 0.138 mT in Kr, and 0.108 mT in Ne. As it was verified in the present study, the hf tensor ani- sotropy ∆A was not that sensitive to the matrix particles surrounding the radical species as was the g-tensor anisot- ropy. The nearest coordination distance of methyl to the matrix particles in solid H2 and Ar was nearly the same: 3.76 Å. The hf anisotropy ∆A in solid H2 is expected to be somewhat smaller than in Ar due to the quantum nature of solid hydrogen while on the contrary, the axial hcp (hex- agonal close packed) crystal structure of H2 favors in- creased ∆A. The best fit, Fig. 1(d), was obtained with ∆g = −0.6·10–4 and τcorr = 2.3 ns. The g-tensor anisotropy turned out to be the smallest among the matrices analyzed. This is because of more efficient averaging of the anisotropy by the signif- icantly increased libration motion of the radical in the H2 quantum crystal. The correlation time is far too shorter than for other matrices, since the trapped radical is much freer to rotate. 2.2. Motional averaging mechanisms of EPR anisotropy Methyl radical reorientational motions about the C3- and C2-symmetry axes average partially the spin-Hamiltonian anisotropy depending mostly on the strength of the matrix- impurity Pauli repulsion [12]. The residual anisotropy may be observed in different amplitudes and widths of any of the hyperfine components, symmetric with respect to the base line, or as anisotropic splitting. The former type of the Table 1. EPR parameters for CH3 in solid gases used in the spectral simulations in Fig. 1. Isotropic hf coupling and g-tensor parame- ters are taken from Ref. 13. Notice the trend of decreasing rotational correlation time with increasing atomic radius of the host Matrix Aiso, mT giso ∆A, mT ∆g ∆H, mT τcorr, ns Ne –2.333 2.002526 0.108 –1.844·10–4 0.0084(L) 28.5 Ar –2.313 2.002322 0.098 –1.73·10–4 0.0086(L) 24 Kr –2.300 2.001655 0.138 1.13·10–4 0.0045(L) 0.0156(G) 17 H2 –2.324 2.002516 0.1 –0.6·10–4 0.0045(L) 0.0196(G) 2.3 500 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 4 Low-temperature tunneling of CH3 quantum rotor in van der Waals solids anisotropy is common of CH3 in molecular hydrogen H2 and solid noble gases Ne, Ar, Kr [12,27], while the later was observed in solid gases consisting of the linear CO, N2O, CO2 molecules [13,14,20]. The rotational averaging is effective even at liquid helium temperatures due exclu- sively to tunneling. The present study is aimed at correlat- ing tunneling rates to barriers of rotation for CH3 in van der Waals solids, applying theoretical approaches de- scribing the experimentally observed residual EPR anisot- ropy of trapped radicals. Lack of direct experimental evidence of anisotropy even at liquid He temperatures is the case for the alpha-protons of free methyl, as only the high resolution 1:1:1:1 ground rota- tional EPR quartet is visible [11]. Pure quantum mechanical inertia rotation treatment required that methyl rotor stops at this temperature with only the above-mentioned spin- rotation state possible [11]. Therefore, all changes of the temperature-dependent hf line splitting were considered to be due to dynamics. Theoretical values of static magnetic parameters of methyl in vacuum coming from first princi- ple quantum chemistry computations can be of importance for this purpose [13]. A great help for the disentanglement of the parameters was offered by the simulation of the high-temperature limit spectra that do not exhibit compli- cations due to motional quantum effects. The effect of molecular reorientation on the EPR lineshape is commonly involved in quantitative methods of molecular mobility data extraction. In low viscosity liq- uids, angular velocities of time scales less than 10−10 s are required to fully average the magnetic parameters of the probe, e.g., the nitroxide. Libration motion is also efficient averaging mechanism at the EPR time scale. The term libra- tion is commonly used for harmonic angular oscillations of molecules in crystals with frequencies of 1011 to 1012 s–1 and amplitudes of approximately 2–3 deg. A similar type of motion in glasses has been evidenced by high-frequency EPR, magnetization transfer, and spin-echo experiments [23,25]. However, low-temperature torsional oscillations greater than 8 deg were detected in the present work for tunneling methyl isolated in solid CO2, vide infra. This motion seems to correlate with unusually wide “hopping precession” angles of the linear CO2 molecules discovered recently by Krainyukova and Kuchta [5] in frozen samples of pure CO2. Due to libration, the anisotropies of methyl spectra were partially averaged as shown earlier [20], gradually decreas- ing with decreasing eccentricity of the matrix molecules in the order: CO2, N2O, CO, N2. Our analysis showed that the greatest uncertainty in obtaining rotational correlation times was associated with the approximation of the obtained poly- nomial coefficients in Eq. (3). It turned out that the relative errors were least for a, while b and c showed somewhat larger error than a. As a result, the EasySpin simulations yielded cautious estimate of τcorr error of about 4 ns relative to the shifts in b and c coefficients. Figure 3 shows the rotational correlation time plotted against the squared Pauli repulsion energy 2 pE . The figure establishes a linear correlation between rotation hindering and repulsion of the trapped CH3 radical by the matrix particles. It is suggested in the present study, that the aver- aged spectrum anisotropy is not only a function of τcorr and the radical–matrix interaction but depends also on the lat- tice symmetry of the matrix. The influence of the matrix symmetry on the anisotropy of the radical EPR parameters is readily seen from the experimental and theoretical stud- ies of trapped H atoms. Free atomic hydrogen is characterized by spherically symmetrical electron wave function and isotropic hf con- stants. When trapped in solids, H atom reveals wave func- tion of reduced symmetry adapting to the electrostatic po- tential of the cage formed by the matrix first couple of coordination spheres. As a consequence, excluding cubic lattice, the hf interaction develops appreciable anisotropy [28–31]. The excellent linear fit in Figure 3 may be acci- dental. Further experimental work with Kr matrices with unstrained crystal lattice, condensed, e.g., at temperatures well above 4.2 K, is required for the verification of the linearity. 3. Rotary tunneling vs. hindering barrier of methyl radical Even strongly hindered methyl groups can perform ro- tary motion at very low temperature in the sense that quan- tum mechanics allows penetration of the torsional barrier. A periodic C3 potential with periodic boundary conditions will be further considered hindering coherent leakage of methyl to the adjacent minima with “distance” 120 deg. The rate of this rotary motion of methyl can be faster than the proton hfi (hyperfine interaction) measured in MHz. The situation can be the reverse for systems with high bar- Fig. 3. Correlation time of the perpendicular rotation of CH3 radical in various matrices plotted against the squared Pauli re- pulsion energy 2 pE , where 0 27.212ε = eV, is the Hartree atomic unit of energy. Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 4 501 Nikolas P. Benetis, Ilia A. Zelenetckii, and Yurij A. Dmitriev riers. The deuteron rotor, e.g., is normally much slower, vide infra, due to the “dramatic” increase of the inertia by factor two. The point here is that one has to use a quantum descrip- tion of the motion in order to reproduce the experimental effects on the EPR lineshape, at least for the lowest tem- peratures close to 5 K. We can use, e.g., approximate har- monic oscillator states localized at the minima of the po- tential energy wells for the description of this motion, see Appendix. Certainly, Hermite polynomials, or any other simple choice like Gaussians [32], cannot be accurate eigenfunctions of a Hamiltonian with an actually periodic potential but they can be used as (zeroth order) basis func- tions for the approximate solution of the problem. Briefly, the approximate localized functions of the smallest C3 group are giving initially an equal number of degenerate basis-functions ( ) ( )( ) ( )k k v v kΨ ϕ ≡ Ψ ϕ−ϑ with location k = 0, ±1, seen in the correlation diagram, Fig. 4. The angles k kpϑ = are given in terms of the period of the potential energy p = 2π/3. The index ν = 0, 1, 2, … of the above functions stands for the vibration level. The eigenvalues of these functions are given by 12 ( ), 2vE B v= β + 33 2 V B β = , (4) where B is the rotational constant and the variable 3(3 / 2) /V Bβ = is expressed in terms of the just men- tioned constant Β and the barrier V3, as seen in Appen- dix A. The 3-fold degeneracy of the above-localized vibra- tor eigenfunctions can be lifted by a periodic potential if the barrier between adjacent wells is not infinitely high. The problem can be solved by both degenerate perturba- tion or by the variational method [33]. The variational principle is formally more appropriate because the basis functions of the localized torsional oscillators cannot be orthogonal as they overlap. As it will be shown later, the overlap, however, is small for all practical purposes. E.g., the overlap of the lowest (ground) torsional level with quantum number ν = 0 is given by 2 ( ) ( 1) 3 0 0 0( ) ( ) exp . 6 k k V S B ±  π ≡ ψ ϕ ψ ϕ = −     (5) This equation is plotted in Fig. 5 and indicates that the overlap is not really significant in the present case. In spite of that, the solution of the tunneling frequency vs. potential problem obtained by this way will not be as accurate as a numerical solution model starting from free rotor in presence of a periodic potential. In practice, it was not easy to make the solutions of the perturbative kind ac- curate enough, see Fig. 6(a). However, the motivation of seeking such a solution, in addition to relatively easily ob- tainable numerical ones, is rather the transparency and the usefulness in the physical interpretation of several effects. Symmetry considerations play a very important role here. Further, one should remember that the analytical expres- sions in the present work are approximate. The assumptions are: i) consideration of parabolic zero order potential; ii) neglect of the mixing with higher vibrational levels. The parabolic potential was obtained by Taylor– Maclaurin series expansion of the periodic C3 potential [ ]3( ) 1 cos3 2 V V ϕ = − ϕ . (6) Finally, the mechanism of the tunneling rotation of the radical can be described as a quantum effect by the time- Fig. 4. Correlation diagram showing roughly the connection be- tween the three different types of rotary/torsional motions. Notice that the three vertical axes cannot be put in a universal scale since the transition frequencies is a multivariate quantity. The ground rotation level has been lifted by the zero-point vibration energy E0 = 3(BV3)1/2/2. The first order correction of the torsional level amounts to C0, see Appendix. The tunneling frequency of the ground vibrational level is designated by 3J0. Going from the left to the right, the second diagram corresponds approximately to the librating radical, while the third diagram to the right is for the radical performing rotation around the C3 axis by tunneling through the barrier with frequency 3J0. In the end of this section, the description of this motion physics is provided. Fig. 5. The overlap S0 of the adjacent vibrator eigenfunctions for the ground torsional level of methyl according to Eq. (5). Notice that the value of this integral is very small already for potential bariers V3 > 100 K and becomes virtually insignificant after 250 K. Those potentials are in compatible order of magnitude for methyl rotor systems studied in the present work (1.000 K = = 2.0837·104 MHz). 502 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 4 Low-temperature tunneling of CH3 quantum rotor in van der Waals solids dependent Schrödinger equation. The diagonalizing trans- formation of the localized vibrator eigenfunctions leading to the torsional oscillator eigenfunctions can be inverted and the former set can also be used in the time dependence of the system. In that case, if the initial state of the system is in one of the potential wells, it will be found in any of the two adja- cent wells at a time equal to the inverse energy difference of the E–A states, signifying the quantum tunneling of the sys- tem over the potential barrier. 3.1. Computation of the tunneling frequency One of the aims of this work is to derive analytical ex- pressions for the tunneling frequency 3J0 vs. the potential barrier V3. Hence, a combination of the perturbation and the variational methods were tested. The value of the overlap integral ( ) ( )1\| k k vS ± ν ν= Ψ Ψ of any two adjacent localized torsional oscillator eigenfunctions was originally omitted in the computation of the perturbation treatment because it is very small for the case here V3 >> B. This is at least veri- fied for the lowest vibrational levels as shown in Fig. 5, which is a plot of the overlap vs. barrier shown in Eq. (5). The inclusion of the overlap in a more accurate analytical expression for the torsional splitting of the ground level ac- cording to the variational theory has also been investigated. When this effect was included it did not help practically, because on the contrary, for low-potential barrier where the overlap is important the analytical expression obtained by perturbation gave better agreement with the numerical result. For high-potential barrier where the disagreement worsens for the perturbation method, the overlap is very small, and the variation method yielded the identical result as the per- turbation. Also, ENDOR has been used to determine tunneling fre- quency. The theoretical treatment of the shift of ENDOR transitions due to tunneling is usually obtained by using the second order perturbation theory of an effective spin- Hamiltonian [34,35]. The potential barrier is then deduced by numerical simulations involving a series of barriers and computing the energy difference between the two lowest levels [34]. Hence, we tried to obtain the most natural ana- lytic expression of the tunneling frequency vs. the potential barrier described further in the text. Figure 6 shows the tunneling frequency of the parallel ro- tation of both protonated and deuteriated methyl radicals vs. potential barrier. The numerical method is based on diagona- lizing the pure rotational Hamiltonian including a hindering periodic C3 potential in the basis set of a two-dimensional free rotor. Similarly, the analysis of the tunneling rotation about the in-plane axes is based on a C2 symmetry potential with periodic boundary conditions. The analytical computation of the C3 case is based on the degenerate perturbation treatment of localized torsional Hermite polynomial oscillator eigenfunctions. Both the bar- riers of the protonated and the deuteriated methyl radicals were computed and compared. One good reason for study- ing also the deuteriated CD3 values is that often the EPR spectra of deuteriated methyl are misinterpreted in literature. Look for the experimental EPR data along with the corre- sponding accurate theoretical predictions shown in Ref. 11. In Fig. 6, the method used in the computations, either numerical, or deriving analytic expressions using the pertur- bation theory on assumed localized oscillators is indicated. The Hermite polynomials are eigenfunctions of these oscil- lators inside each potential well of the two- or three-fold symmetry axis of rotation. The diagonalization of the fol- lowing matrix for the rotational Hamiltonian in presence of the C3 potential V(φ) in Eq. (6) will result in the corrected, albeit approximate, energies for the torsional oscillator ( 1) (0) ( 1) ( 1) (0) ( 1) . v v v v v v v v v v v v v v v v D J J J D J J J D − + − +  Ψ Ψ Ψ    Ψ    Ψ   Ψ  =M (7) Fig. 6. (a) CH3 and CD3 tunneling frequencies about the C3 axis for different values of the barrier V3 keeping the value of the kinetic constants B = 6.752 and 3.376 K for protonated and deuteriated methyl, respectively (1 K = 20.837 GHz). The upper limit of the tunneling frequency for V3 = 0 is formally equal to the rotational constant B. (b) Plot of the numerically obtained tunneling frequency vs. potential barrier of methyl and deuteriated methyl about any in-plane C2 axes of the radical. The computational meth- od was parallel to the approach for the C3 case. The details of the method are described in Appendix C. Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 4 503 Nikolas P. Benetis, Ilia A. Zelenetckii, and Yurij A. Dmitriev The diagonal matrix elements Dν in the isolated oscillator basis set contain the regular quantum vibrator energy Eν and the first order correction Cν due to the potential V(φ), see Appendix. The off-diagonals matrix elements Jν be- tween adjacent oscillators are also shown there. Notice that their nature, the number, and the presence even at the outer off-diagonals positions of matrix Mν in Eq. (7), incorporate the physical meaning of periodic boundary conditions, necessary for the present case. Diagonalizing the above matrix, the eigenvalues EA = = Dν + 2Jν and ΕΕ± = Dν − Jν of the system, corresponding to the totally symmetric A, and the doubly degenerate E- symmetric torsional level, respectively, were obtained. The second order tunneling frequency fν,tun amounts to the dif- ference ( ), = , 2 3 v E A v v v v vf E E D J D J J= ± − − − + = − (8) which is shown in the next equation explicitly in terms of the barrier V3 and the rotational frequency B 2 (2) (2) 3 3 0,tun 0 3 3 exp – 2 6 V V f J B   π = − = ×       2 3 3 3 9 31 exp – . 2 2 2 B B V V    π × − − +           (9) This expression for the tunneling splitting of the ground level is usually designated by 3J0 [30]. In the present work, the tunneling frequency is obtained by including the contri- bution of the full second-order effects concerning the peri- odic potential, as shown in Appendix. There are several treatments in the literature with varying terminology where the symbol J0, or similar, has been used identically as the ( ) ( 1)( ) ( ) ( )k k v vV ±ψ ϕ ϕ ψ ϕ matrix ele- ment and/or as an empirical constant for the tunneling fre- quency in a pure spin-Hamiltonian. To distinguish with the present full second-order degenerate-perturbation treatment, we will use the symbol J0 for the off-diagonal matrix ele- ment ( ) ( 1)( ) ( ) ( )k k v vH ±′ψ ϕ ϕ ψ ϕ for the perturbation part of the rotational Hamiltonian HR = H0 + H′, seen in the Ap- pendix, [36]. According to another interpretation, the tunneling fre- quency is the energy difference of the two lowest levels of the above-hindered rotation Hamiltonian. The relevant nu- merical computations were performed in the more conven- ient basis set of the normalized, Bloch type, exponentials {exp( ) / 2 : Z , [ , ]}.i n n ±− ϕ π ∈ ϕ∈ −π π They were used to span the Hamiltonian matrix in the two-dimensional rota- tional space of a full cycle for a rotor influenced by the sim- plified hindering potential seen in Eq. (6). The matrix repre- sentation of this Hamiltonian was physically blocked by symmetry in three different sub-matrices, an A-symmetry block accompanied by an Ea and an Eb symmetry blocks. The computational details of the matrix elements and the computations of the tunneling frequency vs. barrier V3 are shown in Appendix B and visualized in Fig. 6. This way of working corresponds to separating the three- dimensional (3D) quantum rotator considered by the Popov and coworkers [24] into either a C3 or a C2 methyl-axis rotation under different hindering conditions. They studied in detail how different orientations of the radical rotation axis with respect to a cubic matrix cage determine the strength and symmetry of the hindering barrier. In practice, matrix isolated methyls are embedded in well-defined host cavities restricting the direction of rotary motion not allow- ing free 3D motion. The restriction of the angular momen- tum projection with respect to a laboratory-defined frame reduces significantly the rotational degeneracy even in cases where the rotation appears practically free [17,20]. The nu- merical method of the present work can easily be extended to involve even more sophisticated potentials, exploiting the work of Popov and coworkers [24], provided that the matrix elements of the relevant potential can be computed in the given exponential basis set. The explicit analytical expressions of the present work are not valid for very small V3 < 50 K because of the method of approximation. However, for very low barriers of the order of 20–25 K and lower, it is expected that the “tunneling” frequency will be close to the rotational con- stant B as “free” rotation is approached and formal equality to B will be valid even at liquid He temperatures. In spite the qualitative agreement of the above expression in Eq. (9) with the numerical computation, the two series of values of the tunneling frequency follow each other rather well in the whole interval of the quotient V3/B. The trend of the numerical computations for the two different rotors, proton, and deuteron, are also followed in the whole inter- val of the independent variable V3/B, see Fig. 6. However, the analytical values underestimate the numer- ical tunneling frequency more and more for larger values of the quotient V3/B. One should expect the best agreement for large values of V3 relative to B where the overlap is the least. In reality, this statement holds for the absolute values of the two different computational results, while progressively along with increasing V3, the analytical value of the tunnel- ing frequency worsens in percent units. A formally more appropriate method than perturbation should be the variational method, irrespective of the kind of the used trial functions Ψ. The reason is that the basis functions that were used for the perturbation treatment were not orthonormal, because the eigenfunctions of the well oscillators used are partially overlapping for the po- tential barriers of interest. According to the variational method, minimization of the integral quotient * / * ,RH d dΨ Ψ ϕ Ψ Ψ ϕ∫ ∫ should be performed. Howev- er, consideration of the missing overlap ( 1)) (* ,k k d±Ψ Ψ ϕ∫ of the localized Hermite-polynomial basis used in the computations of the perturbation did not improve the re- sult. At least, according to the variational method, only an upper estimate of the energy can be achieved. 504 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 4 Low-temperature tunneling of CH3 quantum rotor in van der Waals solids As seen also in the plot of Fig. 5, the values of the over- lap integral are negligible for the barriers of interest. A test of a more careful consideration according to the above lines did not improve the result, thus justifying the present inves- tigation. It remains then to give an explanation and a sugges- tion for improving the analytical result. 4. Parallel and perpendicular methyl radical tunneling frequencies The periodic C2 and C3 hindering potentials used in the computations of the present work is an approximation that simply indicates tunneling rotation about the in-plane axes or the higher symmetry axis of methyl radical, respectively. The obtained results on the tunneling-frequency vs. rota- tion-barrier are similar but the magnitude of the barrier and the periodicity of the potential determined the final details. An improved approach for more realistic potentials was presented by Popov, Kiljunen et al. [8,24], quantifying appropriate structural/geometrical parameters of the sys- tem, potentially useful for numerical simulations. Using the group theory, they incorporated the particular cubic cage geometry and the Oh symmetry of the host in combi- nation to the intrinsic C3 methyl symmetry into the final form of the hindering potential, where in addition radial conditions were involved. Figure 7 and Table 2 show results based on the analyti- cal C2 treatment. The correlation times obtained from the EPR simulation of CH3 in matrices of the spherical parti- cles Ar, Kr, Ne and p-H2, presented in Fig. 1(a)–(d), were used to derive the potential barriers. Figure 7 also shows an approximate exponential graphical fitting of the numerical- ly obtained potential barriers from the tunnelling frequen- cies. Diagonalization of the hindered rotation Hamiltonian was used in both the above C2 rotation case as well as for the following C3 case. Table 2. The obtained correlation times considered as invert tunneling frequencies of the CH3 rotor isolated in four solid gases in the first section of this work along with numerically computed potential barriers V3 System τcorr, ns ftun, MHz V3, K Linear host CO, N2O, CO2–CH3 710 1.4 1012 Ne–CH3 28.5 35.1 690.5 Ar–CH3 24 41.7 673.3 Kr–CH3 17 58.8 638.9 Spherical host para-H2, Ne, Ar, Kr ~10 ~100 585.8 pH2–CH3 2.3 435 438.8 Figure 8 shows the computed barriers corresponding to observed tunneling frequencies obtained from the experi- mental spectra simulations for linear and spherical host mol- ecules discussed in the present study and also in Refs. 2, 4, where estimations of possible C3 symmetry barrier values are found for a comparison. The matter of the fact is that the frequency of 1.4 MHz is found for rotation about the in-plane C2 axes and concerns methyl radicals in matrices of linear molecules: CO, N2O, CO2. In this case, a barrier of 1012 K was obtained as shown in Table 2. However also a C3 tunneling of this type of systems gives the same order of magnitude barriers, ca. 946.2 K, as obtained from the graphical representation of Fig. 8. On the other hand, tun- neling frequencies ca. 0.1 GHz are obtained for matrices of Fig. 7. (Color online) Computed tunneling frequency vs. potential in K for C2 rotation (blue dash −), in the range of six different, experimentally determined, tunneling frequencies. The values in Table 2 were obtained by fitting the experimental tunneling fre- quencies the range 250 K < V3 < 850 K to the exponential func- tion ftun = 28715 exp(−0.01V2) MHz. The agreement with the numerically obtained values for C2 symmetry tunneling with the straight line featuring the logarithm of the latter exponential rela- tion was remarkable. Fig. 8. Graphical representation of the numerically computed barriers vs. tunneling frequency for rotational frequency parame- ter of planar methyl B = 6.752 K. Left for tunnel frequency 0.1 GHz = 100 MHz barrier V3 = 534.9 K, middle for frequency 20 MHz barrier V3 = 678 K, and right, for frequency 1.4 MHz barrier V3 = 946.2 K. Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 4 505 Nikolas P. Benetis, Ilia A. Zelenetckii, and Yurij A. Dmitriev spherically symmetrical particles: para-H2, Ne, Ar, Kr. Here, the corresponding barrier for C3 tunneling is approx- imately 534.9 K as seen in the graphical representation of Fig. 8, which is in good agreement with 585.8 K for the C2 tunneling shown in Table 2. The EPR spectra of both CH3 and CD3 adsorbed on the silica gel surface at 77 K were studied by Gardner and Ca- sey [37]. The dependence of the line width on the nuclear spin quantum number was interpreted in terms of the tum- bling of the radicals on the surface and the values of the experimental correlation times/tumbling frequencies were determined as a function of the hyperfine A- and g-tensors anisotropy. Gardner and Casey obtained tumbling frequen- cies, 2.0·107 and 1.3·107 s–1 for CH3 and CD3, respective- ly. Although rapid, these tumbling frequencies still indicate a considerable hindrance to methyl rotation by the silica gel surface, as free rotational frequencies are usually by three orders of magnitude (103) faster. The hindering potential barriers determined by the presently developed barri- er/tunneling relations indicate in particular barriers of ca. 678 K for the protonated and 326.8 K for the deuteriated methyl radical. It seems that the eccentricity of the matrix molecules contributes to both the form and the height of the barrier and should be utilized in the determination of the potential barrier characteristics. It is certainly correlated to the libra- tion motion or the hoppling precession of linear molecules in the sense put forward by Krainyukova and Kuchta [5], which does not occur in case of matrices of spherically symmetrical particles. During the libration or hopping pre- cession, a matrix molecule sweeps a cone which thus is not available to the methyl radicals. Evidently, the larger the molecule eccentricity, the larger is the cone volume, i.e., the barrier width and height. Although this picture is not directly suitable to apply in a mathematical model for the potential energy, it may be considered as a direction to follow in un- derstanding the role of the eccentricity. Namely, the occurrence of such a complicated EPR spectrum of the CH3 radical in solid N2O and CO2 matri- ces, in contrast to solid rare gases and in N2 and CO matrix isolation, depends on the relation between the structure of the hosted radical and the lattice molecules [4]. An appro- priate quantity reflecting our qualitative considerations is the host molecule eccentricity ε taken simply as the ratio of the maximum of the internuclear distance d to the lattice parameter a. The results are summarized in Table 3 together with the relevant parameters supporting the above presump- tion. Indeed, no weak orthorhombic satellites were observed at temperatures above 8 K in CO and N2 matrices [2,14,20] which have relatively small eccentricity, neither such spectra were reported for matrices of spherical symmetrical host particles. On the other hand, the matrices with larger eccen- tricity, such as N2O and CO2, show well discerned weak- line multiplet. Table 3. Estimation of the molecule eccentricity for various cryocrystals at 20 K. The parameters d and a are available from Ref. 1 Matrix Internuclear distance d, Å Lattice parameter a, Å Eccentricity ε = d/a N2 1.098 5.658 0.193 CO 1.128 5.652 0.200 N2O 2.312 5.641 0.410 CO2 2.320 5.554 0.418 5. Conclusion We first obtained rotation correlation times (tunneling frequencies) of the CH3 radical in matrices of spherically symmetrical host particles held at liquid helium tempera- tures. The tunneling rates are correlated to the radical– matrix coupling leading to rotation hindering, mainly gov- erned by the Pauli exclusion forces between trapped radical and matrix molecules. Matrix shifts of spectrum anisotropies in trapped radi- cals testified that matrix effects are more important on ∆g than on ∆A. Quench-condensed Kr crystallizes most prob- ably into axially symmetric hcp local structure, surround- ing the trapped CH3 radical. The theoretical treatment of this study explains how the fast orientational motion of methyl at low temperatures av- erages to different degrees the anisotropy of certain magnet- ic parameters due to rapid rotary tunneling about its C3 and C2 symmetry axes allowing estimation of the potential barri- ers to the rotation. A clearly higher barrier of methyl rotation for the radi- cal isolated in matrices of linear vs. spherical matrix mole- cules was obtained by evaluating the barrier of experimen- tally obtained tunneling frequencies. In particular, the linear CO, CO2, N2O molecular matrices seem to hinder tunneling methyl more efficiently than matrices of spheri- cal molecular particles such as Ar, Kr, Ne and p-H2. The barriers that the above linear molecular matrices oppose are slightly over 1000 K for both C2 and C3 rotational tun- neling, which is almost double the barriers opposed to me- thyl tunneling by the above-mentioned spherical ones. Appendix A. Torsional oscillator equations The zero order eigenfunctions of the torsional oscillator for a three-fold periodic potential, with period p = 2π/3, and periodic boundary conditions, can be approximated by the following localized harmonic oscillator eigenfunctions: 2 /2( ) ( ) ( ) e .kxk v v v kN H x −Ψ ϕ = (A.1) Τhe Hermite polynomials Hν have the scaled angular coor- dinate argument xk, given by ( )k kx = β ϕ−ϑ (A.2) 506 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 4 Low-temperature tunneling of CH3 quantum rotor in van der Waals solids with a specification for the kth well position by the angle k kpϑ = numbering the potential energy minima at the wells by k = 0, ±1, where p = 2π/3 is the period. The k-indepen- dent normalization factors Nν of the above eigenfunctions are given in the following equation: 1/2 . 2 ! v vN v  β =   π  (A.3) The constant β is the square of the angle-scaling factor of the argument xk given by 33 2 V B β = , (A.4) where V3 is the potential barrier and ( )2/ 2 / 2B h I= π (A.5) is the rotation constant. This parameter represents the rota- tional frequency of planar methyl and has approximately the value 6.752 K for the protonated and half that value, 3.376 K, for deuteriated methyl radical species. The de- nominator of Eq. (A.5) gives the parallel component of the methyl rotor moment of inertia I, corresponding to rotation about the highest symmetry C3 axis of a planar methyl. The rotational constants according to Prager and Heidemann [38] are B(CH3) = B(CH4) = 0.655 meV = = 7.601 K (1 eV = 1.1605·104 K), a value corresponding to the nonplanar –CH3 fragment, that agrees with the one adopted in our previous work [11]. Popov et al. [24] and Kiljunen et al. [8] consider instead as rotational constant the double B value, equal to the inverse moment of inertia. The above relations are based in the Taylor expansion of the potential V(ϕ) (6) about its three minima. The result- ing approximate harmonic potential V′(φ) = 9V3φ2/4 is seen in the following differential operator, used as the un- perturbed part of the torsional Hamiltonian at each well: 2 2 2 0 3 1/ (9 / 2) . 2 H B V= − ∂ ∂ϕ + ϕ (A.6) The following energies were obtained for the localized torsional levels: 2 ( 1/ 2).vE B v= β + (A.7) Notice that this is a harmonic oscillator Hamiltonian that approximates the periodic potential energy by a parabola about each minimum. However, because the periodicity is not present in the approximate V′(φ), an additional task is to imply it at a later stage. The perturbation applied to the above zero order Hamil- tonian was 2 3( ) 9 / 4.H V V′ = ϕ − ϕ (A.8) The potential energy V(ϕ) is given in the above Eq. (6). The result obtained after the summation of the terms in Eqs. (A.6) and (A.8) is the identical total rotational Hamil- tonian HR: 2 2/ ( )RH B V= − ∂ ∂ϕ + ϕ (A.9) that comprises the full periodic potential. Appendix B. Derivation of matrix elements The following matrix elements were used in the treat- ment mentioned in the above Appendix A concerning the ground harmonic vibrational level. They are the diagonal (B.1) and the position overlap (B.2) of the square of the angular relocation variable φ. The expression for the variance of the angle φ of the level ν for the localized vibrator given by ( ) 2 ( ) 3 (2 1)( ) ( ) . 3 k k v v v B V + ψ ϕ ϕ ψ ϕ = (B.1) Next, the off-diagonal matrix elements of the square of the angular relocation variable φ is given ( ) ( 1)2 0 0( ) ( )k k±ψ ϕ ϕ ψ ϕ = 2 2 3 3 3 exp 1 . 6 33 V VB B BV    π π = − +          (B.2) The above two matrix elements were further computed for the full hindering potential V(φ): ( ) ( ) 3 0 0 0 3 3( ) ( ) ( ) 1 exp . 2 2 k k V BC V V    = ψ ϕ ϕ ψ ϕ = − −       (B.3) This term is the first-order potential energy correction to the energy E0 in Eq. (Α.7) of the ground vibrator eigenfunction, obtained as a particular case of the following general ν level vibrator correction: ( ) ( )( ) ( ) ( )k k v v vC V= ψ ϕ ϕ ψ ϕ = 2 3 3 0 3 ( 1) 31 exp . 2 2 ! 2 lv l l vV B lV l=      − = − −         β       ∑ (B.4) On the other hand, except for the ground level, the general expressions of the higher level off-diagonal matrix elements of the potential are particularly difficult to obtain. For the ground level we have ( ) ( 1) 0 0 0( ) ( ) ( )k kJ V ±= ψ ϕ ϕ ψ ϕ = 2 3 3 3 3exp 1 exp . 2 6 2 V V B B V     π = − + −             (B.5) The diagonal matrix elements for a general vibrational level ν were first computed by paper and pencil and were reduced for the ground level by putting ν = 0. The matrix elements were then verified by comparing to expressions obtained with “mathematica”. Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 4 507 Nikolas P. Benetis, Ilia A. Zelenetckii, and Yurij A. Dmitriev Appendix C. Numerical treatment The matrix elements of the quantum rotation Hamilto- nian under the influence of a hindering potential, 2 2/ ( ),RH B V= − ∂ ∂ϕ + ϕ in the Bloch type, imaginary exponential basis set { exp( ): Z , [ , ]},N i n n ±− ϕ ∈ ϕ∈ −π π are given by combining the orthogonality relation ( )* e e 2i m i n m,n m,nd m n π − ϕ − ϕ −π ϕ = π δ ⇒ = δ∫ (C.1) with normalization factor N = 1/ 2π and the following matrix elements. The matrix elements of the differential operator in the rotation energy part of HR is given by ( ) 22 2 2 2 ,2/ e e .i m i n m nm n N d n π − ϕ − ϕ −π ∂ ∂ ∂ϕ = ϕ = − δ ∂ϕ∫ (C.2) The following are the off-diagonal, C3 symmetry-dependent matrix elements ( )2cos3 e cos3 ei m i nm n N d π − ϕ − ϕ −π ϕ = ϕ ϕ =∫ ( ), 3 , 3 1 . 2 m n m n− += δ + δ (C.3) The only difference for the C2 case is the modified form of the above integral as follows: ( )*2cos 2 e cos 2 ei m i nm n N d π − ϕ − ϕ −π ϕ = ϕ ϕ =∫ ( ), 2 , 2 1 . 2 m n m n− += δ + δ (C.4) Some additional symmetry related details are discussed in the end of this section. In the above given exponential basis set, the Hamiltoni- an matrix HR in Eq. (A.9) with the simple C3 potential V(φ) in Eq. (6) is found to be block diagonal. In fact, three separate banded tridiagonal matrices, one of A and two of E symmetry, were obtained. By increasing the dimension of the matrix to at least 60 × 60 overall, an acceptable con- vergence of the eigenvalues was obtained. The tunneling frequency of the ground torsional level is the difference between the average EE = (Ea + Eb)/2 of the minimum degenerate eigenvalues of the E block minus the minimum non-degenerate eigenvalue EA of the A block. The compu- tation of the tunneling frequency was repeated as a func- tion of the potential barrier V3 and the C3 rotational con- stant B = 6.752 K. For the deuteriated methyl, half of this value for the rotational constant was used. On the contrary, the in plane C2 rotation of methyl re- quires double these values as rotational constants. This is because the perpendicular moment of inertia of a symmet- ric top disk is half the parallel value. Further, the potential energy function in Eq. (13) is dif- ferent in the simpler C2 tunneling case compared to C3. The Hamiltonian matrix in the C2 case is blocked in two banded tridiagonal matrices, instead of three, one corre- sponding to the totally symmetric A and the other to the anti-symmetric B irreducible representations of the Abelian and cyclic C2 group. Notice that the above numerical method is flexible in another way, allowing the possibility of involving almost arbitrarily realistic potentials. However, in some of these more complex systems the simplifying block structure of the Hamiltonian matrix as for the above simple C3 and C2 potentials may not be possible. Another advantage of the above numerical treatment is that the higher than the ground level tunneling frequencies can also easily be obtained, because all the higher level eigenvalues of the above Hamiltonian are available by the identical diagonalizing procedure. The Hamiltonian matrix is in fact of infinite dimensions as n → ∞ in the basis set {\| }: .n n Z> ∈ Therefore, in the computations the matrix dimensions had to be increased until convergence of the eigenvalues was obtained. Except for the matrix dimension dependent convergence proce- dure, for high potential barriers the tunneling frequency is finally computed as the difference of two very small num- bers leading in an all greater numerical uncertainty. Both the above error sources were most serious for deuteron computations as observed in the erroneous flattening of the numerical tunneling data of deuteron at the high V3 range of Fig. 6(a). Acknowledgments N.P.B. is thankful to professor Nikolaos Kyratzis for as- sistance in certain numerical methods necessary for the best approach for the presentation of some parts in this work. 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Встановлено, зокрема, що спектри ЕПР X- діапазону демонструють лише залишкову анізотропію, що означає усереднення анізотропії навіть при найнижчих темпе- ратурах в експерименті. Моделювання спектрів ЕПР з ураху- ванням динаміки обертального руху використано для корект- ного отримання величин потенційних бар’єрів та частот тунелювання. Частоти тунелювання, які визначаються як ве- личини розщеплення між A та E обертальними рівнями при наявності модельного C3 потенціалу та періодичних граничних умов, розраховано та співвіднесено зі зміною форми спектра ЕПР. Також вивчено тунелювання радикала навколо осей C2, що лежать в площині симетрії радикала. Представлено порів- няння C2 та C3 обертань для протонованих й дейтерованих метильних радикалів. Ключові слова: тверді гази, метильне обертальне тунелюван- ня, аналітична частота тунелювання, частота тунелювання в залежності від перешкоджаючих бар’єрів. Низкотемпературное туннелирование CH3 квантового ротора в ван-дер-ваальсовых твердых телах Nikolas P. Benetis, Ilia A. Zelenetckii, and Yurij A. Dmitriev На основе измеренных экспериментальных спектров ЭПР представлен анализ квантовых эффектов, связанных с тунне- лированием метильных радикалов, захваченных в твердых газах. Получены аналитические выражения для частоты тун- нелирования метильного радикала вокруг осей симметрии с использованием полиномов Эрмита. Эти результаты сравни- Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 4 509 https://doi.org/10.1063/1.2832359 https://doi.org/10.1063/1.2832359 https://doi.org/10.1063/1.3122004 https://doi.org/10.1063/1.3122004 https://doi.org/10.1063/1.1744143 https://doi.org/10.1021/jp984716g https://doi.org/10.1021/jp106598v https://doi.org/10.1021/acs.jpca.5b05648 https://doi.org/10.1021/acs.jpca.5b05648 https://doi.org/10.1021/jp909316b https://doi.org/10.1021/jp909316b https://doi.org/10.1021/acs.jpca.6b04119 https://doi.org/10.1021/acs.jpca.6b04119 https://doi.org/10.1103/PhysRevLett.1.91 https://doi.org/10.1103/PhysRevLett.1.91 https://doi.org/10.1063/1.1744740 https://doi.org/10.1016/j.physb.2004.08.019 https://doi.org/10.1088/0953-8984/21/10/103201 https://doi.org/10.1016/j.physb.2014.01.039 http://www.easyspin.org/ https://doi.org/10.1063/1.1730906 https://doi.org/10.1007/BF03162452 https://doi.org/10.1063/1.2715589 https://doi.org/10.1063/1.2715589 https://doi.org/10.1063/1.2746235 https://doi.org/10.1063/1.2746235 https://doi.org/10.1063/1.473056 https://doi.org/10.1039/B206261E https://doi.org/10.1063/1.446318 https://doi.org/10.1063/1.446318 https://doi.org/10.1063/1.440197 https://doi.org/10.1021/j100205a007 https://doi.org/10.1007/BFb0048204 https://doi.org/10.1063/1.1681986 https://doi.org/10.1016/0301-0104(92)80192-X https://doi.org/10.1016/0301-0104(92)80192-X https://doi.org/10.1016/S0301-0104(97)00301-7 https://doi.org/10.1139/v68-032 https://doi.org/10.1021/cr9500848 Nikolas P. Benetis, Ilia A. Zelenetckii, and Yurij A. Dmitriev ваются с численным расчетом и с данными, полученными моделированием экспериментальных спектров ЭПР. Уста- новлено, в частности, что спектры ЭПР X-диапазона демон- стрируют лишь остаточную анизотропию, что означает ус- реднение анизотропии даже при самых низких температурах в эксперименте. Моделирование спектров ЭПР с учетом ди- намики вращательного движения использовано для коррект- ного получения величин потенциальных барьеров и частот туннелирования. Частоты туннелирования, определяемые как величины расщепления между A и E вращательными уров- нями при наличии модельного C3 потенциала и периодиче- ских граничных условий, были рассчитаны и соотнесены с изменением формы спектра ЭПР. Также изучено туннелиро- вание радикала вокруг осей C2, лежащих в плоскости сим- метрии радикала. Представлено сравнение C2 и C3 вращений для протонированных и дейтерированных метильных ради- калов. Ключевые слова: твердые газы, метильное вращательное туннелирование, аналитическая частота туннелирования, частота туннелирования в зависимости от препятствующих барьеров. 510 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 4 1. Introduction 2. Hindered molecular rotation effect on the EPR parameters 2.1. Determination of methyl radical tunneling frequency 2.2. Motional averaging mechanisms of EPR anisotropy 3. Rotary tunneling vs. hindering barrier of methyl radical 3.1. Computation of the tunneling frequency 4. Parallel and perpendicular methyl radical tunneling frequencies 5. Conclusion Appendix A. Torsional oscillator equations Appendix B. Derivation of matrix elements Appendix C. Numerical treatment Acknowledgments

Low-temperature tunneling of CH₃ quantum rotor in van der Waals solids

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