Ro-vibrational averaging of the isotropic hyperfine coupling constant for the methyl radical

We present the first variational calculation of the isotropic hyperfine coupling constant of the carbon-13 atom in the CH3 radical for temperatures T = 0, 96, and 300 K. It is based on a newly calculated high level ab initio potential energy surface and hyperfine coupling constant surface of CH3 in the ground electronic state. The ro-vibrational energy levels, expectation values for the coupling constant, and its temperature dependence were calculated variationally by using the methods implemented in the computer program TROVE. Vibrational energies and vibrational and temperature effects for coupling constant are found to be in very good agreement with the available experimental data. We found, in agreement with previous studies, that the vibrational effects constitute about 44% of the constant's equilibrium value, originating mainly from the large amplitude out-of-plane bending motion and that the temperature effects play a minor role.


I. INTRODUCTION
][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] The zero-point vibrational corrections (ZPVCs) are proved to be non-negligible for the electron paramagnetic resonance (EPR), 18,20 nuclear magnetic resonance (NMR), 8,15,21 and non-linear optical (NLO) properties. 9,220][11] Moreover, some of the experimentally observed effects, such as temperature dependence and isotope shifts of electric and magnetic properties, are entirely due to the effect of molecular vibrations and rotations. 15everal successful methods for evaluating the rovibrational contributions to various molecular properties were formulated using the perturbation theory (PT) approach over the last few decades. 5,6,24,25The applications of the PT-based approaches are however limited to quasi-rigid molecules vibrating harmonically within a single minimum potential energy surface (PES).For molecules exhibiting large amplitude anharmonic motions, due to the poor convergence of the PT expansion, the ro-vibrational wave functions and subsequent contributions to molecular properties must be obtained by variational methods.This, however, is much more computationally demanding and requires a more sophisticated numerical description of the PES.Thus, it is only applicable to small molecules.Only recently, a few general variational methods have been proposed capable of solving the ro-vibrational proba) Also at Siberian Institute of Physics & Technology, Tomsk State University, Tomsk 634050, Russia.b) Electronic mail: jensen@uni-wuppertal.de lem accurately for very highly excited states.7][28][29][30] One of them, TROVE, 28,30 has been applied to compute the ZPVC-, temperature-, and isotope-dependence of the isotropic spin-spin coupling constants of NH 3 . 15The response theory approach has been developed for the vibrational configuration interaction method allowing accurate calculations of the pure vibrational contributions to the NLO properties. 31,32In these and a few other 10,17,19 studies, the importance of a proper variational treatment of the large amplitude vibrations in obtaining accurate estimates of molecular properties has been emphasized.
In the present work, we report the first comprehensive variational calculations of the ro-vibrational contributions to the isotropic hyperfine coupling constant of the carbon-13 atom in the methyl radical CH 3 , which we henceforth refer to as HFCC.The methyl radical is important in combustion processes and as an intermediate in many chemical reactions, 33 it has been observed in interstellar space, 34 and it is an example of a molecule with large vibrational contribution to HFCC that accounts for up to about 41% of the total value. 18H 3 has been the subject of many theoretical studies [35][36][37][38][39][40][41][42][43][44][45][46] characterizing the electronic structure and vibrational motion.The most recent works 45,46 reported the ab initio calculated PES and dipole moment surface and the variational rovibrational energy calculations for CH 3 .][49] To the best of our knowledge, in all previous studies of the HFCC, the vibrational effects were described by means of PT.In the present work, we have computed the new PES and HFCC surface for CH 3 in the ground electronic state.For the PES, we used the explicitly correlated coupled cluster CCSD(T)-F12 level 50 with the correlation consistent basis set cc-pVQZ-F12, 51 while the HFCC was computed by means of the conventional CCSD(T) with the augmented correlation consistent basis set aug-cc-pVTZ-J. 52,53 We expect these methods to yield an adequate accuracy for at least low energy levels, sufficient to converge the ro-vibrational contributions to HFCC at the room temperature and below.The PES and HFCC surface were used to compute the rovibrational energy levels, ZPVC, and temperature corrections to HFCC by means of the variational method TROVE. 28,30he resulting vibrational energies and the total value of the HFCC at a temperature T = 96 K were found to be in good agreement with experiment, as well as with results of previous theoretical studies.For illustrative purposes, we compared the variationally computed expectation values of HFCC with those obtained from the perturbed-rigid-molecule (PRM) approach.As expected, the results confirm that PRM is not reliable for the expectation values of CH 3 in the excited out-of-plane bending states.

II. COMPUTATIONAL DETAILS
The calculations of the HFCC have been carried out within the framework of the Born-Oppenheimer approximation following a conventional three-step approach.First, the ab intito calculations of the ground state potential energy surface and the electronic contribution to HFCC are performed, which are followed by the calculations of the ro-vibrational energy levels and wave functions.The HFCC values associated with the ro-vibrational states of interest or their Boltzmann distribution are evaluated by averaging the ab initio HFCC function over the corresponding ro-vibrational wave functions.
(1) The electronic energies for the ground electronic state of CH 3 were computed on a grid of 22 640 symmetry-unique molecular geometries employing the open-shell RCCSD(T)-F12b 50,54 level of theory (explicitly correlated F12 restricted coupled cluster included single and double excitations with a noniterative correction for triples) and the F12-optimized correlation consistent polarized valence basis set cc-pVQZ-F12. 51In correlated calculations, the carbon inner-shell electron pair was treated as a frozen core.The diagonal fixedamplitude ansatz 3C(FIX) 55 and a Slater geminal exponent value of β = 1.0 (Ref.56) were used.To evaluate the manyelectron integrals in F12 theory, three additional auxiliary basis sets are required.For the resolution of the identity basis and the two density fitting basis sets, we utilized the corresponding OptRI, 57 cc-pV5Z/JKFIT, 58 and aug-cc-pwCV5Z/MP2FIT 59 basis sets, respectively.Calculations were carried out using the MOLPRO program. 60The analytical representation for the PES was obtained in a least-squares fitting procedure using the functional form from Ref. 61.By varying 248 parameters, we achieved a fitting root-mean-square (rms) deviation of 0.9 cm −1 .The values of the fitted parameters are given in the supplementary material 62 together with a Fortran 90 routine for calculating the PES.
For the coupling-constant surface, the geometry-dependent values of the isotropic hyperfine coupling constant (also known as Fermi contact term) for carbon were obtained 63 as for 19 959 symmetry-unique molecular geometries.In Eq. ( 1), the index N labels a specific nucleus (carbon in our case), µ 0 is the vacuum permeability, µ N is the nuclear magneton, g N is the nuclear g factor, and ρ(N) is the spin density at the carbon nucleus.The hyperfine coupling constant is an important parameter in EPR spectroscopy; it describes the hyperfine splitting and the positions of the resonance lines.A non-vanishing HFCC is due to interaction between the magnetic moments of the unpaired electron and the nuclei in the molecule.It is usually reported in the literature in units of magnetic field strength (G or T) and serves as a measure of the electronic magnetic spin interactions.To obtain the HFCC in Hz, the right hand side of Eq. ( 1) should be multiplied by the conversion factor g e µ B /h (Hz T −1 ), where g e is the g-factor of free electron, µ B is the Bohr magneton, and h is the Planck constant.In the static view of CH 3 as a planar molecule, there is no direct contribution from the unpaired electron to the HFCC and the main contribution comes from spin polarization effects.The out-of-plane vibration allows and adds the direct contribution from the unpaired electron to the equilibrium value of the HFCC.We have calculated the HFCC employing the all-electron unrestricted open-shell CCSD(T) level of theory to account for spin polarization effects and the basis set aug-cc-pVTZ-J 52,53 designed to ensure the proper nuclear-cusp behaviour of the electronic wave function and thus a good description of the HFCC.The calculations were performed with the CFOUR program. 64We have fitted the calculated points to the totally symmetric sixth-order power series expansion 65 in terms of six variables where r k − r e denotes the displacement from the equilibrium value r e of the distance between C and H k , α 1 , α 2 , and α 3 are the instantaneous values of the bond angles ∠(H 2 -C-H 3 ), ∠(H 1 -C-H 3 ), and ∠(H 1 -C-H 2 ), respectively.The factor exp −(r k − r e ) 2 in Eq. ( 2) ensures a physically reasonable asymptotic behaviour of the power series at large distances r k . 66In a least-squares fitting procedure, we determined 185 expansion parameters that reproduce the HFCC data with the rms of 0.11 G.The optimized parameters together with the Fortran 90 function for calculating the HFCC surface are given in the supplementary material. 622) The ab initio PES was used to compute the rovibrational energy levels of CH 3 employing the variational approach and computer program TROVE. 28,30In TROVE, the ro-vibrational Hamiltonian is defined by the power-series expansions of its kinetic energy operator (KEO) and potential energy operator (PEO) in terms of internal coordinates around the equilibrium or reaction-path configuration.In the present work, the expansions of the kinetic and potential parts were truncated after the 6th and 8th order terms, respectively, and the six internal coordinates are three r i = C-H i (i = 1 . . .3) stretching coordinates, two symmetry-adapted bending coordinates ξ 4 and ξ 5 , as given in Eqs. ( 3) and ( 4), and one out-of-plane bending coordinate τ (see Ref. 67 for details).The size of the vibrational basis set is controlled by the polyad number P, where n i are the quantum numbers defined in connection with the primitive basis functions, 28 each describing ith vibrational degree of freedom.They are essentially the principal quantum numbers associated with the local mode vibrations of CH 3 .
The vibrational basis set contains only products of primitive functions for which P ≤ P max .We found that P max = 10 was sufficient to converge the vibrational energies below 7000 cm −1 to better than 0.05 cm −1 and the thermally averaged values of HFCC at a temperature T = 300 K to better than 0.002%.The ro-vibrational basis functions are generated as products of vibrational basis functions and symmetric-top rotational eigenfunctions and the ro-vibrational wave functions are obtained variationally by diagonalizing the full ro-vibrational Hamiltonian matrix. 28Since TROVE uses symmetry-adapted basis functions and the total-angularmomentum quantum number J is a good quantum number, the diagonalization of the Hamiltonian matrix for each irreducible representation of the D 3h symmetry group, and each value of J, is done separately.Another important consequence of molecular symmetry is that the nuclear spin statistical factors 68 for the X 2 A ′′ 2 electronic state of CH 3 are zero for the irreducible representations A ′ 2 and A ′′ 2 , besides for each of the doubly degenerate representations E ′ and E ′′ , only one degenerate component needs to be treated, thus reducing the total computational expenses for CH 3 by a factor of two.For CD 3 , all statistical weight factors are non-zero; thus, only the second argument is viable.
(3) The vibrational and ro-vibrational expectation values of the HFCC were computed for 13 CH 3 and 13 CD 3 using the ab initio calculated coupling constant surface and the TROVE wave functions.The thermal average values for different temperatures were computed by summing over all ro-vibrational states the expectation values multiplied with the corresponding Boltzmann and degeneracy factors.For an ensemble of molecules in thermal equilibrium at absolute temperature T, the thermal average of the isotropic HFCC A iso is given by where g i is the degeneracy of the ith state with the energy E (i) rv relative to the ground state energy, k is the Boltzmann constant, Q is the internal partition function defined as and ⟨A iso ⟩ i is the expectation value of the operator A iso (which represents the HFCC) in the rovibrational state i, The calculation of the quantities in Eqs. ( 7)-( 9) requires the eigenvalues E (i) rv and eigenvectors Φ (i) rv which are obtained variationally with TROVE.
FIG. 1. Convergence of the T = 300 K thermally averaged HFCC vs J max plotted for 13 CH 3 (blue circles) and 13 CD 3 (orange squares) relative to the The degeneracy factor is computed as (2J + 1)g ns , where g ns is the nuclear spin statistical weight taking values in D 3h symmetry group in the order (A ′ 1 , A ′ 2 , E ′ , A ′′ 1 , A ′′ 2 , E ′′ ) as (8, 0, 4, 8, 0, 4) for 13 CH 3 and (2, 20, 16, 2, 20, 16) for 13 CD 3 .(Note that the symmetry of the electronic wave function is A ′′ 2 .)The convergence of the thermal averaged values of HFCC at T = 300 K with respect to the maximal rotational excitation, defined by J max , is shown in Fig. 1.The values are plotted relative to the ZPVC (see Table IV).The energy spectrum of the heavier molecule CD 3 has a higher density than that of CH 3 .In addition, CD 3 has more states allowed by spin statistics.Consequently, in CD 3 , more ro-vibrational states become populated at a given temperature and so, higher J-values must be considered in the theoretical calculations in order to obtain converged values of the thermal averages.The computed values of the partition functions used to normalize the thermally averaged values for T = 300 K(96 K) are 737.08(127.10)for 13 CH 3 and 7519.94(1194.11)for 13 CD 3 .

III. RESULTS
The planar equilibrium geometry of the electronic ground state, X 2 A ′′ 2 , of CH 3 has D 3h geometrical symmetry.The normal modes ν 1 and ν 2 of CH 3 have non-degenerate symmetries A ′ 1 and A ′′ 2 , respectively, and associated principal quantum numbers v 1 and v 2 .0][71][72] Each vibrational state is assigned by the symmetry in D 3h (M) and vibrational quantum numbers ) obtained from the basis function with the largest contribution to the vibrational eigenfunction.The agreement with experiment is generally good, the standard deviation for six states is 3.2 cm −1 which is a little improvement over the previous theoretical results of 4.6 cm −1 (Ref.45) and 7.4 cm −1 (Ref.46).The complete list of computed vibrational energies up to 8000 cm −1 for three isotopologues can be found in the supplementary material. 62is article is copyrighted as indicated in the article.Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 84.135.206.7 On: Thu, 24 Dec 2015 17:02:18 The calculated expectation values of the HFCC for 13 CH 3 given in Table II for a number of vibrational states show a very strong dependence on v 2 , the quantum number of the outof-plane vibration.This is due to both the nonrigid character of the out-of-plane motion and the strong dependence of the coupling constant on the out-of-plane coordinate (see Fig. 2).Even though we need to only consider one minimum of the PES here so that no tunneling motion takes place, the accurate treatment of the nonrigid character of the out-of-plane vibrational mode is very important.This is evident from the comparison (Table II) of HFCC expectation values obtained as described above with the results of a more conventional PRM approach, commonly used to compute vibrational corrections to molecular properties. 6,18The PRM results in Table II were obtained variationally in the present work by expanding the TROVE Hamiltonian in normal coordinates with the KEO, PES, and HFCC parts truncated after zero-, quartic-, and second-order terms, respectively.All expectation values in Table II were obtained variationally but it is obvious that the use of normal coordinates and the restrictive truncation of the various series expansions introduce substantial changes in the expectation values.The PRM results deviate mostly from those of the nonrigid-model TROVE calculation for excited states of the out-of-plane bending mode.For such states, the deviations reach values around 10 G or 10%-20%.A "true" PRM calculation, using perturbation theory to solve the rovibrational Schrödinger equation, would introduce additional approximations and we surmise that it would produce results deviating even more from the nonrigid-model values.A complete list of HFCC vibrational expectation values for 13 CH 3 and 13 CD 3 is given in the supplementary material. 62n Table III, we give the theoretical equilibrium-geometry value of the HFCC for 13 CH 3 , the ZPVC, and the value resulting from the thermal averaging at T = 96 K.These values are compared with the results of the previous theoretical studies and experiment.In the previous theoretical studies, different electronic structure methods and basis sets were used to compute the potential and coupling constant surfaces and the vibrational corrections were treated by means of perturbation theory.As can be seen from the table, the HFCC value is strongly dependent on the ZPVC, which in this work is found to constitute about 44% of the equilibrium value.The temperature correction originating in excited rotationvibration states (i.e., the correction obtained on top of ZPVC) at T = 96 K is 0.02 G and thus tiny; it obviously increases with increasing temperature and attains a value of 1.0 G (see Table IV) for T = 300 K.For 13 CD 3 , the ZPVC is approximately 25% of the equilibrium HFCC value (25.8 G); the additional T = 300 K temperature correction has a small value of 1.7 G.  FIG. 2. The HFCC in 13 CH 3 calculated at the CCSD(T)/aug-cc-pVTZ-J level of theory for molecular geometries with all three C-H bond lengths fixed at 1.0759 Å and all three H-C-H bond angles equal.The HFCC is plotted as a function of ρ, the angle between the three-fold rotational axis and any one of the three C-H bonds.

IV. DISCUSSION AND SUMMARY
Table III confirms that for the theoretical, thermally averaged values of the HFCC of 13 CH 3 , the differences between the value at equilibrium A (eq) iso and the vibrationally/thermally averaged value A (tot) iso are dramatic.As mentioned previously, these differences are solely due to the effect of the ZPVC.Analysis of the contributions from the individual vibrational modes has shown that the dominant vibrational effect originates from the out-of-plane bending mode (corresponding to the "umbrella-flipping" inversion of ammonia NH 3 ).Other vibrational modes contribute only slightly since the associated fundamental and overtone states are hardly populated at T = 96 K.In Fig. 2, we show the dependence of the HFCC in 13 CH 3 on the out-of-plane vibrational coordinate ρ, which is defined as the angle between the three-fold rotational axis and any one of the three C-H bonds.Clearly, the strong dependence of the HFCC on ρ along with the effect of delocalization of the out-of-plane vibrational wave functions, due to the large amplitude character of the vibration, makes the corresponding expectation value and thus the contribution to the ZPVC quite substantial.
Our CCSD(T) equilibrium value A (eq) iso agrees well with the results of previous QCISD(T) 48 and MCSCF 49 (Multiconfigurational Self-Consistent-Field) calculations (Table III), with slightly larger deviation from the B3LYP result, 18 which is known to overestimate the spin polarization effect. 74The TABLE III.The calculated and experimental HFCC (in G) of 13 CH 3 .The values listed are the electronic HFCC at equilibrium geometry, A (eq) iso , the total HFCC value including ZPVC (T = 0 K) and temperature (T = 96 K) correction, A (tot) iso , as well values of the ZPVC (per temperature) effects with respect to the equilibrium.deviations can also be partly attributed to the effect of the different basis sets used in electronic structure calculations.We employed the basis set aug-cc-pVTZ-J, specifically designed for core properties.In several studies, 53,75,76 this basis set has proved to yield coupling constants in good agreement with experiment.

Method/basis
We conclude that with the high-level electronic structure method and comprehensive variational treatment of the rovibrational motion employed in the present work, we were able to obtain reliable values of the HFCC for 13 CH 3 and 13 CD 3 in very good agreement with experiment (Table III).In particular, we calculate realistically the large vibrational contribution to the HFCC, which we found to be 44% of the equilibrium value.In agreement with previous studies, the large vibrational contribution can be attributed to the large amplitude out-of-plane bending motion.For the temperatures considered in this study (T < 300 K), the thermal effects play a minor role.
vibrational energy levels of 12 CH 3 , 13 CH 3 , and13 CD 3 are listed in Table

TABLE I .
13lculated and experimental vibrational energies (in cm −1 ) for12CH 3 ,13CH 3 , and13CD 3 .Γ a Present work Reference 45 b Reference 46 c Experiment d Present work Present work a Irreducible representation spanned by the wave function.b ICMRCI+Q ab initio calculation.c CASSCF-MR-CI ab initio calculation employing the aug-cc-pVTZ basis set.d See Refs.69-72.e Zero-point energy.

TABLE II .
13brational energies, E vib , (in cm −1 ) and expectation values of HFCC (in G) computed for13CH 3 using the TROVE variational and the perturbed-rigid-molecule (PRM) approaches (see the text).
a Irreducible representation spanned by the wave function.
set A This article is copyrighted as indicated in the article.Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 84.135.206.7 On: Thu, 24 Dec 2015 17:02:18