J. Phys. Chem. 1995, 99, 81-84
81
Theoretical Study on the Effect of a-Substituents on the 13C, Hyperfine Splitting Constants of Alkyl Radicals Maurizio Guerra Istituto dei Composti del Carbonio Contenenti Eteroatomi e lor0 Applicazioni, CNR Via Gobetti 101, 40129 Bologna, Italy Received: May IO, 1994; In Final Form: August 23, 1994@
The isotropic 13C hyperfine splitting (hfs) constant, a(13Ca), of symmetrically a-substituted methyl radicals X3c' has been computed as a Boltzmann average over the out-of-plane bending vibration by ab initio calculations at the uMPuDZP//TZP level of theory. Experimental values from ESR spectra are well reproduced by these calculations. The large variations in a(l3Ca) observed experimentally upon a-substitution are due not only to structural changes but also to the electronic effect of the a-substituents. The latter effect is largely dominant with strongly electronegative substituents. The value of a(13C) for X = F (269.8 G) is computed to be about twice that for X = OMe (150 G), whereas both radicals are computed to have a nearly tetrahedral structure. This confirms that in localized radicals care must be exercised in obtaining structural information comparing the experimental hfs constants of radicals bearing a-substituents with different electronegativity.
Introduction
2
The structure of radicals in which the unpaired electron is mainly localized on an atom is usually determined from the s-character of the singly occupied molecular orbital (SOMO) estimated from the experimental isotropic hyperfine splitting (hfs) constants.'q2 This approach is based on the assumption that a direct relation exists between hybridization in the SOMO and geometry at the radical center and that spin-polarization contributions are negligible. The validity of this approximation was supported by ESR and theoretical MO investigations on a-substituted alkyl radicals which showed that the magnitude of the 13Chfs constant, a(13C3, is indeed related to the degree of pyramidalization at the radical center, the out-of-plane bending angle, 6 (Figure l), being predicted both experimentallyZand theoretically3z4to increase with increasing electronegativity of the a-substituents. On the other hand, we have recently shown by means of ab initio calculations at the UMP21 DZP level of theory that the large variations observed experimentally in the z9Sihfs constants of symmetrically a-substituted silyl radicals X3Si' are essentially due to the electronic effect of the a-substituents rather than to structural changes at the radical enter.^ This surprising finding has been attributed to the very low electronegativity of the central silicon atom. In this study, the structures and 13C hfs constants of symmetrically a-substituted methyl radicals X3C' have been evaluated with ab initio calculations to establish whether electronic effects of a-substituents may sizably influence hfs constants at radical centers also in alkyl radicals.
A
Methods and Computational Details Ab initio unrestricted Hartree-Fock (UHF) calculations have been performed on symmetrically a-substituted methyl radicals X3c' with the GAUSSIAN 92 system of programs6 running on a RISC-6000 IBM computer. Theoretical hfs constants have been computed with the DunningMuzinaga full double-c basis set' (DZ) supplemented with polarization functions (P), pfunctions on hydrogens and d-functions on heavy atoms,* employing correlated spin densities obtained from the generalized density matrix9 corresponding to the second-order Moller@Abstractpublished in Advance ACS Abstracts, November 15, 1994.
Figure 1. Definition of the out-of-plane bending angle.
Plesset perturbation energy.'O This level of theory (UMP2DZP) was found to give hfs constants in good accord with experiment.S,ll~lzThe uMp2iDZP wave function correctly describes the doublet state, since its contamination by higher spin multiplets is quite small, the expectation value of s2 being of the order of 0.76. Geometries have been optimized within the constraint of C3 or C3" symmetry by means of an analytical gradient techniq~e'~ using the more flexible 6-3 11** triple-5 basis set14J5 (TZP) but not including the effects of electron correlation. This choice has been adopted for the following reasons. Geometry optimization of radicals bearing methyl groups on a-substituents requires a considerable amount of computer resources at the UMP2DZP level. Replacement of methyl groups by hydrogens was found to produce a sizable variation in the computed hfs constant of the tris(trimethylsily1)silyl r a d i ~ a l . ~The J ~ 6-311G** basis set was found to provide structural parameters, especially bond angles, with accuracy.17 For example, preliminary calculations on halo derivatives have shown that the inclusion of electron correlation (UMP2/TZP) changes only slightly the optimized values of the out-of-plane angle 6, the structural parameter on which the magnitude of a(13Ca) mainly depends. Indeed, this structural parameter has been found to be more sensitive to the flexibility of the basis set than to electron correlation. The optimum value of 6 has been computed to be 17.53', 17.54", and 17.72' for F3C' and 10.30', 10.44", and 12.03" for C13C' at the UHF/TZP, UMP21 TZP, and UMP2DZP level, respectively. Lastly, also vibrational frequencies and inversion barriers computed employing
0022-365419512099-0081$09.0010 0 1995 American Chemical Society
Guerra
82 J. Phys. Chem., Vol. 99, No. 1, 1995
TABLE 1: Optimum (60)and Boltzmann Statistically Averaged (6,)" Out-of-Plane Angles (degree) of X3C' , 6 ( Radicals at the UHF/TZP Level along with Values ) Estimated from the Experimental Values of a(13C3 Reaorted in Table 2 0.0
H H3C Me0 F MeqSi MeS
5.8 9.0 16.5 17.7 7.8 10.3 10.2
8.49 16.32 17.53 6.06 8.83 10.30
c1
TABLE 2: Theoretical (UMP2/DZP//TZP) and of X3C' Radicals" Experimental t ~ ( ' ~ CValues 3 X theory" aexDC Einvd k," J Ts sym ref Hh H&Y' Me@
4.4 6.3 17.7 22.2 4.4 12.4 16.7
F' Me&"' MeS" C1" (I
large basis sets were found to be little affected by electron ~orre1ation.l~ Vibrational effects at a given temperature T have been estimated averaging the computed hfs constants over the thermally populated vibrational states of the pyramidal inversion mode (umbrella mode)
m
The eigenfunctions I+,,, and eigenvalues Em have been taken as solutions of the Hamiltonian %for the inversion mode which has been approximated with a one-dimensionaldouble-minimum potential
+
+
X= -h2 a2qjs& a26 kS2/2 v exp(-cd2)
(2)
where ,D is the reduced mass kept fixed to the value calculated at the equilibrium geometry. For the planar methyl radical the last term of the potential, which determines the barrier height of double-well potentials, has been replaced by a quartic term that accounts for the anharmonicity in the out-of-plane mode,
E=-h a2qjs&
+
+
a26 kd2/2 bs4/2
(3)
The potential energy parameters are uniquely determined by the curvature at the minimum (h)in conjunction with the energy barrier to inversion @inv) for eq 218 and with the fourth derivative at 6 = 0" for eq 3. Derivatives have been computed numerically. Energy barriers have been estimated from energy differences between the minimum and planar structures, the latter being assumed to be the saddle point along the doublewell potential describing the inversion mode. The hfs constant of the central carbon has been expanded in an even-power series of 6
a(6) = & s n
38.3 0.0 0.053 2.435 96 D3h' 49.5 1.83 0.510 10.850 91 C3" 152.7 23.07 1.876 11.858 153 C3 272.8 32.36 2.125 14.803 120 C3" 29.0 3.20 0.468 14.578 333 C3" 54.2 2.42 0.454 12.639 323 C3 116.7 2.29 0.713 14.552 20 C3"
22 23 24 25 26 26 27
Vibrational corrections due to out-of-plane bending mode are given
in parentheses. Inversion barriers (Emv),curvatures at the minima (k,,,),
6, = ((&*))ln.
m
35.3 (10.3) 55.3 (0.9) 150.0(-0.2) 269.8 (-0.2) 32.5 (4.5) 49.2 (5.6) 126.4 (-4.6)
(4)
n
The expansion coefficients have been determined by means of a least-squares fitting of the a(13Ca) values computed as a function of 6 at 2.5" intervals in the range 0-30". A polynomial of 14th degree has been used to achieve a precision up to the second decimal digit. The Hamiltonian m a s been set up in the basis of the eigenfunctions of the harmonic o s ~ i l l a t o rAn .~~ expansion up to the 40th term is sufficient to obtain complete convergence for the thermally populated states. Results and Discussion The optimized values of the out-of-plane angle (do) and the barriers to inversion (,Tinv) of a-substituted methyl radicals X3C computed at the UW/TZP level of theory are reported in Tables 1 and 2 , respectively. The methyl and trifluoromethyl radicals
and structural parametersb are also reported. Bond lengths in angstroms,dihedral and bond angles in degrees, z is the symmetry axis. In Gauss. In kcal mol-'. E In au rad-2. f In au. 8 In K. r(CH) = 1.0735. The anharmonicity constant b is equal to 1.448 au rad-4. jr(CC') = 1.5034, r(C'H) = 1.0887, LHC'C = 111.41. kr(CO) = 1.3463, r(0C') = 1.406, LCOC' = 116.03, w(zC0C') = 57.22, r(C'H), 1.0829, LOC'H = 109.51, w(C0C'H) = 67.24. 'r(CF) = 1.2954. "'r(CSi) = 1.9352, LCSiC' = 113.79, (Sic') = 1.8927, r(C'H) = 1.085, LSiC'H = 111.41. " r(CS) = 1.767, LCSC'= 101.6, o(zCSC') = 41.32, r(SC') = 1.8168, r(C'H) = 1.081. LSC'H = 109.22, w(CSC'H) = 60.08. r(CC1) = 1.7171.
are computed to be planar and nearly tetrahedral, respectively, as in previous MO calculation^^^^ and in accord with the structures determined from the experimental a(13Ca) values (aexp).20~21 Radicals containing a-substituents (namely, 0 and F) much more electronegative than carbon have a nearly tetrahedral structure and a high barrier to inversion. (This class of radicals is denoted as A.) All other radicals have a slightly bent structure and a low inversion barrier (class B). Theoretical hfs constants computed at the UMP2/DZP//TZP level as a Boltzmann average over the pyramidal bending mode are reported in Table 2. The vibrational contribution is large for the planar methyl radical, small for radicals of class B, and negligible for radicals of class A. Interestingly, the vibrational frequency computed for the 12C methyl radical (595 cm-l) is in excellent agreement with the experimental value (607 cm-1).28 UHFISTO-3G calculations on the tert-butyl radical29 showed that the inversion motion is coupled with the rotation of the methyl groups. For radicals of class B bearing groups on the a-substituents, the vibrational correction reported in Table 2 must be therefore considered to be only qualitative although the a(13Ca) value is estimated to be little affected by the conformation of groups attached to the a-substituents. The vibrationally averaged values of a(13Ca)are in good agreement with experiment. However, the difference between the experimental and theoretical values is slightly greater than that found in a-substituted silyl r a d i c a l ~ . ~Displacements ?'~ of the minima by less than 1" make the theoretical hfs values match the experimental values. This lends confidence to the local radical geometries determined at UW/TZP level. Tables 1 and 2 show that there is not a straight relation between the degree of pyramidalization at the radical center and the trend of the computed values of u ( ' ~ C ~ For ) . example, in accord with experiment, the computed value of a(13Ca) is much smaller for X = OMe than for X = F even though the Boltzmann statistically averaged out-of-plane angle 6 , is computed to be of the same magnitude in these two radicals. The same trend emerges from a comparison of the theoretical results for X = C1 with the corresponding ones for X = SMe. Such finding suggests that spin distribution could be influenced significantly by the electronic effects of the a-substituents as found previously in a-substituted silyl radical^.^ Indeed, the variation of u ( ' ~ C ~with ) geometry is smaller than expected in the absence of the electronic effect of the a-substituents. The computed value of a(13Ca) in methyl increases from 25 to 125
J. Phys. Chem., Vol. 99, No. 1, I995 83
13Ca Hyperfine Splitting Constants of Alkyl Radicals
TABLE 3: a-Carbon Electron Distribution in the SOMO of X3C. Radicals along with the Corresponding Electron Transfer to the a-Substituent electron distribution (%) electron X 2s 2P 3d transfer (%)
250/
200 -0Me
150 -H
c1
I
0
I
1
5
l
I
l
15
10
l
I
I
20
6
Figure 2. UMP2DZP values (gauss) of a(13Ca)as a function of the out-of-plane angle 6 (degrees) for X3c' radicals.
201
0-SiMeg
2
2.0
Mersi MeS
I
I
0.0
H3C Me0 F
100 -
50
H
3 Electronegativity
4
2
3 Electronegativity
4
Figure 3. (a) C 2s population (%) in the SOMO and (b) C 2s total
in the constrained tetrahedral structure of X3c' radicals spin density (a) versus the Allred-Rochow electr~negativity~~ of the a-substituents.
G on going from the planar to tetrahedral configuration, whereas aexpin a-substituted methyl radicals ranges from 29 G for the slightly pyramidal (Me3Si)3C' radical to 272.8 G for the nearly tetrahedral F3C' radical. The values of a(13Ca) computed as a function of the out-of-plane bending angle 6 from eq 4 are reported in Figure 2 to estimate the relative importance of the structural and electronic effects. As expected, in the planar structure the influence of the a-substituent is small since only the spin-polarization term contributes to the value of a(13Ca). In bent structures, the variation of a(13Ca) with geometry depends strongly on the electronic nature of the a-substituents. The variation is large with strongly electronegative groups such as methoxy, chlorine, and fluorine and small for the other substituents. In particular, the influence of the electronic effect is estimated to be about twice that of the structural effect in halo derivatives. The computed value of a(13Ca) increases by only 33.2 and 83.0 G on bending the radical center in the methyl radical from the planar structure to that optimized for C13C' and F3C, respectively, whereas Table 2 shows that the increase of a(13C,) on going from methyl to the halo derivatives is as large as 106 and 245 G for X = C1 and F, respectively, excluding the vibrational correction. Interestingly, in the trifluoro derivative a bending of only 6" gives the same value of a( "C,) computed for methyl in the tetrahedral structure (125 G). Indeed, Figure 3 shows that in the constrained tetrahedral structure, which should correspond to sp3 hybridization, the s character of the SOMO as well as the 2s total spin density vary sizably with the electronegativity of the a-substituents. Thus, there is not a direct relation between the percentage of s
18.4 21.5
0.2 0.7 3.4
98.9 66.8 36.0 43.6 57.9 25.3 36.0
0.0 0.1
1.o
1.3 0.0 0.0 0.2
1.1 31.5 44.6 33.6 41.9 74.0 60.5
character in the SOMO and the local radical geometry also in alkyl radicals in contrast with the view commonly accepted by ESR spectroscopist.2 The variation of the s character with electronegativity displayed in Figure 3 is, however, smaller than that found previously for a-substituted silyl radical^.^ This trend is consistent with the smaller electronegativity difference between the central atom and a-substituents occurring in alkyl radicals. In silyl radicals the increase of s character was explained in terms of increasing mixing of the valence p orbital at the radical center A with the o*(A-X) MO which has large s character. According to PMO theory, this mixing is inversely proportional to their energy difference (A&~*(A-x) in the planar form. A E P - , , * ( ~ -is~ expected ) to increase as the electronegativity difference between the central atom A and the substituents X decreases31so that the p-o*(A-X) mixing in the tetrahedral structure should be smaller in alkyl than in silyl radicals. It should be mentioned that the magnitude of a(13Ca) depends also on factors that influence the radial flexibility of the 2s carbon AO. In fact, the value of a(13Ca) depends essentially on the spin density of the 2s inner-valence function as previously found in silyl radicals. For example, in contrast with the trend expected on the basis of electronegativity, Figure 2 shows that also in this case the effect of the chlorine atoms is greater than that of the methoxy groups. Nevertheless, there is a good agreement between the structures of trifluoromethyl determined by ab initio calculations and from experiment, both predicting a nearly tetrahedral structure. It should be, however, considered that in estimating hybridization in the SOMO from experiment it was assumed that the unpaired electron is completely localized at the radical center. By contrast, Table 3 shows that in the SOMO in the optimized structures there is a sizable delocalization of the unpaired electron onto the a-substituents. Hence, the values of the outof-plane angle de, have been here computed taking into account the spin-polarization (awl) and spin-delocalizationcontributions. A reliable estimate of awl has been obtained from the UMP2/ DZP calculations since the present calculations provide 13Chfs constants which are in good accord with experiment and the spin-polarization contribution is expected to decrease very slowly with pyramidal it^.^^ The s character normalized to the electron population in the SOMO at the radical center (e)is given by s = (aexp
- apolYA&
(5)
where A0 is the hfs value for a C 2s orbital with a unitary population. An A0 value of 1115.4 G has been used for c a r b 0 n . l ~ ~The ~ values of 6,, have been estimated from the normalized s populations assuming complete orbital following34
aexp= arccos[2/(s + 2)]"2
(6)
To take into account the vibrational effect, the values of 6exp have been compared in Table 1 with the Boltzmann statistically
Guerra
84 J. Phys. Chem., Vol. 99, No. 1, 1995
averaged values (aav). The agreement between ,a, and 6 , is very poor. The apparently good agreement between theory and experiment in F3C' comes from the incorrect assumption in the experimental approach that the unpaired electron is entirely localized on the central carbon. The consequently minor percentage of C, 2s character estimated from aexpprobably balances the neglect of the electronic effects of the a-fluorine atoms which could increase the s spin density on the central carbon. Conclusion The experimental 13C hfs constant of the central carbon in symmetrically a-substituted methyl radicals is predicted very well by UMP2/DZP//TZP calculations including a statistical average over the out-of-plane bending mode. The s spin density at the central carbon has been found to depend strongly on the electronic effects of the a-substituents so that hybridization in the SOMO cannot be directly related to the geometry at the radical center as found previously for a-substituted silyl radicals. The conclusion can be drawn that caution must be always exercised in using experimental hfs constants to determine local radical geometries in localized radicals. References and Notes (1) Symons, M. C. R. Chemical and Biochemical Aspects of Electron Spin Resonance Spectroscopy; Van Nostrand Reinhold: Melboume, 1978. (2) Nonhebel, D. C.; Tedder, J. M.; Walton, J. C. Radicals; Cambridge University Press: Cambridge, 1979. (3) Morokuma, K.; Pedersen, L.; Karplus, M. J . Chem. Phys. 1968, 48, 4801. (4) Beveridge, D. L.; Dobosh, P. A.; Pople, J. A. J . Chem. Phys. 1968, 48, 4802. (5) Guerra, M. J . Am. Chem. SOC. 1993, 115, 11296. (6) Frisch, M. J.; Trucks, G. W.; Head-Gordon, M.; Gill, P. M. W.; Wong, M. W.; Foresman, J. B.; Johnson, B. G.; Schlegel, H. B.; Robb, M. A,; Replogle, E. S.; Gomperts, R.; Andres, J. L.; RaghavachaA,K.; Binkley, J. S.; Gonzalez, C.; Martin, R. L.; Fox, D. J.; Defrees, D. J.; Baker, J.;
Stewart, J. J. P.; Pople, J. A. GAUSSIAN 92, Revision C; Gaussian, Inc.: Pittsburgh, PA, 1992. (7) Dunning, T. H.; Hay, J. P. In Modern Theoretical Chemistry; Schaefer JII, H. F., Ed.; Plenum Press: New York, 1976. 18) Frisch. M. J.: PoDle. J. A.: Binklev. J. S. J . Chem. Phvs. 1984. 80. 3265,3269. ' (9) Wiberg, K. B.; Hadad, C. M.; Le Page, T. J.; Breneman, C. R.; Frisch, M. J. J. Phys. Chem. 1992, 96, 671. (10) Krishnan, R.; Frisch, M. J.; Pople, J. A. J . Chem. Phys. 1980, 72, 4244. (11) Carmichael, I. Chem. Phys. 1987, 116, 351. (12) Cremonini, M. A,; Luna&, L.; Placucci, G.; Guerra, M. J . Org. Chem. 1992, 57, 5963. (13) Schlegel, H. B. J . Comput. Chem. 1982, 3, 214. (14) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. J . Chem. Phys. 1980, 72, 650. (15) McLean, A. D.; Chandler, G. S. J . Chem. Phys. 1980, 72, 5639. (16) Guerra, M. J . Org. Chem., submitted for publication. (17) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986. (18) Fluder, E. M.; De La Vega, J. R. Chem. Phys. Lett. 1978, 59,454. (19) Chan, S. I.; Stelman, D. J. Chem. Phys. 1963, 39, 545. (20) Karplus, M.; Fraenkel, G. K. J . Chem. Phys. 1961, 35, 1312. (21) Fessenden, R. W.; Shuler, R. H. J . Chem. Phys. 1965, 43, 2704. (22) Fessenden, R. W. J . Phys. Chem. 1967, 71, 74. (23) Wood, D. E.; Sprecher, R. F. Mol. Phys. 1973, 26, 1311. (24) Brunton, G.; Ingold, K. U.; Roberts, B. P.; Beckwith, A. L. J.; Krusic, P. J. J . Am. Chem. Soc. 1977, 99, 3177. (25) Griller, D.; Ingold, K. U.; Krusic, P. J.; Smart, B. E.; Wonchoba, E. R. J. Phys. Chem. 1982, 86, 1376. (26) Schlecker, R.; Henkel, U.; Seeback, D. Chem. Ber. 1977,110,2880. (27) Mishra, S. P.; Symons, M. C. R. Int. J . Radiat. Phys. Chem. 1975, 7, 617. (28) Tan, L. J.; Winer, A. M.; Pimentel, C. G. J. Chem. Phys. 1972,57, 4028. (29) Carmichael, I. J . Phys. Chem. 1985, 89, 4727. (30) Allred, A. L.; Rochow, E. G. J . Inorg. Nucl. Chem. 1958,5, 264. (31) Cherry, W. R.; Epiotis, N. D.; Borden, T. W. Arc. Chem. Res. 1977, 10, 167. (32) Barone, V.; Douady, J.; Ellinger, Y.; Subra, R.; Pauzat, F. Chem. Phys. Lett. 1979, 65, 542. (33) Froese, C . J . Chem. Phys. 1966, 45, 1417. (34) Coulson, A. C. Valence; Clarendon Press: Oxford, 1953. I
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