J. Phys. Chem. 1986, 90, 1980-1983
1980
GENERAL PHYSICAL CHEMISTRY Theoretical Studies on the Barriers for Internal Rotation of the Methyl Groups in the teff -Butyl Radical J. Pacansky* and M. Yoshimine IBM Research Laboratory, S a n Jose, California 951 93 (Received: August 16. 1985) Restricted open-shell Hartree-Fock calculations are used to investigate the potential functions for internal rotation of the methyl groups in the tert-butyl radical. The optimized paths for three types of rotations are reported: Two are the A2 and E rotations permitted by the C,, symmetry of the radical; the third involves rotation of a single methyl group. The calculations reveal that the A2 mode is a sixfold potential function with a 1.69 kcal/mol barrier height, and that the E motion is also described by a sixfold potential function with two different minima one of which is very shallow. The barrier height that must be surmounted for the E mode to go between the well-defined minima, which are separated by a 120' rotation, is 1.86 kcal/mol. A threefold potential function with a 1.51 kcal/mol barrier height is computed for the single methyl rotation. In essence, all of the studies indicate that a reasonable value for the barrier height for internal rotation of methyl groups is 1.5 kcal/mol. This is true regardless of the manner in which the methyl groups are rotated as long as all of the geometrical parameters are optimized as a function of the internal rotation.
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Introduction
Matrix isolation experiments' and ab initio calculations2 have shown that the tert-butyl radical has a nonplanar geometry with a C3tisymmetry, a low frequency for the umbrella motion of the radical center, and, in comparison to alkanes, a relatively low barrier for rotation of a methyl group about an a-CCbond. Due to the importance of the barrier to internal rotation of the methyl groups with respect to the heat of formation of the tert-butyl r a d i ~ a l ~we- ~have extended our previous theoretical studies on this system. Here, we present extensive restricted open-shell Hartree-Fock calculations on the tert-butyl radical whereby all of the geometric parameters are optimized as a single methyl group is rotated about an a-CCbond. Similarly, results are also included for the changes in total energies which result when two and three methyl groups are rotated, respectively, according to the Cj, symmetry of the system. As a consequence, a reasonable estimate for the energetics of the methyl internal rotations in the tert-butyl radical is presented. Computational Details
The calculations in this report use the restricted open-shell Hartree-Fock (ROHF) formalism. The basis sets employed were the computationally efficient 4-3 1G basis,6 and, in addition, the 6-31G* basis' was also used to test for a basis set dependence on the barrier height for internal rotation. All of the calculations were performed using the computer code GAMES* which utilizes the gradient method to optimize geometries. This program also allows the user to fix one, or several, geometrical parameters, and optimize all of the others; hence, the energy paths for internal rotation of one or more methyl groups were readily obtained. Two types of energy paths were calculated: the first is for a rigid internal rotation where all of the geometrical parameters are constrained at their equilibrium values as rotation commences; the second is a relaxed rotation whereby all of the geometrical Chang, J. S. J . Chem. Phys. 1981, 7 4 , 5539. (2) Yoshimine, M.; Pacansky, J. J . Chem. Phys. 1981, 74, 5168. (3) Rossi, M.; Golden, D. M. Inf. J . Chem. Kine?. 1979, 1 1 , 969. (4) Walker, J. A.; Tsang, W. In?. J . Chem. Kinef. 1979, 1 1 , 867. ( 5 ) Parks, D. A.; Quinn, C. P. J . Chem. Soc, Faraday Trans. 1 1976, 7 2 , ( 1 ) Pacansky, J.;
1952. (6) Ditchfield, R.; Hehre, W. J.; Pople, J. A. J . Chem. Phys. 1971, 54, 724. (7) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 1973, 28, 213. (8) Dupuis, M.; Wendoloski, J.. J.; Spangler, D. Nat. Res. Compuf.Chem. Software Cut. 1980, I , QGOI.
0022-3654/86/2090-1980$01.50/0
TABLE I: The UHF and ROHF Optimized Geometries and Total Energies for the tert-Butyl Radical"
total energy R rl r2
0
PI 0 2
El 62
pyramidal angle 4 out-of-plane bending y
UHF
ROHF
ROHF
(4-31G)
(4-31G)
(6-31G*)
-156.450499 1.502 1.084 1.091 118.4 111.4 11 1.5 107.1 108.0 7.4 22.1
-156.446659 1.503 1.084 1.090 117.8 111.4 111.5 107.2 108.0 8.5 25.3
-156.670649 1.504 1.086 1.092 117.4 111.4 111.4 107.2 108.0 9.4 27.7
LI Units: bond lengths in angstroms, bond angles in degrees, and total energies in hartrees.
parameters are optimized as a function of the internal rotation angle. Results and Discussion
In a previous reportZthe optimized geometry of the tert-butyl radical was determined by using unrestricted Hartree-Fock calculations with a 4-31G basis set. Although all symmetry constraints were eliminated from the calculations, nevertheless, a structure with C,, symmetry was found. The infrared spectrum of the tert-butyl radical isolated in an argon matrix was consistent with the Cj, structure. This geometry is drawn in Figure 1, and Table I contains a listing of the optimized geometrical parameters. Also listed in Table I are results of R O H F calculations using a 4-31G and 6-31G* basis set, respectively. An inspection of Table I reveals that the R O H F results given in this report differ little from the previous U H F calculations. The salient features of the radical structure are the following: the a-CH bonds trans to the unpaired electron are slightly longer than the cis P-CH bonds; the a-CC bonds are short; and the radical center is nonplanar. The computer-drawn structure illustrated in Figures 1 and 2 clearly shows the degree of nonplanarity of the carbon skeleton. The enthalpy of formation of the tert-butyl radical has been investigated in a number of lab~ratories.~-~ An important aspect of these studies is an accurate value for the entropy which requires a knowledge of the molecular structure and dynamics of the radical. In particular, the data of upmost importance are the band
0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 9, 1986 1981
Internal Rotation of Methyl Groups
TABLE 11: Total Energies and Geometrical Parameters for the A2 Rigid Internal Rotation 6' b RHF(4-31G) AEa 0.0 8.5 -156.446 659 0.00 h
P'4
_-
30.0 60.0 90.0 120.0
P23
Figure 1. A computer drawing for the optimized geometry of the tertbutyl radical.
8.5 8.5 8.5 8.5
-156.443 160 -156.438 381 -156.443 228 -1 56.446 443
2.20 5.20 2.15 0.14
"Note that AE is not equal to zero at 6' = 120' because the methyl groups do not have local C, symmetry.
TABLE 111: Total Energies and Geometrical Parameters for the Relaxed A2 Internal Rotation 6' Cb RHF(4-31G). hartrees AE. kcal/mol 8.5 0.0 -156.446659 0.0 Yc
h4'
Figure 2. The out-of-plane bending angle, y, for the tert-butyl radical. y is defined as the angle between say the Cl-C4 bond and the plane formed by the C 2 C l C 3 atoms.
12.7 21.7 24.7 30.0 35.3 38.3 47.3 60.0
8.0 7.6 3.1 0.0 -3.1 -7.6 -8.0 -8.5
-1 56.445 740 -1 56.444 924 -156.444 353 -156.443 970 -156.444 353 -156.444924 -156.445 740 -156.446659
0.58 1.09 1.46 1.69 1.46 1.09 0.58 0.00
Figure 3. The A2 internal rotation.
6'
Figure 5. The computed energy paths (ROHF, 4-31G basis set) for the rigid (solid curve) and relaxed (dashed curve) A2 internal rotations. A E is in hartrees and 6' = 6 ] = 62 = 6, is the internal rotational angle in degrees. See Figure 6 for the definition of 6i, Figure 2 for y,and Tables I1 and 111 for a listing of the pertinent geometrical parameters.
H32
0
QH2 Figure 4. The E internal rotation.
centers for vibrational modes below -600 cm-', and the barriers for internal rotation of the methyl groups. Previously, we reported a normal-coordinate analysis9 for the tert-butyl radical, and hence the problems associated with low-frequency vibrational modes have been addressed. The energetics for the internal rotation of the methyl groups is addressed in the text that follows. Three types of internal rotational modes are considered: the first two conform to the C,, point group symmetry, and the third involves the internal rotation of one methyl and hence provides a value for the energy required and the path taken when one methyl group is rotated neglecting symmetry constraints. For a point group with C,, symmetry, the irreducible representations for the internal rotations of the methyl groups are A2 and E. The A2 motion involves a concerted in-phase rotation of each methyl group as described in Figure 3. The E rotation, illustrated in Figure 4, consists of an out-of-phase rotation of two methyl groups with the third one stationary. The computed results (9) Schrade, B.; Pacansky, J.; Pfeiffer, U. J . Phys. Chern. 1984, 88, 4069.
H 2 3 k 2 ,
H4 lw
H43
v
H42
Figure 6. The angle for internal rotation of a methyl group, 6 , = L(H43C4)(C4ClC2) - 76.79', 6, = L(H23C2)(C2ClC3) - 76.79', 6, = L(H33C3)(C3ClC4) - 76.79'.
(ROHF, 4-31G basis set) for the AI rotation are shown in Figure 5 and listed in Table 11. Figure 5 is a plot of AE, the energy
1982 The Journal of Physical Chemistry, Vol. 90, No. 9, 1986 40
1
TABLE IV: Total Energies and Geometrical Parameters for the Rigid E Internal Rotation 6" ROHF(4-31G), hartrees A E , kcal/mol 0.0 -1 56.446 659 0.0
- 30
E
L
P
20
c
U
Q
Pacansky and Yoshimine
10
0
0
30
60
90
120
15.0 30.0 45.0 60.0 75.0 90.0 105.0 120.0
-1 56.445 897 -1 56.444 080 -1 56.442 268 -156.441 467 -156.442 121 -156.443 954 -156.445 769 -156.446516
0.48 1.62 2.76 3.26 2.85 1.70 0.56 0.09
6"
Figure 7. The computed energy path (ROHF, 4-31G basis set) for the E rigid (solid curve) and relaxed (dashed curve) internal rotations. A E is in hartress and 6" = 6,= -8, is the internal rotational angle in degrees. See Figure 2 for the definition of 6i and Tables IV and V for a listing of the pertinent geometrical parameters.
relative to the optimized geometry, vs. 6' = 61 = 6, = a3, where each 6, is the angle for internal rotation about a particular CC bond (see Figure 6 for the definition of 6). The curve indicated by the solid line represents the energy requirements for a rigid A, rotation. Hence, as shown in Figure 5, the consequence of fixing all of the geometrical parameters at their equilibrium values is a threefold rotational potential function with a 5.2 kcal/mol barrier height. However, as shown by the dashed curve in Figure 5, when all of the geometrical parameters are optimized as a function of 6', a sixfold potential function is formed with a lower barrier height, 1.69 kcal/mol. The change from a threefold to a sixfold potential function, accompanied by a lowering of the barrier height, is a direct consequence of allowing the carbon skeleton to planarize. The geometry at the top of the barrier at 6' = 30' has a planar structure; thus as 6' increases from 0' to 30°, the out-of-plane bending angle, y (see Figure 2 for the definition of y), changes from the equilibrium value 25.3' to 0'. For values of 6' > 30' the carbon skeleton inverts and relaxes back to y = -25.3' when 6' = 60'. Consequently, for each 60' A, rotation the out-of-plane bending angle y changes a net 50.6'. The path taken for the E internal rotation is more complicated than for the A, mode. The solid and dashed curves shown in Figure 7 are the energy paths for the rigid and relaxed E internal rotations, respectively. A E is the energy relative to the optimized geometry and 6I' = 6, = -6, is angle for the out-of-phase internal rotation. The rigid rotation results in a threefold potential function with a 3.26 kcal/mol barrier height. The potential function for the relaxed E rotation takes on a vastly different character as shown in Figure 7 . For the relaxed E internal rotation, an asymmetric sixfold potential function is formed with two different minima. One is well-defined and is 1.25 kcal/mol lower than the other which is very shallow. A 1.63 kcal/mol barrier separates the lower minimum at 6" = 0, from the shallow minimum at 6" 90'. In order for the internal rotation to proceed from one well-defined minimum to another, 6" must change by 120' and a barrier height equal to 1.86 kcal/mol must be overcome. As increases from 0' to 60°, the carbon frame planarizes and, when at the top of the lower energy barrier, it inverts to y = -4.6'. As arr approaches 90' the system goes toward the shallow minimum, and concomitantly, y becomes more negative and eventually reaches -19.4'. Past 6'' = 90' to 110' we find that the carbon skeleton reverts back to the planar geometry, and when 6" = 120' finally returns to the original nonplanar geometry at y = 25.3'. Consequently, the carbon skeleton goes through a planar structure twice for the relaxed E rotation, and the structure at 6" = 90' is partially inverted relative to 6" = 0' and 120'. The values for 6", y,and AE for the rigid and relaxed rotations are listed in Tables IV and V . The energetics for the optimized A2 and E internal rotations are influenced by the "gearing" effect of the methyl groups. The A 2 rotation where all of the methyl groups move in phase only requires a 30' rotation for the carbon frame to planarize, and another 30" to invert and return to the equilibrium structure. The
TABLE V: Total Energies and Geometrical Parameters for the Relaxed E Internal Rotation 6" ROHF(4-31G), hartrees A E , kcal/mol Y 25.3 0.0 -1 56.446 658 0.0 16.0 0.39 -4.56 -14.7 -16.8 -19.4 -13.7 0.30 26.5 25.3
30.0 60.0 60.0 75.0 80.0 90.0 105.0 110.0 110.0 120.0
-156.445 161 -156.443 794 -156.444 061 -156.444449 -1 56.444 556 -1 56.444 547 -1 56.443 763 -156.443 691 -1 56.446 350 -1 56.446 658
0.94 1.78 1.63 1.39 1.32 1.33 1.82 1.86 0.19 0.00
TABLE VI: Total Energies and Geometrical Parameters for the Relaxed and Rigid Internal Rotation of One Methyl Group in the tert-Butyl Radical 4-31G basis set 6-31G* basis set 6, E , hartree AE, kcal/mol E , hartree A E , kcal/mol 0.0 -1 56.446 609" 0.0 -156.670649" 0.0 30.0 60.0 60.0 90
-156.445 584" -156.444593n
0.64 1.27
-156.669284' -156.667911' -156.668243' -156.669 208*
0.86' 1.72' 1.51' 0.90'
"Geometry optimized for each value of 6,. 'Total energy computed by changing 6, with all other geometrical parameters constrained at their equilibrium CIGvalues.
-
-
0
30
60
90
120
6
Figure 8. The computed energy paths for the rigid and relaxed internal rotation of a single methyl group in the tert-butyl radical: rigid rotation, F.OHF, 6-31G* basis set ( 0 ) ;relaxed rotation (optimized in C, sym-
metry), ROHF, 4-31G basis set (A), ROHF, 6-31G* basis set
(W).
gearing effect renders complete inversion for the E rotation impossible; while one methyl group remains stationary, the other two, rotated out of phase, must move 120' to return to the equilibrium structure. Thus the carbon skeleton does not invert from y = 25.3' to y = -25.3' as was found for the A, rotation but oscillates around a planar geometry and then returns to y = 25.3'. However, the net effect of the excursions around a planar
J . Phys. Chem. 1986, 90, 1983-1984 carbon frame is to lower the barrier for E internal rotational. We should also add that the energy path shown in Figure 7 for the relaxed E rotation appears to be unusual at 6” 110’. This is due to the fact that every geometrical parameter is being optimized as a function of 6” and hence 6” is not a good reaction coordinate on this part of the potential surface. Thus, at 6” 1loo, although 6” changes little, the energy changes considerably because of the large changes in y. Thermodynamically, the barrier to internal rotation of one methyl group is very important. As shown in Figure 8, the potential function computed by using the ROHF theory with a 4-31G basis set is threefold with a 1.27 kcal/mol barrier height. This calculation was performed by optimizing all of the geometrical parameters as a function of internal rotation of only one methyl group, say al. Polarization effects on the potential function were investigated by using a 6-31G* basis set; in this case a barrier height of 1.72 kcal/mol is calculated for a rigid rotation, and a 1.51 kcal/mol barrier is computed when all of the geometrical parameters are calculated as a function of 6,. Since the barrier height for the 6-31G* and 4-31G basis sets differ little, we take this as an indication that polarization effects may not play a
-
1983
dominant role. The important results here are a threefold potential function for internal rotation for one methyl is found with a barrier height around 1.5 kcal/mol, and whether a rigid or relaxed rotation is taken the height of the barrier changes very little. The gearing effect noted for the A2 and E internal rotations does not appear to be an issue for the single methyl rotation because very little rotation of the other methyl groups was observed during the optimization process. As a consequence, a number of geometrical parameters relaxed to allow the internal rotation of the single methyl group; nevertheless, a 120’ rotation was still required to reach each minimum on the energy path. Summary and Concluding Remarks
The calculations presented in this report optimize all of the geometrical parameters as internal rotation about an a-CC bond commences. Consequently, the energy paths shown for A,, E, and single methyl internal rotations most likely are those taken by the radical upon thermal activation. The lowest barrier found is for internal rotation of a single methyl group, 1 .S kcal/niol, and is the value that should be used in entropy calculations for the tert-butyl radical.
-
COMMENTS The Utility of Ion-Dlpole Orbltlng Potentlal In Obtalnlng the Inner-Sphere Reorganization Energy of Ions In Solutlon Sir: It is now a common practice to decompose the total reorganization energy, R, into inner-sphere, Ri, and outer-sphere, R,, components.’.* Recently, we proposed a formalism for computing R, via the ion-dipole capture (orbiting) potential and a simple model of reorganization p h e n ~ m e n a . ~In order to compare the with experimental results, we utilized results of our theory, Ri,caldr the following equation to extract the experimental inner-sphere reorganization energies, R,,,, from the photoemission data of Delahay: R, = R, R;,m
+
where R, is the total (measured) reorganization energy associated with photoemission in solution and R, is the outer-sphere reorganization energy. Clearly, the agreement between the Rl,caldand R,,,. which we reported earlier is subject to the uncertainties associated with the computation of R, via the nonequilibrium solvent polarization In this Comment we therefore report the results obtained using an alternate method which does not require a knowledge of R,. This method is based on a correlation between the thermal electron exchange in solution and the solution phase photoemission of electrons.6 According to this correlation we have ( I ) Bcckris, J. OM.; Khan, S. U. M. Quantum Electrochemistry; Plenum Press: New York, 1979; Chapter 6. ( 2 ) Cannon, R. D. Electron Transfer Reactions; Butterworth: London, 1980. (3) Tunuli, M. S . ; Khan, S.U. M. J . Phys. Chem. 1985, 89, 4667. (4) Delahay, P.; Dziedic, A. J . Chem. Phys. 1984, 80, 5793. (5) Marcus, R. A. J . Phys. Chem. 1963, 67, 843.
0022-3654/86/2090-1983$01.50/0
Rx = Rm + [Vf, - f z + l ) / V ; +fz+l)lRi,m (2) wheref, is the equilibrium force constant (for the ion-ligand bond stretching) for the reduced species (with charge z ) andf,,, is the corresponding force constant for the oxidized species (with charge z 1). In addition, R, is the total reorganization energy of the thermal electron exchange process. The free energy of activation for the exchange process, AG,*, is related to R, via the work, w , of bringing the two reacting species together from infinite separation to the activated complex as AGx* = w RJ4 (3)
+
+
In order to compare the calculated values AC,,,,lcd* with the corresponding experimental values* ilG,,,,pi* we replace R,.min eq 2 with Rl,calcd and then we use our inner-sphere formalism to computef,, fr+,, and Rl,calcd as follows: &.calcd
= (n/2)f(z
+ 1,
‘r+l)Q2
(4)
(6) (a) Delahay, P.; Dziedic, A. J . Chem. Phys. 1982, 67, 843. (b) In a more rigorous theoretical analysis (results of which are thoroughly substantiated by the experimental data) I have recently (submitted to J . Phys. Chem.) shown that eq 2 is slightly in error. The accurate equation is: Rx = Rm + Kxfr - f , ) / C f ,
+ fo)lR.m
(?a)
where x = 4.2 (theory) or 3.6 (experiment). Results obtained via eq 2a and x = 4.2 are given in the last column of Table I and show an improved theory/experirnent agreement. (7) Chesnavich, W. J.; Su,T.;Bowers, M . T. J . Chem. P h ~ s 1980, . 72* 2641. (8) In ref 1, the AGx,Slpt*values reported in Table I were obtained by using the measured specific rates and assuming K = 1 (where K, the transmission coefficient, measures the nonadiabaticity of the reaction path). However. for the aquo cations used in this work K is now known to be
