16162
J. Phys. Chem. 1996, 100, 16162-16168
Carbon-Carbon Rotational Barriers in Butane, 1-Butene, and 1,3-Butadiene Mark A. Murcko,*,1a Henry Castejon,1b and Kenneth B. Wiberg*,1b Vertex Pharmaceuticals, Inc., Cambridge, Massachusetts 02139, and Department of Chemistry, Yale UniVersity, New HaVen, Connecticut 06520-8107 ReceiVed: July 18, 1996X
The rotational barriers for n-butane, 1-butene, and 1,3-butadiene were calculated at the G2 and CBS-Q theoretical levels. The thermodynamic functions were obtained with explicit calculation of the effect of the rotational modes. The G2 difference in energy between the syn and anti forms of butane at 298 K is 5.1 kcal/mol, which is significantly larger than the experimental estimate. However, it is shown that a reliable experimental estimate of the barrier cannot be made based on the currently available data. The structural changes on rotation are examined and are related to the changes found for C-C bond rotation in ethane. The G2 model reproduces the observed relative energies for both 1-butene and 1,3-butadiene within the experimental uncertainties, and the CBS-Q model also gives generally satisfactory results.
Introduction Potential functions for small molecules have received extensive use in developing molecular mechanics models2 and in conformational studies of polymers.3 Torsional potential functions are of particular importance since they are relative “soft”, and torsional angles may change considerably in the presence of steric or electrostatic forces. Thus, it is important to have accurate values for these functions. The small hydrocarbons n-butane, 1-butene, and 1,3-butadiene are key compounds in developing intramolecular potential functions. Despite considerable study, both experimental4 and theoretical,5 the carbon-carbon rotational barrier in n-butane does not appear to be completely settled. Similarly, with butadiene there is a question as to whether the gauche form is lower in energy than the syn conformer, with experiments using matrix isolation6 and theory7 coming out on different sides of the issue. In an effort to obtain more satisfactory calculated torsional barriers, we have made use of two of the most successful of the model chemistries, Pople’s G28 and Petesson’s CBS-Q.9 In an examination of a large number of organic compounds, both models reproduced the experimental heats of formation with an average root-mean-square error of (1 kcal/mol.10 In a study of rotational barriers, where there is no change in bonding, one might expect that most of the residual errors associated with the methods would cancel, leading to relatively accurate barriers. This study was initiated in order to see how well the model chemistries perform in this context. n-Butane Both model chemistries make use of MP2/6-31G* structures for the higher level calculations. One might ask whether this is adequate to represent the structures of the compounds in this study. The effect of theoretical level on the calculated geometries and relative energies was therefore examined. Increasing the basis set size produced a significant change in calculated bond lengths. Thus, on going from MP2/6-31G* to MP2/6-311++G(2df,2pd), the central C-C bond length decreased from 1.5295 to 1.5221 Å, and the terminal C-C bond length changed from 1.5284 to 1.5220 Å.11 There also were small changes in the bond angles. It is known that the larger X
Abstract published in AdVance ACS Abstracts, September 15, 1996.
S0022-3654(96)02174-0 CCC: $12.00
TABLE 1: QCISD(T)/6-311++G(d,p) Energies for Butane Rotamers Using Different Geometries geometry
anti
gauche
syn
a. Total Energies, hartrees MP2/6-31G* -158.041 65 -158.040 80 -158.032 51 MP2/6-311++(df,pd) -158.041 51 -158.040 67 -158.032 38 MP2/6-311++(2df,2pd) -158.041 17 -158.040 31 -158.032 01 b. Relative Energies, ∆E (kcal/mol) MP2/6-31G* 0.00 0.53 MP2/6-311++G(df,pd) 0.00 0.53 MP2/6-311++G(2df,2pd) 0.00 0.54
5.74 5.73 5.75
basis set structures usually fit the experimental structures more satisfactorily.12 However, the relative QCISD(T)/6-311++G** energies derived from these calculations were essentially independent of the basis set used for the geometry optimization (Table 1). Therefore, the MP2/6-31G* structures are appropriate. The G2 and CBS-Q energies at 0 K for the butane conformers and transition states are summarized in Table 2. They include the zero-point energies and are based on significantly larger basis sets than previous calculations. G2 is effectively QCISD(T)/6-311+G(3df,2pd). CBS-Q makes use of a somewhat lower theoretical levels plus a complete basis set extrapolation and is significantly less computationally demanding than G2. With butane, the CBS-Q calculations required about one-tenth the time needed for the G2 calculations. The relative energies for the gauche and syn rotamers and for the anti-to-gauche transition state were essentially the same for the two models. The only difference between the G2 and CBS-Q curves in Figure 1 results from the relative energies of the anti form. Thus, G2 gives a 0.6 kcal/mol difference between the anti and gauche forms whereas CBS-Q gives 0.9 kcal/mol. The calculated energy differences correspond to 0 K. Except for the internal rotation modes, the conversion of 298 K is straightforward using the mass, the calculated moments of inertia, and the scaled HF/6-31G* vibrational frequencies.13 The three lowest frequencies for the anti and gauche conformers correspond to rotation about C2-C3 and rotation of the two methyl groups. In the 120° and syn transition states, the rotation about the central bond is lost, and the two lowest frequencies correspond to the methyl rotations. The methyl internal rotation modes were treated separately by calculating the appropriate rotor wave functions and energy levels and deriving the partition function for internal rotation from these data.14 The methyl © 1996 American Chemical Society
Rotational Barriers in Butane, 1-Butene, and 1,3-Butadiene
J. Phys. Chem., Vol. 100, No. 40, 1996 16163
TABLE 2: n-Butane Calculated Energies a. Total Energies (hartrees) conformer
τ
CBS-Q
G2
anti “120°” gauche syn
180.0 119.6 63.7a 0.0
-158.080 71 -158.075 07 -158.079 19 -158.070 91
-158.081 17 -158.075 92 -158.080 15 -158.071 87
b. Thermal Corrections, 298 K anti translation rotation vibration Me rotors C2C3 rotor total
120°
gauche
syn
H° - H°0
G° - H°0
H° - H°0
G° - H°0
H° - H°0
G° - H°0
H° - H°0
G° - H°0
1.48 0.89 0.82 0.71 0.43 4.33
-9.88 -6.01 -0.37 -0.49 -0.49 -17.24
1.48 0.89 0.76 0.72
-9.88 -6.01 -0.30 -0.51
-9.88 -6.05 -0.25 -0.42
-16.70
-9.88 -6.01 -0.31 -0.54 -0.48 -17.22
1.48 0.89 0.62 0.64
3.85
1.48 0.89 0.76 0.75 0.40 4.28
3.63
-16.60
c. Relative Energies (kcal/mol) CBS-Q
a
G2
conformer
∆∆H(0 K)
∆∆(298 K)
∆∆G(298 K)
∆∆H(0 K)
∆∆H(298 K)
∆∆G(298 K)
anti 120° gauche syn
0.0 3.54 0.95 6.15
0.0 3.06 0.90 5.45
0.0 4.08 0.97 6.79
0.0 3.29 0.64 5.84
0.0 2.81 0.59 5.14
0.0 3.83 0.66 6.48
The gauche torsional angle has this value at both MP2/6-31G* and MP2/6-311++G(2df,2p).
Figure 1. G2 (solid line) and CBS-Q (dashed line) potential energy curves for rotation about the C2-C3 bond in n-butane.
rotational barriers were calculated at the HF/6-31G* level and were as follows: anti, 3.2 kcal/mol; gauche, 3.0 kcal/mol; syn, 4.1 kcal/mol. The methyl rotational barrier for the 120° transition state was assumed to be the same as for the anti form. The energy levels for the hindered rotation about the central C-C bond have been measured experimentally,4a and the partition function was derived from these data. The thermodynamic functions thus obtained are recorded in Table 2. There are now a number of measurements of the anti/gauche energy difference using a variety of experimental techniques that are in agreement with anti being lower in energy than gauche by 0.7 ( 0.1 kcal/mol.4a This in very good agreement with the G2 calculated energy difference, and the CBS-Q difference is larger by only 0.2 kcal/mol. Thus, both model chemistries give satisfactory results, with a small preference for G2. The anti/gauche barrier has been studied spectroscopically, giving an energy difference of 3.62 kcal/mol,4a which should be corrected for the torsional zero-point energy in the anti form (0.17 kcal/mol) giving ∆Hq(0 K) ) 3.45 kcal/mol. The G2 and CBS-Q calculations gave ∆Hq(0 K) ) 3.29 and 3.54 kcal/
mol, respectively. Here again, the agreement between experiment and theory is very good. The anti/syn energy difference is much more difficult to obtain experimentally because it is so large and requires a long extrapolation of rotational levels. It has been the subject of considerable controversy.5 Allinger’s MM3 force field15 leads to an anti-syn barrier ∆Hq(298 K) ) 4.81 kcal/mol. MP2 theoretical calculations using relatively large basis sets by Raghavachari5d and by Wiberg and Murcko5e suggested a barrier of about 5.5 kcal/mol, but a later calculation by Allinger et al. gave a smaller barrier ∆Hq(298 K) ) 4.89 kcal/mol.5f The Allinger et al. calculations made assumptions about how the energy difference would change with increasing basis set size.5f With their largest basis set CCSD calculation (TZ2(d,p)) the syn-anti energy difference was ∆E ) 5.83 kcal/mol, and on the basis of still larger basis set MP2 calculations, their extrapolated value was ∆E ) 5.25 kcal/mol. However, this extrapolation is unwarranted. QCISD(T) is essentially equivalent to CCSD. With a 6-311+G(d,p) basis set, the QCISD(T) energy difference is ∆E ) 5.75 kcal/mol (Table 1), and with the much larger 6-311+G(3df,2p) basis set used in the G2 calculations it is ∆E ) 5.64 kcal/mol. Thus, the basis set effect is quite small, and their estimate of the energy difference should be increased to a value close to that obtained in the present study. The most recent experimental study gave ∆H(0 K) ) 3.78 kcal/mol for the syn/anti energy difference and 3.11 kcal/mol as the barrier for interconverting the gauche rotamers via the syn transition state.4a It is difficult to assign an uncertainty to these values. The present calculations (G2 and CBS-Q) are in agreement that the gauche-syn barrier is ∆Hq ) 5.2 kcal/mol at 0 K and 4.6 kcal/mol at 298 K, which is considerably larger than the experimental result (3.1 kcal/mol at 0 K). Both of these calculated energy differences would be increased by 0.6 kcal/mol to give the energies with respect to the anti conformer, and they are much larger than the reported experimental values. In order to try to resolve the disagreement between the experimental and theoretical syn/anti energy difference, we have reinvestigated the potential energy function for rotation about
16164 J. Phys. Chem., Vol. 100, No. 40, 1996
Murcko et al. TABLE 4: Calculated Torsional Energy Levels for Butane (cm-1) a. MP2/6-311++G(2df,2pd) Potential Function anti
gauche
transition
observed
calculated
observed
calculated
10 21 32 43 54
121.3 118.8 116.1 113.0 109.8
120.1 118.4 116.2 113.6 110.6
116.6 114.1 111.3
121.1 119.5 117.5
b. Using Modified Potential Function gauche
Figure 2. Potential energy curve for rotation about the C2-C3 bond in butane derived from MP2/6-311+G(2df,2pd) geometry optimizations. The observed energy levels are indicated.
TABLE 3: Calculated MP2/6-311++G(2df,2pd) Internal Rotational Potential Function for Butanea torsional torsional Erel angle (deg) E (hartrees) (kcal/mol) angle (deg) 0.0 30.0 55.0 63.72
-158.060 62 -158.030 32 -158.034 06 -158.068 85
5.74 3.04 0.74 0.57
80.0 119.57 160.0 180.0
E (hartrees)
Erel (kcal/mol)
-158.033 40 -158.064 21 -158.033 83 -158.079 77
1.03 3.49 0.80 0.00
a V (kcal/mol) ) 2.2285 + 0.8692 cos θ + 0.4061 cos 2θ + 1.920 cos 3θ + 0.0519 cos 4θ + 0.0936 cos 6θ - 0.0131 cos 7θ. F (kcal/ mol) ) 0.4744 × 10-2 + 0.6496 × 10-3 cos θ + 0.4731 × 10-3 cos 2θ + 0.1219 × 10-3 cos 3θ - 0.5880 × 10-5 cos 4θ + 0.5710 × 10-5 cos 5θ + 0.1559 × 10-4 cos 6θ - 0.3218 × 10-4 cos 7θ.
C2-C3. An examination of the potential energy curve derived by Herrebout et al.4a shows that the experimental vibrational levels fall far short of the barrier. The spacing of the levels depends on both the barrier height and the width of the well. A change in one may be compensated by a change in the other. If the Herrebout et al. potential function for energies under about 1.8 kcal/mol (i.e., the region for which the torsional levels have been observed) were combined with the G2 energy at 0°, the resulting potential function (Figure 2) reproduces the observed rotational level spacings.16 Thus, it is not possible to obtain a unique potential function from such a limited data set. We have calculated the energy of a series of rotomeric forms of butane making used of the MP2/6-311++G(2df,2pd) theoretical level that closely reproduces the G2 relative energies. In each case, geometry optimization was carried out with the one torsional angle as the only constraint. The potential function derived from these data is given in the Table 3. The other quantity that is needed is the reduced moment of inertia, and this was calculated using Pitzer’s method17 and the calculated geometries. The appropriate Hamiltonian was used with these data to give the calculated energy levels (Appendix). They are compared with the experimental values in Table 4. The agreement is quite satisfactory for the anti rotamer, but the calculated spacings are slightly too large for the gauche rotamer. The small difference would have no effect on the change in free energy on going from 0 to 298 K. In order to better fit the experimental data, the potential energy curve in this region must be somewhat wider. As an example, when the energy of the 55° point was shifted to 53.2° and the 30° point were shifted to 28.85°, the calculated levels were in better agreement with experiment (Table 4). As noted above, it is possible to achieve a very good fit by combining the experimental potential function for the region in which the torsional transitions occur with the
transition
observed
calculated
10 21 32
116.6 114.1 111.3
115.2 114.2 113.0
theoretically derived syn-anti energy difference. We believe the G2 potential function provides the best currently available description for butane, and the ∆H(298 K) thus obtained (5.14 kcal/mol) is not far from the MM3 value of 4.81 kcal/mol. With high-level optimized geometries available, it was of interest to examine the changes in structural parameters on rotation, and they are shown in Figure 3. It can be seen the terminal C-C bond length is essentially unaffected by rotation about C2-C3, but the central bond is strongly affected, with maxima at the eclipsed conformations. This is also observed with ethane where rotation about the C-C bond leads to lengthening of the C-C bond, but essentially no changes in the HCC bond angles or CH bond lengths.18 The source of the rotational barrier is made clear by plots of the electron density for localized C-H bond orbitals in staggered and eclipsed ethane (Figure 4). The repulsion between the electron density distributions is largest for the eclipsed form. The bond orbitals in the latter are best moved apart by increasing the C-C bond length, and changes in the CCH bond angles or the CH bond lengths will not be as effective. The same is true for butane. Figure 3c shows that the CCC bond angle is not affected in going to the 120° transition state, is slightly affected on going to the gauche form, and then strongly affected on going to the syn transition state. This large increase in angle, caused by methyl-methyl repulsion in the syn form, contributes to the higher energy of this rotamer. The corresponding HCC angles are shown in Figure 3d. They change little on going from 180° to 120°, but on going to smaller torsional angle, one of the pairs of nonbonded hydrogens begins to have a significant interaction, leading to an increase in the CCH angle. With a 0° torsional angle, the inner hydrogens have a strong repulsive interaction, and this in turn leads to a change in CCH angle for the outer hydrogens. The C1-C4 distance is shown in Figure 3b. It can be seen that it drops markedly on rotation away from 180°, although the energy does not rise rapidly until after the 60° torsional angle is reached. Thus, the van der Waals repulsion between the terminal methyl groups begins to become important at about 3.0 Å. 1-Butene The G2 and CBS-Q data for 1-butene are shown in Table 5 and Figure 5. There are two sets of data concerning the potential energy curve for rotation about the central C-C bond. One was derived from an analysis of the microwave spectrum,19a and the other was derived from the temperature dependence of the Raman spectrum.17b The microwave study provided structural data for both the syn and gauche conformers as well as
Rotational Barriers in Butane, 1-Butene, and 1,3-Butadiene
J. Phys. Chem., Vol. 100, No. 40, 1996 16165
Figure 4. Electron density distributions for localized C-H bond orbitals for eclipsed (lower) and staggered (upper) ethane.
Figure 3. Changes in structural data during rotation about the C2-C3 bond of butane. In the upper frame, the C1-C2 bond length is given as a solid line, and the C2-C3 length is given as a dashed line. In the bottom frame, curve a gives the C2-C2-H angle to the unique methyl hydrogen in the syn and anti conformer, and curves b and c give the corresponding angles to the other methyl hydrogens. Curves d and e give the C-C-H angles to the methylene hydrogens.
the methyl rotational barriers for the two forms. The gauche conformer was found to be the more stable by 150 ( 150 cal/ mol. This is in agreement with both the previous20 and present calculations. The liquid phase Raman data suggested that the syn conformer is the more stable by 0.22 kcal/mol and found a relatively large entropy difference (3.3 cal/(mol deg)). One possible explanation for the difference between the microwave and Raman data would be that the bands used in the latter determination of the energy difference had been misassigned. In order to examine this possibility, the vibrational frequencies for syn and gauche 1-butene were calculated at the MP2/6-31G* level. The frequencies thus obtained are usually about 5% too large.21 The scaled calculated frequencies are compared with the assigned values in Table 6. The CH stretching modes are not included because they are strongly affected by anharmonicity and frequently are affected by Fermi resonance. There is generally good agreement, although there is a suggestion that a few of the bands have been misassigned (such as ν24). However, the main concern is with ν17, which was used in determining the energy difference. The calculations are in complete accord with the assignment for this mode. There is one important difference between the microwave study and the calculations on one hand and the Raman study on the other. The former refers to the gas phase, and the latter refers to the liquid phase. A solvent effect of 0.2-0.3 kcal/ mol on the conformational energy difference was found with n-butane,22 and it would not be surprising to also find such an effect with 1-butene. In calculating the thermodynamic functions for the 1-butene conformers, the methyl rotational barriers found in the microwave study (gauche, 3.1 kcal/mol; cis, 3.9 kcal/mol) were used along with the torsional energy profile calculated at the G2 level. The energies are given in Table 5. The calculated (0 K) energy difference between the skew and anti forms (1.83 kcal/mol) is in very good agreement with the energy derived from the microwave study (1.75 ( 0.10 kcal/mol), and the calculated syn-skew energy difference (0.36 kcal/mol) is also in good agreement with the microwave data (0.15 ( 0.15 kcal/mol). The Raman study found the gauche and anti rotamers to have similar energies, which is in agreement with the G2 relative energies, whereas there is a much larger difference between the CBS-Q energies. Thus, here again, the G2 model reproduces the experimental data more successfully than CBS-Q. It was also of interest to compare the structures calculated at the MP2/6-311+G(2df,2pd) level with the structural data derived from the microwave study (Table 7). The latter was obtained
16166 J. Phys. Chem., Vol. 100, No. 40, 1996
Murcko et al.
TABLE 5: 1-Butene Calculated Energies a. Total Energies (hartrees) conformer
τ (deg)
CBS-Q
G2
syn gauche skew anti
0.0 54.3 119.0 180.0
-156.869 53 -156.866 16 -156.870 21 -156.867 33
-156.869 27 -156.866 69 -156.869 84 -156.866 93
b. Thermal Corrections anti translation rotation vibration Me rotor C2C3 rotor total
skew
gauche
syn
H° - H°0
G° - H°0
H° - H°0
G° - H°0
H° - H°0
G° - H°0
H° - H°0
G° - H°0
1.48 0.89 0.53 0.36
-9.85 -6.30 -0.17 -0.25
-9.85 -6.35 -0.15 -0.21
-16.57
-9.85 -6.34 -0.31 -0.25 -0.51 -17.26
1.48 0.89 0.51 0.33
3.26
1.48 0.89 0.77 0.36 0.43 3.93
3.21
-16.56
1.48 0.89 0.78 0.33 0.39 3.87
-9.85 -6.35 -0.33 -0.22 -0.47 -17.22
c. Relative Energies CBS-Q
G2
conformer
∆∆H(0 K)
∆∆H(298 K)
∆∆G(298 K)
∆∆H(0 K)
∆∆H(298 K)
∆∆G(298 K)
syn gauche skew anti
0.43 2.54 0.00 1.81
0.37 1.82 0.00 1.14
0.47 3.24 0.00 2.50
0.36 1.98 0.00 1.83
0.30 1.26 0.00 1.16
0.40 2.68 0.00 2.52
TABLE 6: MP2/6-31G* Calculated Vibrational Frequencies for 1-Butene cis mode A′
Figure 5. G2 (solid line) and CBS-Q (dashed line) potential energy curves for rotation about the C2-C3 bond in 1-butene.
A′′
from a limited set of data and required some assumptions about structural parameters. Nevertheless, there is general agreement between the calculated and observed parameters, including the torsional angle for the skew rotamer and the opening of the bond angles on going from the skew to the syn rotamer. a
1,3-Butadiene The G2 and CBS-Q data for 1,3-butadiene are shown in Table 8 and Figure 6. The rotational profile has received considerable study both experimentally6,23 and theoretically.7 The experimental studies making use of matrix isolation generally find that the two minima are at the anti and syn conformers,6 whereas all of the theoretical studies find the two minima to be anti and gauche. However, a more recent experimental gas phase study provided good evidence for a gauche minimum,24 and a molecular dynamics study of butadiene in an argon matrix found that the matrix inverted the relative energies of the gauche and syn rotamers.25 The present study is in good accord with the gas phase experimental data.24 The best experimental value for the anti-gauche barrier is ∆Hq(0 K) ) 5.93 kcal/mol, and the anti-gauche energy difference is ∆H(0 K) ) 2.84 kcal/mol.22 The corresponding
ν7 ν8 ν9 ν10 ν11 ν12 ν13 ν14 ν15 ν16 ν17 ν18 ν19 ν22 ν23 ν24 ν25 ν26 ν27 ν28 ν29 ν30
skew
Durig
MP2
Durig
MP2
1643 1460 1450 1426 1380 1342 1306 1128 1071 988 836 540 311 1469 1264 1177 999 915 784 551
1650 1496 1470 1424 1396 1356 1293 1124 1011 982 831 525 279 1485 1267 1092 997 881 790 542 274 145
1647 1463 1444 1421 1380 1318 1296 1128 1079 1020 854 439 391 1469 1264 1177 993 912 784 634
1650 1492 1472 1426 1395 1321 1284 1177 1020 965 847 419 310 1485 1265 1075 988 886 771 630 234 103
154
103
The calculated values have been scaled by 0.95.20
TABLE 7: Comparison of Calculated and Observed Structural Data for 1-Butene cis
skew
parameter
calc
obsb
r(CdC) r(C2C3) r(C3C4) ∠C-CdC ∠C-C-C τ(CCCC)
1.334 1.498 1.520 125.8 114.9 0.00
1.336 ( 0.008 1.507 ( 0.010 1.536 ( 0.006 126.7 ( 0.4 114.8 ( 0.5 0.00
a
a
calc
obs
1.334 1.495 1.528 124.7 111.8 117.7
1.342 ( 0.009 1.493 ( 0.008 1.536 ( 0.012 125.4 ( 0.2 112.1 ( 0.3 119.9 ( 0.3
MP2/6-311++G(2df,2pd). b Reference 17a.
G2 energy differences are 5.7 and 2.9 kcal/mol, and the CBS-Q energy differences are 6.0 and 3.2 kcal/mol. The only significant difference between the experimental and calculated potential energy functions is found with the syn-anti energy
Rotational Barriers in Butane, 1-Butene, and 1,3-Butadiene
J. Phys. Chem., Vol. 100, No. 40, 1996 16167
TABLE 8: 1,3-Butadiene Energies a. Total Energies (hartrees) conformer
τ (deg)
CBS-Q
G2
anti twist gauche syn
180.0 101.6 37.9 0.0
-155.665 70 -155.656 11 -155.660 67 -155.660 32
-155.664 27 -155.655 23 -155.659 61 -155.658 77
b. Thermal Corrections anti translation rotation vibration C2C3 rotor total
twist
gauche
syn
H° - H°0
G° - H°0
H° - H°0
G° - H°0
H° - H°0
G° - H°0
H° - H°0
G° - H°0
1.48 0.89 0.78 0.36 3.51
-9.81 -5.72 -0.32 -0.29 -16.14
1.48 0.89 0.81
-9.81 -5.80 -0.39
-9.81 -5.82 -0.32
-16.00
-9.81 -5.81 -0.34 -0.43 -16.39
1.48 0.89 0.77
3.18
1.48 0.89 0.79 0.41 3.57
3.14
-15.95
c. Relative Energies CBS-Q
G2
conformer
∆∆H(0 K)
∆∆H(298 K)
∆∆G(298 K)
∆∆H(0 K)
∆∆H(298 K)
∆∆G(298 K)
anti twist gauche syn
0.0 6.02 3.16 3.38
0.0 5.69 3.22 3.01
0.0 6.16 2.91 3.57
0.0 5.67 2.92 3.45
0.0 5.34 2.98 3.08
0.0 5.81 2.67 3.64
Acknowledgment. The studies at Yale were supported by the National Science Foundation. We thank Prof. Alice ChungPhilipps (Miami University) for providing us with her program for calculating the reduced moments of inertia and for helpful discussions. Supporting Information Available: Tables of energies and structural data as a function of basis set for butane, 1-butene, and 1,3-butadiene (6 pages). Ordering information is given on any current masthead page. Appendix The Hamiltonian used for the torsional motion was27
H ) 1/2[pθ2F(θ) + F(θ)pθ2 + 2pθF(θ)pθ] + V(θ) Figure 6. G2 (solid line) and CBS-Q (dashed line) potential energy curves for rotation about the C2-C3 bond in 1,3-butadiene.
difference where the experiment gives 4.0 kcal/mol and the G2 and CBS-Q calculated values are 3.5 and 3.4 kcal/mol, respectively. Conclusion The G2 model chemistry has been found to reproduce the experimental data for the rotational barriers of n-butane, 1-butene, and 1,3-butadiene in a very satisfactory fashion. The computationally less demanding CBS-Q model chemistry was also generally satisfactory, although it gave slightly larger deviations from the experimental values. This is important since for the present systems the times required for the CBS-Q calculations were about one-tenth that for the G2 model. The syn-anti energy difference for n-butane has been found to be significantly larger than that derived from experiment, in agreement with previous theoretical studies. It is shown that in this case the energy difference cannot be accurately determined, making use of the available experimental data. The changes in structural parameters on rotation provide additional information concerning the changes in the intramolecular interactions on rotation about C-C bonds. Calculations. The ab initio calculations were carried out using Gaussian-95.26 The rotational energy levels were calculated using locally developed programs.
The reduced internal rotational constant, F, was calculated as a function of the torsional angle, θ
F(θ) ) F0 + ∑Fi cos(iθ) using the structural parameters obtained in this work along with Pitzer’s formula.17 The potential, also represented as a cosine series, was calculated using ab initio molecular orbital theory. The basis functions chosen for the calculations were {e(imθ}. The eigenvalues of the Hamiltonian were calculated iteratively increasing the number of basis functions until convergence was obtained. The localized energy levels were identified computing the probabilities of the various eigenstates in each potential well. References and Notes (1) (a) Vertex Pharmaceuticals. (b) Yale University. (2) Burkert, U.; Allinger, N. L. Molecular Mechanics; ACS Monograph 177; American Chemical Society: Washington, DC, 1982. (3) Cf.: Mirkin, N. G.; Krimm, S. J. Phys. Chem. 1983, 97, 13887. (4) (a) Herrebout, W. A.; van der Veken, B. J.; Wang, A.; Durig, J. R. J. Phys. Chem. 1995, 99, 578. (b) Murphy, W. F.; Fernandez-Sanchez, J. M.; Raghavachari, K. J. Phys. Chem. 1991, 95, 1124. (c) Gassler, G.; Hu¨ttner, W. Z. Naturforch. 1990, 45A, 113. (d) Rasanen, M.; Bondybey, V. E. Chem. Phys. Lett. 1984, 111, 515. (e) Heenan, R. K.; Bartell, L. S. J. Chem. Phys. 1983, 78, 1270. (5) (a) Radom, L.; Lathran, W. A.; Hehre, W. J.; Pople, J. A. J. Am. Chem. Soc. 1973, 95, 693. (b) Peterson, M. R.; Csizmadia, I. G. J. Am. Chem. Soc. 1978, 100, 6911. (c) Allinger, N. L.; Profeta, S. J. Comput.
16168 J. Phys. Chem., Vol. 100, No. 40, 1996 Chem. 1980, 1, 181. (d) Raghavachari, K. J. Chem. Phys. 1984, 81, 1383. (e) Wiberg, K. B.; Murcko, M. A. J. Am. Chem. Soc. 1988, 110, 8029. (f) Allinger, N. L.; Grev, R. S.; Yates, B. F.; Schaefer, H. F. III J. Am. Chem. Soc. 1990, 112, 114. (g) Tsuzuki, S.; Schaefer, L.; Hitoshi, G.; Jemmis, E. D.; Hosoya, H.; Siam, K.; Tanabe, K.; Osawa, E. J. Am. Chem. Soc. 1991, 113, 4665. Frey, R. F.; Cao, M.; Newton, S. Q.; Schaefer, L. J. Mol. Struct. (THEOCHEM) 1993, 285, 99. (6) Arnold, B. R.; Balaji, V.; Michl, J. J. Am. Chem. Soc. 1990, 112, 1808. Fischer, J. J.; Michl, J. J. Am. Chem. Soc. 1987, 109, 1056. Squillacote, M. E.; Sheridan, R. S.; Chapman, O. L.; Anet, F. A. L. J. Am. Chem. Soc. 1979, 101, 3657. (7) Panchenko, Yu. P.; Abramenkov, A. V.; Bock, C. W. J. Mol. Struct. 1986, 140, 97. Bock, C. W.; Panchencko, Y. N. J. Mol. Struct. 1989, 69. Alberts, I. L.; Schaefer, H. F., III Chem. Phys. Lett. 1989, 161, 375. Szalay, P. G.; Lischka, H.; Karpfen, A. J. Phys. Chem. 1989, 93, 6629. Rice, J. E.; Liu, B.; Lee, T. J.; Rohlfing, C. M. Chem. Phys. Lett. 1989, 161, 277. Weiberg, K. B.; Rosenberg, R. E.; Rablen, P. R. J. Am. Chem. Soc. 1991, 113, 2890. Guo, H.; Karplus, M. J. Chem. Phys. 1991, 94, 3679. (8) Curtiss, L. A.; Raghavachari, K.; Trucks, G. W.; Pople, J. A. J. Chem. Phys. 1991, 94, 7221. Curtiss, L. A.; Carpenter, J. E.; Raghavachari, K.; Pople, J. A. J. Chem. Phys. 1992, 96, 9030. G2 is effectively a QCISD(T)/6-311+G(3df,2p) calculation carried out using the MP2/6-31G* geometry, plus corrections for the zero-point energy and a higher level correction. The latter will cancel in all the comparisons in the present study. (9) Ochterski, J. W.; Petersson, G. A.; Montgomery, J. A. J. Chem. Phys. 1996, 104, 2598. CBS-Q makes use of MP2/6-31G† geometries, a complete basis set explanation based on a MP2/6-311++G(3df,2p) calculation, and higher order corrections based on MP4(SDQ)/6-31+G(2d,p) and QCISD(T)/6-31+G. It also includes correction for the zero-point energy and a higher order correction. The latter will cancel in all the comparisons in this study. (10) Ochterski, J. W.; Peterson, G. A.; Wiberg, K. B. J. Am. Chem. Soc. 1995, 117 11299. (11) More detailed structural data are available in the supporting information. (12) Hehre, W. J.; Radon, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory, Wiley: New York, 1986; Chapter 6. (13) Janz, G. J. Estimation of Thermodynamic Properties of Organic Compounds; Academic Press: New York, 1958.
Murcko et al. (14) Wiberg, K. B.; Rablen, P. R.; Rush, D.; Keith, T. A. J. Am. Chem. Soc. 1995, 117, 4261. (15) Allinger, N. L.; Yuh, Y. H.; Lii, J.-H. J. Am. Chem. Soc. 1989, 111, 8551. (16) The potential function obtained in this case was V (kcal/mol) ) 2.1800 + 0.6665 cos θ + 0.2425 cos 2θ + 1.862 cos 3θ + 0.1747 cos 4θ + 0.1877 cos 5θ + 0.1668 cos 6θ + 1.1076 cos 7θ + 0.0804 cos 8θ + 0.0311 cos 9θ + 0.254 cos 10θ + 0.0160 cos 11θ. (17) (a) Pitzer, K. S.; Gwinn, W. G. J. Chem. Phys. 1942, 10, 428. (b) Pitzer, K. S. J. Chem. Phys. 1946, 14, 239. (18) Bader, R. F. W.; Cheeseman, J. R.; Laidig, K. E.; Wiberg, K. B.; Breneman, C. J. Am. Chem. Soc. 1990, 112, 6530. (19) (a) Kondo, S.; Hirota, E.; Morino, Y. J. Mol. Spectrosc. 1968, 28, 471. (b) Durig, J. R.; Compton, D. A. C. J. Phys. Chem. 1980, 84, 773. (20) Wiberg, K. B.; Martin, E. J. Am. Chem. Soc. 1985, 107, 5036. (21) Cf.: Wiberg, K. B.; Thiel, Y.; Goodman, L.; Leszczynski, J. J. Phys. Chem. 1995, 99, 13850. (22) Reference 4a and Pratt, L. R.; Hsu, C. S.; Chandler, D. J. Chem. Phys. 1978, 68, 4202. (23) Furukawa, Y.; Takeuchi, H.; Hasrada, I.; Tasumi, M. Bull. Chem. Soc. Jpn. 1983, 56, 392. Durig, J. R.; Bucy, W. E.; Cole, A. R. H. Can. J. Phys. 1976, 53, 1832. Carriera, L. A. J. Chem. Phys. 1975, 62, 3851. Lipnick, R.; Garbisch, E. W. J. Am. Chem. Soc. 1973, 95, 6370. (24) Kofranek, M.; Karpfen, A.; Lischka, H. Chem. Phys. Lett. 1992, 189, 281. (25) Engeln, R.; Consalvo, D.; Reuss, J. Chem. Phys. 1992, 160, 427. (26) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Robb, M. A.; Cheeseman, J. R.; Keith, T.; Petersson, G. A.; Montgomery, J. A.; Raghavachari, K.; Al-Laham, M. A.; Zakrzewski, V. G.; Ortis, J. V.; Foresman, J. B.; Cioslowski, J.; Sefanov, B. B.; Nanayakkara, A.; Challacombe, M.; Peng, C. Y.; Ayala, P. Y.; Chen, W.; Wong, M. W.; Andres, J. L.; Repologle, E. S.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Binkley, J. S.; Defrees, D. J.; Baker, J.; Stewart, J. P.; HeadGordon, M.; Gonzalez, C.; Pople, J. A. Gaussian 95, Development Version (Rev. B.1), Gaussian, Inc.: Pittsburgh, PA, 1995. (27) Harmony, M. D. J. Am. Chem. Soc. 1972, 94, 351. Harris, D. O.; Meakin, P. Hirota, E. J. Chem. Phys. 1969, 51, 3775.
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