Molecular mechanics force field for tertiary carbocations

Adri C. T. van Duin, Jan M. A. Baas and Bastiaan van de Graar" Delft University of Technology, Laboratory of Organic Chemistry and Catalysis, Julianalaan 136, 2628 BL Delft, The Netherlands

A new force field for tertiary carbocations has been developed, based on the Delft molecular mechanics (DMM) hydrocarbon force field (A. C. T. van Duin, J. M. A. Baas and B. van de Graaf, J . Chem. Soc., Faraday Trans., 1994, 90, 2881). As in this force field, the Mortier method (W. J. Mortier, S. K. Ghosh and S . Shankar, J . Am. Chem. Soc., 1986, 108, 4315) is used to calculate geometry-dependent charges. The force field was optimized using enthalpies of formation, solvolysis-rate data and ab initio vibration and geometry data. The ab initio vibration data were rescaled using experimental spectra from the isoelectronic alkyl- boranes. To check the influence of electron delocalization on the geometry of tertiary carbocations, the results of the force field were compared with crystal structures.

A correlation coefficient between the calculated and the experimental solvolysis rates for alkyl bromides of 0.988 1 was obtained, showing that the force field gives a good description of carbocation energies.

Carbocations are high-energy intermediates in many organic reactions and their stabilities can be used to obtain good esti- mates for the energy of activation of reactions. It is very diffi- cult to obtain experimental thermodynamic data for these short-lived compounds. Therefore, a computational method to calculate these data would be useful. Ab initio methods can be applied for small compounds; molecular mechanics has the advantage over ab initio methods in that it is also applicable to larger molecules. Molecular mechanics, however, requires a reliable force field, consisting of potential functions and appropriate parameters. The reliability of a force field can only be checked by comparing the calculated properties with experimental data. This introduces a problem, because experi- mental observations of carbocations are quite scarce and often carry a considerable experimental error. This is evident when one compares the reported values of the enthalpies of forma- tion of the tert-butyl (2-methyl-2-propyl) cation. These vary from ca. 707l to 678 kJ mol-'. 2 * 3

However, sources other than the direct measurement of enthalpies of formation can also be used for the optimization of a carbocation force field. It has been shown4,' that solvolysis-rate data can provide a reliable source for carbo- cation energy data, since the reaction rates are related to the stabilities of the carbocation intermediates. Another impor- tant source of data for the optimization of empirical force fields comes from IR and Raman spectroscopy. These data correlate strongly with the parameters in the force fields and thus provide an excellent means of optimizing them. Experi- mental IR spectra of some tertiary carbocation compounds are available,6 but the lack of assignment make them less suit- able for force-field optimization purposes. As an alternative, ab initio calculations can be used to obtain the assigned spectra for some small carbocations. These ab initio data need rescaling before they can be applied. Scale factors can be found by comparing the experimental and ab initio spectra of the isoelectronic alkylboranes.

Ab initio methods can also be used to obtain heights of rotational barriers and geometries of small molecules. These, combined with some crystal structures of tertiary carbocations in HSbF, matrices, yield a sufficiently large data set which can be used to optimize the carbocation force-field parameters.

A main reason for the creation of this force field for tertiary carbocations comes from organic geochemistry. Tertiary car- bocations are proposed as intermediates in isomerizations of geochemical interest,' such as double-bond migration in chol-

estenes. With this force field, the thermodynamical data on these intermediates can be obtained, which allows the predic- tion of the isomerization rates. From experiments on double- bond migration in 5a-cholest-7-ene,* information can be obtained on the relative stability of tertiary cholestane cations. This information was used as a boundary condition in the optimization of the force field.

Calculations were carried out on DECstation 5000/200 and Silicon Graphics Indigo R4400 computers, using the DELPHI molecular mechanics program.'. ' SHANNO' ' and Newton- Raphson minimization techniques were employed in the energy minimization of the molecules in the training set. The force field was optimized using a successive one-parameter optimization, as described before.' Ab initio calculations were performed using the GAUSSIAN92 program' with the 6-31G* basis set.

Initially, as with the DMM hydrocarbon force field,12 the influence of the geometry on the charge distribution was assumed to be a second-order effect and the time-consuming calculation of the derivatives of the charges with respect to the geometry was not done. The neglect of these derivatives in the calculation of the forces in the molecule means that the molec- ular geometry reached after minimization does not exactly correspond with the minimum on the energy surface described by the energy functions in the force field. For uncharged hydrocarbons, this is only a minor discrepancy. For carbo- cations, however, in which chargesharge interactions play a more important role, the energy difference between the geometry found after minimization and the lowest point on the potential surface can be as large as 10 kJ mol-'. Because of the size of this energy difference, analytical derivatives of the charges to the atomic coordinates were introduced in the energy minimization, yielding an exact description of the forces in the molecule. The addition of these derivatives had a considerable influence on the geometries found, especially on the bond lengths. Therefore, the parameters in the carbocation force field had to be redetermined. In this way, two carbo- cation force fields were created, one in which the derivatives of the charges to the geometry are neglected and the other in which they are taken into account. This second force field has the advantage of an exact description of the potential surface, but calculation time increases considerably with molecule size.

J . Chem. SOC., Faraday Trans., 1996,92(3), 353-362 353

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For molecules of about 80 atoms, CPU time increases by a factor of 20 (from 2 to ca. 40 min).

Because of its more consistent description of the forces in the molecule, detailed information will be given only of the force field that includes all charge derivatives. An overview of the results of the other force field will be given in the Dis- cussion. Inclusion of all charge derivatives only slightly influ- enced the results of the parent DMM hydrocarbon force field. l 2

Table 3 Valency-angle parameters

type of atoms on valency angle central C 0,Jdegrees' k,/kJ mol-' rad-*

c-c+-c C 1 2Ob 405.89

111.17 1 1 1.41 107.39

449.36 578.02 578.02

H-C-C+

C-C-Br

106.43 105.24 104.81

296.19 296.19 296.19

Force field 106.32 110.36 1 1 1.03

576.8 576.8 576.8

The force field for tertiary carbocations is an extension of the DMM force field for hydrocarbons.12 This means that the geometry-dependent charge calculation of Mortier et al. l4 is used. Since carbocations are charged species, a realistic charge distribution is important in any force field used for these com- pounds.

An aim of the DMM force field was to restrict the number of parameters in its potential functions. Hence, the carbo- cation force field could be constructed without adding a large number of parameters. This is convenient with respect to the difficulties in finding the experimental data needed to optimize the force field. To create the carbocation force field, 52 param- eters were added to the hydrocarbon force field. These param- eters were optimized using 430 data points contained in the training set (see below).

The energy functions used were the same as in the DMM force field. One additional cross interaction was included, a torsion-bend interaction. To describe the energy of this inter- action, the potential function [eqn. (l)] used by Maple et al." was chosen.

107.29 109.27 106.60

400.9 400.9 400.9

a Angles are converted to radians in the calculation. Not optimized.

Table 4 Torsion-angle parameters

type of torsion angle Vl/kJ mol-' I/,/kJ mol-' V'/kJ mol-'

C + - c - c - c 0.42 - 3.80 - 1.41 c-c+-c-c - 13.04 13.63 5.48

12.10 - 0.82 H-C-C+-C -0.1 1 H-C-C-C+

C-C-C-Br -0.382 - 1.748 5.729 - 1.291 H-C-C-Br -

- - -

E,, = (e - eoNk, cos + k2 cos 20 + k , cos 30) (1)

In eqn. (l), rn denotes the torsion angle, 0 the valency angle in the torsion angle and 8, the equilibrium value of this valency angle. The final values for the parameters are given in Tables 1-10.

Table 5 Out of plane angle parameters ~~ ~

out of plane angle between atoms' k,,/kJ mol - ' rad - ~~ ~

c+, c, C+.'P 518.61 ~

* C+v iP refers to the in-plane projection of the central C'. Training set

To obtain a reliable force field, optimization for different types of data is required. Therefore, the training set consisted of energy and geometry data as well as IR and Raman fre- quencies (Table 11).

The values for the acceptance criteria (Table 11) were altered in some specific cases when the datum seemed less reli- able (higher acceptance criterion) or when a good repro- duction of the datum seemed important for the optimization process (lower acceptance criterion).

Laboratory experiments on double-bond migration in Sa-cholest-7-ene, 1 (Fig. 1) have revealed that the formation of spirocholestenes is kinetically favoured over formation of 14~-methy1-18-nor-5a-cho1estenes.* Proposing tertiary carbo-

Table 6 Van der Waals parameters

atom type &/kJ mol-' R v d W I A b

C+ 0.216 2.0438 13.22 Br 1.1150 2.118 11.85

Table 7 Stretch-bend parameters

bond in valency angle k,,/kJ mol-' A-' rad-'

c- c + (C- c + - C) c-c +(C-c-c +)

c-C(C-c-c +) C-C'(H-C-C')

H-C(H-C-C +) C-Br(R-C-Br)

295.4 141.8 71.1

199.6 146.0 326.2

Table 1 Charge-charge interaction parameters

C+ Br

8.0058 11.97

5.4448 19.73

Table 8 Torsion-stretch parameters

central bond in torsion angle k,&J mol-' A-'

R-C-C+-C -7.31 R-C-C-C+ - 0.27 R-C-C-Br - 1.59

Table 2 Bond parameters

type of bond ROlA a/A - D,/kJ mol-

c-c+ 1.4845 2.3249 299.32 C- Br 1.943 1.6708 289.9 1

354 J . Chem. SOC., Faraday Trans., 1996, Vol. 92

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Table 9 Torsion-bend parameters

torsion angle k,/kJ mol-' rad-' kJkJ mol-' rad-' k,/kJ mol- ' rad- ' c+-c-c-c c-c+-c-c H-C-C+-C H-C-C-C+

- 3.47 - 2.55 - 1.59 - 1.10

- 5.28

11.75

9.7 1 - 5.03

- 2.26 8.10

- 0.65 - 4.28

cations as intermediates in these isomerizations,' means that the 12( 1 3 + 14)abeo-5a-cholestan- 13-ium cation, 3 (Fig. 1) should be lower in energy than the 14P-methyl-18-nor-5a- cholestan-13-ium cation, 4 (Fig. 1). Since an aim of this force field is the prediction of isomerization rates for the com- pounds of geochemical interest, the relative stability of the two carbocations, 3 and 4, has been used as a boundary con- dition in the optimization of the force field.

Table 10 Group increments for enthalpy of formation calculation

central atom in group atoms attached group increment/kJ mol-'

C+ C C C C

C C C C

c , c , c C', H, H, H C', C, H, H C', C, C, H C + , c, c , c C, C, C, Br C, C, H, Br C, H, H, Br H, H, H, Br

598 3940 -31.8001 - 11.6502

5.0133 24.00 10

- 1.42 - 5.57 - 8.39 - 18.87

Table 11 Composition of the training set and acceptance criteria

number of acceptance type of data experiments criterion

~~~~~ ~~~ ~ ~ ~~~~

energy data heats of formation 12 4.184 kJ mol-' solvolysis data 41 4.184 kJ mol-' conformational 4 0.4184 kJ mol-'

energies

geometry data bond lengths 11 0.01 A valency angles 102 2.0" torsion angles 58 2.0-6.0"

IR and Raman frequencies data

202 50-100 cm-'

Energy data

Enthalpies of formation. Table 12 lists some experimental enthalpies of formation. When necessary, POP and TORS terms were calculated for the Compounds.'

Conformations. Four conformations of the tert-butyl cation and two conformations of the tert-pentyl (2-methyl-2-butyl) cation were included in the training set, specifically to opti- mize the torsional constants. The four conformations of the tert-butyl cation are the C,, C3h, c 1 h and C3, conformations, as identified by Sieber et a l l 6 Fig. 2 shows the geometries of the C3h, C,, and c 1 h conformations as calculated by the force field. The C, conformation is almost identical to the C3, con- formation. According to Sieber et al., the c 3 h conformation is, after correction for zero-point energy, ca. 0.4 kJ mol-' more stable than the c , h conformation, and ca. 4 kJ mole' more stable than the C,, conformation. Their calculations show the C, conformation to be almost isoenergetic with the c , h con- formation. The force field calculates the c 3 h conformation to

1 2

3 4

Fig. 1 Some steps in the isomerization of 5a-cholest-7-ene

Table 12 Reproduction of experimental enthalpies of formation

~~~~

tert-but yld tert-pentyl' 2,3-dimethyl-2-butyle 2,3,3-trimethyl-2-butylf 2-methyl-2-pentyl" 3-methyl-3-pen tyl' 2,4-dimethyl-2-pentyP 2-methyl-2-hexyP 1 -methyl-1 -cyclopentyl 1 -methyl- 1 -cyclohexyl 1 -norbornyl' 2-methyl-2-norborn yl"

687.0 660.7 624.7 607.1 634.7 634.3 599.6 608.4 687.8,/*h 697.1 642.7,f 653.1' 832

z 707

0.88 1.26

1.59

2.354

0.349 4.708

687.0 664.8 631.1 603.8 633.2 640.8 592.6 601.1 691.4 65 1 .O 841.2 721.7

" kJ mol-'. ' For several compounds, different values have been reported. The data in this table reflect those observations that agree best with our calculations. Enthalpies of formation have been measured relative to the tert-butyl cation and are corrected for the new value for the enthalpy of formation of this cation. TORS only calculated for the bonds not containing Cf (ref. 12). Ref. 18. Ref. 19. Ref. 20.

= 687.0 kJ mol- ' and A' Hnorbornane = - 52.0 kJ mol- '. Enthalpies of formation of isobutane and norbornane have been derived from Pedley et ~ 1 . ~ '

Ref. 16. ' Ref. 17. Calculated from ref. 21 using 4 Hisobutane = - 134.2 kJ mol-', Af Hteu

J . Chem. SOC., Faraday Trans., 1996, Vol. 92 355

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Clh

1

C3”

Fig. 2 Conformations of the tert-butyl cation

be at the global minimum and the C, conformation is not found (Table 13). The force-field calculated vibration spectrum of the c 1 h conformation shows an imaginary frequency at 43i cm-’, while that of the C,, conformation shows two, at 19i cm - (degenerate) and 89i cm - ’.

Fig. 3 shows the two Clh conformations of the tert-pentyl cation present in the training set. Ab initio calculations using the 6-31G* basis set showed conformation I1 to be 2.51 kJ mol-’ more stable than I. The force-field calculated confor- mation I1 was found to be 2.05 kJ mol- more stable than I.

Solvolysis-rate data. Observed solvolysis rates for various leaving groups and reaction conditions have been converted to standard conditions (343.1 5 K, solvent EtOH, leaving group O T S ) . ~ , ~ To use these reaction-rate data for opti- mization of the force field, a linear relationship between the Gibbs energies of activation and log kexp is assumed. The Gibbs energy of activation can be found by calculating the Gibbs energy of the starting compound and subtracting this from the calculated Gibbs energy of the corresponding carbocation. Bromide is used as the leaving group in the cal- culations. This requires the presence of two force fields, a bromide force field and a tertiary carbocation force field. Tables 1-4, 6-9, 10 and 14 show the final parameters and the composition of the training set for the bromide force field. Details of the optimization and results of the bromide force field will be given in a subsequent paper.

The training set for the carbocation force field (Table 11) contains energy data from different sources. The enthalpies of formation data pertain to the gas phase, while the solvolysis data are derived from carbocations in solution. To avoid dis- crepancies between these different data types, a correction must be made for the stabilization of the carbocations by sol- vation. Because the tert-butyl cation, being the smallest ion, receives the highest stabilization on solution, corrections were made relative to the solvation energy of this compound. We used eqn. (2)23 in which E , is the relative permittivity of the solvent and r the radius of the ion.

AA solv H = - - - Nre2 ( ____ “r2ir ’) - Asolv HBut (2) 4nE,

I I1

Fig. 3 Two C , , conformations of the tert-pentyl cation

Table 13 Calculated energy differences for the conformations of the tert-butyl cation

conformation E - Ec3&ab initioYqb E - EC3&force field)”

C l h 0.4 0.26 C,” 4 1.5

a kJ mol-’. * Ref. 16.

To obtain a value for the radius of the tert-butyl cation, this ion was orientated according to its main axes of inertia. Then the radius was set to be the mean value of half the largest distances between the nuclear positions in the x, y and z direc- tions plus one times the van der Waals radius of the hydrogen atom. For the other cations in the training set, the radius was calculated using eqn. (3), in which M represents the formula weight.

(3)

Fig. 4 shows how the solvolysis rates were incorporated into the parameter optimization. The molecular mechanics calculations produced data for compounds in the gas phase at 298 K. These data were corrected for temperature and solva- tion. This provided the Gibbs energy difference between the alkyl bromide and the corresponding carbocation in solution. These energy differences are correlated to the Gibbs energies of activation for the solvolysis reactions. After determining the best linear fit between the calculated Gibbs energy differences and solvolysis rates, this fit was used to obtain experimental Gibbs energy differences between alkyl bromides and the cor- responding carbocations in solution. From this, experimental enthalpies of formation for the carbocations could be calcu- lated. These were used in the parameter optimization.

After several optimization cycles, the calculated carbocation energies were used to obtain a new fit between the alkyl bromide-carbocation Gibbs energy differences and solvolysis rates, with, hopefully, a better correlation coefficient. This new fit provided new experimental carbocation enthalpies of for- mation for the parameter optimization. This procedure was repeated until the data in the training set were reproduced satisfactorily. Table 15 shows the calculated thermodynamic data for the carbocations and alkyl bromides with corre- sponding solvolysis rates. The energy minima for all structures were checked by the eigenvalues of their force-constants matrix; for structures with more than one conformational energy minimum, the value of the global minimum was used. The solvolysis-rate data were taken from Miiller et al.4924 and refer to solvolysis rates under standard conditions (343 K, solvent EtOH, leaving group OTs). Fig. 5 shows the relation- ship between the calculated alkyl bromide-carbocation Gibbs energy differences and the experimental solvolysis rates.

Miiller and Milin4 obtained, for the same set of compounds, a correlation coefficient of 0.9765 with a a(log k ) of 1.245. They estimated that approximately half of their o(1og k ) value was due to experimental errors.

Table 14 Composition of the training set for the bromide force field

type of data number of

experiments

energy data ~~~~

enthalpies of formation 5 conformational energies 5

geometry data bond lengths valency angles

14 11

IR and Raman data frequencies 104

dipole moments 5

356 J . Chem. Soc., Faraday Trans., 1996, Vof . 92

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343 g.298 -HCZm+HCM3 g,343 h H c d c sd.343 -TScB, sd. 343 I- 4H R,%c 4H R ,%lc * AG R:calc

- - Sd.298 -TSc, sol. 343 J 343

'" RBr. d c * "RBr. calc '" RBr. calc AG RBr,calc g, 298 --ticzw + HCU3 g. 343

best fit for compounds

training set c * a a ~ c b a ~ c Rcorrelalion

sd, 343 $. sol, 343

MG R+-RBr, calc a MG calc

sol. 343

I sd. 343 t. sol, 343 I best fit for compounds

c .--:-A- ..-,

MG R+-RBr, calc a MG calc

L g.2W optimization g. 298 -HCm8 + HCW g. 343

4HR.+kalc * parameters 4H R,+R. exp 4" R :ex$?

Fig. 4 solution

Scheme for the incorporation of solvolysis rates in the parameter optimization. HC means heat content, refers to compounds in

Table 15 Calculation of the solvolysis rates'

R 'f H298, R i AG343, R-Br AG343 . R '

tert-butyl tert-pentyl 3-met hyl-3-pent yl 2,3,3-trimethyl-2-butyl 1, l-di-tert-butyl-l-ethyl tri-tert-butylmethyl l-methyl- l-cyclopent yl l-methyl-l-cyclohexyl 1 -methyl- l-cycloheptyl 1 -tert-butyl- l-cyclopentyl l-tert-butyl- l-cyclohexyl 1-tert-butyl-l-cycloheptyl 1 -norbornyl 2-methyl-2-norbornyl 7-methyl-7-norbornyl 7-tert-butyl-7-norbornyl 7,7-dimethyl- l-norbornyl

1-bicyclo[2.2.2]octy1 2-methyl-2-bicyclo[2.2.2]octyl I -hydrindanyl l-bicyclo[3.2.2]nonyl l-bicyclo[3.3.1 Jnonyl 9-methyl-9-bicyclo[3.3. llnonyl 9-tert-butyl-9-bicyclo[3.3. llnonyl l-bicyclo[3.3.2]decy1 1 -bicycle[ 3.3.31 undecyl 1 O-perhydroquinacyl 1 -noradamant y 1 3-noradamant yl 7-methyl-3-noradamant yl l-adamantyl 2-methyl-2-adamantyl 2-ethyl-2-adaman tyl 2-isobutyl-2-adamant yl 2-tert-butyl-2-adamant y l 2-tert-pentyl-2-adamantyl 2-neopen tyl-2-adaman tyl 6-protoadamant yl 3-homoadamantyl

l-bicycl0[3.2. lloctyl

686.91 664.96 639.53 603.77 542.68 538.26 69 1.45 65 1.03 635.25 600.24 568.87 553.74 841.21 721.67 771.77 674.04 792.10 742.20 737.46 653.27 648.86 693.09 667.60 638.10 557.78 666.85 650.86 772.39 792.42 8 14.83 785.75 669.17 624.92 600.99 533.51 544.10 524.54 500.70 7 16.84 653.14

- 245.40 - 277.22 - 304.88 - 340.58 -391.00 - 376.72 - 232.72 - 282.10 - 279.75 - 309.67 - 350.06 - 344.86 - 166.48 - 195.57 - 193.51 - 257.73 - 228.6 1 - 220.90 - 227.6 1 -256.12 - 262.8 1 - 239.09 - 265.74 - 286.03 - 334.6 1 - 240.53 - 228.86 - 204.89 - 181.77 - 171.16 - 206.13 - 264.87 -291.45 - 3 19.66 - 379.81 - 341.60 - 362.73 -400.14 -217.53 - 256.01

573.85 540.98 507.90 465.80 384.23 365.87 574.73 528.10 504.6 1 459.01 418.89 496.88 732.75 599.68 648.97 527.53 665.32 627.08 624.67 520.67 521.32 569.70 546.32 501.96 393.1 1 537.97 520.67 648.38 678.60 700.80 661.57 553.64 49 1.34 456.09 368.57 381.81 354.48 332.60 596.33 523.98

0 12.6 21.8 29.7 45.6 56.5 20.5 28.5 35.2 40.6 45.2 49.4 27.6 34.3 34.3 48.5 39.8 34.3 34.3 39.8 39.8 39.8 39.8 44.4 55.7 44.4 48.5 43.9 38.9 38.9 43.9 43.9 48.1 51.9 58.2 58.2 61.1 61.1 43.9 48.1

819.25 830.75 834.55 836.08 820.83 799.08 827.95 838.65 819.51 809.26 814.14 791.11 926.85 829.55 876.79 833.79 933.68 882.29 886.59 816.54 823.88 848.54 851.81 832.35 783.37 822.85 798.07 897.21 899.28 910.87 91 1.63 862.44 830.90 827.63 806.53 78 1.57 778.29 793.83 857.79 828.10

3.90 2.49 2.03 1.84 3.7 1 6.37 2.83 1.52 3.87 5.12 4.53 7.35

- 9.29 2.64

-3.15 2.12

- 10.12 - 3.83 -4.35

4.23 3.33 0.3 1

- 0.09 2.30 8.30 3.46 6.50

- 5.65 -5.91 - 7.33 - 7.42 - 1.39

2.47 2.87 5.46 8.52 8.92 7.02

2.82 -0.82

2.38 2.96 3.17 2.45 3.48 6.8 1 3.25 2.05 3.77 5.96 4.65 6.80

2.49

1.83

- 10.45

- 2.50

- 10.49 -5.17 - 4.00

4.58 4.26

0.5 1 3.23 8.18 3.08 6.44

-0.13

-6.16 - 5.28 - 7.28 - 7.96 - 0.4 1

3.10 3.77 4.35 8.40 8.76 4.75

- 0.09 1.91

Energies in kJ mol- '. Asolv H is the enthalpy of solvation of the carbocation relative to the tert-butyl cation. AAG,,,, = AG343, R + - AG343, R-Br

+ AsOlv H . kcalc and kexp are the calculated and the e ~ p e r i m e n t a l ~ , ~ ~ solvolysis rates, respectively.

J . Chem. SOC., Faraday Trans. , 1996, Vol. 92 357

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10

6

7

I p 2 2 g) -2 -

-6

-10 I I I I I I+

776 800 826 860 876 900 926

M G ( R B r-R+)/kJ mol-'

Fig. 5 Relation between calculated AAG (RBr - R') values and experimental solvolysis rates: y = -0.1225~ + 104.3028; R = 0.9881; O(l0g k) = 0.7732

Geometry data

To obtain geometry data to optimize the force field, ab initio calculations were carried out on the tert-butyl (three conformations), tert-pentyl (two conformations), 2,3-dimethyl- 2-butyl, 2,3,3-trimethyl-2-butyl, 3-ethyl-3-penty1, l-methyl-l- cyclopentyl, 1 -methyl- l-cyclohexyl and l-norbornyl cations. To reduce the calculation time for the larger systems, we imposed the expected symmetry planes on the starting geometry (Table 16). According to our force-field calculations, the C,, conformation of the 3-ethyl-3-pentyl cation is at a local energy minimum.

The ab initio calculations produce data on bond lengths, valency angles and torsion angles. There are no problems in using the valency and torsion-angle data, but the bond-length data need some correction before they can be used for the optimization of the force field. The ab initio calculations produce bond lengths for a non-vibrating structure (re values). This means that the calculated bond lengths have, at least, to be corrected for the anharmonicity of the motions of the atoms arising from the zero-point energy vibration^.^^ After comparison of the errors in ab initio calculations for C-C and C=C bonds with the C-C, C=C and C-C+ Morse potentials in the force field, it was decided to add 0.012 8, to the ab initio calculated C-C f bond lengths before including them in the training set. Fig. 6 shows the reproduction of the corrected ab initio bond lengths by the force field. The 11 C-C+ bond lengths in the training set were reproduced with an average error of 0.008 A.

: 1.60 ;F

a $2 0 1.40 .c

1.47

1.46 1.48 1.47 1.48 1.50 1.51 1.52

ab initio Reproduction of corrected ab initio C-C+ bond length data Fig. 6

Table 16 Symmetry constraints used in ab initio calculations ~

carbocation symmetry constraint

3-ethyl-3-pentyl l-methyl- l-cyclohexyl l-norbornyl

'3,

c,, '1,

Fig. 7, 8 and 9 show the reproduction of ab initio data on, respectively, C-C+-C, C-C-C+ and H-C-C+ valency angles. Apart from these C + -containing valency-angle data, some C-C-C and C-C-H valency angles also derived from the ab initio calculations were present in the training set. Table 17 shows the reproduction of the different types of valency angles. The 58 ab initio torsion-angle data were repro- duced with an average unsigned error of 3.0".

125

120 G= Q)

0 $2 -

11s

110 110 115 120 125

ab initio

Reproduction of ab initio C-C+-C valency angle data Fig. 7

120

110

9 = 100

&?

Q)

Q)

0 w-

90

80

80 90 100 110 120

ab inirio

Fig. 8 Reproduction of ab initio C-C-C+ valency angle data

120

11s

U Q) G= - Q) 110

$2 0 .c

105

100 100 110 120

ab initio

Fig. 9 Reproduction of ab initio H-C-C+ valency angle data

Table 17 data

Reproduction of different types of ab initio valency angle

average present in unsigned number

type of valency angle training set error/degrees

C-C+-C, C-C-C+, H-C-C+ 60 1.14 C-C-C, C-C-H 42 0.76 total 102 0.98

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IR and Raman data

The presence of IR and Raman data in the training set is of vital importance to the optimization of the different types of force constants. The short lifetimes of the carbocations inhibit easy measurement of their vibration spectra. However, despite the obvious experimental difficulties, some spectra of com- pounds containing tertiary carbocations have been report-

These spectra are usually obtained after generating the carbocation in a strongly acidic matrix, such as HSbF,, while molecular mechanics usually produce gas-phase data. The effect of the different environment on the vibration fre- quencies is not clear, making these experimental observations less suited for use in the optimization of the force field. Another disadvantage of the experimental spectra is the lack of assignment of the frequencies. Assignments allow the identi- fication of the different types of vibrations and give a more reliable link between the calculated and the experimental values. For these two reasons it was decided not to use the experimental spectra, but to rely on ab initio calculations. Ab initio calculations provide well assigned vibration spectra, giving the symmetry of the normal modes and the IR and Raman intensities.

However, the use of data from ab initio calculations is not without problems. Frequency values obtained from Hartree- Fock calculations contain systematic errors due to the neglect of electron correlation^.^^ This means that ab initio calculated frequencies need to be rescaled before they can be used for optimizing the force field.

Hartree-Fock calculations with the 6-31G* basis set are known to overestimate the value of the frequencies by 10- 12%.2s The easiest way to use ab initio spectra is, therefore, to multiply all frequencies by 0.89, but there is a way to get better values for this scaling factor. There are some well assigned IR and Raman spectra reported for trimethyl- and triethyl-b~rane.~~-, The tert-butyl and 3-ethyl-3-pentyl cations are isoelectronic with trimethyl- and triethyl-borane. We assumed that the scaling factors between the ab initio and the experimental frequencies for the carbocations would be equal to those for the alkylboranes. If we now calculate the ab initio spectra for the alkylboranes, a separate scaling factor is obtained for each vibration in the cations. In this way, we

ed.6.26-28

obtained more reliable values for the vibration frequencies of the tert-butyl and the 3-ethyl-3-pentyl cations. Table 18 shows the results of these calculations for the tert-butyl cation. The imaginary frequency at 71i cm-' in the ab initio tert-butyl cation spectrum is due to the fact that the C,, conformation is not at an energy minimum.

The same rescaling was performed on the frequencies of the 3-ethyl-3-pentyl cation. Since the scaling factors for the tert- butyl and the 3-ethyl-3-pentyl cations looked the same (frequencies in the same regions have approximately the same scaling factors), we applied the same scaling factors to obtain the rescaled spectrum of the 2,3,3-trimethyl-2-butyl cation. The spectrum of this compound was given a lower weight in the optimization than the spectra of the tert-butyl and 3-ethyl- 3-pentyl cations. Fig. 10 shows the results of the frequency calculations. The frequencies were reproduced with an average deviation from the rescaled a6 initio frequencies of 29 cm- '. Charge calculations

Table 19 shows the charges for the l-norbornyl cation as cal- culated by the force-field and the ab initio method (6-31G* basis set, Mulliken charges).

The Mortier parameters x* and y ~ * (Table 1) for the carbo-

- E

200 100 0 -100 -200

0 lo00 2Ooo 9ooo

scaled ab initio frequencykm-' Fig. 10 Results from the frequency calculations

Table 18 Rescaling of the frequencies"*b of the tert-butyl cation

assignment 'BMe3, exp 'BMe3. ai sca VIBU *, ai V ~ B U +, sca v t ~ u + , calc

E" A" E A" A' E E' A' A" E' A' E' E" A" E' A' E' A A" E" A' E'

- -

320 336 67 5 755 864 906

1014 1152 1296 1342 1435 1438 1438 1435 2868 2868 2986 2985 2985 2987

15 104 3 19 375 687 764 927 898

1092 1249 1485 1490 1615 1621 1606 1609 3166 3170 3212 3210 3250 3253

- -

0.9958 1.1146 1.0178 1.0126 1.0729 0.99 1 5 1.0768 1.0839 1.1457 1.1102 1.1256 1.1272 1.1172 1.1213 1.1040 1.1053 1.0758 1.0755 1.0886 1.0890

71i 141 41 5 475 805 894

1055 1087 1245 1409 1499 1504 1576 1601 1622 1624 3179 3189 3237 3237 3346 3347

- -

417 426 79 1 883 983

1096 1106 1300 1309 1355 1401 1420 1452 1449 2880 2885 3009 3010 3074 3073

27 89

417 414 772 865 967 962 987

1292 1302 1429 1535 1530 1511 1536 2898 2882 3010 3009 3047 3050

cm- values of the tert-butyl cation; v,~,,+, cation.

vBMe3, exp , experimental values of trimethylborane; vBMea, oi, ab initio values of trimethylborane; sca = vBMe3. JvBMe3, exp; v , ~ " + , oi, ab initio = ~ , ~ ~ + , ~ ~ / s c a , scaled values of the tert-butyl cation; v , ~ ~ + . ca,c, molecular mechanics values of the tert-butyl

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Table 19 Force field and ab initio charges for the l-norbornyl cation

atom ab initio force field

4 4 1 2 3 4 5 6 7 8 9

10 9

7 11

0.2370 - 0.3866 - 0.3480 - 0.2644 - 0.3649

0.2733 0.2628 0.248 1 0.2409 0.2768 0.2672

0.2058 -0.1218 - 0.0893 -0.0588 - 0.1200

0.1279 0.1258 0.1275 0.1247 0.1285 0.1274

cation could not be optimized specifically using dipole moments, since no dipole data are available for tertiary carbo- cations. However, owing to the large influence that the charges have on the energies of the carbocations, there is still a strong correlation between the training set and the Mortier parameters.

Non-classical behaviour of tertiary carbocations

Table 17 shows that the reproduction of the valency angles containing C + is not as good as the reproduction of the C-C-C and C-C-H valency angles (average error 1.14" and 0.76", respectively). The reason for this difference may be found in the non-classical behaviour of the tertiary carbo- cations.

Molecular mechanics can, in principle, only deal with struc- tures with localized bonds. When delocalization takes place, such as, for example, in the conjugation of double bonds, a separate quantum-chemical scheme is required to calculate the bond orders for the bonds in the conjugated system. Once these bond orders are known, molecular mechanics can be applied.38 Delocalization of electrons can also take place in tertiary carbocations. Fig. 11 shows two possible resonance structures for the 3,5,7-trimethyl-l-adamantyl cation.39

The contribution due to the resonance structure I1 is obvious in the crystal structure of this carbocation:40 the C(l)-C(2) bond length is of the order of 1.43-1.47 A and the C(2)-C(3) bond is 1.61-1.62 A. This means that the bonds next to the ionic centre have a strong double-bond character, while the bonds between the a and p carbon atoms are clearly lengthened. As Table 20 shows, the force field is not able to reproduce all of this non-classical behaviour.

The force field reproduces the valency angles around the cationic centre well, but reproduces only part of the lengthening of the C,-C, bonds [C(2)-C(3) is calculated to be only 0.006 A longer than C(3)-C(4)]. The force-field geometry obviously represents a more classical structure. The failure to reproduce non-classical structures with this force field is even more evident if one compares the force field results for the 1,2,4,7-anti-tetramethyl-2-norbornyl cation with the experimentally measured crystal as is depicted in Fig. 12.

I II Fig. 11 Two resonance structures for the 3,5,7-trimethyl-l-ada- mantyl cation

Table 20 Comparison between crystal and force-field geometries for the 3,5,7-trimethyl-l-adamantyl cation

bond or crystal force-field geometry valency angle" geometry (resonance structure I)

1.43-1.47 8, 1.476 8,

1.51-1.55 8, 1.551 8,

C(l)-C(2) W--C(3) C(3)---C(4) C( 1)-C(2)-C(3) 98-1 01 O 100.1"

1.61-1.62 A 1.557 A

C(2)-C( 1)-c(9) 116-120" 117.4"

" See Fig. 11.

Fig. 12 shows that the force-field geometry is much closer to the classical norbornane structure than the crystal geometry. Again, the deviation from the classical geometry can be explained by the effect of resonance structures.20 This is also observed in the crystal structure of the 2-methoxy-1,7,7- trimethyl-2-bicycloC2.2. llheptyl cation.42

Discussion Comparison of the carbocation parameters with the hydrocar- bon parameters in the DMM force field12 shows that some carbocation parameters have values between the C sp3 and C sp2 parameter values. The V, terms for the carbocation torsion angles with central C+-C bonds are higher (ca. 10 kJ mol-'j than those for C sp3-C sp3 torsion angles (ca. 2 kJ mol-') but much lower than the V2 parameters for torsion angles around C sp2-C sp2 bonds (about 38 kJ mol-I). The values of the force constant for the Cf-C bond stretching (3457 kJ mol-' k2) are between those of the C-C and C=C bond stretching constants (2790 and 5351 kJ mol-' k2, respectively). This agrees with the often proposed reso- nance structures for tertiary carbocations (see, for example, Fig. 11 j in which the original cationic centre bears a double bond.

Fig. 12 Force field (I) and crystal (11) geometries of the 1,2,4,7-anti- tetramethyl-2-norbornyl cation; superimposed in thin lines is the force field geometry of norbornane

360 J . Chem. Sue., Faraday Trans., 1996, Vul. 92

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Table 21 Comparison of the results of the two carbocation force fields

type of data bond length valency angle torsion angle vibration frequency

Rsolvolysis rate 18, /degrees /degrees /cm - ' ~~ ~

with charge derivatives to geometry 0.988 1 0.008 0.98 3.0 without charge derivatives to geometry 0.9866 0.004 1.03 3.3

29 28

The high value for the out-of-plane force constant of the cationic centre indicates a strong tendency for planarity. This seems to agree with the observed geometry for the 3,5,7- trimethyl-1 -adamantyl cation and also explains the low solvolysis rates of alkyl bromides such as l-norbornyl bromide. The corresponding carbocations suffer from severe deviations from planarity at the cationic centres.

The ab initio calculated geometries show a strong corre- lation between the values of the valency angles and those of the torsion in the molecules. This resulted in high values for the torsion-bend force constants in the force field. This torsion-bend interaction proved to be important to the repro- duction of the energy data.

As Fig. 11 and 12 show, it cannot be expected that the force field reproduces the non-classical behaviour of the tertiary carbocations. However, the capability of the force field to reproduce the solvolysis rates shows its ability to describe the energies of the carbocations. This shows that the presence of the resonance structures does not have a large effect on the energies of the carbocations, or that this effect is about equal for all carbocations.

As Table 17 shows, the reproduction of valency angles con- taining an ionic centre is worse than that of the C-C-C and C-C-H angles in the training set (average error of 1-14" and 0.76", respectively). This is probably due to the larger devi- ations from the classical structure close to the ionic centre.

As already mentioned, judging from the results for the solvolysis-rate reproduction, the force field can reproduce the energies of the carbocations. Keeping in mind the large experi- mental errors, the results for the enthalpy of formation calculations are also acceptable. One might be surprised at the good correlation between the calculated energies and the solvolysis rates [ R = 0.9881, o(1og k ) = 0.77321, because the calculated energies carry a cumulative error from the hydro- carbon, the bromide and the carbocation force fields, while the experimental rates also carry an error margin. Miiller and Milin4 estimated this experimental error to contribute about 0.65 to a(log k) .

Apart from providing an excellent means of optimizing the force field, the solvolysis-rate data also showed the impor- tance of the use of entropy data in these types of calculation. The entropy difference between an alkyl bromide and its car- bocation proved to be far from constant, making it an impor- tant factor in the reaction rate. For compounds with methyl groups attached to the cationic centre (for example the tert- butyl, tert-pentyl and 3-methyl-3-pentyl cations), the difference in entropy is much lower (10-20 J mol-' IC') than for the other compounds (30-40 J mol-' K-'). The reason for this is that the barrier for methyl rotation is larger in the alkyl bromide than in the corresponding tertiary carbocation. This increases the entropy of the carbocation relative to that of the alkyl bromide.

The force field reproduces the rescaled ab initio IR and Raman frequencies with an average deviation of 29 cm-' for 202 vibrations. In view of the uncertainties of the rescaling procedure, this is acceptable, although not as good as the reproduction of the vibration frequencies of alkanes and alkenes by the parent DMM force field (average deviation ca. 20 cm - I).

As explained in the Methods section, the addition of 'charge derivatives to geometry' to the energy-minimization scheme has a large influence on the calculated energies and

geometries of the carbocations. For this reason, the present carbocation force field was optimized including this full but time-consuming description of the forces. A second force field was optimized in which these charge derivatives to geometry were neglected, as in the original DMM hydrocarbon force field.12 The results of these two carbocation force fields are compared in Table 21.

As Table 21 shows, neglect of the charge derivatives to geometry slightly worsens the solvolysis-rate results, but the reproduction of the bond lengths improves significantly. Addi- tion of the charge derivatives to geometry causes the charge- charge interactions to have too large an effect on these bond lengths. This cannot be improved without worsening the vibration-data reproduction. Its more adequate description of the forces in molecules makes the force field that includes charge derivatives to geometry better suited for conforma- tional research. When thermodynamic data on sets of large molecules are required, the much higher speed of calculation of the second force field might prevail.

Conclusions As the results for the heat of formation calculation and the solvolysis-rate reproduction show, this force field can describe the energies of tertiary carbocations quite well. Owing to the non-classical behaviour of the carbocations, it was not to be expected that an exact reproduction of geometries would be possible. Judging from the results, the force field seems able to reproduce part of this non-classical behaviour. A good repro- duction of vibration frequencies of tertiary carbocations is found, indicating an adequate description of the forces in these compounds. This makes the force field well suited for calcu- lating thermodynamic data for tertiary carbocations.

This work was supported by a grant from the GOA, the Dutch Foundation for Geological, Oceanographic and Atmo- spheric Research, grant no. 751.355.017.

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Paper 5/05121E; Received 1st August, 1995

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