In the Classroom
Semiempirical and DFT Investigations of the Dissociation of Alkyl Halides Jack R. Waas Department of Chemistry, Bethel University, St. Paul, MN 55112-6999;
[email protected] Computational chemistry is becoming an increasingly useful and desirable tool for undergraduate teaching and research. In the teaching of organic chemistry, it has become possible to use computational methods to illustrate trends in the properties and reactivity of molecules. After some instruction, undergraduate students can perform computational experiments on the same molecules and systems discussed in their classes and textbooks, so as to help the students better visualize and understand the class material (1). In designing such experiments, it is worthwhile for the instructor to run several relevant calculations beforehand, using different methods and levels of theory, so that student results can be anticipated. Interest in such computational experiments has led to recent articles in this Journal involving mechanistic studies of the Michael (2) and Ritter (3) reactions, the reaction of morpholine with tert-butyl acetoacetate (4), and the hydration of alkenes (5, 6). With the recent advent of density-functional theory (DFT) methods (7–10), rigorous calculations have also been recently reported of the conformational energies of substituted cyclohexanes (11) and the enthalpies of homolytic dissociation of C⫺H, N⫺H, and O⫺H bonds (12). Background The dissociation of alkyl halides is the first mechanistic step in SN1 and E1 processes. The process involves the heterolytic, unimolecular dissociation of the alkyl halide, to produce the corresponding carbocation and halide anion. The following general equation describes the process: +
C
C
+
−
(1)
X
X
The ease with which the alkyl halide undergoes dissociation is dependent on the nature of the alkyl halide substrate and the halide X, as well as solvent, temperature, and other factors. For instance, students learn in their organic chemistry lectures that the ease of dissociation increases with increased substitution at the carbon α to the halogen atom and with increased size of the halogen atom. One of the methods by which the dissociations of different alkyl halides can be compared is by their enthalpies of dissociation. The enthalpy of dissociation can be calculated by a Hess’s law treatment of the individual enthalpies of formation for each reactant and product in the above process
( )
( )
∆dissH = ∆f H R + + ∆f H X − − ∆ f H (R X )
(2)
where X− is the halide and R+ is the alkyl group. As an example, to derive the experimental enthalpy of dissociation of tert-butyl chloride in the gas phase, the following version of eq 1 would be used: (CH3)3CCl
(CH3)3C+ + Cl−
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The experimental gas-phase enthalpy of formation values would then be entered in eq 2 as follows: ∆dissH ( CH3 )3C Cl = ∆f H ( CH 3 )3 C +
( )
+ ∆f H Cl −
− ∆f H
( CH3 )3C Cl
= ( 736.4 kJ mol ) + ( −227.4 kJ mol ) − ( −179.9 kJ mol ) = 688.9 kJ mol
This same method can of course also be used with absolute enthalpies calculated using Gaussian or other computational chemistry software. Strictly speaking, however, it is not enthalpies of formation that Gaussian calculates, but absolute enthalpies in hartrees, which can then be converted to kJ兾mol or kcal兾mol as desired. However, using these absolute enthalpy values (rather than enthalpies of formation) in a Hess’s Law treatment still results in an enthalpy difference, which corresponds to ∆rxnH or ∆dissH. Of course it must be stressed that the enthalpy of dissociation is only one of many values that can be used to compare the dissociation of one alkyl halide to another. The enthalpy change gives us no information about the activation energy for the process, the kinetics of the process, or indeed even the spontaneity of the process. However, it is a valid tool for the comparison of similar species in a process such as the dissociation of a corresponding bond in a series of compounds. While calculations of the stabilities (13, 14) and heats of formation (15, 16) of carbocations have been reported, including a method for converting DFT electronic energies of carbocations into enthalpies of formation (17), the use of computational methods (particularly DFT methods) to compare the heterolytic dissociation of alkyl halides in the gas phase has not yet been reported. In this work, the strategy was to determine whether the qualitative statements organic chemistry instructors make to their students about the dissociation of alkyl halides (especially in the context of SN1 and E1 reaction mechanisms) could be demonstrated by calculating the enthalpies of reaction for processes of the type shown in eq 1 using the Gaussian 2003 (18) software package. This could in turn be accomplished by calculating the absolute enthalpies for each alkyl halide RX under study, each carbocation R+, and each halide anion X−, and by subjecting them to the Hess’s law treatment described above. The resulting values, which correspond to enthalpies of dissociation for each alkyl halide RX, can then be compared to literature dissociation enthalpy values whenever possible. The most abundantly available experimental data in the literature refer to experiments performed in the gas phase, although values from experiments performed in aqueous or other solutions would be more easily comparable to the material discussed in the organic chemistry classroom. However,
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obtaining gas-phase values for comparison using Gaussian is relatively straightforward. By comparing these calculated gasphase values to the experimental gas-phase values, we can ascertain which of the computational methods best models the alkyl halide dissociation process. Finally, after determining which computational method gives results for this system that are most comparable to experimental gas-phase enthalpy changes, we can use the same Gaussian computational method to calculate absolute enthalpies for each species in solution. These absolute enthalpies can then be manipulated in the same manner as the gas-phase absolute enthalpies, to calculate values corresponding to enthalpies of dissociation in solution. Methods Absolute enthalpies of a wide range of alkyl halides were calculated using the Gaussian 2003 software package (18) on a Windows XP computer equipped with a Pentium 4 processor at 2.8 GHz and 512 MB of memory. Five different, but common, methods were used to calculate these enthalpies. These included the AM1 (19–29) and PM3 (30) semiempirical methods, as well as the Hartree–Fock (HF) (31–33) and B3LYP (34) methods [the latter two with the 6-31G(d) (35, 36) basis set]. The B3LYP method was also used with the more diffuse 6-31G+(d) (37) basis set, for comparison with the other B3LYP results. Using Gaussian to calculate enthalpies requires the calculation of both electronic energies and vibrational frequencies. Since there are no vibrations or rotations for single atoms, the enthalpies of formation of the halide ions were derived from total energies, Etot. These, in turn, were determined by adding (3兾2)RT (for translational degrees of freedom) to the calculated electronic energies,
where R is the ideal gas constant and T is the Kelvin temperature (38). At 298 K, the value of (3兾2)RT is equal to 3.72 kJ兾mol. However, since enthalpy is defined as follows for a mole of ideal gas (39), (4)
H = E tot + PV = E tot + RT
we can combine eqs 3 and 4 to derive the following expression for monatomic species, H = E elec + ( 3 2 ) RT + RT = E elec + ( 5 2 ) RT
(5)
For polyatomic molecules and ions, calculating thermal enthalpies at 298 K is more complicated. In addition to the electronic energy, the vibrational contribution to the total energy must also be calculated, as well as the translational and rotational contributions. The vibrational energy can be determined by the following equation: 3 N − 6 ∞
∑
∑ N j, n n + 1 2 h νj
j =1 n = 0
(6)
In this equation N( j, n), the number of molecules with vibration j in the nth energy level, is determined by the vibrational frequencies and the temperature. The vibrational 1018
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Data and Analysis The results of the calculations are listed in Table 1. The 12 compounds under investigation include chloromethane, 1-chlorobutane, 2-fluorobutane, 2-chlorobutane, 2-bromobutane, tert-butyl fluoride, tert-butyl chloride, tert-butyl bromide, allyl chloride (AllCl), 2-chloro-2-methyl-3-butene (t-AllCl), 1-chloro-2,4,6-cycloheptatriene (tropylium chloride, tropCl), and chlorobenzene. Alkyl iodides were not studied because of the lack of parameters for iodine. The calculated enthalpy change for the dissociation of t-AllCl is compared with that of t-BuCl, to ascertain the effects of allylic stabilization of the resulting tertiary carbocation, as compared to an unstabilized tertiary carbocation. The calculated enthalpy change for the dissociation of tropCl is compared with that of 2-BuCl, to ascertain the effect of formation of a stabilized
(3)
E tot = E elec + ( 3 2 ) RT
E vib =
frequencies in turn depend on the force constant along each vibration (that is, ∂2E兾∂x2) and the masses of each atom. The translational component of the total energy is then equal to (3兾2)RT, while the rotational component is an additional RT for a linear molecule or (3兾2)RT for a nonlinear molecule (39). Once Gaussian has calculated the thermal vibrational energy of a given species and taken it into account as a part of the total energy, it calculates the absolute enthalpy according to eq 4 above. To complete the determination of the experimental gasphase enthalpies of dissociation, literature values were used for the enthalpies of formation of chloride anion (40), bromide anion (41), chloromethane (42), 1-chlorobutane (43), 2-chlorobutane (44), 2-bromobutane (45), tert-butyl chloride (46), tert-butyl bromide (47), allyl chloride (48), chlorobenzene (49), methyl cation (50, 51), n-butyl, 2-butyl, and tert-butyl cations (52), allyl cation (53), and phenyl cation (54, 55).
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Table 1. Gas Phase Dissociation Values, ∆dissH Cpd
AM1
PM3
HF
B3LYP,6-- B3LYP,6-31G(d) 31G+(d)
Expt
CH3Cl
965
908
871
976
925
939
n-BuCl
830
779
718
745
699
839
2-BuCl
755
717
661
706
660
743
t-BuCl
701
676
613
655
609
689
2-BuF
1083
950
991
1054
819
---
755
717
661
706
660
743 711
2-BuCl 2-BuBr t-BuF t-BuCl
772
658
657
711
657
1023
904
948
1010
775
---
701
676
613
655
609
689
t-BuBr
716
624
607
659
615
656
AllCl
786
738
657
722
660
682
t-AllCl
666
641
559
606
562
---
t-BuCl
701
676
613
655
609
689
tropCl
582
543
420
448
424
---
2-BuCl
755
717
661
706
660
743
PhCl
956
913
826
854
813
906
NOTE: All values in kJ/mol.
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In the Classroom
6π aromatic carbocation as compared to an unstabilized secondary carbocation. According to the trends we commonly teach in undergraduate organic chemistry classes, we would expect the dissociation to become less endothermic (lower enthalpy of dissociation) as the halide leaving group gets larger and also as the carbon bearing the leaving group becomes more substituted. We would also expect the allylic and tropylium substrates to have lower dissociation enthalpies than the corresponding unstabilized substrates. Finally, we would expect chlorobenzene to have one of the highest enthalpies of dissociation of all the compounds in this study, since sp2 hybridized carbons do not tend to undergo loss of a leaving group. The AM1 and PM3 semiempirical methods give results with similar average deviations from eight experimental values (41.8 and 40.1 kJ兾mol, respectively). However the PM3 method gives results that are more closely clustered around the experimental values than the AM1 results. The HF results again agree with all of the expected qualitative trends, with an average deviation of 79.3 kJ兾mol from the experimental values. Note that the HF and B3LYP calculated enthalpy changes for the dissociation of 1chlorobutane are systematically lower than the same enthalpy changes calculated using semiempirical methods. This could be because the calculations converged on a bridged nonclassical structure after optimization by the HF and B3LYP methods, whereas they converged on a classical structure after optimization by the semiempirical methods. The average deviation for the B3LYP method with the 6-31G(d) basis set is 42.4 kJ兾mol. Surprisingly, the B3LYP method with the 6-31G+(d) basis set gives worse results than the same method with the 6-31G(d) basis set. One might have expected that the more diffuse basis set would model the carbocations better, but there appears to be some kind of trend in this particular system that gives results closer to the experimental gas-phase values with the less diffuse basis set. The average deviation from experimental values with the 631G+(d) basis set is 75.3 kJ兾mol, nearly twice the average deviation with the 6-31G(d) basis set. The immediate conclusion is that the PM3 method would be best for this particular system for students in a laboratory situation, because of both the excellent results achieved and the decreased computational expense. Other conclusions from this study include the expected decrease in endothermicity of the dissociation of t-AllCl compared to tbutyl chloride, by a significant quantity [35 kJ兾mol by both AM1 and PM3; 54 kJ兾mol by HF; 49 kJ兾mol by B3LYP/631G(d); and 47 kJ兾mol by B3LYP/6-31G+(d)]. This can be attributed to the allylic stabilization of the carbocation. Tropylium chloride dissociates significantly more readily than 2-chlorobutane [173 kJ兾mol by AM1; 174 kJ兾mol by PM3; 241 kJ兾mol by HF; 258 kJ兾mol by B3LYP/6-31G(d); and 236 kJ兾mol by B3LYP/6-31G+(d)] because the tropylium cation is an aromatic 6π system and highly stabilized compared to a secondary alkyl carbocation control. Finally, chlorobenzene has no significant tendency to dissociate at all, its dissociation being anywhere from 100–300 kJ兾mol more endothermic than chlorides with a similar number of carbon atoms, which agrees with our statement to students that only alkyl halides with the halogen bonded to an sp3 carbon have any tendency to dissociate to form the carbocation. www.JCE.DivCHED.org
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Of course, the trends in the gas-phase calculations might correspond poorly to the trends in solution. In teaching organic chemistry, we generally describe reactions as occurring in solution. Thus, it is advantageous to reexamine the alkyl halide dissociation reaction in solution, by running calculations using a solvent model and comparing the results to the gas-phase results. We used the IEFPCM solvent model (56– 61) to calculate absolute enthalpies of the same species in aqueous solutions, and then in turn calculate values corresponding to the enthalpy of dissociation of each organic halide. Water is an appropriate solvent to choose for these calculations, because SN1 and E1 reactions are generally carried out in a protic solvent. The results are given in Table 2, and compared with the gas-phase data. With the exception of 1-chlorobutane, which is omitted from this table because the 1-butyl cation would not converge using the solvent model, there is a relatively clear trend. The difference between the values in the two columns for any given alkyl halide, corresponding to the heat of dissolution, is relatively constant. The average value of the difference is ᎑554 kJ兾mol, with a standard deviation of 33 kJ兾mol. We can conclude from this that the trend in the data is very similar, whether the experiments are performed in the gas phase or in aqueous solution. Therefore, using gas-phase data to discuss the trends in calculated values corresponding to heats of dissociation of alkyl halides, has some validity even though reactions involving such unimolecular dissociations are generally run in solution. Some experimental rate data for alkyl halide dissociation in solution are available and are consistent with the calculated solution results (62, 63). For example, the 275 kJ兾mol greater calculated enthalpy of dissociation of tert-butyl fluoride compared to tert-butyl chloride is consistent with the 10000-fold increase of solvolysis rate of the chloride compared to the fluoride (64).
Table 2. Comparison of the Gas Phase Dissociation and Solution Values ∆dissH Cpd
B3LYP, 6-31G(d), H2O
B3LYP, 6-31G(d), gas phase
∆solvH
CH3Cl
382
976
᎑594
2-BuCl
155
706
᎑551
t-BuCl
122
655
᎑533
2-BuF
422
1054
᎑632
2-BuCl
155
706
᎑551
2-BuBr
177
711
᎑534
t-BuF
397
1010
᎑613
t-BuCl
122
655
᎑533
t-BuBr
144
659
᎑515
AllCl
᎑562
160
722
t-AllCl
80
606
᎑526
t-BuCl
655
᎑533
tropCl
122 ᎑75
448
᎑523
2-BuCl
155
706
᎑551
PhCl
297
854
᎑557
NOTE: All values in kJ/mol.
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Conclusion The use of five different common computational methods to study the gas-phase dissociation of several alkyl halides allowed the comparison of one method to another. In this particular system, it appears that the B3LYP method [with the 6-31G(d) basis set] gave better results in a quantitative sense than all of the other methods, with the exception of the semiempirical PM3 method. The PM3 method produced gas-phase results that were marginally better than those produced by the B3LYP method or at least equivalent within experimental error. In general, the results of all five methods agreed broadly with empirically known trends in carbocation formation from alkyl halides. IEFPCM calculations with water as the solvent showed a very similar trend to the gas-phase calculations, which further indicated the usefulness of the gas-phase calculations. However, it must be stressed that the “best” method to use for studying any given system can differ from one system to another. Experiments suitable for undergraduates in a laboratory setting can be derived from such calculations and can give the students added insight into the material discussed in their organic chemistry lecture sections. We have already made some trials at implementing a computational experiment derived from this study that is suitable for students in an undergraduate organic chemistry class. We plan on publishing the details of this experiment in a subsequent article. Acknowledgments I wish to thank Rollin King, Dan King, and Don Albright for helpful discussions. Literature Cited 1. Martin, N. H. J. Chem. Educ. 1998, 75, 241–243. 2. Poon, T.; Mundy, B. P.; Shattuck, T. W. J. Chem. Educ. 2002, 79, 264–267. 3. Hessley, R. K. J. Chem. Educ. 2000, 77, 202–203. 4. Cook, A. G.; Kreeger, P. K. J. Chem. Educ. 2000, 77, 90–92. 5. Graham, K. J.; Skoglund, K.; Schaller, C. P.; Muldoon, W. P.; Klassen, J. B. J. Chem. Educ. 2000, 77, 396–397. 6. Hessley, R. K. J. Chem. Educ. 2000, 77, 794–797. 7. Hohenberg, P.; Kohn, W. Phys. Rev. 1964, 136, B864. 8. Kohn, W.; Sham, L. J. Phys. Rev. 1965, 140, A1133. 9. The Challenge of d and f Electrons; Salahub, D. R., Zerner, M. C., Eds.; American Chemical Society: Washington DC, 1989. 10. Parr, R. G.; Yang, W. Density-Functional Theory of Atoms and Molecules; Oxford University Press: Oxford, 1989. 11. Freeman, F.; Tsegai, Z. M.; Kasner, M. L.; Hehre, W. J. J. Chem. Educ. 2000, 77, 661–667. 12. Ingold, K. U.; Wright, J. S. J. Chem. Educ. 2000, 77, 1062– 1064. 13. Del Bene, J. E.; Aue, D. H.; Shavitt, I. J. Am. Chem. Soc. 1992, 114, 1631–1640. 14. Klopper, W.; Kutzelnigg, W. J. Phys. Chem. 1990, 94, 5625– 5630. 15. Mishima, M.; Yamataka, H. Bull. Chem. Soc. Jpn. 1998, 71, 2427–2432. 16. Harris, J. M.; Shafer, S. G.; Worley, S. D. J. Comp. Chem. 1982, 3, 208–213.
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