Schleyer and Houk¹ offer a provocative paper examining the reference compounds that one chooses when trying to evaluate such concepts as ring strain energy and aromaticity. I discuss this at length in Chapter 2 of the book, focusing on the isodesmic, homodesmotic, and group equivalent reactions.

Their work starts with the isodesmic reaction

CH₃CH₂CH₃ + CH₄ → 2 CH₃CH₃

and note that this reaction is endothermic by 2.83 kcal mol^-1. They argue that 1,3-dialkyl interactions are stabilizing, and call this effect “protobranching”.

Gronert^2,3 has recently described the counterargument – that 1,3-dialkyl groups are repulsive – but whether the interaction is attractive or repulsive is not my concern here. Let’s proceed assuming that protobranching is in fact stabilizing.

Schleyer and Houk demonstrate that the stabilization of protobranching is nicely additive. In Table 1 are simple bond separation (isodesmic) reactions of straight-chain alkanes and cycloalkanes. This can then be extended to argue for why branched alkanes are more stable than their straight-chain analogues – namely, branched chains have more 1,3-dialkyl interactions and these are stabilizing. They note that the group separation reaction of iso-butane is more endothermic than that of pentane, yet the difference is neatly ascribed to protobranching.

Table 1. Energy of reactions and energy per protobranch (PB) using experimental heats of formation.

	ΔH	# PB	E per PB
CH3CH2CH3 + CH4 → 2 CH3CH3	2.83	1	2.83
CH3(CH2)2CH3 + 2 CH4 → 3 CH3CH3	5.69	3	2.84
CH3(CH2)3CH3 + 4 CH4 → 6 CH3CH3	14.10	5	2.82
(CH2)6 + 6 CH4 → 6 CH3CH3	7.73	6	2.76
CH(CH3)3 + 2 CH4 → 3 CH3CH3	13.65	6	2.58

Now the interesting aspect is when this concept of protobranching is applied to ring systems. The conventional (homodesmotic) reaction for cyclopropane is

(CH₂)₃ + 3 C₂H₆ → 3 CH₃CH₂CH₃ ΔH = -27.7 kcal mol^-1

Schleyer and Houk argue that protobranching is not balanced in this reaction, and the consequence is that since propane is stabilized by about 2.8 kcal mol^-1, the reaction energy should be reduced by 8.4 kcal mol^-1. Thus the ring strain energy (RSE) of cyclopropane is 19.3 kcal mol^-1. This is essentially the value obtained when one employs the isodesmic reaction to evaluate the RSE of cyclopropane, namely

(CH₂)₃ + 3 CH₄ → 3 C₂H₆ ΔH = -19.2 kcal mol^-1

And this isodesmic reaction has balanced protobrancing (none!) on both sides. The reaction that balances protobranching (two on each side) for obtaining the RSE of cyclobutane is

(CH₂)₄ + 2 CH₄ → 2 CH₃CH₂CH₃ ΔH = -21.0 kcal mol^-1

Protobranching corrections need also be made to the question of aromatic stabilization energy or resonance energy of benzene. For example, since cyclohexane is invoked as one of the reference compounds in the following reaction, the resulting energy must be corrected for six protobranching interactions.

2 C₂H₄ + (CH₂)6 → (CH)₆ + 3 C₂H₆

The question now becomes “Is protobranching real and do we need to correct for it?” Further studies should be performed.

References

(1) Wodrich, M. D.; Wannere, C. S.; Mo, Y.; Jarowski, P. D.; Houk, K. N.; Schleyer, P. v. R., "The Concept of Protobranching and Its Many Paradigm Shifting Implications for Energy Evaluations," Chem. Eur. J. 2007, 13, 7731-7744, DOI: 10.1002/chem.200700602

(2) Gronert, S., "Evidence that Alkyl Substitution Provides Little Stabilization to Radicals: The C-C Bond Test and the Nonbonded Interaction Contradiction," J. Org. Chem., 2006, 71, 7045-7048, DOI: 10.1021/jo060797y.

(3) Gronert, S., "An Alternative Interpretation of the C-H Bond Strengths of Alkanes," J. Org. Chem., 2006, 71, 1209-1219, DOI: 10.1021/jo052363t.

I mentioned “mindless chemistry” in the interview with Fritz Schaefer. This term, the title of the article by Schaefer and Schleyer,¹ refers to locating minimum energy structures through a stochastic search driven solely by a computer algorithm. No chemical rationale or intuition is used; rather, the computer simply tries a slew of different possibilities and mindlessly marches through them.

The approach employed by Schaefer and Schleyer is to use the ‘kick” algorithm of Saunders.² An arbitrary initial geometry is first selected (Saunders even suggests the case where all atoms are located at the same point!) and then a kick is applied to each atom, with random direction and displacement, to create a new geometry. An optimization is then performed with some quantum mechanical method, to produce a new structure. The kick is then applied to this new structure (or to the initial one again) to generate another geometry to start up another optimization. By doing many different “kicks” with different kick size, one can span a large swath of configuration space.

In their first “mindless chemistry” paper, Schafer and Schleyer identified some new structures of BCONS, C₆Be and C₆Be^2-.¹ In their next application,³ they explored the novel molecule periodane, which has the molecular formula LiBeBCNOF, named to reflect its make-up of one atom of every element (save neon) on the first full row of the periodic table. Krüger⁴ located the planar structure 1 (see Figure 1). But Schaefer and Schleyer, employing the “kick” algorithm located 27 structures that are lower in energy than 1, Their lowest energy structure 2 is 122 kcal mol^-1 lower than 1. They advocate for this stochastic search to gain broad understanding of the nature of the potential energy surface and then refining the search using “human logic”.

1

2

Figure 1. Optimized structures of periodane 1 and 2.

(Note – I have only provided a sketch of 2 since the supporting information for the article has not yet been posted on the Wiley web site. I will update this post with the actual structure when it becomes available.)

References

(1) Bera, P. P.; Sattelmeyer, K. W.; Saunders, M.; Schaefer, H. F.; Schleyer, P. v. R., "Mindless Chemistry," J. Phys. Chem. A, 2006, 110, 4287-4290, DOI: 10.1021/jp057107z.

(2) Saunders, M., "Stochastic Search for Isomers on a Quantum Mechanical Surface," J. Comput. Chem.. 2004, 25, 621-626, DOI: 10.1002/jcc.10407

(3) Bera, P. P.; Schleyer, P. v. R.; Schaefer, H. F., III, "Periodane: A Wealth of Structural Possibilities Revealed by the Kick Procedure," Int. J. Quantum Chem. 2007, 107, 2220-2223, DOI: 10.1002/qua.21322

(4) Krüger, T., "Periodane – An Unexpectedly Stable Molecule of Unique Composition," Int. J. Quantum Chem. 2006, 106, 1865-1869, DOI: 10.1002/qua.20948

In the pursuit of further elucidation of just what the concepts “aromatic” and “antiaromatic” mean, Schleyer and Bunz reported the preparation and characterization of a novel antiaromatic compound that is isolable.¹

Bunz synthesized the redox pair of compounds 1 and 2 that differ in the electron count in the pi-system. The former (1) has 14 π electrons and should be aromatic, while the latter (5) has 16 π electrons and should be antiaromatic. The NMR spectrum of both compounds was measured and compared to the computed signals of the parent compounds 3 and 4. The signals match very nicely. The structures of 1 and 2 were further confirmed by x-ray crystallography. 1 and 2 can be interconverted by redox reactions and 2 is stable in air, only slowly oxidizing to 1.

The NICS(0)_πizz values computed for 3 and 4 are shown in Figure 1. (See ref 2 for a discussion on this NICS method and also Chapter 2 of my book.) These values are quite negative for each ring of 3, consistent with its expected aromatic character. On the other hand, the NICS value for each ring of 4 is more positive than the corresponding ring of 3, with the value in the center of the pyrazine ring being positive. These NICS values indicate that 4 is certainly less aromatic than 3, and perhaps even expresses antiaromatic character.

Figure 1. NICS(0)_πzz values for 3 and 4 computed at PW91/6-311G**.

Interestingly, hydrogenation of 3 to give 4 is -14.0, indicating that while 3 appears to be a normal aromatic compound, 4, if it is antiaromatic, exhibits some energetic stabilization. They identify this stabilization as a result of the interaction between the dihydropyrazine ring and the thidiazole ring, evidenced in the exothermicity of the isodemic reaction:

So while 4 may be antiaromatic, it appears to be energetically reasonably stable. It is important to keep in mind though that 4 is not the most stable tricycle isomer; in fact, 5 is 7 kcal mol^-1 lower in energy than 4.

Schleyer and Bunz conclude that antiaromaticity may “not result in a prohibitive energetic penalty.”

References

(1) Miao, S.; Schleyer, P. v. R.; Wu, J. I.; Hardcastle, K. I.; Bunz, U. H. F., "A Thiadiazole-Fused N,N-Dihydroquinoxaline: Antiaromatic but Isolable," Org. Lett. 2007, 9, 1073-1076, DOI: 10.1021/ol070013i

(2) Fallah-Bagher-Shaidaei, H.; Wannere, C. S.; Corminboeuf, C.; Puchta, R.; Schleyer, P. v. R., "Which NICS Aromaticity Index for Planar π Rings Is Best?," Org. Lett., 2006, 8, 863-866, DOI: 10.1021/ol0529546.

InChI

3: InChI=1/C8H4N4S/c1-2-10-6-4-8-7(11-13-12-8)3-5(6)9-1/h1-4H
4: InChI=1/C8H6N4S/c1-2-10-6-4-8-7(11-13-12-8)3-5(6)9-1/h1-4,9-10H
5: InChI=1/C8H6N4S/c1-2-10-6-4-8-7(11-13-12-8)3-5(6)9-1/h1-2H,3-4H2

We noted in Chapter 2.1 some serious errors in the prediction of bond dissociation energies using B3LYP. For example, Gilbert examined the C-C bond dissociation energy of some simple branched alkanes.¹ The mean absolute deviation (MAD) for the bond dissociation energy predicted by G3MP2 is 1.7 kcal mol^-1 and 2.8 kcal mol^-1 using MP2. In contrast, the MAD for the B3LYP predicted values is 13.7 kcal mol^-1, with some predictions in error by more than 20 kcal mol^-1. Furthermore, the size of the error increases with the size of the molecule. Consistent with this trend, Curtiss and co-workers noted a systematic underestimation of the heat of formation of linear alkanes of nearly 0.7 kcal mol^-1 per bond using B3LYP.²

Further evidence disparaging the general performance of DFT methods (and B3LYP in particular) was presented in a paper by Grimme and in two back-to-back Organic Letters articles, one by Schreiner and one by Schleyer. Grimme³ noted that the relative Energy of two C₈H₁₈ isomers, octane and 2,2,3,3-tetramethylbutane are incorrectly predicted by DFT methods (Table 1). While MP2 and CSC-MP2 (spin-component-scaled MP2) correctly predict that the more branched isomer is more stable, the DFT methods predict the inverse! Grimme attributes this error to a failure of these DFT methods to adequately describe medium-range electron correlation.

Table 1. Energy (kcal mol-1) of 2,2,3,3-tetramethylbutane relative to octane.

Method ΔE Expta 1.9 ± 0.5 MP2b,c 4.6 SCS-MP2b,c 1.4 PBEb,c -5.5 TPSShb,c -6.3 B3LYPb,c -8.4 BLYPb,c -9.9 M05-2Xd,e 2.0 M05-2Xc,d 1.4

aNIST Webbook (http://webbook.nist.gov) bRef. 3. cUsing the cQZV3P basis set and MP2/TZV(d,p) optimized geometries. dRef. 4. eCalculated at M05-2X/6-311+G(2df,2p).

Schreiner⁵ also compared the energies of hydrocarbon isomers. For example, the three lowest energy isomers of C₁₂H₁₂ are 1-3, whose B3LYP/6-31G(d) structures are shown in Figure 1. What is disturbing is that the relative energies of these three isomers depends strongly upon the computational method (Table 2), especially since these three compounds appear to be quite ordinary hydrocarbons. CCSD(T) predicts that 2 is about 15 kcal mol^-1 less stable than 1 and that 3 lies another 10 kcal mol^-1 higher in energy. MP2 exaggerates the separation by a few kcal mol^-1. HF predicts that 1 and 2 are degenerate. The large HF component within B3LYP leads to this DFT method’s poor performance. B3PW91 does reasonably well in reproducing the CCSD(T) results.

Table 2. Energies (kcal mol-1) of 2 and 3 relative to 1.

Method 2 3 CCSD(T)/cc-pVDZ//MP2(fc)/aug-cc-pVDZa 14.3 25.0 CCSD(T)/cc-pVDZ//B3LYP/6-31+G(d)a 14.9 25.0 MP2(fc)/aug-cc-pVDZa 21.6 29.1 MP2(fc)/6-31G(d)a 23.0 30.0 HF/6-311+(d) a 0.1 6.1 B3LYP/6-31G(d)a 4.5 7.2 B3LYP/aug-cc-pvDZa 0.4 3.1 B3PW91/6-31+G(d) a 17.3 23.7 B3PW91/aug-cc-pVDZa 16.8 23.5 KMLYP/6-311+G(d,p)a 28.4 41.7 M05-2X/6-311+G(d,p)b 16.9 25.4 M05-2X/6-311+G(2df,2p)b 14.0 21.4

aRef. 5. bRef. 4.

1 xyz file

2 xyz file

3 xyz file

Figure 1. Structures of 1-3 at B3LYP/6-31G(d).

Another of Schreiner’s examples is the relative energies of the C₁₀H­₁₀ isomers; Table 3 compares their relative experimental heats of formation with their computed energies. MP2 adequately reproduces the isomeric energy differences. B3LYP fairs quite poorly in this task. The errors seem to be most egregious for compounds with many single bonds. Schreiner recommends that while DFT-optimized geometries are reasonable, their energies are unreliable and some non-DFT method should be utilized instead.

Table 3. Relative C10H10 isomer energies (kcal mol-1)5

Compound Rel.ΔHf Rel. E(B3LYP) Rel. E(MP2) 0.0 0.0 0.0 5.9 -8.5 0.2 16.5 0.7 10.7 20.5 3.1 9.4 26.3 20.3 22.6 32.3 17.6 31 64.6 48.8 61.4 80.8 71.2 78.8 r2 0.954a 0.986b

aCorrelation coefficient between Rel. ΔHf and Rel. E(B3LYP). bCorrelation coefficient between Rel. ΔHf and Rel. E(MP2).

Schleyer’s example of poor DFT performance is in the isodesmic energy of Reaction 1 evaluated for the n-alkanes.⁶ The energy of this reaction becomes more positive with increasing chain length, which Schleyer attributes to stabilizing 1,3-interactions between methyl or methylene groups. (Schleyer ascribes the term “protobranching” to this phenomenon.) The stabilization energy of protobranching using experimental heats of formation increases essentially linearly with the length of the chain, as seen in Figure 2.

n-CH₃(CH₂)_mCH₃ + mCH₄ → (m + 1)C₂H₆         Reaction 1

Schleyer evaluated the protobranching energy using a variety of methods, and these energies are also plotted in Figure 2. As expected, the G3 predictions match the experimental values quite closely. However, all of the DFT methods underestimate the stabilization energy. Most concerning is the poor performance of B3LYP. All three of these papers clearly raise concerns over the continued widespread use of B3LYP as the de facto DFT method. Even the new hybrid meta-GGA functionals fail to adequately predict the protobranching phenomenon, leading Schleyer to conclude: “We hope that Check and Gilbert’s pessimistic admonition that ‘a computational chemist cannot trust a one-type DFT calculation’¹ can be overcome eventually”. These papers provide a clear challenge to developers of new functionals.

Figure 2. Comparison of computed and experimental protobranching stabilization energy (as defined in Reaction 1) vs. m, the number of methylene groups of the n-alkane chain.⁶

Truhlar believes that one of his newly developed functionals answers the call for a reliable method. In a recent article,⁴ Truhlar demonstrates that the M05-2X⁷ functional performs very well in all three of the cases discussed here. In the case of the C₈H₁₈ isomers (Table 1), M05-2X properly predicts that 2,2,3,3-tetramethylbutane is more stable than octane, and estimates their energy difference within the error limit of the experiment. Second, M05-2X predicts the relative energies of the C₁₂H₁₂ isomers 1-3 within a couple of kcal mol^-1 of the CCSD(T) results (see Table 2). Last, in evaluating the isodesmic energy of Reaction 1 for hexane and octane, M05-2X/6-311+G(2df,2p) predicts energies of 11.5 and 17.2 kcal mol^-1 respectively. These are in excellent agreement with the experimental values of 13.1 kcal mol^-1 for butane and 19.8 kcal mol^-1 for octane.

Truhlar has also touted the M05-2X functional’s performance in handling noncovalent interactions.⁸ For example, the mean unsigned error (MUE) in the prediction of the binding energies of six hydrogen-bonded dimers is 0.20 kcal mol^-1. This error is comparable to that from G3 and much better than CCSD(T). With the M05-2X functional already implemented within NWChem and soon to be released within Gaussian and Jaguar, it is likely that M05-2X may supplant B3LYP as the new de facto functional in standard computational chemical practice.

Schleyer has now examined the bond separation energies of 72 simple organic molecules computed using a variety of functionals,⁹ including the workhorse B3LYP and Truhlar’s new M05-2X. Bond separation energies are defined by reactions of each compound, such as three shown below:

The new M05-2X functional performed the best, with a mean absolute deviation (MAD) from the experimental energy of only 2.13 kcal mol^-1. B3LYP performed much worse, with a MAD of 3.96 kcal mol^-1. As noted before, B3LYP energies become worse with increasing size of the molecules, but this problem is not observed for the other functionals examined (including PW91, PBE, and mPW1PW91, among others). So while M05-2X overall appears to solve many of the problems noted with common functionals, it too has some notable failures. In particular, the error is the bond separation energies of 4, 5, and 6 is -8.8, -6.8, and -6.0 kcal mol^-1, respectively.

References

(1) Check, C. E.; Gilbert, T. M., “Progressive Systematic Underestimation of Reaction Energies by the B3LYP Model as the Number of C-C Bonds Increases: Why Organic Chemists Should Use Multiple DFT Models for Calculations Involving Polycarbon Hydrocarbons,” J. Org. Chem. 2005, 70, 9828-9834, DOI: 10.1021/jo051545k.

(2) Redfern, P. C.; Zapol, P.; Curtiss, L. A.; Raghavachari, K., “Assessment of Gaussian-3 and Density Functional Theories for Enthalpies of Formation of C₁-C₁₆ Alkanes,” J. Phys. Chem. A 2000, 104, 5850-5854, DOI: 10.1021/jp994429s.

(3) Grimme, S., “Seemingly Simple Stereoelectronic Effects in Alkane Isomers and the Implications for Kohn-Sham Density Functional Theory,” Angew. Chem. Int. Ed. 2006, 45, 4460-4464, DOI: 10.1002/anie.200600448

(4) Zhao, Y.; Truhlar, D. G., “A Density Functional That Accounts for Medium-Range Correlation Energies in Organic Chemistry,” Org. Lett. 2006, 8, 5753-5755, DOI: 10.1021/ol062318n

(5) Schreiner, P. R.; Fokin, A. A.; Pascal, R. A.; deMeijere, A., “Many Density Functional Theory Approaches Fail To Give Reliable Large Hydrocarbon Isomer Energy Differences,” Org. Lett. 2006, 8, 3635-3638, DOI: 10.1021/ol0610486

(6) Wodrich, M. D.; Corminboeuf, C.; Schleyer, P. v. R., “Systematic Errors in Computed Alkane Energies Using B3LYP and Other Popular DFT Functionals,” Org. Lett. 2006, 8, 3631-3634, DOI: 10.1021/ol061016i

(7) Zhao, Y.; Schultz, N. E.; Truhlar, D. G., “Design of Density Functionals by Combining the Method of Constraint Satisfaction with Parametrization for Thermochemistry, Thermochemical Kinetics, and Noncovalent Interactions,” J. Chem. Theory Comput. 2006, 2, 364-382, DOI: 10.1021/ct0502763.

(8) Zhao, Y.; Truhlar, D. G., “Assessment of Model Chemistries for Noncovalent Interactions,” J. Chem. Theory Comput. 2006, 2, 1009-1018, DOI: 10.1021/ct060044j.

(9) Wodrich, M. D.; Corminboeuf, C.; Schreiner, P. R.; Fokin, A. A.; Schleyer, P. v. R., “How Accurate Are DFT Treatments of Organic Energies?,” Org. Lett., 2007, 9, 1851-1854, DOI: 10.1021/ol070354w.

InChI

1: InChI=1/C11H10/c1-2-5-7-3(1)4(1)8-6(2)10-9(5)11(7,8)10/h1-10H

2: InChI=1/C12H12/c1-2-4-10-6-8-11-7-5-9(3-1)12(10)11/h1-12H

3: InChI=1/C12H12/c1-2-4-8-11(7-3-1)12-9-5-6-10-12/h1-12H

4: InChI=1/C6H6/c1-4-5(2)6(4)3/h1-3H2

5: InChI=1/C8H10/c1-3-7-5-6-8(7)4-2/h3-4H,1-2,5-6H2

6: InChI=1/C10H10/c1-2-8-5-6-9-4-3-7(1)10(8)9/h1-10H