Here’s one more attempt to discern the failure of DFT to handle simple alkanes (see this earlier post for a previous attempt to answer this question). Tsuneda and co-workers¹ have employed long-range corrected (LC) DFT to the problem of the energy associated with “protobranching”, i.e., from the reaction

CH₃(CH₂)­_nCH₃ + n CH₄ → (n+1) CH₃CH₃

They computed the energy of this reaction for the normal alkanes propane through decane using a variety of functionals, and compared these computed values with experimentally-derived energies. Table 1 gives the mean unsigned error for a few of the functionals. The prefix “LC” indicated inclusion of long-range corrections, “LCgau” indicates the LC scheme with a gaussian attenuation, and “LRD” indicates inclusion of long-range dispersion.

Table 1. Mean unsigned errors of the “protobranching”
reaction energy of various functional compared to experiment.

A number of important conclusions can be drawn. First, with both LC and LRD very nice agreement with experiment can be had. If only LC is included, the error increases on average by over 1 kcal mol^-1. The MO6-2x functional, touted as a fix of the problem, does not provide complete correction, though it is vastly superior to B3LYP and other hybrid functionals. The authors conclude that the need for LC incorporation points out that the exchange functional lacks the ability to account for this effect. Medium-range correlation is not the main source of the problem as large discrepancies in the reaction energy error occur when different functionals are used that are corrected for LC and LRD. Choice of functional still matters, but LC correction appears to be a main culprit and further studies of its addition to standard functionals would be most helpful.

References

(1) Song, J.-W.; Tsuneda, T.; Sato, T.; Hirao, K., "Calculations of Alkane Energies Using Long-Range Corrected DFT Combined with Intramolecular van der Waals Correlation," Org. Lett. 2010, 12, 1440–1443, DOI: 10.1021/ol100082z

How would you characterized the benzotrithiophene 1? Is it planar? How about when methyl groups are attached (2)? Are these compounds aromatic? A joint computational/experimental study by Wu and Baldridge has tackled these questions.¹

It turns out that both of these compounds are non-planar and have C₂ symmetry. Now, 1 is very nearly planar. But 2 is decidedly non-planar. The MO6-2X/DZ(2d,p) structures are shown in Figure 1. The central 6-member ring has long bonds and expresses some bond alternation: the C-C distance for the bond between the thiophene rings is 1.469 Å and that of the bond shared by the two rings is 1.451 Å The exocyclic bonds are short, 1.375 Å. This appears to be [6]-radialene-like. NICS computations confirm this notion. The NICS(0) value for the central ring of 1 and 2 is -1.6, significantly less negative than the value in benzene of -7.2. The NICS_zz values also reflect non-aromatic character of the central ring. The central ring is non-aromatic.

1

2

Figure 1. MO6-2X/DZ(2d,p) structures of 1 and 2.¹

References

(1) Wu, T.-T.; Tai, C.-C.; Lin, W.-C.; Baldridge, K. K., "1,3,4,6,7,9-Hexamethylbenzo[1,2-c:3,4-c:5,6-c]trithiophene: a twisted heteroarene," Org. Biomol. Chem., 2009, 7, 2748-2755, DOI: 10.1039/b902517k.

InChIs

1: InChI=1/C12H6S3/c1-7-8(2-13-1)10-4-15-6-12(10)11-5-14-3-9(7)11/h1-6H
InChIKey=INZUTJPYADZZEL-UHFFFAOYAI

2: InChI=1/C18H18S3/c1-7-13-14(8(2)19-7)16-10(4)21-12(6)18(16)17-11(5)20-9(3)15(13)17/h1-6H3
InChIKey=WYVKJYPLPAEMLN-UHFFFAOYAA

In order to obtain computed NMR chemical shifts, one computes the isotropic magnetic shielding tensor and subtracts this value from that computed for a reference (or standard) compound. Typically, one uses TMS as the standard. Sarotti and Pellegrinet have questioned whether this is a reasonable approach.¹ Since computational methods vary in quality with methodology, basis set, geometry – one might wonder if the use of a single standard for all computed chemical shifts is the best approach.

They computed the ¹³C chemical shielding tensor for 50 organic compounds possessing a wide variety of functional groups and rings – a few examples are given below. They also computed the ¹³C chemical shielding tensor for 11 different simple organic compounds that might be used as NMR references (like TMS, benzene, methanol, and chloroform).

By comparing the computed chemical shifts obtained using the different references and then matching them with experiment, they propose a multi-reference method. For sp³ carbon atoms they propose using methanol as the reference, and for sp² and sp carbons using benzene as the reference. With chemical shifts computed at mPW1PW91/6-311+G(2d,p)//B3LYP/6-31G(d) using the multi-reference model , the average mean difference from experiment is 2.1 ppm, less than half that found when TMS alone is used. The average RMS deviation of 4.6ppm is about half that when TMS is used as the sole standard.

Though the authors mention the solvent effect on chemical shifts, it is surprising that they did not include solvent in their calculations, especially since they are comparing to experimental chemical shifts in deuterochloroform. Nonetheless, I think this is a nice idea and further exploration of this concept (multi-reference fitting) is worth further pursuits.

References

(1) Sarotti, A. M.; Pellegrinet, S. C., "A Multi-standard Approach for GIAO ¹³C NMR Calculations," J. Org. Chem., 2009, 74, 7254-7260, DOI: 10.1021/jo901234h

The chemical shift of the benzene proton is about 7.3ppm, significantly downfield from the range of olefinic protons (5.6-58.ppm). This is rationalized as the standard induced diatropic ring current, found in aromatic species. But what should we make of the chemical shift of the protons in cyclobutadiene at 5.8 ppm? Shouldn’t this be much further upfield?

Schleyer and Mo have applied the block localized wavefunction (BLW) technique to aromatic and antiaromatic chemical shifts.¹ In BLW, self-consistent localized orbitals are produced to describe a particular resonance structure. So, for benzene, BLW describes in effect 1,3,5-cyclohexatriene, lacking any resonance energy.  When chemical shifts are computed with the BLW description, the proton chemical shift is 6.6 ppm, and is even more upfield if the geometry is optimized (in D_3h symmetry) with the BLW method (δ=6.2ppm). Furthermore the NICS(0)_πzz (the tensor component corresponding to the perpendicular direction evaluated in the ring center using just the π orbitals) is -36.3 for benzene and 0.0 for the D_3h BLW variant, strongly indicating the role of cyclic delocalization in affecting chemical shifts.

Now for cyclobutadiene, the proton chemical shift of 5.7 ppm becomes 7.4 in the BLW case. NICS(0)_πzz for cyclobutadiene is +46.9 and +1.6 in the BLW case. The problem is that typical alkenes are poor references for cyclobutadiene – when resonance is turned off, the chemical shift does move downfield – indicating the expected upfield shift for cyclobutadiene. Schleyer and Mo suggest that 3,4-dimethylenecyclobutene is a more suitable reference; its ring protons have chemical shifts of 7.65ppm.

They also describe computations of benzocyclobutadiene and tricyclobutenabenzene and offer straightforward rationalizations of their aromatic vs. antiaromatic behavior.

References

(1) Steinmann, S. N.; Jana, D. F.; Wu, J. I.-C.; Schleyer, P. v. R.; Mo, Y.; Corminboeuf, C., "Direct Assessment of Electron Delocalization Using NMR Chemical Shifts," Angew. Chem. Int. Ed., 2009, 48, 9828-9833, DOI: 10.1002/anie.200905390

InChIs

benzene: InChI=1/C6H6/c1-2-4-6-5-3-1/h1-6H
InChIKey=UHOVQNZJYSORNB-UHFFFAOYAH

cyclobutadiene: InChI=1/C4H4/c1-2-4-3-1/h1-4H
InChIKey=HWEQKSVYKBUIIK-UHFFFAOYAI

3,4-dimethylenecyclobutene: InChI=1/C6H6/c1-5-3-4-6(5)2/h3-4H,1-2H2
InChIKey=WHCRVRGGFVUMOK-UHFFFAOYAP