We noted in Chapter 2.1 some serious errors in the prediction of bond dissociation energies using B3LYP. For example, Gilbert examined the CC 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 coworkers 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 backtoback Organic Letters articles, one by Schreiner and one by Schleyer. Grimme^{3} noted that the relative Energy of two C_{8}H_{18} isomers, octane and 2,2,3,3tetramethylbutane are incorrectly predicted by DFT methods (Table 1). While MP2 and CSCMP2 (spincomponentscaled 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 mediumrange electron correlation.
Table 1. Energy (kcal mol^{1}) of 2,2,3,3tetramethylbutane relative to octane.

Method 
ΔE 
Expt^{a} 
1.9 ± 0.5 
MP2^{b,c} 
4.6 
SCSMP2^{b,c} 
1.4 
PBE^{b,c} 
5.5 
TPSSh^{b,c} 
6.3 
B3LYP^{b,c} 
8.4 
BLYP^{b,c} 
9.9 
M052X^{d,e} 
2.0 
M052X^{c,d} 
1.4 

^{a}NIST Webbook (http://webbook.nist.gov) ^{b}Ref. 3. ^{c}Using the cQZV3P basis set and MP2/TZV(d,p) optimized geometries. ^{d}Ref. 4. ^{e}Calculated at M052X/6311+G(2df,2p).

Schreiner^{5} also compared the energies of hydrocarbon isomers. For example, the three lowest energy isomers of C_{12}H_{12} are 13, whose B3LYP/631G(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)/ccpVDZ//MP2(fc)/augccpVDZ^{a} 
14.3 
25.0 
CCSD(T)/ccpVDZ//B3LYP/631+G(d)^{a} 
14.9 
25.0 
MP2(fc)/augccpVDZ^{a} 
21.6 
29.1 
MP2(fc)/631G(d)^{a} 
23.0 
30.0 
HF/6311+(d)^{ a} 
0.1 
6.1 
B3LYP/631G(d)^{a} 
4.5 
7.2 
B3LYP/augccpvDZ^{a} 
0.4 
3.1 
B3PW91/631+G(d)^{ a} 
17.3 
23.7 
B3PW91/augccpVDZ^{a} 
16.8 
23.5 
KMLYP/6311+G(d,p)^{a} 
28.4 
41.7 
M052X/6311+G(d,p)^{b} 
16.9 
25.4 
M052X/6311+G(2df,2p)^{b} 
14.0 
21.4 

^{a}Ref. 5. ^{b}Ref. 4.

Figure 1. Structures of 13 at B3LYP/631G(d).
Another of Schreiner’s examples is the relative energies of the C_{10}H_{10} 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 DFToptimized geometries are reasonable, their energies are unreliable and some nonDFT method should be utilized instead.
Table 3. Relative C_{10}H_{10} isomer energies (kcal mol^{1})^{5}

Compound

Rel.ΔH_{f}

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


r^{2}

0.954^{a}

0.986^{b}


^{a}Correlation coefficient between Rel. ΔH_{f} and Rel. E(B3LYP). ^{b}Correlation coefficient between Rel. ΔH_{f} and Rel. E(MP2).

Schleyer’s example of poor DFT performance is in the isodesmic energy of Reaction 1 evaluated for the nalkanes.^{6} The energy of this reaction becomes more positive with increasing chain length, which Schleyer attributes to stabilizing 1,3interactions 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.
nCH_{3}(CH_{2})_{m}CH_{3} + mCH_{4} → (m + 1)C_{2}H_{6} 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 metaGGA 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 onetype 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 nalkane 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 M052X^{7} functional performs very well in all three of the cases discussed here. In the case of the C_{8}H_{18} isomers (Table 1), M052X properly predicts that 2,2,3,3tetramethylbutane is more stable than octane, and estimates their energy difference within the error limit of the experiment. Second, M052X predicts the relative energies of the C_{12}H_{12} isomers 13 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, M052X/6311+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 M052X functional’s performance in handling noncovalent interactions.^{8} For example, the mean unsigned error (MUE) in the prediction of the binding energies of six hydrogenbonded dimers is 0.20 kcal mol^{1}. This error is comparable to that from G3 and much better than CCSD(T). With the M052X functional already implemented within NWChem and soon to be released within Gaussian and Jaguar, it is likely that M052X 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 M052X. Bond separation energies are defined by reactions of each compound, such as three shown below:
The new M052X 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 M052X 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 CC Bonds Increases: Why Organic Chemists Should Use Multiple DFT Models for Calculations Involving Polycarbon Hydrocarbons,” J. Org. Chem. 2005, 70, 98289834, DOI: 10.1021/jo051545k.
(2) Redfern, P. C.; Zapol, P.; Curtiss, L. A.; Raghavachari, K., “Assessment of Gaussian3 and Density Functional Theories for Enthalpies of Formation of C_{1}C_{16} Alkanes,” J. Phys. Chem. A 2000, 104, 58505854, DOI: 10.1021/jp994429s.
(3) Grimme, S., “Seemingly Simple Stereoelectronic Effects in Alkane Isomers and the Implications for KohnSham Density Functional Theory,” Angew. Chem. Int. Ed. 2006, 45, 44604464, DOI: 10.1002/anie.200600448
(4) Zhao, Y.; Truhlar, D. G., “A Density Functional That Accounts for MediumRange Correlation Energies in Organic Chemistry,” Org. Lett. 2006, 8, 57535755, 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, 36353638, 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, 36313634, 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, 364382, DOI: 10.1021/ct0502763.
(8) Zhao, Y.; Truhlar, D. G., “Assessment of Model Chemistries for Noncovalent Interactions,” J. Chem. Theory Comput. 2006, 2, 10091018, 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, 18511854, DOI: 10.1021/ol070354w.
InChI
1: InChI=1/C11H10/c12573(1)4(1)86(2)109(5)11(7,8)10/h110H
2: InChI=1/C12H12/c124106811759(31)12(10)11/h112H
3: InChI=1/C12H12/c124811(731)129561012/h112H
4: InChI=1/C6H6/c145(2)6(4)3/h13H2
5: InChI=1/C8H10/c137568(7)42/h34H,12,56H2
6: InChI=1/C10H10/c128569437(1)10(8)9/h110H