Density Functional Calculations 411

Mathieu developed an economical way of calculating heats of formation by performing BP/DN* calculations on molecular mechanics geometries; rms deviations from experiment were about for a variety of compounds [65]. Ventura et al. found DFT to be better than CCSD(T) (a high-level ab initio method) for studying the thermochemistry of compounds with the O–F bond [66].

7.3.2.2b Kinetics

Consider the reaction profiles in Fig. 7.2. In all four cases, the B3LYP/6-31G* and

Let us compare the activation energies for the reactions in Fig. 7.2 with highaccuracy (section 5.5.2.2d) energy difference values. We will compare the 0 K activation enthalpies, which is what these energy differences are (section 2.2), of the CBS-Q and G2 methods, two of the best high-accuracy methods, with the corresponding activa-

CBS-Q, 237; G2, 237; B3LYP, 224; pBP, 200; exp, 282

HCN reaction

CBS-Q, 128; G2, 124; B3LYP, 126; pBP, 121; exp 129

CBS-Q, 164; G2, 161, B3LYP, 164; pBP, 159; exp, 161

Cyclopropylidene reaction

CBS-Q, 22; G2, 22, B3LYP, 25; pBP, 12.5; exp 13 –20

For the isomerization the B3LYP/6-31G* activation energy is a little lower than the CBS-Q and G2 values (which are somewhat lower than the reported experimental value, which as stated above could be significantly in error). For the HCN isomerization the B3LYP/6-31G*, CBS-Q, G2 and experimental values are essentially the same. For theisomerization the B3LYP/6-31G*, CBS-Q, G2 and experimental values are again essentially the same, and for the cyclopropylidene isomerization the B3LYP/6-31G*, CBS-Q and G2 values are again essentially the same and equal to the higher end of the reported experimental values. In all cases the pBP/DN* activation energies are a little less than the CBS-Q, G2 and B3LYP values.

412 Computational Chemistry

Figure 7.3 compares the results of BP86/6-31G*, BP86/6-311G*, and pBP/DN* calculations on the and reaction profiles. The BP86/6-31G* and BP86/6-311G* geometries are very similar and the relative energies are nearly identical, indicating that basis set saturation (section 7.3.1) has been essentially reached. The pBP/DN* geometries and energies resemble the BP86 ones quite closely, matching the BP86/6-311G* results a little more closely than the BP86/6-31G*. These results suggest that B3LYP/6-31 G* activation energies are similar to those from the more timedemanding (below) CBS-Q and G2 methods, and are close to the experimental values; pBP/DN* and BP86/6-31G* (and BP86/6-311G*) activation energies may be a little (perhaps lower than those from B3LYP/6-31G*. Here are the times required for some DFT, CBS-Q, and G2 calculations (optimization + frequencies), in each case starting from an AM1 geometry; the jobs were run with Gaussian 94 [54] on a 600 MHz Pentium III computer:

In a study of alkene epoxidation with peroxy acids, B3LYP/6-31G* gave an activation energy similar to that calculated with MP4/6-31G*//MP2/6-31G* but yielded kinetic isotope effects in much better agreement with experiment than did the ab initio calculation [69]. Even better activation energies than from B3LYP (which it is said tends to underestimate barriers [70,71]) have been reported for the BH&H-LYP functional [71–74]. In a study by Baker et al. [75] of 12 organic reactions using seven methods (semiempirical, ab initio, and DFT), B3PW91/6-31G* was best (average and maximum errors 15.5 and and B3LYP/6-31G* second best (average and maximum errors 25 and Jursic studied 28 reactions and recommended “B3LYP or B3PW91 with an appropriate basis set”, but warned that highly exothermic reactions with a small barrier involving hydrogen radicals “are particularly difficult to reproduce.” [76]. Barriers “above [ca. 40kJ should be reliable. Lower activation energies should be underestimated by [76]. As with thermodynamic energy differences (i.e. energy differences not involving a transition state), consistently obtaining activation energies accurate to with some confidence requires one of the high-accuracy methods.

Density functional transition states and activation energies have their problems. Merrill et al. found that for the fluoride ion-induced elimination of HF from

none of the 11 functionals tested (including B3LYP) was satisfactory, by comparison with high-level ab initio calculations. Transition states were often looser and stabler than predicted by ab initio, and in several cases a transition state could not even be found. They concluded that hybrid functionals offer the most promise, and that “the ability of density functional methods to predict the nature of TSs demands a great deal more attention than it has received to date.” [34]. Note that it is assumed here that