406 Computational Chemistry

the HNC isomerization: 0.035Å (1.383 Å cf. 1.348 Å) and 3.3° (70.8° cf. 67.5°). The consistency of the two DFT methods and their good agreement with MP2 (the “errors” for the HNC reaction transition state should not affect any qualitative conclusions one might draw) suggest that these DFT methods are quite comparable to MP2/6-31G* in calculating transition state geometries.

Hehre has compared bond lengths calculated by the DFT non-gradient-corrected SVWN method, B3LYP, and MP2, using the 6-31G (no polarization functions) and 6-31G* basis sets [59]. This work confirms the necessity of using polarization functions with the correlated (DFT and MP2) methods to obtain reasonable results, and also shows that for equilibrium structures (i.e. structures that are not transition states) there is little advantage to correlated over HF methods as far as geometry is concerned, a conclusion presented in section 5.5.1. Hehre and Lou [48] carried out extensive comparisons of HF, MP2, and DFT (SVWN, pBP, B3LYP) methods with 6-31G* and larger basis sets, and the numerical DN* and DN** bases. For a set of 16 hydrocarbons, MP2/6-311 + G(2d,p), B3LYP/6-311 + G(2d, p), pBP/DN**, and pBP/DN* calculations gave errors of 0.005, 0.006, 0.010, and 0.010Å, respectively. HF/6-311 + G(2d, p) and SVWN calculations also gave errors of 0.010Å. For 14 C–N, C–O and C–O bond lengths B3LYP and pBP (errors of 0.007 and 0.008 Å) were distinctly better than HF and SVWN (errors of 0.022 and 0.014 Å, respectively). The overall indication from the literature and the results in Figs 7.1 and 7.2 and Table 7.1 is that B3LYP/6-31G* calculations give excellent geometries and pBP/DN* calculations give good geometries. Larger basis sets may increase the accuracy, but the increase in time may not make this worthwhile. DFT calculations appear to be “saturated” more quickly by using bigger basis sets than are ab initio calculations: Merrill et al. noted that “Once the double split-valence level is reached, further improvement in basis set quality offers little in the way of structural or energetic improvement.” [34]; Stephens et al. report that “Our results also show that B3LYP calculations converge rapidly with increasing basis set size and that the cost-to-benefit ratio is optimal at the 6-31G* basis set level. 6-31G* will be the basis

The results in section 7.3.2.2b, regarding Fig. 7.3, support the view that the 6-31G* basis nearly saturates gradient-corrected functionals.

7.3.2Energies

7.3.2.1Energies: preliminaries

Usually, we seek from a DFT calculation, as from an ab initio or semiempirical one, geometries (preceding section) and energies. Like an ab initio energy, a DFT energy is relative to the energy of the nuclei and electrons infinitely separated and at rest, i.e. it is the negative of the energy needed to dissociate the molecule into its nuclei and electrons. AM1 and PM3 semiempirical energies (section 6.3.2) are heats of formation, and by parameterization include zero-point energies. In contrast, an ab initio (section 5.2.3.6d) or DFT molecular energy, the energy printed out at the end of any calculation, is the energy of the molecule sitting motionless at a stationary point (section 2.2) on the potential energy surface; it is the purely electronic energy plus the internuclear