approximately 6-311G-type basis set. These workers found average bond length
errors of, e.g., 0.01 A ̊for C–H and 0.009 A ̊for C–C single bonds, and average bond
angle errors of 0.5. El-Azhary reported B3LYP with the 6-31G and cc-pVDZ basis
sets to give slightly better geometries than MP2, but MP2 avoided the occasional large
errors given by B3LYP [ 75 ]. The effect of using different basis sets was minor. In a
comparison of Hartree–Fock, MP2 and DFT (five functionals), Bauschlicher found
B3LYP to be the best method overall [ 76 ]. 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 [ 67 ]. His 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 Hartree–Fock methods as far as geometry is concerned, a conclusion
presented in Section 5.5.1 with regard to correlated ab initio methods. Hehre and Lou
[ 68 ] 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 0.010 A ̊,
respectively. HF/6-311+G(2d,p) and SVWN calculations also gave errors of 0.010 A ̊.
For 14 C–N, C–O and C¼O bond lengths B3LYP and pBP (errors of 0.007 and
0.008 A ̊) were distinctly better than HF and SVWN (errors of 0.022 and 0.014 A ̊,
respectively).
The overall indication from the literature and the results in Fig.7.1and Table7.1
(errors are evaluated above) is that the somewhat old (1994 [ 58 ]) B3LYP functional
gives good geometries. Of the newer functionals tested here, M06 (2007, 2008 [ 45 ,
65 ]) and TPSS (2003 [ 73 ]), the indication from our (admittedly limited) results is
that M06 is about as good as B3LYP and that TPSS is somewhat inferior (but note
that lacking empirical parameters TPSS may be less prone to unexpected (or
catastrophic [ 62 ]; Section7.2.3.4g) failure. Recently published (2007) extensive
general (not just for geometry as the title of this section implies) evaluations of
functionals are those by Sousa et al. [ 44 ], Zhao and Truhlar [ 45 ], and Riley et al.
[ 46 ]. Synopses of these papers are given in Section7.2.3.4.
Besides the functional, the choice of basis set needs to be addressed. Larger basis
sets may tend to increase accuracy, but the increase in time may not make this
worthwhile. DFT calculations have been said to become “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” [ 38 ].
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 set of choice in B3LYP
calculations on much larger molecules [than C 4 H 6 O 2 ]” [ 58 ]. Figure7.3shows the
effect on geometry and relative energies of B3LYP with the modest 6-31G*, the
fairly big 6-311+G**, and the big 6-311++G(2df, 2p) bases. Results given here
support the basis set saturation assertion for geometries, but cast doubt on the ease
7.3 Applications of Density Functional Theory 475