lengths is only 0.033 A ̊(HF/3–21()level for HCHO) and the largest error in bond
angles is only 3.2(HF/3–21()level for H 2 O). The largest error in dihedral angles
(Table5.8), omitting the 3–21G() result for H 2 O 2 , is 8.6 (HF/6–31G for
ClCH 2 CH 2 OH HOCC), but as stated above the reported experimental dihedral of
58.4is suspect.
From Fig.5.23and Table5.7, the mean error in 39 (13þ 8 þ 9 þ9) bond
lengths is 0.01–0.015 A ̊ at the HF/3–21() and HF/6–31G levels, and ca.
0.005–0.008 A ̊ at the MP2/6–31G level. The mean error in 18 bond angles is
only 1.3and 1.0at the HF/3–21()and HF/6–31G levels, respectively, and 0.7
at the MP2(fc)/6–31G level. From Table5.8the mean dihedral angle error at the
HF/3–21() level for nine dihedrals (omitting the questionable ClCH 2 CH 2 OH
dihedral) is 3.0; the mean of eight dihedral errors (omitting the ClCH 2 CH 2 OH
and the HOOH errors) is 2.5. For the other two levels the mean of ten dihedral
angles (including the questionable ClCH 2 CH 2 OH dihedral) is 2.9(HF/6–31G)
and 2.3(MP2/6–31G). If we agree that errors in calculated bond lengths, angles
and dihedrals of up to 0.02 A ̊,3and 4respectively correspond to fairly good
structures, thenallthe HF/3–21(), HF/6–31G and MP2/6–31G geometries, with
the exception of the HF/3–21()HOOH dihedral, which is simply wrong, and the
possible exception of the HOCC dihedral of ClCH 2 CH 2 OH, are fairly good. We
should, however, bear in mind that, as with the HF/3–21()HOOH dihedral, there is
the possibility of an occasional nasty surprise. Interestingly, HF/3–21()geometries
are, for some series of compounds, somewhat better than MP2/6–31G ones. For
example, the RMS errors in geometry for the series H 2 , CH, NH, OH, HF, CN, N 2 ,
H 2 O, HCN, CH 3 , and CH 4 using UHF/3–21G(), MP2/6–31G, and MP2/6–31G{
(a modified basis used in CBS calculations – Section 5.5.2.2b) are 0.012 A ̊, 0.016 A ̊
and 0.015 A ̊, respectively [ 113 ].
The calculations summarized in Tables5.7and5.8are in reasonable accord
with conclusions based on information available ca. 1985 and given by Hehre et al.
[ 114 ]: HF/6–31G parameters for A–H, A/B single and A/B multiple bonds are
usually accurate to 0.01, 0.03 and 0.02 A ̊, respectively, bond angles to ca. 2and
dihedral angles to ca. 3, with HF/3–21G()values being not quite as good. MP2
bond lengths appear to be somewhat better, and bond angles are usually accurate
to ca. 1, and dihedral angles to ca. 2. These conclusions from Hehre et al. hold for
molecules composed of first-row elements (Li to F) and hydrogen; for elements
beyond the first row larger errors are not uncommon.
The main advantage of MP2/6–31G optimizations over HF/3–21()or HF/
6–31G* ones is not that the geometries aremuchbetter, but rather that for a
stationary point, MP2 optimizations followed by frequency calculations are more
likely to give the correct curvature of the potential energy surface (Chapter 2) for
the species than are HF optimizations/frequencies. In other words, the correlated
calculation tells us more reliably whether the species is a relative minimum or
merely a transition state (or even a higher-order saddle point; seeChapter 2). Thus
fluorodiazomethane [ 91 ] and several oxirenes [ 53 ] are (apparently correctly) pre-
dicted by MP2 optimizations to be merely transition states, while HF optimizations
288 5 Ab initio Calculations