Essentials of Computational Chemistry

340 9 CHARGE DISTRIBUTION AND SPECTROSCOPIC PROPERTIES
Table 9.3 Scale factors and post-scaling errors in vibrational frequencies from different
levels of theory a
Level of theory Scale factor RMS error (cm −1 ) Outliers (%) b
AM1 0.9532 126 15
PM3 0.9761 159 17
HF/3-21G 0.9085 87 9
HF/6-31G(d) 0.8953 50 2
HF/6-31G(d,p) 0.8992 53 3
HF/6-311G(d) c 0.9361 32
HF/6-311G(d,p) 0.9051 54 3
HF/LANL2DZ c 0.9393 49
MP2/6-31G(d) 0.9434 63 4
MP2/6-31G(d,p) d 0.9646
MP2/pVTZ e 0.9649 70
QCISD/6-31G(d) 0.9537 37 2
BLYP/6-31G(d) 0.9945 45 2
BLYP/6-311G(d) c 1.0160 38
BLYP/LANL2DZ c 1.0371 47
BP86/6-31G(d) 0.9914 41 2
B3LYP/6-31G(d) 0.9614 34 1
0.9664 f 46
0.9800 g
B3LYP/6-311G(d) c 0.9739 38
B3LYP/pVTZ d 0.9726 42
B3LYP/6-311+G(3df,2p) g 0.9890
B3LYP/6-311++G(3df,3pd) f 0.9542 31
B3LYP/LANL2DZ c 0.9978 45
B3PW91/6-31G(d) 0.9573 34 2
B3PW91/pVTZ d 0.9674 43
VSXC/6-31G(d) 0.9659 48
VSXC/6-311++G(3df,3pd) f 0.9652 37
a Data from Scott and Radom (1996) unless otherwise indicated. b Number of frequencies still in error by
more than 20% of the experimental value after scaling. c From analysis of 511 frequencies in 50 inorganic
molecules (Bytheway and Wong 1998). d Pople et al. (1993). e From analysis of 900 frequencies for 111
molecules comprised of first- and second-row atoms and hydrogen (Halls, Velkovski, and Schlegel 2001).
f From analysis of 110 frequencies for 31 small molecules having only first-row atoms and hydrogen
(Jaramillo and Scuseria 1999). g Bauschlicher and Partridge (1995).
while the hybrid functionals require scale factors consistent with their inclusion of some
HF character. Thus, including HF character results in proportionately too high predictions
in vibrational frequencies, although the scaling procedure is very effective here as well.
Finally, the errors in the semiempirical levels are quite high, and scaling is only modestly
helpful. For those looking for the highest accuracy, the U.S. National Institute of Standards
and Technology (NIST) maintains a web facility that permits users to select a focused set of
molecules from NIST’s computational chemistry database (presumably based upon the user’s
interest in a structurally related unknown) and then to compute least-squares best scaling