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Author's personal copy

Journal of Molecular Structure: THEOCHEM 948 (2010) 102–107

Contents lists available at ScienceDirect

Journal of Molecular Structure: THEOCHEM
journal homepage: www.elsevier.com/locate/theochem

Gas phase isomerization enthalpies of organic compounds: A semiempirical,
density functional theory, and ab initio post-Hartree–Fock theoretical study
Sierra Rayne a,*, Kaya Forest b
a
b

Ecologica Research, Penticton, BC, Canada V2A 8J3
Department of Chemistry, Okanagan College, Penticton, BC, Canada V2A 8E1

a r t i c l e

i n f o

Article history:
Received 27 January 2010
Received in revised form 26 February 2010
Accepted 27 February 2010
Available online 6 March 2010
Keywords:
Isomerization enthalpies
Hydrocarbons
Semiempirical
Density functional theory
Ab initio post-Hartree–Fock

a b s t r a c t
A database of gas phase standard state isomerization enthalpies was constructed for 562 pure and nitrogen-, oxygen-, sulfur-, and halogen-containing hydrocarbon reactions. The PM6 and PDDG semiempirical
methods, B3LYP and M062X hybrid density functionals with the 6-311++G(d,p) and 6-311+G(3df,3p)
basis sets, and the CBS-Q//B3 and G4MP2 ab initio post-Hartree–Fock composite methods were examined
for prediction accuracy within each class of isomerization reactions. At much lower computational cost,
the PM6 and PDDG semiempirical methods offer modest isomerization enthalpy prediction performance
approximately comparable to the B3LYP density functional. The M062X density functional provides
nearly equivalent accuracy to the higher level CBS-Q//B3 and G4MP2 methods across all hydrocarbon
classes. Increasing basis set size from 6-311++G(d,p) to 6-311+G(3df,3p) with the B3LYP and M062X density functionals does not influence their respective isomerization enthalpy prediction accuracies. Using
the 6-311+G(3df,3p) basis set, the M062X functional also achieves near CBS-Q//B3 quality accuracy for
enthalpies of formation using the atomization approach.
Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction
Isomerization enthalpies of organic compounds have attracted
increasing computational interest [1–4] in recent years because
of their insights into molecular structure and bonding, use in predicting synthetic pathways, applicability towards the fields of
polymer science and energy conversion, as well as the ability to
help benchmark theoretical methods. Earlier investigations often
focused on the known hydrocarbon branching errors of density
functional theory (DFT) methods (particularly the B3LYP functional), whereby linear to branched isomerizations (e.g., n-octane
to tetramethylbutane) were incorrectly predicted to be quantitatively less exothermic (and often qualitative errors were present)
than the well established experimental datasets [2,5,6]. Recent
studies have extended these isomerization enthalpy errors to a
broad range of bonding situations, including extensions to oxygen-, nitrogen-, and halogen-containing hydrocarbons and silanes,
as well as various bridged and cyclic structures and functional
group conversions (e.g., acyclic to cyclic, aromatic to aliphatic, carboxylic acid/dione to ester, ketone to ether, diol to peroxide, etc.)
[1,7,8]. In this work, we have investigated the pre-existing isomerization enthalpy validation datasets with levels of theory not previously considered in the literature, as well as compiled larger sets

* Corresponding author. Tel.: +1 250 487 0166.
E-mail address: rayne.sierra@gmail.com (S. Rayne).
0166-1280/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.theochem.2010.02.030

of hydrocarbon isomerization reactions with experimental data,
including new classes of sulfur and halogen substituted derivatives, and have applied various semiempirical, DFT, and higher level ab initio post-Hartree–Fock composite methods to these
broader datasets.
2. Materials and methods
Calculations were conducted using Gaussian-09 [9] on the Western Canada Research Grid (WestGrid; project 100185 [K. Forest])
and the Shared Hierarchical Academic Research Computing Network (SHARCNET; project sn4612 [K. Forest]), both within the umbrella of Compute/Calcul Canada. All calculations used the same
gas phase starting geometries obtained with the PM6 semiempirical method [10] as implemented in MOPAC 2009 (http://
www.openmopac.net/; v. 9.099).
Semiempirical calculations used the AM1 [11], PM3/PM3MM
[12], PM6 [10], and PDDG [5] methods as reimplemented [13] in
Gaussian-09. Density functional theory (DFT) calculations used
the B3LYP [14,15] and M062X [16] hybrid density functionals with
the 6-311++G(d,p) [17–19], 6-311+G(3df,3p) [17–19], and aug-ccpVDZ [20–22] basis sets. Ab initio Hartree–Fock (HF) calculations
were conducted using the 6-311++G(d,p) and aug-cc-pVDZ basis
sets. Complete Basis Set (CBS) calculations used the CBS-Q//B3
[23,24] method. Gaussian-n calculations used the G4MP2 [25,26]
method.

Author's personal copy

S. Rayne, K. Forest / Journal of Molecular Structure: THEOCHEM 948 (2010) 102–107

All optimized structures were confirmed as true minima by
vibrational analysis at the same level. Isomerization enthalpies include zero point and thermal corrections. Only the lowest energy
conformation of each molecule was considered for isomerization
enthalpy calculations. Experimental gas phase standard state
(298.15 K, 1 atm) enthalpies of formation Df H ðgÞ were used to
calculate experimental gas phase standard
state (298.15 K, 1 atm)

isomerization enthalpies Disom H ðgÞ . Df H ðgÞ data were obtained
from Refs. [27–34]. Chemical accuracy is operationally defined
herein as a Disom H ðgÞ error of less than 1.5 kcal/mol based on the
assumption of a typical 1 kcal/mol error in each Df H ðgÞ datapoint
used for a Disom H ðgÞ calculation. Because of the wide range of
Df H ðgÞ errors reported in the literature (from less than 0.5 kcal/
mol up to 5 kcal/mol), a chemical accuracy definition of 1.5 kcal/
mol for Disom H ðgÞ is only for convenience. In practice, each calculated experimental Disom H ðgÞ value will have a unique error defined
by the respective errors of the Df H ðgÞ measurements that comprise
the isomerization reaction, and rigorous chemical accuracy would
need to be defined on a case-by-case basis.

3. Results and discussion
Work by Sattelmeyer et al. [1], Grimme et al. [3], and TiradoRives and Jorgensen [8] established a standard set of Disom H ðgÞ for
various pure hydrocarbons (n = 13; Table S1), nitrogen-containing
hydrocarbons (n = 10; Table S2), and oxygen-containing hydrocarbons (n = 11; Table S3) that have been used to benchmark and
investigate various levels of theory. These studies have collectively
shown that while low mean signed errors (MSEs; 61 kcal/mol) can
be obtained using various levels of semiempirical, Hartree–Fock
(HF) ab initio, and DFT methods, the results only appear favorable
due to error cancellations. Consideration of mean unsigned errors
(MUEs) and root mean square errors (RMSEs) in these reports show
that various semiempirical (AM1, PM3, PDDG), HF, DFT (BLYP,
BHLYP, B2LYP, B3LYP, O3LYP, mPW2PLYP, BMK, BP86, B97D, PBE/
PBE0, TPSS, MPW1k, MPWB1k, and SCC-DFTB), and MP2/SCSMP2 model chemistries generally yield MUEs/RMSEs on the order
of 1–12 kcal/mol, with typically lower MUEs using higher levels
of theory (Table 1).
We obtain similar errors on these datasets with the semiempirical AM1, PM3, PM6, and PDDG methods as implemented in Gaussian-09, as well as the HF level of theory with representative Pople
(6-311++G(d,p)) and Dunning-type (aug-cc-pVDZ) basis sets, but
find near chemical accuracy (MUEs of 61.5 kcal/mol for each of
the three standard isomerization enthalpy sets) can be achieved
using the M062X density functional with the 6-311++G(d,p), 6311+G(3df,3p), and aug-cc-pVDZ basis sets (Table 2). The B3LYP
functional with the 6-311++G(d,p), 6-311+G(3df,3p), and aug-ccpVDZ basis sets does not perform well with either the pure hydrocarbon set (MUEs of 3.2, 3.4, and 2.8 kcal/mol, respectively) or with
the oxygen-containing hydrocarbon dataset (MUEs of 2.2, 2.2, and
1.4 kcal/mol, respectively), but displays remarkably low MUEs of
0.8–0.9 kcal/mol using all three basis sets on the nitrogen-containing hydrocarbon dataset. This B3LYP error profile is similar to the
previous isomerization enthalpy reports with this functional using
other basis sets (see Refs. [1,3,8] and the literature summaries in
Table 1). The aug-cc-pVDZ basis set yields slightly better MUEs
than the Pople-type basis sets with the B3LYP functional, but the
opposite error trending with the M062X functional, and conflicting
error trendings for the HF model chemistry. Increasing the basis set
size from 6-311++G(d,p) to 6-311+G(3df,3p) has no significant effect on the Disom H ðgÞ prediction accuracy for either the M062X or
B3LYP functionals. Similar to the work of Grimme et al. [3] using
higher level MP2 and CCSD(T) methods, we find that the CBS-Q//
B3 and G4MP2 methods display the highest MUEs on the oxy-

103

gen-containing hydrocarbon dataset relative to their corresponding chemical accuracies on the pure hydrocarbon and nitrogencontaining hydrocarbon datasets, potentially highlighting systematic errors in the oxygen-containing hydrocarbon experimental
Df H ðgÞ database. For all three datasets, the CBS-Q//B3 and G4MP2
methods yield the lowest MUEs reported in the literature to date
for these isomerization reactions.
To these standard isomerization datasets of Sattelmeyer et al.
[1], Grimme et al. [3], and Tirado-Rives and Jorgensen [8], we
added additional isomerization reactions in each class to bring
the total number of isomerizations to 311 for the pure hydrocarbons (Table S4; note that the isomerization reactions with the literature Df H ðgÞ data from Schreiner et al. [35] and Zhao and Truhlar
[36] are included in this dataset, as are some isomerization reactions examined by Taskinen [37]), 73 for the nitrogen-containing
hydrocarbons (Table S5), 116 for the oxygen-containing hydrocarbons (Table S6), as well as 39 isomerization enthalpies for sulfurcontaining hydrocarbons (Table S7) and 23 for halogen-containing
hydrocarbons (Table S8), for a total of 562 isomerization enthalpies
across these functional group classes. We omitted the N-methylacetamide ? dimethylformamide,
tetrahydro-2H-pyran-2-one ?
acetylacetone, and hexanoic acid ? methyl pivalate isomerization
reactions from the new composite dataset due to uncertainty
regarding their experimental Df H ðgÞ values. Dimethylformamide
does not have an experimental Df H ðgÞ in the NIST database, and
the reported experimental Df H ðgÞ values for acetylacetone and
methyl pivalate range over 44 and 20 kJ/mol, respectively. Calculations using the PM6 and PDDG semiempirical methods, B3LYP/6311++G(d,p), B3LYP/6-311+G(3df,3p), M062X/6-311++G(d,p), and
M062X/6-311+G(3df,3p) density functional levels of theory, and
the CBS-Q//B3 and G4MP2 methods were then conducted on these
composite datasets. The representative 6-311++G(d,p) and 6311+G(3df,3p) Pople-type basis sets were chosen for the DFT calculations because no clear and consistent increase in accuracy was
obtained across both density functionals for all compound classes
using the validation set results discussed above with the
6-311++G(d,p), 6-311+G(3df,3p), and aug-cc-pVDZ basis sets.
Neither of the semiempirical methods provided chemical accuracy, having Disom H ðgÞ MUEs ranging from about 2 to 3 kcal/mol and
RMSEs from about 3 to 5 kcal/mol (Table 3 and Fig. 1). The B3LYP/
6-311++G(d,p) and B3LYP/6-311+G(3df,3p) methods performed
with modestly better accuracy than the two semiempirical methods for the nitrogen, oxygen, sulfur, and halogen derivatives, yielding MUEs from 1.2 to 2.3 kcal/mol and RMSEs from 1.6 to 2.9 kcal/
mol, but provided much lower accuracy on the pure hydrocarbons
(MUE = 5.5 kcal/mol for both basis sets and RMSE = 6.6–6.8 kcal/
mol, respectively). In comparison, the M062X/6-311++G(d,p) and
M062X/6-311+G(3df,3p) levels of theory provided nearly equivalent accuracies to the more computationally expensive CBS-Q//B3
and G4MP2 methods for all compound classes. The M062X/6311++G(d,p), M062X/6-311+G(3df,3p), CBS-Q//B3, and G4MP2
methods achieved effective Disom H ðgÞ chemical accuracy for all
hydrocarbon classes. Thus, while a modest Disom H ðgÞ accuracy increase can be obtained by using the more expensive CBS-Q//B3
method over the M062X/6-311++G(d,p) and M062X/6-311+
G(3df,3p) levels of theory, there appears to be no Disom H ðgÞ accuracy
increase in adding further computational costs at the G4MP2 level
relative to the cheaper CBS-Q//B3 method. Similarly, there is no error
reduction for Disom H ðgÞ in using the more expensive 6-311+G(3df,3p)
basis set, versus its cheaper 6-311++G(d,p) counterpart, with either
the B3LYP or M062X functionals.
Statistically insignificant (p > 0.05) or generally weak linear
regressions (jrj < 0.75) with low values for the slope (m) of calculated
Disom H ðgÞ error against the experimental Disom H ðgÞ were observed
within each combination of hydrocarbon class and level of theory
(slopes not provided where regressions are not significant):

Author's personal copy

104

S. Rayne, K. Forest / Journal of Molecular Structure: THEOCHEM 948 (2010) 102–107

Table 1
Literature derived calculated Disom H ðgÞ errors for the validation sets of pure hydrocarbons, nitrogen-containing hydrocarbons, and oxygen-containing hydrocarbons developed by
Sattelmeyer et al. [1], Grimme et al. [3], and Tirado-Rives and Jorgensen [8] at various levels of semiempirical, Hartree–Fock ab initio, and density functional theory. Values are in
kcal/mol and presented as mean unsigned error [root mean squared error] (mean signed error).
Pure hydrocarbons

Nitrogen-containing hydrocarbons

Oxygen-containing hydrocarbons

TMa/HF/TZV(2df,2pd)//B3LYP/TZV(d,p)b
TM/B3LYP/TZV(2df,2pd)//B3LYP/TZV(d,p)b
TM/BMK/TZV(2df,2pd)//B3LYP/TZV(d,p)b
TM/B2LYP/TZV(2df,2pd)//B3LYP/TZV(d,p)b
TM/PBE0/TZV(2df,2pd)//B3LYP/TZV(d,p)b
TM/PBE/TZV(2df,2pd)//B3LYP/TZV(d,p)b
TM/BHLYP/TZV(2df,2pd)//B3LYP/TZV(d,p)b
TM/BP86/TZV(2df,2pd)//B3LYP/TZV(d,p)b
TM/B97D/TZV(2df,2pd)//B3LYP/TZV(d,p)b
TM/TPSS/TZV(2df,2pd)//B3LYP/TZV(d,p)b
TM/O3LYP/TZV(2df,2pd)//B3LYP/TZV(d,p)b
TM/BLYP/TZV(2df,2pd)//B3LYP/TZV(d,p)b
TM/mPW2PLYP/TZV(2df,2pd)//B3LYP/TZV(d,p)b
G03c/MPW1k/TZV(2df,2pd)//B3LYP/TZV(d,p)b
G03/MPWB1k/TZV(2df,2pd)//B3LYP/TZV(d,p)b
TM/MP2/TZV(2df,2pd)//B3LYP/TZV(d,p)b
TM/SCS-MP2/TZV(2df,2pd)//B3LYP/TZV(d,p)b
MPd/CCSD(T)/6-31G(d)//B3LYP/TZV(d,p)b
MP/CCSD(T)/cc-pVTZ//B3LYP/TZV(d,p)b

3.6
3.5
1.9
2.1
2.7
2.8
2.8
2.8
2.4
3.0
3.7
4.3
2.0
2.7
2.4
1.4
1.3
1.0
0.7

[5.5]
[5.0]
[2.7]
[3.1]
[3.6]
[3.6]
[4.4]
[3.9]
[3.5]
[4.0]
[5.5]
[5.8]
[3.0]
[3.7]
[3.2]
[2.3]
[1.6]
[1.4]
[1.0]

(0.1)
(0.0)
( 0.9)
(0.3)
( 1.4)
( 1.3)
(0.0)
( 0.9)
(0.2)
( 1.3)
( 1.7)
(0.1)
(0.3)
( 1.3)
( 1.5)
(0.6)
( 0.1)
( 0.6)
( 0.3)

1.8
1.3
1.8
1.2
1.8
1.6
1.3
1.6
1.4
2.2
1.6
1.7
1.3
2.0
1.9
1.7
1.4
2.5
1.5

[2.7]
[2.0]
[2.8]
[1.8]
[2.5]
[2.3]
[2.1]
[2.2]
[2.2]
[2.6]
[2.2]
[2.4]
[1.8]
[2.6]
[2.7]
[2.4]
[2.0]
[3.1]
[2.0]

( 0.4)
( 0.8)
( 1.3)
( 0.5)
( 1.0)
( 1.2)
( 0.5)
( 1.2)
( 1.2)
( 1.9)
( 1.3)
( 1.0)
( 0.5)
( 0.8)
( 1.0)
( 0.3)
( 0.5)
( 1.6)
( 0.7)

3.7
2.6
2.6
1.9
2.8
2.7
2.5
2.5
2.8
3.5
2.6
4.1
2.0
3.5
3.2
2.8
2.7
3.2
2.2

MOPAC/AM1e
MOPAC/PM3e
MOPAC/PDDG/PM3e
G03/B3LYP/6-31G(d)e
SCC-DFTBe

6.6
4.3
2.4
3.1
5.0

[8.9]
[5.1]
[2.8]
[4.7]
[7.0]

( 1.1)
( 0.6)
(0.0)
( 0.2)
(3.3)

4.2
3.8
2.6
2.5
4.7

[5.0]
[4.3]
[3.4]
[3.2]
[7.3]

(1.1)
( 1.3)
( 0.9)
( 1.3)
(0.0)

10.3 [11.8] (5.1)
6.0 [7.1] (2.6)
3.1 [3.3] (0.7)
2.9 [4.8] ( 2.3)
4.7 [5.8] (3.9)

MOPAC/PDDG/PM3//(SCC-DFTB and B3LYP/6-31G(d)e)f
G03/HF/6-31G(d)//(SCC-DFTB and B3LYP/6-31G(d)f)g
G03/B3LYP/6-31G(d)//(SCC-DFTB and B3LYP/6-31G(d)f)g
G03/HF/6-31G(d)//B3LYP/6-31G(d)g
G03/B3LYP/6-31G(d,p)//(SCC-DFTB and B3LYP/6-31G(d)f)g
G03/B3LYP/6-31+G(d,p)//(SCC-DFTB and B3LYP/6-31G(d)f)g

1.7
3.3
3.2
3.1
3.1
3.4

[2.4]
[5.2]
[4.8]
[4.7]
[4.7]
[4.7]

(0.0)
(0.5)
(0.2)
( 0.2)
(0.1)
(0.1)

2.7
2.8
2.9
2.5
1.9
1.1

[3.4]
[3.4]
[3.6]
[3.2]
[2.2]
[1.2]

( 0.1)
( 0.9)
( 1.4)
( 1.3)
( 0.8)
( 0.5)

2.7
3.4
3.0
2.9
2.1
1.7

[4.5]
[3.2]
[2.9]
[2.6]
[3.5]
[3.1]
[3.0]
[3.1]
[3.6]
[4.4]
[3.6]
[4.8]
[2.5]
[4.4]
[3.9]
[3.3]
[2.9]
[4.0]
[2.5]

[3.1]
[3.9]
[4.6]
[4.8]
[2.8]
[2.2]

(1.2)
( 0.4)
(0.6)
(0.3)
(1.4)
( 0.1)
(0.9)
( 0.5)
( 1.3)
( 2.4)
(0.3)
( 1.9)
(0.5)
(2.1)
(1.4)
(1.4)
(0.6)
( 1.2)
(0.5)

(0.9)
(1.1)
( 2.0)
( 2.3)
( 0.8)
( 0.2)

a

Turbomole.
From Ref. [3] and adjusted for error calculation using experimental data from Refs. [1,8].
c
Gaussian-03.
d
MOLPRO.
e
From Ref. [1].
f
Single point energies obtained from SCC-DFTB optimized geometries with the exception of B3LYP/6-31G(d) optimized geometries for propene, 2-methylpropene, and
penta-1,3-diene.
g
From Ref. [8].
b

Table 2
Calculated Disom H ðgÞ errors for the validation sets of pure hydrocarbons, nitrogen-containing hydrocarbons, and oxygen-containing hydrocarbons developed by Sattelmeyer et al.
[1], Grimme et al. [3], and Tirado-Rives and Jorgensen [8] at various levels of semiempirical, Hartree–Fock ab initio, density functional, and ab initio post-Hartree–Fock theory.
Values are in kcal/mol and presented as mean unsigned error [root mean squared error] (mean signed error).

AM1
PM3
PM6
PDDG
HF/6-311++G(d,p)
HF/aug-cc-pVDZ
B3LYP/6-311++G(d,p)
B3LYP/6-311+G(3df,3p)
B3LYP/aug-cc-pVDZ
M062X/6-311++G(d,p)
M062X/6-311+G(3df,3p)
M062X/aug-cc-pVDZ
CBS-Q//B3
G4MP2

Pure hydrocarbons

Nitrogen-containing hydrocarbons

Oxygen-containing hydrocarbons

6.7
4.2
3.6
1.8
3.4
3.0
3.2
3.4
2.8
1.1
1.4
1.5
0.5
0.6

3.9
4.1
3.5
3.1
1.4
1.4
0.8
0.9
0.8
1.0
1.0
1.3
0.7
0.7

8.0 [9.6] (3.9)
5.5 [6.5] (1.3)
3.9 [4.8] (0.3)
3.2 [3.7] ( 0.5)
2.8 [3.2] (0.0)
6.1 [11.0] (0.3)
2.2 [2.4] ( 1.2)
2.2 [2.6] ( 1.5)
1.4 [1.7] ( 0.7)
1.2 [1.7] ( 0.9)
1.5 [1.9] ( 1.2)
1.5 [1.8] ( 0.3)
1.5 [1.8] ( 0.6)
1.1 [1.4] ( 0.6)

[8.6]
[5.0]
[5.5]
[2.3]
[4.9]
[4.7]
[4.3]
[4.6]
[4.0]
[1.7]
[2.0]
[2.0]
[0.8]
[0.8]

( 0.7)
( 0.4)
( 3.2)
(0.1)
(0.4)
(0.1)
(0.1)
(0.1)
( 0.2)
( 0.5)
( 0.7)
( 0.8)
(0.4)
(0.2)

PM6: pure HCs (r = 0.65, p < 10 38, m = 0.16 (kcal/mol)/(kcal/
mol)), nitrogen HCs (r = 0.29, p = 0.01, m = 0.05), oxygen HCs
(r = 0.28, p < 0.01, m = 0.08), sulfur HCs (r = 0.26, p = 0.11),
and halogen HCs (r = 0.26, p = 0.24).
PDDG: pure HCs (r = 0.24, p < 10 4, m = 0.04), nitrogen HCs
(r = 0.03, p = 0.77), oxygen HCs (r = 0.35, p < 10 4, m = 0.11),
sulfur HCs (r = 0.49, p < 0.01, m = 0.47), and halogen HCs
(r = 0.34, p = 0.11).

[4.7] (0.8)
[4.7] ( 1.2)
[5.1] ( 3.3)
[3.5] ( 0.6)
[1.9] ( 0.1)
[1.8] ( 0.1)
[1.0] ( 0.1)
[1.1] ( 0.5)
[1.0] ( 0.5)
[1.4] ( 0.5)
[1.5] ( 1.0)
[1.9] ( 0.6)
[0.9] ( 0.5)
[0.8] ( 0.5)

B3LYP/6-311++G(d,p): pure HCs (r = 0.11, p = 0.05), nitrogen
HCs (r = 0.19, p = 0.11), oxygen HCs (r = 0.14, p = 0.14), sulfur
HCs (r = 0.73, p < 10 6, m = 0.51), and halogen HCs (r =
0.27, p = 0.22).
B3LYP/6-311+G(3df,3p): pure HCs (r = 0.16, p < 0.01,
m = 0.04), nitrogen HCs (r = 0.17, p = 0.14), oxygen HCs (r =
0.32, p < 0.001, m = 0.06), sulfur HCs (r = 0.68, p < 10 5,
m = 0.53), and halogen HCs (r = 0.22, p = 0.31).

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S. Rayne, K. Forest / Journal of Molecular Structure: THEOCHEM 948 (2010) 102–107

Table 3
Calculated Disom H ðgÞ errors for the extended composite isomerization datasets of pure hydrocarbons, nitrogen-containing hydrocarbons, and oxygen-containing hydrocarbons
using the PM6 and PDDG semiempirical methods, the B3LYP/6-311++G(d,p), B3LYP/6-311+G(3df,3p), M062X/6-311++G(d,p), and M062X/6-311+G(3df,3p) density functional
levels of theory, and the CBS-Q//B3 and G4MP2 ab initio post-Hartree–Fock composite methods. Values are in kcal/mol and presented as mean unsigned error [root mean squared
error] (mean signed error).
na

PM6

PDDG

311 3.1 [5.1] ( 0.8) 2.2
Pure HCsb
Nitrogen-containing
73 2.7 [3.7] ( 1.2) 3.2
HCs
Oxygen-containing
116 2.8 [4.1] ( 1.2) 3.1
HCs
Sulfur-containing HCs 39 2.8 [4.2] ( 1.0) 1.9
Halogen-containing
23 2.8 [3.7] ( 0.7) 3.0
HCs
Total
a
b

B3LYP
6-311++G(d,p)

B3LYP
M062X
6-311+G(3df,3p) 6-311++G(d,p)

M062X
CBS-Q//B3
6-311+G(3df,3p)

G4MP2

[3.1] (0.3)
5.5 [6.6] (4.2)
[4.6] ( 0.9) 1.7 [2.4] (0.6)

5.5 [6.8] (4.3)
1.9 [2.7] (0.3)

1.3 [2.1] (0.1)
1.6 [2.5] (0.4)
1.7 [2.3] ( 0.4) 1.7 [2.4] ( 0.5)

1.1 [1.7] (0.4)
1.1 [1.6] (0.3)
1.2 [1.6] ( 0.1) 1.1 [1.5] (0.2)

[4.6] ( 0.9) 2.2 [2.8] (0.5)

2.3 [2.9] (0.3)

1.1 [1.6] ( 0.2) 1.4 [1.8] ( 0.4)

1.2 [1.6] ( 0.3) 1.0 [1.4] (0.1)

1.0 [1.3] ( 0.4) 0.9 [1.2] ( 0.4)
1.1 [1.5] ( 0.6) 1.1 [1.5] ( 0.4)

0.8 [1.1] ( 0.5) 0.8 [1.1] ( 0.6)
1.3 [1.6] ( 0.3) 1.2 [1.5] ( 0.3)

1.3 [1.9] ( 0.1) 1.5 [2.2] (0.0)

1.1 [1.6] (0.1)

[2.9] ( 0.3) 1.8 [2.5] (1.3)
2.0 [2.7] (1.3)
[3.7] ( 0.6) 1.2 [1.6] ( 0.3) 1.2 [1.6] ( 0.2)

562 3.0 [4.6] ( 0.9) 2.5 [3.7] ( 0.2) 3.9 [5.2] (2.6)

4.0 [5.4] (2.5)

1.1 [1.5] (0.1)

Number of isomerization reactions in the composite dataset.
Hydrocarbons.

20

PM6

20

15

15

10

10

5

5

0
14

M062X/6-311++G(d,p): pure HCs (r = 0.31, p < 10 7, m =
0.03), nitrogen HCs (r = 0.44, p < 0.001, m = 0.05), oxygen
HCs (r = 0.14, p = 0.14), sulfur HCs (r = 0.06, p = 0.72), and
halogen HCs (r = 0.42, p = 0.18).
M062X/6-311+G(3df,3p): pure HCs (r = 0.39, p < 10 11,
m = 0.05), nitrogen HCs (r = 0.31, p<0.01, m = 0.04), oxygen
HCs (r = 0.39, p < 10 4, m = 0.05), sulfur HCs (r = 0.15, p =
0.36), and halogen HCs (r = 0.22, p = 0.31).
CBS-Q//B3: pure HCs (r = 0.02, p = 0.78), nitrogen HCs
(r = 0.20, p = 0.09), oxygen HCs (r = 0.15, p = 0.10), sulfur
HCs (r = 0.14, p = 0.39), and halogen HCs (r = 0.23, p = 0.29).
G4MP2: pure HCs (r = 0.14, p = 0.01, m = 0.01), nitrogen HCs
(r = 0.19, p = 0.10), oxygen HCs (r = 0.08, p = 0.39), sulfur HCs
(r = 0.13, p = 0.42), and halogen HCs (r = 0.22, p = 0.31).

PDDG

0
B3LYP/6-311++G(d,p)

40

M062X/6-311++G(d,p)

12
30

10
8

20

6

% Frequency

4

10

2
0
12

0
B3LYP/6-311+G(3df,3p)

M062X/6-311+G(3df,3p)

30

10

25

8

20

6

15

4

10

2

5

0
40

35

0
CBS-Q//B3

35

G4MP2

30
30

25
20

20

15
10

10

5
0
-30 -20 -10

0

10

20

0
-30 -20 -10

0

10

20

Signed error in Δ isom H°(g)
Fig. 1. Histograms showing the signed error in calculated Disom H ðgÞ across all
hydrocarbon classes (n = 562) using the PM6 and PDDG semiempirical methods, the
B3LYP/6-311++G(d,p), B3LYP/6-311+G(3df,3p), M062X/6-311++G(d,p), and M062X/
6-311+G(3df,3p) density functional levels of theory, and the CBS-Q//B3 and G4MP2
ab initio post-Hartree–Fock composite methods. Normal distribution best fits (solid
lines) are shown for comparison.

We do not view these correlations as sufficiently reliable to allow
correction of any of the theoretical methods in order to achieve
improved Disom H ðgÞ accuracy.
As a further check on data quality and sources of error, we calculated gas phase standard state (298.15 K, 1 atm) enthalpies of
formation Df H ðgÞ using the atomization enthalpy approach [38]
for all compounds comprising our Disom H ðgÞ dataset (Table 4,
Fig. 2, and Tables S9–S14). Consistent with previous reports, the
two DFT methods with the 6-311++G(d,p) basis set systematically
overestimated Df H ðgÞ using the atomization approach (although the
M062X/6-311++G(d,p) errors are about one-half those of B3LYP/6311++G(d,p)), whereas the CBS-Q//B3 and G4MP2 methods gave
substantially lower errors. The G4MP2 method provided Df H ðgÞ
chemical accuracy for all functional group classes. With the 6311+G(3df,3p) basis set, the B3LYP Df H ðgÞ errors are decreased by
about half relative to the 6-311++G(d,p) basis set, although they remain well beyond chemical accuracy (i.e., 17 kcal/mol Df H ðgÞ MUE
for all compounds). In contrast, moving from the 6-311++G(d,p)
to 6-311+G(3df,3p) basis set with the M062X functional decreases
the Df H ðgÞ error by about fourfold on average. This increase in basis
set completeness brings the M062X/6-311+G(3df,3p) Df H ðgÞ prediction accuracy with the atomization approach to only about
1.5 kcal/mol less accurate than the CBS-Q//B3 method, and about
3 kcal/mol less accurate on average than the G4MP2 level.
Consequently, where such DFT methods achieve accurate
Disom H ðgÞ estimates with the 6-311++G(d,p) basis set, the favorable
results depend on error cancellations in the Df H ðgÞ values for each
component of the isomerization reaction. Using the 6-311+
G(3df,3p) basis set, the M062X functional achieves accurate
Disom H ðgÞ estimates via the desired mode of accurate Df H ðgÞ estimates. In contrast, the B3LYP functional achieves modest

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Table 4
Calculated Df H ðgÞ errors for all compounds in the extended composite isomerization datasets of pure hydrocarbons, nitrogen-containing hydrocarbons, and oxygen-containing
hydrocarbons using the PM6 and PDDG semiempirical methods, the B3LYP/6-311++G(d,p), B3LYP/6-311+G(3df,3p), M062X/6-311++G(d,p), and M062X/6-311+G(3df,3p) density
functional levels of theory, and the CBS-Q//B3 and G4MP2 ab initio post-Hartree–Fock composite methods via the atomization enthalpy approach. Values are in kcal/mol and
presented as mean unsigned error [root mean squared error] (mean signed error).

Pure hydrocarbons
Nitrogen-containing
hydrocarbons
Oxygen-containing
hydrocarbons
Sulfur-containing
hydrocarbons
Halogen-containing
hydrocarbons
Total
a

na

B3LYP/6311++G(d,p)

B3LYP/6311+G(3df,3p)

M062X/6311++G(d,p)

M062X/6311+G(3df,3p)

CBS-Q//B3

G4MP2

336
102

34.8 [36.7] (34.8)
20.1 [21.8] (20.1)

22.4 [24.4] (22.4)
7.5 [9.2] (7.3)

17.5 [18.1] (17.5)
13.6 [14.2] (13.6)

5.5 [6.1] (5.3)
2.0 [2.5] (0.8)

3.1 [3.5] (3.1)
1.8 [2.1] (0.9)

1.0 [1.4] (0.2)
1.0 [1.3] (0.5)

157

27.0 [28.7] (27.0)

13.7 [15.1] (13.7)

13.8 [14.4] (13.8)

1.9 [2.3] (0.2)

1.1 [1.4] (0.2)

0.9 [1.2] (0.8)

55

28.6 [30.1] (28.6)

16.9 [18.2] (16.9)

14.7 [15.3] (14.7)

2.6 [2.9] (2.5)

0.9 [1.3] (0.5)

37

23.5 [24.7] (23.5)

11.7 [12.5] (11.7)

9.5 [9.9] (9.5)

2.7 [3.3] ( 2.1)

1.9 [2.3]
( 1.8)

1.1 [1.4]
( 0.9)
1.0 [1.6]
( 0.6)

687

29.7 [32.0] (29.7)

17.2 [19.7] (17.2)

15.4 [16.2] (15.4)

3.8 [4.6] (2.8)

2.2 [2.7] (1.7)

1.0 [1.3] (0.3)

Number of compounds in the composite dataset.

6

B3LYP/6-311++G(d,p)

10

5

mental Df H ðgÞ for the various compound class/theoretical method
combinations gave either statistically insignificant (p > 0.05) correlations or a poor quality of fit (jrj < 0.75) with low values for the
regression slope:

M062X/6-311++G(d,p)

8

4

6

3
4

2

2

1
0
6

0
B3LYP/6-311+G(3df,3p)

14
12

5

% Frequency

M062X/6-311+G(3df,3p)

10

4

8

3

6

2

4

1

2

0

0
CBS-Q//B3

30

G4MP2

40

25
30
20
20

15
10

10
5
0

0

20

40

60

80

0

0

20

40

60

80

Signed error in Δ fH°(g)
Fig. 2. Histograms showing the signed error in calculated Df H ðgÞ via the atomization
enthalpy approach across all hydrocarbon classes (n = 687) using the B3LYP/6311++G(d,p), B3LYP/6-311+G(3df,3p), M062X/6-311++G(d,p), and M062X/6311+G(3df,3p) density functional theory methods and the CBS-Q//B3 and G4MP2
ab initio post-Hartree–Fock composite methods. Normal distribution best fits (solid
lines) are shown for comparison.

Disom H ðgÞ prediction performance via Df H ðgÞ error cancellation, even
with the 6-311+G(3df,3p) basis set. The G4MP2 method, similar to
its Gaussian-n precursors, appears capable of accurately estimating
Df H ðgÞ using the atomization method across a broad range of
organic compounds with varying functional groups and molecular
masses. Thus, the G4MP2, CBS-Q//B3, and M062X/6-311+G(3df,3p)
methods achieve quality Disom H ðgÞ predictions due to fundamentally correct underlying Df H ðgÞ estimates.
Similar to our statistical analyses of the Disom H ðgÞ estimates, linear regressions of the calculated Df H ðgÞ error against the experi-

B3LYP/6-311++G(d,p): pure HCs (r = 0.46, p < 10 17, m = 0.12
(kcal/mol)/(kcal/mol)), nitrogen HCs (r = 0.12, p = 0.21), oxygen
HCs (r = 0.29, p < 0.001, m = 0.08), sulfur HCs (r = 0.63,
p < 10 6, m = 0.32), and halogen HCs (r = 0.29, p = 0.08).
B3LYP/6-311+G(3df,3p): pure HCs (r = 0.54, p < 10 25,
m = 0.12), nitrogen HCs (r = 0.07, p = 0.46), oxygen HCs
(r = 0.32, p < 10 4, m = 0.06), sulfur HCs (r = 0.66, p < 10 7,
m = 0.24), and halogen HCs (r = 0.47, p < 0.01, m = 0.05).
M062X/6-311++G(d,p): pure HCs (r = 0.58, p < 10 31,
m = 0.07), nitrogen HCs (r = 0.07, p = 0.49), oxygen HCs
(r = 0.29, p < 0.001, m = 0.04), sulfur HCs (r = 0.64,
p < 10 6, m = 0.15), and halogen HCs (r = 0.11, p = 0.51).
M062X/6-311+G(3df,3p): pure HCs (r = 0.74, p < 10 57,
m = 0.05), nitrogen HCs (r = 0.06, p = 0.53), oxygen HCs
(r = 0.07, p = 0.41), sulfur HCs (r = 0.68, p < 10 8, m =
0.05), and halogen HCs (r = 0.14, p = 0.39).
CBS-Q//B3: pure HCs (r = 0.03, p = 0.64), nitrogen HCs
(r = 0.46, p < 10 5, m = 0.02), oxygen HCs (r = 0.33, p < 10 4,
m = 0.01), sulfur HCs (r = 0.01, p = 0.97), and halogen HCs
(r = 0.01, p = 0.99).
G4MP2: pure HCs (r = 0.30, p < 10 7, m = 0.01), nitrogen HCs
(r = 0.32, p < 0.001, m = 0.01), oxygen HCs (r = 0.31,
p < 10 4, m = 0.01), sulfur HCs (r = 0.06, p = 0.68), and halogen
HCs (r = 0.57, p < 0.001, m = 0.02).
Similar regression of the Df H ðgÞ error against molecular mass for
all compounds within a particular computational method yielded
reasonably strong correlations and modest positive slopes for the
B3LYP/6-311++G(d,p) (r = 0.88, p < 10 226, m = 0.35 (kcal/mol)/(g/
mol)), B3LYP/6-311+G(3df,3p) (r = 0.75, p < 10 125, m = 0.24), and
M062X/6-311++G(d,p) (r = 0.80, p < 10 156, m = 0.13) levels of theory. Much weaker regression qualities of fit and lower slopes were
found for the M062X/6-311+G(3df,3p) (r = 0.27, p < 10 12,
m = 0.03), CBS-Q//B3 (r = 0.25, p < 10 11, m = 0.02), and G4MP2
(r = 0.10, p = 0.01, m = 0.004) methods. We note that the molecular mass scaling error is nearly absent at the M062X/6311+G(3df,3p) level, having a Df H ðgÞ error against molecular mass
slope more than fivefold lower than with the 6-311++G(d,p) basis
set and a substantial reduction in the quality of fit. In contrast, the
mass scaling error is only reduced modestly in terms of both magnitude and quality of fit in moving from the 6-311++G(d,p) to 6311+G(3df,3p) basis set with the B3LYP functional. The findings

Author's personal copy

S. Rayne, K. Forest / Journal of Molecular Structure: THEOCHEM 948 (2010) 102–107

are consistent with the known Df H ðgÞ scaling error of various DFT
methods using atomization approaches which is generally absent
when higher level composite methods or modern DFT functionals
with more balanced basis sets are employed [30,31,37–47].
In conclusion, the PM6 and PDDG semiempirical methods offer
modest Disom H ðgÞ prediction performance approximately comparable to the B3LYP density functional, but at much lower computational cost. For pure hydrocarbons, both semiempirical methods
significantly outperform the B3LYP functional for Disom H ðgÞ estimation. The M062X density functional offers nearly equivalent
Disom H ðgÞ prediction accuracy to the higher level, and much more
expensive, CBS-Q//B3 and G4MP2 methods across all hydrocarbon
classes. Thus, for very large systems where composite methods are
too expensive, pure and functionalized hydrocarbon derivative
isomerization enthalpies can be reliably estimated with the
M062X density functional. With the 6-311+G(3df,3p) basis set,
the M062X functional also provides Df H ðgÞ estimation accuracy
using the atomization method near that of the composite CBS-Q//
B3 method.
Acknowledgements
This work was made possible by the facilities of the Western
Canada Research Grid (WestGrid: www.westgrid.ca; project
100185), the Shared Hierarchical Academic Research Computing
Network (SHARCNET: www.sharcnet.ca; project sn4612), and
Compute/Calcul Canada. Thanks are extended to an anonymous reviewer whose helpful suggestions improved the quality of the
work.

[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]

[30]
[31]
[32]
[33]
[34]

Appendix A. Supplementary material
Supplementary data associated with this article can be found, in
the online version, at doi:10.1016/j.theochem.2010.02.030.
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