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Seemingly Simple Stereoelectronic Effects in Alkane Isomers and the Implications for KohnЦSham Density Functional Theory.

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Communications
Theoretical Chemistry
DOI: 10.1002/anie.200600448
Seemingly Simple Stereoelectronic Effects in
Alkane Isomers and the Implications for
Kohn?Sham Density Functional Theory**
Stefan Grimme*
The concept of stereoelectronic (SE) effects is widely applied
especially in organic chemistry.[1] Substituents are typically
described as electron-density-donating or -withdrawing or
responsible for sterically interacting with, for example,
reagents. The rational application of SE design rules is of
utmost importance in synthetic chemistry where target
structures are often kinetically or thermodynamically
favored. Important SE effects can be introduced by alkylation
(methylation in the following discussion for simplicity) of
strategic positions in a molecule. The resulting energetic
changes can be qualitatively separated into so-called electronic (bonding) and nonbonding (through-space, i.e., attractive van der Waals (vdW) and electrostatic, and repulsive
Pauli) interactions. While a meaningful definition and separation of these individual terms is quite difficult at a rigorous
quantum mechanical level, it is generally believed that
modern quantum chemical methods are at least able to
describe the overall energetic effects quantitatively. It is
shown in this Communication that this does not hold for the
widely used Kohn?Sham density functional theory (DFT)[2]
with all state-of-the-art density functionals; this approach
does not provide even a qualitatively correct picture for an
important class of chemical problems.
The problem has become evident in recent years since
larger molecular systems have been studied more routinely. In
general it appears when a first-row atom or group X (X = B,
C, N) is successively alkylated. Gilbert[3] found increasing
errors of the B3LYP functional (up to 20 kcal mol1) for the
dissociation reaction (1) when R = H is replaced by CH3.
R3 NBR3 ! NR3 ■ BR3
­1я
Similar results were reported later[4, 5] for the CC bond
dissociation reaction (2) and [2+2] cycloadditions and CH2R3 CCR3 ! R3 CC ■ CCR3
­2я
[*] Prof. Dr. S. Grimme
Theoretische Organische Chemie
Organisch-Chemisches Institut der Universit't M)nster
Corrensstrasse 40, 48149 M)nster (Germany)
Fax:(+49) 251-833-6515
E-mail: grimmes@uni-muenster.de
[**] This work was supported by the Deutsche Forschungsgemeinschaft
in the framework of the SFB 424 (?Molekulare Orientierung als
Funktionskriterium in chemischen Systemen?). The author thanks
C. M)ck-Lichtenfeld for technical assistance and helpful discussions.
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2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
Angew. Chem. Int. Ed. 2006, 45, 4460 ?4464
Angewandte
Chemie
alkene insertions with substrates of various sizes[5] and for
radical stabilization energies.[6] Related to the problem are
several observations[7?9] of increasing errors for DFT/B3LYPderived enthalpies of formation with increasing size of the
molecules (hydrocarbons in particular, for example, 0.7 kcal
mol1 per CH2 group[8, 9] in n-alkanes and 25 kcal mol1 for
adamantane[7]).
The origin of these very unusual errors has been subject to
speculations or special ad hoc explanations that range from
?self-interaction error? (for Eq. (2)) to ?problems with dative
CC bonds in general,?[3, 5] to the failure of most density
functionals (DFs) to describe the attractive long-range vdW
interaction (cf. Ref. [10] and references therein). This latter
consideration is quite obvious as increasing alkylation
decreases the average nonbonding distances, which, along
with with a too repulsive vdW potential, would systematically
destabilize the more substituted species (i.e., the left sides in
Equations (1) and (2)). This was also our first working
hypotheses because these intramolecular vdW effects can be
quite large even in medium-sized molecules.[11] However, as
will be shown here, the failure of common DFs to account for
vdW interactions plays only a secondary role; rather, the fact
that all semi-local(hybrid) DFs neglect medium-range electron correlation (EC) effects is of general relevance. It seems
important to stress at this point that we do not question the
usefulness of current DFT in general. However, we think that
especially because DFT (and B3LYP in particular) is being
used more and more as a computational tool by nonexperts, it
is necessary to reveal the strengths and weaknesses of the
theoretical method. We furthermore offer a possible solution
to the problem within the framework of DFT that incorporates parts of conventional wave function methods (B2PLYP
functional[12]).
The examples we use in our analysis are not dissociation
reactions (which have their own special problems) but
seemingly simple isomerizations of branched to linear alkanes. In all these reactions, the number of CH and CC
bonds as well as the formal hybridization states remain the
same. The typical errors for the calculated isomerization
energy branched!linear (DE) are already apparent for
butane and increase with the size of the system. We consider
in detail two isomers (the most branched isomer is labeled a
and the linear b) of butane (1), pentane (2), octane (3), and
undecane (4) (Scheme 1). To separate the SE effect into
electronic and steric parts, the transition state for the gauche?
gauche rotation of n-butane (1 c) (in which the steric
CH3иииCH3 interaction is similar to that in isomers 3 a and
4 a) and of two different hexanes (isomers of octane, 3 c and
3 d) were investigated.
In order to demonstrate that we are not discussing tiny
and probably unimportant effects but a problem of general
importance, the calculated and experimental isomerization
energies for octane are shown in Table 1. Beside traditional
wave function methods that are based on second-order
MЭller?Plesset perturbation theory (MP2 or its improved
version SCS-MP2[13]) all types of modern DFs (pure GGA:
BLYP[14] and PBE;[15] hybrid GGA: B3LYP[16] and BHLYP;[17]
meta-hybrid: TPSSh;[18] GGA with empirical vdW corrections: D-BLYP[10]) were employed.[19]
Angew. Chem. Int. Ed. 2006, 45, 4460 ?4464
Scheme 1.
Table 1: Isomerization energies for octane.[a]
Method
DE(3 b3 a) [kcal mol1]
MP2
exptl
SCS-MP2
B2PLYP
PBE
D-BLYP
TPSSh
BHLYP
B3LYP
BLYP
HF
4.6 (2.7)
1.9 0.5
1.4 (0.5)
3.5 (5.4)
5.5 (7.4)
5.5 (7.4)
6.3 (8.2)
7.2 (9.1)
8.4 (10.3)
9.9 (11.8)
11.5 (13.4)
[a] Errors with respect to experimental results in parentheses. For
reference to the experimental value see the Method Section; the
theoretical data refer to the cQZV3P//MP2/TZV(d,p) level.
A brief inspection of these data reveals a disastrous
situation for DFT: none of the functionals can describe even
the correct energetic order of the isomers (branched lower in
energy than linear). The errors are surprisingly large, ranging
from 7.4 (PBE) to 11.8 kcal mol1 (BLYP). Errors of this
magnitude are unacceptable and disqualify DFT application
in polymer science or in investigations of, for example, alkane
combustion, where branching is an important aspect. From
the theoretical point of view it is further intriguing that the
DFT results are only slightly better than calculations at the
uncorrelated Hartree?Fock (HF) level (error of 13.4 kcal
mol1). This means that the main SE effect (which results
completely from EC) is not described correctly by any of the
DFs, although it is generally believed that at least the main
EC effects are incorporated in current DFs. The finding that
EC is the decisive factor for the relative stability of alkane
isomers is also not well-known in the literature; quite recently
a (wrong) explanation of the effect on the basis of orbital
considerations was published[20] (for a good review on the
2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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4461
Communications
topic including many older works and a quite new concept see
Ref. [21]). It seems also noteworthy that the usual explanation
of these SE effects as resulting from hyperconjugation can be
ruled out because this is mostly included already at the HF
level (by superposition of AC CB, A+B , AB+ valence bond
structures for a covalent AB bond).
The correlated wave function methods have only minor
problems with this system (MP2 ?overshoots? as usual), and
SCS-MP2 in particular yields here (and for other alkanes as
well) results accurate to within experimental error. Also
surprising is the finding that the portion of ?exact? exchange
in a hybrid functional (e.g. 20 and 50 % in B3LYP and
BHLYP, respectively), which usually influences the results
considerably, has almost no effect on DE. So-called meta
functionals involving derivatives of the occupied orbitals
(kinetic energy density) also fail completely; in other words,
TPSSh is not better than its predecessor PBE.
Table 2 shows results for the other systems investigated;
we have included only PBE, B3 LYP, and B2PLYP and note in
passing that other DFs yield similar results. In contrast to 1?3,
the linear isomer of undecane is the most stable as a result of
Table 2: Calculated relative energies DE of alkane isomers.[a]
Isomer
exptl
SCS-MP2
1b
1 c[b]
1.5
5.5
1.5
5.7
2b
2.8
3.0
3b
3c
3d
1.9
1.0
2.0
1.4
0.7
2.1
4b
?
9.4
DE [kcal mol1]
MP2 HF
B3LYP
butane
1.8
0.3
5.7
6.4
pentane
3.6
0.1
octane
4.6 11.5
1.6
3.0
3.8
4.8
undecane
3.4 34.4
PBE
B2PLYP
0.5
5.7
0.8
5.5
1.0
5.7
1.3
1.6
2.1
8.4
2.4
3.5
5.5
1.7
2.1
3.5
0.9
0.8
26.8
21.0
17.9
works in the right direction (stabilizing a with respect to b),
but the BLYP error is reduced by less than 40 % for 3 (see
Table 1); 3) The gauche?gauche transition state for the
rotation around C2C3 in n-butane that is sterically very
similar to the situation in, for example, 3 a (shortest nonbonding CC distances of about 3.2 L compared to 2.9 L in
1 c) is described reasonably well by all DFs; 4) For the energy
difference between 3 c and 3 d, which have the same type of
branching but branching points at different positions, the
DFT errors are opposite to what would be expected when the
vdW interaction is dominant (i.e., 3 d is incorrectly lower in
energy than 3 c).
Clearly the reason for the failure to describe the SE
alkylation effect lies primarily in the electronic (throughbond) part. As already emphasized, it is entirely due to
electron correlation (EC) and our analysis is thus based on a
rigorous partitioning of the EC energy. Owing to the localized
electronic structure of alkanes this is possible by considering
the No(No+1)/2 pair correlation energies [e]ij (No is the
number of occupied orbitals) that sum up to the total
correlation energy. When localized molecular orbitals
(LMO) are used as the basis in the MP2 or SCS-MP2
treatment, the correlation energies [e]ij can be assigned to
different regions in space (electron pairs) and correlation
types (inter- or intrapair correlations, see Scheme 2, where the
dashed arrows indicate ?electron collisions and correlations?). Note that the LMO treatment does not change the
total correlation energy.
[a] With respect to the branched isomer a; cQZV3P//MP2/TZVP level.
[b] Transition state for C2C3 rotation (gauche?gauche) with respect to
n-butane. For the reference value see Ref. [22].
many very unfavorable short contacts between methyl groups.
In general the methods behave similarly, as in the case of
octane. The errors increase with the size of the system, for
example, 1.0, 1.5, 10.3, and about 17 kcal mol1 for B3LYP in
the series 1 to 4. Thus, the error seems to accumulate; this was
also found in the studies of dissociation reactions.[3, 5] In the
calculations of octane, the HF errors are larger than with
B3LYP and the PBE errors are smaller.
For a further analysis of the problem it is important to
answer the question how large the ?steric? contributions to
DE are. It is not rigorously possible to separate these effects
(from a formal point of view the ?steric? part is also of
electronic origin), but there is strong evidence that the vdW
aspect (and the usual DFT problems for that) is not a major
factor: 1) The effect on DE when diffuse basis functions are
added (they are usually necessary to accurately describe vdW
interactions) is small at the SCS-MP2 level (0.5 kcal mol1
difference for DE when the aug-cc-pVTZ is used instead of
the cc-pVTZ AO basis); 2) The empirical vdW correction[13]
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Scheme 2.
The partitioned pair correlation energies and contributions to the isomerization energy of pentane and octane are
listed in Table 3. In Figure 1 contributions to DE are
partitioned according to the estimated distance between the
correlating electrons.
In perusing Table 3, one finds that the intrapair correlation (last column) contributes almost nothing to DE. These
tiny differences between the quite large correlation energy
values (< 0.1 %) strongly support the common picture of the
simple electronic structure (transferrable, localized s bonds)
of alkanes. According to further analysis, the EC effect on the
DE values results entirely from interpair correlations between
orbitals of the same type (CC/CC and CH/CH), which favor
the branched form, compared to CC/CH correlations, which
lower the energy of the linear isomer. This can be simply
2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
Angew. Chem. Int. Ed. 2006, 45, 4460 ?4464
Angewandte
Chemie
Table 3: Pair correlation energies (in m Eh) at the SCS-MP2/TZV(d,p)
level using localized CC and CH orbitals.[a]
Isomer CC/CC
Sum of pair correlation energy
CH/CH
CH/CC
Total
2a
2b
3a
3b
460
482 (13.8)
679
728 (30.7)
128
153 (15.7)
232
289 (35.8)
183
142 (25.7)
308
224 (52.7)
771
777 (3.8)
1219
1241 (13.8)
CC + CH
(intrapair)
449.3
448.8
699.6
699.7
[a] Values in parentheses are the correlation energy contributions to DE
(in kcal mol1). The DE values at the HF/TZV(d,p) level are 0.5 and
10.6 kcal mol1 for pentane and octane, respectively, which yields
together with the values in the fifth column total SCS-MP2 DE values of
4.3 and 3.2 kcal mol1, respectively.
Figure 1. Contribution of the pair correlation energy to the isomerization energy of octane as a function of the distance R between the
center-of-charge of the respective pair of LMOs.
rationalized by the sheer number of (atomic) 1,3-interactions
in the two isomers (CC: 6 (2 a) vs. 3 (2 b), HH: 12 (2 a) vs. 9
(2 b), CH: 12 (2 a) vs. 18 (2 b)). Figure 1 shows that the
correlations that determine DE occur on a medium-range
length scale (1.5?3.5 L), and one may indeed classify these as
correlating 1,3-interactions. This important result can be
considered as the quantum mechanical basis for the recently
proposed 1,3-interaction model for alkane stabilities.[21] Since
we have already pointed out the minor importance of vdW
effects, it comes as no surprise that the long-range correlations actually favor the linear form (see Figure 1).
These findings also very clearly lead to an explanation for
the failure of current DF: the exchange functionals introduce
a localized exchange-hole around each electron that mimics
effects of static EC (the intrapair terms), but this contributes
nothing to DE. Because the correlation functionals are also
too local, the 1,3-correlations and thus the entire SE effect is
more or less absent. The hybrid functionals also do not work
because the exchange hole is not delocalized in saturated
systems. Significantly better (but still qualitatively wrong)
results for all alkanes are obtained from the new B2PLYP
Angew. Chem. Int. Ed. 2006, 45, 4460 ?4464
functional,[12] which includes (beside standard exchange and
correlation components) perturbative (orbital-dependent)
terms similar to MP2 or SCS-MP2.
In summary, it has been shown that the seemingly simple
stereoelectronic alkylation effect (and this also holds for
substitutions at other main-group elements) is entirely due to
electron correlation and that vdW-type interactions (the
?stereo? part) are only moderately (less than 30 % for about
ten close-lying carbon atoms) responsible. However, the
intramolecular vdW interaction should not be ignored as it is
an essential part of the internal energy and its inclusion is
absolutely necessary to obtain reasonable agreement with
experiment.[11] For the explanation of the serious failure of all
common semilocal density functionals, however, the vdW
problem is only of secondary importance. Rather, mediumrange but nonlocal electron correlations originating from
different (perfectly) localized s orbitals contribute almost
exclusively to the energy difference between linear and
branched forms. The different correlation energies are mainly
determined by the spatial distribution of the different bond
types in the isomers. This explains the success of simple
additivity schemes (group, bond, or 1,3-interaction increments) and furthermore holds for most of the DFT problems
mentioned in the introduction.[23]
This, however, is bad news for standard Kohn?Sham DFT
as it seems difficult to obtain the necessary information about
electron pairs and their interactions from simple considerations of the electron density alone. Of course we are not
questioning the exactness and usefulness of the Hohenberg?
Kohn theorems or the Kohn?Sham approach in general, but
just the human ability to find accurate density functionals in
practice. The better results with the new virtual orbitaldependent B2PLYP functional that includes in part the
necessary terms shows, however, the way to go in the future.
In this sense the reaction 3 a!3 b, for which no current DF
provides even the right sign for DE,[24] is suggested as a
mandatory benchmark for new density functionals.
Methods
The (SCS-)MP2 and DFT calculations were performed with slightly
modified versions of the Turbomole suite of programs.[25] . As the AO
basis, the split-valence (SV), triple-zeta (TZV), or quadruple-zeta
(QZV) sets of Ahlrichs et al. [26] or the correlation-consistent sets of
Dunning[27] (cc-pVXZ, X = D, T, Q) were employed. In the (SCS)MP2 and B2PLYP treatments the RI approximation for the twoelectron integrals was used,[28] and all electrons were correlated. The
geometries were optimized at the MP2/TZV(d,p) level, and singlepoint calculations on these structures were performed with a cQZV3P
AO basis set which includes (3d2f/2pd) polarization functions, (2pd)
core-polarization/correlation functions, and an additional single
diffuse s function (a = 0.05) on carbon. This basis set provides results
to within 0.1 kcal mol1 of the basis set limit for DE (as estimated
from aug-cc-pVXZ (X = T, Q) treatments for 3) and furthermore
yields quite small intramolecular basis set superposition errors. The
effect of choosing geometries optimized at a different level than MP2
in the single-point calculations (e.g. DFT) was estimated to be on the
order of 0.2 kcal mol1 for DE in the case of octane. The
experimental DE values were derived from experimental standard
enthalpies of formation[29] by correcting for vibrational zero-point
energies, internal H(0)H(298) contributions, and corrections due to
2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
www.angewandte.org
4463
Communications
thermal population of higher-lying conformers. The harmonic vibrational frequency calculations were performed at the (unscaled) DFTPBE/SV(d,p) level; the MM3(1996)[30] force field was used for the
conformational searches. Since for undecane no experimental data
are available, the accurate SCS-MP2 values were used as reference.
The experimental enthalpies have quoted error bars of (0.2?
0.4) kcal mol1, which yields an estimate for the uncertainty of the
DE(exptl) values used of about 0.5 kcal mol1.
[25]
[26]
Received: February 2, 2006
Revised: March 21, 2006
Published online: May 31, 2006
.
Keywords: ab initio calculations и alkanes и density functional
calculations и isomerization и stereoelectronic effects
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[20] J. Ma, S. Ingaki, J. Am. Chem. Soc. 2001, 123, 1193 ? 1198.
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[23] Our explanation is also in agreement with the findings of Curtiss
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[24] There are mixtures of different GGA components with the LDA
exchange term possible (e.g. 50 % LDA + 50 % B88 + LYP)
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[27]
[28]
[29]
[30]
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