First-principles computational studies of the torsional potential ...

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First-principles computational studies of the
torsional potential energy surface of the sec-butyl
radical
Ya Kun Chen and Yan Alexander Wang
Abstract: First-principles calculations were carried out to investigate the torsional potential energy surface (PES) of the secbutyl
radical. All the wave function methods employed predict a cis-like stable conformation with a dihedral angle of about
47° in addition to the trans-like global minimum conformation and a gauche conformation. However, most of the popular
density functional approaches predict only the latter two local minima and lack the cis conformation that was experimentally
observed. On the other hand, some density functional methods that incorporate the exact exchange and asymptotically corrected
correlation functionals can locate the cis conformation successfully. The basis-set effect was also measured using popular
B3LYP and MP2 Hamiltonians: only moderate shape changes were found for PES profiles upon basis-set variations.
The stationary structures and their Hessians were obtained at both MP2 and B3LYP levels, with or without incorporating
the zero-point energies. Opposite to the relative stability within the Born–Oppenheimer approximation, the cis conformation
is more stable than the gauche conformation upon the zero-point correction, consistent with the experiment observations.
Key words: sec-butyl, torsional potential energy surface, density functional theory, wave function theory.
Résumé : On a effectué des calculs théoriques à partir des principes premiers pour étudier la surface d’énergie potentielle
(SEP) de torsion du radical sec-butyle. Toutes les méthodes de fonction d’onde utilisées prédisent qu’il existe sous une
conformation de type cis stable avec un angle dièdre de 47° à côté d’une conformation de type trans avec minimum global
et d’une conformation gauche. Toutefois, la plupart des approches populaires par la fonctionnelle de la densité ne prédisent
l’existence que des deux derniers minima locaux sans prédire celle de la conformation cis observée expérimentalement. Par
ailleurs, certaines méthodes de la fonctionnelle de la densité qui incorporent l’échange exact et des fonctionnelles de corrélation
corrigées d’une façon asymptotique permettent de localiser avec succès la conformation cis. On a aussi mesuré l’effet
de l’ensemble de base en faisant appel aux hamiltoniens populaires B3LYP et MP2 : toutefois, avec ces changements d’ensembles
de base, on n’a pu observer que des changements modérés de la forme des profils de la surface d’énergie potentielle
(SEP). On a obtenu les structures stationnaires et leurs matrices hessiennes aux niveaux MP2 ainsi que B3LYP, avec
et sans incorporation des énergies du point zéro. Contrairement à la stabilité relative observée dans l’approximation de
Born–Oppenheimer, la conformation cis est plus stable que la conformation gauche lorsqu’on introduit une correction pour
l’énergie au point zéro et ceci est en accord avec les observations expérimentales.
Mots‐clés : sec-butyle, surface d’énergie potentielle de torsion, théorie de la fonctionnelle de la densité, théorie de la fonction
d’onde.
[Traduit par la Rédaction]
Introduction
Torsional potential energy surfaces (PESs) of many chemical
species have been studied numerous times because they
provide the key information for molecular thermodynamic
state functions. At the same time, it is also of great interest
to compare the theoretically predicted PES and available experimental
data to develop and improve computational methods
and to provide deep insight into the nature of torsional
barriers.
Even molecules as simple as ethane have attracted much
theoretical attention for the investigation of the physical
nature of its torsional barrier. The rotational barriers of
ethane and its congeners were mainly attributed to hyperconjugation
interactions by some researchers, challenging the
popular belief in many textbooks of the importance of steric
repulsive interactions. 1 However, using generalized valence
bond theory, a recent theoretical study, in which the torsional
barrier of ethane was decomposed adiabatically into steric repulsion
and hyperconjugation as well as several other insignificant
contributions, manifested that the steric repulsion is
the main factor to the stabilization of staggered conformation
and hyperconjugation is only of a secondary effect. 2
Received 10 February 2011. Accepted 23 June 2011. Published at www.nrcresearchpress.com/cjc on 9 November 2011.
Y.K. Chen and Y.A. Wang. Department of Chemistry, University of British Columbia, 2036 Main Mall, Vancouver, BC V6T 1Z1,
Canada.
Corresponding author: Yan Alexander Wang (e-mail: yawang@chem.ubc.ca).
This article is part of a Special Issue to commemorate the International Year of Chemistry.
Can. J. Chem. 89: 1469–1476 (2011) doi:10.1139/V11-109 Published by NRC Research Press
1469

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1470 Can. J. Chem. Vol. 89, 2011
Without the ambiguity in the physical origin of torsional
barriers, it is widely accepted that conjugation effect dominates
the torsional barrier of the conjugated systems. However,
this well-adopted understanding does not promise good
agreement between experimental results and theoretical predictions
in many conformational studies. The torsional energy
profile of 2,2′-bifuran was studied using various forms of
Hamiltonians, in which density functional theory (DFT)
methods were found inappropriate for the conformation study
of this system. 3 Not only does the form of the Hamiltonian
matter, but the basis set employed also plays a significant
role. When biphenyl PES was studied using several wave
function theory (WFT) methods, the quality of basis sets
was found to be more important than that of the Hamiltonians.
4,5 Besides the individual Hamiltonian or the basis-set
effect, delicate selection of certain combinations of Hamiltonians
and basis sets can be critical for an accurate calculation
of these (semi)rigid molecules. Theoretical study of the torsional
barriers of biisothianaphthene showed that an accurate
and balanced description of both conjugation and steric repulsion
between nonvicinal hydrogens could be only
achieved using high-level Hamiltonians and large basis sets. 6
Therefore, one must be aware of the limitations of popular
theoretical tools; extensive studies are required to avoid false
predictions even for small molecules based on calculations
using a single Hamiltonian with a single basis set.
Other than these well-documented molecules, it might become
even more challenging to study the conformations of
some open-shell and complex species, including radicals
whose PESs are usually so flat that an accurate description
of the electronic energies involved goes beyond the reach of
popular computational methods. For example, in the case of
haloethyl radicals, a theoretical study showed that the DFT
methods with the generalized gradient approximation (GGA)
outperformed the local Møller-Plesset (MP2) method in predicting
the dissociation energy of haloethyl radicals. 7 On the
other hand, large spin contamination was found from DFT
calculations of OOBr and OOCl, contradicting the popular
view that DFT methods suffer less spin contamination than
WFT methods. Sometimes, poor results can come from
many different methods in a single case. In a conformational
study of annulene molecules, both MP2 and DFT
methods were found to fail one way or another. 8 All the uncertainties
and controversies mentioned above suggest that
some popular theoretical methods are still unable to work
equally well for conformation studies of various molecular
systems. Therefore, a convincing conclusion for the conformation
studies often requires systematical benchmarking
comparisons.
In this study, the torsional PES of sec-butyl was theoretically
investigated. The available experimental data were
mainly from hyperfine coupling constant (HFCC) study of
its muoniated sec-butyl formed from muonium addition to 2butene
isomers. 9,10 As a short-lived, light isotope of hydrogen,
the muonium, which is composed of a positive muon
(antimuon) and an electron, is a good probing atom for the
study of various systems. 11
Before any comparison is made, it is important to point out
that all sec-butyl isotopomers share the same PES within the
Born–Oppenheimer (BO) framework, since the electronic
Hamiltonian only parametrically depends on nuclear posi-
tions and has nothing to do with nuclear masses. Thus, a
conformational study within the BO approximation is equally
applicable for both regular (unmuoniated) and muoniated secbutyl
radicals (from muonium addition to 2-butene).
Based on the analysis of muon spin resonance (mSR) data,
the muoniated sec-butyl radical formed from cis-2-butene
has a large 0 K HFCC value on muon and this value is different
from the muon HFCC from muonium addition to
trans-2-butene. Further temperature-dependent HFCC study
also supported the existence of two distinct muoniated secbutyl
radicals, both with large values of HFCCs on muon.
However, calculations using the popular hybrid B3LYP-
DFT Hamiltonian, normally believed to be well suited to
predict HFCCs of small organic molecules, 12 only located
two equilibrium structures on the ground-state torsional
PES: 9,10 one corresponding to the radical formed from muonium
addition to trans-2-butene and the other, a gauche conformation
with HFCC on muon only about one-third of the
experimental value. The absence of the experimentally observed
muoniated sec-butyl formed from cis-2-butene from
B3LYP calculations prompted us to carry out extensive theoretical
studies of the torsional PES of the sec-butyl radical.
Besides providing static conformations, the PES also offers
valuable information for temperature-dependent HFCC studies.
The dynamic (vibrational) and thermodynamic corrections
obtained based on the PES can be used to calculate the
temperature-dependent behaviour of HFCCs by solving torsional
Schrödinger equations. Something worrisome is that
the non-BO effect may in turn be significant in a molecule
containing muon, which may weaken the validity of our
study based on the BO approximation. However, a study of
the reaction rate of Mu + H 2 indicates that the non-BO effect
only changes the reaction barrier height by 3.8%. 13 Therefore,
we can expect much less non-BO effect in the muoniated
sec-butyl radicals in the current study.
To our knowledge, some other experimental results from
the homolysis of butyl radicals are also available. Thermodynamic
and kinetic studies of the decomposition of the sec-butyl
radical into propene and the methyl radical were performed, 14
but detailed conformational information of the sec-butyl
radical could not be inferred from the homolysis study, since
the experiment carried out at high temperature averaged out
the behaviours of all distinct conformations. Therefore, theoretical
calculations have to be done to illuminate possible
conformations of the sec-butyl radical. Thus far, the first ab
initio study on sec-butyl conformations can be traced back to
the Hartree–Fock (HF) calculations in the 1980s, 15 in which
the dynamic correlation effect was not taken into account.
Therefore, a systematical study of the sec-butyl radical with
various correlated Hamiltonians and different basis sets is
much needed.
Computational details
All calculations in this study were carried out using the Gaussian
03 package within the BO framework. 16 Different firstprinciples
electronic Hamiltonians, including spin-unrestricted
HF, MP2, CCSD, CISD, QCISD, CCSD(T), and a variety
of Kohn–Sham DFT methods were utilized. Wave functions
were expanded in terms of atomic-centred one-electron
basis functions including Pople-type basis sets (6–31g(d),
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Chen and Wang 1471
Fig. 1. Hamiltonian and basis-set effects on the potential energy surface of the sec-butyl radical. (a)–(d) Hamiltonian effects using the ccpVDZ
basis set. (e) and (f) Basis-set effects based on B3LYP and MP2 Hamiltonians, respectively.
Energy (kcal/mol)
Energy (kcal/mol)
Energy (kcal/mol)
2.5
2
1.5
1
0.5
TS0
cis
TS1
gauche
trans
0
0 30 60 90 120 150 180
2.5
2
1.5
TS0
Torsional angle
CCSD
CISD
MP2
B3LYP
HF
TS2
TS180
( a ) WFT and B3LYP
( b)
DFT (no cis)
1
(TS1)
(cis)
TS2
0.5
0
gauche
TS180
trans
0 30 60 90 120 150 180
2.5
2
1.5
1
0.5
Torsional angle
( c)
DFT (flat)
BB95
PBEPBE
PW91B95
PW91PW91
6-311++g(3df,2pd)
6-311++g(d,p)/ultrafine grid
6-311++g(d,p)
6-31g(d,p)
AUG-cc-pVTZ
cc-pVTZ
AUG-cc-pVDZ
cc-pVDZ/ultrafine grid
cc-pVDZ
0
0 30 60 90 120 150 180
Torsional angle
( e)
B3LYP
6–311++g(d,p), and 6–311++g(3df,2pd)), 17,18 and Dunning’s
correlation-consistent basis sets (cc-pVDZ, cc-pVTZ, AUGcc-pVDZ,
and AUG-cc-pVTZ). 19 For the DFT calculations,
two-electron integrals were numerically integrated with the
default grid quality in Gaussian 03 (75 radical shells and
302 angular points per shell) unless otherwise specified.
Geometry optimization was performed using the z-matrix
coordinate system. Completely relaxed torsional PESs of the
Energy (kcal/mol)
Energy (kcal/mol)
2.5
2
1.5
1
0.5
TS0
cis
TS1
gauche
TS2
trans
0
0 30 60 90 120 150 180
2.5
2
1.5
1
0.5
Torsional angle
( d)
DFT (cis)
MPwB1K
MPw1K
MPw1B95
BHandH
PBE0
SVWN
6-311++g**
6-311+g**
6-311g**
6-31g**
AUG-cc-pVTZ
cc-pVTZ
AUG-cc-pVDZ
cc-pVDZ
TS180
0
0 30 60 90 120 150 180
Torsional angle
()MP2 f
sec-butyl radical as a function of the dihedral angle (q) of the
four backbone carbon atoms were plotted using selected combinations
of electronic Hamiltonians and basis sets (Fig. 1).
The dihedral angle was scanned from 0° to 180° with a step
size of 5°, while all the degrees of freedom other than the
backbone dihedral angle were allowed to be fully optimized.
The PES profile of q in the range from 180° to 360° is essentially
the mirror image of the profile from 0° to 180°, if
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1472 Can. J. Chem. Vol. 89, 2011
certain symmetry constraints are applied at proper points.
However, for the simplicity and uniformity of the calculations,
the symmetry of the molecule was subject to no constraint
throughout unless otherwise noted.
In addition, the optimized conformations were calculated at
both the MP2 and B3LYP levels to obtain the diagonalized
Hessian matrices to identify the nature of the stationary structures
and to include the zero-point vibrational corrections.
Because the harmonic approximation is usually inappropriate
for the shallow minimum of the cis-structure, anharmonic vibrational
calculations were also carried out at the MP2 level
of theory to obtain the anharmonic zero-point corrections.
In this study, single-reference spin-unrestricted Hamiltonians
were used. Spin contamination, believed to be responsible
for the failure of UMP2 in determining the reaction
barrier of radical formation reactions, 20,21 was found to be
very small for all the spin-unrestricted Hamiltonians in the
current study. This is not surprising because it is well established
that the simple alkyl radical has a localized unpaired
electron at the radical centre. As a result, a single-reference
method should be adequate to treat alkyl radicals well. Thus,
spin-unrestricted single-reference methods were utilized in
this study.
Results and discussion
The torsional PES profiles calculated using different Hamiltonians
and basis sets are shown in Fig. 1, with their lowest
points as the common reference zero point to simplify the
comparison. The cc-pVDZ basis set was utilized in
Figs. 1a–1d because geometry optimization is usually carried
out with a relatively small basis set and cc-pVDZ was found
to be adequate for this purpose. 3
In Fig. 1a, the torsional energy curve obtained from the
HF calculation lies above all other potential energy profiles.
The PES profiles from the CCSD, B3LYP, and MP2 calculations
lie at the bottom across each other, whereas the potential
energy curve obtained from the CISD calculation is
situated in the middle. The PES calculated at the B3LYP
level drops at a slower rate with respect to the dihedral angle
increment when compared with all the WFT methods. The
B3LYP calculation predicts two stable conformations when
the dihedral angle q is less than 180°, whereas all the WFT
methods predict three stable conformations. The most stable
conformation is uniformly predicted to be a trans-like structure
at q ≈ 170° (conformation I in Fig. 2f). A stable gauche
conformation (structure II in Fig. 2d) is also universally predicted
at q ≈ 80°. This gauche local minimum is ~0.3–
0.6 kcal/mol (1 cal = 4.184 J) energetically higher than the
global minimum, depending on the Hamiltonian used.
The existence of a third cis-like conformation (structure III
in Fig. 2b) is worth some elaboration. All the WFT methods
predict the existence of a stable structure on a relatively flat
segment of the PES curves around q ≈ 50°, whereas selected
B3LYP methods predict an accelerated energy drop with increased
dihedral angle in the same range. This cis-like conformation
was earlier recognized as one of the three stable
conformations in the very first theoretical conformation study
of the sec-butyl radical by Chen et al. 15 The correlation effects,
which are important to describe dispersion interactions
for a correct prediction of stable conformations, was not in-
cluded by the HF calculations performed by Chen et al. 15
though. Thus, methods taking account of correlation effects
must be employed here for a reliable conformation study.
Both DFT (B3LYP) and high-level post-HF WFT methods
can incorporate correlation effects, but predict different torsional
profiles for the sec-butyl radical in this study. As verified
by a mSR experiment, the cis-like conformation does
exist and carries a large value of HFCC on the b-muon. 9,10
In this regard, WFT methods outperform B3LYP-DFT methods
decisively.
When q = 0°, the HF calculation predicts that the barrier
TS0 (Fig. 2a), the global maximum and the transition state
connecting the cis-conformation with its mirror image, is
2.3 kcal/mol higher (less stable) than the global mininum,
the trans-like conformation. This indicates that the HF calculation
predicts stronger bulky repulsion (by about 0.4 kcal/mol)
between the two terminal methyl groups of the sec-butyl
radical than any other correlated methods owing to the neglect
of the correlation effects. The barrier separating conformations
II and III, TS1 (Fig. 2c), is located around q ≈
55° using correlated WFT methods and is at q ≈ 60° using
the HF Hamiltonian. This TS1 barrier is very low energetically
with respect to conformation III and can be easily
overcome. The height of the energy barrier between conformations
I and II, TS2 (Fig. 2e), in comparison to conformation
II is about 0.5 kcal/mol from WFT calculations,
whereas it drops to as low as ~0.1 kcal/mol using the
B3LYP method. At q ≈ 180°, the actual torsional barrier
between conformation I and its mirror image was not obtained
directly from the PES curve because the symmetry
was not constrained during the relaxed potential energy
scan. Instead, the C s symmetry was used to help acquire
the torsional barrier at q ≈ 180°. Based on separate calculations,
this TS180 torsional barrier is 0.13 kcal/mol at the
B3LYP/cc-pVDZ level and 0.29 kcal/mol at MP2/cc-pVDZ.
For the sake of simplicity, no further effort was devoted to
obtain this barrier height using other methods.
Besides the comparison between WFT methods and the
representative B3LYP Hamiltonian, DFT methods with various
functionals were also examined using the cc-pVDZ basis
set. Based on the shape of the calculated potential energy
profiles between q = 40° and 60°, where the presence of conformation
III is of interest, the functionals are classified into
three different categories (Figs. 1b–1d).
Figure 1b shows the functionals that predict no minimum
within the range of interest. This group comprises some popular
hybrid or pure GGA functionals. The PES profile obtained
from the VSXC calculation (inset of Fig. 1b) is
erratically different from other functionals. The VSXC functional
fails to describe the potential energy needed to approach
the energy maximum at q = 0° with the big overall
steric repulsion between the terminal methyl groups. The
other functionals in this group yield very similar potential energy
profiles with only two stable conformations (I and II).
At any particular value of the dihedral angle, the energy difference
between any two functionals in this group (VSXC
excluded) is within 0.2 kcal/mol.
In Fig. 1c, the second group of functionals predict relative
flat features around q = 60° before the accelerated energy
drop to conformation II. Usually, geometry optimization has
a hard time to converge to a stationary point if the potential
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Chen and Wang 1473
Fig. 2. Optimized stationary structures on the torsional potential energy surface of the sec-butyl radical.
gradient is very small in a case like this. Thus, further calculations
to verify the existence of a minimum might not be
successful. Note that some of these functionals contain components
like PW X91 or B C95 that were designed for systems
characterized by weak interactions, 22–24 which may partly
compensate for the weakness of the DFT models in Fig. 1b
so that a flat segment around q = 55° is predicted.
In Fig. 1d, calculations using the third group of functionals
predict a shallow potential well lying on the torsional PES
around q = 50°, similar to the PES profiles obtained from
MP2 and other WFT calculations. However, it is well-known
that the SVWN functional, as with any other LDA-type functionals,
should not be trusted unless the electron density
undergoes only a small variation in space. The isolated molecular
systems usually exhibit big density gradients especially
around the nuclei and their electronic structure cannot
be well predicted by LDA calculations. Just as in this case,
the SVWN calculation predicts a distinct PES curve, quite
different from its peers. This indicates that SVWN fails to
capture the nonlocal exchange and correlation effects. Different
from what it is usually referred to, BHandH, comprising
50% exact exchange and 50% LDA-level exchange and LYP
correlation functionals as implemented in Gaussian 03, 16 is
also inappropriate for the conformation study of the sec-butyl
radical because it produces a much deeper potential well
about conformation III. 25 The calculation using the PBE0 (or
PBE1PBE) exchange-correlation functional, a hybrid functional
with about 25% exact exchange, predicts a potential
energy curve with a shallow potential well around q = 50°,
in contrast to the flat surface predicted by the pure PBEPBE
calculation. The other three functionals in this group combine
mPW (modified PW X91) exchange with exact exchange and
PW C91 or meta-B C95 correlation functionals that were designed
for a better long-range behaviour. PBE0 and MPW1K
calculations produce similar potential energy curves to those
from correlated WFT calculations. MPW1B95 and
mPWB1K, on the other hand, predict low-lying curves in the
range of q =0°–60°, a clear indication of the underestimated
repulsion of the terminal methyl groups compared with other
Hamiltonians.
Based on the PES curves of each exchange-correlation
functional combination, it is evident that incorporating the
exact exchange is crucial for locating the cis-like conformation
III. This implies that the nonlocal exchange-correlation
effects may play an important role in this conformation study.
At the same time, correlation functionals should also be carefully
chosen, since mPW3LYP in group one, using a similar
exchange-correlation functional as for mPW1B95, predicts a
PES profile without any sign of the cis conformation III.
The basis-set effect was examined using both B3LYP and
MP2 Hamiltonians. Based on some early studies, 4,5 the quality
of basis functions can be more important than the form of
the Hamiltonians in conformational analysis. The cc-pVDZ
and 6–31g(d) basis sets were believed to be too small to include
correct dispersion interaction, which lead to overstabilization
of the coplanar conformer of biphenyl. 4,5 In a theoretical
study of radicals, calculations using the Pople-type basis
sets with various Hamiltonians approach converged results
only after the size of the basis set went beyond 6–311g(d). 26
To visualize the basis-set effect, the torsional PES profiles
of the sec-butyl radical at the B3LYP and MP2 levels with
different basis sets are plotted in Figs. 1e and 1f, respectively.
Because of its single determinant nature, the Kohn–Sham
DFT method is usually less sensitive to the variation of the
basis set than multideterminant post-HF methods. In Fig. 1e,
the PES profiles appear to be insensitive to the selection of
the basis sets when their sizes grow bigger than cc-pVDZ or
6–31g(d,p), both of which predict the appearance of a small
cusp at q ≈ 55°. Calculations with larger basis sets predict
smooth and high-lying potential energy curves compared with
those with cc-pVDZ or 6–31g(d,p). The energy difference between
calculations using any two basis sets is no more than
0.2 kcal/mol at any given dihedral angle. The convergence of
basis sets is quickly achieved beyond the 6–311++g(d,p)
basis set. With the size of the basis sets increased within
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1474 Can. J. Chem. Vol. 89, 2011
Table 1. Relative energies (in kcal/mol; 1 cal = 4.184 J) and some geometric parameters of the stationary structures of the sec-butyl radical calculated at MP2 and B3LYP levels using
the cc-pVDZ basis set.
Structure Hamiltonian C1–C2 (Å) C2–C3 (Å) C3–C4 (Å) C2–H4 (Å) C1–C2–C3 (°) q (°) Eele (kcal/mol) EZPC (kcal/mol) EZPC,Mu (kcal/mol)
TS0 MP2 1.499 1.505 1.532 1.094 123.7 0.0 1.788 1.547 (1.772) 1.719 (1.906)
B3LYP 1.493 1.500 1.532 1.094 124.4 0.0 1.740 1.694 1.891
III (cis) MP2 1.500 1.503 1.535 1.095 120.9 47.8 0.683 0.814 (0.711) 0.819 (0.735)
B3LYP NA NA NA NA NA NA NA NA NA
TS1 MP2 1.499 1.501 1.539 1.093 120.6 55.2 0.779 0.476 (0.607) 0.572 (0.641)
B3LYP NA NA NA NA NA NA NA NA NA
II (gauche) MP2 1.499 1.501 1.543 1.097 120.4 74.1 0.312 0.512 (0.486) 0.904 (0.854)
B3LYP 1.492 1.497 1.546 1.096 122.0 84.6 0.553 0.788 1.308
TS2 MP2 1.497 1.500 1.539 1.095 121.6 116.7 0.841 0.509 (0.918) 0.719 (1.121)
B3LYP 1.492 1.497 1.542 1.096 122.6 114.0 0.682 0.632 0.897
I (trans) MP2 1.498 1.500 1.532 1.097 120.4 169.0 0.000 0.000 (0.000) 0.000 (0.000)
B3LYP 1.492 1.495 1.532 1.096 121.5 167.9 0.000 0.000 0.000
TS180 MP2 1.497 1.498 1.531 1.094 120.4 180.0 0.299 –0.293 (–0.176) –0.157 (–0.064)
B3LYP 1.491 1.494 1.531 1.094 121.1 180.0 0.126 –0.283 –0.121
Note: Bond lengths are in Å. Bond angles and dihedral angles (q) are in degrees. The relative energies are measured from the global minimal conformation III. The harmonic vibrational zero-pointcorrected
energy values are shown in the last two columns for hydrogen and muoniated isotopomers, respectively. For the MP2 calculation results, the anharmonic vibrational zero-point-corrected energies are
shown in parentheses. NA, not available; Eele, relative energy.
the Pople basis-set family, the PES curve moves upwards
without exception. However, the basis-set effect within the
Dunning family of basis sets is complicated. Adding diffuse
functions to the cc-pVDZ or cc-pVTZ basis sets greatly
raises the relative energies of the gauche conformation (II)
and the TS2 barrier separating conformations I and II. On
the other hand, incrementing splitting valence functions or
polarization functions to the basis set results in a moderate
relative energy rise (to more unstable) of conformation II
and TS2, but a dramatic energy increase of TS0. This implies
that, when q is between 70° and 130°, long-range interactions
may dominate the PES where diffuse basis
functions play an important role. Meanwhile, adding splitting
valence and polarization functions noticeably influences
the PES curves at small torsion angles.
Other than the basis-set effects, the quality of numeric integration
also affects the accuracy of a DFT calculation. A
recent theoretical study of the sec-butyl cation showed that
the quality of numerical integration of a DFT study that uses
grid summation can be important for an accurate prediction
of the equilibrium geometries. 27 Thus, an ultra-fine grid of
higher quality consisting of 99 radical shells and 590 angular
points per shell was also tested, but no significant difference
to the default grid quality was found for the system we
studied here.
Unlike the Kohn–Sham DFT and HF methods, the MP2
method, as with many other correlated WFT methods using
excited configurations, is normally sensitive to the size of
the basis set employed because correlated WFT methods utilize
virtual orbitals that are less optimized during the SCF
procedure. Thus, in most cases, MP2 calculations require a
large basis set to achieve a certain level of accuracy. A study
of biphenyl torsional PES indicated that the unbalanced treatment
of repulsive and attractive dispersion interactions occurred
if only a small Dunning correlation consisted basis
set was used. 4,5 In the present study, except for those using
small basis sets like cc-pVDZ and 6–31g(d,p) with which
the repulsion between the terminal methyl groups is overestimated,
the PES profiles obtained from other Pople basis sets
lie below the potential energy profiles obtained from the
Dunning correlation-consisted basis sets. The congregations
of the PES profiles within each basis-set family suggest a
fast convergence within each basis-set family. On the other
hand, some significant difference persists between the two
sets of PES curves from the two basis-set families, indicating
that either or both of these basis-set families are still far away
from the complete basis-set limit.
To draw a convincing conclusion, calculations using more
accurate Hamiltonians with larger basis sets are always
desired. Limited by our computer resources and the complexity
of the open-shell systems involved, only single-point
CCSD(T)/6–311++g(d,p) calculations using MP2/cc-pVTZoptimized
geometries were carried out for benchmarking purposes.
In the end, the resulting “true” torsional PES profile
calculated from CCSD(T) very closely resembles the MP2
one, which lends confidence to the results from the MP2 calculations.
As occurs here, disagreement between the DFT and WFT
methods also took place in earlier studies. Some good agreements
to the experimental data from DFT calculations were
attributed to the parameterization with which the functionals
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Chen and Wang 1475
were developed. DFT methods were reported to be unsuitable
for the conformation study of 2,2′-bifuran for missing a shallow
gauche conformation on the torsional PES curve. 8 It was
also noted that MP2 and DFT gave very similar geometries,
whereas DFT failed to describe the energetics of the torsional
potential of biisothianaphthene. 6 In addition, many popular
density functionals are believed inappropriate to describe
noncovalent interactions and are difficult to improve systematically.
In contrast to DFT methods, correlated WFT methods
are in principle controllably adequate and, although with
much higher computational cost, can be progressively improved.
High-level correlated post-HF methods can then be
used as a benchmark to justify low-level calculations. Alternatively,
comparison between theoretical predictions and
experimental data provides another way to validate the theories.
Up to today, one of the most convincing ways to tell
whether a theoretical method can be trusted on a specific system
is to make a comparison with available experimental
data. In our current study, all WFT methods predict the existance
of the cis-like conformation III irrespective of the selection
of the basis sets. A comparison with available
experiments has confirmed the existence of the cis-like conformation
III by low-temperature HFCC measurements. 9,10
Thus, the WFT methods and several DFT methods in group
III (Fig. 1d) outperform other theoretical methods by being
more in line with experimental results.
The stable geometries of the sec-butyl radical were optimized
at both B3LYP and MP2 levels with the cc-pVDZ
basis set. The optimized geometric parameters of those stable
conformations are listed in Table 1. These two employed
methods predict very similar bond lengths and bond angles
for the same stationary structure except that the dihedral angle
of the gauche conformation calculated at the B3LYP level
differs by about 10° from the MP2 calculation. The C2–C3
bond length descends almost monotonically as the torsion angle
increases from 0° to 180°. This indicates that the C2–C3
bond grows stronger from the cis-conformation to the transconformation
because of decreased repulsion between the
two terminal methyl groups. The hydrogen–hydrogen repulsion
and hyperconjugation involving C2 and C3 exert only
minor corrections to the PES profiles because the energy barrier
from conformation III to conformation II is very small.
Based on the optimized structures, the C1–C2 bond distance
reaches maximum in TS1. This implies that the internal rotation
about the C1–C2 bond plays an important role in forming
the rotational barrier between conformations II and III.
In an early theoretical study by Chen et al., 15 the shallow
potential well embracing conformation III was merely regarded
as a shoulder extension to the electronically more stable
conformation II. If only the static electronic potential
energy is considered, muoniated sec-butyl formed from mounium
addition to cis-2-butene will not exist or can be easily
turned into the gauche conformation II by an extremely small
fluctuation (e.g., through vibration) to overcome the tiny barrier
TS1, contradicting the experimental observation that the
muon HFCC of this structure remained visibly strong over a
wide temperature range. 9,10 There is no doubt that other effects
stabilizing this molecular species must be taken into
consideration.
In addition to the electronic energy, the zero-point vibrational
correction also affects the relative energy difference
between two structures in reality. More complete thermodynamic
correction must be considered if the experiment is
conducted at a higher temperature. To make a sound comparison
to the experimental data at low temperatures, both harmonic
and anharmonic vibrational corrections of the stable
structures and the corresponding thermal corrections were
calculated and are summarized in Table 1. It turns out that
the zero-point corrected energy of the muoniated isotopomer
of conformation III becomes lower than that of the muoniated
conformation II, opposite to the prediction based on
electronic energies alone. Thus, the large value of the muon
HFCC of the muoniated radical formed from cis-2-butene
can be definitely assigned after including zero-point correction.
Another change in relative stability owing to the inclusion
of the vibrational correction is the transition state TS180
connecting conformation I and its mirror image: the transitionstate
structure becomes the lowest point after the vibrational
correction. Similar effects of the zero-point correction were
also reported in other studies but without detailed discussion.
7
Conclusion
First-principles calculations using spin-unrestricted HF,
MP2, CCSD, and a variety of DFT Hamiltonians with different
basis sets were employed to investigate the stable conformations
of sec-butyl. Several DFT methods only predict two
equilibrium structures with backbone dihedral angles of about
85° and 170°, whereas all WFT methods and some DFT
methods yield an extra cis-like stable conformation with a dihedral
angle of about 55°, followed by a low torsional barrier
towards the gauche conformation, in agreement with existing
experimental results. Inclusion of exact exchange and longrange
behaviour-corrected correlation functionals is crucial
to locate this elusive cis-like conformation for DFT methods.
The selection of the basis sets using the B3LYP or the MP2
Hamiltonian does not affect the number of stable conformations
found but only modifies the shape of the PES profiles.
Noting the stable conformation of any molecular species in
reality is not simply solely determined by electronic energies,
calculations including the zero-point vibrational effect were
further carried out. The zero-point corrected energy shows
that the cis-like muoniated conformer is more stable than the
gauche one, in line with the available experiment data.
Acknowledgement
Financial support for this project was provided by grants
from the Natural Sciences and Engineering Research Council
of Canada (NSERC).
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