MolPhys_107_2537_200..

Molecular Physics
Vol. 107, Nos. 23–24, 10–20 December 2009, 2537–2546
Assessment of theoretical methods for the determination of the mechanochemical
strength of covalent bonds
Maria Francesca Iozzi*, Trygve Helgaker and Einar Uggerud
The Center for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo,
P.O. Box 1033, Blindern, N-0315 Oslo, Norway
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(Received 1 September 2009; final version received 8 October 2009)
The performance of some commonly used quantum-chemical methods in accurately and reliably describing the
influence of applying an external mechanical force has been investigated for a set of small molecules. By applying
coupled-cluster CCSD(T) theory in an extended basis set as benchmark, all methods tested provide a good
qualitative description of the physical process, although the quantitative agreement varies considerably. Hartree–
Fock (HF) theory overestimates both the values of the bond-breaking point and the rupture force, typically by
20–30%. The same applies to density-functional theory (DFT) based on the local density approximation (LDA).
By introducing the generalized gradient approximation (GGA) in the form of the BLYP and PBE functionals,
only a slight overestimation is observed. Moreover, these pure DFT methods perform better than the hybrid
B3LYP and CAM-B3LYP methods. The excellent agreement observed between the CCSD(T) method and
multiconfigurational methods for bond distances significantly beyond the bond-breaking point shows that the
essence of mechanical bond breaking is captured by single-reference-based methods. Comparisons of accurate
numerical bond-dissociation curves with simple analytical forms show that Morse-type curves provide useful
approximate bond-breaking points and rupture forces, accurate to within 10%. By contrast, polynomial curves
are much less useful. The outcome of kinetic calculations to estimate the dissociation probability as a function of
the applied force depends strongly on the description of the potential-energy curve. The most probable rupture
forces calculated by numerical integration appear to be significantly more accurate than those obtained from
simple analytical expressions based on fitted Morse potentials.
Keywords: mechanochemistry; atomic force microscopy; bond-breaking point; rupture force; Morse potential
1. Introduction
There are essentially five different methods for breaking covalent chemical bonds: photolysis, thermolysis,
electrolysis, chemical reaction and mechanical strain.
The latter method, in which a bond is broken as the
result of applying a stretching force, has received much
attention since the introduction of the atomic-force
microscope (AFM) and the force-clamp technique
[1–7]. By anchoring a polymer molecule at one end to
the base of the AFM and at the other end to the
cantilever tip while increasing the base–tip separation
in a controlled fashion, it is possible to monitor
continuously the force acting on the molecule. From
the recorded force-extension profile, it is possible to
follow the stretching of the molecule all the way from
its initial equilibrium state via intermediate states
(resulting from the gradual unveiling of the polymer’s
tertiary structure) up to a transition state, where the
mechanically weakest covalent bond of the backbone
breaks. Mechanochemical bond activation achieved in
*Corresponding author. Email: [email protected]
ISSN 0026–8976 print/ISSN 1362–3028 online
! 2009 Taylor & Francis
DOI: 10.1080/00268970903401041
http://www.informaworld.com
this way can also be combined with other methods of
activation—for example, by changing the temperature
or the chemical environment [8–10].
In their pioneering work, Granbois et al. [1]
determined the force-extension profile of a polysaccharide chain covalently attached to an AFM.
Unfortunately, their experimental data did not allow
for a unique identification of the bond that eventually
breaks. However, these authors realized and demonstrated by quantum-chemical modeling that there is
no simple connection between the bond-dissociation
energy and the mechanochemical strength of a bond.
To assist the interpretation of experimental data, the
bond-breaking probability of each bond in the polymer
must be determined theoretically. Given that bond
breaking is a chemical reaction, an Arrhenius-type
model can be applied to calculate the bond-breaking
rate and the bond-breaking probability distribution,
provided the force-modified potential-energy surface is
known [11–14].
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2538
M.F. Iozzi et al.
An external, constant force that stretches a diatomic
molecule along its axis, modifies the Born–
Oppenheimer potential-energy surface in a well-defined
manner. However, for a polyatomic system, the determination of the force-transformed potential-energy
surface is not a trivial task, since it involves a
multidimensional reaction path. In 2000, Beyer proposed the COGEF (Constrained Geometry simulate
External Force) protocol [15] for determining the
minimum-energy path leading to bond breaking on
the force-transformed potential-energy surface
[4,16,17]. Very recently, a more rigorous but computationally demanding approach was proposed by
Ribas-Arino et al. [18] and Ong et al. [19] for the full
characterization of potential-energy surfaces.
In determining mechanochemical strength and rupture probabilities, two features of the force-modified
potential-energy curve are particularly important—
namely, the breaking-point distance Rbp, where the
second derivative changes sign (the inflection point),
and the rupture force, fmax, which is the negative first
derivative at the inflection point (i.e. the largest value
of the force along the potential-energy curve towards
dissociation). We note that, even though the mechanochemical characteristics of various molecular systems
have been determined on the COGEF potentials
computed using density-functional theory (DFT), the
nature of the bonding interactions at the breaking point,
and the ability of DFT to describe these interactions,
have not yet been clarified.
An accurate theoretical description of bond breaking, involving a wide range of interatomic distances,
requires a consistent treatment of electron correlation.
In particular, any one-determinant-based representation of the wave function, such as those provided
by the Hartree–Fock (HF) and coupled-cluster (CC)
models, fails to describe the static correlation arising
from the multi-configurational character of the dissociating system at large separations. For example,
while coupled-cluster singles-doubles-perturbativetriples (CCSD(T)) theory provides a good description
of dynamic correlation, static correlation is poorly
described, leading to a poor description of bond
breaking. A proper description of static correlation
is provided by complete-active-space (CAS) multiconfigurational self-consistent field (MCSCF) theory.
This approach by itself does not account for much of
the dynamic correlation, which may be included
perturbatively, using either second-order CAS perturbation theory (CASPT2) or second-order n-electron
valence state perturbation theory (NEVPT2).
However, these MCSCF-based models cannot easily
be applied to very large systems. Finally, concerning
DFT, we note that Kohn–Sham theory dissociates
correctly in principle; in practice, Kohn–Sham theory
often provides a poor description of bond breaking
because of inadequacies in the approximations to the
exchange-correlation functional.
From these considerations, several questions arise
concerning the quantum-chemical study of mechanochemical strength. The first question concerns the
ability of the different models to describe correctly the
essential part of the bond-breaking potential—namely,
the bond-breaking point and the rupture force. In the
present paper, we perform a careful benchmarking of
different methods with respect to the reliable calculation of these quantities—in particular, we examine the
performance of HF theory and DFT, suitable for large
molecular systems such as polymers. Secondly, we
consider whether it is possible to predict the breaking
point and rupture force from an order expansion of the
potential-energy surface about the equilibrium geometry or from Morse potentials fitted in different ways.
Besides providing insight into the bond-rupture process,
our study prescribes simple and practical approaches
that can be applied to kinetic modeling of the process.
Part of the present study is concerned with
diatomic molecules, including hydrides of variable
bond polarity and homonuclear molecules containing
single, double and triple bonds. We have also studied
some larger molecular models, suitable for describing
important features of biological polymer stretching,
relevant to experimental work.
2. Computational details
The potential-energy curves of a set of diatomic
molecules (LiH, BH, HF, Li2, N2 and O2) were
determined by performing single-point energy calculations on the systems at a large number of internuclear
distances, followed by a 10-point piecewise polynomial
interpolation using Le Roy’s LEVEL 8.0 code [20]. To
ensure a correct fitting in the border regions of the
potential, appropriate extrapolation procedures were
invoked: at short distances (in the repulsive region), the
potential is extrapolated with an exponential function;
at large distances (in the dissociation region), the potential is extrapolated as a sum of inverse-power terms.
The potential-energy curves were computed at the
HF, CAS, NEVPT2 and CCSD(T) levels of theory. In
addition, Kohn–Sham theory with different approximate exchange–correlation functionals was used: the
local density approximation (LDA), the generalized
gradient approximation (GGA) using the BLYP and
PBE functionals, the B3LYP hybrid functional and the
CAM-B3LYP Coulomb-attenuated hybrid functional.
In all cases, the aug-cc-pVQZ basis set was used.
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Molecular Physics
In the CAS calculations, the numbers of active
electrons and active orbitals, respectively, were (4, 6)
for LiH, (6, 12) for BH, (8, 8) for HF, (6, 10) for Li2,
(6, 12) for N2, and (8,10) for O2.
For the polyatomic molecules, one-dimensional
potential-energy curves were computed for molecules
containing C–C, N–C, N–CO, S–S, and Si–C bonds.
For each X–Y bond, two molecules were considered—
namely, RX–YR with R¼H, CH3. Starting from the
optimized G3 (MP2/6-31G*) geometry [21], the potential curve was computed in a point-wise fashion by
varying the X–Y bond length by a rigid translation
along the bond axis, keeping the remaining geometrical
parameters fixed. In addition to applying this rigidbond-stretching (RBS) procedure, stretching was
simulated with the COGEF procedure, increasing the
distance between the terminal hydrogen atoms while
optimizing the remaining degrees of freedom. The final
potential-energy curves were obtained by the same
extrapolation procedure as for the diatomics, using the
same electronic-structure models but the smaller basis
sets: 6-311G* for the smaller models (R¼H) and
6-31G* for the larger models (R¼CH3). For HSSH, an
extra set of polarization functions was added to the
hydrogen atoms.
For the RX–YR molecules, zero-Kelvin bond dissociation energies (BDEs) were computed as the energy
difference between each molecule and its radical
2539
fragments, having obtained the zero-point corrected
energies at G3 geometries using scaled harmonic HF/
6-31G frequencies.
3. Diatomic molecules
In Table 1 we have listed the values of the rupture force
fmax, the breaking point Rbp, and the rupture elongation E bp ¼ 100(Rbp " Req)/Req (where Req is the equilibrium geometry) for the diatomic molecules
considered in this study. Notably, all molecules, at all
levels of theory, have rupture elongations smaller than
50%. As an illustration, Figure 1 shows the
potential-energy curve for the BH molecule, calculated
at different levels of theory. For reference, a vertical
line indicating the NEVPT2 breaking point has been
drawn at E bp ¼ 37%.
In Table 2 we have given the maximum, average
and minimum values of E bp and fmax at different levels
of theory relative to the corresponding CCSD(T)
value. As argued below, we take the CCSD(T)/
aug-cc-pVQZ values to be close to the true values for
these quantities.
3.1. CCSD(T) results
In a large basis such as aug-cc-pVQZ, the CCSD(T)
model is expected to provide an accurate description of
Table 1. Breaking point Rbp (ppm), elongation E bp (%) and rupture force fmax (nN).
HF
CASa
PT2b
(T)c
LDA
PBE
BLYP
B3LYP
CAMd
Rbp
LiH
BH
HF
Li2
N2
O2
239.1
174.4
126.6
402.3
142.1
153.5
227.7
168.4
126.4
364.2
136.1
138.4
226.9
168.4
124.6
354.8
142.8
150.5
226.3
167.7
125.3
349.7
138.6
154.6
236.5
177.2
128.9
369.2
141.0
155.9
230.7
175.8
128.4
371.7
141.8
157.4
233.9
172.7
128.6
363.6
141.2
159.1
231.0
173.2
128.0
369.8
141.1
156.6
233.5
175.1
128.8
377.0
141.8
156.7
E bp
LiH
BH
HF
Li2
N2
O2
49
43
41
44
33
32
41
37
37
28
24
14
44
37
36
30
30
24
44
36
37
33
26
28
48
41
38
47
29
30
41
40
38
36
28
29
40
39
38
35
28
30
45
40
39
37
29
30
47
42
40
37
30
31
fmax
LiH
BH
HF
Li2
N2
O2
2.6
5.6
13.1
0.9
36.4
22.1
2.1
4.8
11.2
0.6
24.1
11.6
2.3
4.8
10.6
0.7
23.5
12.7
2.4
4.7
10.8
0.9
23.7
13.3
2.4
4.5
10.3
0.8
25.7
14.5
2.3
4.2
10.1
0.8
24.5
13.3
2.2
4.2
9.8
0.8
23.8
12.4
2.4
4.8
10.7
0.8
27.0
14.9
2.6
5.1
11.0
1.0
28.8
16.5
a
The CAS spaces adopted for the MCSCF models: LiH¼(4e"/6o), BH¼(6e"/12o), HF¼(8e"/8o), Li2¼(6e"/10o), N2¼(6e"/12o),
O2¼(8e"/10o).
b
NEVPT2.
c
CCSD(T)
d
CAM-B3LYP.
M.F. Iozzi et al.
the electronic system at equilibrium and, in particular,
an accurate equilibrium structure [22]. On the other
hand, since the wave function is generated by excitations from an HF reference state, the quality of the
description deteriorates as the bond is stretched.
However, it has previously been observed that, for
the OH dissociation of water, the CCSD(T) model
provides an accurate potential-energy curve up to 90%
elongation, failing only at larger separations [23–26].
Therefore, since the inflection point Rbp occurs at an
elongation of less than 50%, we expect the CCSD(T)
model to provide accurate inflection points and rupture forces.
In a coupled-cluster calculation, the norm of the
vector containing the singles amplitude T1 is indicative
of the multi-reference character of the electronic
system (the T1 diagnostic) [27,28]. Systems accurately
represented by single-reference methods have small T1
values (e.g., 0.0096 for H2O at the CCSD/TZ2P level of
theory, from Ref. [28]), while large T1 values are
indicative of strong non-dynamical correlation. In
Table 3, we have listed the T1 values computed for
the diatomic molecules in this study at different bond
distances, at the CCSD/aug-cc-pVQZ level of theory.
Although the multi-reference character of molecules
increases upon bond stretching, all molecules remain
essentially single-configurational at Rbp. As further
evidence of the single-configurational nature of the
wave function at Rbp, we have in Figure 2 plotted CAS
natural-orbital occupation numbers for the dissociation of Li2, N2, and O2. Upon stretching, the plotted
occupation number decreases from two to one, as the
bonding orbital loses an electron to its antibonding
partner. However, at Rbp, the occupation is reduced by
only 13%, 9% and 4% for the three molecules,
confirming the single-configurational nature of the
electronic state at this geometry. In the following, we
shall therefore use the mechanochemical properties
obtained from the CCSD(T) model as our reference for
evaluating the performance of the other models.
BH,aug-cc-pVQZ
–25.1
–25.15
En. (a.u)
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2540
3.2. Comparison with CCSD(T) benchmark results
BLYP
B3LYP
CAM-B3LYP
CCSD(T)
CAS
NEVPT2
–25.2
–25.25
–25.3
100
250
200
B-H (ppm)
150
300
Figure 1. The potential-energy curves computed with different methods. The dotted vertical line corresponds to the Rbp
computed with NEVPT2.
From Table 2, we first note that the uncorrelated HF
model consistently overestimates rupture elongation
and rupture force (typically by about 20% and 30%,
respectively), in agreement with the tendency of the HF
model to overbind. The introduction of static correlation at the CASSCF level of theory overcompensates
this overbinding, giving too small rupture elongations
and forces—in the case of O2, the elongation is
underestimated by a factor of 2, see Table 1. Finally,
when dynamical correlation is introduced at the
NEVPT2 level of theory, an accurate representation
of bond rupture is achieved—indeed, the average
NEVPT2 rupture elongation and force are within one
or two percent of the CCSD(T) values, see Table 2.
Typically, NEVPT2 and CCSD(T) theories give
Table 2. Maximum, average, and minimum values of E bp and fmax relative to the CCSD(T) values in Table 1.
HF
CASa
PT2b
(T)c
LDA
PBE
BLYP
B3LYP
CAMd
E bp
Max
Ave
Min
133
119
111
103
87
50
115
99
86
100
100
100
142
114
103
111
105
93
108
104
91
112
108
102
117
112
107
fmax
Max
Ave
Min
166
134
108
104
96
87
102
98
95
100
100
100
109
102
95
103
96
89
100
93
89
114
105
99
124
113
102
a
CASSCF.
NEVPT2.
c
CCSD(T).
d
CAM-B3LYP.
b
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Molecular Physics
elongations that agree to within 1 pm and rupture
forces that agree to within 0.5 nN, the largest deviations being observed for molecules with double and
triple bonds.
Next, we compare the performance of DFT with
the reference CCSD(T) values. Because of its low
computational cost, DFT is attractive for modeling
large systems—in particular, polymers and biological
molecules of relevance to AFM experiments. From
Tables 1 and 2, we note that LDA strongly overestimates rupture elongation and (less strongly) rupture force, in agreement with the tendency of LDA
to overbind. The performance of DFT is strongly
improved at the GGA level of theory: the PBE and
BLYP exchange–correlation functionals provide only a
slight overestimation of rupture elongation (by about
5%) and a slight underestimation of rupture force
(by about 5%). At the hybrid levels of theory, both
quantities increase and are consistently overestimated—
typically by about 5% with the B3LYP model and 10%
with the CAM-B3LYP model. Comparing the BLYP
and B3LYP results, we note that the BLYP model
Table 3. The T1 diagnostic for a set of diatomic
molecules (aug-cc-pVQZ).
LiH
BH
HF
Li2
N2
O2
Req
Rbp
R1
0.0090
0.0140
0.0093
0.0208
0.0134
0.0154
0.0259
0.0183
0.0141
0.0234
0.0235
0.0245
0.4664
0.1494
0.1394
0.1429
0.0607
0.0595
2541
provides slightly better rupture elongations, whereas
the BLYP and B3LYP rupture forces bracket the true
forces, which are underestimated by the BLYP model
and overestimated by the B3LYP model. From Table 1,
we note that the performance of DFT is best for the
single-bonded hydrides and Li2. With increasing bond
order, the DFT forces become too large, the largest
deviation being around 2 nN, for O2.
3.3. Equilibrium prediction of bond-rupture
quantities
The calculation of the breaking point Rbp and the
rupture force fmax is expensive, requiring the tabulation
of the energy at a large number of bond distances
or, alternatively, the implementation of some iterative
scheme for the optimization Rbp. In the present section,
we shall inquire whether it is possible to predict Rbp
and fmax from equilibrium quantities.
We first consider a simple series expansion of
the potential about the equilibrium bond distance.
Truncating the expansion at third order in
R ¼ R " Req, we may then express the potential in the
manner
1
1
V3rd ðRÞ ¼ V0 þ k2 R2 þ k3 R3 ,
2
6
where V0 is the value of the potential at equilibrium
and k2 and k3 are the harmonic and anharmonic force
constants, respectively. The inflection point is obtained
by setting the second derivative of this expression equal
to zero, yielding
k2
,
k3
ð2Þ
k22
:
2k3
ð3Þ
R3rd
bp ¼ "
Natural orbital occupation
2
f 3rd
max ¼ "
N2
O2
Li2
1.8
1.6
1.4
1.2
1
100
200
300
400
500
600
700
ð1Þ
800
X–X (ppm)
Figure 2. Natural orbital occupation numbers for Li2, N2
and O2. Vertical lines indicate Rbp.
In Table 4 we compare these predictions of the
rupture elongation and the rupture force with the true
quantities, calculated at the NEVPT2 level of theory.
Clearly, the third-order potential Equation (1) gives
a poor description of bond rupture, underestimating
rupture elongation by a factor of 3 and rupture force
by about 40%. Although the description may be
improved by including fourth- and fifth-order terms in
the expansion, high computational cost and numerical
instabilities combine to make this approach, based on
an order expansion at equilibrium, unattractive.
Alternatively, we may model the potential-energy
curve by a Morse potential of the general form
VMorse ðRÞ ¼ V0 ½1 " expð"!RÞ'2 ,
ð4Þ
2542
M.F. Iozzi et al.
Table 4. Elongation E bp (%) and rupture force
fmax (nN): comparison of NEVPT2 results full
potential with predictions based on a third-order
expansions and two Morse potentials.
Num.a
V3rdb
VMkDc
potential, whereas the rupture force is 1.5 times larger
than that predicted by the third-order model, suggesting that the locally fitted Morse model should improve
upon the third-order model.
Indeed, from Table 4, we note that both Morse
models provide reasonably accurate descriptions of
bond rupture, typically predicting E bp and fmax to
within 10% of the true value. The systematic underestimation of both quantities is most likely related to
the fact that the Morse potential gives too strong
binding at large distances [29]. Whereas the globally
fitted Morse potential is slightly more accurate than
the globally fitted potential for BH and HF, the
opposite is true for N2 and O2. For general use, we
recommend the locally fitted Morse potential, which
can more easily be applied to polyatomic molecules,
requiring only information obtained at the molecular
equilibrium geometry.
VMkkd
E bp
BH
HF
N2
O2
37
36
30
24
16
14
12
11
35
32
23
21
32
30
25
23
fmax
BH
HF
N2
O2
4.8
10.6
23.5
12.7
2.9
6.5
15.2
8.1
4.7
10.4
20.9
11.1
4.4
9.8
22.8
12.2
a
Numerically fitted NEVPT2 potential.
Third-order potential Equation (1).
c
Globally fitted Morse potential Equation (5).
d
Locally fitted Morse potential Equation (6).
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b
where V0 determines the depth of the potential and !
its width. In particular, we here consider the following
two parameterizations of the Morse potential:
VMkD ðRÞ ¼ Deq
"
sffiffiffiffiffiffiffiffiffiffi !#2
k2
1 " exp "
,
R
2Deq
$
"
#%2
9k32
k3
R
,
VMkk ðRÞ ¼ 2 1 " exp
3k2
2k3
ð5Þ
ð6Þ
where the first (global) parameterization is designed to
reproduce the dissociation energy Deq ¼ V(1) " V(Req)
and the harmonic force constant k2 of the true
potential, whereas the second (local) parameterization
reproduces the harmonic and anharmonic force
constants k2 and k3, respectively. These two Morse
potentials give the breaking point and rupture forces
rffiffiffiffiffiffiffiffiffiffi
2Deq
MkD
ð7Þ
Rbp ¼ ln 2
,
k2
f MkD
max
rffiffiffiffiffiffiffiffiffiffiffiffi
Deq k2
¼
8
ð8Þ
and
RMkk
bp ¼ " lnð8Þ
f Mkk
max ¼ "
k2
¼ lnð8ÞR3rd
bp ,
k3
3 k22 3 3rd
¼ f :
4 k3 2 max
ð9Þ
ð10Þ
The locally fitted Morse potential Equation (6) thus
predicts a rupture elongation that is ln 8 ( 2.1 times
larger than than that predicted by the third-order
4. Polyatomic molecules
In this section, we consider the breaking of the
covalent bonds C–C, C–N, N–C(O), S–S and Si–C,
all of relevance to AFM experiments. Tables 5 and 6
contain E bp and fmax, respectively, computed at the
NEVPT2 and CCSD(T) levels of theory and by DFT
with different exchange–correlation functionals. As for
the diatomics, we observe a good agreement between
the NEVPT2 and CCSD(T) results. Likewise, the LDA
functional overestimates and the GGA functionals
underestimate the rupture force relative to the coupledcluster results. Among the DFT functionals, the best
agreement with coupled-cluster theory is obtained with
the B3LYP model, whereas the CAM-B3LYP model
gives the poorest agreement.
We observe only a weak correlation between the
computed rupture forces and BDEs, also listed in
Table 6. The dissociation energy and rupture force of
the central X–Y bond are both rather insensitive
to methyl substitutions, going from HXYH to
CH3XYCH3, with the exception of the C–N rupture
force, which decreases from 7.6 to 5.3 nN from the
primary to secondary amine, while the BDE remains
constant.
All results discussed so far have been obtained
using the RBS procedure, performing a rigid relative
translation of the two moieties along the central X–Y
bond axis. Alternatively, the rupture forces can be
estimated by the COGEF procedure, where the
molecular geometry is fully relaxed for each separation
between two selected peripheral atoms (often hydrogens), thereby mimicking the effect of an external force
pulling the molecule apart. The COGEF approach is
Molecular Physics
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Table 5. Elongation E bp (%) for RXYR molecules.
X–Y
R
PT2a
(T)b
LDA
PBE
BLYP
B3LYP
CAMc
Si–C
H
CH3
28
27
27
27
29
27
29
28
28
28
28
29
30
30
C–C
H
CH3
33
—
29
31
31
30
31
30
30
30
31
31
32
32
C–N
H
CH3
27
28
27
28
30
31
29
31
29
31
30
31
31
32
N–CO
H
CH3
33
—
27
27
28
28
28
27
28
27
28
27
29
28
S–S
H
CH3
20
18
20
18
25
24
24
23
24
23
25
24
26
25
a
NEVPT2 with the following CAS spaces: CH3CH3: 2e"/2o; CH3SiH3: 6e"/6o; CH3NH2: 8e"/8o;
CH3CH2NHCH3: 4e"/4o; NH2COH: 8e"/8o; HSSH: 14e"/10o.
b
CCSD(T).
c
CAM-B3LYP.
Table 6. Rupture force (nN) and BDE (kcal mol"1) for RXYR molecules.
X–Y
R
PT2a
(T)b
LDA
PBE
BLYP
B3LYP
CAMc
BDEd
Exp.
Si–C
H
CH3
5.2
5.1
5.3
5.1
5.4
5.2
5.0
4.8
4.7
4.5
5.2
5.0
5.6
5.4
87.01
85.63
86 ) 4 [30]
C–C
H
CH3
7.1
—
7.1
6.9
7.4
7.1
6.9
6.5
6.4
6.1
7.0
6.7
7.5
7.3
87.76
87.16
90.1 ) 0.1 [31]
87.9 ) 0.6 [31]
C–N
H
CH3
7.6
5.2
7.6
5.3
8.0
5.6
7.3
5.2
6.8
5.2
7.6
5.2
8.2
5.7
82.41
82.20
85.2 ) 0.3 [31]
N–CO
H
CH3
9.2
—
9.0
8.6
9.0
8.7
8.2
7.8
7.7
7.2
8.8
8.3
9.4
9.0
99.44
93.66
S–S
H
CH3
3.9
3.8
3.8
3.7
4.4
4.4
4.0
3.8
3.5
3.4
4.1
4.0
4.8
4.6
60.22
62.45
64.7 ) 3.0 [32]
65.2 ) 0.9 [33]
a
NEVPT2 calculations with CAS spaces: CH3CH3: 2e"/2o; CH3SiH3: 6e"/6o; CH3NH2: 8e"/8o; CH3CH2NHCH3:
4e"/4o; NH2COH: 8e"/8o; HSSH: 14e"/10o.
b
CCSD(T).
c
CAM-B3LYP.
d
G3 theory.
more expensive than the RBS approach but describes
the bond-breaking process more faithfully, providing
the minimum-energy path towards dissociation. Note
that, whereas the RBS procedure applies an external
force to a selected bonded atom pair, leaving the
molecule otherwise intact, the COGEF procedure
applies the force to an arbitrary pair of (usually
peripheral) atoms, thereby stretching the entire chain
of atoms connecting the end atoms, eventually rupturing the weakest bond. Clearly, the two approaches
offer complementary insight: local (RBS) and global
(COGEF).
A comparison of the RBS and COGEF models is
presented in Table 7, where we have listed the rupture
force of CH3X–YCH3 calculated at the unrestricted
B3LYP level of theory. We note an overall good
agreement between the RBS and COGEF results, the
rupture forces agreeing to within 5%. In general, we
would expect the COGEF rupture force to be lower
than the RBS force, as happens for the S–S system.
By contrast, for the C–N, C–C and Si–C bonds, the
COGEF force is greater than the RBS force. However,
for these three bonds, the COGEF and RBS calculations have been carried out for slightly different
M.F. Iozzi et al.
molecules (as noted in the table), invalidating such a
direct comparison of the COGEF and RBS results.
All calculations discussed above were carried out
using unrestricted B3LYP theory. For the S–S bond,
we have also carried out restricted B3LYP calculations
with the RBS model—see Table 7. As expected, the
restricted rupture forces are slightly larger than the
unrestricted ones. In passing, we note that no such
difference is observed at the more advanced CCSD(T)
level of theory. Finally, for the S–S bond, two different
basis sets are employed. Upon basis-set extension, the
rupture force increases by 0.2–0.3 nN.
Downloaded By: [University of Oslo Library] At: 13:37 19 December 2009
4.1. Kinetic model
In an AFM experiment, the force is applied in the
manner
f ðtÞ ¼ f0 þ "t,
ð11Þ
where the force-loading rate " is constant. An
Arrhenius-type rate equation may then be used to
describe the probability P that, for an applied force f,
the bond is still intact:
P0 ð f Þ ¼ "
A "E # ð f Þ=kT
e
Pð f Þ,
"
ð12Þ
where A is the Arrhenius pre-factor, E #( f ) is the
activation barrier to bond breaking subject to the
external force f, k is the Boltzmann constant, and T
the temperature. The effect of applying a constant
force along the axis of a diatomic molecule is to
increase the equilibrium bond distance and to reduce
the activation barrier to dissociation—see Figure 3.
For polyatomic molecules, Morse potentials based
on fmax and Deq have previously been used as
minimum-energy paths towards bond breaking [15].
We here treat polyatomic molecules by adding a linear
force term to the RBS potential in the manner
Vf (r) ¼ VRBS(r) " f (t)r. In this approximation, the
Table 7. COGEF and RBS potentials for CH3X–YCH3
except as noted.
X–Y
B3LYP
COGEF
unres
RBS
res
RBS
unres
S–S
6-311þþG(3df,3pd)
6-31G(d)
D95(d,p)
D95(d,p)
D95(d,p)
3.8
3.5
7.2a
6.9a
4.8a
4.2
3.9
3.9
3.7
7.1
6.6
4.9
C–N
C–C
Si–C
a
Computed on the H3C–YCH3 models, with Y ¼ NH, CH2
and SiH2.
applied force produces the same effect as in a diatomic
molecule.
In the form of transition-state theory applied by
Hanke and Kreuzer [13], the Arrhenius pre-factor is
given by A ¼ #$q*/q, where # is the attempt frequency,
$ the accommodation coefficient, and q*/q the ratio of
the internal partition function of the activated complex
to that of the unperturbed molecule.
Assuming that the stretched system is in equilibrium with the external perturbation, we set $q*/q ¼ 1.
The attempt frequency can be viewed as the stretching
frequency along the breaking coordinate—that is, for
RX–YR, the frequency of oscillation about the X–Y
equilibrium geometry. Beyer [15] used for # the
maximum optical phonon of the polymer. We here
set # equal to the fundamental vibrational frequency
of the fictitious diatomic system made up by atoms
centered on the X and Y units, obtained by solving the
one-dimensional Schrödinger equation with the
LEVEL 8.0 code [20].
Setting A equal to the fundamental frequency of
X–Y and determining E#( f ) from Vf(r), we may determine the bond-breaking probability, Pbp( f ) ¼ 1 " P( f ),
by numerical integration of Equation (12). In Figure 4
0
ð f Þ, where the maximum represents
we have plotted Pbp
the most probable rupture force, denoted by fmp. Note
that fmp differs from the rupture force fmax discussed
above.
For completeness, we compare in Table 8 the
results obtained with the fitted PT2 potential with the
predictions based on the two considered Morse-type
potentials, Equations (5) and (6). For these Morse
Energy
2544
E#
r
Figure 3. The RSB potential-energy curve (red curve) and
force-dependent potential Vf (r) (green curve) arising from
the applied linear potential V( f ) ¼ fr (blue curve). The
vertical arrow indicates the transition state (E #) of the
activated bond dissociation.
Molecular Physics
potentials, we compute fmp analytically, from the
approximate expression developed by Hanke and
Kreuzer [13]:
"
"
##
2fmp !AkT
V0
2fmp 2
exp "
1"
¼
,
ð13Þ
1"
4"
!V0
kT
!V0
S-S
C-N
Si-C
P’bp (arbitrary units)
Downloaded By: [University of Oslo Library] At: 13:37 19 December 2009
where the values of V0 and ! are obtained by comparing Equations (5) and (6) with the general expression Equation (4).
The locally and globally fitted Morse potential
provide rather different fmp values, indicating that the
analytical solution is rather sensitive to the shape of
the potential. As compared to the PT2 values, the most
probable rupture force is higher for the single-bond
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
f (nN)
Figure 4. Bond-breaking probability distributions of CH3X–
YCH3 with X–Y ¼ C–N, Si–C, S–S. In a typical force-clamp
experiment at room temperature, the force is applied at a
constant rate of 10 nN s"1.
Table 8. The most probable rupture force fmp (nN): comparison of NEVPT2 results obtained from numerical
integration of Equation (12) and from the analytical expression Equation (13) with the Morse potentials Equations (5)
and (6).
BH
HF
N2
O2
Si–C
C–N
S–S
a
Num.a
VMkDb
VMkkc
1.78
7.10
17.37
8.27
2.69
3.06
1.78
2.29
6.91
14.06
6.66
2.49
7.33
12.94
6.03
Numerical NEVPT2 potential.
Globally fitted Morse potential Equation (5).
c
Locally fitted Morse potential Equation (6).
b
2545
molecules BH and HF, while the opposite occurs in the
case of multiple-bond molecules N2 and O2.
The computed fmp can be directly compared with
experimental measurements. The maximum Si–C rupture force is 2.0 ) 0.3 nN at room temperature and a
force load of 10 nN s"1—that is, significantly lower
than our theoretical result of about fmp ¼ 2.6 nN, which
in turn is much lower than fmax ¼ 4.9 nN. In view of the
crudeness of the applied model and the uncertainties in
the experimental measurements and interpretation, we
do not attempt here to explain the difference between
experimental and calculated values.
The most probable rupture force fmp for each bond
in Figure 4 is lower than fmax but the relative order is
preserved: S–S is the weakest bond and possibly the
first to be broken. Next, the C–N and Si–C bonds
break. Therefore, in the ideal case of a bio-molecule in
a low-force stretching regime, the force will cause the
unfolding of the strains (breaking of the S–S bonds)
before the molecule will detach from the ATM tip
(breaking of the Si–C bond). During the stretching, the
molecule backbone (C–N bond) will remain intact.
5. Conclusions
We have investigated the performance of some standard quantum-chemical methods in describing the
influence of applying an external mechanical force
to a set of small molecules. Importantly, we have
demonstrated that the description of the mechanochemical properties of a molecular system in general
does not require a multireference treatment of the
electrons—in particular, the single-reference CCSD(T)
method provides an accurate description of mechanochemical processes. At a significantly lower cost,
standard GGA exchange–correlation functional were
shown to be useful for computing rupture forces and
breaking points. Comparisons of accurate numerical
bond-dissociation curves with simple analytical forms
show that Morse curves (fitted either to harmonic and
anharmonic constants or to the harmonic force constant and the dissociation energy) provide useful
approximate bond-breaking points and rupture forces,
accurate to within 10%. Finally, we have shown that the
outcome of kinetic calculations to estimate the dissociation probability as a function of the applied force
depends sensitively on the quality of the potentialenergy curve. The most probable rupture forces
calculated by numerical integration appear to be
significantly more accurate than those obtained from
simple analytical expressions based on fitted Morse
potentials.
2546
M.F. Iozzi et al.
Like all bond properties, we expect the rupture force
and bond-breaking point to be significantly affected by
environmental effects such as hydrogen bonding and
solvation. A constrained conformation of the backbone
may, for example, reduce the bond strength and hence
the rupture force; moreover, non-bonding interactions
may assist the breaking process by stabilizing the
di-radical intermediates formed upon bond breaking,
again reducing the rupture force. In the present study,
such effects have not been considered.
Acknowledgements
Downloaded By: [University of Oslo Library] At: 13:37 19 December 2009
This work has received support from the Norwegian
Research Council through a Centre of Excellence Grant
(grant No. 179568/V30), as well as through a grant of computer time from the Norwegian Supercomputing Program.
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