Assessment of theoretical methods for the determination of the mechanochemical strength of covalent...

Molecular PhysicsVol. 107, Nos. 23–24, 10–20 December 2009, 2537–2546

Assessment of theoretical methods for the determination of the mechanochemicalstrength 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

(Received 1 September 2009; final version received 8 October 2009)

The performance of some commonly used quantum-chemical methods in accurately and reliably describing theinfluence of applying an external mechanical force has been investigated for a set of small molecules. By applyingcoupled-cluster CCSD(T) theory in an extended basis set as benchmark, all methods tested provide a goodqualitative 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 by20–30%. 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 hybridB3LYP and CAM-B3LYP methods. The excellent agreement observed between the CCSD(T) method andmulticonfigurational methods for bond distances significantly beyond the bond-breaking point shows that theessence of mechanical bond breaking is captured by single-reference-based methods. Comparisons of accuratenumerical bond-dissociation curves with simple analytical forms show that Morse-type curves provide usefulapproximate bond-breaking points and rupture forces, accurate to within 10%. By contrast, polynomial curvesare much less useful. The outcome of kinetic calculations to estimate the dissociation probability as a function ofthe applied force depends strongly on the description of the potential-energy curve. The most probable ruptureforces calculated by numerical integration appear to be significantly more accurate than those obtained fromsimple 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 break-ing covalent chemical bonds: photolysis, thermolysis,electrolysis, chemical reaction and mechanical strain.The latter method, in which a bond is broken as theresult of applying a stretching force, has received muchattention since the introduction of the atomic-forcemicroscope (AFM) and the force-clamp technique[1–7]. By anchoring a polymer molecule at one end tothe base of the AFM and at the other end to thecantilever tip while increasing the base–tip separationin a controlled fashion, it is possible to monitorcontinuously the force acting on the molecule. Fromthe recorded force-extension profile, it is possible tofollow the stretching of the molecule all the way fromits initial equilibrium state via intermediate states(resulting from the gradual unveiling of the polymer’stertiary structure) up to a transition state, where themechanically weakest covalent bond of the backbonebreaks. Mechanochemical bond activation achieved in

this way can also be combined with other methods ofactivation—for example, by changing the temperatureor the chemical environment [8–10].

In their pioneering work, Granbois et al. [1]determined the force-extension profile of a polysac-charide chain covalently attached to an AFM.Unfortunately, their experimental data did not allowfor a unique identification of the bond that eventuallybreaks. However, these authors realized and demon-strated by quantum-chemical modeling that there isno simple connection between the bond-dissociationenergy and the mechanochemical strength of a bond.

To assist the interpretation of experimental data, thebond-breaking probability of each bond in the polymermust be determined theoretically. Given that bondbreaking is a chemical reaction, an Arrhenius-typemodel can be applied to calculate the bond-breakingrate and the bond-breaking probability distribution,provided the force-modified potential-energy surface isknown [11–14].

*Corresponding author. Email: [email protected]

ISSN 0026–8976 print/ISSN 1362–3028 online

! 2009 Taylor & FrancisDOI: 10.1080/00268970903401041http://www.informaworld.com

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An external, constant force that stretches a diatomicmolecule along its axis, modifies the Born–Oppenheimer potential-energy surface in a well-definedmanner. However, for a polyatomic system, the deter-mination of the force-transformed potential-energysurface is not a trivial task, since it involves amultidimensional reaction path. In 2000, Beyer pro-posed the COGEF (Constrained Geometry simulateExternal Force) protocol [15] for determining theminimum-energy path leading to bond breaking onthe force-transformed potential-energy surface[4,16,17]. Very recently, a more rigorous but computa-tionally demanding approach was proposed byRibas-Arino et al. [18] and Ong et al. [19] for the fullcharacterization of potential-energy surfaces.

In determining mechanochemical strength and rup-ture probabilities, two features of the force-modifiedpotential-energy curve are particularly important—namely, the breaking-point distance Rbp, where thesecond derivative changes sign (the inflection point),and the rupture force, fmax, which is the negative firstderivative at the inflection point (i.e. the largest valueof the force along the potential-energy curve towardsdissociation). We note that, even though the mechan-ochemical characteristics of various molecular systemshave been determined on the COGEF potentialscomputed using density-functional theory (DFT), thenature 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 break-ing, involving a wide range of interatomic distances,requires a consistent treatment of electron correlation.In particular, any one-determinant-based representa-tion of the wave function, such as those providedby the Hartree–Fock (HF) and coupled-cluster (CC)models, fails to describe the static correlation arisingfrom the multi-configurational character of the dis-sociating system at large separations. For example,while coupled-cluster singles-doubles-perturbative-triples (CCSD(T)) theory provides a good descriptionof dynamic correlation, static correlation is poorlydescribed, leading to a poor description of bondbreaking. A proper description of static correlationis provided by complete-active-space (CAS) multi-configurational self-consistent field (MCSCF) theory.This approach by itself does not account for much ofthe dynamic correlation, which may be includedperturbatively, using either second-order CAS pertur-bation theory (CASPT2) or second-order n-electronvalence state perturbation theory (NEVPT2).However, these MCSCF-based models cannot easilybe applied to very large systems. Finally, concerningDFT, we note that Kohn–Sham theory dissociates

correctly in principle; in practice, Kohn–Sham theoryoften provides a poor description of bond breakingbecause of inadequacies in the approximations to theexchange-correlation functional.

From these considerations, several questions ariseconcerning the quantum-chemical study of mechan-ochemical strength. The first question concerns theability of the different models to describe correctly theessential part of the bond-breaking potential—namely,the bond-breaking point and the rupture force. In thepresent paper, we perform a careful benchmarking ofdifferent methods with respect to the reliable calcula-tion of these quantities—in particular, we examine theperformance of HF theory and DFT, suitable for largemolecular systems such as polymers. Secondly, weconsider whether it is possible to predict the breakingpoint and rupture force from an order expansion of thepotential-energy surface about the equilibrium geom-etry or from Morse potentials fitted in different ways.Besides providing insight into the bond-rupture process,our study prescribes simple and practical approachesthat can be applied to kinetic modeling of the process.

Part of the present study is concerned withdiatomic molecules, including hydrides of variablebond polarity and homonuclear molecules containingsingle, double and triple bonds. We have also studiedsome larger molecular models, suitable for describingimportant features of biological polymer stretching,relevant to experimental work.

2. Computational details

The potential-energy curves of a set of diatomicmolecules (LiH, BH, HF, Li2, N2 and O2) weredetermined by performing single-point energy calcula-tions on the systems at a large number of internucleardistances, followed by a 10-point piecewise polynomialinterpolation using Le Roy’s LEVEL 8.0 code [20]. Toensure a correct fitting in the border regions of thepotential, appropriate extrapolation procedures wereinvoked: at short distances (in the repulsive region), thepotential is extrapolated with an exponential function;at large distances (in the dissociation region), the poten-tial is extrapolated as a sum of inverse-power terms.

The potential-energy curves were computed at theHF, CAS, NEVPT2 and CCSD(T) levels of theory. Inaddition, Kohn–Sham theory with different approxi-mate exchange–correlation functionals was used: thelocal density approximation (LDA), the generalizedgradient approximation (GGA) using the BLYP andPBE functionals, the B3LYP hybrid functional and theCAM-B3LYP Coulomb-attenuated hybrid functional.In all cases, the aug-cc-pVQZ basis set was used.

2538 M.F. Iozzi et al.


In the CAS calculations, the numbers of activeelectrons 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-dimensionalpotential-energy curves were computed for moleculescontaining 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 R!H, CH3. Starting from theoptimized G3 (MP2/6-31G*) geometry [21], the poten-tial curve was computed in a point-wise fashion byvarying the X–Y bond length by a rigid translationalong the bond axis, keeping the remaining geometricalparameters fixed. In addition to applying this rigid-bond-stretching (RBS) procedure, stretching wassimulated with the COGEF procedure, increasing thedistance between the terminal hydrogen atoms whileoptimizing the remaining degrees of freedom. The finalpotential-energy curves were obtained by the sameextrapolation procedure as for the diatomics, using thesame electronic-structure models but the smaller basissets: 6-311G* for the smaller models (R!H) and6-31G* for the larger models (R!CH3). For HSSH, anextra set of polarization functions was added to thehydrogen atoms.

For the RX–YR molecules, zero-Kelvin bond dis-sociation energies (BDEs) were computed as the energydifference between each molecule and its radical

fragments, having obtained the zero-point correctedenergies at G3 geometries using scaled harmonic HF/6-31G frequencies.

3. Diatomic molecules

In Table 1 we have listed the values of the rupture forcefmax, the breaking point Rbp, and the rupture elonga-tion Ebp! 100(Rbp"Req)/Req (where Req is the equi-librium geometry) for the diatomic moleculesconsidered in this study. Notably, all molecules, at alllevels of theory, have rupture elongations smaller than50%. As an illustration, Figure 1 shows thepotential-energy curve for the BH molecule, calculatedat different levels of theory. For reference, a verticalline indicating the NEVPT2 breaking point has beendrawn at Ebp! 37%.

In Table 2 we have given the maximum, averageand minimum values of Ebp and fmax at different levelsof 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 forthese 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 Ebp (%) and rupture force fmax (nN).

HF CASa PT2b (T)c LDA PBE BLYP B3LYP CAMd

Rbp LiH 239.1 227.7 226.9 226.3 236.5 230.7 233.9 231.0 233.5BH 174.4 168.4 168.4 167.7 177.2 175.8 172.7 173.2 175.1HF 126.6 126.4 124.6 125.3 128.9 128.4 128.6 128.0 128.8Li2 402.3 364.2 354.8 349.7 369.2 371.7 363.6 369.8 377.0N2 142.1 136.1 142.8 138.6 141.0 141.8 141.2 141.1 141.8O2 153.5 138.4 150.5 154.6 155.9 157.4 159.1 156.6 156.7

Ebp LiH 49 41 44 44 48 41 40 45 47BH 43 37 37 36 41 40 39 40 42HF 41 37 36 37 38 38 38 39 40Li2 44 28 30 33 47 36 35 37 37N2 33 24 30 26 29 28 28 29 30O2 32 14 24 28 30 29 30 30 31

fmax LiH 2.6 2.1 2.3 2.4 2.4 2.3 2.2 2.4 2.6BH 5.6 4.8 4.8 4.7 4.5 4.2 4.2 4.8 5.1HF 13.1 11.2 10.6 10.8 10.3 10.1 9.8 10.7 11.0Li2 0.9 0.6 0.7 0.9 0.8 0.8 0.8 0.8 1.0N2 36.4 24.1 23.5 23.7 25.7 24.5 23.8 27.0 28.8O2 22.1 11.6 12.7 13.3 14.5 13.3 12.4 14.9 16.5

aThe CAS spaces adopted for the MCSCF models: LiH!(4e"/6o), BH!(6e"/12o), HF!(8e"/8o), Li2!(6e"/10o), N2!(6e"/12o),O2!(8e"/10o).bNEVPT2.cCCSD(T)dCAM-B3LYP.

Molecular Physics 2539


the electronic system at equilibrium and, in particular,an accurate equilibrium structure [22]. On the otherhand, since the wave function is generated by excita-tions from an HF reference state, the quality of thedescription deteriorates as the bond is stretched.However, it has previously been observed that, forthe OH dissociation of water, the CCSD(T) modelprovides an accurate potential-energy curve up to 90%elongation, failing only at larger separations [23–26].Therefore, since the inflection point Rbp occurs at anelongation of less than 50%, we expect the CCSD(T)model to provide accurate inflection points and rup-ture forces.

In a coupled-cluster calculation, the norm of thevector containing the singles amplitude T1 is indicativeof the multi-reference character of the electronicsystem (the T1 diagnostic) [27,28]. Systems accuratelyrepresented by single-reference methods have small T1values (e.g., 0.0096 for H2O at the CCSD/TZ2P level oftheory, from Ref. [28]), while large T1 values are

indicative of strong non-dynamical correlation. InTable 3, we have listed the T1 values computed forthe diatomic molecules in this study at different bonddistances, at the CCSD/aug-cc-pVQZ level of theory.Although the multi-reference character of moleculesincreases upon bond stretching, all molecules remainessentially single-configurational at Rbp. As furtherevidence of the single-configurational nature of thewave function at Rbp, we have in Figure 2 plotted CASnatural-orbital occupation numbers for the dissocia-tion of Li2, N2, and O2. Upon stretching, the plottedoccupation number decreases from two to one, as thebonding orbital loses an electron to its antibondingpartner. However, at Rbp, the occupation is reduced byonly 13%, 9% and 4% for the three molecules,confirming the single-configurational nature of theelectronic state at this geometry. In the following, weshall therefore use the mechanochemical propertiesobtained from the CCSD(T) model as our reference forevaluating the performance of the other models.

3.2. Comparison with CCSD(T) benchmark results

From Table 2, we first note that the uncorrelated HFmodel consistently overestimates rupture elongationand rupture force (typically by about 20% and 30%,respectively), in agreement with the tendency of the HFmodel to overbind. The introduction of static correla-tion at the CASSCF level of theory overcompensatesthis overbinding, giving too small rupture elongationsand forces—in the case of O2, the elongation isunderestimated by a factor of 2, see Table 1. Finally,when dynamical correlation is introduced at theNEVPT2 level of theory, an accurate representationof bond rupture is achieved—indeed, the averageNEVPT2 rupture elongation and force are within oneor 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 Ebp and fmax relative to the CCSD(T) values in Table 1.

HF CASa PT2b (T)c LDA PBE BLYP B3LYP CAMd

Ebp Max 133 103 115 100 142 111 108 112 117Ave 119 87 99 100 114 105 104 108 112Min 111 50 86 100 103 93 91 102 107

fmax Max 166 104 102 100 109 103 100 114 124Ave 134 96 98 100 102 96 93 105 113Min 108 87 95 100 95 89 89 99 102

aCASSCF.bNEVPT2.cCCSD(T).dCAM-B3LYP.

–25.3

–25.25

–25.2

–25.15

–25.1

100 150 200 250 300

En.

(a.

u)

B-H (ppm)

BH,aug-cc-pVQZ

BLYPB3LYP

CAM-B3LYPCCSD(T)

CASNEVPT2

Figure 1. The potential-energy curves computed with differ-ent methods. The dotted vertical line corresponds to the Rbp

computed with NEVPT2.



elongations that agree to within 1 pm and ruptureforces that agree to within 0.5 nN, the largest devia-tions being observed for molecules with double andtriple bonds.

Next, we compare the performance of DFT withthe reference CCSD(T) values. Because of its lowcomputational cost, DFT is attractive for modelinglarge systems—in particular, polymers and biologicalmolecules of relevance to AFM experiments. FromTables 1 and 2, we note that LDA strongly over-estimates rupture elongation and (less strongly) rup-ture force, in agreement with the tendency of LDAto overbind. The performance of DFT is stronglyimproved at the GGA level of theory: the PBE andBLYP exchange–correlation functionals provide only aslight overestimation of rupture elongation (by about5%) and a slight underestimation of rupture force(by about 5%). At the hybrid levels of theory, bothquantities increase and are consistently overestimated—typically by about 5% with the B3LYP model and 10%with the CAM-B3LYP model. Comparing the BLYPand B3LYP results, we note that the BLYP model

provides slightly better rupture elongations, whereasthe BLYP and B3LYP rupture forces bracket the trueforces, which are underestimated by the BLYP modeland overestimated by the B3LYPmodel. From Table 1,we note that the performance of DFT is best for thesingle-bonded hydrides and Li2. With increasing bondorder, the DFT forces become too large, the largestdeviation being around 2 nN, for O2.

3.3. Equilibrium prediction of bond-rupturequantities

The calculation of the breaking point Rbp and therupture force fmax is expensive, requiring the tabulationof the energy at a large number of bond distancesor, alternatively, the implementation of some iterativescheme 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 inR!R"Req, we may then express the potential in themanner

V3rd#R$ ! V0 %1

2k2R2 % 1

6k3R3, #1$

where V0 is the value of the potential at equilibriumand k2 and k3 are the harmonic and anharmonic forceconstants, respectively. The inflection point is obtainedby setting the second derivative of this expression equalto zero, yielding

R3rdbp ! " k2

k3, #2$

f 3rdmax ! " k222k3

: #3$

In Table 4 we compare these predictions of therupture elongation and the rupture force with the truequantities, calculated at the NEVPT2 level of theory.Clearly, the third-order potential Equation (1) givesa poor description of bond rupture, underestimatingrupture elongation by a factor of 3 and rupture forceby about 40%. Although the description may beimproved by including fourth- and fifth-order terms inthe expansion, high computational cost and numericalinstabilities combine to make this approach, based onan order expansion at equilibrium, unattractive.

Alternatively, we may model the potential-energycurve by a Morse potential of the general form

VMorse#R$ ! V0 1" exp#"!R$& '2, #4$

1

1.2

1.4

1.6

1.8

2

100 200 300 400 500 600 700 800

X–X (ppm)

Natural orbital occupation

N2O2Li2

Figure 2. Natural orbital occupation numbers for Li2, N2

and O2. Vertical lines indicate Rbp.

Table 3. The T1 diagnostic for a set of diatomicmolecules (aug-cc-pVQZ).

Req Rbp R1

LiH 0.0090 0.0259 0.4664BH 0.0140 0.0183 0.1494HF 0.0093 0.0141 0.1394Li2 0.0208 0.0234 0.1429N2 0.0134 0.0235 0.0607O2 0.0154 0.0245 0.0595



where V0 determines the depth of the potential and !its width. In particular, we here consider the followingtwo parameterizations of the Morse potential:

VMkD#R$ ! Deq 1" exp "

!!!!!!!!!!k2

2Deq

s

R !" #2

, #5$

VMkk#R$ ! 9k322k23

1" expk33k2

R" #$ %2

, #6$

where the first (global) parameterization is designed toreproduce the dissociation energy Deq!V(1)"V(Req)and the harmonic force constant k2 of the truepotential, whereas the second (local) parameterizationreproduces the harmonic and anharmonic forceconstants k2 and k3, respectively. These two Morsepotentials give the breaking point and rupture forces

RMkDbp ! ln 2

!!!!!!!!!!2Deq

k2

r, #7$

fMkDmax !

!!!!!!!!!!!!Deqk28

r#8$

and

RMkkbp ! " ln#8$k2

k3! ln#8$R3rd

bp , #9$

fMkkmax ! " 3

4

k22k3

! 3

2f 3rdmax: #10$

The locally fitted Morse potential Equation (6) thuspredicts a rupture elongation that is ln 8( 2.1 timeslarger than than that predicted by the third-order

potential, whereas the rupture force is 1.5 times largerthan that predicted by the third-order model, suggest-ing that the locally fitted Morse model should improveupon the third-order model.

Indeed, from Table 4, we note that both Morsemodels provide reasonably accurate descriptions ofbond rupture, typically predicting Ebp and fmax towithin 10% of the true value. The systematic under-estimation of both quantities is most likely related tothe fact that the Morse potential gives too strongbinding at large distances [29]. Whereas the globallyfitted Morse potential is slightly more accurate thanthe globally fitted potential for BH and HF, theopposite is true for N2 and O2. For general use, werecommend the locally fitted Morse potential, whichcan more easily be applied to polyatomic molecules,requiring only information obtained at the molecularequilibrium geometry.

4. Polyatomic molecules

In this section, we consider the breaking of thecovalent bonds C–C, C–N, N–C(O), S–S and Si–C,all of relevance to AFM experiments. Tables 5 and 6contain Ebp and fmax, respectively, computed at theNEVPT2 and CCSD(T) levels of theory and by DFTwith different exchange–correlation functionals. As forthe diatomics, we observe a good agreement betweenthe NEVPT2 and CCSD(T) results. Likewise, the LDAfunctional overestimates and the GGA functionalsunderestimate the rupture force relative to the coupled-cluster results. Among the DFT functionals, the bestagreement with coupled-cluster theory is obtained withthe B3LYP model, whereas the CAM-B3LYP modelgives the poorest agreement.

We observe only a weak correlation between thecomputed rupture forces and BDEs, also listed inTable 6. The dissociation energy and rupture force ofthe central X–Y bond are both rather insensitiveto methyl substitutions, going from HXYH toCH3XYCH3, with the exception of the C–N ruptureforce, which decreases from 7.6 to 5.3 nN from theprimary to secondary amine, while the BDE remainsconstant.

All results discussed so far have been obtainedusing the RBS procedure, performing a rigid relativetranslation of the two moieties along the central X–Ybond axis. Alternatively, the rupture forces can beestimated by the COGEF procedure, where themolecular geometry is fully relaxed for each separationbetween two selected peripheral atoms (often hydro-gens), thereby mimicking the effect of an external forcepulling the molecule apart. The COGEF approach is

Table 4. Elongation Ebp (%) and rupture forcefmax (nN): comparison of NEVPT2 results fullpotential with predictions based on a third-orderexpansions and two Morse potentials.

Num.a V3rdb VMkD

c VMkkd

Ebp BH 37 16 35 32HF 36 14 32 30N2 30 12 23 25O2 24 11 21 23

fmax BH 4.8 2.9 4.7 4.4HF 10.6 6.5 10.4 9.8N2 23.5 15.2 20.9 22.8O2 12.7 8.1 11.1 12.2

aNumerically fitted NEVPT2 potential.bThird-order potential Equation (1).cGlobally fitted Morse potential Equation (5).dLocally fitted Morse potential Equation (6).



more expensive than the RBS approach but describesthe bond-breaking process more faithfully, providingthe minimum-energy path towards dissociation. Notethat, whereas the RBS procedure applies an externalforce to a selected bonded atom pair, leaving themolecule otherwise intact, the COGEF procedureapplies the force to an arbitrary pair of (usuallyperipheral) atoms, thereby stretching the entire chainof atoms connecting the end atoms, eventually ruptur-ing the weakest bond. Clearly, the two approachesoffer complementary insight: local (RBS) and global(COGEF).

A comparison of the RBS and COGEF models ispresented in Table 7, where we have listed the ruptureforce of CH3X–YCH3 calculated at the unrestrictedB3LYP level of theory. We note an overall goodagreement between the RBS and COGEF results, therupture forces agreeing to within 5%. In general, wewould expect the COGEF rupture force to be lowerthan the RBS force, as happens for the S–S system.By contrast, for the C–N, C–C and Si–C bonds, theCOGEF force is greater than the RBS force. However,for these three bonds, the COGEF and RBS calcula-tions have been carried out for slightly different

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 5.2 5.3 5.4 5.0 4.7 5.2 5.6 87.01 86) 4 [30]CH3 5.1 5.1 5.2 4.8 4.5 5.0 5.4 85.63

C–C H 7.1 7.1 7.4 6.9 6.4 7.0 7.5 87.76 90.1) 0.1 [31]CH3 — 6.9 7.1 6.5 6.1 6.7 7.3 87.16 87.9) 0.6 [31]

C–N H 7.6 7.6 8.0 7.3 6.8 7.6 8.2 82.41 85.2) 0.3 [31]CH3 5.2 5.3 5.6 5.2 5.2 5.2 5.7 82.20

N–CO H 9.2 9.0 9.0 8.2 7.7 8.8 9.4 99.44CH3 — 8.6 8.7 7.8 7.2 8.3 9.0 93.66

S–S H 3.9 3.8 4.4 4.0 3.5 4.1 4.8 60.22 64.7) 3.0 [32]CH3 3.8 3.7 4.4 3.8 3.4 4.0 4.6 62.45 65.2) 0.9 [33]

aNEVPT2 calculations with CAS spaces: CH3CH3: 2e"/2o; CH3SiH3: 6e

"/6o; CH3NH2: 8e"/8o; CH3CH2NHCH3:

4e"/4o; NH2COH: 8e"/8o; HSSH: 14e"/10o.bCCSD(T).cCAM-B3LYP.dG3 theory.

Table 5. Elongation Ebp (%) for RXYR molecules.

X–Y R PT2a (T)b LDA PBE BLYP B3LYP CAMc

Si–C H 28 27 29 29 28 28 30CH3 27 27 27 28 28 29 30

C–C H 33 29 31 31 30 31 32CH3 — 31 30 30 30 31 32

C–N H 27 27 30 29 29 30 31CH3 28 28 31 31 31 31 32

N–CO H 33 27 28 28 28 28 29CH3 — 27 28 27 27 27 28

S–S H 20 20 25 24 24 25 26CH3 18 18 24 23 23 24 25

aNEVPT2 with the following CAS spaces: CH3CH3: 2e"/2o; CH3SiH3: 6e"/6o; CH3NH2: 8e"/8o;CH3CH2NHCH3: 4e

"/4o; NH2COH: 8e"/8o; HSSH: 14e"/10o.bCCSD(T).cCAM-B3LYP.



molecules (as noted in the table), invalidating such adirect comparison of the COGEF and RBS results.

All calculations discussed above were carried outusing unrestricted B3LYP theory. For the S–S bond,we have also carried out restricted B3LYP calculationswith the RBS model—see Table 7. As expected, therestricted rupture forces are slightly larger than theunrestricted ones. In passing, we note that no suchdifference is observed at the more advanced CCSD(T)level of theory. Finally, for the S–S bond, two differentbasis sets are employed. Upon basis-set extension, therupture force increases by 0.2–0.3 nN.

4.1. Kinetic model

In an AFM experiment, the force is applied in themanner

f #t$ ! f0 % "t, #11$

where the force-loading rate " is constant. AnArrhenius-type rate equation may then be used todescribe the probability P that, for an applied force f,the bond is still intact:

P0# f $ ! "A"e"E ## f $=kTP# f $, #12$

where A is the Arrhenius pre-factor, E #( f ) is theactivation barrier to bond breaking subject to theexternal force f, k is the Boltzmann constant, and Tthe temperature. The effect of applying a constantforce along the axis of a diatomic molecule is toincrease the equilibrium bond distance and to reducethe activation barrier to dissociation—see Figure 3.

For polyatomic molecules, Morse potentials basedon fmax and Deq have previously been used asminimum-energy paths towards bond breaking [15].We here treat polyatomic molecules by adding a linearforce term to the RBS potential in the mannerVf (r)!VRBS(r)" f (t)r. In this approximation, the

applied force produces the same effect as in a diatomicmolecule.

In the form of transition-state theory applied byHanke and Kreuzer [13], the Arrhenius pre-factor isgiven by A! #$q*/q, where # is the attempt frequency,$ the accommodation coefficient, and q*/q the ratio ofthe internal partition function of the activated complexto that of the unperturbed molecule.

Assuming that the stretched system is in equilib-rium with the external perturbation, we set $q*/q! 1.The attempt frequency can be viewed as the stretchingfrequency along the breaking coordinate—that is, forRX–YR, the frequency of oscillation about the X–Yequilibrium geometry. Beyer [15] used for # themaximum optical phonon of the polymer. We hereset # equal to the fundamental vibrational frequencyof the fictitious diatomic system made up by atomscentered on the X and Y units, obtained by solving theone-dimensional Schrodinger equation with theLEVEL 8.0 code [20].

Setting A equal to the fundamental frequency ofX–Y and determining E#( f ) from Vf(r), we may deter-mine the bond-breaking probability,Pbp( f )! 1"P( f ),by numerical integration of Equation (12). In Figure 4we have plotted P 0

bp# f $, where the maximum representsthe most probable rupture force, denoted by fmp. Notethat fmp differs from the rupture force fmax discussedabove.

For completeness, we compare in Table 8 theresults obtained with the fitted PT2 potential with thepredictions based on the two considered Morse-typepotentials, Equations (5) and (6). For these Morse

Ene

rgy

r

E#

Figure 3. The RSB potential-energy curve (red curve) andforce-dependent potential Vf (r) (green curve) arising fromthe applied linear potential V( f )! fr (blue curve). Thevertical arrow indicates the transition state (E #) of theactivated bond dissociation.

Table 7. COGEF and RBS potentials for CH3X–YCH3

except as noted.

X–Y B3LYPCOGEFunres

RBSres

RBSunres

S–S 6-311%%G(3df,3pd) 3.8 4.2 3.96-31G(d) 3.5 3.9 3.7

C–N D95(d,p) 7.2a 7.1C–C D95(d,p) 6.9a 6.6Si–C D95(d,p) 4.8a 4.9

aComputed on the H3C–YCH3 models, with Y!NH, CH2

and SiH2.



potentials, we compute fmp analytically, from theapproximate expression developed by Hanke andKreuzer [13]:

1" 2fmp

!V0! !AkT

4"exp "V0

kT1" 2fmp

!V0

" #2" #

, #13$

where the values of V0 and ! are obtained by com-paring Equations (5) and (6) with the general expres-sion Equation (4).

The locally and globally fitted Morse potentialprovide rather different fmp values, indicating that theanalytical solution is rather sensitive to the shape ofthe potential. As compared to the PT2 values, the mostprobable rupture force is higher for the single-bond

molecules BH and HF, while the opposite occurs in thecase of multiple-bond molecules N2 and O2.

The computed fmp can be directly compared withexperimental measurements. The maximum Si–C rup-ture force is 2.0) 0.3 nN at room temperature and aforce load of 10 nN s"1—that is, significantly lowerthan our theoretical result of about fmp! 2.6 nN, whichin turn is much lower than fmax! 4.9 nN. In view of thecrudeness of the applied model and the uncertainties inthe experimental measurements and interpretation, wedo not attempt here to explain the difference betweenexperimental and calculated values.

The most probable rupture force fmp for each bondin Figure 4 is lower than fmax but the relative order ispreserved: S–S is the weakest bond and possibly thefirst to be broken. Next, the C–N and Si–C bondsbreak. Therefore, in the ideal case of a bio-molecule ina low-force stretching regime, the force will cause theunfolding 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, themolecule backbone (C–N bond) will remain intact.

5. Conclusions

We have investigated the performance of some stan-dard quantum-chemical methods in describing theinfluence of applying an external mechanical forceto a set of small molecules. Importantly, we havedemonstrated that the description of the mechano-chemical properties of a molecular system in generaldoes not require a multireference treatment of theelectrons—in particular, the single-reference CCSD(T)method provides an accurate description of mechan-ochemical processes. At a significantly lower cost,standard GGA exchange–correlation functional wereshown to be useful for computing rupture forces andbreaking points. Comparisons of accurate numericalbond-dissociation curves with simple analytical formsshow that Morse curves (fitted either to harmonic andanharmonic constants or to the harmonic force con-stant and the dissociation energy) provide usefulapproximate bond-breaking points and rupture forces,accurate to within 10%. Finally, we have shown that theoutcome of kinetic calculations to estimate the dissoci-ation probability as a function of the applied forcedepends sensitively on the quality of the potential-energy curve. The most probable rupture forcescalculated by numerical integration appear to besignificantly more accurate than those obtained fromsimple analytical expressions based on fitted Morsepotentials.

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

P’ b

p (a

rbitr

ary

units

)

f (nN)

S-S Si-C C-N

Figure 4. Bond-breaking probability distributions of CH3X–YCH3 with X–Y!C–N, Si–C, S–S. In a typical force-clampexperiment at room temperature, the force is applied at aconstant rate of 10 nN s"1.

Table 8. The most probable rupture force fmp (nN): com-parison of NEVPT2 results obtained from numericalintegration of Equation (12) and from the analytical expres-sion Equation (13) with the Morse potentials Equations (5)and (6).

Num.a VMkDb VMkk

c

BH 1.78 2.29 2.49HF 7.10 6.91 7.33N2 17.37 14.06 12.94O2 8.27 6.66 6.03Si–C 2.69C–N 3.06S–S 1.78

aNumerical NEVPT2 potential.bGlobally fitted Morse potential Equation (5).cLocally fitted Morse potential Equation (6).



Like all bond properties, we expect the rupture forceand bond-breaking point to be significantly affected byenvironmental effects such as hydrogen bonding andsolvation. A constrained conformation of the backbonemay, for example, reduce the bond strength and hencethe rupture force; moreover, non-bonding interactionsmay assist the breaking process by stabilizing thedi-radical intermediates formed upon bond breaking,again reducing the rupture force. In the present study,such effects have not been considered.

Acknowledgements

This work has received support from the NorwegianResearch Council through a Centre of Excellence Grant(grant No. 179568/V30), as well as through a grant of com-puter time from the Norwegian Supercomputing Program.

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