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 Downloaded By: [University of Oslo Library] At: 13:37 19 December 2009 (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]. Downloaded By: [University of Oslo Library] At: 13:37 19 December 2009 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. Downloaded By: [University of Oslo Library] At: 13:37 19 December 2009 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) Downloaded By: [University of Oslo Library] At: 13:37 19 December 2009 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 Downloaded By: [University of Oslo Library] At: 13:37 19 December 2009 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). Downloaded By: [University of Oslo Library] At: 13:37 19 December 2009 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 2543 Downloaded By: [University of Oslo Library] At: 13:37 19 December 2009 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. 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