Article pubs.acs.org/accounts
Quantum Chemical Strain Analysis For Mechanochemical Processes Tim Stauch* and Andreas Dreuw* Interdisciplinary Center for Scientific Computing, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany S Supporting Information *
CONSPECTUS: The use of mechanical force to initiate a chemical reaction is an efficient alternative to the conventional sources of activation energy, i.e., heat, light, and electricity. Applications of mechanochemistry in academic and industrial laboratories are diverse, ranging from chemical syntheses in ball mills and ultrasound baths to direct activation of covalent bonds using an atomic force microscope. The vectorial nature of force is advantageous because specific covalent bonds can be preconditioned for rupture by selective stretching. However, the influence of mechanical force on single molecules is still not understood at a fundamental level, which limits the applicability of mechanochemistry. As a result, many chemists still resort to rules of thumb when it comes to conducting mechanochemical syntheses. In this Account, we show that comprehension of mechanochemistry at the molecular level can be tremendously advanced by quantum chemistry, in particular by using quantum chemical force analysis tools. One such tool is the JEDI (Judgement of Energy DIstribution) analysis, which provides a convenient approach to analyze the distribution of strain energy in a mechanically deformed molecule. Based on the harmonic approximation, the strain energy contribution is calculated for each bond length, bond angle and dihedral angle, thus providing a comprehensive picture of how force affects molecules. This Account examines the theoretical foundations of quantum chemical force analysis and provides a critical overview of the performance of the JEDI analysis in various mechanochemical applications. We explain in detail how this analysis tool is to be used to identify the “force-bearing scaffold” of a distorted molecule, which allows both the rationalization and the optimization of diverse mechanochemical processes. More precisely, we show that the inclusion of every bond, bending and torsion of a molecule allows a particularly insightful discussion of the distribution of mechanical strain in deformed molecules. We illustrate the usefulness of the JEDI analysis by rationalizing the finding that a knot tremendously weakens a polymer strand via a “choking” motion of the torsions in the curved part of the knot, thus leading to facilitated bond rupture in the immediate vicinity of the knot. Moreover, we demonstrate that the JEDI analysis can be exploited to devise methods for the stabilization of inherently strained molecules. In addition to applications in the electronic ground state, the JEDI analysis can also be used in the electronically excited state to determine the mechanical energy that a molecular photoswitch can release into its environment during photoisomerization. This approach allows the quantification of the mechanical efficiency of a photoswitch, i.e., the part of the energy that becomes available for the motion into a specific direction, which enables us to judge whether a photoswitch is capable of performing a desired switching function.
1. INTRODUCTION Mechanochemistry, the use of force to initiate chemical reactions,1,2 looks back on a long tradition3 and is a lively field of research with an enormous amount of applications both in academia and industry.4 Experimental methods to apply forces to molecules are diverse and include single molecule force spectroscopy,5 ultrasound baths (sonochemistry),6 and ball milling techniques.4 Despite the remarkable success of such experiments, the understanding of mechanochemical processes at the molecular level is still limited. During the past two decades, however, this conundrum has at least partially been alleviated by quantum mechanochemical methods.7 The implicit8,9 and explicit10−13 inclusion of mechanical force in quantum chemical geometry optimizations has contributed to this development by providing access to geometries, energies, reaction rates and spectroscopic properties of mechanically deformed molecules. However, with typical quantum mechanochemical methods, it is usually not possible to answer the question which part of a © 2017 American Chemical Society
molecule is particularly susceptible to mechanical stress. Different bonds, bendings, and torsions of a molecule have different stiffnesses and are more or less easily deformed if an external force is applied. It is therefore desirable to determine the force-bearing scaffold of a molecule in order to predict and rationalize, for example, the point of bond rupture in an overstretched molecule. For this purpose, quantum chemical force analysis tools have been developed.7,14−22 The insights gained in these analyses can be used for the development of mechanically resilient and self-healing polymers,23 stressresponsive materials24 and molecular machines.25 One tool for such estimations is the JEDI (Judgement of Energy DIstribution) analysis, which has been developed in our group.20−22 The JEDI analysis is a quantum chemical approach to quantify the mechanical strain energy in a deformed molecule and to investigate its distribution among all bonds, Received: January 19, 2017 Published: March 24, 2017 1041
DOI: 10.1021/acs.accounts.7b00038 Acc. Chem. Res. 2017, 50, 1041−1048
Article
Accounts of Chemical Research bendings and torsions of the system. In this Account, we present the theoretical foundations of force analyses and show that these tools yield particularly valuable insight if the results are independent of the choice of the molecular degrees of freedom included as well as of the mechanochemical intuition of the researchers. Subsequently, we examine the performance of the JEDI analysis in various applications.
2. THEORETICAL ASPECTS OF FORCE ANALYSIS TOOLS During the past few years, a number of force analysis tools have been developed.14−22 Some of them are based on the calculation of internal forces14 or use compliance constants for the quantification of the strength of covalent bonds and noncovalent interactions.15,16 Others have been developed mainly for application in polymer mechanochemistry17,18 or for other large molecules like proteins.19 The interested reader is referred to ref 7 for a comprehensive review on force analysis tools. In the following, we concentrate on the JEDI analysis, which uses the harmonic approximation to calculate an energy contribution for each bond, bending and torsion in a mechanically deformed molecule. The calculation of strain energy instead of forces in a molecule has the advantage that interpretation and visualization of the results is often much more straightforward and insightful. In the literature, the terms strain energy, mechanical stress energy and local work are often used synonymously. We will use the term strain energy throughout this article. Within the harmonic approximation, the strain energy ΔEi in each of the M bonds, bendings, and torsions of a molecule can be calculated via20,21 ΔEi =
where
1 2
M
∑ j
∂ 2V (q ⃗) ∂qi ∂qj
∂ 2V (q ⃗) ⎯ | → ∂qi ∂qj q ⃗ = q0
Figure 1. Potential energy surfaces (PES) that are relevant for the JEDI analysis. (A) In the electronic ground state, the harmonic approximation is applied to the ground state PES (dashed line). Upon deformation, the strain energy within the harmonic approximation, ΔEharm, can be calculated. (B) In the electronically excited state, the harmonic approximation is applied to the excited state PES (dashed line). The energy release upon relaxation from the Franck−Condon point to the minimum on the excited state PES can be quantified within the harmonic approximation (ΔEharm). Details on the generation of the color-coded structures are given in the Supporting Information.
ΔqiΔqj ⎯q q ⃗ =→ 0
(1)
is the Hessian matrix at the equilibrium
M
ΔE harm =
geometry in redundant internal coordinates (RICs), which can be calculated from the Hessian matrix in Cartesian coordinates.26 Δqi is the change in the internal coordinate i upon deformation. To calculate this quantity, a standard relaxed geometry optimization as well as a geometry optimization under an external force9−12 are necessary. In our group, we use custom Python routines interfaced with the Q-Chem27 program package to carry out the required matrix transformations. The set of redundant internal coordinates is generated by first identifying all covalent bonds in the molecule by calculating the interatomic distance matrix and defining a bond when the distance falls below an element-specific cutoff value. The set of bond angles and dihedral angles is generated by using the list of covalent bonds. The identification of those coordinates that store the highest amount of strain energy at a given point of the deformation coordinate affords valuable insight into various mechanochemical processes (Figure 1A). The distinction between bonds, bendings and torsions is beneficial because this set of internal coordinates can be generated using a well-defined set of rules, thus allowing a chemically intuitive discussion of mechanochemical processes. To assess the quality of the harmonic approximation, it is useful to calculate the total harmonic strain energy ΔEharm of a molecule by summing up all individual contributions ΔEi:
∑ ΔEi = i
1 2
M
∑ i,j
∂ 2V (q ⃗) ∂qi ∂qj
ΔqiΔqj ⎯q q ⃗ =→ 0
(2)
ΔEharm can be compared to the difference in total energies between the relaxed and the deformed molecule calculated via standard quantum chemical methods, which we call ΔEab_initio. While ΔEharm usually overestimates ΔEab_initio, we have shown previously that this overestimation is only a result of the harmonic approximation and not of the approach to include each bond, bending and torsion in the force analysis.21 A method to compensate the error of the harmonic approximation in the case of bond rupture will be discussed in section 3.2. In our experience, the quality of the harmonic approximation is typically sufficient for most deformation scenarios. As an exception, the JEDI analysis yields unreasonably large energy contributions in scenarios where torsions flip from one minimum to another, since the Δqi terms in eq 1 and eq 2 become extremely large. Eliminating this error is difficult, so that great care must be exercised if unphysically large energies in the torsional degrees of freedom are encountered. The JEDI analysis can also be carried out in combination with Born−Oppenheimer molecular dynamics (BOMD) simulations, in which the molecule is subjected to external 1042
DOI: 10.1021/acs.accounts.7b00038 Acc. Chem. Res. 2017, 50, 1041−1048
Article
Accounts of Chemical Research
Figure 2. (A) Color-coded distribution of the total strain in stretched 1,3-dimethylcyclopentane using RICs (left) and a Z-matrix in which one of the bonds in the ring is omitted (right). The circles indicate a region that stores different amounts of strain energy, depending on the chosen coordinate system. (B) Color-coded distribution of the total strain in stretched isopropanol. The C−C−C bond angle is included in RICs (left) and in Z-matrix 1 (middle), but not in Z-matrix 2 (right). In both molecules, the arrows indicate the stretching coordinate.
forces.10−12 In this case, each time step is considered a mechanically deformed geometry, which allows the real time observation of the propagation of strain energy. However, random thermal oscillations tend to obscure the results of the JEDI analysis. One possibility to circumvent this problem is to conduct the BOMD simulation at 0 K, so that the motion of the molecule originates solely from the external force. Alternatively, the geometries of the molecule can be averaged over a certain period of time, which yields mean energy distributions during a desired part of the simulation. Moreover, it has to be kept in mind that only the potential energy is considered in these analyses, since it is far from straightforward to transform kinetic energy contributions from Cartesian coordinates into RICs.28 In addition to applications in the electronic ground state, the JEDI analysis can be used in electronically excited states (Figure 1B).22 Such excited state JEDI analyses are typically performed without external force. Since no assumptions are made about the electronic state of the system in the derivation of the JEDI analysis, the same equations as in the ground state can be used in an excited state. The interpretation of the results, however, differs fundamentally. In the ground state, mechanical deformation is described and the distribution of strain energy among the internal coordinates of the molecule is quantified, i.e., energy is brought into the system. In the excited state, by contrast, the relaxation from the Franck−Condon point to a minimum on the excited state potential energy surface (PES) is described via the JEDI analysis. In this case, energy becomes available due to the relaxation of the internal coordinates of the molecule on the excited state PES, which can be used to trigger various transformations in the chemical environment.29 This approach allows us to quantify the mechanical efficiency of a molecular photoswitch, which is defined as the percentage of the energy that becomes available for the motion of the photoswitch into a desired direction (section 3.4). In the case of photoswitches that are based on photochemical cis−transisomerizations,25 for example, the desired motion may be the torsion around the central dihedral angle, since this motion leads to a substantial gain in spatial extension of the photoswitch.30 This, in turn, can lead to the application of forces to the chemical environment and to the generation of
mechanical work. The mechanical efficiency of a photoswitch is typically far below 100%, since other internal coordinates that do not contribute to a change in spatial extension of the photoswitch (e.g., C−H bond lengths) also change during the motion on the excited state PES.
3. UNDERSTANDING MECHANOCHEMICAL PROCESSES In this section, we demonstrate the scope and limitations of the JEDI analysis in a number of applications. In section 3.1, we elaborate on the benefits of using redundant instead of nonredundant internal coordinates for mechanochemical force analyses. Subsequently, we show how the JEDI analysis can be used to rationalize mechanically induced bond rupture events (section 3.2) and to investigate inherently strained molecules (section 3.3). We end this section by illustrating how the mechanical efficiency of a photoswitch can be quantified via the excited state JEDI analysis (section 3.4). 3.1. Illustrative Examples
In the following, we illustrate the differences between the JEDI analysis, which uses redundant internal coordinates (RICs), and a comparable force analysis that uses nonredundant internal coordinates in the form of a Z-matrix. Although the specific case of a force analysis that is based on the harmonic approximation is discussed, many of the results are valid in a general sense and can be transferred to other force analysis tools. In Figure 2A, the total strain in stretched 1,3-dimethylcyclopentane, calculated via force analysis, is shown. The molecule has been stretched using the External Force is Explicitly Included (EFEI)11 method at the BLYP31,32/6-31G(d)33 level of theory with a force of 2.5 nN applied along the vector connecting the carbon atoms of the methyl groups. Force analysis in RICs allows a unique identification of the forcebearing scaffold of the molecule and reveals that most strain energy is stored in direct vicinity of the carbon atoms that are subjected to the force. A moderate amount of strain energy is also stored in the ring, particularly in those coordinates that connect the methyl groups. This finding can be rationalized by 1043
DOI: 10.1021/acs.accounts.7b00038 Acc. Chem. Res. 2017, 50, 1041−1048
Article
Accounts of Chemical Research
Figure 3. (A) Distribution of strain energy among the bonds, bendings and torsions in a knotted polyethylene strand that is stretched end to end by different forces. The “critical bond” is the bond that breaks if the stretching force is strong enough. (B) Time-resolved distribution of potential energy in a knotted polyethylene strand that is stretched end to end by a force of 2 nN during a BOMD simulation. (C) Color-coded distribution of strain energy in a knotted polyethylene strand that is stretched by a force of 2 nN. Reproduced in part from ref 34.
internal coordinates in the context of force analyses. The shortcomings of Z-matrix coordinates, or indeed any other coordinate system that cannot be defined uniquely and that arbitrarily omits certain coordinates, could have been predicted a priori, since 1,3-dimethylcyclopentane includes a ring that stores strain energy. To demonstrate that nonredundant coordinate systems that cannot be defined uniquely cause problems even if no rings are included, a force analysis of isopropanol, which was stretched end to end using the EFEI approach at the BLYP/6-31G(d) level of theory with a force of 2.5 nN, has been carried out (Figure 2B). In the case of RICs, the force-bearing scaffold of the molecule essentially consists of the connecting line between the methyl groups where the force is applied. This is a consequence of C−C bond stretchings and the displacement of the C−C−C bond angle with only minor contributions of the torsions. To investigate the coordinate dependence, two different Zmatrices were conceived. In Z-matrix 1, the C−C−C bond angle, which has been identified as mechanically relevant via the JEDI analysis in RICs, is included, whereas it is omitted in Zmatrix 2. Since Z-matrix 1 includes the mechanically most relevant degrees of freedom, i.e. the two C−C bond lengths and the C−C−C bond angle, a force analysis using this coordinate system yields results that are in qualitative agreement with the force analysis in RICs. The only exception is that contributions from the torsions are now found to be negligible, which leads to minor differences (see the Supporting Information, Figure S2). In Z-matrix 2, however, the neglect of the C−C−C bond angle is compensated by contributions from the C−C−O bond angles and C−C−O−H torsions. As a result, a significant
displacements of the bond length and bond angle coordinates (see the Supporting Information, Figure S1A). If a cyclic molecular structure is described by a Z-matrix, typically one of the bonds in the ring is omitted, since this coordinate is defined implicitly via the rest of the bond lengths, bond angles and dihedral angles. This effect is a result of the nonredundancy of the coordinate system and has some important implications for the force analysis. While still a remarkable amount of strain energy is stored in the region adjacent to the points where the force is applied, one of the bonds that was significantly strained in the force analysis in RICs is now completely relaxed (see the mark in Figure 2A). Instead, several bendings and torsions in the rest of the ring appear strained in the force analysis in Z-matrix coordinates (see the Supporting Information, Figure S1B), so that a remarkable amount of strain energy seems to be stored in a part of the ring that is hardly deformed. This unphysical effect is due to the omission of crucial internal coordinates in the Z-matrix. Although the total harmonic strain energy ΔEharm (eq 2) is the same in RICs and in nonredundant Z-matrix coordinates, the energy dissection differs tremendously. The example above reveals that if a mechanically strained coordinate is omitted in the analysis, no strain energy can be stored in it. This leads to an erroneous and unphysical distribution of strain energy. In other cases it has been observed that, depending on the definition of the coordinate system, the force-bearing scaffold might not have the same symmetry as the molecule14 or the same bond can appear relaxed, stretched or compressed.14,21 The interested reader is referred to ref 7 for a more detailed discussion of the characteristics of nonredundant 1044
DOI: 10.1021/acs.accounts.7b00038 Acc. Chem. Res. 2017, 50, 1041−1048
Article
Accounts of Chemical Research
simulations revealed that during the initial phase of the stretching process most energy is stored in the bonds and bendings in the terminal region of the polymer strand (Figure 3B). Upon tightening of the knot, however, a transfer of strain energy to the torsions in the curved part of the knot can be observed. These findings indicate that the torsions “choke off” the chain in the immediate vicinity of the knot. More generally, the results demonstrate that the scissile bond is not always the only important internal coordinate in a bond rupture process. It has to be kept in mind that bond rupture is a statistical process, which not only depends on force, but also on thermal fluctuations.38 To describe bond rupture more realistically, dynamic JEDI analyses based on BOMD simulations at different temperatures (cf. section 2) are useful. Moreover, the role of quantum effects in mechanochemistry has been addressed in literature.39
propagation of strain energy into the mechanically hardly relevant O−H group of the molecule is observed. This demonstrates that the results of force analyses in Z-matrix coordinates strongly depend on the choice of included bonds, angles, and dihedrals. In summary, force analyses are particularly useful if RICs are used instead of nonredundant coordinates, especially if the latter are defined in a nonunique manner. The major purpose of force analysis tools is to identify the mechanically relevant degrees of freedom, hence an analysis that requires the results as input cannot be particularly insightful by definition. Therefore, we focus on the JEDI analysis in RICs in the following. 3.2. Rationalizing Bond Rupture Events
The JEDI analysis has been used to rationalize bond rupture events in mechanically overstretched molecules.29,34 Intuitively, the consideration of strain energy within the harmonic approximation is an error-prone approach for describing bond rupture events, since at the point of bond rupture anharmonic effects are dominant. Hence, the amount of strain energy in the scissile bond can generally be considered an upper bound. Alternatively, the error of the harmonic approximation can be compensated to a satisfying extent by comparing ΔEharm (eq 2) to ΔEab_initio, which is the energy difference between the relaxed and the mechanically strained structure, calculated via standard quantum chemical methods. The overestimation of this energy difference by ΔEharm within a pulling scenario in which the scissile bond is stretched in an isolated manner can be used to compensate the error of the harmonic approximation within arbitrary pulling scenarios, given that coupling terms to other coordinates are modest. Using the JEDI analysis it has been shown that it is not always sufficient to consider the elongation of the scissile bond as the only important internal coordinate in the investigation of bond rupture processes.34 Although a somewhat surprising result, this observation is in agreement with previous literature.35,36 An example for this effect is provided by the rupture properties of knotted polymer strands. If a polyethylene strand is tangled into a common overhand knot, its mechanical resistance is reduced by approximately 50% and the rupture point is located at the “entry” or “exit” of the knot.34 Previous force analyses have primarily focused on the strain energy in the bonds and bendings of the knotted polymer strand,37 but with the JEDI analysis it has been found that the torsions, particularly those in the curved part of the knot, are the mechanically most relevant degrees of freedom in the system. To investigate the distribution of strain energy in a knotted polyethylene chain, an EFEI stretching coordinate was calculated and the JEDI analysis was carried out at each point (Figure 3A).34 The energy distribution is different for each force value, which is a general feature of the JEDI analysis and can be traced back to the different stiffness of the various RICs and the different displacements of these coordinates at each pulling force. At each force value, however, the torsions store most strain energy, which demonstrates that these coordinates are the mechanically most relevant degrees of freedom (Figure 3C). Due to this effect, the bonds are not particularly strained themselves, which makes a correction of the harmonic strain energy unnecessary. However, care has to be exercised that torsions do not flip to other minima, since otherwise the very large Δqi terms in eq 1 would lead to unphysically high energy contributions. A dynamic JEDI analysis based on BOMD
3.3. Inherently Strained Molecules
Although the JEDI analysis quantifies energy contributions in a molecule, it is not straightforward to calculate the intrinsic strain energy of inherently strained molecules with the JEDI analysis. The reason is that displacements in the internal coordinates Δqi are needed for the calculation of the strain energy (eqs 1 and 2), which necessitates the calculation of a relaxed and a mechanically deformed structure. The former is calculated via a standard geometry optimization and the inherent strain at this geometry is then considered to be zero. Therefore, only additional strain energy that is introduced in the molecule by mechanical distortion can be calculated via the JEDI analysis. For the calculation of inherent strain, other methods such as the calculation of reaction enthalpies in isodesmic and homodesmotic reactions are available.40 As an example of inherently strained molecules, we considered cycloheptyne,41 in which the deviation of the C C−C bond angles from linearity destabilizes the molecule significantly.42 To devise a method for the stabilization of cycloheptyne we investigated the EFEI coordinate in which carbon atoms on opposite sides of the molecule were pulled apart (Figure 4). We found that an adequately chosen stretching coordinate partially linearizes the CC−C bond angles and that this effect leads to a significant stabilization of the molecule. With the JEDI analysis, it was possible to quantify the efficiency of this process. In the case of an ideal coupling of the stretching coordinate to the linearization of the CC−C bond angles, 100% of the strain energy would be stored in these particular degrees of freedom. However, due to the cyclic structure of cycloheptyne, its internal coordinates are heavily coupled. Hence, at a stretching force of 2.5 nN only 42% of the strain energy is stored in the bond angles of the molecule, and the CC−C bond angles store only 8% of the total strain energy. Although stretching the carbon atoms apart on opposite sides of the molecule leads to a linearization of the CC−C bond angles and to a stabilization of the molecule, this process is rather inefficient. The calculation of the inherent strain energy in a highly strained molecule cannot be achieved with the JEDI analysis alone, but the analysis can be used to investigate the effect of forces on such molecules and to devise novel ways of stabilizing them. In this context we found that coupling cycloheptyne to other strained hydrocarbons can potentially be used to stabilize both subunits.41 1045
DOI: 10.1021/acs.accounts.7b00038 Acc. Chem. Res. 2017, 50, 1041−1048
Article
Accounts of Chemical Research
extension of a photoswitch increases tremendously, which has been used extensively in molecular machines.25 For the applicability of such molecular devices it is important that the energy that is brought into the system by the absorption of a photon is converted to mechanical work efficiently. In the case of cis → trans-photoisomerization, the torsion around a double bond is typically most relevant for this energy conversion, since this coordinate makes the major contribution to the tremendous gain in spatial extension. However, during relaxation on the excited state PES other internal coordinates (e.g., C−H bond lengths) that can hardly be exploited for the generation of mechanical work by the photoswitch relax as well. Hence, a certain amount of energy is “wasted” for their relaxation. The mechanical efficiency of a photoswitch is therefore determined only by that part of the energy that becomes available during relaxation of the switching coordinate on the excited state PES. With the excited JEDI analysis it is possible to calculate this mechanical efficiency. To demonstrate the concept of the excited state JEDI analysis, we have considered the first part of the excited state cis → trans-isomerization of stiff-stilbene (1-(1-indanyliden)indan, Figure 5A).29 In the electronic ground state, the central dihedral angle θ of stiff-stilbene amounts to 9° at the B3LYP32,45/cc-pVDZ46 level of theory, which is close to the CASSCF result (7°).47 If the molecule is excited into the first electronically excited singlet state (S1), a relaxation on the excited state PES takes place and θ increases to 31° (as
Figure 4. Color-coded distribution of strain energy among the different kinds of internal coordinates of cycloheptyne that is stretched by a force of 2.5 nN. Reproduced in part from ref 41. Copyright 2016 American Chemical Society.
3.4. Molecular Photoswitches
Photoswitches can convert light into mechanical energy.30,43,44 Upon cis → trans-photoisomerization, for example, the spatial
Figure 5. (A) cis−trans-Photoisomerization of the stiff-stilbene photoswitch. The central dihedral angle θ is colored red. (B) Relevant PESs for the JEDI analysis of the initial phase of the photoisomerization of stiff-stilbene. The molecule is excited from the electronic ground state (S0) to the Franck−Condon point in the first electronically excited singlet state (S1), where θ = 9°. At the minimum in the S1 state, θ = 31°. (C) Color-coded distribution of the energy released by stiff-stilbene upon relaxation in the S1 state. Reproduced in part from ref 29. 1046
DOI: 10.1021/acs.accounts.7b00038 Acc. Chem. Res. 2017, 50, 1041−1048
Article
Accounts of Chemical Research compared to 43° at the CASSCF level)47 at the minimum of the S1 PES. Since this minimum can be approximated as harmonic (Figure 5B), the excited state JEDI analysis can be used to quantify the distribution of the energy that is released during relaxation from the Franck−Condon point to the S1 minimum. In this process 45% of the energy is contributed by the central dihedral angle θ and 55% of the energy is contributed by bonds and bendings that do not lead to a significant change in spatial extension of stiff-stilbene (Figure 5C). Therefore, the mechanical efficiency of stiff-stilbene during the initial part of the cis → trans-photoisomerization is limited to 45%. The energy that is released by the torsion around θ during the initial relaxation on the S1 PES can be used by the chemical environment to induce chemical transformations. Since the JEDI analysis is based on the harmonic approximation, however, an excited state PES that can be approximated by a parabolic potential is required. The quality of the harmonic approximation can once again be assessed by comparing ΔEharm (eq 2) to ΔEab_initio, which is now defined as the energy difference between the Franck−Condon point and the minimum on the excited state PES. Although ΔE harm overestimates ΔEab_initio by only 8% in the case of stiff-stilbene, the harmonic approximation is generally more problematic in electronically excited states than in the ground state. Photoisomerizations are complex processes that may involve relaxation processes on several excited state PESs, and the harmonic approximation is not guaranteed to be reliable for all of them. Moreover, barrierless decays through conical intersections, dissociative decays or practically linear excited state PES shapes cannot be described accurately by the excited state JEDI analysis. The JEDI analysis can therefore be considered a first step toward a comprehensive description of excited state relaxation processes.
environment induces structural changes in the substrate and that mechanical forces play an important role in this process. Force-induced affinity changes, e.g., have been discussed in literature.48 The quantification of the strain energy of the substrate in the binding pocket and the investigation of the distribution of this energy among its internal coordinates will put us into the position of comparing the energetics of different substrates in great detail, which will yield valuable insights for drug design.
■
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.accounts.7b00038. Generation of the color-coded structures, detailed JEDI analysis of 1,3-dimethylcyclopentane, and detailed JEDI analysis of isopropanol (PDF)
■
AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected]. *E-mail:
[email protected]. ORCID
Tim Stauch: 0000-0001-7599-3578 Andreas Dreuw: 0000-0002-5862-5113 Notes
The authors declare no competing financial interest. Biographies Tim Stauch studied Chemistry at the Goethe University of Frankfurt. In 2012, he obtained his M.Sc. degree in the group of Andreas Dreuw and has since been working in the field of Quantum Mechanochemistry. He received his Ph.D. degree in the group of Andreas Dreuw at the Interdisciplinary Center for Scientific Computing of Heidelberg University in 2016. His research interest focuses on the development of computational force analysis tools and molecular force probes.
4. CONCLUSIONS AND OUTLOOK The JEDI analysis is a quantum chemical tool that can be used to rationalize diverse mechanochemical processes with unprecedented spatial resolution. In the electronic ground state, the strain energy stored in each internal coordinate of a mechanically deformed molecule can be calculated based on the harmonic approximation. In the electronically excited state, the energy released by each internal coordinate during relaxation on the excited state PES can be quantified, thus allowing the calculation of the mechanical efficiency of molecular photoswitches. Dynamic JEDI analyses based on BOMD simulations enable the real-time monitoring of the propagation of strain energy in deformed molecules. In the future, we plan to include corrections for anharmonicities in the electronic ground state, in particular in those cases when torsions flip from one minimum to another during stretching. We plan to generalize the excited state JEDI analysis in order to allow for the description of various photoisomerization processes that are not limited to relaxations toward an excited state minimum that can be approximated as harmonic. We are currently testing the applicability of the JEDI analysis in the context of QM/MM calculations. The JEDI equations are very general, since only a Hessian matrix and a difference in geometries is required to calculate the energy distribution in a deformed molecule. In particular, we plan to quantify the distribution of strain energy of a substrate in the binding pocket of an enzyme, since it is possible to argue that the protein
Andreas Dreuw obtained his Ph.D. degree in Theoretical Chemistry from Heidelberg University in 2001. After a 2 year postdoc stay at the UC Berkeley, he joined the Goethe University of Frankfurt first as an Emmy-Noether fellow and subsequently as a Heisenberg-Professor for Theoretical Chemistry. Andreas Dreuw has been holding the chair for Theoretical and Computational Chemistry at the Interdisciplinary Center for Scientific Computing of Heidelberg University since 2011. His research interests include the development of electronic structure methods and their application in Photochemistry, Biophysics, Mechanochemistry, and Material Science.
■
REFERENCES
(1) Beyer, M. K.; Clausen-Schaumann, H. Mechanochemistry: The Mechanical Activation of Covalent Bonds. Chem. Rev. 2005, 105, 2921−2948. (2) Ribas-Arino, J.; Marx, D. Covalent Mechanochemistry: Theoretical Concepts and Computational Tools with Applications to Molecular Nanomechanics. Chem. Rev. 2012, 112, 5412−5487. (3) Takacs, L. The Historical Development of Mechanochemistry. Chem. Soc. Rev. 2013, 42, 7649−7659. (4) James, S. L.; Adams, C. J.; Bolm, C.; Braga, D.; Collier, P.; Frišcǐ ć, T.; Grepioni, F.; Harris, K. D. M.; Hyett, G.; Jones, W.; et al. Mechanochemistry: Opportunities for New and Cleaner Synthesis. Chem. Soc. Rev. 2012, 41, 413−447.
1047
DOI: 10.1021/acs.accounts.7b00038 Acc. Chem. Res. 2017, 50, 1041−1048
Article
Accounts of Chemical Research
(28) Wilson, E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations. The Theory of Infrared and Raman Vibrational Spectra, 1st ed.; Dover Publications Inc.: Mineola, NY, 1955. (29) Stauch, T.; Dreuw, A. Stiff-Stilbene Photoswitch Ruptures Bonds Not by Pulling but by Local Heating. Phys. Chem. Chem. Phys. 2016, 18, 15848−15853. (30) Bléger, D.; Yu, Z.; Hecht, S. Toward Optomechanics: Maximizing the Photodeformation of Individual Molecules. Chem. Commun. 2011, 47, 12260−12266. (31) Becke, A. D. Correlation Energy of an Inhomogeneous Electron Gas: A Coordinate-Space Model. J. Chem. Phys. 1988, 88, 1053−1062. (32) Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density. Phys. Rev. B: Condens. Matter Mater. Phys. 1988, 37, 785−789. (33) Hehre, W. J.; Ditchfield, R.; Pople, J. A. Self-Consistent Molecular Orbital Methods. XII. Further Extensions of Gaussian-Type Basis Sets for Use in Molecular Orbital Studies of Organic Molecules. J. Chem. Phys. 1972, 56, 2257−2261. (34) Stauch, T.; Dreuw, A. Knots “Choke Off” Polymers upon Stretching. Angew. Chem., Int. Ed. 2016, 55, 811−814. (35) Yang, Q.-Z.; Huang, Z.; Kucharski, T. J.; Khvostichenko, D.; Chen, J.; Boulatov, R. A Molecular Force Probe. Nat. Nanotechnol. 2009, 4, 302−306. (36) Li, W.; Gräter, F. Atomistic Evidence of How Force Dynamically Regulates Thiol/Disulfide Exchange. J. Am. Chem. Soc. 2010, 132, 16790−16795. (37) Saitta, A. M.; Soper, P. D.; Wasserman, E.; Klein, M. L. Influence of a Knot on the Strength of a Polymer Strand. Nature 1999, 399, 46−48. (38) Smalø, H. S.; Rybkin, V. V.; Klopper, W.; Helgaker, T.; Uggerud, E. Mechanochemistry: The Effect of Dynamics. J. Phys. Chem. A 2014, 118, 7683−7694. (39) Makarov, D. E. The Effect of a Mechanical Force on Quantum Reaction Rate: Quantum Bell Formula. J. Chem. Phys. 2011, 135, 194112. (40) Wheeler, S. E.; Houk, K. N.; von Ragué Schleyer, P.; Allen, W. D. A Hierarchy of Homodesmotic Reactions for Thermochemistry. J. Am. Chem. Soc. 2009, 131, 2547−2560. (41) Stauch, T.; Günther, B.; Dreuw, A. Can Strained Hydrocarbons Be “Forced” to Be Stable? J. Phys. Chem. A 2016, 120, 7198−7204. (42) Dodziuk, H. Strained Hydrocarbons: Beyond the Van’t Hoff and Le Bel Hypothesis, 1st ed.; Wiley-VCH: Weinheim, 2009. (43) Koumura, N.; Zijlstra, R. W.; van Delden, R. A.; Harada, N.; Feringa, B. L. Light-Driven Monodirectional Molecular Rotor. Nature 1999, 401, 152−155. (44) Hugel, T.; Holland, N. B.; Cattani, A.; Moroder, L.; Seitz, M.; Gaub, H. E. Single-Molecule Optomechanical Cycle. Science 2002, 296, 1103−1106. (45) Becke, A. D. A New Mixing of Hartree-Fock and Local DensityFunctional Theories. J. Chem. Phys. 1993, 98, 1372−1377. (46) Dunning, T. H. Gaussian Basis Sets for Use in Correlated Molecular Calculations. I. The Atoms Boron Through Neon and Hydrogen. J. Chem. Phys. 1989, 90, 1007−1023. (47) Liu, F.; Morokuma, K. Computational Study on the Working Mechanism of a Stilbene Light-Driven Molecular Rotary Motor: Sloped Minimal Energy Path and Unidirectional Nonadiabatic Photoisomerization. J. Am. Chem. Soc. 2012, 134, 4864−4876. (48) Beyer, M. K. Coupling of Mechanical and Chemical Energy: Proton Affinity As a Function of External Force. Angew. Chem., Int. Ed. 2003, 42, 4913−4915.
(5) Neuman, K. C.; Nagy, A. Single-Molecule Force Spectroscopy: Optical Tweezers, Magnetic Tweezers and Atomic Force Microscopy. Nat. Methods 2008, 5, 491−505. (6) Cravotto, G.; Gaudino, E. C.; Cintas, P. On the Mechanochemical Activation by Ultrasound. Chem. Soc. Rev. 2013, 42, 7521−7534. (7) Stauch, T.; Dreuw, A. Advances in Quantum Mechanochemistry: Electronic Structure Methods and Force Analysis. Chem. Rev. 2016, 116, 14137−14180. (8) Nikitina, E. A.; Khavryutchenko, V. D.; Sheka, E. F.; Barthel, H.; Weis, J. Deformation of Poly(dimethylsiloxane) Oligomers Under Uniaxial Tension: Quantum Chemical View. J. Phys. Chem. A 1999, 103, 11355−11365. (9) Beyer, M. K. The Mechanical Strength of a Covalent Bond Calculated by Density Functional Theory. J. Chem. Phys. 2000, 112, 7307−7312. (10) Ong, M. T.; Leiding, J.; Tao, H.; Virshup, A. M.; Martínez, T. J. First Principles Dynamics and Minimum Energy Pathways for Mechanochemical Ring Opening of Cyclobutene. J. Am. Chem. Soc. 2009, 131, 6377−6379. (11) Ribas-Arino, J.; Shiga, M.; Marx, D. Understanding Covalent Mechanochemistry. Angew. Chem., Int. Ed. 2009, 48, 4190−4193. (12) Wolinski, K.; Baker, J. Theoretical Predictions of Enforced Structural Changes in Molecules. Mol. Phys. 2009, 107, 2403−2417. (13) Kochhar, G. S.; Heverly-Coulson, G. S.; Mosey, N. J. Theoretical Approaches for Understanding the Interplay Between Stress and Chemical Reactivity. Top. Curr. Chem. 2015, 369, 37−96. (14) Avdoshenko, S. M.; Konda, S. S. M.; Makarov, D. E. On the Calculation of Internal Forces in Mechanically Stressed Polyatomic Molecules. J. Chem. Phys. 2014, 141, 134115. (15) Brandhorst, K.; Grunenberg, J. Characterizing Chemical Bond Strengths Using Generalized Compliance Constants. ChemPhysChem 2007, 8, 1151−1156. (16) Brandhorst, K.; Grunenberg, J. How Strong Is It? The Interpretation of Force and Compliance Constants As Bond Strength Descriptors. Chem. Soc. Rev. 2008, 37, 1558−1567. (17) Wang, Y.; Zhang, C.; Zhou, E.; Sun, C.; Hinkley, J.; Gates, T. S.; Su, J. Atomistic Finite Elements Applicable to Solid Polymers. Comput. Mater. Sci. 2006, 36, 292−302. (18) Smalø, H. S.; Uggerud, E. Breaking Covalent Bonds Using Mechanical Force, Which Bond Breaks? Mol. Phys. 2013, 111, 1563− 1573. (19) Stacklies, W.; Vega, M. C.; Wilmanns, M.; Gräter, F. Mechanical Network in Titin Immunoglobulin from Force Distribution Analysis. PLoS Comput. Biol. 2009, 5, e1000306. (20) Stauch, T.; Dreuw, A. A Quantitative Quantum-Chemical Analysis Tool for the Distribution of Mechanical Force in Molecules. J. Chem. Phys. 2014, 140, 134107. (21) Stauch, T.; Dreuw, A. On the Use of Different Coordinate Systems in Mechanochemical Force Analyses. J. Chem. Phys. 2015, 143, 074118. (22) Stauch, T.; Dreuw, A. Predicting the Efficiency of Photoswitches Using Force Analysis. J. Phys. Chem. Lett. 2016, 7, 1298−1302. (23) Gossweiler, G. R.; Hewage, G. B.; Soriano, G.; Wang, Q.; Welshofer, G. W.; Zhao, X.; Craig, S. L. Mechanochemical Activation of Covalent Bonds in Polymers with Full and Repeatable Macroscopic Shape Recovery. ACS Macro Lett. 2014, 3, 216−219. (24) Black, A. L.; Lenhardt, J. M.; Craig, S. L. From Molecular Mechanochemistry to Stress-Responsive Materials. J. Mater. Chem. 2011, 21, 1655−1663. (25) Browne, W. R.; Feringa, B. L. Making Molecular Machines Work. Nat. Nanotechnol. 2006, 1, 25−35. (26) Bakken, V.; Helgaker, T. The Efficient Optimization of Molecular Geometries Using Redundant Internal Coordinates. J. Chem. Phys. 2002, 117, 9160−9174. (27) Shao, Y.; Gan, Z.; Epifanovsky, E.; Gilbert, A. T. B.; Wormit, M.; Kussmann, J.; Lange, A. W.; Behn, A.; Deng, J.; Feng, X.; et al. Advances in Molecular Quantum Chemistry Contained in the QChem 4 Program Package. Mol. Phys. 2015, 113, 184−215. 1048
DOI: 10.1021/acs.accounts.7b00038 Acc. Chem. Res. 2017, 50, 1041−1048
