Advances in Quantum Mechanochemistry: Electronic Structure

Review pubs.acs.org/CR

Advances in Quantum Mechanochemistry: Electronic Structure Methods and Force Analysis Tim Stauch* and Andreas Dreuw* Interdisciplinary Center for Scientific Computing, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany ABSTRACT: In quantum mechanochemistry, quantum chemical methods are used to describe molecules under the influence of an external force. The calculation of geometries, energies, transition states, reaction rates, and spectroscopic properties of molecules on the force-modified potential energy surfaces is the key to gain an in-depth understanding of mechanochemical processes at the molecular level. In this review, we present recent advances in the field of quantum mechanochemistry and introduce the quantum chemical methods used to calculate the properties of molecules under an external force. We place special emphasis on quantum chemical force analysis tools, which can be used to identify the mechanochemically relevant degrees of freedom in a deformed molecule, and spotlight selected applications of quantum mechanochemical methods to point out their synergistic relationship with experiments.

CONTENTS 1. Introduction 2. General Considerations 2.1. Influence of Force on the Potential Energy Surface 2.2. Influence of Thermal Oscillations on Bond Rupture 2.3. Relation of Force and Energy 3. Computational Methods for the Description of Mechanically Deformed Molecules 3.1. Simulating Forces via Constrained Geometries 3.2. Approaches that Consider the Force Explicitly 3.3. Bell Theory and Related Approaches 3.4. Mechanochemical Benchmarking 4. Force Analysis Tools 4.1. Coordinate Systems and Their Use in Force Analyses 4.2. Force Analyses Using Only One Coordinate of Interest 4.3. Internal Forces 4.4. Compliance Constants 4.5. Force Analysis Based on the Harmonic Approximation 4.6. Force Analyses from Molecular Dynamics Simulations 4.7. Other Force Analysis Tools 5. Applications of Quantum Mechanochemistry 5.1. Knotted Polymers 5.1.1. Abundance of Knots in Polymers 5.1.2. Unraveling the Mechanical Properties of Polymer Knots Using Force Analysis 5.2. Sonochemistry 5.2.1. Polymers in an Ultrasound Bath 5.2.2. Cycloreversion of Triazoles © 2016 American Chemical Society

5.2.3. Ab Initio Nanoreactor 5.3. Ball Milling and Grinding 5.4. Molecular Force Probes 5.4.1. Spectroscopic Properties of Molecules under Stress 5.4.2. Force Probes in Polymers 5.4.3. Force Probes in Proteins 5.5. Using Photoswitches to Convert Light into Mechanical Energy 5.5.1. Isomerization of Azobenzene under External Force 5.5.2. Stiff-Stilbene 6. Future Directions Author Information Corresponding Authors Notes Biographies Acknowledgments References

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1. INTRODUCTION In mechanochemistry, forces are used to initiate chemical reactions. The origin of mechanochemistry dates back to prehistoric times: mortars and pestles have been used as early as the stone age.1 The mechanical force applied to the material during the grinding process can not only be exploited for crushing solids but also be used for the initiation of chemical processes. The first written documentation of a mechanochemical experiment dates back to 315 B.C., when Theophrastus of Eresus, a student of Aristotle, noted in his book On Stones that grinding of cinnabar in a copper mortar with a copper pestle in the presence of vinegar yields mercury:1,2

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Received: July 18, 2016 Published: October 21, 2016 14137

DOI: 10.1021/acs.chemrev.6b00458 Chem. Rev. 2016, 116, 14137−14180

Chemical Reviews HgS + Cu → Hg + CuS

Review

applying forces to a single molecule is by conducting single molecule force spectroscopy (SMFS) experiments. For this, atomic force microscopy (AFM) is the most commonly used technique. In a typical AFM experiment, a macromolecule (e.g., a protein) is anchored between an AFM tip and a surface.2 The rupture of covalent bonds or the unfolding of secondary structure domains during stretching of a protein are then monitored by the force-versus-extension curves generated in these experiments, which has been used extensively to measure the mechanical response of molecular biological systems. In a groundbreaking piece of work Rief and co-workers measured the mechanochemical properties of titin, a crucial component of muscles and an important research subject in mechanobiology.7 Titin consists of repetitive immunoglobulin domains, each of which is folded into a seven-stranded β-barrel. Unfolding titin yields a sawtooth-like force-versus-extension curve with a periodicity between 25 and 28 nm and maximum forces between 150 and 300 pN. It was shown that the individual immunoglobulin domains unfold successively, and the weakest domain unfolds first, since rupture forces increase in each unfolding step. In the realm of covalent mechanochemistry, Wiita and co-workers studied the reduction of a sulfur− sulfur bond by enzymes under mechanical force using an AFM setup.8,9 Computational studies confirmed that mechanical force has an influence on the rupture of the sulfur−sulfur bond, which in many cases enhances reduction.10−13 Despite this remarkable success, the interpretation of forceversus-extension curves is far from trivial and depends, for example, on the stiffness of the cantilever.14 Computationally, however, it has been possible to calculate force-versus-extension curves of biological polymers that reproduce experiments remarkably well by combining statistical mechanics models of polymers and single molecular elasticities from quantum mechanical calculations.15 Moreover, in most AFM studies, the point of bond rupture cannot be identified uniquely.16,17 In many cases, it is not clear whether a bond in the main chain of the macromolecule breaks or whether rupture occurs instead at one of the bonds connecting the macromolecule to the AFM tip or the surface. Although quantum chemical calculations are limited to small- and medium-sized systems, quantum mechanochemistry can tremendously aid in the interpretation of the experiments by determining the geometry of the stretched molecule and the energetics during stretching as well as by identifying the mechanically relevant degrees of freedom in the molecule via force analyses. Introducing the quantum chemical methods required for these tasks and discussing the implications for the experiments are the main focus of this review. We will refrain from extensive discussions of molecular dynamics (MD) simulations and their relevance for mechanobiology and instead focus on covalent and noncovalent quantum mechanochemistry. Experimental setups, technical aspects, and applications of SMFS have been reviewed extensively.2,18−28 A number of computational studies on SMFS are reviewed in ref 29. Furthermore, a deluge of literature has been published during the past two decades on the experimental30−42 and theoretical43−70 aspects of SMFS in the context of mechanobiology. An alternative experimental approach to apply mechanical forces to single molecules is the use of ultrasound baths, an approach that is known as sonochemistry. While ultrasound baths have traditionally been applied for purposes like cleaning, efficient mixing, and solid activation, a mechanical component has been identified, which can be used to trigger various

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Although mortar and pestle were standard instruments in the laboratories of alchemists, few specifics on mechanochemical reactions were documented until the 19th century.1 We can only assume that scientists predominantly thought of the mortar and pestle as a mere tool to grind solids and were not fully aware of the mechanochemical reactions they were initiating. It was not until 1919 that the term “mechanochemistry” found its way into textbooks.3 In Ostwald’s Handbuch der Allgemeinen Chemie,4 mechanochemistry was established as the fourth of the chemical subclasses, in addition to thermochemistry, photochemistry, and electrochemistry. More information on the history of mechanochemistry can be found in ref 1. Today, mechanochemistry is a lively field with an extraordinarily wide range of applications.5 In its Gold Book, the IUPAC defines a mechanochemical reaction as a “[c]hemical reaction that is induced by the direct absorption of mechanical energy” and proceeds to give a few examples for sources of this mechanical energy.6 Several approaches to apply forces to single molecules are commonly used today, which will be briefly introduced below. The main advantage of using mechanical force to initiate chemical reactions is its vectorial nature, which, at least in principle, allows one to conduct experiments in a more controlled way than by standard thermochemistry. While mechanochemical experiments have been conducted for eons, the systematic description of mechanochemical processes with quantum chemical methods has lagged far behind for a long time. In fact, the field of “quantum mechanochemistry” is not even two decades old. In quantum mechanochemistry, quantum chemical methods are used for the description of mechanochemical processes at the molecular level, thus providing an in-depth understanding of the influence of mechanical forces on molecules and materials. Such a comprehension is desperately needed, since, despite the long history of the field, mechanochemistry is still a largely alchemistic discipline. However, we are currently witnessing an unprecedented gain in understanding of mechanochemical processes, and quantum mechanochemistry has made key contributions to this important development. The quantum chemical description of mechanically deformed molecules, which has been conducted systematically since the turn of the millennium, and above all, the development of computational force analysis tools throughout the past five to ten years has paved the way for a more efficient use of mechanical force in academic and industrial laboratories. This review focuses on recent advances in the field of quantum mechanochemistry with an emphasis on theory and applications of force analysis tools. These computational tools can be used to identify the mechanically relevant degrees of freedom in a molecule, or in other words, to answer the question which part of a molecule is most likely to rupture when the stretching force is sufficiently strong. In addition, methods for the calculation of geometries, energies, transition states, reaction rates, etc. for molecules under external forces will be discussed in detail. Throughout this review, we will point out connections to experiments, since we are convinced that the full potential of mechanochemistry can only be exploited by the synergy between theory and experiment. Several experimental methods have been developed that enable the transmission of forces from a macroscopic system to single molecules in a controlled way. One efficient way of 14138


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chemical processes.71,72 The propagation of an ultrasonic wave through a liquid generates small cavitational bubbles.73−75 Upon collapse, the gas entrapped within these cavities is highly compressed, so that, besides tensile forces, pressures of several hundred atmospheres and temperatures of several thousand Kelvin are generated.72,73 Chemical reactions like bond rupture occur at the bubble interface or inside the bubbles, given that reagents are sufficiently volatile.76 Applications of ultrasound baths in chemical synthesis are diverse and include switching of a reaction mechanism from an aromatic electrophilic to an aliphatic nucleophilic substitution,77 the apparent circumvention of the Woodward−Hoffmann rules,78 the trapping of a diradical transition state,79 and disulfide metathesis.80 Mechanisms and applications of sonochemistry have been reviewed several times.72,81−84 The advantages of conducting mechanochemical syntheses in ultrasound baths include simplicity, energy-efficiency, and wide applicability of the approach as well as the possibility to conduct reactions at low ambient temperatures.72,73 However, as described above, the processes in an ultrasound bath are complex and not purely mechanochemical. They involve several length and time scales in addition to elevated pressures and temperatures. Hence, a systematic optimization of sonochemical reactions based on a thorough understanding of the underlying processes is hardly possible until the present day. For the same reasons, the quantum chemical modeling of sonochemical reactions is a challenging task. Nevertheless, remarkable progress has been made recently in understanding the transduction of mechanical forces through polymers,85,86 the activation of mechanophores (molecules that react to mechanical stress by significant structural changes), and chemical reactivity in an ultrasound bath.87,88 The quantum chemical methods used for this purpose as well as their applications in sonochemistry are presented in this review. Ball milling and grinding techniques are another widely used method to transmit mechanical energy to molecules. The impact of balls in a ball mill and the grinding of a pestle in a mortar generate forces that can be used both for crushing solids and for the initiation of chemical reactions. Recently, for example, a solvent-free two-step mechanochemical synthesis of tetra-substituted porphyrins was presented.89 This application involves grinding with a mortar and pestle as well as treatment in a ball mill and is an example of a mechanochemical carbon− carbon bond formation. As such, it nicely demonstrates the power of mechanochemical synthesis as a cheap, simple, energy-efficient, and typically solvent-free synthetic route as well as its potential as a sustainable and environment-friendly alternative to thermochemistry. Ball milling and grinding techniques as well as their applications for various purposes have been reviewed recently.5,90−92 However, scientists in this field still resort mostly to rules of thumb when it comes to designing and optimizing syntheses conducted in a ball mill or with mortar and pestle. This deficiency is due to a lack of understanding of the mechanochemical processes at the molecular level and the coupling to the macroscopic experimental setups. In this review, we present recent efforts in providing a molecular understanding of mechanochemistry and discuss their potential to predict the outcomes of mechanochemical ball milling and grinding experiments. Mechanochemistry is not limited to observing the effect of external mechanical forces on molecules. A suitably chosen molecule can itself apply forces to its environment, and this concept is exploited in molecular machines and nano-

mechanical devices. These systems are often powered by light, which induces a structural change in a photoswitch.93 Advantageously, light is an abundant resource, and the reaction time of photoswitches usually lies within the picosecond range. The change in spatial extension of a photoswitch upon photoisomerization generates forces that can be used to perform mechanical work in the chemical environment. On the basis of this approach, Hugel and co-workers have designed a single-molecule opto-mechanical device,93 in which azobenzene94,95 is used as a molecular actuator. While related optomechanical devices are based on unidirectional rotational motion,96−100 the cis-trans-photoisomerization of azobenzene offers the advantages of a relatively high yield, reversibility, and a significant change in spatial extension. The mechanical properties of azobenzene and its derivatives are discussed in section 5.5.1. In their study, Hugel and co-workers anchored a polymer of approximately 47 azobenzene units between an AFM tip and a glass surface.93 Optical pumping at 365 nm was used to induce excited state photoisomerization of the azobenzene molecules, resulting in a difference in spatial extension of the polymer of approximately 6 nm. With this setup, the authors were able to exert a force of 200 pN to the AFM tip. Other groups exploited the mechanical work performed by azobenzene to induce structural changes in peptides,101−109 DNA,110−112 and synthetic foldamers.113−115 Photoswitches from the spiropyran family were also shown to be applicable in this context.116,117 Quantum mechanochemistry can make a key contribution to our understanding of the mechanical effects during photoswitching by calculating the efficiency of energy conversion and transmission processes. With the use of computational methods, it has been possible, for example, to juxtapose purely mechanical, photochemical, and combined photomechanochemical switching of the azobenzene photoswitch.118−120 This knowledge paves the way for an optimization of molecular machines by increasing the amount of mechanical work they can perform. Profitably, computations of force-modified molecular properties and force analyses provide detailed insights into the processes at the molecular level. Of course, quantum mechanochemical methods can be used in their own right, without any relation to experiments. While conclusions drawn from an experiment are necessarily limited to the molecule under investigation, general concepts can be derived by quantum mechanochemical theory and calculations of molecules under external forces. This is the key strength of quantum mechanochemistry, and the primary reason why a systematic optimization of many mechanochemical processes has become possible during the past two decades. The major theoretical concepts and computational methods that are contributing to this ongoing development are discussed in this review in great detail. This review is structured as follows: section 2 deals with some general considerations on molecules that are subjected to an external force and their force-modified properties. In section 3, the most widely used electronic structure methods for computing geometries, energies, reaction rates, and other properties of mechanically deformed molecules are introduced. In section 4, we present several mechanochemical force analysis tools that have been developed within the past five to ten years. The purpose of this in-depth treatment is to illuminate the theoretical background of force analysis tools and to emphasize their usefulness in interpreting mechanochemical experiments by identifying the mechanically relevant degrees of freedom in a 14139


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by Freund, who developed a 1D energy landscape model in which a large number of identical bonds are subjected to a constraint.123 While the probability of bond dissociation without external force is extremely small for a strong bond described by a steep potential well, the rupture probability increases rapidly with external force, demonstrating that mechanochemical bond rupture is a statistical process.

molecule. In section 5, we shine spotlights on selected applications of the theoretical methods discussed in section 3 and the force analysis tools introduced in section 4. This demonstrates both the usefulness of quantum mechanochemistry by itself and the benefits of a synergistic treatment with experiments.

2. GENERAL CONSIDERATIONS

2.2. Influence of Thermal Oscillations on Bond Rupture

2.1. Influence of Force on the Potential Energy Surface

As shown in the previous section, mechanical force deforms the PES, thereby changing the activation energy for bond rupture. In the simple case of a covalent bond, a stretching force reduces this energy barrier and the lifetime of the bond. In addition, consideration of a simple Morse potential shows that also temperature has a significant impact on the rupture force of a bond. At room temperature, the available thermal energy amounts to approximately 0.6 kcal/mol,21 which, in the case of a bond that is sufficiently weakened by a mechanical force, can be enough to overcome the activation energy barrier for bond rupture. Physically, the thermal oscillations in a molecule periodically lead to stretching and compression of bonds. Bond rupture is most likely to occur at the moment of highest amplitude, even if the applied force is lower than the “bond rupture force”, which is usually calculated as the force needed to rupture a bond at 0 K in static quantum chemical calculations. The influence of temperature on mechanical bond rupture has already been documented in literature. Lo and co-workers, for example, measured the biotin−avidin bond rupture forces at six different temperatures between 13 and 37 °C in an AFM setup.124 More than 200 individual measurements were carried out at each temperature, and it was found that the unbinding force decreases by a factor of 5 within this temperature range (Figure 2). Although this experiment probed only one type of

When an external force acts on a molecule, the potential energy surface (PES) is affected in a way that chemical reactivity is either enhanced or suppressed. This effect has been described by Kauzmann and Eyring as early as 1940, who investigated the effect of a constant external pulling force, which can be modeled as a linear, falling potential, on a bond with a Morsetype potential.121 If a bond of strength D does not undergo any vibrations, a dissociation energy of magnitude D must be supplied to break it (dotted line in Figure 1). This activation

Figure 1. An external force is represented by a linear falling potential (dashed line), which is added to the Morse potential of the unstretched bond (dotted line), thus resulting in a force-deformed Morse potential (solid line). The new dissociation energy D′ is smaller than D in the case of a stretching force. The scheme was adapted from ref 121.

energy can, however, be reduced by an external force, the extent of which of course depends on the strength of the external force (dashed line in Figure 1). The resulting deformed PES VF(r) can, in the case of a single bond stretched by a constant stretching force F0, be calculated via VF(r ) = Vab initio(r ) − F0Δr

Figure 2. Biotin−avidin unbinding forces as a function of temperature, measured in an AFM experiment. Reprinted from ref 124. Copyright 2002 American Chemical Society.

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noncovalent interaction, a similar behavior is expected in other systems and the results can be assumed to be qualitatively general. Measurements of the unbinding forces of complementary DNA strands at different temperatures between 6 and 36 °C confirm that unbinding forces decrease with increasing temperature.125 For an accurate description of thermal effects and their influence on mechanical bond rupture, static quantum chemical calculations are of limited usefulness. Thermal oscillations can, however, be included explicitly in dynamics calculations at finite temperatures. Smalø and co-workers, for example, have investigated the influence of temperature on rupture forces

where a linear potential involving F0 and the displacement Δr is subtracted from the total energy Vab initio(r). These considerations establish a one-dimensional (1D) picture of mechanochemistry, since only one chemical bond is considered. In this sense, eq 2 constitutes the basis for several quantum mechanochemical methods for the calculation of forcedeformed molecular geometries, which are of course multidimensional in nature (cf. section 3.2). An important consequence of this observation is that every stretched bond has a finite lifetime, with larger pulling forces typically leading to shorter lifetimes.2,21,122 This was also shown 14140


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equal to the number of internal coordinates M. In the case of nonredundant internal coordinates, M = 3N − 6 because translation and rotation are not included in the set of internal coordinates. If redundant internal coordinates are used, however, M > 3N − 6, the implications of which are discussed in section 4.1. The force of a certain magnitude that drives two atoms in a molecule apart is defined in a rather ambiguous way throughout literature. It is illustrative to consider a scenario in which two atoms are stretched apart by forces of 2 nN applied to each of these atoms along the connecting vector between them, pointing outward. One choice is to talk about a total force of 4 nN stretching these atoms apart, since a force of 2 nN is acting on each of these atoms. This definition has been used for the description of stretched isolated molecules in the gas phase or in solution. However, another definition is also in use. If a molecule is, for example, attached to a glass surface and an AFM cantilever exerts a stretching force of 2 nN on the molecule then a force of the same magnitude but opposite direction has to act on the other side of the molecule so that the molecule’s overall translation is zero. Therefore, it is reasonable to talk about a force of 2 nN pulling the atoms apart in opposite directions. This definition of the force is particularly intuitive from a physical point of view, since actio est reactio. Needless to say, this definition of the force is often used in the discussion of AFM experiments. We find this definition physically most reasonable and chose to apply it in recent publications.130−133 Often in the literature it is not made clear which of these two definitions of the force is used by the authors. To avoid confusion, we highly recommend to use the second definition of the stretching force or at least to describe clearly how the force is applied in the corresponding calculation. Throughout this review, we will indicate those studies in which it is clear that the definition actio est reactio was not used. The calculation of the stress energy of a mechanically deformed molecule is an alternative to the interpretation of internuclear forces. Classically, the force is the (negative) derivative of the energy with respect to the displacement, so that the mechanical work performed on a system that is moved along a path q by a force F can be calculated via

and the time needed for bond rupture by applying various theoretical methods to model polymers.126 Using dynamics calculations, two different temperatures were tested and it was shown that both rupture forces and the time needed for bond rupture decrease with increasing temperature. In another dynamics study, Dopieralski and co-workers generated forcetransformed free energy surfaces, which allow the calculation of free energy barriers of mechanochemical transformations in the presence of finite temperature oscillations.127 Applications of this method include the mechanochemical ring-opening of cyclopropane mechanophores127 and mechanochemical disulfide bond cleavage.13 As this review focuses on static quantum mechanochemical methods, however, such dynamical approaches are not discussed in detail here. Computational studies on the effect of different temperatures on rupture forces in mechanochemical pulling scenarios are scarce but desperately needed for comprehending diverse mechanochemical experiments that are conducted at or near room temperature. The neglect of thermal effects in standard static quantum mechanochemical methods may contribute to the effect that bond rupture forces measured in AFM experiments are severely overestimated by static quantum mechanochemical calculations. 2.3. Relation of Force and Energy

A considerable part of this review deals with mechanochemical force analysis tools (cf. section 4). The aim of force analyses is to identify the mechanochemically relevant degrees of freedom in a molecule, or, in other words, its force-bearing scaffold. As will be shown, force analysis tools provide crucial insights into mechanochemical processes at the most fundamental level. In some of these tools, forces acting between different parts of the molecule are quantified and visualized, thus allowing the identification of those regions of the molecule that are particularly strained by the external load. An advantage of force analysis tools that are based on the interpretation of internuclear forces is that the nuclear gradient vector is obtained automatically by every standard quantum chemical and molecular dynamics (MD) program package during geometry optimizations or simulations. Once the gradients are calculated, it is straightforward to visualize and interpret the forces acting in different parts of the molecule. The different partitioning schemes used for this purpose are discussed in section 4. Usually, the gradient is calculated in Cartesian coordinates, but it is straightforward to transform it into internal coordinates by using Wilson’s B-matrix.128,129 The elements of B are defined as Bij =

E=

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where qi and xj denote internal and Cartesian coordinates, ⎯g can respectively. The gradient vector in internal coordinates ⎯→ q then be generated from the gradient vector in Cartesian ⎯g with the generalized inverse of the transposed coordinates ⎯→ x t + B-matrix, (B ) : ⎯→ ⎯g = (Bt )+⎯→ ⎯g q

x

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The interpretation of mechanochemical processes in terms of stress energy or mechanical work has the advantage that, in contrast to force, energy is a scalar quantity and not a vector. Therefore, listing, visualizing and, most importantly, interpreting the results of force analyses in which the distribution of stress energy among various degrees of freedom in a mechanically deformed molecule is quantified is often easier than in the case of force analyses based on partitioning schemes of forces acting between different parts of the molecule. The advantages and disadvantages of considering internuclear forces and stress energy contributions in force analysis tools are discussed thoroughly in section 4.

∂qi ∂xj

∫ − ddEq dq = ∫ F dq

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3. COMPUTATIONAL METHODS FOR THE DESCRIPTION OF MECHANICALLY DEFORMED MOLECULES In this section we introduce the quantum chemical methods that are commonly used to calculate geometries, energies,

The generalized inverse, or pseudoinverse, is a generalization of the inverse of a matrix in that it is usually also defined for nonsquare matrices. Clearly, in the case of the B-matrix this generalization is necessary, since the number of Cartesian coordinates 3N (with N being the number of atoms) is not 14141


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hydrogen atoms over a desired range. Increasing r(H−H) leads to an elongation of the four covalent bonds along the main scaffold of the molecule. The bond breaking point can be determined by finding the inflection point of the COGEF potential, since this is the point of maximum force. If the molecule is stretched further, it is energetically more favorable to relax and shorten all bonds except for the weakest, which is the one that breaks in the pulling scenario. This result is in good agreement with a related study on stretched n-butane.141 In this study, three PES minima, in each of which one bond is longer than the other two, was observed. At the B3LYP138,139/ D95(d,p)140 level of theory, the weakest bond in SiH3CH2CH3 is the Si−C bond and the energy at the bond breaking point amounts to 220 kJ/mol.122 Since four bonds are stretched in the COGEF coordinate (H−Si, Si−C, C−C, and C−H), it was proposed that each bond is activated by approximately 55 kJ/ mol. However, this value is certainly a poor estimate of the energy stored in each bond, since (1) different kinds of bonds with different stiffnesses are compared to each other and (2) other internal coordinates (e.g., bond angles) that are also deformed during the stretching process are not considered in the energy dissection. Therefore, the use of force analysis tools (cf. section 4) to quantify the energy contributions in the different degrees of freedom in the molecule is highly recommended. Typically, the rupture of strong and short bonds requires large forces (Table 1). For the dissociation of the Si−Si bond,

reaction rates and other properties of mechanically deformed molecules. These methods have proven outstandingly useful in the interpretation of mechanochemical processes at the molecular level (cf. section 5). In addition, they provide the foundation for the force analysis tools introduced in section 4. As this review focuses on static quantum chemical methods to describe mechanically deformed molecules, dynamics calculations, in particular Car−Parrinello Molecular Dynamics (CPMD) simulations,134,135 will not be described here. The interested reader is referred to ref 29 for a review on the applications of such ab initio Molecular Dynamics (AIMD) simulations in the context of mechanochemistry as well as to refs 53, 136 and 137 for a discussion of the use of classical MD simulations to stretch and unfold large molecules like proteins. 3.1. Simulating Forces via Constrained Geometries

The application of a mechanical force to a molecule results in a change of the nuclear configuration. Conversely, if a deformed molecular geometry is generated, the force needed to induce this change mechanically can be calculated as the nuclear gradient of the energy at this geometry. This realization led to the development of the computational COGEF (constrained geometries simulate external force) method.122 While the relation between changes in a structural parameter and mechanical force had been realized before,31 the term COGEF was coined by Beyer in 2000. In a typical COGEF calculation, the distance r between two atoms that shall be subjected to an external force is fixed and an otherwise relaxed geometry optimization is carried out. For this calculation, any quantum chemical method that allows constrained geometry optimizations can be used. Varying r within a range of values simulates an isometric pulling scenario and generates the COGEF potential. A typical pulling scenario is shown in Figure 3, in which the stretching of the SiH3CH2CH3 molecule is simulated by incrementing the distance r(H−H) between the terminal

Table 1. Bond Rupture forces Fmax in nanonewtons of Different Bonds in Different Molecules Calculated at the B3LYP138,139/D95(d,p)140 Level of Theory via the Static COGEF Approacha

a

bond type

molecule

Fmax (nN)

H−H H−F Cl−Cl OO N≡N C−C Si−Si O−H O−H

H2 HF Cl2 O2 N2 H3CCH2CH3 H3SiSiH2SiH3 H2O HOCl

8.31 10.79 4.60 14.89 26.27 6.92 3.28 9.77 9.18

Values taken from ref 122.

for example, much lower forces (3.28 nN) are needed than for the rupture of the C−C bond (6.92 nN) in an analogous molecule.122 Moreover, the rupture forces depend critically on the chemical environment: the O−H bond in H2O ruptures at a higher force (9.77 nN) than in the HOCl molecule (9.18 nN). For the selection of a COGEF coordinate, it should be kept in mind that the choice of a different coordinate r changes the rupture force in a nontrivial manner (Table 2). Using the H−H coordinate instead of the O−H bond in H2O as the stretching coordinate, for example, increases the rupture force of the O− H bond from 9.77 to 10.27 nN. Hence, great care should be taken for the COGEF coordinate to indeed model the mechanochemical process of interest. The calculation of bond rupture forces via the COGEF method is an extraordinarily helpful tool to judge the mechanochemical strength of a molecule and to identify the point of bond rupture. COGEF calculations have been applied extensively since the introduction of the approach around the

Figure 3. COGEF potential of the molecule SiH3CH2CH3, in which the distance r between the terminal hydrogen atoms was varied between 4 and 8 Å, calculated at the B3LYP138,139/D95(d,p)140 level of theory. A Morse potential with the same maximum slope as the COGEF potential is included (solid line). From left to right, the structures indicate the equilibrium geometry, the geometry at maximum force, and a geometry at a force that exceeds the bond rupture force. Reprinted with permission from ref 122. Copyright 2000 American Institute of Physics. 14142


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Table 2. Bond Rupture Forces Fmax in nanonewtons for Different Stretching Coordinates r Calculated at the B3LYP138,139/D95(d,p)140 Level of Theory via the Static COGEF Approacha

a

bond

molecule

stretching coordinate

Fmax (nN)

O−H O−H O−F O−F O−Cl O−Cl

H2O H2O HOF HOF HOCl HOCl

O,H H,H O,F H,F O,Cl H,Cl

9.77 10.27 7.00 6.57 5.60 5.42

directly by including the mechanical force instead of indirectly by imposing distance constraints. In principle, this procedure yields the exact distortion of the molecular structure at a given external force acting along a specific vector, thus affording calculations of reactant and transition state geometries and activation energies of mechanochemical reactions. In the following, each of the above methods is introduced. The FMPES method by Ong and co-workers generalizes the 1D picture of Kauzmann and Eyring (section 2.1) to many dimensions and allows the treatment of the PES under external force by quantum chemical methods.142 The FMPES can be constructed via

Values taken from ref 122.

Nattach

VFMPES(r ) = Vab initio(r ) +

turn of the millennium, and it is evident that the COGEF method has played a vital role in shaping the field of quantum mechanochemistry. However, the COGEF method is a static quantum chemical approach that by its nature does not include thermal effects (cf. section 2.2). Therefore, bond rupture forces calculated with the COGEF method are typically too large compared with experiments. Of course, rupture forces depend on the electronic structure method used in the geometry optimizations (cf. section 3.4). In the original paper, the rupture forces were found to differ by more than 5% when a different basis set was chosen. The main strength of the COGEF method, however, is its simplicity. The constrained geometry optimizations and calculations of the nuclear gradient needed to generate the COGEF potential and to calculate the bond rupture force are implemented in every major quantum chemistry program package.

∑

F0(∥rifix − ri∥ −

i

∥rifix

−

ri0∥)

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Here, the force-induced change in the Born−Oppenheimer PES Vab initio(r) is calculated by adding up contributions from each attachment point (AP), which is the point in the molecule where the external force is applied. F0 denotes the constant external force, r0i and ri are the initial and current positions of the AP, respectively, and rfix i is the invariant position of the pulling point (PP). A PP is a point in space toward which the force originating from the AP is directed. As can easily be seen, each of the Nattach AP-PP pairs i contributes a term of the type F0Δr to the total energy. Since the PPs are fixed in the laboratory frame, the molecule reorients in such a way that the external force is colinear with the vector connecting the two attachment points. Ong and co-workers used their knowledge of the FMPES to rationalize the pathways in a paradigmatic mechanochemical reaction, in which the Woodward−Hoffmann rules are apparently circumvented.142 The Woodward−Hoffmann rules are based on orbital symmetry and state that electrocyclic reactions are thermally allowed if they proceed via conrotatory pathways, whereas photochemically induced reactions preferentially proceed in a disrotatory manner.145−147 However, it could be shown that this set of rules can be circumvented in the

3.2. Approaches that Consider the Force Explicitly

As an alternative to the simulation of external forces via constrained geometries, force can also be considered explicitly, thus allowing the simulation of isotensional stretching. In 2009, three such computational methods were introduced, which were named FMPES (force-modified potential energy surface),142 EFEI (external force is explicitly included),143 and EGO (enforced geometry optimization).144 These methods allow the investigation of force-induced molecular distortions

Figure 4. Different pulling scenarios that facilitate the electrocyclic ring opening of cyclobutene. The fixed pulling points (PP) are symbolized by blue spheres. The yellow arrows indicate the molecular attachment points (AP). Left: in this pulling scenario, the thermally forbidden disrotatory pathway can be preferred if mechanical force is pulling two hydrogen atoms on the same side of the molecule apart. Right: the activation energy of the thermally allowed conrotatory pathway can be lowered even further if the force is stretching two hydrogen atoms on different sides of the molecule apart. Reprinted from ref 142. Copyright 2009 American Chemical Society. 14143


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practice, a geometry optimization of a system under the influence of an external force is carried out by adding the force to the gradient along the vector connecting two atoms. The geometry optimization under external force converges when the external force and the internal restoring force of the molecule cancel out, provided that the external force is smaller than the rupture force. The advantage of applying the force along a vector connecting two atoms is that overall translations and rotations of the molecule are ruled out. However, it has to be kept in mind that the direction of the force vector generally changes in every step of the geometry optimization. With the use of the EFEI approach, it was found that forces between 3.1 and 6.5 nN that are applied to the methyl groups of cis-1,2dimethylbenzocyclobutene activate the mechanically induced disrotatory ring-opening, which is forbidden thermodynamically.143 Above the threshold of 6.5 nN, the molecule dissociates. These findings confirm that different pathways can be activated by different forces. The difference between FMPES and EFEI should be stressed: while in the FMPES approach the pulling points are fixed in the laboratory frame, the pulling points in the EFEI method are fixed in the molecular frame. For the calculation of stationary points, however, FMPES and EFEI are equivalent, so that the two methods yield the same minima and minimum energy pathways. In the case of dynamics calculations, however, the use of pulling points that are fixed in the laboratory frame is crucial, since otherwise the surrounding that deforms the molecule mechanically can reorient and translate without any energy penalty. A comprehensive discussion of the particularities of the different pulling setups is given elsewhere.148 A disadvantage of the FMPES and EFEI approaches, in comparison to the COGEF method, is their limited availability in the major quantum chemical program packages. The implementation, however, is straightforward: in the case of the EFEI method, for example, a user-defined constant force is added to the nuclear gradient along the vector connecting two user-defined atoms. If this additional gradient points outward, mechanical pulling is simulated, while in the opposite case the molecule is compressed. In previous literature,143,153,154 the EFEI method was interfaced with the Turbomole155 program package, or alternatively, a modified version of Gaussian09156 was used. However, we would like to point out that the EFEI method is implemented in the Q-Chem157 program package (version 4.3 and newer), allowing for geometry optimizations and AIMD simulations in the ground and excited state under the influence of an external force using various density functionals and wave function-based methods. Moreover, the EFEI method will be featured in a future version of the ORCA program package.158 In most cases, but not always,159 it was found that the differences between the results obtained with COGEF on the one hand and FMPES and EFEI on the other hand are only subtle, leading to the notion that different points of view on the same mechanochemical process are provided by these methods. In fact, it was shown that the EFEI potential is the Legendre transform of the COGEF potential at stationarity.143 The close connection between the FMPES, EFEI, and COGEF methods was also found by Wolinski and Baker, who presented an analogous approach that considers the force explicitly.144 Their approach was also reported in 2009 and was called EGO/N (enforced geometry optimization, N = nuclei) in a subsequent publication. 160 The authors point out that geometry optimization techniques in which the force is considered

case of a mechanically induced ring opening of cyclobutene. The authors demonstrated that a tremendous lowering of the activation energy barrier of the thermally forbidden disrotatory pathway and its preference over the allowed conrotatory pathway can be achieved by mechanical force, provided that an appropriate pulling coordinate is chosen (Figure 4). Different computational studies on this and related reactions have been reviewed recently.148 The interested reader is also referred to ref 149 for an insightful discussion of the question of whether it is reasonable to use the term “Woodward−Hoffmann rules” in the context of mechanochemistry. The concept of the FMPES has aroused considerable interest in the past. Recently, a method to locate transition states (TS) on the FMPES was reported, which avoids the guess of the TS structure that is typically necessary in computational TS searches.150 Moreover, a further generalization of the FMPES concept was introduced by Subramanian and co-workers,151 who point out that in most previous theoretical descriptions of the FMPES the external force was considered to be a constant. However, the inclusion of a spatially varying force is necessary for the description of the transmission of macroscopically uniform stress to the atomic level. Furthermore, in SMFS experiments the constant force approximation is only applicable in the limiting case of “soft” handles through which the force is applied to the molecule.151 Hence, the authors introduce the GFMPES (“Generalized” FMPES) formalism, which is valid for both constant and spatially varying external load. The GFMPES is calculated via VG − FMPES(r ) = Vab initio(r ) +

s = rref

∫s=r

Fext(s) ds

(7)

where Fext(s) is a spatially varying external force. r and rref denote the desired and reference configurations, respectively. In the limit of a constant external force, the potential in eq 7 reduces to the FMPES potential. The G-FMPES approach describes not only the shift of stationary points upon mechanical deformation but also modifications in the curvature of the PES. Hence, also the Hessian changes tremendously if the external force is spatially varying. However, the G-FMPES approach does not allow for the introduction of new minima in the PES upon application of force. More sophisticated and computationally expensive methods have to be used to describe these effects. In their study, the authors applied pseudohydrostatic pressure instead of considering mechanical deformation like pulling explicitly, and the relation between these two scenarios remains unclear. Nevertheless, using the G-FMPES approach, it was shown that compressive stress increases the rotational barrier in ethane and a conformational barrier in a derivative of triazine. On the basis of these results, it was speculated that compressive pressure intensifies steric effects and forces molecules into more compact conformations. The G-FMPES method has also been used to investigate the mechanochemical effects of compression on a Diels−Alder reaction.152 A method similar to the FMPES appraoch is EFEI (external force is explicitly included), which was introduced by RibasArino and co-workers in 2009.143 In the EFEI method, a minimization of the expression VEFEI(x , F0) = Vab initio(x) − F0q(x)

(8)

with respect to the molecular geometry x is carried out. F0 is the constant external force, V ab initio(x) is the Born− Oppenheimer PES and q(x) is a structural parameter. In 14144


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explicitly only deliver stable structures until the bond rupture force is reached. After the bond breaking point, the molecular fragments are pulled further apart, and the geometry optimization fails to converge. The COGEF method, on the contrary, can also be used to calculate the molecular geometry after the bond is ruptured. Wolinski and Baker applied the EGO/N method to study conformational transitions of the pyranose ring and to identify several previously unknown isomers of azobenzene (C12H10N2), some of which are at least metastable. The capability of the EGO method to investigate structural isomerism is based on the formation of new bonds and was later used to generate and study different isomers of stilbene.161 Wolinski and Baker later proposed an extension of their method, which considers the influence of an external force not only on the nuclei (as in FMPES, EFEI, and EGO/N) but also on the electrons.160 In particular, the effect of force on the 1s core electrons was included in the EGO approach, since these electrons are considered by the authors to be rigidly connected to the nucleus. This extended approach is called EGO/NE (NE = nuclei and some electrons). The authors argue that the EGO/N method, which describes only the influence of force on the nuclei, is a zeroth-order approach, as it does not explicitly include the interaction of an AFM tip with the attached molecule. Theories that are used to describe AFM experiments should therefore involve an explicit treatment of the electron density. Hence, it is reasonable to assume that this neglect of electronic treatment is at least partially responsible for the systematic overestimation of AFM rupture forces by zerothorder approaches like EGO/N, FMPES, and EFEI. While thermal oscillations are certainly another key factor in this regard (cf. section 2.2), Bader pointed out that electrons cannot be neglected in the interpretation of AFM experiments.162 An explicit treatment of the effect of force on electrons in the EGO/NE approach is achieved by introducing an external force in the Hamiltonian in analogy to an external electric field. In contrast to traditionally used external electric fields in the Hamiltonian, however, this field acts on the nuclei and electrons in the same and not in the opposite direction. In this treatment, the authors chose to only include those 1s electrons that, in the AO basis, belong to the atom that is subjected to the force, since these electrons are considered to be “rigidly attached” to the nucleus. Although this choice gives rise to a strong basis set dependence, the EGO/NE method delivers results that are remarkably close to experiments. The EGO/NE method yields the same structural changes as the EGO/N method, but the forces needed to induce these changes are usually smaller by a factor of 3. For example, the EGO/N method yields forces between 1.67 and 3.89 nN for pyranose ring transitions. The EGO/NE method, by contrast, yields values between 0.50 and 1.21 nN, which is remarkably close to the AFM experiment, where forces between 0.3 and 1.5 nN were observed. In addition, it is possible to generate forceversus-extension curves that reflect conformational transitions and that can be used for direct comparison with AFM experiments (Figure 5). When plotting the force against the distance between the linkage oxygen atoms of the glycosidic group (“Extension”), a plateau at 0.54 nN, indicating the transition from the Oa-Chair-Oe to the Oe-Boat-Oe conformation, can be observed. The shape of the calculated forceversus-extension curve is in excellent agreement with an AFM experiment on amylose.32

Figure 5. EGO/NE force-versus-extension curve for a model of the pyranose ring. The extension coordinate is the distance between the linkage oxygen atoms of the glycosidic group. Reprinted with permission from ref 160. Copyright 2010 Taylor and Francis.

Wolinski and Baker point out that the number of electrons that are subjected to an external force in the EGO/NE method is limited to two, which is an obvious weakness of the method. They speculate that the factor of 3 by which the force is diminished in comparison to the EGO/N method could originate from the number of particles. If three times the particles are subjected to the same force then only one-third of the total force is needed to induce structural changes. Moreover, quantum chemical methods are typically based on the Born−Oppenheimer approximation,163 which allow the calculation of the molecular energy as a function of the nuclear configuration. In the case of the EFEI method, for example, this allows the calculation of the force-deformed molecular geometry and energetics during a mechanochemical transformation in the rigorous framework of quantum chemistry. If additional forces act on electrons, however, the total energies depend on these forces as well, and it is not clear in how far this still conforms with the basic principles of quantum chemistry. For these reasons, the EGO/NE method cannot be considered a computationally cheap alternative to dynamics calculations that include finite temperature effects explicitly if one is interested in reproducing force values from experiments. This section focused on the application of forces that are constant over time. However, one could of course introduce a time-dependent external force. Such time-dependence, however, is disadvantageous in that stationary points in static geometry optimizations could not be calculated. Moreover, the incorporation of time-dependent forces in dynamics calculations would be highly problematic because this would imply a nonconservative system. 3.3. Bell Theory and Related Approaches

The calculations of mechanochemical pathways, rupture forces, and deformed geometries of molecules under an external force via the COGEF, FMPES, EFEI, or EGO approaches usually 14145


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difficult to find a coordinate that accurately predicts accelerations of a given reaction by mechanical force. While the problem of choosing an appropriate structural parameter is still unsolved, the missing force-dependence of the Δq term has been addressed in the so-called extended Bell theory (EBT).171 Basic Bell theory predicts a linear, first-order dependence of the activation energy on the external force, which was extended to a second-order treatment in EBT by Konda and co-workers. In EBT, the change in activation energy upon application of an external force is calculated as

necessitate a larger number of geometry optimizations. With dependence on the system size and the computational method used in these calculations, the full characterization of the forcedeformed PES can become expensive from a computational point of view. Hence, a number of approximations have been introduced that drastically reduce the computational cost by conducting all necessary calculations at zero force. This family of methods originates from a work by Bell that describes the adhesion of cells.164 Bell theory has primarily found use in the calculation of the change in activation energy of a reaction upon application of an external force. It is assumed that the activation energy barrier depends linearly on the applied force. To a firstorder approximation, the rate of a chemical reaction with an applied external force can be calculated as ⎛ ΔG‡(0) − F Δq ⎞ 0 ⎟⎟ k(F0 , T ) = A exp⎜⎜ RT ⎝ ⎠

ΔΔG‡ = −F0Δq − ΔχF02/2

which originates from a second-order perturbative treatment of the FMPES. In EBT, the additional term that is quadratic in the force includes the mechanical compliance Δχ of the molecule. Δχ can be considered an inverse force constant (cf. section 4.4 for details) and is a measure of how readily a molecule responds to mechanical stress by deformation. The truncation of the Taylor series at second- instead of first-order, however, necessitates the calculation of the Hessian, which is used in the calculation of the mechanical compliance. Nevertheless, EBT is significantly less expensive than the full characterization of the FMPES by the methods introduced in section 3.1 and section 3.2. Kucharski and co-workers compared EBT with basic Bell theory and found that the second-order correction leads to a drastic improvement in the description of forceinduced changes in structure and activation energy in comparison to the first-order treatment. Simple Bell theory was only found to deliver reliable results in the treatment of stiff molecules, since in these cases Δq is small. Moreover, it was found that stretching forces between two atoms accelerate a mechanochemical reaction if the distance between these two atoms increases in the transition state (TS). A basic assumption of EBT is that the PES can be approximated as quadratic in the vicinity of the reactant and transition state, which can hardly be guaranteed in general. Additionally, the quadratic approximation indicates that EBT is also limited to sufficiently low forces, since the activation energy barrier disappears at the bond breaking point and the harmonic approximation breaks down. Analogous observations were made by Bailey and Mosey,172 who found a semiquantitative agreement of the EBT reaction barriers with quantum chemical reference data and a significant improvement over basic Bell theory. The authors point out that sufficiently large forces can alter the reaction pathway, which is not described by the second-order treatment in EBT. A thirdorder expansion, however, is impractical from a computational point of view, since the third-order tensor of the derivatives of the energy with respect to nuclear displacement would need to be calculated. The use of EBT cannot be recommended in general, since qualitatively wrong results are obtained if “unconventional” mechanochemical behavior is encountered (see our discussion of catch bonds and steric strain below). Bailey and Mosey devised a method to calculate the molecular distortion upon application of an external force by using the EBT approximation instead of COGEF, FMPES, EFEI, or EGO. The force can be calculated via

(9)

In this ansatz, the reaction rate depends exponentially on the constant external force F0, demonstrating that chemical reactions are extremely sensitive to force. ΔG‡(0) is the activation energy of the reaction at zero force, A is the Arrhenius pre-exponential factor, and Δq is a structural parameter that has to be chosen properly. Hence, the activation energy barrier of a mechanochemical reaction is described by Bell theory via ΔG‡(F0) = ΔG‡(0) − F0Δq

(11)

(10)

demonstrating that the change in activation energy in Bell theory is proportional to the external force. Tian and Boulatov evaluated the accuracies of different models for the calculation of the change in free energy of activation of the ring-opening reactions of cyclobutene and dibromocyclopropane mechanophores.165 It was found that, while Bell theory yields qualitatively correct results, ΔG‡ is consistently underestimated in those cases where the transition state is distorted toward the reactant geometry. This is a general limitation of the simple Bell model: the term Δq in eq 9 and eq 10 is itself force-dependent, which is not captured by the Bell theory. Another important property of Bell theory was revealed in a study by Kucharski and co-workers, who calculated changes in the activation enthalpy of SN2 reactions via eq 10.166 In this study, the nonbonded separation H3C···OMs in EtOMs (Ms = SO2Me) was taken as q, since this nonbonded coordinate captures both the elongation of a bond and the displacement of bond angles, which constitute crucial structural parameters for the investigated reaction. Although it is a covalent bond breaking in these reactions, the authors argued that this single bond coordinate fails to capture the essential part of the structural changes occurring in the reaction, so that a nonbonded separation was chosen. This view is confirmed by the observation that the elongation of the scissile bond considerably overestimates the force-induced acceleration of the reaction, whereas the nonbonded separation yields accurate results. These findings demonstrate that the predictions afforded by Bell theory strongly depend on the choice of the coordinate q. While the need to properly choose q has been acknowledged in literature,167−169 it is often assumed that the same coordinate q can be used for all reactions of the same mechanism. As pointed out before,170 a clear theoretical or empirical justification to use a simple internal coordinate of the reactant for q simply does not exist. As a result, it may prove

⎯→ ⎯ F ⃗ = H intΔq

(12)

where Hint is the Hessian in internal coordinates. Hence, the molecular distortion can be calculated via 14146


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⎯→ ⎯ Δq = CF ⃗

identifying bonds that are either insensitive to mechanical load177 or that apparently get stronger by an external force.67,178−180 This type of bond is commonly referred to as a catch bond and can lead to a deceleration of chemical reactions by mechanical force. Such an effect can be triggered if competing reaction channels are activated by different forces or if local deformations or large-scale conformational changes occur.67,154 Additionally, it was found that an increase in steric strain due to an external force can lead to a decrease in the reaction rate. Such “unconventional” mechanochemical effects cannot be described by EBT, which delivers qualitatively wrong results.154 The introduction of a second-order term is not enough to account for these effects. Steric strain can be quantified by the empirical Taft equation.181,182 Kuprička and co-workers combined the Taft equation with classic Bell theory, resulting in the Bell-Taft approach.154 In this model, the activation energy of a reaction under external force is calculated via

(13)

173,174 where C = H−1 Within the int is the compliance matrix. harmonic approximation, this procedure yields the molecular distortion and constitutes a computationally less expensive, albeit less accurate, alternative to the full quantum chemical characterization of the FMPES at sufficiently low forces. The change in character of the critical points of the FMPES has been addressed explicitly by Konda and co-workers.175 Their method acknowledges the possibility of switching to an alternative reaction mechanism or to a barrierless process upon application of large forces. The evolution of any given critical point of the PES [i.e., the reactant state (RS) or TS] can be tracked automatically. Moreover, mechanical instabilities, which occur when a critical point ceases to exist at a certain force, are tracked as well. The inclusion of these effects is a significant improvement over Bell theory or EBT, where such situations cannot be described. The method is based on catastrophe theory176 and allows predictions by considering different types of catastrophes. In the fold-catastrophe scenario, in which only one parameter (the force) is varied and the FMPES does not feature any special symmetries, two critical points may coalesce with one another. In the cusp-catastrophe scenario, which can emerge if the FMPES has certain symmetries, two equivalent critical points may merge with a third one. As in the case of EBT, a constant force and a locally quadratic PES is assumed and the equation

r(F ) = r(0) +

∫0

F

‡ ‡ ΔG Bell − Taft (F0) = ΔG (0) − F0Δq − 2.303RTψ ′νx′

where the first two terms originate from basic Bell theory. The factor 2.303 stems from the conversion between the natural and the decadic logarithm, ψ′ is a dimensionless empirical parameter and Taft’s conformation-dependent parameter νx′ is calculated via νx′(α) = νx0′ + η cos(α)

H−1[r(F ′)] l ⃗ dF ′

(16)

(17)

Here, ν′x0 is another empirical parameter. The dimensionless parameter η describes the increase in steric strain due to a change in the torsion angle α and was obtained by a fitting procedure. The carbon atoms in a model system of polyethylene glycol (PEG) were stretched isotensionally, and the changes in activation energies of several substitution reactions were calculated with Bell Theory, EBT, and the Bell-Taft approach. In the case of ethyl propyl ether reacting with ethoxide, for example, an initial increase in the activation barrier for forces up to 500 pN was observed (Figure 6). This behavior can only be qualitatively reproduced by the Bell-Taft approach, whereas the Bell model and EBT predict a decrease in activation energies even at the lowest forces. The observed decrease in activation energy at larger forces is predicted by all three tested models. Hence, the Bell-Taft model is a useful complement to EBT that can be applied in those cases where force induces strain, thereby leading to a deceleration of the mechanochemical reaction at low forces. The simplicity and modest computational demand of the Bell model and its extensions is reflected by the continuous use and steady advancement of the theory. Makarov, for example, included quantum-chemical tunneling effects in Bell theory, resulting in the Quantum Bell formula.183 While the calculation of reaction rates and structural deformations of mechanochemical reactions at zero force is extremely attractive from a computational point of view, it has to be kept in mind that still no unique way of choosing the structural parameter q that is used in Bell theory and its variants is available. Choosing a different coordinate q often changes the results dramatically. Hence, Bell theory remains a computationally inexpensive approximation with limited predictive power. An alternative method to calculate free energy barriers was proposed by the Boulatov group,184,185 who calculated the free energy of activation of a reaction under a given force via

(14)

is propagated numerically to generate the displacement r(F). In eq 14, r(0) is an RS or TS at zero force, F is the absolute value of the force, and l ⃗ is a vector specifying its direction. As before, H−1 is the inverse of the Hessian. The reaction barrier is then calculated as ΔG(F )‡ = V [rTS(F )] − V [rRS(F )] − Fl [⃗ rTS(F ) − rRS(F )] (15)

Like EBT, the advantage of the method is that it requires only a single TS search at zero force. As an example for a foldcatastrophe scenario, the authors present the electrocyclic ringopening of trans-1,2-dimethylbenzocyclobutene. In this example, increasing the stretching force F between the carbon atoms of the methyl groups causes the energy barrier between RS and TS to disappear. This can be monitored conveniently by tracking the eigenvalues of the Hessians during this process. As the catastrophe point is approached, one negative eigenvalue of the TS Hessian approaches zero from below and one positive eigenvalue of the RS Hessian approaches zero from above. In general, this tracking of Hessian eigenvalues allows the identification of mechanical instabilities in various catastrophe scenarios, which cannot be described by EBT. In principle, the integration of eq 14 can be conducted in large force steps far away from an instability. Adaptive integration schemes can ensure that the force steps become smaller in the vicinity of an instability, thus reducing the computational demand. When a molecule is stretched, one intuitively expects that the energy pumped into the system by the mechanical force accelerates chemical transformations. In fact, most works using Bell and related theories are based on this assumption. A bond that is weakened by an external force is called a slip bond. However, during recent years much work has been devoted to 14147


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of the ring-opening reaction of symmetrically substituted trans3,4-dimethylcyclobutenes.185 3.4. Mechanochemical Benchmarking

The interpretation of the results gained via COGEF, EFEI, and related approaches requires knowledge of the reliability of density functional theory (DFT) and wave function based methods in terms of describing mechanochemical processes. Unfortunately, benchmarking studies on the ability of computational methods to accurately predict bond rupture forces and distances are rare. In this section, we summarize the results of two benchmarking studies that help choose appropriate computational methods for quantum mechanochemical calculations. In a study by Iozzi and co-workers, a set of diatomic and polyatomic molecules was mechanically deformed using the COGEF approach and a closely related method, which the authors call rigid bond stretching (RBS).186 Both the COGEF and the RBS method use constrained geometries to simulate mechanical deformation. However, in the RBS approach, single point calculations are carried out instead of constrained geometry optimizations. Thus, the RBS method simulates infinitely fast deformation, whereas infinitely slow deformation is described by the COGEF approach. A well-known problem in computational chemistry is the inability of single-reference methods to describe static correlation at large bond separations, where pronounced multireference character dominates. Although Kohn−Sham DFT187,188 in principle describes the dissociation qualitatively correctly, the approximations to the exchange-correlation functionals that are introduced in practice make DFT functionals also error-prone in the description of long bond elongations. Interestingly, all of the investigated molecules remain singlereference systems up to the bond-breaking point. This remarkable conclusion was drawn by calculating the norm of the vector containing the singles amplitude T1 in a coupledcluster calculation (T1 diagnostic), which is a viable measure of multireference character of a system.189 This result is of great importance for the field of quantum mechanochemistry, as it allows the accurate prediction of bond rupture forces and geometries by computationally cheap single-reference methods, at least in the cases of the molecules investigated in ref 186. Iozzi et al. benchmarked several computational methods against the “gold standard” CCSD(T)190−192/aug-cc-pVQZ193 level of theory. It was found that Hartree−Fock (HF) overestimates rupture elongation in diatomic molecules by approximately 20% and rupture force by more than 30%, which is consistent with the well-known overbinding property of HF (Table 3). For the same reason, local density approximation (LDA) DFT overestimates these values as well. CASSCF overcompensates this effect, resulting in an underestimation of rupture elongation and force. The NEVPT2194,195 method yields remarkably reliable results that differ by only one or two percent from the CCSD(T) results. However, the cost of the NEVPT2 method for the calculation of larger systems makes

Figure 6. (a) Comparison of the Bell model, the Extended Bell model and the Bell-Taft model with quantum chemical (“exact”) isotensional reference data for the reaction of ethyl propyl ether with ethoxide. (b) Modified steric parameter ν′ET(α) that is used in the Bell-Taft model as a function of external force. Experimental values are represented by dotted lines. Reprinted from ref 154. Copyright 2015 American Chemical Society.

ΔG‡(F ) = −NAkT ln

⎛ Ei‡(F ) + TCi‡(F ) − F × qi‡(F ) ⎞ ⎟ ∑ gi‡ exp⎜ − kT ⎝ ⎠

(

∑ gi exp −

Ei(F ) + TCi(F ) − F × qi(F ) kT

) (18)

where NA is Avogadro’s number, k is the Boltzmann constant, T is the temperature, gi is the degeneracy, Ei(F) is the electronic energy of conformer i in its equilibrium with a constraining force F, TCi(F) is a thermodynamic correction, and qi(F) is the constrained distance. Details on these terms have been given in ref 185. The summation is carried out over all conformational minima. Importantly, this static quantum mechanochemical procedure is complementary to dynamic simulations in that it allows the approximate calculation of free energies of activation at finite temperatures using common quantum chemical methods. The treatment of full conformational ensembles yields free energies of activation that are in better agreement with experimental values than many other quantum mechanochemical methods. The calculation of ΔG‡(F) values according to eq 18 has proven advantageous in the description

Table 3. Average Values of the Bond Rupture Elongation ϵBP and the Rupture Force Fmax Relative to the CCSD(T) Values for the Diatomic Molecules LiH, BH, HF, Li2, N2 and O2a ϵBP Fmax a

HF

LDA

CASSCF

NEVPT2

PBE

BLYP

B3LYP

CAM-B3LYP

CCSD(T)

119 134

114 102

87 96

99 98

105 96

104 93

108 105

112 113

100 100

In these cases, the COGEF method and the RBS method are equivalent. The values were taken from ref 186. 14148


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cally relevant degrees of freedom in a molecule can be used in experimental mechanochemistry to design reactions that yield a desired product as well as in mechanobiology to understand allosteric signaling pathways and structural changes in proteins, for example. For this purpose, mechanochemical force analysis tools have been developed, which answer the following questions: Where is the mechanical work stored in a mechanically deformed molecule? Which bonds, angles, and torsions of the molecule are particularly stressed and why? Do these effects change the properties of the molecule to such an extent that force-induced chemical transformations are possible? Various computational force analysis tools developed within the past five to ten years are discussed in this section. In addition to those tools that have been explicitly intended to be applied for mechanochemical purposes, we describe various other approaches that show potential for applications in the context of quantum mechanochemistry. Some of these analysis tools are based on the quantification of forces, while others investigate the distribution of stress energy or mechanical work in the molecules. The relation of force and energy has been examined in section 2.3, and the advantages and disadvantages of both approaches will be discussed throughout this section. Here, only the theoretical foundations of the force analysis tools as well as various applications to toy systems are reviewed, whereas more sophisticated applications are discussed in section 5. Prior to the treatment of force analysis tools, however, it is necessary to introduce coordinate systems that can be used for investigating the distribution of force or mechanical work in molecules.

the application of computationally less expensive methods desirable. In this regard, it was found that general gradient approximation (GGA) DFT functionals deliver reliable results. In particular, PBE196,197 and BLYP138,198 overestimate rupture elongation and underestimate rupture forces by approximately 5% each. Hence, these functionals are a cheap alternative to the costly CCSD(T) method. The very popular functionals B3LYP138,139 and CAM-B3LYP,199 in turn, consistently overestimate rupture forces and elongations in diatomic molecules. However, these properties strongly depend on the system size: in polyatomic molecules, B3LYP was found to deliver the best results, while CAM-B3LYP performed poorly. In another very recent quantum mechanochemical benchmarking study by Kedziora and co-workers, a test set of small molecules was stretched using the COGEF approach.200 The CCSD190,191 method as well as a number of DFT functionals were tested, including the double-hybrid functional B2PLYP.201 It was found that static correlation is not significant until the bond breaking point, which agrees well with the earlier benchmarking study by Iozzi et al. Out of the single-reference methods, unrestricted GGA functionals were shown to perform best. In particular, the functionals N12,202 OLYP,138,203 and PBE196,197 deliver especially reliable results. Methods with scaled exchange, however, often perform poorly. Interestingly, G4 dissociation energies204 are much better reproduced by DFT using the small double-ζ basis set cc-pVDZ205 than by using the larger triple-ζ basis set cc-pVTZ. The aforementioned studies came to the conclusion that computationally inexpensive single-reference methods can be used in the description of many mechanochemical processes, at least until the point of bond rupture. However, great care should be taken in the choice of electronic structure method in quantum mechanochemistry, since the summarized studies were necessarily limited to small sets of molecules. Moreover, the number of DFT functionals and wave function based methods tested in these studies was limited, and the basis set dependence was investigated only coarsely. Therefore, further studies on larger test sets are needed, in which more computational methods and basis sets are tested. Until a comprehensive overview of the accuracy of computational methods for describing mechanochemical processes is available, the combination of method and basis set has to be chosen very carefully.

4.1. Coordinate Systems and Their Use in Force Analyses

In this section, some general features of different coordinate systems and their suitabilities for mechanochemical force analysis are discussed. For the purpose of analyzing the distribution of force in a molecule, an obvious ansatz is to calculate the restoring force Fi in the internal coordinate ri of a molecule as the negative partial derivative of the potential V with respect to ri via

Fi = −

∂V ∂ri

(19)

However, if Fi is calculated for every primitive internal coordinate of a molecule (i.e., every bond length, bond angle, dihedral angle etc.), this analysis is hardly insightful. The reason is that the outcome of this analysis depends on which coordinates are kept fixed in case more than 3N − 6 internal coordinates (or 3N − 5 in the case of linear molecules) are used to describe the molecular structure.206 In this case, the internal coordinates are interdependent, meaning that a change in one coordinate potentially leads to a change in other coordinates. The problem that not all internal coordinates can be varied independently is a result of the redundancy of the coordinate system, since more degrees of freedom are specified than necessary for a complete description of molecular connectivity. In mechanochemistry, the displacement of one coordinate in a redundant coordinate system leads to additional forces in other coordinates. These effects cannot be captured by eq 19, thus leading to physically ambiguous results. To illustrate the problem of interdependent coordinates, it is insightful to consider the two-dimensional function f(x,y) = x2 + y2 + xy. Here, coupling between the x- and y-coordinates is introduced via the xy term (Figure 7). A displacement from the global

4. FORCE ANALYSIS TOOLS The computational methods introduced in section 3 provide access to geometries, minimum energy pathways, and reaction rates of molecules that are deformed by an external force. Therefore, they constitute the basis of quantum mechanochemistry and lend fundamental insights into mechanochemical processes at the molecular level. However, the question of which part of a molecule is mechanochemically most susceptible represents a serious challenge for methods like COGEF, FMPES, EFEI, EGO, and approaches based on Bell theory. Answering this question would allow the identification of the mechanochemically most relevant degrees of freedom in a molecule (or the “force-bearing scaffold”). Accurate predictions about and rationalizations of the bond that is ruptured if a molecule is sufficiently stretched would then be possible. Such insights would be highly valuable in the design of stress-responsive materials, molecular machines, and photoswitches that can be used to trigger structural changes in their environment. Moreover, the identification of mechanochemi14149


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the distribution of mechanical stress in deformed molecules using normal modes has been demonstrated. 215,216 A disadvantage of normal modes, however, is that they are usually delocalized over different parts of a molecule, which makes a localization of mechanical stress difficult. To address this problem, a localization scheme has been proposed, in which a unitary transformation into a set of maximally localized modes is carried out.217 This approach has been used to analyze vibrational spectra, but it would be interesting to test whether the method can be applied in the context of force analysis. Notwithstanding the possibility of localizing normal modes, another property of normal modes that limits their usefulness in force analysis is their rectilinear nature. Using a rectilinear coordinate system to model large scale bendings, torsions, etc. is problematic,218,219 which makes the description of molecular deformations upon application of even moderate external forces via normal modes extremely difficult.215 As an alternative to normal modes, a set of 3N − 6 natural internal coordinates (NICs) can be used as the coordinate system in force analysis tools. The generation of NICs is based on a complicated set of rules partly based on local pseudosymmetry.220 The modest coupling between different coordinates is a profitable feature of NICs and has led to their use in geometry optimizations.220−223 Although this property is extremely useful in the context of mechanochemistry, the removal of redundancies that is carried out in most algorithms that generate NICs is to a certain extent arbitrary, and the coordinates are often highly delocalized. As a result, NICs have never been applied in force analysis tools. Delocalized internal coordinates (DICs) constitute another nonredundant set of 3N − 6 coordinates. DICs are generated by first defining all primitive internal coordinates of a molecule, which include every bond length, bond angle, and dihedral angle of a molecule. This is achieved in all major quantum chemical program packages using a well-defined set of rules. Although a certain degree of arbitrariness is inherent to this step due to the need to specify a cutoff value for the identification of a bond, this procedure hardly ever fails. After the primitive internal coordinates are generated, Wilson’s Bmatrix (eq 3) is used to set up the matrix G = BBT, which is subsequently diagonalized. The DICs are the set of eigenvectors of G with nonzero eigenvalues. DICs enjoy widespread use in geometry optimization algorithms.224 While the coupling between the coordinates is minimal, the problem of the delocalization of DICs over large parts of the molecule limits the utility of DICs in mechanochemical force analysis tools. It is much more intuitive to think of molecular structure and deformation in terms of changes in individual bonds, bendings, and torsions. A coordinate system that is based on these internal coordinates would be much more valuable in force analysis tools. Another possibility is conducting force analyses in Z-matrix coordinates, which comprise N − 1 bond lengths, N − 2 bond angles, and N − 3 dihedral angles, leading to a total of 3N − 6 coordinates. In the case of benzene, for example, typically only five of the carbon−carbon bond lengths in the ring are included in the Z-matrix explicitly, since the sixth bond length is determined by the other bond lengths, bond angles, and dihedral angles. A great strength of the Z-matrix is that force or stress energy is partitioned among maximally localized internal coordinates. Moreover, the curvilinear nature of internal coordinates makes them more appropriate for the description of large-scale displacements than rectilinear coordinates like

Figure 7. Plot of the two-dimensional function f(x,y) = x2 + y2 + xy. If a molecule is displaced from the global minimum P1 to a point P2 in the x direction, an additional force in y direction is generated, since the coordinates x and y are not independent due to the coupling term xy. Relaxation in the y direction leads to the point P3. The minimum energy path is shown in red. Reprinted with permission from ref 207. Copyright 2008 Royal Society of Chemistry.

minimum P1 to the point P2 in the x direction leads to the generation of a force in the y-direction.207 Relaxation in the ydirection leads to the point P3. The additional force at the point P2 is a problem for mechanochemical force analysis tools that aim at a unique quantification of forces at deformed geometries, preferentially without specifying which degrees of freedom are kept fixed. Ideally, a change in any one coordinate should not lead to a change or a force in any other coordinate, which can, however, not be guaranteed in the case of a redundant coordinate system. Since we are confronted with the problem of redundancy, it is not straightforward to calculate the force in all degrees of freedom of molecules in which the number of primitive internal coordinates (i.e., all bond lengths, bond angles, and dihedral angles) exceeds 3N − 6. This is the case, loosely speaking, for every molecule larger than hydrogen peroxide. However, the inclusion of all primitive internal coordinates of a molecule would be highly desirable, since this set is chemically most intuitive and the results are easily interpretable. Approaches to solve or circumvent the problem of redundancy and interdependency of coordinates will be discussed in the following sections. In general, an obvious way to conduct a force analysis is to use a coordinate system that comprises exactly 3N − 6 (or 3N − 5 in the case of linear molecules) coordinates. For example, it is possible to use normal modes, which are well-known from vibrational spectroscopy128 and constitute the orthonormal eigenvectors of the Hessian matrix. The corresponding eigenvalues are the force constants. These quantities are routinely used to estimate molecular stiffness. The interpretation of force constants in terms of bond strengths is, however, not without disadvantages (section 4.4). Normal mode analysis enjoys widespread use in the molecular dynamics (MD) community to study large-scale functional motions of proteins208−211 and is implemented in major MD simulation program packages like AMBER212,213 and GROMACS.214 In the context of force analysis, the nonredundancy of normal modes has some advantages, and the possibility of quantifying 14150


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Cartesian or normal mode coordinates. However, Z-matrices are not always easy to construct, particularly in the case of molecules that contain fused rings. The advantages and disadvantages of the Z-matrix in the context of mechanochemical force analysis are discussed in more detail in section 4.5. At this point, it suffices to say that Z-matrices cannot be constructed uniquely, which leads to an inherent arbitrariness of the obtained results of a force analysis. This discussion has shown that for mechanochemical force analysis the use of redundant internal coordinates (comprising all bonds, bendings, and torsions) would be ideal, since these coordinates are maximally localized and can be defined uniquely. However, the problem of interdependency of the coordinates remains. Before reviewing how this problem has been tackled or circumvented in the force analysis tools introduced during the past few years, we briefly mention another set of coordinates that can potentially be applied in mechanochemical force analyses. Recently, Li and co-workers have worked on generalized coordinates, in which a distinction is drawn between external and internal coordinates.225 External coordinates describe the overall translation and rotation of a system, while its internal vibrational motions are represented by internal coordinates. The authors point out that the coupling between external and internal coordinates complicates the separation of these two sets. However, transformation matrices for this purpose are presented. Li and co-workers use BAT (bond length, angle, and torsion) coordinates, which comprise N − 1 bond lengths, N − 2 bond angles, and N− 3 dihedral angles. This ansatz is related to the Z-matrix, but the authors choose to construct the BAT coordinates using a so-called BKS-tree.226,227 In this approach, an atom is chosen as the starting point and the tree is built step-by-step in a clearly defined way. Nevertheless, due to the nonuniqueness of choosing an initial atom in the construction of the BKS-tree, the generalized forces calculated for the bond angles and the improper dihedrals depend on the definition of the coordinate system. Despite this ambiguity, the authors suggest using their ansatz in force analysis, since their conclusions also hold for systems that are subjected to an external force. The curvilinear nature of the coordinates is certainly helpful in this regard.

Figure 8. Schematic representation of cyclobutane under tensile stress. Forces acting on the atoms are denoted by F⃗1 and F⃗2. The displacement of a carbon atom due to F⃗2 is denoted by dL⃗ . Reprinted with permission from ref 228. Copyright 2016 Elsevier B.V.

under mechanical stress, since in some cases unphysically large forces are observed that unexpectedly do not trigger bond breaking. Hence, the authors conducted a local work analysis by calculating E = ∫ F⃗dL⃗ (eq 5), which yielded a much better description of mechanical deformation and mechanophore activation. This demonstrates the usefulness of both a onedimensional model of mechanical activation (provided the coordinate of interest can easily be identified) and an analysis of stress energy as an alternative to local force analysis. An alternative approach has been used by Huang and coworkers for describing the ring-opening of cyclobutene upon excited state cis → trans-isomerization of 1-(1-indanyliden)indan, commonly referred to as stiff-stilbene.232 The tremendous change in spatial extension of stiff-stilbene during isomerization exerts forces on its environment. It has been suggested that forces generated by stiff-stilbene play a crucial role in the ring opening of cyclobutene. Although thermal effects were found to play a more significant role in this process (section 5.5.2), it is insightful to calculate the force stiff-stilbene can apply to cyclobutene. Huang and co-workers chose the distance between the C6 and C6′ atoms in stiff-stilbene as the coordinate of interest (Figure 9), since these atoms are attached

4.2. Force Analyses Using Only One Coordinate of Interest

The easiest way of circumventing the problems of nonuniqueness and interdependency of coordinates is to consider very few or only one coordinate of interest in the force analysis. Due to its simplicity, this ansatz has been applied extensively in the past. Koo and co-workers, for example, used this approach for determining the key factors for the activation of cyclobutanebased mechanophores embedded in a polymer.228 Activation of these mechanophores by carbon−carbon bond rupture generates fluorescence, which can be used to identify damage precursors in polymers. For their investigation, the authors used a hybrid MD approach by combining the classical Merck molecular force field (MMFF)229,230 and the reactive force field (ReaxFF).231 The carbon−carbon bond lengths of the cyclobutane moieties were chosen as the coordinates of interest. This choice is certainly reasonable because mechanical stretching of one of these covalent bonds leads to rupture of the cyclobutane moiety at sufficiently high forces (Figure 8). Local force analysis reveals that the amplitude of the force indeed contributes to the deformation of the carbon−carbon bonds. However, for different cyclobutane moieties, the results differ dramatically and fail to account for mechanophore activation

Figure 9. Excited state cis → trans-isomerization of stiff-stilbene results in a tremendous change in spatial extension of the photoswitch. In the trans-configuration, stiff-stilbene applies mechanical forces to a substrate. To describe this process, the distance coordinate between the carbon atoms C6 and C6′ was used. Reprinted from ref 232. Copyright 2009 American Chemical Society.

to the cyclobutene moiety via linkers and their separation increases tremendously during isomerization. The strain was estimated by fragmenting the macrocycle and calculating the restoring force via the nuclear gradient. Using this ansatz, a maximum force of 700 pN was found to be generated by stiffstilbene. Force analyses using only one specific coordinate have found various applications, including the activation of dibromocyclo14151


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propane mechanophores by stiff-stilbene,233 analysis of bond cleavage at the Si(100)-SiO2 interface,234 and estimations of the strain in a hindered aryl−aryl bond.235 As in the case of Bell theory and related models (section 3.3), however, choosing the coordinate that is used in the local force or work analysis is not a trivial task.168,236 As a result, scientists have to rely on their mechanochemical intuition to judge which coordinate is relevant for describing a given mechanochemical process. Clearly, different results are obtained for different coordinates, and it is impossible to judge how much force or stress energy is “wasted” on other coordinates that are not included in the analysis. In the case of the cyclobutane moiety discussed above,228 the choice of using the distance between two carbon atoms in the cyclobutane ring is intuitive. However, for the interpretation of the results it has to be kept in mind that, for example, bond angle deformations are not explicitly included in the model. Thus, bendings are not considered to be reservoirs of stress energy, and it is not possible to calculate the total stress in the cyclobutane ring. While a restriction of the force analysis to the discussion of a single coordinate makes the procedure straightforward, the capability of the rest of the molecule to bear force or stress energy is neglected. Moreover, the elongation of the scissile bond was often shown to be an inadequate coordinate for describing mechanical activation even in small systems.168,184,237,238 As in Bell theory, general rules for choosing an appropriate coordinate are unavailable, so that great care has to be exercised in defining a coordinate and interpreting the results.

specifying the coordinate system. Also, the force distribution pattern does not necessarily have the same symmetry as the molecule and apparently unphysical results can be obtained in that internal forces may propagate far into the unloaded part of the system. As shown in Figure 10 (panels b and c), the same

4.3. Internal Forces

Figure 10. (a) Ala14 secondary structure. Hydrogen bonds are indicated by red dashed lines and the points where the pulling force is applied are marked by orange spheres. (b) Internal force map in which hydrogen bonds are included explicitly. Blue and red indicate bond stretching and compression, respectively. Color intensity is a measure of the magnitude of the internal force. (c) Internal force map for a coordinate system in which hydrogen bonds are not included. Reprinted with permission from ref 206. Copyright 2014 American Institute of Physics.

An analysis tool based on the quantification of internal forces was introduced by Avdoshenko and co-workers.206 In this approach, the molecular distortion upon application of an ⎯→ ⎯ external force is calculated to first order according to Δq = CF ⃗ (eq 13), and the discussion is limited to the weak force regime, where the distortion is small. The authors calculate the energy E stored in a mechanically strained molecule to lowest order via E=

∑∫ i2000 K after photoexcitation in the central part of the stiff-stilbene moiety in the macrocycle containing cyclobutene.132 It is reasonable to assume that this energy is dissipated rapidly in the macrocycle and is the key factor for the acceleration of the bond rupture in cyclobutene, although this view has been discussed controversially.425,426 The scissile bond in cyclobutene is thermally labile by nature, and depending on the substitution pattern, only moderate heating is needed for rupture.427−433 The significant influence of thermal oscillations in this reaction has also been acknowledged in previous literature.423 We speculate that the rapid heating upon absorption of a photon is a general working principle of photoswitches and a determining factor in the initiation of various processes like structural changes in proteins104−109 and DNA.110,112 14169


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range of MD and quantum chemical methods for the description of mechanochemistry suggests QM/MM-like approaches. In this context, it would be interesting to simulate the unfolding of proteins using MD simulations under an external force and treat the region of particular interest (e.g., a molecular force probe) quantum chemically. In summary, quantum mechanochemistry has tremendously advanced our understanding of mechanochemical processes at the most fundamental level. However, we are convinced that the full potential of mechanochemistry in academia and industry can only be exploited by extensive collaborations between experimentalists and theoreticians. Both theory and experiment can undoubtedly benefit from each other, provided a common language is found. New quantum mechanochemical methods will always have to be validated by experiments, and mechanochemical experiments will never be fully understood if the underlying processes are not revealed at the molecular level. For these reasons, it is evident that the rapidly expanding field of quantum mechanochemistry looks into a prosperous future.

formation of a liquid phase have to be considered in addition to the purely mechanical effects at the molecular level. Moreover, in contrast to SMFS, the pulling direction is ill-defined in the case of ball milling experiments. As a result, a multiscale theory linking the collision of macroscopic balls in a ball mill to forces that deform single molecules has not been developed and it is hard to judge whether it is even possible to develop a theory that considers all subtleties described above. However, such a theory is direly needed for predicting the outcomes of reactions initiated by ball milling or grinding. Similarly, predicting the result of a chemical reaction in an ultrasound bath is still hardly possible. As described in section 5.2.2, determining the mechanism of the sonochemical scission of triazoles by using quantum mechanochemical methods is a nontrivial task. For deriving a complete physical picture of the underlying processes, a theory linking the collapse of bubbles to the deformation of molecules is needed. We believe that the field of molecular machines will certainly grow in the future, since the possibility of performing work at the molecular level is an exceptionally useful concept. The use of light to generate this mechanical work is particularly convenient, since light is a cheap, abundant, and sustainable resource. Hence, at the heart of many molecular machines is a photoswitch, which undergoes rapid photoisomerization (section 5.5). Azobenzene and stiff-stilbene are just two examples of the photoswitches that are commonly used for this task. Today, stiff-stilbene is widely considered the “strongest” photoswitch, since it is capable of exerting forces in the order of several hundred piconewtons during photoisomerization. However, the mechanical output of today’s photoswitches is limited by local heating and efficient nonradiative decay, which limits the mechanical efficiency of the photoswitch, and poor transmission of mechanical work to the mechanophore through the linkers. To maximize the mechanical output of molecular motors, it is of utmost importance to optimize the mechanical efficiency of the photoswitch. One promising approach in this direction is to maximize the spatial extension of the photoswitch, which causes a larger deformation of the mechanophore during photoisomerization. Also, several studies have shown that the linkers between the photoswitch and its environment are crucial factors in the transmission of stress energy, so that the linkers should be chosen with great care. For the purpose of maximizing the mechanical efficiency of photoswitches, force analysis tools are extremely useful, as they allow the comparison of the mechanical outputs of different derivatives of a photoswitch. Moreover, force analyses afford predictions about the optimal coupling between the photoswitch and its environment. The chemical composition and attachment points that result in an optimal transmission of mechanical work from the photoswitch to its environment can be determined with these methods. However, enhancing the mechanical properties of photoswitches is just one example where force analysis tools are useful. The applicability of force analyses for understanding the mechanochemistry of proteins, polymers, and numerous smaller molecules has been demonstrated in the preceding sections. Macroscopic theories for the description of the wear and tear of materials and crack propagation, on the other hand, have not been discussed in this review. As in the case of ball milling and sonochemistry, multiscale models would surely help relate the microscopic effects to macroscopic observations. Within the realm of computational mechanochemistry, the availability of a wide

AUTHOR INFORMATION Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest. Biographies Tim Stauch studied Chemistry at the Goethe University of Frankfurt. He received his M.Sc. degree in the group of Andreas Dreuw in 2012 and has since been working in the field of Quantum Mechanochemistry. He is currently completing his Ph.D. studies in the group of Andreas Dreuw at the Interdisciplinary Center for Scientific Computing of Heidelberg University. His main research interest lies in the development of molecular force probes and computational force analysis tools, with a special emphasis on the coupling of Photochemistry and Mechanochemistry. Andreas Dreuw received his Ph.D. in Theoretical Chemistry from Heidelberg University in 2001. After a two-year postdoc at UC Berkeley, he joined the Goethe University of Frankfurt first as an Emmy-Noether fellow and then as a Heisenberg-Professor for Theoretical Chemistry. Since 2011, Andreas Dreuw holds the chair for Theoretical and Computational Chemistry at the Interdisciplinary Center for Scientific Computing of Heidelberg University. His research interests comprise the development of electronic structure methods and their application in Photochemistry, Mechanochemistry, Biophysics and Material Science.

ACKNOWLEDGMENTS We acknowledge financial support from the Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences (GSC 220). We are grateful to the Fonds der Chemischen Industrie for funding our research in the field of quantum mechanochemistry in the past. REFERENCES (1) Takacs, L. The Historical Development of Mechanochemistry. Chem. Soc. Rev. 2013, 42, 7649−7659. (2) Beyer, M. K.; Clausen-Schaumann, H. Mechanochemistry: The Mechanical Activation of Covalent Bonds. Chem. Rev. 2005, 105, 2921−2948. 14170


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