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Free energy relationships
CHAPTER 22
Free energy relationships
...in a series of homologous reactions the heats of activation decrease with increasing heats of reaction. ... derive the Tafel equation for hydrogen overvoltage and the series of overvoltages at the metals. Horiuti and Polanyi, (tr. Muller), Acta Physicochem. URSS (1935)
The introduction to this book stressed that “matching experiments” should not be the final goal of a computational effort. To engineer better catalysts, processes, or materials, it is often more important to identify reliable ways of controlling the kinetics. This chapter explores relationships, some empirical and some derived, between kinetics and driving forces. These relationships are useful for predicting trends, e.g. how a change in solvent will change the rate of a chemical reaction, or how a mechanical pulling force will accelerate the unfolding of a protein. The relationships are also useful for interpreting kinetic trends in terms of transition state properties. For example, some free energy relationships indicate the location of a transition state, or the size of a critical nucleus, both from measurable quantities. With some effort we can even go beyond kinetic trends for one reaction, to predict and understand kinetic trends across series of similar reactions. In these efforts, relationships between thermodynamic properties and reactivity have emerged as one of the most fruitful intersections between theory and experiment. For example in organometallic chemistry, we can understand and predict how ligands with different electron withdrawing characteristics influence the rate of a reaction. In catalysis, we can understand and predict how the amount or type of a dopant alters the activity. Modern computing power has made it possible to perform automated and increasingly reliable calculations for hundreds or thousands of slightly different catalysts or substrates. The resulting large data sets have revealed many relationships between molecular structure, electronic structure, and chemical properties including reactivity. Structure-property relationships discovered in this manner often begin as empiricisms and heuristics, but especially in recent catalysis work they have also inspired fundamental advances and ab initio tools for new types of analyses. Reaction Rate Theory and Rare Events Copyright © 2017 Elsevier B.V. All rights reserved.
601
602 Chapter 22
22.1 BEP relations and the Bronsted catalysis law One of the first and still most important relationships between thermodynamics and reactivity is that of Bell1 and Evans and Polanyi (BEP) [2]. BEP postulated that for a series of similar reactions, the more exothermic reactions would have lower activation barriers. BEP relations of the form Ea ≈ Eao + αH o
(22.1.1) H o
are widely used [1,3], especially in catalysis [4–6]. Here is the reaction enthalpy and o o Ea is the activation energy for a (hypothetical) isoergic (H = 0) reaction in the series. The slope α is related to the position of the transition state (early, midway, or late) along the reaction pathway. Aqvist and Warshel [4] and van Santen et al. [7] have reviewed the derivations, applications, and recent advances related to BEP relations in catalysis. The BEP relation is a practical computational tool because it is far easier to measure or compute reaction enthalpies than to accurately compute reaction rates. In heterogeneous catalysis, reaction enthalpies themselves can be further correlated to electronic band structures and to adsorbate properties. These correlations have been combined with BEP relations to create “universal BEP relations” that predict activities of many catalysts for many different reactions. Example: BEP relations in catalysis [8] Computational work in heterogeneous catalysis shows how H o can be estimated from even more readily computed quantities, e.g. d-band characteristics of a metal or alloy [5,9]. These studies first correlate d-band energies to atomic adsorbate energies, e.g. ∗O, ∗H , ∗N , and ∗C. The atomic adsorption energies are then correlated to adsorption energies of more complex and partially saturated adsorbates, e.g. ∗OH , ∗OCH3 , ∗NH2 , ∗CH , ∗CH2 , and ∗CH3 . Surface reaction stoichiometries are used to combine correlations for adsorbate enthalpies into correlations for surface reaction enthalpies. Finally, the reaction enthalpies are used in BEP relations for the breaking of different bond types. The result is a modern “aufbau principle” for building correlations that relate dband characteristics and a small collection of adsorbate properties to activation energies. These correlations-in-series are not extremely accurate, but they are powerful ways of quickly screening the entire periodic table for highly active metals, alloys, dopants, etc. 1 The BEP acronym sometimes refers to Bell-Evans-Polanyi and sometimes refers to Bronsted-Evans-Polanyi in-
stead. Bronsted most definitely contributed to the development of LFERs with his catalysis law. R.P. Bell, who worked in Bronsted’s laboratory, directly used the BEP relation in his own work after it was put forth by Evans and Polanyi. Some sources even point to the much later G.I. Bell. Still others refer to “Bema Hapothle” (Bell-MarcusHammond-Polanyi-Thornton-Leffler) principles which describe changes in transition state structure across series of reactants or solvents. Clearly, there were many contributors to the theory of free energy relationships [1].
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Free energy relationships 603 This approach has now been used to predict the activity of thousands of metals and alloys in silico. BEP analyses have led to (1) an improved understanding of chemical bonding and catalysis on metal surfaces, and to (2) the design of now-patented new catalysts [10,8]. Figure 22.1.1 shows the “universal” BEP relation for dissociation of molecules containing single bonds between C, O, and N atoms on transition metal surfaces.
Figure 22.1.1: Wang et al. developed “universal” BEP relation from computed transition states and dissociation energies for C-C, C-O, C-N, N-O, N-N, and O-O bonds on a variety of transition metal surfaces. The typical error is 0.35eV, a result of correlating such different reactions in a single BEP relation. When different reactants are analyzed separately, the MAE is significantly lower. [Wang et al. Catalysis Lett. 141, 370-73 (2011)]
Recent progress in catalysis illustrates the accelerated fundamental advances and technological discoveries that are possible with relatively simple calculations that focus on trends. Heuristic derivations of the BEP relation (see section 22.2) suggest general guidelines on the situations where a BEP relation will be valid: • • •
BEP relations are most reliable when applied to individual elementary steps. BEP relations tend to be most accurate for reactions in which the transition state is late, i.e. for cases with α near unity so that transition states resemble the products. In practice, there is some ambiguity in what constitutes a related series of reactions. Generally, more closely related groups of reactions will more closely obey the BEP relation.
There are cases where BEP relations fail [11]. In heterogeneous catalysis, BEP relations fail when some members in a series of catalysts are prone to surface reconstructions, preferential segregation of certain alloy components to the surface, or to oxidation. The Bronsted catalysis law [12] relates the kinetics of proton abstraction to an acid dissociation equilibrium. Specifically, consider a series of reactions ki
H Ai + B − −→ H B + A− i
Goi
(22.1.2)
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604 Chapter 22 with equilibrium constant Ki = exp −Goi /kB T . Bronsted’s catalysis law says that, for a series of similar reactions, the equilibrium constants are related to the rate constants by ki ≈ γ Ki α
(22.1.3)
where γ and α are constants. If we assume ki = kB T h−1 Voν−1 exp −G‡i /kB T and take logarithms, equation (22.1.3) becomes G‡i ≈ α Goi − kB T ln kB T Voν−1 / hγ . Now invoke a reference reaction in the series for which Goref = 0. This reference reaction has G‡ref ≈ −kB T ln kB T Voν−1 / hγ . Therefore the Bronsted law (equation (22.1.3)) can also be written as G‡i ≈ G‡ref + α Goi which is similar to the BEP relation but involves free energies. α is called the Bronsted coefficient or Bronsted slope [13,4], ∂G‡ (22.1.4) ∂Go and typically it falls in the range 0 < α < 1. The Bronsted slope has been interpreted as the position of the transition state along the pathway from reactants to products. The reasons for this interpretation will become clear below when free energy relationships are heuristically derived from a Marcus theory model. α≡
22.2 The Marcus equation Chapter 21 noted that electron transfer ideas are applicable to many chemical reactions, especially proton transfer reactions [14,15]. Marcus obtained the parabolic free energy diabats as functions of a charge transfer interpolation coordinate ξ . The reactant state at ξ = 0 and the product state at ξ = 1 are minima for separate free energy parabolas with the same spring constants mω2 . As depicted in Figure 22.2.1, the free energy landscape in state A is approximately 1 GA (ξ ) ≈ mω2 ξ 2 2
(22.2.1)
In state B, the free energy landscape is approximately, 1 GB (ξ ) ≈ mω2 (ξ − 1)2 + Go 2
(22.2.2)
Solving for the point of intersection, ξ‡ ≈
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1 Go + 2 mω2
(22.2.3)
Free energy relationships 605
Figure 22.2.1: The Marcus-BEP relation is based on a simple model with two diabatic states.
Inserting ξ ‡ into GA (ξ ) gives the activation free energy 1 Go 2 2 1 + G = mω 2 2 mω2 ‡
(22.2.4)
For a thermoneutral reference reaction (Goref = 0) we can define 1 G‡ref = mω2 8
(22.2.5)
This allows us to write the activation free energy in terms of Go and the reference activation free energy. The result is the Marcus equation G‡ = G‡ref
Go Go 1+ + 2 8G‡0
(22.2.6)
The reference activation free energy G‡ref is essentially a fit parameter. No member of the reaction series actually needs to have Go = 0 to use the Marcus equation. Cohen and Marcus tested the Marcus equation against data for several reaction series [16]. Figure 22.2.2 shows their test for a large experimental data set on hydrogen abstraction rates from ketones by carboxylate anions and related bases. CH3 COCH RR + A− → CH3 COCRR − + H A
(22.2.7)
Cohen and Marcus tested some twenty reaction families, with results that generally validated the Marcus equation. The Marcus equation has also been validated for proton coupled electron transfers [17] and proton transfer reactions in enzymes [18].
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606 Chapter 22
Figure 22.2.2: A test of the Marcus equation for hydrogen abstraction from ketones by a family of carboxylate anions and other bases. Units on both axes are kcal/mol. [From Cohen and Marcus, J. Phys. Chem. 78, 4249-56 (1968).]
Returning to equation (22.2.3), let us write the transition state location in terms of Go and the activation free energy for the reference reaction. The result is the Marcus-Hammond relation ξ‡ =
Go 1 + 2 8G‡ ref
(22.2.8)
Now using equation (22.2.6) to compute the local slope of the free energy relationship gives α≡
∂G‡ = ξ ‡. ∂Go
(22.2.9)
This equation explains why the slope in BEP relationships and LFERs has been interpreted as the position of the transition state along the reaction pathway. Equations (22.2.8) and (22.2.9) justify the postulates of Leffler and Hammond [19,20]. Specifically, in a series of similar reactions, more exothermic reactions will have earlier transition states along the reaction coordinate and their transition state structures will more closely resemble the reactants. Where chemical accuracy is needed, it should be remembered that most FERs have a heuristic basis. Even theoretically derived FERs like the Marcus equation [16] and the nucleation theorem [21] are based on extremely simple models. The true activation energies and free energies are often widely scattered around the values predicted by an FER. Of course, FERs will fail to predict trends when the reaction path or mechanism changes within a seemingly similar series of reactions, or when the limiting step in a multistep mechanism changes to another other step in the sequence.
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Free energy relationships 607
22.3 Externally controlled driving forces The previous section discussed free energy relationships that correlate rates across a series of similar reactions. This section discusses similar free energy relations that help understand situations where the driving force for a single reaction is externally controlled. Rate theories that account for externally modulated driving forces have a long history, and similar expressions starting with Eyring [22] have emerged in many different contexts [23–25,21]. The simplest models consider a one dimensional free energy profile as a function of the reaction coordinate with a force applied along the reaction coordinate. The applied force alters the free energy landscape by simply tilting the original landscape. For small forces,2 the new free energy landscape qualitatively resembles the original free energy landscape, but the stationary points shift to new locations along q and the barrier heights change as shown in Figure 22.3.1.
Figure 22.3.1: Pulling along the reaction coordinate q tilts the zero-force free energy landscape and shifts the locations and heights of the free energy barriers.
Changes in the barrier to escape state A include terms of order f which are proportional to the unperturbed distance from reactants to transition states along q. Changes in the barrier also include terms of order f 2 which emerge from force-induced changes in the location of the reactant minimum and transition state. The force-perturbed free energy barrier is 1 Ff = F0 − f · (q‡ − qA ) − f 2 {(1/F (q‡ ) − 1/F (qA )} 2
(22.3.1)
To first order in the force f , equation (22.3.1) is consistent with the Hammond-Leffler postulate, i.e. a force that pulls along the direction leading to the transition state lowers the barrier and leads to an earlier transition state. Typically, the first order term in the external force dominates 2 For sufficiently large forces a catastrophe [26] causes the minima lose their mechanical stability. We limit our
discussion to the effects of smaller forces.
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608 Chapter 22 except where the force is strong.3 The slope in a free energy relationship has a physical interpretation which is related to the position of the transition state along the reaction coordinate. Some important special cases of equation (22.3.1) include: •
•
•
Electrochemistry [29,23]: Suppose that an electrochemical reaction M n+ + n e− → M involves transfer of a specific charge n e− from the electrode. A Tafel plot shows the log of the electrochemical reaction rate (current) vs. electrical overpotential. Tafel plots are approximately linear with a slope that is proportional to the number of electrons transferred in the limiting step. Nucleation [30]: Rates of nucleation are highly sensitive to the chemical potential driving force μ, i.e. to supersaturation. The nucleation theorem says that ∂lnJ /∂μ ≈ n‡ . The nucleation theorem is not a linear (or even quadratic) free energy relationship. Nonetheless, the slope at each driving force μ is the distance from reactant to transition state. The nucleation theorem is widely used in experiments to estimate critical nucleus sizes. Single molecule pulling experiments [31,32]: New optical tweezers and atomic force microscopy techniques [33,34] can attach tethers to folded proteins or hybridized DNA strands and pull with constant force until the structures “rupture”. When the tethers are attached at appropriate locations [35], a constant pulling force4 reduces the rupture time in a fashion consistent with equation (22.3.1).
The example below considers a model of mean rupture times in constant force pulling experiments. In this case, the slope of the free energy relationship will give the critical extension at the transition state. Pulling experiments with a constant force Suppose that the reaction coordinate for spontaneous unfolding of a protein or for unzipping of an RNA hairpin [34] is q. So that we may retain a simple one dimensional model, assume that the direction of pulling coincides with the reaction coordinate q. Finally, assume that the free energy as a function of the reaction coordinate with zero pulling force is F0 (q) and that the mobility along q is coordinate independent (and force independent). For a non-zero force f , the free energy as a function of q becomes F (q) = F0 (q) − f · q. We can use the mean first passage time (MFPT) expression from Chapter 18 to compute
3 D. Makarov has shown that important exceptions arise when the pulling direction is not aligned with the reaction
coordinate [27,28]. 4 Pulling experiments can also be done at constant pulling speed. A fascinating new branch of non-equilibrium
statistical mechanics concerns relationships between the reversible work (free energy) and the non-equilibrium work distributions along forward and reverse trajectories [36–39].
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Free energy relationships 609 mean rupture times. The ratio of mean rupture times with and without force is [31,40]
+(F (q)−f ·q)/k T −(F (q)−f ·q)/k T B dq B dq e 0 e 0 τMF P (f ) = ∩ +F (q)/k T ∪ −F (q)/k T B dq B dq 0 0 τMF P (0) ∩e ∪e where subscripts ∪ and ∩ on the integrals indicate integration over values of q near the well bottom and near the barrier top, respectively. Dudko et al. [31] examine a model in which the unbiased free energy surface is F0 (q) = 1.5 · (q/q0‡ )F0‡ − 2.0 · (q/q0‡ )3 F0‡ . The integrals that arise from the model of Dudko et al. can be done analytically giving τMF P (f ) = (1 − f · q0‡ /2F0‡ )−1 exp −βF0‡ 1 − (1 − f · q0‡ /2F0‡ )2 τMF P (0) −1 Instead of MFPTs, define the force dependent rate constant as k = τMF P and take logarithms lnk(f ) − lnk(0) = ln(1 − f · q0‡ /2F0‡ ) + βF0‡ 1 − (1 − f · q0‡ /2F0‡ )2
which for small forces is lnk(f ) = lnk(0) + (βF0‡ − 1/2)
f · q0‡ F0‡
+ ...
The remaining terms are of order (f · q0‡ /F0‡ )2 . The term −f · q0‡ /2F0‡ is unusual in an LFER. Here, it emerged from a force dependence in the prefactor of the MFPT expression. Presumeably, rupture happens while βF0‡ 1/2. Otherwise, the separation of time scales would break down making the simple MFPT formula invalid anyway. Omitting the -1/2 term recovers βF ‡ = βF0‡ (1 − f · q0‡ /F0‡ ) which is the standard ‡
k(f ) = k(0)eβf q0 formula of Bell [25]. The key prediction from Bell’s formula is that dlnk/dβf is q0‡ , i.e. the displacement along the reaction coordinate between the transition state and the minimum at zero force. This, of course, is the same interpretation given in the earlier FERs for chemical reactions.
Additional arenas where free energy relationships with externally controlled driving forces include slip-stick friction, electrophoresis, crack-tip initiation, etc. Undoubtedly, some (if not all) of these phenomena have already been studied using free energy relationships.
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610 Chapter 22
Closing remarks My effort to write a text on rate theories and rare events methods started nine years ago. Of course, the task wasn’t supposed to take nine years. Like most academic pursuits, I began with unwarranted confidence, and surrendered after being humbled by the many gaps in my knowledge. Nevertheless, the task was a pleasure. Hopefully, each member of my audience has found at least some of these pages enlightening and useful. Before closing this endeavor, let me briefly comment on the status of the subject and prognosticate on its future. Recent decades saw tremendous advances: degree-of-rate-control analyses, disconnectivity graphs, empirical valence bond, saddle search algorithms, Landau free energy calculations, kinetic Monte Carlo, reactive flux methods, Grote-Hynes theory, LangerBerezhkovskii-Szabo theory, path sampling, and reaction coordinate optimization methods. Approximately one third of the material in this book did not exist 40 years ago. The recent flurry of theoretical progress is especially remarkable given that the subject began in the 1800s and that its theoretical foundations were established in a separate flurry during the 1930s. The recent progress has been spurred by organizations like the Centre Europeen de Calcul Atomique et Moleculaire (CECAM), the Telluride Summer Research Conferences (TSRC), and the Institute for Pure and Applied Mathematics (IPAM). These institutions and the agencies that support them have helped chemists, physicists, mathematicians, and engineers discover and draw inspiration from each others’ work. The newest rate theories and rare events methods have transformed tasks that were once impossible into readily executed calculations. Opportunities to use these new tools abound, and there is no end in sight to the current period of rapid fundamental discovery. However, there are also major changes afoot. Many current efforts in the computational sciences are focused on simulations at larger and ever longer time scales and on machine learning tools for data driven models. These new tools have already impacted many areas of research, and they will surely impact the theory of reaction rates and rare events. However, large scale computation should not (and as yet cannot) replace physics-based theoretical models and clever rare events methods. In the words of a recent perspective by Marcus [41], “...as a percentage of total theoretical research effort, [analytical theory] is now small compared with computation. I believe the future can be expected to settle on some insightful combination of both, since both are necessary.” Amen
Exercises 1. Following Polanyi, let two potential energy curves VA (q) and VB (q) cross at location q0 with energy Ea as shown in the figure below.
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Free energy relationships 611
2.
3.
4.
5.
Let gA = dVA /dq and gB = dVB /dq be their slopes at the crossing point. Use linear approximations at the intersection to show that when VB (q) is shifted vertically by an amount H the new intersection is at energy Ea ≈ Ea + γ · H . Find the parameter γ in terms of gA and gB . Summarize the Hammett relations [42] for reaction rates and equilibrium constants of reactions involving substituted benzene rings. Write these as relationships between reaction free energies and activation free energies. [Z. Han] Robinson and Holbrook tabulated the high pressure Arrhenius parameters for a series of ten 1,5-hydrogen transfer reactions in 1,3-dienes. Choose an accurate ab initio model chemistry. Compute the energies of the stable reactants and products to develop a BEP correlation for the series of 1,5 hydrogen transfer reactions. Compute the Arrhenius parameters and energies of the reactant and product states for this series. How well does the BEP correlation work? Do you expect a free energy relationship to work better? Why or why not? In the limit of a small force f along the reaction coordinate, what is dlnK(f )/df where K(f ) is the force-perturbed equilibrium constant? Express your answer in terms of the product and reactant positions along the reaction coordinate. The nucleation theorem gives the derivative of the reversible work to create the critical nucleus as dG‡ /dμ = −n‡
See the literature on nucleation [43] for relationships between G‡ and the nucleation rate. At the spinodal, μ = μS and G‡S = 0. Use the nucleation theorem to write the free energy barrier for μ = μS in terms of an integral over estimated nucleus sizes. 6. Let V0 (x, y) = V‡,0 · (1 − 3y 2 − 2y 3 ) + 12 mωx2 (x − cy/mωx2 )2 be the PES in the absence of a pulling force. The reaction coordinate is some linear combination of x and y. (a) Consider a force f that pulls along the x-coordinate so that the potential becomes Vf (x, y) = V0 (x, y) − f · x. Compute limf →0 dV‡ /df . Compare your results to the Bell formula. (b) Suppose that you were unaware of y, and that had began your analysis from the
you −βV (x,y) dy. Show that the stationary 0 free energy at zero force F0 (x) = −kB T ln e points in F0 (x) coincide with the x-coordinates of stationary points in V0 (x, y). When a force is applied, the saddle point on Vf (x, y) moves, as does the maximum in the potential F (x) = F0 (x) − f x. Do they move in consistent directions?
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612 Chapter 22 (c) Makarov examined this situation and concluded that “no one-dimensional model can possibly reproduce the prediction of the two dimensional model...” Clearly, this is true if the one-dimension is x. Assume that x and y are associated with the same mass, perform the usual normal mode analysis to find the reaction coordinate, and repeat the calculations above. Do the transition states and minima from the 1D and 2D models shift in consistent directions? Do the results from the 1D and 2D models become consistent with Bell’s formula? Do we always have the ability to pull along the correct reaction coordinate? See Makarov for equations that describe the surprising effects of pulling forces that are not aligned with the reaction coordinate in higher dimensions [27,28].
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
W.P. Jencks, Chem. Rev. 85 (1985) 511–527. M.G. Evans, M. Polanyi, Trans. Faraday Soc. 34 (1938) 11–24. J.-K. Hwang, A. Warshel, J. Am. Chem. Soc. 109 (1987) 715–720. J. Aqvist, A. Warshel, Chem. Rev. 93 (1993) 2523–2544. B. Hammer, J.K. Norskov, Adv. Catal. 45 (2000). E. Skulason, G.S. Karlberg, J. Rossmeisl, T. Bligaard, J. Greeley, H. Jonsson, J.K. Norskov, Phys. Chem. Chem. Phys. 9 (2007) 3241–3250. R.A. van Santen, M. Neurock, S.G. Shetty, Chem. Rev. 110 (2010) 2005–2048. J.K. Norskov, F. Studt, F. Abild-Pederson, T. Bligaard, Fundamental Concepts in Heterogeneous Catalysis, Wiley, Hoboken, NJ, 2014. C.H. Christensen, J.K. Norskov, J. Chem. Phys. 128 (2008) 182503. J.K. Norskov, T. Bligaard, J. Rossmeisl, C.H. Christensen, Nat. Chem. 1 (2009) 37–46. J.R.B. Gomes, J.M. Bofill, F. Illas, J. Phys. Chem. C 112 (2008) 1072–1080. J.N. Bronsted, K.J. Pederson, Z. Phys. 108 (1924) 185–235. R.A. Marcus, J. Phys. Chem. 72 (1968) 891–899. J.M. Mayer, Acc. Chem. Res. 44 (2011) 36–46. A. Warshel, J. Phys. Chem. 86 (1982) 2218–2224. A. Cohen, R.A. Marcus, J. Chem. Phys. 78 (1968) 4249–4256. J.M. Mayer, Annu. Rev. Phys. Chem. 55 (2004) 363–390. D.N. Silverman, Biochim. Biophys. Acta, Bioenerg. 1458 (2000) 88–103. J.E. Leffler, Science 117 (1953) 340–341. G.S. Hammond, J. Am. Chem. Soc. 77 (1955) 334. D. Kashchiev, J. Chem. Phys. 76 (1982) 5098. H. Eyring, J. Chem. Phys. 4 (1936) 283–292. A.J. Appleby, J.H. Zagal, J. Solid State Electrochem. 15 (2011) 1811–1832. S.N. Zhurkov, Int. J. Fract. Mech. 1 (1965) 311. G.I. Bell, Science 200 (1978) 618. D.J. Wales, Energy Landscapes, Cambridge University Press, Cambridge, UK, 2003. S.S.M. Konda, J.N. Brandtley, B.T. Varghese, K.M. Wiggins, C.W. Bielawski, D.E. Makarov, J. Am. Chem. Soc. 135 (2013) 12722–12729. D.E. Makarov, J. Chem. Phys. 144 (2016) 030901. C. Taylor, S. Wasileski, J.-S. Filhol, M. Neurock, Phys. Rev. B 73 (2006) 165402. D. Kashchiev, G.M. van Rosmalen, Cryst. Res. Technol. 38 (2003) 555–574.
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Free energy relationships 613 [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]
O. Dudko, G. Hummer, A. Szabo, Phys. Rev. Lett. 96 (2006) 108101. G. Hummer, A. Szabo, Proc. Natl. Acad. Sci. USA 107 (2010) 10–15. M.S.Z. Kellermeyer, S.B. Smith, H.L. Granzier, C. Bustamante, Science 276 (1997) 1112–1116. W.J. Greenleaf, K.L. Frieda, D.A.N. Foster, M.T. Woodside, S.M. Block, Annu. Rev. Biophys. Biomol. Struct. 319 (2008) 630–633. J.D. Chodera, V.S. Pande, Phys. Rev. Lett. 107 (2011) 98102. C. Jarzynski, Phys. Rev. Lett. 78 (1997) 2690. G. Crooks, Phys. Rev. E 61 (2000) 2361–2366. R.C. Lua, A.Y. Grosberg, J. Phys. Chem. B 109 (2005) 6805–6811. C. Bustamante, J. Liphardt, F. Ritort, Phys. Today July (2005) 43–48. O.K. Dudko, G. Hummer, A. Szabo, Proc. Natl. Acad. Sci. USA 105 (2008) 15755–15760. R.A. Marcus, Phys. Chem. Chem. Phys. 14 (2012) 13729–13730. L.P. Hammett, J. Am. Chem. Soc. 59 (1937) 96. D. Kashchiev, Nucleation: Basic Theory with Applications, Butterworth-Heinemann, Oxford, 2000.
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Free energy relationships
...in a series of homologous reactions the heats of activation decrease with increasing heats of reaction. ... derive the Tafel equation for hydrogen overvoltage and the series of overvoltages at the metals. Horiuti and Polanyi, (tr. Muller), Acta Physicochem. URSS (1935)
The introduction to this book stressed that “matching experiments” should not be the final goal of a computational effort. To engineer better catalysts, processes, or materials, it is often more important to identify reliable ways of controlling the kinetics. This chapter explores relationships, some empirical and some derived, between kinetics and driving forces. These relationships are useful for predicting trends, e.g. how a change in solvent will change the rate of a chemical reaction, or how a mechanical pulling force will accelerate the unfolding of a protein. The relationships are also useful for interpreting kinetic trends in terms of transition state properties. For example, some free energy relationships indicate the location of a transition state, or the size of a critical nucleus, both from measurable quantities. With some effort we can even go beyond kinetic trends for one reaction, to predict and understand kinetic trends across series of similar reactions. In these efforts, relationships between thermodynamic properties and reactivity have emerged as one of the most fruitful intersections between theory and experiment. For example in organometallic chemistry, we can understand and predict how ligands with different electron withdrawing characteristics influence the rate of a reaction. In catalysis, we can understand and predict how the amount or type of a dopant alters the activity. Modern computing power has made it possible to perform automated and increasingly reliable calculations for hundreds or thousands of slightly different catalysts or substrates. The resulting large data sets have revealed many relationships between molecular structure, electronic structure, and chemical properties including reactivity. Structure-property relationships discovered in this manner often begin as empiricisms and heuristics, but especially in recent catalysis work they have also inspired fundamental advances and ab initio tools for new types of analyses. Reaction Rate Theory and Rare Events Copyright © 2017 Elsevier B.V. All rights reserved.
601
602 Chapter 22
22.1 BEP relations and the Bronsted catalysis law One of the first and still most important relationships between thermodynamics and reactivity is that of Bell1 and Evans and Polanyi (BEP) [2]. BEP postulated that for a series of similar reactions, the more exothermic reactions would have lower activation barriers. BEP relations of the form Ea ≈ Eao + αH o
(22.1.1) H o
are widely used [1,3], especially in catalysis [4–6]. Here is the reaction enthalpy and o o Ea is the activation energy for a (hypothetical) isoergic (H = 0) reaction in the series. The slope α is related to the position of the transition state (early, midway, or late) along the reaction pathway. Aqvist and Warshel [4] and van Santen et al. [7] have reviewed the derivations, applications, and recent advances related to BEP relations in catalysis. The BEP relation is a practical computational tool because it is far easier to measure or compute reaction enthalpies than to accurately compute reaction rates. In heterogeneous catalysis, reaction enthalpies themselves can be further correlated to electronic band structures and to adsorbate properties. These correlations have been combined with BEP relations to create “universal BEP relations” that predict activities of many catalysts for many different reactions. Example: BEP relations in catalysis [8] Computational work in heterogeneous catalysis shows how H o can be estimated from even more readily computed quantities, e.g. d-band characteristics of a metal or alloy [5,9]. These studies first correlate d-band energies to atomic adsorbate energies, e.g. ∗O, ∗H , ∗N , and ∗C. The atomic adsorption energies are then correlated to adsorption energies of more complex and partially saturated adsorbates, e.g. ∗OH , ∗OCH3 , ∗NH2 , ∗CH , ∗CH2 , and ∗CH3 . Surface reaction stoichiometries are used to combine correlations for adsorbate enthalpies into correlations for surface reaction enthalpies. Finally, the reaction enthalpies are used in BEP relations for the breaking of different bond types. The result is a modern “aufbau principle” for building correlations that relate dband characteristics and a small collection of adsorbate properties to activation energies. These correlations-in-series are not extremely accurate, but they are powerful ways of quickly screening the entire periodic table for highly active metals, alloys, dopants, etc. 1 The BEP acronym sometimes refers to Bell-Evans-Polanyi and sometimes refers to Bronsted-Evans-Polanyi in-
stead. Bronsted most definitely contributed to the development of LFERs with his catalysis law. R.P. Bell, who worked in Bronsted’s laboratory, directly used the BEP relation in his own work after it was put forth by Evans and Polanyi. Some sources even point to the much later G.I. Bell. Still others refer to “Bema Hapothle” (Bell-MarcusHammond-Polanyi-Thornton-Leffler) principles which describe changes in transition state structure across series of reactants or solvents. Clearly, there were many contributors to the theory of free energy relationships [1].
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Free energy relationships 603 This approach has now been used to predict the activity of thousands of metals and alloys in silico. BEP analyses have led to (1) an improved understanding of chemical bonding and catalysis on metal surfaces, and to (2) the design of now-patented new catalysts [10,8]. Figure 22.1.1 shows the “universal” BEP relation for dissociation of molecules containing single bonds between C, O, and N atoms on transition metal surfaces.
Figure 22.1.1: Wang et al. developed “universal” BEP relation from computed transition states and dissociation energies for C-C, C-O, C-N, N-O, N-N, and O-O bonds on a variety of transition metal surfaces. The typical error is 0.35eV, a result of correlating such different reactions in a single BEP relation. When different reactants are analyzed separately, the MAE is significantly lower. [Wang et al. Catalysis Lett. 141, 370-73 (2011)]
Recent progress in catalysis illustrates the accelerated fundamental advances and technological discoveries that are possible with relatively simple calculations that focus on trends. Heuristic derivations of the BEP relation (see section 22.2) suggest general guidelines on the situations where a BEP relation will be valid: • • •
BEP relations are most reliable when applied to individual elementary steps. BEP relations tend to be most accurate for reactions in which the transition state is late, i.e. for cases with α near unity so that transition states resemble the products. In practice, there is some ambiguity in what constitutes a related series of reactions. Generally, more closely related groups of reactions will more closely obey the BEP relation.
There are cases where BEP relations fail [11]. In heterogeneous catalysis, BEP relations fail when some members in a series of catalysts are prone to surface reconstructions, preferential segregation of certain alloy components to the surface, or to oxidation. The Bronsted catalysis law [12] relates the kinetics of proton abstraction to an acid dissociation equilibrium. Specifically, consider a series of reactions ki
H Ai + B − −→ H B + A− i
Goi
(22.1.2)
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604 Chapter 22 with equilibrium constant Ki = exp −Goi /kB T . Bronsted’s catalysis law says that, for a series of similar reactions, the equilibrium constants are related to the rate constants by ki ≈ γ Ki α
(22.1.3)
where γ and α are constants. If we assume ki = kB T h−1 Voν−1 exp −G‡i /kB T and take logarithms, equation (22.1.3) becomes G‡i ≈ α Goi − kB T ln kB T Voν−1 / hγ . Now invoke a reference reaction in the series for which Goref = 0. This reference reaction has G‡ref ≈ −kB T ln kB T Voν−1 / hγ . Therefore the Bronsted law (equation (22.1.3)) can also be written as G‡i ≈ G‡ref + α Goi which is similar to the BEP relation but involves free energies. α is called the Bronsted coefficient or Bronsted slope [13,4], ∂G‡ (22.1.4) ∂Go and typically it falls in the range 0 < α < 1. The Bronsted slope has been interpreted as the position of the transition state along the pathway from reactants to products. The reasons for this interpretation will become clear below when free energy relationships are heuristically derived from a Marcus theory model. α≡
22.2 The Marcus equation Chapter 21 noted that electron transfer ideas are applicable to many chemical reactions, especially proton transfer reactions [14,15]. Marcus obtained the parabolic free energy diabats as functions of a charge transfer interpolation coordinate ξ . The reactant state at ξ = 0 and the product state at ξ = 1 are minima for separate free energy parabolas with the same spring constants mω2 . As depicted in Figure 22.2.1, the free energy landscape in state A is approximately 1 GA (ξ ) ≈ mω2 ξ 2 2
(22.2.1)
In state B, the free energy landscape is approximately, 1 GB (ξ ) ≈ mω2 (ξ − 1)2 + Go 2
(22.2.2)
Solving for the point of intersection, ξ‡ ≈
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1 Go + 2 mω2
(22.2.3)
Free energy relationships 605
Figure 22.2.1: The Marcus-BEP relation is based on a simple model with two diabatic states.
Inserting ξ ‡ into GA (ξ ) gives the activation free energy 1 Go 2 2 1 + G = mω 2 2 mω2 ‡
(22.2.4)
For a thermoneutral reference reaction (Goref = 0) we can define 1 G‡ref = mω2 8
(22.2.5)
This allows us to write the activation free energy in terms of Go and the reference activation free energy. The result is the Marcus equation G‡ = G‡ref
Go Go 1+ + 2 8G‡0
(22.2.6)
The reference activation free energy G‡ref is essentially a fit parameter. No member of the reaction series actually needs to have Go = 0 to use the Marcus equation. Cohen and Marcus tested the Marcus equation against data for several reaction series [16]. Figure 22.2.2 shows their test for a large experimental data set on hydrogen abstraction rates from ketones by carboxylate anions and related bases. CH3 COCH RR + A− → CH3 COCRR − + H A
(22.2.7)
Cohen and Marcus tested some twenty reaction families, with results that generally validated the Marcus equation. The Marcus equation has also been validated for proton coupled electron transfers [17] and proton transfer reactions in enzymes [18].
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606 Chapter 22
Figure 22.2.2: A test of the Marcus equation for hydrogen abstraction from ketones by a family of carboxylate anions and other bases. Units on both axes are kcal/mol. [From Cohen and Marcus, J. Phys. Chem. 78, 4249-56 (1968).]
Returning to equation (22.2.3), let us write the transition state location in terms of Go and the activation free energy for the reference reaction. The result is the Marcus-Hammond relation ξ‡ =
Go 1 + 2 8G‡ ref
(22.2.8)
Now using equation (22.2.6) to compute the local slope of the free energy relationship gives α≡
∂G‡ = ξ ‡. ∂Go
(22.2.9)
This equation explains why the slope in BEP relationships and LFERs has been interpreted as the position of the transition state along the reaction pathway. Equations (22.2.8) and (22.2.9) justify the postulates of Leffler and Hammond [19,20]. Specifically, in a series of similar reactions, more exothermic reactions will have earlier transition states along the reaction coordinate and their transition state structures will more closely resemble the reactants. Where chemical accuracy is needed, it should be remembered that most FERs have a heuristic basis. Even theoretically derived FERs like the Marcus equation [16] and the nucleation theorem [21] are based on extremely simple models. The true activation energies and free energies are often widely scattered around the values predicted by an FER. Of course, FERs will fail to predict trends when the reaction path or mechanism changes within a seemingly similar series of reactions, or when the limiting step in a multistep mechanism changes to another other step in the sequence.
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Free energy relationships 607
22.3 Externally controlled driving forces The previous section discussed free energy relationships that correlate rates across a series of similar reactions. This section discusses similar free energy relations that help understand situations where the driving force for a single reaction is externally controlled. Rate theories that account for externally modulated driving forces have a long history, and similar expressions starting with Eyring [22] have emerged in many different contexts [23–25,21]. The simplest models consider a one dimensional free energy profile as a function of the reaction coordinate with a force applied along the reaction coordinate. The applied force alters the free energy landscape by simply tilting the original landscape. For small forces,2 the new free energy landscape qualitatively resembles the original free energy landscape, but the stationary points shift to new locations along q and the barrier heights change as shown in Figure 22.3.1.
Figure 22.3.1: Pulling along the reaction coordinate q tilts the zero-force free energy landscape and shifts the locations and heights of the free energy barriers.
Changes in the barrier to escape state A include terms of order f which are proportional to the unperturbed distance from reactants to transition states along q. Changes in the barrier also include terms of order f 2 which emerge from force-induced changes in the location of the reactant minimum and transition state. The force-perturbed free energy barrier is 1 Ff = F0 − f · (q‡ − qA ) − f 2 {(1/F (q‡ ) − 1/F (qA )} 2
(22.3.1)
To first order in the force f , equation (22.3.1) is consistent with the Hammond-Leffler postulate, i.e. a force that pulls along the direction leading to the transition state lowers the barrier and leads to an earlier transition state. Typically, the first order term in the external force dominates 2 For sufficiently large forces a catastrophe [26] causes the minima lose their mechanical stability. We limit our
discussion to the effects of smaller forces.
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608 Chapter 22 except where the force is strong.3 The slope in a free energy relationship has a physical interpretation which is related to the position of the transition state along the reaction coordinate. Some important special cases of equation (22.3.1) include: •
•
•
Electrochemistry [29,23]: Suppose that an electrochemical reaction M n+ + n e− → M involves transfer of a specific charge n e− from the electrode. A Tafel plot shows the log of the electrochemical reaction rate (current) vs. electrical overpotential. Tafel plots are approximately linear with a slope that is proportional to the number of electrons transferred in the limiting step. Nucleation [30]: Rates of nucleation are highly sensitive to the chemical potential driving force μ, i.e. to supersaturation. The nucleation theorem says that ∂lnJ /∂μ ≈ n‡ . The nucleation theorem is not a linear (or even quadratic) free energy relationship. Nonetheless, the slope at each driving force μ is the distance from reactant to transition state. The nucleation theorem is widely used in experiments to estimate critical nucleus sizes. Single molecule pulling experiments [31,32]: New optical tweezers and atomic force microscopy techniques [33,34] can attach tethers to folded proteins or hybridized DNA strands and pull with constant force until the structures “rupture”. When the tethers are attached at appropriate locations [35], a constant pulling force4 reduces the rupture time in a fashion consistent with equation (22.3.1).
The example below considers a model of mean rupture times in constant force pulling experiments. In this case, the slope of the free energy relationship will give the critical extension at the transition state. Pulling experiments with a constant force Suppose that the reaction coordinate for spontaneous unfolding of a protein or for unzipping of an RNA hairpin [34] is q. So that we may retain a simple one dimensional model, assume that the direction of pulling coincides with the reaction coordinate q. Finally, assume that the free energy as a function of the reaction coordinate with zero pulling force is F0 (q) and that the mobility along q is coordinate independent (and force independent). For a non-zero force f , the free energy as a function of q becomes F (q) = F0 (q) − f · q. We can use the mean first passage time (MFPT) expression from Chapter 18 to compute
3 D. Makarov has shown that important exceptions arise when the pulling direction is not aligned with the reaction
coordinate [27,28]. 4 Pulling experiments can also be done at constant pulling speed. A fascinating new branch of non-equilibrium
statistical mechanics concerns relationships between the reversible work (free energy) and the non-equilibrium work distributions along forward and reverse trajectories [36–39].
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Free energy relationships 609 mean rupture times. The ratio of mean rupture times with and without force is [31,40]
+(F (q)−f ·q)/k T −(F (q)−f ·q)/k T B dq B dq e 0 e 0 τMF P (f ) = ∩ +F (q)/k T ∪ −F (q)/k T B dq B dq 0 0 τMF P (0) ∩e ∪e where subscripts ∪ and ∩ on the integrals indicate integration over values of q near the well bottom and near the barrier top, respectively. Dudko et al. [31] examine a model in which the unbiased free energy surface is F0 (q) = 1.5 · (q/q0‡ )F0‡ − 2.0 · (q/q0‡ )3 F0‡ . The integrals that arise from the model of Dudko et al. can be done analytically giving τMF P (f ) = (1 − f · q0‡ /2F0‡ )−1 exp −βF0‡ 1 − (1 − f · q0‡ /2F0‡ )2 τMF P (0) −1 Instead of MFPTs, define the force dependent rate constant as k = τMF P and take logarithms lnk(f ) − lnk(0) = ln(1 − f · q0‡ /2F0‡ ) + βF0‡ 1 − (1 − f · q0‡ /2F0‡ )2
which for small forces is lnk(f ) = lnk(0) + (βF0‡ − 1/2)
f · q0‡ F0‡
+ ...
The remaining terms are of order (f · q0‡ /F0‡ )2 . The term −f · q0‡ /2F0‡ is unusual in an LFER. Here, it emerged from a force dependence in the prefactor of the MFPT expression. Presumeably, rupture happens while βF0‡ 1/2. Otherwise, the separation of time scales would break down making the simple MFPT formula invalid anyway. Omitting the -1/2 term recovers βF ‡ = βF0‡ (1 − f · q0‡ /F0‡ ) which is the standard ‡
k(f ) = k(0)eβf q0 formula of Bell [25]. The key prediction from Bell’s formula is that dlnk/dβf is q0‡ , i.e. the displacement along the reaction coordinate between the transition state and the minimum at zero force. This, of course, is the same interpretation given in the earlier FERs for chemical reactions.
Additional arenas where free energy relationships with externally controlled driving forces include slip-stick friction, electrophoresis, crack-tip initiation, etc. Undoubtedly, some (if not all) of these phenomena have already been studied using free energy relationships.
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610 Chapter 22
Closing remarks My effort to write a text on rate theories and rare events methods started nine years ago. Of course, the task wasn’t supposed to take nine years. Like most academic pursuits, I began with unwarranted confidence, and surrendered after being humbled by the many gaps in my knowledge. Nevertheless, the task was a pleasure. Hopefully, each member of my audience has found at least some of these pages enlightening and useful. Before closing this endeavor, let me briefly comment on the status of the subject and prognosticate on its future. Recent decades saw tremendous advances: degree-of-rate-control analyses, disconnectivity graphs, empirical valence bond, saddle search algorithms, Landau free energy calculations, kinetic Monte Carlo, reactive flux methods, Grote-Hynes theory, LangerBerezhkovskii-Szabo theory, path sampling, and reaction coordinate optimization methods. Approximately one third of the material in this book did not exist 40 years ago. The recent flurry of theoretical progress is especially remarkable given that the subject began in the 1800s and that its theoretical foundations were established in a separate flurry during the 1930s. The recent progress has been spurred by organizations like the Centre Europeen de Calcul Atomique et Moleculaire (CECAM), the Telluride Summer Research Conferences (TSRC), and the Institute for Pure and Applied Mathematics (IPAM). These institutions and the agencies that support them have helped chemists, physicists, mathematicians, and engineers discover and draw inspiration from each others’ work. The newest rate theories and rare events methods have transformed tasks that were once impossible into readily executed calculations. Opportunities to use these new tools abound, and there is no end in sight to the current period of rapid fundamental discovery. However, there are also major changes afoot. Many current efforts in the computational sciences are focused on simulations at larger and ever longer time scales and on machine learning tools for data driven models. These new tools have already impacted many areas of research, and they will surely impact the theory of reaction rates and rare events. However, large scale computation should not (and as yet cannot) replace physics-based theoretical models and clever rare events methods. In the words of a recent perspective by Marcus [41], “...as a percentage of total theoretical research effort, [analytical theory] is now small compared with computation. I believe the future can be expected to settle on some insightful combination of both, since both are necessary.” Amen
Exercises 1. Following Polanyi, let two potential energy curves VA (q) and VB (q) cross at location q0 with energy Ea as shown in the figure below.
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Free energy relationships 611
2.
3.
4.
5.
Let gA = dVA /dq and gB = dVB /dq be their slopes at the crossing point. Use linear approximations at the intersection to show that when VB (q) is shifted vertically by an amount H the new intersection is at energy Ea ≈ Ea + γ · H . Find the parameter γ in terms of gA and gB . Summarize the Hammett relations [42] for reaction rates and equilibrium constants of reactions involving substituted benzene rings. Write these as relationships between reaction free energies and activation free energies. [Z. Han] Robinson and Holbrook tabulated the high pressure Arrhenius parameters for a series of ten 1,5-hydrogen transfer reactions in 1,3-dienes. Choose an accurate ab initio model chemistry. Compute the energies of the stable reactants and products to develop a BEP correlation for the series of 1,5 hydrogen transfer reactions. Compute the Arrhenius parameters and energies of the reactant and product states for this series. How well does the BEP correlation work? Do you expect a free energy relationship to work better? Why or why not? In the limit of a small force f along the reaction coordinate, what is dlnK(f )/df where K(f ) is the force-perturbed equilibrium constant? Express your answer in terms of the product and reactant positions along the reaction coordinate. The nucleation theorem gives the derivative of the reversible work to create the critical nucleus as dG‡ /dμ = −n‡
See the literature on nucleation [43] for relationships between G‡ and the nucleation rate. At the spinodal, μ = μS and G‡S = 0. Use the nucleation theorem to write the free energy barrier for μ = μS in terms of an integral over estimated nucleus sizes. 6. Let V0 (x, y) = V‡,0 · (1 − 3y 2 − 2y 3 ) + 12 mωx2 (x − cy/mωx2 )2 be the PES in the absence of a pulling force. The reaction coordinate is some linear combination of x and y. (a) Consider a force f that pulls along the x-coordinate so that the potential becomes Vf (x, y) = V0 (x, y) − f · x. Compute limf →0 dV‡ /df . Compare your results to the Bell formula. (b) Suppose that you were unaware of y, and that had began your analysis from the
you −βV (x,y) dy. Show that the stationary 0 free energy at zero force F0 (x) = −kB T ln e points in F0 (x) coincide with the x-coordinates of stationary points in V0 (x, y). When a force is applied, the saddle point on Vf (x, y) moves, as does the maximum in the potential F (x) = F0 (x) − f x. Do they move in consistent directions?
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612 Chapter 22 (c) Makarov examined this situation and concluded that “no one-dimensional model can possibly reproduce the prediction of the two dimensional model...” Clearly, this is true if the one-dimension is x. Assume that x and y are associated with the same mass, perform the usual normal mode analysis to find the reaction coordinate, and repeat the calculations above. Do the transition states and minima from the 1D and 2D models shift in consistent directions? Do the results from the 1D and 2D models become consistent with Bell’s formula? Do we always have the ability to pull along the correct reaction coordinate? See Makarov for equations that describe the surprising effects of pulling forces that are not aligned with the reaction coordinate in higher dimensions [27,28].
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
W.P. Jencks, Chem. Rev. 85 (1985) 511–527. M.G. Evans, M. Polanyi, Trans. Faraday Soc. 34 (1938) 11–24. J.-K. Hwang, A. Warshel, J. Am. Chem. Soc. 109 (1987) 715–720. J. Aqvist, A. Warshel, Chem. Rev. 93 (1993) 2523–2544. B. Hammer, J.K. Norskov, Adv. Catal. 45 (2000). E. Skulason, G.S. Karlberg, J. Rossmeisl, T. Bligaard, J. Greeley, H. Jonsson, J.K. Norskov, Phys. Chem. Chem. Phys. 9 (2007) 3241–3250. R.A. van Santen, M. Neurock, S.G. Shetty, Chem. Rev. 110 (2010) 2005–2048. J.K. Norskov, F. Studt, F. Abild-Pederson, T. Bligaard, Fundamental Concepts in Heterogeneous Catalysis, Wiley, Hoboken, NJ, 2014. C.H. Christensen, J.K. Norskov, J. Chem. Phys. 128 (2008) 182503. J.K. Norskov, T. Bligaard, J. Rossmeisl, C.H. Christensen, Nat. Chem. 1 (2009) 37–46. J.R.B. Gomes, J.M. Bofill, F. Illas, J. Phys. Chem. C 112 (2008) 1072–1080. J.N. Bronsted, K.J. Pederson, Z. Phys. 108 (1924) 185–235. R.A. Marcus, J. Phys. Chem. 72 (1968) 891–899. J.M. Mayer, Acc. Chem. Res. 44 (2011) 36–46. A. Warshel, J. Phys. Chem. 86 (1982) 2218–2224. A. Cohen, R.A. Marcus, J. Chem. Phys. 78 (1968) 4249–4256. J.M. Mayer, Annu. Rev. Phys. Chem. 55 (2004) 363–390. D.N. Silverman, Biochim. Biophys. Acta, Bioenerg. 1458 (2000) 88–103. J.E. Leffler, Science 117 (1953) 340–341. G.S. Hammond, J. Am. Chem. Soc. 77 (1955) 334. D. Kashchiev, J. Chem. Phys. 76 (1982) 5098. H. Eyring, J. Chem. Phys. 4 (1936) 283–292. A.J. Appleby, J.H. Zagal, J. Solid State Electrochem. 15 (2011) 1811–1832. S.N. Zhurkov, Int. J. Fract. Mech. 1 (1965) 311. G.I. Bell, Science 200 (1978) 618. D.J. Wales, Energy Landscapes, Cambridge University Press, Cambridge, UK, 2003. S.S.M. Konda, J.N. Brandtley, B.T. Varghese, K.M. Wiggins, C.W. Bielawski, D.E. Makarov, J. Am. Chem. Soc. 135 (2013) 12722–12729. D.E. Makarov, J. Chem. Phys. 144 (2016) 030901. C. Taylor, S. Wasileski, J.-S. Filhol, M. Neurock, Phys. Rev. B 73 (2006) 165402. D. Kashchiev, G.M. van Rosmalen, Cryst. Res. Technol. 38 (2003) 555–574.
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Free energy relationships 613 [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]
O. Dudko, G. Hummer, A. Szabo, Phys. Rev. Lett. 96 (2006) 108101. G. Hummer, A. Szabo, Proc. Natl. Acad. Sci. USA 107 (2010) 10–15. M.S.Z. Kellermeyer, S.B. Smith, H.L. Granzier, C. Bustamante, Science 276 (1997) 1112–1116. W.J. Greenleaf, K.L. Frieda, D.A.N. Foster, M.T. Woodside, S.M. Block, Annu. Rev. Biophys. Biomol. Struct. 319 (2008) 630–633. J.D. Chodera, V.S. Pande, Phys. Rev. Lett. 107 (2011) 98102. C. Jarzynski, Phys. Rev. Lett. 78 (1997) 2690. G. Crooks, Phys. Rev. E 61 (2000) 2361–2366. R.C. Lua, A.Y. Grosberg, J. Phys. Chem. B 109 (2005) 6805–6811. C. Bustamante, J. Liphardt, F. Ritort, Phys. Today July (2005) 43–48. O.K. Dudko, G. Hummer, A. Szabo, Proc. Natl. Acad. Sci. USA 105 (2008) 15755–15760. R.A. Marcus, Phys. Chem. Chem. Phys. 14 (2012) 13729–13730. L.P. Hammett, J. Am. Chem. Soc. 59 (1937) 96. D. Kashchiev, Nucleation: Basic Theory with Applications, Butterworth-Heinemann, Oxford, 2000.
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