Thermodynamics of Reactions Affected by Medium Reorganization

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Thermodynamics of Reactions Affected by Medium Reorganization Dmitry V Matyushov, and Marshall D. Newton J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b08865 • Publication Date (Web): 04 Dec 2018 Downloaded from http://pubs.acs.org on December 5, 2018

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The Journal of Physical Chemistry

Thermodynamics of Reactions Affected by Medium Reorganization Dmitry V. Matyushov∗,† and Marshall D. Newton∗,‡ †Department of Physics and School of Molecular Sciences, Arizona State University, PO Box 871504, Tempe, Arizona 85287 ‡Brookhaven National Laboratory, Chemistry Department, Box 5000, Upton, NY 11973-5000, United States E-mail: [email protected],Tel:(480)965-0057; [email protected]

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Abstract We present a thermodynamic analysis of the activation barrier for reactions which can be monitored through the difference in the energies of reactants and products defined as the reaction coordinate (electron and atom transfer, enzyme catalysis, etc). The free-energy surfaces along the reaction coordinate are separated into the enthalpy and entropy surfaces. For the Gaussian statistics of the reaction coordinate, the free-energy surfaces are parabolas, and the entropy surface is an inverted parabola. Its maximum coincides with the transition state for reactions with zero value of the reaction free energy. Maximum entropic depression of the activation barrier, anticipated by the concept of transition-state ensembles, can be achieved for such reactions. From Onsager’s reversibility, the entropy of equilibrium fluctuations encodes the entropic component of the activation barrier. The reorganization entropy thus becomes the critical parameter of the theory reducing the problem of activation entropy to the problem of reorganization entropy. Standard solvation theories do not allow reorganization entropy sufficient for the barrier depression. Complex media, characterized by many relaxation processes, need to be involved. Proteins provide several routes for achieving large entropic effects through incomplete (nonergodic) sampling of the complex energy landscape and by facilitating an active role of water in the reaction mechanism.

Introduction

ing electron and atom transfer 7,8 and bond breaking/formation. All nuclear modes and conformational coordinates (of a protein for enzyme catalysis 9 ) are projected on a single collective reaction coordinate. The conceptual basis of all these theories is the idea that reorganization of the medium drives the system out of equilibrium and along the reaction coordinate. The concept of medium reorganization is generally applicable to many problems, considering either medium relaxation due to an external perturbation (such as solvation 10 and ion pairing 11 ) or thermal excitation of collective fluctuations carrying the system over the activation barrier. 12 This is a fundamental property of any condensed material describing near-equilibrium fluctuations of a reaction coordinate affected by a large number of system elements (e.g., by a large number of molecules in the solvent). It naturally enters any theory considering Gaussian structural fluctuations not too far from equilibrium. 13 The mathematical formulation of the theory requires finding the equilibrium constrained free energy function Fi (X) corresponding to a given value of the gap in energies of the products (P) and reactants (R)

There are a number of reactions in condensed materials the progress of which can be described by the free energy/energy diagram. The horizontal axis is the energy reaction coordinate given by the difference of energies of the system in the states of products and reactants. The vertical axis shows the free energy (reversible work) required to bring the system composed of the reactants and the surrounding medium to that specific value of the energy gap (Figure 1). The most common application of this definition of the reaction coordinate, and of the related formalism to calculate the free energy profile, is to reactions significantly affected by electrostatic interaction of the reactants and products with a polarizable medium. The main contribution to the free energy on the vertical axis in Figure 1 then comes from the reversible work of polarizing the medium. Applications include the theory of electron transfer in polar media developed by Marcus, 1 transfer of proton/hydrogen/hydride, 2,3 and more general enzyme catalysis. 4–6 For the latter, Warshel and coworkers have proposed the theory, inspired by the Marcus treatment of electron transfer, in which the energy gap is adopted as the reaction coordinate and thus becomes the central part of the theoretical formulation. The theory is not limited to specific enzymatic reactions and is instead applicable to a broad range of chemical transformations, includ-

X = EP − E R

(1)

There are obviously many microscopic configurations of the system that produce a given value of X.


2

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Fi (X)

R

thermal fluctuation bringing the system to the same state, Pi (X) ∝ exp[−βFi (X)]. The rate constant of an activated transition is then proportional to the probability of reaching the activated state, which is the state of crossing of two free-energy surfaces, FR (X † ) = FP (X † ). For problems related to electron transfer and enzyme catalysis, crossing occurs at the point where energies of the reactants and products become equal, X † = 0 (Figure 1). The reaction rate then becomes

P X† = 0 λR !X"R

λP 2λSt

!X"P X

Figure 1: Schematic representation of the free-energy surfaces of the reactants (R) and products (P). The separation between the first moments of the reaction coordinate ⟨X⟩i , i = R, P defines the reorganization energy λSt ; the curvatures of the free-energy surfaces at their corresponding minima determine the reorganization energies λi . The symmetric (self-exchange) case is illustrated here; X † = 0 marks the transition state in which the free-energy functions cross, FR (X † ) = FP (X † ).

where the free energy of activation is the difference of free energies at the point of crossing at X † = 0 and at the minimum Xi

Entropy related to those configurations is a part of the free energy cost. The free energy profile as a function of X is defined by taking the statistical (Gibbs ensemble) average over all possible configurations of the system, including the medium, at a fixed X. Mathematically, this is expressed in terms of the deltafunction constraining the statistical average ∫ −βFi (X) e = δ(X − ∆H(Γ))e−βHi (Γ) dΓ (2)

The goal of this article is to establish general relations for the free-energy surfaces Fi (X) and the corresponding activation thermodynamics based on the rules of the canonical Gibbs ensemble used in eq 2. Our main focus is on splitting the freeenergy surfaces and the corresponding activation free energies into the enthalpic and entropic contributions. We first proceed to derive general results applicable to any system obeying the rules of Gibbs statistics and then turn to the more specific case of the Gaussian statistics of the reaction coordinate X.

ki ∝ exp[−β∆Fi† ]

∆Fi† = Fi (0) − Fi (Xi )

In this equation Hi (Γ) = Ei , i = {R, P} is the Hamiltonian of the system depending on the entire manifold of the nuclear degrees of freedom Γ (in contrast to the constrained phase space discussed below). Further, ∆H(Γ) = HP (Γ) − HR (Γ) is the instantaneous energy gap depending on the instantaneous nuclear configuration and dΓ denotes integration over the entire phase space of the system. The phase-space element dΓ is defined to cancel the dimension of inverse energy arising from the delta-function. The connection between the free-energy functions Fi (X) and the rate of activated transitions is afforded through Onsager’s principle of microscopic reversibility. 14 It states that the free energy released in relaxing the system from a given nonequilibrium state is the same as the free energy that determines the probability of a spontaneous

Reorganization energies fluctuation relations

(3)

(4)

and

The statistical average producing the free-energy surfaces in eq 2 can be represented as a Fourier integral of the corresponding cumulant generating function 15 by the following relation ∫ ∞ dξ iξβ(X−⟨X⟩i )+Fi (ξ) −βFi (X)+βF0i e (5) e = −∞ 2π Here, Fi (ξ) is an infinite sum of the cumulants of ∆H(Γ) Fi (ξ) =

∞ ∑ (−iβξ)n n=2


3

n!

Kn(i)

(6)


⟨δA2 ⟩ ∝ T . The slope of this scaling, in this case A = X, defines the reorganization energy based on the second cumulant

with the nth-order cumulant given as Kn(i) = ⟨(δ∆H)n ⟩i = ⟨δX n ⟩i

(7)

and δ∆H = ∆H − ⟨∆H⟩i , δX = X − ⟨X⟩i . The angular brackets denote the canonical ensemble average with the corresponding Hamiltonian Hi ∫ −1 ⟨. . . ⟩i = Qi . . . e−βHi (Γ) dΓ ∫ (8) −βHi (Γ) Qi = e dΓ

(i)

λi = 12 βK2 = 21 β⟨δX 2 ⟩i

(10)

In the Gibbs ensemble adopted here, one can establish a connection between the first and second cumulants of the reaction coordinate through the temperature derivative according to the equation ¯ i β (∂⟨X⟩i /∂β)ρ = ±λi − β⟨δ∆Hδ H⟩

(11)

where the temperature derivative is taken at constant density ρ or, alternatively, at constant volume V , given that the number of particles in the system is conserved. Further, here and below we adopt the convention that the upper sign refers to the reactant state (i = R) and the lower sign refers to the prod¯ = (HR + HP )/2 is uct state (i = P). Finally, H the mean of the Hamiltonians in the two states and ¯ =H ¯ − ⟨H⟩ ¯ i in eq 11. δH Equation 11 leads to another fluctuation relation connecting λSt and λi . We cast it in terms of the reorganization entropy 17 ( ) Sλ /kB = β 2 ∂λSt /∂β ρ (12)

Further, F0i = −β −1 ln Qi is the equilibrium free energy in either reactant, i = R, or product, i = P, state. Both the first moments of the reaction coordinate (i) ⟨X⟩i and the second cumulants K2 can be used to determine the central energetic parameter of the theory, the reorganization (free) energy. 1 We will use the notation λSt to define the reorganization energy in terms of the first moments of the energy gap X, and λ for the reorganization energy defined through the second moments. These two reorganization energies generally do not coincide, but, as we discuss below, become equal for the Gaussian statistics of X combined with the Gibbs statistical average in eq 2. The separation between the first moments ⟨X⟩i can be related to optical spectroscopy, where it becomes the separation between the energies of two optical transitions, typically absorption and emission. This is known as the Stokes shift, and that is why we use the superscript “St” to define the reorganization energy as the half of the separation between the two first moments (Figure 1) λSt = 21 |⟨X⟩R − ⟨X⟩P |

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One obtains ¯+K Sλ /kB = β λ

(13)

where the mean reorganization energy is ¯ = 1 [λR + λP ] λ 2

(14)

and the second summand in eq 13 is [ ] ¯ P − ⟨δ∆Hδ H⟩ ¯ R (15) K = (β 2 /2) ⟨δ∆Hδ H⟩

(9)

We have adopted the positive sign of Sλ in eq 14 in accord with experimental 18–20 and theoretical 17,21–23 evidence accumulated thus far. As we note below, λ can be viewed as the negative of a nuclear solvation free energy in a polar solvent, which generally becomes less negative with increasing temperature thus producing Sλ > 0. As is usually the case in solvation theories, 24,25 calculating entropies involves a near complete cancellation of the free energy term (the first summand in eq 13) with the higher order correlations

The first moments ⟨X⟩i ≈ Xi are typically close in magnitude to the minima of the free-energy surfaces Xi , but do not have to coincide with them. The equality occurs for the Gaussian approximation adopted in the Marcus theory discussed below. An alternative definition of the reorganization energy connects this theory to the fluctuationdissipation theorem (FDT), 16 which anticipates that the variance of a macroscopic property A scales approximately linearly with temperature,


4

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One also obtains Xi = ⟨X⟩i and, upon substituting eq 17 to eq 16,

responsible for restructuring of the medium at the solute-solvent interface (the term K in eqs 13 and 15). A somewhat simplified form of Sλ is given below in terms of the perturbation theories. Nevertheless, binary correlations entering K in eq 15 can presently be computed from ensemble configurations produced by numerical simulations. 17,25 This algorithm directly leads to reorganization entropies, bypassing the need for separate simulations and calculations at different temperatures, which involve numerically unreliable free energy slopes. For connection to experiment, one has to keep in mind that most measurements are done at constant pressure and corrections from the constant density (ρ) relations derived here are required to account for the medium expansivity. 17 Our description so far has been quite general and has not anticipated any specific statistics of the reaction coordinate X. The most common reduction of the problem used in theories of activated transitions is the Gaussian approximation, which assumes that all cumulants beyond the second one (n > 2) vanish in the cumulant expansion in eq 6. Before we turn to this approximation, we also note that the Gibbs ensemble used in eq 2 stipulates yet another exact thermodynamic relation 26–28 between the free-energy surfaces Fi (X) FP (X) = FR (X) + X

λSt = λ ∆F0 = 12 [⟨X⟩P + ⟨X⟩R ]

The next relation is the expression for the parabolas minima in terms of the reaction free energy ∆F0 and the reorganization energy Xi = ⟨X⟩i = ∆F0 ± λ

Enthalpy and entropy surfaces

(16)

(X − ⟨X⟩i )2 4λ

(20)

The second relation in eq 19 establishes the connection between the minima of two parabolas and the reaction free energy ∆F0 = F0P − F0R . Note in this regard that the free energy at the minimum, F (Xi ), does not have to coincide 33 with the equilibrium free energy F0i = −β −1 ln Qi . However, the detailed balance requires that the difference of the activation barriers for the forward, ∆FR† = FR (0) − FR (XR ), and backward, ∆FP† = FP (0) − FP (XP ), reactions is equal to the reaction free energy, ∆FR† − ∆FP† = ∆F0 . Since FR (0) = FP (0), this requirement implies that Fi (Xi ) = F0i +Const and a constant shift can always be eliminated.

The enthalpy surfaces 34 Hi (X) along the reaction coordinate X (not to be confused with Hamiltonians Hi (Γ)) can be defined by the thermodynamic relation ∂(βFi (X)) (21) Hi (X) = ∂β

This equality, which also extends to solvation problems, 29,30 implies that the second derivatives d2 Fi (X)/dX 2 should be equal for the reactant and product states at each X. This relation is another form of energy conservation applied to the partial (constrained) free-energy surfaces Fi (X). We discuss its connection to the enthalpy and entropy functions of X below and now turn to the Gaussian (Marcus) approximation for Fi (X). In the Gaussian (Marcus) description, 31 the truncation of the generally infinite 32 cumulant expansion beyond the second cumulant leads to parabolic free-energy surfaces Fi (X) = F0i +

(19)

By combining this definition with eq 2, one gets ∫ βFi (X)

Hi (X) = e

Hi (Γ)δ(X − ∆H(Γ)) e

−βHi (Γ)

(22)

dΓ

where, as above, the Hamiltonians Hi (Γ) are defined on the phase space of the system Γ. It is immediately clear that the energy conservation relation (eq 16) applies to Hi (X) as well

(17)

HP (X) = HR (X) + X

where the condition of equal curvatures of parabolas implies λR = λP = λ (18)

(23)

Taken together, eqs 16 and 23 produce equal en-


5


tropy surfaces in the two states SP (X) = SR (X) = S(X)

Page 6 of 18

the reaction entropy, and (24)

ξ=

Equations 16, 23, and 24 are general thermodynamic results not constrained by any specific statistics of X. In application to electronic transitions, eq 24 is consistent with the Franck-Condon principle. It stipulates that the nuclei do not move on the timescale of an electronic (vertical) transition 1 → 2 lifting the energy of the system by the transition energy X. The change in the entropy ∆S(X) = S2 (X) − S1 (X) must be identically zero. 34 However, in the context of the more general situation of an arbitrary activated transition considered here, the equality of the entropy surfaces implies that entropies between the products and reactants are equal for any reaction in which the difference of energies between them is adopted as the reaction coordinate. Equations 16 and 23 also imply that both Fi (X) and Hi (X) surfaces cross at the activated state of zero energy gap, X † = 0, at which both the Frank-Condon principle for the tunneling of a light particle and the energy conservation are satisfied. 35 Equations 21–24 generally apply to activation thermodynamics of reactions evolving along the energy-gap reaction coordinate. Specific forms for S(X) require Fi (X) as input to the thermodynamic relations. Parabolic free-energy surfaces of the Gaussian model lead to the entropy surfaces Si (X) = −(∂Fi (X)/∂T )ρ obtained by taking temperature derivatives in eq 17 ( ) (X − Xi )2 X − Xi ∂Xi + Si (X) = S0i − Sλ 4λ2 2λ ∂T ρ (25) Here, S0i = −(∂F0i /∂T )ρ is the entropy assigned to the reactants (i = R) or products (i = P). This equation can be simplified by applying the mathematical identity S(X) = (SR (X) + SP (X))/2 (see eq 24) and the definitions of parabolas’ minima in eq 20. After some algebra, one arrives at the following compact form

∆F0 − X λ

(27)

The entropies S0i of electrostatic solvation in polar materials are typically negative, 36,37 implying that the free energy of stabilizing the charge by a polar liquid is less negative at higher temperatures. The total thermodynamic entropy for the reactant and product states includes other contributions: from nonpolar (Lennard-Jones type) interactions and from cavity formation, which is the free energy required to insert the solute into the solvent. 30,38 However, these contributions cancel in ∆S0 if the solute-solvent electrostatics is the main interaction altered by the reaction and the nonpolar solvation and the free energy of cavity formation are independent of whether the reacting system is in the state of reactants or products. Further, since λ can be defined as the negative of the free energy of nuclear solvation of an “electrontransfer dipole”, Sλ is positive for electron-transfer reactions. 17–19,21 Consequently, the function S(X) is an inverted parabola with the maximum Smax = S¯0 + (Sλ2 + ∆S02 )/(4Sλ )

(28)

reached at XS,max = ∆F0 − λ(∆S0 /Sλ )

(29)

Note that neglect of the temperature variation of λ (Sλ → 0 34 ) makes S(X) a linear function of X instead of the inverted parabola. One next obtains the equation for the activation entropy ∆Si† = S(0) − S(Xi ) ) ( ( ) ∆S0 ∆F0 Sλ ∆F02 † 1− 2 + ±1 ∆Si = 4 λ 2 λ (30) Here, ∆SR† refers to the forward reaction, R → P, and ∆SP† refers to the backward reaction, P → R. The forward and backward activation entropies satisfy the detailed balance, ∆SR† − ∆SP† = ∆S0 . The activation entropy is an inverted parabola as a function of the driving force (Figure 2). It reaches the maximum value

S(ξ) = S¯0 + (Sλ /4)[1 − ξ 2 ] + (∆S0 /2)ξ (26) where S¯0 = (S0P + S0R )/2, ∆S0 = S0P − S0R is

† ∆Smax =


6

Sλ2 + ∆S02 ∆S0 ± 4Sλ 2

(31)

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tions of −∆F0 /λ fully determined by two entropic parameters, ∆S0 /Sλ and ∆T λ. The second, and by far most important, parameter is the logarithmic temperature derivative of the reorganization energy

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∆T λ = − (∂ ln λ/∂ ln T )ρ = T Sλ /λ

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This parameter can be typically found in the range ∆T λ ≈ 0.3 − 0.5 for electron transfer reactions in liquid solvents. 17 The entropy of activation reaches its maximum at the reaction free energy given by eq 32 (dashed vertical line in Figure 2), which is distinct from the maximum of the inverted Marcus parabola (dotted vertical line in Figure 2). The shift of entropy’s maximum projects itself into a shifted maximum of −∆HR† (∆F0 ). The negative of the activation enthalpy is an inverted parabola curving into a positive territory in the inverted domain of the reaction, −∆F0 > λ. This is the region of negative activation enthalpies (anti-Arrhenius temperature dependence of the reaction rate). Intramolecular vibrations often significantly affect rates of electron transfer in this region. 39 Nevertheless, negative activation enthalpies have been reported for a number of electron-transfer reactions. 20,40–42 A positive value of the reaction entropy, ∆S0 > 0 (typically found for charge recombination), is required for the scenario shown in Figure 2: at ∆S0 < 0, the maximum of the entropy curve shifts to the negative reaction free energy (eq 32) and the enthalpy and free energy curves nearly coincide. We next analyze the results for reactions with zero reaction free energy, which we will label as self-exchange: ∆F0 = 0 and ∆S0 = 0. This situation is exactly realized when the configurations of reactants and products are identical, but also approximately holds for charge-shift reactions. By defining the scaled parameters x = X/λ, fi (x) = Fi (x)/λ, hi (x) = Hi (x)/λ and s(x) = S(x)/kB , one arrives at a set of equations

%)*#+,

Figure 2: Y = −∆FR† /λ (red), Y = −∆HR† /λ

(black), and Y = T ∆SR† /λ (blue) vs −∆F0 /λ at ∆T λ = 0.3 (eq 35). Calculations are done by using eq 30, 33, and 34 with ∆S0 /Sλ = 0.5. The vertical dotted line marks the maximum of the inverted Marcus parabola −∆F0 = λ at which ∆SR† = ∆FR† = ∆HR† = 0. The vertical dashed line marks the maximum of T ∆SR† given by eq 32. The horizontal dotted line marks zero to highlight the region with negative activation enthalpies.

at the reaction free energy ∆FS,max = λ

∆S0 Sλ

(32)

The classical Marcus inverted parabola −∆Fi† = −

(∆F0 ± λ)2 4λ

(35)

(33)

reaches the maximum at ∆F0,max = ∓λ. From eqs 30 and 33, one obtains the negative of the activation enthalpy ( ) ( ) Hλ ∆F02 ∆H0 ∆F0 † −∆Hi = − 1− 2 − ±1 4 λ 2 λ (34) where Hλ = λ+T Sλ is the reorganization enthalpy and ∆H0 = ∆F0 + T ∆S0 is the reaction enthalpy. The negative of the activation free energy −∆FR† , the activation entropy T ∆SR† , and the negative of the activation enthalpy −∆HR† are all inverted parabolas as functions of the driving force −∆F0 . They are, however, shifted both vertically and horizontally relative to each other (Figure 2). The inverted Marcus parabola −∆FR† (∆F0 ) is obviously not affected by the choice of the entropy parameters. In contrast, the scaled entropy and enthalpy functions, T ∆SR† /λ and ∆HR† /λ, are func-

fi (x) = (x ± 1)2 /4 s(x) = s¯0 + (sλ /4)(1 − x2 ) hi (x) = fi (x) + (βλ)−1 s(x)

(36)

Representative functions based on eq 36 are shown in Figure 3. It is clear that the minima xhi


7


Perturbation theories ! "

There are a number of problems related to reactions in polar media where the reaction leads to a change in the charge distribution of the molecule. The part of the Hamiltonian changing in the reaction is then the electrostatic interaction of the molecule with the polarizable medium. In such cases, one can separate the Hamiltonian Hi into the reference Hamiltonian H0 and the changing electrostatic interaction Vi : 43 Hi = H0 + Vi . To simplify the algebra, we also put ⟨Vi ⟩0 = 0, where the subscript “0” specifies statistical averages performed with the Gibbs distribution defined with the reference Hamiltonian H0 . For these kinds of problems, one can derive the results of the Gaussian approximation by performing the perturbation expansion in Vi and truncating it at the lowest nonvanishing order. 44–46 The reaction coordinate becomes X = ∆V , ∆V = VP −VR and eqs 18–20 follow, with the theory parameters specified in terms of Vi

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Page 8 of 18

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Figure 3: Y = fi (x) = Fi (x)/λ (red), Y = hi (x) = Hi (X)/λ (black), and Y = s(x) = S(X)/kB (blue) for a self-exchange reaction with ∆F0 = 0 and ∆S0 = 0. The scaled reaction coordinate is x = X/λ. Calculations are done by using eq 36 with s¯0 = 0, sλ = 10, and βλ = 40.

of hi (x) are shifted compared to the minima xi of fi (x). One gets xi = Xi /λ = ∓1 from eq 36, while the minima of the enthalpy functions are given as xhi = ∓ (1 − ∆T λ)−1 (37)

⟨X⟩i = −β⟨∆V Vi ⟩0 F0i = − 12 β⟨Vi2 ⟩0

Since ∆T λ > 0 is required by Sλ > 0 (eqs 12 and 35), one has |xhi | > 1 (Figure 3). Another consequence of ∆T λ > 0 is that the activation enthalpy provides an upper estimate of the free energy of activation (eq 34) for the self-exchange case ∆H † = (λ + T Sλ )/4 > ∆F † = λ/4

St

λ=λ =

(39)

1 β⟨∆V 2 ⟩0 2

For instance, the first relation for ⟨X⟩i follows as the first-order perturbation in βVi taken in the statistical average ∫ −1 ⟨X⟩i = Qi ∆V (Γ)e−βHi (Γ) dΓ (40)

(38)

It is important to note that the difference in the enthalpy between the activated state at X † = 0 and at the minimum Xih = λxhi (eq 37) yields the value H(0) − H(Xih ) = (λ/4)(1 − ∆T λ)−1 inconsistent with the thermodynamic route. The enthalpy surfaces Hi (X), therefore, cannot be used to define the activation enthalpy in the same way as this procedure is applied to the free-energy surfaces. The thermodynamic definition as given by eq 34 should be applied instead. We further note that the shapes of the thermodynamic functions shown in Figure 3 are preserved when a nonzero driving force is applied: the free energy and enthalpy surfaces gain vertical shifts between the reactants and products and the maximum of the entropy function shifts horizontally according to eq 32.

where Qi is given in eq 8. The rest of the equations are obtained in similar steps. Since ∆F0 and λ are obtained from eq 39 by algebraic manipulation of the terms of the same order, they should be similar in magnitude and alter with the change in the thermodynamic state of the system to a similar degree. In particular, the temperature dependencies of ∆F0 and λ should be similar. Therefore, in accounting for the activation entropy, one must include both entropies, ∆S0 and Sλ . The perturbation theory for the latter requires accounting for third-order correlations Sλ /kB = 12 β 2 ⟨∆V 2 (1 − βδH0 )⟩0 , where δH0 = H0 − ⟨H0 ⟩0 .


8

(41)

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variance of the energy gap ⟨δX 2 ⟩ as a function of temperature does not pass through zero when extrapolated to T → 0. 48 Another manifestation of the same effect is the distinction between the enthalpy and free energy of activation. By taking the temperature dependence of λ from eqs 42 and 43, one can calculate ∆T λ as defined by eq 35 and substitute it to eq 38. The result is the equation for the activation enthalpy of a selfexchange reaction

A significant advantage of eqs 39 and 41 is that the parameters of the reaction can be calculated from the reference system without direct account of the Coulomb interaction between the reacting system and the surrounding medium. 17 This can be accomplished by either analytical models describing the medium (such as liquid-state theories for liquid solvents) or with computer simulations. For electron transfer in polar liquids, this strategy can be realized to calculate the parameters of electron transfer based on microscopic models of polar liquids. 21,47 This approach suggests that dipolar rotations produce the reorganization energy component λp , which depends on temperature only through density. On the other hand, microscopic density fluctuations of the solvent around the solute lead to the reorganization component λd = σd2 /(2kB T )

∆H † =

(44)

From this relation, the ratio of the activation enthalpy and the activation free energy for selfexchange gives the relative impact of entropydriven (density here) fluctuations on the activation mechanism of the reaction

(42)

λd ∆H † =1+ >1 † ∆F λ

scaling with temperature as 1/T when the average macroscopic density is kept constant 20,21 (σd2 = ⟨δX 2 ⟩d is the energy gap variance produced by the density fluctuations of the solvent around the solute). Density fluctuations, with their probability mostly determined by the entropy of re-packing of the repulsive molecular cores, is a specific example of a general result: ∝ T −1 temperature scaling as shown in eq 42 is expected for any nuclear mode with fluctuations determined by entropic penalty. Since molecular translations and rotations carry different symmetry (isotropic vs angulardependent), the two sources of the thermal bath noise add up in the total reorganization energy λ = λp + λd

λ + λd 4

(45)

where λd is given by eq 42. Note that ∆H † → ∆F † is expected in the dielectric continuum limit of the Marcus theory. We stress again that corrections for solvent expansivity are required when experimental data at constant pressure are considered. 17

Discussion We have developed a general framework for activated transitions when reactions can be monitored by the difference of energies of the reactants and products. As is customary, the energy (or free energy, eq 16) gap, X, is viewed as the reaction coordinate, with the transition state reached at the point of equality between the energies of the products and the reactants, X † = 0. The general framework discussed here applies to a broad class of reactions including electron and atom transfer, electrochemistry, and enzyme catalysis. Relevant for the entropic effects considered here is the configurational flexibility of proteins. In contrast to static structures provided by crystallography, this aspect has recently attracted much attention as a potential component of catalytic action of enzymes. 7,49–51 The ability of the protein to occupy many conformational states with close ener-

(43)

The second term in this equation, representing microscopic density fluctuations (eq 42), disappears in the limit of the solvent radius much smaller than the solute radius, 48 thus leaving only the dielectric continuum term for the reorganization energy. The term arising from entropy-driven density fluctuations brings about the violation of the FDT, which strictly applies only to the macroscopic response. 16 By the same means, other types of shortrange solute-solvent interactions also lead to deviations from the anticipated ⟨(δX)2 ⟩ ∝ T linear scaling. One manifestation of this violation is that the


9


gies 52–54 has promoted the idea of transition state ensembles 55,56 or the subspace of “reactive conformations”. 2 The main conceptual appeal of this proposition in application to catalysis is that conformational flexibility can potentially provide the configurational entropy for the entropic depression of the activation barrier. 56,57 Our formalism provides a general tool to critically analyze this proposition. The framework presented here does not stipulate specific physical mechanisms contributing to the activation entropy, but formulates the expression for the entropy profile of the reaction, S(X), along the energygap reaction coordinate based solely on the properties of the Gibbs ensemble. The main result of this perspective is that any entropic depression of the activation barrier should be encoded into the entropy of the near-equilibrium medium fluctuations shifting the disbalance (energy gap) between the product and reactant energies. This statement is a consequence of Onsager’s principle of microscopic reversibility. 14 It is consistent with experience acquired from analyzing conformational transitions of proteins: a conformation transition often involves a small number of soft modes characterizing near-equilibrium structural fluctuations. 58 Thus the properties of the activation free energy for activated transitions are encoded in the properties of these soft modes describing small deviations from equilibrium. In the Gaussian approximation, our formalism leads to the entropy function in the form of an inverted parabola, with the maximum at XS,max given by eq 29. The maximum of S(X) is shifted relative to the transition state at X = 0 except for selfexchange reactions with ∆F0 = 0 and ∆S0 = 0. For this specific case (i.e., self-exchange), maximum entropy is achieved at the activated state of the reaction and, therefore, maximum entropic depression of the activation barrier is expected. These general considerations suggest that if the mechanism of transition states ensembles applies to enzyme catalysis, it can be efficient for reactions with nearly zero ∆F0 . In fact, most enzymatic reactions proceed with very low reaction free energies. 59 The reorganization entropy Sλ (eq 12) is the main parameter that controls the entropic depression of the barrier regardless of its physical ori-

Page 10 of 18

gin. For reactions of electron transfer in rigid donor-acceptor complexes not allowing much conformational flexibility, polar solvent provides the chief contribution to Sλ . 17 While the effect of the solvent-induced reorganization entropy leads to observable bell-shaped Arrhenius plots for low activation barriers, 20 an entropic depression of the activation barrier is modest for reactions close to self-exchange conditions and can hardly be experimentally distinguished. How this situation can potentially change for flexible systems with significant conformational mobility of the reactants and products is a subject of future studies. In the absence of direct studies, we can offer a crude estimate based on the Marcus expression for the reorganization energy of long-distance electron transfer. 1 Assuming that the donor and acceptor are connected by a flexible linker, one can determine modulations of solvent reorganization energy by fluctuations δR = R − R0 around the average distance R0 . The reorganization energy averaged over the distance fluctuations then becomes ⟨λ⟩ = λ0 − (c0 e2 /R03 )⟨δR2 ⟩

(46)

where e is the elementary charge and c0 ≈ 0.5 is the Pekar factor. 1 Since ⟨δR2 ⟩ ∝ T according to the FDT, 16 flexibility of the donor-acceptor distance contributes a purely entropic term to ⟨λ⟩. We can therefore split Sλ into the solvent component Sλs and the flexibility component (second summand) Sλ /kB = Sλs /kB + βc0 e2 ⟨δR2 ⟩/R03

(47)

The mean-square displacement of the donoracceptor distance can be estimated from the vibration wavenumber ν¯, the thermal wavenumber ν¯T ≃ 200 cm−1 and the Compton wavelength for the proton, Λp = ℏ/(mp c) ≃ 0.21 × 10−13 cm, according to the relation ⟨δR2 ⟩ =

Λp ν¯T mp 2π ν¯2 m

(48)

which is just a convenient re-write of the equipartition theorem, mω 2 ⟨δR2 ⟩ = kB T . In eq 48, mp /m is the ratio of the proton mass and the effective mass of the diatomic forming the bond. For ν¯ ≃ 50 cm−1 characteristic of the boson peak of the pro-


10

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tein 60 one gets ⟨δR2 ⟩ ≃ 3(mp /m) Å2 . At m = 10mp and R0 ≃ 10 Å, one gets a relatively small value ≃ 0.1 for the second summand in eq 47. It is clear that flexibility of rigid molecular systems will not significantly affect ⟨λ⟩. For electron transfer, highly flexible molecular systems, with potential hinge or loop 61,62 motions, are required in order to allow an appreciable entropic depression of the activation barrier. Equally, for proton (hydrogen atom) transfer, 51 ν¯ ≃ 320 cm−1 , m = 14mp , R0 ≃ 3 Å and, from eq 48, ⟨δR2 ⟩ ≃ 5 × 10−3 Å2 . The second summand in eq 47 is again very low, ≃ 0.05, and will not provide a noticeable entropic depression of the activation barrier. 5 If entropic effects due to molecular flexibility have a chance to contribute, alternative mechanisms of producing a large Sλ need to be sought. The donor-acceptor flexibility will also contribute to modulation of the quantum-mechanical matrix element for electron and proton tunneling, 7 which is also an entropic effect. 63,64 If the squared donor-acceptor matrix element decays exponentially with distance, V (R)2 ∝ exp[−γR], one gets an additional term in the rate caused by fluctuations of R 63,65,66 ∝ exp[ 12 γ 2 ⟨δR2 ⟩] (49)

croscopic nature of the liquid fluctuations affecting electron transfer. 48 These microscopic fluctuations lead to the appearance of the second, densityrelated summand in eq 43, which displays an explicit dependence on temperature, ∝ T −1 , not anticipated by the FDT. However, despite a fundamental significance of this result, this source of FDT violation is not sufficient to allow a substantial entropic suppression of the activation barrier. Our analysis shows that much higher values Sλ and, therefore, much more dramatic violations of the FDT, are required to achieve a strong entropic effect. Significant violations of the FDT are anticipated in glassy materials 67 and for driven systems, 68 i.e., systems under a continuous influx of energy. Proteins have long been suspected to carry many of the phenomenological signatures of glassy materials. 53,69 Even though the structural fluctuations of a hydrated protein might appear Gaussian, 13 the intrinsic landscape of the protein is complex, 70 with many minima arising from configurational frustration. 71 Frustration, related to degeneracy, allows the system to accommodate the same folded state through a large number of configurations with nearly equal free energies. 72,73 When projected on a single collective coordinate, like the reaction coordinate X employed in this study, the landscape complexity is reflected in a disconnect between the first and second moments of the reaction coordinate. 74 For the reaction coordinate X, this phenomenology is expressed through the distinction between λSt and λi (Figure 1). Therefore, the breakdown of eqs 18 and 19, combining the Gaussian approximation with the Gibbs statistics, has to be related to a generally non-Gibbsian statistics reflecting the complex nonergodic walk of the system on a rugged energy landscape. 75 While this picture is close to that adopted for structural glass-formers, 69 one has to appreciate the differences as well. The energy landscape of a macroscopic glassy material represents all possible states of a macroscopic system. 76 In contrast, the energy landscape of a hydrated protein represents all possible states of a folded heteropolymer strongly coupled through solvation to the surrounding water thermal bath. Changing temperature alters the balance between intramolecular interactions within the protein and hydration free

We do not consider these additional entropic contribution here, focusing solely on the reaction activation barrier. The reorganization entropy Sλ is the central parameter of our analysis: the appearance of a nontrivial S(X) in eq 26 is the result of Sλ ̸= 0. This property is interesting from a number of perspectives. First, this is the entropic component of nuclear reorganization, and of corresponding medium fluctuations, driving the reaction over the barrier. Second, per eq 10, the reorganization energy λ is the parameter connecting the variance of the reaction coordinate, produced by thermal agitation, with temperature. The macroscopic version of the FDT 16 anticipates ⟨δX 2 ⟩ ∝ T , with the proportionality coefficient, 2λ, depending on temperature only implicitly, that is through other thermodynamic parameters, such as density. Appearance of a nontrivial temperature dependence λ(T ) and the corresponding Sλ ̸= 0 signals violation of the FDT. For electron-transfer reactions in liquids, this violation is the result of the mi-


11


energies of the surface residues. One therefore should anticipate substantial temperature-induced alterations of the protein’s energy landscape and related entropic effects. 77,78 In addition to the intrinsic complexity of the energy landscape of complex materials, the violation of time separation anticipated by the statistical Gibbs ensemble 79 can also lead to the violation of the FDT and the related entropic effects. 80 The standard formulation of the transition-state theory assumes that thermal bath modes are always significantly faster than the activated transition. 2,78 This assumption is violated for many reactions involving proteins, 9,81 which are characterized by dispersive dynamics. 52,70,82 In fact, this phenomenon is not restricted to proteins and can be extended to many glassy materials, such as ionic liquids. 83 An incomplete sampling of the bath configurations on the reaction time-scale 61 requires replacing the Gibbs ensemble average in eq 2 with a statistical non-Gibbsian average constraining the phase space Γ(ki ) to only those degrees of freedom which are faster than the reaction rate ki ∫ −βFi (X,ki ) δ(X − ∆H(Γ))e−βHi (Γ) dΓ e =

be related to advantages of operation at the condition of the dynamic resonance specified by eq 51. Equation 51 also specifies the point of dynamical transition often reported for proteins, 87 when ki (T ) is replaced by the observation window of the in−1 strument, 53,88 ki → τobs . It was hypothesized 35,77 that kinks in the Arrhenius plots of enzymatic reaction rates can be related to phenomenology of dynamical transitions. However, the reaction times for enzymatic reactions displaying kinks in the Arrhenius plots (≈ 0.05 − 1 s 7 ) are immensely longer than the experimental observation time τobs in neutron scattering (≈ 0.1 − 1 ns) and Mössbauer spectroscopy (≈ 140 ns). This disconnect poses the question of what are the relaxation processes in the protein which allow the dynamic resonance in eq 51 to be reached. One of the mechanistic opportunities afforded by proteins as dynamic entities is the active role of water 89,90 in enzymatic reactions not encountered for typical donor-acceptor complexes used in redox chemistry. Penetration of water into the protein interior, 91 caused by altering the charge state of the active site, produces a nonlinear structural reorganization of the medium characterized by a strong temperature dependence 92 and, potentially, by large values of Sλ . In contrast, standard solvation mechanisms analyzed here and following the basic prescriptions of the FDT and the linearresponse approximation 43 do not provide reorganization entropies sufficient for the entropic barrier suppression.

Γ(ki )

(50) The result of this non-Gibbsian statistics is a dependence of Fi (X, ki ) and λi (Figure 1) on the relative time-scales of bath relaxation and of the reaction kinetics (the relevant parameter is the product ki τ of the reaction rate ki and the relaxation time τ 80 ). The reorganization energy λi and the free energy of activation become functions of ki τ (T ) (nonergodic activated kinetics, implying that the activation energy is not given by the Gibbsian free energy 80 ). Since both the reaction rate ki (eq 3) and the relaxation time of the bath τ (T ) often follow the Arrhenius law, such dynamic freezing of the configuration sub-space leads to a strong variation of λi in the region of dynamic resonance 84 ki (T )τ (T ) ≈ 1

Page 12 of 18

Conclusions We have developed a general thermodynamic model for a reaction described by the change of the total energy of the reacting system from the reactant to the product state. The difference of the product and reactant energies defines the reaction coordinate X. Profiles of the free energy, enthalpy, and entropy along X are derived. When fluctuations of the thermal bath are described by the Gaussian distribution, the free-energy surfaces Fi (X) are parabolas and the entropy surface S(X), common to both states, is an inverted parabola. Its curvature, −Sλ /(2λ2 ), is a combination of the reorganization entropy Sλ and the reorganization

(51)

The observable consequences include kinks in the Arrhenius plot of the rate and a high Sλ in the corresponding range of temperatures. The often reported 49,85,86 correlation between reaction rates and time-scales of conformational motions might


12

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energy λ. The highest extent of entropic suppression of the activation barrier is achieved for the self-exchange reactions with ∆F0 = 0 and ∆S0 = 0. Reorganization entropy becomes a critical parameter for achieving entropic suppression of the activation barrier. In this formalism, the entropy of activation is encoded into the entropy of near-equilibrium fluctuations affecting the reaction coordinate X. The enthalpy of activation, obtained experimentally from the slope of the Arrhenius plot, becomes another important parameter of the reaction kinetics. We find that the activation enthalpy exceeds the free energy of activation for self-exchange reactions controlled by the solute-solvent electrostatics (eq 38). The dependence of the negative of the activation enthalpy on the reaction driving force −∆F0 is shifted from the inverted Marcus parabola and passes through a maximum at −∆F0 ̸= λ. Anti-Arrhenius temperature dependence of the reaction rate, characterized by a negative activation enthalpy, is predicted by the model for reactions characterized by a positive reaction entropy, ∆S0 > 0.

(4) Warshel, A.; Sharma, P. K.; Kato, M.; Xiang, Y.; Liu, H.; Olsson, M. H. M. Electrostatic Basis for Enzyme Catalysis. Chem. Rev. 2006, 106, 3210–3235. (5) Adamczyk, A. J.; Cao, J.; Kamerlin, S. C. L.; Warshel, A. Catalysis by dihydrofolate reductase and other enzymes arises from electrostatic preorganization, not conformational motions. Proc. Natl. Acad. Sci. U.S.A. 2011, 108, 14115–14120. (6) Warshel, A.; Bora, R. P. Perspective: Defining and quantifying the role of dynamics in enzyme catalysis. J. Chem. Phys. 2016, 144, 180901. (7) Nagel, Z. D.; Klinman, J. P. A 21st century revisionist’s view at a turning point in enzymology. Nat. Chem. Biol. 2009, 5, 543–550. (8) Pisliakov, A. V.; Sharma, P. K.; Chu, Z. T.; Haranczyk, M.; Warshel, A. Electrostatic basis for the unidirectionality of the primary proton transfer in cytochrome c oxidase. Proc. Natl. Acad. Sci. USA 2008, 105, 7726– 7731.

Acknowledgement This research was supported by the Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Energy Biosciences, Department of Energy (DESC0015641).

(9) Min, W.; Xie, X. S.; Bagchi, B. Twodimensional reaction free energy surfaces of catalytic reaction: Effects of protein conformational dynamics on enzyme catalysis. J. Phys. Chem. B 2008, 112, 454–466.

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Thermodynamics of Reactions Affected by Medium Reorganization

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