Assessing the correctness of pressure correction to solvation theories in the study of electron transfer reactions
AAssessing the correctness of pressure correction to solvation theories in the study ofelectron transfer reactions
Tzu-Yao Hsu and Guillaume Jeanmairet
1, 2 Sorbonne Universit´e, CNRS, Physico-Chimie des ´Electrolytes et Nanosyst`emesInterfaciaux, PHENIX, F-75005 Paris, France R´eseau sur le Stockage ´Electrochimique de l’ ´Energie (RS2E), FR CNRS 3459,80039 Amiens Cedex, France
Liquid states theories have emerged as a numerically efficient alternative to costly molec-ular dynamics simulations of electron transfer reactions in solution. In a recent paper[
Chem. Sci. , 2019, 10, 2130], we introduced the framework to compute energy gap, freeenergy profile and reorganization free energy using molecular density functional theory.However, this technique, as other molecular liquid state theories, overestimates the bulkpressure of the fluids. Because of the too high pressure, the predicted free energy isdramatically exaggerated. Several attempts were made to fix this issue, either based onsimple a posteriori correction or by improving the description of the liquid introducingbridge terms. By studying two model half reactions in water, Cl → Cl + and Cl → Cl − ,we assess the correctness of these two types of corrections to study electron transferreactions. We found that a posteriori corrections, because they violate the functionalprinciple, lead to an inconsistency in the definition of the reorganization free energy andshould not be used to study electron transfer reactions. The bridge approach, becauseit is theoretically well grounded, is perfectly suitable for this type of systems.1 a r X i v : . [ phy s i c s . c h e m - ph ] F e b . INTRODUCTION The widely accepted theory to describe an electron transfer reaction (ETR) in solution wasproposed by Marcus in 1956 . Assuming the solvent can be described by a polarization fieldthat responds linearly to the electric field generated by the solute, he derived an expressionlinking the activation free energy to the reaction free energy. This expression depends onan unique parameter: the reorganization energy (RE). RE measures the cost to distort thesolvent from the configuration in equilibrium with the reactant to the one in equilibrium withthe product, without transferring electron. The polarization varies continuously between theequilibrium values of reactant and product. By computing the free energy of both states forout-of equilibrium polarizations, one can plot the free energies as a function of an appropriatereaction coordinate. This gives rise to the famous Marcus two parabola picture for the freeenergy profiles (FEP).From a simulation point of view, ETRs have been mostly investigated using molecular dy-namics (MD). Warshel proposed to use the energy gap (EG) as the relevant microscopic reactioncoordinate . When the probability distribution of the EG is Gaussian, the FEP are parabolicwith identical curvatures, as predicted by Marcus . The Gaussian behavior has been verifiedin several studies but some systems deviate from the Marcus picture . Resort to numericalsimulations of ETRs is essential to account for such cases, unfortunately computing the FEPwith MD remains costly and require the use of biased sampling techniques .We recently proposed an alternative way to compute FEP and reorganization free energies ofETRs based on molecular Density Functional Theory (mDFT) . Because it allows to computedirectly the solvation free energy through functional minimization, it is computationally moreefficient than MD. Moreover, mDFT allows a fine description of solvation at the molecular leveland a good agreement with MD was found for the Cl → Cl + and the Cl → Cl − ETRs. Thefirst one is found to follow Marcus theory while the second one is not.In previous work, the hyper-netted chain (HNC) approximation was used to describe thesolute-solvent correlations , neglecting the so-called bridge functional. However, because thisfunctional is a Taylor expansion truncated at second-order, it drastically overestimates the freeenergy per solvent molecule at gas density . This increases considerably the cost to create asolvent cavity and thus the predicted solvation free energy. The reference interaction site modeland its 3D variant (3D-RISM), another liquid state theory that have also been extensively usedto tackle ETRs , suffer from the same problem . Because it is a long-time problem,several attempts were made to fix it. By noticing that the gas phase free energy is directly2inked to the pressure, simple a-posteriori corrections were proposed . Essentially,they consist in evaluating the partial molar volume (PMV) created by the solute and applyinga free energy correction proportional to it.A more rigorous approach is to fix the defect of the functional by introducing an approximatebridge term, to go beyond second order. Several models of bridge functional have been proposed,either based on hard sphere theory or weighted density approximation (WDA) .In this paper, we evaluate the impact of those two types of correction on the FEP andreorganization free energies predicted for two model ETRs: Cl → Cl + and Cl − → Cl. For aposteriori correction we selected the widely used PC correction while the bridge functionalis the recently proposed angular independent WDA functional . II. THEORY
We remind here some basics of mDFT which are essential for the comprehension of thispaper. Thorough descriptions can be found in previous reports .In mDFT, solvent and solute molecules are described by rigid models interacting throughclassical forcefield. The solvation free energy of the solute can be computed by minimizing thesolvent functional with respect to the spatially and orientationally dependent solvent density ρ ( r , Ω ). We introduce the following notation x ≡ ( r , Ω ) for concision. The functional to beminimized is F [ ρ ( x )] = F id [ ρ ( x )] + F ext [ ρ ( x )] + F exc [ ρ ( x )] . (1)Its minimum is reached for the equilibrium solvent density, ρ eq . In equation 1, F id corre-sponds to the entropy of the non-interacting liquid F id [ ρ ] = k B T ˆ (cid:20) ρ ( x ) ln (cid:18) ρ ( x ) ρ b (cid:19) − ρ ( x ) + ρ b (cid:21) d x (2)where k B is the Boltzmann constant, ρ b is the homogeneous bulk solvent density and T is thetemperature.The second term of equation 1 is due to solute-solvent interaction and can be expressed as F ext [ ρ ] = ˆ ρ ( x ) V ext ( x ) d x (3)where V ext is the the external energy density exerted by the solute.Solvent-solvent interactions are collected in the last term of equation 1, that is the excessfunctional F exc . Expanding it around the density ρ b of the homogeneous fluid taken as a3eference, and truncating the expansion at second order gives rise to the HNC functional. Allthe higher order correlations are then swept into the unknown bridge functional F B , F exc [ ρ ] = F HNC [ ρ ] + F B [ ρ ] (4)= − k B T ˘ [∆ ρ ( r , Ω ) c ( r − r , Ω , Ω ) × ∆ ρ ( r , Ω )] d r d Ω d r d Ω + F B [ ρ ]where c is the direct correlation function of homogeneous fluid at density ρ b and ∆ ρ = ρ − ρ b11 .We evaluate two types of pressure correction in this paper. The PMV correction takes thefollowing expression F PMVB = ρ b k B T (1 − ρ b c ( k = 0)) ˆ ρ eq ( x ) − ρ b ρ b d x (5)where ˆ c denotes the Fourrier transform of the direct correlation function. Since equation 5 doesnot depend on the density ρ , such a correction does not modify the optimization process. Theequilibrium density remains the same as the one obtained using the HNC functional.For the bridge correction, we take the recently proposed spherical WDA functional F WDAB [ ρ ] = k B Tn b ¨ a (cid:20) ˆ ∆ ρ ( r (cid:48) , Ω ) × w ( | r − r (cid:48) | ) d r (cid:48) ] d Ω d r (6)the value of a is uniquely defined by imposing the correct pressure and w is a Gaussianweighting function w ( r ) = (2 πσ w ) − / exp( − r / σ w ) (7)with σ w = 1 ˚A.We now turn our attention to the study of electron transfer half-reaction of the type Red → Ox + e − , and remind how it is possible to compute FEP and reorganization free energies usingmDFT.We start by introducing a set of external potentials defined as a linear combination betweenthe external potential of the Ox (denoted by 0) and Red (denoted by 1) states φ η = φ + η ( φ − φ ) . (8)where η is a real number.Minimizing the functional in equation 1 using the external potentials in equation 8 generatea set of solvent densities ρ η . While ρ and ρ correspond to the equilibrium solvent densities4or the Ox and the Red states respectively, any other value of η defines a solvent density ρ η that is out of equilibrium for both states. It becomes possible to compute the free energy ofan electronic state α = 0 or 1, solvated in a solvent density ρ η , by evaluating the associatedfunctional: F α [ ρ η ]. The definition of the two solvent reorganization free energies comes outnaturally λ = F [ ρ ] − F [ ρ ] and λ = F [ ρ ] − F [ ρ ] . (9)We then defined the energy gap as (cid:104) ∆ E (cid:105) η = ˆ ρ η ( x ) [ V ( x ) − V ( x )] d x (10)and showed that it is an appropriate reaction coordinate since there is a one to one mappingbetween the energy gap and the solvent density η ↔ φ η ↔ ρ η ↔ (cid:104) ∆ E (cid:105) η . (11)The free energy of any state is therefore a function of (cid:104) ∆ E (cid:105) η and it is possible to compute theFEP associated to state α as a function of this reaction coordinate F α ( (cid:104) ∆ E (cid:105) η ) ≡ F α [ ρ η ] . (12) III. COMPUTATIONAL DETAILS
We study two model ETRs: Cl → Cl + and Cl − → Cl in water. The solute is modeled by oneLennard-Jones site, with σ =4.404 ˚A , and (cid:15) =0.4190 kJ/mol. We consider 3 oxidation statescorresponding to Cl − , Cl (0) and Cl + . We use a Lennard-Jones cut-off of 10 ˚A with long rangecorrections . Water is described using the SPC/E model. We generate a series of biasedexternal potential by varying the atomic charge of the solute between − . ≤ q ≤ /
30 elementary charge. We use a 40 × ×
40 ˚A box with 120 spatial grid points and196 possible orientations per spatial point with periodic boundary condition (PBC). We usethe type C correction of H¨unenberger due to interaction between the solute and its periodicreplica. The temperature is fixed at 298.15 K. IV. RESULTS AND DISCUSSIONSA. partial molar volume correction
The free energy profiles of the two half-reactions calculated using the HNC functional withand without the PMV correction are displayed in figure 1. All minima are shifted to zero to5ase the comparison with the curves computed by Hartnig et al. using MD . At first glance,including the PMV correction do not seem to deeply modify the shape of the curve. However,a closer look reveals that the positions of the minima are shifted. This is especially true for theions where the minima are shifted toward zero. This might appear surprising since the energygap defined in equation 10 solely depends on the density ρ η which is not modified by the PMVcorrection.In fact, the shift of the minima is a consequence of a pathological defect of this correction.This is visible in figure 2 where the FEP of the ions are depicted as a function of the charge q ofthe fictitious solute: q = η for the Cl → Cl + reaction and q = − η for the Cl − → Cl reaction. Theminima of the PMV corrected FEPs no longer correspond to q = 1 and q = −
1. Equivalently,this means that the minimum of the free energy curve is not reached for the equilibrium solventdensity of the solute. This is a violation of the DFT variational principle which is not surprisingsince the PMV correction is not properly integrated in the classical DFT formalism. It is simplyan a posteriori correction that does not influence the optimization. This is acceptable when theobjective is to reproduce some reference solvation free energies but it is problematicto study ETRs. Indeed, the two ways to compute the reorganization free energies, either fromequation 9 or from the graphical definition, i.e. as the difference between the value of the freeenergy of one state at the abscissa corresponding to the minimum of the second state and thevalue at its minimum do not coincide anymore.Reorganization free energies are given in table I. Note that there are two sets of valuesfor the neutral chlorine since the value of λ depends on the choice of the second oxidationstate. Solvent reorganization free energies of Cl and Cl + computed with the HNC functionalare similar indicating that this ETR is well described using Marcus Theory while the Cl − → Clreaction deviates from Marcus theory. These results are consistent with MD simulations .When the PMV correction is added, both ETRs seem to deviate from Marcus theory since thereorganization free energies of ions differ from the one of the associated neutral state. Moreover,the reorganization free energies computed using equation 9 differ by up to 25 kJ/mol from theone computed with the graphical definition.The evolution of the energy gap as a function of q is displayed in figure 3. We recall thatvalues of q between 0 and 1 correspond to the Cl → Cl + while the value between 0 and -1corresponds to Cl − → Cl. When an ETR is following Marcus theory, the energy gap shouldvary linearly. This is the case between q = 0 and q = 1, while the linearity is not respectedbetween q = 0 and q = −
1. This is consistent with the conclusions drawn examining the6
600 -400 -200 0 200 400Energy gap [kJ/mol]-50050100150200250300 F r ee ene r g y [ kJ / m o l ] HNCPMVWDA Cl + Cl -200 0 200 400 600 800 1000Energy gap [kJ/mol]-50050100150200250300 F r ee ene r g y [ kJ / m o l ] HNCPMVWDA Cl - Cl FIG. 1. Free energy profiles of Cl → Cl + and the Cl − → Cl ETRs computed using the HNC functionalwithout (full) and with (dashed) the PMV correction and using the WDA (dotted) bridge functional.The MD data obtained by Hartnig et al are the circles. F r ee ene r g y c u r v e ( kJ / m o l ) HNCPMVWDA -40-2002040 P M V c o rr e c t i on ( kJ / m o l ) Cl + Cl - FIG. 2. Free energy of Cl + and Cl − computed using the HNC functional without (full) and with(dashed) the PMV correction and using the WDA (dotted) bridge functional as a function of thecharge of the fictitious solute. The value of the PMV correction is also displayed as a function of thecharge of the fictitious solute. λ HNC PMV eq 9 PMV graphical WDACl 297 332 287 295Cl −
264 229 230 255Cl 218 220 220 209Cl +
214 212 197 209
TABLE I. Solvent reorganization free energies (in kJ/mol) calculated with the HNC functional, withthe PMV correction using equation 9, with the PMV correction using the graphical definition andwith the WDA functional. reorganization free energies computing using the HNC functional and with the MD results:the Cl → Cl + reaction follows Marcus theory while the Cl − → Cl reaction does not. Sincethe energy gap is not modified by the PMV correction, the linear behavior observed for theCl → Cl + is in contradiction with the different values of λ of table I. This is another proof of8he inappropriateness of the PMV correction to study ETRs. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Atomic charge q [e] -400-2000200400600 E ne r g y gap [ kJ / m o l ] ∆ E HNC and PMV ∆ E HNC-WDAtangent to HNC in -1tangent to HNC in 1
FIG. 3. Vertical energy gap computed using the HNC (solid black) and WDA functional (solid red)as a function of the atomic charge. The dotted and dashed lines are the tangent to the HNC curve in q = − q = 1 respectively. B. Weighted density bridge functional
The same ETRs were studied using the WDA functional of equation 6 and the predictionfor the FEP as a function of the energy gap or of the solute charge are given in figures 1 and 2respectively. Adding WDA bridge has a limited impact on the FEP as compared to the HNCpredictions. More importantly, the minimum of the free energy curve of the cation (resp. anion)is correctly reached for the solute with charge q = 1 (resp. q = − → Cl + follows Marcus picturewhile Cl → Cl − does not. This is also supported by the evolution of the energy gap as a9unction of the atomic charge in figure 3.We now attempt to understand why the reorganization free energy of the ions are reducedwhen the WDA bridge is used. We introduce the free energy difference for a state α as∆ F α ( η ) = F WDA α ( η ) − F HNC α ( η ) (13)= ∆ F ext α ( η ) + ∆ F int α ( η ) (14)where ∆ F ext α is the difference of the external functionals defined by equation 3 and ∆ F int α is thecontribution of the ideal and excess term in equation 1. Since the WDA functional modifiesthe minimization process, the equilibrium solvent densities associated to the same η differ forboth corrections in equation 13.Using equation 9, the reorganization free energy can now be expressed as∆ λ α = ∆ F α (0) − ∆ F α ( α ) (15) ≈ ∆ F elec α (0) − ∆ F elec α ( α ) (16)where ∆ F elec α is the electrostatic part of external functional difference. In equation 16, we as-sume that since the considered species are ions, the main contribution is due to the electrostatic,i.e ∆ F int and the Lennard-Jones contributions can be neglected. As long as we are consideringions, the electrostatic potential is spherical symmetric. The radial component of the solventequilibrium polarization around each ion are displayed in figure 4.The WDA functional reduces the polarization of the solvent in the vicinity of the chargedsolutes, as evidenced by the decrease of the first peaks in figure 4. Since the polarizationis reduced, the ion is destabilized by the WDA functional when immersed in its equilibriumdensity, ∆ F elec α ( α ) >
0. On the contrary, the equilibrium density of the oppositely charged ionis less destabilizing ∆ F elec α ( − α ) <
0. Finally, the WDA correction being angular independentwe can expect the polarization around the neutral solute to be almost unchanged ∆ F elec α (0) ≈ V. CONCLUSIONS
In this paper we studied the appropriateness of two corrections to the HNC functional tostudy aqueous electron transfer reactions with molecular density functional theory. First, we10 r (Å) -0.06-0.04-0.0200.020.04 P [ a . u .] HNCWDA Cl + Cl - FIG. 4. Equilibrium radial polarization density around Cl − and Cl + computed by HNC and WDA examine a simple a posteriori correction of the pressure which has been widely used with differ-ent expression in mDFT and 3D-RISM . Despite its success to predict solvation freeenergy in good agreement with reference simulations and experiments, this correction shouldnot be used to study ETRs. The minima of the FEP does not correspond to the equilibriumsolvent configuration when this correction is used because it modifies the free energy withoutaffecting the functional optimization. The reorganization free energies becomes ill-defined: thegraphical definition do not coincide with the mathematical expression. Moreover, we haveshown that using any of the two definitions of the reorganization free energy lead to a behaviordeviating from the Marcus picture for Cl → Cl + reaction in disagreement with MD simulationsand HNC mDFT calculations. This deviation form Marcus theory is not recovered when ex-amining the evolution of the energy gap, which is a final evidence for the inconsistency of thepressure correction.We then turned to another type of correction, a so-called bridge functional which is tryingto recover some of the contribution due to the terms beyond the quadratic approximationof the HNC functional. We have chosen to use the recent and simple angular independent11eighted density functional that was shown to properly reproduce the solvation free energiesof hydrophobic solutes. Since this bridge has a functional form, the equilibrium density ismodified and this approach does not suffer from the flaws of the PMV correction. Thereare no ambiguities in the definitions of the reorganization free energies, and we recover resultsconsistent with MD simulations and HNC mDFT calculations: Cl → Cl + follows Marcus picturewhile Cl − → Cl deviates from it. Overall, the WDA functional does not modify significantlythe results obtained without correction . This might be because it is an angular independentcorrection, having a low impact on the polarization of the solvent which is the dominant effectfor ions. VI. ACKNOWLEDGEMENT
The authors acknowledge Mathieu Salanne for his constructive comments about the manuscript.This work has been supported by the Agence Nationale de la Recherche, projet ANR BRIDGEAAP CE29.
VII. DATA AVAILABILITY STATEMENT
Data available on request from the authors
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