Theoretical Determination of Standard Oxidation and Reduction

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9066

J. Phys. Chem. B 2005, 109, 9066-9072

Theoretical Determination of Standard Oxidation and Reduction Potentials of Chlorophyll-a in Acetonitrile Anshu Pandey and Sambhu N. Datta* Department of Chemistry, Indian Institute of Technology, Bombay, Powai, Mumbai-400076, India ReceiVed: NoVember 24, 2004; In Final Form: February 1, 2005

QM/MM calculations were performed on ethyl chlorophyllide-a and its radical cation and anion, by using the density functional (DF) B3LYP method to determine the molecular characteristics, and a molecular mechanics (MM) method to simulate the solvating medium. The presence of the solvent was accounted for during the optimization of the geometry of the 85-atom chlorophyll-a system by using an ONIOM methodology. A total of 24 solvent molecules were explicitly considered during the optimization process, and these were treated by the universal force field (UFF) method. Initially, the split-valence 3-21G basis set was used for optimizing the geometry of the 85-atom species, neutral, cation and anion. Electronic energies were then determined for the optimized species by making use of the polarized 6-31G(d) basis set. The ionization energy calculated (6.0 eV) is in very good agreement with the observed one (6.1 eV). The MM+ force field was used to investigate the dynamics of the acetonitrile molecules around the neutral species as well as the radical ions of chlorophyll. The required atomic charges on all the atoms were obtained from calculations on all involved molecules at the DFT/6-31G(d) level. Randomly sampled configurations were used to determine the first solvation layer contribution to the free energy of solvation of various species. A truncated 46-atom model of ethyl chlorophyllide-a was used to evaluate the thermal energies of neutral chlorophyll molecule relative to its two radical ions in the gas phase. Born energy, Onsager energy, and the Debye-Huckel energy of the chlorophyll-solvent aggregate were added as perturbative corrections to the free energy of solvation that was initially obtained through molecular dynamics method for the same complex. These calculations yield the oxidation potential as 0.75 ( 0.32 V and the reduction potential -1.18 ( 0.31 V at 298.15 K. The calculated values are in good agreement with the experimental midpoint potentials of +0.76 and -1.04 V, respectively.

1. Introduction The elucidation of oxidation and reduction potentials of chlorophyll-a has been the subject of many experimental investigations.1a-j One of the key aspects which needs to be explored is the change in the redox behavior in going from an artificial medium to a biological one. Aprotic solvents such as acetonitrile, butyronitrile, propionitrile, dimethylformamide, dimethyl sulfoxide, dichloromethane, and tetrahydrofuran were employed in the cyclic voltammetric work on chlorophylls. Protic solvents such as alcohols tend to protonate the anion radical, complicating the redox behavior of chlorophylls. Initially, there had been contrasting reports on the redox potentials observed for chlorophyll-a in aprotic solvents. It was established by Bard1d that the electrochemistry is extremely sensitive to solvent, electrolyte and the impurities in the electrolytic medium. The experimental determination of redox potential is normally undertaken by using the standard calomel electrode (SCE) as reference, and the observed data are converted to a scale where the normal hydrogen electrode (NHE) potential is zero. The calculation of solvation energy becomes very important in the context of a theoretical determination of the redox potentials. If the stability gained by the interaction with the solvent medium is underestimated by 0.01 hartree, the calculated * Corresponding author. E-mail: [email protected].

redox potential can deviate by about 0.27 V. The direct use of Born and Onsager energies to simulate the free energies of solvation is rendered questionable due to the unusually planar shape of chlorophyll-a in solution. Born and Onsager models are valid for spherical or approximately spherical species. It is possible to deal with the behavior of the solvated chlorophyll molecule through a polarized continuum approach (PCM).2 However, a dynamical consideration of the solvent molecules arranged around the chlorophyll moiety is expected to provide more insight into the role played by the solvent in determining the redox potentials. This aspect would be made clear in the later sections. To our knowledge, a direct calculation of the standard redox potentials of chlorophyll-a has not been carried out in detail. The oxidation potential of chlorophyll-a was calculated in an earlier work3 using the semiempirical INDO/2 methodology. The present article is on the theoretical determination of oxidation and reduction potentials of chlorophyll-a in acetonitrile from density functional treatment (DFT). This report is arranged as follows. Section 2 describes the geometry of the model of chlorophyll molecule used in this work. A description of the methodology adopted is discussed in section 3. The computed results are given in section 4. This section also describes the limitations of the present work. The conclusions drawn from this work are given in section 5.

10.1021/jp0446478 CCC: $30.25 © 2005 American Chemical Society Published on Web 03/31/2005

Oxidation and Reduction of Chlorophyll-a

J. Phys. Chem. B, Vol. 109, No. 18, 2005 9067

Figure 1. System chosen for geometry optimizations. The DFT layer is dark; the lower layer is lightly shaded.

Figure 2. Ethyl chlorophyllide model of chlorophyll-a, Chl-85, with all the side chains retained except the phytyl chain that is replaced by an ethyl group: (a) the organic chemist’s visualization; (b) the three-dimensional species following the crystal data. The molecular geometry was optimized for the calculation of the electronic energy.

Figure 3. Truncated model of chlorophyll-a, Chl-46, with all side chains except the vinyl substituent replaced by hydrogen atoms: (a) structure showing only the geometry around the carbon atoms, as visualized by an organic chemist; (b) the 3-dimensional model for the same species, a truncated version of the crystallographic geometry. The molecular geometry was optimized before determining thermal energy and the associated entropy for each species, neutral, cation and anion.

2. Molecular Geometry Atomic coordinates were initially obtained from the crystallographic data of ethyl chlorophyllide-a dihydrate.4 The coordinates were optimized through a two layer ONIOM5 scheme (Figure 1), by considering the chlorophyll system as the highlevel DFT layer while 24 acetonitrile molecules were considered in the lower level layer. The explicit consideration of an adequate number of solvent molecules around the chlorophyll moiety is able to effectively simulate the environment in which the chlorophyll molecules would be in actual solution. This

technique is adopted in each case, to obtain the geometries of the cation, anion and neutral species of 85 atoms (Figure 2). The computation of thermal energies of the large number of atoms involved in the two ONIOM layers is a formidable task. This problem is circumvented by separately optimizing the geometry of a truncated 46-atom model of chlorophyll (Figure 3) without the consideration of any solvent medium. The newly optimized geometries were then used for thermal energy and entropy calculations. The computation of vibrational modes requires the molecule to be in an optimized form at the same

9068 J. Phys. Chem. B, Vol. 109, No. 18, 2005 level of theory, such that the first derivatives of the potential vanish. This technique is expected to give rise to a reliable difference of thermal energies especially since the additional charge placed onto the molecule is localized largely within the π cloud, and consequently, the bond force constants for the side chain atoms are not expected to change significantly. A smaller 46-atom model is thus expected to generate reliable thermal energy differences. In fact, the thermal energy difference between chlorophyll and its ions is only a minor contributor to the free energy of oxidation and reduction, being less than 0.1 eV. The optimized structures obtained for the solvated systems and those for the nonsolvated 46-atom species are quite different from the crystallographic structures. The most notable difference is that, in the first case, the planarity of the ring is enhanced, with the magnesium atom pushed into the plane of the porphyrin ring, even though no direct ligation of an acetonitrile molecule to the central magnesium atom or any other part of the chlorophyll molecule is observed in all the three cases. See Figure 1. Also, the vinyl substituent is not in the plane of the ring but it is at an angle to the plane. This substituent is expected to be in the plane of the porphyrin ring due to an extensive π conjugation, but it was observed to deviate (Figure 1). 3. Method of Calculation The calculation of the redox potentials of biomolecules that are generally large species is a notoriously difficult job. Different methods have been employed in the past to study the redox behavior of biomolecules. Noodleman et al.6a studied the redox potentials of iron-sulfur clusters in both protein and solvent media. DFT along with the broken symmetry (BS) technique was employed to perform electronic structure calculations.6b For the model cluster calculations in a solvent medium, the medium interaction effects were simulated by treatment of the medium as a dielectric continuum, and considering its polarization due to the solute atomic charges. The electrostatic effects of a protein were considered by considering the interaction of the active site and the protein backbone charges. The authors used a dielectric constant of 4 to screen the protein charges, while the surrounding solvent medium was considered to have a dielectric constant of 80. Olsson et al.6c performed frozen density functional free energy studies of redox potentials of plastocyanin and rusticyanin. These authors considered the active site using density functional theory, while the effects of the protein backbone were considered by taking the electron densities over the backbone to be frozen relative to the nuclear coordinates. A dielectric constant of 60 was used in these calculations.6d In a previous work,6e we reproduced the reduction potential of plastocyanin at pH 7.0, and estimated the entropy of reduction at pH 3.8. An ONIOM methodology was adopted to optimize the protein active site, with the active site being treated at UB3LYP/631G level, and the nearby residues with the UFF force field. In the present work, we have investigated ethyl chlorophyllide-a, which is basically chlorophyll-a with an ethyl group in place of the phytyl chain. The basic task here is to estimate the electronic and thermal energies of the neutral form of chlorophyll-a as well as its cation and anion, taking into account the effects of the solvent. The free energy of each species thus determined is directly used to estimate the redox potentials of chlorophyll in acetonitrile. O’Malley studied the effect of oxidation and reduction of chlorophyll-a on its geometry, vibrational and spin density properties by using hybrid density functional methods, and compared the calculated CdO stretching frequencies with the

Pandey and Datta experimentally known stretching frequencies of ring-V keto group.6f He used an extremely truncated model of chlorophyll-a with only 46 atoms. The vinyl substituent was retained. In a previous work,3,7 we found that the 46-atom model gives only qualitatively correct information. The rather small energy differences (of the order of 1 eV) corresponding to the redox processes cannot be reliably reproduced unless one takes into account all the major substituents, as we require an accuracy of about 0.1 eV. Blomberg et al. performed self-consistent reaction field (SCRF) optimizations on an extremely truncated version of chlorophyll. They used the self-consistent isodensity polarized continuum model (SCI-PCM) treatment in order to mimic the effects of the protein environment, with the adoption of different dielectric constant values for the protein and water environment. The LANL2DZ basis set was used in the optimization of the truncated model geometry.8 Geometry Optimizations. The major challenge in obtaining the correct values of the redox potentials for chlorophyll is to adequately simulate the solvent effects. The importance of the role of the solvent in determining the redox potentials is compounded by the fact that both the oxidized and reduced species are charged, while chlorophyll itself is neutral. Thus, the structure of the solvent molecules around the chlorophyll molecule is expected to change substantially when the species undergoes oxidation or reduction. A major problem encountered in this work is to obtain optimized geometries of the chlorophyll species that would mimic the geometries in solution. Several factors were taken into account before choosing the methodology. Chlorophyll is a largely rigid biomolecule, and the overall procedure should not only reflect the geometry of the chlorophyll species, but also reveal the solvent structure around the chlorophyll moiety. PCM models have been successfully used in the past to obtain geometries of biomolecules in solution. PCM models treat the solvent phase as a polarizable continuum, and the solvation energies are determined through the evaluation of the interaction of the atomic charges on the molecule with the continuum. An alternative to this approach is to explicitly consider solvent molecules around the chlorophyll disk, to mimic solvent effects. The ONIOM methodology offers an attractive route to determine optimized geometries of chlorophyll, the primary advantage being that a number of solvent molecules can be explicitly considered around the chlorophyll center. Second, the use of an ONIOM QM/MM methodology to optimize the chlorophyll geometry bears a greater similarity to the method used later to determine solvation energies. The ONIOM method expresses the energy of the system as n

E(ONIOMn) )

E[Level(i),Model(n + 1 - i)] ∑ i)1 n

E[Level(j),Model(n + 2 - j)] ∑ j)1

(1)

The solvent molecules were modeled as the lower layer, while DFT was used on chlorophyll that formed the higher layer. The lower layer has been treated with the universal force field (UFF) methodology.9 The UFF force field uses a charge equilibration scheme (QEq),10 which enables charges to be generated dynamically in response to the environment and has been shown to reproduce atomic charges accurately for a variety of systems. For the geometry optimization, the computations on the higher

Oxidation and Reduction of Chlorophyll-a layer were performed using the UB3LYP/3-21G methodology. Gaussian 98 on Windows (G98W)11 was used in all these calculations. The reason for our choice of the ONIOM QM/MM procedure is as follows. The anion is observed to have a marginally less interaction with the solvent than the cation, which is a consequence of explicitly placing the solvent molecules around the chlorophyll moiety. From a PCM perspective, this difference may be attributed to the difference in the atomic charge distribution in the ions that in turn would determine solvent polarization. However, PCM calculations performed on the truncated 46-atom model of chlorophyll yielded a free energy of solvation of -0.0436 au for the anion and -0.0414 au for the cation in acetonitrile. This is contrary to the results obtained through the ONIOM modeling approach. Besides, the free energy of solvation for the neutral species turned out to be 0.0076 au, which yields smaller absolute energy differences than those computed by the ONIOM method (-0.0512 vs -0.0642 au for anion - neutral; -0.0490 vs -0.0848 au for cation neutral). Also, it has been experimentally demonstrated that the electrochemistry of chlorophylls shows no clear trend with the variation of the dielectric constant () that determines solvent polarizability in solvation models such as PCM.1i For example, chlorophyll-a has similar oxidation potentials of 0.75 and 0.76 V in propionitrile ( ) 27.7) and acetonitrile ( ) 37.5), while a different potential of 0.84 V in dimethylformamide ( ) 38.3). Charge distribution over the nearby solvent molecules must have an important role in determining the electrochemistry of chlorophylls in solution. Electronic and Thermal Energies. The UB3LYP/6-31G(d) electronic energies were calculated using the optimized structures for the 85-atom species. The geometries of the smaller species, namely H2, involved in the reaction taking place at the normal hydrogen electrode were also optimized by the DFTB3LYP procedure using the 6-31G(d) basis set, and all of the molecular characteristics (such as thermal energy and entropy) were calculated by the same density functional method. Thermal energies of chlorophyll in oxidized, reduced, and neutral states were obtained by using a smaller truncated 46-atom model of chlorophyll-a. The optimization of this species was again undertaken by using density functional theory using a 3-21G basis set for all the chlorophyll atoms in vacuo. Thermal energies and entropies were determined at the same level of theory. All thermal energy corrections include zero point vibrational corrections to energy. Solvation Energy. The solvation free energy of the various chlorophyll species in acetonitrile or any other solvent is especially difficult to estimate to the accuracy required for the determination of redox potentials. This problem has been tackled in the following manner. A great emphasis is placed on determination of the contribution of the first solvation layer that is dynamical in nature. Born and Onsager free energies of solvation were added to the energy of interaction of the species with the primary solvation shell. The Debye-Hu¨ckel energy of ion-ionic atmosphere interaction was also considered. As the acetonitrile molecules in the solvation layer are not expected to be rigidly associated with the chlorophyll moiety but adopt orientations that vary with time, the effects of the nearby solvent molecules can be accounted for by considering several different solvent configurations around the chlorophyll molecule. Molecular dynamics simulations were undertaken for this purpose. A constant temperature ensemble was adopted, and the solvent system that was initially optimized, was allowed to relax for 30 ps (ps). Solvent relaxation time scales typically

J. Phys. Chem. B, Vol. 109, No. 18, 2005 9069 extend over tens of picoseconds, and hence the simulation time of 60 ps is expected to provide an adequate description of solvent dynamics. Six thousand data points were chosen over this duration, and averaged by boltzmann distribution. A time step of 0.001 ps was adopted for the process of simulation. Along with the electrostatic effects, other interactions such as dispersion interactions were considered through the MM+ force field. Periodic boundary conditions were employed to simulate the effects of the bulk of the liquid. Mulliken charges generated by DFT/6-31G(d) were used on all the atoms. The role played by electrostatic interactions was restricted to the acetonitrile molecules in the vicinity of the chlorophyll species by employing a cutoff of 8 Å. Furthermore, electrostatic interactions between the more distant acetonitrile molecules that were close to the faces of the periodic box were not considered at all. This was done essentially to avoid the additional disorder that occurs at the periodic box boundaries due to electrostatic interactions of solvent molecules with the actual solute and the periodic image of the solute. The total averaged solute-solvent interaction energy was taken as the free energy of interaction with the primary shell. Hyperchem Professional Release 7 for Windows12 was used in these calculations. Acetonitrile is a less structured solvent, due to the absence of a hydrogen bond network. Consequently, entropy effects arising from structure breakdown are expected to be limited to the region around the solute. Thus, the remaining solvent has been treated as a structureless continuum. The Born free energy of the ion-dielectric interaction is given by

GBorn ) -

Q2 1 12ao 

(

)

(2)

where ao is the radius of the primary solvation layer and Q is the total charge of the solute-primary layer complex. In our case, the cutoff directly implies ao ) 8.0 Å. The Onsager free energy term is a function of the same ao, and is given by

GOnsager ) -

( - 1) µ2 (2 + 1) ao3

(3)

In the above, µ is the ground-state dipole moment of the solutelayer complex, and  is the dielectric constant of the solvent. We have used a value of 37.5 for  for acetonitrile in our work. The dipole moment of the aggregate has been defined as the sum over individual bond dipole moments. This has been done in order to ensure that a fallacious contribution to this quantity does not arise from the fact that the center of charge for the charged aggregates may not coincide with the origin of coordinate. The additional stabilization provided by the medium through the ion-ionic atmosphere interaction is estimated by using Debye-Hu¨ckel theory. We write

EDH ) -

zi2eo2κ 2

(4)

where Q ) zieo and the Debye-Hu¨ckel reciprocal length κ is given by13

κ)

[ ∑( ) ] 4πeo2 kBT

1/2

Nci

i

1000

zi

2

(5)

Here, eo is the electronic charge, kB is the Boltzmann constant, T is the temperature of the system in kelvin, N is the Avogadro

9070 J. Phys. Chem. B, Vol. 109, No. 18, 2005

Pandey and Datta

number, ci is the concentration of the ith species in molar unit, and zieo is the charge of the ith ion. Free Energy. The standard Gibbs free energy Go of a species is calculated as follows. We write Go per molecule as

Go° ) EDFT + EThermal + (GMM + + GBorn + GOnsager + EDH) + PV - TS (6) Thus, for a specific process,

∆Go

per molecule is written as

∆Go ) ∆EDFT + ∆EThermal + (∆GMM + + ∆GBorn + ∆GOnsager + ∆EDH) + kBT∆n - T∆S (7) where ∆n is the change in the number of species in gas phase. The medium interaction energy terms do not arise for gaseous H2 as well as the electron that is assumed to be gaseous in this work. The PV term is negligibly small for the solvated species. There is no Born energy and no Debye-Hu¨ckel energy for a neutral solute. We have explicitly included the gaseous electron term so that we can compare with the theoretically calculated hydrogen electrode potential. A comparison with the experimental electrode potential is also possible as the conduction electron in the electrode is quasi-free, requiring a negligibly small threshold potential for conduction to commence. It is unlikely that the ideal gas behavior models the conduction electron: hence the ideal gas model is an approximation. The term is in any case small, and its inclusion has no bearing on the final results as it cancels out at the other electrode. The standard oxidation and reduction potentials for redox reactions are written as

Eoxo ) -

1 [∆Goxo + ∆GH+/1/2H2(g)o] eo

(8)

and

Eredo )

1 [-∆Gredo + ∆GH+/(1/2)H2(g)o] eo

(9)

where the ∆Go values are quantities per molecule. The quantity ∆Go[H+(aq) f (1/2)H2(g)] is the standard free energy for the reduction of aquated proton. Zhan et al.14 have determined the absolute hydration free energy of a proton in water. We make use of the same value to calculate the potential of the normal hydrogen electrode. The other species at the electrode, namely, molecular hydrogen, was optimized by using the DFT-B3LYP procedure using the 6-31G(d) basis set. All molecular characteristics such as thermal energy and entropy were also calculated at this level of theory. 4. Computed Results and Discussion Solvent Atomic Charges. The Mulliken B3LYP/6-31G(d) atomic charges computed for acetonitrile (H3C-CtN) are as follows: +0.2083 on each hydrogen, -0.5161 on the methyl carbon, +0.3500 on the nonmethyl carbon, and -0.4589 on the nitrogen atom. Chlorophyll-a. Table 1 shows the molecular characteristics of Chl-85 and its ions. Solvation characteristics and energy values are listed in Table 2. Not surprisingly, the medium interaction energy is found to play a critical role in deciding the redox potentials of the species in solution. Electronic energy differences are found to be of the same order of the corresponding solvation energy differences. Thermal energy differences

TABLE 1: Computed Molecular Characteristics of Chlorophyll-a and Its Ions species

electronic energy (au)a

thermal energy (au)b

entropy (cal/mol-K)b

-TS (au)b,c

Chl-85+ Chl-85 Chl-85-

-2227.9953 -2228.2154 -2228.2678

0.3693 0.3708 0.3680

155.568 153.981 155.601

0.0739 0.0732 0.0739

a UB3LYP/6-31G(d), for the 85-atom species. b UB3LYP/3-21G, for the 46-atom species. c At 298.15K.

are much smaller in comparison and thus the role played by thermal energy in deciding the redox potentials is limited. The use of the in vacuo thermal energies is thus validated. The following reactions are considered in order to determine the standard redox potentials:

Chl(sol) f Chl+(sol) + e-(g) -

-

Chl(sol) + e (g) f Chl (sol)

(10) (11)

The standard free energy changes for the oxidation and reduction have been calculated as

∆Goxo ) Ho(Chl+) - Ho(Chl) + kBT

(12)

∆Gredo ) Ho(Chl-) - Ho(Chl) - kBT

(13)

and

where kBT accounts for the corresponding PV term. The energy of the electron has no bearing on the redox potentials, since the contribution of the electron is canceled out while evaluating the free energy of the cell reaction. The redox potentials can be calculated from eqs 12 and 13, provided that the ∆Go[H+(aq) f (1/2)H2(g)] value is known. Hydrogen Electrode. The free energy of hydration of the proton has been estimated by Zhan et al. to be -262.4 kcal/ mol from a first principles approach.14 The electronic and thermal energies of gaseous hydrogen have been evaluated here. Thus, the task is to calculate ∆Go for the overall process

1 H (g) f H+(aq) + e-(g) 2 2

(14)

It is to be noted that the solvation free energy as obtained by Zhan et al. for a proton does not consider additional stabilization rendered to the proton due to the electrostatic interaction with the oppositely charged ionic cloud. This additional stabilization is accounted for by further addition of Debye-Hu¨ckel energy to the solvation free energy of the proton in solution (Table 3). Redox Potentials. The calculation of the total enthalpy of Chl-85+, Chl-85, and Chl-85- while these remain dissolved in acetonitrile with the experimentally stipulated concentration of an 1:1 electrolyte is explicitly shown in Table 4. The calculated ionization energy in vacuum is 0.2201 au or 6.0 eV. This is in very good agreement with the observed ionization potential of about 6.1 eV,15 considering than the optimization was done in the presence of the associated solvent molecules. Table 5 shows the calculated values of oxidation and reduction potential for chlorophyll in acetonitrile. Two sets of oxidation and reduction potentials are shown. The first set, +0.92 ( 0.32 and -1.34 ( 0.31 V, is obtained from the calculated absolute free energy of reduction ∆Go[H+(aq) f (1/2)H2(g)], -0.1692 au (-4.604 eV). This depends on the free energy of hydration of proton calculated by Zhan et al. and the energy components computed here for H2(g). The calculated

Oxidation and Reduction of Chlorophyll-a

J. Phys. Chem. B, Vol. 109, No. 18, 2005 9071

TABLE 2: Solvation Free Energies for Chlorophyll-a and Its Ions species

primary solvent layer (au)

Born energy (au)

Onsager energy (au)

Debye-Hu¨ckel energy (au)

total (au)

Chl-85+ Chl-85 Chl-85_

-0.5130 ( 0.0089 -0.4626 ( 0.0075 -0.4935 ( 0.0084

-0.0322 0.0000 -0.0322

-0.0014 -0.0003 -0.0002

-0.0011 0.0000 -0.0011

-0.5477 ( 0.0089 -0.4628 ( 0.0075 -0.5270 ( 0.0084

TABLE 3: Calculation of the Gibbs Free Energy for the Smaller Speciesd species

electronic energy (au)

thermal energya (au)

-1.1755

0.0125

H2 H+(aq)

Debye-Hu¨ckel correction (au)

total enthalpyb (au)

entropy (cal mol-1 K-1)

free energy (au)

-1.1620

31.135

-1.1768 -0.4192c

-0.0011

a At 298.15 K (25 °C). This includes the zero-point energy correction. b H ) E + kBT for gaseous species; H ) E for solvated species. c This free energy is the sum of the free energy of hydration (-0.4181 au) from ref 14 and the Debye-Hu¨ckel energy of charge-dielectric interaction.d We have used 1 au ) 27.2116 and 1 eV ) 23.0605 kcal mol-1.

TABLE 4: Calculation of ∆H0 c reaction CHL(acn) f e (g) + CHL (acn) CHL(acn) + e-(g) f CHL-(acn) -

+

∆Eelectronic (au)a

∆Ethermal (au)

∆Gmedium (au)

∆H0 (au)b

0.2201 -0.0524

-0.0015 -0.0028

-0.0848 ( 0.0116 -0.0642 ( 0.0113

0.1347 ( 0.0116 -0.1204 ( 0.0113

a The 46-atom model with the 6-31G(d) basis set yielded the following data: ∆Eelectronic ) 0.2315 and -0.0531 au for the two reactions, respectively. ∆Ho ) (∆Eelectronic + ∆Ethermal + ∆Gmedium) + kBT∆n. Energy of the electron is not considered. c We have used 1 au ) 27.2116 and 1 eV ) 23.0605 kcal mol-1.

b

TABLE 5: Calculation of the Redox Potentials of Ethyl Chlorophyllide-ac midpoint E (V) reaction

∆H° (au)

-T∆S (au)

calcd ∆G (au)

(aq) + e (aq) /2H2(g) CHL(acn) f e-(g) + CHL+(acn) CHL(acn) + e-(g) f CHL-(acn)

0.1347 ( 0.0116 -0.1204 ( 0.0113

0.0008 0.0008

-0.1692 0.1355 ( 0.0116 -0.1196 ( 0.0113

H+

-

1

calcd 0.92 ( 0.32a -1.35 ( 0.31a

0.75 ( 0.32b -1.18 (0.31b

obsd 0.76 -1.04

a Using the calculated ∆G° for reduction at the Hydrogen Electrode. b Using the experimental ∆G° for reduction at the Hydrogen Electrode, -4.43 eV (ref 15). c We have used 1 au ) 27.2116 and 1 eV ) 23.0605 kcal mol-1. Energy of the electron is not considered.

potentials are close to the observed values of +0.76 and -1.04 V.1i However, the absolute free energy for reaction 14 was determined by Reiss and Heller.16 If we use the value ∆Go[H+(aq) f (1/2)H2(g)] ) -4.43 eV as determined by Reiss and Heller, we obtain the second set of potentials, Eoxo ) 0.75 ( 0.32 V and Eredo ) -1.18 ( 0.31 V. The latter set is in very good agreement with the experimental values. The slight differences between the calculated and observed potentials can be largely ascribed to the fact that a degree of uncertainty is still associated with the determination of solvation energy of the species in acetonitrile. This uncertainty arises in part from the consideration of Born and Onsager corrections to the solvation free energies as perturbative additions. Second, the Debye-Hu¨ckel approximation that relies on the linearization of the Poisson-Boltzmann equation is another source of error. The procedure is valid for a low concentration of the electrolyte, and at the experimental concentration, a deviation from eq 5 may be expected. It is thus seen that the solvent interaction with chlorophyll is still the most obvious source of errors in the calculations, and the work may be understood to suffer from these limitations. 5. Conclusions In this paper, we have shown that DFT (B3LYP) calculations on ethyl chlorophyllide-a and its radical ions while using the 6-31G(d) basis set and ONIOM (QM/MM) 3-21G/UFF optimized geometry can lead to the values of oxidation and reduction potentials for chlorophyll-a in acetonitrile in good agreement with the experimentally determined quantities. The differences between the electronic energy of the ions and that

of the neutral species are realistically generated only when a large basis set like 6-31G(d) is employed and post-HartreeFock effects are incorporated through the UB3LYP method. It also requires the substituents on the chlorophyll ring to be retained. For example, the 46-atom models yielded energy differences that vary significantly, (Table 4, footnote). This is in agreement with the observations in ref 7. The precise determination of solvation free energies of chlorophyll in acetonitrile is another major issue. This was successfully dealt with in this work. Chlorophyll is an unusually flat molecule with extensive conjugation that ensures that the excess charge is not localized on a single atom or group, but, rather, it is delocalized over the whole ring. The variation of the observed redox potentials with the dielectric constant of the solvent reveals no simple trend, except when one compares the data for a highly polar solvent and an extremely nonpolar one.1i The QM/MM method is required in such circumstances. The effect of the solvent molecules beyond the first solvation shell can be evaluated by treating the solvent as a dielectric continuum, and then Born and Onsager interaction can be treated as perturbative corrections. Onsager energies of the solutesolvent aggregates are typically small, and hence the treatment of the bulk solvent through a polarizable continuum approach would have a very minor effect on the solvation energy values, and indeed this would remain insignificant in comparison to the error in the primary solvation energies. The treatment of bulk solvent effects as perturbative corrections thus provides an adequate description of the solvent. An explicit consideration of the first solvation layer is also needed to distinguish between the interaction energy of the anion

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