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Accurate Standard Hydrogen Electrode Potential and Applications to the Redox Potentials of Vitamin C and NAD/NADH Toru Matsui,*,† Yasutaka Kitagawa,†,‡ Mitsutaka Okumura,†,‡ and Yasuteru Shigeta*,‡,§ †

Department of Chemistry, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology (JST) Agency, Kawaguchi, Saitama 332-0012 Japan § Department of Physics, Graduate School of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577, Japan ‡

S Supporting Information *

ABSTRACT: We computationally evaluated the standard hydrogen electrode (SHE) potential in aqueous phase and the Gibbs energy of a proton from the experimental pKa values of alcohol molecules. From the “golden standard” CCSD(T)/aug-cc-pVTZ level calculation, we estimated the SHE potential as 4.48 V, which is very close to the IUPACrecommended experimental value of 4.44 V. As applications to the Gaussian-3 (G3) methods, which also reproduce the “golden standard” level calculations, we computed various pKa values and redox potentials for a vitamin series. For vitamin C, we support the experimental result of +0.35 V and predict the pKa value of D-ascorbic acid to be 3.7−3.9. Using a model molecule for nicotinamide adenine dinucleotide (NAD), we reproduced the redox potential and determined the order of the proton/electron addition, based on both the proton affinity and redox potential.

regardless of the methods,8−10 and therefore the ambiguity in choosing the SHE potential often causes serious errors in actual calculations. Thus, it is essentially important to estimate a theoretically consistent SHE potential from the first principles. Although Isse et al.11 and Truhlar’s group12,13 separately computed the SHE potentials as 4.24 and 4.28 V in an aqueous phase, respectively, these values were smaller than the standard SHE potential recommended by IUPAC. In computing the redox potentials of transition metal complexes, many computational studies employed these small SHE potentials in attempts to reproduce the experimental redox potential.14−16 The reason they underestimated the SHE potential is probably due to the inadequate description of the HOMO−LUMO gap by standard hybrid density functional theories (DFTs). In previous studies, we proposed a simple computational method to obtain the Gibbs energy of a proton in solution17,18 and the SHE potential, by referring to a linear relation between the experimental pKa values of alcohol molecules and the computed Gibbs energy differences in their deprotonation reactions.19 As an application, we investigated 10 redox reactions of small molecules at various calculation levels, such as HF, B3LYP, and CCSD(T). In order to prove the generality and applicability of our method, it is important to demonstrate the best calculation method for computing the redox potentials

1. INTRODUCTION Electron transfer in a molecule is a significant phenomenon in biological physics as well as physical chemistry. The redox potential of a molecule is a property concerning the electron transfer ability from arbitrary electron sources. It is important to investigate the redox potentials of proteins that play crucial roles in electron transfer processes in biologically relevant systems, such as photosystems I and II, and the respiratory chain. The estimation of the redox potential of a molecule is required to understand its electronic behavior. Since the redox potential is a similar property to the adiabatic ionization potential,1 many calculations have been performed to estimate it. However, various problems remain in theoretically predicting the redox potential by using quantum chemical methodologies. For example, computing the standard hydrogen electrode (SHE) potential is still a difficult problem, because of the treatment of a proton in solution. Since the absolute redox potential of a half-cell cannot be experimentally obtained, the relative potential to reference half-cells is used to discuss the redox potential. The SHE potential in aqueous phase is normally used as a reference. The International Union of Pure and Applied Chemistry (IUPAC) recommends the use of the value of 4.44 V, reported by Trasatti.2 Although many other experiments3−7 have reported different absolute values of the SHE potential in aqueous phase, ranging from 4.2 to 4.8 V (mainly around 4.2−4.3 V), numerous computational studies have assumed that the SHE potential is a constant value © 2014 American Chemical Society

Received: August 17, 2014 Revised: December 12, 2014 Published: December 16, 2014 369

DOI: 10.1021/jp508308y J. Phys. Chem. A 2015, 119, 369−376

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The Journal of Physical Chemistry A

in which G(H+, aq) stands for the Gibbs energy of a proton in aqueous solution. Since we are interested in only the Gibbs energy in solution, we considered the deprotonation reaction in solution by using a solvation model, such as a polarizable continuum model (PCM).24−27 Therefore, G(H+, aq) must be estimated for this computational scheme. According to our previous studies, a comparable deprotonation process can occur in similar chemical groups.18 Based on this idea, we have proposed a computational scheme for evaluating the pKa value, from the linear relationship between the experimental pKa values and the theoretical Gibbs energy differences, which can be evaluated by using the results obtained from normal-mode analyses at the optimized geometries of analogous molecules. However, the solvent model is not always perfect in many cases. The error caused by using the solvent model is likely to be similar to that in the deprotonation reactions. To eliminate such errors, we introduced the scaling factor s to adjust the Gibbs energy difference between the reactant and the product to the typical experimental values, as shown in eqs 3a and 3b:

for further applications. In this report, we describe several applications of our computational scheme to a vitamin series, such as vitamin C (L-ascorbic acid), and a model molecule for a nicotinamide adenine dinucleotide (NAD), with a structure similar to that of vitamin B3 (niacin). Both molecules are important for electron transport phenomena in biological systems. L-Ascorbic acid, known as vitamin C, is widely used as a reducing agent and an antioxidant. The chemistry of vitamin C includes extensive fields, such as pharmacology, nutrition, and life sciences. Quite recently, vitamin C has been used for a hydrogen-evolving complex, because of its high reducing activity.20 This compound is also known as a coenzyme required in the human body. The function of ascorbic acid depends on the conformation of the α carbon. For example, the pKa value of its stereoisomer D-ascorbic acid, called “erythorbic acid”, is different from that of vitamin C. Despite its broad interest, few computational studies have been reported for vitamin C, except for a DFT study by Bichara et al.21 They reported the geometries of the vitamin C dimer and its infrared spectra for the first time. No computational analyses of the redox potential of vitamin C, or the differences between vitamin C in terms of pKa values and/or redox potentials, have been described so far. Moreover, many different redox potential values have been reported for vitamin C, ranging from +0.06 to +0.40 V22,23 vs SHE. In this study, we will report which value is more reliable by using the gold standard calculations. NAD plays a crucial role in carrying an electron and supplying it to specific enzymes, such as those in the tricarboxylic acid (TCA) cycle. Information about the redox potential of NAD is highly important in biological processes, such as metabolism. It is also interesting to investigate the redox reaction scheme from NAD to NADH. Although the nicotinamide moiety is thought to be the main contributor for the redox reaction, few details have been reported, to the best of our knowledge. The computational research has been hampered for practical reasons, such as the treatment of the SHE potential and the Gibbs energy of a proton. In this report, we aim to elucidate the applicability of our computational scheme to these compounds. At first, we briefly introduce our scheme to compute the redox potential and pKa values in Section 2. In Section 3, we examine the computational method dependencies on the SHE potential, in applications to vitamin C and NAD. Section 4 provides the conclusion of this report.

pK a = s{G(A−, aq) − G(HA, aq)}/(ln 10RT) + sG(H+, aq)/(ln 10RT) = k ΔG0 + C0 k = s /(ln 10RT)

(3b) −1

−1

R and T denote the gas constant, 8.314 J mol K , and the absolute temperature, respectively. ΔG0 means the deprotonation Gibbs energy (G(A−, aq) − G(HA, aq)). According to eq 3b, we obtain the Gibbs energy of a proton as G(H+, aq) = C0/k

(4)

Table 1 lists the molecules used for the linear regression curve of the pKa values. It is almost the same as that in our previous work, except for the inclusion of a water molecule. We computed ΔG0 and applied the experimental pKa values to perform the least-squares analysis for fitting. With given k and C0 values, we can evaluate the pKa value of the other molecule with an −OH group. In principle, we can compute the pKa Table 1. Fitted Parameters, G(H+, aq) (in kcal/mol), and SHE Potential (in V) B3LYP/6-31++G(d,p) HF/a5Zd HF/CBSe MP2/aQZd MP2/CBSe CCSD(T,FC)/aDZc CCSD(T,AE)/aDZ CCSD(T,AE)/aTZ CBS-4M CBS-QB3 G3MP2c G3B3c Exp.

(1)

where ESHE means the SHE potential observed in water and ΔGredox is the reaction Gibbs energy for the redox reaction of the given compound. The reference state is in standard conditions for pressure and temperature; i.e., 1 atm and 25 °C (298.15 K). F and n are the Faraday constant (96.485 kJ mol−1 V−1) and the number of electrons that participate in the redox reaction, respectively. According to our previous study, the SHE potential can be approximately written as ESHE = {2G(H+, aq) − G(H 2 , gas)}/2F

and

C0 = sG(H+, aq)/(ln 10RT)

2. COMPUTATIONAL DETAILS 2.1. Computation of the SHE and Redox Potentials. This section briefly reviews the computational scheme, which was described in detail in our previous study. The redox potential of a compound Eredox can be obtained by Nernst’s law: Eredox = ΔGredox /nF − ESHE

(3a)

c

ka

C0

G(H+, aq)

ESHEb

0.0727 0.0710 0.0710 0.0805 0.0796 0.0767 0.0777 0.0772 0.0762 0.0753 0.0782 0.0770

−79.023 −79.982 −80.000 −89.085 −87.744 −85.212 −86.050 −85.940 −83.780 −83.095 −87.121 −85.520

−1086.79 −1125.66 −1125.71 −1105.16 −1102.19 −1109.47 −1109.39 −1112.32 −1100.48 −1102.74 −1113.11 −1110.68

4.80 3.77 3.77 4.44 4.48 4.52 4.52 4.48 4.70 4.59 4.54 4.53 4.44f

a

(2)

f

370

Unit is mol/kJ. bin V. cTaken from ref 19. dIn this study. eUsing eq 4. Taken from ref 2. DOI: 10.1021/jp508308y J. Phys. Chem. A 2015, 119, 369−376

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The Journal of Physical Chemistry A value of vitamin C with four hydroxyl (OH) groups, by referring to the regression curve. 2.2. Accurate Calculations with the CBS Limit. We employed a strategy to obtain more accurate Gibbs energies, which is similar to that used in our previous study. According to reports by Hobza’s group,28,29 the complete basis set (CBS) limit of HF ECBS(HF) and the electron correlation correction evaluated by a higher level calculation ECBS(corr) are obtained by EX (HF) = ECBS(HF) + A exp( −aX ) EX (corr) = ECBS(corr) + BX −3

(5)

where EX and ECBS are the energies for the basis set with the largest angular momentum X and those at the complete basis set limit, respectively.30,31 A, B, and a are molecule-dependent parameters. We used the aug-cc-pVXZ basis set (aXZ in short), where X = D, T, and Q correspond to X = 2, 3, and 4, respectively. We employed the same equation as eq 5 for the CBS limits of the Gibbs energy GCBS(HF) and GCBS(corr). In our previous study, we had demonstrated the computed SHE potential with the small basis sets aQZ for HF, and aTZ for MP2. In this report, we have improved the basis sets up to a5Z for HF, and aQZ for MP2, in order to obtain the accurate CBS limit. Although we employed the Gibbs energy with MP2 with the aQZ basis set quality, this calculation level is not sufficient for our present purpose, because of the frozen-core (FC) MP2 calculation. To consider the dynamical electron correlation more precisely, we employed the CCSD(T) level with relatively lower basis sets (X = D and T); i.e., a golden standard of quantum chemical calculation with all-electron (AE). The CBS limit of the Gibbs energy with approximate CCSD(T) correlation correction is given by

Figure 1. Experimental pKa value vs computed deprotonation Gibbs energy. Judging from this clear linear dependency, we can draw the linear regression curve by a least-squares fitting procedure.

changes were observed between our previous (up to aQZ) and present (up to a5Z) studies. Moreover, HF gives a very low SHE potential, due to both the high G(H+, aq) and lack of electron correlation energy in G(H2, gas). Using MP2, the computed SHE potential unexpectedly coincided with the global standard SHE potential, 4.44 V. For CCSD(T), no differences were found between CCSD(T, FC)/aDZ and CCSD(T, AE)/aDZ, which both provided 4.52 V for the SHE potential and almost the same parameters. When we increased the basis function to aTZ, CCSD(T, AE) gave a smaller SHE potential, 4.48 V. These results suggest that the absolute SHE potential 4.44 V is supported by the high-level post-HF methods referred to as the “golden standard”, which can reproduce the experimental thermochemical data with the error less than 1 mH (equal to ca. 0.03 V), it is clear that 4.44 V is a reliable value in the reference state. The computed SHE potential is just the difference between the Gibbs energy of a proton and a hydrogen molecule in gas phase, where we do not consider any other experimental conditions. Therefore, we should note here that the computed SHE potential is dependent on the conditions as well as the experimental potential. The CBS methods (CBS-4M, CBS-QB) did not agree with the results of CCSD(T, AE)/aTZ. They overestimated the ESHE, as 4.70 V for CBS-4 M and 4.59 V for CBS-Q, respectively. On the other hand, the G3 theories accurately reproduced CCSD(T), especially in G3B3. Therefore, we employed the G3B3 calculation for further evaluations in redox potential. Many reports have evaluated the solvation Gibbs energy of a proton in aqueous phase, with values ranging from −259.5 to −264 kcal/mol12,36−49 (from −1085.8 kJ/mol to −1104.6 kJ/ mol). In previous studies by other groups, G(H+, aq) has been represented by the summation of the Gibbs energy of a proton in the gas phase G(H+, gas) and the solvation Gibbs energy of a proton to water, Gsolv(H+). According to the Sackur-Tetrode equation,50 our results give G(H+, gas) as −26.28 kJ/mol and Gsolv(H+) as −1086.04 kcal/mol at the CCSD(AE, T)/aTZ level, which are close to the results reported in the literature; i.e., −1085.8 kJ/mol.36−38 We stress here that this solvation energy depends very sensitively on the solvation conditions, such as the temperature and the pH. 3.2. Application to Vitamin C. We applied our computational scheme to ascorbic acid. The entire redox reaction, which involves two electrons and two protons, is shown in Scheme 1.

G(CCSD(T)/aXZ) = GCBS(corr) + E(CCSD(T, AE)/aXZ) − E(MP2(FC)/aXZ)

(6)

This scheme is almost the same as that proposed by Hobza’s group. We also employed the Gibbs energy evaluated from CBS methods, proposed by Peterson et al. (CBS-4Q and CBSQB).32−34 In the present work, we adopted several calculation levels and a series of basis sets, in order to assess the performance of these calculation levels. On the other hand, as a cavity model for the conductor-like PCM (C-PCM) calculations, we only employed universal force field (UFF) parameters, in order to avoid any ambiguity originating from the solvent model. All calculations were performed by the Gaussian 09 program package.35

3. RESULTS AND DISCUSSION 3.1. Improved Gibbs Energy of a Proton and SHE Potential Results. Figure 1 provides the experimental pKa and computed ΔG0 values for given compounds, which clearly show the linear dependency between the experimental pKa and ΔG0 values. As compared with the results obtained by CCSD(T), HF overestimates the deprotonation energy (ΔG0) and MP2 underestimates it. G3B3 almost reproduces the results obtained by CCSD(T, AE). The fitted parameters and the estimated G(H+, aq) values are shown in Table 1. At the HF level, few 371


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In all computational methods, L-Asc1 and D-Asc2 are more stable than the other compounds. The relative energies of LAsc2 and D-Asc1 from the corresponding global minima are so small that these conformations also contribute to the redox potentials and the pKa values. Hence, we considered the Boltzmann distributions of the two conformers for the redox potential and pKa values of the ascorbic acids, in which we employed almost the same computational method as our previous work.51 By applying eq 1 and Scheme 1, the redox potential is given by

Scheme 1. Structures of Vitamin C and Assumed Deprotonation and Redox Reaction

Eredox = (G(Oxidized) + 2G(H+, aq) − G(Reduced))/2

The OH of C3 releases the proton at first due to the existence of a resonant structure by alternating C(2)C(3) to C(1) C(2) and C−O(3)− to C−O(1)− in one proton removal structure (upper right), which stabilizes the system. This is why the pKa value of vitamin C is much lower than that of the other alcohol group. L-Ascorbic acid (vitamin C) has an S type carbon at the C5 position (the carbon numbered “5” in Scheme 1), whereas D-ascorbic acid has an R type carbon. Hereafter, we refer to L-ascorbic acid and D-ascorbic acid as L-Asc and D-Asc, respectively. When we refer to “ascorbic acids”, we generally mean both L- and D-Asc. The ascorbic acids have many conformations, as shown in the work by Bichara et al.21 Therefore, we optimized three conformers, and selected the two local minima shown in Figure 2 for further discussion.

− ESHE

(7)

. The results for the redox potential and pKa values are listed in Table 2. These computational results are in good agreement with the previous studies.52 We overestimated the pKa2 value, because it is difficult for ordinary PCM methods to describe the solvation energy of negatively charged system such as [Asc (reduced)]2−. We have already faced such a deficiency and pointed out that explicit counterions to neutralize the whole system are necessary to reproduce the pKa value of such molecules.18 There are two experimental values for the redox potential, +0.06 V and +0.35 V. Our results ranged from +0.40 to +0.50 V, thus supporting the value of +0.35 V. The redox potential of L-Asc is almost the same as that of D-Asc. Regarding the pKa value, the ascorbic acids have very low pKa values as compared with the other compounds with OH groups, because their molecular structures have possible resonance structures in the deprotonated state, resulting in their stability due to the delocalized negative charge. G3B3 and G3MP253−55 reproduce the experimental pKa value56 for L-Asc, whereas B3LYP somewhat overestimates it. On the other hand, the obtained pKa value of D-AscAve is smaller than that of LAscAve by 0.6 pKa units, by any computational method. A hydrogen bond exists between OH and the oxygen atom of a proton donor in L-Asc2 and D-Asc2. When the proton of the ascorbic acids is released, the oxygen atom of the proton donor becomes more negatively charged. Since the hydrogen bond is thus stronger, the deprotonated states of L-Asc2 and D-Asc2 become more stable than that of L-Asc1 and D-Asc1, meaning that the estimated pKa values of L(D)-Asc2 are smaller than that of L(D)-Asc1. It is also possible that the change of the dipole moment might depend on the pKa value. In the Supporting Information, we draw the correlation plot between the change of the dipole moment and the pKa value. However, no remarkable

Figure 2. Molecular structure of vitamin C: L-ascorbic acid, labeled “LAsc”) (a,b) and D-ascorbic acid (labeled “D-Asc”) (c,d). L-Asc1 (a) and D-Asc2 (d) are more stable than L-Asc2 (b) and D-Asc1 (c), respectively. The atom shown in blue is the proton, which is deprotonated at first.

Table 2. pKa Value and Redox Potential (vs SHE, in V) of L- and D-Ascorbic Acids B3LYP ΔG L-Asc1 L-Asc2 D-Asc1 D-Asc2 L-AscAve D-AscAve L-AscExpb a

G3MP2

a

pKa1

pKa2

0.00 7.07 10.12 6.94

4.92 4.19 4.94 4.20 4.88 4.36 4.10

13.75 13.18 13.35 13.24 13.72 13.33 11.74

redox

ΔG

a

pKa1

pKa2

0.48 0.49 0.40 0.48 0.48 0.46 -

0.00 2.55 6.69 4.43

4.50 3.63 4.27 3.81 4.27 3.94 4.10

13.39 13.00 12.36 12.96 13.29 12.53 11.74

G3B3 redox

ΔG

a

pKa1

pKa2

redox

0.50 0.55 0.42 0.54 0.51 0.51 -

0.00 5.77 7.95 6.95

4.46 3.46 4.23 3.46 4.37 3.77 4.10

13.30 12.71 12.83 12.71 13.25 12.78 11.74

0.48 0.54 0.42 0.53 0.49 0.49 -

Relative energies from L-Asc1 are shown in kJ/mol for each computational method. bTaken from ref 56. 372


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with the experimental value. Although B3LYP slightly underestimates the experimental value, the error is less than 0.15 V. On the other hand, the “Previous scheme”, which employed a constant ESHE (4.44 V) and G(H+, aq) (−1112.53 kJ/mol) in eq 1, never reproduced the experimental value because of the underestimation of the SHE potential for every method. This fact clearly indicates that the CBS limit calculations with the highly accurate treatment of the electron correlation effects with our scheme are necessary for the evaluation of both the pKa value and redox potential for this kind of molecule. Judging from these results, the Model system is minimal, but sufficient, for further discussions with the G3B3 theory. 3.3.2. Order of Proton/Electron Addition at the Nicotinic Amide (NAD Model). In this section, we discuss two topics: (1) the order of proton/electron addition, and (2) which part accepts the proton in NAD, by using G3B3 within the model system. Scheme 2 shows the three possible pathways to add both the electrons and proton.

correlation is found. This might be because the PCM we employed here ignores the hydrogen bond between a molecule and the water solvent. 3.3. Redox Potential of NAD/NADH. 3.3.1. Redox Potentials for Model/Real Systems. Finally, we considered the redox potentials of nicotinamide adenine dinucleotide (NAD) and its model compound. When we replaced the adenine dinucleotide moiety with a methyl group, as illustrated in Figure 3a, the model molecule became small enough to

Scheme 2. All Possible Redox Reaction Pathways from NAD (1) to NADH (6), for Which the Deoxyribose and the Adenine Parts Are Represented as R for Simplicity Figure 3. (a) Real molecule for NAD and its model molecule. (b) Molecular structures of NAD (oxidized) and NADH (reduced).

consider by G3B3. In modeling NAD, we note here that the redox reaction of NAD mainly occurs in the nicotinamide moiety. As shown in Figure 3a, NAD consists of the nicotinamide, the deoxyribose, the phosphate, and the adenine moieties. The observed reduction potential of an adenine is less than −2.0 V.57 Since both the deoxyribose and phosphate moieties do not participate in any reduction process in ordinary experimental conditions, we can only consider contributions from the nicotinamide moiety for the redox reaction for simplicity. The redox reaction for NAD is shown in Figure 3b.58 As mentioned in the Introduction, it is difficult to compute the redox potential of this half-cell reaction, because the numbers of protons and electrons are different. The entire NAD molecule consists of at least 80 atoms, and thus exceeds the present feasibility of G3B3. In order to investigate the validity for the model system, we computed the redox reaction for both the real NAD molecule (Real) and the model molecule (Model) at the B3LYP/6-31++G(d,p) level, as shown in Table 3, where

We evaluated the electron affinity by using eqs 4 and 5, with the proton affinity GPA estimated as G PA = G(deprotonated) + G(H+, aq) − G(protonated) (8)

The negative value for GPA means that the protonated state is less stable than the deprotonated state. The redox potentials and proton affinities are listed in Table 4. Judging from the table, the reaction 1 → 2 → 5 → 6, i.e., an electron comes first and proton addition is followed by another electron transfer, is most likely to occur among the three reaction schemes. However, a very low redox potential, −1.17 V, is estimated for the first electron accepting reaction, which might not readily occur in a single NAD, meaning that some other factors may

Table 3. Computed Redox Potential (vs SHE) of Two Models for Model and Real Molecules for NAD (in V) B3LYP/6-31++G(d,p) G3MP2 G3B3 Exp. a b

previous

model

real

−0.22 −0.18 −0.17

−0.45 −0.29 −0.28

−0.45 N/Aa N/Aa −0.32b

Table 4. Computed Redox Potential (vs SHE, in V) and Proton Affinity GPA (in eV) of Each Reaction

We could not calculate the value because of computational limitation. Taken from ref 59.

B3LYP/6-31++G(d,p)

the experimental redox potential for NAD is −0.32 V.59 These B3LYP results suggest that the nicotinamide moiety is responsible for almost the entire redox reaction, since there is no significant difference between the Model and Real systems. According to the results from the model system, we estimated −0.28 (−0.29) V with G3B3 (G3MP2) and −0.45 V with B3LYP, respectively, indicating that the G3 theories agreed well

1→2→4→6 1→2→5→6 1→3→5→6

G3B3

first

second

third

first

second

third

−1.39 −1.39 −3.14

−2.78 +0.26 +2.02

+3.28 +0.23 +0.23

−1.17 −1.17 −3.72

−2.46 −0.16 +2.40

+3.08 +0.77 +0.77

a

Numbers in italic denote the redox potential. bNumbers in block denote the proton affinity. 373


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The Journal of Physical Chemistry A participate to initiate the first step. Once the one-electron reduction proceeds, the accompanying reactions can occur easily. For the position of the proton in NADH, we compared the relative energies of all of the possible protonated structures. Figure 4a shows the spin density for the one-electron reduced

for NAD, we have shown that the nicotinamide moiety plays an important role in the redox reaction. According to the results obtained by G3B3, we specified the order of electron/proton addition in the redox reaction from NAD to NADH. The last electron addition determines the position of the additional proton. Therefore, we have confirmed that the G3 theories are very effective to evaluate reactions concerning both electron and proton transfers. Unfortunately, this computational scheme is limited to aqueous phase reactions so far, because we employed the experimental pKa values of alcohol molecules in water. In other words, the solvent dependencies have not been examined yet. Moreover, it is widely known that many biological systems utilize the proton-coupled electron tranfer (PCET) reactions for electron and proton transports through proteins, for enzymatic reactions, and so on (see reviews in refs 60−63 and the references within). In this study, we have proposed a new technique to analyze a mechanism for PCET from the view of the static molecular properties. We will tackle these issues in the future.

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ASSOCIATED CONTENT

S Supporting Information *

Input files for the alcohol molecules, detailed data of the Gibbs energy corrections in accurate calculations, and optimized geometries for L- and D-Asc and several structures from NAD to NADH. This material is available free of charge via the Internet at http://pubs.acs.org.

Figure 4. (a) Spin density of (at B3LYP/6-31++G(d,p) level) and (b) relative Gibbs energy for proton addition. We set the zero-point as the Gibbs energy of a molecule when the proton binds like NADH.

■

NAD (2). Assuming that the electron coupled proton transfer takes place in the process of the reaction from NAD to NADH, the additional proton attacks the carbon atom with the spin density shown in red. Figure 4b also shows which part of the nicotinamide moiety can accept a proton in compounds 5 and 6, judging from the relative Gibbs energies for the corresponding protonated states (see the figure at each site). In the case of compound 5, the C atoms next to the N atom and the amide O atom could accept a proton more easily than the proper C atom does, as shown in Figure 3, since their relative energies are negative. Once compound 5 accepts an electron, i.e., to become compound 6, the structure protonated at the proper C atom is the most stable structure. These facts indicate that the proton bound to the other atom in 5 is transferred to the proper C atom, during the reaction from 5 to 6.

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (T.M.). *E-mail: [email protected] (Y.S.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS T.M. is thankful for a research fellowship for young scientists, from the Japan Society for the Promotion of the Science (JSPS). This study was supported by Grants-in-Aid for Innovative Areas (Nos. 26102525, 25104716, and 26107004), and for Scientific Research (B) (No. 23350064) from JSPS.

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4. SUMMARY By using more reliable post-HF calculations, we have reinvestigated our computational scheme to obtain more accurate values of the Gibbs energy of a proton in aqueous solution and the SHE potential. As a result, we estimated the SHE potential to be 4.48 V by the gold standard CCSD(T)/ aug-cc-pVTZ level, which is close to the global standard SHE potential. Moreover, the value obtained by the G3B3 theory was the closest to these calculated results, and is the most applicable to computing the redox potentials, when CCSD(T) could not be applied to the systems. By using this scheme, we demonstrated the detailed calculations for the redox potential and pKa values of ascorbic acids, including vitamin C. The position of the OH bound to the α-carbon determines the 0.6 pKa unit difference between D-ascorbic acid and vitamin C (Lascorbic acid). On the other hand, the redox potentials are almost the same for the L- and D-ascorbic acids, and were estimated to be about +0.50 V. Regarding the redox potential

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