With the rapid development of functional materials, novel micro/nanostructures of the materials are highly demanded to obtain advanced properties, which can be achieved by the rational design of solution-phase chemical reactions 1 2 3 4 5 6 . Therefore, it is of significance to thoroughly understand the reactivity of chemical species and the mechanism of chemical reactions to further realize the solution reaction design. Among many chemical reactivity indices, one quantity of importance is the electrophilicity, ω, introduced by Parr and co-workers 7 . They defined ω as

ω = μ 2 2 η

where μ is the chemical potential and η is the chemical hardness of an N-electron system with total energy, E, defined as μ = (∂² E/∂² N) _v(r) and η = (∂² E/∂² N) _v(r). This index has been found to be helpful in analyzing the reactivity behaviors of a variety of compounds as well as the reaction mechanisms of diverse chemical processes 8 9 . As an important contribution to the ω, Gazauez et al. 10 argued that from a chemical perspective, it would make sense to differentiate the response of a system to the electron acceptance from the electron donation grounded on that the left and right derivatives of the total energy, E _DFT(N), of an N-electron system with respect to the integer electron number, N, are different. By introducing an electron bath of nonzero chemical potential, μ _bath, with which the chemical species can exchange electrons, they proposed electroaccepting [ω ⁺] and electrodonating [ω ^-] powers as the following equation:

ω ± = μ ± 2 - μ bath 2 2 η ±

where the chemical potential, μ ^±, and the chemical hardness, η ^±, were defined as

μ + = - I + 3 A 4 η + = η = I - A 2

μ - = - 3 I + A 4 η - = η = I - A 2

where I and A are the ionization potential and the electron affinity, respectively. A larger value of ω ⁺ corresponds to a larger capability of accepting charges, whereas a smaller value of ω ^- implies a larger capability of donating charges.

Although some chemical phenomena have been rationalized by establishing the quantitative structure-reactivity relationships using these reactivity indices in the gas phase 11 , the presence of solvent is bound to affect the reactivity behaviors of chemical substances. Therefore, studies on the reactivity indices such as ω and ω ^± in solution are quite necessary to reveal the accurate reactivity of chemical species in solution and further predict and design the solution phase reactions 12 . While several theoretical calculations about the solvent effect on the ω for various chemical species have been performed 12 13 14 , the solvent effect on the ω ^± which are regarded as better descriptors of the donor-acceptor type interactions 10 has not received much attention to date. In this work, the solvent effect on the ω ^± is estimated by establishing a linear relationship between the change of the ω ^± due to the solvation and the hydration free energy, ΔG _hyd. The values of aqueous electroaccepting power, ω ⁺ _s, of 39 metal cations are quantitatively calculated, which are proven to be appropriate descriptors for the reduction reactions.

In a previous study, Perez et al. 14 examined the solvent effect on the electrophilicity index, ω, by introducing a first-order finite variation in the ω due to the solvation

Δ ω g → s = μ g η g Δ μ g → s - 1 2 μ g η g 2 Δ η g → s = Δ ω 1 , g → s + Δ ω 2 , g → s

where Δμ _g→s and Δη _g→s are the variations in μ and η from the gas to solution phase, respectively.

They rearranged the first contribution as

Δ ω 1 , g → s = μ g η g Δ μ g → s = Δ E g Δ N g υ r Δ N g Δ μ g Δ μ g → s ≅ Δ E ins = 2 Δ G solv

where ΔE _ins is the insertion energy of the solute going into the solvent which is suggested as twice the solvation energy.

The second contribution in Equation 4 is rewritten as

Δ ω 2 , g → s = - 1 2 μ g η g 2 Δ η g → s = - 1 2 μ g η g μ g η g Δ μ Δ N = - 1 2 μ g η g Δ N μ g η g Δ μ . = - 1 2 μ g η g Δ N Δ E ins   = Δ N g , max Δ N Δ G solv

Finally, they deduced the expression of Δω _g→s

Δ ω g → s = Δ ω 1 , g → s + Δ ω 2 , g → s = 2 + Δ N g , max Δ N Δ G solv = γ Δ G solv

where Δω _g→s showed a linear dependence on the solvation energy, ΔG _solv, with a regression slope, γ. They used 18 well-known electrophilic ligands including hard electrophiles such as Li⁺ and Na⁺ to test this linear correlation and obtained good results (R = 0.9925, γ = 1.00765 at B3LYP/6-311G**and R = 0.9918, γ = 0.96843 at HF/6-311G**levels of theory).

Herein, we reconstruct the second contribution in Equation 4 which will directly lead to a quantitative expression for Δω _g→s with a definite slope value, γ.

Δ ω 2 , g → s = - 1 2 μ g η g 2 Δ η g → s = - 1 2 Δ N g , max 2 Δ η g → s = - 1 2 Δ N g , max 2 Δ μ s Δ N s , max - Δ μ g Δ N g , max ≅ - 1 2 Δ N g , max 2 Δ μ s - Δ μ g Δ N g , max = - 1 2 Δ N g , max 2 0 - μ s - 0 - μ g Δ N g , max = - 1 2 Δ N g , max 2 μ g - μ s Δ N g , max = - 1 2 Δ N g , max - Δ μ g → s = - 1 2 - μ g η g - Δ μ g → s = - 1 2 Δ ω 1 , g → s = - Δ G solv

Substitution of Equations 5 and 8 into Equation 4 leads to the expression of Δω _g→s.

Δ ω g → s = Δ ω 1 , g → s + Δ ω 2 , g → s = 2 Δ G solv - Δ G solv = Δ G solv

Therefore, the global electrophilicity, ω _s, in solution can be calculated by

ω s = ω g + Δ ω g → s = ω g + Δ G solv .

It should be noted that one key assumption in our approach is ΔN _{s, max} ≈ ΔN _{g, max} which could be justified by the data of Table 1 in Perez's work 14 . Our result, γ = 1, has turned out to be fairly consistent with Perez's regression value, i.e., γ = 1.00765 and γ = 0.96843, which thus approve the reasonableness of our approach to dealing with Δω _{2, g→s}.

Further, we try to extend our approach to examine the solvent effect on the ω ^±. For the charged ions, we suppose that the chemical potential, μ ^± _bath, of the electron bath equals that of the parent atoms of ions since the charged ions become neutral atoms after accepting or donating the maximum amount of electrons. In addition, as the solvent only has little effect on the chemical potential, μ, of the neutral species 12 13 15 , there exists a relationship as μ ^± _bath = μ _{s, atom} ≈ μ _{g, atom}. The ion exchanges electrons from the bath to the point that its chemical potential, μ ^±, equals the value μ ^± _bath with the maximum amount of electron flow:

Δ N ± max = μ ± bath - μ ± η ± .

The first-order variation in the ω ^± leads to the following equation:

Δ ω ± g → s = μ ± g η ± g Δ μ ± g → s - 1 2 μ ± g 2 - μ ± bath 2 η ± g 2 Δ η ± g → s = Δ ω ± 1 , g → s + Δ ω ± 2 , g → s .

The first part of Equation 12 in terms of the variation in μ ^± is given by

Δ ω ± 1 , g → s = μ ± g η ± g Δ μ ± g → s = Δ E g Δ N ± g υ r Δ N ± g Δ μ ± g Δ μ ± g → s ≅ Δ G hyd

where the energy change Δω ^± _1,g→s due to the variation of the chemical potential from the gas to solution phase can be represented by ΔG _hyd 16 17 .

The second part of Equation 12 in terms of the variation in η ^± is given by

Δ ω ± 2 , g → s = - 1 2 μ ± g 2 - μ ± bath 2 η ± g 2 Δ η ± g → s = - 1 2 μ ± g 2 - μ ± bath 2 η ± g 2 Δ Δ μ ± Δ N ± = - 1 2 μ ± g 2 - μ ± bath 2 η ± g 2 Δ μ ± s Δ N s , max - Δ μ ± g Δ N g , max ≅ - 1 2 μ ± g 2 - μ ± bath 2 η ± g 2 μ ± bath - μ ± s - μ ± bath - μ ± g Δ N g , max = 1 2 μ ± g 2 - μ ± bath 2 η ± g 2 η ± g Δ μ ± g Δ μ ± g → s = - 1 2 μ ± bath + μ ± g η ± g Δ μ ± g → s = - 1 2 μ ± bath + μ ± g μ ± g μ ± g η ± g Δ μ ± g → s = - 1 2 μ ± bath + μ ± g μ ± g Δ ω ± 1 , g → s

Combining Equation 13 with Equation 14 yields

Δ ω ± g → s = Δ ω ± 1 , g → s + Δ ω ± 2 , g → s =   1 - 1 2 μ ± bath + μ ± g μ ± g Δ ω ± 1 , g → s = 1 - 1 2 μ ± bath + μ ± g μ ± g Δ G hyd .

Therefore, the electroaccepting and electrodonating powers in solution, ω ^± _s, can be calculated by

ω ± s = ω ± g + Δ ω ± g → s = μ ± g 2 - μ bath 2 2 η ± g + 1 - 1 2 μ ± bath + μ ± g μ ± g Δ G hyd .

According to Pearson's viewpoint that cations are electron acceptors and anions are electron donors 16 , we pay attention to the ω ⁺ _s for cations and ω ^- _s for anions. By using Equation 16, the ω ⁺ _s values for 39 metal cations with charges from +1e to +3e are calculated and summarized in Table 1. From Table 1, we find that the solvation weakens the capacity of cations to accept electrons due to the negative values of Δω⁺, in agreement with the previous conclusions 12 13 14 15 16 . Unfortunately, it is impossible to quantitatively calculate the ω ^- _s values for anions so far due to the absence of experimental electron affinities needed in Equation 3b. Herein, these values can be qualitatively estimated:

ω - s = ω - g + Δ ω - g → s = ω - g + 1 - 1 2 μ - bath + μ - g μ - g Δ G hyd > ω - g + 1 - 1 2 μ - g + μ - g μ - g Δ G hyd   > ω - g .

Since a larger value of ω ^- _s implies a smaller capability of donating charges, we can conclude from Equation 17 that the solvation also weakens the capacity of anions to donate electrons, which agrees with the general viewpoints 12 13 14 15 16 .

Many liquid-phase chemical reactions involve the electron-transfer steps, and a key thermodynamic variable that describes the tendency of chemical species in solution to gain or lose electrons is the redox potential. The quantum-chemical computation approach to electrochemistry has become available very recently 18 . However, the estimation of redox potential by the quantum-chemical calculations is a great challenge due to the complexity of the processes involved in a typical electrochemical reaction 19 . For example, the complicated diffusion and adsorption processes on the electrode surface which should be necessarily taken into account in the quantum-chemical modeling of the reduction-oxidation reaction lead to the considerable system size and thus require strong computing power. Therefore, previous studies mainly focus on the one-electron reduction reactions between different oxidation states of transition metals to avoid modeling of an electrode-solution boundary 18 19 20 . In this work, we try to use the ω ⁺ _s to describe the many-electron reduction reaction including both main- and sub-group metal cations. According to the reaction formula M ^Z+ (aq) + z/2H₂ (g) = M (aq) + zH⁺ (aq), the absolute reduction potential, E°_abs, can be calculated by

E abs ∘ = E SHE = 0 ∘ + E abs ∘ SHE

where E°_abs(SHE) is the conventional reduction potential and E°_{SHE = 0} is the absolute standard hydrogen electrode potential. Note that the ω ⁺ _s is the energy lowing associated with a maximum amount of electron flow between two species; it is reasonable to establish a correlation between zE°_abs and ω ⁺ _s. A good relationship shown in Figure 1 approves that our ω ⁺ _s can act as an appropriate descriptor of the many-electron energy change. Moreover, this method is more simple and convenient compared to the quantum-chemical approach to the estimation of the E°_abs.

Except for the reduction reaction, the ω ^± _s can also be expected to qualitatively and quantitatively predict other properties of ions in connection with ligand binding, hydrolysis processes, and stability of coordination compounds, etc. In addition, compilation of experimental data on solvation energies in nonaqueous solutions will make it possible to evaluate the corresponding electroaccepting and electrodonating powers, ω ^±, which will undoubtedly lead to the deeper understanding of the chemical reactivity of ions in these media.

By reconstructing a first-order variation of the ω due to the solvation, the linear relationship between the change in the ω and the solvation energy is reproduced, which suggests that our method is theoretically reasonable. The solvent effect on the electroaccepting and electrodonating powers, ω ^±, for charged ions is examined, and a definite quantitative expression for the aqueous ω ^± _s is established. It is found that the solvation weakens the capability of both electron-accepting power of cations and electron-donating power of anions. A good relationship between the ω⁺ _s and E°_abs shows the validity of the electroaccepting powers in determining the chemical reactivity of the ions in aqueous solution. It is expected that our ω ^± _s will be helpful to achieve a better understanding of chemical properties of ions in solution and further be used in many aspects of solution chemistry such as the design of solution-phase reactions according to these indices.

The authors declare that they have no competing interests.

KL participated in the design and coordination of the study and corrected the manuscript. ML assisted in the calculation of the data and prepared the manuscript initially. DX conceived the study, supervised, and corrected the manuscript. All authors read and approved the final manuscript.