Complexation and bonding studies on [Ru(NO)(H2O)5]3+ with nitrate ions by using density functional theory calculation

Complexation reactions of ruthenium–nitrosyl complexes in HNO3 solution were investigated by density functional theory (DFT) calculations in order to predict the stability of Ru species in high-level radioactive liquid waste (HLLW) solution. The equilibrium structure of [Ru(NO)(NO3)3(H2O)2] obtained by DFT calculations reproduced the experimental Ru–ligand bond lengths and IR frequencies reported previously. Comparison of the Gibbs energies among the geometrical isomers for [Ru(NO)(NO3)x(H2O)5−x](3−x)+/− revealed that the complexation reactions of the ruthenium–nitrosyl complexes with NO3− proceed via the NO3− coordination to the equatorial plane toward the Ru–NO axis. We also estimated Gibbs energy differences on the stepwise complexation reactions to succeed in reproducing the fraction of Ru–NO species in 6 M HNO3 solution, such as in HLLW, by considering the association energy between the Ru–NO species and the substituting ligands. Electron density analyses of the complexes indicated that the strength of the Ru–ligand coordination bonds depends on the stability of the Ru species and the Ru complex without NO3− at the axial position is more stable than that with NO3−, which might be attributed to the difference in the trans influence between H2O and NO3−. Finally, we demonstrated the complexation kinetics in the reactions x = 1 → x = 2. The present study is expected to enable us to model the precise complexation reactions of platinum-group metals in HNO3 solution.

Dependences of a NO3− and a H2O on C HNO3 tot .

Gibbs energy calculation
Standard Gibbs energy, G, can be described as sum of total energy, E tot , and thermal correction to the Gibbs energy term, G corr (T) (eq. S1). The G corr (T) can be divided into thermal correction to enthalpy term, H corr (T), and entropy term, S(T) (eq. S2). The which are less than 60 cm −1 [1,2]. The derivation of formulas was referred to "Thermochemistry in Gaussian" by Ochterski [3]. and equals to k B {ln(2s + 1)}. The S rotation (T) is described in eq. S8, where Θ r (t) and σ r denote characteristic rotational temperature of t = x, y, z rotational axes and rotational symmetry number, respectively. The S translation (T) is described in eq. S9, where m and P denote molecular weight and pressure, respectively.  4 ] 2+ whose values were employed in Figure 3. This method is based on Mulliken population analysis [4]. The DOS values of the i th MO, N(i), is calculated by eq. S19, where P μν and S μν denote the density matrix and the overlap matrix between basis functions ψ μ and ψ ν , respectively.  Table S5.

Fitting methods of Ru fraction
We show two fitting models to simulate the dependency of the Ru fraction on total HNO 3 concentration by using the calculated ΔG x stepwise and ΔG x stepwise ' values for eqs. 10 and 11 in manuscript. Fitting model 1 is using the activities of H 2 O and NO 3 − , denoted as a H2O and a NO3− , respectively, based on the experimentally reported data [5]. We estimated the a H2O values by multiplying 55.39 mol L −1 (concentration of pure H 2 O) with values of "Rational H 2 O activity" in Table 4 of Ref. 5 for 0-12 mol L −1 of total HNO 3 concentration (C HNO3 tot ). The a HNO3 values were estimated by using the values of "Degree of dissociation" (α) and "Hypothetical activity coefficient" (y h ), which means activity coefficients of fully ionized nitric acid, in Table 4 of Ref.
For simplicity fitting model 2 is using the activities assuming the activity coefficients of H 2 O and NO 3 − as 1. We limit to the solution condition that total Ru concentration is smaller than C HNO3 tot and C H2O tot enough to be ignored (such as the experimental condition of Ref. 6 as well as HLLW solution) to give eqs. S11-S13. By combining eqs. S11-S13, acid dissociation constant of HNO 3 (K a ), and percentage by mass of HNO 3 in C HNO3 tot (W HNO3 ), we obtained the activities of NO 3 − and H 2 O as eqs. S17 and S18, respectively. Figure S1 shows the dependences of the activities of NO 3 − and H 2 O on C HNO3 tot for the two fitting models. S11 C HNO3 tot ≈ C HNO3 + C NO3− (S11) C H2O tot ≈ C H2O + C H3O+ (S12)

Transition states searching by relaxed surface scan
We modeled the transition state structures by using constrained geometrical optimization.
Octahedral wedge geometries in which the distances between Ru atom and the leaving H 2 O/ entering NO 3 − were fixed to 2.5 Å were created by using the equilibrium structures of the complex a. We considered the start geometries with up-side and down-side entries of NO 3 − ligand. Based on the octahedral wedge structures obtained by the constrained optimization, we scanned the potential surface of the distance between the Ru atom and the leaving H 2 O from 2.0 Å to 3.0 Å by intervals of 0.1 Å with structural relaxation in which the sum of the distances between Ru atom and the leaving H 2 O and between Ru atom and the entering NO 3 − were fixed to 5.0 Å. The structural relaxations were performed by the same method to the geometry optimization method in this study. The relaxed surface scanning based on the total energies by the single-point energy calculations are shown in Figure S3. The local maxima were obtained at 2.5 Å for the up-side entry and 2.6 Å for the down-side entry.