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Free-energy calculations. The exchange effect of silver isotopes (107Ag and 109Ag) was computationally studied in [Ag25(DMBT)18] and [Ag29(BDT)12(PPh3)4]3− clusters by calculating free energy and thermochemistry parameters such as ZPE, enthalpy (H), and entropy (S) using density functional theory (DFT), as implemented in real-space grid-based projector augmented wave (PAW) package (22). The PAW setup Ag(4d105s1), S(3s23p4), P(3s23p3), C(2s22p2), and H(1s1) was considered to include only the valence electronic structure for the constituent atoms including the scalar-relativistic effects for Ag. Further, a reduced model was used considering −CH3 instead of the benzene rings in DMBT, BDT, and TPP ligands to reduce the high computational time of frequency calculations. The real-space calculation in finite difference mode, along with Perdew-Burke-Ernzerhof (PBE) functional, was applied for the geometry optimizations with a grid spacing of 0.2 Å, and the minimization criterion was the residual forces of 0.05 eV/Å, without considering any symmetry constraints. The atomic masses of Ag isotopes were taken as 106.905 and 108.905 for 107Ag and 109Ag, respectively. The vibrational modes were calculated only for Ag, S, and P atoms using the finite difference approximation of the Hessian matrix by considering the two displacements (+Δ and −Δ) per atom in each Cartesian coordinate. Further, the calculated vibrational energies were used to calculate the thermodynamic quantities like H, S, and Gibbs free energy (G).

The calculation of G is made in the ideal gas approximation. It includes the electronic energy (Epot), ZPE (EZPE), translational, rotational, and vibrational components of H and S, which are based on DFT calculations. An additional entropy of mixing component was calculated separately from statistical mechanics.

Enthalpy (H) is calculated within the atomistic simulation environment as

H = Epot + EZPE + Cv_trans + Cv_rot + Cv_vib, and entropy is S = Strans + Srot + Selec + Svib

Hence, the Gibb’s free energy at temperature T and pressure P is calculated asEmbedded Image

The structural isomers of each isotopically substituted cluster arising from the different possible ways of arranging n Ag isotopic substituent atoms among the total number of Ag atoms are all degenerate as far as their total electronic energy is concerned, with a small difference of only 0.01 eV for both Ag29 and Ag25 in the value of G of the parent and isotopic substituent clusters, in terms of their enthalpic and the vibrational entropic components (see table S2).

We may write the reaction equations for the 1:1 ratio of mixing as follows, and in doing so, we assume that both products, (m,n) = (12,13) and (13,12) for Ag25 and (m,n) = (14,15) and (15,14) for Ag29, are equally likely to form.Embedded Image

For the case of [Ag29(BDT)12(TPP)4]3 −Embedded Image

For (m,n) = (12,13), (13,12) in [Ag25(DMBT)18] and (m,n) = (14,15), (15,14) in [Ag29(BDT)12(TPP)4]3− substituent cases (1:1 molar ratio), we have computed the reaction molar Gibbs free energy (ΔGreact) at standard temperature (298 K) and pressure (1 atm). The reaction free energies (ΔGreact) are zero for both Ag29 and Ag25 clusters (see table S2), and the overall free energy of reaction is given byEmbedded Image

For [Ag25(DMBT)18],Embedded ImageEmbedded Imageand similarly for Ag29.

The expression for mixing or configurational entropy is simply that of mixing two different ideal gases, which is known from statistical mechanics, and is given byEmbedded Imagewhere p is the mole fraction of 107Ag, (1 − p) is the mole fraction of 109Ag, and nmol is the total number of moles of the mixture (17). We note that the mixing entropy is independent of the cluster size and only depends on the mixing ratio; hence, we expect the half-and-half-mixture to have the largest mixing free energy. For the 1:1 mixture, we have p = 0.5, and hence, ΔSmix is Rln2 (nmol = 1) and ΔGmix = −TSmix = −RTln2, where R is the gas constant in joules per mole and nmol = 1. This analysis reveals that the entropy of isotopic mixing is the largest and most significant contribution to the Gibbs free energy. Because of the fractional mixing ratio, the mixing entropy is always positive, and therefore, ΔGmix is always negative and is larger than the other terms in the free energy. Hence, ΔGmix, being the largest contribution to the overall reaction, causes ΔG to be always negative, which makes the reaction spontaneous.

The mixing ratio x = 0.5 corresponds to the nearest integer numbers of exchanged Ag atoms to half of the total number of Ag atoms in the cluster, because both clusters have an odd number of Ag atoms, for example, (25/2) = 12.5, hence, (13,12) or (12,13), where these compositions both have the identical maximum degeneracy in arrangements as a function of the number of substituents n in the Ag25 cluster. Similarly, for the Ag29 cluster, (29/2) = 14.5 and, hence, (14,15) or (15,14) are the most entropically favorable compositions in Ag29.

Molecular docking. To understand the intermolecular interactions in [Ag25(SR)18] clusters, molecular docking studies were performed using AutoDock4.2 and its associated tools (23). DFT-optimized geometry and partial charges of [Ag25(SR)18] were used for this study. We used [Ag25(SR)18] as both “ligand” and “receptor.” Receptor grids were generated using 126 × 126 × 126 grid points in xyz, with a grid spacing of 0.375 Å, and map types were created using AutoGrid-4.2. The grid parameter file (.gpf) was saved using MGL Tools- The docking parameter files (.dpf) were generated using MGLTools- The results of AutoDock generated an output file (.dlg), and the generated conformers were scored and ranked as per the interaction energy. Ten lowest-energy conformers were obtained. We used the Lamarckian genetic algorithm for the output file using MGLTools-1.4.6. The binding free energy of the FFGMG of the dimeric cluster adduct was −23.7 kcal/mol. Similar study was done with [Ag29(S2R)12(TPP)4]3− clusters, where [Ag29(S2R)12(TPP)4]3− was used as both ligand and receptor. In this case, the binding free energy of FFGMG of the dimeric adduct was −7.8 kcal/mol.

Calculation of theoretical isotope patterns with varying composition of 107Ag/109Ag. We calculated the theoretical isotope patterns of [Ag25(DMBT)18] and [Ag29(BDT)12(TPP)4]3− by varying the abundance of each isotope (107Ag/109Ag) in them by 1% change so that the composition is (x,y), that is, {(100,0), (99,1), (98,2).......(0,100)}, where x and y are the abundance of 107Ag and 109Ag, respectively. The experimental spectra were compared with the calculated spectra to find the best match and hence confirm the composition.

Details of fitting the kinetic data. The triexponential fitting in Fig. 3 was performed using the Origin 8.5 software package. The equation y = k1exp(−t*a) + k2exp(−t*b) + k3exp(−t*c) was used for the triexponential fits. The parameters k1, k2, k3, a, b, and c were varied during the fitting, and t was used as the independent variable. Both monoexponential and biexponential fits were inadequate, and only a triexponential fit could successfully fit the data points.

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