Intra- and Interchain Interactions in (Cu1/2Au1/2)CN, (Ag1/2Au1/2)CN, and (Cu1/3Ag1/3Au1/3)CN and Their Effect on One-, Two-, and Three-Dimensional Order

Mixed-metal cyanides (Cu1/2Au1/2)CN, (Ag1/2Au1/2)CN, and (Cu1/3Ag1/3Au1/3)CN adopt an AuCN-type structure in which metal-cyanide chains pack on a hexagonal lattice with metal atoms arranged in sheets. The interactions between and within the metal-cyanide chains are investigated using density functional theory (DFT) calculations, 13C solid-state NMR (SSNMR), and X-ray pair distribution function (PDF) measurements. Long-range metal and cyanide order is found within the chains: (−Cu–NC–Au–CN−)∞, (−Ag–NC–Au–CN−)∞, and (−Cu–NC–Ag–NC–Au–CN−)∞. Although Bragg diffraction studies establish that there is no long-range order between chains, X-ray PDF results show that there is local order between chains. In (Cu1/2Au1/2)CN and (Ag1/2Au1/2)CN, there is a preference for unlike metal atoms occurring as nearest neighbors within the metal sheets. A general mathematical proof shows that the maximum average number of heterometallic nearest-neighbor interactions on a hexagonal lattice with two types of metal atoms is four. Calculated energies of periodic structural models show that those with four unlike nearest neighbors are most favorable. Of these, models in space group Immm give the best fits to the X-ray PDF data out to 8 Å, providing good descriptions of the short- and medium-range structures. This result shows that interactions beyond those of nearest neighbors must be considered when determining the structures of these materials. Such interactions are also important in (Cu1/3Ag1/3Au1/3)CN, leading to the adoption of a structure in Pmm2 containing mixed Cu–Au and Ag-only sheets arranged to maximize the numbers of Cu···Au nearest- and next-nearest-neighbor interactions.

(Cu1/2Au1/2)CN was precipitated on addition of acid to a solution of the parent cyanides in aqueous potassium cyanide, as previously described. 1 LT-Copper cyanide (0.86 g, 9.6 mmol), gold cyanide (2.14 g, 9.6 mmol) and potassium cyanide (2.50 g, 38.4 mmol) were dissolved in 50 mL of deionised water. The resulting solution was vigorously stirred under a steady flow of nitrogen gas and 1 M nitric acid (40 mL, 40 mmol) was added to precipitate a pale-yellow solid. The precipitate was then washed in deionised water (500 mL), filtered and dried in air overnight. The sample was further dried prior to use by heating under vacuum at 80 °C for 14 h.
(Ag1/2Au1/2)CN was precipitated on addition of Ag + ions to a solution containing [Au(CN)2] − , as described previously. 1 A solution of [Au(CN)2] − was prepared by dissolving gold cyanide (1.59 g, 7 mmol) with potassium cyanide (0.46 g, 7 mmol) in 50 mL of deionised water. A silver nitrate solution, prepared by dissolving AgNO3 (1.2 g, 7 mmol) in 50 mL water, was added to the rapidly stirred [Au(CN)2] − solution, immediately producing an off-white precipitate. After stirring for a further 10 minutes, the solid was filtered, rinsed well with water and dried as described above.
(Cu1/3Ag1/3Au1/3)CN was precipitated on addition of acid to a solution of the parent cyanides in aqueous potassium cyanide, as previously described. 2 LT-Copper cyanide (0.179 g, 2 mmol), silver cyanide (0.268g, 2 mmol), gold cyanide (0.446 g, 2 mmol) and potassium cyanide (0.651 g, 10 mmol) were added to 8 mL of deionised water. The resulting solution was vigorously stirred under a steady flow of nitrogen gas and 1 M nitric acid (11 mL, 11 mmol) was added to precipitate a cream solid. The precipitate was then repeatedly washed in deionised water (20 mL aliquots), filtered and dried as described above.

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Figure S1a. PXRD patterns measured at room temperature of (a) HT-CuCN, (b) (Cu1/2Au1/2)CN, and (c) AuCN and indexed on the basis of the AuCN structure type. AuCN has a hexagonal unit cell with a ~ 3.39 and c ~ 5.07 Å at room temperature.

Interactions on a Hexagonal Lattice is Four
Consider the regular hexagonal tiling of the plane, and suppose that each hexagon is colored either black or white. We investigate the average number, defined in a natural way, of oppositely colored neighboring tiles, and we show that this average cannot exceed 4.
Let X denote the set of all tiles. Given a two-coloring c of the tiling and a finite set ⊆ we write ( ) for the average number, taken over tiles ∈ , of tiles y adjacent to x such that ( ) ≠ ( ).
For each tile ∈ we define
Calculations were carried out first without approximating possible weak interactions (aurophilic and/or argentophilic), then two DFT-based van der Waals' approaches were considered. The three calculation types are labelled no-vdW, vdW-1 and vdW-2, respectively.
Calculated energies are negative. In the following tables, by "scaled" energy it is meant that the lowest energy is taken as a reference: E(scaled) = |E0| -|Ei|; where E0 is the lowest energy per formula unit of a given structure, and Ei is the energy per formula unit of each of the model The energy differences between some structural models are very small, so more than one model could possibly be considered as being energetically stable with respect to the others.
Including vdW interactions does not alter the stability of the model structures with respect to each other.
The effect of interchanging C and N atoms in the metal-cyanide chains is also explored. Interchanging the C and N atoms does not alter the stability of the model structures with respect to each other. However, the bonding sense -NC-Au-C≡N-M-is found to be energetically more stable than -CN-Au-N≡C-M-.

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Computational Details: Total-energy calculations of the studied models were performed using the projector-augmented wave formalism [1,2] of the Kohn-Sham density functional theory [3,4] within the generalized gradient approximation (GGA), implemented in the Vienna ab-initio simulation package (VASP) [5]. [Note: Relativistic effects are included in the construction of the projector augmented wave-based pseudopotentials used in the VASP code [1,2,5,6]. VASP performs a fully relativistic calculation for the core-electrons and treats valence electrons in a scalar relativistic approximation.] The GGA was formulated by the Perdew-Burke-Ernzerhof (PBE) density functional [7]. The Gaussian broadening technique was adopted and all results were well converged with respect to kmesh and energy cutoff for the plane-wave expansion. A plane-wave energy cutoff of 700 eV was used, and the integrations over the Brillouin zone were sampled using grids of k-points generated by the Monkhorst-pack method [8]. The break condition for the self-consistent field (SCF) loop was set to 10 -8 eV. Given the weak and dispersive nature of the aurophilic and/or argentophilic effects, stemming from closed d-shell dispersive weak interactions (where applicable), they can be described by van der Waals' type interactions. Standard Kohn-Sham-based DFT lacks the description of static, long-ranged, dispersion forces [9]. Therefore, a vdW-type correction should be considered within the present DFT framework. In VASP, various vdW-based schemes, accounting for the London-like R -6 behavior, originating from nonlocal electron correlation [10,11], are available. Two vdW approximations, based on the representative significance of their conceptual implementation, were explored. The first vdW approach is based on the Grimme method [12] (vdW-1). The second is a density functional where the non-local correlation functional approximately accounts for dispersion interactions [13,14] -presently we adopt the one called optPBE-vdW [15] (vdW-2). The validity of such treatment can further be supported by other works in the literature [16][17][18][19].

)CN formula unit) between each model (A -J) and model (J) with chain ordering in the sense -N≡C-Au-C≡N-Cu-.
(Cu1/2Au1/2)CN with chain ordering in the sense -N≡C-Au-C≡N-Cu- Note: Without applying a vdW correction, the most stable model is J (P6/mmm{6}) and the least stable is A (P6/mmm{2}). This conclusion does not change when vdW effects are included using the two different approximations, vdW-1 and vdW-2. Figure S3. The scaled energies per formula unit of (Cu1/2Au1/2)CN for the different crystallographic models (A -J) as a function of number of nearest-neighbor heterometallic atoms calculated using the three DFT schemes: no-vdW (blue), vdW-1(orange) and vdW-2 (black) ( Table S1).

N≡C-Au-C≡N-Ag-.
(Ag1/2Au1/2)CN with chain ordering in the sense -N≡C-Au-C≡N-Ag- Note: In the absence of a vdW correction, the most stable model is (I) (Amam{8}) and the least stable is (A) (P6/mmm{2}). This does not change when including vdW effects using the two different approximations, vdW-1 and vdW-2. In addition, applying vdW interactions to model (A) stablilizes the structure greatly. Figure S4. The scaled energies per formula unit of (Ag1/2Au1/2)CN different crystallographic models (A -J) as a function of number of nearest-neighbor heterometallic atoms calculated using the three DFT schemes: no-vdW (blue), vdW-1(orange) and vdW-2 (black). (Table S2).
Note: The relative energy of (Ag1/2Au1/2)CN with N = 0 calculated without van der Waals' interactions is extremely high (+~0.54 eV) and the point is therefore excluded from the plot. The bonding sense -N≡C-Au-C≡N-Cuin the chains is found to be energetically more stable than -C≡N-Au-N≡C-Cu-, with or without the inclusion of vdW interactions.     SI-25         Table S10.).
Note: R factors, RDr, calculated using the following: