Structure and chemical bonding in high-pressure potassium silver alloys

Abstract

The high-pressure structures of K-Ag alloys were examples of pressure-induced electron transfer from the electropositive potassium to the electronegative silver. We re-examined the crystal and electronic structures of KAg₂, K₂Ag, and K₃Ag using powder X-ray diffraction and theoretical calculations. Our findings establish a connection between the morphologies of these three phases and the precursor face-centered cubic Ag. For K₂Ag, we discovered a disordered structure that better matches the X-ray pattern. Valence electron density distributions obtained from the maximum entropy method, along with charge density calculations, provide a comprehensive understanding of the evolution of chemical bonding in these systems. It was found that K atoms share their valence electrons during alloy formation, contributing to K-Ag and Ag-Ag bonds in K₂Ag and KAg₂, while no Ag-Ag bonds are present in K₃Ag. These results indicate the Zintl-Klemm model may be too simplistic to describe the structure and bonding in high-pressure binary intermetallic compounds.

Introduction

Discovered in the late 1990s, K-Ag alloys were the first examples of potassium forming intermetallic compounds with silver at relatively low pressure^1,2, as K and Ag do not form compounds at ambient conditions. The fast kinetics of the solid-state reactions at higher pressure was attributed to the large electronegativity difference between the two elements facilitating a charge transfer from K to Ag and forming compounds like K₂Ag and K₃Ag¹. This description is similar to the formation of Zintl phases³. Early theoretical calculations found that the band structures of K₂Ag and K₃Ag are remarkably similar to the hypothetical Ag lattices with the K atoms removed⁴. The observation suggested the stability of both alloys as derived from the interaction of Ag 5p orbitals filled by electrons donated by the K atoms. Since then, the Zintl-Klemm concept⁵ has been invoked to elucidate the structure and chemical bonding of various binary intermetallic compounds at high pressures^6,7,8,9, such as the metal superhydrides¹⁰ showing unprecedentedly high superconductivity critical temperatures. Despite the simplicity and apparent successes of the concept, the fundamental assumption of a large electron transfer between elements presenting large electronegativity differences has not been thoroughly scrutinized to establish a sound theoretical foundation to explain the experimental results.

With this study, we wish to understand better the crystalline structure and bonding of a prototypical series of binary alloys of K and Ag. This is accomplished by analyzing the crystalline structures and the electron density distribution obtained from experiments and comparing them to our theoretical calculations. For this purpose, we recorded and analyzed the X-ray diffraction patterns of KAg₂, K₂Ag and K₃Ag compounds synthesized from mixtures of K and Ag subjected to low pressures at room temperature. Electron density maps were derived by the maximum entropy method (MEM)¹¹ using the intensity of the Bragg reflections and the Fourier synthesis obtained from Rietveld refined powder X-ray diffraction patterns¹². In earlier studies^13,14,15, we have shown that the MEM analysis of powder X-ray diffraction patterns measured under high pressure in diamond anvil cells can effectively unveil how electron density distributions (ED) change under pressure. For instance, we observed the hybridization of Si s, p, and d orbitals, leading to the metallization of densified silicon¹³. In this study, we aim to gain qualitative insights into how the electron density distribution changes, hence addressing how the chemical bonding evolves with the application of pressure in K-Ag alloys. Plane-wave density functional theory calculations were performed to support our analysis. The electron density distribution and wavefunction computed were analyzed by employing density-based and projected localized atomic basis methods to characterize the nature of electron interactions using conventional and the familiar bonding descriptors.

The key findings of this investigation are succinctly summarized as follows. K atoms donate their valence electrons to the system. The number of electrons shared by each K atom does not change substantially with pressure upon the formation of the respective K-Ag alloys, namely, KAg₂, K₂Ag, and K₃Ag. In the case of KAg₂, the valence electrons donated by K are utilized to form K-Ag and Ag-Ag covalent bonds. For K₂Ag, we reinterpreted the previous studies using our X-ray diffraction patterns and found a K and Ag disordered hexagonal structure, different from the previously suggested ordered structure. For K₃Ag, it is found that the K atoms occupied both the tetrahedral and octahedral voids of the face-centred cubic (FCC) lattice formed by Ag atoms, strengthening the K-Ag bonds, while no Ag-Ag bonds are present. The electron density distribution derived from the experimental data indicates that the K-K interactions are weak yet discernible for all structures identified. Our observations and results diverge from the idealized complete electron transfer supported by the Zintl-Klemm model of alloy formation.

The outline of the paper is as follows. First, we provide details on the powder X-ray diffraction experiments and essential information regarding the Rietveld full-pattern refinements and the analysis by MEM. Then, we elaborate on the density functional calculations and the methodologies employed to characterize chemical bonding. Discussion on the structures and evolutionary pathways of the dense alloys. A comprehensive analysis of experimental and theoretical charge density distribution is provided to assess the chemical bonding. We conclude with a critical comparison between experimental and theoretical results.

Methods

Powder X-ray diffraction using synchrotron radiation

Samples of potassium (Strem Chemicals; ampouled, 99% min.) and silver (Sigma-Aldrich; powder, 99.99% min.) were prepared and loaded in gasketed diamond anvil cells (P < 25 GPa at 295 K) in appropriate different molar ratios in an inert (Ar) atmosphere presenting less than 1 ppm of oxygen and moisture. Most samples were constrained in Re gaskets. Tests indicated that T301 stainless steel gaskets were also appropriate. Once loaded without a pressure-transmitting medium, reactants were checked for purity by X-ray diffraction below the reactivity threshold pressure and, if adequate, were further compressed and decompressed in a series of X-ray diffraction experiments to study phase stability and changes. Pressures were measured at room temperature by the photoluminescence of Al₂O₃:Cr³⁺ microspheres (about 10 μm in diameter), loaded with the samples. Typically, the spectral broadening of the photoluminescence developed at higher pressures, increasing uncertainty to about 0.6 GPa above 10 GPa.

X-ray diffraction images were collected at the Wiggler High Energy beamline at the Brockhouse Sector for X-ray Diffraction and Scattering of the Canadian Light Source using photons at λ = 0.3497 Å, following a proven experimental method¹⁶. Given the inhomogeneous nature of the microscopic samples, X-ray diffraction maps were recorded from each sample with a spatial resolution of about 20 μm. Hence, at a given sample location corresponding to a specific K/Ag ratio, X-ray diffraction images were recorded as a function of pressure. The images were processed for calibration and data reduction¹⁷ and converted to X-ray diffraction patterns. In most instances, the X-ray diffraction patterns consisted of mixed crystalline phases of unreacted K and/or Ag and pressure-induced compounds. A preliminary analysis of the X-ray diffraction patterns was conducted using XRDA¹⁸ to identify the alloy phases and provide initial cell parameters for further Rietveld analysis.

Rietveld refinements and analysis by MEM

Phase identification and indexing of the X-ray diffraction patterns were further conducted using the cell indexing module in the General Structure Analysis System II (GSAS-II) software¹⁹. Initial background points computed using the automatic algorithm were fitted using a 20-term Chebyshev polynomial in GSAS-II. Additional points were added manually when necessary to achieve a more accurate description of the irregular backgrounds. The peak profile was described by a pseudo-Voigt function. 3D Fourier maps were calculated by the internal modules of GSAS-II.

The Le Bail method implemented in Jana2020²⁰ was employed to extract the structure factors needed for the MEM analysis, accomplished using Dysnomia²¹. MEM calculations were initiated from uniform densities and employed the zeroth-order single-pixel approximation (ZSPA)²¹ as the optimization algorithm. Except for the analysis of K₂Ag, which will be addressed later, the convergence of the MEM calculation was achieved for all reported results.

Computation details

All structural optimizations, charge densities, wavefunctions and electronic band structures were calculated using the Vienna ab initio Simulation (VASP) code²² and Projector Augmented Plane Waves (PAW) potential²³ and Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional²⁴. The PAW potentials employed 3p⁶s¹ and 4d¹⁰ 5s¹ as valence states for K and Ag, respectively. The k-point mesh used in the calculations were 21 × 21 × 10, 7 × 7 × 10, and 7 × 7 × 7 for KAg₂, K₂Ag and K₃Ag, respectively. The default energy cutoff of 249.8 eV for the PAW potentials was used. The Wannier90 code²⁵ interfaced with VASP was used to obtain the band structure from a GW²⁶ calculation and disentangle the electronic bands to generate the Wannier orbitals. One of the objectives of this study is to unveil the electron interactions of the alloys in the well-known chemical picture, addressing, for instance, atomic charges, orbital hybridization, and bond order. The electronic distributions can be analyzed with the charge density or the projection of localized basis sets for K and Ag atoms to the plane-wave wavefunctions. Bader’s quantum theory of “atoms-in-molecules” (QTAIM)²⁷ was used for charge density analysis. The topology of electron density is the physical manifestation of the forces acting within the system. The electron density distribution is determined by the interaction between two nuclei and chemical bonding. The associated topological properties at the critical points, e.g., the bond critical point (BCP), provide a good measure of the strength of the interaction. Topological analysis was performed with the Critic2 program²⁸ using the charge density calculated by VASP. In addition, net atomic charge and bond order were calculated using the density-derived electrostatic and chemical (DDEC6) method implemented in the Chargemol program²⁹. This new method partitions electron density into chemically meaningful components and delineates the electron density of the system into atomic contributions, providing insight into charge distribution, bonding patterns, and related properties. Natural bond orbital (NBO)^30,31 is a powerful method to reveal chemical bonding by deriving local orbitals (i.e., the Lewis structure) from the electron density of the system. However, to obtain a real space description of the bonding, it is necessary to project localized atomic orbitals onto the plane wave. The reliability of the projection is subject to the quality of the atomic basis sets and the parameters used in the computation of localized properties. In our calculations, the quadruple-zeta Gaussian atomic orbital (AO) basis sets, def2-QZVP³² were used. Orbitals with very small exponents (diffuse) were removed. The quality of the NBO projection was accessed by ensuring the total spread after projection is less than 1.0 × 10⁻². The calculation of the natural atomic orbital (NAO) occupation reflects the hybridization of the atom.

Results

Analysis of X-ray diffraction patterns

X-ray diffraction patterns of crystalline KAg₂, K₂Ag, and K₃Ag^1,2 were observed from compression/decompression experiments starting from mixtures of elemental K and Ag. The X-ray diffraction patterns at 4.13 (KAg₂), 4.41 (K₂Ag), and 5.88 GPa (K₃Ag) are shown in Fig. 1. The X-ray diffraction patterns indicate the presence of residual K mixed from the respective K-Ag alloy phases. The K contamination, however, does not impede the quality of the patterns for Rietveld refinements and MEM analysis. The different phases identified are discussed in detail below.

Fig. 1: Representative X-ray diffraction patterns for K-Ag alloys.

KAg₂ (top) K₂Ag (middle) and K₃Ag (bottom).

KAg₂

KAg₂ is the lowest pressure compound among the K-Ag alloys identified from the X-ray diffraction data. The phase is observed between 1.56 GPa and 4.2 GPa at room temperature. The results of our analysis confirmed the unit cell reported². It has a hexagonal P6₃/mmc space group with lattice parameters a = 5.7237 Å, c = 9.7411 Å and Z = 4 at 2.26 GPa. In the unit cell, the Ag atoms occupy the 2a and 6 h sites and the K atoms are located at the 4 f sites. The X-ray diffraction pattern (Fig. 2) shows the presence of cubic phase I of K (Im$\bar{3}m$, K-I). A Rietveld refinement on the combined patterns was successful with R_w = 0.91% with the KAg₂ to K-I ratio of about 10:1 (see Fig. 2).

Fig. 2: Comparison of the Rietveld refined and measured X-ray diffraction pattern of KAg₂ at recorded 2.36 GPa.

The pure potassium contribution to the pattern belongs to the low-pressure body-centered cubic phase (Im$\bar{3}m$ K-I). The KAg₂ was obtained by releasing the pressure from a higher-pressure phase. Inevitably, different phases in small quality are present in the sample, resulting, for instance, in a weak line at 2θ = 9.7^o.

An important feature of KAg₂ (Fig. 3a) that was overlooked in an earlier study² is that its crystalline structure can be described by (Ag)₄ clusters constructed from two-dimensional (2-D) layers of corner-shared triangular (Ag)₃ units in the ab plane (plane A, Fig. 3b) linked through coplanar apical Ag atoms (plane B, Fig. 3c) in the c-direction. The in-plane Ag-Ag distances are identical at 2.84 Å. The distance from the apical Ag to the Ag in the plane is slightly longer at 2.92 Å. The K-Ag distances range from 3.36–3.49 Å. The shortest K-K separation is 3.39 Å.

Fig. 3: Structural motifs of crystalline KAg₂.

The crystal structure is constructed from the precursor face-centered cubic (FCC) Ag. a shows interconnected Ag-tetrahedra (open circles) and K atoms (solid, purple). The Ag atoms are arranged in (b) 2-D quadrilateral (plane B) and hexagonal (Kagome) Ag planes (plane A). c shows the Ag sites replaced by K.

When viewed down the c-axis, the A, B planes of Ag stacking resemble that of the precursor FCC Ag with the Ag atoms in the center of the hexagons removed (Fig. 3c). In KAg₂, these “voids” are replaced with pairs of K atoms above and below and alternate the positions between planes A and B.

A Le Bail refinement²⁰ of the X-ray diffraction pattern was performed to extract the Bragg reflection intensities. Using a set of 19 reflections, convergence in the MEM analysis²¹ was reached. The electron density (ED) distribution obtained is consistent with the crystalline structure and the Fourier map resulting from the Rietveld refinement (Fig. 4). The MEM-derived valence electron topology (Fig. 4b) shows that the electron densities of the Ag atoms in the planar triangular clusters are spread along the c-direction. The electron density at the K atoms is also slightly distorted from the spherical distribution. The former observation suggests that there may be a covalent interaction between the Ag atoms. The electron density distribution revealed from the 3D Fourier maps, assuming spherical atomic scattering factors, also shows slight distortions around the K and Ag atoms (Fig. 4a). Indeed, an examination of the (004) plane of the Ag₃ clusters in both the MEM electron density distribution and the Fourier map indicates the accumulation of electrons between the Ag atoms (Fig. 4c, d, respectively).

Fig. 4: Charge Density Distribution in KAg₂.

Charge density distribution obtained (a) from Fourier analysis of the Rietveld refined results using spherical atomic scattering factors using an iso-surface value of 13.0 e/${{{\rm{a}}}}_{0}^{3}$ and (b) from the MEM analysis using an iso-surface value of 6.5 e/${{{\rm{a}}}}_{0}^{3}$. Electron density distribution of KAg₂ in the (004) plane from (c) Fourier map and (d) MEM-derived. Note the contour scales for the two plots are different. The units of the scales are e/ų.

K₂Ag

The synthesis of K₂Ag and its crystalline structure was first reported in ref.1. At 4 GPa, the unit cell was reported to be the hexagonal P6/mmm space group with Z = 1 and lattice parameters a = 5.5434 Å and c = 3.770 Å. In the hexagonal unit cell, Ag atoms are at the 1a (0,0,0) position, whereas the K atoms are at the 2d (1/3,2/3,1/2) site. Both atomic species form layers stacked along the c-axis. In the original study, Atou et al.¹. noted the absence of (00 l) reflections in the X-ray diffraction pattern, which, with the assumption of preferred orientation, led to the conclusion of the P6/mmm space group.

In the present study, high-resolution X-ray diffraction patterns recorded at 2.24 and 3.22 GPa (upon pressure decrease) show the existence of the K₂Ag phase. Apart from a minor contribution of the K-I phase, the patterns would be seen in agreement with the proposed P6/mmm unit cell¹ if the (001) reflection could be ignored. However, other absent reflections, particularly at higher scattering angles, indicate this is not the case. Table 1 lists the predicted but unobserved reflections below 15^o (2θ) with the P6/mmm space group¹.

Table 1 Predicted but unobserved reflection for P6/mmm space group

A unit cell indexing performed using GSAS-II and JANA2000 predicted a P6₃/mmc space group, which matches better the X-ray diffraction pattern with the same lattice parameters. The new space group assignment eliminates the discrepancies related to the unobserved reflections as given in Table 1 (except for (301) line observed at 13.52°) and other high-angle reflections. However, by assuming the P6₃/mmc space group, Ag atoms would need to occupy the 2a sites and K atoms the 4 f sites. In this case, the resulting crystalline structure gives unrealistically short interatomic distances. There are two possible solutions to circumvent this issue to explain the observed X-ray diffraction patterns: (1) partial occupancy of the atomic sites or (2) to invoke the existence of a superlattice by doubling the cell along the c-axis.

Before further addressing the pattern refinement problem, we analyzed the electron density distribution derived from MEM analysis in both P6/mmm and P6₃/mmc space groups. No convergence was achieved using either space group. Nevertheless, upon examination of the crude EDs, we found hints of atomic disordering along the c-axis in both space groups, indicated by a continuous electron density distribution in the [001] direction. The results lead us to advance the possibility of a superlattice. We then constructed a superlattice unit cell by doubling the c-axis for both space groups. In both models, a very good Le Bail fit with R_p less than 1% was obtained. However, the P6/mmm supercell predicts an additional peak at a low scattering angle that was not observed in the experiment.

As shown in Fig. 5, ED maps obtained in both space groups clearly show a continuous electron density between the Ag atoms along the c-axis. The “extended” charge distribution is too large for Ag-Ag bonding. We speculate that there may be a “disordered” structure with Ag partial occupancy halfway along the c-axis and between the two equivalent Ag atoms in Wyckoff position 2b. In addition, the electron density distribution around the K sites in the middle of the unit cell also appears to show disorder. Based on this observation, we propose four possibilities for the structure of K₂Ag: a supercell (i) with no partial occupancy (i.e., with all atoms in their respective Wyckoff symmetry positions) (ii) with the (0,0,1/4) site partially occupied with Ag (iii) with K atoms displaced from the ideal Wykoff sites, or (iv) with one K and all Ag sites allowed to be partially occupied. We performed a Rietveld refinement on the above-mentioned models without considering the preferred orientation. Table 2 reports the R_w-factor for the refined models in the P6₃/mmc space group.

Fig. 5: ED maps derived from MEM analysis.

Supercells in P6₃/mmc (left) and P6/mmm (right) space groups with non-distorted atomic positions were used. In both space groups, the Ag atoms are located at the corners and the edges of the unit cell, and the K atoms are inside the unit cells. The units of scale are e/ų.

Table 2 R_w agreement factor for the Rietveld refinement of the four proposed structural models in the P6₃/mmc space group

The Rietveld refinements show gradual but significant improvements in R_w when a supercell with doubling of the c-axis is adopted and disordering K and Ag atoms when partial occupancy is considered. The results lend support to the “continuous” electron distribution between the Ag atoms due to the disordering as revealed by MEM analysis. The best fit achieved is when partial occupancy of disordered K and Ag sites is considered. It is noteworthy to indicate that low-angle, high-intensity reflections bias the goodness-of-fit parameter, R_w, of the Rietveld refinement. However, a qualitative comparison of the calculated and experimental patterns supports the results of the Rietveld analysis. With no partial occupancy, some unobserved reflections appeared at high angles and their intensities did not fit well. In contrast, when partial occupancy is considered, those reflections disappear, and concomitantly, fits to the intensity of the high-angle reflections improves. Even with the best structural model, not all high-angle features are adequately accounted for. As mentioned in the earlier study, the discrepancy could be due to a preferential orientation of the sample. Indeed, we obtain a significant improvement using a preferential orientation model that reduces R_w to 0.83%. Furthermore, the refined isotropic thermal parameters are also very reasonable. A comparison of the measured and fitted X-ray diffraction patterns is shown in Fig. 6. The final model gives a stoichiometry of K₂Ag_0.94, almost identical to the ideal composition. The structural parameters obtained from the Rietveld refinement of the final model are presented in Table 3.

Fig. 6: Comparison of Rietveld refined and measured X-ray diffraction pattern of K₂Ag.

Refinement was performed in the P6₃/mmc supercell with disordered Ag and K atoms (for details see K₂Ag section). For this refinement, a preferred orientation is considered leading to an R_w = 0.83%.

Table 3 Parameters obtained from Rietveld refinement of the data presented in Fig. 6, considering the disordered K₂Ag model in the P6₃/mmc space group

A similar analysis was performed considering the P6/mmm space group. In this case, the agreement is significantly inferior. Moreover, a diffraction peak at around 2.5^o, predicted for the supercell, is not accounted for in the experimental X-ray diffraction pattern. In addition, many predicted high-angle reflections are also not present in the data, resulting in a poor fit. We rule out the possibility of a unit cell with the P6/mmm space group on this basis. Our analysis thus indicates that the best structural model consistent with the measured diffraction patterns of K₂Ag has a P6₃/mmc supercell, with a = 5.604 Å, c = 7.757 Å and Z = 2. The 3D Fourier map (Fig. 7a) with ED obtained from the MEM analysis (Figs. 5 and 7b) shows a remarkable resemblance.

Fig. 7: Comparison of 3D Fourier map and electron density distribution obtained from MEM of KAg₂.

a 3D Fourier map for the disordered structure of K₂Ag and (b) superposition of the rescaled Fourier map with the electron density distribution (grey colour) obtained from the MEM analysis.

It is worth noting that although the Ag-Ag distance is longer than in the elemental phase, K₂Ag maintains the morphology of 2D honeycomb layers, which stack in a (disordered) …A-A-A… manner with layers intertwined with K atoms.

K₃Ag

K₃Ag is the highest-pressure K-Ag phase observed in this study, up to 13.2 GPa. It typically appears at pressures above 5.5 GPa. It has a cubic Fm$\bar{3}$m structure and the unit cell parameter decrease from a = 7.89 Å at 5.5 GPa to a = 7.41 Å at 13.2 GPa¹. The X-ray diffraction patterns show the presence of a small amount of the K-I phase, which is taken into account during the full-pattern refinement without complications. Le Bail and Rietveld refinements were conducted on five diffraction patterns measured at different pressures. A comparison of the refined and measured pattern at 5.5 GPa is shown in Fig. 8. In this case, the goodness-of-fit factor, R_w equals 0.84%. In the unit cell, the Ag atoms occupy the 4a site and the K atoms the 4b and 8c sites. The crystalline structure can simply be described with K atoms inserted into the octahedral (4b) and tetrahedral (8c) sites of the FCC Ag lattice and expanding the unit cell parameter from 4.042 Å of pure Ag at 5.5 GPa to 7.89 Å for the K₃Ag unit cell at the same pressure.

Fig. 8: Rietveld refined and measured X-ray diffraction pattern of K₃Ag.

The pattern (Fm$\bar{3}$m) was recorded at 5.5 GPa.

The MEM analysis converged readily on all X-ray diffraction patterns. The 3D ED maps derived from the Rietveld refined patterns at five pressure points are presented in Fig. 9. The 3D electron density distribution around the K and Ag atom locations shows no apparent deviation from a spherical distribution. A detailed examination of the charge density on the (100) and (110) planes, encompassing the Ag-Ag and Ag-K atoms in the unit cell, shows hints of K-K and K-Ag interactions as observed in Figs S1–S10. However, no discernible interactions between Ag atoms are found. This observation is at odds with a previous suggestion⁴ based on theoretical band structures whereas Ag accepts electrons from K to populate its 5p orbitals leading to the formation of Ag-Ag bonds.

Fig. 9: MEM-derived ED of K₃Ag at the indicated pressures.

The ED at 5.50 GPa has an iso-surface value of 0.77 e/${{{\rm{a}}}}_{0}^{3}$. All other pressures have an isosurface value of 2.0 e/${{{\rm{a}}}}_{0}^{3}$.

Theoretical analysis

Electronic structure

Although there are ambiguities in the structure of K₂Ag, nonetheless, the models proposed above are all based on the same basic structural motif, i.e., the stacking of hexagonal 2D-Ag layers with K atoms situated between the layers, as derived from the P6/mmm space group. As it is not possible to compute the electronic structure of the P6₃/mmc disordered solid, we employed the ordered P6/mmm structure in the calculation for comparison. The GW disentangled band structures together with the dominant Wannier orbitals in the indicated electronic band regions for KAg₂, K₂Ag, and K₃Ag are depicted in Fig. 10. Examination of the electron band dispersions indicates substantially different chemical interactions for the three K-Ag alloys considered.

Fig. 10: GW electronic band structures and dominant Wannier orbitals.

(top) KAg₂, (middle) K₂Ag, and (bottom) K₃Ag.

In KAg₂, the band structure shows mixing of the participating K and Ag atoms. The Ag 4d bands are not distinguishable and are heavily involved in the bonding. Disentanglement into K and Ag atoms is more complicated. The Ag 4d band is broad and extends from −5.5 eV to −4.0 eV. The shape of the dominant Ag 4d Wannier orbitals (Fig. 10, top) is distorted from the ideal atomic orbitals, indicating hybridization with other orbitals. K-Ag bonding is observed from −3 eV to the Fermi level. Above the Fermi level, K orbitals with “d” character start to appear. To accurately disentangle the bands in the lower valence state in K2Ag, we employed a projection-based method with the inner and outer disentanglement energy windows set to −6 eV and 0 eV, respectively. As depicted in Fig. 10 (middle), the disentangled Wannier orbital shows the band mixing is primarily due to the 4d and 5 s orbitals of the Ag atom. The localized bands between −4 eV and −5.2 eV correspond to 4d orbitals of the Ag atoms, which do not participate strongly to the bonding. Similarly, the bands around the Fermi level are disentangled by freezing the electron states up to the Fermi level, with the inner and outer energy windows set to −2 eV and 4 eV, respectively. The Wannier functions around the Fermi level are mostly for Ag 4p. The Wannier orbitals indicate that the Ag 4d and 5 s electrons are strongly mixed with the K orbitals, showing a significant chemical bonding. In K₃Ag, the Ag atom dominates the lower energy valence states (Fig. 10 bottom). The 4d bands between −4.5 and 5.0 eV are very narrow and completely isolated from the 5 s with a gap of 1 eV. Using inner and outer energy windows of −1 eV and 2 eV, respectively, with the frozen state set to the Fermi level (i.e. only states up to the Fermi level are included in the wannierization), the Wannier orbitals extracted from the upper valence state and lower conduction bands reveal a significant contribution from K atoms to the bonding states. The Wannier orbital shows a K s, p, and d (${d}_{{z}^{2}}$ and ${d}_{{x}^{2}-{y}^{2}}$) hybrid orbital.

The analysis of the calculated projected electron density of states (PDOS) corroborates the qualitative description of the chemical bonding in the K-Ag alloys. The Ag and K projected density of states for the three alloys are shown in Fig. 11. With 8 Ag atoms (each with 10 d-orbitals) and four K atoms in the unit cell of KAg₂, the 4d density of states dominates the lower valence level from −7 to −3 eV. The 4d contribution relative to 5 s and 5p is relatively significant up to the Fermi level (Fig. 11). In comparison, the PDOS of Ag in K₂Ag is more localized between −4.4 and 4.0 eV, albeit mixed with a small amount of 5 s, and did not extend into the upper valence region. The dominance of Ag 5 s between −4.0 to −0.8 eV is discernible. The PDOS of Ag is a mixture of 5 s and 5p orbitals close to the Fermi level (Fig. 11). Clear separation of the 4d and 5 s bands is displayed in K₃Ag (Fig. 11). Contributions from the 5p orbitals are becoming more significant around −0.8 eV.

Fig. 11: Plane-wave projected electron density of states (PDOS).

Orbitally resolved Ag and K PDOS in the valence band of KAg₂, K₂Ag and K₃Ag. Note that different scales were used for the PDOS plots.

The K PDOS in KAg₂ shown in Fig. 11 indicates strong participation of the 3p orbitals in the entire valence region. In particular, the high PDOS between −5.5 and −3.0 eV overlaps with the Ag 4d and 5 s bands (Fig. 11), showing, unambiguously, Ag-K bonding. The PDOS of K profiles from −3.0 eV to the Fermi level exhibit dissimilarities to the PDOS of Ag (Fig. 11), indicating the likelihood of predominantly K-K interactions. In K₂Ag, the distribution of the PDOS of K (Fig. 11) of predominantly 3 s is quite broad between −4.4 eV and −2.0 eV and corresponds well with the 4d and 5 s bands of the Ag atom, indicating K (4 s) – Ag (4d, 5 s) bonding. The predominantly 4 s PDOS profile near the Fermi level from −1.60 eV is not similar to the Ag, so it likely indicates K-K interactions. The K “d” character becomes stronger above the Fermi level. The PDOS of K in K₃Ag is less complicated (see Fig. 11). It overlaps with the Ag 5 s from −4.2 to 2.5 eV (cf. Fig. 11). The peak at −0.8 eV matches the Ag 5p in the PDOS of Ag suggesting Ag-K bonding. It is noteworthy to highlight that the PDOS of the Ag atom near or at the Fermi level decreases as the K content increases in the alloy.

Discussion

Examining the computed electronic structures provides an overall picture of a change in the nature of chemical bonding in KAg₂, K₂Ag, and K₃Ag. Particularly, a progressive localization of the Ag 4d band is most noticeable. Atomic Ag has an electron configuration [Kr]4d¹⁰5 s¹ and the d-shell is filled formally. However, the 5s to 4d orbital proximity facilitates the sd hybridization. This is evident in the case of KAg₂. On increasing K content, the Ag 4d orbitals become more localized and do not participate in the bonding. The anticipated charge transfer from the electropositive K to Ag certainly affects the valence bonding. The conventional method of charge partitioning implemented in most plane wave codes is subjected to arbitrariness when choosing the atomic volume. Therefore, unbiased methods of assigning atomic charges based on the calculated density are preferred. For this purpose, we computed the atomic charge based on QTAIM and the more recent DDEC6 algorithms. For comparison, we also evaluate the NBO orbital occupancy using quadrupole zeta (def-QZVP) atomic basis sets. Table 4 summarizes the computed atomic charges.

Table 4 Computed atomic charges for K and Ag for the different K-Ag alloys

The atomic charges calculated from the plane wave electron densities (viz. QTAIM and DDEC6) are consistent with each other. As expected, the K atoms lose the 4 s electrons to the system. The number of electrons lost for each K atom seems to decrease with the increasing K content in a compound and the number of electrons gained by Ag increases. The NAO charges computed from the projection of localized atomic orbitals to the plane wave wavefunctions show a similar trend. The calculated natural orbital occupancies for Ag in KAg₂, K₂Ag, and K₃Ag are 4d¹⁰5s^1.205p^0.07, 4d¹⁰5s^1.645p^0.64 and 4d¹⁰5s^1.555p^0.11, respectively; and for K, 4s^0.154p^0.133d^0.02, 4s^0.294p^0.243d^0.02 and 4s^0.224p^0.293d^0.02, respectively. The results show K sp and Ag dsp hybridization. The NBO populations are consistent with the density analysis, which shows that the total number of electrons being transferred from K to Ag increases with K content, but the population of the Ag 5p orbitals does not increase significantly. This observation indicates that although Ag is more electronegative, it does not fully accommodate all the available K valence electrons when the potassium concentration is high.

The bonding in the alloys can be characterized by the bond critical points (BCP) from the analysis of the electron density topology with QTAIM and the bond order, which can be used as an indicator for bond strength, as computed from the DDEC6 method. In all cases, the Morse sums are satisfied in the QTAIM analysis.

As illustrated in Fig. 12, many BCPs are found in KAg₂. All BCPs are located and situated almost halfway along the K-K, K-Ag, and Ag-Ag bond paths. Therefore, the QTAIM analysis clearly reveals that the interactions are found between all the atoms in KAg₂. Bond orders (BO) calculated by DDEC6 using the same charge density support this assignment (Table 5). Although not a genuine two-electron bond, the calculated Ag-Ag BO values of 0.39–0.47 are reasonably high. The planar triangular Ag-Ag BO of 0.47 is higher than that of the in-plane-apical Ag-Ag bond which is 0.39. This is consistent with the difference for the Ag-Ag bond lengths (Table 6). The Ag-K and K-K bond orders are comparably weaker, around 0.09 and 0.03, respectively. Thus, electrons from K atoms are not completely shared with Ag atoms to form anions, as described in the simple Zintl-Klemm picture. Instead, the electrons are involved to some extent in forming different chemical bonds.

Fig. 12: Bond critical points for KAg₂.

Ag-Ag, Ag-K, and K-K bond critical points (BCP) found by Critic2²⁸ for KAg₂.

Table 5 Bond order in KAg₂, K₂Ag and K₃Ag calculated by the DDEC6 algorithm²⁹

Table 6 List of experimental atomic separations in KAg₂, K₂Ag and K₃Ag

Following the electron density topological analysis, three distinct types of bond critical points between Ag-Ag, Ag-K, and K-K atom pairs are found in K₂Ag (Fig. 13). The density (ρ(r_BCP)), which can be estimated, is an indicator of bond strength. In this case ρ(r_BCP) for Ag-Ag, Ag-K and K-K are computed to be 0.015, 0.008, and 0.007 (e/a₀³), respectively. These values can be correlated with the calculated bond order of 0.30, 0.1 and, 0.07, given in the same order.

Fig. 13: Bond critical points for K₂Ag.

The hexagonal P6/mmm at 4 GPa was used in the calculation. Colour code: purple = K atom, silver = Ag atom and black = bond critical points (BCP).

BCP and bond order analyses were performed on K₃Ag at 6.4 GPa. The primitive cell used in the calculation and the BCP are illustrated in Fig. 14. Ag-K and K-K BCPs are found, but no Ag-Ag BCP is found. Two crystallographically distinct K atoms are located in the tetrahedral and octahedral sites of the Ag FCC lattice. There are noticeable differences in the K-Ag interaction for the two sites. The ρ(r_BCP) of 0.012 e/a₀³ at the tetrahedral site has a slightly smaller atomic charge +0.506e is higher than the K at the octahedral site with ρ(r_BCP) of 0.008 e/a₀³ and a charge of +0.519e. BO calculations confirmed this assessment. The calculated BO for K-Ag at the tetrahedral site is 0.195 and a much smaller BO of 0.070 at the octahedral site. QTAIM analysis also revealed K-K interactions with a relatively small ρ(r_BCP) of 0.008 e/a₀³. The corresponding BO is 0.076.

Fig. 14: Bond critical points of for cubic K₃Ag.

The calculations were performed in the primitive cell at 6.4 GPa.

In passing, it is informative to elucidate the chemical bonding in the K-Ag alloys in the viewpoint of the crystal structures. A tabulation of the Ag-Ag, Ag-K, and K-K separations in the crystal structure of KAg₂, K₂Ag (P6/mmm), and K₃Ag are summarized in Table 6. Interestingly, the Ag-Ag, Ag-K, and distances in the two lower pressure phases, KAg₂ at 1.57 GPa and K₂Ag at 2.36 GPa, are very similar, even after considering the pressure difference. So, it is not surprising that the atomic interactions revealed from the theoretical analysis, that is, the presence of Ag-Ag, Ag-K, and K-K bonds, are also similar. Only in the higher-pressure K₃Ag are the bonds somewhat weakened and the ionicity of K and Ag becomes larger. The Ag-Ag separation increases abruptly from 2.91 Å in K₂Ag to 5.58 Å in K₃Ag with the pressure change of 3.2 GPa. The long distance between the Ag atoms prevents orbital overlap. The earlier postulation suggests that interactions between the Ag 5p orbitals⁴ are thus invalid. Instead, it is observed from our experimental results and theoretical analysis that the acceptance of the K electrons into the Ag 5 s and 5p orbitals facilitates Ag-K bonding.

Conclusion

The crystalline structures, electronic structure, and chemical bonding in K-Ag alloys formed under the application of pressure¹ have been studied by high-resolution powder X-ray diffraction using synchrotron radiation and the analysis of the electronic structures. We found a systematic structural trend with the insertion of K into the Ag FCC lattice. When K atoms are in contact with Ag atoms, they lose their valence electron and become smaller ions that diffuse progressively into the FCC Ag framework, forming the different alloy structures. We also have re-evaluated the structure of K₂Ag. Through full-pattern Rietveld refinements, we found that a disordered structure with a supercell of P6₃/mmc symmetry fits better the observed X-ray diffraction pattern. Electron density distribution maps obtained from MEM analyses indicated Ag-Ag and Ag-K bonding in the alloys. This observation is supported by the examination of the nature of disentangled Wannier orbitals, QTAIM topological analysis^27,28 of the charged density and the projection to localized atomic orbitals NBO³⁰ and DDEC6²⁹ analyses. All results indicate that interactions of K 4 s, 4p, and 3d with mix Ag 5 s, 5p, and 4d orbitals dominate the valence bands of KAg₂ and K₂Ag. In comparison, the Ag 4d orbitals are localized in K₃Ag. Succinctly, electron transfer from K to the 5 s and 5p orbitals of Ag atoms helps to maintain Ag-K bonding in these structures. Contrary to a previous suggestion⁴, no Ag-Ag bonding via the overlap of Ag 5p orbitals is found. Nevertheless, the Zintl-Klemm charge-transfer concept⁵ is broadly followed. This phenomenon is also observed in the hydrogen-rich superhydrides, as demonstrated in the study on the bonding of strontium hydrides¹⁰. However, a straightforward application of this concept to elucidate high-pressure structures of intermetallic compounds⁶ should be exercised with caution.