# Modeling the Shape of Ions in Pyrite-Type Crystals

## Abstract

**:**

_{2}, with chalcogen ions X = O, S, Se and Te. The latter are found to exhibit the shape of ellipsoids being compressed along the <111> symmetry axes, with two radii r

_{||}and r

_{⊥}describing their spatial extension. Based on this ansatz, accurate interatomic M–X distances can be derived and a consistent geometrical model emerges for pyrite-structured compounds. Remarkably, the volumes of chalcogen ions are found to vary only little in different MX

_{2}compounds, suggesting the ionic volume rather than the ionic radius to behave as a crystal-chemical constant.

## 1. Introduction

_{c}superconductors [3], multiferroic compounds [4] or the modelling of technical devices, like high-k field-effect transistors [5], to mention only a few examples. The approach assumes that ions of a chemical element and specific valence attain the same spherical volume in distinct crystalline compounds AB, such that the interatomic distance d

_{AB}between the nearest neighbors is determined by the addition of radii r

_{A}+ r

_{B}.

_{h}) or 43m (T

_{d}). Here, consistent ionic radii could be derived that are exhibited in all of those crystals with only little variations [6]. It was the concept’s tempting promise that one geometric parameter would be conserved in all crystal surroundings, establishing a crystal-chemical conservation law. As it turned out, however, the respective ionic radii had to be split into different values according to bonding geometry, coordination number, etc., in order to arrive at sufficiently accurate interatomic distances. The extensive analysis of some hundred oxide and fluoride compounds led Shannon and Prewitt (1969) and Shannon (1976) to a set of ionic radii from the majority of relevant ions that still represent the state-of-the-art data [7,8]. Although these tables allowed for an accurate modelling of bond distances for the compounds from which they derive, many bond distances could only be predicted with little accuracy, in particular, when crystals with highly polarizable ions or complex crystallographic structures were considered.

_{2}[2]. Later, Birkholz and Rudert fully exploited the formalism to transition metal disulfides crystallizing in the pyrite structure. They showed that the sulfur ion in the MS

_{2}series, with M = Mn, Fe, Co, Ni, Cu, can consistently be described by a rotational ellipsoid with varying main axes r

_{||}and r

_{⊥}[11]. The significance of this more complex consideration of bonding geometry for the T

_{c}maximum in solid solution of superconducting pyrite-type di-selenides and di-tellurides has recently been demonstrated by Guo and co-workers [12,13]. In order to study more crystals with respect to the applicability of the approach, the complete set of pyrite-type compounds MX

_{2}with monovalent chalcogenide ions X

^{−}will be investigated in this work.

## 2. Motivation of Ellipsoidally Shaped Ions in Crystals

_{core}(r) + ρ

_{val}(r,θ,φ). The approach shares common aspects with techniques for the determination of electron densities (ED) from X-ray diffraction [17,18,19]. The question of ionic shape, however, is substantially easier than a full ED determination, since it only seeks the course of the outermost iso-contour surface of the electron density instead of its entire distribution in the unit cell.

_{nl}stands for the radial function that determines the ions spatial extension and κ

_{nl}is the contraction-expansion parameter having the dimensionality of an inverse length. Strictly speaking, R

_{nl}does not only depend on the radial coordinate r, but on its product with κ

_{nl}having R

_{nl}= R

_{nl}(κ

_{nl}r). The parameter κ is usually associated with an effective nuclear charge Z

_{eff}via κ = Z

_{eff}mq

^{2}/(2πε

_{0}ћ

^{2}n) in order to account for screening (m electron mass, q elementary charge, ε

_{0}permittivity of vacuum). Moreover, κ is related to the bonding energy of the electronic level via En = −κ

^{2}ћ

^{2}/(8m) [20].

_{nl}of electronic state nl is given by the sum over a small number of basis functions of the form:

_{nl}(κ

_{n}

_{l}r)

^{n}exp(−κ

_{n}

_{l}r/2)

_{nl}is a normalization factor depending only on n and l. The determination of the set of κ parameters and their relative weights in the sum over all Equation (1) terms is the central task of the Hartree–Fock procedure [16,21,22,23,24]. Since this work is only concerned with the outermost course of the electron density of an ion, it will be assumed that considering one single term suffices to describe the ED at large distances from the nucleus. This will be the term with the smallest κ value.

^{l}, which are the spherical harmonics:

_{lm}

_{±}that derive from spherical harmonics via for |m| = 1 and y

_{l}

_{0}= for m = 0. Then, the Cartesian coordinate system has to be oriented along the main symmetry axes of the lattice site, like the z-axis along the major axes of rotation, etc.

_{nl}→ R

_{nl}

_{µ}. This becomes evident from the observation that symmetry adapted basis functions spanning different irreps, like and , are not transformed into each other under the symmetry operations allowed by the PG. For instance, a p

_{z}orbital extending along the symmetry axis of a 3 (C

_{3}) lattice site does not transform to p

_{x}or p

_{y}under any symmetry operation contained in the group.

_{nl}

_{µ}without violating the constraint of length conservation under symmetry operations. Accordingly, fully symmetry-adapted basis functions are given by the product function R

_{nl}

_{μ}, and there are various possibilities by which a distinction of radial functions may be introduced. In the framework used in self-consistent field theories and as given in Equation (1), the natural choice is to allow for different κ

_{µ}for basis functions of different irreps R

_{nl}

_{µ}(κ

_{µ}r). The κ splitting would correspond to a splitting of formerly degenerate levels into states of different bonding energy, since both are related as given above. No assumption will be made on the magnitude of the splitting; however, it will just be allowed for basis functions of different irreps in accordance with lattice site symmetry.

_{l}

_{µ}may amount to zero, one or two. The sum over all h

_{l}

_{µ}gives the number of valence electrons, Σh

_{l}

_{µ}= N

_{val}. The consequences of κ splitting upon the shape of ions are straightforward. While only one average radial parameter r

_{s}had to be used in the concept of ionic radii, in the approach outlined here, an envelope function r

_{s}(θ,φ) must be introduced that derives from identifying a surface of constant charge density:

_{val}(r

_{s},θ,φ) = ρ

_{s}

_{val}q enclosed by the r

_{s}(θ,φ) surface. The magnitude of f is typically set to values around 90%. For the concept proposed here, it is assumed that the precise value of f is chosen such that it assures the iso-density surface r

_{s}(θ,φ) of neighboring ions to touch at a common point in the space between them. These points will be named contact points in the following.

_{s}and x

_{s,i}now stand for the radial and Cartesian coordinates of the iso-density surface. It is easily shown that Equation (7) describes a general ellipsoid for κ

_{1}≠ κ

_{2}≠ κ

_{3}. In addition, a careful analysis of Equation (7) reveals that the spherical character of the r

_{s}(θ,φ) should increase with quantum number n, i.e., the period of the ion within the periodic table. The iso-contour surface of electron density in the outermost regions of a p valence shell ion at triclinic, monoclinic and orthorhombic lattice sites accordingly is that of a general ellipsoid.

_{1}= κ

_{2}≠ κ

_{3}, which is obeyed by p valence ions on trigonal, tetragonal and hexagonal lattice sites. It simplifies further to a sphere on cubic lattice sites, where κ

_{1}= κ

_{2}= κ

_{3}holds. In the latter case, all terms in Equation (7) can be placed before the sum that ends up with Σ(x

^{2}+ y

^{2}+ z

^{2}), such that the whole equation just depends on the radial coordinate r and, thus, describes the shape of the sphere. Strictly speaking, these results would be valid only for either closed shell systems, all h

_{i}= 2, or half-closed shells, all h

_{i}= 1. It is realized from Equation (7) that any aspherical deformation would be amplified by a mixed occupation, i.e., valence shells with varying occupation numbers.

_{−}compared to that of the neutral atom V

_{at}is larger than for cations: |V

_{−}− V

_{at}| > |V

_{at}− V

_{+}|. This rule holds at least for the most important ions of the second to fourth period, although the asphericity of cations may become equally important for higher periods.

## 3. Application to Pyrite-Structure Dichalcogenides

_{2}that crystallize in the pyrite structure; see Table 1. It will not be endeavored to give a theoretical calculation of ionic wave functions according to the rules outlined in the previous section. Nor will the numerical values of κ parameters or the fraction f of enclosed valence electrons be determined. Instead, the ellipsoidal modelling of anions will be investigated by making use of highly precise structural data that have been determined for these compounds by diffraction experiments and which are also listed in Table 1. As in the crystal radii approach, the ionic shapes will be chosen such that neighboring ions share a common contact point via their iso-density surfaces.

**Table 1.**Lattice parameters of pyrite-type compounds MX

_{2}. Unit cell edges a and positional parameters u were taken from cited references and used to calculate X–X and M–X bond length d

_{XX}and d

_{MX}. Metal ionic radii r

_{MSP}were selected from SPS tables for VI-fold coordinated divalent metal ions [7,8]; spin-states were chosen in agreement with experimental data, i.e., high-spin for Mn

^{2+}and low-spin for Fe

^{2+}, etc. Ellipsoidal radii r

_{||}, r

_{⊥}and ionic volume V

_{X}are given in the last columns as calculated from d

_{XX}and Equations (8) and (9). Lengths are given in nm and volumes in nm

^{3}.

MX_{2} | a | u | Ref. | d_{MX} | d_{XX} | r_{MSP} | r_{||} |
r_{⊥} | V_{X} |
---|---|---|---|---|---|---|---|---|---|

MgO_{2} | 0.48441 | 0.4114 | [26] | 0.2083 | 0.1487 | 0.072 | 0.0743 | 0.1415 | 0.00623 |

ZnO_{2} | 0.4871 | 0.413 | [27] | 0.2099 | 0.1468 | 0.074 | 0.0734 | 0.1413 | 0.00614 |

CdO_{2} | 0.5313 | 0.4192 | [28] | 0.2308 | 0.1487 | 0.095 | 0.0744 | 0.142 | 0.00628 |

FeS_{2} | 0.54160 | 0.38484 | [29] | 0.2263 | 0.2161 | 0.061 | 0.1080 | 0.1676 | 0.0127 |

CoS_{2} | 0.55385 | 0.38987 | [30] | 0.2325 | 0.2113 | 0.065 | 0.1056 | 0.1704 | 0.0129 |

NiS_{2} | 0.56852 | 0.39454 | [31] | 0.2398 | 0.2077 | 0.069 | 0.1038 | 0.1744 | 0.0132 |

CuS_{2} | 0.57891 | 0.39878 | [32] | 0.2453 | 0.2030 | 0.073 | 0.1015 | 0.1766 | 0.0133 |

MnS_{2} | 0.6104 | 0.4011 | [33] | 0.2593 | 0.2091 | 0.083 | 0.1046 | 0.1810 | 0.0143 |

FeSe_{2} | 0.5783 | 0.386 | [34] | 0.2419 | 0.2284 | 0.061 | 0.1142 | 0.1836 | 0.0161 |

CoSe_{2} | 0.58593 | 0.379 | [35] | 0.2437 | 0.2456 | 0.065 | 0.1228 | 0.1804 | 0.0167 |

NiSe_{2} | 0.59629 | 0.383 | [35] | 0.2488 | 0.2417 | 0.069 | 0.1208 | 0.1820 | 0.0168 |

MnSe_{2} | 0.6417 | 0.393 | [36] | 0.2702 | 0.2379 | 0.083 | 0.1189 | 0.1908 | 0.0181 |

MgTe_{2} | 0.70212 | 0.3875 | [26] | 0.2941 | 0.2736 | 0.072 | 0.1368 | 0.2257 | 0.0292 |

MnTe_{2} | 0.6943 | 0.386 | [36] | 0.2904 | 0.2742 | 0.083 | 0.1371 | 0.2103 | 0.0254 |

_{2}in Figure 1. The cubic pyrite structure contains four formula units MX

_{2}per unit cell with anions residing on three (C

_{3}) and cations on 3 (C

_{3i}) lattice sites; further structural details are given in the figure legend and [37]. For plotting Figure 1, cation radii r

_{MSP}were taken from the SPS listing [7,8], while the radius of sulfur anions was naturally chosen as half the sulfur dumbbell distance, r

_{S}= d

_{SS}/2. It turned out that the calculation of the interatomic Fe–S distance d

_{Fe–S}by simply summing up r

_{MSP}and r

_{S}resulted in a 25% deviation from the experimental value. This insufficiency of the spherical model was not only obtained for FeS

_{2}, but equally observed for all compounds listed in Table 1. It demonstrates the inadequacy of the spherical modelling for pyrite-type crystals.

**Figure 1.**Spherical modelling of the pyrite unit cell (space group Pa3). While Fe

^{2+}ions (blue) form an fcc lattice, sulfur ions (yellow) reside on <111> axes, as described by positional parameter u. In FeS

_{2}u amounts to 0.38484 in fractional coordinates of the unit cell [38]. The body diagonal is shown as an arrow. S ions form dumbbells, and there are four S

_{2}dumbbells in the unit cell. Only the dumbbell at (uuu) and (ūūū) appears undivided, while the remaining S

_{2}dimers all extend from the cell shown into neighboring cells. In order to visualize the different site symmetries, a mesh is inscribed into S ion spheres with poles lying on <111> symmetry axes. All ions are modelled by spheres with radii r

_{MSP}for Fe

^{2+}and half the S–S bond distance d

_{SS}/2 for S

^{−}. With this setting, Fe and S spheres have contacts with none or only one neighbor, respectively. This result also holds for the other compounds listed in Table 1. Such a spherical packing would be unstable in terms of packing theory [2], demonstrating the failure of the ionic radius concept in the case of pyrite-type compounds.

_{MSP}values were again taken from the SPS listing [7,8]. Only those pyrite-structure dichalcogenides were selected to enter into Table 1, for which a radius value for a six-fold-coordinated M

^{2+}was given in the SPS listing. Hence, the availability of these data served as a selection criterion for the compounds to be investigated here.

^{3}hybridization was assumed to account for their electronic configuration as deduced from their tetrahedral coordination. Since the electron density is assumed to scale with the square of electron wave functions, the hybridization does not affect the splitting of p levels into symmetry-adapted states. It is well known that the p orbitals in a three symmetry span a one-dimensional A and a two-dimensional E representation. Accordingly, two contraction-expansion parameters κ

_{||}and r

_{⊥}are needed to describe the spatial extension of chalcogen ion ellipsoids, which correspond to one radius r

_{||}along the <111> symmetry axis and another radius κ

_{⊥}perpendicular to it. Rotational ellipsoids of initially unknown principal extension where accordingly placed on X lattice sites in the second step of the geometrical construction.

_{||}and r

_{⊥}until the first contact with the iso-surfaces of lattice neighbors was obtained. This construction is easily performed for r

_{||}, since its magnitude is simply given by half the dumbbell distance, r

_{||}= d

_{XX}/2. The geometrical derivation of r

_{⊥}, however, is more complex. It can be seen from the bonding geometry shown in Figure 2 that the metal-chalcogen distance d

_{MX}should be decomposed in the metal radius r

_{MSP}and half of the diameter of the chalcogen ellipsoid, with the diameter chosen to lie in the direction of the M–X internuclear axis [11]. Since the ellipsoid diameter depends on both r

_{||}and r

_{⊥}, the equation:

_{⊥}can be derived, if the other quantities are known. From a geometric point of view, the solution of Equation (8) for known values of d

_{MX}, r

_{MSP}, r

_{||}and α corresponds to varying r

_{⊥}until the X ellipsoid comes in contact with the neighboring M spheres. The obtained X ellipsoid with the radii r

_{||}and r

_{⊥}is visualized in Figure 2 for FeS

_{2}, where the neighbors of the first bonding spheres of an S

_{2}dumbbell are shown.

**Figure 2.**Bonding coordination of the sulfur dumbbell in FeS

_{2}at (uuu) and (ūūū). (

**a**) Within the crystallographic unit cell and (

**b**) as S

_{2}Fe

_{6}cluster with internuclear axes. According to the presented approach, ellipsoidal deformations are allowed to occur for S anions on three (C

_{3}) symmetry sites, but might be neglected for Fe cations. The bond partner of sulfur dimers S

_{2}form two interpenetrating tetrahedrons with the neighbor S residing on top and three Fe

^{2+}forming the basis. The tetrahedrons itself are compressed as are the sulfur ellipsoids. The latter exhibit a smaller radius parallel to the <111> axis r

_{||}compared to the perpendicular radius r

_{⊥}. Comparable geometries are found for the other pyrite-type compounds listed in Table 1.

_{M}and r

_{X}, but derive from a geometrically more complex scheme that has to comply with the crystal symmetry and the relative orientation of valence orbitals. Only this procedure is in accordance with Neumann’s principle.

_{⊥}entering quadratically, as both perpendicular directions are equivalent.

## 4. Results and Discussion

_{||}, r

_{⊥}and V

_{X}, were analyzed with this ansatz for all MX

_{2}investigated in this work, and the results are also compiled in Table 1. It can be seen that the magnitude of the equatorial radius r

_{⊥}exceeds that of the polar radius r

_{||}in all cases. In terms of contraction-expansion parameters, this result is equivalent to κ

_{||}> κ

_{⊥}. Accordingly, chalcogen ions exhibit the shape of oblate ellipsoids being compressed along <111> axes.

_{⊥}/r

_{||}is seen in Table 1 to increase in the sequence O > S > Se > Te. This development is fully in accordance with the statement that r

_{⊥}/r

_{||}should diminish with principal quantum number n for an otherwise unchanged energy splitting, as may occur for a nearly constant crystal-structure surrounding when going from one pyrite-structured dichalcogenide to another. Figure 3 displays the aspherical modelling for the prototype compound of the pyrite structure, which is the first visualization of the FeS

_{2}crystallographic unit cell that is adequately modelled with ellipsoidal sulfur ions.

_{3}. This procedure makes the new picture difficult to grasp. On the other hand, the model is more in accordance with the crystalline state, where chemical bonding is not a phenomenon between neighboring atoms, but between one ion and all other ions of the crystallite.

_{X}. Their values, as calculated from r

_{||}and r

_{⊥}according to Equation (8), are given in Table 1. It can be seen that ionic volumes compare very well, when compounds with equal chalcogen ions are compared (O

^{−}, S

^{−}, Se

^{−}and Te

^{−}). It must be remembered that the accuracy of metal ionic radii is only on the order of some percent [7,8], which suggests that the scatter in V

_{X}values may practically vanish, if small corrections on r

_{MSP}would apply.

**Figure 3.**A new look at the pyrite structure: crystallographic unit cell of pyrite FeS

_{2}with Fe and S ions modelled by spheres and ellipsoids, respectively. Radial parameters r

_{MSP}for Fe

^{2+}and r

_{||}and r

_{⊥}for S

^{−}were used as given for FeS

_{2}in Table 1. The mesh inscribed into the sulfur ions now reveals the ellipsoidal compression along <111> directions. In this model, the number of contact points of S and Fe ions become four and six, respectively, and the ellipsoidal deformation is concluded to enable a stable packing.

_{X}as obtained from each group of MX

_{2}. The data are found to exhibit a monotonic increase from O to Te, as expected. The respective standard deviations δV

_{X}have also been given, which are perceived to be on the order of 1.7%. These variations are rather small when compared to other approaches for the geometrical modelling of pyrite-type compounds. They will even diminish when transformed from a volume scale to a length scale to compare their accuracy with data derived from the ionic radius concept.

_{x}of chalcogen ions, which is derived from and stands for the radius a spherical X

^{−}ion would have. Sphere-equivalent radii r

_{x}were calculated for each compound investigated here, and averages were formed for the groups of dioxides, disulfides, etc.

_{x}are seen to range from 0.3% to 2.3%. This is a rather small scatter compared to the conventional crystal radii approach [7,8] and again signifies the reliability of the approach introduced here. Furthermore, the calculation of interatomic distances now yields only small deviations from experimental data in the few percent range, when their calculation is based on V

_{X}values given in Table 2 and the d

_{MX}decomposition according to Equation (8).

_{X}is underlined by a comparison with the volumes of chalcogen atoms V

_{at}and divalent ions V

_{di}. Accurate data for them can be obtained from the literature, where they were either obtained through theoretical calculation or were determined from sphalerite-type compounds [7,8,39].

**Table 2.**Spherically-averaged radii and volumes of chalcogen atoms and ions with radii given in nm, volumes in 10

^{−3}nm

^{3}and relative errors δr

_{x}and δV

_{X}in %. Numbers in parenthesis give the standard deviations for the last digit(s) of the preceding value. Averaged data of monovalent ions r

_{x}and V

_{X}were derived as outlined in the text, while spherical volumes of chalcogen atoms V

_{at}and tetrahedrally coordinated divalent ions V

_{di}were calculated from their radii given in [39] and [7,8], respectively.

X | r_{x} | δr_{x} | V_{X} | δV_{X} | V_{at} | V_{di} |
---|---|---|---|---|---|---|

O | 0.1141(4) | 0.3 | 6.22(6) | 1.0 | 1.2 | 11.0 |

S | 0.1469(21) | 1.4 | 13.3(6) | 4.3 | 4.71 | 26.1 |

Se | 0.1593(15) | 1.0 | 16.9(7) | 4.3 | 6.71 | 32.5 |

Te | 0.1867(43) | 2.3 | 27(2) | 7 | 10.77 | 45.2 |

_{X}is seen to follow the same trend as V

_{at}and V

_{di}. Moreover, V

_{X}values consistently range between 50% and 60% of divalent ions V

_{di}. The trend in V

_{X}and the absolute values can thus be considered to be very reliable.

_{X}values rather than a single radius as a crystal-chemical constant, which are attained by the specific ion in different crystalline solids. The distinction is irrelevant for ions on cubic lattice sites for which the concept of crystal radii has initially been developed. It becomes important, however, when ions on positions with lower site symmetry are considered. In these cases, rather, the crystal volume seems to be conserved, while the ion’s extension in different directions will depend on existing symmetry axes and their relative orientation. The shape of a p valence shell ion has to be modelled by a general ellipsoid for ions on low-symmetry sites. Monovalent chalcogen ions O

^{−}, S

^{−}, Se

^{−}and Te

^{−}in pyrite-type compounds exhibit remarkably consistent volumes when modelled by ellipsoids in accordance with the presented formalism.

**Figure 4.**The volume of chalcogen atoms and ions as a function of atomic number Z (data are given in Table 2). In the plot, V

_{at}stands for the volume of covalent atoms (from [39], MW), V

_{X}for monovalent ions as derived in this work and V

_{di}for divalent ions (from [7,8], SPS). Continuous and dashed lines merely serve as a visual guide.

## 5. Conclusions

_{2}gave two remarkable results. First, re-establishing an additivity rule for interatomic distances d

_{MX}has succeeded. However, d does not simply derive from the addition of two radial parameters r

_{||}and r

_{⊥}, but a decomposition of d in accordance with the lattice site symmetry must be performed. Equation (8) exemplarily shows how this ensues in the case of the pyrite structure. In addition, the ionic volume, as in rock salt and sphalerite structures, re-appeared as a conserved quantity by applying the geometrically correct bond distance decomposition. A possible conservation of ionic volume seems of fundamental importance for crystalline bonding in general and would allow the prediction of crystal structures from a finite set of parameters.

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Birkholz, M. Modeling the Shape of Ions in Pyrite-Type Crystals. *Crystals* **2014**, *4*, 390-403.
https://doi.org/10.3390/cryst4030390

**AMA Style**

Birkholz M. Modeling the Shape of Ions in Pyrite-Type Crystals. *Crystals*. 2014; 4(3):390-403.
https://doi.org/10.3390/cryst4030390

**Chicago/Turabian Style**

Birkholz, Mario. 2014. "Modeling the Shape of Ions in Pyrite-Type Crystals" *Crystals* 4, no. 3: 390-403.
https://doi.org/10.3390/cryst4030390