On the crystal chemistry of inorganic nitrides: crystal-chemical parameters, bonding behavior, and opportunities in the exploration of their compositional space

The scarcity of nitrogen in Earth's crust, combined with challenging synthesis, have made inorganic nitrides a relatively unexplored class of compounds compared to their naturally abundant oxide counterparts. To facilitate exploration of their compositional space via a priori modeling, and to help a posteriori structure verification not limited to inferring the oxidation state of redox-active cations, we derive a suite of bond-valence parameters and Lewis acid strength values for 76 cations observed bonding to N3−, and further outline a baseline statistical knowledge of bond lengths for these compounds. Examination of structural and electronic effects responsible for the functional properties and anomalous bonding behavior of inorganic nitrides shows that many mechanisms of bond-length variation ubiquitous to oxide and oxysalt compounds (e.g., lone-pair stereoactivity, the Jahn–Teller and pseudo Jahn–Teller effects) are similarly pervasive in inorganic nitrides, and are occasionally observed to result in greater distortion magnitude than their oxide counterparts. We identify promising functional units for exploring uncharted chemical spaces of inorganic nitrides, e.g. multiple-bond metal centers with promise regarding the development of a post-Haber–Bosch process proceeding at milder reaction conditions, and promote an atomistic understanding of chemical bonding in nitrides relevant to such pursuits as the development of a model of ion substitution in solids, a problem of great relevance to semiconductor doping whose solution would fast-track the development of compound solar cells, battery materials, electronics, and more.


Bond-valence parameters for mixed-anion polyhedra
Bond-valence parameters were refined with the GRG method for Ba 2+ , Cr 3+ and Cu 2+ with and without their mixedanion (N 3and O 2-) coordination polyhedra to test whether use of mixed-anion coordination polyhedra leads to the same bond-valence parameters as those derived with only one type of anion. For Ba 2+ , R o = 2.432 Å and B = 0.405 Å, n = 53, including 9 coordination polyhedra with one or more oxygen anion. Removing those 9 coordination polyhedra, R o and B refine to 2.433 and 0.406 Å, respectively. For Cr 3+ , R o = 1.796 Å and B = 0.403 Å, n = 26, including 14 coordination polyhedra with one or more oxygen anion. Their removal from the optimization gives R o = 1.796 Å and B = 0.406 Å. For Cu 2+ , R o = 1.577 Å and B = 0.515 Å, n = 17, including 13 coordination polyhedra with one or more oxygen anion. Their removal gives R o = 1.585 Å and B = 0.490 Å. These results show that using mixed-anion coordination polyhedra for refining new bond-valence parameters is justified, provided that the bond-valence parameters used to convert the concerned bond lengths into constants that do not enter the refinement are known to be of high quality. Thus, mixed-anion coordination polyhedra were used in deriving the new set of cation-N 3bond-valence parameters below (Table 1).

Bond-valence parameters for cations observed in multiple coordination numbers
To refine both bond-valence parameters R o and B for an ion pair (see equation 1), such ion pair must be observed in at least two different coordination environments, either in terms of cation coordination number, or varying mixedanion ratio. Where this is not observed, either R o or B must be fixed, and the other parameter refined. In our dataset, 45 cations are observed in more than one coordination number. Of those, 37 have enough data (and data of high enough quality) for confident derivation of their bondvalence parameters, and an additional 3 (Gd 3+ , P 5+ and S 6+ ) have varying mixed-anion ratios that allow refinement of both R o and B despite occurring in only one coordination number. Together, they account for 1168 of the 1436 coordination polyhedra in our final dataset. Bond-valence parameters for these 40 cations were derived optimizing a 2:1 ratio between equations (3) and (4), as described above, and are given in Table 1.
We then plotted the ratio of bond-valence parameter R o and the observed mean bond-length for the cations 〈R ij 〉 CN (weighting each coordination number equally, using only coordination polyhedra where the cation is bonded solely to N 3-) to the nth ionization energy of the cation, a relation first identified by Gagné & Hawthorne. 2 This is shown in Fig. S1, with best-fit equation .
(eq. S1) 〈 〉 = 7.28 10 -3 + 0.526 Excellent agreement is observed with R 2 = 0.86, which increases to R 2 = 0.92 when removing data points for Ag + and Li + (Gagné & Hawthorne found R 2 = 0.75 for O 2-, for 90 cations 2 ). Thus, the validity of using this relation to extrapolate values of R o for cations observed in only one coordination number is confirmed. Figure S1: Relation between the ratio of bond-valence parameter R o and the coordination-based mean bond-length for the cation as a function of ionization energy (kj mol -1 ).
In some cases, deviation between the value of R o predicted by eq.5 and that obtained from the GRG refinement results from the shallowness of the global minimum of the RMSD, and is an artifact of small sample size. For example, although Cs + (n = 24) has the largest deviation between refined (1.979 Å) and predicted (2.315 Å) R o , the associated RMSD values are 0.074 v.u and 0.098 v.u., respectively, compared to an ideal bond-valence sum of 1 v.u. for Cs + . Most values of R o in the range quoted above could be suitable provided that the B parameter is properly refined.
1 Electronic Supplementary Material (ESI) for Chemical Science. This journal is © The Royal Society of Chemistry 2021

Bond-valence parameters for cations observed in one coordination number
Bond-valence parameters for the 36 ion pairs observed in only one coordination number, or for which not enough data are available to refine both bond-valence parameters with confidence, were derived in the following way: (1) Fix R o to the value predicted by equation (S1). Let B refine via the GRG method. If the value for B falls within a range similar to that of ions with similar crystal-chemical behavior, accept the bond-valence parameters. Otherwise, move on to (2); (2) Fix B to the family average (e.g. 0.399 Å for the transition metals) or to a value compatible with ions of similar crystal-chemical behavior. Let R o refine and see if it falls within a reasonable range (typically within a 6.0% difference, the average deviation between observed and predicted values of R o for the 40 cations for which both R o and B were refined). If not, move on to (3); (3) Fix B to its mean value for all families combined (0.422 Å) and let R o refine. This is typically done where there are insufficient data available to make a reasonable estimate of B (e.g. for actinides U 4+ and U 6+ ).
As discussed in Gagné & Hawthorne, 2 fixing the value of R o is less forgiving than fixing that of B and should be done with caution. For the 36 cations considered here, we fixed R o for 4 and B for 32 ion pairs (the method of derivation is identified in Table 1).

Anion sum verification
For simplicity, bond-valence parameters are usually derived by optimizing the valence-sum rule for cations (the work of Krivovichev & Brown 3 being a notable exception). However, we emphasize that the valence-sum rule applies equally to cations and anions, and that good agreement for cation bond-valence sums in no way implies good agreement for anion bond-valence sums (and vice versa). It is imperative that new bond-valence parameters be checked against both these quantities, on a large set of crystal structures, before they should be widely accepted. Of the 70+ publications that have given bond-valence parameters since the inception of the model (https://www.iucr.org/__data/assets/file/0007/126574/ bvparm2016.cif), such verification has only been done by Krivovichev & Brown for the Pb 2+ -O 2ion pair, 3 by Gagné & Hawthorne for their comprehensive set of bond-valence parameters for cations bonded to O 2-, 2 and by Sidey & Shteyfan for the P 5+ -S 2ion pair. 4 We assembled a set of structures with the goal of evaluating as many bond-valence parameters as possible from Table 1 (see Table S1), i.e., unless no structure could be evaluated using solely the bond-valence parameters derived in this work. The RMSD for the anion bond-valence sums (BVS) over the resulting set of 52 crystal structures, covering 135 anion coordination polyhedra, is 0.209 v.u.
The simple mean deviation is 0.172 v.u., or 5.7% compared to an ideal BVS of 3 v.u. For comparison, the mean cation RMSD for ion pairs with 10 or more coordination polyhedra is 0.122 v.u. (0.120 v.u. over all ion pairs weighted by their number of coordination polyhedra). This is very similar to the value reported by Gagné & Hawthorne for O 2-, 2 of 0.126 v.u. for n ≥ 10, (0.124 for all data weighted by their number of coordination polyhedra). These authors also reported a RMSD of 0.104 v.u. over 511 anion coordination polyhedra of O 2-(5.2% compared to an ideal BVS of 2 v.u.).
These numbers allow us to make two observations. First, the RMSD of the anion BVS is slightly larger than that of cations. This result is somewhat expected; aside from not being the subject of the optimization, few structures could be evaluated for anion BVS, and those that could were not necessarily ideally suited to the task (e.g. 0.3 < R 1 < 0.06). Second, we observe slightly higher RMSD values for both cation and anion BVS for structures of N 3vs O 2-. This is likely a consequence of sampling a wide range of metastable nitride structures, 5 as opposed to thermodynamically-stable minerals which make up a greater fraction of the oxide and oxysalt data of Gagné & Hawthorne. 2 Meta-stable structures generally entail lessthan-perfect mapping of bond-length constraints in threedimensional space (which can be calculated a priori 6 ) under the constraints of symmetry and periodicity, leading to higher variations in mean bond lengths across structure types in comparison to thermodynamically-stable structures. 7