Transferable Aspherical Atom Modeling of Electron Density in Highly

Apr 4, 2017 - Comparative electron density study of NaNO3 and KNO3 probes the performance of the invariom approach in describing chemical bonding feat...
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Transferable Aspherical Atom Modeling of Electron Density in Highly Symmetric Crystals: A Case Study of Alkali-Metal Nitrates Yulia V. Nelyubina,*,† Alexander A. Korlyukov,†,‡ Konstantin A. Lyssenko,† and Ivan V. Fedyanin† †

A.N. Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, Vavilova Street, 28, Moscow 119991, Russia ‡ Pirogov Russian National Research Medical University, Ostrovitianova Street, 1, Moscow 117997, Russia S Supporting Information *

ABSTRACT: A comparative electron density study (from X-ray diffraction and periodic quantum chemistry) of sodium and potassium nitrates is performed to test the performance of a transferrable aspherical atom model, which is based on the invarioms, to describe chemical bonding features of ions occurring in sites of different symmetry typical of inorganic salts and in different crystal environments. Relying on tabulated entries for the isolated ions (although tailor-made to account for different site symmetries), it takes the same time to employ as the spherical atom model routinely used in Xray diffraction studies but provides an electron density distribution that faithfully reveals all the interionic interactionseven the weakest ones (such as between the nitrate anions or a K···N interaction found in the metastable form of KNO3) yet important for properties of inorganic materialsas if obtained from high-resolution X-ray diffraction data.

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INTRODUCTION Electron density (ED) studies from X-ray diffraction (XRD) have become over the past decade1 a useful tool in modern material and biosciences,2 from explaining properties of inorganic materials3 to deeper understanding functions of proteins4 and even to predicting behavior of liquids.5 To make full use of this experimental technique, high-resolution data are usually required; them being rarely available for many reasons (weak scattering power of a crystal, high thermal motion, disorder, twinning, etc.) prompts an ED researcher to rely on transferable aspherical atom models,6 such as based on invarioms.7 Invarioms are aspherical scattering factors (in contrast to spherical ones routinely used in crystal structure determination8) that are calculated for an atom in a given covalent environment within the multipole formalism9 used in conventional ED studies from high-resolution XRD. Available from the invariom database10 (although other aspherical atom libraries also exist, a theoretical UBDB11 and an experimental ELMAM12) that now stores more than 4000 entries for a large variety of chemical environments typically found in organic compounds13 (including salts14), the invarioms allow obtaining the same15 or even more accurate14b,16 details on chemical bonding peculiarities in a crystal but from XRD data of inferior quality. Collecting a high-resolution data set is usually not a problem with inorganics, but it could still suffer from extinction, strong absorption, or other sample-related issues treatable by the use of invarioms. They, however, did not yet find their way into metal-containing inorganic compounds (some metal complexes were probed, though17), as only the covalent environment © 2017 American Chemical Society

which, for example, atoms in the nitrate anion have but the metal atom lacksis taken into account while constructing an invariom.7 Another concern is that inorganic metal salts are usually highly symmetric, with their atoms occupying special positions posing different restrictions on the components of the invarioms, which thus must be tailor-made14a for each site symmetry. An alternative is to use multipolar parameters obtained from periodic quantum chemistry.18 Refinement of these against available experimental structure factors in the same manner as that of the invarioms makes the resulting ED in a crystal to avoid “contamination” by experimental errors and also to contain information on asphericity of atoms as a response to a particular crystalline environment; however, it also requires to perform such time-consuming calculations for each crystal structure individually. The latter raises the question of whether this approach provides enough advantage over the invarioms, and both of them over the spherical atom model that renders them worthy of additional computational efforts for, for example, alkali-metal salts known to adopt highly symmetric structures; alkali metals chosen as a good approximation to point-charge ions like chloride for which the invariom model did very well.14a To answer it, we obtained an experimental ED distribution by refining multipolar parameters against high-resolution XRD data collected for two nitrate saltsNaNO319 and KNO320 (Figure 1)in their most stable21 low-temperature crystal phases, which have calcite (space group R3̅c, Z′ = 1/6) and Received: February 7, 2017 Published: April 4, 2017 4688

DOI: 10.1021/acs.inorgchem.7b00340 Inorg. Chem. 2017, 56, 4688−4696

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Lorentz-polarization effects using the SADABS package; semiempirical absorption correction from equivalents was applied using SADABS.30 The structures of NaNO3 and KNO3 were solved by the direct method using the SHELXS program for structure solution31 and were then refined by the full-matrix least-squares technique against F2 in anisotropic approximation using the refinement program SHELXL.32 For NaNO3, the refinement converged to wR2 = 0.0522 and goodnessof-fit (GOF) = 1.007 for 624 independent reflections (R1 = 0.0190 was calculated against F for 587 observed reflections with I > 2σ(I)). For KNO3, the refinement converged to wR2 = 0.0588 and GOF = 1.002 for 2460 independent reflections (R1 = 0.0219 was calculated against F for 2011 observed reflections with I > 2σ(I)). Additional crystallographic information is available in the Supporting Information. For KNO3−III (will be introduced later), the standard-resolution XRD data (λ(Mo Kα) = 0.71072 Å, 2θ < 55°) at 123 K are available from ref 21b (see also Table S2 of Supporting Information). Its crystals are trigonal, space group R3m: a = 5.4325(2), c = 8.8255(7) Å, V = 225.56(2) Å3, Z = 3 (Z′ = 1/6), dcalc = 2.233 gcm−3. High-resolution XRD data sets for NaNO3 and KNO3 were modeled using the Hansen−Coppens formalism9 as implemented in the program package XD.33 In the conventional multipole refinements, positional and thermal parameters were first refined against high-angle data (sin θ/λ > 0.7), followed by the refinement of multipoles up to the octupole level for nitrogen and oxygen atoms; the appropriate symmetry restraints were imposed on the metal and nitrate moieties those of the −3 and 32 symmetry in NaNO3 and of the mirror plane in KNO3, respectively. The monopole term was then refined for all the atoms, together with first-order κ values and second-order κ values, all preceded and followed by “coordinates + ADPs and multipoles” refinement cycle. For NaNO3, this refinement converged to R = 0.0148, Rw = 0.0145, GOF = 1.11 for 547 merged reflections with I > 3σ(I). For KNO3, it converged to R = 0.0169, Rw = 0.0130, GOF = 0.99 for 1994 merged reflections with I > 3σ(I). For the multipole refinement of NaNO3 and KNO3 based on the invarioms, the multipolar populations (up to hexadecapolar level) and κ parameters for nitrogen and oxygen atoms of the isolated nitrate anion occupying a site with symmetry 32 (for NaNO3) or a mirror plane (for KNO3) were calculated as reported previously,14a using the computational tools kindly provided by Dr. Habil Birger Dittrich. The metal atoms were treated as spherical isolated species. The monopole populations were adjusted to keep the net charges of the alkali metal and the nitrate ions equal to their formal values of +1 and −1, as implied in the invariom concept.34 The resulting “symmetry-corrected” invarioms were then kept fixed, while the positional and displacement parameters of all the atoms, together with a scale factor, were refined against F for all the experimental data using statistical weights based on 1/σ(Fobs). For NaNO3, this refinement converged to R = 0.0153, Rw = 0.0153, GOF = 1.19 for 547 merged reflections with I > 3σ(I). For KNO3, it converged to R = 0.0195, Rw = 0.0160, GOF = 1.22 for 1994 merged reflections with I > 3σ(I). For KNO3−III, the invarioms that account for the site symmetry (3m) of the nitrate anion were refined against the standard-resolution XRD data (2θ < 55°) available from,21b thereby converging to R = 0.0087, Rw = 0.0093, GOF = 2.24 for 79 merged reflections with I > 3σ(I). In the third type of the multipole refinement, the multipolar populations and κ parameters were obtained from quantum chemical calculations of crystalline NaNO3 and KNO3. For this, wave functions were first calculated in their experimental geometries within the periodic density functional theory (DFT) approach with atomcentered Gaussian-type basis sets as implemented in Crystal14 software package,35 using the combination of B3LYP functional36 with POB-TZVP basis set specially fitted for solid-state calculations.37 Shrinking factors 8 8 8 were used for Monkhorst−Pack grid, yielding in total 65 and 125 k-points in irreducible Brillouin zone for trigonal (NaNO3 and KNO3−III) and orthorhombic (KNO3) structures. The theoretical structure factors were then computed up to the resolution d = 0.356 Å (sin θ/λ up to 1.407 Å−1). Multipole refinement against these structure factors was performed in a static model for NaNO3, KNO3, and KNO3−III, by using the XD program package33 with the

Figure 1. General view of NaNO3 (left) and KNO3 (right) in representation of atoms by thermal ellipsoids (p = 50%), showing the environment of the metal atom and shortest contacts between the nitrate anions (see below).

aragonite (space group Pnma, Z′ = 1/2) structures, respectively. In these salts, the nitrate anion occupies two different special positions with symmetry 32 and m: the twofold symmetry axes that pass along each of the N−O bonds and intersect with a threefold axis at the nitrogen atom in NaNO3 or the mirror plane that cuts through the bond N(1)−O(2) in KNO3. Adapting the invarioms to account for the site symmetries of the nitrate anion while treating the alkali-metal cations as spherical isolated species, as is implied in the original invariom concept,7 and then recovering their “in-crystal” analogues from periodic quantum chemical calculations of the crystals NaNO3 and KNO3 provided us with two types of transferable aspherical atom models of EDsone that uses multipolar parameters for the isolated ions and the other, for a particular salt. Together, they were compared to the experimental ED and to its procrystal model22 (built by non-interacting overlapping spherical atoms) to see how well they can reproduce chemical bonding peculiarities in crystalline NaNO3 and KNO3. Those were analyzed by Bader’s Atoms in Molecules (AIM) theory,23 a very useful toolalthough still subjected to debates and controversies24for identifying all bonding interactions (or, as put more recently, privileged exchange channels25) in a crystal by the presence of bond critical points (bcps) in the ED and quantifying them by, for example, a semiqualitative relation26 between the value of local potential energy v(r) in a bcp and an interaction energy (Eint).27 Among other things,2 such an analysis allowed resolving a contradiction between density and thermodynamic stability (which should be both higher or lower for a polymorph28) of true calcite and aragonite CaCO3.29

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EXPERIMENTAL SECTION AND COMPUTATIONAL DETAILS

Crystals of NaNO3 (see Table S1 of Supporting Information) were trigonal, space group R3̅c: a = 5.06110(10), c = 16.5351(3) Å, V = 366.798(16) Å3, Z = 6 (Z′ = 1/6), dcalc = 2.309 gcm−3. Crystals of KNO3 (see Table S1 of Supporting Information) were orthorhombic, space group Pnma: a = 6.2732(2), b = 5.39100(10), c = 9.1288(2) Å, V = 308.725(13) Å3, Z = 4 (Z′ = 1/2), dcalc = 2.175 gcm−3. Highresolution XRD data were collected at 120 K with a Bruker APEX2 DUO diffractometer [λ(Mo Kα) = 0.71072 Å, ω-scans, 2θ < 120°] in three batches, a low-angle (2θ = −32°), a middle-angle (2θ = −62°), and a high-angle (2θ = −92°) batch, in an ω-scan mode (Δω = 0.5°) with a detector to a sample distance of 4.1 cm at exposure times of 1 s for low-angle reflections, 3 s for middle-angle reflections, and 10 s for high-angle reflections, respectively. Raw data were integrated by using the program SAINT and then scaled, merged, and corrected for 4689

DOI: 10.1021/acs.inorgchem.7b00340 Inorg. Chem. 2017, 56, 4688−4696

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Inorganic Chemistry same level of the multipole expansion and all the appropriate symmetry restraints, as in the refinement against XRD data above. The resulting theoretical multipolar and κ parameters were then kept fixed, while the positional and displacement parameters of all the atoms, together with a scale factor, were adjusted against F for all the experimental data, as in the invariom refinement. For NaNO3, this refinement converged to R = 0.0160, Rw = 0.0159, GOF = 1.23 for 547 merged reflections with I > 3σ(I). For KNO3, it converged to R = 0.0182, Rw = 0.0144, GOF = 1.10 for 1994 merged reflections with I > 3σ(I). For KNO3−III, the refinement with fixed multipolar and κ parameters from periodic quantum chemistry against the standardresolution XRD data (2θ < 55°) available from ref 21b converged to R = 0.0093, Rw = 0.0100, GOF = 2.41 for 79 merged reflections with I > 3σ(I). In each of the three refinements for NaNO3 and KNO3, the residual electron density maps were flat, with the largest and lowest values below 0.19 and −0.22 e Å−3 (both observed for KNO3). In all cases, the Hirshfeld test38 yielded a highest difference in mean square displacement amplitudes along covalent bonds of 2 × 10−4 Å2, indicating a proper deconvolution of ED from the thermal motion. For KNO3−III, the largest and lowest residuals were below 0.12 and −0.15 e Å−3, and the displacement amplitude along the covalent bond was 6 × 10−4 Å2 only. Topological analysis of the resulting ED functions ρ(r) for NaNO3 and both KNO3 was performed using the WINXPRO program package;39 for the theoretical EDs not projected onto multipoles, the topological code (TOPOND) as incorporated in Crystal1435 was used. In all cases, the interaction energies were estimated by means of a semiquantitative relation between the energy of an interaction and the value of the potential energy density v(r) in its bcp,26 which was repeatedly shown to give accurate estimates in many cases.27 Potential energy density v(r) was evaluated through the Kirzhnits’s approximation40 for kinetic energy density function g(r). Accordingly, the g(r) function is described as (3/10)(3π2)2/3[ρ(r)]5/3 + (1/72)|∇ρ(r)|2/ρ(r) + 1/6∇2ρ(r), giving in conjunction with the virial theorem (2g(r) + ν(r) = 1/4∇2ρ(r))23 the expression for v(r). To calculate the energy of the two phases of KNO3, the same theoretical background was used as for the theoretical structure factors. The values resulting from the two types of calculationswith the atomic positions optimized and with unit cell parameters fixed at their experimental values, and with all these parameters freely optimized (upon which the unit cell volume shrinks by 2.5 and 5.1% for KNO3− II and KNO3−III, respectively)differ by 0.3 kcal/mol.

Figure 2. Deformation EDs for the nitrate anion in NaNO3 (top) and KNO3 (bottom) as obtained by the conventional multipole refinement against high-resolution XRD data (A), with the use of invarioms (B), and of multipolar parameters from periodic quantum chemistry (C). Contours are drawn through 0.1 e Å−3; the negative contours are dashed.

is, however, twice as low and may be thus attributed to their “natural spread” (0.1 e Å−3);41 the difference between the two salts does not go beyond 0.11 e Å−3. The Laplacian values vary within 9 e Å−5, which significantly exceeds the reported “transferability index” (3−4 e Å−5),41 and they nearly change their sign from negative to positive; although the latter is typical for polar bonds,42 here it is observed only for the EDs based on the multipolar parameters from periodic quantum chemistry. When compared between the two salts, the largest difference in the Laplacian of the N−O bonds is lower (6 e Å−5) and occurs in the EDs built by the invarioms. As they ignoreby design the effect of the alkali-metal counterions, this modest variation can only stem from the invariom approximation having some issues with ions of identical geometry but located at different symmetry elements. Note that in the case of the pro-crystal (spherical atom) model of ED distribution in NaNO3 and KNO3, the ED values in the bcps for the N−O bonds decrease significantly (e.g., by 0.4 e Å−3 from those obtained by the conventional multipole refinement), and their Laplacian becomes positive (up to 10.1 e Å−5). With both these parameters nearly equal in the two salts (within 0.02 e Å−3 and 0.6 e Å−5, respectively), such a description leaves out of account the influence of different crystal environment on the nitrate anion in NaNO3 and KNO3. Although the latter is also inherent in the invariom approximation of the nitrate’s ED, the aspherical nature thereof gives a more realistic ED distribution that is second only to those obtained from the high-resolution XRD data (Table 1). Indeed, on the one hand, the invariom approach and the spherical atom model, respectively, treat the ions in the two salts as having charges equal to their formal values (+1 for alkali-metal cations and −1 for the nitrate anion) and as neutral entities, either way ignoring how these charges are influenced by the crystal packing effects in the chosen phases of NaNO3 and KNO3. On the other hand, for an alkali-metal salt they should not be very different (although for some of the multivalent anions a reduction by 30% was reported43) from those for the isolated ions due to a strongly ionic character of bonding, as supported here by the charges estimated (by

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RESULTS AND DISCUSSION For NaNO3 and KNO3, the EDs obtained either by the conventional multipole refinement of high-resolution XRD data or by using theoretical multipolar parameters feature very similar distributions in the nitrate anion with different site symmetries and different alkali-metal cations (Figure 2): they all have an approximate D3h symmetry with the expected ED concentrations along the covalent bonds and near each oxygen atom where its two lone pairs should be located; all agreeing with the ED delocalization over the nitrate anion. Although the degree to which the ED is accumulated/depleted in these areas is different between the three models, the relevant ED maps for the two salts are nearly the same pairwise (for the ED variation along the N−O bonds in the different ED models of NaNO3 and KNO3, see Figures S1 and S2 in Supporting Information). For the six versions of the nitrate anion, a topological analysis within the AIM approach (see Table 1 below) provides identical sets of bcps, which are located almost in the midpoint of the N−O bonds. The largest difference in the corresponding ED values (0.32 e Å−3) is found between the two models of NaNO3 that use theoretical multipolar parameters. Their deviation from the experimental ED distribution for each salt 4690

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Table 1. Features of Interatomic Interactions in NaNO3 and KNO3 As Revealed by AIM Analysis of EDsa Obtained from the Conventional Multipole Refinement against High-Resolution XRD Data (first entry) and with the Use of Invarioms (second entry) and Multipolar Parameters from Periodic Quantum Chemistry (third entry). The Values from the Spherical Atom Model Are Given in Brackets interaction R,c Å N(1)−O(1)

nb 3

1.2495(2) Na(1)−O(1)

6

2.38214(12) O(1)···O(1′)

2

3.1190(2)

N(1)−O(1)

2

1.2540(4) N(1)−O(2)

1

1.2530(6) K(1)−O(1)

2

2.8574(3) K(1)−O(1A)

2

2.8071(3) K(1)−O(2B)

1

2.8033(5) K(1)−O(1C)

2

2.8463(3) K(1)−O(2C)

2

2.88717(19) O(1)···O(1A)

1

3.2214(6) N(1)···N(1B) 3.1382(5)

2

d1/d2,c Å

ρ(r), e Å−3

0.631/0.618 0.598/0.650 0.614/0.634 [0.579/0.669] 1.089/1.293 1.091/1.292 1.088/1.295 [1.095/1.288] 1.560/1.559 1.560/1.560 1.560/1.560 [1.560/1.559] 0.629/0.624 0.598/0.653 0.632/0.619 [0.583/0.670] 0.631/0.622 0.585/0.556 0.618/0.631] [0.583/0.670] 1.473/1.384 1.465/1.393 1.471/1.387 [1.476/1.383] 1.461/1.346 1.462/1.347 1.454/1.355 [1.458/1.348] 1.460/1.344 1.452/1.353 1.452/1.353 [1.457/1.347] 1.468/1.381 1.459/1.388 1.466/1.381 [1.473/1.378] 1.484/1.405 1.472/1.416 1.485/1.403 1.491/1.402] 1.610/1.611 1.611/1.611 1.611/1.611 [1.611/1.610] 1.592/1.546 1.567/1.571 1.568/1.570 [1.568/1.570]

NaNO3 3.02 3.17 2.84 [2.75] 0.12 0.09 0.10 [0.11] 0.05 0.03 0.04 [0.04] KNO3 3.11 3.08 2.83 [2.73] 3.10 3.05 2.84 [2.74] 0.08 0.07 0.07 [0.08] 0.08 0.07 0.08 [0.08] 0.08 0.08 0.08 [0.08] 0.08 0.07 0.08 [0.08] 0.07 0.07 0.07 [0.07] 0.03 0.02 0.03 [0.04] 0.04 0.05 0.04 [0.05]

∇2ρ(r), e Å−5

−v(r), a.u.

Eint, kcal/mol

−2.97 −9.56 −0.83 [9.50] 2.41 2.52 2.54 [2.35] 0.56 0.58 0.58 [0.56]

1.4943 1.5930 1.3576 [1.3222] 0.0148 0.0130 0.0139 [0.0137] 0.0035 0.0028 0.0030 [0.0032]

4.6 4.1 4.4 [4.3] 1.1 0.9 1.0 [1.0]

−4.92 −6.86 −0.17 [10.11] −3.67 −3.92 −0.33 [10.05] 1.36 1.46 1.39 [1.33] 1.42 1.44 1.48 [1.43] 1.43 1.51 1.50 [1.43] 1.41 1.51 1.43 [1.34] 1.29 1.40 1.29 [1.23] 0.39 0.41 0.40 [0.37] 0.56 0.64 0.55 [0.68]

1.5637 1.5297 1.3500 [1.3063] 1.5590 1.5166 1.3600 [1.3106] 0.0079 0.0080 0.0079 [0.0081] 0.0084 0.0080 0.0087 [0.0088] 0.0085 0.0085 0.0088 [0.0088] 0.0082 0.0083 0.0082 [0.0082] 0.0075 0.0076 0.0072 [0.0074] 0.0021 0.0018 0.0020 [0.0022] 0.0032 0.0036 0.0029 [0.0041]

2.5 2.5 2.5 [2.5] 2.6 2.5 2.7 [2.8] 2.7 2.7 2.8 [2.8] 2.6 2.6 2.6 [2.6] 2.4 2.4 2.3 [2.3] 0.7 0.6 0.6 [0.7] 1.0 1.1 0.9 [1.3]

a

In all cases, the charge leakage that results from numerical integration over atomic basins is below 0.002 e. The sum of atomic volumes in NaNO3 and KNO3 (60.92−61.12 Å3 and 77.10−77.29 Å3) reproduces well the volume of an independent part of the unit cell (61.13 and 77.18 Å3) with a relative error less than 0.3%. bNumber of all the interactions of this type formed by the alkali metal ion (for M−O bonds) or by the nitrate anion (for anion−anion interactions). cR stands for the interatomic distance from X-ray diffraction data; d1 and d2 are the distances from the located bcp to the interacting atoms (identified as critical points (3, −3) or maxima in the ED).

Depending on whether it came from the multipole refinement of the XRD data or of the theoretical structure factors, the ionic

integrating over atomic basins defined according to the AIM formalism23) from the other two types of an aspherical ED. 4691

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them, the Na−O bonds (of ∼4.4 kcal/mol), vary the most between the ED models: the largest difference (0.5 kcal/mol) is observed between the invariom-based ED and the one obtained from high-resolution XRD data. The weaker K−O bonds (2.3− 2.8 kcal/mol) show a better agreement (within 0.3 kcal/mol), which however gives a very similar relative error (of ca. 12%) as for the Na−O bonds. Note that all these estimates of the M−O bond energy fit well (R2 > 0.95) into a power-law dependence on the bond length from similar ED studies of a mixed sodium copper carbonate (for Na−O)47 and of a potassium fumarate (for K−O);48 another power-law dependence is observed when the energy and interatomic distance for the O···O interactions between the nitrate anions are put together with those for some organic nitrates49 (R2 > 0.91). On average, the interaction energies from the experimental ED are nicely reproduced by the periodic quantum chemistry (by 0.10 kcal/mol), the second to the experimental ED is the invariom-based ED (by 0.11 kcal/mol), and the spherical atom model is predictably the last one (by 0.12 kcal/mol), but it also matches these interaction energy values very well. As has been previously shown for CaCO3,29 the calcite structure (NaNO3) turns out to have a higher total energy of the M−O bonds; although less numerous (only six per one alkali-metal cation), they sum to 24.6−27.8 kcal/mol depending on the ED model used. Those in aragonite-like KNO3 (nine per one alkali-metal cation) give 22.7−23.1 kcal/mol in total. However, the energy of all the anion−anion interactions formed by one nitrate anion in a crystal is also higher in NaNO3 (5.3−6.5 vs 3.1−4.0 kcal/mol), so that the same order of densities as in the two phases of CaCO3calcite being less dense than aragonite29does not hold here (2.309 vs 2.175 gcm−3), which is no wonder, as here they belong to different compounds. However, KNO3 has another ordered phase at the atmospheric pressure,50 the trigonal phase III (a ferroelectric21b with space group R3m), for which single-crystal XRD data at a similar temperature (123 K) are available.21b KNO3−III is metastable and quickly transforms to the phase II, yet these phases can coexist at the same temperature range21b and simultaneously cocrystallize from a solution at some concentration.50 Its crystal structure also features the potassium cation bound to six nitrate anions (see Figures 1 and 3) by nine K−O bonds that are, in average, slightly shorter (2.813 vs 2.844 Å), agreeing with its slightly higher density (2.233 g cm−3 at 123 K). In KNO3−III, the nitrate anions are not stacked on top of each other, as in the above phase II, but alternate with the potassium cations; them being slightly off-center of this axis gives rise to its ferroelectric behavior.21b

charges are ±0.72 or ±0.87 e in NaNO3 and ±0.81 or ±0.85 e in KNO3 (see Table S3 of Supporting Information). These are, of course, the result of interionic interactions, as is the difference between them and from the other two ED models, since the interactions between the ions are taken into account neither by the invariom approximation nor by the spherical atom model. Topological analysis performed to identify all such interactions in NaNO3 and KNO3 (Figure 1) has revealed the same set of bond critical points (bcps), which provide a concise summary of the bonding within a crystal,23 in every ED distribution discussed above, whether was it obtained by the conventional refinement against high-resolution XRD data, with the use of the invarioms or the multipolar parameters from periodic quantum chemistry, or produced by a pro-crystal, spherical atom model (Table 1). In all cases, the bcps appear not only for the expected Na−O and K−O bonds (six and nine in total, respectively) but also for a few short contacts between the nitrate anions (exactly mirroring their pattern in true calcite and resembling it in aragonite29). Although these contacts involve two like-charged species, they are nevertheless stabilizing, as confirmed by, for example, similar interactions found responsible for the higher density of the less stable phase of CaCO3.29 Unlike the latter, however, KNO3 lacks some of the O···O interactions that occur between the carbonate anions in the aragonite phase;29 those have the bcps and the bond paths that go straight between the two metal ions.29 Their absence in KNO3 may be attributed to the potassium cation that is too large (21.2 vs 13.3 Å3 for calcium ion in CaCO3; Table S3) compared to the anion (56.1 vs 42.8 Å3 for carbonate ion in CaCO3) for these interactions to occur; it is mostly the different ratio of the ionic volumes that places the two salts into different topological families as in alkali halides,44 rather than just an increase in the shortest O···O distance24b in KNO3 (Table 1) by ∼0.5 Å. Other anion−anion interactions remaing in NaNO3 and KNO3 are the ones that pack the carbonate anions in CaCO3 into stacks, wherein they rotate by 30° upon the transformation of its aragonite to calcite phase.45 ED parameters in the bcps pertaining to all the interionic interactions found in NaNO3 and KNO3 show the anticipated difference of their features in the two salts but also their amazing similarity in the four types of the ED distributions (Table 1). They all have low ρ(r) and low positive ∇2ρ(r) values, which vary in narrow ranges of, respectively, 0.09−0.12 e Å−3 and 2.35−2.54 e Å−5 for the Na−O bonds, 0.07−0.08 e Å−3 and 1.23−1.51 e Å−5 for the K−O bonds, and 0.02−0.05 e Å−3 and 0.37−0.68 e Å−5 for the anion−anion interactions. Together with positive electron energy density (he(r)) at a bcp,46 those are typical for closed-shell interactions such as ionic bonds and van der Waals contacts. The variation in the above ρ(r) and ∇2ρ(r) values by up to 0.03 e Å−3 and 0.19 e Å−5 arising from the use of four different ED models is very much below the corresponding “transferability indexes” (0.1 e Å−3 and 3−4 e Å−5).41 Of the three (a)spherical approximations to the ED distribution in NaNO3 and KNO3 (second to fourth entries in Table 1), the closest to the experimental one is, on an average, from the periodic quantum chemistry, although the others also perform surprisingly well. Energy of these interactions estimated by the Espinosa’s relation based on the potential energy density v(r) at their bcps26 found in the different EDs is 2.3−4.6 kcal/mol for the M−O bonds and below 1.3 kcal/mol for the interactions between the anions in NaNO3 and KNO3. The strongest of

Figure 3. General view of KNO3−III with selected interionic interactions (left) and deformation EDs for the nitrate anion as obtained from the standard-resolution XRD data with the use of the invarioms (center) and of the multipolar parameters from periodic quantum chemistry (right). 4692

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Table 2. Features of Interatomic Interactions in KNO3−III As Revealed by AIM Analysis of EDsa Obtained with the Use of Invarioms (first entry) and Multipolar Parameters from Periodic Quantum Chemistry (second entry). The Values from the Spherical Atom Model of ED Are Given in Brackets. The Fourth Entry Contains Data from Quantum Chemical Calculation without the Use of the Multipole Model interaction R,c Å N(1)−O(1)

nb 3

1.253(2) K(1)−O(1)

6

2.780(2) K(1)−O(1A)

3

2.8778(7) O(1)···O(1′)

2

3.268(3) O(1)···O(1”)

4

3.379(3) K(1)···N(1) 3.846(2)

1

d1/d2,c Å 0.660/0.590 0.611/0.636 [0.584/0.670] 0.605/0.649 1.435/1.346 1.446/1.344 [1.446/1.334] 1.351/1.432 1.459/1.419 1.490/1.388 [1.487/1.395] 1.431/1.447 1.642/1.642 1.633/1.634 [1.634/1.633] 1.661/1.661 1.708/1.674 1.703/1.681 [-] 1.695/1.702 1.947/1.893 1.879/1.965 [1.902/1.944] 1.874/1.972

ρ(r), e Å−3 KNO3−III 3.12 3.00 [2.75] 2.96 0.08 0.08 [0.09] 0.08 0.08 0.06 [0.08] 0.08 0.03 0.02 [0.03] 0.03 0.02 0.02 [-] 0.03 0.01 0.01 [0.02] 0.01

∇2ρ(r), e Å−5

−v(r), a.u.

Eint, kcal/mol

-8.05 -2.9 [10.34] -15.91 1.60 1.57 [1.53] 1.55 1.48 1.28 [1.25] 1.38 0.37 0.36 [0.34] 0.36 0.36 0.30 [-] 0.34 0.12 0.15 [0.13] 0.13

1.6086 1.4772 [1.3228] 0.7973 0.0093 0.0090 [0.0095] 0.0090 0.0083 0.0067 [0.0076] 0.0082 0.0018 0.0016 [0.0020] 0.0025 0.0017 0.0013 [-] 0.0021 0.0005 0.0006 [0.0009] 0.0005

2.9 2.8 [3.0] 2.8 2.6 2.1 [2.4] 2.6 0.6 0.5 [0.6] 0.8 0.5 0.4 [-] 0.6 0.2 0.2 [0.3] 0.2

a

In all cases, the charge leakage that results from numerical integration over atomic basins is below 0.001 e. The sum of atomic volumes (75.21− 75.26 Å3) reproduces well the volume of an independent part of the unit cell (77.19 Å3) with a relative error less than 0.1%. bNumber of all the interactions of this type formed by the potassium cation (for K−O bonds) or by the nitrate anion (for anion−anion interactions). cR stands for the interatomic distance from X-ray diffraction data; d1 and d2 are the distances from the located bcp to the interacting atoms (identified as critical points (3, −3) or maxima in the ED).

As the XRD data collected for KNO3−III are of a standard resolution, it thus provides an ultimate test for all the above approximations to the ED distribution (spherical and two aspherical ones with multipolar parameters taken from the invarioms or from the quantum chemical calculation of its crystal), with the ED straight from the periodic quantum chemistrythat is, not projected onto multipolesas a reference (instead of the experimental ED distribution from high-resolution XRD data owing to their unavailability). For NaNO3 and KNO3, for example, the latter provides chemical bonding features (Table S4 of Supporting Information) that are consistent with those from other aspherical EDs (Table 1) and are quantitatively close to the experimental ones: an average deviation in the ρ(r) and ∇2ρ(r) values is below 0.03 e Å−3 and 4.6 e Å−5, respectively. In KNO3−III, the nitrate anion occupies the site with symmetry 3m and is nonplanar,21b the latter representing an additional challenge for the invariom approximation that treats it as an isolated moiety, that is, as perfectly planar. Despite this, the resulting ED distribution for it (Figure 3) nicely resembles those in NaNO3 and KNO3−II (Figure 2). Of the three models of the ED in KNO3−III, the invariom-based one provides the closest match to the periodic quantum chemistry (Table 2), even though the charges of the ions (0.85 e) are better reproduced with the theoretical multipolar parameters (0.95 e; Table S5 of Supporting Information). The two of them have

similar ED values at the N−O bond’s bcp (Table 2), which are higher by ∼0.4 e Å−3 than in the spherical (pro-crystal) model (for the ED variation along the N−O bond in the different ED models of KNO 3 −III, see Figure 3S in Supporting Information), and the Laplacian that becomes positive when the non-interacting spherical atoms are used instead of the invarioms. If compared with the above salts, the ED and Laplacian values vary by no more than 0.16 e Å−3 and 4 e Å−5, which are not very far from the above “transferability indexes” (0.1 e Å−3 and 3−4 e Å−5)41 and keep with the change in the N−O bond length from 1.2495(2) (in NaNO3) to 1.2530(6) Å (in both phases of KNO3). The AIM search23 for the interionic interactions in KNO3− III reveals the identical sets of bcps (Table 2), except for the pro-crystal ED, which does not reproduce the longer O···O contact found in all other cases. The bonds K−O, for which the different ED models provide more consistent results, have similar ρ(r) and ∇2ρ(r) values (0.06−0.08 e Å−3 and 1.25−1.60 e Å−5, respectively) as in the phase II. The same is also true for their energies estimated by the Espinosa’s relation26 to be 2.1− 3.0 kcal/mol in an agreement with their length. For instance, put together with the appropriate values for the phase II (see Table 1), they nicely fit into the above power-law dependence (R2 > 0.95). Depending on the model used, the total energy of the K−O bonds in KNO3−III (20.8−24.3 kcal/mol) is nearly the same (in the pro-crystal and the invariom-based ED) or 4693

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reveals all the interionic interactions in their crystals, as if obtained from the conventional multipole refinement of highresolution XRD data but with no actual need for the latter. As inorganic crystalline materials, including rather simple alkalimetal nitrates, have a complex bonding pattern where even the weakest interaction (below 0.3 kcal/mol) matters,29,47 the invariom approach can help to better understand their properties, thus contributing to the successful design of new ferroelectric, piezoelectric, conductive, luminescent, and other important materials.

lower (in the ED from theoretical multipolar parameters) than in the phase II; the smallest deviation of 0.4 kcal/mol from the value obtained in the reference quantum chemical calculation (23.9 kcal/mol) is attained by the invariom approximation. The higher density of KNO3−III, therefore, mostly owes to the anion−anion interactions that are weaker (ρ(r) and ∇2ρ(r) of 0.02−0.03 e Å−3 and 0.30−0.37 e Å−5, Eint below 0.5 kcal/mol and fits well into the above dependence for the O···O interactions) than in II yet more numerous. Excluding the pro-crystal model with its lack of the four weakest O···O interactions formed by each oxygen atom (Table 2), their energy per one nitrate anion totals to 6.3−9.9 kcal/mol, which is thrice as high as in the phase II above. Together with the energy of all the K−O bonds formed by the cation in KNO3− III, it gives 29.0−34.2 kcal/mol. Of the three approximations to the ED, the aspherical oneswith multipolar parameters taken either from the invarioms or from the quantum chemical calculationthus produce the total energy of interionic interactions for a formula unit that is larger in KNO3−III. Under the spherical atom model, the same value appears to be slightly higher in the phase II (27.1 vs 27.0 kcal/mol), in a good agreement with it being more thermodynamically stable and the two phases being close in energy51 (according to our quantum chemical calculations, by 0.3 kcal/mol). This, however, is primarily a result of the weaker anion−anion interactions missing in the pro-crystal of KNO3−III, even though they are faithfully reproduced by the invariom approximation based on the XRD data of a standard resolution, as expected of weak interatomic interactions.15d What was unexpected when comparing the different ED models of KNO3−III is that an even weaker K···N interaction (below 0.3 kcal/mol) kept emerging in all of them. It connects the slightly pyramidalized nitrogen atom21b to the nearest metal cation along the line K···N···K (K···N 3.846(2) and 4.980(2) Å) and may thereby mediate the ferroelectric behavior of KNO3−III, which owes to the nitrate anion being slightly offcenter of this axis. This extremely weak interaction, pinpointed by a bcp and an associated bond path23 as a privileged exchange channel,25 helps keeping the nitrate anion in the position closer to one of the cations but yet allows it to jump to the other cation upon the reversal of the electric field.21b

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.7b00340. Supplementary tables and outputs of multipole refinement for NaNO3, KNO3, and KNO3−III (PDF) Crystallographic information (CIF) Crystallographic information (CIF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Yulia V. Nelyubina: 0000-0002-9121-0040 Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Funding

This study was financially supported by the Russian Foundation for Basic Research (Project No. 16−03−00691) and by Foundation of the President of the Russian Federation (Project No. MK-6224.2016.3). Notes

The authors declare no competing financial interest. Further details of crystal structure investigation may be obtained from FIZ Karlsruhe, 76344 Eggenstein-Leopoldshafen, Germany (fax: (+49)7247−808−666; e-mail: crysdata@fizkarlsruhe.de), on quoting the deposition numbers CSD 432570 and 432571.

■

CONCLUSION By comparing the ED distributions from XRD and periodic quantum chemistry for two alkali-metal nitrates, we tested the performance of the approach that uses multipolar parameters for an isolated ion (the invarioms) to describe its chemical bonding features when it occurs in sites of different symmetry and in different crystal environments. The latter are not accounted for in the invarioms, as they are in theoretical multipolar parameters derived from structure factors calculated for a given crystalline phase, but their features are reproduced equally well. Yet the invariom approach enjoys the benefits from the use of the tabulated entries for any ion or its fragments, which can be calculated once and for all. For inorganic compounds that tend to form highly symmetric crystals, however, these invarioms must be tailor-made to account for different site symmetries (but only if the symmetry element coincides with an atom); on the bright side, some of them occur more frequently than the others.52 Although for alkali-metal salts, the spherical atom model performs rather well too, the invariom approach takes the same time to employ but provides an ED distribution that faithfully

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REFERENCES

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DOI: 10.1021/acs.inorgchem.7b00340 Inorg. Chem. 2017, 56, 4688−4696

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DOI: 10.1021/acs.inorgchem.7b00340 Inorg. Chem. 2017, 56, 4688−4696