The floppiness of it all: bond lengths change with atomic displacement

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The floppiness of it all: bond lengths change with atomic displacement parameters; the flexibility of various coordination tetrahedra in zeolitic frameworks. An empirical structural study of bond lengths and angles. Werner H. Baur, and Reinhard X. Fischer Chem. Mater., Just Accepted Manuscript • DOI: 10.1021/acs.chemmater.8b04919 • Publication Date (Web): 08 Mar 2019 Downloaded from http://pubs.acs.org on March 11, 2019

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Chemistry of Materials

The floppiness of it all: bond lengths change with atomic displacement parameters; the flexibility of various coordination tetrahedra in zeolitic frameworks. An empirical structural study of bond lengths and angles. Werner H. Baur† and Reinhard X. Fischer‡. † ‡

Western Springs, IL 60558, USA Fachbereich Geowissenschaften der Universität, Klagenfurter Straße 2, 28359 Bremen, Germany

ABSTRACT: We have evaluated more than 7,000 crystal structures of zeolites in terms of the values of their bond lengths T-O (T = Si, Al, P, Zn, Be, Ge, B, As, Ga, Co) and their variability, as well as the flexibility of their bond angles O-T-O. Out of these known crystal structure descriptions of zeolites we have selected 1179 which have estimated standard deviations of their T-O bond length of 0.01 Å or less. For the most common bond lengths we obtain 1.603(11) Å for 1,323 mean tetrahedral Si-O, 1.736(8) Å for 416 Al-O, and 1.522(9) Å for 228 P-O. It is unsettling that the spread of each of these values is large: about 0.07 Å within the population studied by us. Furthermore these values disagree by several hundredth of an Ångström from some of the mean values of Si-O, Al-O and P-O compiled from nonzeolitic types of compounds. This is at variance with the widespread conviction that such T-O distance values should be relatively constant across different types of inorganic compounds. Ever since Cruickshank (Acta Crystallogr. 1956, 9, 757) it has been known that high atomic displacement factors shorten observed bond lengths. Corrections for this effect were applied in the past, but have become lately rarer. We find that much of the variance observed by us in the bond lengths is due to the fact that topologically different zeolite framework types have different atomic displacement parameters of their oxygen atoms. Thus it makes no sense to search for mean tetrahedral bond lengths in TO4 tetrahedra. Instead a particular mean bond length, e.g. Si-O, can be only characteristic for an Si-O bond for one given framework type as we show for the topologically different framework types CAN, FER, MFI, NAT, and SOD. Even after bond length corrections for differing displacement parameters the mean Si-O bonds range from 1.601 Å for FER to 1.629 Å for SOD. The observed angles O-T-O cover a range from 94.5° to 129.1° averaging around the tetrahedral angle of 109.5°. The largest deviations from the tetrahedral angle occur for tetrahedra with long mean T-O distances of the T-atoms. Our results, properly applied, can be useful as input for distance least squares (DLS) calculations on zeolites and for checking on the results of crystal-structure refinements or of theoretical calculations.

1. INTRODUCTION Zeolites are porous frameworks of coordination tetrahedra of oxygen atoms around atoms such as Si or Al connected at their vertices. The pores are large enough to admit atoms and small molecules. The recognition of this fact opened up the possibility of using zeolites in a variety of applications, starting with ion exchange, on to catalysis as well as to their use as molecular sieves. However, it was early recognized as well that these zeolitic frameworks were not rigid and would change their shape depending upon which atoms or molecules were introduced into their pores. They would also respond flexibly when exposed to varying temperatures and pressures. How and how much these zeolitic frameworks change under diverse conditions has been described in numerous crystal structure determinations beginning about 80 years ago. What has not been done so far is to make a comparative study of zeolites documenting the extent of their flexibil-

ity when accommodating in their pores various atoms or molecules. The possibility to do such studies developed over the years with the slow and gradual improvement of diffraction techniques. Today one can determine the atomic positions, and consequently the interatomic distances in a crystal structure within a few thousandth of an Ångström. Of course bond angles are likewise known with high accuracy. Furthermore one can do this for zeolitic crystal structures of ever increasing complexity. Such systematic studies can allow us to understand better the mechanics of zeolitic flexibility and thus help us making zeolites more useful. In two back-to-back papers published in 19131,2 the Braggs explained how they solved the crystal structures of halite, NaCl, and of diamond. Since these were the first two crystal structures to be determined by X-ray diffraction the bulk of these papers naturally was concerned with the methods used by them. In addition W.L. Bragg

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says1: "...in sodium chloride the sodium atom has six chlorine atoms equally close...", and on diamond they both write2: "The union of every carbon atom to four neighbours in a perfectly symmetrical way might be expected in view of the persistent tetravalency of carbon" and furthermore: "These lattices are so situated in relation to each other that, calling them A and B, each point of lattice B is surrounded symmetrically by four points of lattice A, arranged tetrahedron-wise and vice versa.” Thus, the Braggs do not call them coordination numbers, CN, or coordination polyhedra even though the concept and the term "coordination number" had been introduced already in 18933 by the founder of coordination chemistry, Alfred Werner, twenty years before the Braggs solved their first structures. Werner deduced the coordination numbers, CN, and some of the arrangements (octahedral, tetrahedral, planar four-coordination) around the cations studied by him by observing the various numbers of isomers occurring for given chemical compositions. Jensen retraced a few years ago4 the in-between steps of how the terminology of coordination chemistry was transferred from Werner to Victor Moritz Goldschmidt5 and herewith to crystal chemistry, so we don't have to repeat the details here. In any event Goldschmidt's "Laws of Crystal Chemistry" of 19265,6 established the nomenclature that we are still using today. The crystal structure determinations in the first years of the application of X-ray diffraction involved mostly simple crystal structures of high symmetry, with the atoms located in highly symmetric sites without any adjustable parameters or at most one such parameter. Consequently, the coordination polyhedra were found to be ideal and the interatomic distances were reasonably precisely determined, depending mostly on the quality of the measurements of the unit cell dimensions. As successively crystal structures of lower symmetry were determined in which atoms were located in more general positions with numerous adjustable parameters the interatomic distances were known less precisely and most likely with errors in the range of 0.05 to 0.1 Å. The coordination polyhedra could very well be ideal, but nobody would know for sure because of the lack of accuracy in the determinations. The methods of the day were insufficient for that. An example would be rutile, TiO2, the tetragonal structure of which was determined in 1916 by Vegard7. There is one free parameter to be determined there for the location of the oxygen atom and the resulting two symmetrically inequivalent Ti-O distances can have different values for their Ti-O distances. These measured in the Vegard study 1.9 and 2.0 Å. That was close enough to indicate that they possibly were of equal length. It took until the fifties of the 20th century before the crystal structure refinement methods were sufficiently improved so that the estimated standard deviations (e.s.d.) of bond lengths in inorganic crystal structures fell below 0.01 Å. In 1955 two papers8,9 showed, using different methods (single crystal8 and powder diffraction9), that in rutile type TiO2 four equatorial bonds within the coordination octahedron were shorter (by respectively 0.044(7) and 0.038(5) Å) than two

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axial Ti-O bonds normal to the equatorial bonds. Most likely "...[T]hese were the first structure refinements of inorganic crystals with adjustable positional parameters where the mean estimated standard deviations of the bond lengths within one coordination polyhedron were smaller than 0.01Å."10 Today e.s.d.s of 0.01 Å and much less are standard in routine single crystal structure determinations. Currently 245 topologically different types of zeolite frameworks have been accepted as valid by the Structure Commission of The International Zeolite Association11 (IZA-SC; 239 are listed on the website, another 6 were accepted in the last round in August 2018, personal communication by Rob Bell). Each of these has been assigned a framework type code (FTC). Quite a few of the crystal structures of substances corresponding to these codes had been determined long before the Structure Commission was established in 197711. The first of these was the structure of nosean, Na8Si6Al6O24SO4, a sodalite-type mineral, published in 1929 by Frans Maurits Jaeger12. The corresponding FTC is SOD. Jaeger discusses there already in a rather modern way the possible exchange of nonframework atoms. One year later Pauling claimed13 to have been the first to have solved a sodalite type crystal structure. That claim was false14. However, important was Pauling's observation that the framework of the sodalite structure, Na8Si6Al6O24Cl2, obviously collapses upon itself by a rotation of the silicate tetrahedra because its unit cell would be 9.4 Å if its framework were fully expanded, but in fact measures only 8.87 Å13. Thus these two early papers12,13 already touch upon two of the several topics which were to become important in the investigation of zeolites in the future: ion exchange in zeolites and the flexibility of their frameworks. Pauling considered the atomic positions in sodalite "...to be accurate to about 0.05 Å"13. We calculate from his coordinates distances of 1.58 Å for Si-O and of 1.76 Å for Al-O in reasonable agreement with modern values and in agreement with his estimate of their accuracy. Pauling solved in the same year15 the crystal structure of natrolite, Na2Al2Si3O10 ∙ 2H2O, without stating the numerical values of the coordinates, but providing a sufficiently clear diagram of the structure and pointing out that the fibers consisting of SiO4 and AlO4 coordination tetrahedra must be rigid, but can rotate about their long axes and thus the structure can collapse. Taylor and collaborators, from W. L. Bragg's laboratory in Manchester, published in 193316 the complete structural description of natrolite and several related zeolites with the low accuracy available to them at the time. It took another 27 years for a modern structural study of natrolite to appear: Walter Meier17 did this at Pauling's suggestion as a postdoc at the California Institute of Technology in 1960. At this point it is useful to emphasize that the flexibility and distortions pointed out by Pauling in the sodalite and natrolite type crystal structures are due to the changes in the T-O-T angles between the coordination tetrahedra. The tetrahedra themselves can in principle remain completely rigid and unchanged while the overall shape of the


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frameworks change because the tetrahedra change their relative positions to each other. What happens is that their hinges T-O-T adjust. What we are investigating in this contribution, however, are the shapes of the coordination tetrahedra themselves, that is the T-O distances and the O-T-O angles as they occur for one chemical composition, let us say SiO4 when that unit is placed in topologically different zeolite frameworks. This approach differs from, e.g., the rigid unit model (RUM) where "...the SiO4 tetrahedra are able to move as rigid objects without distorting significantly,..."18. So the focus in the RUM approach is on changes in the Si-O-Si hinges between the tetrahedra, wheras we investigate the changes of tetrahedral distortions. Depmeier19 studied in detail the distortions of the tetrahedra in cubic sodalite frameworks (SOD) when the Al content was varying whereas we are analyzing pure AlO4 and SiO4 coordination tetrahedra located in topologically different environments. In another very detailed investigation Depmeier20 presented a model of the tilt and the distortions of tetrahedra occupied statistically by Al and Si atoms in LTA-type zeolites whereas we consider only tetrahedra occupied by one type of cation, but do not limit ourselves to one kind of framework. Another study of LTA-type frameworks21 considered exclusively the important role of the T-O-T hinges in maintaining the noncollapsible character of this type of zeolite. Distortions of the coordination tetrahedra were not taken into account in this approach. We estimate that currently many more than 7,000 crystal structure descriptions of all types of zeolites are available in the literature. It is now possible, using these data, to see to which degree the frameworks of zeolitic structures can be distorted by the effects of various enclosed ions or molecules, and by pressure and temperature changes. It is often assumed that the coordination tetrahedra TO4 around the T atoms (Si, Al, P and others of similar sizes) are rigid. But we don't know very well the empirically observed actual limits to changes in the T-O distances and the O-T-O angles of these coordination tetrahedra in differently stuffed frameworks. In addition we can test now the question whether or not the mean T-O distances themselves are as constant as we usually assume when comparing one chemical compound with another compound or classes of compounds to other classes of compounds. The basis of characteristic ionic or other radii for different elements is the very idea that mean interatomic distances remain constant from chemical compound to chemical compound under certain conditions, such as equal coordination numbers for both cation and anions. Is that true among the zeolitic crystal structures? Is it true when comparing zeolites with other types of crystal structures? Our aim is to provide reliable and well documented data of 1. mean T-O distances for the most common tetrahedral coordinations in zeolite frameworks irrespective of which FTC the frameworks belong to;

2. individual O-T-O angles for the same; 3. the extreme values of individual and mean T-O distances and of the individual angles O-T-O of the same. Such data can be useful as input into distance least squares (DLS) calculations on zeolite crystal structures, as guidelines for judging the quality of theoretical zeolite structure simulations and of crystal-structure refinements of zeolites. Modern Rietveld refinements are usually performed with distance and angle constraints or restraints because often the powder diffraction data by themselves are insufficient to define the investigated crystal structures properly. It would be useful to rely in such cases for comparative purposes on well documented empirical data. This is particularly true given the fact that investigations into zeolites are presently a very active field of study. A search for "zeo*" in the Web of Science22 yields 91,015 hits. For reasons why zeolites are such a hot field of study see a recent review23. 2. DATA Before we cover the available data on the geometry of zeolite frameworks we wish to define what a zeolite is. This is a surprisingly difficult proposition24-27. A simple definition is given on the website of the IZA-SC11: "Classically, zeolites are defined as aluminosilicates with open 3dimensional framework structures composed of cornersharing TO4 tetrahedra, where T is Al or Si. Cations that balance the charge of the anionic framework are loosely associated with the framework oxygens, and the remaining pore volume is filled with water molecules. The nonframework cations are generally exchangeable and the water molecules removable. This definition has since been expanded to include T-atoms other than Si and Al in the framework, and organic species (cationic or neutral) in the pores." We have considered in our present study only materials exhibiting one of the 239 framework topologies accepted by the IZA-SC and listed on their website11. The bulk of the data used here is referenced in our Landolt-Börnstein volumes on materials with zeolite-type crystal structures28,29. The details of these structures are contained in our databank ZeoBase30. The data used for the statistics presented here are from the 1179 zeolite single crystal structure determinations in our database with mean e.s.d.s of their T-O distances of less than 0.01 Å as documented in Tables S001 and S002. One could also employ Liebau's31 (see p. 15 in ref. 31) standard of only using single crystal structure determinations with R values below 8%. But we judge that basing the limit on an e.s.d. of less than 0.01 Å is more stringent. All interatomic distances and angles from zeolites used here have been recalculated from the fractional coordinates and other relevant information given in the original papers using ZeoBase30 and/or Sadian32. All datasets entered into ZeoBase30 have been checked for reliability, and for internal and external consistency. About 40% of them have been edited, corrected or emended. Sometimes these corrections were trivial and could be fixed without contacting the authors: atomic coordinates with inverted digits were



rectified (when a comparison of bond lengths reported in the papers deviated from what we calculated using the coordinates supplied in the same papers), or missing or wrong space group designations in the original papers were supplied by us wherever it was clear from the context what they really should have been. In other cases we needed input from the original authors, for instance whenever whole coordinate tables were garbled. There exist a few zeolite type crystal structures in which some of the nominally four coordinated T-atoms are actually five or six coordinated. In these cases both the individual and the mean bond lengths are different from those in the purely tetrahedral frameworks. These have been left out of all statistics presented here, as well as have all refinements which were performed using distance restraints or constraints. The derived data used in the discussions here are presented in histograms, plots and tables. The references to the individual numerical values are listed in detail in the supporting information as is additional material not im-

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mediately needed to understand the discussions in the paper. 3. THE SHAPES OF THE COORDINATION TETRAHEDRA IN ZEOLITES 3.1. Synopsis of zeolitic coordination tetrahedra. The interatomic distances and angles describing the geometry of the most common coordination tetrahedra occurring in zeolitic compounds are listed in Table 1. The corresponding T-O distances and O-T-O angles of these tetrahedra are presented as histograms in Figure 1 as are the identifications of each data point in the figure (Table S003). The various amounts of chemically different coordination tetrahedra reflect their availability in the literature. That in turn reflects the scientific and/or commercial interest in the various chemical types of zeolites. The elements Si and Al are clearly most common in our sample followed by P.


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Figure 1.Histograms of individual and mean T-O distances and of O-T-O angles of the tetrahedra presented in Table 1. The detailed data used for each of the histograms in Figure 1 are listed in Table S003. Labels on the ordinate refer to the frequency of occurrences. All histograms are produced using ZeoBase30.

Table 1. Shapes of most common coordination tetrahedra in zeolites (ind = individual values within tetrahedron; ind mean = mean over individual values in tetrahedron): T-O, ind; T-O, ind mean; O-O, tetrahedral edges; O-T-O, ind, tetrahedral angles. The T-O distances presented here are only strictly valid for zeolite crystal structures and possibly for nonzeolitic tetrahedral framework structures. They would be about 0.01 to 0.05 Å longer for coordination tetrahedra in nonframework structures (see text). The corresponding histograms are shown in Figure 1, the data used for each of these histograms in Figure 1 are listed in Table S003. Tetrahedral T-O distances,edges O-O, and angles O-T-O

Mean value (mean deviation)

Range of values

Delta of range

DI* mean

# of data points

a. SiO4 data from 401 zeolite crystal structures, in 1,323 tetrahedra and including 36 different FTC Si-O, ind

1.603(13) Å

1.547-1.674 Å

0.127 Å

0.008

5,292

Si-O, ind mean

1.603(11) Å

1.567-1.639 Å

0.072 Å

0.007

1,323

O-Si-O, ind

109.5(1.4)°

101.0-117.7°

16.7°

0.013

7.938

O-O, ind

2.617 (28) Å

2.496-2.794 Å

0.298 Å

0.011

7,938

b. SiO4 data from 53 zeolite crystal structures, including only those with SiO2 frameworks, in 534 tetrahedra including 10 different FTC Si-O, ind

1.594(9) Å

1.547-1.632Å

0.085 Å

0.005

2,136

Si-O, ind mean

1.594(6) Å

1.567-1.616 Å

0.049 Å

0.004

534

O-Si-O, ind

109.5(0.9)°

104.8-113.6°

8.8°

0.008

3,204

O-O, ind

2.602 (18) Å

2.508-2.698 Å

0.190 Å

0.007

3,204



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c. AlO4 data from 292 zeolite crystal structures, in 416 tetrahedra and including 24 different FTC Al-O, ind

1.736(12) Å

1.671-1.785 Å

0.114 Å

0.007

1,664

Al-O, ind mean

1.736(8) Å

1.694-1.768 Å

0.074 Å

0.004

416

O-Al-O, ind

109.4(3.1)°

94.6-123.7°

29.1°

0.029

2,496

O-O, ind

2.833(57)

2.550-3.069 Å

0.519 Å

0.020

2,496

d. PO4 data from 111 zeolite crystal structures, in 228 tetrahedra and including 31 different FTC P-O, ind

1.522(12) Å

1.472-1.565 Å

0.093 Å

0.008

912

P-O, ind mean

1.522(9) Å

1.488-1.552 Å

0.064 Å

0.006

228

O-P-O, ind

109.5(1.6)°

103.6-115.6°

12.0°

0.014

1,368

O-O, ind

2.485(28) Å

2.371-2.582 Å

0.211 Å

0.011

1,368

e. ZnO4 data from 31 zeolite crystal structures, in 53 tetrahedra and including 11 different FTC Zn-O, ind

1.941(16) Å

1.876-2.006 Å

0.130 Å

0.008

212

Zn-O, ind mean

1.941(8) Å

1.902-1.958 Å

0.056 Å

0.004

53

O-Zn-O, ind

109.4(4.8)°

95.7-126.3°

30.6°

0.044

318

O-O, ind

3.163(94) Å

2.905-3.480 Å

0.575 Å

0.030

318

f. BeO4 data from 30 zeolite crystal structures, in 32 tetrahedra and including 10 different FTC Be-O, ind

1.625(14) Å

1.561-1.668 Å

0.107 Å

0.008

136

Be-O, ind mean

1.625(8) Å

1.607-1.643 Å

0.036 Å

0.005

34

O-Be-O, ind

109.5(2.3)°

101.9-116.2°

14.3°

0.021

199

O-O, ind

2.652(38) Å

2.510-2.760 Å

0.250 Å

0.014

199

g. GeO4 data from 25 zeolite crystal structures, in 43 tetrahedra and including 10 different FTC Ge-O, ind

1.744(12) Å

1.702-1.789 Å

0.087 Å

0.007

172

Ge-O, ind mean

1.744(9) Å

1.723-1.775 Å

0.052 Å

0.005

43

O-Ge-O, ind

109.4(2.9)°

99.2-116.7°

17.5°

0.027

258

O-O, ind

2.846(52) Å

2.667-2.990 Å

0.323 Å

0.018

258

h. BO4 data from 13 zeolite crystal structures, in 15 tetrahedra and including 5 different FTC B-O, ind

1.474(10) Å

1.445-1.512 Å

0.067 Å

0.007

60

B-O, ind mean

1.474(5) Å

1.465-1.490 Å

0.025 Å

0.004

15

O-B-O, ind

109.5(2.1)°

103.7-113.5°

9.8°

0.019

90

O-O, ind

2.406(29) Å

2.316-2.468 Å

0.152 Å

0.012

90

i. AsO4 data from 9 zeolite crystal structures, in 20 tetrahedra and including 5 different FTC As-O, ind

1.668(10) Å

1.645-1.703 Å

0.058 Å

0.006

64

As-O, ind mean

1.668(7) Å

1.651-1.688 Å

0.037 Å

0.004

16

O-As-O, ind

109.5(2.2)°

102.4-117.2°

14.8°

0.020

96

O-O, ind

2.722(40) Å

2.574-2.818 Å

0.244 Å

0.015

96

j. GaO4 data from 12 zeolite crystal structures, in 19 tetrahedra and including 7 different FTC Ga-O, ind

1.827(24) Å

1.758-1.895 Å

0.137 Å

0.013

76

Ga-O, ind mean

1.827(24) Å

1.773-1.868 Å

0.095 Å

0.013

19

O-Ga-O, ind

109.4(3.6)°

98.0-122.1°

24.1°

0.033

114

O-O, ind

2.979(73) Å

2.689-3.252 Å

0.563 Å

0.024

114

k. CoO4 data from 8 zeolite crystal structures, in 11 tetrahedra and including 6 different FTC Co-O, ind

1.958(14) Å

1.927-2.003 Å

0.076 Å

0.007

44

Co-O, ind mean

1.958(4) Å

1.942-1.968 Å

0.026 Å

0.002

11

O-Co-O, ind

109.4(5.8)°

94.5-129.1°

34.6°

0.053

66

O-O, ind

3.190(113) Å

2.892-3.538 Å

0.646 Å

0.035

66

* DI = distortion indices, definitions from

Baur33.


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It has proved to be useful to measure and compare the distortions of coordination tetrahedra33 by calculating for them the average deviations from their means for the T-O bond length [DI(TO)], the O-T-O angles [DI(OTO)], and the length of the tetrahedral edges [DI(OO)]: DI(TO)=(∑4𝑖=1 ∣TOi -TOm∣)/4TOm

(1)

DI(OTO)=(∑6𝑖=1 ∣OTOi -OTOm∣)/6OTOm DI(OO)=(∑6𝑖=1 ∣OOi -OOm∣)/6OOm

(2) (3)

where m indicates mean values. One can define analogously distortion indices for other types of coordination polyhedra as well33. 3.2 Si-O bond lengths in SiO4 tetrahedra. The geometry of the silicate coordination tetrahedron as observed for 1323 tetrahedra in 401 zeolite crystal structures is presented in Table 1, where the mean value of all the individual bond length Si-O is found to be 1.603(13) Å. Inasmuch as silicate tetrahedra have been often investigated in the past we shall begin with a comparison of the values of the Si-O distances obtained by us with some of the results of previous efforts (Table 2) and theoretical calculations34. That table includes bond lengths obtained from nonframework silicates (lines 1, 3, 4, 8, 9 and 11), from pure SiO2 zeolitic frameworks (line 10) and also from frameworks in which the tetrahedral positions are occupied either by Si or by other atoms. The overall value of 1.603(11) Å is astonishingly close to the 1.60(1) Å which J. V. Smith35 found already in 1954 for the mean Si-O distance (Table 2). (and corresponds to the DFT calculated34 value of 1.608(10) Å within the error margins). Naturally at that time most of the crystal structure determinations used by Smith were of low accuracy and apparently none of them were from a zeolite. Smith's aim was to get reliable values of tetrahedral Si-O and AlO bond lengths in order to estimate from observed (Si,Al)-O distances the Si:Al ratio in crystals with statistical occupancies of one site by both Si and Al. This was at the time especially interesting for the study of feldspars, which are aluminosilicate frameworks in principle similar to zeolites, but with smaller pores. This procedure is of course also useful in the study of zeolites when, as often is the case, individual tetrahedral sites are occupied statistically by both Si and Al atoms. In a second look at tetrahedral distances in 196336 Smith and Bailey proposed, based on newer results, 1.61 Å for the average Si-O distance in frameworks and 1.63 Å for isolated tetrahedra in nonframework silicates (Table 2). The former number is again in excellent agreement with the values obtained here (Tables 1 and 2) for frameworks. Smith and Bailey's second number for nonframework SiO bonds agrees well with the average found in 293 structures in 197037 (Table 2, line 4). Smith and Bailey36 call this decrease in mean Si-O bond lengths with the increase in the number of linkages from the tetrahedrally coordinated Si atoms to neighboring tetrahedral coordinations the tetrahedral linkage effect. They indicate that such effect appears to be larger for tetrahedral Al-O bonds. The Si-O distances obtained from Shannon's effective ionic radii38

are similar. A distance of 1.61 Å would apply to a framework of pure SiO2 composition (CN of two for the oxygen atoms), and of 1.62 Å for a chemical composition in which part of the tetrahedral sites were occupied by cations with oxidation states of less than 4. Here we are assuming arbitrarily a CN of three for the oxygen atoms (Table 2, lines 5 and 6). We can separate the data collected by Baur on tetrahedral Si-O distances39 into those based on framework and nonframework structures, again showing a difference in bond lengths (lines 7 and 8). The mixed data collected by Griffen40 fall with their mean correctly between the values based on Baur39. The compilation of Wragg et al.41 encompasses only zeolites with a framework composition of SiO2, however, some of the frameworks occlude non-bonding neutral molecules. In two very recent articles Gagné & Hawthorne42,43 study the various influences on a multitude of bond lengths including the Si-O bond. They cover both framework and nonframework structures (line 11). Our data (lines 12 to 14) are taken exclusively from zeolite crystal structures and at first blush do not appear to have characteristics very different from those of the other data sets. The second rule of Pauling's "Principles determining the structure of complex ionic crystals"46-48 essentially states that the charges on oppositely charged ions in a crystal structure are balanced locally. Smith49 observed considerable deviations from Pauling's rule within the SiO4 tetrahedron in melilite, Ca2MgSi2O7, and suspected "This deviation may be related to the variation in Si-O distances since the longer the distance the larger is the sum of electrostatic valence bonds". Baur showed37,50,51 that this applies to numerous diverse, chemically different compounds. However, it is obvious from the data in Table 2 that even when no deviations from Pauling's electrostatic valence rule exist, i.e. in the cases of pure SiO 2 frameworks (lines 10 and 13), there are variations in observed SiO bond lengths. These are slightly smaller than those for nonframework silicates or for frameworks containing Si and other tetrahedral ions, but they still do vary. The bond lengths Si-O within each individual SiO4 tetrahedron vary of course much more than the corresponding averaged values as can be seen by comparing columns 4 with columns 5 in Table 2. A look at Table 2 shows also that the mean values of the bond length Si-O vary by 0.033 Å for the ten compilations summarized here (Table 2, line 15). When one is used to relying on the effective ionic radii of Shannon38 one might think that bond lengths can be predicted to within 0.01 or 0.02 Å (lines 5 and 6). This is apparently not true for the 5292 tetrahedral Si-O bonds presented here, see Figure 1.a1. The spread of values for the individual bonds within the tetrahedra reaches 0.186 Å (line 14) and amounts to roughly a 5.8% deviation from the overall mean for the ten compilations compared here in Table 2.



Page 10 of 28

Table 2. Selected empirical tetrahedral Si-O distances for framework and nonframework structures, chronologically ordered. Included are bond lengths obtained from nonframework silicates, from pure SiO2 zeolitic frameworks and also from frameworks in which the tetrahedral positions are occupied either by Si or by other ions. CN = coordination number; comp = composition; eir = from effective ionic radii; fr = from framework structures; ind = individual values within tetrahedron; ind mean = mean over individual values in tetrahedron; mixed = from both framework and nonframework structures; nonfr = from nonframework structures; zeol = from zeolite crystal structures; Δ = difference. Reference, first author only. Tetrahedral distances

Mean value (Å)

Range of values (Å) of ind.

Range of values (Å) of ind. mean

# of structures

Reference

1

Si-O, mixed

1.60(1)

not available

1.59-1.615

7*

#35, Smith, 1954

2

Si-O, fr

1.61

1.566-1.661

1.59-1.628

16

#36, Smith, 1963

3

Si-O, nonfr

1.63

1.50**-1.67

1.595-1.642

22

#36, Smith, 1963

4

Si-O, nonfr

1.625

1.55-1.72

not available

293

#37, Baur, 1970

5

Si-O, eir, CN of O = 2

1.61

not available

not available

23***

#38, Shannon, 1976

6

Si-O, eir, CN of O = 3

1.62

not available

not available

23***

#38, Shannon, 1976

7

Si-O, fr

1.610(9)

not available

1.584-1.629

18

#39, Baur, 1978

8

Si-O, nonfr

1.626

not available

1.601-1.654

137

#39, Baur, 1978

9

Si-O, mixed

1.621

not available

not available

75

#40, Griffen, 1979

10

Si-O, fr, comp of framework is SiO2, zeol

1.597(26)

1.540-1.67

1.563-1.621

35

#41, Wragg, 2008

11

Si-O, mixed

1.625(24)

1.560-1.726

1.595-1.666

334

#42, #43, Gagné, 2017/18

12

Si-O, fr, all data, s. Table 1, zeol

1.603(11)

1.547-1.674

1.567-1.639

401

this work, 2018

13

Si-O, fr, SiO2 comp of framework, s. Table 1, zeol

1.594(9)

1.547-1.632

1.567-1.616

53

this work, 2018

14

Overall values: grand mean SiO of all 10 compilations (1-4, 712), total ranges

1.615

1.540-1.726

1.563-1.666

Δ

Δ

0.095-0.17

0.025-0.072

15

Overall ranges: of all 10 compilations of Si-O and Δ of extreme ranges

1.597-1.63

* Mostly framework: 5 frameworks out of 7 crystal structures. ** This value is from a very poor refinement44 that was later superseded45. *** One framework out of 23 crystal structures.

When we look at the averages of the four Si-O distances in each tetrahedron in all samples (Table 2, col. 5) the range of the extreme values are, not surprisingly, smaller, but still appreciable, ranging from 0.025 to 0.072 Å (line 15). Figure 1.a2 shows this for our sample consisting of 1323 tetrahedra. While Figure 1.a1 displays a faint resemblance to a normal distribution, Figure 1.a2 has twin peaks with maxima approximately at 1.595 and 1.615 Å. The 1.595 Å peak is close to the mean value of 1.597 Å for the Si-O distances reported for zeolites with SiO2 framework compositions41 ten years ago. We show data for the corresponding silica zeolites in our sample in Table 2 (line 13): a comparison of lines 10 and 13 shows the similarity of the results. The input into the two tables overlaps, but our

data include structure publications since 2008 and do not contain structures based on Rietveld refinements as the older data do41. Some of the discussions in the preceding paragraph should be compared to the similarly skeptical approach advocated by Gagné & Hawthorne in the summary and conclusion section of their paper. 42 Strictly speaking we should expect Gaussian distributions only in cases where we are measuring an identical datum repeatedly using the same method. In practice of course we make such comparisons even if the measurements apply to similar and not identical data using various related methods (e.g. X-ray and neutron diffraction).


Page 11 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


What we compare here are bond lengths between chemically identical atoms, but located within wildly different crystalline environments, e.g. SiO2 compositions vs. nonframework silicates. When Bragg, the son, wrote the first paper on crystal chemistry52, though he did not use the term, he established a set of atomic (not ionic) radii which he assumed could reproduce the interatomic distances between atoms in a variety of compounds (though he excluded from that already the metals). Goldschmidt5 established ionic radii a few years later. When Shannon & Prewitt published their effective ionic radii53 they limited their validity to oxides and fluorides and assumed different values for the radii of differently coordinated oxygen atoms. Thus over time, as the available data multiplied, and the understanding of the subject matter increased, the various empirical compilations of radii became more and more specialized. Nevertheless, when we ignore the radii and just compare the observed Si-O distances in our sample we see an appreciable scatter in their values both for the individual bond lengths within a coordination tetrahedron and for the mean bond lengths of the tetrahedra. Numerous reasons for such variations have been discussed in the literature (e.g. in references #39, #42 and #54). One possible reason has not been considered very much: experimental error. Here we try to avoid faulty determinations by using only crystal structures determined with high precision. However, when dealing with more than a thousand crystal structures it is impossible to investigate them all with equal attention. Possibly some of the outliers (and some of the "inliers" as well) are not as precisely determined as one would wish. This could be as simple as making a mistake in the calculation of the e.s.d. To err is human. But we consider it unlikely that our data suffer much from poor crystal structure determinations. When we relaxed the selection criteria and included in individual plots and histograms T-O distances with mean e.s.d.s of up to 0.02 Å the scatter increased appreciably and blurred the pictures. Thus we stuck with the original choice of 0.01 Å. What may be more of a problem are the chemical compositions at the T-sites. Thus it is fairly routine today to find out the Al and Si contents of a sample. However, this does not mean that all the occupancies of the individual tetrahedral sites are known. The tendency of Al and Si atoms to replace each other in these sites may mean, that Si occupied sites housing some Al atoms have apparently longer T-O bond lengths, while predominantly Al occupied sites may contain a small amount of Si, making them appear to have shorter bond lengths. This could be checked by solid state NMR measurements, but this does not happen in all crystal structure determinations. Faced with numerous structure determinations used in this paper we did not attempt to find out which of them were accompanied by reliable NMR studies as well. One could take the fact that the mean Si-O distances in zeolite frameworks of SiO2 composition (Table 1b) are shorter than the mean Si-O distances in (mostly) aluminosilicates (Table 1a) as an indication that the latter are contaminat-

ed with Al. The same is true for the observed ranges between the extremes of the Si-O distances and the angles O-Si-O for these groups. However, for this to be proven, we would have to test the statistical significance of such a correlation by looking also for the correlations with numerous other possible explanations as was done for instance in references #39 and #42. That is beyond the scope of this study. There is one more influence, which has to be considered: Slovokhotov55,56 has pointed out that geometrical parameters gathered from diffraction data deposited in large crystallographic databases may have unexpected biases. In several examples he shows55 how bond lengths and angles of particular small organic molecules are affected. One example he uses is benzene, C6H6, where C-C bond lengths calculated from 186 crystal structure determinations obtained from the Cambridge Structural Database57 tend to be longer for low temperature determinations as compared to room temperature results. However, there is considerable overlap of the bond lengths distributions at all temperatures, nevertheless, the averages differ. This results in C-C bond length distributions broad in comparison to the error in the experimental values of the individual bond lengths. This is what Slovokhotov55 calls the "blind area" in the histograms where the "data points are randomly scattered within 0.02-0.04 Å". Incidentally all X-ray determinations yield lower C-C distances than neutron diffraction results on crystals or electron diffraction results for free molecules of benzene55. This might be called a not completely unexpected result of comparing data obtained by different experimental methods. We have used indiscriminately data of zeolite crystal structures determined at various temperatures and pressures. The effects of these two variables on the bond lengths are in zeolites usually smaller than their e.s.d. are. The temperature effects observed by Slovokhotov55 for the small, often loosely bonded, organic molecules considered by him are certainly larger than for coordination tetrahedra held more or less tightly within the zeolitic frameworks investigated here. But still this might blur some of our histograms. In a second recent publication Slovokhotov56 took on another potential problem with structural databases: they can "...display unusual statistical properties like nonGaussian shape, polymodality, and heavy tails." that usually are associated with social systems. Thus Slovokhotov gives as an example the volatility of stock prices58. He suggests that social factors influencing the contents of structural databases could be "financial support, exchange of information, competition etc." 56. When inspecting the numbers of zeolites for which we have structural investigations in ZeoBase30, it is obvious that among others the FTC types MFI and FAU are heavily represented, most likely because they are useful as catalysts. That would be a "social" factor in Slovokhotov's sense. It shows up clearly in our data by the heavy preponderance of MFI-type data in samples including pure SiO2 compositions. Figure 1.b2 is based on 53 crystal structures of which 20 structures are of MFI topology and have a mean Si-O distance of



1.594(6) Å. Of the 534 data points 408 are from MFI-type structures, see also Table S003. Thus in this case roughly 40% of the structures provide about 3/4 of the Si-O data points because the MFI topology has a complex crystal structure with 12 symmetrically independent T atoms when crystallizing in space group Pnma and 24 T atoms in space groups P21/n or P212121, while the other structures in this sample are not nearly as complex. If we leave out the MFI-data in the histogram shown in Figure 1.b2 it contains many fewer data but their distribution looks still pretty much the same and their mean bond length is with 1.597(6) Å very close to the value for the bonds displayed in Figure 1.b2. Might it be that in SiO2 zeolites the Si-O bonds are really shorter than in zeolites containing other tetrahedral ions and that this is not just a property of the MFI-type frameworks? Could this be a pure SiO2 framework shrinkage effect in zeolites? At that point one can wonder whether or not in all tetrahedral frameworks of SiO2 composition, including those that are denser than zeolites, the Si-O bond lengths are on average just under 1.600 Å? In low cristobalite59 the mean Si-O distance is 1.605 Å, in low tridymite60 it is 1.597 Å, in coesite61 1.611 Å, in low quartz62 1.609 Å. The mean for all four of these crystal structures is 1.606 Å, which is higher than the averages given for the SiO2 only zeolites in Figure 1.b2 (1.594 Å) and for the SiO 2 zeolites compiled by Wragg et al. 41 (1.597 Å), but is lower than the grand mean of all Si-O distances considered in Table 2, line 14. However, all four of these values fall within the distributions of mean bond lengths in compounds with an SiO2 composition. 3.3 O-Si-O bond angles and O-O edges in SiO4 coordination tetrahedra. The O-Si-O angles for all 401 crystal structures containing SiO4 groups (Table 1, Figure 1.a3) vary from 101.0° to 117.7° for a difference between the extremes of 16.7°. However, when we look at the silicate tetrahedra from 53 zeolites with pure SiO2 frameworks, the range is only from 104.8° to 113.6° (Figure 1.b3) and the difference between the extremes is almost one half of the overall range. The corresponding distortion indices DI(OTO)33 are 0.013 and 0.008 (Table 1) making them the tetrahedra with the smallest DI(OTO) values among all those considered in this study. In pure SiO2 frameworks no cations are needed to balance the missing charges resulting from a replacement of some of the Si ions by tetrahedral ions with smaller oxidation states. This balancing requires the presence of extraframework cations in the pores of the zeolites and these apparently distort the individual SiO4 tetrahedra more than it happens in pure SiO2 zeolites. The lengths of the O-O edges of the tetrahedra vary of course with the O-Si-O angles and consequently their distortion indices change in concert with those of the O-Si-O angles (Table 1).

Page 12 of 28

Al-O was based on six pre-1947 determinations. In the "Second Review of Al-O and Si-O Tetrahedral Distances"36 the value is given as 1.75 to 1.80 Å. Later compilations yielded values closer to Smith's lower bound of the range indicated in reference #36: their average is 1.749 Å (Table 3, lines 5 to 7). However, all these earlier values are based on mixed data, that is on coordination tetrahedra from both framework and nonframework crystal structures. The value of 1.736 Å obtained here (line 8 of Table 3) for the Al-O distances in zeolitic structures is smaller by 0.013 Å. A similar difference of 0.016 Å was observed for nonframework (1.626 Å) and framework (1.610 Å) mean Si-O bond lengths, see Table 2, lines 7 and 8. But in the case of the Si-O bonds we were comparing data for framework structures in general, not just zeolite structures, with nonframework cases. In order to get an estimate of the tetrahedral bond length Al-O in nonframework structures by themselves we need more reliable data than the old value based on two crystal structures given by Smith and Bailey36 55 years ago. Thus we selected data from 40 tetrahedra in 30 nonframework structures containing AlO4-tetrahedra (Table S004, and Table 3, line 9) and obtained 1.767 Å (Table 3), or 0.031 Å longer than for the zeolitic framework case, 1.736 Å (Table 3, line 8) and more than the difference observed for the corresponding comparison for Si-O (Table 2, lines 7 and 8). The extremes, that is the deltas of the ranges of Al-O, both of the individual values, 0.114 Å (Table 1, Figure 1.c1), and of the individual means, 0.074 Å, differ by similar amounts as in the case of Si-O (Table 1). The same is true of the distortion indices DI(TO) for the individual bonds. Given that the Al-O bonds are longer than the Si-O bonds by ca. 8% (or 0.133 Å) and therefore should be weaker than the Si-O bonds one might have thought they would deform more easily than Si-O bonds do, but this is not the case. What is true, however, is that the O-Al-O angles differ in the various zeolite structures almost twice as much as the O-Si-O angles do (Table 1, Figure 1.c3). This is shown by the values of the deltas of their ranges and by their distortion indices DI(OTO) as given in Table 1. Apparently the energy needed to deform the O-Al-O angles and the O-O edges of the AlO4 tetrahedra is smaller than for the SiO4 tetrahedra, but the energy needed for deforming the T-O distances in these two cases appears to be similar.

3.4 Al-O bond lengths and O-Al-O angles in AlO4 coordination tetrahedra. The mean value of the Al-O bond length in 416 AlO4 coordination tetrahedra is 1.736(8) Å (Table 1, Figure 1.c2). Most of the zeolite crystal structures contributing to this value are aluminosilicates (Table S001). The value of 1.78(2) Å assigned by Smith 35 to


Page 13 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 3. Selected empirical tetrahedral Al-O distances for framework and nonframework structures, chronologically ordered. CN = coordination number; comp = composition; eir = from effective ionic radii; fr = from framework structures; ind = individual values within tetrahedron; ind mean = mean over individual values in tetrahedron; mixed = from both framework and nonframework structures; nonfr = from nonframework structures; zeol = from zeolite crystal structures; Δ = difference. Reference, first author only. Tetrahedral distances

Mean value (Å)



# of structures

Reference

1

Al-O, mixed

1.78(2)

not available

1.64-1.85

6*

#35, Smith, 1954

2

Al-O, fr

1.746

1.692-1.820

1.720-1.780

9

#36, Smith, 1963

3

Al-O, nonfr

1.786

1.721-1.802

1.770-1.802

2

#36, Smith, 1963

4

Al-O, eir, CN of O = 3

1.75

not available

not available

8

#38, Shannon, 1976

5

Al-O, mixed

1.750

not available

not available

29

#40, Griffen, 1979

6

Al-O, mixed

1.752

not available

not available

160**

#54, Baur, 1981

7

Al-O, mixed

1.746(13)

1.685-1.833

1.714-1.806

181***

#43, Gagné, 2018

8

Al-O, fr, s. Table 1, zeol

1.736(8)

1.671-1.785

1.694-1.768

292

this work, 2018

9

Al-O, nonfr, s. Table S003

1.767

1.699-1.849

1.729-1.805

20****

this work, 2018

10

Al-O, overall values: grand mean of lines 8 and 9, total ranges, # of structures

1.752

1.671-1.849

1.694-1.805

312

this work, 2018

Δ

Δ

11

Overall ranges of all 8 compilations of Al-O and Δ of extreme ranges

1.736-1.786

0.081-0.178

0.032-0.21

* Four out of 6 crystal structures are not framework structures. ** In this case number of tetrahedra. Number of structures most likely about half as large. *** Private communication from Olivier Gagné **** References 63 to 81.

3.5 P-O bond lengths and O-P-O angles in PO4 coordination tetrahedra. The extreme values of the P-O distances, the tetrahedral O-O edges and the O-P-O angles are clearly smaller for the PO4 tetrahedron when compared with the corresponding values for the SiO4 tetrahedron (Table 1). The distortion indices for the two cases do not reflect that in the same degree. The 1323 SiO 4 tetrahedra (Table 1a) have a slightly smaller DI(OTO) value than the 228 phosphate tetrahedra show (Table 1d). The P-O bond lengths when compared with the Si-O and Al-O bond lengths are clearly shorter and the oxidation state of the P atoms is larger than in the other two cases, thus it is to be expected, that the P-O distances change less than the bond lengths in the other two TO4 coordination tetrahedra. The overall mean P-O value for the zeolite crystal structures studied here is 1.523(12) Å, as compared with 1.537 Å for P-O distances observed in numerous phosphate groups in nonframework crystal structures (Table 4, lines 1, 2, 3, 6, 7 and 8). Thus five different compilations of P-O

bond lengths in nonframework crystal structures yield an identical result of 1.537 Å, the one compilation which includes two cases of framework structures result in a value of 1.536 Å (line 6 in Table 4). Our compilation of PO bond length gives an average value of 1.523 Å, or 0.014 Å less for the zeolites than for the nonframework structures. This is a puzzling aspect of Table 4. 3.6 Differences and scatter among Si-O. Al-O, and P-O bond lengths. This is analogous to what was found in the case of the mean Si-O distance in zeolites, 1.603 Å (Table 2, line 8) compared with an Si-O of 1.626 Å in nonframework structures (Table 2, line 12) for a difference 0.023 Å. Similar is the case of the Al-O distances in zeolites of 1.736 Å (Table 3, line 8) and the nonframework AlO distances of 1.767 Å (Table 3, line 9) for a difference of 0.031 Å. The overall result of comparing the mean bond lengths in phosphates, silicates and aluminates is that these bond lengths are in zeolites by 0.014 Å to 0.031 Å shorter than in nonframework compounds.



Page 14 of 28

Table 4. Selected empirical tetrahedral P-O distances for framework and nonframework structures, chronologically ordered. CN = coordination number; comp = composition; eir = from effective ionic radii; fr = from framework structures; ind = individual values within tetrahedron; ind mean = mean over individual values in tetrahedron; mixed = from both framework and nonframework structures; nonfr = overwhelmingly from nonframework structures; zeol = from zeolite crystal structures; Δ = difference. Reference, first author only. Tetrahedral distances

Mean value (Å)



# of structures

Reference

1

P-O, nonfr

1.537

1.44-1.64

not available

174

#37, Baur, 1970

2

P-O, nonfr

1.537(9)

not available

1.515-1.558

62

#82, Shannon, 1973

3

P-O, nonfr

1.537

1.412-1.662

1.506-1.572

129

#33, Baur, 1974

4

P-O, eir, CN of O = 2

1.52

not available

not available

not available

#38, Shannon, 1976

5

P-O, eir, CN of O = 3

1.53

not available

not available

not available

#38, Shannon, 1976

6

P-O, nonfr, except 2 are fr

1.536

not available

not available

51

#40, Griffen, 1979

7

P-O, nonfr

1.537

1.439-1.625

1.459-1.602

244

#83, Huminicki, 2002

8

P-O, nonfr

1.537(8)

1.430-1.696

1.503-1.579

1626*

#84, Gagné, 2018

9

P-O, fr, s. Table 1, zeol

1.522(9)

1.472-1.565

1.488-1.552

108

this work, 2018

10

Overall values: grand mean P-O of all 7 compilations, total ranges

1.535

1.412-1.696

1.459-1.602

Δ

Δ

0.093-0.266

0.043-0.143

11

Overall ranges of all 7 compilations of P-O and Δ of extreme ranges

1.523-1537

* Private communication from Olivier Gagné

Within the zeolites the trends of the interatomic distances within the periodic table are being faithfully followed in period 3 from left to right with the individual TO distances from Al-O to P-O diminishing from 1.736 Å to 1.522 Å and the corresponding individual O-O tetrahedral edges from 2.833 Å to 2.485 Å (Table 1). However, the distortion indices DI(TO) remain essentially constant for these three T-O distances. 3.7 Coordination tetrahedra around Zn, Be, Ge, B, As, Ga and Co. The number of crystal structures used for the statistics in the cases of SiO4, AlO4 and PO4 (Table 1) ranges from 111 to 401. Only 8 to 31 structures each were used in obtaining the data in Table 1 and the histograms in Figures 1e to 1k showing the geometry of the coordination tetrahedra around Zn, Be, Ge, B, As, Ga and Co. This may seem to be a small sample compared with our SiO4, AlO4 and PO4 data until we realize that Shannon's effective ionic radii38 for Si, Al, and P are based on respectively 15, 8 and 21 crystal structures and those still are a very important basis of many discussions concerning bond distances. Our compiled data for the coordination tetrahedra around Zn, Be, Ge, B, As, Ga and Co are insufficient to study them individually as thoroughly as we did the first

three cases of the SiO4, AlO4 and PO4 tetrahedra. But they may be of interest for those who need comparative T-O bond lengths and O-T-O angle data when performing Rietveld analyses, are doing DLS-calculations or want to check the quality of theoretical simulations of zeolite crystal structures containing these seven types of coordination tetrahedra. They are also useful for comparing the geometry and distortions of the coordination tetrahedra as a function of the mean bond lengths T-O (ranging from 1.474 Å for B-O to 1.941 for Zn-O, Table 1) and the oxidation states (OS) of the central cations (ranging from +2 for Be, Co and Zn to +5 for P and As, Table 1). The correlation between the DI(OTO) and the mean T-O as taken from Table 1, but excluding 1b which is a subset of 1a, shows a Pearson R 2 (see Figure 2) of 0.845 for a linear regression (Figure 2a). For DI(OTO) and the oxidation states Pearson R2 has a value of 0.449 (Figure 2b). This is not as high as the dependence on mean T-O, but still a significant relationship. Obviously tetrahedra with longer mean T-O distances display more distortion of their O-T-O angles than those with shorter mean T-O distances, while the strength of the T-O bond, which increases with the oxidation number of the central cation is of less importance.


Page 15 of 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Chemistry of Materials linear regression 0.887DI(OTO)meas.

(a)

DI(OTO)pred

=

0.003

+

The fact that the points for BO4 and PO4 are so closely to each other in the DI(OTO) vs mean T-O plot indicates by itself that the T-O distance largely determines the DI(OTO) values. When the oxygen atoms in a tetrahedron are forced together by short T-O bonds the repulsive forces between them are stronger as when the T-O bonds are longer as is the case for instance for a ZnO4 tetrahedron. A plot of DI(OTO)predicted vs. DI(OTO)measured with R2 = 0.887 where DI(OTO)predicted is derived from a multiple linear regression calculation. We have demonstrated the close relationship between mean T-O and DI(OTO) here for coordination tetrahedra in zeolitic frameworks, but it is holding true as well for nonzeolitic inorganic compounds including for other coordination numbers as shown by Gagné and Hawthorne85 for the similar cases of six-fold coordinated Na+, , ten-fold coordinated K+ and eight-fold coordinated Ca2+ in their dependence on bond-length distortion.

(b)

(c)

Figure 2: Regression analyses of distortion indices DI(OTO) as a function of T-O distances TO and oxidation states OS. Values of DI(OTO)meas, T-O and OS are from Table 1, of DI(OTO)pred from regression equation. The squared Pearson coefficient is defined as (∑(𝒙−𝒙 ̅)(𝒚−𝒚 ̅))𝟐

𝑹𝟐 = ∑(𝒙−𝒙̅)𝟐 ∑(𝒚−𝒚̅)𝟐 . (a) DI(OTO) vs. T-O. Regression line calculated as DI(OTO =-0.094+0.071 TO. (b) DI(OTO) vs. OS. Regression line calculated as DI(OTO)=0.0510.007OS. (c) DI(OTO)pred vs DI(OTO)meas, from multiple

3.8 The site symmetry of coordination tetrahedra in crystal structures of zeolites. In a study of 129 high precision crystal structure determinations of phosphate compounds33 it was observed that the site symmetry of the PO4 groups was usually 1. The highest site symmetry of a regular tetrahedron, however, is 4 3 m, which was not observed for any of these phosphate groups33. Similarly we are seeing a complete absence of this highest site symmetry among the coordination tetrahedra present in the zeolite crystal structures considered here. When describing crystal structures of zeolites we are using terms such as “tetrahedral coordination” without specifying that in fact these tetrahedra are not regular but instead are distorted with distortion indices33 different from 0.0. In principle nothing could stop a coordination tetrahedron located on a site symmetry 1 to have practically geometric dimensions corresponding to a local noncrystallographic site symmetry of 4 3 m. If this were the case we would find for such tetrahedra values of 0.0 for the distortion indices DI(OTO), DI(OO), and DI(TO). We have not found them. A single TO4 tetrahedron in a 4 3 m site connected to four neighboring tetrahedra means that their oxygen atoms must be on threefold axes resulting in T-OT angles of 180° in all four directions thus continuing the high symmetry of the arrangement. Such a configuration has only been described for MTN type zeolites (see chapter MTN in ref. 29) but subsequently it was shown86 that the apparent high symmetry is caused by twinning and the actual symmetry is I 41/a. This changes the T-O-T angles from 180° to 149° in the lower symmetry. It is true that we use terms such as tetrahedral, or for that matter octahedral, coordination, but usually these are not regular polyhedra. In a very recent paper by Carreras et al.87 these authors discuss the varying shapes and symmetries of molecules from a different viewpoint and propose methods to measure symmetry continuously. This might be a way to handle our data as well.



4. T-O BOND LENGTHS AND ATOMIC DISPLACEMENT PARAMETERS

Cruickshank88,89 pointed out in 1956 and 1961 that the angular oscillations of molecules in crystal structures determined by diffraction methods can displace the apparent maxima of the electron density from the true positions of the atoms towards the center of the oscillation. He studied this for the case of anisotropic displacement parameters. Busing and Levy90 attacked this problem in 1964 a little differently and in addition considered displacements due to isotropic displacement parameters as well. They proposed a method to calculate for a given bond, based on measured displacement parameters, a correction to the apparent bond distance as a possible lower bound, an upper bound and a correction based on the joint distribution of the relative motions of two atoms (the riding motion). For about 20 years after that such corrections were applied often in the literature to the results of crystal structure determinations, but as of late this is happening less and less. Liebau31,91 used the general idea of investigating the relationship of displacement parameters with the corresponding observed bond lengths to study their correlation in various silica phases by using the Biso of the involved oxygen atoms. For 25 silica polymorphs he obtained a regression equation which explained 67% of the total variation in the mean Si-O distances: Si-Omean = 1.6220 - 0.0075(11) BO Å,

(4)

where Si-Omean is the mean Si-O bond length for one crystal structure, and BO is the 8π2Ueq displacement of the oxygen atom Liebau also gives a regression equation for the 85 individual Si-O bonds in his sample: Si-Oind = 1.6157 - 0.0070(6) BO Å,

(5)

explaining 63% of the total variation in the individual SiO distances. Furthermore Liebau31,91 suggested to use the following relationship for correcting experimental Si-O bond length values to actual real Si-O: Si-Oind(real) = Si-Oind(exp) + 0.007 BO Å.

(6)

This amounts to a simplified way to get an empirically corrected estimate of the real mean distance Si-O by looking for its value at a B of the oxygen atom when it equals zero instead of making a calculation such as Cruickshank89 or Busing and Levy90 did, who used the actual individual displacement factors for their corrections. Liebau is careful to emphasize that Biso "accounts for both the influence of thermal motions and of static disorder". He also reminds the reader that the slope of 0.007 has been derived from a sample of pure SiO2 frameworks and should not be applied to other silicates as long as it is not proven to hold true in other silicates as well. What has to be added to that is that measuring atomic displacement parameters is fraught with uncertainty because in the refinement process various uncorrected experimental or instrumental factors which are functions of sinθ may be absorbed by the displacement parameters, making them more unreliable than often hoped for. Liebau's work on the relationship between mean bond lengths Si-O and the displacement parameters of their

Page 16 of 28

oxygen atoms did not lead to a flood of analogous studies. Among the few we mention here three. One is the investigation of Wragg et al.41 who obtained in their review of zeolitic SiO2 frameworks a plot of mean Si-O distances vs. Biso similar to the plot prepared by Liebau31,91. Another one is by Boisen et al.92 who analyzed data from several silica structures and a few clathrasils and found that the Si-O bond lengths depended heavily on Biso of the oxygen atoms, but that this did not explain completely the observed variation in the Si-O bond lengths. In addition Gagné and Hawthorne85 pointed out the correlation between atomic displacement and mean bond length for six-coordinated Na polyhedra (see their Figure 21 in ref. 85). Our result (Figure 3) corresponds to Liebau's equation (1), and is in essential agreement with it, even though the data used for our plot are only partly overlapping with Liebau's data : Si-Omean = 1.6153(16) - 0.0080(8) Beq Å

(7).

Our plot is based on 35 mean Si-O distances found in 11 single crystal structure determinations of clathrasil compounds with an R square of 0.73. Especially remarkable among these structures is the MTN-type refinement by Knorr and Depmeier86, which is the first instance for this topological type with ordered oxygen atoms present. The values for the four compounds map nicely in different areas of the plot in accordance with their mean atomic displacement factors. Liebau31,91 used for his correction only the displacement parameters of the oxygen atoms. Downs et al.93 introduced a “simple rigid bond model correction” (srb) based on isotropic displacement parameters (Biso or Beq) employing both the B of the cations and the anions in a bond as discussed in ref. 34. The srb corrected bond length Rsrb is calculated according to 𝑅𝑠𝑟𝑏 = √𝑅2 +

3 8𝜋2

[𝐵𝑒𝑞 (𝑌) − 𝐵𝑒𝑞 (𝑋)]

(8)

with R = observed distance, isotropic displacement parameter Beq (or Biso), in an XYn coordination polyhedron. When we apply the srb correction93 using eqn. (8) the picture is quite different showing no significant dependence of the Si-O distances on the B value. The regression line (Si-Omean = 1.6117(19) - 0.0004(10) Beq Å2) essentially is equivalent to a constant value within the margin of error. The simple beauty of Figure 3 disappears once we look at all the 1282 data points from 36 different types of zeolites in Figure 4 for uncorrected (Fig. 4a) and srb corrected (Fig 4b) distances. The distribution of the entries is much more fuzzy in Figure 4 and it is obvious that despite of the similar overall trend different zeolite types display different mean values of Si-O distances and B values. The big blob of light green diamonds close to the center of the distribution represents the 576 separate entries for the MFI type. In order to simplify the picture we show in Figure 5 just the data points for the heavily populated CAN (104 entries), FER (79 entries), NAT (89 entries) and SOD (53 entries) zeolite types. This plot shows clearly that the data for each topologically different zeolite type


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have not only different mean values of Si-O and B, but also that the slopes of Si-O over B are distinct among the various topological types for the uncorrected distances. Again, the srb corrected data are much closer to distances being more or less independent of the B values. Three of them (NAT, CAN, and FER) even show a positive slope which, however, might not be significant due to still insufficient statistics (number of data points). The corresponding numerical values are presented in Table 5. A list of all srb corrected distances is given in Table S008. The fact that the mean bond lengths observed for the uncorrected FER and MFI zeolite structure types are the shortest among those displayed here is thus simply a consequence of them having the largest mean atomic displacement parameters for their oxygen atoms. We accept this as an empirical observation and do not offer any reasons why this is so. A cursory look at Figure 5 shows that the intersections of the regression lines of the framework populations of individual FTC with the Si-O axis are different for each of the four zeolite types considered here which also applied to the srb corrected distances. If Liebau's predictive equation (4), see above, were of general applicability all slopes had to be the same. Or even if they were not identical the intersections of all of them at B equal zero should be at the same value of mean Si-O. Neither is the case as we can see from Table 5: the predicted values of Si-O range from 1.601 to 1.629 Å, the slopes vary from -0.001 to -0.010 for Si-O over B. Inspecting Figure 3 after having seen Figure 4 makes one wonder whether the nice linear arrangement in the former is not just by coincidence and due to the overall paucity of data. Even within the four FTC shown in Figure 3 we can distinguish different slopes and B values for each of the different zeolite types.

This means that any average values of bond lengths, such as given here in Table 1, or obtained from atomic or ionic radii38, have to be taken with a grain of salt, that is one has to be aware that they could be off the true values one obtains at B equals zero by up to 0.03 Å , provided the numbers shown in Table 5 are typical of nonzeolitic inorganic substances as well. Many of the facts about variations in bond lengths presented above in Tables 1 through 4 might be caused by differences in atomic displacement parameters. Before our present study such relationships were only explored for a small selection of frameworks of SiO2 composition31,41,91,92 while we consider relatively large amounts of data. However, a limitation of our study is that we employed only data from zeolites. Nevertheless, it is likely that the dependence of cation-anion distances on specific structure types is generally true also for nonzeolitic crystal structure types. This would explain some of the results obtained by Slovokhotov55,56, see above, and by Gagné and Hawthorne42, who recently examined mean bond-length variations for 55 cation configurations for ions bonded to oxygen for correlations with various properties of the crystal structures. The data used were not only from zeolites but from inorganic crystals in general. The only statistically significant correlations found were with bond length distortions in 42 of the cases. The SiO 4, AlO4, and PO4 tetrahedra were not among these 42. It was suggested by Gagné and Hawthorne42 that "...structure type has a major effect on mean bond length, the magnitude of which goes beyond that of the other variables analyzed...". We can add to that, that even after applying the bond length correction due to observed mean Bvalues, as we have done here in Table 5, there are still differences in mean bond lengths Si-O which we can only attribute to structure type influences. Each zeolite framework type appears to have a characteristic value of Si-O mean as given in Table 5 (see there the rows CAN, FER, MFI, NAT and



Page 18 of 28

(a)

(b)

Figure 3. Plot of 35 mean tetrahedral Si-O distances [Å] versus mean Beq [Å2] of the oxygen atoms in six precisely determined zeolitic clathrasil frameworks of SiO2 composition belonging to four different FTC types (AST, DDR, MTN, SGT). For detailed data see Table S006 (a) Si-O distance vs. B, no srb correction,. (b) Si-O distance vs. B, with srb correction.


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(a)

(b)

Figure 4: Plot of 1282 mean tetrahedral Si-O distances [Å] versus mean Beq [Å2] of the oxygen atoms in 381 precisely determined zeolitic frameworks belonging to 36 different FTC types. For detailed data see Table S007 (a) without srb corrections (b) with srb corrections.



Page 20 of 28

(a)

(b)

Figure 5: Plot of 325 mean tetrahedral Si-O distances [Å] vs. mean value of B[Å2] of the oxygen atoms in 200 precisely determined zeolitic crystal structures of CAN, FER, NAT and SOD types. The values of the y-intercepts and the slopes of the lines are given in Table 5. The line "all" refers to the regression line of all data points taken together. For detailed data see Table S005. (a) without srb bond corrections. (b) with srb bond corrections.

SOD). Our observation is proof that Gagné and Hawthorne's42 suggestion that mean bond lengths may vary with structure type has considerable merit. We have shown that there are zeolite types with values of bond lengths and displacement parameters characteristic for these specific framework types. It remains to be seen if this is true for nonzeolitic structure types as well. Because of the underlying principle of course this is not limited to mean Si-O distances. We show in Figures 6 and

7 examples of plots for mean Al-O and P-O bond lengths versus B. In both cases we can recognize how the data points of different zeolite types intermingle with each other within the general trend, but with distinct slopes and with different mean values. They can be easily recognized within the plots as having distinct characteristics. The values of B in our sample range up to 8.6 Å2. This means that bond length corrections according to the method of Busing and Levy90 or Downs et al.93 would


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Table 5. Tetrahedral T-O mean [Å] (with and without srb correction) vs. mean value of B[Å2] of oxygen atoms in five topologically different zeolite structure types and for SiO4, AlO4 and PO4 tetrahedra; R2 = Pearson R2; slope = slope of regression line [Å/Å2]; srb = simple rigid bond model; corr. = correction; and predicted (or characteristic) values of T-O mean in column 5. For detailed data see Table S005. Population

T-O, ind mean

B, mean

36 different FTC, see Figure 4

1.603(11)

2.1(7)

CAN

1.611(4)

FER

R2

Predicted

Slope

T-O mean at B = 0

T-O/B

# of structures

# of data points

0.53

1.624(1)

-0.0104(3)

381

1282

1.4(3)

0.10

1.617(2)

-0.0040(12)

97

104

1.593(4)

2.9(10)

0.30

1.601(2)

-0.0028(5)

18

79

MFI

1.593(5)

2.6(4)

0.32

1.609(1)

-0.0064(4)

30

576

NAT

1.619(4)

1.3(6)

0.01

1.615(1)

-0.0005(7)

32

89

SOD

1.617(7)

1.2(6)

0.59

1.629(2)

-0.0102(12)

53

53

CAN,FER,NAT, SOD, see Figure 5

1.610(10)

1.7(8)

0.50

1.624(1)

-0.0080(4)

200

325

Overall ranges of values

1.593-1.619

1.2-2.9

0.10-0.59

1.601-1.629

-0.001-0.010

36 different FTC, see Figure 4

1.615(7)

2.1(7)

0.14

1.623(1)

-0.036(3)

379

1279

CAN

1.620(4)

1.4(3)

0.01

1.617(2)

0.002(1)

97

104

FER

1.611(4)

2.9(10)

0.34

1.604(1)

0.0027(4)

18

79

MFI

1.609(4)

2.6(4)

0.03

1.608(1)

0.0005(4)

30

576

NAT

1.625(4)

1.3(6)

0.16

1.621(1)

0.0030(7)

31

88

SOD

1.624(5)

1.2(6)

0.17

1.628(2)

-0.004(1)

53

53

CAN,FER,NAT, SOD, see Figure 5

1.620(6)

1.7(8)

0.07

1.623(1)

-0.0019(4)

199

324

Overall ranges of values

1.609-1.625

1.2-2.9

0.03-0.34

1.604-1.628

-0.0360.0030

1.736(7)

1.4(5)

0.16

1.744(1)

-0.0055(6)

278

394

1.743(7)

1.4(5)

0.00

1.743(1)

0.0003(9)

278

394

1.522(9)

2.9(12)

0.48

1.538(1)

-0.0052(4)

111

228

1.539(5)

2.9(12)

0.08

1.535(2)

0.0011(3)

111

228

Si-O, without srb bond corr.

Si-O, with srb bond corr.

Al-O, without srb bond corr. 24 different FTC, see Figure 6 Al-O, with srb bond corr. 24 different FTC P-O, without srb bond corr. 31 different FTC, see Figure 7 P-O, with srb bond corr. 31 different FTC



Page 22 of 28

(a)

(b)

Figure 6: Plot of 394 mean tetrahedral Al-O distances [Å] vs. mean value of B[Å2] of the oxygen atoms in 278 precisely determined zeolitic frameworks belonging to 24 different FTC types. For detailed data see Table S005. (a) without srb bond corrections. (b) with srb bond corrections.

have increased many of the T-O values considered here. We can take the number of citations94 to Busing and Levy’s paper over time as an indication that applying that correction is going out of fashion. Their paper has been cited a respectable 1078 times since 1964 which makes it Busing's most cited paper. It was cited a maximum of 71 times in 1971 alone. In the last full year available right now in the Web of Science94, 2017, however, it was still cited 9 times which happens to few papers more than 50 years

old. However, only some of these 9 papers cited it because a bond length correction was made. In most recently published crystal structure determinations a correction for displacement parameters has not been considered, neither has it in the past in papers dealing with mean bond lengths or ionic radii38,42,33 except for the studies by Liebau31,91 and those immediately following him41,92 which all were limited to Si-O bond lengths in compounds of SiO2 composition.


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(a)

(b)

Figure 7: Plot of 228 mean tetrahedral P-O distances [Å] vs. mean value of B[Å2] of the oxygen atoms in 111 precisely determined zeolitic frameworks belonging to 31 different FTC types. For detailed data see Table S005. (a) without srb bond corrections. (b) with srb bond corrections.



Page 24 of 28

Figure 8. Mean T-O bond distances within TO4 tetrahedra in zeolitic crystal structures. The green lines indicate the mean values, the brown areas indicate the observed range of these values.

5. CONCLUSION When we began this work our understanding of the factors influencing the mean bond lengths of cation-anion distances in inorganic compounds was basically at the level described by Gagné & Hawthorne42 in 2017. This meant that we thought the main influence on the bond lengths was the distortion of the T-O distances within the polyhedra and that this would play only a minor role for our sample consisting of polyhedra with small coordination numbers and with mostly strong bonds, where distortion would not be a predominant influence. We also expected the deviations between the mean values obtained by us for given T-O mean distances would agree closely with values reported previously in the literature for compilations of nonzeolitic collections of bond length data. However, as shown in the first part of this paper the mean T-O bond lengths from different compilations can deviate from each other by up to 0.05 Å (see Tables 2 to 4). Some of these deviations must be due to the influence of the mean displacement parameters of the oxygen atoms in the T-O bonds, which apparently change from crystal structure type to crystal structure type, thus making it impossible to define "true" mean bond lengths and ionic radii for large classes of bonds or atoms. In terms of bond lengths each zeolite framework type is different. It

is likely that this is also true for nonzeolitic structure types. There is even an indication that zeolitic frameworks as a group differ from other structure types as is evident from the puzzling shortening of their mean P-O bond lengths shown in Table 4. Presumably their mean atomic displacement parameters are on average larger than they are in other phosphates. The mean T-O bond lengths of ten coordination tetrahedra occurring in zeolites presented in Figure 8 are not corrected for displacement parameters inasmuch as almost all crystal structures described these days are uncorrected for temperature effects and/or statistical disorder of individual atoms or groups. We want our data to be useful for currently practicing scientists. Except for a wealth of statistical material about the geometry of tetrahedral groups in zeolitic structures our most important general findings are: 1. the strong relationship between mean T-O bond lengths and the angular distortion indices DI(OTO) of the various tetrahedra; 2. the dependence of the mean T-O bond lengths on the various mean B values of a given zeolite framework type, that means basically on the topology of the zeolite type;


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3. the total absence of the highest site symmetry 4 3 m among the coordination tetrahedra (around Si, Al, P, Zn, Be, Ge, B, As, Ga, and Co) present among the zeolite crystal structures considered here. These three observations are likely to be true for nonzeolitic inorganic structure types as well. For points 1. and 2. this has been suggested already by Gagné & Hawthorne84 based on entirely different data. Consequently our reliance on very precise atomic and ionic radii valid across many different crystal structure types may be misplaced. They vary according to structure type.

■ ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: xxxxxx. References to the crystal structures from which the data employed in the tables and figures in this paper were taken and detailed listings of the individual values of bond lengths, bond angles O-T-O and B values used in the tables, plots and histograms. ■ AUTHOR INFORMATION Corresponding Authors †E-mail: [email protected] ‡E-mail: [email protected] ORCID Werner H Baur: 0000-0002-0804-0416 Reinhard X Fischer: 0000-0002-2643-3387 Author Contributions W.H.B. and R.X.F.: Equal contributions. Notes The authors declare no competing financial interest. ■ ACKNOWLEDGMENTS We thank Olivier C. Gagné and F. C. Hawthorne for preprints of several of their papers, Michael Fischer for discussions of bond corrections, and Johannes Birkenstock for his assistance with the multiple regression analysis. We also thank Olivier C. Gagné for a review of the manuscript of this paper.

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The floppiness of it all: bond lengths change with atomic displacement

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