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Continuous chirality analysis of hexacoordinated tris-chelated metal complexes
Crystal Engineering 4 (2001) 179–200 www.elsevier.com/locate/cryseng
Continuous chirality analysis of hexacoordinated tris-chelated metal complexes Santiago Alvarez
a,*
, Mark Pinsky c, d, Miquel Llunell b, David Avnir c
a
Departament de Quı´mica Inorga`nica, Centre de Recerca en Quı´mica Teo`rica, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain b Departament de Quı´mica Fı´sica, Centre de Recerca en Quı´mica Teo`rica, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain c Institute of Chemistry and The Lise Meitner Minerva Center for Computational Quantum Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel d Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem 91904, Israel Received 13 November 2000; accepted 26 January 2001 (Refereed)
Abstract We use the notion that symmetry can be evaluated on a continuous scale for the quantitative chirality analysis of tris-chelated metal complexes. After reviewing the Continuous Symmetry Measures methodology and its resulting Continuous Chirality Measure, we analyze the chirality of these complexes in terms of the specific contributions from the innermost ML6 shell and from the ligand shells. The chirality behavior of seven families of tris(chelate) complexes was analysed. The results of the analysis point to a universal behavior and were found to be consistent with those for a molecular model. The maximum expected chirality for the octahedral to trigonal-prismatic route is at a twist angle of 23° and, remarkably, we found a complex that occupies this point, and have thus identified the most chiral hexacoordinated complex: (AsPh4)2[W(S2C6H4)3]. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Chirality; Stereochemistry; Coordination compounds; Molecular symmetry; Quantitative chirality
* Corresponding author. Tel.: +34-93-4021269; fax: +34-93-4907725. (S. Alvarez); tel.: +972-26585332; fax: +972-2-6520099 (D. Avnir). E-mail addresses: [email protected] (S. Alvarez), [email protected] (D. Avnir). 1463-0184/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 1 4 6 3 - 0 1 8 4 ( 0 1 ) 0 0 0 1 3 - 2
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1. Background 1.1. Measuring symmetry and chirality on a continuous scale The concept of symmetry has attracted virtually all domains of intellectual activity and has strongly influenced the sciences and the arts. It has functioned as a condensed language for the description and classification of order within shapes and structures; as an identifier of inherent correlations between structure and physical properties of matter; and as a guideline in artistic and practical aesthetic design. Our study of symmetry is based on the recognition that the majority of molecular structures do not represent exact, ideal symmetries. Distortions due to crystal packing, Jahn–Teller effects, or ligand and substituent effects are the rule not the exception. To realize it one should refine the resolution of observation — spatial or temporal — up to the point where it becomes evident. It appears that symmetry has served in many such instances as an approximate, idealized descriptive language of the physical world, beyond the atomic scale. While it is true that an imprecise language helps in grasping complex situations and in identifying first order trends, the danger of missing the full picture because of a vague description is always awaiting the user of the current symmetry language. This motivation has led us to propose that it is natural to evaluate on a quantitative scale how much of a given symmetry there is in a structure [1– 3]. Thus, we are treating symmetry as a structural property of continuous behavior, complementary to the classical discrete point of view. A continuous symmetry scale should be able to express quantitatively how far is a given distorted structure from ideal symmetry, at any temporal resolution, at any spatial resolution, and relative to any symmetry. Towards this goal, we have designed a general symmetry measurement tool, based on a definition that is minimalist. Our answer to the question “How much of a given symmetry there is in a given structure?” is then: Find the minimal distances that the points of a shape have to undergo, in order for it to attain the desired symmetry. To translate this definition into practice, we have developed the Continuous Symmetry Measure (CSM) methodology and computational tool, described in the next section. Using this measurement procedure it became possible to evaluate quantitatively how much of a given symmetry exists in a non-symmetric configuration, what is the nearest symmetry of a particular configuration, and what is the actual shape of the nearest symmetric structure. Closely related is the Continuous Chirality Measure (CCM) which evaluates the distance to the nearest achiral symmetry; much of our activity was devoted to this special aspect of symmetry measurements. We have demonstrated the feasibility and versatility of this approach on two levels. The first one is purely geometric. Here we developed solutions for specific problems such as: 앫 Evaluation of the degree of bilateral symmetry [4]; 앫 measurement of the symmetry content of distorted classical Platonic polyhedra [5]; 앫 assessment of the symmetry content of objects which contain an element of randomness in their construction [6];
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앫 analysis of the concepts of left/right handedness and mirror symmetry [7]; 앫 evaluation of symmetry distributions within populations [8] and more. The second level concentrated on applications of the symmetry measure to real problems in structural chemistry at large. Examples include symmetry analyses of molecules [9–14], of (macroscopic) crystals [15], of dynamically changing structures [16], of small (3–12 molecules) clusters [17] and of large disordered aggregates [18]; of enzymes and their activities [19,20], and more. Three major findings emerged from these studies: 1. The symmetry measure describes the molecular world in a well behaved way: Its trends of change agree with intuition and reflect what is visible to the eye and what has been expected on a qualitative level, before measurement was possible. 2. Hitherto unknown quantitative correlations between symmetry and physical/chemical/biochemical properties have been revealed. 3. Far more than before, the importance of a global-shape descriptor — symmetry — for the quantitative observation and analysis of structural chemistry, in distinction from the classically specific geometry descriptors, has been revealed. In conclusion of this Background, we mention that the CSM methodology is applicable, beyond the molecular level analysis, to most other domains of the natural sciences, of the social sciences, and of the arts, where symmetry is an issue, either as a real feature or as an abstract one. Examples are the quantitative analysis of the symmetry of hand axes of early man [4], and symmetry analysis of facial features [21]. Finally, for other approaches [22–28] to chirality analysis, see our reviews [1,29]. 1.2. Computational aspects: the continuous chirality measure (CCM) The evaluation of the chirality content of an object by the CCM approach is based on estimation of the degree of the nearest achiral symmetry content within the CSM framework [2,30]. As explained above, the measure is a function of the minimal distance that the vertices of the object have to undergo to attain the desired symmetry (achirality, in our application here). Thus, given the original size-normalized vectors of the locations of the n vertices of the molecule, {Pi}, given the desired G-symmetry, and given the (size-normalized) coordinates, {Pˆ i}, of the (searched) symmetric (achiral) structure, the CCM value, S⬙(G), is calculated according to the definition:
冘 n
1 |P ⫺Pˆ |2 S⬙(G)⫽ ni⫽1 i i
(1)
The symmetry transform, seeking the set {Pˆ i} which is at minimal distance from {Pi}, is the heart of the method. Several algorithms have been developed towards this goal, and the interested reader can find them in previous papers [2,5,30,31]. To calculate the CCM, S⬙(G) is evaluated for every (relevant) achiral point group, and the minimal value is chosen:
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S⬘(G)⫽min[S⬙(G)]
(2)
In practice, one needs not to screen over all achiral G groups, because quite often one can identify the potential nearest achiral symmetry element simply by inspection; in most cases, and in all cases analysed here, the nearest point group that is achiral contains (at least) one mirror plane. Finally, we found it convenient to expand the scale by a factor of 100: S(G)⫽100·S⬙(G)
(3)
and these are the values reported here. The minimal value of S(G) (in short, S) is zero if the object has the desired Gsymmetry, which, in our application means that it is achiral. The maximal value of S⬘(G) is limited by 1 (100 on the expanded scale) due to size normalization, which is performed to make the CSM (and the CCM) invariant to scale changes. It is accomplished by division of the vertex distances of the object from the center of mass by the root mean square (r.m.s.) length of all these vectors: Pi⫽
Poriginal i r.m.s
(4a)
where
r.m.s⫽
冪
冘 n
|Poriginal |2 i (4b)
i⫽1
n
This size normalization assures the scale of S(G) to be bound by an upper limit of 100 because:
冘 冘 n
100 S(G)⫽
冘 冘 n
|Poriginal −0|2 100n i ⫽
i⫽1
冪
n
|Poriginal |2 i
i⫽1
2
|Poriginal |2 i
i⫽1
n
n
⫽100
(5)
|Poriginal |2 i
i⫽1
n
This maximal value is obtained if, e.g., one wishes to find the degree of pentagonality of a hexagon: The nearest pentagon to a hexagon is the collapsed pentagon into a single center point, the distance of which is, by definition, 100. However, for achiral G’s it never reaches 100, because the distance to the mirror plane is always smaller than the distance to a central point. One of the main features of the CCM approach, which we use in the next Section, is that it not only computes the chirality content, but also provides the actual shape of the nearest achiral object, i.e., the structure of the set of {Pˆ i}. In most of its previous applications, the CSM method was used for evaluating the
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chirality of the molecule as a whole. However, in what follows we analyse large families of molecules. As such, it is then more relevant to analyse the chirality of molecular fragments, which are common to all members of the family, and which therefore provide a wider perspective. Analyzing the chirality of the whole molecule for all the family members will mask trends, if such exist, because of the wide variability in the specific structures of the substituents and ligands. 2. Chirality analysis of experimental structural data: hexacoordinated trischelated complexes In this section we exemplify the above concepts and methodology by analysing the chirality of metal complexes of the general formula [M(chel)3], where chel represents a bidentate ligand. Interest in the well established [32–37] chirality of such complexes stems mainly from their ability to induce chirality in catalytic processes. In view of a recent study in which we have shown [38] that the hexacoordinate ML6 complexes with monodentate ligands are chiral for twisted geometries (1) intermediate between the octahedron (q=60°) and the trigonal prism (q=0°), it becomes interesting to ask, what happens when we combine in the same molecule tris-chelation with a twisted coordination sphere? Is there a synergetic enhancement of chirality or do they cancel out [30]? Thus, for the study of the chirality of tris(chelate) complexes it is useful to consider the atoms in the molecule as pertaining to successive shells, as exemplified in 2 for the case of tris(dithiolene) complexes. The first shell comprises the metal and the coordinated donor atoms, the second shell is formed by the spacers that connect each pair of donor atoms in a bidentate ligand, and the third shell is formed by the rest of the ligand atoms. Thus, we shall refer to the chirality measures as S1 (first shell only), S2 (second shell only), S1+2 (first and second shells together, and Sf for that of the full molecule (devoid of the hydrogen atoms). Accordingly, it is important to differentiate the twist angle of the first shell, q1, from that of the second shell, q2.
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2
The analysis below will reveal the following points: (i) In all hexacoordinated complexes, the first shell is chiral at geometries intermediate between octahedral and trigonal prismatic; (ii) tris(chelate) complexes are chiral even if the coordination polyhedron is octahedral (i.e., q1=0°), because of the inequivalence of the X-M-X bond angles belonging to a bidentate ligand and to two different ligands (first shell), or because of the lower symmetry imposed by the skeleton of the bidentate ligands (second shell); and (iii) for a trigonal prismatic coordination sphere, neither the first nor the second ligand shell is chiral. Next, we analyze the contributions of the first and second shells to the chirality of tris(chelate) complexes, both separately and in combination. 2.1. The chirality of tris(dithiolene) and tris(diselenolene) complexes: models and experimental data analysis We shall use [M(S2C2)3] as a basic molecular model of a metal tris(dithiolene) complex (schematically shown in 3; see 1 for details of the model), and will compare the chirality-response of this model to structural variations, with the corresponding experimental data for tris(dithiolene) and tris(diselenolene) complexes. Having in
1 The rotation was accompanied by gradually changing the bond angles around the metal in order to fulfill the definition of ideality, namely equal edges within each of the polyhedra. Bond angles varied from 90° in the octahedron to 81.79° in the trigonal prism, and bond to edge lengths ratio (normalized bite) from 1.414 in the octahedron to 1.309 in the trigonal prism. The full model used for calculations of S1+2 consisted in a M(S2C2)3 group with bond distances representative of those found in crystal struc˚ (increasing with q1). Chirtures of tris(dithiolene) complexes: M-S=2.34, S-C=1.76 and C-C=1.35–1.55 A ality measures S1 correspond to the MS6 fragment of this a model, and S2 to the collection of the three C2 groups.
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mind this model, we shall later on analyze the chirality of other families that are characterized by other types of bidentate ligands. These include tris(chelate) complexes with planar chelate rings, namely those in which the bidentate ligands are either all bipyridines, dithiocarbamates, β-diketonates or catecholates; and finally, we will present a preliminary analysis for the analogous compounds in which the chelate rings are puckered, such as the tris(dithiolate) and tris(ethylenediamine) complexes.
3
2.1.1. The chirality of the first shell As mentioned above, the structures of the MX6 core in many [M(chel)3] complexes can be described as octahedra that are distorted by twisting two parallel faces of the octahedron around a trigonal symmetry axis. This distortion route is known as the Bailar pathway [39], and is depicted in 4 for the tris-chelated case, where the thick lines represent the edges occupied by the bidentate ligands. At the two extremes of that path one has either a perfect octahedron (q1=60°, Oh symmetry), or a perfect trigonal prism (TP, q1=0°, D3h symmetry, all edges equal to each other). In-between (0⬍q1⬍60°), the molecular symmetry is lowered to D3. On the CCM scale, the outcome is that while the S value of the MX6 group is zero for the two ideal polyhedra, in between it has non-zero values, and should pass through at least one maximum. The S1 curve in Fig. 1 shows the theoretical line for a model of this twist route1 and it is seen that, indeed, a maximum chirality value (the chiramer [31]) appears at q1=23°. In Fig. 2(a), where the experimental data of tris(dithiolene) and tris(diselenolene) complexes are superimposed on this theoretical line, one can see that the fit is remarkable. The fact that all structures fall along that line, suggests that they represent the Bailar pathway from Oh to TP; indeed, the analysis of the bonding parameters of the actual structures confirms that these are of D3-type, as illustrated in Fig. 3 for three representative compounds. Since this behavior was also found among homoleptic complexes with alkyl, aryl or thiolato ligands [38], we believe that the chirality behavior of S1 characterizes generally most of the Bailar-twisted hexacoordinated complexes. Below we show (Fig. 7) that this is indeed the case for practically all complexes analysed in this study.
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Fig. 1. Chirality measures as a function of the twist angle (see 1) for a model tris(chelate) complex M(S2C2)3 (see 2). Shown are the chirality variation of the first shell S1 (defined in 2, thick line); that of the second shell, S2, and that of the first and second shells combined, S1+2. The distance of the first shell to an ideal trigonal prism is also shown (dashed line).
4 Next, let us comment on the maximum at q1=23° of the S1 line (Figs. 1 and 2(a)): It is seen that this maximum is a non-differential point, and this observation deserves a further comment: The source of non-differentiability is inherent to the search for the closest symmetric structure, and reflects situations where there are two (or more) different nearest types of achiral shapes [16]. In other words, the change of the closest achiral structure occurs along the crossing line of two chirality value hypersurfaces (surfaces of the chirality as a function of all the structural parameters that
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Fig. 2. Chirality measures as a function of the twist angle, calculated from structural data of tris(dithiolene) complexes (triangles): (a) for the MS6 shell, S1, (b) for the (C2)3 shell, S2, and (c) for the combination of the first two ligand shells, MS6C6 (S1+2). The continuous lines correspond to the molecular model M(S2C2)3 as in Fig. 1. See Supplementary Material for data and references: www.elsevier.nl/gejng/10/15/30/38/55/88/show/index.hH.
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Fig. 3. Structures of the inner shells and rotation angles q1 and q2 of the tris(chelate) complex showing the largest value for the chirality measure S1, [W(C6H4S2)3]-, and for those presenting the largest values of S2 and S1+2, [M(acac)3] (M=Mn, Rh).
affect it), each belonging to a different achiral structure. A pathway leading from one hypersurface to the other is therefore non differentiable at the crossing line of these surfaces. Since the actual shapes of the nearest achiral structures can be determined by the CCM methodology (the set of {Pˆ i}, see previous Section), one can also determine the structural nature of these hypersurfaces and what happens at the nondifferential crossing point. It is found that from q=60° (the ideal octahedron) and to the left, the nearest achiral structure is a distorted octahedron that is bent along the z-axis (see 5) and contains only two symmetry planes. Between the ideal TP on the left of Fig. 1 (q1=0°) and the maximum of the S1 line at q1=23°, the nearest achiral structure is always a D3h-TP varying in height from point to point. At the peak, one has crossing of the two surfaces, i.e., the chirality measure represents an equal minimal distance to two achiral structures: a TP and a bent z-axis octahedron 5. This non-differentiability can be removed, if the restriction for nearest achirality is replaced with a restriction for a specific nearest achiral shape. For instance, if one asks, what is the degree of chirality with respect to the nearest TP, then indeed a continuous line would be obtained. To conclude this section we note that the
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maximum at 23° is nicely exemplified by the structure [40] of (AsPh4)2[W(S2C6H4)3], with S1=2.40, the most chiral core-shell identified in this study (shown in projection in Fig. 3).
2.1.2. The chirality of the second shell We move now to the chirality measure of the second shell alone, S2, that in our molecular model [M(S2C2)3] is constituted by three separate C2 fragments (see 2). Although each C2 spacer, separately, is achiral, their mutual spatial arrangement is helical. The theoretical S2 values for this model are shown in Fig. 1. In analysing this behavior, it is important to notice that while the correlation is still with the twist angle of the MS6 core, q1, the actual twist of the spacers, q2, is smaller, as illustrated in Fig. 4: Here, a first shell twist of q1=60° is translated into a much smaller twist angle in the second shell of 15°. In fact, it becomes possible to correlate S2 directly with q1 instead of q2, because the two twist angles are linearly correlated: q2=⫺0.1+0.25 q1 (correlation coefficient: r2=0.986). Returning to Fig. 1, we see that at q1=0°, where the bidentate ligands align with the sv symmetry planes of the trigonal prism (thus are also symmetric with respect to sh), S2=0, as is S1. However, unlike S1, S2 increases continuously with the twist angle. This continuous increase results because, unlike S1, the nearest achiral structure to the collection of the three fragments is only of one type at the interval of rotation angles of Fig. 1, namely a parallel arrangement of the three C-C fragments aligned along the edges of a trigonal
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Fig. 4. The relation between the polyhedron twist angles of the first and second shells in the M(S2C2)3 model (continuous line) and in the experimental structures of tris(dithiolene) complexes (squares).
prism. Finally we comment on the theoretical behavior at twist angles higher than 60° (see 6): when the octahedron (q1=60°) is distorted by an anticlockwise rotation (6, right) towards an alternative trigonal prism corresponding to q1=120°, S2 continues to increase. This is in contrast to what is found for the first shell, in which both clockwise and anticlockwise rotations of the octahedron are equivalent. The experimental behavior for the family of transition metal tris(dithiolene) and tris(diseloenolene) complexes (Fig. 2(b)), follows the model nicely. Our molecular model provides a good representation of the whole range of experimental structures for these complexes. The good fit indicates that the linear correlation between q1 and q2 is kept; and this, in turn, is apparently due to the conformational rigidity of the five-member chelate rings (imposed by the sp2 carbon atoms at the bridges). 2.1.3. The chirality of the first and second shells, combined We are ready now to look at the combined chirality measure, S1+2, the theoretical behavior of which is shown in Fig. 1 for the same model. Let us compare the evolution of S1+2 to that of S2 and S1 upon rotation (Figs. 1 and 3). First, it is seen that S1+2 behaves in a similar way to S2, namely, it increases continuously from zero as the rotation angle departs from q1=0°, although in a steeper way. The similarity in behavior is a reflection of referring to the same type of nearest achiral object in both cases (that has the sh and sv reflection planes of the nearest trigonal prism) throughout the full range of the rotation shown. We term chirality measures that refer to the same type of improper symmetry group (such as the S1+2 and S2 measures in our case) as commensurate. On the other hand, incommensuration exists between S1 and both S2 and S1+2 at rotation angles larger than 23°, as already explained above in the discussion about the S1 maximum at this angle. It is seen that in the commensurate zone (0°⬍q1⬍23°), S1 increases quite steeply with the angle but S2 is much less sensitive. Because of the commensuration, S1+2 behaves as some average of S1 and S2, but we do not have yet the tools to derive theoretically S1+2 from its components.
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Yet on a qualitative level we can explain now why S1+2 rises so steeply in the incommensurate zone (q1⬎23°): S1+2 in this zone is still an average of two lines: S2 and the very steep line that would have been obtained for S1 if the chirality measure (which looks for the minimal distance to any relevant achiral object) would have been replaced with the distance to a trigonal prism (a specific achiral object); this is shown as a dashed line in Fig. 1. Finally, we comment on the crossing points of the lines in Fig. 1: The structures that correspond to such points are isochiral, a concept that is possible only if chirality is treated as a quantitative measurable quantity. S1 and S2 cross at 41°, namely S1=S2 at this angle, and it is seen that the additivity of the S values is not straightforward: S1+2 is not their sum. From that point of view, the other crossing point, between S1 and S1+2 (at q1=30°) is particularly interesting because it can be visualized as indicating that at that angle, S1 contributes nothing to S1+2. Experimental results supporting the theoretical S1+2 prediction appear in Fig. 2(c), showing the correlation between the chirality measures of the tris(dithiolene) (and diselenolene) complexes as obtained from their crystal structural data,2 and their twist angles (q1). It is seen that this family covers a wide range of twist angles between the octahedron and the trigonal prism, including the commensurate and incommensurate regimes. Note the case of (NEt4)[W(S2C6H4)3] [41], which appears to be the one that is closest to the crossover point (q1=29.6°) of S1+2 and S2. 2.1.4. The chirality of the full molecule: Sf Our final stage is to ask whether the chirality measure of the first two shells correlate with that of the full molecule, i.e., of Sf. To obtain a preliminary answer, we calculated the chirality measures for the complete molecules (with the exception of the hydrogen atoms) of several representative dithiolene complexes distributed along the Bailar path, and plotted them alongside the S1+2 values (Fig. 5). It is seen that for these complexes one always gets S1+2⬎Sf. We will return to this observation in the next section. 2.2. An overview of the chirality behavior of tris-chelated complexes 2.2.1. The analyzed families How general are the models and the chirality behavior detailed in the previous Sections? To answer this question, we analyzed six additional families of important tris(chelate) complexes. Four of them form planar chelate rings, having as ligands bipyridine, β-diketonates, dithiocarbamates or catecholates. Two families explored have ligands that form puckered chelate rings, namely aliphatic dithiolates and ethylenediamines. The various ligands differ in the donor atoms and in the size of the spacers among them, as summarized in Table 1. To these we add as a seventh type
2
Only four molecules (CSD refcodes pabzta, kabcar, kaxmur and selkoj) were found that strongly deviate from the general pattern shown in Fig. 2, and these were disregarded because they present distortions from the octahedron other than the trigonal twist.
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Fig. 5. Chirality measures for representative tris(dithiolene) complexes along the Bailar pathway, as a function of the twist angle. Shown are S1+2 (black circles) and Sf, the chirality values of the full molecules (except the hydrogen atoms; white circles). The upper continuous line corresponds to S1+2 for the M(S2C2)3 model, and the lower line to a least-squares fitting of the experimental Sf values. Table 1 Families of tris(chelate) complexes studied, indicating their main structural features, the definitions of the first and second shells, and the specific members of each family selected for the calculation of the chirality measure of the full molecule Family
General formula
Donor
1st shell 2nd shell Selection
R
Dithiolenes
[M(X2C2R2)3]
X
MX6
(C2)3
C⬵N
Dithiocarbamato Bipyridine
[M(S2CNR2)3]
S
MS6
(C)3
[M(N2C10R6)3]
N
MN6
(C2)3
β-diketonato Catecholato
[M(O2C3R2R’)3] [M(O2C6R4)3]
O O
MO6 MO6
(C3)3 (C2)3
Ethylenediamine Dithiolato CrCl3
[M(N2R4C2H4)3] [M(S2C2R4)3] “CrCl6Cr3”
N S Cl
MN6 MS6 CrCl6
(C2)3 (C2)3 (Cr)3
Maleonitriledithiolene Benzodithiolene Diethyldithiocarbamate Unsubstituted bipyridine Acetylacetonate Unsubstituted catecholate
C4H4 Et H Me, H H
CrCl3 the bidentate ligand of which is ClCrCl. The ClCrCl group forms from the six chlorides that surround each Cr(III) ion in an approximately octahedral geometry, sharing edges with three neighboring octahedra [42]. All other details of the complexes and their chirality values are collected in the Supplementary Material. 2.2.2. The universal behavior of the chirality of the first shell: As seen in Fig. 6, which displays the first shell chirality behavior of all nine families, the model we proposed is universal. In Fig. 7 we show the S1 behavior of
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Fig. 6. Chirality measures of the first shell, S1, as a function of the twist angle, q1, calculated from structural data of seven families of tris(chelate) complexes (summarized in Table 1) and for CrCl3. The continuous line corresponds to the molecular model of M(S2C2)3. See Supplementary Material for data and references.
each family separately. It is seen, for instance, that most of the structures are found with rotation angles between 30 and 60°, indicating a preference for an octahedral stereochemistry around the metal atoms, but also showing that Bailar distortions are quite common. The interesting exceptions are the Y(III) dipivaloylmethanato [43,44], a Cd(II) acetylacetonate [45] and tris(9,10-phenanthrenequinone)molybdenum [46], all with nearly perfect trigonal prismatic geometry. Another particular example is that of a Ti(III) dinuclear complex with a scaffolding bis(catecholato) ligand [47], that presents one of the most achiral first shells (S1=2.14 at q1=23.3°). The dithiocarbamates appear to be the only family for which the octahedral geometry cannot be attained: all structures present twist angles between 31 and 46°. The β-diketonates constitute the only family among those studied here that present anticlockwise twist angles (i.e., q1⬎60°). The different behavior of these two families is associated to the different normalized bites of those ligands (average edge to bond ˚ , respectively, for first row transition metal length distances of 1.21(4) and 1.41(5) A atoms, according to our structural database survey), since it is a well established trend that in tris(chelate) complexes a small normalized bite favors a small twist angle and viceversa [48]. In summary, the chirality of the first shell for all the studied tris(chelate) complexes shows the same dependence on the twist angle q1 as the molecular model used in the previous section, regardless of the nature and size of the metal and donor atoms, the number of spacer atoms, or the planar or puckered nature of the chelate rings. 2.2.3. The chirality of the spacers, S2 Unlike S1, the behavior of S2 is not expected to follow the M(S2C2)3 model, because each bridge follows the twist of the first shell in a different way. Nevertheless, in the case of planar chelate rings, still the two twist angles q1 and q2 are correlated (e.g., q2=⫺0.9+0.34·q1, r2=0.926 for catecholates; q2=0.1+0.50·q1, r2=0.970 for β-diketonates) as discussed above for the dithiolenes. In contrast, when
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Fig. 7. Breakdown of Fig. 6. into some specific ligand families: (a) bipyridines, (b) dithiocarbamates, (c) β-diketonates, (d) catecholates, (e) ethylenediamines, (f) dithiolates.
flexibility in the spacers is allowed by moving from sp2 to aliphatic sp3 carbon atoms, the twist angles of the two shells are found to be uncorrelated as in the families of dithiolato and ethylenediamine complexes. Consider as examples two metal complexes with aliphatic dithiolates [49,50] that present intermediate q1 twist angles of 38 and 30°, but very small S2 values (0.05 and 0.23, respectively) because their C2 groups are nearly aligned along the edges of trigonal prisms (q2 twist angles of 2 and 6°, respectively; see Fig. 8 for [Ti(S2C2H4)3]2⫺) [49]. This behavior appears also for other aliphatic spacers. For instance, in most of the tris(ethylenediamine) complexes with 38°⬍q2⬍56°, the corresponding q2 angles are between 0 and 30°, resulting in low S2 values. As a result of the different relationship that each type of ligand presents between q1 and q2, even the rigid planar families do not follow the model closely (Fig. 9) but, interestingly, are characterized by higher chirality values. The chirality measures of the second shell in the bipyridine complexes that have the same set of C2 spacers
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Fig. 8. Projection view of the [Ti(S2C2)3] core of the [Ti(S2C2)3]2- anion showing the trigonal faces of the rotated octahedron formed by the six sulfur atoms and the helix formed by the bidentate ligands (left), together with a perspective view of the trigonal prismatic alignment of the three C-C bonds (middle) and a side view showing the two faces of the S6 distorted octahedron above and below the C6 trigonal prism (right). q1 and q2 are the twist angles between the trigonal faces of the S6 octahedron and of the C6 prism, respectively.
as our model seem to be closer to the model line than the β-diketonate (C3 spacers) and the dithiocarbamato (C spacers) derivatives. However, we now show that the procedure for fitting a model is general: By changing the parameters of the model to parameters that reflect the specific characteristics of the spacers,3 excellent fits are obtained, as shown in Fig. 9. Among the complexes with planar chelate rings studied, the C3 spacers of several β-diketonato complexes with twist angles ⬇70° z [51] constitute the most chiral second shells (illustrated by [M(acac)3], M=Mn or Rh, Fig. 3), a fact that is associated with a large bite. It corresponds to an anticlockwise distortion of the octahedral coordination sphere, for which we have shown above that chirality measures must increase with z. We note also that the second shell of the dithiocarbamato complexes is chiral at all rotation angles, since the C spacers of the three ligands remain coplanar throughout the Bailar twist. 2.2.4. The combined chirality, S1+2 The excellent fit to the adjusted models is seen also for the S1+2 plots in Fig. 10. Again, the β-diketonato complexes, result in large S1+2 chirality measures (S1+2⬇7.6 for [Mn(acac)3], Fig. 3). An interesting case is that of the S1+2 value of CrCl3 [52] which is topologically equivalent to the dithiocarbamates with its bidentate ClCrCl 3
The following parameters were used in the molecular models: In the dithiocarbamato model, the S˚ upon rotation, the C-S distance was set to 1.712 A ˚ , and the SS distance was kept constant at 2.845 A ˚ (for q1=0°) C-S bond angle fixed at 112.4°, while the M-S distance was allowed to vary between 2.387 A ˚ (for q1=53°), implying changes in the S-M-S bond angles from 70.7 to 77.7°. In the bipyridine and 2.318 A ˚ , the C-N and C-C distances to 1.369 and 1.475 A ˚, model, the N-N distance was kept fixed at 2.616 A ˚ (for and the C-C-N bond angles at 114.7°, while the M-N distance was allowed to vary between 2.090 A ˚ (for q1=61°), implying changes in the N-M-N bond angles from 73.9 to 91.0°. In q1=0°) and 1.938 A ˚ , the C-O and C-C distances at the acetylacetonato model, the O-O distance was kept fixed at 2.809 A ˚ , and the C-C-O and C-C-C bond angles at 126 and 124°, while the M-O distance was 1.272 and 1.390 A
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Fig. 9. Chirality measures for the second shell (S2) of tris(β-diketonate), tris(bipyridine) and tris(catecholate) complexes (black triangles, white squares and white circles, respectively) as a function of the twist angle, obtained from the experimental crystal structural data. The corresponding values for the tris(dithiocarbamate) complexes are zero in all cases and are not shown. The experimental values are compared with the curve for the M(S2C2)3 model in (a) and with those for M(N2C2)3 and M(O2C3)3 models consistent with the structural parameters in the diketonato and bipyridine complexes 3 (upper and lower curves, respectively) in (b).
ligands: the CrCl6 shell is nearly octahedral (q1=53°) and achiral (S1=0.07) and the second shell formed by three neighboring Cr atoms is strictly achiral (S2=0.00, just as the C atoms in dithiocarbamato complexes). However, since S1 and S2 are incommensurate (see Section 2.1, above), when the two achiral shells are combined, pronounced chirality appears (S1+2=6.63). Again, it is worth noting that the different geometries of the topologically equivalent chelate rings in CrCl3 and in dithiocarbam˚ , respectively) result in a ates (cf. Cr-Cr and S-S distances of 3.18 and 2.83(5) A significantly smaller S1+2 value for the former than expected from the dithiocarbamate molecular model in Fig. 10. ˚ (for q1=0°) and 1.923 A ˚ (for q1=74°), implying changes in the O-Mallowed to vary between 2.146 A O bond angles from 81.8 to 93.8°.
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Fig. 10. S1+2 chirality measures along the Bailar pathway for the families of tris(dithiocarbamato) (crosses), tris(bipyridine) (triangles), tris(catecholato) (squares) and tris(β-diketonato) (circles) complexes, as deduced from the experimental structural data. The continuous lines correspond (from top to bottom) to the M(S2C)3, M(N2C2)3 and M(O2C3)3 models.3
From the results for the different models, two generalities regarding S1+2 can now be drawn: 1. For donor atoms of about the same size and same twist angle, S1+2 decreases with increasing number of spacer atoms (i.e., from C2 in bipyridine or catecholate to C3 in diketonate; and from C1 in dithiocarbamate to C2 in dithiolene). 2. For ligands with the same number of spacer atoms, S1+2 decreases with the size of the donor atoms (i.e., smaller for the dithiolenes compared to bipyridine or catechol) or with the size of the spacers (cf. Cr and C spacers in CrCl3 and dithiocarbamates, respectively).
2.2.5. The chirality of the full molecule: Sf Finally, we ask whether the evaluation of the chirality of a part of a molecule represents the chirality behavior of the whole molecule. For that purpose we looked for a possible correlation between the chirality measure of the full molecule without hydrogens (Sf) and S1+2. Such correlation was indeed found: Calculating the chirality values for the most common members of these families (identified in Table 1 under the heading Selection), it was found that Sf is linearly correlated with S1+2 in all cases (Fig. 11). Specifically, it is seen that the benzodithiolene and maleonitriledithiolene complexes obey the same relationship between the two chirality measures; that in the cases of dithiolene, catecholato and dithiocarbamato complexes, Sf is always smaller than S1+2, whereas in the other cases (bipyridine and acetylacetonate) it is always larger than S1+2; and that the catecholato and dithiolene complexes present practically the same linear correlation between the two values. This is an important result because it suggests that S1+2 provides an adequate quantitative estimate of the
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Fig. 11. The chirality measures of the full molecules (with the exception of hydrogen atoms), Sf, for different families of tris(chelate) complexes as a function of S1+2. The chelating ligands (Table 1) are: bipyridine (white squares), β-diketonates (white circles), dithiolenes (black circles), catecholates (black squares) and dithiocarbamates (white triangles). The continuous lines are least-squares linear fittings of the experimental data to which the condition was imposed to have a zero intercept; the dashed line corresponds to the case Sf=S1+2.
chirality of the whole, and can therefore serve to relate experimental properties to quantitative geometric chirality.
Acknowledgements This work has been supported by the Direccio´ n General de Ensen˜ anza Superior (DGES), project PB98-1166-C02-01. Additional support from Comissio´ Interdepartamental de Cie`ncia i Tecnologia (CIRIT) through grant SGR99-0046 is also acknowledged. S. A. thanks the Dozor Foundation (Israel) for a Visiting Professorship that acted as a catalyst for the collaboration reported in this paper and M. Verdaguer for discussions. D. A. acknowledges support of the US-Israel Binational Science Foundation, and continuing useful discussions with Prof. K. Lipkowitz.
Appendix A The X-ray structural data used to calculate the symmetry measures were retrieved from the Cambridge Structural Database [53] (CSD, version 5.19), restricted to crystal structures with no disorder and R⬍10%. The symmetry measures were calculated with the computer program symm developed by the Jerusalem group. For the analysis of large number of structural data from the CSD, the interface program csmFctrl developed by the Barcelona group was used.
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Continuous chirality analysis of hexacoordinated tris-chelated metal complexes Santiago Alvarez
a,*
, Mark Pinsky c, d, Miquel Llunell b, David Avnir c
a
Departament de Quı´mica Inorga`nica, Centre de Recerca en Quı´mica Teo`rica, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain b Departament de Quı´mica Fı´sica, Centre de Recerca en Quı´mica Teo`rica, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain c Institute of Chemistry and The Lise Meitner Minerva Center for Computational Quantum Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel d Institute of Earth Sciences, The Hebrew University of Jerusalem, Jerusalem 91904, Israel Received 13 November 2000; accepted 26 January 2001 (Refereed)
Abstract We use the notion that symmetry can be evaluated on a continuous scale for the quantitative chirality analysis of tris-chelated metal complexes. After reviewing the Continuous Symmetry Measures methodology and its resulting Continuous Chirality Measure, we analyze the chirality of these complexes in terms of the specific contributions from the innermost ML6 shell and from the ligand shells. The chirality behavior of seven families of tris(chelate) complexes was analysed. The results of the analysis point to a universal behavior and were found to be consistent with those for a molecular model. The maximum expected chirality for the octahedral to trigonal-prismatic route is at a twist angle of 23° and, remarkably, we found a complex that occupies this point, and have thus identified the most chiral hexacoordinated complex: (AsPh4)2[W(S2C6H4)3]. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Chirality; Stereochemistry; Coordination compounds; Molecular symmetry; Quantitative chirality
* Corresponding author. Tel.: +34-93-4021269; fax: +34-93-4907725. (S. Alvarez); tel.: +972-26585332; fax: +972-2-6520099 (D. Avnir). E-mail addresses: [email protected] (S. Alvarez), [email protected] (D. Avnir). 1463-0184/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 1 4 6 3 - 0 1 8 4 ( 0 1 ) 0 0 0 1 3 - 2
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1. Background 1.1. Measuring symmetry and chirality on a continuous scale The concept of symmetry has attracted virtually all domains of intellectual activity and has strongly influenced the sciences and the arts. It has functioned as a condensed language for the description and classification of order within shapes and structures; as an identifier of inherent correlations between structure and physical properties of matter; and as a guideline in artistic and practical aesthetic design. Our study of symmetry is based on the recognition that the majority of molecular structures do not represent exact, ideal symmetries. Distortions due to crystal packing, Jahn–Teller effects, or ligand and substituent effects are the rule not the exception. To realize it one should refine the resolution of observation — spatial or temporal — up to the point where it becomes evident. It appears that symmetry has served in many such instances as an approximate, idealized descriptive language of the physical world, beyond the atomic scale. While it is true that an imprecise language helps in grasping complex situations and in identifying first order trends, the danger of missing the full picture because of a vague description is always awaiting the user of the current symmetry language. This motivation has led us to propose that it is natural to evaluate on a quantitative scale how much of a given symmetry there is in a structure [1– 3]. Thus, we are treating symmetry as a structural property of continuous behavior, complementary to the classical discrete point of view. A continuous symmetry scale should be able to express quantitatively how far is a given distorted structure from ideal symmetry, at any temporal resolution, at any spatial resolution, and relative to any symmetry. Towards this goal, we have designed a general symmetry measurement tool, based on a definition that is minimalist. Our answer to the question “How much of a given symmetry there is in a given structure?” is then: Find the minimal distances that the points of a shape have to undergo, in order for it to attain the desired symmetry. To translate this definition into practice, we have developed the Continuous Symmetry Measure (CSM) methodology and computational tool, described in the next section. Using this measurement procedure it became possible to evaluate quantitatively how much of a given symmetry exists in a non-symmetric configuration, what is the nearest symmetry of a particular configuration, and what is the actual shape of the nearest symmetric structure. Closely related is the Continuous Chirality Measure (CCM) which evaluates the distance to the nearest achiral symmetry; much of our activity was devoted to this special aspect of symmetry measurements. We have demonstrated the feasibility and versatility of this approach on two levels. The first one is purely geometric. Here we developed solutions for specific problems such as: 앫 Evaluation of the degree of bilateral symmetry [4]; 앫 measurement of the symmetry content of distorted classical Platonic polyhedra [5]; 앫 assessment of the symmetry content of objects which contain an element of randomness in their construction [6];
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앫 analysis of the concepts of left/right handedness and mirror symmetry [7]; 앫 evaluation of symmetry distributions within populations [8] and more. The second level concentrated on applications of the symmetry measure to real problems in structural chemistry at large. Examples include symmetry analyses of molecules [9–14], of (macroscopic) crystals [15], of dynamically changing structures [16], of small (3–12 molecules) clusters [17] and of large disordered aggregates [18]; of enzymes and their activities [19,20], and more. Three major findings emerged from these studies: 1. The symmetry measure describes the molecular world in a well behaved way: Its trends of change agree with intuition and reflect what is visible to the eye and what has been expected on a qualitative level, before measurement was possible. 2. Hitherto unknown quantitative correlations between symmetry and physical/chemical/biochemical properties have been revealed. 3. Far more than before, the importance of a global-shape descriptor — symmetry — for the quantitative observation and analysis of structural chemistry, in distinction from the classically specific geometry descriptors, has been revealed. In conclusion of this Background, we mention that the CSM methodology is applicable, beyond the molecular level analysis, to most other domains of the natural sciences, of the social sciences, and of the arts, where symmetry is an issue, either as a real feature or as an abstract one. Examples are the quantitative analysis of the symmetry of hand axes of early man [4], and symmetry analysis of facial features [21]. Finally, for other approaches [22–28] to chirality analysis, see our reviews [1,29]. 1.2. Computational aspects: the continuous chirality measure (CCM) The evaluation of the chirality content of an object by the CCM approach is based on estimation of the degree of the nearest achiral symmetry content within the CSM framework [2,30]. As explained above, the measure is a function of the minimal distance that the vertices of the object have to undergo to attain the desired symmetry (achirality, in our application here). Thus, given the original size-normalized vectors of the locations of the n vertices of the molecule, {Pi}, given the desired G-symmetry, and given the (size-normalized) coordinates, {Pˆ i}, of the (searched) symmetric (achiral) structure, the CCM value, S⬙(G), is calculated according to the definition:
冘 n
1 |P ⫺Pˆ |2 S⬙(G)⫽ ni⫽1 i i
(1)
The symmetry transform, seeking the set {Pˆ i} which is at minimal distance from {Pi}, is the heart of the method. Several algorithms have been developed towards this goal, and the interested reader can find them in previous papers [2,5,30,31]. To calculate the CCM, S⬙(G) is evaluated for every (relevant) achiral point group, and the minimal value is chosen:
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S⬘(G)⫽min[S⬙(G)]
(2)
In practice, one needs not to screen over all achiral G groups, because quite often one can identify the potential nearest achiral symmetry element simply by inspection; in most cases, and in all cases analysed here, the nearest point group that is achiral contains (at least) one mirror plane. Finally, we found it convenient to expand the scale by a factor of 100: S(G)⫽100·S⬙(G)
(3)
and these are the values reported here. The minimal value of S(G) (in short, S) is zero if the object has the desired Gsymmetry, which, in our application means that it is achiral. The maximal value of S⬘(G) is limited by 1 (100 on the expanded scale) due to size normalization, which is performed to make the CSM (and the CCM) invariant to scale changes. It is accomplished by division of the vertex distances of the object from the center of mass by the root mean square (r.m.s.) length of all these vectors: Pi⫽
Poriginal i r.m.s
(4a)
where
r.m.s⫽
冪
冘 n
|Poriginal |2 i (4b)
i⫽1
n
This size normalization assures the scale of S(G) to be bound by an upper limit of 100 because:
冘 冘 n
100 S(G)⫽
冘 冘 n
|Poriginal −0|2 100n i ⫽
i⫽1
冪
n
|Poriginal |2 i
i⫽1
2
|Poriginal |2 i
i⫽1
n
n
⫽100
(5)
|Poriginal |2 i
i⫽1
n
This maximal value is obtained if, e.g., one wishes to find the degree of pentagonality of a hexagon: The nearest pentagon to a hexagon is the collapsed pentagon into a single center point, the distance of which is, by definition, 100. However, for achiral G’s it never reaches 100, because the distance to the mirror plane is always smaller than the distance to a central point. One of the main features of the CCM approach, which we use in the next Section, is that it not only computes the chirality content, but also provides the actual shape of the nearest achiral object, i.e., the structure of the set of {Pˆ i}. In most of its previous applications, the CSM method was used for evaluating the
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chirality of the molecule as a whole. However, in what follows we analyse large families of molecules. As such, it is then more relevant to analyse the chirality of molecular fragments, which are common to all members of the family, and which therefore provide a wider perspective. Analyzing the chirality of the whole molecule for all the family members will mask trends, if such exist, because of the wide variability in the specific structures of the substituents and ligands. 2. Chirality analysis of experimental structural data: hexacoordinated trischelated complexes In this section we exemplify the above concepts and methodology by analysing the chirality of metal complexes of the general formula [M(chel)3], where chel represents a bidentate ligand. Interest in the well established [32–37] chirality of such complexes stems mainly from their ability to induce chirality in catalytic processes. In view of a recent study in which we have shown [38] that the hexacoordinate ML6 complexes with monodentate ligands are chiral for twisted geometries (1) intermediate between the octahedron (q=60°) and the trigonal prism (q=0°), it becomes interesting to ask, what happens when we combine in the same molecule tris-chelation with a twisted coordination sphere? Is there a synergetic enhancement of chirality or do they cancel out [30]? Thus, for the study of the chirality of tris(chelate) complexes it is useful to consider the atoms in the molecule as pertaining to successive shells, as exemplified in 2 for the case of tris(dithiolene) complexes. The first shell comprises the metal and the coordinated donor atoms, the second shell is formed by the spacers that connect each pair of donor atoms in a bidentate ligand, and the third shell is formed by the rest of the ligand atoms. Thus, we shall refer to the chirality measures as S1 (first shell only), S2 (second shell only), S1+2 (first and second shells together, and Sf for that of the full molecule (devoid of the hydrogen atoms). Accordingly, it is important to differentiate the twist angle of the first shell, q1, from that of the second shell, q2.
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2
The analysis below will reveal the following points: (i) In all hexacoordinated complexes, the first shell is chiral at geometries intermediate between octahedral and trigonal prismatic; (ii) tris(chelate) complexes are chiral even if the coordination polyhedron is octahedral (i.e., q1=0°), because of the inequivalence of the X-M-X bond angles belonging to a bidentate ligand and to two different ligands (first shell), or because of the lower symmetry imposed by the skeleton of the bidentate ligands (second shell); and (iii) for a trigonal prismatic coordination sphere, neither the first nor the second ligand shell is chiral. Next, we analyze the contributions of the first and second shells to the chirality of tris(chelate) complexes, both separately and in combination. 2.1. The chirality of tris(dithiolene) and tris(diselenolene) complexes: models and experimental data analysis We shall use [M(S2C2)3] as a basic molecular model of a metal tris(dithiolene) complex (schematically shown in 3; see 1 for details of the model), and will compare the chirality-response of this model to structural variations, with the corresponding experimental data for tris(dithiolene) and tris(diselenolene) complexes. Having in
1 The rotation was accompanied by gradually changing the bond angles around the metal in order to fulfill the definition of ideality, namely equal edges within each of the polyhedra. Bond angles varied from 90° in the octahedron to 81.79° in the trigonal prism, and bond to edge lengths ratio (normalized bite) from 1.414 in the octahedron to 1.309 in the trigonal prism. The full model used for calculations of S1+2 consisted in a M(S2C2)3 group with bond distances representative of those found in crystal struc˚ (increasing with q1). Chirtures of tris(dithiolene) complexes: M-S=2.34, S-C=1.76 and C-C=1.35–1.55 A ality measures S1 correspond to the MS6 fragment of this a model, and S2 to the collection of the three C2 groups.
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mind this model, we shall later on analyze the chirality of other families that are characterized by other types of bidentate ligands. These include tris(chelate) complexes with planar chelate rings, namely those in which the bidentate ligands are either all bipyridines, dithiocarbamates, β-diketonates or catecholates; and finally, we will present a preliminary analysis for the analogous compounds in which the chelate rings are puckered, such as the tris(dithiolate) and tris(ethylenediamine) complexes.
3
2.1.1. The chirality of the first shell As mentioned above, the structures of the MX6 core in many [M(chel)3] complexes can be described as octahedra that are distorted by twisting two parallel faces of the octahedron around a trigonal symmetry axis. This distortion route is known as the Bailar pathway [39], and is depicted in 4 for the tris-chelated case, where the thick lines represent the edges occupied by the bidentate ligands. At the two extremes of that path one has either a perfect octahedron (q1=60°, Oh symmetry), or a perfect trigonal prism (TP, q1=0°, D3h symmetry, all edges equal to each other). In-between (0⬍q1⬍60°), the molecular symmetry is lowered to D3. On the CCM scale, the outcome is that while the S value of the MX6 group is zero for the two ideal polyhedra, in between it has non-zero values, and should pass through at least one maximum. The S1 curve in Fig. 1 shows the theoretical line for a model of this twist route1 and it is seen that, indeed, a maximum chirality value (the chiramer [31]) appears at q1=23°. In Fig. 2(a), where the experimental data of tris(dithiolene) and tris(diselenolene) complexes are superimposed on this theoretical line, one can see that the fit is remarkable. The fact that all structures fall along that line, suggests that they represent the Bailar pathway from Oh to TP; indeed, the analysis of the bonding parameters of the actual structures confirms that these are of D3-type, as illustrated in Fig. 3 for three representative compounds. Since this behavior was also found among homoleptic complexes with alkyl, aryl or thiolato ligands [38], we believe that the chirality behavior of S1 characterizes generally most of the Bailar-twisted hexacoordinated complexes. Below we show (Fig. 7) that this is indeed the case for practically all complexes analysed in this study.
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Fig. 1. Chirality measures as a function of the twist angle (see 1) for a model tris(chelate) complex M(S2C2)3 (see 2). Shown are the chirality variation of the first shell S1 (defined in 2, thick line); that of the second shell, S2, and that of the first and second shells combined, S1+2. The distance of the first shell to an ideal trigonal prism is also shown (dashed line).
4 Next, let us comment on the maximum at q1=23° of the S1 line (Figs. 1 and 2(a)): It is seen that this maximum is a non-differential point, and this observation deserves a further comment: The source of non-differentiability is inherent to the search for the closest symmetric structure, and reflects situations where there are two (or more) different nearest types of achiral shapes [16]. In other words, the change of the closest achiral structure occurs along the crossing line of two chirality value hypersurfaces (surfaces of the chirality as a function of all the structural parameters that
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Fig. 2. Chirality measures as a function of the twist angle, calculated from structural data of tris(dithiolene) complexes (triangles): (a) for the MS6 shell, S1, (b) for the (C2)3 shell, S2, and (c) for the combination of the first two ligand shells, MS6C6 (S1+2). The continuous lines correspond to the molecular model M(S2C2)3 as in Fig. 1. See Supplementary Material for data and references: www.elsevier.nl/gejng/10/15/30/38/55/88/show/index.hH.
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Fig. 3. Structures of the inner shells and rotation angles q1 and q2 of the tris(chelate) complex showing the largest value for the chirality measure S1, [W(C6H4S2)3]-, and for those presenting the largest values of S2 and S1+2, [M(acac)3] (M=Mn, Rh).
affect it), each belonging to a different achiral structure. A pathway leading from one hypersurface to the other is therefore non differentiable at the crossing line of these surfaces. Since the actual shapes of the nearest achiral structures can be determined by the CCM methodology (the set of {Pˆ i}, see previous Section), one can also determine the structural nature of these hypersurfaces and what happens at the nondifferential crossing point. It is found that from q=60° (the ideal octahedron) and to the left, the nearest achiral structure is a distorted octahedron that is bent along the z-axis (see 5) and contains only two symmetry planes. Between the ideal TP on the left of Fig. 1 (q1=0°) and the maximum of the S1 line at q1=23°, the nearest achiral structure is always a D3h-TP varying in height from point to point. At the peak, one has crossing of the two surfaces, i.e., the chirality measure represents an equal minimal distance to two achiral structures: a TP and a bent z-axis octahedron 5. This non-differentiability can be removed, if the restriction for nearest achirality is replaced with a restriction for a specific nearest achiral shape. For instance, if one asks, what is the degree of chirality with respect to the nearest TP, then indeed a continuous line would be obtained. To conclude this section we note that the
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maximum at 23° is nicely exemplified by the structure [40] of (AsPh4)2[W(S2C6H4)3], with S1=2.40, the most chiral core-shell identified in this study (shown in projection in Fig. 3).
2.1.2. The chirality of the second shell We move now to the chirality measure of the second shell alone, S2, that in our molecular model [M(S2C2)3] is constituted by three separate C2 fragments (see 2). Although each C2 spacer, separately, is achiral, their mutual spatial arrangement is helical. The theoretical S2 values for this model are shown in Fig. 1. In analysing this behavior, it is important to notice that while the correlation is still with the twist angle of the MS6 core, q1, the actual twist of the spacers, q2, is smaller, as illustrated in Fig. 4: Here, a first shell twist of q1=60° is translated into a much smaller twist angle in the second shell of 15°. In fact, it becomes possible to correlate S2 directly with q1 instead of q2, because the two twist angles are linearly correlated: q2=⫺0.1+0.25 q1 (correlation coefficient: r2=0.986). Returning to Fig. 1, we see that at q1=0°, where the bidentate ligands align with the sv symmetry planes of the trigonal prism (thus are also symmetric with respect to sh), S2=0, as is S1. However, unlike S1, S2 increases continuously with the twist angle. This continuous increase results because, unlike S1, the nearest achiral structure to the collection of the three fragments is only of one type at the interval of rotation angles of Fig. 1, namely a parallel arrangement of the three C-C fragments aligned along the edges of a trigonal
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Fig. 4. The relation between the polyhedron twist angles of the first and second shells in the M(S2C2)3 model (continuous line) and in the experimental structures of tris(dithiolene) complexes (squares).
prism. Finally we comment on the theoretical behavior at twist angles higher than 60° (see 6): when the octahedron (q1=60°) is distorted by an anticlockwise rotation (6, right) towards an alternative trigonal prism corresponding to q1=120°, S2 continues to increase. This is in contrast to what is found for the first shell, in which both clockwise and anticlockwise rotations of the octahedron are equivalent. The experimental behavior for the family of transition metal tris(dithiolene) and tris(diseloenolene) complexes (Fig. 2(b)), follows the model nicely. Our molecular model provides a good representation of the whole range of experimental structures for these complexes. The good fit indicates that the linear correlation between q1 and q2 is kept; and this, in turn, is apparently due to the conformational rigidity of the five-member chelate rings (imposed by the sp2 carbon atoms at the bridges). 2.1.3. The chirality of the first and second shells, combined We are ready now to look at the combined chirality measure, S1+2, the theoretical behavior of which is shown in Fig. 1 for the same model. Let us compare the evolution of S1+2 to that of S2 and S1 upon rotation (Figs. 1 and 3). First, it is seen that S1+2 behaves in a similar way to S2, namely, it increases continuously from zero as the rotation angle departs from q1=0°, although in a steeper way. The similarity in behavior is a reflection of referring to the same type of nearest achiral object in both cases (that has the sh and sv reflection planes of the nearest trigonal prism) throughout the full range of the rotation shown. We term chirality measures that refer to the same type of improper symmetry group (such as the S1+2 and S2 measures in our case) as commensurate. On the other hand, incommensuration exists between S1 and both S2 and S1+2 at rotation angles larger than 23°, as already explained above in the discussion about the S1 maximum at this angle. It is seen that in the commensurate zone (0°⬍q1⬍23°), S1 increases quite steeply with the angle but S2 is much less sensitive. Because of the commensuration, S1+2 behaves as some average of S1 and S2, but we do not have yet the tools to derive theoretically S1+2 from its components.
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Yet on a qualitative level we can explain now why S1+2 rises so steeply in the incommensurate zone (q1⬎23°): S1+2 in this zone is still an average of two lines: S2 and the very steep line that would have been obtained for S1 if the chirality measure (which looks for the minimal distance to any relevant achiral object) would have been replaced with the distance to a trigonal prism (a specific achiral object); this is shown as a dashed line in Fig. 1. Finally, we comment on the crossing points of the lines in Fig. 1: The structures that correspond to such points are isochiral, a concept that is possible only if chirality is treated as a quantitative measurable quantity. S1 and S2 cross at 41°, namely S1=S2 at this angle, and it is seen that the additivity of the S values is not straightforward: S1+2 is not their sum. From that point of view, the other crossing point, between S1 and S1+2 (at q1=30°) is particularly interesting because it can be visualized as indicating that at that angle, S1 contributes nothing to S1+2. Experimental results supporting the theoretical S1+2 prediction appear in Fig. 2(c), showing the correlation between the chirality measures of the tris(dithiolene) (and diselenolene) complexes as obtained from their crystal structural data,2 and their twist angles (q1). It is seen that this family covers a wide range of twist angles between the octahedron and the trigonal prism, including the commensurate and incommensurate regimes. Note the case of (NEt4)[W(S2C6H4)3] [41], which appears to be the one that is closest to the crossover point (q1=29.6°) of S1+2 and S2. 2.1.4. The chirality of the full molecule: Sf Our final stage is to ask whether the chirality measure of the first two shells correlate with that of the full molecule, i.e., of Sf. To obtain a preliminary answer, we calculated the chirality measures for the complete molecules (with the exception of the hydrogen atoms) of several representative dithiolene complexes distributed along the Bailar path, and plotted them alongside the S1+2 values (Fig. 5). It is seen that for these complexes one always gets S1+2⬎Sf. We will return to this observation in the next section. 2.2. An overview of the chirality behavior of tris-chelated complexes 2.2.1. The analyzed families How general are the models and the chirality behavior detailed in the previous Sections? To answer this question, we analyzed six additional families of important tris(chelate) complexes. Four of them form planar chelate rings, having as ligands bipyridine, β-diketonates, dithiocarbamates or catecholates. Two families explored have ligands that form puckered chelate rings, namely aliphatic dithiolates and ethylenediamines. The various ligands differ in the donor atoms and in the size of the spacers among them, as summarized in Table 1. To these we add as a seventh type
2
Only four molecules (CSD refcodes pabzta, kabcar, kaxmur and selkoj) were found that strongly deviate from the general pattern shown in Fig. 2, and these were disregarded because they present distortions from the octahedron other than the trigonal twist.
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Fig. 5. Chirality measures for representative tris(dithiolene) complexes along the Bailar pathway, as a function of the twist angle. Shown are S1+2 (black circles) and Sf, the chirality values of the full molecules (except the hydrogen atoms; white circles). The upper continuous line corresponds to S1+2 for the M(S2C2)3 model, and the lower line to a least-squares fitting of the experimental Sf values. Table 1 Families of tris(chelate) complexes studied, indicating their main structural features, the definitions of the first and second shells, and the specific members of each family selected for the calculation of the chirality measure of the full molecule Family
General formula
Donor
1st shell 2nd shell Selection
R
Dithiolenes
[M(X2C2R2)3]
X
MX6
(C2)3
C⬵N
Dithiocarbamato Bipyridine
[M(S2CNR2)3]
S
MS6
(C)3
[M(N2C10R6)3]
N
MN6
(C2)3
β-diketonato Catecholato
[M(O2C3R2R’)3] [M(O2C6R4)3]
O O
MO6 MO6
(C3)3 (C2)3
Ethylenediamine Dithiolato CrCl3
[M(N2R4C2H4)3] [M(S2C2R4)3] “CrCl6Cr3”
N S Cl
MN6 MS6 CrCl6
(C2)3 (C2)3 (Cr)3
Maleonitriledithiolene Benzodithiolene Diethyldithiocarbamate Unsubstituted bipyridine Acetylacetonate Unsubstituted catecholate
C4H4 Et H Me, H H
CrCl3 the bidentate ligand of which is ClCrCl. The ClCrCl group forms from the six chlorides that surround each Cr(III) ion in an approximately octahedral geometry, sharing edges with three neighboring octahedra [42]. All other details of the complexes and their chirality values are collected in the Supplementary Material. 2.2.2. The universal behavior of the chirality of the first shell: As seen in Fig. 6, which displays the first shell chirality behavior of all nine families, the model we proposed is universal. In Fig. 7 we show the S1 behavior of
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Fig. 6. Chirality measures of the first shell, S1, as a function of the twist angle, q1, calculated from structural data of seven families of tris(chelate) complexes (summarized in Table 1) and for CrCl3. The continuous line corresponds to the molecular model of M(S2C2)3. See Supplementary Material for data and references.
each family separately. It is seen, for instance, that most of the structures are found with rotation angles between 30 and 60°, indicating a preference for an octahedral stereochemistry around the metal atoms, but also showing that Bailar distortions are quite common. The interesting exceptions are the Y(III) dipivaloylmethanato [43,44], a Cd(II) acetylacetonate [45] and tris(9,10-phenanthrenequinone)molybdenum [46], all with nearly perfect trigonal prismatic geometry. Another particular example is that of a Ti(III) dinuclear complex with a scaffolding bis(catecholato) ligand [47], that presents one of the most achiral first shells (S1=2.14 at q1=23.3°). The dithiocarbamates appear to be the only family for which the octahedral geometry cannot be attained: all structures present twist angles between 31 and 46°. The β-diketonates constitute the only family among those studied here that present anticlockwise twist angles (i.e., q1⬎60°). The different behavior of these two families is associated to the different normalized bites of those ligands (average edge to bond ˚ , respectively, for first row transition metal length distances of 1.21(4) and 1.41(5) A atoms, according to our structural database survey), since it is a well established trend that in tris(chelate) complexes a small normalized bite favors a small twist angle and viceversa [48]. In summary, the chirality of the first shell for all the studied tris(chelate) complexes shows the same dependence on the twist angle q1 as the molecular model used in the previous section, regardless of the nature and size of the metal and donor atoms, the number of spacer atoms, or the planar or puckered nature of the chelate rings. 2.2.3. The chirality of the spacers, S2 Unlike S1, the behavior of S2 is not expected to follow the M(S2C2)3 model, because each bridge follows the twist of the first shell in a different way. Nevertheless, in the case of planar chelate rings, still the two twist angles q1 and q2 are correlated (e.g., q2=⫺0.9+0.34·q1, r2=0.926 for catecholates; q2=0.1+0.50·q1, r2=0.970 for β-diketonates) as discussed above for the dithiolenes. In contrast, when
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Fig. 7. Breakdown of Fig. 6. into some specific ligand families: (a) bipyridines, (b) dithiocarbamates, (c) β-diketonates, (d) catecholates, (e) ethylenediamines, (f) dithiolates.
flexibility in the spacers is allowed by moving from sp2 to aliphatic sp3 carbon atoms, the twist angles of the two shells are found to be uncorrelated as in the families of dithiolato and ethylenediamine complexes. Consider as examples two metal complexes with aliphatic dithiolates [49,50] that present intermediate q1 twist angles of 38 and 30°, but very small S2 values (0.05 and 0.23, respectively) because their C2 groups are nearly aligned along the edges of trigonal prisms (q2 twist angles of 2 and 6°, respectively; see Fig. 8 for [Ti(S2C2H4)3]2⫺) [49]. This behavior appears also for other aliphatic spacers. For instance, in most of the tris(ethylenediamine) complexes with 38°⬍q2⬍56°, the corresponding q2 angles are between 0 and 30°, resulting in low S2 values. As a result of the different relationship that each type of ligand presents between q1 and q2, even the rigid planar families do not follow the model closely (Fig. 9) but, interestingly, are characterized by higher chirality values. The chirality measures of the second shell in the bipyridine complexes that have the same set of C2 spacers
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Fig. 8. Projection view of the [Ti(S2C2)3] core of the [Ti(S2C2)3]2- anion showing the trigonal faces of the rotated octahedron formed by the six sulfur atoms and the helix formed by the bidentate ligands (left), together with a perspective view of the trigonal prismatic alignment of the three C-C bonds (middle) and a side view showing the two faces of the S6 distorted octahedron above and below the C6 trigonal prism (right). q1 and q2 are the twist angles between the trigonal faces of the S6 octahedron and of the C6 prism, respectively.
as our model seem to be closer to the model line than the β-diketonate (C3 spacers) and the dithiocarbamato (C spacers) derivatives. However, we now show that the procedure for fitting a model is general: By changing the parameters of the model to parameters that reflect the specific characteristics of the spacers,3 excellent fits are obtained, as shown in Fig. 9. Among the complexes with planar chelate rings studied, the C3 spacers of several β-diketonato complexes with twist angles ⬇70° z [51] constitute the most chiral second shells (illustrated by [M(acac)3], M=Mn or Rh, Fig. 3), a fact that is associated with a large bite. It corresponds to an anticlockwise distortion of the octahedral coordination sphere, for which we have shown above that chirality measures must increase with z. We note also that the second shell of the dithiocarbamato complexes is chiral at all rotation angles, since the C spacers of the three ligands remain coplanar throughout the Bailar twist. 2.2.4. The combined chirality, S1+2 The excellent fit to the adjusted models is seen also for the S1+2 plots in Fig. 10. Again, the β-diketonato complexes, result in large S1+2 chirality measures (S1+2⬇7.6 for [Mn(acac)3], Fig. 3). An interesting case is that of the S1+2 value of CrCl3 [52] which is topologically equivalent to the dithiocarbamates with its bidentate ClCrCl 3
The following parameters were used in the molecular models: In the dithiocarbamato model, the S˚ upon rotation, the C-S distance was set to 1.712 A ˚ , and the SS distance was kept constant at 2.845 A ˚ (for q1=0°) C-S bond angle fixed at 112.4°, while the M-S distance was allowed to vary between 2.387 A ˚ (for q1=53°), implying changes in the S-M-S bond angles from 70.7 to 77.7°. In the bipyridine and 2.318 A ˚ , the C-N and C-C distances to 1.369 and 1.475 A ˚, model, the N-N distance was kept fixed at 2.616 A ˚ (for and the C-C-N bond angles at 114.7°, while the M-N distance was allowed to vary between 2.090 A ˚ (for q1=61°), implying changes in the N-M-N bond angles from 73.9 to 91.0°. In q1=0°) and 1.938 A ˚ , the C-O and C-C distances at the acetylacetonato model, the O-O distance was kept fixed at 2.809 A ˚ , and the C-C-O and C-C-C bond angles at 126 and 124°, while the M-O distance was 1.272 and 1.390 A
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Fig. 9. Chirality measures for the second shell (S2) of tris(β-diketonate), tris(bipyridine) and tris(catecholate) complexes (black triangles, white squares and white circles, respectively) as a function of the twist angle, obtained from the experimental crystal structural data. The corresponding values for the tris(dithiocarbamate) complexes are zero in all cases and are not shown. The experimental values are compared with the curve for the M(S2C2)3 model in (a) and with those for M(N2C2)3 and M(O2C3)3 models consistent with the structural parameters in the diketonato and bipyridine complexes 3 (upper and lower curves, respectively) in (b).
ligands: the CrCl6 shell is nearly octahedral (q1=53°) and achiral (S1=0.07) and the second shell formed by three neighboring Cr atoms is strictly achiral (S2=0.00, just as the C atoms in dithiocarbamato complexes). However, since S1 and S2 are incommensurate (see Section 2.1, above), when the two achiral shells are combined, pronounced chirality appears (S1+2=6.63). Again, it is worth noting that the different geometries of the topologically equivalent chelate rings in CrCl3 and in dithiocarbam˚ , respectively) result in a ates (cf. Cr-Cr and S-S distances of 3.18 and 2.83(5) A significantly smaller S1+2 value for the former than expected from the dithiocarbamate molecular model in Fig. 10. ˚ (for q1=0°) and 1.923 A ˚ (for q1=74°), implying changes in the O-Mallowed to vary between 2.146 A O bond angles from 81.8 to 93.8°.
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Fig. 10. S1+2 chirality measures along the Bailar pathway for the families of tris(dithiocarbamato) (crosses), tris(bipyridine) (triangles), tris(catecholato) (squares) and tris(β-diketonato) (circles) complexes, as deduced from the experimental structural data. The continuous lines correspond (from top to bottom) to the M(S2C)3, M(N2C2)3 and M(O2C3)3 models.3
From the results for the different models, two generalities regarding S1+2 can now be drawn: 1. For donor atoms of about the same size and same twist angle, S1+2 decreases with increasing number of spacer atoms (i.e., from C2 in bipyridine or catecholate to C3 in diketonate; and from C1 in dithiocarbamate to C2 in dithiolene). 2. For ligands with the same number of spacer atoms, S1+2 decreases with the size of the donor atoms (i.e., smaller for the dithiolenes compared to bipyridine or catechol) or with the size of the spacers (cf. Cr and C spacers in CrCl3 and dithiocarbamates, respectively).
2.2.5. The chirality of the full molecule: Sf Finally, we ask whether the evaluation of the chirality of a part of a molecule represents the chirality behavior of the whole molecule. For that purpose we looked for a possible correlation between the chirality measure of the full molecule without hydrogens (Sf) and S1+2. Such correlation was indeed found: Calculating the chirality values for the most common members of these families (identified in Table 1 under the heading Selection), it was found that Sf is linearly correlated with S1+2 in all cases (Fig. 11). Specifically, it is seen that the benzodithiolene and maleonitriledithiolene complexes obey the same relationship between the two chirality measures; that in the cases of dithiolene, catecholato and dithiocarbamato complexes, Sf is always smaller than S1+2, whereas in the other cases (bipyridine and acetylacetonate) it is always larger than S1+2; and that the catecholato and dithiolene complexes present practically the same linear correlation between the two values. This is an important result because it suggests that S1+2 provides an adequate quantitative estimate of the
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Fig. 11. The chirality measures of the full molecules (with the exception of hydrogen atoms), Sf, for different families of tris(chelate) complexes as a function of S1+2. The chelating ligands (Table 1) are: bipyridine (white squares), β-diketonates (white circles), dithiolenes (black circles), catecholates (black squares) and dithiocarbamates (white triangles). The continuous lines are least-squares linear fittings of the experimental data to which the condition was imposed to have a zero intercept; the dashed line corresponds to the case Sf=S1+2.
chirality of the whole, and can therefore serve to relate experimental properties to quantitative geometric chirality.
Acknowledgements This work has been supported by the Direccio´ n General de Ensen˜ anza Superior (DGES), project PB98-1166-C02-01. Additional support from Comissio´ Interdepartamental de Cie`ncia i Tecnologia (CIRIT) through grant SGR99-0046 is also acknowledged. S. A. thanks the Dozor Foundation (Israel) for a Visiting Professorship that acted as a catalyst for the collaboration reported in this paper and M. Verdaguer for discussions. D. A. acknowledges support of the US-Israel Binational Science Foundation, and continuing useful discussions with Prof. K. Lipkowitz.
Appendix A The X-ray structural data used to calculate the symmetry measures were retrieved from the Cambridge Structural Database [53] (CSD, version 5.19), restricted to crystal structures with no disorder and R⬍10%. The symmetry measures were calculated with the computer program symm developed by the Jerusalem group. For the analysis of large number of structural data from the CSD, the interface program csmFctrl developed by the Barcelona group was used.
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