Topological studies of the charge density of some

Topological studies of the charge density of some group 2 metallocenes M(~~-C,H,), (M = Mg or Ca) Ian Bytheway, Paul L.A. Popelier, and Ronald J. Gillespie

Abstract: Ab initio quantum chemical calculations using both the Hartree-Fock and the B3LYP density functional theory methods have been performed for the group 2 metallocenes M(~~-c,H,), (M = Mg or Ca). The topology of the calculated charge have been analysed using a new critical point search algorithm in order to understand why density (p) and its Laplacian (vZp) M ~ ( ~ ~ - c , H , is ) , linear while C ~ ( ~ ~ - C , H is , ) bent. , The Laplacian of the core of the Ca atom in c ~ ( ~ ~ - c , His, )perturbed ~ by the polarizing field of the cyclopentadienyl ligands and the bent geometry is a consequence of the interactions between the distorted ,, concentration maxima in the Mg core occur along the vectors core and the ligand atoms. In the case of M ~ ( ~ ~ - c , H , )charge connecting the metal to the centroids of the cyclopentadienyl ligands irrespective of whether or not the molecule is linear, and the preferred geometry is linear as expected. The results of these calculations demonstrate that the geometries of the group 2 metallocenes can be understood in terms of the repulsive interactions between the ligands and between the ligands and the distorted core of the metal atom. Key words: atoms in molecules, group 2 metallocenes, VSEPR, charge density, Laplacian of p.

Resume : Utilisant les mkthodes de Hartree-Fock ainsi que de la thCorie de la densitt fonctionnelle B3LYP, on a effectut des calculs ab initio de chimie quantique sur des mCtallocbnes du groupe 2, M(~~-c,H,),(M = Mg ou Ca). Afin d'essayer de , ) , IinCaire alors que le Ca(-q5-C5H,), est repliC, on a analysC la topologie de la densit6 de comprendre pourquoi le M ~ ( ~ ~ - C , Hest charge calculCe (p) et de son laplacien (vZp)en utilisant un nouvel algorithme pour la recherche du point critique. Le laplacien du , ) ,perturb6 par le champ polarisant des coordinats cyclopentadiCnyles et sa gComCtrie noyau de l'atome de Ca du C ~ ( ~ ~ - C , Hest replite est une consCquence des interactions entre le noyau deform6 et les atomes du coordinat. Dans le cas du Mg(-q5-c,H,),, la concentration maximale de la charge dans le noyau du Mg se produit le long des vecteurs reliant le mCtal aux centroi'des des coordinats cyclopentadiCnyles que la moltcule soit linCaire ou pas et on s'attend alors B une gComCtrie linCaire qui soit privilCgiCe. Les rksultats de ces calculs dtmontrent qu'il est possible de comprendre les gComCtries des mttallocbnes du groupe 2 en termes d'interactions rkpulsives entre les coordinats et entre les coordinats et le noyau dCformC de l'atome de mCtal. Mots elks : atomes dans les moltcules, mCtallocbnes du groupe 2, VSEPR, densit6 de charge, laplacien de p. [Traduit par la ridaction]

Introduction The factors governing molecular geometry are important and fundamental to our understanding of chemistry - why do molecules prefer some shapes and not others? In trying to answer this question one is struck by the fact that despite the vast number of different molecules, there exist common geometrical motifs that persist even when the constituent atoms of a molecule are changed. Clearly, rules which allow this vast body of

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Received September 27, 1995.

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This paper is dedicated to Professor Richard F. W. Bader on the occasion of his 65th birthday.

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' Author to whom correspondence may be addressed.

I. Bytheway. Department of Physical and Theoretical Chemistry, University of Sydney, Sydney 2006, NSW, Australia. P.L.A. Popelier. University Chemistry Laboratory, Lensfield Road, Cambridge CB2 lEW, U.K. R.J. Gi11es~ie.lDepartment of Chemistry, McMaster University, Hamilton, ON L8S 4M1, Canada. Telephone: (905) 529- 9140, ext. 23307. Fax: (905) 522-2509. E-mail: [email protected]

Can. J. Chem. 74: 1059-1071 (1996). Printed in Canada / Imprim6 au Canada

information to be classified are of importance to chemistry as they allow us to understand what is known, and make predictions about what is not. Perhaps the simplest set of rules that can be used to understand and classify molecular geometries are those of the valence shell electron pair repulsion (VSEPR) model (1-3). The VSEPR model requires that we consider the strength of mutual repulsions between the different types of electron pairs present within the molecule in order to obtain its preferred, or energetically most favourable, geometry. The intuitive nature of the VSEPR model is one of its most useful features, and one that has allowed its application to a wide variety of molecules, ranging from those containing main group atoms (1-3) to those containing transition metals (4, 5). The successes of the VSEPR model should, of course, prompt us to study the model in order to understand why it is often so successful, and, likewise, we should seek the reasons for its failures because uncovering these reasons should lead to a deeper understanding of molecular geometry. The fundamental postulate of the VSEPR model, that molecular shape is determined by repulsions between bonding and nonbonding pairs of electrons, is explained by the Pauli exclusion principle. Electrons are not, however, really local-

Can. J. Chern. Vol. 74, 1996

ized to form pairs (6,7), but this idea remains valid if we consider regions in the valence shell where charge is locally concentrated arising from the localization of the Fermi hole (8, 9). Regions where charge is concentrated, which are revealed by the Laplacian of the charge density ( ~ ' p ) (8-10) have played an important role in understanding the VSEPR model, resulting in it being reformulated in terms of electron-pair domains (2, 3, 11, 12) i.e., the regions in the valence shell where charge is locally concentrated. For many molecules of the main groups, there is excellent agreement between the expected locations of the electron pairs of the VSEPR model and the locations of maximum charge concentration (8-10). Therefore, the physical basis for the VSEPR model is provided by the Laplacian of the charge density (8, 9) in that preferred molecular geometries may be described in terms of the repulsions between these regions of charge concentration rather than between localized pairs of electrons. For example, the geometries of ClF, (1) and SF, (2) are both derived from the trigonal bipyramid, in which the bonding and lone pair electrons of the VSEPR model appear as regions of maximum charge concentration and their preferred geometries explained accordingly (8, 9). This relationship between the VSEPR model and V2p is observed for many main group molecules (9), so that one might then ask what the Laplacian of the charge density reveals for molecules which have geometries not accounted for by the VSEPR model. One such set of counter-examples to the VSEPR model are the YSF, (Y = 0 , NH, or CH,) (3), which are trigonal bipyramidal as predicted; however, taking into consideration the decreasing electronegativity of the series Y = 0 , NH, CH, it might be expected that both equatorial and axial F-S-F bond angles would decrease. Experimentally, it is found that the equatorial bond angle does decrease (13, 14) as expected but the axial bond angles in fact increase. Analysis of V'p for these molecules provides a clear picture of the S-Y bonding region, from which it can be seen that the charge concentra-

an increased amount of charge concentrated in the equatorial plane (resulting in decreased equatorial F-S-F bond angles) and a decreased amount of charge concentrated in the axial plane (resulting in increased axial F-S-F bond angles). Thus the observed geometries can be thought of as arising from the repulsions between the different types of bonding charge concentrations as revealed by the Laplacian of the charge density. The explanation of these apparent exceptions to the VSEPR model in terms of V2p is encouraging and led to the consideration of others, namely the difluorides of group 2. It has been known for some time that the halides of Ca, Sr, and Ba (4) are bent (16, 17) and analysis of V'p for these molecules has shown (18) that this bending is a consequence of distortions of the metal atom by the presence of the fluoride ligands. It is usual to think of the central atoms in these molecules as s ~ h e r ical but, as was suggested some time ago (l9), metals may be sufficiently polarizable that their cores are distorted in the presence of some ligands. The importance of considering such distortions of the core charge density of transition metals has been noted previously, for example in explaining the deviation of the bond angles in VOCl, (20) (5) and CrO,Cl, (21) (6) from the VSEPR predicted angles. In the case of CaF,, partial condensation of the core electrons into pairs results in a distorted ca2+core that contains regions of charge concentration, i.e., it is no longer spherical. Interaction between the ligands and these regions of core charge concentrations (CCC) results in the observed nonlinear geometry for this molecule.

6 F

A closely related set of molecules are the metallocenes of group 2, for which gas phase electron diffraction experiments have shown that MgCp, (22) and M ~ c ~ \23, * , 24) are linear, , BaCp (25, 26) are all while c a c p t 2 (23, 24), s ~ c ~ * ,and bent. In the solid phase MgCpz (27) is also linear, while CaCp, (28), CaCp*, (29), and BaCp*, (30) are all bent, where Cp = cyclopentadienyl, 715-c5~, and cp*= pentamethyl-cyclopentadienyl, r 1 5 - ~ 5 ( ~ ~Theoretical ,)5. studies of the bonding and photoelectron spectra (31-33) have been reported for lanthanide metallocenes and the importance of electron correlation in the description of transition metal metallocenes has also been established (3636). More recently, ab initio calculations of group 2 and lanthanide metallocenes (37) showed that the bending potential in these molecules is small ( 0 ) by broken contours. Critical points in p are denoted by the same symbols as those used in Fig. 3. (a) Linear MgCp2 showing the location of the maxima in -V2p (denoted by the filled circles near the Mg nucleus) in the magnesium core. Note also that the Mg-C bond critical points are in regions of charge depletion. (b) V2p plotted in the same plane as that used in Fig. 3b. Neither of the magnesium CCC are located in this plane. ( c )A plot of V2p in the plane of the cyclopentadienyl ligand, showing the covalent C-C and C-H bonds, and the region of charge depletion in the center of the ring. (4A plot of V2p for the bent MgCp, molecule in the same plane as that shown in (a). Note that the CCC are still directed along the Mg-X vector as in the linear molecule.

concentration is better described as a torus as the contour value shown is for a vZpvalue of 0.395 au while the values at the maxima are only 0.402 au. The resolution of the torus of charge concentration into five maxima is a consequence of the D,, symmetry of the molecule. In contrast to the bent molecule then, there are now regions of charge concentration (i.e., both

tori) along all 10 Ca-C bonds in the linear molecule, which results in a less favourable geometry. Previous studies of p and v * have ~ shown that their topologies remain the same upon inclusion of electron correlation and only small quantitative changes at various critical points were noted (54, 55). With this in mind, the topologies of V2p

Bytheway et al

Fig. 5. Plots of V2p for the HF optimized CaCp, molecule. (a) In the same plane as that used previously in Figs. 4(a) and 4(d) for MgCp,. The region of charge concentration in the calcium core can be seen, although its topology is not evident. (b) In the same plane but focussing upon the calcium core region. Four maxima in -V2p were found in this region, two are well defined by the contour chosen (0.39 au) regions and two in the long, narrow region in the middle left of the plot.

Fig. 6. Schematic representations of the arrangement of charge concentration maxima in the calcium core in both CaF, and CaCp, molecules relative to the fluoride and cyclopentadienyl ligands (X denotes the centroid of the Cp ligand). The approximate positions of charge concentration maxima are denoted by asterisks and are in the plane of the page in all cases except for the B3LYP optimized geometry of CaCp, (bottom right) in which case + and - symbols denote charge concentration maxima above and below the plane of the page, respectively.

Further bending

X CaCp, HF Optimized Geometry X-Ca-X = 156'

CaF, HF Optimized Geometry F-Ca-F = 156'

CaCp, HF Wave Function X-M-X = 148' (Fixed)

CaCp, B3LYP Optimized Geometry X-M-X = 148'

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Can. J. Chem. Vol. 7 4 , 1996

Fig. 7. Plots of VZpfor the linear CaCp, molecule. (a) In the same plane as that used in Fig. 40. (b) Focussing on the the calcium core region in this plane. The intersection of this plane with the tori of charge concentration maxima is apparent in this plot. ( c ) Plotted in the plane containing five charge concentration maxima with the C atoms of the nearest Cp ligand labelled. The five maxima are not apparent even though a contour of 0.395 was used and the maxima have values of 0.402 au.

obtained from the B3LYP charge densities were also examined and values of the various (3, -3) charge concentration maxima are given in Table 2. As might be expected, the values of VZpcalculated by the two methods are in good agreement with each other, as are the distances from the calcium nucleus to the (3, -3) critical points. There is, however, a small difference in the atomic graphs of Ca given by the two methods. Analysis of the wave function obtained from the B3LYP

geometry optimization of CaCp, revealed a square pyramidal arrangement of charge concentration maxima in the atomic graph of the Ca atom. A summary of the different atomic graphs obtained for the various calculations of CaCp2 are shown in Fig. 6, along with that obtained previously for CaF, (18). These diagrams show how the various atomic graphs are related through the joining and splitting of charge concentration maxima.

Bytheway et al

Fig. 8. Plots of V2p for the B3LYP optimized CaCpz molecule. (a) A view of the molecule plotted in the plane containing three of the calcium CCC maxima. No C or H nuclei are in this plane. ( 6 )The calcium core shown in this same plane. The five charge concentration maxima are arranged in the form of a square pyramid, with the apex to the right of the Ca nucleus, and two diametrically opposed maxima shown in the left. ( c )A plot for this molecule in the plane containing the Ca and two C atoms, i.e., that used in previous plots. V2p in the core region is similar (in gross terms) to that shown in Fig. 5a.

A plot of v 2 p in the plane defined by tht apical and two diametrically opposed charge concentrations is shown in Fig. 8a, which shows the relationship between the location of the core charge concentration maxima and the cyclopentadienyl ligands - note that no nuclei lie in this plane. Figure 86 is a magnification of the Ca core in the same plane as that used in Fig. 8a, where the apical charge concentration maximum is to

the right of the Ca nucleus. Finally, a plot of v2p in the same plane as that used in Fig. 5a is given in Fig. 8c, which shows the gross similarity between the HF and B3LYP results. Results for the B3LYP calculations of the linear CaCp, molecule are also similar to the HF results, revealing two tori of charge concentration intersecting the Ca-C bond paths. In this case, however, the 10 charge concentration maxima are

Can. J. Chem. Vol. 74, 1996

eclipsed with respect to the nearest cyclo entadienyl ligand but, as found previously, the values of -V p are only marginally larger than the surrounding region of charge concentration. Thus a plot of V2p in the plane of the five charge concentrations nearest a C p ligand obtained from the linear B3LYP wave function is almost indistinguishable from that given in Fig. 7c. Although this finding is slightly different than the H F results, the location of critical points is an artifact of the imposed symmetry, and the explanation for the preferred bent geometry is the same.

P

Conclusions T h e work presented here is part of the continuing exploration of the physical basis for the VSEPR model. We have presented charge density analyses for both magnesium and calcium dicyclopentadienes, and have shown that their preferred geometries can b e understood using arguments similar to those used to explain the geometries of both MgF, and CaF, (18). Metallocene formation results in sufficient distortion of the metal atom such that it can no longer be considered spherical, as is usually assumed in the VSEPR model. It is these interactions between the ligands and the distorted core of the metal atom that play an important part in determining the preferred arrangement of the cyclopentadienyl ligands. Thus the preferred geometry is one in which the various types of interactions (i.e., ligand-ligand, CCC-ligand, and CCC-CCC) are minimized, in keeping with the spirit of the VSEPR model. It is worth noting, however, that the possibility of attractive interactions between the cyclopentadienyl ligands in these group 2 metallocenes has been suggested (38,39). In these calculations the group 2 metal atom was assumed to b e spherical (39) so that the introduction of a C p C p attractive term is then necessary in order to recover the bent nature of these molecules. O n the basis of the results presented here it is not possible to dismiss the possibility of such dispersive (attractive) interactions between the C p ligands in these molecules, although our explanation has the advantage that it is applicable to situations where such ligand-ligand attractions are not expected (e.g., the heavier group 2 dihalides). In terms of a molecular mechanics model, it seems reasonable to suggest that a potential function that mirrors these CCC-ligand repulsions be incorporated in place of the ligand-ligand attractive potential. The testing ground for these ideas lies in the realm of experiment because, as w e have seen, it is the exceptions to the VSEPR model that have prompted further study, and challenged its underlying assumptions. Within the context of this current work, it would be of interest to see how preferred geometries are affected as the size of the coordinated ligands is increased.

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