Extending the VSEPR model through the properties

Extending the VSEPR model through the properties of the Laplacian of the charge density PRESTON J. MACDOUGALL' A N D MICHAEL B. HALL Department of Chemistry, Texas A&M University, College Station, 7X 77843, U.S.A. AND

RICHARD F. W. BADERA N D JAMESR. CHEESEMAN Department of Chemistry, McMaster University, Hamilton, Ont., Canada L8S 4Ml Received April 3 , 1989

Can. J. Chem. Downloaded from www.nrcresearchpress.com by 113.31.80.216 on 06/04/13 For personal use only.

This paper is dedicated to Professor Ronald J . Gillespie on the occasion of his 65th birthday

PRESTON J . MACDOUGALL, MICHAEL B. HALL,RICHARD F. W. BADER, and JAMES R. CHEESEMAN. Can. J. Chem. 67, 1842 (1989). The Laplacian of the electronic charge density demonstrates the presence of local concentrations of charge in the valence shell of an atom in a molecule. These local maxima faithfully duplicate in number, location, and size the spatially localized electron pairs of the VSEPR model. Thus the Laplacian of the charge density provides a physical basis for the Lewis and VSEPR models. The VSEPR model can fail to predict the observed geometry when a transition metal is the central atom in the system. In such a case there is no basis within the model itself to predict the existence of the nonbonded pair or pairs of electrons whose presence would account for the observed geometry. By appealing to the Laplacian of the charge density in one such system, namely VOC13, one observes that the penultimate shell of charge concentration of this transition metal atom exhibits nonbonded concentrations of charge with the locations and properties required to account for the observed geometry of the molecule. Thus the properties of the Laplacian distribution, which are model independent, can be used to extend the applicability of the VSEPR model to those cases where the usual arguments fail to account for the number and type of charge concentrations that exist within the outer shell of an atom. Key words: Laplacian of the charge density, VSEPR model, transition metal. PRESTON J . MACDOUGALL, MICHAEL B. HALL,RICHARD F. W. BADERet JAMES R. CHEESEMAN. Can. J. Chem. 67, 1842 (1989). La distribution de Laplace de la densitt de la charge tlectronique dtmontre qu'il existe des concentrations locales de charges dans la couche de valence d'un atome d'une moltcule. Ces maxima locaux sont une rtplique exacte du nombre, de la position et de la grandeur des paires d'tlectrons localistes dans l'espace du modtle VSEPR. La distribution de Laplace de la densitt de charge fournit donc une base physique pour les modkles de Lewis et VSEPR. Lorsque l'atome central d'un systtme est un Cltment de transition, il existe des cas oh le modkle VSEPR ne permet pas de prtdire la g6omttrie observte. Dans un tel cas, il n'existe pas debase dans le modtle pour prtdire l'existence de la paire non-like ou des paires d'tlectrons dont la presence pourrait expliquer la gtomttrie observte. En faisant appel i la distribution de Laplace de la densitt de charge d'un tel systtme, i savoir le VOC13, on peut observer que l'avant-dernitre couche de la concentration de charge de cet tltment de transition prtsent des concentrations de charge non-likes dont les positions et les propritt6s correspondent B celles requises par la gtom6trie obsen6e pour la mol6cule. Les proprittts de la distribution de Laplace, qui sont indtpendantes du modtle, peuvent donc Etre utilistes pour permettre d'appliquer le modtle VSEPR aux cas oh les arguments usuels ne permettent pas d'expliquer le nombre ou le type de concentrations de charge qui existent dans la couche exttrieure d'un atome. Mots clis : distribution de Laplace de la densit6 de la charge, modtle VSEPR, tltment de transition. [Traduit par la revue]

Introduction The valence-shell electron pair repulsion or VSEPR model (1) is a natural extension of the localized electron pair model of Lewis (2). It has become the most successful and widely used model for the prediction of the geometries of closed-shell molecules AX,. The VSEPR model has two basic assumptions: (1) the valence charge density of the central atom A is spatially localized into pairs of electrons, and (2) the geometrical arrangement of the ligands X about atom A is that which maximizes the separations between the pairs, assuming the pairs to be on the surface of a sphere centred on the A nucleus. In general, not all the pairs are equivalent and three subsidiary postulates are required to determine the relative sizes of the electron pair domains: (a) nonbonding pairs have larger domains than bonding pairs; (b) bonding pair domains decrease in size with increasing electronegativity of the ligand, and increase in size with increasing electronegativity of the central atom; (c) double ' ~ u t h o rto whom correspondence may be addressed. Printed in Canada / ImprimC au Canada

and triple bond domains are larger than single bond electron pair domains. The Laplacian distribution of the electronic charge density, the quantity V2p, has been shown to provide the physical basis for the Lewis and VSEPR models (3). It does this by demonstrating the presence of local concentrations of charge in the valence shell of an atom in a molecule. These local charge concentrations, local maxima in -V2p, faithfully duplicate in number, location, and size the spatially localized electron pairs of the VSEPR model (3, 4), recovering all of the properties ascribed to them as put forth in the postulates of the model. The existence and the arrangement of the local charge concentrations within the valence shell of an atom are a conseauence of the partial condensation of the valence electrons into pairs (3b, 4). The condensation is caused by a corresponding spatial localization of the Fermi hole, a localization that arises from the ligand field acting in concert with the Pauli exclusion principle. Because of this linking of the concepts of the VSEPR model to a physical property of a system, the model is reduced to but a

MACDOUGALL ET AL.

TABLE1. Structural data of oxyhalidesa Bond lengths and bond anglesb Compound

Compound

Bond lengths and bond anglesb -

POF3

P-F P 4 LFPF

1.524 1.436 101.3

POCI~

P A 1 P 4 LClPCl

1.993 1.449 103.3

F S 4 LFSF LOSO

1.53 1.40 97 123

S-Cl S 4 LClSCl LOSO

1.99 1.43 111 120


so2F2

S02C12

V0Cl3

V 4 V Z l LOVCl LClVCl

1.57 2.14 108 111

Cr02Clz

Cr-0 CrZl LOCrCl LOCrO LClCrCl

1.57 2.12 109.5 105 113

C r 4 Cr-F LOCrO LFCrF

1.58 1.74 102 119

"Reference 8. bValues in angstroms and degrees.

I

single postulate. This postulate is that the most stable molecular geometry of a molecule AX, corresponds to maximally separating, according to size and magnitude, the local maxima in the valence shell of charge concentration (VSCC) of the atom A as defined by the Laplacian of the charge density. An advantage of finding a physical basis for a model is that one can then investigate those cases that are perhaps of greatest interest in providing a deeper understanding of the phenomena covered by the model, namely, the exceptions to the model. The VSEPR model does not fail for closed-shell systems composed of main group elements. Exceptions to the model can be found, however, for systems containing transition group elements. This brief note is an initial study of one such exception, as provided by the molecule VOC13. The basic tetrahedral geometries of main group compounds such as POF3 and POC13 are explained on the basis of the presence of four electron domains in the valence shell of phosphorus, three containing singly bonded pairs and one containing the two bonded pairs of the double bond. The subsidiary postulate regarding the relative sizes of single and double bond domains then correctly predicts the XPO angle to be greater than the XPX angle and for this latter angle to be less than tetrahedral in these molecules. In agreement with these predictions, the XPX angle is 101.3" in the fluoride and 103.3" in the chloride. Table 1 gives the geometrical parameters for these and other related molecules. In the compounds S02X2,the XSX angle is smaller than the OSO angle. Also given in this table are data for transition metal compounds that exhibit contrasting behaviour. In the Cr02X2 systems, the XCrX angle is greater than the OCrO angle. In the molecule VOC13, the OVCl angle is actually slightly less than tetrahedral, being equal to 108", thereby yielding a value of 11 1" for the ClVCl angle. This opening of the ClVCl angle to a value greater than the tetrahedral value cannot if One assumes, as is be explained by the VSEPR for the corresponding main group com~oundsythat the shell of the central atom has but four electron domains. The model is saved if one could argue that five electron domains exist in the outermost electron shell of a transition element, with

the fifth domain being located in the nonbonded position along the threefold axis, that is, opposite the oxygen atom. It is shown here that the Laplacian of the charge density for the vanadium atom in VOC13 does indeed exhibit five local maxima or local charge concentrations in its outermost shell of charge concentration, with the largest of these concentrations being found in the nonbonded position along the threefold axis. The positions of the three "bonded maxima" associated with the chlorine atoms are, however, not as anticipated on the basis of analogous behaviour with the main group compounds. A corresponding polarization of the outermost shell of charge concentration is shown to be present in the molecule VC13. The properties of the Laplacian distribution, which link the tenets of the model to observed properties of the charge distribution, do more than simply extend the domain of application of the model. In addition, the Laplacian of p delineates the differences in the properties of the electronic charge distributions that exist between main group and transition metal atoms, differences that are responsible for their differing chemical behaviour. Gillespie et al. (5) have stated that "the 3d electrons cannot be clearly identified as either valence shell electrons or as core electrons. This special nature of the 3d electrons is responsible for most of the unusual properties of the transition metals."

Calculations The RHF SCF calculations for VC13 and VOC13 were caried out using the Huzinaga basis sets (6): V(532121s/52 1lp/221d), 0(5211s/321p/ ld), Cl(5321 Is1521 lp/ 1 4 . ' The calculations included an optimization of the geometries. The calculated geometrical parameters for VOC13 are R(V-0) = 1.501 A, RLV--€I) = 2.147 A, LClVCl = llO.SO, LOVCl = 108.4". The agreement with the experimental values is good (see Table 2~ single polarization function was added to each atom's basis (d for 0 , C1 and p for V). The V basis was that optimized for the (3d4(4s)' atomic state. To satisfactorily describe the diffuse charge on the ligands, diffuse s and p functions were added to their bases (exp. = 0.08 for 0 , 0 . 0 5 for Cl).


CAN. J. CHEM. VOL. 67, 1989

ClF,

FIG.1. Relief maps of -VZp for the equatorial (upper) and the axial (lower) planes of CIF3. 'The chlorine atom exhibits three shells of charge concentration. The equatorial plane shows the presence of two nonbonded and one smaller bonded charge concentration. In the axial plane there are three bonded charge concentrations and a fourth apparent maximum indicated by an arrow. This peak is actually another view of the (3, -1) critical point between the two nonbonded maxima, which is also denoted by an arrow. The VSCC of C1 possesses five local charge concentrations, three bonded and two larger nonbonded maxima in -VZp.

I), particularly the bond angles. The theoretical model certainly recovers the angular behaviour that is the topic of discussion. The calculated parameters for VC13 are R(V-41) = 2.184 A , LClVCl = 120.0". The calculated energies are E(VOC1,) = -2395.66771 au, E(VC1,) = -2320.89293 au.

The VSEPR model and the local charge concentrations of the Laplacian of p The spherical surface on which the electron pairs are assumed to be localized in the VSEPR model is identified with the sphere of maximum charge concentration found in the valence shell charge concentration, the VSCC, of the central atom (3), and the localized pairs of electrons are identified with the local concentrations of charge present on the surface of this sphere. This is illustrated for the chlorine atom in the molecule ClF3 in Fig. 1. Since a local charge concentration corresponds to a maximum in

-V2p, this and other relief diagrams illustrate the negative of the Laplacian of p, a maximum in this function being a maximum in charge concentration. There are pairs of regions, one negative, one positive, for each quantum shell, with the innermost region being a spike-like maximum in charge concentration. This shell structure is evident in the figure, with three shells being present on chlorine and two on fluorine. The extrema in p or -V2p are points where the first derivative of the scalar field vanishes. Whether such a point is a maximum, minimum, or a saddle is determined by the signs of the associated curvatures of the function at the critical point, i.e., by the eigenvalues of the Hessian of the function. If all three eigenvalues or curvatures are negative, the function exhibits a local maximum at that point, a (3,-3) critical point. If all three curvatures are positive, the function exhibits a local minimum, a (3,+3) critical point. The other two possible critical points with three nonzero curvatures are correspondingly labelled as (3, - 1 ) and (3,+ 1). The outer or valence shell of charge concentration of C1 exhibits five maxima in -V2p, or local charge concentrations. There are two nonbonded and three bonded concentrations, with the size and magnitude of these maxima decreasing in the following order: nonbonded > equatorial bonded > axial bonded. The angle between the two nonbonded maxima is relatively large and equal to 148", while the axial bondednonbonded angle is opened to 97". In higher energy geometries of this molecule, the angles between the bonded and nonbonded maxima are all smaller than those found for this favoured geometry. Similar results are obtained for other molecules containing third-row elements such as ClF30, SF4, S F 4 0 , and ClFS (3a, 4). The Laplacian distribution of the vanadium atom is first investigated in the simpler, more symmetrical compound VC13. This molecule is known experimentally only in its polymeric form. The Laplacian distribution of this molecule is shown in Fig. 2. Also shown in this figure are the intersections of the interatomic surfaces as defined by the zero flux boundary condition that defines a quantum subsystem, the basis of the theory of atoms in molecules (7). All properties of such a subsystem, which is identified with an atom in a molecule, are defined by quantum mechanics. In this brief account we shall refer to only the average electron population. This is obtained by an integration of p over the basin of the atom. The charge on the C1 atom in ClF3 for example is 1.56e. In VC13, the charge on the vanadium atom is 1.992e, 0.664e having been transferred to each C1 atom. The vanadium atom has the electron configuration [ k ] 4 s 2 3 d 3and two of the five valence electrons of V have been transferred to the ligands in VCl,. Because of this charge transfer, the vanadium nuclear-electron attractive force increases in dominance within the atomic basin and the orbital ordering approaches more closely the true one-electron case. Thus the 3d orbitals are closer in spatial extent and energy to the s and p orbitals of the third quantum shell in the bound than they are in the free vanadium atom. This ordering of the orbital levels and the transfer of valence density are in agreement with the presence of three quantum shells on the vanadium atom (Fig. 2). As is discussed below, still stronger evidence for the mixing of the 3d orbitals with the 3s and 3p orbitals of the bound vanadium atom is provided by the polarizations of this shell. Each C1 atom exhibits the anticipated three complete quantum shells, Fig. 2. The polarization of the VSCC of each of these atoms towards the positively charged vanadium atom is evident from the contour diagram of the Laplacian and there is a bonded charge concentration in the VSCC of each C1 atom of magnitude

+

+


MACDOUGALL ET AL.

FIG.2. Contour and relief maps of the Laplacian of p for VCI,. The plane shown contains the vanadium nucleus and one of the chlorine nuclei. The dashed contours denote negative, the solid contours positive values of VZp. The interatomic surfaces and the bond paths are also shown, the bond critical points being indicated by the small dots. The large dots indicate the positions of maxima in the i-VSCC of vanadium and in the VSCC's of the ligands. Note the extreme polarization of the i-VSCC of the vanadium,atom. The relief map shows that the vanadium atom does not possess a fourth or valence shell of charge concentration. The peak in the i-VSCC of vanadium nearest the C1 is a (3, - 1) critical point. Directly behind it is the nonbonded maximum induced in the i-VSCC by the negatively charged C1 ligand. The two remaining and largest of the peaks in the i-VSCC are the axial nonbonded charge concentrations. 0.63 au; 1 au = 1 atomic unit = e/ao3. Because of its small relative magnitude, its presence is not evident in the relief map. Since vanadium does not possess a fourth shell of charge concentration, there are no bonded charge concentrations on vanadium. However, the shell of charge concentration associated with the third quantum shell of vanadium exhibits very marked polarizations. It is the thrusting of this inner shell of charge concentration of a transition metal atom into the forefront of chemical interaction with the ligands that accounts for the differing behaviour of a transition metal atom from main group elements.

1845

The third shell of charge concentration on V shall be referred to as the inner-valence shell charge concentration, or i-VSCC. There are cases where a first-row transition metal atom still possesses a fourth shell of charge concentration, an example being Mn2(CO)lo (3b). In such a case, where the bonding to the transition metal atom involves two quantum shells, the second shell is distinguished by refemng to it as the outer or o-VSCC. The i-VSCC of vanadium exhibits a (3,-1) critical point on each of the V - 4 1 bond paths (with a magnitude of 17 au), with its unique positive curvature lying in the plane of the nuclei. Thus this critical point appears as a two-dimensional maximum in the display shown in Fig. 2, since its two negative curvatures lie in the plane of the diagram. On the side of the i-VSCC opposite each such (3,- 1) critical point is a (3,-3) critical point, or local charge concentration of magnitude 22 au. 'The i-VSCC is clearly very susceptible to polarization by the negatively charged ligands, and a local charge concentration is induced in this shell opposite each of the chlorine atoms. Two more maxima are present in the i-VSCC on the threefold axis on each side of the molecular plane. They are by far the largest of the local charge concentrations found in this shell, with a magnitude of 39 au. The existence of concentrations of electronic charge along distinct spatial axes indicates that the electrons in the third shell of vanadium possess significant angular momentum, and they cannot be described as occupying a closed, albeit slightly polarized, inner shell. Thus the charge concentrations in the i-VSCC of vanadium form a trigonal bipyramid, reminiscent of the set of sp3d hybrid orbitals of the valence bond model. In agreement with the observation that the o-VSCC of vanadium is absent and that the 3d orbitals have reverted to the third quantum shell, the i-VSCC of vanadium can be described as possessing the configuration 3s23p63d2.There are five pairs of electrons in the third quantum shell of the vandium atom and, correspondingly, its i-VSCC exhibits five local concentrations of charge. Figure 3 shows corresponding diagrams of the Laplacian distribution for VOC13. The vanadium atom in this molecule possess a charge of +2.405e, with each chlorine bearing a charge of -0.554e and the oxygen one of -0.743e. The VSCC of oxygen exhibits a bonded charge concentration with a magnitude of 2.64 au and a single, slightly larger, nonbonded maximum of magnitude 3.09 au. The bonded charge concentration present in the VSCC of each C1 is slightly reduced from its value in VC13, to 0.60 au. The i-VSCC of vanadium is topologically equivalent to that found in VC13 -the same number and type of critical points are prcsent in each case - but their relative magnitudes have changed in significant ways. The extent of charge concentration in the i-VSCC of vanadium is decreased relative to that found in VC13 because of the further loss of charge to the oxygen atom. The magnitude of the (3, - 1) critical point along the bond path to each of the chlorines is decreased to 10 au and the opposing maximum induced in the i-VSCC by the presence of the negatively charged ligand is reduced in magnitude to 19 au. It is the axial charge concentrations, however, that undergo the largest changes. The magnitude of the axial charge concentration lying on the bond path to the oxygen, now a bonded charge concentration in the i-VSCC of vanadium, is reduced from 39 au to 9.5 au while the magnitude of the remaining axial concentration, a nonbonded charge concentration, is reduced by a lesser amount, to 21 au. Thus the charge concentration along the threefold axis, while reduced in magnitude from that present in VC13, is strongly polarized away from the oxygen into the corresponding nonbonded region. One anticipates that the addition of oxygen to the ligand

CAN. J. CHEM.

VOL. 67. 1989


This downward motion is equilibrated for a ClVCl angle greater than tetrahedral by the approach of the bonded maxima on the chlorine ligands to the axial nonbonded maximum on the metal and by the approach of the nonbonded maxima associated with the chlorines towards the bonded maximum opposite oxygen within the i-VSCC of vanadium.

Conclusions It has been suggested ( 3 b , 4 )that a modified VSEPR model be given in terms of the Laplacian of the charge density, one which is free of the assumptions regarding the existence of some number and arrangement of localized pairs of electrons with prescribed sets of properties. In the modified version of the model, this information is provided by the Laplacian of p, an observable and physical property of a system. The prediction of a molecular geometry then requires but a single postulate, that the minimum energy geometry is the one that maximizes the separations between the local concentrations of charge in the outer shell of charge concentration of the central atom. It is possible in some systems, particularly those involving transition metal atoms, that charge concentrations in both the o-VSCC and i-VSCC play a role-in determining the geometry. The domain of applicability of the model is then extended to all systems, since the Laplacian distribution has a definable set of properties in every system. What remains to be done is to continue to categorize the properties of the Laplacian of the charge density and discover in what ways it can deviate from our models of localized electron pairs. The vanadium systems that have been discussed in this brief note illustrate how this can be done. With these examples, there can be no doubt as to the fundamental nature of the basic assumption underlying the VSEPR model, that the most stable molecular geometry is in some manner determined by maximizing the separations between local concentrations of electronic charge. The Laplacian of p extends the physical basis of the model and removes the constraints of its range of applicability, but Gillespie's basic assumption remains true.

FIG.3. Contour and relief maps of VOC13 for a plane containing the vanadium, the oxygen, and one of the chlorine nuclei. Details as for Fig. 2. Note the increase in the angle between the axial nonbonded and one of the nonbonded maxima opposite a C1 from the value of 90" found in VCI3. The i-VSCC of vanadium in this plane exhibits three local charge concentrations -the bonded maximum nearest oxygen and the larger axial nonbonded maximum directly behind it with the nonbonded induced maximum lying between them. The peak nearest the C1 corresponds to a (3, - 1 ) critical point. sphere with its double-bonded domain will cause the ClVCl angle to decrease from its value of 120" in VC13. However, it is clear from the polarization of the i-VSCC of the vanadium that the ClVCl angle will not be reduced to less than the tetrahedral value, as would be found for a main group element, because of the presence of the axial nonbonded charge concentration, the largest of all the charge concentrations in the i-VSCC of vanadium. Because of the dominance of the nonbonded charge concentration in the i-VSCC of vanadium, the angle between it and the nonbonded charge concentration opposite each of the chlorine atoms is increased from the value of 90" in VC13 to 106. 1". This upward shift in these critical points away from the axial nonbonded concentration is coupled to the downward motion of the C1 ligands that are responsible for their presence.

Acknowledgement The authors wish to thank Professor Gillespie for suggesting the problem addressed in this paper. They also wish to dedicate the paper to him and to his ideas. P.J.M. and M.B.H. thank the National Science Foundation (U.S.A.) for partial support of this work. P.J.M. thanks N.S.E.R.C. of Canada for the award of a postdoctoral fellowship. 1. R. J. GILLESPIE. Molecular geometry. Van Nostrand Reinhold, and R. S. NYHOLM. Q. Rev. London. 1972; R. J. GILLESPIE

Chem. Soc. 11,239 (1957). 2. G. N. LEWIS.J . Am. Chem. Soc. 38,762 (1916). 3. (a) R. F. W. BADER, P. J. MACDOUGALL, and C. D. H. LAU.J. Am. Chem. Soc. 106, 1594 (1 984); (b) P. J. MACDOUGALL. The Laplacian of the electronic charge distribution. Ph.D. Thesis, McMaster University, Hamilton, Ont. 1989. R. J. GILLESPIE, and P. J . MACDOUGALL. J. 4. R. F. W. BADER, Am. Chem. Soc. 110,7329 (1988). D. A. HUMPHREYS, N. C. BAIRD,and E. A. 5. R. J. GILLESPIE, ROBINSON. Chemistry. Allyn and Bacon, Boston. 1986. p. 763. 6. S. HUZINAGA. Gaussian basis sets for molecular calculations. Elsevier, Amsterdam. 1984. Adv. Quantum 7. R. F. W. BADERand T. T. NGUYEN-DANG. Chem. 14,63 (1981). 8. A. F. WELLS.Structural inorganic chemistry. 5th ed. Oxford University Press, London. 1984.