Henry A. Bent
University of Minnesota
I
Minneapolis
Tangent-Sphere Models of Molecules 17. Uses in teaching
Chemistry witnessed this past year a striking event. Synthesized for the first time were compounds of the "inert" gases. Like most events in chemistry, this event was exciting in proportion to the degree that it was unexpected. It served, thus, as a dramatic reminder of a fundamental fact: chemistry is (still) an experimental science. Even today, for example, several decades after the advent of quantum mechanics, chemistry does not possess a satisfactory mathematical theory of systems more complex than the isolated hydrogen molecule. Several reasons for this were examined in Part I. It was concluded there that possibly the Pauli Principle is more important than previously supposed. To explore the chemical implications of this possibility, there was introduced a model in which tangent-spheres were used to represent a combined atom's valenceahell electron pairs-or, perhaps more properly, the "Fermi holes" associated with these electron pairs. This model, it was seen, is capable of representing the salient stereochemical features of conventional, covalent honds. It is found that the model can also be used to represent highly strained bonds and multicentered honds, atoms with expanded octets and atoms with contracted octets, intermolecular interactions and intramolecular interactions, and the effects of electronegative groups, lone pairs, and multiple bonds upon molecular geometry, bond properties, and chemical reactivity. These uses of the models are described below, following a summary, in the first three sections, of model uses discussed previously in Part I. Isomorphism with Valence-Bond Diagrams
Every line or pair of dots in the valence-bond representation of a molecule becomes in the model a sphere, and vice versa. I n effect the model is a three-dimensional reconstmction of the two-dimensional valencebond diagram.' Conversely, a valence-bond diagram may be viewed as a two-dimensional projection of a three-dimensional model in which (1) lines are used A ~reliminaryaccount of this work was presented at the Gordon Research Conference in Inorganic Chemistry, New Hsmpton, N. H., August 610,1962. This study was supported by a grant of a Faculty Summer Research Committee Appointment from the Graduate School of the University of Minnesota, by a du Pont Granbin-Aid to the Department of Chemistry of the University of Minnesota, and in part by the National Science Foundation. This support is grateful acknowledged. This is an effective way to introduce the models to beginning students who already have had some experience with valencehond diagrams. The models may he viewed as adding literally another dimension to their knowledge of atereochemistry. With very young students, however, we suspect it might be easier and more "natural" to introduce the models first and their twodimensional projection diagrams second.
for protonated spheres (honds to hydrogen atoms) and for spheres adjacent to two tetrahedral holes (the single bond in ethane, Fig. 5 (Part I), for example, and the components of the multiple honds in ethylene and acetylene, Figs. 38 and 39); (2) dots are used for unprotonated spheres bordering on a single tetrahedral hole (lone pairs); and (3) letters (H, C, N, 0, etc.) are used to indicate locations of protons and other more complex atomic cores. Gross Details of Slereochemislry
One immediate consequence of the tangent-sphere model is the tetrahedral geometry about atoms involved in four single bonds (Figs. 3 and 5; also Figs. 8-15 (Part I)), the planar geometry about atoms involved in one double bond (Fig. 38), and the linear geometry about atoms involved in a triple hond (Fig. 39) or two double honds (Fig. 21, Part I). Finer Details of Stereochemistry
More subtile details of molecular conformation that can be illustrated and discussed with the aid of tangentsphere models include: the staggered and eclipsed configurations of ethane; the trans and gauche confomers of n-butane; the equatorial and polar bonds and the chair and boat confomers of cyclohexane (and the more complex confomers of higher perhydroaromatics); the puckered cyclobutane ring; the orientation of groups adjacent to double honds and the equilibrium shapes of such molecules as acetaldehyde and propylene, l,3-hutadiene, l,3,5-hexatriene, and 1,3,5,7-cyclooctatetraene; and steric interactions in skew n-butane and such other molecules as 2-butene, the isomers of flnoro-propene, and the mono- and di-methyl suhstituted cyclohexanes (80, $ 1 ) . Strained Bonds
Tangent-sphere models can be used to represent not only normal multiple bonds, Figures 38 and 39, hut as well the strained, bent honds in such molecules as cyclopropane, Figure 25, cyclopropene, Figure 26. and benzyne, Figure 27. Multicentered Bonds (22)
The model of ethylene, Figure 38, can serve equally well as a model of B2H,, if one supposes that the heavy atoms in the two tetrahedral holes are boron atoms and that all six spheres are protonated. The relation of EDITOR'S NOTE: The first part of this paper appeared in the 40, 446 (1963). Literature September issue of Tnra JOURNAL, citations and figures are numbered consecutively for the whole paper; citations 1-21 and Figures 1-24 appeared in the first pad, Volume 40, Number 10, October 1963
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Figure 25. Tangent-sphere model d cyclopropane. Dotted liner represent rpherer whose senterr lie behind the plane of the paper, heavy lines spheres whore senterr lie above this plane. The inner three circles repre3ent corban-corbon bonds, the outer 3ix corbon-hydrogen bondr At the right is the corresponding valence-bond diagram.
The corresponding tangent-sphere representation of BAH,, . .. is illustrated in Fieure 28. The B,H,, . ."structure contains four three-centered boron-hydrogen-boron bonds.z Addition of a trigonal set, representing the ion CH8+, to one component of the double bond of ethylene converts the model shown in Figure 38 into the tangentsphere representation of a non-classical carbonium ion whose valence-bond structure as shown below and in Figure 29 contains one three-centered carbon-carboncarbon bond.
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Figure 26. Tangent-sphere model and valence-bond diogram of syclo(top propene. Clockwise fmm 12 o'clock the circler represent C-C center), C-H, C-C, CHn (bottom center), C-C, C-H.
This three-centered bond is similar to the three-centered aluminum-carbon-aluminum bonds in the dimer of aluminum trimethyl and to the three-centered boronboron-boron bonds in B,Cla.
Figvre 27. Tangent-sphere model and valence-bond diagram of the triple-bond representation of benzyne. The model ir formed from fragk e d containing, from top to bottom (or ments of flve c l o ~ e - ~ ~ ~layers bottom to top). 1,4, 4, 4, and 1 spheres. Only the =ix outer tetrahedral holes (two on the right, two on the left, and one each at the top and bob tom) are occupied by carbon nuclei. It may be significant to note lhot it i s apparently imposible to construct an equal-sphere-sire model of o rirmembered ring containing four double bondr.
Figvre 28. Tangent-sphere model mnd valence-bond diagrom of &Alo. The model's four tetrahedral holes, which lie N, E, S, and W from thecenter, correspond to the gaps in the volence-bond diogrom at the right. The six stippled circler represent two-centered boron-hydrogen bond,, the four open ones three-centered boron-hydrogen-boron bondr. The crorr hotched circle reorerents the two-centered boron-boron bond.
this structure to the ethylene structure may be illustrated in two-dimensions as follows (cf., ref. (Sf), p. 306).
I n a similar manner the valence-bond representation of B4HI0can be derived from the following (probably hypothetical (93)) bicyclic structure of the dirner of nitric oxide. 524
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Figure 29. Tangent-lphere model of the nonclarricd carbonium ion CsHit. In the superimposed valence-bond diogrom the threecentered sorbon-sorbon-carbon bond i s indicated by three intersecting lines, which form on inverted "Y." In the literature this bond is usually indicated by o dashed triangle. Transition from this bridged structure to the claaical structure CH3CHeHnt can be viruolired or occurring through a lperhopr solvent, SN?-initiatedl downward motion aaoinrt the carboncarbon twc-centered bond (the bottom-most sphere) ofihe two protonoted spheres of the methylene group an the right, or the one on the left. Formotion of the secondary carbonium ion CHsCH+CH8from the primary cwbonium ion CH3CHlCHltcon be visualized ar occurring in two steps of the type just described through the intermediate formation of o three-centered carbon-hydrogen-carbon bond.
Figure 30. Two views of the bonding poirr in B4Clr. The four stippled rpherer represent two-centered boron-chlorine bonds, the four openspheres three-centered boron-boron-boron bondr. Each boron atom has in ils one boron-chlorine bond and three boron-boronvalence shell four pair.; boron bondr Not shown are the unshared pairs on the chlorine otomr
The tangent-sphere representation of BPCln, excluding the lone pairs on the chlorine atoms, contains an inner tetrahedral set formed from four mutually taugent spheres. On each of the four trigonal faces of this tetrahedron rests another sphere (Fig. 30). Created in this way are four symmetrically placed tetrahedral holes. These holes contain the boron nuclei and their inner-shell electrons. The four outer spheres represent two-centered boron-chlorine bonds. The four inner A three-centered bond is an electron pair that is in the valence shell of three atoms.
spheres represent three-centered boron-boron-boron bonds. Most of the known boron hydrides can be represented by tangent-sphere models. Figure 31 is an illustration of a symmetrical structure for the unknown compound BOHIS. The presumed instability of this structure probably arises in part from the close approach to each other of the top three hydrogen atoms. The tangent-sphere representation of lithium hydride is interesting to consider. It consists of a cubicclose-packed array of spheres, each sphere of which represents an electron pair and at whose center is a
Figure 31. Tangent-sphere model and valence-bond diegram of a hypothetical structure for BpHla. Open spheres are protonoted. The three a t the M p ond the six outer ones a t the bonom represent two-centered boron-hydrogen bondr; the six inner open spheres represent threecentered boron-hydrogen-boron bondr. Stippled spheres represent twocentered boron-boron bondr, block ones three-centered boron-bwonboron bondr For a leading reference on the boron hydrider, see Hoffmonn and Liprcomb, I . Chem. Phys., 37, 2872 (19621.
proton. These protonated spheres represent the hydride ions. Surrounding each hydride ion are six lithium ions in the octahedral holes of the protonated electron-pair lattice.= The hydride ions represent, therefore, seven-centered lithium-lithium-lithiumlithium-lithium-lithium-hydrogen bonds. This result is not entirely surprising. Starting with the neon atom and moving to the left through the hydrides HF, HzO, NHs, CH*, B2H6,and (BeH,),, one finds that the number of electron pairs in one-centered orbitals (the unshared pairs) steadily decreases; the figures are, respectively, 4, 3, 2, 1, 0, 0, 0. The corresponding figures for the number of electron pairs per heavy atom in two- and three-centered orbitals are, respectively, 0, 1, 2, 3,4, 2, 0 and 0, 0, 0,0, 0, 1, 2. The trend seems clear. In moving toward lithium hydride increasing use is made of multicentered orbitals. Altepatively, one may note that the core radius (0.60 A) of lithium is much larger than the core radii of the other first-row elements (0.15, 0.11, 0.09, and 0.07, respectively, for carbon, nitrogen, oxygen, and fluorine (Pauling)). In this respect, lithium is like the secondrow elements-particularly magnesium (core radius 0.65 A)-where, so to speak, the valence-shell electrons touch the atomic cores before they touch each other and where, as a result, expanded octets are not uncommon. In the case of lithium, however, the formal charge on the atom would generally be excessive when its valence-shell contains six electron pairs unless, as inlithium hydride, these electrons are widely shared.
8 Each Li+ ion consists of a nucleus, charge +3e, and a 1s' shell (not mentioned above)whose volume i8 about 3% of the volume of an H-ion.
Electron Conflgurotion About Atoms That Do Not Obey the Octet Rule
From the standpoint of the tangent-sphere model, the stereochemical features listed under "Gross Details" are an immediate consequence of the coordination by an atom of four localized electron pairs. The examples so far discussed all fall into this category. Figure 32 illustrates the useful fact that the tangent-sphere model can also be used to obtain the orbital configuration about atoms with electron-pair coordination numbers 1 He, Li+), 2(BeH,), 3(BH3, CH,+, CHZ (singlet state)), 5(PC15, SF4, C1F3, XeF2, I%-), and 6 (SF6,NSFa,BrF5,XeF,, IC1,-).
Figure 32. Summary of the rtereochemirtry about combined otomr for the electron-poi. coordination numbers commonly encountered in practice. Coordination about on atomic core of one, two, three, four. Rve, or six locolired electron pairs in the monner illustrated corresponds to the use. respectively, of the hybrid o r b i t d ~s, rp, rpZ, spa, drps, or d2sppJ.
Isoelectronic Molecules
The methane model, Figure 3 (Part I), may be thought of as representing also the electronic structure of ammonia and water. In the latter instances one or more of the valence-shell spheres would not be protonated. Calculations suggest that this does not greatly alter the effective size of a sphere (16a). Similarly, Figure 5 may be used to illustrate the electronic structures of ethane, methyl amine, methyl alcohol, methyl fluoride, hydradne, hydroxyl amine, hydrogen peroxide, and so forth. It is interesting to note that the chlorine derivative of the recently reported (and probably planar) molecule XeF, (24) is isoelectronic with the wellknown planar ion IC14-. In the same sense, XeFz is equivalent to IC12-. Intermolecular Interoctions
Inspection of the tangent-sphere models suggests that the interaction of an unshared pair on one molecule with the force field of a nucleus in another molecule should probably occur most readily off one of the external "pocket" sites in the second molecule. If the interaction between the attacking pair and the nucleus attacked is sufficiently strong, the nucleus may move from its normal position within its tetrahedral hole in the direction of the approaching lone pair on the nucleophilic reagent. This shift in nuclear position relative to the electron cloud of the system can be crudely summarized as follows.
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The models thus provide a wncrete and perhaps not altogether imprecise picture in three dimensions of that branch of two-dimensional pencil-and-paper chemistry that for want of a better term is often referred to as "arrow bending" or "curly arrow chemistry." Pictured in Figure 33 is the tangent-sphere representation of a conventional SN2displacement reaction. The same form of pair-pocket interaction can be used to account for such facts as the structure of crystalline iodine and the structures of the iodine complexes of amines, ethers, and ketones (25); the presence of linear chains in crystalline halogen cyanides; the occurrence in the halogen cyanides as well as in certain other molecular crystals of abnormally short intermolecular contacts; and the shapes of the ions Is-, Figure 34, and L2-(26). Intramolecular Interactions
Strong interaction between an electron pair and an adjacent internal pocket may lead after a simple nuclear displacement of the type mentioned to the formation of a new chemical bond with simultaneous rupture of a n old one, i.e., to ,trans-elimination. Weaker interactions of the same kind may manifest themselves as lengthened vicinal bonds (27) or, if the vicinal bond is one of the C-H bonds of a methyl group, to an a p parent tilt of the methyl group's symmetry axis toward the electron pair ( B ) , with a concomitant increase in the corresponding valence angle. (In dimethyl ether, for example, the COC angle is greater than tetrahedral.) I n orbital language this interaction may be described as the overlap of an electron pair with the internal lobe of the antibonding orbital of a trans-oriented vicinal bond.4 These conclusions are supported by the recent suggestion of N. Muller and D. Can (J.Phys. Chem., 67, 112 (1963)) that the most important single parameter influencing the magnitude af the spin-spin coupling constants between IPFand directly bonded LsCnuclei is the "degree of double character" of the C-F bond.
Starter Kit
Many of the facts of structural chemistry cited above can be demonstrated at the cost of only a few cents with 12 small spheres preassembled as a tetrahedral group (called below "4"), a digonal group ("2'7, and two trigonal groups ("3", "3"). From these four vest pocket-sized components can be assembled-with the aid of appropriate valence-bond diagrams and a little manual dexterity-models of the tangent-sphere representations of the families of isoelectronic structures that contain such molecules as methane (4), ethane (4+3), propane (4+3+3). ethylene (4+2 or 3+3), carbon dioxide (3+3+2), carbon suboxide (3+3+2+4), propylene (4+2+3), cis-and trans-2-butene (4+2+3+ 3), acetone (4+2+3+3), acetylene (3+2), diacetylene (3+2+4), cyclopropane (4+3+2), cyclopropene (3+ 3+2), and cyclobutane (4+3+3+2). The 1Psphere kit can also be used to illustrate such things as the digonal, trigonal, trigonal bypyramidal, and octahedral arrangements of electron pairs about atoms (Fig. 32);
Figure 34. Tangent-sphere representotion o f the pentaiodide ion. As the formulotion 1-(i9l2 for this ion ruggert, i s may b e viewed ar the interaction of an i o d i d e ion with t w o iodine moleculer: 1-1,. .I-. .I-I. The dotted liner in thir formulotion of t h e complex represent interoctionr between electron poirr in t h e valence shell of the iodide ion (the stippled spheres .above) w i t h t h e external pockets of the iodine molecules. The solid liner reprerent the original, now slightly weokened iodine-iodine bonds within t h e iodine ligands (the black rpherer above). T w o such pairpocket interaction$ ( . .) on the $.me tetrahedral donor atom (1-1 produce a non-lineor, v-shaped complex. It is pictured a b o v e as seen f r o m the open e d g e of the "v.'. Similar conriderations o p p l y t o the ion lrP,whose modified v.lenre.bond reprerentotion is 1.1.. . I - . .I-I.. .I-. .1.1
.
.
.
+
.
Stippled rpherer represent 0-H and HsC-CI = HO--CHs CI-. o f a W d d e n inversion; the r e a d i o n H 0 - + Figure 33. Tangencsphere C-H bondr, open spheres lone pairs. Cross-hatched spheres represent bondr between carbon and chlorine (stages 1 and 21 or c w b o n ond o x y g e n (stager 4 and 51. The rolid d o t represents the carbon core. The dashed line thmugh t h e methyl group indicates a plone through which t h e sorbon core eventually Stoge I : the readants S t o g e 2: bockride attack on the methyl group b y t h e hydroxide ion; t h e carbon core is beginning to move t o w a r d one of polre.. the previously unrhored pairs on t h e o x y g e n atom. Stage 3: the transition state; thir could a i m b e o model of the triiodide ion, or of xenon difluoride. S t o g e 4: "mirror imoge" o f stage 2 ; t h e carbon c a e is now more closely a m x i a t e d with t h e incoming poir on o x y g e n than with t h e outgoing pair on chlorine. Stage 5 : the product,.
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the equilibrium conformation of methyl groups adjacent to double bonds and the origin of the relatively low barrier to rotation in cis-2-butene (930); tilted methyl groups; and such simple hut fundamental chemical reactions as borane plus ammonia, reversible hydration of olefins, the ring- and olefin-producing reactions of the carbonium ion CH3CH2CH2+(39), and carbene insertion reactions. The Hellmann-Feynmon Theorem
A consideration of the electrostatic forces on atomic nuclei in molecules adds another dimension to the uses of the tangent-sphere models. All forces on atomic nuclei in molecules can he considered as purely classical electrostatic attractions and repulsions involving Coulomb's law. The force on a nucleus, according to the Hellmann-Feynman theorem (?), is simply the electrostatic force that would be exerted on this nucleus by the other nuclei in the molecule and by the molecule's electron cloud considered as a classical charge cloud that is prevented from collapsing by obeying Schrijdinger's equation (7a).= To use the Hellmann-Feynman theorem it is not necessary to have a detailed knowledge of the wave function. A knowledge of the ordinary 3-space electron density function is sufficient. Today, of course, even this is not known with any great precision for most molecules of chemical interest. One might well question, therefore, the use in a qualitative sense (see below) of a theorem that, strictly speaking, applies rigorously only to a mathematically exact electron density function-which we do not have. The answer must be that our procedure is not the procedure of applied mathematics but the procedure of a growing science. Qualitative applications of the HellmannFeynman theorem, such as will be described below, do seem to fall into a consistent, and therefore in some senses, useful pattern. This fact established, one might hope thereby to be able in the future to progress further, step by step, toward a satisfying description of molecular structure. Effect of Lone Pairs on Bond Angles
The angle between single bonds that meet at an atom with unshared electrons is generally less than tetrahedral, 107' in ammonia, for example, and 104.5" in water. This fact has been described by saying that lone pairs behave as &seeking centers (30, 31). Such behavior can be rationalized as follows. The carbon nucleus in methane experiences no net electrostatic force upon i t - d u e to repulsions of the protons and to attractions of the bonding pairs-when it is in the center of its tetrahedral hole. (Calculations (16a) suggest that nuclei of first-row elements can he assumed to be at the centers of their ls2 shells, which in the present context may therefore be regarded as part of the nucleus.) In ammonia, however, the center of the tetrahedral hole-for simplicity we suppose here that all valence-shell spheres are the same size-is not a position of electrostatic equilibrium for the nitrogen In the context of the tangenesphere models, i t may be noted that minimizing the energy of a molecule with respect t o the nuclear positions holding constant the positions and radii of the tangent-apheres yield8 the Hellmmn-Feynman theorem. Minimizing the energy with respect t o the sphere radii holding eonstant the nuclear positions yields the virisl theorem.
nucleus. To reach such a position the nitrogen nucleus must move away from the protons toward the lone pair. In effectthe nitrogen atom nucleus is pushed toward the lone pair by the hydrogen atom nuclei. This diminishes the H-N-H angle and requires that less p character and more s character be included in the description of the lone pair orbital if this orbital is to be described in terms of spherical harmonics centered a t the nitrogen nucleu~.~ Effect of Electronegolive Groups on Bond Angles and Bond Lengths
Both bond angles and bond lengths are affected by electronegative groups. In compounds of the type A-B-C the length of bond A-B often decreases as the electronegativity of substituent C increases (32);
y,;
,--Y- - - -
Figure 35. Summary d the dirplosement of the electron cloud about on .tom relative to the .tom'$ nucleus lthe small sphere) when the electranegativity of the wbrtituentr bound by the two domward-pointing valencier increaser. The central atom's p character in these two valencies increaser and, correlpond'ngly, nts p charooer :n the two upword-directed V O I ~ ~ Cdecreorel. ~I In drawing this Rgure In. aswmpt;on has been mooe that the rodiol fbnction onouotsd w m o r, orb'td is not os contrortea 05 that of an r orbital
similarly, in compounds of the type AB2 and AB3 the valence angle BAB frequently decreases as the electronegativity of substituent B increases (33). These effects have been summarized in a rule that states that the p character of an atom tends to concentrate in orbitals that the atom uses in bonds toward electronegative groups (30, 31). This rule is illustrated diagrammatically in Figure 35. Figure 35 might represent, for example, the change in the electron cloud about an oxygen atom that occurs in going from OH2 (bond angle 104.5") to OF2 (bond angle 103.S0). In this case the small sphere in the cube center represents the oxygen atom core, the two lower spheres represent bonding pairs of electrons (in either O-H or O-F bonds), and the two upper spheres represent the unshared pairs on the oxygen atom. In going from OH2to OF%the bonding pairs move away from the 6 Dr. Kimhall has remarked that "In discussing bond angles [to such elmw~rts35 r ~ r h o n ,nitrogeu, aud o w p e n ] it is an importnlrt f k ~ thal l the ewe (inclmhllg the la pllr is s m d etlollgll r c n ~ e rcf t t wtr:~lle,lrm ~ ~ without tc, m o w w i ~ h i nt h e .hde1s t ~ l w overlap. The motion of the core within the hole is the principal, but not the onlv motion. zffectine the bond snele. ' h e less
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oxygen nucleus and the angle subtended by them a t the oxygen nucleus decreases; correspondingly the unshared pairs move in toward the oxygen nucleus and the angle subtended by them at the oxygen nucleus increases. The four arrows summarize the diisplacement of the electron cloud relative to the oxygen nucleus. The dotted lines indicate the new interorbital angles. If the wave function of a molecule can be satisfactorily expressed in terms of spherical harmonics centered at the nuclei, one may say that the change from OHI to OFz corresponds to shifting oxygen p character from the orbitals occupied by the lone pairs on oxygen to the bonding orbitals of the oxygen atom. To this point, as well as in all other discussions of molecular structure based upon the use of atomic orbitals, the frame of reference adopted has been that of an observer standing a t the location of an atomic nucleus and looking a t the electron cloud about him from this fmed point. The tangent sphere model suggests a second frame of reference: the electron cloud itself. From the point of view of an observer anchored to the electron cloud, the change in the position of the oxygen nucleus relative to the electron cloud when going from OH2 to OFz can be represented in Figure 35 by a single arrow: one on the oxygen nucleus pointing upward. A displacement of this kind seems reasonable. Electronegative atoms are atoms with large effective nuclear charges. These relatively unscreened charges exert on adjacent nuclei strong forces of repulsion that tend to drive the nuclei away from the electronegative atoms. Displacements resulting from these forces alter bond angles and bond lengths in a manner that is consistent in every respect with the previously cited rnle regarding the distribution of atomic p character in molecules. Bond angles involving electronegative groups and bond lengths to substituents opposite electronegative groups will both tend to be smaller than normal, as k often observed. Shapes of SF* and ClFa
The sulfur-fluorine bonds in SF4 are slightly shorter than the sum of the conventional covalent radii for sulfur and fluorine. It therefore seems reasonable to suppose that the sulfur-fluorine bonds in this molecule are conventional electron-pair bonds. This implies that the sulfur atom in SFI has an electron-pair coordination number of five (four pairs in bonds to fluorine and one pair unshared). The structure of this molecule has been determined recently by Tolles and Gwinn (54). Their results are given in Figure 36. The molecule can be considered to be a trigonal bipyramid in which one of the equatorial positions is occupied by a lone pair. The structure of this molecule appears to he consktent with the expectations of the tangent-sphere model. Location of the lone pair at an equatorial position is reasonable since this places the "s-seeking pair" (50,Sl) as close as possible to the sulfur nucleus. The bonds to the axial fluorines (F.) are slightly longer than those to the equatorial fluorines (Fe). This, too, is to be expected if the plausible assumption is made that the sulfur nucleus lies in the plane defined 528
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by the centers of the spheres that represent the equatorial electron pairs? That the actual ratio of bond lengths (S-F)& (S-F.,,.,,i.~) is less than the calculated value is to be expected, also; for unlike the axial fluorine nuclei, which exert practically no net force on the sulfur nucleus, the equatorial fluorine nuclei exert a large resultant force on the sulfur nucleus in the direction of the lone pair. A displacement of the sulfur nucleus toward the lone pair would increase the axial-fluorinesulfur distance less rapidly than the equatorial-fluorinesulfur distance and would therefore diminish the value of the ratio of the first distance to the second. Such a displacement would also make the angle F,SF. less than 120°, as observed, and would cause the angle FaSF. to depart from 180" in the observed direction. S i a r considerations apply to the molecule CIF,, Figure 37.
Figure 36. tetrafluoride
Strvdure
(34).
of
sulfur
Figure 37. trifluoride.
Structure of chlorine
Effect of Multiple Bonds on Bond Angles and Bond Lengths
The angle between two single bonds that meet at an atom bound by a double bond to another atom is generally greater than tetrahedral-118' for the adjacent C-H bonds in H2C=CH2,for example. Also, the length of the single bond from a substituent R to an atom X as a rnle decreases as the remaining three bonds to atom X change from being all single bonds (R-X-)
/
to one single bond and the components of a
\ 7 This an&& can be csrried a step further. Insofar as the equal-sphere-siae approximation for the valence-shell electron pairs of an atom is useful, one would expect the axial and equatorial bond lengths in SFc a molecule in which the central atom has an electron pair coordination number (EPCN) 5, to he, respectively, greater than and less than the S-F bond length in SFn(EPCN 6). The latter length 0.02 A. . is listed as 1.56-1.58 his, as expected, (S - FSxial)EPCN 5 > (S - F)EPCN6 > (S - F..IEPCN 5 The difference between the first two is about 0.07 A. S~rn~laxly, the Xe-F bonds in linear XeFs (an EPCN 5 configurittion that place8 the three unshared pairs in xenon's valence shell as close as possible to the xenon core-at the slight expense of the bonding paim) we longer than the Xe-F bonds in XeF, (EPCN 6), agaln by about 0.07 1. The experirnent%lvalues recently reported far these Xe-F bond lengths are 2.00 A in XeFp (LEw, H. A. AND ADRON. P. A. . I Am. . Chem. Soc.. 85. 241 (1963)) and 1.93 A in XeF4 (~.EMPLETON, D. H., ZALKIN, A,,FORRESTER, J. D. AND WILLIAMSON, S. M., J. Am. Chem. Soc., 85,242 (1963)).
*
double bond (R-X-)
/
Mechanism of the Inductive Effect
to the three components of a
\\ triple hond (R-X=). The C-H bond lengths in ethane, ethylene, and acetylene, for example, are 1.114, 1.086, and 1.055 A, respectively. These facts can be reconciled with each other in the following fashion. The carbon-carbon internuclear distance decreases from 1.53 to 1.34 to 1.20 A in going from ethane to ethylene to acetylene. Correspondingly the coulombic force of repulsion between the two carbon nuclei increases. I n the molecules just cited this mutual repulsion tends to drive the carhon nuclei into their respective C-H bonds. This outward thrust tends to make the valence angle opposite a double bond greater than normal, Figure 38. At the same time it tends to
The effect of electronegative substituents and multiple bonds on molecular structure can be summarized as follows. The location of atomic cores within their tetrahedral holes can be altered in two ways: by changing the nuclear charges on adjacent atoms (see section on "Effect of Electronegative Groups on Bond Angles and Lengths") or by changing the distances to adjacent atoms (see section on "Multiple Bonds). Taken together these two items provide a concrete picture of the inductive effect. Consider, for example, the molecular fragment R-C,-C2-C3. As the electronegativity of substituent R increases,carbon-core C,ispushed toward carbon-core Cz; and as the distance between carbon-core C, and carbon-core C2 decreases, carhoncore C2 is pushed toward carbon-core C8; etc. This description of the inductive effect is analogous in every detail to one given earlier by the author in which use was made of the rule concerning the distribution of atomic p character in molecules (88). The Variable Effect of Multiple Bonds on Bond Lengths
C b
C.H.
HCH
1095.
118.
C-H
1.0%
1.086
JCH
125
65
w
DM,
a36
Fig. 3 8
C-H
1.055
Jw
248 1.35
r
Fig. 39
Figure 38. Tangent-sphere model of ethylene. S m d dots represent protons, larger dots carbon nuclei (rmoll enclosing dashed circles represent 1 s elechonrl. The carbon nuclei push each other toward the adfacent C-H bonds, with the rervltr indicated. Straight liner illurtrote the iromomhim that exists between a molecule'^ tonaent-r~here model and . . valence-bond diogram. Figure 39. Tongent-sphere model d acetylene showing the effect of the outword displacement of the carbon nuclei on several properties of the rorbon-hydrogen bond icf. Fig. 381.
make the corresponding C-H distance less than normal, the more so the closer the two carbon nuclei are t o each other, Figures 38 and 39. Also, as the carboncarbon internuclear distance decreases and the buttressing effect of one carbon nucleus on the other increases, the external C-H bonds are brought ever more under the influence of the carbon nuclei and the apparent electron-withdrawing power or inductive constant, v, of the carbon atoms in these valencies increases (55) as does also the carbon(l3)-proton coupling constant, Jm, whose m a g ~ t u d ecritically depends on the interaction between the electrons of the C-H bond and the carbon nucleus (56) This discussion also illustrates the appropriateness of a long-standing supposition (57); namely, that when describing the electron distribution inmolecules in terms of spherical harmonics centered a t atomic nuclei, it is proper to suppose that the radial function associated with an s-orbital is more contracted than that of the corresponding gorbitals. &Thecarbon-carbon spin-coupling constants reported recently by K. Frei and H. Bernstein ( J . C h . Phys., 38, 1216 (1963)) provide another illustration of the manner in which the properties of single bonds are influenced by the outward thrust of adjacent, multiply banded carbon rn~clei.
The previous discussion provides an explanation for the curious fact that the change from a single bond to a double bond or from a double bond to a triple bond alters the length of an adjacent carbon-substituent hond relatively little when the substituent is hydrogen, somewhat more when the substituent is a methyl group, and still more when the substituent is chlorine (81,59). For as the carbon nucleus moves toward the substituent, the substituent will tend to ride further along the bond--corresponding t o the picture of the inductive effect just cited-the larger the force constant of the carbon-substituent bond. One expects to find, therefore, an inverse correlation between the stretching force constant of a bond, which measures the resistance of the bond to changes in length, and the susceptibility of the bond to changes in length with a change in adjacent unsaturation. The larger the stretching force constant, the smaller should be the change in bond length. Typical stretching force constants for C-H, C-CH3, and C-C1 bonds, are, respectively, 5.0,4.5, and 3.4 millidynes/angstrom. Conclusion
With the passage of time it seems likely that the discussion given above will prove to have only touched the surface of the possible applications of the tangeutsphere models. Already it has been shown (16a, b) that with some simplifying assumptions regarding the nature of the charge distribution within each sphere-.g., that it is uniform-it is possible to introduce Planck's constant into the treatment of complex molecules with a simplicity that exceeds and with an accuracy that rivals that of conventional, more complex treatments. Largely this is because the tangeut-sphere treatment avoids the use of overlapping one-electron orbitals (16, Part I) and, hence, the very great and perhaps entirely artificial mathematical problems associated with exchange integrals (9, Part I). On the other hand, the tangent-sphere treatment does partake of that part of current molecular orbital theory that is generally conceded to be sound: namely, Volume 40, Number 10, October 1963
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the requirement that the electron cloud be considered not as parts in isolation but as a whole. This happens quite automatically since the spheres by supposition are tangent to one another. Remove one of them and the shape of the set that remains, or the location of the atomic nuclei within the set, will generally change. The implications of this statement appear to be consistent with what is known about the shapes of radicals and of molecules in excited states. Another feature of the model not discussed above concerns the role of charge correlation between electrons of two different spin sets. Probably always to a small extent and, following Linnett (Id), perhaps sometimes t o a large extent the centers of the Fermi holes of two spin-paired electrons will not be coincident. Some chemical implications of this extension of the model have been explored elsewhere (40,41). The qualitative features of the energetics of the model have been summarized by Strong (19): An assembly of atoms will be most stable when the electrons can be as large as possible, consistent with keeping electrons away from each other but close to the nuclei, while keeping the nuclei away from each other. HF ; Figure 40. Tongent-sphere model of the symmetrical hydrogen bond Lmge dots reprered the in HF;. fluorine corer. The m o l l dot representsthe proton.
been carefully skirted in the present discussion. Also, while one may agree that the models do work in some cases, it should be emphasized that it is not really known why they work as well as they seem to. In short, what has been described here is a model, not a chemical theory. Properly the models should he viewed as a supplement to, not a substitute for, quantum mechanics in an area where the latter has encountered severe computational problems. Perhaps some young reader of THIS JOURNAL may someday see how to achieve ahappy union between thesemodels of the experimental facts of structural chemistry and the underlying principles of theoretical chemistry.
Literature Cited (22) EISTERT,B., Z. Phys. Chemie, B 52,202(1942); EBERHARDT, W. H., CRAWFORD, B., JR., AND LIPSCOMB, W. N., J . Chem. Phys., 22, 989 (1954); Longuet-Higgins, H. C., U v a ~ tRe".. . 11.121 (19571. (23) Cf. FATELEY, W. F., BENT,H. A,, AND CRAWFORD, B., JR., J . Chent. Phys., 31,204(1959). H. H., SELIG, H., AND MALX,J. G., J. Am. Chem. (24) CLAASSEN, Soe., 84, 3593 (1962); CHERNICK, C. L., ET AJ.., Science, 138,136 (1962). (25) HassEL, O., AND R@MMING, C., Quad. Reu., 16, l(1962). E. E., AND WIEBENGA, E. H., Recueil, 78, 724 (26) HAVINGA, (1959). (27) BENT,H. A,, J . Chem. Phys., 36,1090 (1962). (28) PIERCE,L., AND HAYASHI, M., J . Chem. Phys., 35, 479 (1961). I., J. Am. Chem. Soe., 84, 3962 (29) SKELL,P. S., AND STARER,
. . . .
(lQfi7I ,*""-,.
Hydrogen bond formation (Fig. 40) has been described by Kimball (18). As the nuclear charge on the heavy atom increases in the sequence CHa, NHB, HzO, HF, the protons are pushed closer to the surface of the shrinking electron pairs. Conversely, as the central nuclear charge decreases in the sequence NH&+, CH,, BH4-, the protons approach the centers of their expanding electron pairs; these protonated pairs therefore become more hydridic and in the presence of sufficiently strong Lewis acids, such as A13+and BH8, may be accepted, proton and all, into the valence shell of another atom, forming thereby, as in aluminum borohydride and diborane, protonated, three-centered bonds. Many interesting problems, however, remain to be considered. Aromatic systems, for example, have
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(30) (31) (32) (33)
(34) (35) (36) . . 1.17)
BENT,H. A., J. CAEM.EDUC.,37,616 (1960). BENT,H. A,, Chem. Rev., 61,275(1961). BENT,H. A., J. C h . Phys., 33,1260 (1960). MELLISH, C. E., AND LINNETT,J. W., Trans. Far. Soc., 50, 657 (1954); W A L ~ AA., D., in "Progress in Stereochemistry," Vol. I, KLYNE,W., ed., Academic Press, Ine., New York, 1954, chap. 1. TOLLES,W. M., AND GWINN,W. D., J. Chem. Phys., 36, 1119 (1962). TAFT,R. W., JR., in "Steric EfTeets in Organic Chemistry," NEWMAN, M. S., ed., John Wiley and Sons, Inc., N. Y., 1956, chap. 13. MULLER. N.. AND PRITCHARD, D. E., J. Chem. P h.m .. 31,768 . (1959): MTTT~LIR R.~ N S... J . Ph118. C h . . 41. 318 (19371: WALSH.
(39) BROWN, M. G., Trans. Far. Soe., 55,694(1959) (40) BENT,H. A., J.Znorg. Chem., 2,747 (1963). (41) BENT.H. A,, in "Orpanic Sulfur Compounds," KHARASCH, N.,ed., erga am on Press, N. Y., In press
