Henry A. Bent
University of Minnesota Minneopolis
II
Tangent-Sphere Models of Molecules I.
Theory a n d consfrvcfion
Quantum mechanics has achieved some strikmg successes. Exact calculations have been made of the complex spectroscopic properties of the hydrogen atom and the hydrogen-molecule ion. I n addition, computations of extraordinary accuracy have been made of the bond dissociation energy of the hydrogen molecule and of the ionization energy of the helium atom, the latter-with the aid of a 1078-term functionto an accuracy of one part in two billion ( 1 ) . At this level of computational accuracy, relativistic corrections for the variation in mass with velocity and for the spin magnetic dipole interaction between electrons become important. Equally striking, however, is the loss of accuracy in calculations on more complex systems (8). Reminiscent of the time in the development of the DebyeHuckel limiting law for ionic solutions when physical chemistry was sometimes humorously referred to as the chemistry of slightly impure water, rigorous quantum chemistry today might he called in the same spirit, a quantitative theory of the hydrogen molecule, and of slightly less complicated systems. But as Platt has pointed out (S), quantitative calculations on the hydrogen molecule, however exact, yield essentially no information whatsoever regarding stereochemistry. I t is not surprising, therefore, to find that rigorous quantum chemistry has so far failed to become a workable, everyday theory of chemical structure. There seem to be a t least two possible reasons for this. First, the calculation of the properties of a manyelectron system based upon a straightforward, mathematical solution of the Schrodinger equation is impossible. Hartree has pointed out that merely to print the result of such a calculation for a single stationary state of a single stage of ionization of a single atom as complex as iron would require more matter for ink and paper than is contained in the entire universe (4). Clearly such an answer even if computable would he far too bulky to use. Moreover, Dirac has shown that it would contain an enormous amount of superfluous information (5). To compute exactly the energy of an N-electron system, for example, the full wave function expressed as a function of the separate coordinates of each electron is not necessary. Wilson has shown that
a knowledge of the ordinary 3-space electrou density as a function of, say, the nuclear charges is sufficient (6). Also hanging over the calculation is the haunting possibility that the entire effort may be operationally unsound, in as much as the wave function for a mauyelectron system is not a physical observable or even very closely related to one.' This has led to a situation in which massive efforts have been made to solve by approximate methods a philosophically questionable prohlem whose solution, if it could be obtained, probably could not be used and, if it could be used, probably would be much more complex than necessary. The second reason why rigorous quantum chemistry has failed to become a workable and broadly applied theory of structural chemistry concerns a time-honored approximation that is several years older than wave mechanics itself. Except in a small number of highly specialized calculations-mainly those on hydrogen and helium-the wave function of an i k l e c t m n atomic or molecular system has always been taken to be a product (or sum of products) of A' "one-electron orbitals." But again these are not physically ohservable and in a many-electron system there is in reality no such thing as a "one-electron orbital." Indeed, the idea of overlapping (albeit orthogonal) "oneelectrou orbitals" is contrary to the intrinsic character of the Pauli principle. The Pauli priuciple is essentially a statement concerning the collective particle behavior of a system of electrons, whereas the idea of "oneelectron orbitals," to be truly valid in a physical sense, as opposed to being valid or useful in a con~putational sense, must presume that the system exhibits in an essential manner the characteristics of a collection of independent particles. Both Coulomb forces and the effects of the Pauli principle, however, prevent electrons in atoms and molecules from heharing as independent particles. I n summary, one is struck by several facts. The most highly successful applications of quautum mechanics to chemistry have almost invariably been concerned with the properties of the ground states of very simple oue- and two+Jectrou systems (H, Hs+, H2, and He); and always in the latter instances to achieve high computational accuracy the idea of one-electron orbitals has been abandoned. Equally significant, perhaps, is
A preliminary account of this work was presented a t the Gordon Research Conference in Inorganic Chemistry, New Hampton, N. H., August 6-10,1962. This study was supported by a grant of a Faculty Summer Research Appointment from the Graduate School of the University of Minnesota, by a du Pont GrantAn-Aid to the Depart. ment of Chemistry of the University of Minnesota, and in part by the National Science Foundation. This support is gratefully acknowledged.
' One might consider, for example, how the electron density function p is related to 4. In a simple, one-electron system, p is obtained from + by simply multiplying by its complex conjugate +*. The product ++*, times the electronic charge, is the electron density function p. To obtain P for a manyelectron system, one must, in addition, integrate the product ++* over the complete range of the electronic coordinates, spatial and spin, oi every electron but one.
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the fact that in none of the systems treated with great accuracy are there even two electrons of parallel spin. I n a sense, quantum mechanics has not yet treated with the accuracy demanded by chemists the properties of systems for which the Pauli principle may be important. A moral may perhaps be drawn from this. It is that in passing from one- and two-electron systems to complex many-electron systems one should not be surprised if there should emerge in the description of the electron clouds of the complex systems features not previously emphasized in accurate computations on the simpler systems. The discussion of the electronic structure of molecules that follows will he based upon a direct use of the total electron density distribution in molecules and upon two fundamental principles that presumably govern (or describe) the nature of this distribution: the Pauli principle and the Hellmann-Feynman theorem (7) (see also Part 11). The Hellmann-Feynman theorem is a consequence of the Schrodinger equation; the Pauli principle is not. The procedure usually adopted in discussions of molecular structure is to begin with the Schrodinger equation and to introduce a t some later stage the Pauli principle. It is sometimes convenient, as will be illustrated here, to introduce these two fundamental statements in the opposite order. Pauli Principle and the Fermi Hole
We seek for the Pauli principle a statement whose validity is not contingent upon the validity of the idea of "onedectron orbitals." Such a statement is the statement that the wave function $ must be antisymmetric with respect to an interchange in the coordinates of any two electrons. For example, for the interchange of coordinates (spatial and spin) of electrons 1 and 2, the Pauli principle states that
tion of a potential barrier by a light particle. In other words, there may exist in configuration space places where the potential energy exceeds the total energy; the momentum is then imaginary and the kinetic energy negative. Nonetheless, coulombic repulsion between electrons does introduce considerable correlation in their motion. I n the hydrogen molecule, for example, the average distance between electrons is greater than the internuclear distance. The statement summarized in equation (2) is sometimes expressed by saying that "two electrons of parallel spin cannot be in the same place a t the same time." In a sense this is a stronger statement concerning the nature of electron correlation in molecules than is the usual statement of the exclusion principle in which it is said that electrons with parallel spins cannot occupy the same orbital. For many orthogonal orbitals-the hydrogen-like 2s and 2p orbitals, for example, or the usual u and T components of a multiple bond--overlap extensively in space. Of course, any properly antisymmetrized product function will satisfy equation (2); however, it is important to realize, as has been emphasized in THIS J O U R Nby~ L Kimball and Loebl (Q), that the mathematical procedure of rendering antisymmetric a product of one-electron orbitals destroys the simple orbital picture one might have started with whenever the starting one-electron orbitals overlap in space. If the physical basis of the orbital picture is to he retained a t all, equation (2) suggests that the orbitals used ought to be localized, nonoverlapping orbitals. The physical significance of equation (2) is illustrated schematically in Figure 1. Such a figure was suggested first in 1933 by Wigner and Seitz in a discussion of the
In this expression where rt represents the spatial coordinates of electron i (say its Cartesian coordinates xr, yi,zi)and ti represents its spin coordinate. A consequence of equation (I) is that the wave function must vanish whenever electrons 1 and 2 have the same coordinates (spatial and spin). For example, if XI and xp are equal to some common value, x, it follows from (1) that and, hence, that
It should be noted that the Pauli principle does not imply that the wave function vanish when two electrons of opposite spin (6 # $2) are at the same place (r, = r,) for then XI f x2.' Nor, in fact, does the Schrodinger equation require that $ vanish when rl = r2even though the electron-electron repulsion term becomes infinite. This matter has been discussed for the hydrogen molecule by James and Coolidge (8). Close approach of two electrons to each other is analogous to penetra1 In the nucleus of an atom, the Pauli principle would permit two protons of opposite spin and two neutrons of opposite spin to share the same region of space.
Fig. 1
Fig. 2
Fig. 3
Figure 1. The Fermi hole, ofter Wigner and Seitr (101. Figure 2. Configuration of maximum probability for four spin-paired valence-shell electronr, according to the Pouli principle (1 3, 14). The otomic nvcleusir located ot the center of the tetrahedron. Figure 3. Tangent-sphere model of the electron doud about an ommis corethotisrurrounded by four localized electron pain.
electronic structure of metallic sodium (10). The area cross-hatched represents the electron cloud of a manyelectron system. The small circle represents the location of a specific electron within whose nearneighborhood the probability of finding another electron of the same spin is very small. In effect about each electron of a given spin exists an excluded volume or, as a Slater has called it, a "Fermi hole" that consists of a deficiency of charge of the same spin as the electron in question (11). Of ~- considerable chemical interest is the auestion: How large are these Fermi holes? ~~
The Tangent-Sphere Model
Consider the molecule CHI. In the carbon atom's valence shell are four electrons whose spins are parallel to each other. Taken in pairs or otherwise these, Volume 40, Number 9, September 1963
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electrons can uever he in the same place at the same time (equation (2)). I n other words, taken together the four electrons constitute in effect a collection of hard objects in the Aristotelian sense (12). Would it not therefore be physically reasonable to represent each electron of the collection not as a point particle as one does in setting up the Schrodinger equation hut as a sphere whose nonvauishing radius represents the effective chemical radius of the Fermi hole of the electron? If this suggestion is adopted, one is led again to the previous question: I n effect, how much of the valence shell is occupied by these Fermi holes? In seeking an answer to this question, it is couvenient to designate as the free volume the volume of the valence shell not occupied by the Fermi holes. Broadly speaking there are three possibilities to consider: the sum of the Fermi volumes may be very much less than the free volume; the sum of the Fermi volumes may be comparable to the free volume; or the sum of the Fermi volumes may be greater than the free volume. The first possihility corresponds to picturing the mutual behavior of the valence electrons, a t least so far as Pauli effects are concerned, as corresponding to that of a collectionof alrnost-independent particles analogous to a dilute van der Wads gas. I n these terms the second possibility corresponds to a compressed van der Waals gas and the third possibility to a condensed van der Waals gas. The third possihility perhaps appears the least credible of the three. I n fact, it turns out to he a very useful model of molecular structure. This novel and a t first glance seemingly implausible fact can be rationalized-to some extent3-in the following manner. Consider again the four electrons of parallel spin in the valence shell of the carbon atom of methane and suppose that in the space occupied by these four electrons we wish to draw four spheres such that the probability of finding a t least one but no more than one electron in each of the four spheres is a maximum. We wish, that is, to'draw four spheres such that the product P1P2P3Pa is a maximum, where PIis the prohability of finding one and only one electron of a given spin in the first sphere, Pzis the probability of finding one and only one electron of the same spin in the second sphere, etc. Two questions immediately arise. First: Where should the sphere centers be placed? Theoretical calculations based upon the Pauli principle (IS) and chemical evidence (14) suggest tbe same answer: At the corners of a regular tetrahedron, Figure 2. While i t is true that charge correlation would favor the same configuration, the point to be emphasized here is that it may not be necessary (or correct) to invoke conlomhic repulsions between electrons in order to explain this tetrahedral arrangement."he Pauli principle by itself teuds to make this SEinstein said " . . . there is no completely logical way to a new idea." Nonetheless, it is one of the goals of theoretical chemistry to provide for chemical models aiogicd justification based upon deductive arguments from accepted principles. If one does invoke coulombic repulsions between electrons as the primary reason for the tetrahedral arrangement, then, in view of the fact that it is generally believed that spin does not affeot the form of the eleotrostatic law of force between two electrons, there arises the further question: Why do not electrostatic repulsions between electrons of opposite spin manifest themselves in a.similar fashion?
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configuration the most probable one. The second question is this: How large should the spheres be? Make them very small and the probability of finding even one electron in the entire set will he very small. On the other hand the probability of finding in a very large sphere only one electron is also very small. Clearly there must exist some optimum size for the spheres (15). Probably this size will be such that the spheres do not overlap significantly. They could be tangent to each other, however. We assume they are, Figure 3. I n this way we arrive a t the tangent-sphere model. I n this model the electron pairs in the valence shell of a combined atom are represented (in general) as tangent spheres. The one chemist who seems seriously to have considered this model previously is George IGmball, who with several students has carried through calculations on the hydrides of the second-row elements (16). A brief account of their work has been given by Platt (17). The energetics of the model have been discussed in a lecture by Kimhall(18) and in a succinct and lucid statement in THIS JOURNAL by Strong (19).5 Construction
A model of the tangent-sphere representation of the electron cloud of a methane molecule is easily made by first gluing together three identical spheres, Figure 4 (convenient for personal use are cork halls about '/% in. diameter); then add directly above the hole in this trigonal set a fourth sphere (this procedure is indicated in Figure 4 by the sign). This forms a tetrahedron (cf. Fig. 3). The four spheres of the tetrah~droii
"+"
Figure 4. Construction of the tongent-sphere model d methane. The rigniRer the presence in the completed model of o tangent sphere directly above the trigonol hole. Figure 5. Tongent-sphere model of ethane. The seven spheres shown corrermnd to the seven lines in the conventional valence-bond structure of ethane. The model includes two tetrahedral holes There contoin the carbon nvclei ond their inner-shell electron$. The sinole rphere between the upper ond lower trigonal sets is common to bothcarbon-core-containing tetrohedrd holes. This rphere represents the electron pair of the corbon-corbon bond. For convenience the corbon-corbon and carbon-hydrogen spheresare drawn the ramerire.
"+"
.
represent the four electron pairs of the C-H bonds in CH,. I n the interstice betweeu these spheres-it is a hole with an electron-pair coordination number of 4, hereafter to be referred to as a tetrahedral hole-is to be imagined the carbon nucleus surrounded by its inner, k-shell pair. This pair may he represented by a fifth sphere very much smaller in size than the four valence-shell pairs; this has not heen done, however, in most of the drawings in this paper. The proton of a hydrogen atom of a C-H bond in CHais to be imagined Dr. Kimball has kindly pointed out to the author that the tangenesphere model used by him and his students may be considered to be "a special case of the molecular orbital model, derivable from a single antisymmetrized product of one-electron orbitals (Sleter determinant). Its unique properties stem from the abmdonment of the usual atomic orbitals in favor of nonoverlapping orbitals."
as lying about halfway between the center of the sphere that represents the C H bond and this sphere's outer edge on a line that passes through the sphere center and the carbon nucleus. Figure 5 is a drawing of the tangent-sphere model of ethane. In viewing such models it is helpful to remember that each sphere in the three-dimensional model corresponds to one of the lines (or sometimes a pair of dots) in the two-dimensional valence-bond diagram of the corresponding molecule. In the present model the three top and three bottom spheres represent the electron pairs of the carbon-hydrogen bonds; as before, each of these spheres is to be imagined as having imbedded within it a proton at approximately the location previously described for methane. The ethane model can be made by stacking a trigonal set of three on top of a tetrahedral arrangement of four. A more elegant method of achieving the same result is described below.B The ethane model may be viewed as composed of fragments of layers of closely packed spheres. The three spheres that represent the C H electrons of the bottom methyl group in Figure 5, for example, constitute a fragment of a layer of close-packed spheres. Upon this fragment is stacked a one-sphere layer (the carbon-carbon bond) followed by another layer of three for the C-H electrons of the second methyl group. Since the stable configuration of the two methyl groups with respect to each other as viewed along the carboncarbon bond is the staggered configuration shown in Figure 5-this configuration is about 3 kcal/mole more stable than an eclipsed configuration, which is obtained by rotating one of the methyl groups 60' about its three-fold axis of symmetry-the stackmg arrangement just described may be denoted as ABC. This is the kind of stacking of close-packed layers that occurs in cubic close packing.
second methyl group opposite the first pair lie in a plane, Figure 6. These five spheres are represented in Figure 7 as they might be assembled on a flat surface from a trigonal set and a digonal set with the aid of two buttressing trigonal sets. Placement of a sixth sphere Figure 8. Tongent-circle represent.. lion of the principal plone of proPlacement of two rpherer pone. obove the trigon01 holes labeled ond a third sphere below the hole labeled "-"completes the model.
"+"
Fig. 9
Fig. 10
Construction of the tram-confomer of n-butone. The three inner Figure 9. rpherer in the middle row represent the electron p a i n of the carbon-sorbon from the That their centen foll on 0 straiaht line can be seen 01bondr. projection diagrom. Figure 10. Condrustion of the gauche- or skew-confomer of n-butone. The center of the spheres that reprerent the carbon-carbon bondr do rot lie on a shoiaht line h e .. also.. the ~roiestiondiaoroml. The orrow indicoter the ro-called "skew butane" interaction 120).
. .
-
Figure 11. Construction of iko-butone. The sorbon-carbon bonds ore represented by the three inner spheres.
above the top trigonal hole and of a seventh beneath the lower one (indicated by the and "-" signs, respectively) completes the model of ethane. This simple technique can be readily extended to more complex molecules. In propane, for example, the centers of seven of the ten valence-shell spheres lie in a plane. The sphere locations in this plane, the "principal plane" of propane, are indicated in Figure 8. Figures 9 and 10 illustrate the construction with the aid of buttressing spheres of the trans and gauche confomers of normal butane. Construction of isobutane is illustrated in Figure 11. The principal planes of these three molecules contain as shown the centers of nine of the thirteen valence-shell spheres of C4H,,. The structures of trans and gauche n-butane as seen in projection along the central carbon-carbon bond are shown below.
"+"
Figure 6. Tongent-sphere model of ethane showing o rot of Rve spheres whorecenten lie in a plone. Figure 7. Two-dimensional projection of a planar, Rve-sphere set from the tangent-sphere model of ethane (Fig. 6). The spheres whose centers ore connected by the equilateral triangle ore port of o preauembled trigonol set; similarly the line indicates a preosrembled digond set.
Because the tangent-sphere model of ethane is a fragment of a cubic-close-packed lattice, the sphere centers of a pair of spheres of one methyl group, the carbon single-bond sphere, and the pair of spheres of the In Figure 5 the carban-carbon bond sphere is drawn the same size as the protonated spheres. Strictly speaking these spheres are not the same size. The optimum radius of a carbon-carbon hond sphere--as determined, for example, by minimizing (subject to the tangent-sphere condition) the total molecular energy with respect to the radius of the carbon-carbon hond represented as a uniformly charged spherical charge cbud-is generally not the same as the optimum radius of the adjacent protonated spheres. Since, however, the difference is usually relatively small, and since model construction by the methods described herein is greatly facilitated when all spheres are the same size, we have in the illustmtiotions in this paper used the aerotharder, equal-sphere-size approximation. This approximation is probably least applicable to molecules with multiple bonds.
trans confomer
gauche or skew confamer "-buton.
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Figure 12 is the tangent-circle representation of the principal plane of the chair confomer of cyclohexane. The carbon ring is puckered. Three of the carbon nuclei, those in the tetrahedral holes formed from the trigonal holes labeled "f," lie in a plane parallel to and above the principal plane while the other three nuclei lie in a parallel plane beneath the principal plane (20). The orientation of the six spheres about each carboncarbon bond is the same as that in ethane in its equilibrium configuration (Fig. 7). This is not the case for the boat confomer of cyclohexane (see below).
V
perhydroanthracene and perhydrophenanthrene (21). I n the simpler case of cyclohexane a model of the less stable boat confomer can be made from the structural unit shown in Figure 8 by placing two such units opposite each other with additional spheres acting as spacers a t the positions marked The principal plane of cyclohutane is indicated in Figure 15. Figure 16 illustrates the construction of the tangentsphere model of ethylene. The principal plane in ethylene contains the centers of four spheres: two from thedouble bond and two from trans C H bonds. I n the tangent-sphere representation of ethylene the two components of the double bond are represented as equivalent bent bonds. I n Figure 17 is pictured the principal plane of propylene. I n the completed model the orientation of the six spheres bordering on the carbon-carbon single --bond is the same as that in ethane (Fig. 7). This means that as the methyl group rotates about its
"+."
Fig. 1 3
Fig. 1 2
Figvre 12. Construction of the choir confomer of cyclohexone. The inner six spheres represent the elechons of the carbon-carbon bandr, the outer six the electrons of the equatorial carbon-hydrogen bonds. To mmplete the model,rix more ~pheresmurtb e added, three above the.principo1 plone and three below it; these represent the electrons of the bonds to the polor lor o x i d ) hydrogen .tom% Figure 13. Construction of n-pentane in its all-Inns conformdion. The six $.heres of lhe " m . i n c.i ~ a lline" (the middle row1 ore the bockbone of the structure. Once there ore in place it i r relolively simple to place properly the model's remaining ten spheres.
Fig. 1 6
Fig. 1 7
Fig. 1 8
Figure 16. Principal plane of ethylene. The two inner circler which touch each other represent the components of the sorbon-carbon double bond. Figure 17. Principal plone of propylene. The spheres shown represent. from the left, re~pectively,a C-H bond of the methyiene group, the two components of the carbon-carbon double bond, the carbon-carbon single bond, ond two of the C-H bondr of the methyl group. Figure 18. Pdncipol plone of tronr-1.3-butodiene. Large circler ore "red to locote the centers of the smaller tongent circler, which reprertnt,from lefl (or right), these bonds: C-H (methylene), C=C, C-C, C=C, C-H (methylenel.
Figure 14. Tongent-circle reprerentation of the principal plone o f transdecalin. Both six-membered rings are in their choir conflgurotionr (cf. Fig. 121. The hydrogen atoms on the two carbon atom~sammonto theringr are trans to each other; one i, above lhe principal plane, the other below it. The model contains two principal liner lthe second and fourth rows1 upon which its construction con be bored. Figure 15. Tangtnt.sirsie reprerentation of the principal plane o f cycloThe inner four cirbutane. Note thot the four-membered ring .is puckered. . cler represent the carbon-carbon bondr, the outer four the equatorid corbon-hydrogen bandr
threefold axis one of its C-H bonds, the one in the completed model that points toward the reader in Figure 17, reaches its position of closest approach to the adjacent methylene C-H bond when the methyl group as a whole has reached its most stable orientation. Possibly this is one reason why the harrier to rotation of the methyl group in propylene is only about twothirds as great as it is in ethane. Figure 18 shows the principal plane of 1,3-butadiene for the molecule in its trans configuration (adjacent double bonds trans to each other with respect to the
Figure 13 shows that in making models of the normal paraffins in their all-trans configurations a straight-edge can be used to advantage in lining up the trigonal components. Alternatively, the "principal line" in these models can be preassembled on the groove of an inexpensive slide rule and the construction of the principal plane initiated from that point. The principal plane of trans-decalin (20) is illustrated in Figure 14. This plane can be constructed in a manner analogous to that described previously for the chair confomer of cyclohexane (Fig. 12). It can also be constructed from the components of n-pentane (see Fig. 13). One advantage of the latter method of construction is that it can be extended with suitable modifications to the construction of models of the less stable confomers of decalin and of the corresponding confomers of such more complex perhydroaromatics as
Figure 19. Projection diagrams for three sonfomers o f 1.3-butodiene. Proiestions ore along the carbon-carbon single band. Curved liner to the CH2 groups represent the equivalent-orbital components of the double bonds. Relative confomer rtabilitier are probably trans govche cis (Since this remark was written, the microwave spectrum of fluoroprene ICH9=CHCF=CHJ has been reported b y D. R. Lide, Jr. IJ. Chem. Phys., 37,2074 (1 96211. The molecule, ar expected, hor a planar tranaconflgurolion in the ground rtote. No rpeciflc rearch wa. mode for the spectrum of a nonplanar isomer. The dr isomer wot searched for but not found.)
Fig. 1 4
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Fig. 15
Trans
Gauche
(Cis)
>
>>
single bond joinmg them). As in ethane, electron pairs on opposite sides of the carbon-carbon single bond are staggered with respect to each other. The same is trne for the corresponding gauche or skew configurations but not for the cis configuration, Figure 19. Figure 20 shows the principal plane of the all-trans confomer of 1,3,6hexatriene.
Figure 20. Principal plane of dl-Irons 1,3,5-hexotriene. Circler, from left or right, represent respectively the bonds C-H (methylene), C=C, C-C, C=C, C-C, C d , C-H (methylenel.
Figure 21. Tangent-sphere model o f allene (CH4=CHJ superimposed upon ib vdence-bond diogrom. The three heavy dots represent the mrbon cores; the four light dots, the protons. As inferred b y van't Hoff in 1875 the molecvle ir nonplmor. (Cf. Skeil, P. S,. and Wertmtt, 1. D., I. Am. Chem. Sor, 85, 1023 (19631, for a description of the isomers of bis-ethanoollene that are formed b y the low-temperature addition of Cs, o dicarbene,to oleflns.)
To summarize, construction of tangent-sphere models requires a supply of spheres, glue, and a flat surface. A box half-full of spheres makes a useful "jig" for holding partially compieted models in desired~o~ientations. Useful, also, are preassembled trigunal and diagonal sets and a valence-bond diagram of the molecule under consideration. For lecture demonstrations models made from 34-in. diameter Styrofoam spheres are suitable. Lucite or Plexiglas spheres can be obtained in attractive colors' for color coding bonds; the '/=An. size glues easily with Duco or some similar quick-drying, ketone- or ester-based cement. For student use '/h. cork balls8 work very well. They are light, inexpensive, and easy to handle. Figure 24. Tangenl-sphere model of P+ Stippled spheres represent unshared poirr, open 3pherer phosphorus-pharphorusringle bondr. Each pho3phorow atom hor in its valence shell one unrhared pair ond three pairs with which i t forms single, bentbondr tails three neighbors.
Additional molecules whose models are interesting to make are allene (Fig. 21), carbon suboxide, acetone (Fig. 22), 1,3,5,7-cyclooctatetraene(Fig. 23), cyanogen, tetracyanoethylene, bicyclo-(2.2.2.)-octane,P4 (Fig. 24), P40s P401a, and diamond. Literature Cited
Figure 22. Tangent-sphere model of the equilibrium configuration of acetone superimposed upon the molecule's valence-bond diagram. The four heavy dots reprerent the unshmed pairs on the oxygen atom. Removol o f there two p a i n leaves behind the tangent-sphere representotion o f the electron cloud o f propone,or of ethyl alcohol, dimdhyl ether,oxygen difluoride, and other molecules isoelectronic with propane. The orientation with respect to each other o f the two trigond sets at either end of a carbon-carbon single bond is the same as in ethane. The orientation of the methyl groups with respecttothe double bond irthe sameor in propylene.
Figure 23. Tangenbrphere model and valence-bond diagram o f 1.3.5.7cyclwdotetraene
Stippled gpherer represent carbon-hydrogen bondr, open spheres corbancarbon bondr. The model i 3 a fragment of flve layers of a cubic-closepacked lattice. Intere$tingly, the corbon-carbon ringle bonds in this nonplonor,tub-shaped molecule are perhaps 0.014.02 A shorter than the corbon-corbon ringle bond in planar 1.3-butadiene. This fact, which i s momolous from the view point of current theory ICf. Bmtianren and Traetteberg Ip.1471, Cruickshonk (p. 1551, ond Coulron, C. A., Tetrohedran, 1 7 256 (196211, con be accounted for, in port, by considering the vector dispiocementt o f the doubly-bonded carbon nuclei from the centerr of their tetrahedral holes. These displacements are described below in more detail in the section on the effects of multiple bonds on moleeulor geometry.
(1) PEKERIS,C. L., Phys. Rev., 112, 1649 (1958). (2) . . STEWART,E. T., Ann. R e ~ t s .Chem. Soe.,. (Lond.), . .. 58,. 7 (1961): (3) PLATT,J. R.,"Handhuch der Physik," Springer-Verlag, Berlin, 1961, Vol. 37/2, p. 173. D. R., "The Citlculation of Atomic Structures," (4) HARTREE, John Wiley and Sons, Inc., New York, 1957, pp. 16-7. (5) DIRAC,P. A. M., Camb. Phil. Soc., 26, 376 (1930); ibid., 27,240 (1931); see also ref. (51, p. 177. (6) WILSON,E. B., JR., J. Chem. Phys., 36,2232 (1962). (7) (a) FEYNMAN, R. P., Phys. Rev., 56, 340 (1939); (b) BERLIN, T. J., 3. Chem. Phys., 19, 208 (1951); LONGUETHIGGINS,H. C., AND BROWN,D. A,, J. Inorg. Nw1. G. Chem., 1, 60 (1955); BADER,R. F. W., AND JONES, A,, Can. J . Chem., 39, 1253 (1961). (8) JAMES, H. M., AND COOLIDGE, A. S., J. C h m . Phys., 1, 825 (1933). (9) KIMBALL, G. E., AND LOEBL,E. M., J. CEEM.EDUC.,36. 233 (1959). (10) WIGNER,E., AND SEITZ,F.,Phys. Reu., 43, 804 (1933); ibid,,46,509 (1934). J. C.. Phus.Reu.. 81.385 (1951). (11) SLATER. i n j RUEDENBERO, K.: Revs. r oh. phis., 34,326 (1962); KAUZMANN, W., "Quantum Chemistry," Academic Press, Ine., New York, 1957, p. 320.
' Ace Plastic Co., 91-38 Van Wyck Expressway, Jamaica 35, N. Y.: Auburn Plastic Eneineerine. -, 4916-24 South Loomis St.. Chicago 9, Ill. Cork Products Co., Inc., 239 Park Ave., South, New York 3, N. Y. EDITOR'SNOTE: The second part of this paper will appear in the October, 1963 issue. In it Professor Bent will discuss uses of the tangent sphere models as devices for depicting a wide variety of structural features and concepts. Figures 2540 and literature citations 2 2 4 1 will appear with the second article.
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