M. G, EVANS 9 A SURVEY OF THE PRINCIPLES DETERMINING THE STRUCTURE AND PROPERTIES OF MOLECULES PART I.-THE FACTORS RESPONSIBLE FOR MOLECULAR SHAPE AND BOND ENERGIES BY SIR JOHN LENNARD-JONES AND J. A. POPLE Received 19th March, 1951 In this paper a general account of the basic factors determining the structure of molecules is presented. It is pointed out that previous theories used to ex- plain the structure of molecules are inconsistent in certain respects and tend to obscure the nature of the forces operating. Here i t is shown that a satisfactory qualitative picture of the structure of a molecule can be obtained by considering only electrostatic repulsions operating in conjunction with the antisymmetry principle. This is applied t o the relative position of bonds and lone pairs inI 0 STRUCTURE OF MOLECULES molecules and then t o the interaction of two electrons in a bond.In the con- cluding sections, i t is shown how the same qualitative ideas can be applied to strained hydrocarbons and to simple free radicals. 1. Introduction.-The wave theory of chemical valency has had a curious history. The first applications were naturally made to the simplest diatomic molecules. Even for such systems different methods of calculation were devised, some based on the original treatment of the hydrogen molecule by Heitler and London,l others based on the concept of orbitals extending throughout the whole Most of the attempts to deal with the theory of bonds have been approximate in treatment and only in one or two cases, such as in the elaborate treatment of the hydrogen molecule by James and C~olidge,~ and by Hylleraas,* have the calculations been pushed through to a stage of satisfactory accuracy. Such calculations have, however, served to establish the validity of wave mechanical methods and to inspire the belief that, when properly applied, they are capable of accounting satisfactorily for all molecular properties.The surprising feature about the subsequent developments is that the main successes have been achieved not with simple molecules but with conjugated hydrocarbon systems. It is true that there have been attempts to deal in detail with molecules like ~ a t e r , ~ methane and ethylene,7 but even in such comparatively simple systems the quantitative treatment is heavy and elaborate.It was the study of the oxygen molecule by the molecular orbital theory that led Hiickel to consider the molecules of formaldehyde 8 and ethylene and then to show how benzene and other conjugated systems could be dealt with on an approximate basis.10 This led the way to extensive calculations of the resonance energies and bond lengths of a wide variety of conjugated hydrocarbons.11 The essential step in all this theoretical work WAS the recognition of the part played by n-electrons in molecules of this type. The assumption was made that they could be treated separately from all the other electrons whether these were in ordinary single bonds ( 0 bonds) or in the inner shells of the atoms. While this assumption has never been rigorously justified, it is believed to be true because of the difference in the symmetry properties of the n-electrons and those of the rest of the molecule.I t is not proposed to deal in this short paper with the various successes and limitations of the theory of conjugated hydrocarbons, because this field has been adequately reviewed in a Discussion recently organized by the Royal Society.12 Some indication will, however, be given in the paper which follows this l3 of the reasons for the success of the theory in this field. It appears that by a curious coincidence the equations used in the theory are the same as those obtained by a more complete treatment, though the equations and the parameters which appear in them have a different interpretation. The theory of the structure of saturated hydrocarbons cannot be said to have attained a comparable degree of success.All attempts to deal Heitler and London, 2. Physik, 1927, 4, 255. Lennard- Jones, Trans. Faraday Soc., 1929, 25, 668. James and Coolidge, J . Chem. Physics, 1933, I, 825. Hylleraas, 2. Physik, 1930, 65, 209. 5 Van Vleck and Sherman, Rev. Mod. Physics, 1935, 7, 167. 6 Van Vleck, J . Chem. Physics, 1933, I , 177, 219 ; 1934,~~ 20. 7 Penney, Pvoc. Roy. Soc. A , 1934, 14, 166. Hiickel, 2. Physik, 1930, 60, 423. lo Hiickel, 2. Physik, 1931, 72, 310. 11 Pauling and Wheland, J. Chem. Physics, 1933, I, 362 ; Pauling and Sherman, l2 Lennard- Jones, Coulson, Longuet-Higgins, et al., Proc Roy. SOC. A , 1951 l3 Lennard- Jones and Hall, this Discussion. Hiickel, 2. Physik, 1931, 70, 204. J . Chem. Physics, 1933, I , 606, 679.(to be published shortly).SIR JOHN LENNARD-JONES AND J. A. POPLE 1 1 with them have been based on the method of electron pairs. In this treatment the polyvalent atoms such as carbon are first “ prepared ” in the sense that the occupied atomic orbitals are superimposed to give other orbitals with directional properties. These are then used in precisely the same way as Heitler and London dealt xith the interaction of hydrogen atoms. An electron in each directed orbital is assumed to interact with an electron of another atom to form a bond. In the calculation of the interaction, however, the terms in the normalization factors, usually described as overlap integrals, are neglected. This is a serious limitation because the directed orbitals are adjusted to give the maximum overlap with the orbitals of the interacting atom and i t is inconsistent to neglect quantities which have been given a maximum value.The omission of these quantities from the formulae usually used in the theory of electron pairs renders i t difficult to understand clearly the physical interpretation of the various terms which ossur in the formulae or to see what the effect would be of a change in the directed orbitals of the atoms concerned. In other words, the formulae leave obscure which properties of a molec- ular system remain invariant when the system of representation in terms of directed atomic orbitals is changed, and which properties vary in a significant way. This distinction is necessary if we are to understand the factors which determine the shapes and energies of molecules.In view of the unsatisfactory nature of existing theories, this paper will be concerned more with the general principles which determine the structure and properties of molecules than with a detailed account of calculations which have been made for particular molecules. The account is based on a series of papers, published recently,14-21 which have aimed at providing a more systematic formulation of the molecular orbital theory of molecular structure than has hitherto been available. 2. The Factors Determining the Relative Positions of Bonds .-The electronic wave function of a molecule has to be a solution of the appropriate Schrodinger equation and must be antisymmetric for inter- changes of the co-ordinates (including spin) of any pair of electrons.This antisymmetry principle does not appear important in the problem of two electrons in a bond for then the antisymmetry occurs in the spin part of the wave function, but for systems of three or more electrons it intro- duces restrictions on the relative distribution of the electrons in space. To assess the part played by the antisymmetry principle in deter- mining the structure of molecules, we need an approximate quantum- mechanical wave function which can be interpreted in simple chemical terms. The most convenient way of doing this is to assign each electron to an orbital, associated with CI or /I spin functions and combine these into a determinant, so that the total wave function is automatically’ antisymmetric.If the space orbitals are $1 . . . z,h~ and each orbital is doubly occupied, the total wave function is where only the diagonal elements of the determinant have been included inside the brackets. This is the general form of the molecular orbital approximate wave function.l*~ 15 Now since a determinantal function such as (I) is only multiplied by a numerical factor if the rows or columns are replaced by 1inea.r com- binations of themselves, it is usual to restrict the orbitals to be normalized l4 Lennard-Jones, Proc. Roy. SOL. A , 1949, 198, I. 15 Lennard-Jones, Proc. Roy. SOC. A , 1949, 198, 14. 16 Hall and Lennard-Jones. Proc. Roy. SOC. A , 1950, 202, 156. 17 Lennard-Jones and Pople, Proc. Roy. SOC. A , 1950, 202, 166. 18 Pople, Proc. Roy. SOC. A , 1950, 202, 323.19 Hall, Proc. Roy. SOC. A , 1950, 202, 336. 2 o Hall and Lennard-Jones. Proc. Roy. SOC. A , 1951, 205, 357. 21 Hall, Proc. Roy. soc. A , 1951, 205, 541.I 2 STRUCTURE OF MOLECULES and orthogonal. This is not a further approximation; it is merely a convenient mathematical device for simplifying the physical interpretation. Even with this restriction, however, the orbitals . . . *N are not fixed uniquely. It is still possible to apply certain linear transformations to #l . . . $N without altering the total wave function Y . This leads to alternative ways of describing the electronic structure of molecules in terms of orbitals. Two particular sets of orbitals have been found to be of particular significance. The first set, called molecular orbitals, each have a sym- metry determined by the nuclear framework and are generally spread throughout the molecule.It can be shown that these are particularly significant in connection with spectroscopic processes l6 ; if an electron is removed from a molecule, i t should be regarded as removed from a molecular orbital. The other important set of orbitals, called equivalent orbitals, give rise to more localized distributions of charge corresponding to the various bonds or lone pairs of the molecule. A symmetrical molecule such as methane, for example, could be described either in terms of orbitals spread throughout the molecule OY in terms of four equivalent orbitals, identical with each other except for orientation. It is possible to find the transformation from the molecular to the equivalent orbitals, or vice versa, by group theory.15 To find the way in which electrostatic attractions and repulsions influence the relative positions of the various parts of a complex molecule, we divide the total energy into the following parts : (i) the kinetic energy of the electrons, (ii) the potential energy due to the electrostatic repulsion of the (iii) the potential energy due to the electrostatic repulsion of the (iv) the potential energy due to the attractive forces between the nuclei, electrons, nuclei and the electrons. This can be done using the determinantal wave function (I), and explicit expressions can be obtained for the energies in terms of the molecular orbitals or of the equivalent orbitals.16,17 We find that each of the four parts is invariant under the transformation from molecular to equivalent orbitals, as we should expect.Although it might appear at first sight as though no advantage could be gained by a transformation, further exam- ination shows that the equivalent orbital description permits a closer understanding of the part played by electron repulsions, that is of the con- tribution (iii) to the energy. Suppose we denote the molecular orbitals by I,&, # z . . , and the cor- responding equivalent orbitals by xl, xZ . . . . notation for two electron integrals If we use the following - (4 we find that the electron-electron potential energy divides into two parts which are not separately invariant, although their sum is. These parts may be writtenSIR JOHN LENNARD-JONES AND J.A. POPLE r3 for molecular orbitals, and PY v J for equivalent orbitals. In these expressions z’ means summation over both suffixes omitting the terms for which m = n. Now, if we examine the individual terms in these expressions, we see that the first term in J (or j ) gives the repulsive potential energy between two electrons in the same molecular (or equivalent) orbital. The second term represents the repulsive potential energy between electrons in different orbitals. We may therefore refer to J or i as the Coulomb part of the interelectronic potential energy. Finally, the part (- K ) or (- k ) gives a contribution which cannot be so easily interpreted ; it is referred to as the exchange energy. This latter part only differs from zero because the distributions $I (or X) overlap one another to a certain extent ; if there were no overlap, there would be no exchange energy.We expect, therefore, the exchange part to be small when the orbitals are well localized and so to be less im- portant for equivalent orbitals than for molecular orbitals. This is amply confirmed for the ( z s ) ( z p ) configuration of an atom such as beryl- lium, each orbital being singly occupied. The ‘‘ molecular orbitals ” are the 2s and z p functions themselves, and the equivalent orbitals are localized on opposite sides of the nucleus. The latter are the digonal hybrids used by Pauling. It is found that, using molecular orbitals, the exchange energy (- K ) is about 28 yo of ( J - K ) , whereas if equivalent orbitals are used, the corresponding paxt (- K) is only 2-4 Yo of the total.The general conclusion to be drawn from this discussion is that molec- ular structure can most conveniently be described in terms of pairs of electrons in equivalent orbitals, interacting according to ordinary electro- static forces. The exchange part of energy’, though not negligible, will then be small relative to the Coulomb part. On this view there are two important factors determining molecular shape. One is the exclusion principle, which demands that the occupied equivalent orbitals shall be orthogonal to each other. The other in the principle of minimum energy. This requires that subject to all other conditions the repulsion of the electrons shall be a minimum. We are now in a position to discuss the relative orientation of bonds in hydrocarbons, and the significance of the tetrahedral valency of carbon.The structure and stability of the methane molecule can best be under- stood by comparison with the neon atom which has the same number of electrons. The structure of neon is usually represented by the configura- tion (1s)2(2s)2(2~,)2(2PZ)8. These are the symmetrical “ molecular ” orbitals of the atom. The last four, however, can be tra.nsformed into four equivalent orbitals in tetrahedral directions (usually described as tetrahedral hybrids). With neon i t is the wZative orientation of these four orbitals that is significant; the total electron density, or course, is spherically symmetric. Now i t can be shown that if we use the equiv- alent orbital description, go yo of the electron-electron repulsion energy arises from the Coulomb part of (4).The methane molecule can be regarded as obtained from a neon atom by removing four unit positive charges from the nucleus along the directions of the four tetrahedral equivalent orbitals. As these positive charges will, to some extent, draw the electrons in the corresponding equivalent orbital with them, charge distributions in these orbitals will become more localized and the Coulomb part of the interaction of pairs of electrons in equivalent orbitals will account for more than g o yo of the total electron-electron potential energy. We may thus conclude that both the electrostatic m, 12I4 STRUCTURE O F MOLECULES repulsion betweenthe electrons in different orbitals and the antisym- metry principle operate in such a way that the pairs of electrons get as far away as possible from other pairs.These are the chief factors underlying the tetrahedral structure of methane. In addition, we have the repulsion between the protons which has a similar effect, and, of course, a partly cancelling term arising from the attraction of a proton for the electrons of other bonds. All these are significant terms in the total energy. The point we wish to make, however, is that from the arguments outlined above, the exchange part is expected to be comparatively unimportant. Electrostatic repulsions also play a vital part in determining the structure of molecules with lone pair electrons, such as water and am- monia.l* Although these are not hydrocarbons, it is worth while ex- amining their structure briefly, for we shall find in a later section that certain hydrocarbon free radicals are somewhat analogous.The ammonia molecule has the form of a triangular pyramid with three equivalent N-H bonds inclined to each other at an angle of about 107O, so that the structure is nearly tetrahedral. The equivalent orbital picture of this molecule describes it as having three equivalent orbitals associated with the N-H links, each doubly occupied, and a fourth orbital, also doubly occupied, symmetrically related to them, corresponding to the lone pair of electrons. Again the tetrahedral structure is preferred because of the repulsion between pairs of electrons. It has been found 18 that it can best be described as having pairs of electrons in two equivalent orbitals associated with the OH links and pairs of electrons in two other different equivalent orbitals projecting out (backwards) from the oxygen atom so that the whole forms an approximately tetrahedral system.Some other ways in which the repulsion between bonding orbitals or lone pairs is likely to be an important factor in determining molecular structure may be noted. The rotation about a single carbon-carbon bond in molecules such as ethane will depend on the repulsion between electrons in C-H bonding orbitals on different carbon atoms. We should expect the ‘ I staggered ” configuration (Dsd) to be the most stable form, as this would allow the bonding electrons to keep farther apart than in the D3h configuration. Similar factors will operate if the C-H bonds are replaced by lone pairs, as in the hindered rotation of the OH bonds about the C-0 single bond in alcohols.3. The Factors Determining the Energy of a Bond.-In the previous section we have seen how the molecular orbital theory in its equivalent orbital form gives a satisfactory qualitative picture of the interaction be- tween electrons in different bonds or lone-pair orbitals. Where the theory is incomplete is in the way it treats the interaction of a pair of electrons in the same equivalent orbital. This is of great importance, for it is the interaction of two electrons in a bonding orbital which determines the energy of that bond. We consider the case of two electrons in a symmetrical bond (e.g. a homonuclear diatomic molecule, or the C-C bond in ethane).Accord- ing to the molecular orbital theory described above, the distribution of the two electrons is described by a wave function where f, is a n equivalent or localized orbital embracing both nuclei. The factor in brackets represents the spin part of the wave function. If the nuclei are similar, the function fs will, in general, be symmetrical in the two nuclei. According to this wave function each electron moves in the smoothed-out field of the other. In an actual molecule, of course, the motions of the two electrons are strongly interrelated, for each repels the other. The approximation (5) is quite inappropriate for large inter- nuclear separations, for it predicts too large a probability of dissociation The water molecule also has this type of structure.= fS(1)fS(2){a(1)fl(2) - a(2)fl(1)>J * - ( 5 )SIR JOHN LENNARD-JONES AND J. A. POPLE 15 into ions. To take into account the interrelation of the two electrons in the same orbital it is necessary to improve the representation ( 5 ) by adding other terms. Thus the tendency of the electrons to keep apart and there- fore to be on opposite sides of the centre of symmetry of the bond can be represented by adding a term involving antisymmetric functions fa If, in particular, certain symmetric and antisymmetric linear combinations of atomic functions +a and +b are used for f, and fa the function (6) takes the form which is just the form of electron pair function originally used for the hydrogen molecule by Heitler and London.1 This is a particular case of the generalized molecular orbital function (6).Another way of increasing our understanding of the chemical bond, and particularly of the effect of electrostatic forces, is to divide up the energy of the system in a way similar to that used for the many electron problem in The results of doing this for the hydrogen molecule are given in Table I. Also given in this Table are the corresponding energies for two separate hydrogen atoms. = {f*(Ilf8(4 - fa(Ilfu(2)) { + ) 8 ( 4 - 42)8(1)) - - (6) ?P = {+U(I)+b(4 + + b ( I ) + U ( 2 ) H . ( I ) B ( 4 - 42)8(1)), * - (7) 2. TABLE I.-ENERGIES FOR THE HYDROGEN MOLECULE 22 AND Two HYDROGEN ATOMS Total energy . . I - 1-16 Contribution to 1 (Atom%nits) 1 (Atomic 'H Units) 1 Binding (kcal.) Energy - 1'0 I - 9s) I I I Kinetic energy .Nuclear repulsion . Electron repulsion . Electron-nuclear 1-15 0.59 3'63 94 455 - 2-0 - I018 368 co I16 STRUCTURE OF MOLECULES Although figures such as those of Table I have only been worked out for two electrons in the presence of two bare nuclei (assumed to be protons), there is not much doubt that similar results apply for other chemical bonds ; that is, three of the contributions listed in the table above will be positive and only one negative. This analysis of the results is inter- esting, because i t shows that to achieve accuracy in the calculations it is necessary to represent a bond by such a wave function that not only is the average electrostatic attraction between electrons and nuclei as large as possible, but also the average electrostatic repulsion of the electrons is as low as possible.It is the second part which causes difficulty in the calculations. To obtain it accurately other terms must be included in the wave function (6) to represent to the full the tendency of the electrons to avoid each other. It is hoped to publish a fuller discussion of this problem elsewhere. 4. Strained Bonds in Hydrocarbons.-In all the systems discussed so far, we have been able to describe the electronic structure in terms of equivalent orbitals which have been concentrated mainly on the line joining the two nuclei connected by the bond. These may be described as straight or normal bonds. In many molecules, including some hydro- carbons, however, there are bonds whose electrons are most probably to be found off the internuclear line.In other words, they are bent or strained. A simple example is the molecule cyclopropane C,H,, in which the three carbon nuclei form an equilateral triangle. As the angle between adjacent C-C lines is only 60°, we can see that the bond orbitals mainly concentrated along the internuclear lines would be abnormally close to each other. Both the exclusion principle and electrostatic repulsion between the electrons tend to open out the angle between the bonding orbitals, so that i t is considerably larger than the angle between the internuclear lines. This effect has been investigated quantitatively by Coulson and Moffitt. 23 Another simple example of a strained bond system is the ethylene molecule. This is of particular interest, as it affords a striking illustration of the relation between the two types of molecular description referred to earlier.In recent years it has been customary to describe the double bond of ethylene in terms of two orbitals, one symmetrical in the plane of the molecule (0-bond) and the other with a node in this plane (n-bond). Alternatively, we can describe the structure in terms of two equivalent orbitals, These two orbitals are concentrated in regions on opposite sides of the molecular plane. It should be emphasized that these two pictures of the double bond are equally valid, and the choice of the one to be used should be deter- mined by the type of property under discussion. The equivalent orbital picture is more useful when we wish to discuss the electron distribution or the parts of the molecule which are likely to be reactive.Our description of the double bond in terms of two bent orbitals has retained the approximate tetrahedral relation of the four valencies of carbon. There is a modification, however. The two orbitals of the double bond emanating from one carbon atom are drawn together by the field of the other carbon nucleus. The angle between them, therefore, will be some- what less than that characteristic of tetrahedral valency’. It may be expected, as a result, that the p-character of these orbitals near the carbon atoms will be increased. It then follows from the orthogonality conditions 2s Coulson and Moffitt, Phil. Mag., 1949, 40, I. One further point about the ethylene molecule is worth noting.SIR JOHN LENNARD-JONES AND J.A. POPLE 17 that the bonds of the CH orbitals should have more s-character and conse- quently be inclined at an angle greater than the tetrahedral value. This is confirmed by the experimentally determined value of the HCH angle in ethylene which is in the neighbourhood of IZOO. The triple bond between two carbon atoms can also be described in terms of bent orbitals. The triple bond is usually described in terms of a single o-bond, and two r-bonds with nodes in two perpendicular azimuthal planes. These three bonds can be superimposed so as to pro- duce three equivalent bonds, localized about different planes through the CC axis. The angles between these planes is IZOO. This picture therefore corresponds to the model often shown in text-books of chemistry’ of three bent springs connecting the carbon atoms as in acetylene.It can be shown that these two sets of orbitals are related to each other by a transformation which leaves the total determinantal wave function invariant. The triple bond differs from the double bond in one respect, however. Whereas the equivalent orbitals of ethylene are uniquely defined, there is a rotational degree of freedom in the corresponding orbitals of acetylene. It is only the relative azimuthal angles of the three bonds that is important. The actual position of any one may be taken in an arbitrary azimuthal plane, but the system as a whole is axially symmetric. 5 . The Structure of Radicals.-Up to this point we have been con- cerned with molecular systems in which the structure was determined by the interaction of pairs of electrons, each pair occupying a different orbital.Although this gives a satisfactory description of the ground state of almost all stable molecules, it will not apply to free radicals with an odd number of electrons or to a system with an even number of electrons not com- pletely distributed in pairs. The theory of the structure of these radicals has not yet been dealt with satisfactorily. We shall not attempt to de- velop such a theory here, but we give only some general indication of the lines along which it might proceed, using simple hydrocarbon radicals as examples. Before we can discuss the effect that electrostatic repulsion has on the structure of such systems, it is necessary to review briefly the modifications of the orbital theory when some orbitals are singly occupied.Two cases will be considered :- (i) One orbital singly occupied.-It turns out that this configuration can be described by a single-determina.nt wave function, provided that the singly occupied orbital belongs to one of the irreducible representations of the symmetry group of the molecule.16 It must, in fact, be a molecular orbital. (ii) Two orbitals singly occupied-This type of configuration can give rise to two distinct molecular states, a triplet and a singlet. The triplet state can be represented by a single-determinant function if all the singly occupied orbitals are associated with the same spin. The singlet state cannot be represented by one determinant, but this is not a serious limita- tion as the triplet always has a lower energy.I f we examine the details of possible transformations within the determinant representing the triplet state, we find that it is possible to transform from molecular to equivalent orbitals within the set of doubly occupied orbitals and within the set of singly occupied orbitals, but not by using linear combinations of the two. The significance of this becomes clearer when simple examples are con- sidered. This system has an odd number of electrons and will belong to class (i) above. If we suppose that the three C-H bonds are equivalent, then the radical can either be planar with the carbon at the centre of the equilateral tri- angle of the three hydrogens, or it can have a pyramidal form like ammonia.In either case the three bonds will be represented by three localized equiv- alent orbitals and there will be one other “ lone electron” in a singly As a first example we will consider the methyl radical CH,.18 STRUCTURE OF MOLECULES occupied orbital. If the radical is planar, this singly occupied orbital will have a node in the molecular plane, and if it is pyramidal, the orbital will probably be directed much as the lone pair in ammonia. We cannot say which configuration will have the lower energy without making detailed calculations, but it is worth noting that the same factors which cause ammonia to be non-planar are operative also in the CH, radical, viz. the electrostatic forces between electrons in different equivalent orbitals, but to a less extent. As there is only one lone electron, this may not suffice to counteract the mutual repulsion of the three CH bonds and so the CH, radical may be planar or very nearly planar. If a further hydrogen atom be removed from CH, we are left with the methylene radical with six outer electrons. The electrons may be arranged in three doubly occupied orbitals (singlet state) or in two doubly and two singly occupied orbitals (triplet state). Again it is not possible to predict which of these is the more stable without lengthy calculations. We may, however, say something about the electronic arrangements in both states. The triplet state is analogous to the ground state of the water molecule, except that the two lone pair orbitals in water are replaced by two singly occupied orbitals. The configuration which has the lowest energy will be determined by electrostatic repulsions between the orbitals. We may note that the repulsion between electrons in non-bonding orbitals is smaller -than in water and as this is one of the principal factors which prevents water from becoming linear, we may expect the corresponding barrier for CH, to be smaller and possibly non-existent ; that would imply a linear CH, radical. The singlet state of CH, will have its shape determined by similar factors. As we now have three localized orbitals (two bonding and one lone pair) tending to get as far apart from each other as possible, the equilibrium configuration is likely t o be one in which the three orbitals point from the centre towards the vertices of an equilateral triangle. The shape of CH, would then be triangular. Department of TheoreticaZ Chemistry, University of Cambridge.