18 STRUCTURE OF MOLECULES A SURVEY OF THE PRINCIPLES DETERMINING THE STRUCTURE AND PROPERTIES OF MOLECULES PART 11.-THE IONIZATION POTENTIALS AND RESONANCE ENERGIES OF HYDROCARBONS BY SIR JOHN LENNARD-JONES AND G. G. HALL Received 19th March, 1951 The theory of molecular orbitals provides a suitable method of calculating the ionization potentials of molecules. In this paper the method is applied to saturated hydrocarbons and i t is found to account satisfactorily for the change in the ionization potentials of paraffin molecules as the length of the chain is increased. The theory is applied also to unsaturated hydrocarbon molecules and an examination is made of methods previously used to calculate resonance energies. It is shown that the results of earlier work can be given a different interpretation from that usually accepted.In this new theory ionization potentials play an important part.SIR JOHN LENNARD-JONES AND G. G. HALL I 9 1. Introduction.-In the preceding paper 1 a discussion has been given of the properties of chemical bonds in terms of equivalent orbitals. The object of the paper was to discuss the factors which determine the disposition of bonds relative to one another and the strength of the bonds; that is, the paper was concerned mainly with the shapes and energies of molecules. This treatment was shown to be one aspect oi a generalized theory of orbitals, given elsewhere, which lends itself to trans- formations from one type of orbital to another according to the problem considered. The equivalent orbitals used for the study of bonds were ob- tained as transforms of molecular orbitals and were so designed as to be confined to particular regions of a molecule.Such a point of view is useful in the study of the chemistry of molecules, for then attention is usually focused on the localized properties, but when molecules are sub- ject to the interaction of light or a beam of impinging electrons, it is the whole of the molecule which is concerned, and not any particular part. The interpretation of the results of spectroscopy must thus be sought in terms of molecular orbitals, for these have the appropriate property of being symmetrically related to the whole system. In this paper we con- sider methods of calculating the ioniza.tion potentials of molecules and show that while these quantities are properties of the whole molecule, they can none the less be expressed using equivalent orbitals.The theory is applied to some saturated hydrocarbons and satisfactorily accounts for the change in ionization potentials as the length of the paraffin chain is increased. An attempt is then made to extend the theory to calculate the ioniza- tion potentials of unsaturated hydrocarbons, but the experimental evidence available is not sufficient to justify detailed calculations. This leads to an examination of the methods previously used to calculate the total energies (and thus by inference the resonance energies) of conjugated hydrocarbons. Equations are obtained for these energies which have precisely the same form as those used in earlier theories but their interpretation is different ; in fact, it appears that the success of earlier work is due to a series of coincidences in that approximate methods led to equations of the correct type and the deficiencies of the theory were concealed by the use of certain parameters (usually denoted by a and p).These parameters were obtained not by direct calculation but by a comparison with experiment. They thus provided connecting links between one set of experimental results and another, and served to correlate the properties of conjugated mole- cules among themselves. The method developed here has the advantage of providing a precise interpretation of the parameters ( a and j3) and, though these may not be calculable directly, it may be possible to trace their variation from one molecule to another and so to resolve some of the difficulties which surround the theory of unsaturated molecules in its present form.It may, for example, provide a starting point for an ex- tension of the theory to heterocyclic molecules, which have not so far been amenable to satisfactory treatment. 2. Molecular Orbitals and Equivalent Orbitals.-In the orbital theory of valency the electronic structure of a molecule is described in terms of orbitals each of which describes the motion of one electron alone. These orbitals have to be combined with one another in the form of a deter- minant to obtain a wave function for the molecule as a whole which conforms to the Pauli principle. In order to be the best possible, these orbitals must satisfy equations which, for a state with paired spins, can be written (H + v + 4 d J n = 2 EmndJ,,, ??& 1 Lennard- Jones and Pople, this Discussion.2 Lennard-Jones, Proc. Rqy. SOC. A , 1949, 198, I , 14.20 STRUCTURE O F MOLECULES where H is the Hamiltonian for an electron in the field of the bare nuclei, V and A are the Coulomb and exchange operators representing the effect of the remaining electrons and Em, is defined as These equations do not define the orbitals uniquely, so that the motion of the electrons can be described equally accurately using several different types of orbital.* One possible type is the equivalent orbital which is explained and illustrated in the previous paper.l In many respects this is the most convenient description of a molecule, but for certain purposes it is essential to use the molecular orbital description.A molecular orbital 4, is defined by the condition that Em, = 0, w + n. * ( 3 ) From this definition it can be proved that each molecular orbital belongs to one or other of the irreducible representations of the symmetry group of the molecule. This means that they cannot usually be localized in a particular part of the molecule but are spread throughout it. On the other hand, equivalent orbitals are identical with each other except for their position and orientation in space. They satisfy the orbital equations above, for the operators V and A are unaltered by a transforma- tion from molecular orbitals to equivalent orbitals ; (in fact, these operators are invariant under any orthonorm transformation).The quantities Em, are replaced by em,, defined as where x, and X, denote any two orbitals in the equivalent orbital set. These quantities enan have the property of depending only on the velative orientation of X, and x,. Thus, if X, is equivalent to X, and X, to xu and if x, and x, have the same relative positions as X, and xY, then emn = ew. . * ( 5 ) The diagonal elements en% are all equal for a set of equivalent orbitals, and the non-diagonal elements are equal when they involve pairs of orbitals similarly distributed and similarly related in position. The total energy of the molecule can be expressed equally conveniently using either kind of orbital. In terms of molecular orbital qua.ntities the total energy is where the last term is the sum of the internuclear repulsions and Hnn = [JnH+ndx. .J In the equivalent orbital description it is where (7) Methods have been given elsewhere of passing from one set of such symbols to another. a Hall and Lennard-Jones, Proc. Boy. Soc. A , 1950, 202, 155.SIR JOHN LENNARD-JONES AND G. G. HALL 21 This new orbital theory not only provides two alternative descriptions of a molecule in terms of orbitals but also shows how these orbitals may be transformed into one another. Thus the equivalent orbital description is directly related to the molecular orbital one, and any statement or equation made in one set of terms can be translated into the other set. If, for example, the magnitudes of the equivalent orbital parameters e, are known, the molecular orbital parameters En, may be found by dia- gonalizing the matrix ern,. 3.Ionization Potentials of Saturated Molecules.-The removal of an electron from a molecule is one of the simplest phenomena involving primarily the electronic structure of the molecule. When considered on the orbital theory 3 it is found that the effect of ionization is to remove an electron from a molecu1a.r orbital without serious changes to the remaining orbitals. Ionization is not, therefore, the removal of an electron from one particular part of a molecule such as a lone pair or a bond, but rather from a molecular orbital spread over the whole molecule. The vertical ionization potential corresponding to a molecular orbital I& is equal to - En, to a good approximation. Since these vertical ionization potentials are directly observable quantities, the form of the molecular orbitals and the magnitude of their parameters become a matter of considerable ex- perimenta 1 significance.Conversely, the importance of the experimental ionization potentials in elucidating the electronic structure of molecules can hardly be over-emphasized (see Mulliken 4)). Quantities such as heats of formation, internuclear distances and dipole moments depend on many occupied orbitals. They only yield information about individual orbitals after an extensive analysis. Ionization potentials, on the other hand, give direct and unambiguous information about single molecular orbitals. There is a very close parallel between the theory of molecular vibrations and the orbital theory of ionization potentials.Vibration frequencies and their normal co-ordinates correspond to ionization potentials and their molecular orbitals. The matrix of force constants corresponds to the equivalent orbital matrix em,,. This suggests that, just as vibrational spectra can be analyzed and the magnitude of the frequencies calculated in terms of force constants, the ionization potentials can be interpreted and calculated using the ern, parameters.6 The ionization potentials of methane may’, for example, be calculated in the following way. In the equivalent orbitals corresponding to the four CH bonds. Because of the equivalence, the em, matrix has the form a b b b b b a b b b b a and the ionization potentials are found by diagonalizing this matrix. diagonalization of this matrix leads to the equation The I emn - E8,n I = 0, .* (11) just as the diagonalization of the force constant matrix leads to a deter- minantal equation for the vibration frequencies. For methane this equation becomes a-E b b b b a-E b b b b a-E b b b b a-E = 0. * (14 Mulliken, J . Chern. Physics, 1935, 3, 517. Hall, Proc. Roy. SOC. A , 1951, 205, 541.2 2 STRUCTURE OF MOLECULES The roots of this equation are (a + 3b) and (a - b), the latter appearing three times, so that the ionization potentials are The values of the parameters u and b are not known initially, just as the force constants of a vibrating system are not known initially, but when a sufficient number of molecules have been analyzed it should be possible to predict suitable constants for any particular molecule.4. Saturated Hydrocarbons.-In practice it is much more difficult to determine ionization potentials than vibration spectra, because usually only one ionization potential, that of the most loosely bound electron, is available. Until it becomes possible to measure the inner ionization potentials experimentally, it will be necessary, therefore, to consider homo- logous series and not individual molecules. Unfortunately very few of these series have been studied experimentally to date. Another difficulty is that the outer ionization potentials are not very sensitive to changes in structure, so that very accurate determinations are required. The easiest series to consider in this way is the normal paraffin s e r i e ~ . ~ All its CC bonds and all its CH bonds may be taken as chemically equivalent, so that the number of enan parameters required is small.Since i t is only the form of the equations that is known, these parameters have to be deter- mined from the observed ionization potentials. Once the parameters are known, the ionization potentials may be calculated for the whole series. Table I shows the agreement between the calculated and the observed6 potentials. TABLE I.-~ALCULATED AND OBSERVED IONIZATION POTENTIALS &(Al) =- (u + 3b) ; 12{T2) =- ( a - b). . - (13) 11.394 12.72~ I I'sg3 I I 6-08, Propane . Butane . Pentane . Hexane . Heptane . Octane . Nonane . Decane . Calculated 11'2 I4 10.795 10'554 10.41~ 10.32~ 10.26~ 10'224 10-1g4 Observed 11'21 10.80 10.55 10.43 10.35 10.24 10'2 I 10.19 From the same parameters some of the inner ionization potentials may also be calculated.This application of the method will be of increasing importance as the inner potentials come to be studied experimentally. Not all the potentials can be calculated, for some of the molecular orbitals have symmetry properties such that their potentials involve a different set of parameters. As an illustralion some of the ionization potentials of normal octane are given in Table 11. TABLE II.-SDME CALCULATED IONIZATIOX POTENTIALS OF NORMAL OCTANE 10.795 I 1.87~ 12.36~ 15'46, I I - 7 1 ~ 13'534 - - These ionization potentials, arranged in order of increasing magnitude, correspond to the removal of electrons from different molecular orbitals. The first and last rows refer to molecular orbitals involving mainly (but not exclusively') the CC links, while the second and third rows refer to orbitals involving mainly the CH, groups.Honig, J . Chem. Physics, 1948, 16, 105.SIR JOHN LENNARD-JONES AND G. G. HALL 23 5. The Ionization Potentials of Conjugated Hydrocarbon Molecules.- It is characteristic of conjugated molecules that, although their structure may still be described in terms of equivalent orbitals, the equivalent orbitals representing the double bonds are not localized between two atoms. In butadiene, for example, in addition to the equivalent orbital representing the central CC single bond, there are four equivalent orbitals corresponding in pairs to the outer CC double bonds of the classical formula. These double bond equivalent orbitals are not localized strictly between a pair of carbon atoms but spread partly over the central bond. This means that the e-parameters of these equivalent orbitals are not constant from the double bond of one molecule to the double bond of another molecule and so the ionization potentials cannot be calculated in the same way as for saturated molecules.The difficulty of finding suitable parameters has been overcome 8 by considering, instead of the ground state, an excited state of the molecule, referred to as the standard excited state, in which the T electrons are in singly occupied orbitals with parallel spins. These singly occupied w orbitals can be transformed among themselves into equivalent orbitals localized around a single atom, so that the ionization potentials may be calculated in the same way as for saturated molecules.Provided the internuclear distances are the same, the ionization potentials and molec- ular orbitals of the ground state are equal, to a good approximation, to those of the lower members of the set corresponding to the excited state, so that the ionization potentials of the ground state can be found in this indirect way. To illustrate the method the ionization potentials of benzene may be considered. In the standard excited state there is one singly occupied equivalent orbital, antisymmetrical in the plane, around each carbon atom. If the parameters are denoted by e = en,, f = enn*l and the remainder taken to be negligibly small, the ionization potentials are found by solving the secular equation e-E f f f e--E f f e--E f f 8--E f f e-E f f f e-E - - 0.. (14) The roots of this are e + 2f; e + f (twice) ; e - f (twice) ; e - 2f, . (15) so that the ionization potentials of the electrons in the occupied orbitals of the ground state are Unfortunately the ionization potentials for conjugated molecules are not yet sufficiently accurate to justify detailed calculations. It will be neces- sary, for example, to take into consideration the variation of the parameter f with internuclear distance and this will call for a large amount of em- pirical data. 6. The Calculation of the Energies of Conjugated Hydrocarbon Molecules.-The contribution of the v electrons to the total energy of a conjugated molecule can be expressed, according to the orbital theory, as the sum of two terms.The first term is the sum of the ionization poten- tials and so can be calculated by the method of 5 5. The second term cannot be calculated directly, but, if the molecular orbitals are known, it 7 Hall and Lennard-Jones, Proc. Roy. SOC. A , 1951, 205, 357. W L U ) = - (8 + 2f) ; = - (e + f). - . (16) Hall (to be published shortly).24 STRUCTURE OF MOLECULES can be expressed 9 in terms of a second set of parameters hmn. These h,, are the matrix elements of the bare nuclei Hamiltonian with respect to the equivalent orbitals X, and x, of the standard excited state and so can be taken as constant whenever the emn are constant. For convenience we may include the internuclear repulsion into the h,, so that the energy, as in (8), becomes additive.Since the molecular orbitals are easily found in terms of these equivalent orbitals, the total 7 electronic energy can be calculated in terms of two sets of parameters. It has become usual not to compare this energy with experimental quantities directly but to do a similar calculation for a state corresponding to a single structural formula with definite single and double bonds. The difference in energy between the actual state and this hypothetical " reference state '' is regarded as the empirical resonance energy. We note in passing that this procedure is not very satisfactory either experimentally or theoretically. The present method of calculating resonance energies has several features in common with the molecular orbital method used by Huckel l o for conjugated hydrocarbons and extended by Lennard- Jones l 1 (for a general account and further references, see Lennard- Jones 1 2 ) .The molecular orbitals were expressed as linear combinations of atomic orbitals and the coefficients in the expansions determined by minimizing the energy. This led t o a determinantal equation for the energies of the molecular orbitals and it happens to have the same form as the one used in $ 5 to calculate ionization potentials. The quantities appearing in this determinant, usually denoted by a and fl, were not calculated theor- etically but were expressed in terms of experimentally known quantities.'l Despite the great difference between this procedure and that used above it can be shown that, as applied to conjugated hydrocarbon mole- cules, the equations used are similar and, since both methods use experi- mental data to determine their parameters, their results will coincide. The definition of the parameters according to the theory developed here is a = +(en, + hnn) ; P = i(enn+i + hnn+i)- * * (17) The coincidence between the old theory and the new one does not take place generally but depends on such accidents as the neglect of second- neighbour interactions and the use, in the older theory, of only two para- meters a and fl.For hetero-molecules, therefore, the two methods yield different results, Thus the agreement between the theories is confined to that part of the older theory which has given the best agreement with experiment. In earlier theories it was found very difficult to give an exact definition of a and 8.They were regarded as matrix elements of a suitable self- consistent Hamiltonian with respect to the atomic orbitals, but this Hamiltonian was not defined precisely. This lack of exact definition led to some ambiguities. For example, the definition of fl differed according as it was assumed that the atomic orbitals were orthogonal lS or not. Thus, whereas Hiickel estimated fi as I 5-20 kcal. /mole, Lennard- Jones allowed for the variation of fi with distance and found fl = 35 kcal./mole for Y = 1.33 A, and Mulliken and Rieke, by making allowance for the overlap integral and hyperconjugation, find p = 44.5 kcal./mole at the same distance. It is probable that this theory will lead to a value rather larger than these. We note in passing that to relate the orbital theory to the earlier molec- ular orbital theory we must distinguish between the interpretation of a and j? for conjugated hydrocarbons as matrix elements with respect to equivalent orbitals and the corresponding interpretation for saturated Hall (to be published shortly).l o Hiickel, 2. Physik, 1931, 70, 204. l1 Lennard-Jones, Proc. Roy. SOC. A , 1937, 158, 280. l2 Lennard-Jones, Proc. Roy. SOC. A , 1951 (in press). l3 Mulliken and Rieke, J . Amer. Chem. SOC., 1941, 63, 1770.SIR JOHN LENNARD-JONES AND G. G. HALL 25 molecules. The equivalent orbitals, in the latter case, may be CC bonds or CH bonds (or, in hetero-molecules, lone pairs). It thus appears as though in such molecules the parameter a is closely related to a bond property, but b, which is a matrix element relative to two equivalent orbitals meeting on a common centre, is rather to be associated with a particular atom.In conjugated molecules, on the other hand, the equiv- alent orbitals are localized in the neighbourhood of a particular atomic centre, so that a should be associated with an atom and with two neighbouring atoms. The differences between this orbital theory and the previous molec- ular orbital one are best illustrated by showing that many of the as- sumptions and approximations of the older theory, as discussed by Coulson and Dewar,l* are now unnecessary. In the previous theory the approximations were made of expanding the molecular orbitals as linear combinations of a small number of atomic orbitals and of assuming that these atomic orbitals were orthogonal.The errors introduced by these approximations are undoubtedly appreciable. On the other hand, in this theory the molecular orbitals are expressed as transforms of an orthonormal set of equivalent orbitals. This does not involve any approximations, for the determinantal wave function for the molecule as a whole is invariant under such transformations. For con- jugated molecules, however, there is an approximation involved in equat- ing the molecular orbitals of the ground state with those of the standard state described above, but the error thus introduced can be estimated using perturbation theory. Frequently the older theory used a wave function for the molecule as a whole in the form of a product of orbitals.Such a wave function does not satisfy the Pauli principle and consequently omits the exchange con- tribution to the energy and over-estimates the probability of the electrons being simultaneously around one atom. To reduce these errors a deter- minant of T orbitals has sometimes been used for conjugated molecules, as for example, by Goeppart-Mayer and SkIar,l6 by Craig l6 and by Parr and Mulliken,17 but even this is not antisymmetrical with respect to the interchange of u and T electrons. When the total wave function is made fully antisymmetrical, the theory loses much of its simplicity and does not lead to the usual secular equation.l% 5 None of these difficulties arise in the present form of the theory, since it is based on a wave function of the correct determinantal form.Then again, in the older theory there was some difficulty in isolating the contributions to the total energy from the internuclear repulsions, the electrons in symmetrical orbitals (U bonds) and those in antisymmetric orbitals (T bonds). This seems to have been due to confusion between the Schrodinger equation for the molecule as a whole and the equation for the T-orbitals. Now that the equations for the orbitals have been derived explicitly, this difficulty disappears. In both theories the approximation is made of using a single deter- minant as a wave function in contrast with the electron pair theory which uses a linear combination of determinants. The closeness of this ap- proximation is to be judged by the flexibility of the wave function and it is obviously better to use orbitals chosen a so as to make the approximation as good as possible than to use orbitals in which only a certain number of parameters can be varied.Indeed, it is probable that, unless a large number of determinants are used, the wave function of the electron pair theory, in which only the coefficients of the determinants are varied, will not be as good an approximation as the wave function of this theory. l4 Coulson and Dewar, Faraday SOC. Discussion, 1947, 2, 54. l5 Goeppart-Mayer and Sklar, J. Chem. Physics, 1938, 6, 645. l6 Craig, Proc. Roy. SOC. A, 1950, 200, 474. l7 Parr and Mulliken, J. Chem. Physics, 1950, 18, 1338. 18Moffitt, Proc. Roy, SOC. A, 1949, 196, 510.26 BOND RESONANCE ENERGIES The previous molecular orbital theory had the severe limitation that, jn order to be self-consistent in the sense introduced by Coulson and Rushbrooke,lg the v electron distribution had to be uniform.This meant that hetero-molecules could not be considered quantitatively, and that variations in the parameters due to changes in bond length, etc., could only be treated approximately. This particular difficulty does not arise in this theory, since the equations for the orbitals of the standard excited state are self-consistent in the iullest sense. As already mentioned, there is an approximation involved in the use of the standard state, but the error can be estimated and is unlikely to be large. In practice, owing to the difficulty of obtaining the necessary ionization potentials, it is necessary to make a number of working assumptions as to the magnitude and vari- ation of the parameters emn and h,,, but these are not necessary to the validitj of the theory and can be refined considerably as more data become available. Finally, it may be remarked that hyperconjugation appears in a new aspect in the equivalent orbital theory. In the theory developed by Mulliken,l3 for example, hyperconjugation is a consequence of the expan- sion of molecular orbitals as linear combinations of atomic orbitals. Thus, in ethane, there are four antisymmetrical orbitals of the 7~ type corre- sponding to localized molecular orbitals of each of the CH, groups. The interaction of these T type orbitals gives to the C-C bond a certain triple bond character. The disadvantage of this treatment is that it becomes difficult to understand what a normal CC link is. On the other hand, in this orbital theory the phenomenon known as hyperconjugation is included in the general treatment. There is no need to separate out this factor from the others which determine the energy of a molecule. Department of Theoretical Chemistry, University of Cambridge. 1 9 Coulson and Rushbrooke, Proc. Camb. Phil. SOC., 1940, 36, 193.