Molecular orbitals arid the Hartree field By W. Moffitt, New College, Oxford {Communicated by F. E. Simon, F.R.S.—Received 6 August 1948— Revised 11 November 1948) The aim of this paper has been to introduce self-consistency into a general, but necessarily rather oversimplified, method of molecular orbitals. Thus the non-linearity of the equa tions (19) has enabled us to deal in a systematic way with the charge distribution and bond properties of different conjugated molecules in a variety of configurations. Further analysis is required before term values can be predicted. The electro-affinity scale which may be set up for both <t and n electron pair bonds by means of this method turns out to be identical with that of Mulliken. A general account of the configurational theory of molecular structure has been given in an introductory section.
Molecular orbitals and the Hartree field 511 belongs, and the magnetic quantum number mja) designates some row of this (2Z(a) + l)-fold degenerate representation. The principal quantum number serves to distinguish the various 0(a) with the same transformation properties but with different ‘energy parameters' and m(8a) represents the spin co-ordinate. It is assumed that this set of functions contains no accidental degeneracies: = da) only holds when NW = N(a) and = Z(a). For an w-electron system, the set of all product functions . <j>(n\ri), which may be formed by choosing the (a) = (N^a\ Z(a), m|a), m(sa)) so as to be consistent n with Pauli’s exclusion principle and a given total zeroth-order energy = 2 a—1 and by permuting the electrons m amongst the 0(a), may be called a configuration. Since e depends only on the and Z(a), a given configuration may be uniquely specified more simply by the notation .[0/*f.$«]«. (a= 1,.,/*); a now only represents the twofold symbol ( , la) and is the number of electrons in this configuration which are associated with the functions 0®, so that /» n = 2 n®. The spectral states and their term symbols which arise from these con- ct = l figurations may easily be found (e.g. Slater 1929), and give a satisfactory account of the essential features of atomic spectra.
512 W. Moffitt and further, that they belong to irreducible representations of the molecule’s spatial symmetry group (point group), r, just as a for atoms, is a twofold symbol. It in dicates, first, the transformation properties or symmetry characteristics of — as did la for the <j)a—and secondly, it contains an ordering symbol, analogous to Na in the atomic case, which distinguishes molecular orbitals belonging to the same irreducible representation but with different energy parameters or ‘ vertical ioniza tion potentials ’. t = nTis the number of electrons of the ^-electron system which are V associated with \Jfr, so that n — ^ n T. If the representation to which \[rT belongs is T=1 <7T-fold degenerate, it is clear that nT < 2 gT,by Pauli’s principle; similarly, in the case of atoms, of course, a — ^n* 2(2 la+ 1), and for the same reason.
Molecular orbitals and the 513 bond properties and to estimate the ‘vertical ionization potentials’ of their various electrons. In order to explain the more quantitative aspects of ultra-violet spectra —just as for atoms—antisymmetrization is necessary (Goeppert-Mayer & Sklar 1938) and may be attained most simply by an adaptation of Dirac’s vector model to molecules. The intensities of electronic spectra have, however, been calculated relatively successfully (Mulliken 1939) using simple LCAO ^T’s without this com plication.
514 W. Moffitt Fortunately, we do not need to specify the explicitly, and it will be sufficient that the vT satisfy the following requirements: (a) !>( = *'''• i (b) It is necessary that the $>\ should not lose their property of being essentially atomic orbitals. Therefore, as the atoms go separately to infinity while retaining the charge associated with them in the molecule {vide infra), so v\ must become v*—which is the field of the nucleus i, the non-7r-bonding electrons of i and the other 7T valence electrons associated with i. f>\ then becomes <f>£, satisfying and is a proper atomic ‘ valence state ’ function whose screening constant ZJ is deter mined by v£ and whose term value is — (c) The v\ transform, under the operations of the molecule’s symmetry group, like the atomic nuclei to which they refer, i.e. they have the symmetry of the corre sponding atomic sites in the molecule.
Molecular orbitals and the Hartree field 515 It is primarily with these quantities qiy pis that Coulson & Longuet-Higgins (1947) have associated the properties of conjugated systems, though their method of determining the y\ differs from that which we shall develop.
516 W. Moffitt using equations (2) and (3), <$ - JVjX 'fidn = + and (k+j) are the so-called Coulomb and resonance integrals respectively. In performing this process of minimization, we must ensure that the molecular orbitals remain orthonormal, namely, that conditions (5) are satisfied SylrS = $Tp- In order to carry through the analysis, certain simplifying assumptions will now have to be made. This is necessary because the functional forms 6f e\ and /?£,•, which will depend in no simple manner on the yl and the equilibrium internuclear distances are unknown. These assumptions have been chosen so as to exhibit the maximum number of physically significant factors consistent with relative analytical sim plicity. In particular the compromise between ionic and covalent bonding, and the necessity of using self-consistent Coulomb integrals are included, while the usual properties of calculations on conjugation problems are retained. The eventual justification of any such assumptions will be in the measure of agreement attained between predicted and observed properties of molecules. No attempt will therefore be made to evaluate the integrals e\, y a zeroth-order treatment is very probably too approximate to warrant the attack of such quadratures in the first place. (It is because of this that we have introduced the polarized atomic functions <f>\, e.g. (5), which simplify the formal development of our approximation, but which would complicate considerably the direct evaluation of these integrals.) We shall assume, first, that the resonance integrals depend only on the nature of atoms i and,;, so that = fiy for all r. (This assumption would, however, have to be modified if both pir and dn electrons of the same atom were considered as participating in the conjugation—as is possibly the case for the sulphur atom in thiophene, for example.) Since the e\ will occur only as differences {vide infra for CO, A 1II), our suppositions concerning these will be relative only: it would be rash to use these e*’s for the prediction of ‘ vertical ionization potentials ’ or orbital term values without further analysis. Our second assumption is now that e< = %(«<)> = 2 (nP - &*) (yi)2, (12)' p where the function is the same as that defined in equation (9) above and, more specifically, takes a linear form *li{x) = Vi + Vi(x)-(13) To a first approximation, therefore, it is assumed that, with an appropriate choice for the fiy, we may take e\ = yjJ. This is certainly the simplest of feasible forms for the el when dealing with atoms of different electro-affinities; it is the more plausible when it is shown {vide infra) that this choice leads to an absolute electro-affinity scale.
Molecular orbitals and the Hartree field 517 which is identical with that set up so successfully on other theoretical grounds by Mulliken in 1934. If, however, we were to introduce these values in (11), we should count the interactions between the various electrons twice. Accordingly, with the linear form (13) for the functions %(#), it is clear that when considering the total energy of a configuration we must use not the values (12)' but rather el = ifcflK). (12) If we could treat all the y\ as independent variables we should have, using (8), (9), (12) and (13), W ; = «+*/>+*).
518 W. Moffitt has five tt electrons and may be treated as arising from the configuration [ where fr, ijr' are both doubly degenerate, the former being a bonding and the latter an antibonding function. It is assumed that the er electronic charge distribution is predominantly homopolar and may be neglected in the variational problem. In LCAO approximation, therefore, f = 7c0c + 7o0o> f = 7c0c + 7o0'o- By orthonormality we have 7c + 7o = 1 = 7c + 7o> 7c7c + 7o7o = 0, and the equations (19) may in this case be written as ^(7c£c + 7oAco) + /^7c + /I/7c = (7c£c + 7oAco) + ^*7c + ^-,7c = 4(7o£o + 7cAco) + ^7o + ^ 7o = (7o £o + 7cAx>)4"A+ ^70 = On eliminating A, A' and A", it is found that 47c7o(£c- £o) + 7c7o(£c~ £o) + M7o — 7c)Aco + (7o — 7c)Aco = from which we may further eliminate y0, y'c and y'Q by means of the orthonormality conditions and obtain the equation 7cV(1- 72c){4(^c-^ o)-(^ -^ o)} + 3Aco(1- 272c) = 0. (A) Since the five n electrons may be considered as responsible for the bonding between the two nuclei C++ and 0+++, it is readily shown that ic = ~4 + (Ic~ Ec()3yc + 7c ~ 1) ~ ~~ ^c + (Iq (27c)> £0 = ~^o + vrc — -^o) (37o + 7o — 2) — — Iq + (^o ~ ^o) (1— 27c)» £c == ~-^c + (-^c- -®c) (4ye - 1)> fo = -Io + (Io-Eo) (472o - 2) = -70 + (70 - E0) (2 - 4y2c).
Molecular orbitals and the Hartree field 519 all the XTp with p=f=r. Let us therefore .determine the A.rp. On multiplying (19) by A* and summing over all i, using (5), it is found that A"> = »rs{(r«r5)fi+S'Wr5)A/j • (*i) Now it is always possible to find a set of functions 0\ with the symmetry properties of the $1, and a set of symmetrical operators (symmetrical in the sense that they transform under the identical representation of the molecule’s point group), such that jdl^f>T0ldv = and Then = S CT = S 1 i transform like \Jrp, ijrr respectively, and (21) becomes A Tp = nrjgM&'pdv.