The molecular orbital theory of chemical valency* XIX. The charge density function By P. P. Manning! Department of Theoretical Chemistry, University of Cambridge (Communicated by C. A. Coulson, —Received 8 March 1955) An account of the properties of the electronic charge density function in the first and higher approximations in molecular orbital theory is given. Application to conjugated molecules allows a simple understanding of the relation between bond order and bond length in both symmetrical and unsymmetrical bonds. The final section shows how any diagonal element of the charge-density matrix may be expressed as a basic constant modified by a term taking account of any difference in the number of electrons with each spin and by further terms which may be regarded as charge drifts from one orbital to another.
416 P. P. Manning of the bond orders (see, for example, Coulson & Longuet-Higgins 1947; Manning 19546). This provides a simple understanding of the variation of bond lengths in conjugated molecules and the correlation these have with the mobile bond order. In the last section an account of the relation between the determinantal coefficients and the diagonal terms in the charge-density function is given.
The molecular orbital theory of chemical valency. XIX 417 Regarding these quantities as elements of the charge-density matrices Q“ and respectively, we see that these can be written in block diagonal form, each block representing a different symmetry type. Usually there is no advantage to be gained by doing this to the fullest possible extent because those orbitals which belong to the irreducible representations of the symmetry group are generally delocalized throughout the molecule and do not provide a simple interpretation of the factors governing the charge distribution. However, in the case of conjugated molecules, it is convenient to separate the orbitals into sets respectively symmetric and anti symmetric in the plane of the conjugating nuclei.
418 P. P. Manning generalization of the Fock operator. For a partly filled band the one-centre orbitals will be more localized than the Wannier functions because they are still transforms of all the Bloch orbitals in the band, though some of these will be virtual orbitals in the one-determinant approximation, while Wannier restricted the transformations to occupied orbitals. This would apply to a covalent crystal such as diamond or to a metal; a simple ionic crystal such as sodium chloride is a case in which there would be a close similarity.
The molecular orbital theory of chemical . XIX 419 This gives Qbb = 1-977, Qaa = 0-023. (3-08) Though the hydrogen molecule is a rather special case it seems very likely that Qau for symmetrical bonds in other saturated molecules will be of this order of magnitude.
420 P. P. Manning and the corresponding functions if it is the equivalent orbital of the one-determinant theory. From (3*11) it is clear that the occupation numbers of the orbitals av a2 and a3 are not small in either approximation, and it is unlikely that taking account of electron correlation by the use of more than one determinant will result in greatly different values in the two cases. This is a well-known feature of conjugated systems (see, for example, Coulson 1947), the bond orders depending on the system as a whole and often varying considerably from one bond to another.
The molecular orbital theory of chemical valency. XIX 421 Suppose the inner-shell orbitals, |JV0 in number, are doubly occupied in all determinants and that the remaining N electrons are described by N valence shell orbitals. To this approximation no explicit account of the inner shells is necessary. The number of electrons with a spin is Na and with /? spin is Suppose the real hybrid one-centre orbitals are •••> an(l that the space-spin orbitals constructed from these are <f>s = Xs<*> (*= 1,2,., A7). (4-01) Now divide the determinants in the wave function into two classes as they do, or do not, contain some particular space-spin orbital, say <f)s. If is a determinant containing <j>s together with a number of other space-spin orbitals represented by the single suffix r, and if is one in which <fis has been replaced by (j>t, possibly with a change in ordering of orbitals in the determinant, the wave function can be writtenas (4-02) (r«) <(+s) (r,t) where the first summation is over all possible sets ( ofNx—l space orbitals with a spin and N p with /? spin excluding (j>s, and the second summation of the last term is over similar sets but excluding both <f>s and For the wave function to be normalized we must have S Bl+ S s 1. (4*03) (r,) <(+s) (r,t) The occupation number of (j>8, the spin component of that for ys, is (4-04) (r.) (r.) <(+s) (r«t) on using (4-03). Equation (4*04) has the disadvantage of bearing no clear relation to the part the orbital plays in the molecule and accordingly we try to re-write it in the form N = (405) The function T“sr contains Na orbitals with a spin (apart from inner-shells) so that there are N -X V orbitals from the a set which are not included. If, therefore, G% is to be related to the replacement of <f)s in by <f>t, where r contains neither <fis nor <fit, each term in the first summation in (4-04) must be equally related to one of the G% for Np different values of t. This gives the contribution of the first term in (4-04) to G% as (r.)-'V Similarly, contains Na orbitals with cl spin, any one of which may be regarded as replacing <j)s in for a range of sets r. Hence each term in the second summation in (4-04) contributes equally to Na of the G%, the contribution being.
422 P. P. Manning Proceeding in this way for each orbital <f>8 in turn we obtain a matrix G* with elements (4*06) and such that (4-05) is satisfied.