A further step is made in the development of the orbital theory of molecular structure, the object being to allow for the tendency of two electrons in the same space orbital to keep apart because of the repulsive field between them. The description of a molecule is generalized in such a way as to bracket together electrons in pairs, the simple theory of molecular orbitals being used as a starting point for a wider treatment. If two orbitals are used for each electron pair (paired electron orbitals), it is shown that the theory contains as special cases both previous results of the present series of papers and also the valence-bond wave function which is shown to be an alternative particular approximation to the general function.
The high-temperature neutron diffraction measurements of UO
2
and CaF
2
made recently by Willis are examined by means of the generalized formulation of the structure factor which recognizes the antisymmetric properties of atoms occupying non-centrosymmetric lattice sites. Numerical analysis of the neutron data in terms of an anharmonic vibration formalism developed for effective one-particle potential fields of the type
V
j
(r) =
V
0
,
j
+ ½
α
j
(
x
2
+
y
2
+
z
2
) +
β
j
xyz
+ quartic terms shows that certain unusual features of the observations are readily accounted for by the antisymmetric (
β
) components associated with the tetrahedral point symmetry of the anionic sites. Values are derived for the
α
j
and
β
j
parameters in both UO
2
and CaF
2
, and their ability to reproduce the experimental observations over a wide range of elevated temperature is demonstrated. The parameters derived for CaF
2
are compared with theoretical estimates obtained from an Einstein-type model of the vibrating lattice, using the various expressions for the interionic interaction in fluorite that have been proposed by Benson & Dempsey. Despite evident limitations in this model, it is possible to conclude that the Pauling radii for Ca
2+
and F
-
cannot reproduce the physical behaviour of this system.
An integral form of the quantum-mechanical virial theorem, applicable to ionic as well as neutral systems, is established. This theorem is used to derive an approximate expression for the potential curve for a doubly positive diatomic ion in terms of the corresponding curve for a related neutral molecule. The appearance potentials of a number of these ions are calculated and compared with values obtained from electron-impact measurements. Satisfactory agreement is found, but a crucial test of the theory is vitiated by uncertainties in the experimental value of either the appearance potential of the doubly charged ion or the dissociation energy of the corresponding neutral molecule. The accuracy of the approximations underlying the theory is estimated by a series of calculations on two-electron systems.
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