Directed valencies, theory

Directed valence Directed valence theory uses two-center bonding orbitals, with hybrid combinations of... 

For AB6 molecules octahedral sp3d2 hybrids are used. AB5E molecules must, naturally, be spy. For AB4E2 molecules there is nothing in the directed valence theory itself to show whether the lone pairs should be cis or trans. The assumption that they must be trans leads to consistent results. 

The concepts of directed valence and orbital hybridization were developed by Linus Pauling soon after the description of the hydrogen molecule by the valence bond theory. These concepts were applied to an issue of specific concern to organic chemistry, the tetrahedral orientation of the bonds to tetracoordinate carbon. Pauling reasoned that because covalent bonds require mutual overlap of orbitals, stronger bonds would result from better overlap. Orbitals that possess directional properties, such as p orbitals, should therefore be more effective than spherically symmetric 5 orbitals. 

Due to the simplicity and the ability to explain the spectroscopic and excited state properties, the MO theory in addition to easy adaptability for modern computers has gained tremendous popularity among chemists. The concept of directed valence, based on the principle of maximum overlap and valence shell electron pair repulsion theory (VSEPR), has successfully explained the molecular geometries and bonding in polyatomic molecules. 

The foregoing discussion of valence is. of course, a simplified one. From ihe development of the quantum theory and its application to the structure of the atom, there has ensued a quantum theory of valence and of the structure of the molecule, discussed in this hook under Molecule. Topics thal are basically important to modem views of molecular structure include, in addition to those already indicated the Schroedinger wave equation the molecular orbital method (introduced in the article on Molecule) as well as directed valence bonds bond energies, hybrid orbitals, the effect of Van der Waals forces and electron-dcticiem molecules. Some of these subjects are clearly beyond the space available in this book and its scope of treatment. Even more so is their use in interpretation of molecular structure. [However, sec Crystal Field Theory and Ligand.)... 

The problem of directed valence is treated from a group theory point of view. A method is developed by which the possibility of formation of covalent bonds in any spatial arrangement from a given electron configuration can be tested. The same method also determines the possibilities of double and triple bond formation. Previous results in the field of directed valence are extended to cover all possible configurations from two to eight s, p, or d electrons, and the possibilities of double bond formation in each case. A number of examples are discussed. 

In the previous papers no attempt has been made to discover all the possible stable electron groups which lead to directed valence bonds, nor have the possibilities of double bond formation been completely explored. In the present paper both of these deficiencies in the theory have been removed. 

In otder to construct such sets of orbitals, it is most convenient to make use of group theory. Each set of equivalent directed valence orbitals has a characteristic symmetry group. If the operations of this group are performed on the orbitals, a representation, which is usually reducible, is generated. By means of the character table of the group7 this representation, which we shall call the a representation, can be reduced to its component irreducible representations. The s, p, and d orbitals of the atom also form representations of the group, and can also be divided into sets which form irreducible representations.8... 

That molecules do have definite bonds, and that these tend to correspond in direction and number to the conventional bonds of simple valence theory, is indicated by the quantum theory of atoms-in-molecules (AIM, or QTAIM) [2], This is based on an analysis of the variation of electron density in molecules. 

Intramolecular rearrangements provide dramatic evidence to the inadequacy of the theories of directed valence bonds. The Beckmann rearrangement of ketoximes into acid amides, represented by the scheme ... 

It is important to realize that, although various types of experiments can give the value of a dipole moment, none is capable of determining the direction in which the negative or positive end lies. In molecules such as NF3 and C6H5CH3, in which the dipole moment is so small that simple arguments will not predict the direction, only accurate valence theory calculations can provide the answer. These calculations themselves are reliable only for fairly small molecules (see Section 5.2.3 for the case of CO). 

The choice of a single function from either set (36) or (37) does not permit such a useful physical interpretation, and may indeed lead to difficulties as the internuclear distance is varied. Thus if one chooses just the perfectly paired function from the set (36), as -R-> 00 one finds each N atom is described by a curious non-stationary state - the so-called valence state of the atom, about which there has been so much discussion in the literature.18 The choice of the set of functions (36) in which orbitals participating in a bond are directly coupled to each other is just the VB theory as proposed by Slater and Pauling,19 whereas the set (37) formed from atoms in specific L-S coupled states corresponds to the spin-valence theory employed by Heitler.20... 

This figure is identical to Fig. 8.6, which means that we have arrived at a result similar to the valence-bond description made previously. It was, in fact, only after the introduction of equivalent molecular orbitals (which later evolved into almost localized m.o.s) by J.E. Lennard-Jones in 1949 to provide the m.o. equivalent of directed valence bonding (ref. 113) that m.o. theory found widespread application to structural problems in chemistry. 

X-ray (XPS) [58-61] and ultraviolet photoelectron spectroscopy (UPS) [62-65] provide rich information about the valence bands of extended systems. The further development of angle-resolved UPS (ARUPS) can even be used to directly observe the band structures. Then they can provide a tool to directly check theory. 

The present book is an attempt to rectify this omission. Ideally, as we have just argued, we should present a coherent account of the whole field of chemistry in which the results of structural studies appeared in their rightful place among those of the many other means of determining chemical constitution. This, however, would be a formidable task, and, indeed, an unnecessary one when there already exist so many works on chemistry, valency theory and other aspects of the solid state. We shall therefore presuppose the reader to have some knowledge of general chemical principles, and we shall confine ourselves to a discussion of those properties of solids, directly related to crystal structure, which are not normally considered in detail in chemical works. 

It may well seem that the introduction of the concept of hybridization to account for the valency of the carbon atom is both arbitrary and artificial, and this would indeed be the case if it had no other basis. As with so many other aspects of the application of quantum mechanics to valency theory, however, the justification is mathematical, and here, where we are concerned only with those aspects of valency theory which have a direct bearing on crystal structure, we shall unfortunately often be compelled to quote conclusions without seeking a rigorous justification for the mathematical arguments the reader is referred to the many works on the theory of valency already available. 

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