Molecular orbitals in polyatomic molecules

The molecular orbital theory of polyatomic molecules follows the same principles as those outlined for diatomic molecules, but the molecular orbitals spread over all the atoms in the molecule. An electron pair in a bonding orbital helps to bind together the whole molecule, not just an individual pair of atoms. The energies of molecular orbitals in polyatomic molecules can be studied experimentally by using ultraviolet and visible spectroscopy (see Major Technique 2, following this chapter). 

As in diatomics (Section 4.3), the molecular orbitals in polyatomic molecules may be expressed as linear combinations of atomic orbitals (AOs) centered on the nuclei. A minimal basis set of AOs contains all of the AOs that are occupied in the separated atoms [1, 2]. In the bent ozone molecule, for example, the separated O atoms have the ground state configuration (ls) (2s) (2p). The minimal basis set for ground-state O3 therefore consists of the Is, 2s, and three 2p orbitals centered on each of the oxygen nuclei (Fig. 7.1). The orientations selected for the 2p AOs in the basis set are of course arbitrary (aside from the constraint that basis AOs centered on any nucleus must be linearly independ-... 

The idea can be extended to a polyatomic molecule and generalized so that each molecular orbital in a molecule is a linear combination of all the atomic orbitals occupied by the electrons in the constituent atoms. 

The existence of more than two nuclei in polyatomic molecules means that molecular orbitals are polycentric, extending, in principle, over the whole molecule. It is in this sense that molecular orbitals in polyatomic systems are said to be delocalized. Within the linear combination of atomic orbitals (l.c.a.o.) approach, valence m.o.s are approximated by linear combinations of the valence a.o.s of all the intervening atoms. Just as for diatomic molecules, this is an approximation because only a finite set of basis fimctions are being used. The eventual inclusion of virtual a.o.s of energy close to the valence orbital energies can help to improve the results. 

In Chapters 4 and 6 we have distinguished between a molecular orbitals and 7T molecular orbitals in diatomic molecules on the basis of the symmetry of the m.o.s with respect to rotation around the intemuclear axis whereas a a orbital has cylindrical symmetry, a tt orbital changes sign upon a rotation of 180°. In Chapters 7 and 8 we have extended the notion of a orbitals to polyatomic molecules by referring to the local symmetry with respect to each X-Y intemuclear axis. We will now study systems of tt m.o.s in polyatomic species. 

In the previous two Sections we only considered some of the molecular orbitals in the molecules we discussed. We now look briefly at the complete set of molecular orbitals for a simple polyatomic molecule, H20. 

For both types of orbitals, the coordinates r, 0 and cji refer to the position of the electron relative to a set of axes attached to the centre on which the basis orbital is located. Although STOs have the proper cusp behaviour near the nuclei, they are used primarily for atomic- and linear-molecule calculations because the multi-centre integrals which arise in polyatomic-molecule calculations caimot efficiently be perfonned when STOs are employed. In contrast, such integrals can routinely be done when GTOs are used. This fiindamental advantage of GTOs has led to the dominance of these fimetions in molecular quantum chemistry. 

Hurley, A. C., Lennard-Jones, J., and Pople, J. A., Proc. Roy. Soc. [London) A220, 446, The molecular orbital theory of chemical valency XVI. A theory of paired electrons in polyatomic molecules." Use of two-electron functions T fa, x5) as a basis. 

Molecular orbital bonding concepts in polyatomic molecules a novel pictorial approach. D. K. Hoffman, K. Ruedenberg and J. G. Verkade, Struct. Bonding (Berlin), 1977, 33, 57-96 (14). 

Hoffmann, D. K., Ruedenberg, K., Verkade, J. G. Molecular Orbital Bonding Concepts in Polyatomic Molecules - A Novel Pictorial Approach. Vol. 33, pp. 57-96. 

Use molecular orbital theory to describe bonding in polyatomic molecules such as the benzene molecule, Section 3.12. 

In a molecule, the one-electron eigenfunctions (Mulliken and Hund s molecular orbitals, M. 0.) are determined by a core field U(x,y,z) and have well-defined symmetry type yn. However, the actual calculation of such M. 0. is very difficult in polyatomic molecules, whereas the Hartree-Fock method can be applied to monatomic entities when large electronic computers are available (13,14). I have written a book about several of the principal problems regarding the concept of one-electron functions... 

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