Polyatomic molecules molecular orbitals

Mulliken, R.S. (1935c) Electronic structures of molecules. XI1. Electroaffinity and molecular orbitals, polyatomic applications. J. Chim. Phys., 3, 586-591. 

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. 

The molecular orbital (MO) approach to the electronic structure of diatomic, and also polyatomic, molecules is not the only one which is used but it lends itself to a fairly qualitative description, which we require here. 

The bonding in molecules containing more than two atoms can also be described in terms of molecular orbitals. We will not attempt to do this the energy level structure is considerably more complex than the one we considered. However, one point is worth mentioning. In polyatomic species, a pi molecular orbital can be spread over die entire molecule rather than being concentrated between two atoms. 

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). 

We shall illustrate these rules first with H2 and then with other diatomic molecules. The same principles apply to polyatomic molecules, but their molecular orbitals are more complicated and their energies are harder to predict. Mathematical software for calculating the molecular orbitals and their energies is now widely available, and we shall show some of the results that it provides. 

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). 

Another important polyatomic molecule is benzene, C6f I6, the parent of the aromatic compounds. In the molecular orbital description of benzene, all thirty C2s-, C2p-, and Hls-orbitals contribute to molecular orbitals spreading over all twelve atoms (six C plus six H). The orbitals in the plane of the ring (the C2s-, C2px, and ( 2/ -orbitals on each carbon atom and all six Hls-orbitals) form delocalized o-orbitals that bind the C atoms together and link the H atoms to the C atoms. The six C2pz-orbitals, which are perpendicular to the ring, contribute to six delocalized tt-orbitals that spread all the way around the ring. However, chemists... 

According to molecular orbital theory, the delocalization of electrons in a polyatomic molecule spreads the bonding effects of electrons over the entire Energy molecule. 

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

Despite the obvious limitation of the LCAO procedure as revealed by the Hj and H2 problems it still is the most popular scheme used in the theoretical study of polyatomic molecules. There is a bewildering number of approximate methods, commonly distinguished in terms of cryptic acronyms, designated as either ab initio or semi-empirical, but all of them based on the LCAO construction of molecular orbitals. The precise details can be found in many books and reviews. The present summary uses the discussion of Richards and Cooper [92] as a guide. 

Abstract. The development of modern spectroscopic techniques and efficient computational methods have allowed a detailed investigation of highly excited vibrational states of small polyatomic molecules. As excitation energy increases, molecular motion becomes chaotic and nonlinear techniques can be applied to their analysis. The corresponding spectra get also complicated, but some interesting low resolution features can be understood simply in terms of classical periodic motions. In this chapter we describe some techniques to systematically construct quantum wave functions localized on specific periodic orbits, and analyze their main characteristics. 

Mulliken introduced the term "orbital" distinct from "orbital wave function" in 1932 in the second of fourteen papers carrying the general title, "Electronic Structures of Polyatomic Molecules and Valence." Mulliken defined atomic orbitals (AOs) and molecular orbitals (MOs) as something like the... 

Mulliken, Life, 90. On the "orbital," Mulliken wrote in 1932 "From here on, one-electron orbital wave functions will be referred to for brevity as orbitals. The method followed here will be to describe unshared electrons always in terms of atomic orbitals but to use molecular orbitals for shared electrons." In Robert Mulliken, "Electronic Structures of Polyatomic Molecules and Valence," Physical Review 41 (1932) 4971, on 50. 

In summary, the molecular orbitals of a linear molecule can be labeled by their m quantum number, which plays the same role as the point group labels did for non-linear polyatomic molecules, and which gives the eigenvalue of the angular momentum of the orbital about the molecule s symmetry axis. Because the kinetic energy part of the... 

Just as in the non-linear polyatomic-molecule case, the atomic orbitals which constitute a given molecular orbital must have the same symmetry as that of the molecular orbital. This means that o,%, and 8 molecular orbitals are formed, via LCAO-MO, from m=0, m= 1, and m= 2 atomic orbitals, respectively. In the diatomic N2 molecule, for example, the core orbitals are of o symmetry as are the molecular orbitals formed from the 2s and 2pz atomic orbitals (or their hybrids) on each Nitrogen atom. The molecular orbitals fonned from the atomic 2p i =(2px- i 2py) and the 2p+j =(2px + i 2py ) orbitals are of Jt symmetry and have m = -1 and +1. 

The molecular electronic wave functions ipe] are classified using the operators that commute with Hei. For diatomic (and linear polyatomic) molecules, the operator Lz for the component of the total electronic orbital angular momentum along the internuclear axis commutes with Hel (although L2 does not commute with tfel). The Lz eigenvalues are MLh,... 

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