Polyatomic molecules molecular orbital symmetry

The probabilities of absorption by structures larger than an atom are determined in essentially the same way. For diatomic and linear polyatomic molecules the orbital momentum selection rule becomes AA = 0, 1, where A is the symbol for molecular total orbital angular momentum. For systems with appropriate symmetry, the rules are g m, -I- <-> H-, —. For non-linear polyatomic molecules... 

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. 

Polyatomic molecules. The same term classifications hold for linear polyatomic molecules as for diatomic molecules. We now consider nonlinear polyatomics. With spin-orbit interaction neglected, the total electronic spin angular momentum operator 5 commutes with //el, and polyatomic-molecule terms are classified according to the multiplicity 25+1. For nonlinear molecules, the electronic orbital angular momentum operators do not commute with HeV The symmetry operators Or, Os,. .. (corresponding to the molecular symmetry operations R, 5,. ..) commute... 

The Jahn-Teller theorem is important in considering the electronic states of polyatomic molecules. Jahn and Teller proved in 1937 that a nonlinear polyatomic molecule cannot have an equilibrium (minimum-energy) nuclear configuration that corresponds to an orbitally degenerate electronic term. Orbital degeneracy arises from molecular symmetry (Section 1.19), and the Jahn-Teller theorem can lead to a lower symmetry than... 

To form the molecular orbitals for polyatomic molecules AX , we first carry out linear combinations of the orbitals on X and then match them, taking into account their symmetry characteristics, with the atomic orbitals on the central atom A. 

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... 

Whatever the level of approximation used in the calculation, in general the molecular orbitals (m.o.s) for polyatomic molecules have contributions from the orbital functions of several atoms. They are non-localized, in the sense that they do not just involve each atom and a neighbour, as might be suggested by the traditional structural formulae. An exception is provided by the occurrence of localized m.o.s as non-bonding orbitals, namely for symmetry considerations, as is the case for the non-bonding electron pair of 7T symmetry in the H2O molecule studied in the previous chapter. 

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. 

The molecular orbital model as a linear combination of atomic orbitals introduced in Chapter 4 was extended in Chapter 6 to diatomic molecules and in Chapter 7 to small polyatomic molecules where advantage was taken of symmetry considerations. At the end of Chapter 7, a brief outline was presented of how to proceed quantitatively to apply the theory to any molecule, based on the variational principle and the solution of a secular determinant. In Chapter 9, this basic procedure was applied to molecules whose geometries allow their classification as conjugated tt systems. We now proceed to three additional types of systems, briefly developing firm qualitative or semiquantitative conclusions, once more strongly related to geometric considerations. They are the recently discovered spheroidal carbon cluster molecule, Cgo (ref. 137), the octahedral complexes of transition metals, and the broad class of metals and semi-metals. 

To our knowledge, apart from a brief and elementary outline of a new approach developed by the present authors (5), no simple systematic didactic method for accomplishing the aforementioned goals has been reported (particularly for polyatomic molecules). While excellent introductory descriptions of bonding concepts exist (6,7), no attempts seem to have been made to find a pictorial substitute for a substantial portion of group theory as applied to molecular orbitals or to elaborate in detail the equivalence of the localized and delocalized bonding views on an elementary level. Our approach has been developed and tested in a freshman chemistry course for majors at Iowa State University for a number of years. In the present paper we give and justify a more elaborate discussion of this pictorial method which leads to delocalized and localized MO s for a wide variety of polyatomic molecules. The key concept is that delocalized and localized MO s can be deduced from an appropriate extension of the characteristics of AO s. More specifically, the symmetry and directional characteristics of MO s are obtained from the symmetry and directional characteristics of AO s. 

For molecules, just as for many-electron atoms, one uses equation (2.8) to describe the series. As before, the quantum defect p characterises the whole Rydberg series but now absorbs the influence of the molecular core. In practice, its value still depends largely upon the atomic symmetry of the Rydberg orbital (more detail on its properties for polyatomic molecules will be given in section 3.11). 

The MO model of C2 predicts a doubly bonded molecule, with all electrons paired, but with both highest occupied molecular orhitals (HOMOs) having tt symmetry. C2 is unusual because it has two tt bonds and no cr bond. Although C2 is a rarely encountered allotrope of carbon (carbon is significantly more stable as diamond, graphite, fullerenes and other polyatomic forms described in Chapter 8), the acetylide ion, C2 , is well known, particularly in compounds with alkali metals, alkaline earths, and lanthanides. According to the molecular orbital model, 2 should have a bond order of 3 (configuration TT TT a-g ). This is supported by the similar C—C distances in acetylene and calcium carbide (acetylide) . ... 

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