Molecular orbital homonuclear diatomic case

Usually the electronic structure of diatomic molecules is discussed in terms of the canonical molecular orbitals. In the case of homonuclear diatomics formed from atoms of the second period, these are the symmetry orbitals 1 og, 1 ou, 2ag,... 

The limitation of the above analysis to the case of homonuclear diatomic molecules was made by imposing the relation Haa = Hbb> as in this case the two nuclei are identical. More generally, Haa and for heteronuclear diatomic molecules Eq. (134) cannot be simplified (see problem 25). However, the polarity of the bond can be estimated in this case. The reader is referred to specialized texts on molecular orbital theory for a development of this application. 

There exists no uniformity as regards the relation between localized orbitals and canonical orbitals. For example, if one considers an atom with two electrons in a (Is) atomic orbital and two electrons in a (2s) atomic orbital, then one finds that the localized atomic orbitals are rather close to the canonical atomic orbitals, which indicates that the canonical orbitals themselves are already highly, though not maximally, localized.18) (In this case, localization essentially diminishes the (Is) character of the (2s) orbital.) The opposite situation is found, on the other hand, if one considers the two inner shells in a homonuclear diatomic molecule. Here, the canonical orbitals are the molecular orbitals (lo ) and (1 ou), i.e. the bonding and the antibonding combinations of the (Is) orbitals from the two atoms, which are completely delocalized. In contrast, the localization procedure yields two localized orbitals which are essentially the inner shell orbital on the first atom and that on the second atom.19 It is thus apparent that the canonical orbitals may be identical with the localized orbitals, that they may be close to the localized orbitals, that they may be identical with the completely delocalized orbitals, or that they may be intermediate in character. 

The molecular orbital diagram for the nitrogen monoxide molecule is shown in Figure 4.6. The orbitals are produced from the same pairs of atomic orbitals as in the cases of the homonuclear diatomic molecules of Section 4.2. 

Heteronuclear diatomic molecules are naturally somewhat more complicated than the homonuclear comprehensive comparisons with homonuclear molecules were given by Mulliken [15]. The atomic orbital coefficients in the molecular orbitals ofheteronu-clear diatomic molecules are no longer determined by symmetry alone, and the electrons in the molecular orbitals may be shared equally between atoms, or may be almost localised on one atom. The molecular orbitals can still be classified as a or n, but in the absence of a centre-of-symmetry the g/u classification naturally disappears. Some heteronuclear molecules contain atoms which are sufficiently similar that the molecular orbitals resemble those shown in figure 6.7. In many other cases, however, the atoms are very different. This is particularly the case for hydride systems, like the HC1 molecule,... 

The simple molecular orbital theory of bonding in homonuclear diatomic molecules can be used to estimate the electron affinities of clusters. In these cases, there can be different geometries. The Cn clusters have been studied most extensively. In the case of the triatomic molecules, there are now two distances and one angle that... 

A buildup principle analogous to that for atoms exists for molecules. The order of filling molecular orbitals from the valence shells in the case of homonuclear diatomic molecules, where x is the bond axis, is as follows ... 

Let us return to the coefficients c and c2 of Equation 1.1, which are a measure of the contribution which each atomic orbital is making to the molecular orbital (equal in this case). When there are electrons in the orbital, the squares of the c- values are a measure of the electron population in the neighbourhood of the atom in question. Thus in each orbital the sum of the squares of all the c-values must equal one, since only one electron in each spin state can be in the orbital. Since lr 11 must equal lc2l in a homonuclear diatomic like H2, we have defined what the values of c and c2 in the bonding orbital must be, namely 1/ /2 = 0.707 ... 

Let us first recall the simple case of a homonuclear diatomic molecule B-B, each atom B with one atomic orbital of energy a and with a coupling matrix element p and overlap S. The molecular orbital energies, corresponding to bonding and antibonding combinations = (p (p )l / 2 2S), are ... 

When the atoms in a heteronuclear diatomic molecule are close to one anoth in a row of the periodic table, the molecular orbitals have the same relative order of energies as those for homonuclear diatomic molecules. In this case you can obtain the electron configurations in the same way, as the next example illustrates. 

There is a different way to consider this coupling, and it becomes useful for polyatomic molecules Use symmetry when possible. Each molecular orbital can be given a symmetry label that is one of the irreducible representations of the molecular point group. In the case of the homonuclear diatomic, the point group is As we might... 

NIR studies have been applied also to heteronuclear diatomics like CO and NO [95-97]. On alkali and alkaline earth cations the results are essentially similar to those obtained with homonuclear probes. In the case of CO and NO strengthening of the internal bonding is observed, caused by the withdrawal of electron density from slightly antibonding molecular orbitals in correspondence with quantum chemical calculations [70,72]. In spite of the uncertain Do determination on cobalt and copper ion-exchanged zeolites A the CO dissociation energy turns out to be decreased, which may be understood by backdonation of electronic charge into the carbonyl tt orbital [97,98]. 

In contrast to homonuclear diatomics, the molecular orbitals of heteronuclear diatomics cannot be constructed from a linear combination of two identical atomic orbitals. The atomic orbitals may differ in energy and type and hence do not contribute equally to the molecular orbitals. In the general case,... 

In the case of ethylene the a framework is formed by the carbon sp -orbitals and the rr-bond is formed by the sideways overlap of the remaining two p-orbitals. The two 7r-orbitals have the same symmetry as the ir 2p and 7T 2p orbitals of a homonuclear diatomic molecule (Fig. 1.6), and the sequence of energy levels of these two orbitals is the same (Fig. 1.7). We need to know how such information may be deduced for ethylene and larger conjugated hydrocarbons. In most cases the information required does not provide a searching test of a molecular orbital approximation. Indeed for 7r-orbitals the information can usually be provided by the simple Huckel (1931) molecular orbital method (HMO) which uses the linear combination of atomic orbitals (LCAO), or even by the free electron model (FEM). These methods and the results they give are outlined in the remainder of this chapter. 

---