Bonding and antibonding configurations

We first consider the descriptions provided by the closed-shell CSFs of symmetry. From Table 5.1, we obtain the following expressions for the energies of the bonding and antibonding configurations  

These expressions may be given a simple physical interpretation. Thus, the first term in (5.2.20) represents the kinetic energies of the two electrons in orbital 4 as well as the interactions of these electrons with the nuclei. The second term represents the Coulomb repulsion between the charge distributions of the two electrons. Similar considerations apply to (5.2.21). 

State Kinetic Attraction Electron repulsion Nuc. rep. Total 

As discussed in Section 2.7.3, the two-electron densities represent the probability of simultaneously locating two electrons at two given positions in the molecule. According to (5.2.17), the two-electron density functions of the bonding and antibonding configurations are given by 

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Applying to the vacuum state the two-body creation operators introduced in Section 2.3.3, we obtain two closed-shell CSFs of symmetry (the bonding and antibonding configurations)... 

In Figure 5.4, we have plotted the density functions (5.2.16) and (5.2.17) on the molecular axis for the bonding and antibonding configurations. The one-electron densities have been plotted along the z axis - that is, as functions of (0,0, z). Likewise, the two-electron densities have been... 

We have seen that the closed-shell bonding and antibonding configurations each provide an uncorrelated description of the electronic system but that a superposition of these configurations introduces correlation. In the ground state, the electrons tend to be located around opposite nuclei, whereas, in the excited state, there is a tendency for the electrons to be located around the same nucleus. Further insight into the correlation problem may be obtained by isolating the pure covalent and ionic states, where the electrons are either always located around opposite nuclei or always located around the same nucleus. 

To determine the covalent and ionic states, we expand the bonding and antibonding configurations in terms of the localized AOs Isa and Isb rather than in terms of the delocalized MOs lo and l 7 . We therefore write the creation operators for the la and 1[Pg.154]

At equilibrium, therefore, the covalent and ionic states are both dominated by the bonding configuration. In Section 5.2.9, we shall see that the bonding and antibonding configurations contribute equally to the covalent and ionic states at infinite intemuclear separation. 

In general, a truly uncorrelated many-particle state is always represented by a product of one-particle functions. Conversely, any superposition of such products represents a stale where the motion of the particles is correlated. Nevertheless, a Slater determinant - which leiniesents an antisymmetric superposition of spin-orbital products - may in some cases represent a tmly uncorrelated electronic state. Thus, in the closed-shell bonding and antibonding configurations of the hydrogen molecule, the Pauli principle is satisfied by an antisymmetrization of the spin part of the wave function... 

Let us also consider the energies of the bonding and antibonding configurations at infinite separation. From (5.2.59) and (5.2.60), we obtain... 

Figure 3.2 Symmetric and antisymmetric wave fnnctions, describing the two electrons in a pair of atoms and leading to bonding and antibonding electron configurations.

The electron configurations are [Ar] 3d5 4s1 for Cr and [Ar] 3d10 4s2 for Zn. Assume that the 3d and 4s bands overlap. The composite band, which can accommodate 12 valence electrons per metal atom, will be half-filled for Cr and completely filled for Zn. The melting points will depend on the occupancy of the bonding and antibonding MOs. 

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