Covalent and ionic states

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 t orbitals in terms of the operators for the localized I a and Isb orbitals  

Here and are the creation operators for electrons of spin a in 1 a and Isb, respectively. The 15a and 15b orbitals are nonorthogonal, but we recall from Section 1.9 that nonorthogonality affects only the anticommutation relations between creation and annihilation operators (1.9.12) - not the anticommutations between two creation operators and between two annihilation operators. 

Using the anticommutation relations for nonorthogonal orbitals, the reader may verify that the normalization constants are given by 

Whereas the covalent state (5.2.39) represents an electronic structure with the electrons located always around different nuclei, the ionic state (5.2.40) represents a situation where the electrons are located around the same nucleus. 

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Next we evaluate the PDLD + EVB surface for the enzymatic reaction using eq. (5.17). The resulting surface is shown in Fig. 5.6. As seen from the ligure, the protein can reduce Aby stabilizing the ionic state more than water. In fact, in the specific case of papain the protein inverts the stabilization of the covalent and ionic states relative to their order in solution. 

There can be resonance between covalent and ionic states. In the molecule H H a complete shift of the electron pair to the left would have the effect to make the left-hand H atom a negative ion, leaving the right-hand one as a positive ion. Next to the state H H there will be two others, H H+ and H+H , which closely resemble the electrostatic model for the H2 molecule. Since the three states are resonating, the states H H+ and H+H will make a contribution to the bonding energy, too in this case, however, their contribution will be relatively small, because the energy of the covalent state H H certainly is much lower than that of the ionic states. H H+ and H+H-. 

These results can be understood using a simple model where the accessed state is a mixed state resulting from the avoided crossing of the covalent and ionic states, the covalent one bearing the oscillator strength. The simulation shown in Figure 4-3 results from the interaction of one weakly bound van der Waals potential and the ionic Hg+-Cl2 curve. 

In bonds between different atoms the dipole moment may have intermediate values owing to the superposition of covalent and ionic states. In such cases the bond is described by the function... 

The results of an EVB/FEP for proton transfer from Cys to His were presented by Warshel (1991), based on a calibration of the protein reaction surface with solution results, which amount to 6 kcal/mol difference (in favor of SH relative to ImH+) due to the pKa values of the two conjugate acids. The protein is found to invert the stabilization of covalent and ionic state relative to their order in solution. This is a result of the stronger solvation in the enzyme, compared to water, and due to the orientation of protein dipoles. 

Fig. 25. Dependence of the classical deflection function for M + XY collisions on the asymptotic energy difference AE = /(M) —, 4(XY) between the covalent and ionic state. (Young et al., 1974.)...

This representation of the ground state wave function IPn is quite analogous to the usual manner of representing resonance between covalent and ionic states. By extending the anology, the excited state of the complex may be represented similarly and the corresponding wave-function is given by... 

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