The Molecular Orbital Energy

To appreciate the MO picture of chemical bonds, the level of detail given above is not actually required. We will not usually even be concerned with separating the Hamiltonian into its kinetic and potential components. However, we do need to consider the role of the overlap of basis functions in the chemical bond energy. Using the example of H2 with the SALC representation for the bonding orbital lo-g+ (Equation (7.6) with Mg from Equation (7.15)), we can estimate the orbital energy from the expectation value of the Hamiltonian  

The first two terms here involve only a single basis funetion these represent the energy an electron would have if confined to either of the single s functions to simplify the discussion, the value of sueh integrals will be written as Q. The second two integrals involve both 

In the H2 molecule there will, of course, also be electron-electron interactions, which tend to be repulsive but for a large class of molecules, a working, qualitative, understanding of bonding can be obtained by assuming that the nuclear-electron interaction is dominant. 

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All m oleciilar orbitals are com biiiations of the same set of atom ic orbitals they differ only by their LCAO expansion coefficients. HyperC hem computes these coefficients, C p. and the molecular orbital energies by requiring that the ground-state electronic energy beat a minimum. That is, any change in the computed coefficients can only increase the energy. 

FIGURE 3.32 The molecular orbital energy-level diagram for the homonuclear diatomic molecules on the right-hand side of Period 2, specifically 02 and F2. 

The molecular orbital energy-level diagrams of heteronuclear diatomic molecules are much harder to predict qualitatitvely and we have to calculate each one explicitly because the atomic orbitals contribute differently to each one. Figure 3.35 shows the calculated scheme typically found for CO and NO. We can use this diagram to state the electron configuration by using the same procedure as for homonuclear diatomic molecules. 

FIGURE 3.37 The molecular orbital energy-level diagram for methane and the occupation of the orbitals by the eight valence electrons of the atoms. 

FIGURE 3.39 The molecular orbital energy-level diagram for the ir-orbitals of benzene. In the ground state of the molecule, only the net bonding orbitals are occupied. 

FIGURE 3.40 The molecular orbital energy-level diagram for SFf, and the occupation of the orbitals by the 12 valence electrons of the atoms. Note that no antibonding orbitals are occupied and that there is a net bonding interaction even though no d-orbitals are involved. 

Figure 6.6 shows the molecular orbital energy diagrams for a few homonudear diatomic molecules. The stability of the molecules can be estimated from the number of electrons occupying bonding orbitals compared with the number of electrons in the antibonding orbitals. (Antibonding orbitals are sometimes denoted with the subscript, as in 2jt. )... 

Figure 3.7 shows both of the molecular orbital energy diagrams that result for diatomic molecules of second-row elements. 

Draw the molecular orbital energy level diagram for SO. What properties do you predict for this molecule ... 

FIGU RE 16.11 The molecular orbital energy level diagrams for 02 and N2. 

Liu B, Bazan GC (2006) Optimization of the molecular orbital energies of conjugated polymers for optical amplification of fluorescent sensors. J Am Chem Soc 128 1188-1196... 

In Fig. 16 the profile of the molecular orbital energies of propene at different densities is reported. The results have been obtained by an ab initio molecular... 

Here P°jj,)V is a constant (having energy units) characteristic of the bonding interaction between and %v its value is usually determined by forcing the molecular orbital energies obtained from such a qualitative orbital treatment to yield experimentally correct ionization potentials, bond dissociation energies, or electronic transition energies. 

Fig. 3.14 The occupancy of the molecular orbital energy levels across the first row diatomic molecules. (After Cotton and Wilkinson (1980).)...

Fig. 6.16 Molecular Orbital pictures and qnalilalive energies of linear and bent AB molecules. Open and shaded areas represent differences in sign (+ or ) of the wave functions. Changes in shape which increase in-phase overlap lower the molecular orbital energy- From Gimarc.

Corrected values (the molecular orbital energies were decreased by a constant value). 

Snyder et ai.364 have also calculated the molecular orbital energies of anionic forms of thymine by ab initio method. b Mainly localized on 0-7. 

In Figure 6-5 the molecular-orbital energy levels for the linear COa molecule (Figure 6-2) are displayed on the left-hand side. By bending the O—C—O angle a little, we have the energy levels shown on the right-hand side. 

It is the purpose of this paper to develop a molecular orbital theory for square planar metal complexes. Both the spectral and magnetic properties of typical square planar complexes of Ni2+, Pda+, Pts+ and Au8+ will be considered m order to arrive at consistent values for the molecular orbital energies. However, in contrast to previous workers, effort will be concentrated on the assignments of the charge transfer bands of representative square planar complexes. 

Finally, there are tig and tau rb and tt ligand orbital combinations which do not interact with metal orbitals. The molecular orbital energy level scheme expected for the bonding situation described above is shown in Fig. 2. 

The calculations of the molecular orbital energies e (which we employ in the same manner as is usual in the semiempirica] methods for organic molecules) and the evaluation of the coefficients of the atomic orbitals and sets of atomic orbitals in Table I in the final molecular orbital requires the solution of secular determinants (one for each irreducible representation) of the form 6 c =0, where Htj has its usual... 

The molecular orbital energies are in units of /3 at the corners of the polygon. The nonbonding level corresponds to the horizontal dashed line drawn through the center of the circle. 

Exercise 21-13 For a regular pentagon inscribed in a circle with a corner down, use trigonometry to calculate the molecular-orbital energies as in Figure 21-13. 

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