HOMO orbital energies

Despite the strong MO mixings indicated by the Ae splittings, one may question to what extent the MO energy variations are reflected in measurable physical properties. As described in Section 3.2.4, the interactions of filled NBOs lead to symmetric second-order energy shifts with no net effect on total energy, wavefunc-fion, and other properties. However, the assumptions of Koopmans theorem imply that the vertical ionization potential (IP) is related to HOMO orbital energy by... 

If one is interested in only approximate values of the orbit energies of the TT HOMOs, a reduced procedure may be used based on the truncated secular Equation 104. This shortened procedure reproduces the tt HOMO orbital energies within 0.3 eV. 

The calculated binding energies per unit cell Eb, HOMO orbital energies Stop, and atomic Lowdin charges q for a number of cycbc clusters embedded in an infinite Madelung field, C8.1, C12.1, C24.2, and C32.2, are given in Table 6.6 [321]. 

Quantum chemical descriptors such as atomic charges, HOMO and LUMO energies, HOMO and LUMO orbital energy differences, atom-atom polarizabilities, super-delocalizabilities, molecular polarizabilities, dipole moments, and energies sucb as the beat of formation, ionization potential, electron affinity, and energy of protonation are applicable in QSAR/QSPR studies. A review is given by Karelson et al. [45]. 

The most extensive calculations of the electronic structure of fullerenes so far have been done for Ceo- Representative results for the energy levels of the free Ceo molecule are shown in Fig. 5(a) [60]. Because of the molecular nature of solid C o, the electronic structure for the solid phase is expected to be closely related to that of the free molecule [61]. An LDA calculation for the crystalline phase is shown in Fig. 5(b) for the energy bands derived from the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) for Cgo, and the band gap between the LUMO and HOMO-derived energy bands is shown on the figure. The LDA calculations are one-electron treatments which tend to underestimate the actual bandgap. Nevertheless, such calculations are widely used in the fullerene literature to provide physical insights about many of the physical properties. 

Fig. 10.3. Orbital coefficients for HOMO and next highest n orbital for some substituted benzenes. (From CNDO/2 ealculations. Ortho and meta eoefficients have been averaged in the case of the unsymmetrical methoxy and formyl substituents. Orbital energies are given in atomic units.)...

Ari — [(yfuMO i HOMo) (l LUMO JfHOMo)]/ where and x" re the orbital energies of the reactant and rr-complex. 

Orbital energy is usually the deciding factor. The chemical reactions that we observe are the ones that proceed quickly, and such reactions typically have small energy barriers. Therefore, chemical reactivity should be associated with the donor-acceptor orbital combination that requires the smallest energy input for electron movement. The best combination is typically the one involving the HOMO as the donor orbital and the LUMO as the acceptor orbital. The HOMO and LUMO are collectively referred to as the frontier orbitals , and most chemical reactions involve electron movement between them. 

Electron-donor and electron-acceptor substituents selectively interact with different ring orbitals. Compare the HOMO and LUMO of azobenzene with the corresponding orbitals of the two substituted molecules. Which orbitals show signficant substituent contributions What are the nature of these contributions, bonding or antibonding Try to relate this to the effect which the substituents have on orbital energies and on the HOMO-LUMO gap in azobenzene. 

Repeat your analysis for the LUMO of ethene, 1,3-butadiene, 1,3,5-hexatriene and -carotene, except now focus on each orbital s net antibonding character. (Assume that LUMO energy rises as net antibonding character increases.) What effect does conjugation have on LUMO shape and energy Are your predictions for the HOMO-LUMO energy gap consistent with the experimental data ... 

Lewis acids catalyze Diels-Alder reactions. Do they enhance overlap between diene and dienophile orbitals and/ or do they reduce the HOMO/LUMO energy difference ... 

Consider now the rr-system in benzene. The MO approach will generate linear combinations of the atomic p-orbitals, producing six rr-orbitals delocalized over the whole molecule with four different orbital energies (two sets of degenerate orbitals). Figure 7.3. The stability of benzene can be attributed to the large gap between the HOMO and LUMO orbitals. 

The fact that features in the total electron density are closely related to the shapes of the HOMO and LUMO provides a much better rationale of why FMO theory works as well as it does, than does the perturbation derivation. It should be noted, however, that improvements in the wave function do not necessarily lead to a better performance of the FMO method. Indeed the use of MOs from semi-empirical methods usually works better than data from ab initio wave functions. Furthermore it should be kept in mind that only the HOMO orbital converges to a specific shape and energy as the basis set is... 

These concepts play an important role in the Hard and Soft Acid and Base (HSAB) principle, which states that hard acids prefer to react with hard bases, and vice versa. By means of Koopmann s theorem (Section 3.4) the hardness is related to the HOMO-LUMO energy difference, i.e. a small gap indicates a soft molecule. From second-order perturbation theory it also follows that a small gap between occupied and unoccupied orbitals will give a large contribution to the polarizability (Section 10.6), i.e. softness is a measure of how easily the electron density can be distorted by external fields, for example those generated by another molecule. In terms of the perturbation equation (15.1), a hard-hard interaction is primarily charge controlled, while a soft-soft interaction is orbital controlled. Both FMO and HSAB theories may be considered as being limiting cases of chemical reactivity described by the Fukui ftinction. 

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