HOMO and LUMO

In view of this, early quantum mechanical approximations still merit interest, as they can provide quantitative data that can be correlated with observations on chemical reactivity. One of the most successful methods for explaining the course of chemical reactions is frontier molecular orbital (FMO) theory [5]. The course of a chemical reaction is rationali2ed on the basis of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), the frontier orbitals. Both the energy and the orbital coefficients of the HOMO and LUMO of the reactants are taken into account. 

As an example, we shall discuss the Diels-Alder reaction of 2-methoxybuta-l,3-diene with acrylonitrile. Figure 3-7 gives the reaction equation, the correlation diagram of the HOMOs and LUMOs, and the orbital coefficients of the correlated HOMO and LUMO. 

FMO theory requires that a HOMO of one reactant has to be correlated with the LUMO of the other reactant. The decision between the two alternatives - i.e., from which reactant the HOMO should be taken - is made on the basis of which is the smaller energy difference in our case the HOMO of the electron rich diene, 3.1, has to be correlated with the LUMO of the electron-poor dienophile, 3.2. The smaller this HOMO-LUMO gap, the higher the reactivity will be. With the HOMO and LUMO fixed, the orbital coefficients of these two orbitals can explain the regios-electivity of the reaction, which strongly favors the formation of 3.3 over 3.4. 

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]. 

I he electron density distribution of individual molecular orbitals may also be determined and plotted. The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are often of particular interest as these are the orbitals most cimimonly involved in chemical reactions. As an illustration, the HOMO and LUMO for Jonnamide are displayed in Figures 2.12 and 2.13 (colour plate section) as surface pictures. 

These absorptions are ascribed to n-n transitions, that is, transitions of an electron from the highest occupied n molecular orbital (HOMO) to the lowest unoccupied n molecular orbital (LUMO). One can decide which orbitals are the HOMO and LUMO by filling electrons into the molecular energy level diagram from the bottom up, two electrons to each molecular orbital. The number of electrons is the number of sp carbon atoms contributing to the n system of a neuhal polyalkene, two for each double bond. In ethylene, there is only one occupied MO and one unoccupied MO. The occupied orbital in ethylene is p below the energy level represented by ot, and the unoccupied orbital is p above it. The separation between the only possibilities for the HOMO and LUMO is 2.00p. 

Using QMOBAS, TMOBAS, or Mathcad and the method from Computer Project 6-2, calculate the energy separation between the HOMO and LUMO in units of p for all compounds in Table 6-1 and enter the results in Table 6-2. Enter the observed energy of ultraviolet radiation absorbed for each compound in units of cm . The reciprocal wavelength is often used as a spectroscopic unit of energy. 

HOMO and LUMO energies FMO reactivity indices Refractivity Total energy Ionization potential Electron affinity Energy of protonation Orbital populations Frontier orbital densities Superdelocalizabilities... 

HOMO and LUMO, also known as Frontier orbitals, are important in interpreting results of a calculation (see Frontier Molecular Orbitals on page 141). You can use these molecular orbitals to compute the lowest excited electronic singlet state of molecules and the ground states of radicals. 

You can interpret the stereochemistry and rates of many reactions involving soft electrophiles and nucleophiles—in particular pericyclic reactions—in terms of the properties of Frontier orbitals. This applies in particular to pericyclic reactions. Overlap between the HOMO and the LUMO is a governing factor in many reactions. HyperChem can show the forms of orbitals such as HOMO and LUMO in two ways a plot at a slice through the molecule and as values in a log file of the orbital coefficients for each atom. 

Thermodynamic properties such as heats of reaction and heats of formation can be computed mote rehably by ab initio theory than by semiempirical MO methods (55). However, the Hterature of the method appropriate to the study should be carefully checked before a technique is selected. Finally, the role of computer graphics in evaluating quantum mechanical properties should not be overlooked. As seen in Figures 2—6, significant information can be conveyed with stick models or various surfaces with charge properties mapped onto them. Additionally, information about orbitals, such as the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), which ate important sites of reactivity in electrophilic and nucleophilic reactions, can be plotted readily. Figure 7 shows representations of the HOMO and LUMO, respectively, for the antiulcer dmg Zantac. 

Electron donor molecules are oxidized in solution easily. Eor example, for TTE is 0.33V vs SCE in acetonitrile. Similarly, electron acceptors such as TCNQ are reduced easily. TCNQ exhibits a reduction wave at — 0.06V vs SCE in acetonitrile. The redox potentials can be adjusted by derivatizing the donor and acceptor molecules, and this tuning of HOMO and LUMO levels can be used to tailor charge-transfer and conductivity properties of the material. Knowledge of HOMO and LUMO levels can also be used to choose materials for efficient charge injection from metallic electrodes. 

Fig. 11. Mechanism of electron transfer from an excited dye molecule to a silver haUde crystal where HOMO and LUMO are highest occupied and lowest...

The ring opening of 2//-azirines to yield vinylnitrenes on thermolysis, or nitrile ylides on photolysis, also leads to pyrrole formation (B-82MI30301). Some examples proceeding via nitrile ylides are shown in Scheme 92. The consequences of attempts to carry out such reactions in an intramolecular fashion depend not only upon the spatial relationship of the double bond and the nitrile ylide, but also upon the substituents of the azirine moiety since these can determine whether the resulting ylide is linear or bent. The HOMO and second LUMO of a bent nitrile ylide bear a strong resemblance to the HOMO and LUMO of a singlet carbene so that 1,1-cycloadditions occur to carbon-carbon double bonds rather than the 1,3-cycloadditions needed for pyrrole formation. The examples in Scheme 93 provide an indication of the sensitivity of these reactions to structural variations. 

Calculations for Ceo in the LDA approximation [62, 60] yield a narrow band (- 0.4 0.6 eV bandwidth) solid, with a HOMO-LUMO-derived direct band gap of - 1.5 eV at the X point of the fee Brillouin zone. The narrow energy bands and the molecular nature of the electronic structure of fullerenes are indicative of a highly correlated electron system. Since the HOMO and LUMO levels both have the same odd parity, electric dipole transitions between these levels are symmetry forbidden in the free Ceo moleeule. In the crystalline solid, transitions between the direct bandgap states at the T and X points in the cubic Brillouin zone arc also forbidden, but are allowed at the lower symmetry points in the Brillouin zone. The allowed electric dipole... 

The same conclusions are drawn by analysis of the frontier orbitals involved in cycloadditions. For the most common case of the Diels-Alder reaction, which involves dienophiles with electron-attracting substituents, the frontier orbitals are l/2 of the diene (which is the HOMO) and n of the dienophile (which is the LUMO). Reaction occurs by interaction of the HOMO and LUMO, which can be seen from the illustration below to be allowed. 

Fig. 11.14. Orbital coefficients for HOMO and LUMO n MOs of some common 1,3-dipoles. [From K. N. Houk, J. Sims, R. E. Duke, Jr., R. W. Strozier, and J. K. George, J. Am. Chem. Soc. 95 7287 (1973).]...

Compare the dipole moment of ethylene and formaldehyde. Then compare the HOMO and LUMO in both molecules. Use the data from the formaldehyde example earlier in this chapter. 

Here are the HOMO and LUMO for ethylene (some non-significant lines have been removed from the output) ... 

In ethylene, both the HOMO and LUMO are formed primarily from p orbitals from the two carbons. The carbons lie in the YZ-plane, and so the p,j orbitals lie above and below the C-C bond. In the HOMO, the orbitals have like signs, and so they combine to form a bonding n molecular orbital. In contrast, in the LUMO, they have opposite signs, indicating that they combine to form an antibonding Tt molecular orbital. 

In order to specify the proper electronic state, ozone calculations should be performed as unrestricted calculations, and the keyword Gue s=Mix should always be included. This keyword tells the program to mix the HOMO and LUMO within the wavefunction in an effort to destroy a-P and spatial symmetries, and it is often useful in producing a UHF wavefunction for a singlet system. Running a UHF GuesssMix Stable calculation confirms that the resulting wavefunction is stable, and it predicts the same energy (-224.34143 hartrees) as the previous Stable=Opt calculations. 

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. 

Although the most important, the electrostatic potential is not only the quantity which when mapped onto an electron density surface may provide useful chemical information. Maps of certain key molecular orbitals, in particular, the HOMO and LUMO, may also lead to informative models. Consider, for example, a map of the (absolute) value of the lowest-unoccupied molecular orbital (LUMO) in cyclohexanone, two views of which are shown below. 

Solvent can alter a dye s color. One inteipretation is that light absorption moves an electron from one part of the molecule to another with a resulting change in overall polarity. Examine the HOMO and LUMO of azobenzene, 4-hydroxyazobenzene and 4-amino-4 -nitroazobenzene. Which, if any, of the molecules would be expected to change color in different solvents How does excitation change the polarity of these molecules Explain how you reached your conclusions. 

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. 

Next, examine the highest-occupied and lowest-unoccupied molecular orbitals (HOMO and LUMO) of dichlorocarbene. Were the reaction a nucleophilic addition , how would you expect CCI2 to approach propene Were the reaction an electrophilic addition , how would you expect CCI2 to approach propene Which inteqDretation is more consistent with the geometry of the transition state ... 

HOMO and LUMO of dichlorocarbene characterize the molecule s electrophilic and nucleophilic behavior, respectively. 

HOMO and LUMO of 1,3,5-hexatriene show origin and destination of excited electron. 

Examine and describe both the highest-occupied and lowest-occupied molecular orbitals (HOMO and LUMO, respectively) oiground state acetone. On which atom(s) is the HOMO primarily concentrated Is it in the o system or in the 7t system Repeat your analysis for the LUMO. 

Woodward and Hoffmann pointed out that the Diels-Alder reaction involved bonding overlap of the highest-occupied molecular orbital (HOMO) on the diene and the lowest-unoccupied molecular orbital (LUMO) on the dienophile. Display the HOMO for 2-methoxybutadiene. Where is it localized Display the LUMO for acrylonitrile. Where is it localized Orient the two fragments such that the HOMO and LUMO best overlap (A clearer picture is provided by examining-the HOMO map for 2-methoxybutadiene and the LUMO map for acrylonitrile.) Which product should result ... 

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