LUMO HOMO orbitals

Frontier Orbital theory supplies an additional assumption to this calculation. It considers only the interactions between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). These orbitals have the smallest energy separation, leading to a small denominator in the Klopman-Salem equation. The Frontier orbitals are generally diffuse, so the numerator in the equation has large terms. 

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... 

Frontier orbitals (Section 30.1) The highest occupied (HOMO) and lowest unoccupied (LUMO) molecular orbitals. 

Carrier generators in molecular conductors have been associated for a long time to a partial charge transfer between the HOMO (or LUMO) electronic band and other chemical species. These systems are known as two-component molecular conductors. Tetrathiofulvalene derivatives are versatile systems for the formation of molecular organic conductors due to their electron donor capacity by transferring one u-electron from the HOMO orbital, and to their planar shape that promotes their stacking as a consequence of the n-n orbital overlap. The electronic properties of these salts are essentially determined by the packing pattern of the donor molecules which, in turn, depends on the counter-ion. 

Figure 16.1 The chemical hardness of an atom, molecule, or ion is defined to be half. The value of the energy gap between the bonding orbitals (HOMO—highest orbitals occupied by electrons), and the anti-bonding orbitals (LUMO—lowest orbitals unoccupied by electrons). The zero level is the vacumn level, so I is the ionization energy, and A is the electron affinity, (a) For hard molecules the gap is large (b) it is small for soft molecules. The solid circles represent valence electrons. Adapted from Atkins (1991).

Figure 8 Tuning of HOMO (t2g) and LUMO (tt ) orbital energy in various ruthenium polypyridyl...

Fig. 4 Schematic representation of (1) the energy of electron donor (D) or electron acceptor (A) units (regardless as to whether molecules or electrodes), (2) the HOMO and LUMO molecular orbitals, and (3) the energy gap AE between D/A and the molecular orbitals, (a) AE is changed by changing the electronic structure of the molecular bridge, (b) AE is changed by changing the energy levels of the donor or acceptor units...

Fig. 3 Energy diagram for an M-A-M diode showing elastic and inelastic tunneling processes (top). The HOMO (n) and LUMO (71 ) orbital energies and a few vibrational levels are indicated. Applied bias energy (eV) is just sufficient to allow inelastic tunneling with excitation of the first vibrational level, eV = hv. Also shown (bottom) are the I(V) curve, conductance- / curve, and the IETS spectrum that would result from both elastic processes and the first inelastic channel. (Reproduced by permission of the American Chemical Society from [19])...

From this, it follows that the eigenvalues of the LUMO and HOMO orbitals obtained by a local functional are approximately shifted from the exact values by... 

With this simplification in mind, the stabilization energy AE can be given by equation 15, homo and lumo being orbital energies, C A, C A and Cjy, Cp being the relevant orbital coefficients at the carbon centers to which the new bonds are being formed fi AD and f A,D, are the resonance integrals for the overlap at the sites of interaction. 

It is interesting to note that the difference between the potentials of the first oxidation and the first reduction constitutes an experimental measurement of the energy separation of the HOMO/LUMO frontier orbitals. This follows that (as mentioned in the Introduction as well as in Chapter 1, Section 2.4) one assumes that in the oxidation process the electron is removed from the occupied orbital of highest energy (HOMO), whereas in the reduction process the electron is added to the unoccupied orbital of lowest energy (LUMO). In the present case, this separation is equal to AEo = +1.28 - (-1.04) = 2.32V (and hence 2.32 eV). This value is in accord with the value of 2.6 eV theoretically determined for the separation of the HOMO hu and the LUMO tiu. The relatively stable cation [C60]+ has been characterized in solution.16... 

Models of CO adsorption show that top site binding is governed by the CO HOMO (5cr orbital) donating electrons into the metal unoccupied states, with simultaneous back-donation of electrons from the metal s occupied dxz and dyz states into the CO LUMO 2tt orbital). Therefore, it follows that the standard chemisorption model, which considers shifts in the total d-band center, can be inaccurate for systems in which individual molecular orbitals, involved in bonding with the adsorbate, shift differently due to external interactions. In particular, we have shown that the formation of hybrid orbitals with the support material can lead both to downward shifts in the metal d-band center, which do not affect the adsorption of molecules to the metal surface, and to upward shifts that are vitally important. 

The above is based on the calculation of a collective r for the whole molecule. This value changes the HOMO of either the diene or dienophile, as is necessary. This equation is accurate to about 0.5 eV on either side of the known values [15]. The value of ttotal is inserted into the HOMO-LUMO calculation as the parameter r Y), Note that in its pure form, this equation only yields values for the HOMO orbitals. Corrections are used for the calculation of the LUMO values. Table 1 contains examples of the Wiswesser Line Notation and the raw r values used in the computation of orbital energies. 

An electron donating functional group raises the energy of the HOMO orbital of a system about twice as much as it raises the LUMO. 

The conformational orientation adopted by the allenylidene group =C=C= CR R also merits comment. Thus, in half-sandwich complexes, mainly derived from [M(77 -CxHy)L2] (M = Fe, Ru, Os) metal fragments, a marked preference of the allenylidene group to adopt a vertical orientation in which the ipso carbon atoms of the R /R substituents are contained in the molecular plane (pseudo mirror plane bisecting the half-sandwich metal fragment) is observed. Preference for this conformation arises from the dominant metal ( xy-Cp back donation of the metal-HOMO into the allenylidene-LUMO n orbital (see Fig. 13). In contrast to this general trend, an unusual horizontal orientation of the allenylidene group was... 

The electrochemical properties of TNT-EMFs, M3N C2n n > 39) differ from those of the empty cage fullerenes (see Fig. 6) due to the interaction of the metal cluster with the carbon cage and because the structure of these carbon cages are generally different. As a consequence, the reductive processes are electrochemically irreversible but chemically reversible. The oxidative processes occur at lower potentials because the HOMO orbital is mainly localized on the trimetallic nitride clusters and the HOMO-LUMO gaps in solution are smaller [25,58]. The endohedral metallo-fullerenes M C2n show similar behavior but even smaller HOMO-LUMO gaps [59]. 

The fabrication of diodes on silicon substrates was demonstrated using the supramolecular interactions between a 5,10,15,20-tetra(3-fluorophenyl)porphyrin and Ceo fullerene with a rectification ratio of 1,500 (see Fig. 11). The rectifying behavior is explained by theoretical calculations which show that the LUMO orbital is located mainly on the fullerene whereas the HOMO orbital is located on the porphyrin moiety [99]. 

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