Molecular orbital localization

The photoluminescence of lattice oxide ions of transition-metal oxides mixed or supported on conventional carriers has also been reported (160b). The luminescence is shown to occur from oxo complexes (M04)" (M = V, Mo, W, Cr) in which the transition-metal ion exists in a high oxidation state with a d° electronic configuration. Since the d orbitals of the transition-metal ion are not occupied and therefore the d-d transitions impossible, S0 -)-S charge-transfer electronic transitions occur in the oxo complexes upon absorption of light. The result is that an electron is transferred from a filled molecular orbital localized mainly on the O2 anions to a d orbital of the transition-metal ion. This leads to the formation of an excited singlet electronic state S, with two unpaired electrons, in which the total electron spin,... 

Stizza et al. (73,274) have investigated amorphous vanadium phosphates, which are also of interest in relation to a XAS study of the butane-maleic anhydride (V, P)0 catalysts (99a). From the V K edge useful information is obtained about the distortions in the vanadium coordination sphere [molecular cage effect on the pre-edge intensity (312)] and on the vanadium oxidation state. Notably, V4+ is silent to most spectroscopic methods. A mixed V4+-V5+ valence state can be measured from the energy shift of the sharp core exciton at the absorption threshold of the Is level of vanadium due to Is -f 3d derived molecular orbitals localized within the first coordination shell of vanadium ions. 

After cis-addition of a monomer the geometry of an active center is reconstructed with a change of the positions of the vacancy and the alkyl. From the estimation of the contribution of the atomic orbitals to the molecular orbital localized at the Ti—C bond, it was concluded that the metal-alkyl bond is preserved during the reaction route. Cossee regarded the similar reactions to the concerted-type reactions which are characterized by synchronous redistribution of the electron density between dissociated and newly formed bonds which accounts for the low activation energies of these reactions. 

A similar study for H2O (ref. 111), now including both bonding and nonbonding m.o.s, leads to two equivalent molecular orbitals localized on O (which are approximately sp hybrids) and to two equivalent molecular orbitals localized at each 0-H bond. The latter are, approximately, the result of the superposition of the Is orbital of H with an sp hybrid orbital of O (along with a small contribution from the Is orbital of the other H atom). For an sp orbital, the square of the coefficient of 2p is four times that for 2s. 

C atoms with the nucleus of this atom the atomic number of the latter becomes 8 (oxygen isotope). The bonding molecular orbitals localized in the corresponding C-H bonds become non-bonding, localized in O H2C=0. The remaining molecular orbitals (a and tt) do not change significantly, except for the polarity of the CO bond. 

We wish to compare the valence band density of states (DOS) of f.c.c. and h.c.p. metals with and without stacking faults. We therefore adopt a mixture of the f.c.c. and h.c.p. structures as a representative of the stacking fault structure of either of these structures. To calculate the DOS we summed up the squares of the coefficients of molecular orbital wave functions and convoluted the summed squares with the Gaussian of full width 0.5 eV at half maximum. For these DOS calculations we chose the metals Mg, Ti, Co, Cu and Zn. The model clusters employed here for both the f.c.c. and the h.c.p. structures were made of 13 atoms i.e., a central atom and 12 equidistant neighbor atoms. These structures are shown in Fig. 1. We reproduced the typical electronic structures in bulk materials by extracting the molecular orbitals localized only on the central atom from all the molecular orbitals which contributed - those localized on ligand atoms as well as on the central atom. To perform calculations we take the symmetry of the cluster as C3, and the number... 

Close inspection of the solution phase redox peaks for the four species under analysis obtained in a thin layer cell in deaerated non-aqueous media (not shown here) yielded values for the onset of the reduction of the first Co site very similar to one another. Hence, it can only be surmised that the differences in specificity are related to the presence of the peripheral substituent and seemingly unrelated to the redox properties of the metal sites. Evidence that the differences in the specificity between the meso-substituted and non-meso-substituted materials are due to subtle electronic effects was provided by quantum mechanical calculations. These showed that upon addition of the weso-substituents, the electronic charge density associated with the HOMO (highest occupied molecular orbital) localized on the dioxygen moiety is markedly reduced, thereby weakening its Lewis acid character and hence its ability to coordinate a proton (see Figure 3.67). 

It is also interesting to use molecular orbitals localized in core, lone pairs, and bond regions, rather than fully delocalized canonical orbitals. A good choice of the remaining n(n - l)/2 Lagrangian multipliers should lead to those localized orbitals, but such an a priori choice is hardly feasible. Most often, localized orbitals are determined by applying an adequate unitary transformation on previously obtained canonical orbitals ... 

As the next step, using a generalized version of Little s original propxal, we have to take into account that the atomic cores of the TCNQ molecules (containing besides the nuclei and the Is electrons also the valence shell electrons in the PFP approximation) provide a localized polarizable "side chain" electron system which may substantially reduce through its screening the repulsion between the Jtf -electrons. For the description of these electrons we applied LCAO molecular orbitals localized each on... 

We note that much care has to be taken when Koopman s theorem is used to estimate the transfer integrals in asymmetric dimers, as has been extensively discussed elsewhere [58,59], In such instances, part of the electronic splitting can simply arise from the different local environments experienced by the two interacting molecules, which create an offset between their HOMO and LUMO levels prior to their interaction due to polarization and/or electrostatic effects. In order to evaluate the effective couplings, this offset can be accounted for by performing calculations using molecular orbitals localized on the individual units as basis set [59] or by applying an electric field to promote the resonance between the electronic levels, as done by Jortner and coworkers [56], This artifact can also be prevented when... 

This situation results in crystals of the first type the molecular orbitals can be localized around each of the islets. On the other hand, as shown in a simple case in Appendix C, the molecular orbitals localized around each islet are practically identical to those that would be obtained for isolated molecules corresponding to the group of nuclei of each islet thus one can consider the crystal as being formed by the packing of molecules of finite size e.g.,... 

It is known that in a molecular crystal, the entities begin to turn about their own axes, when the temperature increases. This rotation, which occurs long before the melting point, explains the discontinuity in specific heat and the narrowing of the NMR lines (e.g., in benzene at 110 K). This fact is not incompatible with a description of crystal as a giant molecule. In a small molecule (e.g., ethane), the molecular orbital localization does not forbid an internal rotation. The localized molecular orbitals follow the nuclei in their rotation. Likewise, in molecular crystal, when the temperature increases, various localized molecular orbital sets can rotate, without dislocation for the crystal. 

To illustrate molecular orbital localization in a molecular crystal, we shall consider the simple case of two neighboring H2 molecules, whose nuclei are aligned. 

Appendix C An example of molecular orbital localization Appendix D Mineral hardness Improved Mohs scale... 

The mean-field approximation of this chapter offers us the orbital model of the electronic structure of molecules within the RHF approach. In this picture, the electrons are described by the doubly occupied molecular orbitals. Localization of the orbitals gives flie doubly occupied inner shell, lone pair, and bond MOs. The first and second are sitting on atoms, and the third on chemical bonds. Not all atoms are bound with all, but instead die molecule has a pattern of chemical bonds. 

The Mj3 atom cluster is a unique cluster. It is highly symmetric and has only one bulk atom. The highly symmetric environment of its central atom locahzes its valence electrons. The central atom orbitals mainly interact with symmetrical combinations of the atomic orbitals of the atoms of the outer sphere. This hmits their interaction to only a few molecular orbitals localized on the outer atoms. 

For two independent H2 molecules using molecular orbitals localized on the two monomers, (i.e., l, 2, I2,22) we have seen that the lEPA gives the exact result for the correlation energy of the dimer. We now repeat our analysis using a set of equivalent delocalized molecular orbitals. Since orbitals 11 and 12 as well as 2 and 22 are degenerate, we can take an arbitrary linear combination of them and retain the same HF description. In particular... 

The molecular orbitals (localized as well as canonical) can be classified as to the number of nodal surfaces going through the nuclei. A o- bond orbital has no nodal surface at all, a 7T bond orbital has a single nodal surface, a 5 bond orbital has two such surfaces. 

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