Three-orbital model

The relevant Hamiltonian for the gas-phase solute molecules can be treated by the same three-orbitals four-electron model used in Chapter 2. Since the energy of 3 is much higher than that of , and d>2 (see Table 2.4), we represent the system by its two lowest energy resonance structures, using now the notation fa and fa as is done in eq. (2.40). The energies of these two effective configurations are now written as... 

The sp hybrid orbitals of carbon were considered as a mix of the 2s orbital with three 2p orbitals. To provide a model for ethylene, we now need to consider hybrid orbitals that are a mix of the 2s orbital with two 2p orbitals, giving three equivalent sp orhitals. In this case, we use just three orbitals to create three new hybrid orbitals. Accordingly, we find that the energy level associated with an sp orbital will be below that of the sp orbital this time, we have mixed just two high-energy p orbitals with the lower energy orbital (Figure 2.13). The... 

Fukui and Fujimoto (1966a) suggest that an overlap or bond-order criterion is another way in which secondary interactions may be examined. If benzene is assumed to be a crude model for the transition state of the [3,3] sigmatropic reaction, it is easy to show that P for the three orbitals is 2(p z +pz + p ) = — 0 33. The negative bond order indicates repulsion. Therefore, the concerted [3,3] sigmatropic reaction prefers the chair transition state in which atoms 2 and 5 are as far apart as possible. 

In 1913, the Danish physicist Niels Bohr developed a model of the atom that explained the hydrogen emission spectrum. In Bohr s model, electrons orbit the nucleus in the same way that Earth orbits the Sun, as shown in Figure D.2. The following three points of Bohr s theory help to explain hydrogen s emission spectrum. 

What model does provide a valence electron/composition connection for [Ru6(CO)i8]2 Let s try this. Limit the metal to three orbitals for ligand binding and three orbitals for cluster bonding. Thus, we force it to act like a six rather... 

Perhaps you are thinking that the metal atoms in Fe6C(CO)i6]2 are now using more than three orbitals for cluster bonding. How else can they interact effectively with the interstitial atom But notice that the interaction described in Figure 3.8 is no different in principle than the addition of four H atoms to a square pyramidal B5H5 cluster or the addition of 4 H+ to [B5H5I4 as in the one electron MO method the electrons are added last. For the iron cluster, then, the equivalent model would be the insertion of a C4+ ion into the center of a [Fe6(CO)i6]6- cluster. No additional metal orbitals are needed. 

In the vast majority of cases in which six coordination is observed, the bonding can be viewed as arising from the interaction of all three cr -orbitals with a halide anion, i.e., all three in S. Because the three orbitals are all trans to the primary E-X bonds, such a situation leads naturally to octahedral coordination. Moreover, in cases in which the primary and secondary bonds are the same length, i.e., where A = 0 and a three-center, four-electron bonding model is appropriate, a regular octahedron is the result. Such a structure is clearly at odds with simple VSEPR theory, which is predicated on the lone pair(s) occupying specific stereochemical sites, but stereochemical inactivity of the lone pair tends to be the rule rather than the exception in six-coordinate, seven-electron pair systems Ng and Zuckerman (102) have reviewed this topic for p-block compounds in general. 

Most clusters, however, caimot be described adequately in terms of two-center, two-electron bonds because the coimectivity of the vertices exceeds the number of valence orbitals that are available for bonding. Early efforts to rationalize such systems, such as Lipscomb s styx approach and Kettle s Topological Equivalent Orbital Method, are described by Mingos and Johnston. In these more difi cult cases, the simple valence-bond picture is inappropriate examples are deltahedral clusters composed of B-H vertices or conical M(CO)3 fragments, both of which usually have only three orbitals available for skeletal bonding. Theoretical models for describing these systems will be discussed in the next section. 

The most effective approach to interpreting the barriers for a wide range of compounds lies in the consideration of the relative interactions within the Dewar, Chatt, Ducanson model of metal alkene bonding. An extended Hiickel MO approach has explored the interactions of the valence orbitals and examined the important interactions. A comprehensive extended Hiickel MO treatment of ethylene bonding and rotational barriers by Albright, Hoffmann et a/. presents an excellent analysis and the reader is referred to their paper for further discussiou. We have found that the following approach, which considers oifly three orbitals on the metal and the n and y orbitals of the alkene, provides the essential elements for understanding the barriers to rotation. Naturally, steric effects and secondary interactions with other orbitals modulate these primary iuteractious. 

Abstract In a solid with orbital degree of freedom, an orbital configuration does not minimize simultaneously bond energies in equivalent directions. This is a kind of frustration effect which exists intrinsically in orbital degenerate system. We review in this paper the intrinsic orbital frustration effects in Mott insulating systems. We introduce recent our theoretical studies in three orbital models, i.e. the cubic lattice orbital model, the two-dimensional orbital compass model and the honeycomb lattice orbital model. We show numerical results obtained by the Monte-Carlo simulations in finite size systems, and introduce some non-trivial orbital states due to the orbital frustration effect. 

FIGURE 6.47 (a) Ball-and-stick model (bottom) and molecular orbitals for bent triatomic molecules. The central atom has three sp hybrid orbitals (not shown) that would lie In the plane of the molecule. From the three orbitals perpendicular to this plane, three 77 orbitals can be constructed. (b) Correlation diagram for the 77 orbitals. 

Figure 3.3 describes the shapes of the three possible p orbitals within a given level. Each has the same shape, and that shape appears much like a dumbbell these three orbitals differ only in the direction they extend into space. Imaginary coordinates x, y, and z are superimposed on these models to emphasize this fact. These three orbitals, termed p py, and p, may coexist in a single atom. Their arrangement is shown in Figure 3.4. 

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