Linear tt Systems

Allyl complexes (or complexes of substituted allyls) are intermediates in many reactions, some of which take advantage of the capability of this ligand to function in 

The [Mn(CO)5] ion displaces CF from allyl chloride to give an 18-electron product containing ir) -C3H5. The allyl ligand switches to trihapto when a CO is lost, preserving the 18-electron count. 

FIGURE 13-26 Examples of Molecules Containing Linear tt Systems. 

Identify the transition metal in the following 18-electron complexes  

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The molecular orbital of lowest energy has no nodes, the next has one node, the next two nodes, and so on. The nodes are arranged symmetrically with respect to the center of a linear tt system. 

The simplest case of an organic molecule having a linear tt system is ethylene, which has a single tt bond resulting from the interactions of two 2p orbitals on its carbon atoms. Interactions of these p orbitals result in one bonding and one antibonding tt orbital, as shown ... 

One more example should suffice to illustrate this procedure. 1,3-Butadiene may exist in cis or trans forms. For our purposes, we will treat both as linear systems the nodal behavior of the molecular orbitals is the same in each case as in a linear tt system of four atoms. The 2p orbitals of the carbon atoms in the chain may interact in four ways, with the lowest energy tt molecular orbital having all constructive interactions between neighboring p orbitals, and the energy of the other tt orbitals increasing with the number of nodes between the atoms. 

The one-dimensional system has a lot of similarities with the planar n-systems treated using the Hiickel model in Chapter 3. In the case of solids, the Hiickel model becomes the tight-binding model. Let us first assume that we have a onedimensional metal containing a great number of atoms. Each atom provides one loosely bound electron, as in a linear tt-system. Our experience tells us that instead of a linear system, we could use a cyclic system. The wave functions are different just at the end points, and our system is assumed to be so large that the end points do not matter. 

The mapping Xss = tt to, p) represents the steady state zero output submanifold and Uss = 7 eo,p) is the steady state input which makes invariant this steady state zero output submanifold. Condition (48) expresses the fact that this steady state input can be generated, independently of the values of the parameter vector p, by the linear dynamic system... 

In this equation, N is equal to the number of unit cells in the crystal. Note how the function in Eq. 5.27 is the same as that of Eq. 5.19 for cyclic tt molecules, if a new index is defined ask = liij/Na. Bloch sums are simply symmetry-adapted linear combinations of atomic orbitals. However, whereas the exponential term in Eq. 5.19 is the character of the yth irreducible representation of the cychc group to which the molecule belongs, in Eq. 5.27 the exponential term is related to the character of the Mi irreducible representation of the cychc group of infinite order (Albright, 1985). This, in turn, may be replaced with the infinite linear translation group because of the periodic boundary conditions. It turns out that SALCs for any system with translational symmetry are con-stmcted in this same manner. Thus, as with cychc tt systems, there should never be a need to use the projection operators referred to earher to generate a Bloch sum. 

In the LCAO method, the molecular orbitals (MO) 1// . of a tt system are represented by (64) as a linear combination of atomic orbitals (AO) g, (usually p-orbitals perpendicular to the molecular plane). 

The procedure for obtaining a pictorial representation of the orbitals of cyclic tt systems of hydrocarbons is similar to the procedure for the linear systems described above. The smallest such cyclic hydrocarbon is cycZo—C3H3. The lowest energy tt molecular... 

The molecular orbital model as a linear combination of atomic orbitals introduced in Chapter 4 was extended in Chapter 6 to diatomic molecules and in Chapter 7 to small polyatomic molecules where advantage was taken of symmetry considerations. At the end of Chapter 7, a brief outline was presented of how to proceed quantitatively to apply the theory to any molecule, based on the variational principle and the solution of a secular determinant. In Chapter 9, this basic procedure was applied to molecules whose geometries allow their classification as conjugated tt systems. We now proceed to three additional types of systems, briefly developing firm qualitative or semiquantitative conclusions, once more strongly related to geometric considerations. They are the recently discovered spheroidal carbon cluster molecule, Cgo (ref. 137), the octahedral complexes of transition metals, and the broad class of metals and semi-metals. 

Sixteen electrons, so molecule is linear sp hybridization of central N gives two cr bonds, with Ip orbitals on outer nitrogen atoms (four electrons). Lone pairs on both 2s orbitals on outer nitrogen atoms (four atoms), tt system as in Figure 16.21 with eight electrons. Total bond order = 4 bond order 2 per N—N bond. N3 and N3 should be bound. N3 and N3 are paramagnetic. 

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