Nodes, molecular orbital

Similar results are obtained for other cyclic tt systems two of these are shown in Figure 13-22. In these diagrams, nodal planes are disposed symmetrically. For example, in cyclo-C r the single-node molecular orbitals bisect the molecule through opposite sides the nodal planes are oriented perpendicularly to each other. The 2-node orbital for this molecule also has perpendicular nodal planes. 

Fig. 8.31. The benzene molecule. The hybridization concept allows us to link the actual geometiy of a molecule with its electronic structure (al. The sp hybrids of the six carbon atoms form the six o CC bonds, and the structure is planar. Each caibon atom thus uses two out of its three s[7 hybrids the third one lying in the same plane protrudes toward a hydrogen atom and forms the a CH bond. In this way, each caibon atom uses its three valence electrons. The fourth one resides on the 2p orbital that is perpendicular to the molecular plane. The six 2p orbitals form six rr molecular orbitals, out of which three are doubly occupied and three are empty (b). The doubly occupied ones are shown in panel (b). The (fio of the lowest energy is an all-in-phase linear combination rf the 2p atomic orbitals (only their upper lobes are shown). The and

Note that the higher the eneigy of a molecular orbital (in our case, they are identical to the Bloch functions), the more nodes molecular orbitals have. Let us take the example of benzene (N — 6, cf. Fig. 9.8, this time for carbon atoms) and consider only those molecular orbitals that can be written as linear combinations of the carbon 2p, where z is the axis orthogonal to the plane of the molecule. The wave vectors f) chosen as corresponding to... 

The Is orbital interacts with the n orbital to form two new molecular orbitals (A and B), but there is no interaction between Is and 7t in this geometry. Accordingly, n becomes C, the third molecular orbital for the transition state. To order the energy of A, B, C, we need only count nodes. Molecular orbital A has no new nodes, but B and C each have one additional node (Fig. 11.64). 

The Extended Hiickel model treats all valence electrons within the spirit of the TT-electron model. Each molecular orbital is written as an LCAO expansion of the valence orbitals, which can be thought of as being Slater-type orbitals (to look ahead to Chapter 9). Slater-type orbitals are very similar to hydrogenic ones except that they do not have radial nodes. Once again we can understand the model best by considering the HF-LCAO equations... 

The lowest-energy tt molecular orbital (denoted lowest energy, 4>2< has one node between nuclei and is also bonding. Above ip] and i j2 in energy are the two antibonding tt VIOs, ii/3 and 1//4. (The asterisks indicate... 

Active Figure 14.2 Four - molecular orbitals in 1,3-butadiene. Note that the number of nodes between nuclei increases as the energy level of the orbital increases. 

Figure 15.3 The six benzene tt molecular orbitals. The bonding orbitals >p2 and t 3 have the same energy and are said to be degenerate, as are the antibonding orbitals tf/4 and 5. The orbitals and 4 have no tt electron density on two carbons because of a node passing through these atoms.

What do molecular orbitals and their nodes have to do with pericyclic reactions The answer is, everything. According to a series of rules formulated in the mid-1960s by JR. B. Woodward and Roald Hoffmann, a pericyclic reaction can take place only if the symmetries of the reactant MOs are the same as the symmetries of the product MOs. In other words, the lobes of reactant MOs must be of the correct algebraic sign for bonding to occur in the transition state leading to product. 

The electron density i/ (0)p at the nucleus primarily originates from the ability of s-electrons to penetrate the nucleus. The core-shell Is and 2s electrons make by far the major contributions. Valence orbitals of p-, d-, or/-character, in contrast, have nodes at r = 0 and cannot contribute to iA(0)p except for minor relativistic contributions of p-electrons. Nevertheless, the isomer shift is found to depend on various chemical parameters, of which the oxidation state as given by the number of valence electrons in p-, or d-, or /-orbitals of the Mossbauer atom is most important. In general, the effect is explained by the contraction of inner 5-orbitals due to shielding of the nuclear potential by the electron charge in the valence shell. In addition to this indirect effect, a direct contribution to the isomer shift arises from valence 5-orbitals due to their participation in the formation of molecular orbitals (MOs). It will be shown in Chap. 5 that the latter issue plays a decisive role. In the following section, an overview of experimental observations will be presented. 

Figure 1.24 How two isolated carbon p orbitals combine to form two n (pi) molecular orbitals. The bonding MO is of lower energy. The higher energy antibonding MO contains an additional node. (Both orbitals have a node in the plane containing the C and H atoms.)...

Figure 13.2 Combination of three atomic p orbitals to form three n molecular orbitals in the allyl radical. The bonding n molecular orbital is formed by the combination of the three p orbitals with lobes of the same sign overlapping above and below the plane of the atoms. The nonbonding n molecular orbital has a node at C2. The antibonding n molecular orbital has two nodes between Cl and C2, and between C2 and C3. The shapes of molecular orbitals for the allyl radical calculated using quantum mechanical principles are shown alongside the schematic orbitals.

A very good example is the conductance of a dianthra[a,c]naphtacene starphenelike molecule presented in Fig. 20, interacting with three metallic nano-pads. The EHMO-NESQC T(E) transmission spectrum per tunnel junction looks like a standard conjugated molecule T(E) with well-identified molecular orbitals and their resonances. For the Fig. 20 case all the T(E) are the same. One can note a small deviation after the LUMO resonance, due to a little asymmetry in the adsorption site between the three branches on the nano-pads [127]. A lot of asymmetric star-like three-molecular-branches system can be constructed, in particular in reference to chemical composition of the central node. This had been analyzed in detail [60]. But in this case, each molecule becomes a peculiar case. The next section presents one application of this central-node case to construct molecule OR and molecule XOR logic gates. 

Fig. 15 A molecular OR gate, whose chemical structure maps the electrical circuit diagram shown in Fig. 20a. Two Aviram-Ratner molecular rectifier chemical groups have been bonded to a central chemical node. This intramolecular circuit with one simple node can be easily designed, because the node Kirchoff node law is valid here. Note that the molecular orbital of each partner can be still identified on the 2 T(E) because of their weak interactions through the CH2 bridge. This is not always the case. The obtained logic surface demonstrates an OR function for well-selected values of the input voltage, but with two logical level 1 outputs which would have to be corrected using an additional output circuit...

The muon and 29Si hyperfine parameters provide compelling evidence in support of the BC model. In the simple molecular-orbital model proposed by Cox and Symons (1986) the muon is located at the center of a Si—Si bond near a node in the unpaired electron spin density, which is... 

---