Phase, of an orbital

The wave diagrams need further discussion to establish one fine point—the phase of an orbital. 

In this chapter, then, we shall learn what is meant by the phase of an orbital, and what bonding and antibonding orbitals are. We shall see, in a non-mathematical way, what lies behind the Hiickel 4 + 2 rule for aromaticity. And finally, we shall take a brief look at a recent—and absolutely fundamental—development in chemical theory the application of the concept of orbital symmetry to the understanding of organic reactions. 

The same is true for orbitals. A nodal plane, such as that in the 2p orbitals, divides the orbital into two parts with different phases, one where the phase of the wavefunction is positive and one where it is negative. The phases are usually represented by shading—one half is shaded and the other half not. You saw this in the representation of the 2p orbital above. The phase of an orbital is arbitrary, in the sense that it doesn t matter which half you shade. It s also important to note that phase is nothing to do with charge both halves of a filled 2p orbital contain electron density, so both will be negatively charged. 

The cyclobutene-butadiene interconversion can serve as an example of the reasoning employed in construction of an orbital correlation diagram. For this reaction, the four n orbitals of butadiene are converted smoothly into the two n and two a orbitals of the ground state of cyclobutene. The analysis is done as shown in Fig. 11.3. The n orbitals of butadiene are ip2, 3, and ij/. For cyclobutene, the four orbitals are a, iz, a, and n. Each of the orbitals is classified with respect to the symmetiy elements that are maintained in the course of the transformation. The relevant symmetry features depend on the structure of the reacting system. The most common elements of symmetiy to be considered are planes of symmetiy and rotation axes. An orbital is classified as symmetric (5) if it is unchanged by reflection in a plane of symmetiy or by rotation about an axis of symmetiy. If the orbital changes sign (phase) at each lobe as a result of the symmetry operation, it is called antisymmetric (A). Proper MOs must be either symmetric or antisymmetric. If an orbital is not sufficiently symmetric to be either S or A, it must be adapted by eombination with other orbitals to meet this requirement. 

The side-on overlap of two p orbitals forms an MO that is no longer symmetrical about the inter-nuclear axis. If we rotate about this axis, the phase of the orbital changes. The orbital is described as having n symmetry—a n orbital is formed and the electrons in such an orbital make up a K bond. Since there are two mutually perpendicular pairs of p orbitals that can combine in this fashion, there are a pair of degenerate mutually perpendicular n bonding MOs and a pair of degenerate mutually perpendicular n antibonding MOs. 

The formation of the new o-bond(s) must occur by an appropriate overlap of the same phases of these orbitals. If only one a-bond is forming, as in electrocyclic reactions, then only the overlap of the HOMO of the open chain reactant is considered. Such an overlap can occur in one of the two fundamental ways suprafacial mode or antarafacial mode (see Fig. 8.14). If two or more a-bonds are formed during the reaction, as in cycloaddition reactions, then the overlap of the HOMO of one reactant with the LUMO of the second reactant must be considered (see section 8.3). 

The combination of hydrogen Is atomic orbitals to form MOs. The phases of the orbitals are shown by signs inside the boundary surfaces. When the orbitals are added, the matching phases produce constructive interference, which give enhanced electron probability between the nuclei. This results in a bonding molecular orbital. When one orbital is subtracted from the other, destructive interference occurs between the opposite phases, leading to a node between the nuclei. This is an antibonding MO. 

Note that here 8, (j> are coordinates of the electron in the molecule-fixed coordinate system 8, do not specify, nor are they affected by, the orientation of the molecule-fixed coordinate system relative to the laboratory-fixed system. Since the dependence of an orbital angular momentum basis function of the one-electron ( A)) or many-electron ( A)) type can be expressed in terms of a factor e A( + °) or e A( +< °), where phase factor, the effect of crv xz) on a molecular orbital becomes... 

An example will show the application of some of the ideas introduced above. Let us start with the simple two-ccntcr-two-orbital problem described exhaustively in Chapter 2. In the language of perturbation theory these two orbitals experience a degenerate interaction for the case of H2 where the energies of each atomic orbital are the same. The result is an in-phase (bonding) combination and an out-of-phase (antibonding) combination, between the centers A and B. A more complicated example arises when there are two orbitals on A and one on B as when the orbitals of linear H3 are constructed from those of II2 + H (3.10). This is shown in Figure 3.1, where the relative phases of the orbitals have been chosen so that Sij and are positive. 

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