The 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. 

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... 

The cyclobutene-butadiene interconversion can serve as an example of the construction of an orbital correlation diagram. For this reaction to occur, the four tt orbitals of butadiene must be converted smoothly into the two t and two a orbitals of the ground state of cyclobutene. The tt orbitals of butadiene are < i, < 2, <1 3, and < 4. For cyclobutene, the four orbitals are ct, tt, ct, and tt, with each of them classified with respect to the symmetry 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 symmetry to be considered are planes of symmetry and rotation axes. An orbital is classified as symmetric, S, if it is unchanged by reflection in a plane of symmetry or by rotation about an axis of symmetry. If the orbital changes sign (phase) at each lobe as a result of the symmetry operation, it is called antisymmetric, A. Proper molecular orbitals must be either symmetric or antisymmetric. If an orbital is neither S nor A, it must be adapted by combination with other orbitals to meet this requirement. 

From this point forward, we will draw the lobes of an orbital with colors (red and blue) to indicate the phase of xp for each region of space. 

The sign of an orbital (or any wave function) is completely arbitrary—it is just a phase factor. 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

Scheme 3c, d illustrates the orbital phase relation when (j) has (j) mix. An orbital energy level according to the theory of two-orbital interaction. The orbital lies below (j>. 

The orbital phase continuity conditions stem from the intrinsic property of electrons. Electrons are fermions, and are described by wavefnnctions antisymmetric (change plus and minus signs) with respect to an interchange of the coordinates of an pair of particles. The antisymmetry principle is a more fnndamental principle than Pauli s exclusion principle. Slater determinants are antisymmetric, which is why the overlap integral between t(a c) given above has a negative... 

Orbital phase continuity for the participation of an electron-donating geminal bond... 

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