The Phase of Orbitals

The orbitals we have just shown are three-dimensional waves. We can understand an important property of these orbitals by analogy to one-dimensional waves. Consider the following one-dimensional waves  

Just as a one-dimensional wave has a phase, so does a three-dimensional wave. We often represent the phase of a qnantum-mechanical orbital with color. For example, the phase of a Is and 2p orbital can be represented as follows  

In these depictions, blue represents positive phase and red represents negative phase. The Is orbital is all one phase, while the 2p orbital exhibits two different phases. The phase of quantum-mechanical orbitals is important in bonding, as we shall see in Chapter 10. 

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The importance of considering the phase of orbitals is that only orbitals of the same phase will overlap, and so result in a bonding situation orbitals of different phase lead to a repulsive, anti-bonding situation. 

Fig. 2.1 (a) The a donation OC metal here, the transition metal d 2 orbital is shown as the acceptor but it could be some mixture of s, and d 2. Note that there is lone-pair a electron density on both 0 and C. That on C is the larger and so has the greater overlap with the empty metal orbital. In this, and all other figures in this chapter, filled orbitals are shaded the phases of orbitals are given explicitly, (b) The % back-donation metal - CO the metal orbital is almost pure d, the CO orbital is an empty antibonding % orbital. Note that for a linear triatomic OCM system there is second, equivalent, interaction to that shown above (it is like that shown but rotated 90 about the OCM axis and so is located above and below the plane of the paper). 

For the butadiene-cyclobutene interconversion, the transition states for conrotatory and disrotatory interconversion are shown below. The array of orbitals represents the basis set orbitals, i.e., the total set of 2p orbitals involved in the reaction process, not the individual MOs. Each of the orbitals is tc in character, and the phase difference is represented by shading. The tilt at C-1 and C-4 as the butadiene system rotates toward the transition state is different for the disrotatory and conrotatory modes. The dashed line represents the a bond that is being broken (or formed). 

The quote is from the third volume of Henri Poincare s New Methods of Celestial Mechanics, and is a description of his discovery of homoclinic orbits (see below) in the restricted three-body problem. It is also one of the earliest recorded formal observations that very complicated behavior may be found even in seemingly simple classical Hamiltonian systems. Although Hamiltonian (or conservative) chaos often involves fractal-like phase-space structures, the fractal character is of an altogether different kind from that arising in dissipative systems. An important common thread in the analysis of motion in either kind of dynamical system, however, is that of the stability of orbits. 

The orbital phase theory includes the importance of orbital symmetry in chanical reactions pointed out by Fukui [11] in 1964 and estabhshed by Woodward and Holiimann [12,13] in 1965 as the stereoselection rule of the pericyclic reactions via cyclic transition states, and the 4n + 2n electron rule for the aromaticity by Hueckel. The pericyclic reactions and the cyclic conjugated molecules have a conunon feature or cychc geometries at the transition states and at the equihbrium structures, respectively. 

Scheme 4 Conditions tor the continuity of orbital phase ( ) electron accepting orbitals are in phase.

Here we derive the conditions of orbital phase for the cyclic orbital interactions. The A B delocalization is expressed by the interaction between the ground configuration C Q and the electron-transferred configuration tBp(A B) (Scheme 3). A pair of electrons occupies each bonding orbital in which is expressed by a single Slater determinant 0 ... 

Similar arguments lead to the prediction that the cross conjugate TMM dication should be more stable than the linear conjugate BD dication. The cyclic orbital interaction is favored by the continuity of orbital phase in the TMM dication, but the orbital phase is discontinuous in the BD dication. 

The SOI concept is akin to the unsymmetrization of orbitals. The only difference is in the sites of the subsidiary interactions, which occur between the non-reacting centers (positions 3 and 4 in Fig. 3a) in SOI and between the reacting and non-reacting centers (sites 2 and 3 in Fig. 3b) for the unsymmetrization of orbitals (Fig. 1). The orbital phase environment around the reaction centers is a general idea... 

On the other hand, when n lies close to Jt-HOMO, the interaction between 7t-HOMO and n is strong. Both orbitals contribute considerably to FMO. The combined orbital, Jt-HOMO - n is a component of FMO. The Jt orbital interacts with the n more strongly than with the jt-HOMO dne to the spatial proximity. The phase of is determined by the relation with n rather than jt-HOMO so as to be out-of-phase with n (viz. in phase with Jt-HOMO). As a result, is the opposite... 

If the radical orbitals are mainly of the p-character (e.g., a p-orbital of carbon in TMM), they interact with each other through the n bond, labeled fl (Fig. 4a). This kind of ir-conjugated diradicals with p-type radical orbitals is classified as the ir-type diradicals. Interestingly, in the language of orbital phase theory, the simplest 1,3-localized diradical, TM (2) also belongs to the ir-type family provided its radical orbitals are dominated by the p-character. The only difference between the TMM (Fig. 4a, left) and TM (Fig. 4a, right) lies in the fact that the radical orbitals of TM interact with each other through the a bond (labeled X) instead of the n bond in TMM. 

The history of orbital phase can be traced back to the theory of chemical bond or bonding and antibonding orbitals by Lennard-Jones in 1929. The second milestone was the discovery of the importance of orbital symmetry in chemical reactions, pointed ont by Fnkni in 1964 (Scheme 3) and established by Woodward and... 

The ability to selectively excite a particular ion (or group of ions) by irradiating the cell with the appropriate radiofrequencies provides a level of flexibility unparalleled in any other mass spectrometer. The amplitude and duration of the applied RF pulse determine the ultimate radius of the ion trajectories. Thus, by simply turning on the appropriate radiofrequency, ions of a single m/z may be ejected from the cyclotron. In this way, a gas-phase separation of analyte from matrix is achieved. At a fixed radius of the ion trajectories the signal is proportional to the number of orbiting ions. Quantitation therefore requires precise RF control. 

Conrotatory movement results in the apposition of orbital lobes with opposite phase—an anti-bonding situation, while disrotatory movement results in the apposition of orbital lobes with the same phase—a bonding situation, leading to formation of the cyclohex-adiene (7) in which the two Me groups are cis. 

The phasing of the molecular orbitals (shown as +/-) is a result of the wavefunctions describing the orbitals. + shows that the wavefunction is positive in a particular region in space, and - shows that the wavefunction is negative. 

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