Spin pairings

The Roothaan equations just described are strictly the equations for a closed-shell Restricted Hartree-Fock (RHF) description only, as illustrated by the orbital energy level diagram shown earlier. To be more specific  

The notation here means that electron 1 occupies a spatial orbital /i with spin up (no bar on top) electron 2 occupies spatial orbital / with spin down (a bar on top), and so on. An RHF description 

Conversely, an unrestricted Uartree-Fock description implies that there are two different sets of spatial molecular orbitals those molecular orbitals, /j , occupied by electrons of spin up (alpha 

Notice that the orbitals are not paired /j does not have the same energy as /jP. An unrestricted wave function like this is a natural way of representing systems with unpaired electrons, such as the doublet shown here or a triplet state  

This last Restricted Hartree-Fock (RHF) state, if allowed to go unrestricted, would probably result in the following UHF state  

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It is useful to represent the polyelectronic wave function of a compound by a valence bond (VB) structure that represents the bonding between the atoms. Frequently, a single VB structure suffices, sometimes it is necessary to use several. We assume for simplicity that a single VB stiucture provides a faithful representation. A common way to write down a VB structure is by the spin-paired determinant, that ensures the compliance with Pauli s principle (It is assumed that there are 2n paired electrons in the system)... 

The system does not change the spin-pairing scheme during the process. In this case, A)j. remains put throughout the reaction, and only the intemucleai distances or angles change. Such transformations are called intraanchor reactions. 

The spin-pairing scheme of the product, B)j., is different from that of the reactant. This happens if at least two pairs of electrons have exchanged partners. In other words, at least three electi ons need to be involved. 

In the transition state region, the spin-pairing change mnst take place. At this nuclear configuration, the electronic wave function may be written as... 

The most stable nuclear configuration of this system is a pair of H2 molecules. There are three possible spin coupling combinations for H4 corresponding to three distinct stable product H2 pairs H1 H2 with H3 H4, H1 H3 with H2 H4, and H1 H4 with H2 H3. Each H atom contributes one electron, the dot diagrams indicate spin pairing. The three combinations are designated as Hfl), HOT), and H(III), respectively. They may be interconverted via square transition states, Figure 2. 

The electronic wave functions of the different spin-paired systems are not necessarily linearly independent. Writing out the VB wave function shows that one of them may be expressed as a linear combination of the other two. Nevertheless, each of them is obviously a separate chemical entity, that can he clearly distinguished from the other two. [This is readily checked by considering a hypothetical system containing four isotopic H atoms (H, D, T, and U). The anchors will be HD - - TU, HT - - DU, and HU -I- DT],... 

Generalizing on [12], we construct a loop by using a sequence of three elementary reactions. It is emphasized that the reactions comprising the loop must be elementary ones There should not be any other spin pairing combination that connects two anchors. This ensures that the loop in question is indeed the smallest possible one. Inspection of the loops depicted in Figure 4 shows that the H3 and H4 systems are entirely analogous. We include the H3 system in order to introduce the coordinates spanning the plane in which the loop lies, and as a prototype of all three-electron systems. 

A chemical reaction takes place on a potential surface that is determined by the solution of the electronic Schrddinger equation. In Section, we defined an anchor by the spin-pairing scheme of the electrons in the system. In the discussion of conical intersections, the only important reactions are those that are accompanied by a change in the spin pairing, that is, interanchor reactions. We limit the following discussion to these class of reactions. 

The results of the derivation (which is reproduced in Appendix A) are summarized in Figure 7. This figure applies to both reactive and resonance stabilized (such as benzene) systems. The compounds A and B are the reactant and product in a pericyclic reaction, or the two equivalent Kekule structures in an aromatic system. The parameter t, is the reaction coordinate in a pericyclic reaction or the coordinate interchanging two Kekule structures in aromatic (and antiaromatic) systems. The avoided crossing model [26-28] predicts that the two eigenfunctions of the two-state system may be fomred by in-phase and out-of-phase combinations of the noninteracting basic states A) and B). State A) differs from B) by the spin-pairing scheme. 

This is an example of a Mobius reaction system—a node along the reaction coordinate is introduced by the placement of a phase inverting orbital. As in the H - - H2 system, a single spin-pair exchange takes place. Thus, the reaction is phase preserving. Mobius reaction systems are quite common when p orbitals (or hybrid orbitals containing p orbitals) participate in the reaction, as further discussed in Section ni.B.2. 

In general, at least three anchors are required as the basis for the loop, since the motion around a point requires two independent coordinates. However, symmetry sometimes requires a greater number of anchors. A well-known case is the Jahn-Teller degeneracy of perfect pentagons, heptagons, and so on, which will be covered in Section V. Another special case arises when the electronic wave function of one of the anchors is an out-of-phase combination of two spin-paired structures. One of the vibrational modes of the stable molecule in this anchor serves as the out-of-phase coordinate, and the loop is constructed of only two anchors (see Fig. 12). 

Here the prototype is H4—as only three spin-pairing arrangements are possible, this system is simple to analyze. It turns out to be very frequently encountered in practice, even in rather complex systems. 

The classic example is the butadiene system, which can rearrange photochemi-cally to either cyclobutene or bicyclobutane. The spin pairing diagrams are shown in Figure 13. The stereochemical properties of this reaction were discussed in Section III (see Fig. 8). A related reaction is the addition of two ethylene derivatives to form cyclobutanes. In this system, there are also three possible spin pairing options. 

Although this reaction appears to involve only two electrons, it was shown by Mulder [57] that in fact two jc and two ct elections are required to account for this system. The three possible spin pairings become clear when it is realized that a pair of carbene radicals are formally involved. Figure 14. In practice, the conical intersection defined by the loop in Figme 14 is high-lying, so that often other conical intersections are more important in ethylene photochemistry. Flydrogen-atom shift products are observed [58]. This topic is further detailed in Section VI. 

Figure 17, Possible spin-pairing schemes for CHDN, involving n electrons only.

In the more general case of nonsymmetric systems, we have shown that one can use reaction coordinates connecting two different spin-paired anchors. These two approaches should be equivalent We shall show that this is indeed the case by discussing some examples. 

A simple VB approach was used in [75] to describe the five structures. Only the lowest energy spin-pairing structures I (B symmehy) of the type (12,34,5 were used (Fig. 21). We consider them as reactant-product pairs and note that the transformation of one structure (e.g., la) to another (e.g., Ib) is a thr ee-electron phase-inverting reaction, with a type-II transition state. As shown in Figure 22, a type-II structure is constructed by an out-of-phase combination of... 

Figure 21. The five equivalent spin-paired structures of CPDR.

The system provides an opportunity to test our method for finding the conical intersection and the stabilized ground-state structures that are formed by the distortion. Recall that we focus on the distinction between spin-paired structures, rather than true minima. A natural choice for anchors are the two C2v stmctures having A2 and B, symmetry shown in Figures 21 and 22 In principle, each set can serve as the anchors. The reaction converting one type-I structirre to another is phase inverting, since it transforms one allyl structure to another (Fig. 12). 

Other spin-pairing forms that may in principle be used to construct a loop are shown in Figure 24. 

Structures III and IV that have different spin-pairing schemes are expected to be higher in energy than type-I because of the strain introduced by the cyclopropyl rings. They may be anchors for secondary conical intersections around the most symmetric one. 

Simple VB theory [75] uses for the basis set five low-lying structures that differ in their spin pairing characteristics, as shown in Figure 26. Similar to the case of the radical, the degenerate 2 f e lowest singlet state of... 

The key to the correct answer is the fact that the conversion of one type-V (or VI) structures to another is a phase-inverting reaction, with a 62 species transition state. This follows from the obseiwation that the two type-V (or VI) stiucture differ by the spin pairing of four electrons. Inspection shows (Fig. 28), that the out-of-phase combination of two A[ structmes is in fact a one,... 

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