Baer M., Billing G.D. (eds.) - The role of degenerate states in chemistry (Adv.Chem.Phys. special issue, Wiley, 2002)

on a single reaction coordinate, may be termed the single coordinate model. In the second category, the coupling between the excited and the ground state is of central importance. This is the two coordinate model, in which concerted motion along two coordinates is essential for understanding the photochemical process. One coordinate carries the reactant to the product, the other couples the excited state with the ground state. The situation is identical to the case of the Jahn–Teller effect, where motions along two coordinates are equally important in removing the degeneracy.

A corollary of the single coordinate model is the idea that the photochemical process can be analyzed by a correlation diagram based on a single coordinate. Both MO [19,52] and VB [30,31,39] models have been proposed, and were widely used. The MO-based orbital symmetry conservation rules offered by Woodward an Hoffmann (WH) [52], gained special popularity. These rules, though not based on a solid theoretical basis, were of paramount importance in systematizing organic reactions. They are easy to apply and work very well for thermal reactions [53,54]. For photochemical reactions, the correlation between an occupied MO and an unoccupied one (in the ground state) is considered. In that sense, the WH rules assume that the reaction proceeds on the excited-state surface and results in an electronically excited product. It is usually accepted, that the rules for photochemical reactions and thermal ones are mutually exclusive: reactions that are ‘‘forbidden’’ thermally, are ‘‘allowed’’ photochemically.

The VB model considers the entire system, rather than molecular orbitals; At its heart is the passage of the system from one bonding mode to another [17,18,31,32,45,47] (in our nomenclature, change in spin pairing). This model was considered by some authors to be superior to the MO approach for thermal reactions, as it accounts naturally for the energy barrier found in thermal reactions [44]. As pointed out by Oosterhoff and co-workers [47] for the example of butadiene ring closure, the WH approach considers an antisymmetric excited state, while their model shows that a symmetric excited state is important. They show that for this state, the disrotatory ring-closure mode has an energy minimum along the reaction coordinate on the excited-state potential surface. The system will therefore move along this route, rather than the conrotatory one (preferred on the ground state. In both the WH and the Oosterhoff models, the reaction is taking place on the excited-state surface, and the return to the ground state is assumed to occur somehow (in Oosterhoff’s theory, near the minimum). Thus, both models, based on essentially two-state paradigm, do not consider explicitly the equivalent importance of two coordinates.

The concept of biradicals and biradicaloids was often used in attempts to account for the mechanism of photochemical reactions [2,20,129–131]. A biradical (or diradical) may be defined as [132] an even-electron molecule that has one bond less than the number permitted by the standard rules of valence.

Biradical or biradicaloid minima are introduced in simple MO theory in cases where a pair of nearly degenerate approximately nonbonding orbitals is occupied with a total of only two electrons in the ground state [2, p. 187; 30, 133]. The approximate degeneracy of the two MOs can usually occur only at certain nuclear geometries, referred to as biradicaloid geometries. In both VB and MO theories, the spacing between the ground and excited state surfaces of biradicals are small, so that these species are considered as efficient funnels for radiationless transitions [2,133,134]. Salem and co-workers used biradical structures to construct correlation diagrams to analyze photochemical reactions. Their method differs from the WH approach in two respects [30, p. 136]. First, they use resonance structures, rather than the WH orbital configurations; Second, they chose as a symmetry element a reaction plane that does not cut through bonds being made. In distinction with Oosterhoff, Salem explicitly considered the direct passage from an excited reactant to a ground state product. His correlation diagrams are essentially represented by Figure 5—they are plotted against a phase preserving coordinate. As discussed there, it corresponds to a cut across the conical intersection along the phase preserving coordinate. Likewise, the WH and Oosterhoff correlation diagrams are a cut across the conical intersection along a phase-inverting coordinate (see Fig. 6). The conical intersection model uses both for the full account of photochemical reactions.

Biradicals were central building blocks in the construction of MO-based correlation diagrams, in which orbital symmetry was assumed to be preserved (as in the WH treatment) [129,135,136]. For example, the H4 system was studied by Gerhartz et al. [34] in some detail. The biradical approach leads to a 3 3 CI treatment, which allows the construction of correlation diagrams along certain reaction coordinates. The perfect tetrahedral geometry was recognized as a touching ‘‘point’’ of S1 and S0. It was also admitted that neither the 3 3 CI approach, nor the simple VB one are satisfactory: In MO, more than three configurations are required, in VB, ionic terms in addition to covalent ones [129].

The biradical model suggests a connection between the single coordinate model, emphasizing reactions on a single energy surface, and the two-coordinate model, in which the coupling between states is important.

These couplings between the states were introduced as perturbations that mix the ground and excited state (the g and d parameters [2,136]). For example, the electronegativity difference between orbitals is parametrized by d, their energy difference, as a function of the nuclear configuration. In biradicals such as twisted ethylene, even a weak polarizing perturbation d causes an essentially complete uncoupling of the two zwitterionic configurations (in VB language). This leads to a highly polar character of S1, near the perpendicular configuration, a phenomenon known as ‘‘sudden polarization’’ [82,137].

By using these parameters, the single coordinate scheme becomes, in practice, a two-coordinate one, the coordinate along which the d parameter

changes serving as the second one. Comparison with the spin-pairing model shows that the biradicaloid model reaches the same conclusions, using essentially perturbation methods (via the parameters g and d). For example, in the case of butadiene photocyclization, the g perturbation leads to cyclobutene, and the d perturbation, to bicyclobutane. On the other hand, no perturbations are required in the spin-pairing model. It postulates two independent nuclear coordinates from the outset, and finds the asymmetric conical intersection in a conceptually simple way.

Robb and Bernardi and their co-workers [44] used the Evans VB-based model as the starting point for their analysis. Their model is based on the coordinate leading from the reactant to the product (in our notation—the phaseinverting coordinate). However, they recognized the importance of the coupling between the ground and excited state, and introduced specifically a coordinate that leads to the minimization of the energy gap between the two (the gradient difference coordinate [Eq. (25)]. This coordinate was found computationally, but the physical meaning assigned to it was not specified.

The method outlined in this chapter continues the trend proposed by [44], and uses the phase-change concept to provide the physical meaning. It acknowledges the fact that two spin-paired species only are not sufficient to account for a photochemical reaction. A third one is required. The third anchor provides a simple interpretation to the second coordinate, used in the Bernardi–Robb model, as well as in the biradicaloid one. It leads to the same qualitative results as the orbital correlation method and the biradical hypotheses, but it does not introduce ad hoc assumptions. The conical intersection is not a biradical in the simple sense, but the fact that electrons can be re-paired in its neighborhood in several different ways, makes it appear as if the electrons are independent. However, in general these ‘‘biradicals’’ are not intermediates, and cannot be trapped or observed directly by spectroscopic methods. This accounts for the general failure to isolate many proposed ‘‘intermediate’’ biradicals. The model also explains naturally how apparently photochemically ‘‘forbidden’’ reaction products are obtained in practice. The results obtained are based on the phase change rule, which is a direct consequence of Pauli’s principle. The method can be used to make simple qualitative applications, enabling the prediction of photochemical products and their stereospecificity. On the other hand, it can be used to guide high-level quantum chemical calculations on fairly complex systems. It can therefore be used by both experimentalists and computational chemists.

APPENDIX A. PHASE INVERTING REACTIONS

I.THE MODEL (see 28)

Consider a molecule consisting of more than three atoms, with an even number of valence electrons, 2n ðn 2Þ. The basic assumption of the model is that the

system will tend to form as many valence electron pairs between the atoms as possible. Each pair of electrons that are spin paired, form a bond between the atoms from which they originated. If the pair happens to be situated on a single atom, the pair is counted as a nonbonding pair (as in the lone pair of ammonia).

In the reaction, compound A transforms to B; the total number of electron pairs is preserved, but at least four electrons are assumed to change spin partners.

The task is now to calculate the structure and energy of the system in the transition state between A and B. Its wave function is assumed to be constructed from a linear combination of the two. It is convenient to use VB terminology for this purpose. Let the wave function of A be denoted by a VB function jAi and that of B by jBi.

The jAi wave functions is written in the standard fashion:

With fA containing a normalization factor and all permutations over the atomic orbital wave functions i ð1 ¼ 1; 2; . . . 2nÞ. Likewise, if all electron pairs were exchanged in a cyclic manner, the product wave function, fB, has the form:

jBi ¼ ð12n 12nÞð2n 12n 2 2n 12n 2Þ ð32 32Þ ðA:3Þ

If the exchange was done in a different pattern, an corresponding expression would result. If only part of the pairs were re-paired, some of the factors of Eq. (A.3) stay put, and some change. In that case, one can always transform jBi to a form jB0i where all the unchanged factors are grouped together. Each transposition multiplies the determinant by 1. Therefore, if an even number of transpositions are needed, the signs of jAi and jB0i are equal. If an odd number is required, jAi and jB0i have opposite signs. The actual wave function of the system is constructed from the combination of the two VB structures jAi and jBi. Two combinations are possible, an in-phase one jAi þ jBi, and an out-of-phase one jAi jBi. We disregard possible different coefficients, since we are only interested in the sign of the combination. Their energies are given by

498 yehuda haas and shmuel zilberg

According to Eq. (A.4), if HAB < 0, the ground state will be the in-phase combination, and the out-of-phase one, an excited state. On the other hand, if HAB > 0, the ground state will be the out-of-phase combination, while the inphase one is an excited state. This conclusion is far reaching, since it means that the electronic wave function of the ground state is nonsymmetric in this case, in contrast with common chemical intuition. We show that when an even number of electron pairs is exchanged, this is indeed the case, so that the transition state is the out-of-phase combination.

In order to show this, we have to evaluate the matrix element HAB ¼ hAjHjBi. We begin by writing out the orbital part of the A wave function explicitly

The sign of the last term depends on the parity of the system. Note that in the first and last term (in fact, determinants), the spin–orbit functions alternate, while in all others there are two pairs of adjacent atoms with the same spin functions. We denote the determinants in which the spin functions alternate as the alternant spin functions (ASF), as they turn out to be important reference terms.

Next, assuming that all pairs were exchanged in a cyclic pattern, we transform the VB function jBi [Eq. (A.3)] to a form similar to jAi by making n 1 transpositions of the form (2n,2), (2n 1; 3), and so on. Each transposition multiplies the function by 1, obtaining

It is evident that the only determinants that appear in both jAi and jBi are the ASFs.

The cross-term hAjHjBi in Eq. (A.4) can now be evaluated. This term may be written as (omitting the normalization constant):

Where the Coulomb integral Q is

Q ¼ h1 2 3 4 2n 1 2njHj1 2 3 4 2n 1 2ni ¼ h1 2 3 2n 1 2njHj1 2 3 2n 1 2ni

The exchange integrals Kij contain terms such as hi; i þ 1jgji þ 1; ii þ 2Si;iþ1 hijhji þ 1i [138]. The second term, representing the attractive interaction between two nuclei and the electronic overlap charge between them, is the dominant one and completely outweighs the first repulsive term. Ki;iþ1 therefore has the same sign as the Coulomb integral Q. (For a similar derivation, see [139].) In Eq. (A.7), HAB;CL is the cross-term obtained by classic VB theory [140], in which only contributions from electron pairwise transposition permutations were considered. Bonding in these systems is due mainly to the exchange integrals Ki;iþ1 between orbitals in the same cycle [138]. Pauling [140] showed that the most important contributions are due to neighboring orbitals, justifying the neglect of the smaller terms in Eq. (A.7).

Because of the orthogonality of the spin functions, and since we assume no spin–orbit coupling, only the first and last terms in Eqs. (A.5) and (A.6) will contribute to the Coulomb integral in HAB;CL. The Coulomb integrals together with the exchange ones between neighbors, Kiiþ1, are larger than all other terms and determine which will be the ground state. The sign of their contributions is determined as follows:

When n is odd, the first ASF in both jAi and jBi is positive, while the second is negative. The two resulting Coulomb integrals are equal contributing together 2Q.

When n is even, the two terms in jAi have equal signs, as in jBi, but the sign in jBi is opposite to that in jAi. Therefore, the total contribution to the energy is 2Q.

Since Q is negative, and HAB;CL for the ground state must be a negative sign, it follows that the ground state for the odd parity case is the in-phase combination, while for the even parity case, the out-of-phase wave function is the ground state.

This classical VB picture, using only pairwise electron transpositions in the permutations, in which the spins of the two electrons of every bond are paired, is sometimes termed ‘‘the perfect pairing approximation’’ [138]. Figure 7 shows a schematic representation of the different contributions to the in-phase and out- of-phase combinations. Note that the energies of the twin excited states shown in Figure 7 can also be calculated from Eq. (A.7). Their destabilization with respect to the HAA þ HBB reference due to the cross-term will be larger than the stabilization of the ground state, due to the different contributions of the overlap integral in the denominator of Eq. (A.4).

If only part of the electron pairs were exchanged in the reaction (the usual case), the argument follows similar lines. The important factor is the parity of the exchanges: for an even number of electron pairs exchanged, the out-of-phase combination is always of lower energy than the in-phase one. Therefore, the transition state is phase inverting. For an odd number of electron pairs exchanged, the reaction is phase preserving.

Acknowledgments

We thank Professor S. Shaik, Professor B. Dick, Professor L. S. Cederbaum, and Dr. W. Fuss for many enlightening discussions and suggestions. This research was supported by The Israel Science Foundation founded by The Israel Academy of Sciences and Humanities and partially by The VolkswagenStiftung. The Farkas Center for Light Induced Processes is supported by the Minerva Gesellschaft mbH.

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