Spin polarization allyl radical

Radicals with adjacent Jt-bonds [e.g. allyl radicals (7), cyclohexadienyl radicals (8), acyl radicals (9) and cyanoalkyl radicals (10)] have a delocalized structure. They may be depicted as a hybrid of several resonance forms. In a chemical reaction they may, in principle, react through any of the sites on which the spin can be located. The preferred site of reaction is dictated by spin density, steric, polar and perhaps other factors. Maximum orbital overlap requires that the atoms contained in the delocalized system are coplanar. 

Unpaired spin in a delocalized 7T-system tends to follow a parity-type distribution, with alternation of the sign of 7T-spin density due to polarization. As shown in Scheme 2, the odd alternant allyl radical has its major spin density confined to the 7T-atoms at each end within the simple Hiickel molecular orbital... 

SAD Spin-alternant determinant. The VB determinant with one electron per site and with alternating spins. Other terms describing the same determinant are the quasiclassical (QC) state, and the antiferromagnetic (AF) state. In nonalternant hydrocarbons, where compete spin alternation is impossible, the determinant is called MS AD, namely, the maximum spin-alternating determinant. The SAD MSAD are the leading terms in the wave function of molecules with one electron per site, for example, conjugated hydrocarbons. In radicals (e.g., allyl radical) the SAD is the root cause of spin polarization (i.e., negative spin densities flanked by positive ones). See Chapters 7 and 8. 

Electron Paramagnetic Resonance (EPR) can be used to measure the spin-densities in radicals. It is then assumed that the hyperfine coupling constants for the hydrogen atoms are proportional to the spin-density of the adjacent carbon atom [70]. Measurements on the allyl radical [71] give with such an analysis ratio of —0.282 between the spin-densities of the central and the end carbon atom. The CASSCF value is 0.311. One would suspect that methods that include spin polarization of the a skeleton would give better values. The UHF value is, however, — 0.717. What is the reason for this large difference Let us take a closer look at the CAASCF wave function. It contains three terms ... 

The reaction mechanism has tJso been investigated by DFT calculations, lending support for the stepwise mechanism [111, 112]. The first key intermediate is the diruthenium nitrenoid species (S = 1/2). In the transition state of the C-H bond scission, the coordinated nitrenoid interacts with the allylic H atom, polarizing the C-H bond. Subsequent H atom transfer to the N atom generates a diradical species comprising a diruthenium amide tethered to an allylic radical (S = 3/2). Of interest, the unpaired electrons are primarily localized at the two Ru centers (and the allylic fragment), with litde spin density (0.14) at the N atom. Finally, radical rebound forms the cyclized amine. 

Large amounts of spin polarization are expected in radicals where (1) the SOMO has nodes at some nuclei and (2) some of the formally paired electrons occupy subjacent bonding MOs of relatively high energy (e.g., k MOs) that are easily polarized. Good examples of this combination of features are odd-alternant hydrocarbon radicals (e.g., allyl and benzyl) where (1) the SOMOs have nodes at every other carbon atom and (2) electrons in subjacent n MOs, whose spin is opposite to that of the electron in the SOMO, can relatively easily be confined at these nodal carbons. 

Figure 2 ROHF and UHF Jt MOs and the highest o h MO of allyl radical (from an STO-3G calculation). Note the strong spin polarization in 7t, and its near absence in the less polarizable Och MO. The values for the pairs of hydrogens at the terminal carbons are averaged.

As discussed above, UHF wavefunctions mirror the spin polarization that is seen experimentally, but they do so at the cost of introducing contamination from higher spin states. We now use allyl radical to illustrate how this occurs. Employing Dirac s ket notation for Slater determinants (introduced in... 

Since (S ) is close to 1 in allyl, has an uncomfortably high level of spin contamination. If one compares the observed hyperfin. couplings in allyl radical to those computed from a UHF wavefunction, one finds that spin contamination causes the amount of spin polarization to be exaggerated. In longer odd-alternant hydrocarbon radicals, spin contamination can become quite spectacular As shown in Figure 3, (S ) for the UHF wavefunctions of odd-alternant polyenyl radicals increases by 0.38 units for every pair of CH groups added, instead of remaining constant at (S ) = 0.75, as it would for a pure doublet wavefunction. 

Figure 5 ROHF and UHF MOs for twisted allyl radical. Note that there is much less spin polarization in 7t of twisted allyl than in Ki of planar allyl (see Figure 2). Numbers denote the coefficients of the p AOs on carbon.

However, even if one is not interested in modeling ESR spectra, preventing electrons of opposite spin from having different spatial distributions imposes a constraint on ROHF wavefunctions that has energetic consequences. For example, if one compares the ROHF energy of the planar allyl radical in C2J, symmetry (cf. below), where spin polarization is quite important, to that of the twisted species, where spin polarization of the electrons in the n bond is almost absent, one obtains a rotational barrier that is far too low (see Table 1), compared to the experimental value of 15 kcal/mol. 

We now return to the allyl radical to illustrate how a CASSCF wavefunction incorporates spin polarization, which is absent from ROHF wavefunctions, without introducing spin contamination, which is present in UHF wave-functions. Shown schematically in Figure 7 is the lowest energy configuration,... 

Equation [12] demonstrates that the terms in Eq. [7] that introduce spin polarization into the UHF wavefunction for the k electrons in the allyl radical simultaneously contaminate it with the quartet wavefunction. In fact, the greater the amount of spin polarization in q UHF coeffi-... 

In each of the two configurations in Eq. [21], there is an a electron localized at two carbons of square CB and a p electron localized at the other two carbons. As in the allyl radical, this uneven distribution of spin in the NBMOs causes each of the electrons in the bonding n MO to tend to localize in such a way as to avoid the electron of opposite spin in an NBMO. With inclusion of this type of electron correlation in the wavefunction, the polarized bonding n MOs resemble those depicted in Figure 15c. The MOs shown are appropriate for the occupancy of Vj/ja and by electrons of the same spin and and V /3<... 

In Eq. [22] /2 contains an a electron in the first configuration and a P electron in the second. Consequently, the electron spins in Xj/ja and xiijb in the second configuration in Eq. [22] are opposite to those in the first configuration. Thus, unlike the spin polarization in the allyl radical, that in the lowest singlet... 

Because p and d AOs have nodes at the nuclei, Ps(X) can be nonzero only if the SOMO that contains the unpaired a electron has some s character. However, as discussed in the section on UHF calculations, spin polarization in subjacent MOs leads to nonzero local a and P spin density in a MOs, so even when the unpaired electron occupies an MO with no s character, hfcs can still be observed. The clearest manifestation of this phenomenon is the observation of hydrogen hfcs in planar 7t systems, such as the allyl and benzyl radicals, discussed earlier. One might conclude that only methods that account for CT spin polarization by Jt electrons are capable of yielding useful predictions of ESR spectra. However, the observation by McConnell that hydrogen hfcs in planar hydrocarbons are usually proportional to the 7t spin populations on the carbons to which they are attached makes it possible to arrive at quite reasonable estimates of relative ax based on simple Hiickel- or PPP-type calculations on just the Jt electrons. 

Structure in the esr spectra of allyl and benzyl radicals caused by spin polarization while SCF calculations can (26). ... 

The bonding pi orbital tti, the nonbonding pi orbital nz, and the antibonding pi orbital 713 of an allyl radical (a) in the absence of spin polarization and (b) in the presence of spin polarization. The up- and down-spin arrows are used in (b) to indicate the up- and down-spin spatial orbitals, respectively. 

The spin density distribution of the two ions is expected to give rise to characteristic CIDNP effects the 2B2 species would induce strong polarization for the allylic methyl groups the 2A, radical cation would cause the bridgehead methyl groups to be polarized. The fact that both signals showed sizeable CIDNP effects [361] was explained by the simultaneous involvement of both radical cations. 

In the presence electron-rich alkenes such as 2,3-dimethylbut-2-ene, irradiation of CA gives the allylethers 59 and 60, whereas with BQ, a substantial amount of the spiro-oxetane is also formed.The product distribution of the allyl ethers is rationalized by steric effects on the H+ abstraction and on the recombination of radicals as well as spin densities. The crucial role of solvent polarity in CA photochemistry is well illustrated by the results of a study into the reaction between the quinone and cyclohexanone enol trimethylsilyl ether 61 using time-resolved (ps) spectroscopy. The influence of the solvent occurs following the formation of the radical ion pair (CA - 61+-). The CA- species is short lived in nonpolar solvents and cyclohex-2-en-l-one and 62 are the reaction products, whereas in acetonitrile, the lifetime is much longer, which allows diffuse separation of the radical ion pair and transference of the TMS to the solvent. The resulting ketyl radical couples to CA - yielding 63. 

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