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In addition to a-proton splittings, alkyl-substituted 7c-radicals show splittings from the P-protons in their EPR spectra. Generally, P-protons are not in the nodal plane of

the greater distance between C« and Hp as compared with C« and Ha, spin-polarisation is relatively unimportant. The unpaired electron couples to p-protons as a result of

overlap of the SOMO with the p-C-H bonding orbital, and this gives a spin population in the i s orbital of the hydrogen atom, which produces a positive coupling constant. This is known as hyperconjugation and in valence-bond terms there is a contribution to the resonance hybrid from the canonical structure 4 (Scheme 3).5a

Scheme 3

In terms of molecular orbital theory, the Hp-H, C^-2sl2p and Ca~2p have been shown to contribute to the SOMO (Scheme 4).

c — a

/ 4

Hp-lj, C^-2s/2p, Ca~2p

all contribute to SOMO

Scheme 4

The magnitude of p-proton splitting is proportional to cos^G, as shown in the Heller- McConnell equation [eqn. (6)],^ where A is the spin-polarisation parameter, which is

small {ca. 1 G) and often neglected, and B is the hyperconjugation parameter, which is much larger {ca. 58.5 G).

a(H|,) = {A + Bcos^G) p"c„ (6)

The dihedral angle 0 between the Ca~2pn orbital and the p-C-H bond is defined as shown in Scheme 5 and p^ca is the unpaired electron population in the Ca-2pj[ orbital. When 0 is zero, i.e. when the p-C-H bond is in the same plane as that of the Ca~2pn orbital the p- proton coupling constant is at a maximum.

Scheme 5

Molecular orbital (MO) calculations also support the form of the Heller-McConnell equation. It has been reported that Fermi contact integrals (FI) obtained from ab initio

MO calculations using the UMP2 density matrix give hyperfine couplings which are in good agreement with experiment.^ Roberts and Steel have carried out such calculations for the ethyl radical, and it can be seen that a plot of FI for H* [which is proportional to a(H*)] against cos^0 gives a perfect straight line that passes almost through the origin, and this indicates that A is close to zero/ It has been shown that even when the radical centre is slightly non-planar, the Heller-McConnell equation still provides a good basis for semi-quantitative discussion of the conformations of alkyl radicals.^

By considering the cyclohexyl radical, it can be seen that the Heller-McConnell equation can be successfully applied to alkyl radicals. MO calculations predict dihedral angles of ca. 11° and ca. 71° for the axial and equatorial p-protons, respectively, in the cyclohexyl radical (Scheme 6), which implies that <3(Heq)/a(Hax) = (cos^ 7I)/(cos^ II) = O.I I. By experiment, the splitting constants are found to be 41.5 G and 5.3 G, i.e. they are in a ratio of 0.13, close to the calculated value.

ax'

ax

View from here

Scheme 6

The baiTiers to rotation about Ca-Cp bonds of substituted ethyl radicals of the type YCH2CH2* and the equilibrium conformations may be obtained from the temperature

dependence of the P-proton coupling constants and line-width effects. The cos^G term in the Heller-McConnell equation is an average over the hindered rotation about the C(i-Cp bond [eqn. (7)].

a(H|,) = {A + B<cos^e>) p”ca (7) The observed P-splittings can exhibit a marked temperature dependence as the

conformation population changes with temperature, because different conformations have different energies. Substituted ethyl radicals YCH2CH2’ can either exist in an

eclipsed (structure 5) or staggered conformation (structure 6) at low temperatures, and

the low temperature limits of c/(Hp) are B/4 for structure 5 and 3B/4 for structure

As the temperature increases the amplitude of torsional motion about the Ca-Cp bond will increase until free rotation is approached, and for a freely rotating radical the average dihedral angle will be 45°, corresponding to a coupling constant of BI2. Thus, the value of a(Hp) for radicals which have a preference for the eclipsed equilibrium conformation 5 will increase with increasing temperature toward a free rotation value of

BH and the value of r/(Hp) for radicals which adopt the staggered conformation 6 will decrease with increasing temperature toward the value of B!2 (Scheme 8).5b

BI2

BI4

T/K

However, if the equilibrium geometry of the radical is not symmetrical, and adopts the conformation as shown by structure 7, then there will also be a second structure, of equal energy, which is obtained by rotating by (180 - 2(|)) about the Ca-Cp bond. The two P-protons in structures 7 and 8 give rise to different coupling constants since they are non-equivalent, and interconversion between 7 and 8 exchange and H®. The EPR spectrum will consist of a doublet of doublets [(a) in Scheme 10]^ when rotation about the C«-Cp bond is slow on the EPR timescale and a 1:2:1 triplet with line spacing equal to -t- c/(H®)]/2 will be observed when rotation is fast on the EPR timescale [(c) in Scheme 10]. In the latter case, and H® are magnetically equivalent.

Scheme 9

When exchange takes place at an intermediate rate on the EPR timescale, i.e.

exchange between and takes place at a rate comparable with [a(H^) - rf(H^)] expressed in frequency units, the spectrum will appear as shown in (b) in Scheme 10, with the wing lines remaining sharp, while the central component is broadened. When the two inner lines in the slow exchange spectrum have just coalesced to form a single broad line, the rate constant for exchange between 7 and 8 is (7r/v/2) [<^(H^) - r/(H^)] x (2.8 X 10^) s '\ The spectrum can also be computer simulated to give the rate constant for other rates of interconversion between 7 and 8.

(a) Slow exchange (b) Intermediate rate of exchange (c) Fast exchange [a(H^) + a{W)]l2 Scheme 10