IMPUESTO SOBRE EL ALCOHOL Y BEBIDAS DERIVADAS

4.3.2.1 The isotropic hyperfine coupling

In a planar alkyl radical, values of typically -60 to -70 MHz are found experi- mentally for the isotropic HFC value of α protons - the seminal example being the radiation-induced malonic acid radical [76]. Because the proton is located in the nodal plane of the 2pz orbital in which the unpaired electron resides,

there is no direct contribution of the unpaired electron to the isotropic HFC interaction (cf. Eq. (4.15)). Partial sp3 hybridisation would lead to a positive spin density and, hence, a positive isotropic HFC. Moreover, perfectly planar structures also yield such negative aα

isovalues. Weissman et al. have suggested

4.3. The HFC interaction in organic radicals

but calculations show the resulting values for aα

iso are at least an order of

magnitude smaller than experimentally observed [78]. It was among others McConnell who finally showed that the isotropic α proton HFC value arises mainly from a spin-polarisation mechanism through σ-p-electron exchange interactions [79, 80]. An intuitive explanation is presented in Figure 4.6: by virtue of the Pauli-exclusion principle, the C-H σ bond electrons must have opposite spins. Due to the atomic exchange interaction with the (polarised) electron in the carbon 2pz orbital, configurations I and II in Figure 4.6 have

different energies.9 _{The result is a net positive spin density at the carbon side}

of the σ orbital and a net negative spin density at the hydrogen side. The latter accounts, via Eq. (4.16) for the observed negative isotropic HFC values of α protons. Note that, since the isotropic HFC value in a hydrogen atom is∼1420 MHz, the spin density in the hydrogen 1s orbital must be∼ −65

1420 ≈ −0.05.

Based on several experimental studies (see, e.g. the references in Ref. [76]), a linear dependence of the α proton isotropic HFC value on the (unpaired) electron density ρπ_{at the carbon 2p}

zorbital was proposed:

aα

iso =Qρπ (4.27)

where Q is a constant. This is the well-known McConnell equation. McConnell and Chesnut derived this equation theoretically under certain conditions. The derivation and the expression for Q are rather involved and we refer to Ref. [80] for details. The McConnell equation (4.27) has given rise to an impressive amount of fruitful research but caution is needed in applying it. There are essentially two things to keep in mind:

1. Q in Eq. (4.27) is not truly a constant.

Its value depends on the geometry and the charge state of the radical as well as on the chemical nature of the radical. Fessenden and Schuler reported a comprehensive set of measurements on alkyl radicals from which the values and concepts essentially became the standard for applying the McConnell relation in analytical problems [81]. For an aliphatic π radical they proposed Q ≈ 63 MHz. However, in the literature often Q≈72 MHz is assumed, even if this value was originally proposed for aromatic rather than aliphatic π radicals [82]. A concise overview of Q values for a range of different radical types is presented in Ref. [4].

2. Equation (4.27) is not always valid. The proportionality between aα

iso and ρπ does not always hold. The two 9_{This can be understood in the context of Hund’s rules: in the atomic ground state the} valence electron spins in a carbon atom are aligned.

4.3. The HFC interaction in organic radicals

Figure 4.6: Spin polarisation in a -C•_αHα- fragment: the (filled) σ orbital is spin polarised

through σ-p electron-exchange interactions, resulting in a net negative spin density at the proton.

most drastic assumptions in its derviation are (i) perfect planarity of the spin centre and (ii) neglect of spin density in adjacent bonds and on adjacent atoms. The influence of radical centre bending is dramatic: the isotropic HFC value can vary by as much as 200 % (e.g. from -65 MHz to +65 MHz, assuming ρπ _{= 1) when the centre is significantly}

bent. This was shown both experimentally (by Dobbs et al., who used

13_{C hyperfine couplings to determine the bending of a centre [83, 84])}

and by means of DFT calculations [85, 86]. Spin density not located in the carbon 2pz orbital can also be quite important: spin density at

β sites will give a positive contribution to aα_iso (see Section 4.3.3). The

contribution to aα

isoresulting from a β oxygen can easily be a significant

portion of that arising from the spin-polarisation mechanism, as argued by Bernhard [87]. The latter also pointed out that this effect may account for a part of the variation in Q values discussed above. The effect is even more pronounced when the radical centre deviates from planarity [84, 88].

Finally, we note that the DFT calculations carried out during the doctoral research revealed that hydrogen-bonds with the surrounding lattice can easily cause shifts in the isotropic HFC values of 5-10 MHz.

4.3. The HFC interaction in organic radicals

of Eq. (4.27). As will be discussed in the next section, the dipolar HFC values of an α proton offer a much more reliable indicator of ρπ_.

4.3.2.2 The dipolar hyperfine coupling

α-proton dipolar HFC tensors have been determined for a vast number of

organic radicals. The first relatively accurate report was that of the α proton HFC in a malonic acid radical [76]:

Adip,x, Adip,y, Adip,z

= (32,−30,−2)MHz (4.28) where the <x> axis was found to be along the Cα-Hα bond and the <z> axis

along the symmetry-axis of the 2pz orbital in which the unpaired electron

resides. These features indeed are the trademark of an α proton in a π-electron radical, and they follow immediately from the two-centre approximation (Section 4.3.1.2).10 Gordy proposed values of 72 pm for Rp and 108 pm for

RCH [75]. Assuming ρπ =1, Eq. (4.26) yields

Adip,x, Adip,y, Adip,z

= (38.7,−36.0,−2.8)MHz (4.29) If a more realistic spin density ρπ _{=0.85 is assumed, we get}

Adip,x, Adip,y, Adip,z

= (32.9,−30.6,−2.4)MHz (4.30) which is in remarkably good agreement with the experimental values (4.28). The (+a , 0 ,−a) pattern is reproduced, calculated and experimental values agree well in size, and for the eigenvector directions we find

~

V+a k Cα-Hαbond (4.31)

~

V0 k LEO axis (4.32)

~

V−a ⊥ Cα-Hαbond and LEO axis (4.33)

When the carbon is sp2 hybridised in the parent molecule, the radical centre is expected not to reorient much upon radical formation and the directions at the right-hand side can be calculated directly from the atomic coordinates. When the carbon is sp3hybridised in the parent molecule, usually the X-C•Hα-

Y radical centre is approximately planar (sp2 hybridised) and the Hα proton

10_{In the seminal paper by McConnell and Strathdee [89], the A}

dip matrix elements of Eqs. (4.11) are calculated by integrating over the 2pz electron orbital by employing a Slater 2p-orbital wavefunction. The derivation is rather lengthy but yields results comparable to the much simpler two-centre approach.

4.3. The HFC interaction in organic radicals

reorients. In this case, good approximate directions are ~

V+a k in-plane bisector of the X-C•-Y angle (4.34)

~

V0 ⊥ X-C•-Y plane (4.35)

~

V−a ⊥ both of the above (4.36)

Of course, when there is severe structural reorganisation - as can be the case in e.g. sugar ring opening events - it is in general impossible to predict how the radical is oriented without using (advanced) quantum-chemical calculations. In principle, each of the Adip,i (i = x, y, z) values allows determination of

ρπ through Eqs. (4.26) provided accurate values for RCH and Rp are known.

Gordy and, later, Bernhard showed that Adip,x is in general least dependent

on the precise conformation of the radical and proposed following relation between ρπ_{and A}

dip,x:

Adip,x =Qdipρπ (4.37)

where Qdip= 38.7 MHz (cf. Eq. (4.29)). The study by Bernhard indicates this

relation holds well for rather large deviations from planarity (≤20◦) and in a broad range of RCHvalues (100 - 115 pm) - although Qdipstrongly depends on

Rp. Therefore, Eq. (4.37) is considerably more accurate than the McConnell

relation (4.27) to determine ρπ_{. A DFT-based study by Erling and Nelson}

on radical structures C•_αHα(R1)(R2) (R1,R2 = H, Me, COOH, NH2 and OH)

corroborates this: bending of the Cα centre from a planar to a tetrahedral

configuration only leads to minor changes in the Adip,x component (less than

6 %) (whereas the isotropic component can vary by ∼ 200 %, as mentioned in Section 4.3.2.1). Large discrepancies between ρπ _{values obtained from}

Eq. (4.37) and Eq. (4.27) may indicate a significant deviation from planarity at the radical centre.

Eq. (4.26) holds for any value of ρπ_{, but in practice large deviations from 1}

typically imply significant spin densities in the σ bonds of the carbon or in orbitals centred on adjacent atoms, which will of course also contribute to the dipolar HFC values. However, because of the r−3_{dependence the contribution}

of the carbon ρπ _{spin density still is the dominant contribution for relatively}

small values of ρπ_{. For instance, our own experimental data on the dominant}

stable sucrose radicals T2 and T3 (Table 5.3, page 117) shows that Eq. (4.26) still holds well for ρπ _{≈ 0.5. For a more in-depth discussion and quantitative}

estimations of these contributions, the reader should consult Ref. [87].

4.3. The HFC interaction in organic radicals

from planarity. In the literature it is commonly assumed that the ~V+a

eigenvector is still parallel with the C-H bond (leaving the orientation of the LEO axis undetermined). The DFT-study by Erling and Nelson corroborates this assumption: the~V+adirection does not deviate by more than 6◦(and often

less than 3◦) from the Cα-Hα bond direction upon bending from a planar to

a tetrahedral configuration [86]. Therefore, the ~V+a eigenvector direction is

a reliable indicator of the Cα-Hα bond direction, even in the case of extreme

bending. Consequently, the~V0 eigenvector can easily deviate by 10◦or more

from the LEO axis upon bending. The~V0 eigenvector direction is, however,

still a valuable criterion in the search for a suitable model. Using values for ρπ_{obtained by Eq. (4.37), Bernhard calculated R}

CHand RPfor

the α protons in numerous alkyl radicals as well as electron-gain and electron- loss aromatic radicals from the experimental Adip,xand Adip,yvalues via

RCH =  −ρπµ0 4π ggNµBµN Adip,y 1₃ _A dip,y−Adip,x 3Adip,y 1₂