IMPUESTO SOBRE LA CERVEZA (cuotas tributarias devengadas)
IMPORTACIONES DE ALCOHOL Y BEBIDAS ALCOHÓLICAS EN EL BIENIO 2004-2005
RP = −ρπµ0 4π ggNµBµN Adip,y 13 2A dip,y+Adip,x 3Adip,y 12 (4.39)
which can be obtained from Eq. (4.26) [87]. These RCH and RP values are
remarkably constant within one class of radicals (alkyl, electron-gain, electron- loss) and the differences between classes are significant enough to distinguish between them.
4.3.3 β protons
4.3.3.1 Isotropic hyperfine coupling
Consider a -Cα•-CβHβ- radical fragment with the spin density residing in the
2pz orbital on the α carbon (Figure 4.7). Experimentally, the β protons in this
type of radical fragments are found to have isotropic HFC values between 0 and ∼126 MHz, depending on the dihedral angle θ (Figure 4.7). Unlike for α protons, (secondary) spin-polarisation effects usually only play a minor role for β protons. Their isotropic HFC arises mainly from a mechanism called hyperconjugation: the 2pzorbital partially delocalises onto the β carbon
(yielding a π-like molecular orbital) and directly overlaps with the Cβ-Hβ σ
bond. The effect is a net positive spin density at the Hβ proton, which results
in the observed positive isotropic HFC. It is clear that the amount of overlap between the (delocalised) 2pzorbital and the σ orbital depends on their relative
4.3. The HFC interaction in organic radicals
Figure 4.7: A -Cα•-CβHβ- radical fragment with the spin density residing in the 2pz
orbital on the α carbon, viewed along the Cα-Cβ bond. The angle between the plane
containing the LEO and the Cα-Cβbond and the plane containing the Cα-Cβbond and
the Cβ-Hβbond is called the dihedral angle θ.
On the basis of theoretical arguments and experimental data, Heller and McConnell proposed the following dependence of the β-proton isotropic HFC value aβisoon the spin density in the 2pzorbital (ρπ) and on the geometry of the
radical structure [90, 91]:
aβiso =ρπ B0+B2cos2θ (4.40)
which is known as the Heller-McConnell relation. θ is the dihedral angle between the LEO and the Cα-Hα bond, viewed along the Cα-Cβ bond (Figure
4.7) and B0and B2are empirical constants. The second term on the right-hand
side is due to hyperconjugation. It is maximal when θ = 0◦and vanishes when
θ = 90◦. The first term arises from spin polarisation through the intervening
bonds.
The Heller-McConnell relation (4.40) can more generally be applied to radical fragments of the type -X•-Y-Hβ-, where X and Y need not necessarily be
carbon atoms. The values of B0 and B2 vary strongly with the nature of X
and Y, the only general rule being that B0 is substantially smaller than B2.
Considerable experimental effort has been made to determine values for B0
and B2for different -X•-Y-Hβ- radical types. A concise overview is presented
in Ref. [4]. B0 is in general assumed to be between -14 MHz and +14 MHz,
and for alkyl (X ≡C, Y ≡C), hydroxyalkyl (X ≡C, Y ≡O) and alkoxy (X ≡ O, Y ≡C) radicals, values for B2of∼126 MHz [92],∼73 MHz [93] and∼336
4.3. The HFC interaction in organic radicals
MHz respectively are commonly employed in the literature.11 For all three
types of radicals, it is acceptable to neglect the B0term altogether to obtain an
indication of θ from Eq. (4.40), as is indeed often done in the literature.
The Heller-McConnell equation (4.40) can in principle be used to obtain an estimate of ρπ when the local radical geometry is known. In practice it is
mostly used the other way round: obtaining a value for θ by inserting a value for ρπobtained from, e.g., an α-proton HFC tensor. In this context two remarks
should be made:
1. based on ab-initio molecular orbital calculations, Sevilla et al. found that in non-planar radical centres a term B1cos θ should be added to the right-
hand side of Eq. (4.40) with typically 5≤B2/B1 ≤10 [95, 96]. This extra
term was chosen empirically (to give the best fit) and is not based on theoretical arguments.
2. hyperconjugation can be enhanced (or diminished) quite drastically by the so-called Whiffen-effect [97] when the spin density does not reside on one, but on both sides of the Cβ-Hβ fragment. In this case, ρπ should be
changed to p
ρπ1 +pρπ22in Eq. (4.40), where ρπ1 and ρπ2 are the adjacent
spin densities. Even when ρπ
2 is (very) small compared to ρπ1, the effect
can be quite substantial.12
4.3.3.2 The dipolar hyperfine coupling
βprotons typically are 190-230 pm separated from the α carbon in a -C•α-CβHβ-
radical. Figure 4.5 indicates that this distance is too small to employ the pure point-dipole approximation. Experimentally, it is found that [4]
• the deviation from axial symmetry can be quite big, especially when θ → 0◦.13
• the anisotropy depends strongly on the value of the dihedral angle
θ. For θ → 0◦ and θ → 90◦, the difference between maximum and
minimum HFC values is typically∼11 MHz and∼17 MHz respectively in alkyl radicals [98] and typically∼15 MHz and∼23 MHz respectively in alkoxy radicals.
11The latter value was taken from Ref. [94] and is significantly higher than the 263 MHz reported in [4].
12E.g., for ρπ
1 =0.85 and ρπ2 =0.05, we find p
ρπ1 +pρπ2=1.31. 13E.g., eigenvalues of∼(+7,-6,-1) MHz are not seldom encountered.
4.3. The HFC interaction in organic radicals
• the eigenvector associated with the largest principal value (denoted~V2b for brevity) still roughly corresponds to the Cα· · ·Hβ direction [90]. For
θ ≈ 0◦ the correspondence is very good, while for θ → 90◦, ~V2bmoves
somewhat towards the Cβ· · ·Hβdirection [99].
• In the case of a hydroxyalkyl radical C•OH, the deviation from axial symmetry can be further enhanced and the anisotropy increases signifi- cantly (the difference between maximum and minimum principal values being∼30 MHz when ρπ ≈ 1 [98, 100]).V~
2bis approximately parallel to
Cα· · ·Hβ for all values of θ [101].
Some of these observations can be understood in the context of the two-centre approximation (Section 4.3.1.2), even if we can only employ it when the proton is in the nodal plane of the 2pzorbital (which corresponds to θ=90◦):
• Eq. (4.26) yields ( Adip,x, Adip,y, Adip,z)≈( +10 , -6 , -4 ) MHz for a β proton
at 215 pm from Cα, when Rp = 72 pm and ρπ =0.85 are used. This is in
good accordance with typical experimental values for alkyl radicals. • the increase in anisotropy for a β hydroxy proton in a hydroxyalkyl
radical is in part due to the presence of spin density on the intervening oxygen atom, but can also be in part attributed to a difference in RCH
in Eq. (4.26). The former value typically is 215 pm in alkyl radicals, but 195 pm in a hydroxyalkyl radical.14 The variation of the anisotropic HFC
values with RCH according to Eq. (4.26) is given in Figure 4.8.
• likewise, the increase in anisotropy for a β proton in an alkoxy radical (as compared to an alkyl radical) can in part be attributed to a difference in ROH (cf. previous item), but also to a difference in Rpin Eq. (4.26). Rp
is typically 72 pm in an alkyl radical, but 52 pm in an alkoxy radical (the oxygen 2pz orbital is more ’compact’). The variation of the anisotropic
HFC values with Rpaccording to Eq. (4.26) is given in Figure 4.9.
The dipolar HFC values in a -Cα•-CβHβ- radical fragment deviate from Eq. (4.26)
mainly because of spin density in the Cβ-Hβ σ bond and on Cβ, resulting
from hyperconjugation and secondary spin-polarisation effects. Because of the r−3 dependence of the dipolar HFC values, these spin densities can be quite important, even if they are small compared to ρπ.15 The largest
deviations from the two-centre approximation can be expected for θ = 0◦, since hyperconjugation effects are strongest then. We note that Derbyshire [90]
14Assuming tetrahedral configuration of the C
βcarbon and typical values for the C-C, C-H,
C-O and O-H bonding distances of 152 pm, 110 pm, 142 pm and 95 pm respectively. 15This causes Eq. (4.26) to be less reliable for β than for α protons.
4.3. The HFC interaction in organic radicals
Figure 4.8:The dipolar HFC values for the HFC interaction between a β proton H and an unpaired electron in a 2pzorbital centred on a carbon C, as a function of RCH (the distance between C and H) in the two-centre approximation (Figure 4.4). Eq. (4.26) was used with g= 2.0023, ρπ =0.85 and R
p = 72 pm. This is in fact a ’zoomed in’ graph of Figure 4.5 .
Figure 4.9: The dipolar HFC values for the HFC interaction between a β proton and an unpaired electron in a 2pzorbital centred on a carbon, as a function of Rp(the distance between the effective spin centres and the carbon) in the two-centre approximation (Figure 4.4). Eq. (4.26) was used with g=2.0023, ρπ=0.85 and R
CH=215 pm. Note that for Rp→0 pm the point-dipole approximation is retrieved.
4.3. The HFC interaction in organic radicals
derived theoretical expressions for the dipolar components in which both the spatial distribution and the non-vanishing spin densities in the Cβ-Hβ bond
and on Cβ are taken into account. Some test calculations on typical radical
fragments indicate that the results are very similar to those of the two-centre approach.