EVOLUCIÓN DE LA RECAUDACIÓN POR EL IMPUESTO SOBRE PRODUCTOS INTERMEDIOS (*)

We can derive a spin Hamiltonian for the HFC interaction in organic radicals by treating it as a classical interaction between the point-dipole magnetic moments generated by the electron and the nucleus. In this approximation, the energy of the interaction between magnetic moments~µ1at position~r1and

~µ2at position~r2is E= − µ0 4πr3  3(~µ1·~r)(~µ2·~r) r2 − ~µ1· ~µ2  (4.5) where µ0 is the vacuum permeability,~r = ~r2−~r1 and r = |~r|. The magnetic

moments are (cf. Eq. (2.4))

~µS = −gµB~S (4.6)

~µI = gNµN~I (4.7)

where g is the electronic g factor, gNthe nuclear g factor, µBthe Bohr magneton

and µNthe nuclear magneton. This yields

E= µ0ggNµBµN 4πr3 " 3(~S·~r)(~I·~r) r2 − ~S· ~I # (4.8)

4.3. The HFC interaction in organic radicals

Expansion in Cartesian coordinates yields E = µ0 4πr5ggNµBµN  3x2−r2
SxIx+ 3y2−r2
SyIy + 3z2−r2
SzIz+3xy SxIy+SyIx
(4.9) +3yz SyIz+SzIy
+3xz(SxIz+SzIx)

The quantum-mechanical spin Hamiltonian is obtained by integrating the right-hand side of this equation over the orbital part of the electronic wave- function and replacing the vectors~S and~I by their respective operators ˆ~S and

ˆ

~_{I. This yields}6

ˆ

HHF,dip = A0xxSˆxˆIx+A0yySˆyˆIy+Azz0 SˆzˆIz+A0xy SˆxˆIy+SˆyˆIx
(4.10)

+A0_yz SˆyˆIz+SˆzˆIy
+A0xz SˆxˆIz+SˆzˆIx

where A0_ii = µ0 4πggNµBµN Z Vψ ∗_{(x, y, z)}3i2−r2 r5 ψ(x, y, z)dV (4.11) A0_ij = A0_ji= = µ0 4πggNµBµN Z Vψ ∗_{(x, y, z)}3ij r5ψ(x, y, z)dV

with i, j= x, y, z and i6=j. In matrix notation we can write ˆ

HHF,dip=~Sˆ·Adip·~Iˆ (4.12)

where Adip is called the dipolar hyperfine coupling tensor or anisotropic

hyperfine coupling tensor and is given by

A_dip =         A0_xx A0_xy A0_xz A0_xy A0_yy A0_yz A0_xz A0_yz A0_zz         (4.13)

From Eqs. (4.11) it is clear that this tensor is anisotropic (in general), symmetric and traceless. Consequently, it can always be cast into a diagonal form.7 We

6_{The reason for adding the subscript dip and the supercript 0 will become clear below.} 7_{As in Chapter 2, the diagonal elements in the tensor’s eigenframe (which are the principal} values of the tensor) are distinguished from those in other reference frames by denoting them as A0_i instead of A0_ii(i=x, y, z).

4.3. The HFC interaction in organic radicals

note that, as intuitively expected, this dipolar interaction averages to zero in solutions due to the rapid random movement of the molecules [72].

We have so far ignored the fact that the r−3 factor in the integrands of Eqs. (4.11) diverge to∞ for r→0. The integration will still yield a finite value for any orbital wavefunction ψ with a non-vanishing angular momentum because such a wavefunction decays exponentially to 0 for r → 0. Thus, only for s orbitals a problem arises. The cause of this problem is that the point- dipole approximation – which is at the basis of our derivation – breaks down for r →0 and Eqs. (4.11) are simply not valid for s orbitals.

The spherical symmetry of s orbitals implies that the hyperfine interaction between an s-orbital electron spin~S and a nuclear spin~I must be isotropic, so that the corresponding spin Hamiltonian should have the form

ˆ

H_HF,iso = a_iso~Sˆ·~Iˆ (4.14) For a one-electron system, Fermi obtained (in a rigorous quantum-mechanical approach) [73]: aiso= 2µ0 3 ggNµBµN   Ψ(~0)    2 (4.15) where   Ψ(~0)  

 is the amplitude of the electronic wavefunction at the nucleus. This isotropic hyperfine interaction is also referred to as the Fermi-contact interaction since in a one-electron system

Ψ(~0)   

2

is the probability of finding the electron at the nucleus. Eq. (4.15) can also be retrieved in a semi-classical approach, treating the nucleus as a spinning charged shell of finite size and calculating the energy of interaction between the electron-spin magnetic moment and the magnetic field generated by the nucleus within the shell (see e.g. Ref. [74]). This approach gives an intuitive grasp of the physics involved. For multi-electron systems, Eq. (4.15) has to be generalised to

aiso =

2µ0

3 ggNµBµNρS(~0) (4.16) where ρS(~0)is the spin density at the nucleus (cf. Eq. 3.12). Note that Eqs. (4.11)

and (4.16) correspond to Eq. (3.47) for a system with one electron and if~R= ~0 . In summary, the HFC interaction between an electron spin~S and a nuclear spin ~_{I in organic radicals is adequately described by following spin Hamiltonian:}

ˆ

HHF =~Sˆ·A·~Iˆ (4.17)

with

4.3. The HFC interaction in organic radicals

where 1 is the 3x3 unit matrix, the matrix elements of Adip are given by

Eqs. (4.11), and aisois given by Eq. (4.16). The matrix elements of A are denoted

Aij.

The statement made above that the isotropic hyperfine interaction vanishes for all orbitals except s orbitals only holds for a one-electron system. In multi- electron atoms and in molecules, hyperfine interactions will in general have both an isotropic and dipolar component due to among others spin-exchange interactions. Several examples are discussed in Sections 4.3.2 and 4.3.3.

Before proceeding to the specific cases of α protons, β protons, . . . , two different approximations for the dipolar hyperfine coupling are presented: the pure point-dipole approximation and the two-centre approximation.

4.3.1.1 The pure point-dipole approximation

In this approximation, the spatial distribution of the electron is completely neglected: the electron is assumed to be localised at the Xα nucleus (X =

C or O). If we put the origin of our reference frame at Xα, the point-dipole

approximation boils down to replacing ψ(x, y, z)by the Dirac-delta function

δ(~0)in Eqs. (4.11). If we chose the <x> axis along the line connecting Xα and

the proton (so that the coordinates of Hβare (x,0,0) and r =x), Eq. (4.11) yields

Adip=    +2b 0 0 0 −b 0 0 0 −b    (4.19) with b= µ0 4π ggNµBµN r3 (4.20)

If the spin density at the Xα nucleus differs from unity (ρ6=1), an extra factor

ρshould be introduced in Eq. (4.20):

b=ρµ0

4π

ggNµBµN

r3 (4.21)

Note the following properties:

1. the dipolar hyperfine interaction is diagonal in this reference frame. 2. the eigenvalues exhibit the axial symmetric (+2b,−b,−b) pattern charac-

teristic of the classical point-dipole interaction.

3. the eigenvector corresponding to the 2b eigenvalue (V~_2b) points from Xα

to the proton. The other two eigenvectors cannot be related to the radical geometry (which is logical considering the approximation made), their

4.3. The HFC interaction in organic radicals

only limitation being that they have to be perpendicular to ~V_2b and to each other.

The position of the proton with respect to Xα can be deduced from Adip:

Eq. (4.20) yields the distance and ~V_2b yields the direction. This makes the dipolar part of the HFC tensor a very useful analytical tool.

In practice, the point-dipole approximation is only valid when the distance between the proton and the Xαnucleus is large compared to the spatial spread

of the electron density. Also, Eqs. (4.19) and (4.21) strictly speaking only hold when the g tensor is isotropic [25]. For the (very) small g-tensor anisotropies encountered in this work, however, they can safely be employed.

4.3.1.2 The two-centre approximation

Assuming the electron is localised in a 2pz orbital on Xα (X = C or O),

we can partially account for the spatial delocalisation of the electron by considering two ’effective spin centres’, located at a distance Rpfrom X, along

the symmetry axis of the 2pz orbital (Figure 4.4). This ’two-centre approach’,

originally due to Gordy [75], is quite suited for gaining insight in α proton HFC interactions (see Section 4.3.2).

Assuming the proton is located in the nodal plane of the 2pz orbital, we can

calculate the dipolar HFC values when the field is directed along the <x>, <y> and <z> axis (Figure 4.4), which are, because of the symmetry of the system, also the principal axes. It follows from Eq. (4.5) that the classical interaction energy between two magnetic moments~µ1and~µ2is

E= −µ0 4π µ1µ2 r3 (3 cos 2 θ−1) (4.22)

when~µ1 k ~µ2 k ~B is assumed. Here, θ is the angle between~B and~r (which

connects the point-dipoles). In the present case, the magnetic moments arise from an electron spin ~S and a proton spin~I (cf. Eqs. (4.6) and (4.7)). The magnitudes of the magnetic moments are

µ_S = ρ

π

2 gµBMS (4.23)

µI = gNµNMI (4.24)

where the factor ρ₂π appears because each of the effective spin centres carries half of the spin density.8 Because the dipolar interaction energy is also given

8_{Note that the supercript π is used, even if a 2p}

zorbital is considered (cf. the discussion in Section 4.2.2 ).

4.3. The HFC interaction in organic radicals

Figure 4.4: The two-centre approach for the dipolar HFC interaction between a proton (H) and a nucleus X (X = C or O). Rpis the distance between an effective spin centre and X. The proton is assumed to lie in the nodal plane of the 2pzorbital. The <y> axis (not drawn) is perpendicular to the plane of the paper, pointing out of it.

by E = AdipMSMI (Chapter 2), the dipolar HFC value for one effective spin

centre is ρπ 2 µ0 4π ggNµBµN R3 (3 cos 2 θ−1) (4.25)

When ~B is oriented along a principal direction, θ is the same for the two effective centres: θ = π

2 for~B k<y>, cos θ = RXH

R for~B k<x> and cos θ = RP

R

for~Bk<z>. Since R= qR2

XH+R2P (Figure 4.4), we finally obtain

        Adip,x Adip,y Adip,z         = ρπµ0 4π ggNµBµN R2_XH+R2_P
3₂          2R2 XH−R2P R2 XH+R2P −1 2R2 P−R2XH R2 XH+R2P          (4.26)

Note that for Rp =0, Eq. (4.19) is retrieved.

In Figure 4.5 the variation is shown of the dipolar HFC values for an alkyl radical (X = C) as a function of the distance RCHfor both the pure point-dipole

approximation and the two-centre approximation. The spatial extensiveness of the 2pzorbital has two main consequences: (i) the degeneracy of the Ayand

4.3. The HFC interaction in organic radicals

Figure 4.5: Dipolar HFC values for the hyperfine interaction between a proton and an unpaired electron in a 2pz orbital centred on a carbon (i) in the pure point-dipole approximation, using Eqs. (4.19) and (4.21) (green dotted line) and (ii) in the two- centre approximation (Figure 4.4), using Eq. (4.26) (full lines), with g = 2.0023, ρ = ρπ=0.85 and Rp=72 pm. The latter two values are typical for alkyl radicals.

Az values is lifted and (ii) the magnitude of the values is decreased. Figure 4.5

also indicates that the point-dipole approximation is in general ’safe’ to use for RXHdistances of more than 300 pm.