Biomedical Engineering Reference
In-Depth Information
Here
I
is a nuclear angular momentum operator,
S
an electron angular momentum operator
and
R
α
i
the vector joining nucleus α and electron
i
. The
g
's and β's are the corresponding
g
-factors and magnetons.
Experimentalists work with a
spin Hamiltonian
which in the latter case would be written
h
α
a
α
S
z
I
α
z
(17.33)
This spin Hamiltonian operates only on spin wavefunctions, and all electronic details are
absorbed into the
coupling constant a
α
. These can be written very neatly in terms of the
spin density
Q
(
r
)as
4π
3
g
e
β
e
g
α
β
α
h
a
α
=
Q
(
R
α
)
(17.34)
S
z
where the spin density has to be evaluated at the position of nucleus α. The spin density is
the difference between the α- and the β-spin electrons, and unlike the total electron density
it can have negative as well as positive regions.
To give an example, the vinyl radical H
2
C=CH (Figure 17.8) can be produced by cleavage
of a C-H bond in ethane, and it has been well studied in the gas phase.
Figure 17.8
The vinyl radical
AROHF-6-311G** calculation gives the HOMO shown in Figure 17.9. In the language
of descriptive organic chemistry, this corresponds to the odd electron occupyingaCsp
3
hybrid, but of course there are negative regions as well as positive. The spin density is
just the square of the HOMO and so is positive everywhere. The experimental evidence
is that coupling constants can be positive or negative and so the ROHF model is not
satisfactory.
In the UHF model, the α- and β-spin electrons occupy spatially different orbitals and so
the spin density can indeed be negative in regions of space.AUHF/6-311G** calculation on
the vinyl radical gives regions of space where the spin density is negative and in particular
it is negative at the position of the second carbon atom.
Some results of calculations and experiments are given in Table 17.7. I have put the
experimental negative signs in brackets because the sign of the coupling constant cannot
easily be deduced from studies in solution. Agreement with experiment is not particularly
