**
Claude Chachaty**

**
e-mail :
****
claude.chachaty@wanadoo.fr**

*Keywords** :** NMR, ESR, spin, hyperfine
coupling, spectra, simulations, automated fitting.*

In spite of its outstanding scientific
potential, **APL** is up to now ignored
or scarcely exploited by research workers. During 15 years as the head of the
Magnetic Resonance Laboratory of the **APL**
in his works [1-3] and continues to promote its scientific applications.

The Magnetic Resonance Spectroscopy (**MRS**) includes two main branches, the
Nuclear Magnetic Resonance (**NMR**) and
the Electron Spin Resonance (**ESR**) also
called Electron Paramagnetic Resonance (**EPR**).
The **NMR** is a priviledged method for
the identification and conformational
analysis of organic and biological molecules and is well known for its medical
application, the Magnetic Resonance Imaging. The **ESR/EPR** which is the main subject of this topics, is the specific
method for studying paramagnetic molecules i.e. molecules possessing at least
one unpaired electron, namely the free radicals resulting from the breaking of
a chemical bond, triplet fundamental (e.g. the oxygen of air) or lowest excited
states and some metal coordination complexes. Most of these species are very
reactive and are initiators or intermediates in a large number of chemical and
biological processes : oxidation, combustion,
polymerization, radiation damaging, photosynthesis etc… An important
application common to the **NMR **and **ESR **is** **the molecular ** **dynamics which
provides thorough information on some physical properties of condensed matter.

**II**-**Principles**

**
** Electrons and most of the nuclei
possess a spin angular momentum, denoted _{}and _{}, respectively, as well as magnetic moments _{} and _{}where _{}are the spectroscopic factors, the latter being
specific of each nucleus, _{} the
relevant gyromagnetic ratios, _{} the Bohr
and nuclear magnetons and _{} the
Planck’s constant divided by _{}. In a magnetic field _{}, the spins and magnetic moments undergo a
precession of angular frequency _{} about _{}. For a
spin quantum number S or I, multiple of ½, the spins and
magnetic moments take 2S+1 or 2I+1 orientations defined by the projections _{}or _{} of _{}or _{} on _{}. To each magnetic quantum number _{} corresponds an energy level.

A magnetic resonance experiment consists in
submitting a small sample (0.1-1 ml) placed in a very strong and homogeneous
magnetic field _{}to a rotating radiofrequency (**NMR**) or microwave (**ESR**) magnetic field _{} perpendicular to _{} (_{}). The resonance phenomenon corresponds to a
transition between adjacent energy levels which occurs when the angular frequency
of _{} is equal
to _{} and
involves the absorption of a photon of energy _{}, _{} being
the spectrometer frequency and h the Planck’s constant. For technical reasons,
the resonance is obtained by varying _{} at
constant field (**NMR**) or _{} at
constant frequency (**ESR**) and the **ESR** spectra are usually recorded as the
first derivative.

The nuclear and electron spins are seldom isolated and generally experience local magnetic fields due to other spins. The 2S+1 fundamental energy levels of a spin S interacting with a spin I are splitted into 2I+1 sublevels and the resulting (2S+1)(2I+1) levels are :

E¬(B0×(ge×be×MS)°.-gn×bn×MI)+a×MS°.×MI

where **a**
is the hyperfine coupling constant expressed in energy units. The allowed **ESR** transitions between these levels
follow the selection
rule _{}. The above expression is easily extended to
any number of spins of any quantum number and is an usual approximation when its first
term is much larger than the second one. Figure 1 shows a simple application of
these principles to a S=1/2, I=1/2 system, the H· atom, the smallest and one of the most reactive free radical.

Figure 1 :
Energy levels and ESR resonance lines of the hydrogen atom. Hyperfine coupling
constant a = 1.42 GHz or 508 gauss (1 gauss = 0.1 mT), spectrometer frequency _{} = 9.24
GHz. Allowed transitions : IÛIV
and IIÛIII.

**III-Applications to the ESR
spectroscopy.**

The interpretation of the **ESR **spectra** **in terms of
identification of the paramagnetic species we are dealing with, of the
hyperfine coupling parameters and sometimes of dynamical behaviour is generally
not feasible without the help of computer simulations. A
visual comparison of the experimental spectum with the simulated one tell
us if the starting assumptions made about this species are likely or not. The **hresol** function listed below is a
simplified version for the simulation of high resolution **ESR **spectra of radicals in solution.

''

''

''

''

©

''

The
figure 2 shows the spectrum of the benzyl (_{}) radical generated by this function.

Figure 2. Simulated ESR spectrum (first derivative) of the _{} radical in
fluid solution. The electron spin is coupled to 3 pairs of equivalent
protons and a single proton.

The interactions between the spins and the
magnetic field and between the spins are of the form _{} and _{}, respectively,
where **g** and **A** are symmetric second rank tensors
whose components are the sum of an isotropic term (g factor and hyperfine
coupling constant)

and an anisotropic one. In fluids, the latter is
averaged to zero by fast molecular motions but is partially or not averaged in
anisotropic systems as solids and liquid crystals.

The functions for fitting the spectra of spin S=1/2 species (free radicals, copper and vanadyl ions) in anisotropic media proceed by the following steps :

1 - Parameters

Invariant : spectrometer frequency,
spectral width and nuclear spin quantum
numbers. Adjustable : principal values of
A, g and
s (linewidth)
tensors, width of Gaussian line broadening and rate of rotational m_{}otion if any.

2 – Angular dependence of the
transitions probabilities and linewidths _{}, the angles
q and
f defining the orientation of _{} in the
frame of th_{} e g tensor.

3 – Calculation of resonance fields .

4 – For each transition, summation of spectra over all orientations:

_{}

where B is the scanning magnetic field, F the form function and N the normalization factor.

For a Lorentzian form function F(x) = 1/(1+x^{2}),
the relevant APL expression is:

(d1
d2 d3)„½¨B th phi

U„B°.-,Br
ª F„÷‘U×1+U×U„U÷‘U„(d1½1)°.×,sigma

S„S÷+/S„F+.×,P×(1±th)°.×d3½1

**5** – The overall spectrum
obtained by summing _{} over _{} is
convoluted by a Gaussian and derived numerically. _{}

**6** – If the agreement with the
experimental spectrum is not satisfactory, return to **1** to reajust the parameters. This step may be automated by means of
an optimization function based on the Levenberg-Marquardt’s algorithm [4] to
minimize the variance between the experimental and computed spectra.
Figure 3 shows an example of
an automated fitting
using the method outlined above.

.Figure 3. Experimental (solid line)
and computed (·····) spectra of an ESR spin-probe, a nitroxide radical, in a model
phospholipid membrane before (a) and after (b) addition of cholesterol. This
membrane is constituded by phospholipid bilayers separated by water and behaves
as a liquid crystal.The broadening and increased asymmetry of the lines from
(a) to (b) are significant of an increase of the membrane rigidity and
molecular ordering upon cholesterol addition, which may be quantitavely
estimated [5].

**IV-Conclusion**

The
theory of magnetic resonance is for a large part founded on matrix algebra, one
of the strong points of **APL**, making
quite easy the programming of spectral simulations and
fitting of experimental data. For this reason the author has chosen **APL** rather than other programming
languages currently used by the scientific community (Fortran,
Basic, C, Pascal) in spite of its small diffusion and of some problems of
portability.

The
Magnetic Resonance software is written in APL2 (IBM) and APL+WIN (APL2000). Descriptions
of the workspaces are given in the sites www.garpe.org and ftp://ierc.scs.uiuc.edu/pub/SoftwareDatabase.

**References**

[1 ] C. Chachaty et G.
Langlet

*Logiciels
d'étude conformationnelle par RMN, de molécules flexibles.*

Journal de Chimie-Physique et de
Physico-Chimie Biologique, **82**, 613
(1985).

[2] C. Chachaty

* Simulations de spectres de résonance
magnétique appliquées à la dynamique de*

* molécules en
milieux anisotropes.*

ibid., **82**, 621 (1985).

[3] C. Chachaty and E. Soulié

*Determination of electron
spin** resonance parameters by
automated fitting of the spectra.*

Journal de Physique III
France, **5**,** **1927** **(1995).

[4] D.W. Marquardt

*An algorithm for
least-squares estimation of nonlinear parameters.*

Journal
of the Society of Industrial Applied Mathematics, **11**, 431 (1963).

[5] C. Wolf and C. Chachaty

* Compared effects of cholesterol and 7-dehydrocholesterol
on sphingomyelin- *

* glycerophospholipid
bilayers studied by ESR.*

Biophysical Chemistry, **84**, 269 (2000).