) of the
processed waveform (minus the deadtime fill points)
is calculated. Experimental data filtering allows extraction
of experimental parameters from saved experimental data files and processing of
the modulation amplitude waveform for simulation analysis. For two-pulse ESEEM,
the extracted experimental parameters are experiment type, τ value, and B0 field. For three-pulse ESEEM, the
additional experimental parameter, T,
is extracted. An example data
filtering routine that is compatible with our experimental control software is
provided with the package. The
ESEEM waveform is then processed as follows. If necessary, baseline decay of the echo
envelope, owing to spin-lattice (T1)
or spin-spin (T2)
relaxation processes, is removed by using an empirical decay function. This is accomplished by fitting the
waveform to a user specified monoexponential, biexponential, or stretched exponential function, and then
subtracting the decay function from the experimental waveform. Any points recorded within the
spectrometer deadtime are removed. The deadtime portion of the waveform is reconstructed as part of the optimization during the
simulation. The average noise
variance ( ) of the
processed waveform (minus the deadtime fill points)
is calculated.
Numerical
simulations of ESEEM are performed for one or more electron-nuclear couplings
with the identity of each nucleus supplied by the user, corresponding to the
parameters: gN and I.
This information can often be gleened from a
qualitative interpretation of the spectrum, or with the aid of multiple
microwave frequency/B0 experiments. In the case of
unidentified nuclei or numbers of nuclei, the most likely models can be assessed from the statistical criteria in OPTESIM for
separate simulations that incorporate different types and numbers of coupled
nuclei. A model for the combination
of the ESEEM from multiple nuclei is also specified. In the most simple case of all nuclei coupled to the same electron spin, and a single nuclear
isotope for each coupling, the envelope modulation for two-pulse and
three-pulse experiments is combined according to the product rule
expressions. When the experimental
ESEEM is known to arise from sub-populations of electron spins with different
nuclear coupling(s), the “sum rule” for combination of the individual ESEEM is
used, with a weighting factor for each population, to generate the total
ESEEM. Practical cases where the
sum rule and weighting are necessary are incomplete or heterogeneous isotopic
substitution of a nuclear site, conformational isomers of the same paramagnetic
molecule, and different paramagnetic molecules. OPTESIM allows users to specify
the particular combination rule by using a numerical expression. For example, in the case of two-pulse
ESEEM from a molecule with uniform isotope content at coupled nuclear site 1,
and heterogenous isotopic content at coupled nuclear
site 2, the combination of the individual modulation is specified by “E1*[C2,a*E2+C2,b*E2], where the Ei are
the envelope modulation and the Ci,a/b are the normalized isotope
substitution factors at site 2. The
weighting can be included in the simulation as an adjustable parameter. In cases of three-pulse experiments,
this combination is done by separately combining the modulation from the α and β manifolds.
The computation of the powder average ESEEM is achieved by using the following steps:
The simulated spectrum is matched to the filtered experimental spectrum after the following transformation,
where 
and 
are the transformed and original
simulation envelope modulation, respectively. The four real constants 
are calculated by minimizing the squared
residuals of the match between the transformed amplitude and the filtered
experimental data. The ci are
defined, as follows: c1 is the scaling factor for
the simulated waveform, c3 compensates
for modulation amplitude decay, and c2 and c4 are
offset terms for adjusting the waveform prior to, or subsequent to, the decay
operation. The scaling constants, c1, c2, and c4,
are used when the target ESEEM originates from a sample in which the
contributions to the ESEEM from nuclear couplings, other than the target
couplings of the simulation, are not known. The minimum squared residual is set as
the objective function for optimization over the adjustable parameters. For one model of number and type of
nuclear couplings and method of combination, users can specify multiple experimental
spectra under different experimental conditions (for example, τ value, B0 value for the same ge value) for a global optimization. A standardized definition of the objective function in OPTESIM
allows users to take full advantage of available Matlab optimization routines.
A Java-RMI based distributed computational framework is included in OPTESIM. This framework allows users to build a distributed system of ESEEM simulation on their own PC computer hardware resources. The orientation sampling vectors for the powder average are first divided on the local computer and sent to different remote PCs according to their computational speed. The results are then collected and averaged on the local PC.