Simulation of field- and frequency-sweep cw EPR spectra in the slow-motional regime.
chili(Sys,Exp) spec = chili(...) [x,spec] = chili(...) ... = chili(Sys,Exp,Opt)
See also the user guide on how to use chili
.
chili
computes cw EPR spectra in the slow-motional regime. The simulation is based on solving the Stochastic Liouville equation in a basis of rotational eigenfunctions. chili
supports arbitrary spin systems.
chili
takes up to three input arguments
Sys
: static and dynamic parameters of the spin system
Exp
: experimental parameters
Opt
: options and settings
If no input argument is given, a short help summary is shown (same as when typing help chili
).
Up to two output arguments are returned:
x
: magnetic field axis, in mT, or frequency axis, in GHz
spec
: spectrum
If no output argument is given, chili
plots the spectrum.
chili
can simulate field-swept spectra as well as frequency-swept spectra. For field-swept spectra, specify Exp.mwFreq
(in GHz), for frequency-swept spectra specify Exp.Field
(in mT).
chili
has two parallel implementations. One is fast, but restricted to S=1/2 with up to two nuclei. The other one works for general spin systems, but is much slower. chili
automatically chooses the faster method, unless instructed otherwise (see options).
Sys
is a structure containing the parameters of the spin system. For the fast method, the used parameters are g
, gFrame
, Nucs
, A
, and AFrame
. The nuclear quadrupole interaction (specified in Q
and QFrame
) is neglected. For the slow method, all spin Hamiltonian parameters are taken into account. See the documentation on spin system structures for details.
For simulating a multi-component mixture, Sys
should be a cell array of spin systems, e.g. {Sys1,Sys2}
for a two-component mixture. Each of the component spin systems should have a field weight
that specifies the weight of the corresponding component in the final spectrum.
Sys
should contain dynamic parameters relevant to the motional simulation. One of the field tcorr
, logtcorr
, Diff
or logDiff
should be given. If more than one of these is given, the first in the list logtcorr
, tcorr
, logDiff
, Diff
takes precedence over the other(s).
tcorr
[txy tz]
: anisotropic diffusion with axial diffusion tensor
[tx ty tz]
: anisotropic diffusion with rhombic diffusion tensor
For example,
Sys.tcorr = 1e-9; % isotropic diffusion, 1 ns correlation time Sys.tcorr = [5 1]*1e-9; % axial anisotropic diffusion, 5 ns around x and y axes, 1 ns around z Sys.tcorr = [5 4 1]*1e-9; % rhombic anisotropic diffusion
The axes x, y, and z refer to a molecule-fixed frame in which the diffusion tensor is diagonal (the "diffusion frame", see DiffFrame
field).
Instead of tcorr
, Diff
can be used, see below. If tcorr
is given, Diff
is ignored. The correlation time tcorr
and the diffusion rate Diff
are related by tcorr = 1./(6*Diff)
.
logtcorr
tcorr
, logDiff
and Diff
are ignored.tcorr
for least-squares fitting with esfit.
Diff
[Dxy Dz]
gives axial tensor [Dxy Dxy Dz]
[Dx Dy Dz]
Diff
is ignored if logtcorr
, tcorr
or logDiff
is given.
logDiff
Diff
. If given, Diff
is ignored.Diff
for least-squares fitting with esfit.
DiffFrame
[a b c]
containing the Euler angles, in radians, describing the orientation of the rotational diffusion tensor in the molecular frame. DiffFrame
gives the angles for the transformation of the molecular frame into the rotational diffusion tensor eigenframe. See frames for more details.
In addition to the rotational dynamics, convolutional line broadening can be included using Sys.lw
or Sys.lwpp
.
lwpp
GaussianPP
.[GaussianPP LorentzianPP]
.lwpp
takes precedence over lw
.
lw
GaussianFWHM
.[GaussianFWHM LorentzianFWHM]
.lwpp
takes precedence over lw
.
If there is an ordering potential, it should be given in Sys.lambda
.
lambda
[lambda20 lambda22 lambda40 lambda42]
, corresponding to the four linear combination coefficients λ2,0, λ2,2, λ4,0, and λ4,2 for the ordering potential
U(Ω) = - kB T ΣLλL,0DL00 - kB T ΣLλL,2(DL0,2+DL0,-2), where the DLM,K are Wigner D functions.
If you give less than five numbers, the omitted ones are assumed to be zero.
The frame of the ordering potential is assumed to be collinear with that of the rotational diffusion tensor.
For details about this type of ordering potential, see K.A. Earle & D.E. Budil, Calculating Slow-Motion ESR Spectra of Spin-Labeled Polymers, in: S. Schlick: Advanced ESR Methods in Polymer Research, Wiley, 2006.
For concentrated solutions, it is possible to include Heisenberg exchange:
Exchange
The experiment structure Exp
contains all parameters relating to the experiment. These settings are identical for all cw EPR simulation functions (pepper, chili, garlic). See the page on cw EPR experimental parameters.
Opt
, the options structure, collects all settings relating to the algorithm used and the behaviour of the function. The most important settings are:
Verbosity
chili
prints to the screen. If Opt.Verbosity=0
, is is completely silent. 1 prints details about the progress of the computation.
LLKM
[evenLmax oddLmax Kmax Mmax]
chili
automatically picks a medium-sized basis. This is adequate for many, but certainly not all, cases. In general, the basis needs to be larger for slower motions and can be smaller for faster motions. It is stronly advised to vary these settings to check whether the simulated spectrum is converged.
nKnots
Sys.lambda
are large.
LiouvMethod
'Freed'
and 'general'
. The first method is very fast, but limited to one electron spin with S=1/2 and up to two nuclei. Also, the nuclear quadrupole interaction is neglected. On the other hand, the general method works for any spin system, but is significantly slower. By default, chili
uses the fast method if applicable and falls back to the general method otherwise.
PostConvNucs
Sys.Nucs = '14N,1H,1H,1H'
and Opt.PostConvNucs = [2 3 4]
, the only the nitrogen is used in the SLE simulation, and all the protons are included via post-convolution.
Post-convolution is useful for including the effect of nuclei with small hyperfine couplings in spin systems with many nuclei that are too large to be handled by the SLE solver. Nuclei with large hyperfine couplings should never be treated via post-convolution. Only nuclei should be treated by post-convolution for which the hyperfine couplings (and anisotropies) are small enough to put them in the fast-motion regime, close to the isotropic limit, for the given rotational correlation time in Sys.tcorr
etc.
The cw EPR spectrum of a slow tumbling nitroxide radical can be simulated with the following lines.
Sys.g = [2.008 2.0061 2.0027]; Sys.Nucs = '14N'; Sys.A = [16 16 86]; % MHz Sys.tcorr = 1e-9; % = 1 ns Exp.mwFreq = 9.5; chili(Sys,Exp);
chili
solves the Stochastic Liouville equation (SLE) using a basis set of normalized Wigner rotation functions DLK,M(Ω) with -L ≤ K,M ≤ L to represent the orientational distribution of the spin system. The number of basis functions is determined by maximum values of even L, odd L, K and M. The larger these values, the larger the basis and the more accurate the spectrum.
chili
computes the frequency-swept EPR spectrum, and then converts it to a field-swept spectrum using a first-order approximation. This is appropriate for most organic radicals. It is somewhat inaccurate for transition metal complexes, e.g. Cu2+ or VO2+. For the diffusion, both secular and nonsecular terms are included.
If the spin system has S=1/2 and contains no more than two nuclei, chili
by default uses a fast method to construct the Liouvillian matrix that is based on explicit expressions for the matrix elements (Opt.LiouvMethod='Freed'
). For all other cases, a more general matrix-level method is used to construct the Liouvillian matrix.
Post-convolution works as follows: First, the SLE is used to simulate the spectrum of all nuclei except those marked for post-convolution. Next, the isotropic stick spectrum due to all post-convolution nuclei is simulated and convolved with the SLE-simulated spectrum to give the final spectrum.
For full details of the various algorithms see