This tutorial explains how to simulate slow-motion cw EPR spectra of S=1/2 systems using EasySpin's function chili. For full information, see the reference page on chili.

• Invoking the simulation function chili
• Specifying the static parameters
• Dynamic parameters
• The orienting potential
• Experimental settings
• MOMD vs single orientation
• Simulation parameters

• Systems with more than one nucleus

Slow-motion cw EPR spectra of S=1/2 systems are computed by the EasySpin function chili.

chili(Sys,Exp)

It is called with two arguments. The first argument Sys tells chili all about the static and dynamic parameters of the spin system. The second argument Exp gives the experimental parameters.

If no output argument is given, chili plots the computed spectrum. But it can also return one or two outputs. (Don't forget the semicolon at the end of the line to suppress output to the command window.)

Spec = chili(Sys,Exp);
[Field,Spec] = chili(Sys,Exp);

The outputs Field and Spec are vectors containing the magnetic field axis and the spectrum, respectively. If these are requested, chili does not plot the spectrum.

Doing a simulation only requires a few lines of code. A simple example is

Sys = struct('g',[2.008,2.006,2.003],'Nucs','14N','A',[20,20,85]);
Sys.tcorr = 4e-8;
Exp = struct('mwFreq',9.5);
chili(Sys,Exp);

The first line defines the spin system, a nitroxide radical with anisotropic g and A tensors. The second line gives the rotational correlation time of the radical. The third line specifies experimental parameters, here only the microwave frequency (The magnetic field range is chosen automatically). The fourth line calls the simulation function, which plots the result. Copy and paste the code above to your MATLAB command window to see the graph.

Of course, the names of the input and output variables don't have to be Sys, Exp, Field and Spec. You can give them any name you like, as long as it is a valid MATLAB variable name, e.g., FremySaltSolution or QbandExperiment. For convenience, thoughout this tutorial, we will use short names like Sys and Exp.

The first input argument specifies the static parameters of the paramagnetic molecule. It is a MATLAB structure with various fields giving values for the spin system parameters.

Sys.g = [2.008,2.006,2.003];
Sys.Nucs = '14N';
Sys.A = [20,20,80];  % MHz

The first line defined the g tensor of the spin system via its three principal values. chili always assumes a single unpaired electron spin S=1/2.

The field Sys.Nucs contains a string giving all the magnetic nuclei in the spin system, a nitrogen-14 in the above example. Use a comma-separated list of isotope labels to give more than one nucleus. E.g., Sys.Nucs = '14N,1H,1H' specifies one nitrogen and two different protons. See the section on multinuclear systems about details of the slow-motion simulation in that case.

Sys.A gives the hyperfine couplings in MHz (Megahertz), with three principal values on a row for each of the nuclei listed in Sys.Nucs. The following defines a hydrogen atom with a 10 MHz coupling to the unpaired electron and a ¹³C atom with a 12 MHz coupling.

Sys.Nucs = '1H,13C';
Sys.A = [10 12]; % MHz

Remember that chili (and other EasySpin functions, too), take the hyperfine coupling values to be in MHz. Often, values for hyperfine couplings are given in G (Gauss) or mT (Milltesla), so you have to convert these values. For g = 2.00232, 1 G corresponds to 2.8025 MHz, and 1 mT corresponds to 28.025 MHz. The simplest way to convert coupling constants from magnetic field units to MHz is to use the EasySpin function mt2mhz:

A_MHz = mt2mhz(A_mT);    % mT -> MHz conversion
A_MHz = mt2mhz(A_G/10);  %  G -> MHz conversion (1 G = 0.1 mT)

The orientations of the tensors relative to the molecular frame are defined in terms of Euler angles in 3-element array (see the function erot.

Sys.gpa = [0 0 0];    % Euler angles for g tensor
Sys.Apa = [0 pi/4 0]; % Euler angles for A tensor

For more details, see the documentation on spin systems.

The spin system structure also collects parameters relating to the dynamics of the paramagnetic molecules.

There are several alternative ways to specify the rate of isotropic rotational diffusion: either by specifying tcorr, the rotational correlation time in seconds

Sys.tcorr = 1e-7;   % 10^-7 s = 100 ns

or by givings its base-10 logarithm

Sys.logtcorr = -7;   % 10^-7 s = 100 ns

or by specifying the principal value of the rotational diffusion tensor (in s^-1)

Sys.Diff = 1e9;  % 1 ns^-1

Diff = 1/6/tcorr;

Sys.Diff = [1 2]*1e8;  % in Hertz

The lw field has the same meaning As for the other simulation functions garlic and pepper, the field Sys.lw can be used to specify a Gaussian and a Lorentzian broadening (FWHM, in mT)

Sys.lw = [0.05 0.01];   % [GaussianFWHM, LorentzianFWHM] in mT

For the reliability of the simulation algorithm we recommend to always use a small residual Lorentzian line width in Sys.lw

Sys.Exchange = 1e8;     % 100 MHz

A short example of an nitroxide radical EPR spectrum with exchange narrowing is

Nx = struct('g',[2.0088, 2.0061, 2.0027],'Nucs','14N','A',[16 16 86]);
Nx.tcorr = 1e-7; Nx.lw = [0 0.1]; Nx.Exchange = 1e8;
Exp = struct('mwFreq',9.5,'CenterSweep',[338, 10]);
chili(Nx,Exp);

chili can also include an orienting potential in the simulation. It is specified in the field lambda in the spin system structure. Up to five coefficients can be given, λ_2,0, λ_2,2, λ_4,0, λ_4,2, and λ_4,4, in that order. For example,

Nx.lambda = [0.3, -0.2];

indicates λ_2,0 = 0.3 and λ_2,2 = -0.2.

All experimental settings are given in the second input argument Exp, again a MATLAB structure. There are five possible fields:

Exp.mwFreq = 9.5;              % GHz, mandatory

Exp.CenterSweep = [330 20];    % mT, optional, default: automatic
Exp.Range = [320 340];         % mT, optional, default: automatic

Exp.Harmonic = 1;              % optional, default: 1
Exp.nPoints = 1024;            % optional, default: 1024

Exp.mwFreq gives the spectrometer frequency in GHz. It is the only field that has to be specified, all others are optional.

There are two ways to specify the magnetic field sweep range. Either the center and the sweep width (in mT) are given in Exp.CenterSweep, or the lower and upper limit of the sweep range (again in mT) are given in Exp.Range. In many cw EPR spectrometers, the field range is specified using a center field and a sweep width, so Exp.CenterSweep is usually the more natural choice.

Exp.CenterSweep and Exp.Range are only optional. If both are omitted, chili automatically chooses a field range large enough to accomodate the full spectrum. If both are given, chili takes the values given in Exp.CenterSweep and ignores those in Exp.Range.

The optional field Exp.Harmonic specifies the detection harmonic. It has three possible settings:

Exp.Harmonic = 0;  % absorption spectrum
Exp.Harmonic = 1;  % first derivative of the absorption spectrum (default)
Exp.Harmonic = 2;  % second derivative of the absorption spectrum

The default value for Exp.Harmonic is 1. Note that Exp.Harmonic>0 only computes the appropriate derivative of the absorption spectrum. Broadening effects due to strong field modulation have to be added separately, see below.

Yet another optional field is Exp.nPoints, giving the number of points in the simulated spectrum. If it is not given, the default 1024 is assumed. You can set any value, unlike in many EPR spectrometers, where usually only powers of 2 are available (1024, 2048, 4096, 8192).

In a frozen solution of spin-labelled protein, the protein molecules assume all possible orientations. For slow-motion spectra, this orientational distribution has to be taken into account if a orienting potential is present. If not, it is sufficient to compute only one orientation, as the spectra from all orientations are identical.

The summation of slow-motion spectra over all possible protein ("director") orientations is called the MOMD (microscopic order macroscopic disorder) model. In chili, MOMD can be activated by setting Exp.MOMD=1.

When chili uses the MOMD model for the simulation, it takes the number of orientations to include from Opt.nKnots. Often, Opt.nKnots does not have to be changed from its default setting, but if the spectrum does not appear to be smooth, Opt.nKnots can be increased. Of course, this also increases the simulation time. For quick simulations, Opt.nKnots should be minimized.

With the setting Exp.MOMD=0, only a single orientation is computed, even when an orienting potential is given. The orientation is taken from Exp.psi, which is the director tilt angle in radians, i.e. the angle between the magnetic field and the main axis of the orienting potential (which is fixed to the protein).

Opt.Verbosity = 1;     % log information

Opt.LLKM  = [24 20 10 10];

CuPc = struct('g',[2.0525 2.0525 2.1994],'Nucs','63Cu,14N','n',[1 4]);
CuPc.A = [-54 -54 -608; 52.4 41.2 41.8];
CuPc.logtcorr = -7.35;
Exp = struct('mwFreq',9.878,'CenterSweep',[330 120],'nPoints',5e3);
Opt.LLKM = [36 30 8 8];
chili(CuPc,Exp,Opt);