This document discusses using a master equation approach to simulate electron spin resonance (ESR) spectral lineshapes. It compares using 6-state and 48-state stochastic models to represent rotational diffusion, an important relaxation process in ESR. Simulated spectra from both models capture the main spectral features but the 48-state model provides more detail, especially at higher dispersion. The results help establish criteria for selecting appropriate models to faithfully reproduce ESR lineshapes over a wide range of transition rates.

1. Model Selection in a Master Equation Approach
to Spectral Lineshape Simulation in
Magnetic Resonance
1,2
Oleksandr Kazakov, 1,3
Keith Earle
1
-University at Albany, Physics Department,1400 Washington Ave., Albany, NY 12222 (USA)
2
-Wajam, Inc.4115 Boul St-Laurent #300 Montr´eal, QC H2W 1Y8 (Canada)
3
-ACERT, Cornell University, Ithaca, NY 14853 (USA) (email: kearle@albany.edu)
Introduction
Simulations of magnetic resonance spectra are often performed by choosing a model for spectral
relaxation and discretizing the model for computational convenience. One relaxation model of partic-
ular relevance for Electron Spin Resonance (ESR) is rotational diffusion. Historically, eigenfunction
expansions have been the method of choice for the numerical discretization step[1]. An alternative
approach[2], based on a master equation discretization of the relaxation process has features in com-
mon with contemporary molecular dynamics methods[3]and offers some complementary perspectives
to the eigenfunction approach typically used. We have shown in previous work that spectra may be
profitably analyzed by using concepts derived from information geometry[4]. In the work reported
on here, we explore the question of how many discrete states are necessary to faithfully reproduce an
ESR lineshape.We compare our simulations to experimental spectra of nitroxide spin labels whose
lineshape is modelled by isotropic rotational diffusion over a broad range of diffusion rates. These
results will help to establish model selection criteria for further development of molecular dynamics-
based approaches to lineshape simulation.
Theoretical Background
The line shape calculation in the Kubo and Anderson approach was generalized by Blume [2]. His
expression for the spectral profile is the following:
P(ω) =
2
Γ(2I0 + 1) m1m0,m1 m0
I1m1|H(−)|I0m0
ab
pa I0m0I1m1a|L−1|I0m0I1m1b I0m0|H(+)|I1m1 (1)
In this expression I0 is the nuclear spin of the ground state and I1 is the spin of the excited state,a
and b are states of the stochastic model function f(t) and pa is the a priori probability of stochastic
transition between states. L is a Liouville super matrix which maps allowed transitions into each
other. Its matrix elements can be found from the following operator expression:
L = p(ω)ˆ1 − W − i
j
V ×
j Fj (2)
One part of the problem is to determine an appropriate W that models the stochastic process that in-
duces transitions between quantum states. Another part involves quantifying the significant electron
and nuclear spin interactions and couplings to the local environment. In many cases for EPR exper-
iments, the electron spin Zeeman and Hyperfine interactions suffice for treating the most significant
interactions, particularly for nitroxide spectra. Hamiltonian can be written as:
ˆHeff = ˆS · gS · B + ˆS · A · ˆI (3)
Space part is represented by gS and A tensors that are defined in their own and unique principal axis
frames(P) that are fixed to each other by the molecular frame. Magnetic field and spin operators
as well as their product are defined in laboratory frame. In order to compute EPR spectra or reso-
nance frequencies both tensors have to be defined in the laboratory frame. Relation between reference
frames is given on Fig.1.
In this work, we also model the magnetic interactions by a dipolar Hamiltonian
H = −g0βH · (S(1) + S(2)) +
g2
0β2
r3
(S(1) · S(2) − 3(S(1) · n)(S(2) · n)). (4)
Here, S(1) and S(2) are the spin angular momenta of the magnetic resonance active dipole moments,
g0 is the rotationally invariant contribution to the Zeeman interaction, and β is the Bohr Magneton.
In addition, r12 is the displacement vector pointing from the first to the second dipole, r = |r12| and
n = r12/r. In the presence of motion of the dipoles, this orientation will be a stochastic function of
time. The stochastic process is modeled by a master equation approach in this work. It is important
to note that the second term of the Hamiltonian in Equation 4 has the same form as the anisotropic
contributions to the hyperfine interaction in Equation 3. The time series defined by the stochastic
process leads to fluctuations in the energy levels in the system, and thus lifetime broadening effects.
These qualitative considerations can be made quantitative within the context of the models studied by
our group, and typical results are discussed below.
Figure 1: Example of reference frames that define structural and dynamic properties of the spin-labelled molecules[5].
Computationally, the problem involves algebraic inversion of large sparse matrices whose size is de-
termined by the quantum mechanical and stochastic indices La,b,m0,m0,m1,m1
.
Results
Figure 2: In order to properly represent stochastic interaction a suitable number of allowed quantum state transitions have
to be determined. In this work we compare 6 and 48 transition states models. The 6 state model treats transitions among
orientations defined by the vertices of an octahedron with allowed transitions shown in 2a. In 2b, we show adjacency
matrix for a decorated hypersphere constructed from the direct product of a base icosahedron and a circle decorated by
the vertices of a square.
Figure 3: Computed high field EPR spectra arising from the isotropic stochastic modulation of the quadrupole moment of
an 14
N nucleus contained within the nitroxide spin label. The red line is the spectrum arising from the 6 state model. The
blue line is the simulation arising from 48 state model. Transition rates are (a)W=0.1,(b) W=0.3,(c)W=1 and (d)W=10
Figure 4: Example of High Field EPR line shapes for interacting I = 1/2 nucleus with S = 1/2 electron spin(left)
and I = 1 nucleus with S = 1/2 electron(right) with anisotropic g and A tensors representing 6 state model. Spectral
contributions from Zeeman transitions are split into additional lines by the Hyperfine interaction. Transition rates are
(a)W=0.1,(b) W=1,(c)W=10 and (d)W=100. At high transition rates all interactions average out and spectral lines are
centered at two for I = 1/2 and three for I = 1 Zeeman frequencies.gx = 2.0089, gy = 2.0061,gz = 2.00032 and scaled
Ax = 5.5,Ay = 4.5, Az = 34.
Figure 5: Computed high field EPR spectra for 14
N (5a) and 15
N (5b) nitroxide spin-labels arising from 6(blue line) and
48(red line) state models in medium transition rate regime. Transition rate W=5.
Conclusions
Stochastic models can be used for simulating magnetic resonance line shapes over the wide range of
transitions rates. The Kubo-Anderson model has been extended to treat coupled electron and nuclear
spins suitable for the simulation of nitroxide EPR spectra. The goodness of dynamic system repre-
sentation was compared on the basis of 6 and 48 stochastic state models. It has been observed that
both the 6 and 48 state models capture the main spectral features, but the 48 state model captures
more spectral detail when the features are more broadly dispersed. With these preliminary results in
hand we can now adapt the methods developed in the Earle group to assess parameter sensitivity more
quantitatively.
References
[1] J. Freed, et al. J. Phys. Chem. 75 3385 (1971).
[2] M. Blume, Phys. Rev. 174(2) 351 (1969).
[3] D. Sezer, et al. J. Am. Chem. Soc. 131 2597 (2009).
[4] K. Earle, et al. Appl. Magn. Reson. 37 865 (2010).
[5] Franck et al,The Journal of Chemical Physics. 142 21 (2015).