EPR Spectra Simulation of Spin 3/2, 5/2, 7/2, 9/2 Systems

Hanqing Wu,   hanqing@csd.uwm.edu

Department of Chemistry, University of Wisconsin-Milwaukee

Milwaukee, WI 53201, USA








         The field-swept absorption spectrum of a powder distribution of spins is simulated by numerical integration and by using of matrix diagonalization.  The spin  Hamiltonian for  high-spin (S = n/2, n=3,5,7,9) systems is given by this equation:

where D and E are the axial and rhombic zero-field splitting parameters, respectively. There are n(n+1)/2 possible transitions between i and j levels (i, j = 1 to (n+1), i not equal j). It is needed to set the i and j values, not all transitions are necessary to simulate. Figure A, figure B shows the simulated EPR spectra of S=3/2, 5/2, 7/2, 9/2 systems with D=1 cm-1, E=0 systems, and the transition between 1 and 2.  Different conditions with different spin system and different D and E values are also can be simulated.


       Electron paramagnetic resonance (EPR), also known as electron spin resonance (ESR) and electron magnetic resonance (EMR), is the name given to the process of resonant absorption of microwave radiation by paramagnetic ions or molecules, with at least one unpaired electron spin, and in the presence of a static magnetic field (see "What is EPR"). It has a wide range of applications in chemistry, physics, biology, and medicine: it may be used to probe the "static" structure of solid and liquid systems, and is also very useful in investigating dynamic processes.

      In many application of EPR spectroscopy, only spectra of randomly oriented samples (powders or glasses) can be obtained. Very few powder spectra of S=5/2 systems have been analyzed. Some EPR simulation programs are only used for S=5/2 or 3/2 system. In this poster, various spin systems (S=3/2, 5/2, 7/2, 9/2) will be considered and simulated by only one universal EPR simulation program, it takes the advantage of previous EPR simulation programs, and may be applied for on-line EPR simulation on internet like other softwares.

      Computer simulation is based on the spin Hamiltonian in the following equation:


where all zero-field splitting terms higher than second order are neglected. D and E are axial and rhombic zero-field splitting parameters. Part of the simulated EPR spectra will be presented here.


       Figure 1 shows the comparison among different spin systems (S=3/2, 5/2, 7/2, 9/2) when D= 1 cm-1 and E/D=0.0, only 1 to 2 levels transition can be obtained, other transitions ( from i to j, i and j equal 1 to 2S+1, i not equal j) are not much contributed to the absorption (or dabsorption/dH). When E/D varies, the absorption position is shifted and other transitions (like 3 to 4 levels) are also contributed to the absorption.


 Figure 1.  EPR spectra of Spin 3/2, 5/2, 7/2, 9/2 system when D=1 cm-1, E/D=0.0, 1 to 2 transition.

1. S=3/2 system

        Figure 2 shows the comparison of simulated EPR spectra of different E/D values at S=3/2, D=1.0 cm-1 condition for 1 to 2 transition (the lowest Kramers doublet transition), we can see the signal (positions) or the effective g values are very sensitive to the E/D values (see another figure). The EPR spectrum of the nitric oxide complex of ferrous rPAH formed by exposure of the pterin-reduced enzyme to ascorbate/nitrite under anaerobic conditions (see the original paper by T. Joseph Kappock et al) has g = 4.12 and g = 3.88, which is essentially identical to those reported for the NO adducts of isopenicillin N-synthase (g = 4.09, g = 3.95), the non-heme iron site of photosystem II (g = 4.09, g = 3.95), protocatechuate 4,5-dioxygenase (g = 4.09, g = 3.91), catechol 2,3-dioxygenase (g = 4.16, g = 3.83), soybean lipoxygenase, and the model complex [Fe(EDTA)(NO)] (g = 4.10, g = 3.90). The EPR spectrum with E/D = 0.02 in Figure 2 is the right one for these Fe-NO complexes.

       Figure 3 and Figure 4 show the absorption of 1 to 2 transition and 3 to 4 transition respectively, and the intensity or strength of 3 to 4 transition is about 10% of  that of 1 to 2 transition. 

       The D value set as 1 cm-1 has same result as D > 1 cm-1 for X-band EPR simulation. If the rhombic parameter E supposed to be zero, the value of the zero-field splitting parameter D can be evaluated from the intensity dependence of the transitions, according to the Boltzmann distribution of the two Kramers' level, (+-1/2) and (+-3/2) associated with the ground spin state S=3/2.  


Figure 2.  Simulated EPR spectra of S=3/2 within small range of E/D ( from 0.01 to 0.03) at D=1.0 cm-1 for 1 to 2 transition


  Figure 3. Simulated EPR spectra of S=3/2 with different E/D values at D=1.0 cm-1 for 1 to 2 transition.

  Figure 4. Simulated EPR spectra of S=3/2 with different E/D values at D=1.0 cm-1 for 3 to 4 transition.

2.  S=5/2 system

         Figure 5 is an example of EPR simulation for S=5/2, D=1.0 cm-1, E/D=0.03 condition, different transitions (1 to 2, and 3 to 4 are marked on the figure). Only the 1 to 2 transition largely contribute the whole simulated EPR spectra (marked as "sum" in Figure 5), transition 5 to 6 is almost negligible (not ploted in Figure 5). Many heme or non-heme proteins have a high-spin (S=5/2) electron paramagenetic resonance (EPR) signal composed of at least one component. If the D and E/D values are given for S=5/2 species, the simulated EPR spectra can be obtained.


   Figure 5.  Simulated EPR spectra of S=5/2 with D=1.0 cm-1 and E/D=0.03

           When E/D value increase, the intensity or strength of 3 to 4 transition will also increase. After E/D > 0.1, the 3 to 4 transition becomes largely contribution to the absorption.The effective g value is much sensitive to the E/D value, see Figure 6 in the following (the relationships between g values and E/D value can be see from another figure published elsewhere).

    Figure 6. Simulated EPR spectra of S=5/2 within narrow range of E/D at D=1.0 cm-1 for transition from 3 to 4.

          By using the edited program, different linewidths and lineshapes (Lorentzian and Gaussian shapes) can be selected. Figure 7 is an example of EPR spectra at different selection of linewidth.

        Figure 7. Simulated EPR spectra of S=5/2 with different linewidth (W values, here set  Wx = Wy = Wz = W) at D=1.0 cm-1 and E/D=0.15 for transition from 3 to 4.

       Figure 8 shows that when the D values is large than the frequency, the absorption is almost the same, when the D value close to the frequency, the absorption position is shifted. When E/D=0.33, the intensity of transition from 1 to 2 is negilible(see another figure).

      Figure 8. Simulated EPR spectra of S=5/2 with different D values at E/D=0.33 for transition from 3 to 4.

3. S=7/2 system

       Spin S=7/2 system may be obtained in mutiple metal systems. Figure 9 is an example of EPR simulation for S=7/2 with D=1.00 cm-1 and E/D=0.1, we can see the intensity of transition from 3 to 4 is largely contribute to the simulated EPR spectra ("sum" in Figure 9), other transition is negilible for contribution to the simulated EPR spectra.

     Figure 9.  Simulated EPR spectra of S=7/2 with D=1.0 cm-1 and E/D=0.1

     The same results can obtained, when E/D changes, the absorption position and strength or intensity are also changed (see Figure 10 below). The g~5 signal can be seen from Figure 10.

     Figure 10.  Simulated EPR spectra of S=7/2 with different E/D values at D=1.0 cm-1

4. S=9/2 system

        S=9/2 system may be obtained from the coupling between S=5/2 and S=2. Different conditions for simulation of S=9/2 system can also be selected as above (see another figure for relationships between g values and E/D value). Figure 11 still shows the sensitivty of E/D values affect the simulated EPR spectra, and Figure 12 shows one of many simulated EPR spectra at  particular condition (D=1.0 cm-1, E/D=0.1). The signal at g~7 is appeared on the simulated EPR spectra

  Figure 11. Simulated EPR spectra of S=9/2 with different E/D values at D= 1.0 cm-1

  Figure 12. Simulated EPR spectra of S=9/2 with E/D=0.1 at D=1.0 cm-1


1.  When E/D=0.0, D > frequency ( consider X-band here), only transition from 1 to 2 can be observed known from EPR simulation, and the g value is at 2S+1 (for instance S=5/2, the g value is 6). When E/D=1/3, g=4.3 signal can be observed for S=5/2 system.

2.  When E/D increase from 0 upto 1/3, other transitions (like transition from 3 to 4) are also observable, when E/D close to 1/3, the transition from 1 to 2 is negibilable; The simulated EPR spectra is very sensitive to the E/D value for all the spin systems.

3. Super spin systems (S=7/2,9/2) can also contribute to the absorption of g value between 5 and 7, in which the S=5/2 system with large D value (large than the frequency) and small value of E/D is normally appeared. The S=3/2 system with large D value and small value of E/D can appear the EPR sighnal between 3 and 5.


1. Data Input and Data Output

2. Part of the program and the subroutines used


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5. T.J. Kappock, P.C. Harkins, S. Friedenberg & J.P. Caradonna, Spectroscopic and Kinetic Properties of Unphosphorylated Rat Hepatic Phenylalanine Hydroxylase Expressed in Escherichia coli: Comparison of resting and activated states",  J. Biol. Chem. 270, 30532-30544 (1995). Abstract Full Text [Medline]

6. Mark J. Nelson, The Nitric Oxide Complex of Ferrous Soybeans Liopxygenase-1, J. Biol. Chem. 262, 12137-12142(1987).

7. Hanqing Wu, EPR Spectra Simulation of Anisotropic Spin 1/2 System, WATOC96, E-Posters #1 at  http://www.ch.ic.ac.uk/watoc/abstracts/.

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