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Application of a new rotation matrix in dynamic solid state ESR and NMR lineshape simulation
Computers Chem. Printed in Great
Vol.
17. No. 3, pp. 319-321, rights reserved
1993
0097~&m/93
Britain. All
$6.00
f 0.00
Copyright 0 1993 Pergamon Press Ltd
APPLICATION
NOTE
APPLICATION OF A NEW ROTATION MATRIX DYNAMIC SOLID STATE ESR AND NMR LINESHAPE SIMULATION
IN
SHENGDAI Department
of Chemistry, (Xeceived
Abstrati-A lineshapes.
University
of Tennessee,
Knoxville,
TN 37996-1600, U.S.A.
18 May 1992; in revised form 26 May 1993)
new rotation matrix was applied to the computer simulation of dynamic ESR and NMR A cubic jump model was chosen to illustrate the use of this rotation matrix.
INTRODUCTION
Computer simulation of experimental ESR and NMR lineshapes is very important in interpreting experimental spectra and particularly in extracting dynamic information about a spin system (Alexander et al., 1974; Wemmer et al., 198 1; Braram ef al., 1976; Mehring, 1983; Greenfield et al., 1987; Cory & Ritchey, 1989; Kevan & Schlick, 1986; Matsushita et al., 1989). Many dynamic processes affect experimental lineshapes. These include Brownian motion, diffusion process, and discrete rotation of an anisotropic spin center. Only the discrete rotation will be discussed in this paper. The most widely used method for computer simulation of this process has involved the use of Euler rotation matrix to conduct coordinate transformation among different sites (Braram et ai., 1976; Mehring, 1983; Greenfield et al., 1987; Cory & Ritchey, 1989; Kevan 8~ Schlick, 1986). Although determination of Euler angles for a rotation around a coordinate axis is simple, it is difficult and tedious to calculate in the case of the rotation around an arbitrary axis. This also introduces some complications in computer programming. In order to solve the rotation matrix about a noncoordinate axis, another formalism is developed in this paper for the rotational jump simulation. The essence of the method is that it is possible to write down directly a matrix representing a rotation through any angle about any arbitrary axis passing through the original coordinate. In the case that the rotation axis coincides with a coordinate axis, the usual Euler rotation matrices are of course reproduced. PRINCIPLES This rotation matrix formalism is not new and has been proven very convenient and useful in such fields as mechanics and crystallographic coordinate transformation. The derivation has been nicely given in the literature (Palazzolo, 1976; Mathews, 1976;
Koehler & Trickey, results:
1978). Here we only give the final
1, = cos ui cos cj + [.+ sin B cos uk - cos oI x cos ajcos S](l - 6,) + 6,-(sin* u,cos Q),
(1)
where k is a dummy index such that i #k #j, and u, , u2, q are the angles between the rotation axis and original coordinates. Also here S = the Kronecker delta symbol, and + = the Levi-Cevita density. Here 8 is tbe rotation angle and is considered positive when measured counterclockwise. Figure 1 gives a diagrammatic illustration of the angular parameters involved in the above equation. In the following, we illustrate the application of the above rotation matrix by the simulation of an ESR lineshape for a discrete rotation of a g-anisotropic system. The calculation of the spectrum for a g-anisotropic radical species undergoing a dynamic conformation rotation is normally based on the formalism of the modified Block equations (McConnell, 1958). These equations are obtained by adding kinetic terms that express the changes in magnetization due to interconversions among different rotamers or sites. The equation for the complex transverse magnetization of rotamer A, MA, is given by (McConnell, 1958)
with 1 a,4 =T-i(w,-co). 24
(3)
TZ,, is the spin-spin relaxation time of rotamer A; CD” is the resonance frequency for rotamer A; y is the magnetogyric ratio; H, is the microwave field; M,,, is the magnetization of rotamer A along the z-axis and it is used in place of MZA, in absence of saturation and for relatively short spin-lattice relaxation times T,; and k, = l/r,; here t, is the mean lifetime of rotamer
319
Application Note
320
A. The last summation term represents a sum over ah rotamers (x), except that for which the equation is written. The total complex transverse magnetization M is sum over all rotamers. M = i
M,.
(4)
The imaginary part of M gives the absorption line shape. A general solution of the magnetization A4 which includes interconversion between N equally populated rotamers has been given. The imaginery part G of the transverse magnetization is given by (Sullivan & Bolton, 1970)
x=1
G=
wheref, = (N + a,r)-I. given by
if,) i-1 The resonant
(5)
N(1 -
frequency
o, is
where y = 17.6 x IO6 s-‘G-’ (McCalley et al., 1972). The expression for g,,+) is (Wertz & Bolton, 1972; Libertini & Griffith, 1970)
where u and u are unit vectors in the direction of an applied magnetic field H and its transpose. The underlined d and gz are the rotation matrix given in equation (1) and second rank g tensor respectively. The corresponding total powder spectrum (S) can be obtained by integrating solid angles or numerically approximated as (VanderHart & Gutowsky, 1968):
In the following, a cubic jump model was chosen to illustrate the use of this rotation matrix. This model involves an interconversion between three
I.2
3240.2
3290.2
3340.2
3290.2
FIELD (gauss) --c
Fig. 2. Simulated ESR spectra of a spin - f system with an axial symmetric g-tensor Cp,= gv = 2.0023 and g, = 2.0058) at various rotation lifetimes (7s). Microwave frequency is assumed to be 9500 MHz.
rotamers by rotation about an axis making equal angles with the three principal axis of a g tensor. Clearly in this case, u, = u2 = c3 = cos-’ (3 - “I) and 0 = 120. From equation (I), the rotation matrix is given by 0 1 0 0
0
1.
[1 0
0I
Figure 2 gives the calculated ESR spectra changing with the mean lifetime t. As expected, the theoretical spectra turn to an isotropic line when the rate of rotation motion is increased. Finally, it should be pointed out that this rotation matrix can be also used to replace traditional rotation matrices in other coordinate transformation problems in NMR and ESR. Program uvaikzbility-The complete program is available from the author on request.
REFERENCES Alexander S., Baram A. % Luz Z. (1974) Mol. Phys. 27,441. Braram A., Luz Z. & Alexander S. (1976) J. Chem Phys. 64,
Fig. 1. Definition of angles used in the rotation matrix.
4321. Cory D. G. & Ritchey W. M. (1989) J. Magn. Reson. 81, 383. Greenfield M. S., Ronemus A. D., Vold R. L. & Vold R. R. (1987) J. Magn. Reson. 72, 89. Kevan L. & Schlick S. (1986) J. Phys. Chem. 90, 1998.
Application Koehler T. R. & Trickey S. B. (1978) Am. J. Phys. 46, 650. Libertini L. J. & Griffith 0. H. (1970) .I. C/rem Phys. 53, 1359. Mathews J. (1976) Am. J. Phys. 44, 1210. Matsushita M., Momose T., Kato T. & Shida T. (1989) Chem. Phys. .Let&. 161, 461. McCalley R. C., Shimshick E. J. & McConnell H. M. (1972) Chem. Phys. L&L 13, 115. McConnell H. M. (1958) J. Chem. Phys. 28, 430. Mehring M. (1983) Principles of High Resolution NMR in Soli&. Berlin, Springer.
Note
321
Palazzlo A. (1976) Am. J. Phys. 44, 63; 44, 490 (errata). Sullivan P. D. & Bolton J. R. (1970) ddv. Magn. Resort. 4, 39. VanderHart D. L. & Gutowsky H. S. (1968) J. Chem. Phys. 49, 261. Wemmer D. E., Ruben D. J. & Pines A. (198 I) J. Am. Chem. sot. 103, 28. Wertz J. E. & Bolton J. R. (1972) Electron Spin Resonance Elemenlary Theory and PracGcal Applications. McGrawHitt, New York.
Vol.
17. No. 3, pp. 319-321, rights reserved
1993
0097~&m/93
Britain. All
$6.00
f 0.00
Copyright 0 1993 Pergamon Press Ltd
APPLICATION
NOTE
APPLICATION OF A NEW ROTATION MATRIX DYNAMIC SOLID STATE ESR AND NMR LINESHAPE SIMULATION
IN
SHENGDAI Department
of Chemistry, (Xeceived
Abstrati-A lineshapes.
University
of Tennessee,
Knoxville,
TN 37996-1600, U.S.A.
18 May 1992; in revised form 26 May 1993)
new rotation matrix was applied to the computer simulation of dynamic ESR and NMR A cubic jump model was chosen to illustrate the use of this rotation matrix.
INTRODUCTION
Computer simulation of experimental ESR and NMR lineshapes is very important in interpreting experimental spectra and particularly in extracting dynamic information about a spin system (Alexander et al., 1974; Wemmer et al., 198 1; Braram ef al., 1976; Mehring, 1983; Greenfield et al., 1987; Cory & Ritchey, 1989; Kevan & Schlick, 1986; Matsushita et al., 1989). Many dynamic processes affect experimental lineshapes. These include Brownian motion, diffusion process, and discrete rotation of an anisotropic spin center. Only the discrete rotation will be discussed in this paper. The most widely used method for computer simulation of this process has involved the use of Euler rotation matrix to conduct coordinate transformation among different sites (Braram et ai., 1976; Mehring, 1983; Greenfield et al., 1987; Cory & Ritchey, 1989; Kevan 8~ Schlick, 1986). Although determination of Euler angles for a rotation around a coordinate axis is simple, it is difficult and tedious to calculate in the case of the rotation around an arbitrary axis. This also introduces some complications in computer programming. In order to solve the rotation matrix about a noncoordinate axis, another formalism is developed in this paper for the rotational jump simulation. The essence of the method is that it is possible to write down directly a matrix representing a rotation through any angle about any arbitrary axis passing through the original coordinate. In the case that the rotation axis coincides with a coordinate axis, the usual Euler rotation matrices are of course reproduced. PRINCIPLES This rotation matrix formalism is not new and has been proven very convenient and useful in such fields as mechanics and crystallographic coordinate transformation. The derivation has been nicely given in the literature (Palazzolo, 1976; Mathews, 1976;
Koehler & Trickey, results:
1978). Here we only give the final
1, = cos ui cos cj + [.+ sin B cos uk - cos oI x cos ajcos S](l - 6,) + 6,-(sin* u,cos Q),
(1)
where k is a dummy index such that i #k #j, and u, , u2, q are the angles between the rotation axis and original coordinates. Also here S = the Kronecker delta symbol, and + = the Levi-Cevita density. Here 8 is tbe rotation angle and is considered positive when measured counterclockwise. Figure 1 gives a diagrammatic illustration of the angular parameters involved in the above equation. In the following, we illustrate the application of the above rotation matrix by the simulation of an ESR lineshape for a discrete rotation of a g-anisotropic system. The calculation of the spectrum for a g-anisotropic radical species undergoing a dynamic conformation rotation is normally based on the formalism of the modified Block equations (McConnell, 1958). These equations are obtained by adding kinetic terms that express the changes in magnetization due to interconversions among different rotamers or sites. The equation for the complex transverse magnetization of rotamer A, MA, is given by (McConnell, 1958)
with 1 a,4 =T-i(w,-co). 24
(3)
TZ,, is the spin-spin relaxation time of rotamer A; CD” is the resonance frequency for rotamer A; y is the magnetogyric ratio; H, is the microwave field; M,,, is the magnetization of rotamer A along the z-axis and it is used in place of MZA, in absence of saturation and for relatively short spin-lattice relaxation times T,; and k, = l/r,; here t, is the mean lifetime of rotamer
319
Application Note
320
A. The last summation term represents a sum over ah rotamers (x), except that for which the equation is written. The total complex transverse magnetization M is sum over all rotamers. M = i
M,.
(4)
The imaginary part of M gives the absorption line shape. A general solution of the magnetization A4 which includes interconversion between N equally populated rotamers has been given. The imaginery part G of the transverse magnetization is given by (Sullivan & Bolton, 1970)
x=1
G=
wheref, = (N + a,r)-I. given by
if,) i-1 The resonant
(5)
N(1 -
frequency
o, is
where y = 17.6 x IO6 s-‘G-’ (McCalley et al., 1972). The expression for g,,+) is (Wertz & Bolton, 1972; Libertini & Griffith, 1970)
where u and u are unit vectors in the direction of an applied magnetic field H and its transpose. The underlined d and gz are the rotation matrix given in equation (1) and second rank g tensor respectively. The corresponding total powder spectrum (S) can be obtained by integrating solid angles or numerically approximated as (VanderHart & Gutowsky, 1968):
In the following, a cubic jump model was chosen to illustrate the use of this rotation matrix. This model involves an interconversion between three
I.2
3240.2
3290.2
3340.2
3290.2
FIELD (gauss) --c
Fig. 2. Simulated ESR spectra of a spin - f system with an axial symmetric g-tensor Cp,= gv = 2.0023 and g, = 2.0058) at various rotation lifetimes (7s). Microwave frequency is assumed to be 9500 MHz.
rotamers by rotation about an axis making equal angles with the three principal axis of a g tensor. Clearly in this case, u, = u2 = c3 = cos-’ (3 - “I) and 0 = 120. From equation (I), the rotation matrix is given by 0 1 0 0
0
1.
[1 0
0I
Figure 2 gives the calculated ESR spectra changing with the mean lifetime t. As expected, the theoretical spectra turn to an isotropic line when the rate of rotation motion is increased. Finally, it should be pointed out that this rotation matrix can be also used to replace traditional rotation matrices in other coordinate transformation problems in NMR and ESR. Program uvaikzbility-The complete program is available from the author on request.
REFERENCES Alexander S., Baram A. % Luz Z. (1974) Mol. Phys. 27,441. Braram A., Luz Z. & Alexander S. (1976) J. Chem Phys. 64,
Fig. 1. Definition of angles used in the rotation matrix.
4321. Cory D. G. & Ritchey W. M. (1989) J. Magn. Reson. 81, 383. Greenfield M. S., Ronemus A. D., Vold R. L. & Vold R. R. (1987) J. Magn. Reson. 72, 89. Kevan L. & Schlick S. (1986) J. Phys. Chem. 90, 1998.
Application Koehler T. R. & Trickey S. B. (1978) Am. J. Phys. 46, 650. Libertini L. J. & Griffith 0. H. (1970) .I. C/rem Phys. 53, 1359. Mathews J. (1976) Am. J. Phys. 44, 1210. Matsushita M., Momose T., Kato T. & Shida T. (1989) Chem. Phys. .Let&. 161, 461. McCalley R. C., Shimshick E. J. & McConnell H. M. (1972) Chem. Phys. L&L 13, 115. McConnell H. M. (1958) J. Chem. Phys. 28, 430. Mehring M. (1983) Principles of High Resolution NMR in Soli&. Berlin, Springer.
Note
321
Palazzlo A. (1976) Am. J. Phys. 44, 63; 44, 490 (errata). Sullivan P. D. & Bolton J. R. (1970) ddv. Magn. Resort. 4, 39. VanderHart D. L. & Gutowsky H. S. (1968) J. Chem. Phys. 49, 261. Wemmer D. E., Ruben D. J. & Pines A. (198 I) J. Am. Chem. sot. 103, 28. Wertz J. E. & Bolton J. R. (1972) Electron Spin Resonance Elemenlary Theory and PracGcal Applications. McGrawHitt, New York.