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ESR rigid-lattice lineshapes for a three-electronic-spin system. A triradical or “three-spin label”
JOURNAL
OF MAGNETIC
RESONANCE
65,62-8 1 (1985)
ESR Rigid-Lattice Lineshapesfor a Three-Electronic-SpinSystem. A Triradical or “Three-Spin Label” R.E.COFFMANANDA.PEZESHK Chemistry Department, University of Iowa, Iowa City, Iowa 52242 Received April 8, 1985 The ESR rigid-lattice lineshape is calculated for a randomly oriented system of molecules each bearing three weakly interacting electronic spins. Each spin has a unique g tensor and associated Zeeman interaction and ah three spins interact through an isotropic exchange interaction (J& . S2) and general magnetic point dipole-dipole interaction. The lineshape is calculated by the transition frequency method, and the resulting spectra are averaged over four octants of the unit sphere. The results show significant lineshape dependence on the value of the dihedral angle in the complexes, as well as the usual dependence on the dipolar distances. Based on these observations, we propose that a “three-spin label” would have a strong response to local conformational strain effects in membranes and macromolecules. Such a label, along with an appropriate lineshape theory, may have unique applications in paramagnetic resonance molecular structure-function studies. 0 1985 Academic Press, Inc.
INTRODUCTION
Molecules with three independent electronic spins present interesting problems in regards to the calculation of their ESR spectra. Such problems arise for certain copper complexes having two nitroxyl containing ligands and in biological macromolecules having multiple independent paramagnetic moieties. Three-spin systems containing a transition metal and two nitroxyl ligands which have been studied in the past may be grouped into the following three classes: Class A are those compounds for which the nitroxyl radicals are attached to a salicylaldehyde Schiff base ligand (1) class B are those compounds for which the nitroxyl radicals are attached to 2-hydroxy-napthaldehyde Schiff base ligands (2, 3), and Class C are the copper complexes of cyclocarboxylic acids containing nitroxyl radicals (4, 5). A “pure” three-spin system in a macromolecule (other than an S = i system) does not appear to have been identified yet, although examples of macromolecules with three or four paramagnetic moieties are well known (6). Previous quantitative considerations of three-spin ESR lineshapes begin with Kottis (7) who considered a three-site triplet problem in tribenzotriptycene including dipolar interactions, and Lemaistre and Kottis (8) who studied ESR lineshapes of spin dimers and trimers in disordered matrices, from the point of view of the effects of exciton transfer in competition with exchange and dipolar effects on the lineshape. More recently, Sagdeev et al. (5) have approximated the general shape of a three-spin AB2 system from special case considerations of spin-spin effects on the absorption spectrum turning points. In this paper, we present a general treatment of the three-spin problem 0022-2364185 $3.00 Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.
62
ESR
LINESHAPE
FOR
THREE-ELECTRON
SYSTEM
63
from the point of view of the most important spin-Hamiltonian operators which govern the largest terms in the energy and which determine the structural variables which are strongest affected by conformational changes of the molecule. These are, of course, the Zeeman operators, and the exchange and general magnetic dipolar operators. The ESR lineshapes in a single molecule with three weakly interacting electronic spins will generally be affected by slow tumbling effects as well as the spin-spin interactions which are dependent on the internal molecular structural variables. The dependence on slow tumbling appears in the lineshape problem as a coupling of the lineshape equations for solid angle C!with neighboring angles O’, a”, etc. One effect of this coupling is to increase dramatically the dimension of all matrices for the threespin problem, so we make the simplifying assumption that this coupling is turned ufl so that the lineshape calculation for any one D is independent of all other Q’s. This will be true at a sufficiently low temperature (or equivalently), a sufficiently high viscosity, so that a simpler rigid-lattice calculation may be used for the lineshape properties of the system. In many cases, this is reasonable especially if the sample at hand exists either as a powder or a very viscous sample at the temperature of interest. For such a glassy or polycrystalline sample, we calculate the ESR lineshape for all orientations of the magnetic field using a method sufficiently general as to account for all of the many magnetic interaction effects. The inclusion of the slow tumbling effects will be treated as a separate problem. There are two general approaches to this type of rigid-lattice calculation for which the principal problem is to find eigenvectors and eigenvalues of the spin Hamiltonian since all lineshape properties are assumed a priori in a given spin-packet lineshape function: (1) Use perturbation theory of order “n” for the energy eigenvalues and eigenvectors, find the eigenfields, calculate the approximate intensities, then convolute with a field-dependent lineshape function, and integrate over all (necessary) angles until convergence has been achieved. This is the standard perturbation procedure, which one may easily accomplish for large spin spaces, but which may have poor convergence properties outside of certain ranges for some Hamiltonian-related parameters. (2) Use matrix diagonalization of the complete Hamiltonian matrix, find the transition frequencies and intensities, then convolute with a frequency-dependent lineshape function, and integrate over all (necessary) angles. This method will numerically converge for all Hamiltonian parameters, but is strongly limited by the dimensions of the spin space. The latter method has a better basis in physical theory, since it deals directly with the energies (transition frequencies) of all of the effects which affect the lineshape, although it is limited in regards to the number of electronic and nuclear spins that can be handled. A distinct advantage is that the spin-Hamiltonian operators can be treated in full generality, and there are no restrictions due to failure of perturbation theory expansions. Another advantage is that it is distinctly easier to interpolate the transition frequencies and intensities than the associated eigenfields and intensities. (An example: one often finds “looping fields,” but never looping transition frequencies!). In addition, the “eigenfields” need never be directly found from the transition frequencies, since the frequency dependent lineshape function peaks automatically at the transition fields. The three-spin problem would also be extremely difficult to treat by perturbation methods, and would certainly involve considerable numerical com-
64
COFFMAN
AND
PEZESHK
plexity. We therefore work in a fixed quantization basis in the following treatment since we do not use perturbation theory, and the axis of quantization is chosen to be conveniently fixed within the molecule. (The molecular reference frame coincides with the laboratory frame.) The basic problem considered in the following treatment is a molecule containing a Cu(I1) atom and two weakly interacting free radicals, e.g., nitroxyl radicals. For simplicity, as well as for the sake of simpler and faster computations, we ignore all hyperfine tensors and we concentrate only on basic electronic interactions: the Zeeman operators with associated g tensors, and the spin-spin interactions. The principal advantage of the “third” spin (the Cu(I1) spin) arises from the anisotropy of its associated g tensor. Since it is a transition metal, the anisotropy of the g tensor causes the spectrum to be sufficiently spread out in magnetic field that the structural effects due to small (dipolar) spin-spin interactions are easily detected. A practical reason why a “threespin” label may be more useful than a “two-spin” label has to do with binding: There is more likelihood that a membrane binding site (as an example) may bind in one way to two of the unpaired spin-bearing moieties, and in another way to the third spin-bearing group. Thus there is more possibility to observe unusual binding effects, or strain effects within the paramagnetic probe due to local conformational changes, with a three-spin label. COORDINATE
SYSTEMS
AND
SPIN
HAMILTONIANS
The principal spin-Hamiltonian operators to be considered are the Zeeman operators, the exchange interaction, and the point dipole-dipole interaction. A coordinate system is chosen as shown in Fig. 1. The coordinate system is defined by the g tensor
FIG.
1. Coordinate
system
for the three-spin
system.
ESR LINESHAPE
FOR THREE-ELECTRON
SYSTEM
65
of spin 1 (at the origin). The x, y, z directions are the directions along which g(l) is diagonal. The tensors g’*’ and gC3)are generally nondiagonal. The spherical angles [I2 and v12 are the general conventional spherical angles of quantum chemistry (see (4, 9)). The angles r13 and v13 are similarly defined. The meanings of R12,R13,and Rz3 are obvious. The direction of the vector R23 is defined by direction cosines 123, m23, and n23. Having defined the locations of molecules 1, 2, and 3, we now associate a Hamiltonian operator with each of them as well as a set of interactions: &“=~,+~2+~3+~,2+~,3+~23.
[II
Now, we choose X1 to be the Zeeman operator whose g tensor, gl , is diagonal in the reference frame. The g tensors for spins 2 and 3 are generally not diagonal:
The rotation angles and rotation matrices for the tensors g2 and g3 are given later on in this paper. The interaction operators are choosen to be the sum of an exchange plus a magnetic dipole-dipole interaction for each pair of spins: 312 213 x23
=
z”,x(l,
-t
2)
= ~ex(l, 3) + = ~ex(2, 3) +
fldd(l, xdd(l, zdd(2,
2)
3) 3).
[31
(The purpose of the repetition above is to emphasize the total number of interactions in this problem.) A specific form is needed for both the exchange and dipolar operators. As is well known, both operators are at least bilinear spin operators with nine real, nonsymmetric coefficients: 2~4 12 2)
=
zdd(
=
1~ 2)
[da1
C JijslisZj LJ
c
WI
Di$li&j
LJ
where, however, the D, and Jij (i, j = x, y, z) coefficients have considerably different properties. The dipolar D,] also may be estimated from a classical model; there is no such (classical) model for the exchange tensor elements Jij. We refer to these operators, collectively, as the spin-spin interaction operators. MODELS
FOR
THE
SPIN-SPIN
OPERATORS
The exchange operator, Eq. [4a], may be written in a symmetrized form: [51
66
COFFMAN
AND
PEZESHK
where the components of the dyadic I’ and vector D are simply related to the components Jij of the defining operator tensor operator. The relations are easily found to be of the form
rxy = WJxy + J,A - * *
WI
D, = 1/2(Jyz - J,), . . . .
[71
and
So, it is the off-diagonal elements Of J;j which govern the magnitudes of D and r. We shall henceforth neglect the symmetric anisotropic and antisymmetric, anisotropic exchange terms in what follows for these reasons: First, we expect the general inequalities [JoI > IDI > ]I’1 to hold (10). (In some cases, however, D may vanish due to symmetry, but not for the case treated here. Moriya (10) lists some general symmetry related properties of the vector quantity D.) Second, the anisotropies in the exchange operator Eq. [4a] are expected to be much smaller for the metal-free radical exchange than for metal-metal exchange (which is the most often studied case). We can easily deduce this result from the master equation given by Nagamiya, Yosida, and Kubo (II) for the components of the Jij tensor which are there expressed in terms of the spin-orbit couplings of both components of the exchange pair, the respective g tensors, etc. We have thus deduced the result that (since XR 4 X,): J,j(M* * *R) G 1/2Jij(A4* * *n/r),
i#j.
PI
But these are just the terms which contribute to D and r of Eq. [5]. Hence, there is considerable reason to expect that these terms may be small and can be neglected for the metal-free radical exchange case. A better calculation is needed, however, to make this result more quantitative. Thus, we write for the simplified form of the exchange operator:
Zex = Jo(R)S - S2
[91
as a good approximation for the case at hand. Furthermore, in order to better model the physical interactions that we wish to study, we assume that Jo(R) has this simple dependence on R: Jo(R) = AemBR
[lOI
where A( = 1.35 X 10’ cm-‘) and /3(= 1.80 A-‘) are chosen as estimates from statistical observations on the nature of the limiting form of superexchange at long range (22). The dipolar interaction requires calculation of the dipolar interactions Z& 1, 2), Zdd( 1, 3) and Z&2, 3). The latter interaction requires evaluation of the basic dipolar interaction zdd
=
Pl --
* P2
R3
3(111- w2 R3
- 3
[Ill
for two cases: first where one g tensor is diagonal and the other is nondiagonal, and second for the case where both g tensors are nondiagonal. We give here the result for the latter case, since it is most general. Substituting P = -pg * S above for fi2 and p3 we obtain, after collection of terms, the result:
ESR
D:j
LINESHAPE
=
FOR
g~'{g$
THREE-ELECTRON
SYSTEM
67
3(f2g$ + IWZgR)+ I?Zg~~‘)}
-
+ g$‘{g::’
- 3(mlgj;:) + m*gl:) + mng$j)}
+ g$‘{g$’
- 3(nl&)
+ nmg$
+ n2g(,:))}
1121
where i, j = X, y, z and D, = (P*/R&)D:j (see Eq. [4b]). The dipolar Dij coefficients for zdd( 1, 2) and A?& 1, 3) each consist of only one term of the three large bracketed terms in Eq. [ 121, since the tensor g(l) is assumed diagonal. To evaluate Dij for GF!dd(1, 2) we replace (2) by (l), and (3) by (2) in Eq. [ 121 above, so that we are now working with tensors go) and g(*). (The only nonzero g$? is g(l) XX,g,;(1)IS g$$, etc.) The Dij are easily found for the two cases A?dd(1, 2) and Xdd( 1, 3), and both gc2)and gc3’ are easily related to their principal axes through the Eulerian rotation matrices. The direction cosines I I*, m12, nr2 are given by the usual relations in terms of the standard spherical angles: 112= ml2 n12
sin(C~2)cos(m2)
=
sinKl2bMm2)
=
cW,2).
[I31
Similar formulae apply for 113, m12, and ~2~3.The distance Rz3 and direction cosines 123, m23, and 1~23 follow by application of a little analytic geometry: R23 = [Rf, + R:3 - ~R~~R]~cos
(Y]
iI41
where cos(d =
112113 +
ml2ml3
+
[I51
n12%3
and 123, m23, and n23 are given by the relations 123 =
@,31,3
m23
=
(R13ml3
n23
=
(R1m
-
RddlR23 -
&mdl&
iI61
&ndl&.
The expressions of Eqs. [8]-[ 161 completely define the point dipole-dipole interaction operator of Eq. [ 1 I] in the Cartesian form of Eq. [4b]. We need, however, to define the general g tensor in terms of its diagonal components and the Eulerian angles which define the rotation of a given tensor from the reference frame of spin 1 to the reference frame in which g2 (or g3) is diagonal. g TENSORS,
COORDINATE
SYSTEMS,
AND
ROTATIONS
The g tensor for g1 is defined to be diagonal in the reference frame S(x, y, z). The tensors g2 and g3 are both generally nondiagonal in this reference frame, and we express their components g,,(i, j = x, y, z) in terms of the diagonal elements grii(i = x, y, z) and angles (Y,/3, y, and also gzii(i = X, y, z) and angles (Y’,p’, y’. Thus, the symmetrical tensor g2 (and similarly g3) consists of six unique quantities. These relations are best made clear by considering the Zeeman operators z2 and X3 which, being scalar operators, have the same “value” in any coordinate system. The interaction P2
68
COFFMAN
AND
PEZESHK
22 = /3B-g2-S2
1171
is more conveniently expressed in matrix language as
22
g2x.Y
&y
= PVL By, & 1
gzyx
g2YY
g2yz
[
g2rx
g2zy
g2zz-
which we write conveniently
-s,s 2Y s _ 22-
g2.a
[I81
as z2
=
1191
BBT&2S2
where S2 is defined in reference frame 8(x, y, z). Now, in another reference frame 8(x’, yf, z’), related to 8(x, y, z) by the unitary matrix of Eulerian orthogonal rotations 4i‘ we have s = Ts and B = ?i%. The operator Z2, expressed in this frame, is ‘372 = p(lBygg?&.
PO1
If the Eulerian angles are so chosen that
=p(EP)t!g(D)s = = =m@, P, rnsDT% AY) =ma’, P’, Y’k22JD)sp(~ P’, Y?
then 22
pr!LqT3pqs,
Thus and
prBjg&l.
P21
G
VW
&3
PW
The relations between coordinate systems, principle values and angles are listed in Table 1. The general definitions of all three tensors requires 15 unique numbers (each & assumed to be symmetric). There are two ways of defining the Eulerian rotation matrices: one definition uses the zxz’ sequence of orthogonal rotations (13) and the other uses the zyz’ sequence of orthogonal rotation (14). Since the latter is the conventional definition in quantum chemistry, this is the definition used in our calculations. The “sense” of the angles CY’, ,L3,y are defined to be positive for a rotation from 8(x, y, z) to 8(x’, y’, z’).
TABLE Coordinate Coordinate
Systems,
Diagonal
system
&x,
b Rotation
angles from
Elements,
S -
8’
I and Rotation
Y> 2)
G’,
Angles Y’, 4
for g, , g,, and g3 8(x”,
y”, z”)
%I =,%Y”’
%F
%Y”’
0, 60
a, 0, Y
a’, P’, Y’
ESR ESR
LINESHAPE
LINESHAPE
FOR
CALCULATED
BY
THREE-ELECTRON THE
69
SYSTEM
TRANSITION
FREQUENCY
METHOD
The operator of Eq. [l] is now suitably defined with respect to Zeeman, exchange, and dipolar operators. It remains only to find the matrix elements. This is fairly straightforward for the Zeeman and exchange operators, but somewhat cumbersome for the dipolar operators. Both the Zeeman and dipolar operators are conveniently evaluated after transformation to forms more suitable to evaluation of matrix elements. The complete set of matrix elements for Zeeman, dipolar, and exchange operators is given in the Appendix. The numerical procedure begins with evaluation of the matrix H (complex Hermetian), and one solves MC=az ]241 where c is the eigenvector matrix, and the diagonal elements of E are the eigenvalues. From E, at any field value “I?,” we determine the transition frequency vij(B, n, =
n) - dB, Q)llh
lcj(B,
WI
and then we determine the selection rules from c and the radiation Hamiltonian This operator is defined by zrad. zrad
=
PBr.g,.Sr
where B, is chosen perpendicular i-s?rad, we evaluate
+ PB,.g2’Sz
+ PB,.g,.&
WA
to the static field Bo. Finding the matrix @ad of p”2 = c%&,dc
v71
and then find the intensity factors P, defined by pij = (Jp’/2)ij(p’/2)~
P81
which gives the relative intensity of the i --+ j transition for the given value of field B and field orientation Q = 8, 4. A field or frequency lineshape (not including Boltzmann population factors, although this is easily included) is now found by evaluating the sum over all transitions: S(V, B; Q) = C Pij(B, ~)G[v - Y~,(B,O)]
~291
iq
where v = v. may be fixed and B varied (field swept spectrum) or v may be varied and B held fixed (frequency swept spectrum). The function G(v - vi,) is a convolution of a Lorentzian with a Gaussian function: G(v - vi,) = lr
g(V
-
X)h(X
-
Vij)dX
[301
where g(v - x) = the homogeneous lineshape function (typically Lorentzian) and h(x - vi,) = an inhomogeneous lineshape function (typically Gaussian). Assuming that the intrinsic width of the Gaussian function, h, is much larger than the intrinsic width of g (i.e., the linewidths are determined mainly by local strain effects rather than relaxation), which is equivalent to the mathematical approximation g(v - x) x S(v - x), then
70
COFFMAN
G(u - Uij) - Jtm S(U-
AND
X)h(X
PEZESHK
-
U,)dX
= h(~
-
Vii).
--a,
For this purpose, we choose h(U -
Uij)
= GTzexp{-2rr*T$(v
-
Vii>'}
1321
which is normalized on the interval (-co, +co). In this approximation, the linewidth of each spin packet is given by a single parameter: l/T*. This is a good approximation for the case at hand, since the lineshape is determined mainly by the envelope of the entire set of anisotropic lines, rather than by the linewidth of any set of single inhomogeneously broadened lines. In practice, this sum (i.e., Eq. [29]) is evaluated by diagonalizing the field at a small number of field points and then linearly interpolating the v;j(B, Q) and Pij(B, Q) between these points, for each orientation Q. For the sake of program efficiency, the sum over all transitions is also restricted to only those transitions which have significant intensity. In addition, the numerical evaluation of the matrix triple product of Eq. [24] is relatively slow, and one must pay attention to the efficiency of this operation. The result of evaluation of Eq. [29] is a single crystal spectrum versus magnetic field for fixed frequency u = v. and magnetic field orientation Q = 0, 4. The final step is to spherically average (or do a spherical integration of; there is a subtle, but important difference) the lineshape function S(V, B, Q). We represent the result by: S(u, B) = s S(u, B; O)dQ
[331
where the range of integration may be over either one, two, or four octants of the unit sphere. Integration over four octants of the unit sphere (hemisphere) is equivalent to integration over the complete sphere, as is seen by a simple argument: consider any one spectrum in one hemisphere S(V, B). This spectrum must be identical to S(V, -B) (since reversing the current in an electromagnet cannot change the ESR spectrum!). Thus, if S(V, B, a) and S(V, B, -L?) are single crystal spectrum pairs in opposite hemispheres, then
s
hemisphere
S(u, B; Q)dQ =
s hemisphere’
Ls(u,B; -fl)&.
1341
Thus, since the integral over any one hemisphere is equal to its opposite, the integral over the full sphere = twice the integral over any one hemisphere. This proves the assertion. The matrix elements were carefully checked against other calculations of dipolar and exchange lineshapes for the simpler case of two electronic spins. The basic integration program used was modeled on a calculation for the spectrum of a single Cu(II)containing molecule (also dimension 8 X 8), for which the details of the numerical processing were proved by comparison both to experiment and by comparison to perturbation theory calculations for Cu(I1) lineshapes. Finally, the “three-spin” calculation itself was carefully checked by allowing either RI2 or RI3 to become large (ca. 20 A or larger) thus turning off all dipolar and exchange interactions except between
ESR
LINESHAPE
FOR
THREE-ELECTRON
SYSTEM
71
the other (l-2) pair. The resulting lineshape was as expected, which compared to an independent calculation of the remaining spin-spin interacting pair. RESULTS
The numerical steps which control the speed of the calculation are diagonalization of the Hamiltonian matrix w (Eq. [24]), evaluation of the selection rule matrix p”’ (Eq. [27]), and the final spectrum summation (Eq. [29]), all of which must be done for each of the angles in the spherical integration of Eq. [33]. For the case described here (three spins: S, = 1, S, = 1, S, = $), the complex, Hermetian matrix w is of dimension 8 X 8. Numerical quadrature on the unit sphere was accomplished by using a finite element integration method which is well suited to this type of ESR spherical averaging: each solid angle Q = (0, 4) is determined by the process of subdivision of the regular icosahedron set of apical angles (1.5). The spectra for each angle are then numerically added allowing for an appropriate weight factor. A total of 5 185 angles for a hemisphere gave good convergence both for the three-spin spectra and for numerical integration of a test function over the unit sphere. The complete process described by Eqs. [24]-[33] can be accomplished on the IBM 3033 in about 5-12 min. The results were used to explore the nature of the pure spin-spin interaction between three spins, ignoring hyperfine structure. The basic question that we investigated is: how sensitive are the intramolecular magnetic interactions to variation of the dihedral angle in a proposed “three-spin label”? That is, how sensitive is the spectrum for a simple arrangement of three spins, one on each molecular fragment, as depicted in the simplified model of Fig. 2, to variation of the angle (Y, as well as to variation of the basic distances RI2 and Ri3? The results of the calculations described in the preceding sections are presented in the following figures. Figures 3 and 4 present the calculated ESR absorption and absorption derivatives for a “typical” Cu(I1) complex having two free radicals with distances RI2 = RI3 = 7.0 A versus the dihedral angle “CL” The results show a strong dependence of the lineshape on this angle provided that (Ydoes not become so large that the distance parameter R13“turns off” the two to three dipolar spin-spin interaction. Figure 5 shows the absorption derivative for a similar calculation, but with RI2 = RI3increased to 9.0 A. For the sake of comparison, Fig. 6 displays the calculated absorption derivative spectra versus R = RI2 = RI3 for cyfixed at the “magic angle” value of { = 54.74”, 9 = 45”. Here too, there is a marked dependence of the spectral lineshape on the geometric parameters.
FIG. 2. Definition of the dihedral one unpaired electronic spin.
angle “a”
for the three-spin
problem.
Molecules
I, 2, and 3 have each
‘==
60
KGAUSS RG. 3. ESR absorption versus magnetic field B being held fixed. Interspin distances R 12 = R,, = 1 (Cu(Il))g,, = 2.060, g,z,,, = 2.076, g,,, = 2.353; g,,, = 2.003,~~ = B = y = O,g,, = 1.997, g,, = = 9.113 GHz. Linewidth parameter T2 = 5.4 X
for varying values ofthe dehedral angle a, other parameters 7.0 A. The g-tensor values were chosen as follows: for spin for spins 2 and 3 (free radicals) gnz = 2.001, g,, = 2.002, 1.998, gzzj = 1.999, (Y’ = 8’ = y’ = 0. Microwave frequency IO-’ s.
72
ESR
LINESHAPE
FOR
THREE-ELECTRON
1”“1’~~‘,“‘~,~“~,~~~‘,~~“,‘~~~,~’~’,~~”,~’~’, 24-O 232 2.64 2.76 286 3.00
3.12
73
SYSTEM
3.a
3.38
3.44
KGAUSS FIG. 4. Derivative of the absorption displayed in Fig. 3.
3.60
COFFMAN
AND
PEZESHK
o!= 80
d=so
k--
A
I’~~~,“~~,‘.“,..‘I,...‘,‘...,.‘..,’,..,’,,’,..,,,
2.40 252
2.64
2.72
2.82
3.00
3.12
3.24
KGAUSS FIG. 5. Absorption derivative versus field for various nterspin distance RN1 = R,, = 9.0 8, for each spectrum.
representative All parameters
3.32
3.42
3.80
values of the dihedral angle 01 with and frequency as in Fig. 3.
R=6
KGAUSS FIG. 6. Absorption derivative versus field for various values of RI1 = RI3 = “R” with the dihedral ande fixed at the “magic-angle” value of LY= 54.74”.
15
76
COFFMAN
AND
PEZESHK
CONCLUSIONS
The three-spin-label lineshape, for molecules of the approximate sizes considered here, shows a significant response to variation of the dihedral angle LYprovided the distance R23 is not too large. Such a label would clearly be sensitive to external forces which create small conformational changes in the overall label geometry. A similar result must also follow from consideration of dynamical lineshapes, although such calculations will be considerably more difficult, and one must use a method for rotational diffusion coupling of spectra of different Q’s which works for a completely general spin Hamiltonian (16, 17). Both experiments and calculations including dynamical effects will be the subjects of future studies. The rigid-lattice lineshapes for the three-spin problem, presented here, omit hyperfine structure which is frequently important. In the case ofthe nitroxyl-copper(nitroxyl complexes, we certainly need to include the hyperfme structure of the Cu(I1) moiety, and we would hope that the nitroxyl hyperfine structure (I = 1, nitrogen) would be less than the linewidth in most cases. In this case, for S, = f, I, = ;, Sz = f, & = 1, the w matrix will be of dimension 32 X 32 which means that the complete calculation will take hours, instead of minutes, on the IBM 3033 or equivalent. One might consider the alternative of only calculating the interactions between two identical nitroxyl radicals within the same molecule (with no Cu(I1) present) including all exchange and dipolar effects. The dimensions of such a problem would, however, be 36 X 36, which is superficially more difficult but which might present an easier calculation for slow tumbling if one is able to use the powerful methods of Freed, Bruno, and Polnaszek (18) for the slow tumbling lineshape analysis. Further calculations, if performed on a “supercomputer,” (using vector rather than scalar numerical processing) may speed up the total calculation by a factor of 100 or more. Taking this future development into consideration, we conclude that calculations of such dipolar structure as considered here for three electronic spins are indeed possible, the truly limiting factor being the availability of sufficiently fast computer hardware. APPENDIX:
MATRIX
ELEMENTS
FOR
THREE
SPINS
Zeeman Matrix Elements The operator zz( 1) = /3B. g1 . S1 may be written in the form Z,(l)
= ,!3B(G;S: + G:S;
+ 2G$,,)
where (assuming g, diagonal) G; = 1/2(lgg + img$) G; = 1/2(1&j - img$) = CONJG(G:) G? = 1/2ng!i-). Here 1, m, and y1are the direction cosines of the external B field along the x, y, z axes (1 = sin 0 cos 4, m = sin 0 sin 4, n = cos ~9). Now, both g2 and g3 are assumed generally nondiagonal in the reference frame which diagonalizes gr . So, let
ESR
LINESHAPE
FOR
THREE-ELECTRON
77
SYSTEM
Gi = 1/2{(lg$$ + mg$ + ng$:) + i(Zg$d + rng$$ + ng$)} Gi = CONJG(G;) G$’ = l/2{ 1giT + mg$ + rzg!:)}. Then zz(2) assumes the simple form A?“,(2) = PB{G;S:
+ G:Sy + 2G&}
and similarly for E’“,(3). With these definitions of the Zeeman operators, matrix elements of the basic Zeeman operators for any one spin assume this simple form:
$112 -l/2
+1/2
-l/2
,L3BG” /3BG+
PBG-PBG’
The full 8 X 8 Zeeman matrix is given in Table 2. Exchange Matrix Elements Matrix elements for the representative scalar exchange operator between spins 1 and 2 are easily found from xe, = JOS, * s’2 = JoS,zS2r + 1/2Jo(S,+S2- + SI-ST,) and the matrix elements for the (1, 2) pair are I+, +>
I+, +>
I+, ->
I-9 +>
I-, ->+
1/4Jo
- 1/4Jo
1/2Jo
1/2Jo
- 1/4Jo 1/4Jo
The resulting exchange matrix for the three-spin case is given in Table 3. The Dipolar Matrix Elements The dipolar matrix elements are the most complicated set of matrix elements: the total dipolar operator xdd
=
zdd(
1, 2) +
zdd(
1,
3)
+
zdd(2,
3)
had 64 - 8 = 56 generally nonzero matrix elements. We find the dipole-dipole matrix elements first for any general pair, and then insert the matrix elements for each pair into the correct locations in the total 8 X 8 dipolar interaction matrix. To keep track of the notation, we define the dipolar coefficients for each pair by: pdd(
19 2)
xdd(
1 I 3) =
zdd(2,
where i, j = x, y, z.
3)
=
=
2 DijS, i,i 2 ii
Ei$‘,;&,
c Fiis,iS3j i,i
3’2,
I+, I+, I+, I+, I-, I-, I-. I-.
+, +, -, -1 +, +, -/ -.
+) -) +) -) +) -) +) -)
I-, -, -)
I-, -,+j
I-, +, -)
I-, +, f)
IJIZ + Jn + J&4
1+1/2, +1/2 t l/2)
BBC;
VII
Ifl/Z,
BBC:
I+, -, +)
JoI2
Jd2
JI,
Jd4
i-l/2, -l/2)
BBC:
BBC;
SDG:
I+, -, -)
BBC,
L?B(Gpt G: - G;)
@[GP t G: + Go]
I+, +, +)
1+1/2,+1/2,-I/2)
I+, +, -)
1+1/z, +1/2, +1/z)
[-Ju
Jd2
+ Jn - JnV4
52312
1+1/2, -1/2,il/2)
Three-Spin
BBC:
SBG:
,9B(Gt - G; + G:)
BBC;
1+1/2, -l/2,+1/2)
3
Jd2
Jd2
+l/2,+1/2)
Elements
43/2
5~12
+1/2. -l/2)
[-Jn + J,, - h31/4
l-l/2,
BBC;
SB(-CT t G: - G:,
PCs
BBC;
J13/2
L-512 - Ju + JnV4
I-I/Z,
Matrix
BBC:
BBC;
@B(pG: + G; + CT,
OBG;
I-i/2,+1/2,-I/2)
Jut2
[FJu - Jn + h31/4
Elements
I-1/2. +1/2, fl/2)
Matrix
2
Exchange
lt1/2, -l/2.-1/2)
Isotropic
TABLE
BBC:
/3B(G: - G! - G!,
BBGS
SBGi
-l/2)
Zeeman
1+1/2,-l/2,
Three-Spin
TABLE
fl/2)
BBC:
- G; t CT,
BBGi
BBC;
-l/Z,
IJIZ - Ju - Jnll4
Jd2
JnP
l-112, -1/2,fl/2)
@c-G:
I-1/2,
-l/2)
-I/2,
-I/2)
t G; + G:)
BBC;
BBGi
iSBG;
-l/2,
[Ju + Ju + Jt31/4
l-l/2,
+B(G:
I-I/2,
8 2
ESR
LINESHAPE
FOR
THREE-ELECTRON
SYSTEM
79
80
COPFMAN
AND
PEZESHK
We simplify the subscript notation by defining: D, , = D, , D12 = D2, * . . OS3 = Dg, and similarly for the Eij and Fij. The numerical values are found from Eq. [ 121 for all these pairs of interactions by means of a subroutine which requires input of the gtensor elements and direction cosines for each pair of spins. The basic form of the dipole-dipole interaction matrix for spin pair (1, 2) is as follows: ++ ++
+-
l/4(0,
11409
--
-+ - iDs)
1/4(D3 - iD6)
l/4(01 - Dd -i/4(Dz
+-
1/4(0, + iD8)
- 1/4D9
l/4(01 + Ds)
+ 04)
-1/4(D3 - iD6)
+i/4(Dz - 04) -+
l/4(03 + iD6)
l/4(01 + Ds) -i/4(D2
--
l/4(&
- Ds)
+1/4(D2 + 04)
- 1/4(D7 - iDs)
- l/44
- D4)
- 1/4(D3 + iD6)
- l/4(0,
+ iDg)
+ 1/4D9
-
The total dipolar interaction matrix consisting of all DI * * . Dg, El . . * Es, and FI - . . F9 elements is displayed in Table 4. The sum of all three matrices: Jk! = Table 2 + Table 3 + Table 4 gives the total Hamiltonian matrix used in Eqs. [24]-[27]. ACKNOWLEDGMENTS This research was supported by NIH Grant GM24480. Stevens for a helpful discussion. We thank Ms. Karen Waln and preparation of the manuscript.
One of us (R.E.C.) thanks Professor K.W.H. for assistance in many steps of the calculation
REFERENCES 1. A. A. MEDZHILIOV, L. N. DIRICHENKO, AND G. I. LIKHTENSHTEIN, Izv. Akad. Nauk, SSSR, Ser. Khim., 968 (1969) (p. 629 in English translation). 2. Yu. G. MAMEDOVA, A. A. MEDZHIDOV, AND L. N. KOLOMINA, Russ. J. Inorg. Chem. 17, 1548 (1972). 3. A. A. MEDZHIDOV, A. M. MUSAEV, AND E. G. ROZANTSEV, Izv. Akad. Nauk. SSSR, Ser. Khim, 538 (1977) (p. 483 in English translation). 4. A. RASSAT AND P. REY, Bull. Sot. Chim. Fr., 8 15 (1967). 5. R. Z. SAGDEEV, Yu. N. MOLIN, R. A. SADIKOV, L. B. VOLODARSKY, AND G. A. KUTIKOVA, J. Magn. Reson. 9, 13 (1973). 6. W. H. ORME-JOHNSON, N. R. ORME-JOHNSON, C. TOUTON, M. EMUIPTAGE, M. HENZL, J. RAWLINGS, K. JACOBSON, J. P. SMITH, W. B. MIMS, B. H. HUYNH, E. MUNCK, AND G. S. JACOB, in “Molybdenum Chemistry of Biological Significance” (W. E. Newton and S. Otsuka, Eds.), p. 85, Plenum, New York, 1980. 7. P. KOTTIS, J. Chem. Phys. 47, 509 (1967). 8. J. P. LEMAISTRE AND P. Korrrs, J. Chem. Phys. 68,273O (1978); 76,872 (1982).
ESR LINESHAPE
FOR THREE-ELECTRON
SYSTEM
81
9. E. V. CONDON AND G. H. SHORTLEY, “The Theory of Atomic Spectra,” Cambridge Univ. Press, Cambridge, England, 1963. 10. T. MORIYA, Phys. Rev. 120,91 (1960). II. T. NAGAMIYA, K. YOSIDA, AND R. KUBO, Adv. Phys. 4, I (1955). 12. R. E. COFFMAN AND G. R. BUE~TNER, J. Phys. Chem. 82,2387 (1979). 13. H. GOLDSTEIN, “Classical Mechanics,” Addison-Wesley, Cambridge, Mass., 1953. 14. M. E. ROSE, “Elementary Theory of Angular Momentum,” Wiley, New York, 1957. 15. International Union of Crystallography, “International Tables for X-Ray Crystallography,” Vol. 2, “Mathematical Tables,” Kynoch Press, Birmingham, England, 1952-1962. 16. R. G. GORDON AND T. MESSENGER,in “Electron Spin Relaxation in Liquids” (L. T. Muus and P. W. Atkins, Eds.), p. 341, Plenum, New York, 1972. 17. R. G. GORDON, Adv. Magn. Reson. 3, 1 (1968). 18. J. H. FREED, G. V. BRUNO, AND C. F. POLNASZEK, J. Phys. Chem. 75,3385 (1971).
OF MAGNETIC
RESONANCE
65,62-8 1 (1985)
ESR Rigid-Lattice Lineshapesfor a Three-Electronic-SpinSystem. A Triradical or “Three-Spin Label” R.E.COFFMANANDA.PEZESHK Chemistry Department, University of Iowa, Iowa City, Iowa 52242 Received April 8, 1985 The ESR rigid-lattice lineshape is calculated for a randomly oriented system of molecules each bearing three weakly interacting electronic spins. Each spin has a unique g tensor and associated Zeeman interaction and ah three spins interact through an isotropic exchange interaction (J& . S2) and general magnetic point dipole-dipole interaction. The lineshape is calculated by the transition frequency method, and the resulting spectra are averaged over four octants of the unit sphere. The results show significant lineshape dependence on the value of the dihedral angle in the complexes, as well as the usual dependence on the dipolar distances. Based on these observations, we propose that a “three-spin label” would have a strong response to local conformational strain effects in membranes and macromolecules. Such a label, along with an appropriate lineshape theory, may have unique applications in paramagnetic resonance molecular structure-function studies. 0 1985 Academic Press, Inc.
INTRODUCTION
Molecules with three independent electronic spins present interesting problems in regards to the calculation of their ESR spectra. Such problems arise for certain copper complexes having two nitroxyl containing ligands and in biological macromolecules having multiple independent paramagnetic moieties. Three-spin systems containing a transition metal and two nitroxyl ligands which have been studied in the past may be grouped into the following three classes: Class A are those compounds for which the nitroxyl radicals are attached to a salicylaldehyde Schiff base ligand (1) class B are those compounds for which the nitroxyl radicals are attached to 2-hydroxy-napthaldehyde Schiff base ligands (2, 3), and Class C are the copper complexes of cyclocarboxylic acids containing nitroxyl radicals (4, 5). A “pure” three-spin system in a macromolecule (other than an S = i system) does not appear to have been identified yet, although examples of macromolecules with three or four paramagnetic moieties are well known (6). Previous quantitative considerations of three-spin ESR lineshapes begin with Kottis (7) who considered a three-site triplet problem in tribenzotriptycene including dipolar interactions, and Lemaistre and Kottis (8) who studied ESR lineshapes of spin dimers and trimers in disordered matrices, from the point of view of the effects of exciton transfer in competition with exchange and dipolar effects on the lineshape. More recently, Sagdeev et al. (5) have approximated the general shape of a three-spin AB2 system from special case considerations of spin-spin effects on the absorption spectrum turning points. In this paper, we present a general treatment of the three-spin problem 0022-2364185 $3.00 Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.
62
ESR
LINESHAPE
FOR
THREE-ELECTRON
SYSTEM
63
from the point of view of the most important spin-Hamiltonian operators which govern the largest terms in the energy and which determine the structural variables which are strongest affected by conformational changes of the molecule. These are, of course, the Zeeman operators, and the exchange and general magnetic dipolar operators. The ESR lineshapes in a single molecule with three weakly interacting electronic spins will generally be affected by slow tumbling effects as well as the spin-spin interactions which are dependent on the internal molecular structural variables. The dependence on slow tumbling appears in the lineshape problem as a coupling of the lineshape equations for solid angle C!with neighboring angles O’, a”, etc. One effect of this coupling is to increase dramatically the dimension of all matrices for the threespin problem, so we make the simplifying assumption that this coupling is turned ufl so that the lineshape calculation for any one D is independent of all other Q’s. This will be true at a sufficiently low temperature (or equivalently), a sufficiently high viscosity, so that a simpler rigid-lattice calculation may be used for the lineshape properties of the system. In many cases, this is reasonable especially if the sample at hand exists either as a powder or a very viscous sample at the temperature of interest. For such a glassy or polycrystalline sample, we calculate the ESR lineshape for all orientations of the magnetic field using a method sufficiently general as to account for all of the many magnetic interaction effects. The inclusion of the slow tumbling effects will be treated as a separate problem. There are two general approaches to this type of rigid-lattice calculation for which the principal problem is to find eigenvectors and eigenvalues of the spin Hamiltonian since all lineshape properties are assumed a priori in a given spin-packet lineshape function: (1) Use perturbation theory of order “n” for the energy eigenvalues and eigenvectors, find the eigenfields, calculate the approximate intensities, then convolute with a field-dependent lineshape function, and integrate over all (necessary) angles until convergence has been achieved. This is the standard perturbation procedure, which one may easily accomplish for large spin spaces, but which may have poor convergence properties outside of certain ranges for some Hamiltonian-related parameters. (2) Use matrix diagonalization of the complete Hamiltonian matrix, find the transition frequencies and intensities, then convolute with a frequency-dependent lineshape function, and integrate over all (necessary) angles. This method will numerically converge for all Hamiltonian parameters, but is strongly limited by the dimensions of the spin space. The latter method has a better basis in physical theory, since it deals directly with the energies (transition frequencies) of all of the effects which affect the lineshape, although it is limited in regards to the number of electronic and nuclear spins that can be handled. A distinct advantage is that the spin-Hamiltonian operators can be treated in full generality, and there are no restrictions due to failure of perturbation theory expansions. Another advantage is that it is distinctly easier to interpolate the transition frequencies and intensities than the associated eigenfields and intensities. (An example: one often finds “looping fields,” but never looping transition frequencies!). In addition, the “eigenfields” need never be directly found from the transition frequencies, since the frequency dependent lineshape function peaks automatically at the transition fields. The three-spin problem would also be extremely difficult to treat by perturbation methods, and would certainly involve considerable numerical com-
64
COFFMAN
AND
PEZESHK
plexity. We therefore work in a fixed quantization basis in the following treatment since we do not use perturbation theory, and the axis of quantization is chosen to be conveniently fixed within the molecule. (The molecular reference frame coincides with the laboratory frame.) The basic problem considered in the following treatment is a molecule containing a Cu(I1) atom and two weakly interacting free radicals, e.g., nitroxyl radicals. For simplicity, as well as for the sake of simpler and faster computations, we ignore all hyperfine tensors and we concentrate only on basic electronic interactions: the Zeeman operators with associated g tensors, and the spin-spin interactions. The principal advantage of the “third” spin (the Cu(I1) spin) arises from the anisotropy of its associated g tensor. Since it is a transition metal, the anisotropy of the g tensor causes the spectrum to be sufficiently spread out in magnetic field that the structural effects due to small (dipolar) spin-spin interactions are easily detected. A practical reason why a “threespin” label may be more useful than a “two-spin” label has to do with binding: There is more likelihood that a membrane binding site (as an example) may bind in one way to two of the unpaired spin-bearing moieties, and in another way to the third spin-bearing group. Thus there is more possibility to observe unusual binding effects, or strain effects within the paramagnetic probe due to local conformational changes, with a three-spin label. COORDINATE
SYSTEMS
AND
SPIN
HAMILTONIANS
The principal spin-Hamiltonian operators to be considered are the Zeeman operators, the exchange interaction, and the point dipole-dipole interaction. A coordinate system is chosen as shown in Fig. 1. The coordinate system is defined by the g tensor
FIG.
1. Coordinate
system
for the three-spin
system.
ESR LINESHAPE
FOR THREE-ELECTRON
SYSTEM
65
of spin 1 (at the origin). The x, y, z directions are the directions along which g(l) is diagonal. The tensors g’*’ and gC3)are generally nondiagonal. The spherical angles [I2 and v12 are the general conventional spherical angles of quantum chemistry (see (4, 9)). The angles r13 and v13 are similarly defined. The meanings of R12,R13,and Rz3 are obvious. The direction of the vector R23 is defined by direction cosines 123, m23, and n23. Having defined the locations of molecules 1, 2, and 3, we now associate a Hamiltonian operator with each of them as well as a set of interactions: &“=~,+~2+~3+~,2+~,3+~23.
[II
Now, we choose X1 to be the Zeeman operator whose g tensor, gl , is diagonal in the reference frame. The g tensors for spins 2 and 3 are generally not diagonal:
The rotation angles and rotation matrices for the tensors g2 and g3 are given later on in this paper. The interaction operators are choosen to be the sum of an exchange plus a magnetic dipole-dipole interaction for each pair of spins: 312 213 x23
=
z”,x(l,
-t
2)
= ~ex(l, 3) + = ~ex(2, 3) +
fldd(l, xdd(l, zdd(2,
2)
3) 3).
[31
(The purpose of the repetition above is to emphasize the total number of interactions in this problem.) A specific form is needed for both the exchange and dipolar operators. As is well known, both operators are at least bilinear spin operators with nine real, nonsymmetric coefficients: 2~4 12 2)
=
zdd(
=
1~ 2)
[da1
C JijslisZj LJ
c
WI
Di$li&j
LJ
where, however, the D, and Jij (i, j = x, y, z) coefficients have considerably different properties. The dipolar D,] also may be estimated from a classical model; there is no such (classical) model for the exchange tensor elements Jij. We refer to these operators, collectively, as the spin-spin interaction operators. MODELS
FOR
THE
SPIN-SPIN
OPERATORS
The exchange operator, Eq. [4a], may be written in a symmetrized form: [51
66
COFFMAN
AND
PEZESHK
where the components of the dyadic I’ and vector D are simply related to the components Jij of the defining operator tensor operator. The relations are easily found to be of the form
rxy = WJxy + J,A - * *
WI
D, = 1/2(Jyz - J,), . . . .
[71
and
So, it is the off-diagonal elements Of J;j which govern the magnitudes of D and r. We shall henceforth neglect the symmetric anisotropic and antisymmetric, anisotropic exchange terms in what follows for these reasons: First, we expect the general inequalities [JoI > IDI > ]I’1 to hold (10). (In some cases, however, D may vanish due to symmetry, but not for the case treated here. Moriya (10) lists some general symmetry related properties of the vector quantity D.) Second, the anisotropies in the exchange operator Eq. [4a] are expected to be much smaller for the metal-free radical exchange than for metal-metal exchange (which is the most often studied case). We can easily deduce this result from the master equation given by Nagamiya, Yosida, and Kubo (II) for the components of the Jij tensor which are there expressed in terms of the spin-orbit couplings of both components of the exchange pair, the respective g tensors, etc. We have thus deduced the result that (since XR 4 X,): J,j(M* * *R) G 1/2Jij(A4* * *n/r),
i#j.
PI
But these are just the terms which contribute to D and r of Eq. [5]. Hence, there is considerable reason to expect that these terms may be small and can be neglected for the metal-free radical exchange case. A better calculation is needed, however, to make this result more quantitative. Thus, we write for the simplified form of the exchange operator:
Zex = Jo(R)S - S2
[91
as a good approximation for the case at hand. Furthermore, in order to better model the physical interactions that we wish to study, we assume that Jo(R) has this simple dependence on R: Jo(R) = AemBR
[lOI
where A( = 1.35 X 10’ cm-‘) and /3(= 1.80 A-‘) are chosen as estimates from statistical observations on the nature of the limiting form of superexchange at long range (22). The dipolar interaction requires calculation of the dipolar interactions Z& 1, 2), Zdd( 1, 3) and Z&2, 3). The latter interaction requires evaluation of the basic dipolar interaction zdd
=
Pl --
* P2
R3
3(111- w2 R3
- 3
[Ill
for two cases: first where one g tensor is diagonal and the other is nondiagonal, and second for the case where both g tensors are nondiagonal. We give here the result for the latter case, since it is most general. Substituting P = -pg * S above for fi2 and p3 we obtain, after collection of terms, the result:
ESR
D:j
LINESHAPE
=
FOR
g~'{g$
THREE-ELECTRON
SYSTEM
67
3(f2g$ + IWZgR)+ I?Zg~~‘)}
-
+ g$‘{g::’
- 3(mlgj;:) + m*gl:) + mng$j)}
+ g$‘{g$’
- 3(nl&)
+ nmg$
+ n2g(,:))}
1121
where i, j = X, y, z and D, = (P*/R&)D:j (see Eq. [4b]). The dipolar Dij coefficients for zdd( 1, 2) and A?& 1, 3) each consist of only one term of the three large bracketed terms in Eq. [ 121, since the tensor g(l) is assumed diagonal. To evaluate Dij for GF!dd(1, 2) we replace (2) by (l), and (3) by (2) in Eq. [ 121 above, so that we are now working with tensors go) and g(*). (The only nonzero g$? is g(l) XX,g,;(1)IS g$$, etc.) The Dij are easily found for the two cases A?dd(1, 2) and Xdd( 1, 3), and both gc2)and gc3’ are easily related to their principal axes through the Eulerian rotation matrices. The direction cosines I I*, m12, nr2 are given by the usual relations in terms of the standard spherical angles: 112= ml2 n12
sin(C~2)cos(m2)
=
sinKl2bMm2)
=
cW,2).
[I31
Similar formulae apply for 113, m12, and ~2~3.The distance Rz3 and direction cosines 123, m23, and 1~23 follow by application of a little analytic geometry: R23 = [Rf, + R:3 - ~R~~R]~cos
(Y]
iI41
where cos(d =
112113 +
ml2ml3
+
[I51
n12%3
and 123, m23, and n23 are given by the relations 123 =
@,31,3
m23
=
(R13ml3
n23
=
(R1m
-
RddlR23 -
&mdl&
iI61
&ndl&.
The expressions of Eqs. [8]-[ 161 completely define the point dipole-dipole interaction operator of Eq. [ 1 I] in the Cartesian form of Eq. [4b]. We need, however, to define the general g tensor in terms of its diagonal components and the Eulerian angles which define the rotation of a given tensor from the reference frame of spin 1 to the reference frame in which g2 (or g3) is diagonal. g TENSORS,
COORDINATE
SYSTEMS,
AND
ROTATIONS
The g tensor for g1 is defined to be diagonal in the reference frame S(x, y, z). The tensors g2 and g3 are both generally nondiagonal in this reference frame, and we express their components g,,(i, j = x, y, z) in terms of the diagonal elements grii(i = x, y, z) and angles (Y,/3, y, and also gzii(i = X, y, z) and angles (Y’,p’, y’. Thus, the symmetrical tensor g2 (and similarly g3) consists of six unique quantities. These relations are best made clear by considering the Zeeman operators z2 and X3 which, being scalar operators, have the same “value” in any coordinate system. The interaction P2
68
COFFMAN
AND
PEZESHK
22 = /3B-g2-S2
1171
is more conveniently expressed in matrix language as
22
g2x.Y
&y
= PVL By, & 1
gzyx
g2YY
g2yz
[
g2rx
g2zy
g2zz-
which we write conveniently
-s,s 2Y s _ 22-
g2.a
[I81
as z2
=
1191
BBT&2S2
where S2 is defined in reference frame 8(x, y, z). Now, in another reference frame 8(x’, yf, z’), related to 8(x, y, z) by the unitary matrix of Eulerian orthogonal rotations 4i‘ we have s = Ts and B = ?i%. The operator Z2, expressed in this frame, is ‘372 = p(lBygg?&.
PO1
If the Eulerian angles are so chosen that
=p(EP)t!g(D)s = = =m@, P, rnsDT% AY) =ma’, P’, Y’k22JD)sp(~ P’, Y?
then 22
pr!LqT3pqs,
Thus and
prBjg&l.
P21
G
VW
&3
PW
The relations between coordinate systems, principle values and angles are listed in Table 1. The general definitions of all three tensors requires 15 unique numbers (each & assumed to be symmetric). There are two ways of defining the Eulerian rotation matrices: one definition uses the zxz’ sequence of orthogonal rotations (13) and the other uses the zyz’ sequence of orthogonal rotation (14). Since the latter is the conventional definition in quantum chemistry, this is the definition used in our calculations. The “sense” of the angles CY’, ,L3,y are defined to be positive for a rotation from 8(x, y, z) to 8(x’, y’, z’).
TABLE Coordinate Coordinate
Systems,
Diagonal
system
&x,
b Rotation
angles from
Elements,
S -
8’
I and Rotation
Y> 2)
G’,
Angles Y’, 4
for g, , g,, and g3 8(x”,
y”, z”)
%I =,%Y”’
%F
%Y”’
0, 60
a, 0, Y
a’, P’, Y’
ESR ESR
LINESHAPE
LINESHAPE
FOR
CALCULATED
BY
THREE-ELECTRON THE
69
SYSTEM
TRANSITION
FREQUENCY
METHOD
The operator of Eq. [l] is now suitably defined with respect to Zeeman, exchange, and dipolar operators. It remains only to find the matrix elements. This is fairly straightforward for the Zeeman and exchange operators, but somewhat cumbersome for the dipolar operators. Both the Zeeman and dipolar operators are conveniently evaluated after transformation to forms more suitable to evaluation of matrix elements. The complete set of matrix elements for Zeeman, dipolar, and exchange operators is given in the Appendix. The numerical procedure begins with evaluation of the matrix H (complex Hermetian), and one solves MC=az ]241 where c is the eigenvector matrix, and the diagonal elements of E are the eigenvalues. From E, at any field value “I?,” we determine the transition frequency vij(B, n, =
n) - dB, Q)llh
lcj(B,
WI
and then we determine the selection rules from c and the radiation Hamiltonian This operator is defined by zrad. zrad
=
PBr.g,.Sr
where B, is chosen perpendicular i-s?rad, we evaluate
+ PB,.g2’Sz
+ PB,.g,.&
WA
to the static field Bo. Finding the matrix @ad of p”2 = c%&,dc
v71
and then find the intensity factors P, defined by pij = (Jp’/2)ij(p’/2)~
P81
which gives the relative intensity of the i --+ j transition for the given value of field B and field orientation Q = 8, 4. A field or frequency lineshape (not including Boltzmann population factors, although this is easily included) is now found by evaluating the sum over all transitions: S(V, B; Q) = C Pij(B, ~)G[v - Y~,(B,O)]
~291
iq
where v = v. may be fixed and B varied (field swept spectrum) or v may be varied and B held fixed (frequency swept spectrum). The function G(v - vi,) is a convolution of a Lorentzian with a Gaussian function: G(v - vi,) = lr
g(V
-
X)h(X
-
Vij)dX
[301
where g(v - x) = the homogeneous lineshape function (typically Lorentzian) and h(x - vi,) = an inhomogeneous lineshape function (typically Gaussian). Assuming that the intrinsic width of the Gaussian function, h, is much larger than the intrinsic width of g (i.e., the linewidths are determined mainly by local strain effects rather than relaxation), which is equivalent to the mathematical approximation g(v - x) x S(v - x), then
70
COFFMAN
G(u - Uij) - Jtm S(U-
AND
X)h(X
PEZESHK
-
U,)dX
= h(~
-
Vii).
--a,
For this purpose, we choose h(U -
Uij)
= GTzexp{-2rr*T$(v
-
Vii>'}
1321
which is normalized on the interval (-co, +co). In this approximation, the linewidth of each spin packet is given by a single parameter: l/T*. This is a good approximation for the case at hand, since the lineshape is determined mainly by the envelope of the entire set of anisotropic lines, rather than by the linewidth of any set of single inhomogeneously broadened lines. In practice, this sum (i.e., Eq. [29]) is evaluated by diagonalizing the field at a small number of field points and then linearly interpolating the v;j(B, Q) and Pij(B, Q) between these points, for each orientation Q. For the sake of program efficiency, the sum over all transitions is also restricted to only those transitions which have significant intensity. In addition, the numerical evaluation of the matrix triple product of Eq. [24] is relatively slow, and one must pay attention to the efficiency of this operation. The result of evaluation of Eq. [29] is a single crystal spectrum versus magnetic field for fixed frequency u = v. and magnetic field orientation Q = 0, 4. The final step is to spherically average (or do a spherical integration of; there is a subtle, but important difference) the lineshape function S(V, B, Q). We represent the result by: S(u, B) = s S(u, B; O)dQ
[331
where the range of integration may be over either one, two, or four octants of the unit sphere. Integration over four octants of the unit sphere (hemisphere) is equivalent to integration over the complete sphere, as is seen by a simple argument: consider any one spectrum in one hemisphere S(V, B). This spectrum must be identical to S(V, -B) (since reversing the current in an electromagnet cannot change the ESR spectrum!). Thus, if S(V, B, a) and S(V, B, -L?) are single crystal spectrum pairs in opposite hemispheres, then
s
hemisphere
S(u, B; Q)dQ =
s hemisphere’
Ls(u,B; -fl)&.
1341
Thus, since the integral over any one hemisphere is equal to its opposite, the integral over the full sphere = twice the integral over any one hemisphere. This proves the assertion. The matrix elements were carefully checked against other calculations of dipolar and exchange lineshapes for the simpler case of two electronic spins. The basic integration program used was modeled on a calculation for the spectrum of a single Cu(II)containing molecule (also dimension 8 X 8), for which the details of the numerical processing were proved by comparison both to experiment and by comparison to perturbation theory calculations for Cu(I1) lineshapes. Finally, the “three-spin” calculation itself was carefully checked by allowing either RI2 or RI3 to become large (ca. 20 A or larger) thus turning off all dipolar and exchange interactions except between
ESR
LINESHAPE
FOR
THREE-ELECTRON
SYSTEM
71
the other (l-2) pair. The resulting lineshape was as expected, which compared to an independent calculation of the remaining spin-spin interacting pair. RESULTS
The numerical steps which control the speed of the calculation are diagonalization of the Hamiltonian matrix w (Eq. [24]), evaluation of the selection rule matrix p”’ (Eq. [27]), and the final spectrum summation (Eq. [29]), all of which must be done for each of the angles in the spherical integration of Eq. [33]. For the case described here (three spins: S, = 1, S, = 1, S, = $), the complex, Hermetian matrix w is of dimension 8 X 8. Numerical quadrature on the unit sphere was accomplished by using a finite element integration method which is well suited to this type of ESR spherical averaging: each solid angle Q = (0, 4) is determined by the process of subdivision of the regular icosahedron set of apical angles (1.5). The spectra for each angle are then numerically added allowing for an appropriate weight factor. A total of 5 185 angles for a hemisphere gave good convergence both for the three-spin spectra and for numerical integration of a test function over the unit sphere. The complete process described by Eqs. [24]-[33] can be accomplished on the IBM 3033 in about 5-12 min. The results were used to explore the nature of the pure spin-spin interaction between three spins, ignoring hyperfine structure. The basic question that we investigated is: how sensitive are the intramolecular magnetic interactions to variation of the dihedral angle in a proposed “three-spin label”? That is, how sensitive is the spectrum for a simple arrangement of three spins, one on each molecular fragment, as depicted in the simplified model of Fig. 2, to variation of the angle (Y, as well as to variation of the basic distances RI2 and Ri3? The results of the calculations described in the preceding sections are presented in the following figures. Figures 3 and 4 present the calculated ESR absorption and absorption derivatives for a “typical” Cu(I1) complex having two free radicals with distances RI2 = RI3 = 7.0 A versus the dihedral angle “CL” The results show a strong dependence of the lineshape on this angle provided that (Ydoes not become so large that the distance parameter R13“turns off” the two to three dipolar spin-spin interaction. Figure 5 shows the absorption derivative for a similar calculation, but with RI2 = RI3increased to 9.0 A. For the sake of comparison, Fig. 6 displays the calculated absorption derivative spectra versus R = RI2 = RI3 for cyfixed at the “magic angle” value of { = 54.74”, 9 = 45”. Here too, there is a marked dependence of the spectral lineshape on the geometric parameters.
FIG. 2. Definition of the dihedral one unpaired electronic spin.
angle “a”
for the three-spin
problem.
Molecules
I, 2, and 3 have each
‘==
60
KGAUSS RG. 3. ESR absorption versus magnetic field B being held fixed. Interspin distances R 12 = R,, = 1 (Cu(Il))g,, = 2.060, g,z,,, = 2.076, g,,, = 2.353; g,,, = 2.003,~~ = B = y = O,g,, = 1.997, g,, = = 9.113 GHz. Linewidth parameter T2 = 5.4 X
for varying values ofthe dehedral angle a, other parameters 7.0 A. The g-tensor values were chosen as follows: for spin for spins 2 and 3 (free radicals) gnz = 2.001, g,, = 2.002, 1.998, gzzj = 1.999, (Y’ = 8’ = y’ = 0. Microwave frequency IO-’ s.
72
ESR
LINESHAPE
FOR
THREE-ELECTRON
1”“1’~~‘,“‘~,~“~,~~~‘,~~“,‘~~~,~’~’,~~”,~’~’, 24-O 232 2.64 2.76 286 3.00
3.12
73
SYSTEM
3.a
3.38
3.44
KGAUSS FIG. 4. Derivative of the absorption displayed in Fig. 3.
3.60
COFFMAN
AND
PEZESHK
o!= 80
d=so
k--
A
I’~~~,“~~,‘.“,..‘I,...‘,‘...,.‘..,’,..,’,,’,..,,,
2.40 252
2.64
2.72
2.82
3.00
3.12
3.24
KGAUSS FIG. 5. Absorption derivative versus field for various nterspin distance RN1 = R,, = 9.0 8, for each spectrum.
representative All parameters
3.32
3.42
3.80
values of the dihedral angle 01 with and frequency as in Fig. 3.
R=6
KGAUSS FIG. 6. Absorption derivative versus field for various values of RI1 = RI3 = “R” with the dihedral ande fixed at the “magic-angle” value of LY= 54.74”.
15
76
COFFMAN
AND
PEZESHK
CONCLUSIONS
The three-spin-label lineshape, for molecules of the approximate sizes considered here, shows a significant response to variation of the dihedral angle LYprovided the distance R23 is not too large. Such a label would clearly be sensitive to external forces which create small conformational changes in the overall label geometry. A similar result must also follow from consideration of dynamical lineshapes, although such calculations will be considerably more difficult, and one must use a method for rotational diffusion coupling of spectra of different Q’s which works for a completely general spin Hamiltonian (16, 17). Both experiments and calculations including dynamical effects will be the subjects of future studies. The rigid-lattice lineshapes for the three-spin problem, presented here, omit hyperfine structure which is frequently important. In the case ofthe nitroxyl-copper(nitroxyl complexes, we certainly need to include the hyperfme structure of the Cu(I1) moiety, and we would hope that the nitroxyl hyperfine structure (I = 1, nitrogen) would be less than the linewidth in most cases. In this case, for S, = f, I, = ;, Sz = f, & = 1, the w matrix will be of dimension 32 X 32 which means that the complete calculation will take hours, instead of minutes, on the IBM 3033 or equivalent. One might consider the alternative of only calculating the interactions between two identical nitroxyl radicals within the same molecule (with no Cu(I1) present) including all exchange and dipolar effects. The dimensions of such a problem would, however, be 36 X 36, which is superficially more difficult but which might present an easier calculation for slow tumbling if one is able to use the powerful methods of Freed, Bruno, and Polnaszek (18) for the slow tumbling lineshape analysis. Further calculations, if performed on a “supercomputer,” (using vector rather than scalar numerical processing) may speed up the total calculation by a factor of 100 or more. Taking this future development into consideration, we conclude that calculations of such dipolar structure as considered here for three electronic spins are indeed possible, the truly limiting factor being the availability of sufficiently fast computer hardware. APPENDIX:
MATRIX
ELEMENTS
FOR
THREE
SPINS
Zeeman Matrix Elements The operator zz( 1) = /3B. g1 . S1 may be written in the form Z,(l)
= ,!3B(G;S: + G:S;
+ 2G$,,)
where (assuming g, diagonal) G; = 1/2(lgg + img$) G; = 1/2(1&j - img$) = CONJG(G:) G? = 1/2ng!i-). Here 1, m, and y1are the direction cosines of the external B field along the x, y, z axes (1 = sin 0 cos 4, m = sin 0 sin 4, n = cos ~9). Now, both g2 and g3 are assumed generally nondiagonal in the reference frame which diagonalizes gr . So, let
ESR
LINESHAPE
FOR
THREE-ELECTRON
77
SYSTEM
Gi = 1/2{(lg$$ + mg$ + ng$:) + i(Zg$d + rng$$ + ng$)} Gi = CONJG(G;) G$’ = l/2{ 1giT + mg$ + rzg!:)}. Then zz(2) assumes the simple form A?“,(2) = PB{G;S:
+ G:Sy + 2G&}
and similarly for E’“,(3). With these definitions of the Zeeman operators, matrix elements of the basic Zeeman operators for any one spin assume this simple form:
$112 -l/2
+1/2
-l/2
,L3BG” /3BG+
PBG-PBG’
The full 8 X 8 Zeeman matrix is given in Table 2. Exchange Matrix Elements Matrix elements for the representative scalar exchange operator between spins 1 and 2 are easily found from xe, = JOS, * s’2 = JoS,zS2r + 1/2Jo(S,+S2- + SI-ST,) and the matrix elements for the (1, 2) pair are I+, +>
I+, +>
I+, ->
I-9 +>
I-, ->+
1/4Jo
- 1/4Jo
1/2Jo
1/2Jo
- 1/4Jo 1/4Jo
The resulting exchange matrix for the three-spin case is given in Table 3. The Dipolar Matrix Elements The dipolar matrix elements are the most complicated set of matrix elements: the total dipolar operator xdd
=
zdd(
1, 2) +
zdd(
1,
3)
+
zdd(2,
3)
had 64 - 8 = 56 generally nonzero matrix elements. We find the dipole-dipole matrix elements first for any general pair, and then insert the matrix elements for each pair into the correct locations in the total 8 X 8 dipolar interaction matrix. To keep track of the notation, we define the dipolar coefficients for each pair by: pdd(
19 2)
xdd(
1 I 3) =
zdd(2,
where i, j = x, y, z.
3)
=
=
2 DijS, i,i 2 ii
Ei$‘,;&,
c Fiis,iS3j i,i
3’2,
I+, I+, I+, I+, I-, I-, I-. I-.
+, +, -, -1 +, +, -/ -.
+) -) +) -) +) -) +) -)
I-, -, -)
I-, -,+j
I-, +, -)
I-, +, f)
IJIZ + Jn + J&4
1+1/2, +1/2 t l/2)
BBC;
VII
Ifl/Z,
BBC:
I+, -, +)
JoI2
Jd2
JI,
Jd4
i-l/2, -l/2)
BBC:
BBC;
SDG:
I+, -, -)
BBC,
L?B(Gpt G: - G;)
@[GP t G: + Go]
I+, +, +)
1+1/2,+1/2,-I/2)
I+, +, -)
1+1/z, +1/2, +1/z)
[-Ju
Jd2
+ Jn - JnV4
52312
1+1/2, -1/2,il/2)
Three-Spin
BBC:
SBG:
,9B(Gt - G; + G:)
BBC;
1+1/2, -l/2,+1/2)
3
Jd2
Jd2
+l/2,+1/2)
Elements
43/2
5~12
+1/2. -l/2)
[-Jn + J,, - h31/4
l-l/2,
BBC;
SB(-CT t G: - G:,
PCs
BBC;
J13/2
L-512 - Ju + JnV4
I-I/Z,
Matrix
BBC:
BBC;
@B(pG: + G; + CT,
OBG;
I-i/2,+1/2,-I/2)
Jut2
[FJu - Jn + h31/4
Elements
I-1/2. +1/2, fl/2)
Matrix
2
Exchange
lt1/2, -l/2.-1/2)
Isotropic
TABLE
BBC:
/3B(G: - G! - G!,
BBGS
SBGi
-l/2)
Zeeman
1+1/2,-l/2,
Three-Spin
TABLE
fl/2)
BBC:
- G; t CT,
BBGi
BBC;
-l/Z,
IJIZ - Ju - Jnll4
Jd2
JnP
l-112, -1/2,fl/2)
@c-G:
I-1/2,
-l/2)
-I/2,
-I/2)
t G; + G:)
BBC;
BBGi
iSBG;
-l/2,
[Ju + Ju + Jt31/4
l-l/2,
+B(G:
I-I/2,
8 2
ESR
LINESHAPE
FOR
THREE-ELECTRON
SYSTEM
79
80
COPFMAN
AND
PEZESHK
We simplify the subscript notation by defining: D, , = D, , D12 = D2, * . . OS3 = Dg, and similarly for the Eij and Fij. The numerical values are found from Eq. [ 121 for all these pairs of interactions by means of a subroutine which requires input of the gtensor elements and direction cosines for each pair of spins. The basic form of the dipole-dipole interaction matrix for spin pair (1, 2) is as follows: ++ ++
+-
l/4(0,
11409
--
-+ - iDs)
1/4(D3 - iD6)
l/4(01 - Dd -i/4(Dz
+-
1/4(0, + iD8)
- 1/4D9
l/4(01 + Ds)
+ 04)
-1/4(D3 - iD6)
+i/4(Dz - 04) -+
l/4(03 + iD6)
l/4(01 + Ds) -i/4(D2
--
l/4(&
- Ds)
+1/4(D2 + 04)
- 1/4(D7 - iDs)
- l/44
- D4)
- 1/4(D3 + iD6)
- l/4(0,
+ iDg)
+ 1/4D9
-
The total dipolar interaction matrix consisting of all DI * * . Dg, El . . * Es, and FI - . . F9 elements is displayed in Table 4. The sum of all three matrices: Jk! = Table 2 + Table 3 + Table 4 gives the total Hamiltonian matrix used in Eqs. [24]-[27]. ACKNOWLEDGMENTS This research was supported by NIH Grant GM24480. Stevens for a helpful discussion. We thank Ms. Karen Waln and preparation of the manuscript.
One of us (R.E.C.) thanks Professor K.W.H. for assistance in many steps of the calculation
REFERENCES 1. A. A. MEDZHILIOV, L. N. DIRICHENKO, AND G. I. LIKHTENSHTEIN, Izv. Akad. Nauk, SSSR, Ser. Khim., 968 (1969) (p. 629 in English translation). 2. Yu. G. MAMEDOVA, A. A. MEDZHIDOV, AND L. N. KOLOMINA, Russ. J. Inorg. Chem. 17, 1548 (1972). 3. A. A. MEDZHIDOV, A. M. MUSAEV, AND E. G. ROZANTSEV, Izv. Akad. Nauk. SSSR, Ser. Khim, 538 (1977) (p. 483 in English translation). 4. A. RASSAT AND P. REY, Bull. Sot. Chim. Fr., 8 15 (1967). 5. R. Z. SAGDEEV, Yu. N. MOLIN, R. A. SADIKOV, L. B. VOLODARSKY, AND G. A. KUTIKOVA, J. Magn. Reson. 9, 13 (1973). 6. W. H. ORME-JOHNSON, N. R. ORME-JOHNSON, C. TOUTON, M. EMUIPTAGE, M. HENZL, J. RAWLINGS, K. JACOBSON, J. P. SMITH, W. B. MIMS, B. H. HUYNH, E. MUNCK, AND G. S. JACOB, in “Molybdenum Chemistry of Biological Significance” (W. E. Newton and S. Otsuka, Eds.), p. 85, Plenum, New York, 1980. 7. P. KOTTIS, J. Chem. Phys. 47, 509 (1967). 8. J. P. LEMAISTRE AND P. Korrrs, J. Chem. Phys. 68,273O (1978); 76,872 (1982).
ESR LINESHAPE
FOR THREE-ELECTRON
SYSTEM
81
9. E. V. CONDON AND G. H. SHORTLEY, “The Theory of Atomic Spectra,” Cambridge Univ. Press, Cambridge, England, 1963. 10. T. MORIYA, Phys. Rev. 120,91 (1960). II. T. NAGAMIYA, K. YOSIDA, AND R. KUBO, Adv. Phys. 4, I (1955). 12. R. E. COFFMAN AND G. R. BUE~TNER, J. Phys. Chem. 82,2387 (1979). 13. H. GOLDSTEIN, “Classical Mechanics,” Addison-Wesley, Cambridge, Mass., 1953. 14. M. E. ROSE, “Elementary Theory of Angular Momentum,” Wiley, New York, 1957. 15. International Union of Crystallography, “International Tables for X-Ray Crystallography,” Vol. 2, “Mathematical Tables,” Kynoch Press, Birmingham, England, 1952-1962. 16. R. G. GORDON AND T. MESSENGER,in “Electron Spin Relaxation in Liquids” (L. T. Muus and P. W. Atkins, Eds.), p. 341, Plenum, New York, 1972. 17. R. G. GORDON, Adv. Magn. Reson. 3, 1 (1968). 18. J. H. FREED, G. V. BRUNO, AND C. F. POLNASZEK, J. Phys. Chem. 75,3385 (1971).