MAGNETIC RESONANCE IN CHEMISTRYMagn. Reson. Chem. 2003; 41: 806–812Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/mrc.1251

Calculating NMR paramagnetic relaxationenhancements without adjustable parameters:the spin-3/2 complex Cr(III)(AcAc)3

Jeremy Miller,1 Nathaniel Schaefle2 and Robert Sharp2∗

1 Department of Chemistry, State University of New York–Oneonta, Oneonta, NY 13820, USA2 Department of Chemistry, The University of Michigan, Ann Arbor, MI 48109, USA

Received 8 May 2003; Revised 6 June 2003; Accepted 14 June 2003

NMR paramagnetic relaxation enhancement (NMR-PRE) produced by the electron spin S = 3/2 complexCr(III)(acac)3 (acac = acetylacetonato) has been simulated by spin dynamic (SD) simulation methods inorder to test current theory of NMR-PRE. This system provides a particularly demanding test of theory,since the Zeeman and zero field splitting (zfs) contributions to the electron spin Hamiltonian are ofcomparable magnitude in the range of magnetic field variation of the data, and Brownian reorientationof both the zfs tensor and the interspin vector play important roles in the relaxation mechanism. ForCr(III)(acac)3, all of the sensitive parameters of theory were known from independent experiments, sothat a calculation was possible without variation of parameters. The R1 field dispersion profile (fdp),reported by Wang et al. (J. Magn. Reson. 1987; 73: 277), exhibit a single pronounced dispersive featurebetween 0.23 and 2.1 T. SD simulations showed that this feature results physically from the change inspatial quantization of the electron spin motion that occurs in the intermediate regime of field strengthswhere the Zeeman and zfs energies are comparable. Thus, the observed R1 dispersion for Cr(III)(acac)3 hasa qualitatively different physical origin than the dispersions of Solomon–Bloembergen–Morgan theory,where R1 dispersions result from Zeeman breaking of level degeneracies. The shape of the experimentalfdp suggests that the point-dipole approximation is invalid for Cr(III)(acac)3 due to delocalization ofelectron spin density onto the p-system of the acac ligand. After accounting for the altered geometry, goodagreement with experiment was obtained. Copyright 2003 John Wiley & Sons, Ltd.

KEYWORDS: NMR relaxation; NMR paramagnetic relaxation enhancement; NMR-PRE; Cr(III)(acac)3

INTRODUCTION

The theory of NMR relaxation produced by paramagneticsolutes in the Zeeman limit was developed by Solomon1

and Bloembergen and Morgan2,3 (SBM theory) around 1960.This theory, refined in various ways in succeeding years,has been widely employed to extract chemical, dynamicand electron spin information from NMR paramagneticrelaxation enhancement (NMR-PRE) data. SBM theoryassumes that the electron spin Hamiltonian HS is a ZeemanHamiltonian (HS D HZeem), as occurs for electron spinS D 1/2 species such as Cu(II) and Ti(III). It may also bean approximate description for S > 1/2 ions; however,this is more problematic, since there may then be apermanent zero field splitting (zfs) interaction in the electronspin Hamiltonian. Unless the zfs interaction is zero byreason of the molecular point group symmetry or thezfs-level structure is averaged to zero by rapid Brownian

ŁCorrespondence to: Robert Sharp, Department of Chemistry, TheUniversity of Michigan, Ann Arbor, MI 48109, USA.E-mail: rrsharp@umich.eduContract/grant sponsor: National Science Foundation;Contract/grant number: CHE-0209616.

reorientation, Hzfs influences the electron spin motions and,in consequence, the NMR-PRE. It is useful to considerNMR-PRE phenomena with respect to two limiting cases:(1) the high field or Zeeman limit, where the electronicZeeman Hamiltonian is much larger than the zfs Hamiltonian(HZeem × Hzfs); and (2) the low field or zfs limit, whereHzfs × HZeem. SBM theory applies to the former, andcorresponding zfs-limit formulae have been derived for thelatter.4 – 7

The intermediate regime, where Hzfs ³ HZeem, is moredifficult to describe, especially when the zfs contributionto the spin level structure is modulated by Brownianreorientation. The present study describes a test of currenttheory in this physical situation. Proton R1 field dispersionprofile (fdp) data for the complex ion Cr(III)(acac)3 (acac Dacetylacetonato), measured by Wang et al.,8 have beencalculated using spin dynamics (SD) simulation methods.In these calculations, all of the sensitive parameters of theorywere measured in independent experiments. These include:(1) the zfs D parameter; (2) the reorientational correlationtime �R; (3) the interspin distance rIS; and (4) the angle �between the interspin vector rIS and the unique axis of thezfs tensor. Our objectives are to understand the physical

Copyright 2003 John Wiley & Sons, Ltd.

Calculating relaxation enhancements 807

0

1000

2000

3000

4000

5000

6000

10.2

Zeem

R1p

(s-1

)

B0 (T)

Figure 1. Magnetic field dispersion profiles of R1 of the methylprotons of Cr(III)(acac)3, taken from Wang et al.8 Profiles weremeasured at T D 253, 263, 273, 283, 293, 303, 313, and 323 K,increasing with arrow. Zeeman-limit profile (dashed line) wascalculated assuming re D 465 pm, �R D 140 ps, �S D 1.4 ns.

origin of the dispersive features in the experimental fdp andto test the accuracy of SD simulation methods in a systemwhere Brownian reorientation of the zfs tensor is rapid andHzfs ³ HZeem.

The experimental data are shown as a function offield strength at seven temperatures in Fig. 1. At alltemperatures, the profiles decrease with increasing B0 inthe range 0.23–1.0 T, then are almost field independent forB0 > 1.0 T. At the lowest temperatures, a shallow minimumis present in the vicinity of 1.0 T. We show below thatthese dispersive features result physically from the changeof spatial quantization of the electron spin motion as thespin system passes through the intermediate regime offield strengths where HZeem ³ Hzfs. Thus, they have afundamentally different origin than the dispersions of SBMtheory, which result from the Zeeman splitting of leveldegeneracies.

THEORETICAL AND COMPUTATIONALMETHODS

The electron spin Hamiltonian HS can be written

HS D Hzfs C HZeem �1�

Hzeem D geˇeB0S�1�0 �2�

Hzfs D 6�1/2hcD OS�2�0 C 21/2hcE� OS�2�

C2 C OS�2��2� C h.o.t. �3�

where the spin operators of HZeem and Hzfs are expressedin spherical tensor form in different coordinate systems, theformer in the laboratory coordinate system, the latter in themolecule-fixed principal axis system of the zfs tensor. Thequantities ge, ˇe, B0, h, and c are the electron g-value (assumed

to be 2.00), the Bohr magneton, the polarizing magneticfield, Planck’s constant, and the speed of light. The ESRparameters D and E (cm�1� are coefficients of the cylindricaland orthorhombic parts of the quadratic zfs tensor. For spinsS ½ 2, Hzfs may contain terms of higher even orders in thespin operators, but these are absent for Cr(III), which is ahigh-spin d3 ion with S D 3/2.

The molecular point group is D3 (the crystal structure9 isshown in Fig. 2), and thus the zfs tensor is cylindrical:

Hzfs D 6�1/2D OS20 �4�

The D parameter, measured by ESR in benzene10 andtoluene11 solution, is 0.592 š 0.002 cm�1. The electron spinlevel diagram for Cr(III)(acac)3 is shown in Fig. 3 formolecular orientations parallel and perpendicular to B0. Inthe low-field limit, the spin level diagram consists of themS D š1/2 and mS D š3/2 Kramers doublets separated by aninterdoublet splitting equal to 2D. The presence of a Zeemanfield splits the doublets by an amount that depends on bothjB0j and on the angle ˇ between B0 and the molecular z0-axis.In the parallel orientation, level crossings occur at ωS D ωD

and ωS D 2ωD. Level crossings do not occur in non-parallelorientations. The intermediate regime can be taken, roughly,as the vicinity where ωS ³ ωD, which occurs near B0 ³ 0.6 T.

The fact that Hzfs and HZeem do not, in general, commuteintroduces a number of complications into the theory. Thefirst of these is that the electron spin wavefunctions dependon the magnetic field strength, varying from the Zeemanbasis functions quantized along B0 in the Zeeman-limitto zfs-limit eigenfunctions polarized along the molecule-fixed zfs principal axes at low fields. In liquids, Brownianmotion introduces stochastic time dependence into HS and

Z′

Figure 2. Crystal structure9 of Cr(III)(acac)3.

Copyright 2003 John Wiley & Sons, Ltd. Magn. Reson. Chem. 2003; 41: 806–812

808 J. Miller, N. Schaefle and R. Sharp

0 2

β=0

β=π/2

±3/2

±1/2

B0 (T)1

ε

Figure 3. Electron spin level diagram of S D 3/2 in the paralleland perpendicular molecular orientations for D D 0.60 cm�1

and E D 0 (cylindrical zfs tensor).

into the spin eigenfunctions. In addition to the stochasticmodulation of the electron spin dynamics, Brownian motionof the interspin vector rIS modulates the electron–nucleardipolar interaction, and these motions (of HS and rIS� arecorrelated.

The formal theory of the NMR-PRE is developedin Ref. 12. Two levels of the theory were employed incalculations. The first is the ‘constant HS’ approximation,in which the time dependence of HS is ignored (HS D H�0�

S ).The spin averages are calculated as in a powder.13 – 16 Theinterspin vector rIS undergoes Brownian reorientation withcorrelation time ��1�

R , whereas for second-rank molecule-fixedspherical harmonics the correlation time is ��2�

R D 3�1��1�R . We

write �R � ��2�R , as is usual in the NMR literature. Expressing

R1p in the laboratory frame (Ref. 12, eqn (2.11)) gives

R1p D 5�1c2dr�6

IS

∫ 1

02Re

{[Gz�t� � G?�t�]

ð exp�iωSt � t/�R�}

ea dt �5�

Gz�t� D 3hSz�t� Ð Sz�0�i �6�

G?�t� D hSC�t� РS��0�i C 6hS��t� РSC�0�i �7�

cd D geˇe��0/4�� �8�

where ωI and �0 are the nuclear Larmor frequency and thepermeability of space, Re stands for ‘the real part of’ andea denotes ensemble average. The curly brackets denotean ensemble average over molecular degrees of freedom.The time correlation functions Gr�t� of the electron spinmotion can be evaluated as a trace over the eigenfunctionsof H�0�

S :

Gr�t� D 3

{∑i

hij exp��iH�0�S t�Sr

ð exp�iH�0�S t�Srjii exp��t/��i�

S �

}oa

�9�

where the curly brackets represent an average over molecularorientations and oa denotes orientational average. To providethis average, the calculation averages 92 molecular orienta-tions corresponding to the vertices and face centers of thetruncated octahedron (buckeyball). The electron spin relax-ation times ��i�

S in Eqn (9) result from collisional modulationof the zfs tensor. They can be used as constant parameters orelse evaluated as field-dependent quantities using the theoryin Refs 17 and 18. According to Eqns (5)–(9), the ‘constant HS’approximation includes the time dependence of rIS but notthat of HS. This approximation is useful because it is directlyinterpretable in terms of the electron spin level diagram.Sharp13 describes the earliest formulation of our ‘constantHS’ algorithms, which have subsequently been refined, forexample to include rhombic and other higher order zfs terms.

The second level of theory includes a description ofthe correlated motions of HS and rIS. This introduces con-siderable complexity into the calculation, but is of criticalimportance for a quantitative description. Work in our labo-ratory employs SD simulation methods,19,20 which calculatethe Gr�t� as thermal ensembles of Brownian trajectories ofcoupled spin and molecular motions. The problem has alsobeen approached by Kowalewski and coworkers using thestochastic Liouville equation.21 – 24 In SD simulations, trajec-tories are composed of sums of terms, each a product ofthe form

T�t� D hexp��iHS�t�t�Sr

ð exp�iHS�t�t�SriY2,p��0, ϕ0; t�Y2,p0��, ϕ; 0� �10�

The Y2,m��, ϕ; t� are second-rank spherical harmonics of thepolar angles �, ϕ, of rIS in the laboratory frame. The motionof the Y2,m��, ϕ; t� is correlated with that of the electron spinHamiltonian HS�t�. Time correlation functions defined bythese products are calculated as ensemble averages, usingthe quantum mechanical equation of motion to propagatethe spin variables and a classical model (see below) to sim-ulate molecular reorientation. The simulations are believedto be capable of a realistic description of NMR-PRE phe-nomena subject, in our implementation, to two importantapproximations: (1) the point–dipole approximation for theelectron spin; and (2) an assumption of isotropic molecularreorientation.

The algorithms are implemented in a Fortran 90 computerprogram, Parelax2, in the following manner. To provideorientational averaging, the molecular orientation (�, ϕ) atthe start of each trajectory is set equal (as for the ‘constantHS’ algorithm) to the polar angles of one of the 92 verticesor face centers of the truncated icosahedron. Starting fromthis initial orientation, the electron spin Hamiltonian isevaluated and the spin operators propagated quantummechanically in the time domain. At a sequence of randomtimes during the trajectory, the molecular frame undergoes ajump reorientation about a random axis. The reorientational

Copyright 2003 John Wiley & Sons, Ltd. Magn. Reson. Chem. 2003; 41: 806–812

Calculating relaxation enhancements 809

algorithm that implements this is governed by a parameterns equal to the number of Brownian jumps per �R interval.The reorientational jump angles are generated as Gaussiandeviates of zero mean and standard deviation . Themagnitude of depends on ns and must be selected togive the desired correlation time �R. Test calculations wereused to determine the appropriate relationship, which, forreasonably small steps (ns ½ 10), is well described by

D �√

0.101/ns

By choice of ns, finite step sizes can be simulated. Thereorientational algorithms of Parelax2 are believed to providea realistic description of Brownian reorientation for isotropicrigid molecules. No attempt is made to provide the levelof motional detail offered by full molecular dynamicssimulations as in the work of Odelius et al.25,26 and Krukand Kowalewski27 in studies of Ni(II)(H2O)2C

6 .Following each reorientational jump, HS�t� is recalcu-

lated, and the spin propagator, exp��iHS�t�t�, is evaluatedfrom the series definition using double-precision Fortran 90(single precision results in loss of the operator norm). In thisway, ensembles of the required time correlation functionsare constructed. The minimum number of trajectories in theensemble is 92 (corresponding to the number of startingorientations in the spatial average). Typical runs involveensembles of 12 ð 92 trajectories for a single R1p value andtypically contain <5% noise. The full set of SD simulationsin Fig. 5 required about 6 h using an e-mac.

SIMULATIONS

Physical parametersThe calculations focus on the T D 253 K profile of Fig. 1,where the dispersive features are most pronounced. Fourphysical parameters (Table 1) are used in the calculations.At magnetic field strengths outside the Zeeman limit, theNMR-PRE depends on the molecular geometry through twoquantities: the electron–nuclear interspin distance rIS; andthe polar angle � between the interspin vector rIS� and thethree fold molecular rotation axis z0. Internal methyl rotationhas been shown to be rapid in Pd(acac)2, and we assume thesame for Cr(III)(acac)3. Averaging over this motion gives, forthe effective interspin distance

re D �hr�6IS i��1/6 D 465 pm

and for the effective �

�e D h�r�6IS i/hr�6

IS i D 1.03 rad

Table 1. Physical parameters used in calculationsand SD simulations

Parameter Value Ref.

D (cm�1� 0.59 10E (cm�1� 0.00�R at 253 K (ps) 138 8�S at 253 K (ns) 1.2 11re (pm) 465 9�e (rad) 1.03 9

The reorientational correlation time �R has been measuredfor the diamagnetic surrogate Co(III)(acac)3 in CHCl3 as afunction of temperature by Wang et al.,8 who measuredapproximately equal values from both methine and methyl13C R1 data. The measured �R values at 253 K and 323 K are138 ps and 35 ps respectively, and the dependence of ��1

R

at intermediate temperatures is approximately exponential.Doddrell et al.28 inferred significantly shorter �R values basedon an analysis of methyl 1H R1 values, but this measurementis less direct than that of Wang et al.,8 and we use thelatter. The methine C—H vector lies in the transverse (x0 –y0)molecular plane, and the 13C R1 involves motions aboutboth longitudinal and transverse axes. Stochastic motionof the zfs tensor involves reorientation of z0 about (x0,y0), thus introducing an assumption of isotropic molecularreorientation into the analysis.

The electron spin relaxation time �S is one or two orders ofmagnitude longer than �R and is not a sensitive parameter oftheory. A lower limit can be estimated from the ESR linewidthdata of Ref. 11, which gives the value �S ½ 1.2 ns at 253 K. Noattempt was made to account for magnetic field dependenceof this quantity, nor for possible multi-exponential behavior,since the contribution of �S to the dipolar correlation time issmall.

Physical interpretation of the fdpThese parameters suffice for simulations of the R1 fdps.Simulated fdps calculated in the ‘constant HS’ approximationare shown as a function of �e in Fig. 4. As is well known,5,6,24,29

the shape of the fdp depends strongly on the nuclearorientation with respect to the zfs principal axes in theintermediate and zfs-limit range of Zeeman field strengths.The Zeeman-limit fdp is shown as a dashed line. Two well-developed dispersive features are present in the calculations.

0

5000

10000

15000

0.01 0.1 1

B0

Zeem

ωS=ωDωSτd=1

θ=π/2

θ=0

R1p

(s-1

)

Figure 4. R1p field dispersion profiles for S D 3/2 calculatedassuming the ‘constant HS’ approximation described in thetext. The calculations used the physical parameters of Table 1,except �e, which was varied as follows: �e D 0.0, 0.3, 0.6, 0.9,1.2, �/2, increasing with the arrow. The dashed curve isSBM theory.

Copyright 2003 John Wiley & Sons, Ltd. Magn. Reson. Chem. 2003; 41: 806–812

810 J. Miller, N. Schaefle and R. Sharp

A low-field feature, centered in both the ‘constant HS’ andZeeman profiles at the field strength where ωS�d D 1, arisesfrom the one-quantum (1Q) matrix elements of the transversespin operators and results from breaking of the degeneracyof the spin levels with increasing B0. In the Zeeman-limittheory, contributions to this dispersion arise from both themS D š1/2 and š3/2 levels, whereas in the constant HS

theory only the mS D š1/2 levels contribute, since the 1Qcontributions between mS D š1/2 and mS D š3/2 levels aresuppressed by the zfs.

At higher field strengths (0.1 < B0 < 2.0 T), relaxationefficiency results almost entirely from diagonal spin matrixelements. The high-field dispersive feature results fromeffects on these matrix elements of the change of spatialquantization that occurs in the regime of field strengthswhere ωS D ωD (D 2�hcD). The mid-field position of thehigh-field dispersion, unlike that of the low-field feature,is independent of the dipolar correlation time and dependsonly on D. The shape of the dispersion is determined solelyby the angle �e.

It should be mentioned that the characteristic form of fdpdepends critically on the electron spin quantum number,29

with integer spins exhibiting rather different propertiesthan half-integer spins. The functional forms of the fdpsof integer spins are strongly influenced by the low symmetry(orthorhombic and fourth order) components in the zfstensor, because these terms break the degeneracies of thenon-Kramers doublets.23,24,30 – 33 The Kramers doublets ofhalf-integer spins are not split by crystal field componentsof any symmetry or magnitude, and for this reason thecharacteristic R1p profiles of integer and half-integer spinsare dissimilar. The effect of changing spin quantization on thequalitative shape of the fdp in the intermediate regime hasbeen described theoretically.4,5,7,12,14,20,24,29 The phenomenonhas been demonstrated experimentally for the S D 5/2 ionsMn(II)20 and Fe(III),34 but not, as far as we are aware, forS D 3/2.

SD simulationsFigure 5 compares the ‘constant HS’ profiles with the resultsof SD simulations. The 253 K data set from Fig. 1 isalso shown. Since the Zeeman-limit profile (dashed linesin Figs 1 and 4) is essentially flat across the experimentalrange, it is clear that the observed dispersion results fromthe changing spatial quantization of the electron spin, asdescribed above. The effect of motion of HS depresses therelaxation efficiency and is responsible for the local minimumnear 1 T. SD simulations for �e � 0.4 rad were in excellentagreement with the data (about 10% discrepancy, with noadjustment of parameters). Surprisingly, however, largervalues of �e corresponding to the point dipole approximation(�e D 1.03 rad) did not give acceptable agreement. Avariety of trial calculations (not shown) were carried outto explore possible reasons for the apparent discrepancy.These confirmed that the shape of the high-field dispersionis critically determined by �e, and good agreement with thedata was obtained only for �e � 0.4 rad. SD simulationswere also performed assuming specific conformations of themethyl protons. Conformations consistent with the crystal

0

5000

10000

0.1 1.0

R1p

(s-1

)

B0 (T)

Figure 5. SD simulations of R1p field dispersion profiles forS D 3/2 using the parameters of Table 1 and the �e valuesgiven in the legend of Fig. 4. Dashed lines show the ‘constantH

0S calculations of Fig. 4.

structure (which does not locate hydrogen atoms) have�IS and rIS within the ranges, 0.83 � �IS � 1.25 rad and432 � rIS � 499 pm. As indicated by the results in Fig. 5,however, good agreement with experiment was obtainedonly in simulations with �e � 0.4 rad.

Another factor that was investigated as a possibleexplanation for the unexpectedly small inferred value of�e is the influence of reorientational step size in the SDsimulations. As described above, conditions correspondingto both large- and small-step reorientational diffusion (thelatter corresponding to the classical diffusion limit) arepossible in the simulations through the choice of theparameter ns (the number of reorientational steps per �R

interval). The simulations shown in Fig. 6 were performedassuming �e D 0.85 rad with ns values of 67, 33, 20, 10, and 3.The corresponding 1– jump angles are given in the figurelegend. Simulations with ns ½ 10 differed only slightly fromthe small-step limit, whereas ns D 3 showed significantenhancement of relaxation efficiency. For all step sizesstudied (ns ½ 3), the profile shape was almost independentof step size. Thus, large-step Brownian diffusion does notexplain the small value of �e needed to reproduce the shapeof the experimental fdp.

The results suggest that the point-dipole approximationis not accurate for Cr(III)(acac)3, presumably due to delocal-ization of spin density onto the �-system of the acac ligands.Such delocalization affects both rIS and �e, and thus it cannotbe claimed at present that a fully satisfactory explanation ofthe data has been obtained without variation of parameters.However, the physical origin and shape of the observed R1

dispersion, including the local minimum near 1.0 T and theapproximate magnitude of R1, are very well reproduced inthe simulations. Clearly, the effects of both the permanent zfsinteraction and of Brownian modulation of HS are of centralimportance in the relaxation mechanism.

Copyright 2003 John Wiley & Sons, Ltd. Magn. Reson. Chem. 2003; 41: 806–812

Calculating relaxation enhancements 811

0

2500

5000

7500

0.01 0.1 1

R1p

(s-1

)

B0 (T)

Figure 6. Dependence of the simulated R1p field dispersionprofiles on reorientational step size. Parameters are those ofTable 1. The SD simulations were performed for parametervalues of ns D 67 (♦), 33 (�), 20 (��, 10 (°�, and 3 (ž), wherens is the number of reorientational steps per �R interval. Thesevalues correspond to 1– jump angles D 0.071, 0.100,0.129, 0.183, and 0.317 rad.

It is interesting that the shape of the fdp is so sensitive to�e in the intermediate regime of field strengths. This allows amore direct and positive examination of the point-dipoleassumption than is provided by distance considerationsalone. Failure of the point-dipole approximation in metalacetylacetonates has been proposed by Gottlieb et al.35

based on calculations of spin density on the acac ligands.Supporting experimental evidence was also found36 in theparallel behavior of isotropic hyperfine chemical shifts and(1H/13C) R1 ratios. Failure of the point-dipole approximationin acac complexes appears to result from spin delocalizationinto the �-system. Ab initio calculations37,38 of -bondedcomplexes, including hexa-aqua metal ions, have indicatedthat distance corrections due to departures from the point-dipole approximation are insignificant for atoms more thantwo bonds from the metal.

However, recent analyses of large R1 data sets forparamagnetic proteins of known structure have suggestedfailures of the point-dipole approximation over longerdistances. Measurements39 of 15N, 13C, and 1H R1 values inCu(II) plastocyanin from Anabaena variabilis indicate that thepoint-dipole approximation is not satisfied for heteronucleiwithin about 10 A of the metal center. Likewise, Wilkenset al.40 have found that the point-dipole approximation isinadequate for accurate calculations of 15N R1p data sets fromthe high-spin Fe(III) rubredoxin of Clostridium pasteurianum.

Accuracy and scatter of SD simulationsSystematic tests of the SD algorithms were run in limitingcases where comparison with analytical theory is available;these include the Zeeman limit and the slow reorientationlimit. A number of comparisons of this kind are run routinely

0

5000

10000

15000

0.001 0.01 0.1 1 10

Zeem

Slow Reor

R1p

(s-1

)

B0 (T)

Figure 7. Tests of the SD algorithms. Dashed line is theZeeman-limit calculation of Fig. 3, and closed circles are SDsimulations performed under equivalent physical conditions.Solid line is the slow reorientation calculation, and open circlesare the corresponding SD simulations performed underequivalent physical conditions, as described in the text.

as part of the self-check routine of Parelax2. Figure 7 showsthe results of two such comparisons. In one set of calculations(open circles), the SD simulations of Fig. 5 were repeatedwith the zfs D parameter set to zero (equivalent to theZeeman limit). The simulations agreed well with SBM theory(dashed line) with �5% scatter. In a second set of calculations(filled circles), the results of SD simulations performedunder slow reorientation conditions were compared withthe ‘constant HS’ calculations of Fig. 4. These simulationsused the parameters of Table 1 with �e D 0, but the SDreorientational algorithm was turned off (long �R�. Thedipolar correlation time was set equal to 125 ps in bothcalculations, giving equivalent physical conditions. With thereorientational algorithm turned off, the SD simulations arenoise free, and agreement was essentially perfect.

CONCLUSIONS

(1) The physical mechanism of the NMR-PRE for Cr(III)(acac)3 was analyzed. Both the permanent zfs interaction andthe Brownian modulation of HS are of central importance inthe mechanism, and a quantitative understanding of theshape of the R1 fdp must include both of these factors. Thedispersive features in the measured fdp result physicallyfrom the change in spatial quantization of the electronspin motion that occurs in the intermediate regime of fieldstrengths between the Zeeman and zfs limits. Thus, theobserved R1 dispersion for Cr(III)(acac)3 has an entirelydifferent physical origin from the dispersions of the classicalZeeman-limit (SBM) theory, where they result from Zeemanbreaking of level degeneracies.

(2) Attempts were made to calculate the R1 fdp at253 K using only parameters (�R, �s, D, rIS, �e) known

Copyright 2003 John Wiley & Sons, Ltd. Magn. Reson. Chem. 2003; 41: 806–812

812 J. Miller, N. Schaefle and R. Sharp

from independent experiments. From these simulations,it was concluded that the point-dipole approximation ofthe electron spin is not appropriate for Cr(III)(acac)3.SD simulations performed using an angle �e D 1.03 rad,corresponding to a point spin located on the Cr(III) nucleus,produced profiles inconsistent with the data with respectto both shape and magnitude. However, excellent generalagreement with experiment was obtained for �e � 0.4 rad.The shape of the fdp, including the presence of a local R1

minimum near 1.0 T, and the approximate magnitude of R1

were reproduced well with no variation of parameters otherthan �e.

(3) The effect of varying the reorientational step size inthe SD simulations on the shape of the fdp and the magnitudeof R1 was investigated. Simulated profiles were only slightlydependent on the step size for physically realistic choices ofthis parameter.

AcknowledgementsThis material is based upon work supported by the National ScienceFoundation under grant no. CHE-0209616.

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