214 Bulletin of Magnetic Resonance

Stochastic Averaging Revisited

D.H. Jones, N.D. Kurur, and D.P. Weitekamp

Arthur Amos Noyes Laboratory of Chemical PhysicsCalifornia Institute of Technology

Pasadena, CA 91125, USA

Contribution No. 8724

1 Introduction

The extraction of molecularpotentials and rates by modeling thedependence of spectra on thermodynamicstate is one of the major contributions ofmagnetic resonance to molecular physics.We argue here that this entire endeavor isconceptually suspect due to the implicitfactorization of spin and spatial degrees offreedom in calculating stochastic averages.All such averages have heretofore neglectedspin—dependent energies in the potential fornuclear motion and are therefore not resultsof equilibrium statistical mechanics. Wedevelop an averaging procedure fromequilibrium statistical mechanics and findthat it predicts large, previouslyunrecognized contributions to motionallyaveraged spin Hamiltonians which aredirectly proportional to spatial terms in theenergy. We discuss the present state of theexperimental evidence for the acceptedaverage, with particular emphasis onindirect scalar couplings averaged byconformer equilibria, and find that thequality of presently available data cannotresolve the issue conclusively. The weaknessof the theoretical basis of the acceptedtheory is also outlined.

2 The Traditional Stochastic Average

For over forty years it has beenaccepted1"36 that spin states are transportedbetween spatial states with spin-independent rates. This unexaminedassumption was clearly stated in the seminalwork of Bloembergen, Purcell and Pound:1"The atom or molecule is simply a vehicle

by which the nucleus is conveyed from pointto point. We thus neglect the reaction ofmagnetic moments of the nuclei upon themotion." This notion is the" basis for allexisting formalisms1"36 for calculating themagnetic resonance lineshapes of spinsystems undergoing spatial rate processes —most importantly, chemical exchange. Awell—known consequence is that the averagevalue of a spin—Hamiltonian parameter(chemical shift, scalar coupling, dipolarcoupling, etc.) in the fast—exchange limit isgiven by

= SpnX:n n, (1)

where Xn is the value of this parameter inthe spin Hamiltonian of the nth spatialmanifold and pn is viewed as the probabilityof the system being in that manifold,irrespective of spin state. The sum may beover molecular eigenstates or over' largegroups of them, as when n indexes molecularconformers. The traditional prescription,which neglects spin energies, is to expressthe molecular partition function q as a sumof parts qn associated with each indexedmanifold. The probability for each manifoldis pn = qn/<l> and the ratio of two suchprobabilities is (in the absence of workterms)

Pn/pn' = exp[-AAnn' /RT] (2a)

= exp[(-AUnn'+TASnn')/RT], (2b)

where AAnn'= -RTln(qn/qn') is thedifference in the molar Helmholtz freeenergies of the manifolds. In Eq. 2b, the freeenergy differences have been divided into

Vol. 14, No. 1-4 215

differences in energy AUnn' and entropyASnn'- The connection to molecularenergies is AUnn' = N^(En-En ') , where En

is the common spatial contribution to freeenergy of the nth manifold of spin states andN 4 is Avogadro's number.

3 An Alternative Stochastic Average

The actual molecular energies may bewritten as E? = En+E (n), where 7 indexesa spin eigenstate within the nth manifold.Since spin Hamiltonians are constructed astraceless, the spin—dependent contributionsE (n) sum to zero in each manifold.Nevertheless, the absence of the spin energyterms in the pn indicates unambiguouslythat Eq. 1 is not derivable withoutapproximation from equilibrium statisticalmechanics.

We address the following questions.What is the exact equilibrium expression forthe stochastically averaged parameters? Forwhich systems will it differ measurably fromEq. 1? Do existing experiments decide theissue conclusively?

To proceed, we specialize to the casethat the spin Hamiltonians in allsignificantly occupied spatial manifolds aremutually commuting. Then the spineigenbasis {I7)} is independent of spatialstate and is the basis needed to describe thestochastically averaged spectrum. Thespatially-averaged energy of a particularsuch spin eigenstate 17) is

E =SE2exp(-A2/RT)/Sexp(-A2/RT) (3a)I n n

= SpM (3b)n

Note that Eq. 3 uses the completemanifold free energy, A 2 = NAEZ-TSn,including spin terms, according to theprescription of equilibrium statisticalmechanics. Each such summation isrestricted to a particular spin state and thedistribution among spatial states for eachspin state is assumed to be the equilibriumdistribution at the lattice temperature.Thus p2 is the conditional probability of

being in the nth spatial manifold, given thatthe spin state is [7). As in the traditionalformulation, no specification of thedistribution of population among spin statesis needed. Our hypothesis is that thespectral line positions for sufficiently fastexchange between spatial manifolds are theBohr frequencies, ^ ^ ( E ^ - E ^ / hcorresponding to differences between theaverage energies of Eq. 3. Thecorresponding motionally averagedspin—Hamiltonian parameters are thosewhich generate this spectrum.

As a simple illustration, consider aspin Hamiltonian with only two distincteigenvalues, ±Xn/2 (in Hz), for each n.Then the proposed alternative to Eq. (1) is(with 7 = ±)

(X) =

(= S [(p£ -

n

(4a)

(4b)

(4c)

For the usual high—temperature case wherehXn << kT, the approximation pn 2

' " "̂ is adequate, and the differencebetween the formulations of Eqs. 1 and 4 isessentially the sum over n of the first termin "brackets of Eq. 4c. This differencevanishes at infinite or zero temperature andat all-temperatures if either the En or the Xnare all degenerate. Cases without suchdegeneracy are the ones where (X) mightprovide information on molecular potentials.In these the difference term is of orderXnEn/kT. Thus, it is not in general a smallcorrection and can in fact dominate thetemperature dependence over some ranges ofthe variables (Fig. 1). This new term in thespin Hamiltonian may be viewed as arisingfrom the dependence of average molecularconfiguration on the spin Hamiltonian, incontrast to the usual analysis which includesonly the dependence of the spin Hamiltonianon molecular configuration. Alternativelystated, because the distribution of moleculesamong the spatial manifolds (eg. conformers)

216 Bulletin of Magnetic Resonance

i "i

t! Im

depends on spin, spatial energy differencesbetween manifolds contribute to thestatistically averaged frequencies for spintransitions. The product of a smallspin—dependent probability differencemultiplied by a large spatial energydifference gives a readily measuredcontribution to the magnetic resonance lineposition.

4 Test Systems

In both formulations it is possibleideally to predict the fast exchangeobservations without adjustable parameters.This would constitute a purely experimentaltest, the quality of which depends only onthe experimental uncertainties. In practicethere seems to be no case where NMRspectra of individual molecular eigenstateshave been obtained separately and also as athermal average.

Thus, it seems necessary to look tothose situations where n indexes conformers.The experimental concept is simple andwell-known. At low temperature, the spinHamiltonian for each conformer can bedetermined, since, in the limit of negligiblechemical exchange between them, separatespectra are seen for each. The relative areasof these spectra at each slow—exchangetemperature provide the relative populationsand thus the free—energy difference betweenconformers. A linear fit to the temperaturedependence of this free—energy differenceallows it to be decomposed into two terms,which can be viewed as an energy differenceand an entropy difference if one additionallyassumes that the temperature dependence ofthese is negligible over the experimentalrange.

Since a conformer is a set ofmolecular eigenstates, additional dependenceon thermodynamic state (eg. temperaturedependence) is possible due to averagingwithin this set. Theoretically, one hasprecisely the same problem in deciding howto take this average as for the averaging overconformers. However, if one can measure anysuch temperature dependence within theslow—exchange regime and extrapolate tofast—exchange, then the stochastic theoryneed not enter at this level.

Thus, we arrive at a set of criteriawhich need to be met for a compelling, fullyexperimental test of any theory relating

slow-exchange and fast—exchange spectra:i) The system must have

state—dependent rates such thatmeasurements in both the slow— andfast-exchange regimes are possible. Forfluids this typically requires barriers betweenconformers on the order of 10 kcal/mole.

ii) The state dependence of theconformer spin Hamiltonians and thefree-energy differences must be measured inthe slow—exchange region to allowextrapolation through the fast—exchangeregion. This is often the major source ofuncertainty because of the smalltemperature range corresponding toslow-exchange. The difference inthermodynamic state between these regimesideally is small or even zero, so as tominimize the propagation of errors due tophenomenological extrapolation. Usingdifferent NMR transitions or field conditionsto measure the same spin Hamiltonianparameter can help in this regard; since thecriterion for motional collapse varies withthe transition observed, there is no minimum

8.

6. -

81 4.oo

•ao

GO

0.

1 1

- 'V^f1 1

1 __l

—

1 1

0. 200. 400. 600. 800. 1000.

Temperature (K)

Figure 1. Traditional (dashed) andalternative (solid) stochastic averages for asimple two—conformer, two—spin system.Simulations have the general features ofsome substituted ethanes: a gauche couplingof 2.0 Hz, a trans coupling of 20.0 Hz, thetrans conformer with a free energy 200cal/mol greater than the doubly—degenerategauche conformer.

Vol. 14, No. 1-4 217

difference in thermodynamic state betweenfast and slow exchange.

iii) Some or all of the fast-exchangedata should fall outside the error bars on thepredictions of one of the theories, therebydisproving it. The accepted and alternativetheories embodied in Eqs. 1 and 3,respectively, have identical predictions formutual exchange and whenever the occupiedconformers are degenerate in spatial energyor in spin Hamiltonians. For two—siteproblems, the theories will typically differmeasurably when the conformer free energiesdiffer by > 102 cal/mole. When thisdifference exceeds 103 cal/mole, sensitivitywill usually preclude observing theslow-exchange spectrum of the minorconformer. Figure 1 is a numericalcomparison of the two theories for the simplecase of two conformers and a two—spinsystem. It demonstrates that the predictionsof the theories are different by a magnitudethat should be measurable. There is also aqualitative difference: the sign of thetemperature dependence of thefast—exchange spin Hamiltonian parameteris opposite to that of the accepted theoryover part of the temperature range. Thisbehavior is inconsistent with Eq. 1 regardlessof how the pn are calculated.

The above conditions are notextremely restrictive; a substantial fractionof the molecules whose conformer equilibriahave been studied by solution-state NMRfall into this range of free—energy differences.Since the accepted theory has been inincreasing use for four decades, it might beexpected that it would have substantial anddiverse experimental support. While it isdifficult to have confidence in thecompleteness of a search through such alarge literature, we are as yet unaware ofany data set that meets the criteria above.Thus, no theory has presently beenevaluated by this seemingly reasonablestandard.

The only theories ever consideredpreviously are of the form of Eq. 1.Numerous authors have noted failures in itsapplication,19"21 but these have usually beenplausibly attributed to inadequacies in thedata, most commonly uncertainties ofconformer assignment or unmeasuredtemperature dependence of a conformer spinHamiltonian.

5 Substituted Ethanes

Substituted ethanes in solution14"24

are the most studied systems and includecases which nearly meet the criteria of theprevious section. Figure 2 illustrates, withthe example of 1-fluoro-l,1,2,2—tetrachloroethane, the trans and gaucherotational isomers which interconvert atconvenient rates. Since the two gaucheconformers are mirror—images, there areonly two magnetically distinct conformers,one with the H and F atoms trans to oneanother and the other a degenerate pair ofgauche rotamers at the other two staggeredpositions of the dihedral angle of rotationabout the carbon—carbon single bond.Increasing temperature carries the systemfrom the slow—exchange limit to thefast—exchange limit without a change incomposition or phase. The fast—exchange,three—bond vicinal coupling (Jrrp) is knownto be temperature dependent and this hasbeen attributed to the averaging, accordingto Eqs. 1 and 2, between distinct values Jtand Jg in the trans and gauche conformers,respectively.15"16 Unlike chemical shifts,scalar couplings do not require a nominallytemperature—independent referenceresonance to compensate for the usualuncontrolled shifts of internal field withtemperature. Also, scalar couplings aregenerally believed to be less sensitive tointermolecular interactions which couldprovide a confounding mechanism oftemperature dependence.

For the particular case of1—fluoro—1,1,2,2—tetrachloroethane, a test as

E(<t)

Figure 2. Potential energy versus dihedralangle (<J>) for the conformers of 1—fluoro—1,1,2,2—tetrachloroethane.

218 Bulletin of Magnetic Resonance

described of the stochastic averaging theoriesis prevented by the failure to resolve thecoupling Jg.37 This quantity has anexperimental upper bound of 2 Hz. If it isassumed that Jg is independent oftemperature, then the data are consistentwith the accepted theory.16 However atemperature dependence of 0.005-O.01 Hz/Kbetween 150 K, where slow-exchangeobservations have been made on the 19Fresonances, and 300 K, where the *Hspectrum is motionally averaged, wouldallow our alternative stochastic average inthe form

(5)

to fit the data as well or better.Reason to suspect such a temperature

dependence can be found from a closereading of the literature on the relatedsystem 1—fluoro—1,1,2,2—tetrabromo—ethane.22 A value of Jg = 2.4 ± 0.3 Hz indimethylether at 188 K can be measuredfrom published data,39 but has been reportedas 1.7 Hz at 180 K in the same solvent.23 InCFCI3 this coupling has been tabulated as1.15 Hz from 171 to 178 K,22 but our recentfits of the spectrum (at 171 K only) inreference 23 indicate Jg = 1.5 ± 0.3 Hz.Thus the reported absence of temperaturedependence to three significant figures isdubious and further experimental work isneeded.

6 Discussion

The present conformer model is by nomeans the most refined version of Eq. 3possible, but has the advantage of employingonly quantities that are experimentallymeasured on the same sample. Continuousclassical—mechanical forms of the presenttheory are readily written down and thedifferences from the accepted theory persist.Alternatively, the quantum partitionfunctions could be modeled. Suchmodifications would introduce unmeasuredparameters. Such extensions might bewarranted after improvements in theexperimental data base.

It is of interest to note that the sametheoretical prediction for (Jjrp) obtained byuse of Eqs. 3 and 5 also results if the sameconformer model is evaluated using Eq. 4

with Xn = Jn and + and — indicating,respectively, triplet and singlet zero-fieldeigenstates. Thus, no measurable fielddependence of (Jjjp) is predicted by thealternative theory, which is not obviousbecause of the entanglement of different spinHamiltonian parameters in the spin energies.

The theoretical justification for Eq. 1and related propositions is also weaker thanhas been appreciated. Such an averagefollows from dynamic models1"36 based onthe assumption that spin states insuperposition are transported betweendifferent spatial states in perfect concert.Any such model exists in a truncatedLiouville space that excludes superpositionsof states that differ in both their spin andspatial factors. Whether such a truncatedspace suffices to describe magnetic resonancelineshapes is an open question. What isclear is that such a space cannot describe theapproach to equilibrium of the total system,since this requires spin—dependent ratesbetween spatial manifolds. Thus a fulldynamic solution is needed in this completeLiouville space. One result of such a fullsolution will be the equilibrium average spinenergies of Eq. 3. Less clear is under whatdynamic assumptions either these energies orthose that follow from Eq. 1 will describe thefast—exchange spectrum. In any case theproblem is richer than has been appreciated.

The accepted idea of how to calculatea stochastically averaged spectrum isuniversal in the literature of magneticresonance, underlying the interpretation ofaverage chemical shifts, dipolar couplings,tunnel splittings, quadrupole couplings, andhyper-fine interactions. In most situationsthe number of unknowns is such that it isnot possible to verify the form of thestochastic average, but valuable informationcould be obtained if the correct form wereknown. If the traditional ideas are generallyincorrect, many thousands of experimentsneed to be reinterpreted with forms of thepresent theory to in fact obtain thequantitative information on molecularstructure that they were designed to yield.Ultimately, the choice of theory will bedecided by a preponderance of data. Thepresent work indicates clearly that the issuemust be reopened and is a first step in thereexamination of the experimental basis ofEq. 1. Accurate experimental measurements

Vol. 14, No. 1-4 219

of systems such as those discussed here willbe needed in order to discriminate betweenthe stochastic theories, as opposed to fittingfree parameters according to one or the othertheory.

Precisely the same issue of how tocalculate a stochastic average also arises inthe (ab initio) theoretical calculation of ameasurable spin Hamiltonian fromexpectation values of the underlyingmolecular eigenstates, which are almostnever sufficiently long—lived to measureindividually by magnetic resonance.Application of the correct statisticalprescription will often be needed to in facttest experimentally whether the quantum-mechanical part of the calculation isadequate.

This work was supported by theNational Science Foundation(CHE-9005964). DHJ holds a Department ofEducation Graduate Fellowship. DPW is aCamille and Henry DreyfusTeacher—Scholar.

*N. Bloembergen, E.M. Purcell, and R.V.Pound, Phys. Rev. 73, 679 (1948).2E.R. Andrew and R. Bersohn, J. Chem.Phys. 18, 159 (1950).3E.L. Hahn and D.E. Maxwell, Phys. Rev.88, 1070 (1952).<H.S. Gutowsky, D.W. McCall, and C.P.Slichter, J. Chem. Phys. 21, 279 (1953).5D.M. Graham and J.S. Waugh, J. Chem.Phys. 27, 968 (1957).6J.N. Shoolery and B.L. Crawford, Jr., J.Mol. Spectrosc. 1, 270 (1957).7J.A. Pople, W.G. Schneider, and H.J.Bernstein, Can. J. Chem. 35,1060 (1957).8J.A. Pople, Mol. Phys. 1, 3 (1958).9H.M. McConnell, J. Chem. Phys. 28, 430(1958).l0J.I. Kaplan, J. Chem. Phys. 28, 278(1958); 29, 462 (1958).US. Alexander, J. Chem. Phys. 37, 967, 974(1962)12C.S. Johnson, J. Chem. Phys. 41, 3277(1964).l3G.S. Johnson, Adv. Magn. Reson. 1, 33(1965).14J.C. Schug, P.E. McMahon, and H.S.Gutowsky, J. Chem. Phys. 33, 843 (1960).15R.J Abraham and H.J. Bernstein, Can. J.Chem. 39, 39 (1961).

16T.D. Alger, H.S. Gutowsky, and R.L.Void, J. Chem. Phys. 47, 2818 (1967).17H.S. Gutowsky, G.G. Belford, P.E.McMahon, J. Chem. Phys. 36 (1962).18R.A. Newmark and C.H. Sederholm, J.Chem. Phys, 43, 602 (1965).19R.A. Newmark and C.H. Sederholm, J.Chem. Phys. 39, 3131 (1963).20S. Ng, J. Tang, and C.H. Sederholm, J.Chem. Phys. 42, 79 (1965).21R.R. Dean and J. Lee, Trans. Faraday Soc.65, 1 (1969).22W.S. Brey, Jr., and K.C. Ramey, J. Chem.Phys. 39, 844 (1963).23G. Govil and H.J. Bernstein, J. Chem.Phys. 47, 2818 (1967).24G. Govil and H.J. Bernstein, J. Chem.Phys. 43, 285 (1968).25R.M. Lynden-Bell, Prog. NMR Spec. 2,163 (1967).26G. Binsch, Top. Stereochem. 3, 97 (1968).27G. Binsch, J. Amer. Chem. Soc. 91, 1304(1969).28F. Heatley and G. Allen, Mol. Phys. 16, 77(1969).29W.-C. Lin, J. Chem. Phys. 52, 2805(1970).3°A. Vega and D. Fiat, J. Magn. Reson. 19,21 (1975).31L.M. Jackman and F.A. Cotton, eds.,Dynamic Nuclear Magnetic ResonanceSpectroscopy (Academic Press, New York,1975).32S. Szymanski, M. Witanowski, and A.Gryff-Keller, Ann. Rept. NMR Spec. 8, 227(1978).33G.-Binsch and H. Kessler, Angew. Chem.,Int. Ed. Engl. 19, 411 (1980).34K.G,R. Pachler and P.L. Wessels, J.Molec. Struc. 68, 145 (1980).ssj. Kaplan and G. Fraenkel, NMR ofChemically Exchanging Systems (AcademicPress, New York, 1981).3«J. Jeener, Adv. Magn. Reson. 10, 1 (1982)."Previously reported38 values for Jg of « 3.7Hz were taken from fluorine spectra showingan apparent gauche doublet. The splittingwas later discovered to be a chlorine isotopicshift on the fluorine resonance.3«D.H. Jones, N.D. Kurur, and D.P.Weitekamp, 33r d Experimental NMRConference, Pacific Grove, CA, April 2,1992.39G. Govil and H.J. Bernstein, Mol. Phys.14,197(1968).