Molecular properties determined from the relaxation of long-lived spin states

The populations of long-lived spin states, in particular, populations of singlet states that are comprised of antisymmetric combinations of product states, I S − I S , are characterized by very long lifetimes because the dipole-dipole interaction between the two “active” spins I and S that are involved in such states is inoperative as a relaxation mechanism. The relaxation rate constants of long-lived singlet states are therefore determined by the chemical shift anisotropy CSA of the active spins and by dipole-dipole interactions with passive spins. For a pair of coupled spins, the singlet-state relaxation rate constants strongly depend on the magnitudes and orientations of the CSA tensors. The relaxation properties of long-lived states therefore reveal new information about molecular symmetry and structure and about spectral density functions that characterize the dynamic behavior. © 2007 American Institute of Physics. DOI: 10.1063/1.2778429


INTRODUCTION
Recently, it has been shown that the population of a long-lived state ͑LLS͒ can have a very small relaxation rate constant R LLS ͑and hence a very long lifetime T LLS =1/R LLS ͒ when relaxation due to dipolar interactions is quenched. 1,2his occurs, for example, when two spins I and S are temporarily made to behave as if they were magnetically equivalent, which can be achieved either by taking the sample out of the static field or by suppressing the effects of the chemical shifts with suitable decoupling methods. 1,3,4If the molecule contains only two magnetic nuclei with nonvanishing spins I and S, the long-lived state ͑LLS͒ corresponds to a so-called singlet state ͑SS͒, i.e., a linear combination of product states that is antisymmetric with respect to the permutation of the two spins: ͉␣ I ␤ S ͘ − ͉␤ I ␣ S ͘.In this case, we may speak of a singlet-state relaxation rate constant R S = R LLS or, alternatively, of a singlet-state relaxation time constant T S =1/R S = T LLS =1/R LLS .In a system with three or more spins, 5 say I, S, and R, it is, however, preferable to speak of a longlived state Q LLS , which is defined as the eigenstate of the Liouvillian with the smallest eigenvalue ͑other than the eigenstate Q 0 that corresponds to the sum of all populations, which must remain conserved at all times and must therefore be associated with an eigenvalue 0 =0͒. 6In this paper, we use the general notations R LLS and T LLS .Two ways have been proposed to preserve the populations of long-lived ͑sin-glet͒ states: ͑i͒ the zero-field method, which requires taking the sample out of the magnetic field 1 and ͑ii͒ the high-field method, which requires the application of a suitable radio-frequency irradiation to mask the difference in chemical shifts between the two spins, so that they become in effect magnetically equivalent. 2,7When the zero-field method is used, the remaining dominant relaxation mechanisms of the long-lived spin states are the intra-or intermolecular dipoledipole ͑DD͒ interactions between either of the two active spins I and S and passive spins R, which may be either nuclear or electronic ͑e.g., Curie spins with rapid electronic relaxation 8 ͒.If the other magnetic nuclei in the molecule ͑notably the remaining protons͒ are substituted by isotopes with low gyromagnetic ratios ͑e.g., by deuterons͒ and if the samples are carefully degassed, the lifetimes of the longlived spin states can be greatly extended.In the high-field method, an additional source of relaxation appears, due to fluctuations of the local fields at the sites of the two active spins that arise through the combination of rotational diffusion and chemical shift anisotropy ͑CSA͒.The lifetimes of long-lived states will therefore increase with decreasing static field strength.Long lifetimes T LLS in effect provide a "memory" to the spin system that is much longer than its spin-lattice relaxation time constant T 1 .This property allows one to study very slow dynamic processes, such as translational diffusion 9,10 and conformational rearrangements, 3 that occur on time scales up to tens of seconds.
Although CSA relaxation is obviously detrimental for the lifetimes T LLS of the long-lived ͑singlet͒ states ͑and hence limits the time scales of the slow processes that can be studied͒, this effect can be put to good use.Indeed, the measurement of the relaxation rate constants R LLS can be used for the characterization of CSA tensors.Except at very high static fields, the CSA relaxation of protons is usually overshadowed by large dipolar interactions, so that the param-eters of proton CSA tensors are often difficult to determine with reasonable accuracy, and it is often necessary to exploit the field dependence of CSA autorelaxation rate constants and/or CSA/DD cross-correlation effects.It will be shown in this paper that the CSA relaxation of long-lived ͑singlet͒ states critically depends on molecular symmetry properties.In two-spin systems, both the principal components of the two CSA tensors and their relative orientation affect the relaxation rate constants of long-lived states.The CSA contribution to relaxation vanishes if the two tensors have identical principal components and parallel principal axes.This relaxation mechanism is similar to that due to random field fluctuations in the absence of a static magnetic field. 11The relaxation rate constant R LLS of a long-lived state involving two active spins I and S is affected by DD interactions I-R i and S-R i with passive spins R i ͑with i =1,2, ... ,N.͒The resulting DD contributions R LLS DD to the relaxation rate constants of long-lived states depend on the molecular symmetry.If N = 1, one can extract valuable information on the ͑average͒ position of the third spin R relative to the first two.If spin R is mobile, averaging effects may allow one to gain novel insight into internal dynamics.This paper focuses on the interpretation of the experimentally determined relaxation rate constants R LLS of long-lived states in terms of CSA ten-sors and DD interactions, which in turn may provide insight into molecular symmetry, structure, and dynamics.

THEORY AND EXPERIMENTS
First, we shall address the effects of CSA interactions on the relaxation of a long-lived ͑singlet͒ state in a system with only two spins I and S. The principal axes and orientations of the CSA tensors characteristic for a planar molecule are defined in Fig. 1.All the calculations assume isotropic rotational diffusion.The tensor operators for the dipolar I-S interaction and for the CSA interactions of spins I and S, along with the relevant frequencies, are given in Table I.
The autorelaxation time constants of the normalized operators Q that span the basis of the density operator may comprise contributions from DD and CSA mechanisms: The contribution from the dipolar interactions is: 12 In the extreme narrowing regime ͑ c Ӷ 1/ 0 ͒, the spectral density function is almost white, 13 i.e., J͑0͒ϷJ͑ 0 ͒ Ϸ J͑2 0 ͒, while for macromolecules ͑ c ӷ 1/ 0 ͒, one has J͑0͒ ӷ J͑ 0 ͒ϷJ͑2 0 ͒Ϸ0.Numerical simulations were run for two typical correlation times.To mimic the behavior of small molecules, we chose c = 0.05 ns, which yields the ratio J͑0͒ : J͑ The other symbols have their usual meaning.For two protons separated by 2 Å, D is 94.3 kHz.The tensor operators T j ͑K͒±q for the CSA interactions of spins K = I or S are defined using The long-lived state eigenoperator Q LLS , which decays with a relaxation rate constant R LLS , corresponds to the difference between the population of the singlet state and the mean of the populations of the three triplet states. 6,10With Eq. ͑3͒, one obtains the following expression for the rotationinvariant normalized operator of the long-lived state: Since Q LLS commutes with all tensor operators that represent the I-S DD interaction, it is not affected by dipolar relaxation between the two active spins, i.e., ⌫ Q LLS DD = 0 if there are no further spins.On the other hand, the contribution of the CSA interactions of spins I and S to the relaxation of Q LLS is where P 2 is the Legendre polynomial and ͑ ii I , jj S ͒ is the angle between the principal axes ii I and jj S of the CSA tensors of the two spins, and the tensor operators T j ͑K͒ with K = I or S are given in Table I = zz S ͒, using the notation ⌬ I,S = ⌬ xx I,S -⌬ zz I,S , the CSA contributions to the autorelaxation rate constant of long-lived ͑singlet͒ state involving spins I and S are given by where = ͑ xx I , xx S ͒ is the angle between the principal axes xx I and xx S .
If the fluctuations of the DD and CSA interactions are correlated, there can be a partial conversion ͑cross relax-ation͒ of IS singlet-state populations into a difference of Zeeman polarizations ͑I z − S z ͒.This type of cross relaxation is effectively suppressed by the coherent terms in the Hamiltonian. 13

Similar CSA tensors with nonparallel principal axes
In some cases, it can be inferred from molecular symmetry that the principal values of the CSA tensors of spins I and S must be similar.In this case, it may be possible to determine the relative orientations of the two tensors from the relaxation rate constants of long-lived states.If the two CSA tensors are similar and axially symmetric, we can further simplify the notation by using ⌬ = ⌬ I = ⌬ S , and Eq.͑6͒ becomes where = ͑ I , S ͒ is again the angle between the principal axes xx I and xx S .The autorelaxation rate constant of Eq. ͑7͒ depends only on the chemical shift anisotropy ⌬ and on the relative orientation of the two tensors ͓Fig.2͑a͔͒.In the extreme narrowing limit, we have This expression can be compared with the CSA contributions to the longitudinal relaxation rate constants R I =1/T I of the individual spins, Therefore, the ratio between the CSA contributions R LLS CSA to the autorelaxation rate constant of a long-lived state and the CSA contributions R 1 CSA to the longitudinal relaxation rate constants of the individual spins yields thus providing direct information about the relative orientation of the two CSA tensors, i.e., about the angle subtended between their principal axes ͓Fig.2͑b͔͒.Note that one must exclude the ͑field-independent͒ dipolar contributions R LLS DD , R 1 DD ͑I z ͒, and R 1 DD ͑S z ͒ to the relaxation rate constants.Once the angle and the spectral density J͑0͒ are known, the anisotropy ⌬ can in principle be determined from Eq. ͑8͒.

Different CSA tensors with parallel principal components
If the principal components of the CSA tensors are parallel and the CSA tensors have cylindrical symmetry ͑e.g., if xx I yy I = zz I and xx S yy S = zz S ͒, the rate constant given in Eq. ͑6͒ only depends on the difference Thus, for two protons with ⌬ I − ⌬ S = 5 ppm in the extreme narrowing limit ͑ c = 0.05 ns͒, the relaxation rate constants calculated from Eq. ͑11͒ are R LLS CSA = 0.0012 s −1 at 400 MHz and 0.0045 s −1 at 800 MHz.These values increase to R LLS CSA = 0.0472 and 0.1876 s −1 for c = 5 ns.The rate constants in Eq. ͑11͒ vanish regardless of the static field if the two CSA tensors, besides having parallel orientations, are also equal in magnitude.The calculations using Eq.͑11͒ were confirmed numerically using the GAMMA program, 14 as described in the Appendix below.
In a two-spin system where the CSA tensors are known to have cylindrical symmetry ͑ xx yy = zz ͒ and a known relative orientation , the variation of the relaxation rate constant R LLS CSA of the long-lived ͑singlet͒ state as a function of the B 0 field can provide a measure of the difference between the anisotropies of the two spins, ⌬ I − ⌬ S ͑Fig.3͒.The profiles of R LLS CSA in the slow motion limit are very similar to those for extreme narrowing in Fig. 3.

Dipolar interactions with an external passive spin R
The symmetry properties of a long-lived state also determine the effects of dipolar interactions I-R and S-R between the two active spins I and S that are involved in the longlived state and an external passive spin R. If the position of spin R deviates from the bisecting plane, the dipolar contributions to R LLS increase parabolically ͓trajectory A in Fig. 4͔, provided the distance between spin R and the center of the line connecting spins I and S is maintained constant.For this circular trajectory, the sum of the dipolar interactions I-R and S-R is not constant since it depends on the sum of the inverse distances to the sixth power ͑1/r IR 6 +1/r SR 6 ͒.On the other hand, if we move spin R along a trajectory so as to keep the sum of these dipolar interactions constant ͓see trajectory B in Fig. 4͔, the rate constant R LLS DD increases with the angle ⍀.This behavior is similar to that observed for two CSA interactions as a function of the relative orientation in Eq. ͑6͒.It can be seen that in both cases, the symmetry has similar effects on the relaxation of the two-spin order 2I z S z and of the long-lived ͑singlet͒ states.The latter have, in the fast tumbling limit, lower relaxation rate constants, i.e., R zz Ͼ R LLS ͑Fig.4͒.The dependence of the relaxation properties of a long-lived state on the geometry of a three-spin system should allow one to extract information regarding the position of spin R with respect to spins I and S. The long intrinsic lifetimes of long-lived states may afford an experimental determination of structural information in cases when the third spin is relatively remote from the first two ͑i.e., when the interactions to be determined are weak͒.

Experimental evidence
In order to obtain information about molecular symmetry and structure using Eqs.͑8͒ and ͑9͒, the CSA contributions R 1 CSA ͑I z ͒ and R 1 CSA ͑S z ͒ to the longitudinal relaxation rate constants of the two active spins need to be known.In practice, this may be difficult for protons since dipolar interactions tend to dominate CSA effects, often by more than an order of magnitude, even in high magnetic fields. 13However, the longitudinal relaxation rate constants R 1 CSA ͑I z ͒ and R 1 CSA ͑S z ͒ may be measured in a similar molecule where the spin under investigation ͑say, spin I͒ has a similar environment, while the dipolar interaction with the other proton spin ͑S͒ is removed by substitution of the latter, for instance, by a chlorine atom.
The longitudinal and long-lived ͑singlet͒ state relaxation rate constants have been measured in two related molecules ͑Fig.5͒: pentachlorobenzene ͑which contains an isolated I =1/2 proton that is predominantly relaxed through CSA͒ and trichlorophenol ͑with two coupled protons that experience both dipolar and CSA interactions͒.We shall assume for simplicity that the CSA tensors of the protons are the same in the two molecules.The measurements were carried out at static fields of 9.4 and 14.1 T ͑400 and 600 MHz for protons͒, yielding the rate constants given in Table II.As has been shown in the previous sections for longlived states created on pairs of spins I and S, while the dipolar interaction between the I and S nuclei is effectively silenced, these states can still be relaxed via dipolar interactions with external spins.In the case of the trichlorophenol molecule in water solution, some contributions to the relaxation of the intramolecular long-lived state are expected to arise from interactions with spins in the solvent and from intermolecular dipole-dipole interactions.The characteristic correlation times of these interactions are extremely short, and their spectral density functions are therefore expected not to show any dependence on the magnetic field.
The only field-dependent contribution to relaxation is therefore related to the magnitudes of the CSA tensor operators of the two spins ͑Table I͒, which increase linearly with the field.The same holds for the contributions to the longitudinal relaxation rate constant of the hydrogen nucleus in pentachlorobenzene.Consequently, in order to eliminate possible contributions from interactions other than the CSA's, we have considered the ratio of differences between rate constants measured at different fields, The experimental ratio was found to be F = 3.7± 1. TABLE II.Experimental relaxation rate constants of longitudinal magnetization R 1 ͑I z ͒ in pentachlorobenzene ͑containing a single isolated proton͒ and experimental relaxation rate constants of long-lived ͑singlet͒ states R LLS in trichlorophenol ͑with two coupled protons͒ at 400 and 600 MHz.To measure R LLS at two fields, the rf field amplitude used for continuous-wave irradiation in the mixing time was 1 / ͑2͒ = 1.2 kHz at 400 MHz and 1.7 kHz at 600 MHz, so that the ratio ⌬⍀ / 1 between the isotropic chemical shift difference ⌬⍀ = ⍀ I − ⍀ S and the amplitude 1 was the same at both fields ͑⌬⍀ / 1 = 0.0814͒.are relatively large, due to the propagation of errors on the ratio of differences between experimental values.Using the same procedure, the relaxation rate constants R LLS of the long-lived states in trichlorobenzaldehyde ͑threespin system͒ and trichlorophenol ͑two-spin system͒ were also determined ͑Fig.5͒ in order to investigate the effects of DD interactions between the active spins I and S and a third passive spin R on the relaxation rate constant of a long-lived state.

CONCLUSIONS
It has been shown that CSA contributions to relaxation properties of long-lived ͑singlet͒ states depend on the symmetry of the molecules and on the relative orientations of the principal components of the two CSA tensors.Dipolar interactions between the active spins I and S and a third spin R critically depend on molecular symmetry.Understanding these relationships can be valuable for dynamic and structural studies by NMR.

ACKNOWLEDGMENTS
The authors are indebted to Malcolm Levitt for providing the manuscript of Ref. 6 prior to publication.This work was supported by the Swiss National Science Foundation ͑FNRS͒, the Ecole Polytechnique Fédérale de Lausanne ͑EPFL͒, the Swiss Commission for Technology and Innovation ͑CTI͒, the French CNRS, and the Laurence Berkeley Laboratory ͑USA͒.The authors thank Luminita Duma, Karthik Gopalakrishnan, and Fabien Ferrage for many insightful discussions, Martial Rey for technical assistance, and Euro Solari for his help with sample preparation.

APPENDIX: SIMULATIONS
To calculate contributions of various relaxation mechanisms to the Liouvillian, we considered two protons separated by a distance ͉r IS ͉ = 2 Å with two axially symmetric CSA tensors ͑ xx The simulations were performed considering a difference between the chemical shifts of 0.19 ppm, a scalar coupling J IS = −12 Hz, and a radio-frequency amplitude 1 / ͑2͒ = 5 kHz, applied half-way between the chemical shifts.The decay of the populations of the long-lived ͑singlet͒ states was monitored for 1000 different relaxation intervals.The decays were then fitted to exponential functions using three fitting parameters ͑initial amplitude, decay rate constant, and dc off-set͒.When the effect of a third spin R was calculated, this spin was considered at an offset of 4 Hz without any scalar couplings, i.e., J IR = J SR =0.

FIG. 1 .
FIG. 1. Principal values and relative orientations of CSA tensors in a system of two spins I and S such as two aromatic protons.The zz components ͑not shown͒ of the two CSA tensors are assumed to be both perpendicular to the plane of the figure and hence to the internuclear vector r IS .The xx I and xx S

FIG. 2 . 1 CSA ͑I z ͒ = R 1 CSA
FIG. 2. ͑a͒ Theoretical CSA contributions R LLS CSA to the relaxation rate constants of long-lived ͑singlet͒ states in the system of Fig. 1 at 400 MHz ͑lower curve͒ and 800 MHz ͑upper curve͒, considering two CSA tensors with cylindrical symmetry ͑ xx yy = zz ͒ and with the same anisotropy ⌬ xx I = ⌬ xx S = 10 ppm for both spins I and S, as a function of the angle = ␣ − ␤ between the orientations of the xx I and xx S components.The rate constants were calculated using Eq.͑5͒, with a correlation time c = 0.05 ns.͑b͒ Theoretical ratio of R LLS CSA to the CSA contribution to the longitudinal relaxation rate constants R 1 CSA ͑I z ͒ = R 1 CSA ͑S z ͒ of the individual spins as a function of the angle .This ratio does not depend on the static field B 0 .

FIG. 3 .
FIG. 3. Theoretical CSA contributions R LLS CSA to the relaxation rate constants of long-lived ͑singlet͒ states in the spin system of Fig. 1 at 400 MHz ͑lower curve͒ and 800 MHz ͑upper curve͒, considering two CSA tensors with cylindrical symmetry ͑ xx yy = zz ͒ and parallel principal components ͑ =0͒ as a function of the anisotropy ⌬ S = ⌬ xx S , which is varied in the range of 0 Ͻ⌬ S Ͻ 2⌬ I , with ⌬ I = ⌬ xx I = 15 ppm.The rate constants were calculated using Eq.͑5͒ for a correlation time c of 0.05 ns.
FIG.4.Calculated DD contributions to the relaxation rate constants of long-lived states ͑R LLS DD , filled diamonds͒ and of longitudinal two-spin order 2I z S z ͑R ZZ DD , stars͒, due to dipolar interactions I-R and S-R with a third passive spin R, at 400 MHz, for a correlation time c = 0.05 ns.In trajectory A ͑left panel͒, the distance of spin R from the midpoint between spins I and S was kept constant ͑3 Å͒, while the angle ⍀ was varied between −90°and 90°.In trajectory B ͑right panel͒, the third spin R was displaced on a trajectory so as to keep ͑1/r IR 6 +1/r SR 6 ͒ = 0.016 Å −6 .The rate constants were calculated as explained in the Appendix.
S ͒ with different anisotropies xx I − zz I = 15 ppm and xx S − zz S = 10 ppm, with correlation times c = 0.05 and 5 ns.The theoretical relaxation rate constants of Eq. ͑11͒ were compared with GAMMA simulations incorporating coherent evolution under the full spin Hamiltonian ͑expressed in units of ប as usual͒, in addition to dipolar and CSA relaxation mechanisms, H = I I z + S S z + 2J IS I • S + 1 ͑I x + S x ͒. ͑A1͒

TABLE I .
Tensor operators A 2 ±q that are relevant for dipolar IS interactions, with a =1/2, b = ͱ 3/2, and D