Longitudinal relaxation in dipole-coupled homonuclear three-spin systems : Distinct correlations and odd spectral densities

A system of three dipole-coupled spins exhibits a surprisingly intricate relaxation behavior. Following Hubbard’s pioneering 1958 study, many authors have investigated different aspects of this problem. Nevertheless, on revisiting this classic relaxation problem, we obtain several new results, some of which are at variance with conventional wisdom. Most notably from a fundamental point of view, we find that the odd-valued spectral density function influences longitudinal relaxation. We also show that the effective longitudinal relaxation rate for a non-isochronous three-spin system can exhibit an unusual inverted dispersion step. To clarify these and other issues, we present a comprehensive theoretical treatment of longitudinal relaxation in a three-spin system of arbitrary geometry and with arbitrary rotational dynamics. By using the Liouville-space formulation of Bloch-Wangsness-Redfield theory and a basis of irreducible spherical tensor operators, we show that the number of relaxation components in the different cases can be deduced from symmetry arguments. For the isochronous case, we present the relaxation matrix in analytical form, whereas, for the non-isochronous case, we employ a computationally efficient approach based on the stochastic Liouville equation. C 2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4937377]


I. INTRODUCTION
A few years after Solomon's seminal analysis of dipolar cross relaxation in two-spin systems, 1 Hubbard investigated longitudinal relaxation in systems of three or four dipolecoupled spins. 2 In multi-spin systems, correlations between distinct dipole couplings (usually referred to as cross correlations) come into play, and Hubbard showed that their effect is to make the relaxation of the total longitudinal magnetization in the extreme-narrowing (EN) regime weakly bi-exponential and slightly slower. 2 Subsequent studies 3-10 confirmed Hubbard's results and extended them to non-EN conditions, where longitudinal relaxation is tri-exponential, and to anisotropic rotation models, where distinct correlations can have a more pronounced effect.
All of these studies considered a system of three geometrically equivalent spins, located at the vertices of an equilateral triangle, as in the widely occurring -CH 3 and -NH 3 groups.Relatively little attention has been devoted to less symmetric nuclear geometries, where the three dipole couplings differ in magnitude.The first attempt in this direction, pertaining to three isochronous spins at the vertices of an isosceles triangle, predicted that longitudinal relaxation in the EN regime involves seven exponentials. 113][14][15] In two rarely cited papers, 12, 13 Schneider examined the relaxation (also outside the EN regime) of three a) bertil.halle@bpc.lu.se isochronous spins at the vertices of equilateral, isosceles, and right-angled triangles.In contrast to all other authors, he retained the odd-valued spectral density function (OSDF) and showed that it influences longitudinal relaxation in the dispersive regime when at least two of the three dipole couplings differ in magnitude. 12,13To our knowledge, this is the only explicit demonstration in the literature that the OSDF affects relaxation.][18][19][20][21][22][23] For example, Abragam 16 dismisses the OSDF with the following assertion: "It can be shown that the imaginary term results in a very small shift in the energy levels of the system which can be included in a redefined unperturbed Hamiltonian and thus dropped from the relaxation equation."It is true that the imaginary part of the relaxation superoperator R, associated with the OSDF, can be expressed as a commutator superoperator, −i Im{R} = −i [H ′ , . ..], just like the Liouvillian associated with the time-independent spin Hamiltonian. 22,24(Karthik and Kumar, 22 following Jeener, 18 refer instead to the anti-Hermitian part of R, but for all cases considered here this is identical to the imaginary part of R.) It is also true that if there were no other terms in the equation of motion, the total evolution superoperator would be unitary (since H ′ is Hermitian), resulting in purely coherent evolution without dissipation.][27] Consistent with these results, Pfeifer argued in general terms that the OSDF can affect relaxation outside the EN regime, except when the relaxation function is strictly exponential so that a unique relaxation time can be defined. 28owever, this general conclusion must be amended in at least two respects.First, for the isochronous three-spin system, both cross relaxation between nonequivalent spins (with different dipole couplings) and correlations between distinct dipole couplings (equal or unequal in magnitude) can give rise to multi-exponential relaxation.The relaxation effect of the OSDF is only associated with the latter type of non-exponentiality.Second, the relaxation effect of the OSDF vanishes when the relaxation superoperator is invariant under permutation of all spins.Such invariance requires both geometric and dynamic equivalence, as for three spins at the vertices of an equilateral triangle undergoing sphericaltop rotational diffusion.For lower geometric or dynamic symmetry, longitudinal relaxation is affected by the OSDF.
To clarify the subtle and sometimes misunderstood manifestations of distinct correlations, cross relaxation (in the Solomon sense), and the OSDF in multi-spin systems, we present here a comprehensive analysis of longitudinal dipolar relaxation in a three-spin system.Using the Liouville space formulation of Bloch-Wangsness-Redfield (BWR) theory 15,29 with a basis of irreducible spherical tensor operators to exploit inherent symmetries, we obtain in analytical closed form the 10 × 10 relaxation supermatrix that governs the longitudinal relaxation of three isochronous spins with arbitrary nuclear geometry and without restrictions on the motional model (as long as the bath is isotropic).Depending on the number of unequal dipole couplings (one, two, or three), on the dynamic symmetry, and on the Larmor frequency, the relaxation of the total longitudinal magnetization involves between two and ten exponential components.In practice, it is often useful to characterize longitudinal relaxation by a single effective rate 4  R 1 , which we refer to as the integral relaxation rate.To illustrate the full range of relaxation behavior, we compute the relaxation dispersion profile  R 1 (ω 0 ) for several cases of interest, including two anisotropic rotation models.While the emphasis is on the isochronous three-spin system, we also examine the effect of different chemical shifts for the three nuclei.We find that chemical shifts can give rise to an unusual inverted step in the dispersion profile, resulting from symmetry-breaking nonsecular decoupling that principally affects distinct correlations.We establish the range of validity of the isochronous relaxation theory and we show that the effect of chemical shifts on longitudinal relaxation is always negligible outside the motional-narrowing (MN) regime.
This paper is organized as follows.In Sec.II, starting from the BWR master equation, we develop the theory of longitudinal relaxation in an isochronous three-spin system of arbitrary geometry and without restrictions on the motional model other than the overall isotropy of the molecular system.In Sec.III, we specialize to isotropic motions and investigate, for each of the three geometric cases (with one, two, or three distinct dipole couplings), the eigenmode rates and weights and the integral relaxation rate over the full frequency range.We also examine the time dependence of the longitudinal magnetization.In Sec.IV, we explore two anisotropic motional models: axial internal rotation superimposed on spherical-top rotation and symmetric-top rotation with arbitrary orientation of the rotational diffusion tensor.In Sec.V, we use the stochastic Liouville equation to examine the effect of chemical shifts on the longitudinal relaxation behavior, finding an unusual inverted relaxation dispersion at the nonsecular decoupling frequency.Finally, in Sec.VI, we summarize the principal results of this work.

A. Liouville space formulation of BWR theory
We consider three isochronous spin-1/2 nuclei, labeled I, S, and P, subject to Zeeman and mutual intramolecular dipole couplings.If we neglect scalar couplings, the static Hamiltonian is where ω 0 is the common Larmor frequency.In Sec.V, we consider the more general case where the three nuclei have different Larmor frequencies.
The fluctuating dipolar Hamiltonian, with vanishing ensemble average, is with (X denotes either of the three spin pairs IS, IP, or SP) Here, the T 2 M (X) are orthonormal three-spin irreducible spherical tensor operators (Sec.II C), the D 2 M 0 (Ω X ) are rank-2 Wigner functions, 30 Ω X (t) ≡ (θ X (t), ϕ X (t)) are the fluctuating spherical polar angles that specify the orientation of the internuclear vector r X with respect to the lab-fixed frame, and the dipole coupling frequency is defined as ω D, X ≡ (3/2)(µ 0 /4π) γ 2 /r 3 X .Since the geometric arrangement of the three nuclei is arbitrary, the three dipole couplings may differ in magnitude.
We start from the semi-classical BWR master equation 16 where  σ(t) = exp(i L Z t) σ(t) and  L D (t) = exp(i L Z t) L D (t) exp(−i L Z t) are, respectively, the density operator (relative to its equilibrium value) and the dipolar Liouvillian in the interaction representation, and the Liouvillians are the usual derivation superoperators L Z ≡ [H Z , . ..] and L D (t) ≡ [H D (t), . ..].The angular brackets denote an equilibrium ensemble average over the molecular degrees of freedom (the so-called "lattice" or "bath").
The set of three-spin irreducible spherical tensor operators T K Q , to which the T 2 M (X) in Eq. ( 3) belong, are eigenoperators of the Zeeman Liouvillian (Sec.II C), The BWR equation ( 4) can therefore be expressed, in the Schrödinger representation, as The relaxation superoperator is given by and the complex-valued spectral density functions Clearly, the real and imaginary parts of J XY (ω) are even and odd functions of frequency, respectively: j XY (−ω) = j XY (ω) and k XY (−ω) = −k XY (ω).We refer to these parts as the even spectral density function (ESDF) and the OSDF.The complex-valued spectral density functions have the general properties, The second relation follows from Eq. ( 12) and the equality G Y X (τ) = G XY (τ), which in turn follows from Eq. ( 9) and the assumption that the stationary process obeys the principle of detailed balance, which ensures that the propagator is invariant under time reversal.Throughout Sec.II, we place no further restrictions on the time correlation functions G XY (τ).Specific motional models will be introduced in Secs.III and IV, where we also present numerical results.Although we have not yet specified the motional model, it is convenient to loosely define a correlation time τ c as the time scale for the essential decay of the time correlation functions G XY (τ).This allows us to distinguish three frequency regimes: the EN regime (ω 0 τ c ≪ 1), the dispersive regime (ω 0 τ c ∼ 1), and the adiabatic regime (ω 0 τ c ≫ 1).There are no fixed and universal demarcation lines between these regimes.For example, the value of ω 0 τ c below which relaxation can be considered to be independent of ω 0 depends on the chosen (geometric and dynamic) model and on the desired accuracy.For some purposes, it is more appropriate to refer to the zero-field limit rather than to the EN regime.
The foregoing results are well-known; their derivation is sketched here merely to establish our notation and to exhibit the underlying assumptions.

B. Spin inversion and conjugation symmetries of the relaxation superoperator
The density operator must remain Hermitian at all times 32 so the two operators on the right-hand side of Eq. ( 6) must also be Hermitian. 29In other words, physical consistency demands that (L Z σ) † = −L Z σ and (R σ) † = R σ.The first of these identities is readily verified by noting that since both H Z and σ are Hermitian.To prove the second identity, we use Eq.(13a) and note that (C XY M σ) † = C XY −M σ by virtue of Eq. (11).To complete the proof, we substitute these results into the expression for (R σ) † obtained from Eq. (10) and then interchange M and −M in the symmetric sum.
The general physical requirement that the operator Rσ is Hermitian does not imply that the relaxation superoperator R itself is Hermitian.Since (S T ) † = T † S † for arbitrary superoperators S and T , 18 Eq.( 11) yields Inserting this result into the adjoint of Eq. ( 10), interchanging the dummy variables X and Y , and using Eq.(13b), we find In view of Eq. ( 12), this result shows that the relaxation superoperator is the sum of Hermitian and anti-Hermitian parts.The Hermitian part is the real part of R, associated with the ESDF, j XY (Mω 0 ).The anti-Hermitian part is i times the imaginary part of R, associated with the OSDF, k XY (Mω 0 ).Therefore, R is Hermitian only in the EN regime, where k XY (ω) ≪ j XY (ω), and in the adiabatic regime, where j XY (ω), k XY (ω) ≪ j XY (0).
We now consider the effect on R of spin inversion, spin conjugation, and spin inversion conjugation. 15,29The superoperators, collectively denoted by X, associated with each of these symmetry operations are unitary and selfinverse, 29 and, for our three-spin system, they can be factorized as The spin inversion superoperator acting on I-spin operators is defined as 15,29 with I y = [I y , . ..], so it transforms the spherical spin operators according to Table I.The spin conjugation superoperator is usually defined in terms of its action on the shift operators, 15,29 and the spin inversion conjugation operation is simply the combination of the first two operations, W = VY = YV. 15,29The transformation rules in the last two columns in the upper part of Table I are readily obtained from these definitions.
To establish the behavior of the relaxation superoperator R under these symmetry operations, we first use the explicit form of the operators T 2 M (X) and the transformation rules for single-spin operators in the upper part of Table I to obtain the rules on the fourth row of the table.The rules for the superoperator C XY M , shown in the next row, are then obtained with the aid of Eq. (11).Finally, we use Eqs.( 10), (13a), and ( 14) to obtain the transformation rules for R in the last row of Table I.We thus find that the relaxation superoperator is invariant under spin inversion conjugation, whereas it is invariant under spin inversion and spin conjugation separately only in the EN and adiabatic regimes, where the OSDF can be neglected.

C. Spin operator basis and spin modes
As a basis for three-spin Liouville space, we use the 64 irreducible spherical tensor operators (ISTOs) T K Q (k I k S { K } k P ), constructed by two consecutive couplings of the set of four orthonormal single-spin ISTOs for each spin, e.g., to obtain 30,32 where K is the rank of the intermediate tensor operator obtained by first coupling spins I and S. Here, and in the following, the rank and projection indices are written in upper case for three-spin ISTOs and in lower case for single-spin ISTOs.The ISTOs T 2 M (X) appearing in dipolar Hamiltonian (3) belong to this basis set, e.g., T 2 0 (I S) = T 2 0 (11{2}0) = (3I z S z − I • S)/ √ 3, and Eq. ( 5) follows directly from Eq. ( 19) and the fundamental commutation relation 30 The main virtue of the ISTO basis is that the 2K + 1 operators T K Q of a given rank K transform according to the irreducible representation D K of the three-dimensional rotation group, 30,33 leading to selection rules for the relaxation supermatrix (Sec.II D 2) that greatly simplify the relaxation problem.In addition, some of the ISTOs have well-defined parity under spin inversion, spin conjugation, and spin inversion conjugation, leading to additional selection rules (Sec.II D 3).Applying the transformation rules in the upper part of Table I to Eqs. ( 18) and ( 19) and using the symmetry properties of the 3 j symbol, 30 we find where N s = k I + k S + k P is the number of single-spin operators (not counting identity operators) involved in the basis operator.Equation (20b) is the generalization to three-spin ISTOs of the familiar conjugation relation, T k † q = (−1) q T k −q , for single-spin ISTOs. 30,33As seen from Eq. (20c), all 64 basis operators have definite parity with respect to spin inversion conjugation, being odd (antisymmetric) if they involve one or three spins and even (symmetric) if they involve two spins.In the following, we refer to basis operators that are odd or even under spin inversion conjugation as OSIC or ESIC operators, respectively.In contrast, only the 20 zero-quantum (Q = 0) ISTOs have definite parity under spin inversion and spin conjugation.Equation (20a) shows that these basis operators are either even (for even rank K) or odd (for odd K) under spin inversion, and Eq.(20b) shows that they are either Hermitian (for even K + N s ) or anti-Hermitian (odd K + N s ).
As we shall see in Sec.II D, the symmetry properties of the ISTO basis operators under rotation and spin inversion conjugation ensure that longitudinal relaxation in the threespin system can be described, for arbitrary nuclear geometry, in the subspace of the ten zero-quantum OSIC operators listed in Table II.(The identity operator also belongs to this category, but it can be omitted on physical grounds.)Based on their parity with respect to Y and V, these ten operators fall in two groups: seven Hermitian operators with odd rank K and three anti-Hermitian operators with even rank.
For convenience, we index the basis by a single number n, denoting the basis operators by B n .The basis operators obtained from Eq. (19) are orthonormal in the sense, Because the basis is complete, we can expand the density operator in spin modes (sometimes called state multipoles), 32 defined as σ n ≡ (B n | σ).Using Eqs. ( 5), (6), and (21), we find that the zero-quantum spin modes obey the BWR master equation since the coherent term The relaxation supermatrix elements are obtained from Eq. (10) as with the coefficients D. Symmetry properties of the relaxation supermatrix in the ISTO basis

Transposition and complex conjugation
According to Eq. ( 11), the coefficients in Eq. ( 23) are given by where the last form results from the cyclic permutation invariance of the trace.As shown in Appendix A of the supplementary material, 34 all matrix elements C XY M, n p are realvalued in the ISTO basis.It then follows from Eqs. (12)  and ( 23) that the real part of the relaxation supermatrix is associated with the ESDF and the imaginary part with the OSDF.While this correspondence was established for the relaxation superoperator in Sec.II B, it is not true for the relaxation supermatrix in any basis.Indeed, in a basis of Hermitian operators, the relaxation supermatrix is real even though the spectral density is complex. 29BLE II.The zero-quantum OSIC basis operators Noting that C XY M, n p is real, we can use Eq. ( 24) to show that By interchanging the dummy variables X and Y in Eq. ( 23) and using Eqs.(13b) and (25), we obtain showing that (both the real and imaginary parts of) the relaxation supermatrix is symmetric in the ISTO basis.
Another symmetry property of the relaxation supermatrix in the ISTO basis can be demonstrated by first using Eqs.( 10)- (13) to show that (R A) † = R A † for an arbitrary operator A and then using this result and the cyclic permutation invariance of the trace in the definition of the supermatrix element as follows: where the underlined subscript n denotes the adjoint basis operator B † n .This relation is particularly useful in the zero-quantum subspace, for which Eq. (20b) shows that the basis operators are either Hermitian or anti-Hermitian.It then follows from Eq. ( 27) that relaxation supermatrix elements connecting two Hermitian or two anti-Hermitian basis operators are real (R * n p = R n p ), whereas matrix elements connecting basis operators of different conjugation parity are imaginary (R * n p = −R n p ).Since J XY (Mω 0 ) is the only complex-valued quantity in Eq. (23), it is clear that the OSDF has the effect of coupling odd-rank (Hermitian) and even-rank (anti-Hermitian) spin modes.

Rotational symmetry
Because the superoperator R is ensemble averaged over the isotropic molecular system, it must exhibit the cylindrical symmetry of the spin system in the external magnetic field.According to the Wigner-Eckart theorem, 15,29,30,32,33 the relaxation supermatrix in the ISTO basis must therefore be block-diagonal in the projection index Q, The evolution of the longitudinal magnetization modes σ 1 (t), σ 2 (t), and σ 3 (t) (Table II) can, therefore, be fully described within the subspace of the 19 zero-quantum operators (omitting the identity operator).In the EN regime, where the Zeeman Hamiltonian may be neglected, the relaxation superoperator R becomes fully rotationally invariant.The Wigner-Eckart theorem then implies that the relaxation supermatrix in the ISTO basis is block-diagonal not only in the projection index Q but also in the rank index K, 15,29 In the EN regime, rotational symmetry thus reduces the invariant subspace to the nine rank-1 zero-quantum basis operators.

Spin inversion conjugation symmetry
As seen from Table I and Eq.(20c), the relaxation superoperator is invariant and the ISTO basis operators have definite parity under spin inversion conjugation.According to the basic orthogonality theorem of group theory, 15,33 of which the Wigner-Eckart theorem is a special case, the relaxation supermatrix in the ISTO basis can, therefore, have nonzero elements only between basis operators of the same parity (that is, belonging to the same irreducible representation of the symmetry group).Among the 19 zero-quantum operators, 10 are OSIC (including the ones representing the longitudinal magnetizations) and 9 are ESIC.Longitudinal relaxation in a three-spin system can, therefore, be fully described within the subspace of the ten zero-quantum OSIC operators in Table II, corresponding to the three longitudinal magnetizations and seven zero-quantum coherences (ZQCs).
Since N s is odd for these ten basis operators, Eq. (20b) shows that the seven odd-rank operators are Hermitian, whereas the three even-rank operators are anti-Hermitian, as can be verified from Table II.In view of Eq. ( 27), it then follows that the 10 × 10 relaxation supermatrix that governs longitudinal relaxation has 7 × 7 and 3 × 3 real symmetric blocks along the diagonal (Fig. 1, top left).The elements in these two blocks are linear combinations of some or all of the ESDFs j XY (0), j XY (ω 0 ), and j XY (2ω 0 ).The seven odd-rank spin modes are coupled to the three even-rank modes via the purely imaginary "off-diagonal" 7 × 3 and 3 × 7 blocks (Fig. 1, top left).The elements of these blocks are linear combinations of the OSDFs k XY (ω 0 ) and k XY (2ω 0 ).In other words, the only effect of the OSDFs is to couple odd-rank and even-rank spin modes; they have no effect on the odd-rank and even-rank blocks.If the OSDF is neglected, this coupling disappears and the longitudinal relaxation is then fully described by the real symmetric 7 × 7 block associated with the odd-rank basis operators (Fig. 1, bottom left).This is the case in the EN regime, where, in addition, selection rule (29) holds so that only the six rank-1 modes of this subspace can couple.
To compute the relevant 10 × 10 block of the relaxation supermatrix from Eq. ( 23), we need 45 C XY M matrices.On account of symmetry relations (25) and which follows from Eqs. ( 13) and (27) and is valid only in the zero-quantum subspace, there are only 18 unique C XY M matrices to compute.The details of this straight-forward but tedious task can be found in Appendix A of the supplementary material. 34By combining these matrices, given in exact analytical form in Appendix A, 34 with Eqs. ( 22) and ( 23), the longitudinal relaxation behavior is readily calculated for any geometrical arrangement of the three spins and for any motional model.

Nuclear permutation symmetry
In the context of spin relaxation, nuclear permutation symmetry refers to the spectral density functions. 15,29For dipolar relaxation in an isochronous multi-spin system, the relaxation superoperator R is invariant under permutation (or interchange) of two nuclei if these nuclei are related by a symmetry operation of the molecular point group (geometric symmetry) and if the dipole couplings between each of these nuclei and any other nucleus are modulated in the same way by the molecular motion (dynamic symmetry).These two requirements can be concisely expressed in terms of the spectral density functions J XY (ω).For our three-spin system, R is invariant under I ↔ S interchange if J IP,IP (ω) = J SP,SP (ω) and J IS,IP (ω) = J IS, S P (ω).Similarly, R is invariant under permutations of all three nuclei if J IS,IS (ω) = J IP,IP (ω) = J SP,SP (ω) and J IS,IP (ω) = J IS, S P (ω) = J IP, S P (ω).
We shall use the conventional spin system notation, normally based on the static spin Hamiltonian, to label the three possible types of geometric symmetry or equivalence.In the A 3 system, the three nuclei reside at the vertices of an equilateral triangle and are therefore geometrically equivalent.In the A 2 A ′ system, the nuclei define an isosceles triangle.If spin P is at the apex, then spins I and S are geometrically equivalent.Finally, in the AA ′ A ′′ system, all three internuclear vectors have different lengths so there is no geometrical equivalence.
For isotropic dynamic models (Sec.III), such as sphericaltop rotational diffusion, all dipole couplings are modulated in the same way (full dynamic symmetry).Geometric equivalence then implies nuclear permutation symmetry.For anisotropic rotation models, the lower dynamic symmetry may reduce or abolish the nuclear permutation symmetry even though geometric symmetry is present (Sec.IV).
For the isotropic dynamic model considered in Sec.III, the relaxation superoperator is thus invariant under permutation of geometrically equivalent nuclei.For the A 3 and A 2 A ′ systems, the relaxation supermatrix, therefore, becomes blockdiagonal in a basis of operators that have definite parity under permutation of equivalent nuclei. 17,29The ISTO basis operators in Table II are constructed by first coupling spins I and S and then coupling the resultant to spin P (Sec.II C).Accordingly, they have definite parity under interchange of spins I and S, but not under permutations of all three spins.This basis is, therefore, adapted to the geometric symmetry of the A 2 A ′ system (with spin P at the apex), but not to the higher symmetry of the A 3 system.

E. Self-and distinct correlations
A relaxation supermatrix element R n p with n p describes cross relaxation between spin modes n and p.For example, n and p might be the longitudinal magnetizations of two different spins (as in the two-spin Solomon equations 1 ), or they might represent a longitudinal magnetization and a ZQC.Both auto-relaxation rates R nn and cross relaxation rates R n p (n p) may have contributions from self-correlations (terms with X = Y ) and from distinct correlations (terms with X Y ).In the literature, these contributions are usually referred to as auto-and cross correlations, but we prefer the descriptors "self" and "distinct" to distinguish them from "auto-mode" and "cross-mode" relaxation.Note that the term "distinct" here refers to the spin pairs X and Y rather than to the dipole couplings ω D, X and ω D,Y , which may or may not be distinct.
The role of self-and distinct correlations is illuminated by the selection rules (derived in Appendix B 34 ) where the indices n and p refer to the basis operator ordering in Table II.The first rule shows that cross relaxation between longitudinal magnetization modes (n = 1-3) and ZQCs (p = 4-10) is induced entirely by distinct correlations.
The second rule shows that cross relaxation among the longitudinal magnetization modes (in the A 2 A ′ and AA ′ A ′′ systems) is induced entirely by self-correlations.
If distinct correlations are omitted, the relaxation behavior simplifies considerably (Fig. 1).According to selection rule (31a), the self-matrices C X X M are block-diagonal, with a 3 × 3 longitudinal magnetization block and a 7 × 7 ZQC block (Fig. 1, top right).The longitudinal self-relaxation is, therefore, fully determined by the 3 × 3 block, for which Eq. ( 30) yields C X X −M = C X X M .We then obtain from Eqs. ( 12), (13), and ( 23) where the matrices refer to the 3 × 3 longitudinal magnetization block.In the absence of distinct correlations, longitudinal relaxation is thus at most tri-exponential and the OSDF has no effect.In other words, the OSDF affects longitudinal relaxation only via distinct correlations, consistent with our earlier conclusion (Sec.II D 3) that the OSDF only affects cross relaxation between odd-rank and even-rank spin modes.Neglect of the OSDF does abolish the cross relaxation (induced by self-correlations) between odd-rank and even-rank ZQCs (Fig. magnetizations, which are decoupled from the ZQCs in the absence of distinct correlations.Substituting the C X X M matrices from Appendix A 34 into Eq.( 32), we find with the familiar auto-and cross relaxation combinations of (self) spectral densities 16

F. Eigenmode decomposition
The time evolution of the total longitudinal magnetization, σ z ≡ σ 1 + σ 2 + σ 3 , after a nonselective excitation, with σ n (0) = δ n1 + δ n2 + δ n3 , may be decomposed into eigenmode contributions as The normalized mode amplitudes or weights, some of which may be zero for symmetry reasons, are given by The eigenvalues Λ k and eigenvectors {V nk , n = 1, 2, . . ., 10} are obtained by diagonalizing the symmetric 10 × 10 relaxation matrix by the similarity transformation where  V is the transpose of the complex orthogonal matrix V.Note that the defining relation,  V = V −1 , for an orthogonal matrix is the same whether V has real or complex elements. 35n both cases, the columns (eigenvectors) v k of V are orthonormal in the sense  v k v l = δ k l , which differs from the normal (Hermitian) inner product  v * k v l in a complex vector space.If the OSDF is neglected, as allowed in the EN and adiabatic regimes, the relaxation matrix R is real symmetric (Sec.II D 1) so the eigenvalues Λ k are real. 35In general, however, the spectral density is complex and R is complex symmetric.A sufficient condition for a complex symmetric matrix to be diagonalizable by a complex orthogonal similarity transformation is that all its eigenvalues are distinct, 35 which is the case whenever R is complex.(Degenerate eigenvalues do occur in the EN and adiabatic regimes, where R is real, and at all frequencies for the A 3 system with isotropic motion (Sec.III) since nuclear permutation symmetry then cancels the effect of the OSDF.Because a real symmetric matrix is always diagonalizable, 35 these degeneracies do not pose a problem.Furthermore, in all cases, eigenvalues with nonzero weights C k are distinct.)In the dispersive regime, R can have up to three complex-conjugate pairs of eigenvalues, Λ k = λ k ± i µ k , and associated weights, C k = a k ± i b k .The remaining eigenvalues and weights are real.By combining any complex-conjugate pairs, we can express the evolution function in terms of real quantities, where N λ ≤ 10 is the number of exponential components with nonzero weight.For real eigenvalues c k = C k = a k , whereas complex-conjugate eigenvalue pairs have oscillating weights: . Whereas λ k is always positive, a k , b k , and µ k may have either sign.Thus, although Eq. ( 38) implies that the initial weights c k (0) sum up to 1, some of them may be negative.
The number, N λ , of exponential components in Eq. ( 38) cannot exceed the dimension of the invariant subspace to which the longitudinal magnetizations belong.In Table III, we give N λ for the different nuclear geometries and frequency regimes, with and without distinct correlations.Here, we have assumed an isotropic motion (Sec.III), so nuclear permutation symmetry is governed solely by geometry (Sec.II D 4).For anisotropic motions, N λ can be larger (Sec.IV).In the absence of distinct correlations, N λ ≤ 3 (Sec.II E), also for anisotropic motions.For self-relaxation induced by isotropic motions, N λ equals the number of distinct dipole couplings.For isotropic motions, the OSDF only affects the longitudinal relaxation in the dispersive regime and then only for the less symmetric geometries A 2 A ′ and AA ′ A ′′ .If the OSDF is neglected, N λ = 5 and 7, respectively, for these cases.When the OSDF is included, one (A 2 A ′ ) or one -three (AA ′ A ′′ ) complex-conjugate eigenvalues occur, which split into two real eigenvalues in certain frequency ranges, thereby increasing the number of exponential components by one for the A 2 A ′ system and by one, two, or three for the AA ′ A ′′ system.But these bifurcating eigenmodes have very small weights.
For the A 3 system with isotropic motion, the relaxation superoperator is invariant under permutation of the three geometrically and dynamically equivalent nuclei.Because only three orthonormal odd-rank modes exhibit this invariance (Sec.III B), N λ = 3 in the dispersive regime.In the EN regime, selection rule (29) forbids coupling to the rank-3 mode (Sec.III B) so N λ = 2.For A 3 case, the deviation from single-exponential relaxation is entirely due to motional correlations between dipole couplings of distinct spin pairs (with one shared spin).In contrast, for the A 2 A ′ and AA ′ A ′′ systems, multiexponential relaxation is caused both by distinct correlations and by cross relaxation between the longitudinal modes of nonequivalent spins.For the A 2 A ′ system, there are five odd-rank and one even-rank mode with I ↔ S interchange symmetry (Sec.III C), so N λ = 6 in the dispersive regime.If the OSDF is neglected, N λ = 5 since coupling between oddrank and even-rank modes is then no longer allowed (Sec.II D).However, also with a complex spectral density, N λ = 5 in a limited frequency range where two distinct real eigenvalues merge into a complex conjugate pair.In the EN regime, where selection rule (29) only allows couplings within the rank-1 subspace, two of the modes are decoupled so that N λ = 4.The reduction of the invariant spin operator subspace, and thus of N λ , as the various symmetries are taken into account is presented in graphical form in Fig. 2, which contains most of the information from Table III.Like Table III, Fig. 2 only applies to the case of isotropic motion, which yields the maximum nuclear permutation symmetry.

G. Integral and initial relaxation rates
If the longitudinal relaxation does not deviate strongly from a single exponential, it can be characterized, with little loss of information, by a single effective rate, defined as Even when relaxation is markedly multi-exponential, this integral longitudinal relaxation rate may be useful (at the expense of some information loss) since it can be measured and computed under all conditions.By integrating the matrix form of BWR master equation ( 28) and applying the nonselective initial condition, we find Alternatively, we can insert the eigenmode expansion (35)  into Eq.( 39) to obtain where for complex-conjugate eigenvalue pairs.Consequently,  R 1 is always real.The real weights c ′ k sum up to 1, but some of them may be negative.The initial longitudinal relaxation rate is defined as By setting t = 0 in BWR master equation (22) and applying the nonselective initial condition, we find FIG. 2. Reduction of the spin operator subspace required to describe longitudinal relaxation in three-spin systems with isotropic motion.To the right of each operator block is given the number of basis operators that exhibit all the symmetries that follow from the properties listed above it.This is also the maximum number of exponential components in the relaxation function.
where the last form follows from Eqs. ( 23) and (31b).The initial rate is thus unaffected by distinct correlations (and the OSDF).Combining this result with Eqs. ( 33) and (34), we obtain where R I S 1 is the relaxation rate for the isolated two-spin IS system 1 The initial rate may thus be regarded as the average twospin rate, obtained by taking the arithmetic average of the relaxation rates for three isolated two-spin systems and then multiplying by 2 since, in the three-spin system, each spin is dipole-coupled to two other spins.
Although the initial rate R 0 1 is unaffected by distinct correlations, it differs, in general, from the integral selfrelaxation rate  R self 1 , obtained by substituting R self from Eq. ( 33) into Eq.( 40).In contrast to the initial rate,  R self 1 generally involves the zero-frequency spectral densities, j X X (0).Only for the A 3 system with motional models that do not violate the dynamic equivalence of the three spins is

III. LONGITUDINAL RELAXATION BY ISOTROPIC MOTIONS
In this section, we illustrate the theory developed in Sec.II by quantitatively examining the longitudinal relaxation behavior for the A 3 , A 2 A ′ , and AA ′ A ′′ spin systems in the special case where the dipole couplings are modulated by an isotropic motion.(Anisotropic motions are considered in Sec.IV.)If the motion is isotropic in the molecular frame, the time correlation functions G XY (τ) defined in Eq. ( 9) take the simple exponential form where β XY is the fixed angle between internuclear vectors X and Y , that is, cos This form is valid for the spherical-top rotational diffusion model and for the strong-collision model, with the correlation time τ c being the rotational correlation time, (6 D R ) −1 , or the mean survival time, respectively.The spectral density function in Eq. ( 12) can then be factorized as with the real-valued geometric factor and the complex-valued reduced spectral density function In Sec.III A, we examine the deviation from singleexponential relaxation and in Subsections III B-III D, we present the complete frequency dependence (dispersion) of the integral relaxation rate  R 1 , the integral self-relaxation rate  R self 1 , and the initial rate R 0 1 , as well as of the eigenmode rates λ k and weights c k , for the three spin systems.We also examine the effect of the OSDF.The numerical results were computed from the 10 × 10 relaxation matrix defined by Eqs. ( 23) and ( 47)-( 49) and the C XY M matrices in Appendix A. 34 All relaxation rates are presented in reduced form, in units of ω 2 D,IS τ c versus the reduced frequency ω 0 τ c .As geometric parameters, we use the interior angles of the triangle, denoted by β I , β S , and β P .We thus specify β P = β IP, S P for the A 2 A ′ system, and β I = β IS,IP and β S = π − β IS, S P for the AA ′ A ′′ system.The remaining dipole couplings are then, for the A 2 A ′ system, and for the AA ′ A ′′ system,

A. Non-exponential relaxation
To examine the deviation from single-exponential relaxation of the normalized longitudinal non-equilibrium magnetization, σ z (t)/σ z (0), we compare the multi-exponential decay obtained from eigenmode expansion (38) with the singleexponential decay exp(−  R 1 t), with the integral longitudinal relaxation rate  R 1 obtained from Eq. ( 40) or (41).We define a deviation function as As noted many years ago, 2,4 the deviation from singleexponential relaxation is insignificant for the A 3 spin system with isotropic motion.The largest deviation, with δ max = 0.002, is found in the EN regime.Such small deviations are certainly beyond experimental detection.For the A 2 A ′ spin system, the deviation from single-exponential relaxation is more pronounced, as shown in Fig. 3 for β P = 120 • .Again, the deviation is larger in the EN regime (δ max = 0.13) than in the dispersive regime (δ max = 0.05 at ω 0 τ c = 1).If the OSDF is omitted, the deviations become slightly larger (δ max = 0.14 and 0.06, respectively).The largest deviations are seen for the AA ′ A ′′ spin system, as illustrated in Fig. 4 for a geometry with β I = 80 • and β S = 40 • .Here, the maximum deviation δ max is 0.23 in the EN regime and 0.16 at ω 0 τ c = 1.
In all cases, the initial decay is faster for σ z (t)/σ z (0) than for the exponential function exp(−  R 1 t).In other words, the calculations show that R 0 1 >  R 1 .Since distinct correlations affect  R 1 but not R 0 1 (Sec.II G), this observation is consistent with a slowing down of longitudinal relaxation by distinct correlations, as can be demonstrated in a general way for the A 3 system with isotropic motion. 4As seen from Eq. (39), the areas under the two decay curves shown in each panel of Figs. 3 and 4  must decay more slowly than exp(−  R 1 t) at longer times, leading to a crossover (Figs. 3 and 4).An exponential fit to the multi-exponential decay is, therefore, likely to yield an effective rate that is close to the integral rate  R 1 .However, if the deviation from single-exponential relaxation is substantial, it may be better to determine the integral rate directly from the experimental data.

B. Relaxation dispersion in the A 3 system
For the A 3 spin system, the three nuclei are geometrically equivalent and if the motion is isotropic, as assumed in this section, the relaxation superoperator is invariant under permutation of all three nuclei (Sec.II D 4).The spin operator basis in Table II is not fully symmetry-adapted for this case, because we have broken the nuclear permutation symmetry by first coupling spins I and S and then coupling their resultant to spin P (Sec.II C).However, the ten basis operators in Table II can be transformed into a fully symmetry-adapted basis comprising three operators that are even and seven that are odd under permutations of the three nuclei.In this basis, the relaxation supermatrix is block-diagonal and longitudinal relaxation is fully described within the even subspace, spanned by the following fully symmetry-adapted orthonormal basis operators (distinguished by an overbar from the operators in Table II), 17 FIG.4. Decay of the longitudinal magnetization, σ z (t)/σ z (0), versus reduced time, t × ω 2 D,IS τ c , for the AA ′ A ′′ spin system with β I = 80 • and β S = 40 • and isotropic motion.The multi-exponential decay obtained from Eq. (38) (solid curve) is compared with the single-exponential decay exp(−  R 1 t) (dashed curve).
Consequently, longitudinal relaxation in the A 3 system is in general tri-exponential.However, in the EN regime, selection rule (29) ensures that the rank-1 modes σ1 and σ2 do not couple to the rank-3 ZQC σ3 (Table II).Therefore, longitudinal relaxation is bi-exponential in the EN regime.Since all three operators in Eq. ( 52) are of odd rank, Eq. (30) shows that CXY −M = CXY M in the fully symmetryadapted basis.It then follows from Eq. ( 23), which is valid for any orthonormal basis, and Eqs. ( 12) and (13a), which yield J XY (Mω 0 ) + J XY (−Mω 0 ) = 2 j XY (Mω 0 ), that the OSDF does not affect longitudinal relaxation in the A 3 system with isotropic motion.(As we shall see in Sec.IV B, this is not always true if the motion is anisotropic.)The three eigenvalues Λ k of R (and of R) and the associated weights C k are, therefore, real.From Eqs. ( 23), ( 30), ( 47 matrices in Appendix A, 34 we obtain the relaxation matrix R in the fully symmetry-adapted basis (Appendix C 34 ), in full agreement with the results presented by Werbelow and Grant. 17igure 5 shows the eigenmode rates λ k and the corresponding nonzero weights c k (=a k ) in Eq. (38), obtained by diagonalizing the relaxation matrix R as in Eq. (37).Here, and in the following, the eigenmodes are numbered in order of decreasing absolute real weight |Re{C k }| at ω 0 τ c = 1.The rank order of the three eigenvalues is independent of frequency, but λ 1 exhibits avoided crossings with λ 3 and λ 2 at ω 0 τ c ≈ 0.28 and 1.09, respectively, where the weights of the corresponding eigenmodes are equal.Each eigenmode is a linear combination of the longitudinal magnetization mode σ1 and the ZQCs σ2 and σ3 , but the dominant eigenmode (different in each frequency regime) is essentially a σ1 mode.
To reconcile the very nearly exponential longitudinal relaxation in the A 3 system (Sec.III A) with the distinctly different eigenvalues in Fig. 5, the following observations can be made.In the EN regime, one eigenmode, which is essentially a longitudinal mode, dominates strongly (c 3 = 0.9908).In the dispersive regime, two eigenmodes have equal weight (close to 0.5) at the avoided-crossing frequencies, but then the corresponding eigenvalues are nearly equal so the longitudinal relaxation remains almost exponential.One eigenvalue has a high-frequency plateau, λ 3 = (9/20) ω 2 D,IS τ c for ω 0 τ c ≫ 1, but the corresponding weight is effectively zero in the adiabatic regime.The weight of the minor adiabatic FIG. 5. Eigenmode rates λ k (in units of ω 2 D τ c ) and weights c k for the A 3 spin system with isotropic motion versus the reduced Larmor frequency ω 0 τ c .eigenmode is not negligible (c 1 = 0.0412) but λ 1 is close to λ 2 for ω 0 τ c > 1, so, again, the deviation from exponential longitudinal relaxation is insignificant.
The dipolar relaxation of the total magnetization of two isochronous spins-1/2 is isomorphic with the quadrupolar relaxation of a single spin-1.However, because of the occurrence of distinct correlations, this isomorphism does not carry over to three or more spins.The longitudinal relaxation of a single quadrupolar spin-3/2 is bi-exponential in the dispersive regime and exponential in the EN regime, 36 and the relaxation rates only involve the spectral densities j(ω 0 ) and j(2ω 0 ).In contrast, the total longitudinal magnetization of three equivalent spins-1/2 relaxes tri-exponentially in the dispersive regime and bi-exponentially in the extremenarrowing regime (Table III, Fig. 5).Moreover, the component rates also depend on the zero-frequency spectral density j(0).Inserting the inverse of R into Eq.(40), we find, for the integral relaxation rate, where we have used the short-hand notation j M ≡ j(Mω 0 ).
Although  R 1 depends on j 0 , this is not a linear dependence that would give rise to a high-frequency plateau in the dispersion profile  R 1 (ω 0 ).The j 0 dependence derives from the auto-and cross relaxation rates of the ZQCs σ2 and σ3 , which couple to the longitudinal mode σ1 via cross relaxation rates (produced by distinct correlations) that go to zero at high frequencies since they do not involve j 0 (Appendix C 34 ). 17This is seen by rearranging Eq. (53) into and noting that δ = j 1 /64 when ω 0 τ c ≫ 1.Therefore, at high frequencies,  R 1 does not exhibit a plateau but goes to zero as ω −2 0 and is then numerically very close to . This behavior is evident from Fig. 6, which also shows that the distinct correlations slow down the longitudinal relaxation by at most 0.8%.

C. Relaxation dispersion in the A 2 A ′ system
The ISTO basis in Table II allows us to directly exploit the I ↔ S interchange symmetry of the A 2 A ′ system with spin P at the apex and with isotropic motion (Sec.II D 4).Forming the symmetric magnetization mode σ I S = (σ 1 + σ 2 )/ √ 2, we see from Table II that five of the remaining spin modes are even (invariant) under I ↔ S interchange, while three modes (n = 5, 8, and 9) are odd.Longitudinal relaxation in the A 2 A ′ system can, therefore, be fully described within the invariant subspace of the six even basis operators.In the EN regime, the interchange symmetric operators B 7 and B 10 with rank K > 1 can be omitted on account of selection rule (29), leaving an invariant subspace spanned by four fully symmetry-adapted basis operators.In other words, longitudinal relaxation in the A 2 A ′ system with isotropic motion involves (at most) six FIG. 7. Eigenmode rates λ k (in units of ω 2 D,IS τ c ) and weights c k for an A 2 A ′ spin system with β P = 120 • and isotropic motion versus the reduced Larmor frequency ω 0 τ c .exponential components in the dispersive regime, but only four components in the EN regime (Table III).
Figure 7 shows, for an A 2 A ′ system with β P = 120 • , the six eigenmode rates λ k and the corresponding nonzero weights c k for the four eigenvalues that remain real at all frequencies.Each of these four eigenvalues exhibit one avoided crossing and the corresponding eigenmodes are each dominant in some frequency range (Fig. 7).In contrast to the A 3 system, the OSDF now affects longitudinal relaxation in the dispersive regime, where the real eigenvalues λ 4 and λ 5 coalesce to a complex conjugate pair at ω 0 τ c ≈ 0.80 and then split up again at ω 0 τ c ≈ 2.70.These two eigenmodes have significant but not dominant weights, e.g., c 4 + c 5 = 0.06 at ω 0 τ c = 0.7 and c 4,5 (0) = 0.12 at ω 0 τ c = 0.9.We also note that the two eigenmodes (c 3 and c 1 ) that have nonzero weights in the adiabatic regime correspond to eigenvalues without a highfrequency plateau (due to j 0 ).If the OSDF is discarded, the eigenvalues λ 1 , λ 3 , and λ 6 and the corresponding weights are hardly affected, but the complex conjugate eigenvalues λ 4,5 and the real eigenvalue λ 2 are now replaced by two real eigenvalues with an avoided crossing.(Note that Fig. 7 shows a crossing for the real parts of these eigenvalues; the complex eigenvalues are distinct at all frequencies.)The number of exponential components in the dispersive regime is thus five without the OSDF, but five or six (depending on frequency) when the OSDF is included.In the EN regime (where the FIG. 8. Dispersion of the integral relaxation rate  R 1 , its self-correlation part  R self 1 , and the initial relaxation rate R 0 1 (all three in units of ω 2 D,IS τ c ) for an A 2 A ′ spin system with β P = 120 • and isotropic motion.The upper panel shows the relative differences between  R 1 and  R self 1 (blue), between  R self 1 and R 0 1 (black), and between  R 1 with and without the OSDF (magenta).
This article is copyrighted as indicated in the article.OSDF is negligible), there are four components: c 6 , c 1 , c 3 (shown in Fig. 7), and c 5 (corresponding to the larger of the two bifurcated eigenvalues; c 4 = 0 in the EN regime).
The only previous theoretical study of relaxation in the A 2 A ′ system over the full frequency range is that of Schneider, who also included the OSDF. 13Comparing our numerical results for C k and Λ k with those tabulated by Schneider, we find agreement to the last quoted decimal in the EN and adiabatic regimes and essential agreement (nearly always to the second decimal place in C k and to the first decimal place in Λ k ) in the dispersive regime.
Figure 8 shows the dispersion profiles of the total (  R 1 ) and self-(  R self 1 ) integral relaxation rates and of the initial relaxation rate (R 0 1 ).As compared to the A 3 system, distinct correlations have a much larger effect in the A 2 A ′ system, reducing  R 1 by as much as 36% (in the EN regime) as compared to R 0 1 , which is unaffected by distinct correlations (Sec.II G).Unlike in the A 3 system,  R self 1 is not identical to R 0 1 , being 17.5% smaller in the EN regime.Figure 8 also shows that the OSDF increases  R 1 by up to 1.5% in the dispersive regime.The maximum OSDF effect for the A 2 A ′ system, 2.6%, is found for β P = 108 • .

D. Relaxation dispersion in the AA ′ A ′′ system
As expected, the most complicated relaxation behavior is obtained when all three dipole couplings are distinct so there is no nuclear permutation symmetry at all.Then, all ten spin FIG. 9. Eigenmode rates λ k (in units of ω 2 D,IS τ c ) for an AA ′ A ′′ spin system with β I = 80 • and β S = 40 • and isotropic motion versus the reduced Larmor frequency ω 0 τ c .Shown in separate panels are the four major real eigenvalues (bottom) and three minor complex conjugate eigenvalues (top).modes in Table II contribute and longitudinal relaxation can, in principle, involve up to ten exponential components.In the EN regime, there can be at most six components since selection rule (29) restricts the invariant subspace to the six rank-1 modes (Table II).We illustrate the generic AA ′ A ′′ relaxation behavior with results for a specific nuclear geometry with β I = 80 • and β S = 40 • (and thus β P = 60 • ).
As for the A 2 A ′ system, four eigenvalues (λ 1 -λ 4 ) are real at all frequencies (Fig. 9, bottom).However, there are now three complex-conjugate pairs of eigenvalues (Fig. 9, top).Two of them bifurcate at both low and high frequencies but further away from ω 0 τ c = 1 than for the A 2 A ′ system (λ 5,6 bifurcates at ω 0 τ c = 0.0783 and 5.94, λ 7.8 at ω 0 τ c = 0.0493 and 18.8), whereas the third complex-conjugate eigenvalue (λ 9,10 ) only bifurcates in the adiabatic regime (at ω 0 τ c = 62.4),where this mode has vanishingly small weight.The weights of the four real eigenmodes are shown in two formats in Fig. 10.These four modes contribute in the EN and dispersive regimes, but only the two modes that lack a high-frequency eigenvalue plateau (Fig. 9) contribute in the adiabatic regime.In the EN regime, the two additional modes c 7 and c 8 have negligibly small weights that decrease gradually to zero as the zero-field limit is approached.Hence, Table III and Fig. 2 quote N λ = 4 (rather than 6) for the AA ′ A ′′ system in the EN regime.
In the frequency range 0.0783 < ω 0 τ c < 5.94, where there are three complex-conjugate eigenvalues, longitudinal relaxation involves seven exponential components.The influence of the three complex-conjugate modes is the largest at ω 0 τ c = 0.57, where their combined weight is −0.11.Accordingly, the sum of the four real mode weights is 1.11 at this frequency (Fig. 10, top).Outside this frequency range, one or more eigenvalues has bifurcated, so the relaxation function contains eight, nine, or ten exponential components, but the combined weight of modes c 5 -c 10 then never deviates by more than −0.01 from zero (Fig. 10, top).If the OSDF is omitted, there are seven distinct real eigenvalues which maintain their rank order at all frequencies and exhibit six avoided crossings.The relaxation function then contains seven components in the dispersive regime and four in the EN regime.
Figure 11 shows the dispersion profiles of the total (  R 1 ) and self-(  R self 1 ) integral relaxation rates and of the initial relaxation rate (R 0 1 ).Distinct correlations have an even larger effect than for the A 2 A ′ system, reducing  R 1 by as much as 58% (in the EN regime) as compared to R 0 1 .The OSDF increases  R 1 by at most 0.29% in the dispersive regime.Other AA ′ A ′′ geometries yield larger OSDF effects, which, however, do not exceed the A 2 A ′ maximum of 2.6%.

IV. ANISOTROPIC MOTIONS
The analysis in Sec.III was restricted to the simplest possible time correlation function, with the single-exponential FIG.11.Dispersion of the integral relaxation rate  R 1 , its self-correlation part  R self 1 , and the initial relaxation rate R 0 1 (all three in units of ω 2 D,IS τ c ) for an AA ′ A ′′ spin system with β I = 80 • and β S = 40 • and isotropic motion.The upper panel shows the relative differences between  R 1 and  R self 1 (blue), between  R self 1 and R 0 1 (black), and between  R 1 with and without the OSDF (magenta).form of Eq. ( 46).However, the relaxation theory in Sec.II is valid for any motional model as long as the bath is isotropic.Replacing Eq. ( 46) by an anisotropic motional model will obviously affect the results of Sec.III in a quantitative way.More importantly, if the dynamic symmetry is broken (Sec.II D 4), the relaxation behavior may also be altered qualitatively.For the A 3 and A 2 A ′ systems, where nuclear permutation symmetry reduces the invariant subspace in the case of isotropic motion (Fig. 2), anisotropic motions may introduce additional relaxation components and can enhance the influence of the OSDF on longitudinal relaxation, also in the A 3 system.In the following, we examine two anisotropic motional models, one of which breaks the dynamic symmetry and one which does not.

A. Axial internal rotation
First, we consider the situation where, in addition to spherical-top rotational diffusion (as in Sec.III), there is a statistically independent internal rotational diffusion of the spin system about an axis perpendicular to the nuclear plane.In the A 3 case, this model might represent a spherical macromolecule with a methyl group that rotates freely about its threefold axis.As shown in Appendix D, 34 the time correlation function for this model is with τ R = 1/(6 D R ) and τ int = 1/(4 D int ).The spectral density functions now cannot be split into geometric and purely dynamic factors, as in Eq. ( 47).Instead, we obtain from Eqs. ( 12) and (55) where J R (ω) and J int (ω) are given by Eq. ( 49) with τ c replaced by τ R or (1/τ R + 1/τ int ) −1 , respectively.For the A 3 system, all dipole vectors are affected in the same way by the internal rotation so the nuclear permutation symmetry is not affected.Formally, this conclusion follows by noting that the factor within square brackets in Eq. ( 56) is the same for all self correlations, since β X X = 0, and for all distinct correlations, since cos(2 β XY ) = −1/2 for β XY = 60 • or 120 • .For the A 2 A ′ system, the internal rotation does not affect the dynamic symmetry of the two geometrically equivalent nuclei I and S, as seen by noting that cos(2 ) so that J IS,IP (ω) = J IS, S P (ω).Consequently, internal rotation about an axis perpendicular to the nuclear plane does not affect the nuclear permutation symmetry for any of the three spin systems.
Even though internal rotation does not alter the relaxation behavior qualitatively, it can have substantial quantitative effects.For example, Fig. 12 shows the three eigenvalues and associated nonzero weights for the A 3 system when τ R = 100 τ int , as might be the case for a methyl group in a macromolecule.(Here, we depart from our eigenmode numbering convention in order to maintain correspondence with the case of isotropic motion.)As in the isotropic case (Fig. 5), λ 2 < λ 1 < λ 3 at all frequencies, although the two avoided crossings are not evident on the scale of Fig. 12.However, as compared to the isotropic case, the boundaries of the EN and adiabatic regimes, where the relaxation function changes from tri-exponential to bi-exponential, move out to much lower and higher frequencies, respectively.Whereas in the isotropic case each eigenmode dominates in one frequency regime, now two of the modes dominate in two separate frequency intervals (Fig. 12).Moreover, the weight of eigenmode c 2 vanishes "accidentally" at two frequencies (ω 0 τ c ≈ 4.50 and 20.31), where relaxation becomes bi-exponential.Except at these frequencies, longitudinal relaxation becomes much more non-exponential than in the isotropic case (Fig. 13).For example, δ max = 0.27 at ω 0 τ c = 1 as compared to 0.0014 in the isotropic case.We note also that the tabular results reported by Schneider 13 for the A 3 system and the spectral density function in Eq. ( 56) agree quantitatively with our calculations.
Because the internal rotation is two orders of magnitude faster than the overall tumbling, the relaxation dispersion exhibits two well-resolved steps (Fig. 14).The retarding effect of distinct correlations is modest for the highfrequency internal-rotation step (−5.4% at ω 0 τ c = 10) but very large for the low-frequency tumbling step (a factor 8 at ω 0 τ c = 0.01).For the A 3 system  R self 1 = R 0 1 , as for isotropic FIG. 13.Decay of the longitudinal magnetization, σ z (t)/σ z (0), versus reduced time, t × ω 2 D τ R , for the A 3 spin system modulated by sphericaltop tumbling and internal rotation with τ R = 100 τ int .The multi-exponential decay obtained from Eq. ( 38) (solid curve) is compared with the singleexponential decay exp(−  R 1 t) (dashed curve).

B. Symmetric-top tumbling
The simplest motional model that breaks the nuclear permutation symmetry in the A 3 and A 2 A ′ systems is rigid- body symmetric-top rotational diffusion.Remarkably, this simple model, with arbitrary orientation of the principal axis of the rotational diffusion tensor with respect to the nuclear plane, does not seem to have been investigated even for the A 3 system.For the A 3 system, Hubbard examined the special case where the principal rotation axis is perpendicular to the nuclear plane, 6 in which case the three spins are "scrambled" as in the axial rotation model considered in Sec.IV A. The nuclear permutation symmetry is therefore not broken.Hubbard, 7 and later Werbelow and Marshall, 8 also considered, for the A 3 system, a principal rotation axis with arbitrary orientation, but only in the simultaneous presence of internal rotation about an axis perpendicular to the nuclear plane.Because of the internal motion, the three spins are again "scrambled," so the nuclear permutation symmetry is unaffected and the relaxation function has at most three components. 7Even if the rate of internal rotation is set to zero, these results do not reduce to the results for rigid-body symmetric-top rotation, because rotational symmetry about an axis perpendicular to the nuclear plane has been imposed in the derivation.
We show in Appendix E 34 that, for the rigid-body symmetric-top rotational diffusion model, the spectral density function in Eq. ( 12) is a sum of three terms of the same type as in Eq. ( 47), with the real-valued geometric coefficients D XY, N given in Eq. (E.7) of the supplementary material. 34The complex-valued spectral densities J N (ω) are given by Eq. ( 49), but with the correlation times 37 where τ 0 ≡ 1/(6 D R, ⊥ ) and γ ≡ D R, ∥ /D R,⊥ .Rather than pursuing the general case, we shall illustrate the relaxation behavior for symmetric-top rotation by examining two special cases of the model.If the principal axis of the rotational diffusion tensor is perpendicular to the nuclear plane, Eq. (57) reduces to (Appendix E 34 ) as previously shown by Hubbard. 6This result differs from Eq. ( 56) only in the interpretation of the correlation times.Consequently, the relaxation behavior is qualitatively the same as for the internal rotation model in Sec.IV A. In particular, symmetric-top rotational diffusion with the principal axis perpendicular to the nuclear plane does not break the nuclear permutation symmetry in the A 3 and A 2 A ′ systems.In the EN regime, the self and distinct spectral densities for the A 3 system are obtained from Eq. (59) as In the special event that D R,⊥ = (4/7) D R, ∥ , the distinct spectral density thus vanishes, making longitudinal relaxation singleexponential, as first noted by Hubbard. 6o illustrate the full scope of anisotropic rotation effects, we consider the case where the principal axis of the rotational diffusion tensor lies in the nuclear plane.For this model, Eq. ( 57) yields (Appendix E 34 ) Here,  α X ≡ α X + θ, where α X specifies the orientation of the internuclear vector r X relative to r I S (so α I S = 0, α I P = β I , and α S P = π − β S ) and θ specifies the orientation of the principal rotation axis (also relative to r I S ).
Because of its lower dynamic symmetry, this model alters the nuclear permutation symmetry for the A 3 and A 2 A ′ systems (Sec.II D 4).However, if the principal rotation axis is either parallel with or perpendicular to the internuclear vector r I S , that is, if θ = 0 or π/2, the I ↔ S interchange symmetry is not affected.For any other orientation θ, the I ↔ S interchange symmetry is broken.As a result, the A 2 A ′ system with symmetric-top rotation behaves qualitatively as the AA ′ A ′′ system with spherical-top rotation.For the A 2 A ′ system, the number of relaxation components in the dispersive regime therefore increases from 5-6 for spherical-top rotation to 7-10 for symmetric-top rotation (Table III).For the A 3 system, symmetric-top rotation with θ = 0 or π/2 leaves only I ↔ S interchange symmetry, so the A 3 system with symmetric-top rotation behaves qualitatively as the A 2 A ′ system with spherical-top rotation.Indeed, Eq. (61) yields for θ = 0, and for θ = π/2, FIG. 16.Decay of the longitudinal magnetization, σ z (t)/σ z (0), versus reduced time, t × ω 2 D τ 0 , for the A 3 spin system modulated by symmetric-top rotational diffusion with γ = 10 and the principal rotation axis in the nuclear plane.The multi-exponential decay obtained from Eq. ( 38) (solid curve) is compared with the single-exponential decay exp(−  R 1 t) (dashed curve).
These results show that, for θ = 0 or π/2, the A 3 system does not have full nuclear permutation symmetry, but only I ↔ S interchange symmetry (Sec.II D 4), like the A 2 A ′ system with isotropic motion.This symmetry breaking has two important consequences.First, the number of relaxation components increases from 2 to 4 in the EN regime and from 3 to 5 or 6 in the dispersive regime (Table III).Second, the OSDF now affects longitudinal relaxation also in the A 3 system.For the A 2 A ′ system, the orientations θ = 0 and π/2 yield the same number of relaxation components (the same as for isotropic motion), but the eigenmode rates (and the integral relaxation rate) are quantitatively different for the two orientations.For the A 3 system, on the other hand, these two orientations yield quantitatively the same relaxation behavior.This is not obvious from the spectral densities in Eqs. ( 62) and (63), which, although the same in the EN regime, differ in general.In fact, the A 3 relaxation behavior is quantitatively the same for any orientation θ of the principal diffusion axis in the nuclear plane.For the A 3 system, the relaxation supermatrix R(θ) is related to R(0) by a similarity transformation so the eigenvalue spectrum is independent of θ. 35 (However, this is not true for R self (θ).)Although the eigenvectors depend on θ, the component weights C k in Eq. ( 36) do not.In other words, a less symmetrical orientation of the diffusion axis not only does not lead to further symmetry breaking (and additional relaxation components), but it has no effect at all.Even out-of-plane orientations do not alter the qualitative relaxation behavior further (as long as the diffusion axis is not perpendicular to the nuclear plane; see above), although there are quantitative changes.
To illustrate these results quantitatively, we consider the A 3 system with rotational anisotropy γ = 10.The relaxation rates λ k and weights c k of the six contributing eigenmodes are shown in Fig. 15.The two eigenvalues with the smallest weight, λ 5,6 , form a complex conjugate pair in the dispersive regime, which splits up into two real eigenvalues in the EN and adiabatic regimes.In the EN regime, four components make significant contributions (the fourth one, not shown in FIG. 17. Dispersion of the integral relaxation rate  R 1 , its self-correlation part  R self 1 , and the initial relaxation rate R 0 1 (all three in units of ω 2 D τ 0 ) for the A 3 spin system modulated by symmetric-top rotational diffusion with γ = 10 and the principal rotation axis along the IS internuclear vector.
This article is copyrighted as indicated in the article.Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.235.27.31On: Wed, 16 Dec 2015 14:59:55 Fig. 15, is c 5 = 0.038), as compared to only two components for isotropic rotation (Fig. 5, Table III).As a result, the relaxation function is markedly non-exponential (Fig. 16) with δ max = 0.082 as compared to 0.002 in the isotropic case.In the dispersive regime, there are five relaxation components (Fig. 15), as compared to three in the isotropic case (Fig. 5, Table III).However, because the four major (real) eigenvalues are of similar magnitude (Fig. 15), relaxation is nearly exponential (Fig. 16) with δ max = 0.008 at ω 0 τ 0 = 1.Distinct correlations now have a substantial effect (Fig. 17), reducing  R 1 by up to 25%, as compared to 0.8% in the isotropic case (Fig. 6).Despite the large rotational anisotropy (γ = 10) used here, the dispersion shape is only slightly more extended than expected for a single correlation time because the dispersion profiles of the three spectral density functions J N (ω) in Eq. ( 61) overlap.The OSDF increases  R 1 by at most 0.02% in the dispersive regime, but this effect becomes more pronounced for larger rotational anisotropy.For example, for γ = 200, the OSDF increases  R 1 by up to 1.9% for the A 3 system and by up to 22.5% for the A 2 A ′ system with β P = 103 • and θ = 7 • .For still larger γ, the OSDF effect increases further.

V. NON-ISOCHRONOUS SPINS
For the sake of simplicity and clarity, we assumed at the outset that the three spins are isochronous.We now remove this restriction, replacing the Zeeman Hamiltonian in Eq. ( 1) by with the "chemical shifts" defined with reference to spin I so δ I = 0 and δ X ≡ (ω X − ω I )/ω I for X = S or P. The analytical complexity of the non-isochronous three-spin BWR theory can be avoided by realizing that, within the MN regime (ω D τ c ≪ 1), the spherical-top rotational diffusion and strong-collision models produce the same relaxation behavior.This must be so because both models yield the time correlation function in Eq. ( 46), albeit with different interpretations of the correlation time.Moreover, for the strong-collision model, the orientational part of the stochastic Liouville equation (SLE) can be solved analytically, thereby allowing the integral relaxation rate to be obtained with modest computational effort. 38y a straight-forward extension of the two-spin SLE theory, 38 we can, in full analogy with Eq. (40), compute the integral relaxation rate as where R SLE is the supermatrix representation of the relaxation superoperator Here, E is the identity superoperator, L Z and L D are the Liouvillians corresponding to the Hamiltonians in Eqs. ( 64) and ( 2), respectively, and the angular brackets signify an isotropic orientational average.As a bonus, the SLE theory of integral relaxation rates computed with BWR and SLE theories, the former with or without inclusion of the OSDF, for an isochronous A 2 A ′ system with β P = 108 • , ω D,IS = 10 4 rad s −1 , τ c = 10 −7 s, and isotropic motion.
is valid also outside the MN regime, although the strong collision and spherical-top rotational diffusion models are then no longer equivalent.Because the relaxation superoperator R SLE is isotropically averaged, it must reflect the cylindrical symmetry of the spin system.Consequently, selection rule (28) applies so we only need to retain the 19 × 19 zero-quantum block of R SLE (Sec.II D 2).This is true also outside the MN regime.Within the MN regime, the relaxation supermatrix R SLE must be identical to the supermatrix R BWR that would be obtained from the non-isochronous BWR theory, although this is not obvious from the corresponding superoperators.Furthermore, because chemical shifts can only break nuclear permutation symmetry, the non-isochronous BWR relaxation superoperator R BWR is still invariant under spin inversion conjugation.Therefore, within the MN regime, we need only consider the 10 × 10 block of R SLE , corresponding to the invariant subspace spanned by the basis operators in Table II.
Before examining the effect of chemical shifts, we shall use the SLE theory to check our conclusion, based on BWR theory, that the OSDF affects longitudinal relaxation.65) and (66), which implicitly incorporate any effect of the OSDF, should agree with  R BWR 1 computed with the aid of Eqs. ( 23) and ( 47)-(49) provided that the imaginary part of J(ω) is included.As seen from Fig. 18, this prediction is confirmed quantitatively.Because ω D,IS τ c = 10 −3 in this calculation, the relative difference between  R BWR 1 and  R SLE 1 due to the MN approximation is less than 0.001%.
Using the SLE computational scheme, Eqs. ( 65) and ( 66), but with parameter values in the MN regime (ω D,IS τ c = 10 −4 ), we now examine the effect on the integral relaxation dispersion  R 1 (ω 0 ) of chemical shifts of different magnitudes.We set δ P = 2 δ S so that all three spins have different Larmor frequencies and we let δ P vary from 0.1 to 1000 ppm ( 1 H shifts rarely exceed 10 ppm).The results in Fig. 19 show that chemical shifts increase  R 1 by up to 17.6% for an AA ′ A ′′ system with β I = 80 • and β S = 40 • .Figure 20 shows how the shift effect depends on the nuclear geometry.For the A 3 geometry, the maximum effect is only 0.42% for δ S = 5 ppm and δ P = 10 ppm, but for the A 2 A ′ and AA ′ A ′′ systems the maximum shift effect is larger (19.4% and 17.4%, respectively) and depends on the triangle angles.(We still use the spin system notation solely to indicate geometric symmetry.) For non-isochronous spins, the relaxation dispersion profiles are non-monotonic (Fig. 19).This unusual feature appears because not all of the ISTOs T 2 M (X) in the dipolar Hamiltonian (3) are eigenoperators of the Zeeman Liouvillian corresponding to Eq. (64).For homonuclear spins (so that δ S , δ P ≪ 1), the only effect of this complication is that each of the superoperators C XY M in Eq. ( 11) becomes a (double) sum of superoperators (derived from eigenoperators of L Z ) multiplied by oscillating factors exp(i ω XY M N N ′ t) with frequencies ω XY M N N ′ that are linear combinations of the shifts δ S and δ P .If this frequency, of order δ ω 0 , is much larger than the corresponding "partial relaxation rate," of order ω 2 D τ c , then the modulated term is effectively cancelled and only terms with ω XY M N N ′ = 0 survive.The Larmor frequency where this "nonsecular decoupling" (NSD) sets in is thus given by where ω D and δ characterize the magnitudes of the dipole couplings and shifts, respectively.For example, for ω D = 10 4 rad s −1 , τ c = 10 −8 s, and δ P = 100 ppm, Eq. (67) yields ω NSD = 10 4 rad s −1 (Fig. 19).
For a given pair of dipole couplings X and Y , C XY M is a sum of 14 terms.In the general case, where δ S δ P , the number of terms for which ω XY M N N ′ = 0 is six for self-correlations (X = Y ) but only two for distinct correlations (X Y ).The NSD thus mainly suppresses distinct correlations, which tend to slow down relaxation, so an "inverted dispersion," where  R 1 increases with frequency, appears at ω 0 ≈ ω NSD (Fig. 19).The greater susceptibility of distinct correlations to NSD also explains the very small chemical-shift effect on the A 3 system (Fig. 20), where  R 1 is only marginally influenced by distinct correlations (Fig. 6).
According to Eq. ( 67), the NSD frequency increases linearly with the correlation time.When τ c is so long that ω NSD approaches the main dispersion at ω 0 ≈ 1/τ c , the chemicalshift effect is diminished (Fig. 21) and when ω NSD is far above the main dispersion, so that δ ≪ (ω D τ c ) 2 , chemical shifts do not affect longitudinal relaxation.Outside the MN regime, where ω D τ c 1, chemical shifts can therefore safely be ignored when considering the longitudinal relaxation of homonuclear spin systems.
Also in the presence of chemical shifts, the relaxation supermatrix is symmetric (R pn = R n p ) and the odd-rank (n = 1-7) and even-rank (n = 8-10) blocks are real-valued while the mixed odd-even blocks are pure imaginary (Fig. 1).However, the imaginary supermatrix elements are now caused by chemical shifts as well as by the OSDF.In fact, the chemical shifts only affect the imaginary elements.Specifically, they affect the coupling between odd-rank ZQCs (n = 4-7) and even-rank ZQCs (n = 8-10).In the isochronous A 2 A ′ system, the I ↔ S interchange symmetry prohibits coupling between the symmetric ZQCs σ 4 , σ 6 , and σ 7 and the anti-symmetric ZQCs σ 8 and σ 9 .In the presence of chemical shifts, the I ↔ S interchange symmetry is broken above the NSD frequency, leading to mixing of these modes.Just like anisotropic rotation (Sec.IV B), chemical shifts make the A 2 A ′ system behave qualitatively as an AA ′ A ′′ system, with up to ten relaxation components in the dispersive regime.Presumably, the OSDF affects longitudinal relaxation also in the A 3 system with isotropic motion if the nuclear permutation symmetry is broken by chemical shifts.However, without actually solving the non-isochronous BWR problem, we cannot demonstrate this particular OSDF effect, which is likely to be small.For the A 2 A ′ and AA ′ A ′′ systems, on the other hand, our calculations show that the combined effect on  R 1 of typical proton chemical shifts (of order 1 ppm) and the OSDF is 1-2 orders of magnitude larger than the effect of the chemical shifts alone.

VI. CONCLUSIONS
We have revisited the problem of longitudinal relaxation in a dipole-coupled homonuclear three-spin system, first addressed by Hubbard in 1958. 2 Nearly all subsequent studies of this problem have been concerned with the special, but important, case of three geometrically equivalent, isochronous spins.In contrast, our treatment is valid for arbitrary geometry.By formulating the BWR theory in Liouville space and making full use of symmetry, we establish the number of exponential relaxation components for all nuclear geometries and for isotropic as well as anisotropic motions.We characterize the relaxation behavior with an eigenmode expansion and with the integral relaxation rate, both of which are examined over the full frequency range.We also investigate the effect of chemical shifts on the integral relaxation rate by means of the stochastic Liouville equation.This is a computationally efficient approach because the orientational part of the stochastic Liouville equation can be solved analytically for the strong-collision model, which is equivalent to the spherical-top rotational diffusion model in the motional-narrowing regime.
The main results of this study are as follows.
(1) Using an irreducible spherical tensor operator basis, we show that longitudinal relaxation in a three-spin system interacting with an isotropic bath can be fully described within an invariant subspace spanned by ten zero-quantum operators that, like the relaxation superoperator, are invariant under spin inversion conjugation.These basis operators correspond to the three longitudinal magnetizations and seven zero-quantum coherences.The 10 × 10 relaxation supermatrix, valid for arbitrary nuclear geometry and motional model, is obtained in analytical form.(2) Contrary to conventional wisdom, [16][17][18][19][20][21][22][23] we find that the odd spectral density contributes, via distinct correlations, to longitudinal relaxation in the dispersive regime if the three spins are geometrically or dynamically nonequivalent.For the A 2 A ′ and AA ′ A ′′ geometries with dynamically equivalent spins, we reproduce the results of Schneider, 13 but we also find that the odd spectral density influences longitudinal relaxation for the A 3 system in the presence of symmetry-breaking motions.For strong rotational anisotropy, the OSDF can enhance the integral relaxation rate by more than 25%.(3) The symmetric-top rotational diffusion model with arbitrary orientation of the principal rotation axis has not previously been investigated for multi-spin systems.
For this model, we find that longitudinal relaxation in the A 3 geometry involves up to six exponential components.In contrast, previous studies of the A 3 geometry, restricted to motional models that do not break the nuclear permutation symmetry, have found at most three relaxation components. 17(4) Chemical shifts break the nuclear permutation symmetry, thereby increasing the number of relaxation components.An inverted relaxation dispersion step is predicted at the frequency where the differential precession rate matches the relaxation rate.Above this frequency, nonsecular decoupling preferentially eliminates contributions from distinct correlations, thereby increasing the integral relaxation rate.The effect of chemical shifts disappears when the nonsecular decoupling frequency exceeds the main dispersion frequency, as is always the case for homonuclear spin systems outside the motionalnarrowing regime.

APPENDIX A: THE C XY M MATRICES
Here we show how to evaluate the coefficient matrices C XY M that, together with Eqs.
(12) and ( 23), define the relaxation supermatrix in the ISTO basis.We prove that these matrices are real-valued and we present in explicit form the 10 × 10 C XY M matrices needed to describe longitudinal relaxation in a three-spin system.
Our starting point is Eq. ( 24), we may be expressed as Using Eq. ( 19) and the single-spin ISTO conjugation relation 1 T k † q = (−1) q T k −q , we obtain for the first commutator in Eq. (A.1), with Using the general expression for single-spin ISTO commutators 2 and noting that, for spins-1/2, the tensor rank must be 0 or 1, we find where we have defined the sign function (equal to 0, +1 or −1) where (A.9) In the same way, we obtain for the second commutator in Eq. (A.1), Tr{T k q T k q } = δ kk δ q,−q (−1) q , (A.12) as follows from the orthonormality ( 21) of the single-spin ISTOs.The trace over a product of three single-spin-1/2 ISTOs is given by 2 Tr{T k q T k q T k q } = δ q ,−q−q (−1) and the trace over a product of four single-spin-1/2 ISTOs is given by 2 Tr{T k q T k q T k q T k q } = δ q+q ,−q −q (−1) k+k +k +k +q+q where the summation range, λ = 0, 1, follows from the triangular conditions on the 6j symbols. 1 We can now evaluate the nine traces in Eq. (A.11).For example, × [(2k I + 1)(2l I + 1)(2k S + 1)(2l S + 1)(2k P + 1)(2k P + 1)(2l P + 1)(2l and similar expressions for the other eight traces. It is evident from these expressions that all the coefficients C XY M,np are real-valued.This may be shown without evaluating the traces by the following arguments.According to Eqs. (A.6) -(A.9) and (A.11), C XY M,np can be expressed as a linear combination, with real-valued coefficients, of products Z I Z S Z P of single-spin traces of the form (A.30)  The desired spectral density function is now obtained by combining Eqs.(12) and (E.4), with the result J XY (ω) = [I z (S × P) + S z (I × P)] • e z a Parity under spin inversion.b Parity under spin conjugation.c Identity operators have been omitted.d e z denotes the unit vector along the z axis.

FIG. 1 .
FIG. 1. Schematic representation of the 10 × 10 relaxation supermatrix in the zero-quantum OSIC subspace, comprising three longitudinal modes (Mz) and seven ZQCs.The effects of removing distinct correlations and/or the OSDF are shown by shading supermatrix elements that then become identically zero.

FIG. 6 .R self 1 (
FIG. 6. Dispersion of the integral relaxation rate  R 1 and its self-correlation part  R self 1 (both in units of ω 2 D τ c ) for the A 3 spin system with isotropic motion.The upper panel shows the relative difference between the two rates.This article is copyrighted as indicated in the article.Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.235.27.31On: Wed, 16 Dec 2015 14:59:55

FIG. 10 .
FIG. 10.Eigenmode weights c k for an AA ′ A ′′ spin system with β I = 80 • and β S = 40 • and isotropic motion versus the reduced Larmor frequency ω 0 τ c .Only the four major eigenmodes with real weights and real eigenvalues are shown.In the top panel, the cumulative weights are shown on a linear scale so the colored areas correspond to the relative mode contributions.This article is copyrighted as indicated in the article.Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.235.27.31On: Wed, 16 Dec 2015 14:59:55

FIG. 12 .
FIG. 12. Eigenmode rates λ k (in units of ω 2 D τ R ) and weights c k versus the reduced Larmor frequency ω 0 τ R for the A 3 spin system modulated by spherical-top tumbling and internal rotation with τ R = 100 τ int .This article is copyrighted as indicated in the article.Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.235.27.31On: Wed, 16 Dec 2015 14:59:55

FIG. 14 .R self 1 (
FIG. 14. Dispersion of the integral relaxation rate  R 1 and its self-correlation part  R self 1 (both in units of ω 2 D τ R ) for the A 3 spin system modulated by spherical-top tumbling and internal rotation with τ R = 100 τ int .This article is copyrighted as indicated in the article.Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.235.27.31On: Wed, 16 Dec 2015 14:59:55

FIG. 15 .
FIG. 15.Eigenmode rates λ k (in units of ω 2 D τ 0 ) and weights c k versus the reduced Larmor frequency ω 0 τ 0 for the A 3 spin system modulated by symmetric-top rotational diffusion with γ = 10 and the principal rotation axis in the nuclear plane.This article is copyrighted as indicated in the article.Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.235.27.31On: Wed, 16 Dec 2015 14:59:55

(A. 10 )
A comparison of Eqs.(A.6) and (A.10) shows that A I , A S and A P can be obtained from A I , A S and A P in Eqs.(A.7) -(A.9) by adding a prime to all quantum numbers except M and taking the adjoint of all five single-spin ISTOs (remembering to invert their order).Combination of Eqs.(A.1), (A.6) and (A.10) yields C XY M,np = Tr{A I A I } + Tr{A I A S } + Tr{A I A P } + Tr{A S A I } + Tr{A S A S } + Tr{A S A P } + Tr{A P A I } + Tr{A P A S } + Tr{A P A P } .(A.11) Substitution from Eqs. (A.7) -(A.9) and the analogous expressions for the primed quantities shows that the traces factorize into partial traces over products of two, three or four single-spin ISTOs.The first of these is simply

3 I
A.16) S5 with n = 2, 3 or 4. Evaluating the traces in the angular momentum eigenbasis and noting that m |T k q (I)|I m = (−1) I−m (2k + 1) -valued quantity, it follows that all matrix elements C XY M,np are real-valued in the ISTO basis.From the foregoing expressions, we can obtain the 45 different 63 × 63 matrices C XY M .Because of symmetry relations (Sect.II D), we only need to compute 18 of these 45 matrices.Moreover, to describe longitudinal relaxation, we only need the first 10 × 10 block of C XY M , corresponding to the ten zero-quantum (Q = 0) basis operators (Table II) with odd spin inversion conjugation parity (Sect.II D).These 18 submatrices are given in Eqs.(A.18) -(A.35) with the spin pairs (X and Y ) indexed as follows: 1 = IS, 2 = IP and 3 = SP .

TABLE I .
Spin inversion and conjugation symmetries of operators and superoperators.

TABLE III .
Number, N λ , of exponential components in different frequency regimes for longitudinal relaxation in a three-spin system with isotropic motion.