Nuclear magnetic relaxation by the dipolar EMOR mechanism : General theory with applications to two-spin systems

In aqueous systems with immobilized macromolecules, including biological tissue, the longitudinal spin relaxation of water protons is primarily induced by exchange-mediated orientational randomization (EMOR) of intraand intermolecular magnetic dipole-dipole couplings. We have embarked on a systematic program to develop, from the stochastic Liouville equation, a general and rigorous theory that can describe relaxation by the dipolar EMOR mechanism over the full range of exchange rates, dipole coupling strengths, and Larmor frequencies. Here, we present a general theoretical framework applicable to spin systems of arbitrary size with symmetric or asymmetric exchange. So far, the dipolar EMOR theory is only available for a two-spin system with symmetric exchange. Asymmetric exchange, when the spin system is fragmented by the exchange, introduces new and unexpected phenomena. Notably, the anisotropic dipole couplings of non-exchanging spins break the axial symmetry in spin Liouville space, thereby opening up new relaxation channels in the locally anisotropic sites, including longitudinal-transverse cross relaxation. Such cross-mode relaxation operates only at low fields; at higher fields it becomes nonsecular, leading to an unusual inverted relaxation dispersion that splits the extreme-narrowing regime into two sub-regimes. The general dipolar EMOR theory is illustrated here by a detailed analysis of the asymmetric two-spin case, for which we present relaxation dispersion profiles over a wide range of conditions as well as analytical results for integral relaxation rates and time-dependent spin modes in the zero-field and motional-narrowing regimes. The general theoretical framework presented here will enable a quantitative analysis of frequency-dependent water-proton longitudinal relaxation in model systems with immobilized macromolecules and, ultimately, will provide a rigorous link between relaxation-based magnetic resonance image contrast and molecular parameters. C 2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4942026]


I. INTRODUCTION
Soft-tissue contrast in clinical magnetic resonance imaging derives largely from spatial variations in the relaxation behavior of water protons.Yet, a rigorous theory relating the water 1 H relaxation rate to microscopic parameters is still not available.The lack of theoretical underpinning is also a limitation in biophysical studies of, for example, water-protein interactions and intermittent protein dynamics by field-cycling measurements of the water 1 H magnetic relaxation dispersion (MRD) in protein gels.Previously, such data have been interpreted with semi-phenomenological models [1][2][3] involving questionable assumptions about the relaxation-inducing motions. 4,5Earlier water 1 H MRD studies of biopolymer gels from this laboratory 5,6 made use of a nonrigorous extension of the multi-spin Solomon equations to conditions outside the motional-narrowing regime.
Nuclear spins residing permanently in immobilized macromolecules give rise to solid-state type NMR spectra, whereas spins that are only transiently associated with the macromolecules, because they exchange chemically or a) bertil.halle@bpc.lu.se physically with the solvent phase, exhibit liquid-state NMR properties provided that the immobilized macromolecules are isotropically distributed so that anisotropic nuclear spin couplings are averaged to zero.In such locally anisotropic samples, exchange plays a dual role.On the one hand, exchange transfers magnetizations and coherences between macromolecule-bound spins and solvent spins.On the other hand, exchange randomizes the orientation of anisotropic nuclear interaction tensors, thereby inducing spin relaxation.For this relaxation mechanism, known as exchange-mediated orientational randomization (EMOR), the motional-narrowing regime coincides with the fast-exchange regime.For the EMOR mechanism, the conventional Bloch-Wangsness-Redfield (BWR) perturbation theory of nuclear spin relaxation 7 breaks down when, as is frequently the case, the mean survival time of the macromolecule-bound spin is comparable to, or longer than, the inverse of the anisotropic nuclear spin coupling that it experiences in the bound state.We have therefore embarked on a program to develop a general non-perturbative theory, based on the stochastic Liouville equation (SLE), 8,9 that can describe relaxation by the EMOR mechanism over the full range of exchange rates and spin coupling strengths.
The EMOR SLE theory was first developed for quadrupolar relaxation 10,11 and it has been extensively applied to water 2 H MRD studies of colloidal silica, 12 polymer gels, 13,14 cross-linked proteins, [15][16][17][18] and cells. 19As compared to quadrupolar relaxation, which only involves single spins, dipolar relaxation is theoretically more challenging.The EMOR SLE theory for dipolar relaxation of a homonuclear spin pair exchanging as a unit 20 is isomorphic with the corresponding theory for quadrupolar relaxation of a single spin-1, 11 but for heteronuclear spins, multispin (>2) systems and/or fragmentation of the spin system by exchange, qualitatively new phenomena appear in the dipolar relaxation.In a previous report, 20 hereafter referred to as Paper I, we developed the EMOR SLE theory for a (homonuclear or heteronuclear) spin pair that exchanges as an intact unit, a situation that we now refer to as symmetric exchange.Contrary to our earlier expectations, 20 the case of asymmetric exchange, where only one of the two dipole-coupled spins undergoes exchange, differs fundamentally from the symmetric case.In particular, since the non-exchanging spins are not isotropically averaged, the longitudinal and transverse magnetizations are dynamically coupled in the anisotropic sites.Such cross-mode relaxation, distinct from the cross-spin relaxation familiar from the Solomon equations, 21 gives rise to an inverted relaxation dispersion at low field.
Here, we develop the general dipolar EMOR SLE theory, valid for spin systems of arbitrary size and for symmetric as well as asymmetric exchange.To illustrate the general theory, we present explicit results for the asymmetric two-spin case, which is contrasted with the previously treated symmetric two-spin case. 20These results are directly applicable to, for example, a macromolecular hydroxyl proton in chemical exchange with water protons (asymmetric case) or to an internal water molecule in physical exchange with bulk water (symmetric case).
This paper is organized as follows.In Sec.II, we present the dipolar EMOR formalism for an arbitrary spin system, with general and two-spin results in separate subsections.As compared to Paper I, the formalism has been modified and extended in order to accommodate asymmetric exchange.In Sec.III, we discuss the zero-field regime, which is of special significance for asymmetric exchange, and the motional-narrowing regime, where we obtain explicit results for the asymmetric two-spin case that serve to rationalize the unexpected inverted relaxation dispersion.In Sec.IV, we illustrate the theory by numerical results for the two-spin case, emphasizing the new phenomena that emerge for asymmetric exchange.Further physical insight is provided by an analysis of the time evolution of the relevant spin modes.Lengthy derivations and tables are relegated to six appendices. 22

General case
We consider a system of spin-1/2 nuclei, some or all of which exchange between a solid-like anisotropic (A) state and a liquid-like bulk (B) state.The spins need not be isochronous (or even homonuclear), but the Zeeman coupling is taken to be the same in states A and B. (In any case, longitudinal relaxation is not affected by exchange-modulation of the Zeeman coupling.)The A state comprises a large number, N, of sites distinguished by their fixed orientations.Collectively, the N site orientations approximate an isotropic distribution.Each A site hosts a spin system with m A ≥ 2 mutually dipolecoupled spins.A subset (or fragment) of this spin system, comprising m B spins (with 1 ≤ m B ≤ m A ), exchanges with the B state.The exchange is said to be symmetric if m B = m A and asymmetric if m B < m A .We refer to the m B exchanging spins as labile spins and the m A − m B nonexchanging spins as nonlabile.The general theory developed here is valid without further restrictions on m A and m B .
To identify different exchange cases, we use the notation "(spins in state A)-(spins in state B)."For example, I S-I is a two-spin system with one labile spin and I SP-I S is a three-spin system with two labile spins.The I S-I case might refer to a macromolecular hydroxyl proton (I) dipole-coupled to a nearby aliphatic proton (S).The I SP-I S case might refer to the two protons (I S) of a water molecule temporarily trapped in a protein cavity, where the water protons are dipole-coupled to a nearby aliphatic proton (P).Note that both of these cases involve asymmetric exchange since the spin system is fragmented, even though no covalent bonds are broken in the latter case.We shall only consider cases where a single type of spin system is present in each state, but we note that it is straightforward to extend the theory to cases where more than one subset of spins exchange independently, possibly at different rates.
The orientations of all internuclear vectors involving at least one labile spin are taken to be instantaneously randomized upon exchange, thereby inducing dipolar relaxation.This assumption in the EMOR model is justified if the mean survival time of the labile spin(s) in the A sites is long compared to the time required for orientational randomization when the labile spin(s) has been transferred to state B. This is the case, for example, for chemical exchange of labile macromolecular protons with bulk water and for physical exchange of trapped (internal) water molecules with bulk water. 20,23We can then ignore all dipole couplings among the labile spins in state B. If so desired, the small and frequencyindependent relaxation contribution from fast modulation of dipole couplings in state B can be added to the final expression for the overall relaxation rate. 20t any time, a fraction P A of the labile spins reside in state A, while a fraction P B = 1 − P A reside in state B. The nonlabile spins are only present in state A. The general dipolar EMOR theory developed here is valid without restrictions on P A .However, some of our results are only valid in the dilute regime, where P A ≪ 1.[6]

Two-spin case
In Paper I, we analyzed the symmetric exchange case I S-I S. Here, we consider the more complicated and interesting asymmetric exchange case I S-I.In a typical application, spin I is a labile proton, e.g., in a hydroxyl Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions.Downloaded to IP: 130.235.252.19On: Thu, 25 Feb group, exchanging with bulk water protons.This is actually an I S-I 2 case, but since the I-I dipole coupling in state B plays no role (see above), the results are the same as for the I S-I case.The only difference between the I S-I and I S-I 2 cases lies in the interpretation of the I-spin fraction: The Zeeman (H Z ) and dipolar (H D ) Hamiltonians for the two-spin system are given by Eqs.(2.2) and (2.3) of Paper I, with the dipole frequency ω D defined as 3/2 times the usual dipole coupling constant, i.e., ω D ≡ (3/2) [µ 0 /(4π)] γ I γ S /r 3 I S .

General case
Formally, we can regard the total system as a mixture of N + 1 species, labeled by α = 0, 1, 2, . . ., N, with α = 0 referring to state B and α ≥ 1 to site α in state A. Thus, P 0 = P B and P α = P A /N for α ≥ 1.All spin systems belonging to a given species α have the same spin Hamiltonian H α , with To an excellent approximation, the individual spin systems can be regarded as mutually noninteracting and uncorrelated.5][26] In the absence of exchange, the spin systems associated with the N + 1 species evolve independently according to the Liouville equation The Liouville-space representation σ α of the species density operator σ α is a column vector of dimension where m α (= m A or m B ) is the number of spins in species α and −1 comes from omitting the superfluous identity basis operator.The spin density operator of the total system is represented as a column vector in a composite Liouville space 24,25 of dimension D = D B + D A N, formed as the direct sum of the spin operator spaces of the N + 1 species.Thus, An element of the D-dimensional column vector σ can be expressed in the following equivalent ways: where B n α is a member of a complete set of orthonormal spin basis operators for species α, To make full use of symmetry, we represent spin Liouville space in a basis of irreducible spherical tensor operators (ISTOs) T K Q (λ) of rank K, quantum order Q, and additional quantum numbers λ. 27 In the composite space, the N + 1 independent Liouville equations (1) can be expressed as with a block-diagonal Liouvillian supermatrix, where L α is a D α × D α matrix with elements L α n α p α and 0 is the D α × D β null matrix.

Two-spin case
Whereas D A = D B = 15 for the symmetric I S-I S case, 20 we have D A = 15 and D B = 3 for the asymmetric I S-I case.The one-spin (state B) and two-spin (state A) ISTOs are given in Appendix A of the supplementary material. 22For the I S-I case, n B refers to one of the three B-state basis operators, while n A refers to one of the 15 A-state basis operators.All these operators are normalized in the same two-spin (I S) space according to Eq. ( 4).

General case
In the presence of exchange, the composite spin density operator evolves according to the SLE The exchange superoperator W describes the transfer of one or more spins from one site to another. 24,25An exchange from site α to site β instantaneously switches the spin Hamiltonian from H α to H β .If this stochastic modulation is sufficiently frequent, it induces relaxation.For asymmetric exchange, which breaks up the spin system into fragments, A → B exchange has an additional effect: all multispin correlations within the spin system that have developed as a result of dipole couplings between labile and nonlabile spins in state A are lost. 28For symmetric exchange, where the whole spin system exchanges as an intact unit, all multispin correlations are retained even though the dipole couplings are modulated.
To describe both of these effects, we decompose the exchange superoperator as The "molecular" operators T m and K m act on the site kets |α⟩, so their composite-space supermatrix representations are block-diagonal with respect to the spin operators.These operators define the kinetic model (site-to-site transition probabilities), regardless of whether the spin system is fragmented or not.The superoperators T s and K s act on spin operators, so (as for L) their composite-space supermatrix representations are block-diagonal in the site basis.These superoperators distinguish labile from nonlabile spins and they account for decorrelation of multispin modes by exchange fragmentation of the spin system. 28The composite-space supermatrix representation of W factorizes as The first term in Eq. ( 9) describes the exchange-mediated transfer of mode p β in site β into mode n α in site α.Conversely, the second term represents transfer of mode n α in site α into mode p β in site β.
The matrix representation of the transition rate operator T m in the site basis is where π α β is the transfer probability from site β to site α.The second step in Eq. ( 10) follows from the model assumption 10,20 that direct exchange between sites belonging to state A is not allowed, so that all π α β = 0 except π 0, β ≥1 = 1 and π α ≥1,0 = 1/N.The form of the site operator K m then follows from probability conservation as 29 ⟨α| Combination of Eqs. ( 9)-( 11) yields for the four types of matrix element where we have suppressed the superfluous 0 site index for state B. The spin supermatrix elements (n α |T s | p β ) and (n α |K s | p β ) in Eq. ( 9) can be regarded as selection rules; their values are either 0 or 1.The element (n B |T s | p A ) in Eq. (12c) equals 1 if A → B exchange converts spin mode p A into spin mode n B ; otherwise it equals 0. In other words, (n B |T s | p A ) = 1 only if the mode with sequence number n B in the B-state operator basis is the same as the mode with sequence number p A in the A-state operator basis, and if the exchange transfers all spins that are involved in this mode.Thus, where ∆ T (n B , p A ) is a sum of products of Kronecker deltas for the exchange-linked modes.
where ∆ K (n A ) is a sum of Kronecker deltas over nonexchanging modes, composed exclusively of operators associated with nonlabile spins.In view of Eqs. ( 12)-( 14), the exchange supermatrix in the composite space can be expressed as where τ A and τ B are the mean survival times 30 in the two states.Furthermore, 1 B is the given by Eq. ( 13), T ′ is the D A × D B matrix transpose of T, and K is a diagonal D A × D A matrix with elements given by Eq. (14b).The elements of the matrices T and K are thus either 0 or 1.The nonzero elements of T connect labile-spin modes that are interconverted by exchange, and the vanishing diagonal elements of K correspond to A-state modes that only involve nonlabile spins.For symmetric exchange,

Two-spin case
For the symmetric I S-I S case, 20 all spins are labile so there is no exchange fragmentation.Consequently, T = K = 1, the 15 × 15 identity matrix.
For the asymmetric I S-I case, exchange interconverts the three one-spin modes in state B, n B = 1, 2, and 3 (Table S2), and the corresponding three one-spin modes in state A, p A = 1, 6, and 10 (Table S1). 22Consequently, The three one-spin modes in state A that do not involve the labile I spin are n A = 2, 7, and 11 (Table S1), 22 so Therefore, K differs from the 15 × 15 identity matrix only in that K 22 = K 77 = K 11,11 = 0.

General case
The SLE ( 7) is a differential equation involving superoperators acting in the composite Liouville space.As previously shown for quadrupolar spins 10,11 and for the symmetric dipolar I S-I S case, 20 the SLE for the EMOR model admits an exact analytical solution that only involves spin superoperators, without explicit reference to site operators.Because we now develop the general dipolar EMOR theory using a somewhat modified formalism, the full derivation of the analytical SLE solution is presented in Appendix B. 22 In this subsection, we merely display the definitions that are needed for the following development.
A Laplace transform, converts the SLE (7) into an algebraic equation with formal solution where E is the identity superoperator in composite space and  U (s) is referred to as the resolvent superoperator.Macroscopic spin observables are related to a density operator summed over all sites, where σ α (t) = ⟨α|σ(t)⟩ and σ B (t) ≡ σ 0 (t).Combination of Eqs.(19) and (20) yields In Paper I, σ(t) referred to a single spin system and to obtain ⟨σ(t)⟩ we then had to weight with the relative population, P α , of spin systems in different sites α.Now, this relative weight is subsumed into σ(t) so that ⟨σ(t)⟩ is simply a sum over sites, as in Eq. ( 20), without the need to account explicitly for the fact that the macroscopic sample contains different numbers of spin systems in A and B sites.According to this new convention, the initial density operator σ α (0) is proportional to the relative population, P α , for excitation under high-field conditions or by a fast field switch. 10We express this as so that Eq. (20) yields Here, η A and η B are spin operators that depend on the initial condition of the spin system (selective or nonselective excitation) and, for heteronuclear spin systems, on the relative magnetogyric ratios (Sec.II D 2).
Combination of Eqs. ( 21) and (22) yields where η β equals η B for β = 0 and η A for β ≥ 1 and P β equals P B for β = 0 and P A /N for β ≥ 1.For asymmetric exchange, when different spin operator bases are used for states A and B, it is convenient to express the spin operator basis representation of Eq. ( 24) in terms of partitioned matrices, with the column vectors ′ (the prime denotes transposition).In Eq. ( 25), the site-averaged density operator column vector has been partitioned into (for notational simplicity, we omit the angular brackets) Without further approximations, we show in Appendix B 22 that where and Here L Z and L D (Ω) are the Liouvillian supermatrices (in the appropriate spin basis) corresponding to the Zeeman and dipolar Hamiltonians, respectively.Because G A is isotropically averaged, it must reflect the axial symmetry of the spin system in the external magnetic field.For a basis of ISTOs T K Q (λ), it then follows from the Wigner-Eckart theorem 27 that G A and the resolvent submatrices in Eq. ( 28) are block-diagonal in the projection index Q.(For asymmetric exchange, where D B < D A so that the matrices  U BA and  U AB are rectangular, also the "Q-blocks" are rectangular.)Longitudinal relaxation can therefore be fully described within the zero-quantum subspace.The dimension of this subspace is 5 for two spins and 19 for three spins.However, in the individual sites, the zero-quantum modes do not evolve independently of the remaining modes.The matrix within square brackets in Eq. ( 30) is not block-diagonal in Q so it must be evaluated in the full D A -dimensional spin Liouville space of state A.

Two-spin case
With the spin operators indexed as in Tables S1 and S2 for the I S-I case, 22 the elements of the initial-condition vectors η B and η A in Eq. ( 25) are for nonselective excitation For I-selective excitation, we have instead while for S-selective excitation Here, κ ≡ γ S /γ I , the ratio of the magnetogyric ratios.For the I S-I S case, η A n is the same as for the I S-I case and η B n = η A n for all three excitation modes.In Paper I, we used a slightly different notation.

General case
The integral longitudinal relaxation rate is defined as the inverse of the time integral of the reduced longitudinal magnetization and it may be expressed in terms of spin density operator components as where the sums run over spin modes (or basis operators) corresponding to the longitudinal magnetization of the observed spin(s) and Eq. ( 23) was used to obtain the last form.In the following, we specify the observed spin by a subscript, e.g.,  R 1, I .According to Eq. ( 25), where we have introduced the shorthand notation The values of η B p B and η A p A depend on the initial conditions for the relaxation experiment, as exemplified by Eqs. ( 31)-( 33) for the two-spin case.In field-cycling experiments, excitation is always nonselective.In conventional relaxation experiments, selective excitation of the labile spins can be accomplished with a soft RF pulse if the nonlabile spins have a much wider NMR spectrum than the labile spins.For selective excitation, we only consider the case where the excited spin is also observed.The excitation mode is indicated by a superscript, e.g.,  R non 1, I or  R sel 1, I .In the dilute regime, where P A ≪ 1, the detailed balance condition 11 P A τ B = P B τ A and Eq.(28) show that the matrix elements U XY n p are of the following orders of magnitude: In the dilute regime, we only need to retain the matrix elements of leading order in P A in the denominator of Eq. ( 34).The condition P A ≪ 1 is specified by a superscript, e.g.,  R dil 1, I .

Two-spin case
For the I S-I case, with the spin modes indexed as in Tables S1 and S2, 22 Eq.( 34) yields Using Eqs. ( 31)-( 33), (35), and (38) and noting that P A + P B = 1, we obtain for nonselective excitation and for selective excitation In the dilute regime, Eq. ( 37) allows us to reduce these expressions to In Eq. (41a), we only display the superscript "dil" since Eq. ( 30).This reduction is outlined in Appendix C 1; 22 here we merely quote the results for the rates in Eqs.(40b) and (41), with the shorthand notation As expected,  R sel 1, S is independent of P A , whereas in the dilute regime, the nonselective rates are rigorously proportional to P A .
The corresponding expressions for the integral relaxation rate in the I S-I S case, most of which are given in Paper I, are readily obtained in the same manner.For convenience, these expressions are collected in Appendix C 2 22 with the same notation as used here for the I S-I case.

General case
In the absence of an external magnetic field, the macroscopic system is rotationally invariant (isotropic).The Wigner-Eckart theorem 27 then implies that supermatrices that are averaged over the A sites are block-diagonal in the rank index K as well as in the projection index Q if they are represented in the ISTO basis T K Q (λ) ≡ B n .Furthermore, the nonzero matrix elements do not depend on Q.For example, We refer to the conditions under which this selection rule is valid as the zero-field (ZF) regime.In the ZF regime, the Larmor frequencies are much smaller than the rate of evolution induced by the dipole coupling, so we can set L Z ≡ 0.
In Paper I, we defined a low-field (LF) regime through the inequality where ω D is the dipole coupling frequency, as defined in Sec.II A 2, and ω I is the Larmor frequency.In the ultraslowmotion regime, where , that is, the Larmor precession is much slower than the coherent dipolar evolution.In the motional-narrowing (MN) regime, where (ω D τ A ) 2 ≪ 1, inequality (45) implies that (ω I τ A ) 2 ≪ 1, which is the so-called extreme-narrowing condition.Physically, extreme narrowing corresponds to a situation where the local field (produced by the dipole coupling) is randomized by exchange (on the time scale τ A ) before any significant Larmor precession has taken place.
For symmetric exchange, such as the I S-I S case treated in Paper I, the LF condition (45) also defines the ZF regime.In other words, the integral relaxation rate is independent of ω I in the frequency range defined by inequality (45).In contrast, for asymmetric exchange, such as the I S-I case, the Zeeman coupling can be neglected only if the Larmor precession is slow compared to the cross-mode relaxation in the A sites (Sec.III B 2).For asymmetric exchange, the ZF regime is therefore defined by the more restrictive inequality Figure 1 depicts the LF and ZF regimes for the case of asymmetric exchange, as defined by inequalities ( 45) and ( 46).In the MN regime (the lower half of Fig. 1), the ZF regime corresponds to |ω In the ultraslow-motion regime (the upper half of Fig. 1), the ZF regime corresponds to The blue curve in Fig. 1, at the boundary between the LF and adiabatic regimes, indicates the frequency of the main dispersion step for the integral longitudinal relaxation rate.The red curve, at the boundary between the ZF and LF regimes, indicates the frequency of an inverted dispersion step that only appears for asymmetric exchange (Sec.III B 2).
In the ZF regime, selection rule (44) implies that G A and other site-averaged supermatrices consist of (m A + 1) 2 blocks.For a given rank K, the 2K + 1 blocks corresponding to different values of the projection index Q are identical.Furthermore, all blocks are symmetric.In the ZF regime, the evolution of the longitudinal magnetizations can therefore be fully described within the rank-1 zero-quantum subspace.
FIG. 1.For asymmetric exchange, the LF and ZF regimes are distinct, as shown here.For symmetric exchange, the coincident LF and ZF regimes both extend up to the blue boundary.

Two-spin case
For two-spin (I S) systems in the ZF regime, selection rule (44) implies that G A has one diagonal element corresponding to the rank-0 singlet operator T 0 0 (11), three identical 3 × 3 rank-1 blocks spanned by the operators 10), ( 01) or (11), and five identical rank-2 diagonal elements corresponding to T 2 Q (11).Since the matrix is symmetric, there are only 9 unique elements.The evolution of the longitudinal magnetizations in state A is fully described by the T 1 0 (k I k S ) block, spanned by the basis operators (Table S1) 22 Since longitudinal relaxation in the ZF regime can be fully described within the rank-1 zero-quantum subspace, many results can be obtained in analytical form.For example, in Appendix D, 22 we derive, for the I S-I case, closed-form expressions for  σ A 1 (s),  σ A 2 (s), and  σ A 4 (s), from which the time evolution of these spin modes can be obtained by an inverse Laplace transformation.In Appendix D, 22 we also derive expressions for the integral relaxation rate in Eqs. ( 39) and (40).For example, In the MN regime, Eq. (48a) can be written as This is as expected, because, for the EMOR model, the MN regime is also the fast-exchange regime.If relaxation in the A-sites were governed by an internal motion independent of the exchange kinetics, then the Luz-Meiboom equation, R 1 = P A /(τ A + 1/R A 1 ), should hold in the dilute + MN regime. 31However, if Eq. (48a) is cast on this form, we find that ], which agrees with the foregoing expression only in the MN (fast-exchange) regime.This inconsistency arises because the Luz-Meiboom equation is not valid for the EMOR model, where the intrinsic relaxation is induced by the exchange itself so that the MN condition (ω D τ A ) 2 ≪ 1 is automatically violated as soon as we leave the fast-exchange regime.

General case
In the MN regime, where (ω D τ A ) 2 ≪ 1, the composite spin density operator evolves according to the "stochastic Redfield equation" (SRE) where W is same exchange superoperator (8) as in SLE (7) and R is the relaxation superoperator prescribed by the BWR perturbation theory. 7The Liouvillian L 0 is associated with the time-independent part of the spin Hamiltonian that is unaffected by the exchange process.For symmetric exchange in general and for asymmetric exchange with only one nonlabile spin (m A = m B + 1), the static Hamiltonian only contains the Zeeman coupling so L 0 = L Z .For example, this is true for the I S-I S and I S-I cases.However, for asymmetric exchange with m A ≥ m B + 2 (e.g., I SP-I), so the A sites contain at least two nonlabile spins, L 0 includes the static dipole coupling(s) besides the Zeeman coupling.
Like SLE (7), SRE (49) can be solved in site space, in full analogy with the treatment in Appendix B. 22 Specifically, Eqs. ( 25) and ( 28) remain valid, but Eq. ( 30) is replaced by where the angular brackets indicate the same isotropic orientational average as in Eq. ( 30) and R α is the orientationdependent relaxation supermatrix for site α.As noted in Sec.II D 1, the isotropically averaged supermatrix G A (s) is block-diagonal in the projection index Q.To determine the integral longitudinal relaxation rate, as described in Sec.II E, we only need the Q = 0 block of the supermatrix with In the EMOR model, exchange plays two roles: it transfers spin modes between the A and B states and it induces relaxation by randomizing the orientation of the dipole vector(s) in the A sites.In SLE (7), both of these roles are played by the exchange superoperator W. In the SRE (49), on the other hand, the first role is played by the exchange superoperator W, which describes the transfer (or decorrelation) of local spin modes σ α n (t), while the second role is played by the orientation-dependent relaxation superoperator R α , which describes relaxation induced by orientational randomization of the dipole vector(s) in site α.Because of the dual role played by exchange in the EMOR model, the MN condition (ω D τ A ) 2 ≪ 1 not only ensures that the BWR theory is valid, but it also corresponds to the fastexchange limit of the SRE because To derive the integral relaxation rate in the MN regime from SRE (49), we must therefore implement the MN condition twice: first in obtaining the relaxation supermatrix R α from the BWR theory 7 and then in implementing the fast-exchange limit by expanding the matrix inverse (Λ α ) −1 to first order in ∥R α ∥ τ A , that is, to second order in ω D τ A .
][34] As noted above, rotational symmetry ensures that we only need to consider the Q = 0 block of the isotropically averaged supermatrix G A (0).In contrast, the relaxation supermatrix R α in Eq. ( 52) pertains to a site α with a particular orientation so it is not block-diagonal in Q.3][34] According Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions.Downloaded to IP: 130.235.252.19On: Thu, 25 Feb to the basic orthogonality theorem of group theory, 35 the supermatrices i L Z and R α in the ISTO basis can then have nonzero elements only between basis operators of the same parity.If we order the basis operators so that the odd operators (including the single-spin longitudinal operators) precede the even operators, then the supermatrices i L Z and R α are blockdiagonal.For exchange cases with less than two nonlabile spins, so that L α 0 = L Z , it then follows (since matrix inversion does not alter the block structure and since K is diagonal) that also the supermatrix Λ α in Eq. ( 52) is block-diagonal.As long as we are concerned with longitudinal relaxation, we therefore need to consider only the odd-parity zero-quantum subblock of the G A (0) supermatrix.This partitioning of the Q = 0 subspace on the basis of SIC parity is helpful also for exchange cases with two or more nonlabile spins, even though the odd and even subblocks are then coupled because the superoperator i L D associated with the static dipole coupling(s) has odd SIC parity. 34

Two-spin case
If we reorder the 15 basis operators in Table S1 22 so that the six odd-parity (single-spin) operators (including I z and S z ) precede the nine even-parity (two-spin) operators, the supermatrix Λ α in Eq. ( 52) is block-diagonal.Longitudinal relaxation is fully described by the odd 6 × 6 block.Ordering the odd basis operators as {I z , I + , I − , S z , S + , S − }, we can then partition Λ α into 3 × 3 submatrices as where, for the I S-I case, Here, 1 is the 3 × 3 identity matrix and Q is a diagonal matrix with diagonal elements [0, 1, −1].To obtain Eq. (54), we also noted that, according to Eq. ( 17), K I I = 1 and For the elements of the four relaxation submatrices, given explicitly in Appendix E, 22 we use a notation where, e.g., ).These local (orientation-dependent) relaxation rates are of four kinds.First, there are longitudinal (R I I z z and R S S z z ) and transverse (R I I ±± and R SS ±± ) auto-spin auto-mode rates.Second, there are longitudinal (R I S z z and R S I z z ) and transverse (R I S ±± and R S I ±± ) cross-spin auto-mode rates.Auto-spin and cross-spin rates also occur in the two-spin Solomon equations, 7 but they are then isotropically averaged.Third, there are the auto-spin cross-mode rates R I I z± , R I I ±z , and R I I ±∓ , and the corresponding rates for spin S. Finally, there are cross-spin cross-mode rates, like R I S z± .All these rates pertain to a particular site α and they therefore depend on the orientation of the dipole vector in that site as detailed in Appendix E. 22 The cross-mode rates couple the longitudinal and transverse magnetizations of the same or different spins.This coupling is a consequence of the spatial anisotropy of the A sites.Indeed, we show in Appendix E 22 that all cross-mode rates vanish after isotropic averaging.As shown below, such averaging occurs in the symmetric I S-I S case, where the same I-S pair exchanges rapidly among the A sites (via the B site), but not in the asymmetric I S-I case, where the two spins are no longer correlated after the exchange.As shown explicitly in Appendix E, 22 the cross-mode rates are only effective in the ZF regime, as defined by Eq. ( 46).At higher fields, they become nonsecular, that is, the longitudinal and transverse magnetizations are decoupled by the Larmor precession, which then is much faster than the (local) crossmode relaxation.
All the local relaxation rates, except for the longitudinal auto-mode rates R I I z z , R S S z z , and R I S z z = R S I z z , involve both the even (real) and the odd (imaginary) parts of the complex spectral density function (Appendix E 22 ).However, the odd spectral density function (OSDF) has no effect on longitudinal relaxation in the two-spin cases.In the I S-I S case, isotropic averaging cancels the cross-mode rates so only the longitudinal auto-mode rates are relevant.In the I S-I case, the cross-mode rates are only effective in the ZF regime, where the OSDF is negligibly small compared to the real part.So, in either case, the OSDF only affects the evolution of the transverse spin modes, giving rise to the well-known second-order dynamic frequency shift. 36However, for larger spin systems, the OSDF can also affect the longitudinal modes, e.g., in the I SP-I SP case. 34eturning now to the derivation of the integral relaxation rate for the I S-I case, we invert the partitioned matrix in Eq. (53), obtaining for the I I block where Eq. (54) was used in the second step.Here we have also defined the diagonal matrix and the "cross relaxation" matrix We now expand the inverse in Eq. ( 55) to second order in ω D τ A and perform the isotropic site average, to obtain We are primarily interested in the I-spin integral relaxation rate in the dilute regime.This rate only involves the matrix element Combination of Eqs.(42a) and ( 56)-(59) yields where . The first term within square brackets in Eq. ( 60) is the well-known 7,21 longitudinal auto-spin relaxation rate, averaged over the isotropic distribution of A sites (Appendix E 22 ), Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions.Downloaded to IP: 130.235.252.19On: Thu, 25 Feb with the reduced spectral density function given by (62) The second term in Eq. ( 60) accounts for cross-spin as well as cross-mode relaxation.Cross-mode relaxation only comes into play in the ZF regime, where all relaxation channels are secular, meaning that all oscillating factors in Eq. (E.28) 22 can be replaced by unity.According to Eq. ( 57 ±z , all of which are multiplied by oscillating factors exp(± i ω S t) in Eq. (E.28). 22Two conclusions follow.First, the longitudinal-transverse cross-mode relaxation contribution to  R dil 1, I in the zero-field regime involves the four crossspin cross-mode rates R I S z± and R S I ±z and the four auto-spin-S cross-mode rates R SS z± and R S S ±z , but it does not involve the four auto-spin-I cross-mode rates R I I z± and R I I ±z .Second, for a heteronuclear spin pair, with ω S ω I , the entire cross- Outside the ZF regime, cross-mode relaxation is nonsecular and can therefore be neglected.The four relaxation submatrices are then diagonal so Eqs.( 57) and (60) yield where the orientation-dependent auto-spin and cross-spin rates in the second term are given in Appendix E. 22 In the LF + MN regime, which is also the extreme-narrowing regime (Sec.III A 1), these rates are given by Eqs.(E.35a) and (E.35b). 22ubstitution into Eq.(63) yields after isotropic averaging with the numerical constant Result ( 63) is not valid in the ZF regime, where there is cross-mode coupling.In the ZF + MN regime, Eq. (48a) reduces to which is a factor 2.008 343 . . .smaller than the result in Eq. (64).It is clear, therefore, that longitudinal-transverse cross-mode coupling slows down the longitudinal relaxation of the labile I-spin.As the Larmor frequency increases from below ω 2 D τ A to above this value, the integral relaxation rate  R dil 1, I thus exhibits an inverted dispersion step.The locus of this dispersion step is indicated by the red curve in Fig. 1 (the boundary between the ZF and LF regimes).
It is instructive to contrast these results for the asymmetric I S-I case with the corresponding results for the symmetric I S-I S case. 20For the symmetric case, Eq. (41a) is replaced by (Appendix C 2 22 ) depending on whether we observe spin I only or both spins.Spin Liouville space is now spanned by the same 15 basis operators (Table S1) 22 for both states A and B. Furthermore, T = K = 1 and Eqs.(28a) and ( 50) yield (in the dilute regime) and where we have invoked the MN approximation by expanding G A (0) to second order in ω D τ A .In contrast to the asymmetric exchange case, relaxation now enters only via the isotropically averaged relaxation supermatrix ⟨R α ⟩.This has two important consequences.First, all cross-mode relaxation rates vanish (Appendix E 22 ).Second, because the relaxation supermatrix is now isotropically averaged, we can invoke the Wigner-Eckart theorem to establish that ⟨R α ⟩ is block-diagonal in Q.
To describe longitudinal relaxation, we therefore only need to consider the 5 × 5 Q = 0 block.Moreover, because R α is invariant under SIC, this block decomposes into an odd-parity 2 × 2 block (spanned by I z and S z ) and an even-parity 3 × 3 block (spanned by the basis operators B 3 , B 4 , and B 5 in Table S1). 22Although the Q = 0 block of L Z is not diagonal for ω I ω S , we only need the odd-parity sub-block, which is the 2 × 2 null matrix.We thus obtain from Eqs. ( 68) and (69), in agreement with the results (using a slightly different notation) of Paper I. In particular, for symmetric exchange of a pair of homonuclear (κ = γ S /γ I = 1) and isochronous (ω I = ω S ) spins, both rates in Eq. ( 71) reduce to the familiar form In the MN regime, rotational and SIC symmetries ensure that longitudinal relaxation can be fully described within Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions.Downloaded to IP: 130.235.252.19On: Thu, 25 Feb the two-dimensional zero-quantum odd-parity subspace corresponding to the longitudinal spin modes I z and S z .This is true for symmetric as well as for asymmetric exchange.The crucial difference between these exchange cases in the MN regime is that the intrinsic relaxation rates in the A sites are isotropically averaged only for the symmetric I S-I S case.For the asymmetric I S-I case, the orientation-dependent relaxation in the A sites must, in general, be described in the six-dimensional odd-parity subspace corresponding to the single-spin longitudinal and transverse local modes.However, outside the ZF regime, the longitudinal local spin modes I α z and S α z are decoupled from the transverse local spin modes I α ± and S α ± .

IV. NUMERICAL RESULTS FOR TWO-SPIN SYSTEMS
In this section, we illustrate the theoretical results obtained in Secs.II and III by numerical calculations.Except in Sec.IV B, we consider a homonuclear and effectively isochronous (ω S = ω I ) spin pair.The dipole coupling frequency is set to ω D = 1 × 10 5 rad s −1 , corresponding to an internuclear separation of r I S = 2.245 Å for two protons, and the fraction bound I spins is P A = 10 −3 , corresponding to the dilute regime.Following the standard convention, we take ω I to be positive.We focus on the asymmetric I S-I case, highlighting differences compared to the symmetric I S-I S case.

A. Cross-mode relaxation
Figure 2 shows dispersion profiles  R dil 1, I (ω I ) for the integral longitudinal relaxation rate of spin I in the dilute regime for four different values of the mean survival time τ A in the A sites.As expected, the SLE and BWR results, computed from Eqs. (42a) and (60), respectively, coincide in panels (a) and (b), where (ω D τ A ) 2 ≪ 1.In panel (c), where ω D τ A = 1, the approximate BWR rate exceeds the exact SLE rate by 20% in the ZF limit.In panel (d), where (ω D τ A ) 2 = 100, the corresponding discrepancy is a factor of ∼21.
The inverted and normal dispersion steps in panels (a) and (b) are centered at the frequencies ω I = ω 2 D τ A and ω I = 1/τ A , respectively, corresponding to the red and blue curves, respectively, in Fig. 1.The slower relaxation in the ZF regime is a consequence of longitudinal-transverse crossmode relaxation in the anisotropic A sites (Sec.III B 2).In the LF regime (and above), cross-mode relaxation is abolished by nonsecular decoupling.The ZF drop in  R dil 1, I (ω I ) is close to 50% in the MN regime (Sec.III B 2), but it becomes less pronounced in the slow-motion regime.The ZF rate  R dil 1, I (0) agrees with Eq. (48a) at all τ A values and in panel (d) it is already within 5% of the ultraslow-motion limit  R dil 1, I (0) = (2/3) P A /τ A .In accordance with the frequencydependent MN condition (ω D τ A ) 2 ≪ 1 + (ω I τ A ) 2 for isochronous spins, the BWR and SLE results coincide in the highfrequency part of the main dispersion in panels (c) and (d).As seen from panel (d), the SLE profile merges with secular BWR result (63), without cross-mode relaxation, at high frequency.(In panels (c) and (d), the BWR and secular BWR profiles cross over just below the main dispersion.)In the slow-motion regime, the effect of cross-mode coupling on the exact SLE profile is only evident as a distortion of the main dispersion shape.As τ A is increased further,  R dil 1, I (0) decreases as 1/τ A in accordance with Eq. (48a), but the locus of the main

B. Heteronuclear spins
Although our focus is on homonuclear spin systems, all results in Secs.II and III are valid also for heteronuclear spin systems.To illustrate the different relaxation behaviors of homonuclear and heteronuclear spin systems, we shall compare, for the I S-I case, the dispersion of the integral relaxation rate  R dil 1, I (ω I ) for a homonuclear spin pair (ω S = ω I ) with that for a heteronuclear spin pair with ω S = − 0.1014 ω I .The homonuclear case might represent a serine side-chain with the I spin in the labile hydroxyl proton and the S spin in one of the adjacent methylene protons (chemical shifts have no significant effect).The heteronuclear case might represent an 15 N-labeled lysine side-chain, with the I spin in one of the labile amino protons and the S spin in the directly bonded nitrogen atom so that κ = γ S /γ I = −0.1014.In both cases, we excite nonselectively but observe only the I spin.For simplicity, the dipole coupling ω D is taken to be the same for the two cases.
In Fig. 3, the homonuclear dispersion profiles (red) are compared with the heteronuclear profiles (blue) for the same parameter values as in Fig. 2. The red profiles are thus the same in the two figures.In the MN regime (panels (a) and (b)), the principal difference between the two profiles is that the inverted dispersion occurs at a 10-fold higher frequency in the heteronuclear profile.This observation is consistent with our earlier conclusion (Sec.III B 2) that the effect of cross-mode relaxation vanishes when ω S exceeds ω 2 D τ A , that is, when ω I 10 ω 2 D τ A .Consequently, the inverted dispersion in the heteronuclear profile has only one step (that is, there is no dispersion step at ω I ≈ ω 2 D τ A ) and it has the same shape as in the homonuclear profile.In addition to the shift of the inverted dispersion, there is also a small difference in the main (normal) dispersion step, where differences in the spectral densities j(ω S ), j(ω I − ω S ), and j(ω I + ω S ) result in a slightly steeper dispersion in the heteronuclear profile.As a result, the heteronuclear profile lies ∼15% below the homonuclear profile at the high-frequency end of the main dispersion (but this is not visible on the scale of Fig. 3).
Outside the MN regime (panels (c) and (d)), the effects of having ω S ω I are more complicated.For ω D τ A = 1 (panel (c)), the prominent maximum in the homonuclear profile is replaced by a small shoulder in the heteronuclear profile.For ω D τ A = 10 (panel (d)), the heteronuclear profile has a downshifted and more nearly "Lorentzian" main dispersion and a distinct high-frequency dispersion step.In all four panels, the heteronuclear profile lies ∼15% below the homonuclear profile at the high-frequency end of the dispersion profile.
In Fig. 3, we also examine the effect of the relative sign of the magnetogyric ratios γ I and γ S by comparing dispersion profiles for γ S = − 0.1014 γ I (blue solid curves), corresponding to 1 H− 15 N, and γ S = +0.1014γ I (magenta dashed curves).The sign of γ S can affect the spin dynamics in three ways: via the Larmor frequency ω S = −γ S B 0 , via the dipole coupling frequency ω D ∝ γ I γ S , and via the factor κ = γ S /γ I in the initial condition, reflecting the equilibrium magnetization.Following standard practice, we regard ω I as positive, even though γ I > 0 in the most relevant situation, where I refers to the proton spin.
The dispersion profiles in Fig. 3 pertain to the dilute regime and nonselective excitation, so the initial FIG. 3. Dispersion of the integral longitudinal relaxation rate of spin I for exchange case I S-I and homonuclear spins with ω S = ω I (red solid curves, same as in Fig. 2) and heteronuclear spins with ω S = −0.1014ω I (blue solid curves) or ω S = 0.1014 ω I (magenta dashed curves).Other parameter values as in Fig. 2. All dispersion profiles were computed from the SLE result in Eq. (42a).
Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions.Downloaded to IP: 130.235.252.19On: Thu, 25 Feb I-spin magnetization is strongly dominated by the equilibrium magnetization in state B. Consequently (the sign of) κ has no significant effect on the dispersion profiles.
In the MN regime (panels (a) and (b)), the spin dynamics only depend on the square of ω D so the only effect of reversing sign of γ S is to interchange the spectral densities j(ω I − ω S ) and j(ω I + ω S ). 37As seen from panels (a) and (b), this effect is rather small; the maximum effect (barely visible in Fig. 3) occurs in the adiabatic regime (ω I τ A ≫ 1), where  R dil 1, I (ω I ) is ∼14% smaller for γ S = +0.1014γ I than for γ S = −0.1014γ I .
Outside the MN regime (panels (c) and (d)), a sign reversal in γ S makes the dispersion profile more smooth, without any pronounced shoulder.As in the MN regime, this effect is entirely due to the sign reversal of ω S .Although the evolution of some spin modes depends on the sign of ω D (for example, see Eqs. (D.9f) and (D.18) 22 ), the evolution of the longitudinal magnetizations only involves even powers of ω D .As in the MN regime,  R dil 1, I (ω I ) is ∼14% smaller for γ S = + 0.1014 γ I than for γ S = − 0.1014 γ I at the high-frequency end of the dispersion profile.

C. Symmetric versus asymmetric exchange
In Fig. 4, we compare the dispersion profiles for the I S-I and I S-I S cases with the same parameter values as in Figs. 2  and 3.The red I S-I profiles are thus the same in Figs.2-4.Because of isotropic averaging (Appendix E 22 ), there is no cross-mode relaxation in the A sites for the symmetric case so the integral relaxation rate is constant throughout the extreme-narrowing regime (no inverted dispersion at the boundary between the ZF and LF regimes).In addition, the dispersion midpoint occurs at a lower frequency for the symmetric case.
The numerical calculations confirm the analytical prediction, based on Eq. (48a) and Eqs.(5.2) and (5.3) of Paper I, that the ZF rate ] larger for the I S-I S case than for the I S-I case.In the MN regime (panel (a)), the ratio  R dil 1, I (I S-I S)/  R dil 1, I (I S-I) is thus 5 in the ZF regime, while it is ∼2.5 in the LF regime.In the ultraslow-motion regime (ω D τ A ) 2 ≫ 1 (panel (d)), the ZF rates for the symmetric and asymmetric exchange cases converge to the same value,  R dil 1, I (0) = (2/3) P A /τ A , but the dispersion for the asymmetric case is upshifted in frequency and deviates more from "Lorentzian" shape.
For the comparison in Fig. 4, we use the same P A value for the two exchange cases.If we want to compare the contributions to the observed bulk water proton relaxation rate from a single labile proton (I) and from the two protons in an internal water molecule (I 2 ), we should compare the exchange cases I S-I 2 and I 2 − I 2 .As noted in Sec.II A 2, we should then (in the dilute regime) divide P A by a factor 2 for the asymmetric I S-I 2 case.Consequently, in the MN regime, an internal water molecule contributes 10-fold more than a labile proton to  R dil 1, I (0) and ∼5-fold more in the LF regime, other things (notably, τ A and ω D ) being equal.However, the mean survival time τ A is typically longer for labile protons than for internal water molecules. 16 to larger τ A will be more pronounced because ω D is generally smaller for a labile proton in a macromolecule than for the water protons.In the ultraslow-motion regime, (ω D τ A ) 2 ≫ 1, the symmetric and asymmetric cases yield the same ZF rate,  R dil 1, I (0) = (2/3) P A /τ A (black dashed-dotted curve in Fig. 5).

D. Time evolution of spin modes
Further physical insight can be obtained by analyzing the evolution in time of the longitudinal magnetization and other spin modes.We shall perform such an analysis for the ZF regime, where longitudinal relaxation can be fully described in terms of the four rank-1 zero-quantum spin modes σ B 1 (t) , and σ A 4 (t) (Sec.III A 2).Here we use a simplified notation where, for example, I B z (t) ≡ Tr{I z σ B (t)}.Using Eqs.(D.9) and (D.18) 22 and performing the inverse Laplace transform, we obtain the results shown in Fig. 6 for the I S-I case with nonselective excitation.As before, we use P A = 10 −3 so we are in the dilute regime, where As seen from Fig. 6, to an excellent approximation, with the integral relaxation rate  R dil 1, I (0) given by Eq. (48a).The total I-spin longitudinal magnetization is thus very nearly exponential even outside the MN regime.Consequently, very little information is lost in charactering the decay of I z (t) by the integral relaxation rate  R dil 1, I (0). Figure 6(a), with (ω D τ A ) 2 = 10 −4 , pertains to the MN regime, where the longitudinal magnetizations are decoupled from modes with even SIC parity so σ A 4 (t) ≡ 0. Because the MN regime is also the fast-exchange regime, the ratio of the longitudinal magnetizations in the two states is maintained at the initial value throughout the relaxation process, FIG. 6.Time evolution of the normalized spin modes I B z (t)/P B (red solid curve), I A z (t)/P A (blue), S A z (t)/P A (magenta), and σ A 4 (t)/P A (green) for the I S-I case in the zero-field regime and with nonselective excitation.The black dashed curve coinciding with I B z (t)/P B in all three panels is the exponential decay in Eq. ( 73 The time evolution of the longitudinal modes can then be described by two coupled equations (Appendix F 22 ) For P A = 1, Eq. ( 75) is of the same form as the twospin Solomon equations. 7,21However, whereas ρ I = ρ S and σ I S = σ S I in the Solomon equations in the ZF regime, this symmetry is broken in the asymmetric I S-I model.In Appendix F, 22 we show that The solution to Eq. ( 75) subject to nonselective initial conditions, I z (0) = 1 and S A z (0) = P A , is then (Appendix F 22 ) with  R dil 1, I (0) given by Eq. (48a).The validity of this analytical result is numerically verified in Fig. 6(a).
With the aid of these analytical results, the time evolution in Fig. 6(a) can be understood in detail.On a short time scale, of order (ω 2 D τ A ) −1 , the total I z magnetization hardly changes since only a tiny fraction P A ≪ 1 of the labile I spins is relaxed by dipole coupling to an S spin.Meanwhile, the S A z magnetization of the nonlabile S spin relaxes at a rate ρ S , but, because of cross relaxation with spin I, it does not approach its equilibrium value (which, by our convention, is zero) but a non-equilibrium steady-state value (corresponding to a negative spin temperature).The steady-state magnetization is obtained from Eq. (75b) by setting I z (t) = 1 and dS A z (t)/dt = 0, whereby S A z (t)/S A z (0) = −σ S I /ρ S = −1/5.On a longer time scale, of order (P A ω 2 D τ A ) −1 , the I-spin and S-spin magnetizations relax to their (zero) equilibrium values at a common rate  R dil 1, I (0).By inserting the steadystate S A z magnetization, −(1/5) P A I z (t), in Eq. (75a), we see that this rate is given by P A (ρ I − σ I S /5) = P A [(2/9) − (4/45)] ω 2 D τ A = (2/15) P A ω 2 D τ A , consistent with Eqs.(48a), (76a), and (76c).The net magnetization flux due to cross relaxation is thus from the S spin to the I spin, thereby retarding I-spin relaxation.In contrast, for the symmetric I S-I S case, a similar analysis shows that the I-spin and S-spin magnetizations relax at a common rate P A (ρ The 5-fold slower I-spin relaxation in the ZF + MN regime for asymmetric exchange as compared to symmetric exchange (Fig. 4(a)) is thus seen to be a consequence of a smaller auto-spin rate ρ I (because of longitudinal-transverse cross-mode relaxation in the A sites) and a reversed magnetization flow from the negative steady-state S A z magnetization.The latter effect should operate also outside the ZF regime, when local cross-mode relaxation does not occur.
The inverse of the S-spin integral relaxation rate  R non 1, S (0) equals the time integral of the expression within brackets in Eq. (77b).Noting that ρ S ≫  R dil 1, I (0) in the dilute regime, we thus obtain in agreement with Eq. (D.22b). 22utside the MN regime (Figs.6(b) and 6(c)), when the mean survival time τ A in an A site is no longer short compared to the time scale 1/ω D of coherent dipolar evolution, all three A-state modes show oscillatory features superimposed on the initial decay, which occurs on the time scale of τ A when (ω D τ A ) 2 ≫ 1.However, as long as we are in the dilute regime, the I B z magnetization still relaxes exponentially as in Eq. ( 73).Because the SIC parity selection rule does not apply outside the MN regime, the odd-parity longitudinal magnetizations couple with the even-parity zero-quantum coherence σ A 4 (t), which builds up when I A z and S A z start to decay and ultimately decays on the same time scale as I B z .As seen from Eq. (47c) or Eqs.(D.9d) and (D.18), 22 σ A 4 (t) is purely imaginary so Fig. 6 shows Im{σ A 4 (t)}.

V. CONCLUSIONS
The non-perturbative stochastic theory of longitudinal relaxation by the dipolar EMOR mechanism, based on an analytical partial solution of the SLE, was first developed for the symmetric I S-I S case. 20Here, we have substantially generalized the theoretical framework to spin systems of arbitrary size with symmetric or asymmetric exchange.The asymmetric case, where the spin system is fragmented by the exchange, is of considerable interest since it applies to chemical exchange of labile macromolecular protons as well as to physical exchange of internal water molecules involved in intermolecular dipole couplings.Of course, the distinction between symmetric and asymmetric exchange is irrelevant for the single-spin quadrupolar EMOR mechanism. 10,11rom a theoretical point of view, asymmetric exchange has two important consequences: (1) a decorrelation of spin modes involving both labile and nonlabile spins and (2) a lowering of the rotational symmetry of the local relaxation matrix.Both of these effects contribute to making the relaxation of a labile spin less efficient when its dipole-coupled partner is nonlabile.The effect of local symmetry reduction is most readily appreciated and most clearly manifested in the MN regime, where exchange is much faster than relaxation in the individual A sites.In the symmetric case, the local relaxation matrix is exchange-averaged over the isotropic distribution of A sites and it therefore exhibits the axial symmetry of the applied magnetic field.As a direct consequence of this axial symmetry, relaxation can only couple spin modes of the same quantum order.Longitudinal relaxation therefore only involves longitudinal magnetizations and zero-quantum coherences of odd parity under spin inversion conjugation (of which there are none for a two-spin system).In the asymmetric case, because the nonlabile spins are not exchange-averaged, local relaxation can couple local spin modes of the same parity but of different quantum order.
In particular, the observed longitudinal relaxation is affected by local longitudinal-transverse cross-mode relaxation in the A sites.To the best of our knowledge, such cross-mode relaxation phenomena have not previously been described in the literature.
As an illustration of the general dipolar EMOR theory and, in particular, of the previously unrecognized 20 effects of asymmetric exchange, we presented here a detailed analysis of the asymmetric two-spin case I S-I.A variety of analytical results were obtained for the ZF and MN regimes and numerical results were presented under more general conditions, including the case of a heteronuclear spin pair.We demonstrated how local longitudinal-transverse cross-mode relaxation slows down the observed relaxation of the labile I spin in the ZF regime and that an unusual inverted dispersion step occurs at higher fields, where the cross-mode relaxation channel becomes nonsecular.This inverted dispersion splits the extreme-narrowing regime into two sub-regimes, referred to here as the zero-field and low-field regimes.We also presented a detailed analysis of the time evolution of the spin modes in the ZF regime.
The general theoretical framework developed here will enable a quantitative analysis of frequency-dependent water-proton longitudinal relaxation in model systems with immobilized macromolecules and, ultimately, will provide a rigorous link between relaxation-based magnetic resonance image contrast and the molecular-level properties of the tissue.In a forthcoming report, we apply this framework to a threespin system I SP, analyzing in detail the I SP-I, I SP-I S, and I SP-I SP exchange cases.

APPENDIX A: BASIS OPERATORS
Here we list the 15 irreducible spin tensor operators (ISTOs) 1 T K Q (k I k S ) that constitute a complete (together with the identity operator) orthonormal basis for the spin Liouville space of two spins I and S. For the symmetric IS−IS case, the same two-spin basis (Table S1) is used for the A and B states.For the asymmetric IS −I case, the single-spin basis in Table S2 is used for state B. All the operators in Tables S1 and S2 are normalized in the same two-spin Liouville space.For example, ( The two-spin basis used in paper I 2 differs from that in Table S1 in that the basis operators B 3 and B 5 were taken to be linear combinations of the ISTOs T 0 0 (11) and T 2 0 (11).
a Parity of B n under spin inversion conjugation.
b Identity operators have been omitted.Equations (B.5) and (B.8) can now be solved for the two unknowns, with the result

TABLE S2. Spin basis operators B
which allows us to obtain the site-averaged resolvent as The superoperator U(s) acts in a composite spin Liouville space of dimension D = D B + D A .In Eq. ( 25), we partitioned the D × D supermatrix representations of U(s) into four blocks.To express Eq.(B.10) in terms of partitioned matrices, we partition the identity matrix as where 1 B and 1 A are D B × D B and D A × D A identity matrices, respectively, and 0 is a According to Eq. ( 13), where the D B × D A matrix T (and its transpose T ) was defined in connection with Eq. (15).Furthermore, where, according to Eq. (B.9), Finally, where, according to Eq. (B.7), and the The Liouvillian for the B state, with only a Zeeman coupling, is  16), ( 29) and (C.3), it follows that in the Q = 0 block of the matrix T G B (0) T all elements are 0, except the (11) element g 11 0 0 0 0 g 21 0 0 0 0 g 31 0 0 0 0 g 41 0 0 0 0 and

APPENDIX D: ZERO-FIELD REGIME
Here we derive analytical expressions, valid for the IS − I case in the zero-field (ZF) regime, for the Laplace transform of the spin modes σ A 1 (t), σ A 2 (t), σ A 4 (t) and σ B 1 (t), with the subscripts referring to the basis operators in Tables S1 and S2.We also obtain analytical expressions, valid in the ZF regime, for the integral relaxation rates in Eqs.
In the ZF regime, state A can be described in the subspace spanned by the rank-1 zero-quantum operators B 1 , B 2 and B 4 in Table S1, so Eq. ( 25) can be written as with the resolvent submatrices given by Eq. ( 28), where G A now is a 3 × 3 matrix.
The s-dependent quantities g np used here are given by Eq. (B.7) as where D(Ω) is the unitary (D † = D −1 ) rotation superoperator and In the last step of Eq. (D.11), we noted that D(Ω) K s D † (Ω) = K s since K s is rotationally invariant.Note that the full 15×15 M 0 matrix must be used here, since the Wigner-Eckart theorem only applies after isotropic orientational averaging.
Identifying the basis operators B n with the ISTOs T K Q (k 1 k 2 ) in Table S1 and using the transformation rules 1 and the orthonormality of the Wigner functions, 1 we obtain from Eqs. (D.10) -(D.13) Equation (D.14) yields for the six matrix elements occurring in Eq. (D.9), The matrix representation of M 0 (s; Ω) in the ISTO basis of Table S1 is where we have defined Inverting the block-diagonal matrix M 0 in Eq. (D.16) and inserting the results into Eq.(D.15), we obtain where we have defined The time evolution of the four rank-1 zero-quantum spin modes can now be obtained by inserting the matrix elements g np (s) from Eq. (D.18) into Eq.(D.9) (or the analogous expressions for selective excitation) and performing the inverse Laplace transform numerically.

APPENDIX E: MOTIONAL-NARROWING REGIME
Here we obtain the elements of the four orientation-dependent relaxation submatrices R α II , R α IS , R α SI and R α SS , appearing in Sec.III B 2. We start from the semi-classical Bloch-Wangsness-Redfield (BWR) master equation where  S3) of L Z , such that and where the eigenvalues Ω M λ are linear combinations of ω I and ω S (Table S3).It then follows that where Combination of Eqs.(E.1) and (E.5) yields with the time correlation function where the dipole vector orientation Ω α (τ ) is modeled as a stationary random process.
Note that the time correlation function is not averaged over the initial orientation Ω α (0), which is fixed by the nuclear geometry in site α.In the EMOR model, the orientation of the dipole vector is randomized upon exchange, as expressed by the propagator where τ A is the mean survival time in the site.Combining Eqs.(E.7) and (E.8) and making use of the orthogonality of the spherical harmonics C 2 M (Ω α ), 1 we find with With the time correlation function (E.9), we can write the master equation (E.6) as The time-dependent relaxation superoperator is given by with a complex-valued reduced spectral density function Our development is more general than the standard treatment in two respects.First, we retain the imaginary part, k(ω), of the spectral density, which, contrary to conventional wisdom, 3 can affect longitudinal relaxation under certain conditions. 4Because k(−ω) = − k(ω), we refer to k(ω) as the odd spectral density function (OSDF).Second, we do not invoke the secular approximation to eliminate terms with oscillating factors in Eq. (E.12), since we want to describe relaxation over the full frequency range.
The integral relaxation rate is most conveniently obtained from a time-independent relaxation superoperator.For this purpose, we transform the master equation (E.11) to the Schrödinger representation as 5][6] In the motional-narrowing regime, longitudinal relaxation can therefore be fully described within the subspace of the six single-spin basis operators in Table S1.These operators have odd parity with respect to spin inversion conjugation, whereas the remaining nine two-spin operators have even parity.The supermatrix representation of the operator master equation (E.14) in the subspace of the six single-spin basis operators B n takes the form where σ α (t) is the column vector of the six single-spin modes σ α n (t) = (B n | σ α (t)) = Tr{B † n σ α (t)} and the time-independent relaxation supermatrix R α is given by where we have defined a coefficient supermatrix C M M λλ with real-valued elements The oscillating factors in Eq. (E.12) are absent from Eq. (E.16) because the single-spin basis operators are eigenoperators of the Zeeman Liouvillian and Ω M λ + Ω M λ = Ω n − Ω p for all nonzero values of C M M λλ ,np .
Ordering the basis operators for the single-spin subspace as {I z , I + , I − , S z , S + , S − }, we can write the supermatrix L Z in Eq. (E.15) as where 0 is the 3 × 3 null matrix, and Similarly, we partition the relaxation supermatrix as The elements of the relaxation submatrices are obtained from Eqs. (E.10), (E.16) and (E.17), with spin operators and eigenfrequencies from Tables S1 and S3.Thus, (E.26c) The elements of the submatrices R α SS and R α SI are obtained from the corresponding elements of R α II and R α IS , respectively, by interchanging I and S everywhere, which amounts to the following three substitutions: j I ↔ j S , k I ↔ k S and k − ↔ −k − .We note that the OSDF affects all rates, except the longitudinal auto-mode rates R II zz , R SS zz and R IS zz = R SI zz .Whereas the master equation (E.15) in the Schrödinger representation is most convenient for obtaining the integral relaxation rate, the effect of nonsecular decoupling is more apparent in the interaction representation.For this purpose, it is more convenient to order the basis operators in the relevant subspace as {I z , S z , I + , S + , I − , , S − }.In this subspace, the supermatrix representation of master equation (E.11) becomes In the ZF regime (Sec.III A), where the frequencies ω I , ω S and ω I ± ω S are much smaller than the relaxation rates (of order ω 2 D τ A ), the complex exponential factors in Eq. (E.28) can all be replaced by unity.The cross-mode relaxation rates then come into play, coupling the evolution of the longitudinal and transverse magnetization components.This happens in the asymmetric IS−I case because the I-S dipole coupling is not isotropically averaged.At higher fields, where ω I , ω S ω 2 D τ A , the oscillating factors cancel all relaxation supermatrix elements, except possibly those involving the difference frequency ω I − ω S (for a homonuclear spin pair).The relaxation supermatrix then becomes block- since all cross-mode rates vanish when isotropically averaged, as is evident from their dependence on the azimuthal angle α in Eqs.(E.22) and (E.23).This result, including the familiar Solomon equations, is usually derived from BWR theory by invoking the secular approximation.Fundamentally, however, the decoupling of the longitudinal and transverse magnetizations is a consequence of isotropic averaging.Cross-mode coupling is therefore absent also at zero field, which is not obvious if the secular approximation is invoked.The same result can be obtained from the time-independent relaxation supermatrix R α in Eq. (E.21) (with a different basis ordering).As a consequence of the vanishing of all cross-mode rates upon isotropic averaging, the four submatrices in Eq.
(E.21) become diagonal.It is then evident from Eq. (E.15) that the longitudinal and transverse modes are decoupled.
The isotropically averaged auto-mode rates coincide with the familiar longitudinal and transverse auto-spin and cross-spin relaxation rates for a two-spin system, In the extreme-narrowing regime, where ω I , ω S 1/τ A , all the even spectral densities in Eq. (E.24) are equal to τ A and all the odd spectral densities in Eq. (E.25) can be neglected.The relaxation submatrices then take the form with ρ S given by Eq. (F.32b) and R dil 1,I (0) by Eq. (F.20a).
and  U AA (s) are, respectively, D B × D B , D B × D A , D A × D B , and D A × D A submatrices of the spin basis representation of the site-) | β⟩ P β .
), the cross relaxation matrix element Γ I I z z = (1 |Γ α I I | 1) only has contributions from the first row of R α I S and from the first column of R α S I .The four off-diagonal elements in this group of six elements involve the cross-spin cross-mode rates R I S z± and R S I where the familiar expressions for the longitudinal autospin rates ρ I ≡ R I I z z and ρ S ≡ R S S z z and cross-spin rate σ ≡ R I S z z = R S I z z are given in Eq. (E.32) of Appendix E. 22 Combination of Eqs.(67) and (70) then yields  R dil/non 1, I

Figure 5 FIG. 4 .FIG. 5 .
FIG.4.Dispersion of the integral longitudinal relaxation rate of spin I for exchange case I S-I computed from the SLE result in Eq. (42a) (red solid curves, same as in Figs.2 and 3) and for exchange case I S-I S computed from the corresponding SLE result in Paper I (blue dashed curves).Parameter values as in Figs.2 and 3.Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions.Downloaded to IP: 130.235.252.19On: Thu, 25 Feb FIG.6.Time evolution of the normalized spin modes I B z (t)/P B (red solid curve), I A z (t)/P A (blue), S A z (t)/P A (magenta), and σ A 4 (t)/P A (green) for the I S-I case in the zero-field regime and with nonselective excitation.The black dashed curve coinciding with I B z (t)/P B in all three panels is the exponential decay in Eq. (73) and the black dashed curve coinciding with S A z (t)/P A in panel (a) is the biexponential function in Eq. (77b).The black dashed-dotted curve in panel (a) is the exponential decay of I B z (t)/P B = S A z (t)/P A for the symmetric I S-I S case.Parameter values: P A = 10 −3 and ω D τ A = 0.01 (a), 1 (b), or 5 (c).

a
Identity operators have been omitted.b The operators B n are normalized in the two-spin Liouville space.

Σ
27)   where, for clarity, we have represented the six spin density operator elements by the corresponding basis operators.The time-dependent relaxation supermatrix isI t R II z+ e −iω S t R IS z+ e iω I t R II z− e iω S t R I t R SI z+ e −iω S t R SS z+ e iω I t R SI z− e iω S t R SS z− e iω I t R II +z e iω I t R IS +z R II ++ e i∆t R IS ++ e i2ω I t R II +− e iΣt R IS +− e iω S t R SI +z e iω S t R SS +z e −i∆t R SI ++ R SS ++ e iΣt R SI +− e i2ω S t R SS +− e −iω I t R II −z e −iω I t R IS −z e −i2ω I t R II −+ e −iΣt R IS −+ R II −− e −i∆t R IS −− e −iω S t R SI −z e −iωS t R SS ≡ ω I + ω S , (E.29a) ∆ ≡ ω I − ω S .(E.29b)

30 )
Nonsecular decoupling thus cancels all cross-mode rates so the longitudinal modes (I z and S z ) evolve independently from the transverse modes (I ± and S ± ).For the symmetric IS−IS case in the motional-narrowing regime, the same IS spin pair samples all anisotropic A sites on a time scale that is short compared to the relaxation in each site because, when the exchange time is also the correlation time (as in the EMOR model), the motional-narrowing regime coincides with the fast-exchange regime.The relaxation behavior is then governed by the isotropic average of the relaxation supermatrix in Eq. (E.28), which becomes
are the spin density operator and the dipolar Liouvillian for site α, both in the interaction representation.The Liouvillians are the usual derivation superoperators L Z ≡ [H Z , ...] andL α D (t) ≡ [H α D (t), ...], with H Z = ω I I z + ω S S z and the dipolar Hamiltonian H α D (t) as given by Eq. (2.3) of paper I.2The angular brackets with subscript α denote an equilibrium ensemble average over the molecular degrees of freedom in site α.
M(11)appearing in H α D (t) are decomposed into eigenoperators A M λ (Table