Nuclear Spin Relaxation Driven by Intermolecular Dipolar Interactions

5676

J. Phys. Chem. B 2006, 110, 5676-5689

Nuclear Spin Relaxation Driven by Intermolecular Dipolar Interactions: The Role of Solute-Solvent Pair Correlations in the Modeling of Spectral Density Functions Diego Frezzato, Federico Rastrelli, and Alessandro Bagno* Dipartimento di Scienze Chimiche, UniVersita` di PadoVa, Via Marzolo 1, 35131 PadoVa, Italy ReceiVed: October 20, 2005; In Final Form: January 9, 2006

Nuclear spin relaxation provides useful information related to the dynamics of molecular systems. When relaxation is driven by intermolecular dipolar interactions, the relevant spectral density functions (SDFs) also have significant contributions, in principle, from distant spins all over the dynamic range typically probed by NMR experiments such as NOESY. In this work, we investigate the intermolecular dipolar spin relaxation as driven by the relative diffusion of solvent and solute molecules taking place under a central force field, and we examine the relevant implications for (preferential) solvation studies. For this purpose, we evaluate the SDFs by employing a numerical approach based on spatial discretization of the time-propagation equation, and we supply an analytical solution for the simplest case of a steplike mean-field potential. Several situations related to different solute-solvent pair correlation functions are examined in terms of static/dynamic effects and relaxation modes, and some conclusions are drawn about the interpretation of NOE measurements. While we confirm previous results concerning the spoiling effect of long-range spins (Halle, B. J. Chem. Phys. 2003, 119, 12372), we also show that SDFs are sufficiently sensitive to pair correlation functions that useful, yet rather complicated, inferences can be made on the nature of the solvation shell.

1. Introduction Nuclear spin relaxation has long been recognized as an invaluable source of information. Indeed, because of the strong connection with time-dependent processes, techniques based on NMR relaxation still constitute an essential methodology for the investigation of the dynamics of molecular systems.1 There is, moreover, a further impressive range of current applications of NMR relaxation techniques, spanning investigations of (macro)molecular structure,2 solute-solvent interaction,3-5 and studies of complex matrices in materials science and eventually magnetic resonance imaging. Although nuclear spin relaxation stems, in principle, from manifold dynamic processes, the relaxation of spin-1/2 nuclei in solution is generally driven by time-dependent fluctuations of magnetic dipolar interactions. Moreover, in physical systems, a network of mutual dipolar couplings is often arranged in such way that the involved spins cannot relax independently of each other. In this situation, cross-relaxation pathways will exist among spins whereby a selective perturbation of one spin can affect the populations of other coupled partners. Known as the nuclear OVerhauser effect (NOE), the dynamic polarization resulting from cross-relaxation manifests itself in terms of amplified (or sometimes reduced) intensities of spectral lines with a general dependence on spin-spin proximities. Intramolecular cross-relaxation rates are widely used to estimate internuclear distances and hence supply input geometry restraints in the modeling of (macro)molecules.2,6 It is somewhat less obvious to the NMR-user community at large that intermolecular dipole-dipole interactions may also generate weak yet often detectable NOE enhancements. After the initial experimental evidence,7-11 this phenomenon has attracted the attention of several research groups, who realized the potential * To whom correspondence should be addressed. Fax: +39 0498275239. E-mail: [email protected].

of intermolecular NOEs for the study of solute-solvent interactions in biochemical3 and chemical4,5 systems. Nonetheless, in contrast to the intramolecular case, the analysis of intermolecular NOEs is hampered by both experimental and theoretical difficulties.5 Broadly speaking, even though both relaxation mechanisms originate from random modulations of dipolar couplings, they differ in many relevant aspects. Intramolecular relaxation results from molecular tumbling motions in which the interacting spins sit at a fixed relative distance. On the other hand, intermolecular relaxation is driven by translational diffusion processes, in which not only the relative spin positions change with time but also the number of interacting spins changes with the distance between them. The complexity in the formal treatment of this phenomenon led to interpretation of the experimental results on the basis of simplified “force-free” models which are hardly representative of realistic solute-solvent interactions. In the context of NOE studies, these models lead to the solute-solvent cross-relaxation rate, σIS, being proportional to a zero-frequency factor, nS/DISrIS, where I and S represent solute and solvent spins and the symbols stand for the spin number density, relative diffusion coefficient, and distance of closest approach, respectively. Despite the simplified formulation, these models have been successfully employed to interpret intermolecular NOE data, albeit only in a qualitative or semiquantitative way. Thus, there is a fairly large number of works dealing with (i) solvation studies of small molecules and macromolecules,3,4 often supplemented by molecular dynamics simulations, and (ii) preferential solvation phenomena.5 In the first case, the solvation of specific molecular sites is investigated, whereas the second case is concerned with the competition between two (or more) chemical species in the solvation of a probe molecule. However, NOE measurements alone cannot furnish information on some important issues such as the energy of solute-solvent interactions and the depth of

10.1021/jp0560157 CCC: $33.50 © 2006 American Chemical Society Published on Web 02/25/2006

Nuclear Spin Relaxation the preferential solvation layer. In any event, there is little doubt that solvation phenomena are governed by the energetics of solute-solvent interactions. An extremely important contribution, for this scenario, has been recently added by Halle,12 who called attention on the fact that long-range dipolar couplings involving distant solvent spins (up to 30 Å from the solute) are not negligible in the determination of the intermolecular cross-relaxation rates. Clearly, as distant couplings contain little if any information on the phenomenon being sought, one is cautioned against overinterpreting the results (e.g., in intermolecular 1H-1H NOE studies of protein hydration). Also, there have been reports in which the data on preferential solvation (albeit determined indirectly) seem to indicate that these effects, as measured by NMR, may have been overestimated so far.13 The work of Halle, especially, casts considerable doubt on the applicability of intermolecular NOE measurements; as such, it has potentially far-reaching consequences and has raised objections that clearly deserve an answer. Regardless of the complexity of the real chemical systems and of the experimental procedures, the analysis of intermolecular NOEs is based on the assumption that the measured relaxation rates are sensitive to the features of the solvation shell. In formal terms, this is equivalent to invoking a “solvation sensitivity” of the spectral density (or time-correlation) functions characteristic of the intermolecular dipolar interactions. The aim of the present work is to explore such sensitivity in situations that incorporate a model interaction potential. In principle, a direct connection between experimental observables and microscopic details of the liquid phase can be achieved by means of molecular dynamics (MD) simulations. On the other hand, the need for consideration of widely extended systems and for coverage of a very long time scale of inspection (several microseconds of simulation), still limit the methods to simple archetypal situations.14,15 An alternative approach consists of adopting a diffusive model to describe the solute-solvent relative dynamics in the time window explored by NMR spectroscopy and to parametrize it in terms of (i) dissipative parameters (i.e., mutual diffusion coefficients) and (ii) equilibrium distribution, specifying the configuration of a targeted solvent molecule with respect to a fixed solute molecule. The equilibrium distribution is usually expressed in terms of a pair correlation function, which contains all information about the mean-field interaction potential between solute and solvent in the liquid phase. The stochastic approach has been successfully adopted in the past decades and has been instrumental in filling the gap between spectroscopic data and the physics of solute-solvent interactions. The first attempts to model nuclear spin relaxation, as generated by translational spin motions, were the work of Torrey and Resing,16 who treated such dynamics as a random walk among sites on lattices. Subsequent refinements on the modeling of such jumplike motions and continuous diffusion descriptions more appropriate for liquids are the work of several authors,17-19 including the recent work of Halle,12 who first faced the general problem of establishing the sensitivity of NMR measurements to the contribution of the solvent spins close to the solute. In such work, solute and solvent molecules are treated as hard spheres and possible solute-induced dynamic perturbations are accounted for by allowing a reduction in the solvent diffusion coefficient within a layer surrounding the solute: an actual description of solvation, from the solute-solvent energetics, is not considered in the model. In the present work, we intend to go beyond such an idealized situation, by pursuing the modeling

J. Phys. Chem. B, Vol. 110, No. 11, 2006 5677

Figure 1. Geometric representation of the interacting molecules: the spherical solute molecule carrying the spin I and the spherical solvent molecule hosting a centered spin S.

of diffusive dynamics to determine how the spectral density responds to the short-range features of pair correlation functions from different mean-field potentials. This work is organized as follows. First, we outline the theory of dipolar spin-spin relaxation in liquid phases: the relevant time-correlation function and its spectral density connected to NMR relaxation rates are presented, and the stochastic model for their specification is elaborated to provide closed expressions to be evaluated for general pair correlation functions. Subsequently (section 3), calculations are made for the model cases of spin-centered solute and solvent molecules interacting through steplike and Kihara’s mean-field potentials. As differences between the generated spectral profiles are explored, we also provide a detailed analysis of how the specific interaction potential affects the spectral features on the basis of the system response to motions occurring on different time scales. Finally, in section 4, we focus our attention on the possibility of discriminating, on the basis of measured relaxation rates, between the different profiles of the solute-solvent pair correlation function. Technical appendices are supplied in which we present the tools for numerical calculations and provide the relevant derivations. In particular, in Appendix A, we propose an exact analytical solution for the case of a steplike pair correlation function, which may be regarded as the simplest model of solvation where the extra stabilization of solvent molecules only occurs within a solute-centered segregation layer. Appendices B-D are provided as Supporting Information. 2. Dipolar Spin Relaxation in Liquids Let us consider the dipolar interactions existing between a spin I located on a solute molecule and NS spins S carried by solvent molecules in the liquid phase. In our model, the solute spin, I, may have an eccentric location within its spherical host molecule, while solvent molecules are all represented by spincentered spheres. The geometry of this system is depicted in Figure 1: in particular, with respect to a Cartesian frame of reference tethered to the central (solute) molecule and with fixed orientation, r and r′ indicate the components of the center-tocenter and spin-to-spin vectors, respectively, while the displacement, G defines the spin eccentricity. Accordingly, in spherical coordinates we set r ) (r, Ω) and r′ ) (r′, Ω′), where Ω and Ω′ denote the pairs of azimuthal and polar angles which specify the orientation of vectors r and r′. As a result of the aforementioned model, dipolar spin relaxation is driven by the relative translation of solute and solvent molecules, as well as by the rotational dynamics of the solute molecule itself (in case of eccentricity of spin I). Basic assumptions in our modeling are that (i) such dynamics can be described in terms of stochastic diffusive processes and (ii) both the solute and solvent molecules are spherically symmetric

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Frezzato et al.

objects. More precisely, rotation of the central molecule is considered to be a force-free diffusion process, while the relative translational diffusion takes place in a central force field. By assuming uncorrelated motions for the solvent molecules, the following time-correlation function is sufficient to describe the relaxation of dipolar coupling in liquids

G(2)(t) ) NS〈F2,0(r′0)*F2,0(r′)〉,

F2,0(r′) ) (r′)-3Y2,0(Ω′)

(1)

where Y2,0 represents spherical harmonics of the second rank. Within the motional-narrowing regime, the spectral density of function G(2)(t) allows one to establish a direct connection with measurements of nuclear magnetic relaxation rates.20 The spectral density function (SDF) we are interested in is the real part of the Fourier-Laplace transform of G(2)(t)

J(2)(ω) ) Re{

∫0∞ dt e-iωtG(2)(t)}

(2)

The link to nuclear magnetic relaxation is then obtained by turning to the frequency-dependent rates

R(2)(ω) ) 4π KIS J(2)(ω),

KIS )

(

)

µ0 pγ γ 4π I S

2

(3)

where KIS is the dipolar coupling constant, µ0 is the vacuum permeability, and γI and γS are the magnetogyric ratios of the two nuclei. A proper linear combination of R(2)(ω), evaluated at specific frequencies, provides the relaxation rates observed for the specific NMR experiment under consideration. Of particular interest for the organic and biomolecular chemists are the longitudinal- and cross-relaxation rates typically observed by inversion-recovery and NOESY experiments. In this context, our aim is to investigate the sensitivity of the spectral profile, R(2)(ω), to the solvation features (i.e., to the (equilibrium) distribution of the solvent molecules in closest proximity to the solute). As demonstrated by Halle in ref 12, a decoupling between rotational and translational processes can be worked out. Accordingly, eq 1 is converted into the following sum of factorized rotational (R) and translational (T) contributions

G(2)(t) )

1

∞

(t) ∑(l + 1)(l + 2)(2l + 3)F2lG(l)R (t) G(l+2) T 6 l)0

(4)

(L) (L) where the quantities G(L) R (t) ) exp[-ωR t], with ωR ) L(L + 1)DR, are (zero-time-normalized) correlation functions for the isotropic rotational diffusion of the central molecule and DR is the rotational diffusion coefficient. Then, the correlations

- (L+1) YL,0(Ω) G(L) T (t) ) NS〈FL,0(r0)*FL,0(r)〉, FL,0(r) ) r

(5)

account for the contribution to the relaxation from the relative translational diffusion of the NS solvent spins with respect to the central one. By inserting eq 4 in eq 2, one gets

∞

(ω - iω(l) ∑(l + 1)(l + 2)(2l + 3)F2l J(l+2) T R)

6 l)1

G(L) T (t) ) NS

∫ dr ∫ dr0 FL,0(r0)*FL,0(r)p(r0)p(r0|r, t)

(7)

where p(r) is the spatial equilibrium distribution and p(r0|r, t) is the conditional probability to find the solvent molecule at location r at time t0 + t if is was placed at r0 at time t0 (the dependence on t0 is lost in stationary conditions). The evolution of the conditional probability is then governed by the diffusion equation

∂ p(r |r, t) ) -Γr p(r0|r, t) ∂t 0

(8)

where Γr is the diffusive operator which can be expressed in the form

Γr ) - DT∇‚p(r)∇p(r)-1

(9)

Moreover, the conditional probability is constrained by the initial condition p(r0|r, 0) ) δ(r - r0) and by the stationary limit p(r0|r, t f ∞) ) p(r). Since we deal with spherical interacting particles, the equilibrium distribution is spherically symmetric and depends only on the modulus r, thus yielding the normalization condition

∫0∞ dr 4πr2p(r) ) 1

(10)

When changed to spherical coordinates, eq 8 is transformed into

∂ p(r , Ω | r, Ω, t) ) - Γr,Ω p(r0, Ω0|r, Ω, t) ∂t 0 0

(11)

The stochastic operator in the above expression is then decoupled as Γr,Ω ) Γr + ΓΩ, with the rotational part given by

DT 1 ∂ 1 ∂2 ∂ ≡ 2 Jˆ 2 ΓΩ ) - DTr-2 sin θ - DTr-2 2 2 sin θ ∂θ ∂θ sin θ ∂φ r (12)

J(2)(ω) ) J(2) T (ω) + 1

the solvent molecules is accounted for.21 In general, to obtain the SDF required to interpret the NMR data (eq 2), we need to evaluate the function J(L) ˆ ) for all ranks L g 2. T (ω Dipolar Relaxation due to Mutual Translational Diffusion. Hereafter, we will focus on the problem of a formal specification of the time-correlation function G(L) T (t), which is done by modeling the stochastic solute-solvent diffusion process in the liquid at thermal equilibrium. The derivation will adopt the standard tools already presented by Freed and Huang in ref 17. The relative diffusion of solvent molecules with respect to the central solute core is modeled as a stationary Markov process22 in the stochastic variable r. If DIT and DST are the translational diffusion coefficients of the solute and solvent molecules, the coefficient of mutual diffusion is then DT ) DIT + DST. We further assume that DT does not depend on r (i.e., we assume that the viscous drag experienced by solute and solvent molecules is invariant as their separation changes, although this limitation is not essential in our treatment and can be removed in favor of the complete derivation where some dependence DT(r) is invoked). The correlation function, eq 5, is then given as

(6)

ˆ ) is the real part of the Fourier-Laplace transform where J(L) T (ω of function G(L) T (t), evaluated at the generally complex-valued frequency ω ˆ . It should be stressed that a series expansion similar to eq 6 can be also recovered if the eccentricity of spins S in

where Jˆ 2 is the modulus squared operator of the angular momentum,22 and the radial part expressed as

∂ ∂ Γr ) -DTr-2 r2p(r) p(r)-1 ∂r ∂r

(13)

Integration of eq 11 with respect to p(r0, Ω0|r, Ω, t) and

Nuclear Spin Relaxation


substitution into eq 7 upon the change to spherical coordinates, yields the following form of the dipolar correlation function

∫0∞dr r2∫dΩ r- (L+1)Y2,0(Ω)*e-Γ

G(L) T (t) ) NS

r-(L+1) ×

r,Ωt

Y2,0(Ω)p(r) (14) π with ∫dΩ(‚‚‚) ≡ ∫2π 0 dφ ∫0 dθ sin θ(‚‚‚). Since the spherical harmonics are eigenfunctions of the operator Jˆ 2 with Jˆ 2YL,m(Ω) ) L(L + 1)YL,m(Ω), we also have ΓΩYL,0(Ω) ) L(L + 1)DTr-2YL,0(Ω), and the following relation is easily verified for a general function f(r)

e- Γr,Ωtf(r)YL,0(Ω) ) YL,0(Ω)e-Oˆ r tf(r) (L)

(15)

where the operator O ˆ r(L) is given by -2 + Γr O ˆ (L) r ) L(L + 1)DTr

(16)

By specifying eq 15 for f(r) ) r-(L+1)p(r) and considering the normalization condition ∫ dΩ YL,0(Ω)*YL,0(Ω) ) 1, eq 14 reduces to

G(L) T (t) ) NS

∫0∞dr r- (L-1)e-Oˆ

r-(L+1)p(r)

(L)t

r

(17)

The distribution function p(r) can be related to the intermolecular pair correlation function, g(r), defined as the ratio between the actual distribution probability at a distance r and the value corresponding to the force-free (i.e., purely random) case: g(r) ) p(r)/p(∞). With respect to solute-solvent interactions, deviations of g(r) from the unitary value account for the solvation features around the solute molecule. By imposing the normalization condition of p(r) within a spherical sample of radius much larger than the intermolecular interaction length scale, one finds p(r) ) g(r)/V, V being the sample volume. Substitution into eq 17 then yields

G(L) T (t) ) nS

∫0∞dr r- (L-1)e-Oˆ

r-(L+1)g(r)

(L)t

r

(18)

where nS ) NS/V is the S-spin number density, the dependence on the sample size being thus removed. Furthermore, to introduce a link with the energetics of pair interactions, g(r) can be expressed by means of the mean-field radial potential V(r) as

g(r) ) e-(V˜ (r)-V˜ (∞))

(19)

where V ˜ (r) ) V(r)/kBT is the dimensionless scaled potential and V ˜ (∞) denotes its offset value in the long-range force-free limit. Finally, we recall that the L-rank SDF of function G(L) T (t) is

ˆ ) ) Re{ J(L) T (ω

∫0∞dt e-iωˆ tG(L) T (t)}

(20)

where the frequency, ω ˆ , can be generally complex-valued, as required in expression 6 for the evaluation of J(2)(ω). At this stage, we have defined the formal procedure to relate the spectral profiles, R(2)(ω), required in the description of NMR observables (eq 3) to the pair correlation function g(r). The aim of the present work can now be cast more precisely: (i) to establish if the spectral profiles of R(2)(ω) are quantitatively sensitive to the short-range features of g(r) (i.e., if NMR data can provide information on the solvation around the central molecule) and (ii) to get an insight into the kind and amount of achievable information. In this context, we will assume that the

Figure 2. Pair correlation functions g(r) in the cases of (a) force-free and (b) steplike mean-field potentials.

mutual diffusion coefficient DT is known a priori, and we will focus only on the dependence of dipolar relaxation on g(r). Generally speaking, the pair correlation g(r) affects the spectral profiles at two levels. First, g(r) is related to the spreading of the magnetic anisotropies (from dipolar interactions) because of the equilibrium distribution of solvent spins: in fact, it determines the mean-squared amplitude of fluctuations of the dipolar interactions, which ultimately contributes as a scaling factor to the magnitude of the NMR relaxation rate. More precisely, g(r) affects the zero-time value G(2)(0) of the timecorrelation function in eq 1 or, equivalently, all the zero-time values, G(L) T (0), of correlations in eq 18. In the following, such an incidence of g(r) on the relaxation rates will be referred to as the static effect. Second, as shown above, g(r) also directly affects the time evolution of the nonequilibrium distribution of solvent spins by altering the time scale of all processes which drive the decay of the dipolar correlation. Thus, even if the amplitude of fluctuations were fixed (same static effect), different local features of g(r) might well be responsible for different spectral profiles of R(2)(ω). Accordingly, such an effect will be hereafter referred to as dynamic. To examine the response of the spectral profiles, R(2)(ω), to the changes in the solvation features on both static and dynamic grounds, one has to consider representative situations. Unfortunately, the L-rank correlation function of eq 18, or likewise its spectral density, can be evaluated analytically only for few special profiles of g(r). Ayant et al.18 have derived closed forms of J(L) T (ω) for the hard-sphere case corresponding to g(r) ) 0 if r < ra and g(r) ) 1 elsewhere, with ra the distance of the closest approach between the solute and solvent spins24 (see Figure 2a). Clearly lacking any interaction potential for r g ra, such a model does not consider the energetics of solvation. Following Ayant’s approach, in Appendix A we give the analytical solution for a steplike pair correlation function defined as g(r) ) 0 if r < ra, g(r) ) R > 0 for ra e r < rb, and g(r) ) 1 for r g rb (for the sake of completeness, in our derivation we also allow the mutual diffusion coefficient to assume different


Frezzato et al.

values when r < rb and r g rb, even if all considerations made in the following are referred to the case of uniform value for DT). The situation is depicted in Figure 2b. Intuitively, such a steplike profile provides the simplest modeling of solvation around a central impenetrable core of radius ra, where the surrounding molecules are stabilized inside a segregation layer of thickness ∆ ) rb - ra, with stabilization energy (in units of kBT) given by ˜ ≡ V ˜ (∞) - V ˜ (r) ) ln R. For general profiles of g(r), the solution has to be achieved via numerical methods. Since translational diffusion is unbounded, standard tools to evaluate the L-rank correlation function from eq 18, such as the expansion of the integral onto a set of orthonormal functions of r, are not applicable. Nonetheless, the problem can be faced by means of the discretization of eq 18 (i.e., formally, by solving the time evolution ruled by the operator O ˆ (L) r via discretization of the radial dimension). In Appendix B we provide the detailed description of such an approach, which has been employed to generate the SDFs described in the text. 3. Model Calculations To limit the number of parameters entering the model, in the following we refer to the ideal case of spin I located at the center of the solute molecule (F ) 0). Thus, eqs 4 and 6 reduce to

F ) 0:

G(2)(t) ) G(2) T (t),

J(2)(ω) ) J(2) T (ω)

(21)

and the dipolar relaxation is entirely driven by mutual diffusion of central and solvent molecules. Such a simplification is motivated by the fact that our aim is to investigate the sensitivity of the spectral profiles to the interaction potential between central and solvent molecules, and the F ) 0 situation allows one to get a first insight into the problem. On the other hand, even if we focus on the case where L ) 2, we stress that the methodology presented in section 2 and elaborated in Appendices A and B is fully applicable to the evaluation of J(L) ˆ ) (for general ranks L) appearing in eq 6. T (ω To distinguish between the static and dynamic effects of the g(r) function, the SDF relative to G(2) T (t) is best expressed in the following partitioned form (2) (2) J(2) T (ω) ) GT (0)jT (ω)

(22)

where

∫0∞ dr r-4g(r)

G(2) T (0) ) nS

(23)

is the amplitude of the fluctuations of the dipolar interactions, (2) (2) while j(2) T (ω) ) JT (ω)/GT (0) is the spectral density scaled with respect to such amplitude. Clearly, both factors, G(2) T (0) and j(2) T (ω), contributing to the spectral density depend on the molecular mean-field pair interactions through g(r) but in different ways: G(2) T (0) is a static factor determined by the equilibrium distribution of solvent spins, while j(2) T (ω) is sensitive to the dynamic effects induced by the peculiar pair correlations. To explore how the spectral profiles are sensitive to the features of g(r), we refer to the practical case of glucose dissolved in water. Some of the parameters will then be fixed (DT ) 3 × 10-5 cm2 s-1 for the relative diffusion coefficient and nS ) 2/30 spins Å-3 for the spin number density of water protons).12 Moreover, for all cases we fix the distance of closest approach between the solute and solvent molecules (once a

proper criterion to define it is chosen) and set it equal to 4.5 Å. Then, we refer to model interactions between solute and solvent molecules which generate pair correlation functions, g(r), increasing beyond the distance of closest approach up to a maximum value and then monotonically decreasing toward the unitary value in the force-free limit (we ignore features such as modulations of g(r) and secondary maxima). The mean-field interaction potentials are then parametrized to give a realistic maximum value of g(r), say between 1.0 and 3.0. Finally, rather than J(2) T (ω), we consider the profiles of the frequency-dependent relaxation rate R(2)(ω) defined in eq 3 through a multiplication by 4πKIS. The advantage of such rescaling is that R(2)(ω) can be compared, in magnitude, to actual relaxation rates extracted from a variety of NMR experiments. Also, instead of (2) (2) j(2) T (ω) we equivalently discuss R (ω)/GT (0). The value employed for the proton-proton dipolar coupling is KIS ) 5.7 × 1011 Å6 s-2. Computational Details. FORTRAN codes have been created to implement the tools described in Appendix B for numerical evaluation of the spectral density, J(2) T (ω), for a general pair correlation function g(r). Two model profiles of g(r) are considered here, as generated by a steplike potential function and by Kihara’s potential function.26 The radial dimension is discretized, from rmin to rmax, into intervals with widths of ∆rn with n ) 1, 2, . . ., N. A very dense homogeneous partitioning with ∆rn ≡ ∆1 is made from rmin up to a turning distance, rtrans, while an inhomogeneous partitioning with rapidly increasing widths given by ∆rn ) ∆rn-1 × fampl (fampl > 1) is employed beyond rtrans. Generally speaking, ∆1 and rtrans should be tuned to achieve an accurate mapping of both the g(r) details and the radial decay of the dipolar interactions. To avoid artifacts in case g(r) continuously decreases to zero as r decreases, rmin should be close enough to the center of the solute core to ensure g(rmin) = 0. Then, rmax has to be extended up to include all solvent spins that significantly contribute to the decay of the dipolar correlation function. As rmax is increased, further slow relaxing contributions from the long-range displacements of spins are brought into the SDF profiles. On the other hand, a radial truncation at rmax introduces a low-frequency cutoff on the relaxation modes contributing to J(2) T (ω). Thus, rmax should be further extended as the lower value of ω in the spectral window of interest is decreased. Moreover, in our coarse-grained approach, the peculiar kind of long-range nonlinear discretization determines the number and weight factors of the characteristic frequencies describing such slow modes of relaxation and has to be chosen to reach convergence on J(2) T (ω) in the lowfrequency range. As described in Appendix B, the spatial discretization yields the matrix representation of eq 18. The Lanczos algorithm25 with nsteps iterations is then employed to allow the expansion of J(2) T (ω) in terms of a continued fraction (eqs S28 and S29) which can be quickly evaluated at each frequency. The spectral profiles discussed hereafter have been calculated for frequencies ν ) ω/2π ranging from 105 Hz up to 1011 Hz. Convergence has been checked with respect to individual variations of the parameters rmin, rmax, rtrans, ∆1, fampl, and nsteps, which have been adjusted so as to reach indistinguishable profiles in the whole frequency range (the adopted values are provided for the specific cases). On the other hand, the convergence with respect to rmax could be only apparent since a large increase of rmax is required to attain only small variations of the SDF in the low-frequency range. Thus, to establish if the spatial sampling is sufficiently extended, we check the accuracy of the numerical calculations by adopting the steplike



case as a reference, for which the analytical solution is available (see Appendix A). Then, we implicitly assume that the same value of rmax can be employed in all calculations even for different profiles g(r), since the ignored long-range motions should be only slightly affected by the details of g(r) close to the central core. Pair Correlation Function from a Steplike Potential. To deal with a continuous function g(r) possessing a steplike behavior, we employ the following potential

[ ( )

]

˜ S rb r - rb V ˜ (r) ) arctan + arctan β β π/2 + arctan(rb/β)

(24)

where β is a control parameter: for β f 0, the function rapidly rises from 0 up to V ˜ (∞) ) ˜ S at the step-point rb. As a matter of fact, such a V ˜ (r) has a continuous profile which is flat for r < rb and r > rb (note that eq 24 is just one among the many possible functions which can be built to model a steplike function). The corresponding pair correlation function g(r) is then

g(r) )

{

0 e

-(V ˜ (r)-V ˜ (∞))

r < ra r g ra

(25)

which tends toward the constant value R ) e˜ S for ra e r < rb, the unitary value for r > rb, and rapidly drops around rb. Clearly, ra is the distance of closest approach between solute and solvent, and the (inner) reflecting surface in the discretization procedure (see Appendix B) is naturally chosen as the sphere of radius rmin ≡ ra. Model calculations presented in Figures 3 and 4 show the profiles of g(r), the relaxation rate, R(2), and the same rate scaled with respect to the amplitude of fluctuations of dipolar interactions. Here we explore how the stabilization energy ˜ S and the width of the segregation layer ∆ ) rb - ra (ra being fixed) affect the spectral profiles. Profiles in Figure 3 have been obtained with the following parameters: ra ) 4.5 Å, rb ) 6.0 Å (∆ ) 1.5 Å), and two values of the stabilization energy, ˜ S, 1.0 (solid lines) and 0.5 (dashed lines). Calculations have been repeated by decreasing the control parameter β to get indistinguishable profiles. For the employed parameters, the resulting amplitudes of fluctuations -4 Å-6 for are G(2) ˜ S ) 1.0 and G(2) T (0) ) 4.86 × 10 T (0) ) 3.35 -4 -6 × 10 Å for ˜ S ) 0.5. Calculations of the SDF have been performed by employing rmin ) 4.5 Å, rmax = 100rmin ) 450 Å, rtrans ) 5 rmin, ∆1 ) 0.011 Å, and fampl ) 1.1. Such a discretization yields a representative matrix of dimensions 1666 × 1666. Lanczos procedure has been applied with nsteps ) 15 000. The analytical solution (see eqs 53 and 54 of Appendix A) is superimposed with circles to the profiles obtained from numerical calculations. As was previously discussed, the largest errors occur in the low-frequency subdomain (not shown) where the truncation on rmax implies the neglect of slow-relaxing modes and in turn is responsible for an unrealistic leveling off of the profile. However, the maximum deviation (negative) at 105 Hz is found to be less than 0.3% in both cases, allowing us to affirm that the calculation is accurate. The distinct profiles in Figure 3b reveal that the relaxation rate is indeed sensitive to the change of g(r). The percentage variation (with respect to the lower profile) is on the order of 25% at zero frequency and grows as the frequency increases: about 30% at 600 MHz and 35% at 2 × 600 MHz (such frequencies are in the range sampled by NOE measurements for proton-proton interactions). One may ask if such sensitivity to g(r) is only the result of the different amplitudes of the dipolar

Figure 3. Pair correlation function (a), spectral profiles of relaxation rates (b), and normalized relaxation rates (c) for the steplike potential: dependence on the stabilization energy (in kBT units). Profiles refer to ra ) 4.5 Å, rb ) 6.0 Å (∆ ) 1.5 Å), and to stabilization energies, ˜ S, of 1.0 (solid lines) and 0.5 (dashed lines). Other parameters are nS ) 2/30 Å-3 and DT ) 3.0 × 10-5 cm2 s-1. Superimposed circles represent the analytical solution.

fluctuations (i.e., a static effect) or if g(r) also affects the profiles on dynamic grounds. To answer this question, we have examined the quantity R(2)/G(2) T (0): we see from Figure 3c that the profiles still differ in the two cases, implying that the features of the intermolecular interactions also affect the relative dynamics among solute and solvent spins. Let us now consider the opposite situation where the stabilization energy is fixed, while the width of the segregation layer changes. The profiles in Figure 4 are referred to ˜ S ) 1.0, ra ) 4.5 Å, rb ) 6.0 Å (solid line), and rb ) 6.5 Å (dashed line). Numerical evaluation of the SDF has been performed with the same values of parameters rmin, rmax, rtrans, ∆1, fampl, and nsteps employed to get the profiles in Figure 3. From Figure 4b we see that such a change of g(r) leads to distinct spectral


Frezzato et al. employed to generate a function g(r) rapidly growing beyond a “hard-core” distance δ. The potential is defined as

V ˜ (r) ) 4˜ K

[(σr -- δδ) - (σr -- δδ) ], 12

6

r>δ

(26)

where σ is the “soft-core” distance parameter and ˜ K quantifies the interaction strength (note that eq 26 is nothing but a shifted Lennard-Jones potential, which is recovered for δ ) 0). In the force-free limit, one has V ˜ (∞) ) 0, while the minimum value V ˜ m ) - ˜ K is located at rm ) δ + 21/6(σ - δ). The corresponding pair correlation function is then

g(r) )

Figure 4. Pair correlation function (a), spectral profiles of relaxation rates (b), and normalized relaxation rates (c) for the steplike potential: dependence on the width of the segregation layer. Profiles refer to ˜ S ) 1.0, ra ) 4.5 Å, rb ) 6.0 Å (solid lines), and rb ) 6.5 Å (dashed lines). Other parameters are nS ) 2/30 Å-3 and DT ) 3.0 × 10-5 cm2 s-1. Superimposed circles represent the analytical solution.

profiles of the relaxation rate (percentage variations in the frequency range within 2 × 600 MHz are on the order of 8-10%). On the other hand, Figure 4c shows that the increasing width of the segregation layer (yet with a fixed distance of closest approach) does not alter much the behavior of R(2)/ G(2) ˜ S and ra affects the T (0). Thus, a change of ∆ with fixed relaxation rate mainly in a static way (amplitudes of fluctuations -4 Å-6 for r ) 6.0 Å and G(2)(0) ) are G(2) b T (0) ) 4.86 × 10 T 5.24 × 10-4 Å-6 for rb ) 6.5 Å). However, we remark that this self-compensating effect is not a general feature: slight deviations of R(2)/G(2) T (0) at different ∆ values appear when shorter distances of closest approach are considered. Pair Correlation Function from Kihara’s Potential. Kihara’s form26 for the mean-field solute-solvent potential is

{

0 e-(V˜ (r)-V˜ (∞))

r ra. 4. Obtainable Information from the Spectral Profiles In section 3 we have shown that the spectral profiles of the relaxation rate R(2)(ω), and therefore selected NMR observables, are quantitatively sensitive to the solvation features of the solute molecule (i.e., to the profile of the pair correlation function g(r)). However, this dependence is rather complex. In fact, g(r) is found to influence the relaxation rates by determining both the amplitude of fluctuations of the dipolar interactions (static effect) and the time scales of the relaxation process itself (dynamic effect). The exploratory work presented in section 3 is focused on the model situation of a spin-centered solute molecule. However, the tools we have developed can be applied to the general case of eccentric location of spins and allow for the numerical evaluation of the spectral profiles from an assumed or known g(r). We now focus on the inverse problem, that is, to establish what information can be obtained, in practice, from NMR measurements which probe the experimental profile of R(2)(ω) within a certain frequency window. If the functional form of the pair correlation function were known as further input information, one could fit its parameters to the experimental profile R(2)(ω). On the other hand, if the form of g(r) is not known a priori, one first asks if the interpretation of R(2)(ω) is unique or if multiple solutions are in principle possible; in other words, one asks if under some conditions several forms of g(r) may give R(2)(ω) profiles which are very similar and therefore indistinguishable within experimental error. To achieve a first level of understanding, we focus again on the situation of a centered spin I. Let us face the problem by first isolating the dynamic effect of the g(r) on the spectral profiles. We have highlighted that the distance of closest approach between solute and solvent strongly affects the NMR relaxation rates. Let us simplify the analysis by assuming that ra is known a priori (e.g., on the basis of steric hindrance). Moreover, to focus on the dynamic effects, we also fix the value of G(2) T (0): we consider pair correlation functions which have the same static effect on the relaxation rates. Under such conditions, can different distributions of the same amount of solvent molecules interacting with the central solute molecule lead (to a good approximation) to the same profile R(2)(ω)? Indeed, a positive answer would imply that (i) a degree of arbitrariness about the functional form of g(r) is inherent in the data interpretation and that (ii) under such constraints the

J. Phys. Chem. B, Vol. 110, No. 11, 2006 5685 features of g(r) do not significantly affect the dipolar spin relaxation on dynamic grounds. The constraint of an equal amount of solvent molecules interacting with the solute is given by the invariance of the average number of solvent molecules exceeding the “force-free” distribution. In formal terms, the three constraints to which we refer are summarized as follows

ra: fixed ∆NS ) nS

∫0∞ dr [g(r) - gff(r)]r2: ∫0∞ dr

G(2) T (0) ) nS

(33a)

fixed

(33b)

g(r) : fixed r4

(33c)

where gff(r) is the force-free pair correlation function (unit step) and ∆NS is the excess number of solvating spins; the spin number density, nS, is assumed as known and fixed in all the compared situations. If different forms of g(r) satisfying the conditions of eq 33 led to similar profiles of R(2)(ω), then the same quantities defined by eqs 33a-c would constitute the information obtainable from data analysis. To provide a direct answer we resort to numerical investigations. Let us consider again Kihara’s potential with parameters ˜ K ) 1.0, σ ) 4.65 Å, and δ ) 3.10 Å yielding G(2) T (0) ) 4.84 × 10-3 Å-6, ra ) 4.5 Å, and rb ) 6.0 Å. The steplike function which satisfies the conditions in eq 33 is characterized by ra ) 4.5 Å, rb ) 6.0 Å, and ˜ S ) 0.4462. Both functions g(r) are drawn in Figure 8a. In Figure 8b and c, we show the related spectral profiles R(2)(ω) and R(2)(ω)/G(2) T (0), which appear to be almost indistinguishable in the whole frequency window. Thus, these two different pair correlation functions can, under the appropriate constraints, virtually reproduce the same experimental data. Calculations have also been done also with a much higher stabilization energy: employed parameters are ˜ K ) 5.0 and the same soft- and hard-core distances, σ ) 4.65 Å and δ ) -3 Å-6, r ) 4.5 Å, and r 3.10 Å yielding G(2) a b T (0) ) 4.73 × 10 ) 5.4 Å. In this case, the spectral profile obtained from g(r) generated by Kihara’s potential is very close to that obtained from a steplike g(r) satisfying the conditions of eq 33. To get a deeper insight, we analyze the density of modes w(λ) associated with the two pair correlation functions in such a case of strong solute-solvent interaction. The profiles are presented in Figure 9 and reveal that only the modes at very high frequency (which globally contribute with a low cumulative weight) are affected by the change of function g(r). Similar comparisons (not shown here) have been done by parametrizing Kihara’s potential to generate a much greater distance of closest approach (parameters were ˜ K ) 1.0, σ ) -5 Å-6, 10.2 Å, and δ ) 8.2 Å, which give G(2) T (0) ) 2.73 × 10 ra ) 10.0 Å, and rb ) 11.9 Å). Moreover, comparisons have also been done between pair correlation functions generated by Morse’s and steplike potentials. In all the cases, we got the same outcome. Under the conditions of eq 33, substitution of g(r) generates, to an excellent approximation, the same profiles of the relaxation rate R(2)(ω). Calculations have been also performed for different values of the diffusion coefficient DT (much smaller or greater than the reference value 3.0 × 10-5 cm2 s-1) to see if the compared spectral profiles significantly differ within the explored frequency window (105-1011 Hz). No limitations are found on the possibility to substitute the functional form of g(r) under the conditions of eq 33, even if the diffusion coefficient is much increased or reduced.


Frezzato et al.

Figure 9. Mode density w(λ) for Kihara’s potential with ˜ K ) 5.0, σ ) 4.65 Å, and δ ) 3.10 Å, corresponding to ra ) 4.5 Å and rb ) 5.4 Å (solid line) and for the steplike function satisfying the conditions in eq 33 (dashed line). Other parameters are nS ) 2/30 Å-3 and DT ) 3.0 × 10-5 cm2 s-1.

Figure 8. Comparison between spectral profiles (b and c) corresponding to the pair correlation functions (a) generated by Kihara’s potential (solid lines) and the steplike function (dashed lines) satisfying the conditions of eq 33. Employed parametrization of Kihara’s potential is ˜ K ) 1.0, σ ) 4.65 Å, and δ ) 3.10 Å (corresponding to ra ) 4.5 Å and rb ) 6.0 Å). Parameters of the steplike function are ra ) 4.5 Å, rb ) 6.0 Å, and ˜ S ) 0.4462. Other parameters are nS ) 2/30 Å-3 and DT ) 3.0 × 10-5 cm2 s-1.

On the basis of such evidence, we can draw the following conclusion: under the conditions of eq 33, several pair correlation functions can reproduce the same spectral profile, thus leading to an ambiguity in the functional form of g(r). The underlying information that we can extract is thus limited to the distance of closest approach ra, to the amount of solvent interacting with the solute molecule, and to the amplitude of fluctuations of the dipolar interactions, G(2) T (0). As a consequence, the possibility to substitute the effective (yet unknown) pair correlation function with arbitrary functions satisfying the conditions in eq 33 allows one to adopt the most convenient form of g(r). For example, one might adopt the steplike (for which the analytical solution is available) or any

other functional form deemed more appropriate for the case in point, possibly on the basis of other evidence. In the ideal case of a centered solute spin I, the following procedure could be devised to interpret the profiles R(2)(ω), assuming that both nS and DT are known: 1. Choose the distance of closest approach, ra. 2. Fit the spectral profile, R(2)(ω), using the analytical solution for a steplike pair correlation function (see eqs 53 and 54 in Appendix A) with the distance rb and the amplification factor R as the fitting parameters. 3. Use the extracted parameters rb and R to evaluate G(2) T (0) and ∆NS. 4. Adopt a functional form for g(r) and parametrize it to reproduce the same values of ra, G(2) T (0), and ∆NS. A further step would be to generalize this analysis to the case of an eccentric location of spin I and to explore if, under proper conditions such as those of eq 33, different forms of g(r) still lead to very similar spectral densities, J(L) ˆ ), entering the L (ω summation in eq 6. In fact, if rank-dependent conditions analogous to eq 33c are considered, the eccentricity of spin I would prevent the possibility of freely choosing the form of g(r), thereby removing the degree of arbitrariness in the data interpretation. In this case, analysis of NMR data would also allow, in principle, the full characterization of the functional form of the solute-solvent pair correlation by means of the tools outlined in this work. Investigations along this line are currently in progress. 5. Conclusions and Outlook We have approached the classic description of intermolecular dipolar relaxation in liquids by including solute-solvent pair correlations in the general diffusion problem. We have provided the tools for computing the spectral profiles, R(2)(ω) (connected to measurable relaxation rates), for general pair correlations, and we have also given the analytical solution for the model case of a simple steplike distribution of solvent molecules around a central solute core. The spectral profiles are found to be affected by the type of mean-field interaction potential and by its parametrization. Thus,



the main assumption of NMR data being sensitive to the solvation features is confirmed, and the outlined results may be useful in the interpretation of NMR experiments involving translational diffusion-controlled relaxation. For example, the calculated spectral profiles can be easily related to NMR observables such as the cross-relaxation rates, σ, measured either in the static (NOE) or in the rotating (ROE) frames of reference

σNOE ) [0.6R(2)(ωI + ωS) - 0.1R(2)(ωI - ωS)] (34a) σROE ) [0.3R(2)(ωI) + 0.2R(2)(ωI - ωS)]

(34b)

where subscripts to the angular Larmor frequency ω indicate the observed (I) and perturbed (S) spins.2 On the other hand, a successful fitting depends on the quality and number of experimental data (relaxation rates), which need to be sampled in the widest possible frequency range. In this respect, magnetic relaxation dispersion (MRD)29-31 is probably a more powerful technique than intermolecular NOE, since the cross-relaxation rates observed with the latter technique result from a linear combination of spectral profiles sampled at only two Larmor frequencies (see eq 34). We have shown that, in the simplest case of solute and solvent molecules treated as spin-centered spheres, proper constraints set upon widely different pair correlation functions yield virtually identical spectral profiles in the whole frequency window sampled by NMR investigations. Such an outcome can be seen on one hand as a limitation of the NOE technique, but on the other hand, it allows one to adopt whatever solvation model (e.g., the steplike distribution of solvent) to fit the data and obtain the basic inVariants. These invariants are, as just mentioned, only a fraction of the complex information contained in the real g(r). Nevertheless, one should not get the impression that, for example, intermolecular NOEs are devoid of information: preferential solvation can still be characterized just by assuming an identical functional form of g(r) for both solvents and fitting the data to one such function for each solvent. In this same context, however, a general caveat applies. The application of NOE to preferential solvation studies seems to be quite critical, in that the SDFs generated by competing solvent molecules (with supposedly similar ra values) inevitably contain long-range dipolar contributions that mask the relevant phenomenon and flatten out the sensitivity of the experimental technique. Moreover, given the strong dependence of the SDF on the distance of closest approach, any inference on the features of g(r) from NOE data is additionally limited by possible uncertainty about this parameter. Finally, in the case of real molecules, steric hindrance would bring in an anisotropy of pair correlation functions, thus adding a further complication that our simple model ignores a priori. Acknowledgment. We acknowledge the kind support from Prof. Bertil Halle (Lund University, Sweden), who provided helpful notes concerning the analytical treatment reported in Appendix A, and we are grateful to Prof. Giorgio J. Moro (Padova University, Italy) for discussions and comments about the manuscript. Supporting Information Available: Appendices B-D give detailed description of the methods for numerical evaluation of eqs 18 and 20, derivation of the mode density w(λ) in the forcefree case, and approximations of J(2) T (ω) at low and high frequencies. This material is available free of charge via the Internet at http://pubs.acs.org.

Appendix A: Analytical Solution of Equation 20 for a Steplike Pair Correlation Function In this appendix we provide the explicit form of the L-rank spectral densities J(L) T (ω) appearing in eq 6. In doing so, we refer to the basic work of Ayant et al.18 and to its recently reviewed version by Halle,12 both approaching the problem of intermolecular dipolar relaxation by means of spin-carrying molecules treated as hard spheres. The pair correlation function resulting from a force-free diffusion problem is depicted in Figure 2a. Basically, we extend the results of refs 18 and 12 to the case of a step-concentration gradient sketched in Figure 2b. From a physical point of view, the probability to find solvent spins within a layer of width ∆ ) rb - ra around the solute molecule is uniformly scaled by factor R with respect to the same probability found in the bulk solution. To conform to the notation of ref 12, it will prove convenient to further define the propagator f (r0|r, t) as the joint probability of finding the central and solvent molecules separated by vector r0 at the generic time t0, and by vector r at time t0 + t

f (r0|r, t) ≡ p(r0)p(r0|r, t), p(r0) )

{

R/V ra e r0 < rb 1/V r0 g rb (35)

where p(r0) is the equilibrium distribution for the separation vector and p(r0|r,t) is the conditional probability already introduced in Section 2. The time evolution of f (r0|r,t) is governed by eqs 8 and 9, that is

∂ f (r0|r, t) ) ∇‚DT(r)p(r)∇p(r)-1f(r0|r, t) ∂t

(36)

where a dependence of the relative diffusion coefficient on the stochastic variable is kept for the sake of completeness. According to the cited authors, we adopt the simplest modeling DT(r) ) D(1) T for molecular separations below rb and DT(r) ) D(0) T elsewhere. We shall now investigate separately the cases where r0 < rb and r0 g rb by exploiting the two different initial conditions which follow directly from eq 35

f(r0|r, 0) )

{

Rδ(r - r0) ra e r0 < rb δ(r - r0) r0 g rb

(37)

The physical space of interest is thus partitioned into two homogeneous regions whereby the Fourier-Laplace transformed propagators ˜f (0) and ˜f (1) obey the equations

R (∇2 - κ12)f˜ (1)(r0|r,ω) ) - (1)δ(r - r0) DT

(38)

1 (∇2 - κ02)f˜ (0)(r0|r,ω) ) - (0)δ(r - r0) DT

(39)

with κn ) (iω/D(n) T ), and the following boundary conditions apply

∂ (1) ˜f (r0|r,ω)|r)rb ) 0 ∂r

(40)

˜f (0)(r0|r,ω)|rf∞ ) finite

(41)

Moreover, because of the symmetry of the problem, the transformed propagators can be expanded in terms of Legendre polynomials, PJ, so we may write


Frezzato et al.

∞

˜f (r0|r,ω) ) (n)

∑

F(n) J (r|r0)PJ(cos

η)

(42)

J)0 (n) (n) F(n) J (r|r0) ) RJ (r0)iJ(κnr) + βJ (r0)kJ(κnr)

(43)

for n ) 0, 1, where η is the angle between vectors r and r0 and iJ and kJ are modified spherical Bessel functions of the first and second kind,27 respectively. The F(n) J functions also depend on the frequency ω, but this specification is dropped for simplicity. To complete the definition of the problem, the effect of the step-concentration gradient must be also included in the continuity conditions imposed at the segregation frontier r ) rb to obtain (0) F(1) J (rb|r0) ) RFJ (rb|r0)

(44)

∂ (1) ∂ (0) D(1) F (r|r0)|r)rb ) D(0) F (r|r0)|r)rb T T ∂r J ∂r J

(45)

In this form, the unknown functions that specify the transformed propagators are F(n) J (r|r0). Substitution of eq 42 into eqs 38 and 39 reveals that such functions are continuous at r ) r0, yet their first derivatives are not. Specification of the behavior of F(n) J (r|r0) at the singularity r ) r0 allows one to recover the full dependence on r. To this purpose, the key point of Ayant’s derivation consists of reinterpreting eqs 38 and 39 in terms of Poisson’s equations for an equivalent electrostaticlike problem. With reference to the r0 < rb case, eq 38 becomes 2

∇2Φ(r) ) -4πq(r), q(r) ) -

κ1 R δ(r - r0) Φ(r) + 4π 4πD(1) T (46)

where the transformed propagator ˜f (1) has been replaced by the electriclike potential Φ(r) generated by the corresponding “charge” q(r) (a similar equation, without the multiplier R and (1) with κ0 and D(0) T in place of κ1 and DT , can be written after eq 39 for the case r0 g rb). In particular, q(r) is made of two contributions: a localized one, R/(4πD(1) T ), situated at r0, and a distributed one, -(κ12/4π)Φ(r), defined through the potential itself. In turn, the localized contribution is best thought of as originating from a surface charge density, σ, such that

∫0

2πr20

π

dη sin η σ(η) )

R 4πD(1) T

(47)

with σ(η) assumed to be nonnegligible only for η f 0. At this point we note that, while the potential Φ(r) is continuous in r0, the electriclike field, -∇Φ, is not. In particular, moving along the radial direction and invoking the Gauss theorem, the following discontinuity is found across the surface r ) r0

-

|

|

∂Φ ∂Φ + ) 4πσ(η) ∂r r)r0+ ∂r r)r0-

(48)

Moreover, σ can be customarily expanded in a basis set of Legendre polynomials as ∞

σ(η) )

F(1)< (r0|r0) ) F(1)> (r0|r0), J J

(50)

∂ (1)< 2J + 1 [F (r|r0) - F(1)> (r|r0)]r)r0 ) R 2 (1) J ∂R J 4πr D 0

(51)

T

>

(1) < where F(1) J (r|r0) is a shorthand notation for FJ (r ( |r0) when > < f 0. Similar conditions are derived for functions F(0) J (r|r0) (0) in the case r0 g rb, without the factor R and with DT in place of D(1) T . Then, consideration of the continuity conditions (eqs 50 and 51) within eq 43 allows one to derive the unknown (n) functions R(n) J (r0) and βJ (r0), thereby obtaining the full specification for ˜f (r0|r,ω). Ultimately, as a corollary to eq 7, the required spectral density J(L) T (ω) is obtained from the Fourier-Laplace transformed propagator as

J(L) T (ω) ) NS

∫ dr ∫ dr0 FL,0(r0)*FL,0(r) ˜f (r0|r,ω)

(52)

Because of of the space partition we made depending on r0, after lengthy elaboration eq 2.37 of ref 12 rearranges to

J(L) T (ω) )

(

2L-1 1 R + (1 - R)λ + 2L-3 2L - 1 ζ02 D(0) T ra

nS

RλL-1ζ1QL-1(ik) -

{

[

1 2L-2 λ ζ0kL-1(ζ0/λ)XL + R ζ1YL TkL

] })

2λL+1 k (ζ /λ) VL ζ0 L-1 0

(53)

(1) 1/2 2 where nS ) NS/V, λ ) ra/rb, γ ) (D(0) T /DT ) , ζn ) (iωra / (n) 1/2 DT ) , and the auxiliary quantities are

QL-1(ik) ) iL-1(ζ1)kL-1(ζ1/λ) - kL-1(ζ1)iL-1(ζ1/λ) Skk L ) γk′L(ζ0/λ)kL(ζ1/λ) - RkL(ζ0/λ)k′L(ζ1/λ) SLki ) γk′L(ζ0/λ)iL(ζ1/λ) - RkL(ζ0/λ)i′L(ζ1/λ) SikL ) γi′L(ζ0/λ)kL(ζ1/λ) - RiL(ζ0/λ)k′L(ζ1/λ) SiiL ) γi′L(ζ0/λ)iL(ζ1/λ) - RiL(ζ0/λ)i′L(ζ1/λ) TkL ) k′L(ζ1)SkiL - i′L(ζ1)Skk L TLi ) k′L(ζ1)SLii - i′L(ζ1)SLik k UL-1 ) kL-1(ζ1) - λL-1kL-1(ζ1/λ) i ) iL-1(ζ1) - λL-1iL-1(ζ1/λ) UL-1 i k + i′L(ζ1)UL-1 VL ) k′L(ζ1)UL-1

XL ) iL-1(ζ0/λ)TkL + kL-1(ζ0/λ)TiL

∑aJPJ(cos η), J)0 aJ )

and by taking into account that ˜f (1), namely the potential Φ(r) is similarly expanded, as in eq 42, then eqs 47 and 48 yield the required conditions

2J + 1 2

∫0π dη sin η PJ(cos η)σ(η)

i k YL ) SLkkUL-1 + SkiL UL-1 .

(49) We finally observe that, by assuming a thoroughly homogeneous

Nuclear Spin Relaxation (0) diffusion coefficient such that D(1) T ) DT ≡ DT, the value γ ) 1 applies in the previous equations.

References and Notes (1) Hertz, H. G. Prog. NMR Spectrosc. 1967, 3, 159 and references therein. (2) Neuhaus, D.; Williamson, M. The Nuclear OVerhauser Effect in Structural and Conformational Analysis; VCH: Weinheim, Germany, 1989. (3) Otting, G. Prog. NMR Spectrosc. 1997, 31, 259. (4) Brand, T.; Cabrita, E. J.; Berger, S. Prog. NMR Spectrosc. 2005, 46, 159. (5) Bagno, A.; Rastrelli, F. Saielli, G. Prog. NMR Spectrosc. 2005, 47, 41. (6) Wu¨thrich, K. NMR of Proteins and Nucleic Acids; Wiley: New York, 1986. (7) Kaiser, R. J. Phys. Chem. 1965, 42, 1838. (8) Balaram, P.; Bothner-By, A. A.; Dadok, J. J. Am. Chem. Soc. 1972, 94, 4015. (9) Glickson, J. D.; Dadok, J.; Marshall, G. R. Biochemistry 1974, 13, 11. (10) Pitner, T. P.; Dadok, J.; Marshall, G. R. Nature 1974, 250, 582. (11) Pitner, T. P.; Glickson, J. D. J. Am. Chem. Soc. 1975, 97, 5917. (12) Halle, B. J. Chem. Phys. 2003, 119, 12372. (13) Bentley, T. W.; Koo, I. S. Org. Biomol. Chem. 2004, 2, 2376 and previous works by the same authors. (14) Odelius, M.; Laaksonen, A.; Levitt, M. H.; Kowalewski, J. J. Magn. Reson., Ser. A 1993, 105, 289. (15) Grivet, J. P. J. Chem. Phys. 2005, 123, 034503. (16) Torrey, H. C. Phys. ReV. 1953, 92, 962; Torrey, H. C. Phys. ReV. 1954, 96, 690; Resing, H. A.; Torrey, H. C. Phys. ReV. 1963, 131, 1102.

J. Phys. Chem. B, Vol. 110, No. 11, 2006 5689 (17) Huang, L.-P.; Freed, J. H. J. Chem. Phys. 1975, 63, 4017. (18) Ayant, Y.; Belorizky, E.; Alison J.; Gallice, J. J. Phys. (Paris) 1975, 36, 991. (19) Albrand, J. P.; Taieb, M. C.; Fries, P. H.; Belorizky, E. J. Chem. Phys. 1983, 78, 5809. (20) Abragam, A. The Principles of Nuclear Magnetism; Clarendon: Oxford, U.K., 1961. (21) Ayant, Y.; Belorizky, E.; Fries, P.; Rosset, J. J. Phys. (Paris) 1977, 38, 325. (22) Gardiner, C. W. Handbook of Stochastic Methods; Springer: Berlin, 1994. (23) Rose, M. E. Elementary Theory of Angular Momentum; Wiley: New York, 1948. (24) Freed also derived an analytical expression of the SDFs for the force-free case under the assumption of finite jump dynamics (Freed, J. H. J. Chem. Phys. 1978, 68, 4034) and then recovered Ayant’s expressions in the limit of small-step diffusion. (25) Moro, G.; Freed, J. H. In Large Scale EigenValues Problems; Cullum, J., Willoughby, R. A., Eds.; Elsevier: Amsterdam, 1986; p 143. (26) Kihara, T. ReV. Mod. Phys. 1953, 25, 831. (27) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover Publications Inc.: New York, 1970. (28) Widder, D. V. An Introduction to Transform Theory; Academic Press: New York, 1971; p 126. (29) Venu, K.; Denisov, V. P.; Halle, B. J. Am. Chem. Soc. 1997, 119, 3122. (30) Halle, B.; Denisov, V. P. Methods in Enzymology 2001, 338, 178. (31) Halle, B. Philos. Trans. R. Soc. London B 2004, 359, 1207.

Nuclear Spin Relaxation Driven by Intermolecular Dipolar Interactions

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