^{1,a)}

### Abstract

We present a two-particle Monte Carlo method for computing the outer-sphere (OS) dipolar time correlation function (DTCF) of the relative position of a nuclear spin on a diamagnetic molecule with respect to a nuclear or electronic spin on a molecule when both molecules are anisotropic and undergo translational and rotational diffusion. As a first application, we question the validity of the appealing interspin procedure [L. P. Hwang, Mol. Phys.51, 1235 (1984); A. Borel *et al.*, Chem. Eur. J.7, 600 (2001)] based on the solutions of a Smoluchowski diffusion equation, which conserve the interspin radial distribution function in the course of time. We show that the true random spatial motion of the interspin vector obtained by simulation can be very different from that given by the Smoluchowski solutions and lead to notable retardation of the time decay of the OS-DTCF. Then, we explore the influence of the solvation properties of on the decay rate of the DTCF. When is significantly larger than , its rotation accelerates the decay only weakly, even if follows in its Brownian tumbling. By contrast, viscous solvation layers in OS pockets of can yield an important local slowdown of the relative translational diffusion of , leading to a decay retardation of the DTCF, which adds to that due to the shape anisotropy of . When is a -based contrast agent, this retardation leads to a notable increase of the OS contribution to relaxivity even at rather high imaging field.

This work is a contribution to the EC COST Action D-38 and European EMIL network.

I. INTRODUCTION

II. OUTER-SPHERE RELAXATION

III. SITE-SITE DISTRIBUTION FUNCTIONS AND MOLECULAR ANISOTROPY

A. Link with the molecular pair distribution function

B. Stegosauruslike molecule

IV. TWO-PARTICLE MONTE CARLO SIMULATION

A. Translational diffusion

B. Additional rotational diffusion

C. Local solvation slowdown

V. RESULTS

A. Retardation effects due to the molecular anisotropy of a nonrotating stegosaurus

B. Rotational drive

C. Local retardation effects

D. Outer-sphere relaxivity optimization

VI. DISCUSSION AND CONCLUSION

### Key Topics

- Diffusion
- 40.0
- Anisotropy
- 34.0
- Particle distribution functions
- 24.0
- Monte Carlo methods
- 16.0
- Rotational correlation time
- 12.0

## Figures

Change of the random trajectory of a hard point molecule of center due to the anisotropic plates of a stegosaurus molecule of center defined by Eqs. (6) and (7) and made of a central hard sphere of radius , a green equatorial hard plate, and three blue meridian hard plates of extension . Here, the stegosaurus has a fixed orientation in the course of time, practically, a very slow rotation of correlation time . Starting from the same initial position in a molecular frame rigidly bound to , the green and red dots are the successive positions of for typical random trajectories generated without (green) and with (red) the geometrical restraints of the plates. The lengths of the vertical bars are proportional to the distances written below between and traveling along the trajectories. The black bar represents the initial distance. The random displacements of with time can be viewed by playing movies_Fig1.gif (Ref. 58).

Change of the random trajectory of a hard point molecule of center due to the anisotropic plates of a stegosaurus molecule of center defined by Eqs. (6) and (7) and made of a central hard sphere of radius , a green equatorial hard plate, and three blue meridian hard plates of extension . Here, the stegosaurus has a fixed orientation in the course of time, practically, a very slow rotation of correlation time . Starting from the same initial position in a molecular frame rigidly bound to , the green and red dots are the successive positions of for typical random trajectories generated without (green) and with (red) the geometrical restraints of the plates. The lengths of the vertical bars are proportional to the distances written below between and traveling along the trajectories. The black bar represents the initial distance. The random displacements of with time can be viewed by playing movies_Fig1.gif (Ref. 58).

Effects of the anisotropic plates of the stegosaurus of Fig. 1 on the (a) MC OS-DTCF and (b) associated spectral density of the relative translational motion of centered spins and on and . These functions reflect the changes of the relative Brownian trajectories of due to three different kinds of plates of , which are assumed to be the following: (k1) the reference hard sphere of radius without plates, (k2) the stegosaurus of Fig. 1 with meridian plates, and (k3) a stegosaurus with meridian plates. For the hard sphere, the values of from the MC simulation superimpose with their analytical ABHF counterparts.

Effects of the anisotropic plates of the stegosaurus of Fig. 1 on the (a) MC OS-DTCF and (b) associated spectral density of the relative translational motion of centered spins and on and . These functions reflect the changes of the relative Brownian trajectories of due to three different kinds of plates of , which are assumed to be the following: (k1) the reference hard sphere of radius without plates, (k2) the stegosaurus of Fig. 1 with meridian plates, and (k3) a stegosaurus with meridian plates. For the hard sphere, the values of from the MC simulation superimpose with their analytical ABHF counterparts.

Change of the random trajectory of the molecule by rotational (r) driving of the stegosaurus having the initial orientation as defined in Fig. 1. Here, the stegosaurus drives in its tumbling (rotational effects RE2 of Sec. V B). (a) Moderately fast tumbling rate with . (b) Fast tumbling rate with . The green random trajectory of already shown in Fig. 1 is transformed into the red ones by the geometrical restraints of the plates and the rotational driving. The meaning of the other symbols is as in Fig. 1. The random displacements of with time can be viewed by playing movies_Fig3(a).gif and movies_Fig3(b).gif (Ref. 58).

Change of the random trajectory of the molecule by rotational (r) driving of the stegosaurus having the initial orientation as defined in Fig. 1. Here, the stegosaurus drives in its tumbling (rotational effects RE2 of Sec. V B). (a) Moderately fast tumbling rate with . (b) Fast tumbling rate with . The green random trajectory of already shown in Fig. 1 is transformed into the red ones by the geometrical restraints of the plates and the rotational driving. The meaning of the other symbols is as in Fig. 1. The random displacements of with time can be viewed by playing movies_Fig3(a).gif and movies_Fig3(b).gif (Ref. 58).

Effects of the rotational (r) driving of a molecule by the stegosaurus of Fig. 1 on the (a) MC OS-DTCF and (b) associated spectral density of the relative translational motion of centered spins and on and . The stegosaurus drives in its tumbling (rotational effects RE2 of Sec. V B) at the tumbling rates of Fig. 3.

Effects of the rotational (r) driving of a molecule by the stegosaurus of Fig. 1 on the (a) MC OS-DTCF and (b) associated spectral density of the relative translational motion of centered spins and on and . The stegosaurus drives in its tumbling (rotational effects RE2 of Sec. V B) at the tumbling rates of Fig. 3.

Change of the random trajectory of the molecule due to retarding solvation layers in the holes of the tumbling stegosaurus of Fig. 1. The stegosaurus starts tumbling with the same initial orientation as in Fig. 1. Here, the slow rotation of the stegosaurus with does not change the position of (indirect rotational effects RE1 of Sec. V B). The green random trajectory of already shown in Fig. 1 is transformed into the red ones by (a) the sole geometrical restraints of the plates (reference situation), (b) the same plate restraints and a rather modest local slowdown in the stegosaurus holes with , (c) the same plate restraints and a stronger local slowdown with The meaning of the other symbols is as in Fig. 1. The random displacements of with time can be viewed by playing movies_Fig5(a).gif, movies_Fig5(b).gif, and movies_Fig5(c).gif (Ref. 58).

Change of the random trajectory of the molecule due to retarding solvation layers in the holes of the tumbling stegosaurus of Fig. 1. The stegosaurus starts tumbling with the same initial orientation as in Fig. 1. Here, the slow rotation of the stegosaurus with does not change the position of (indirect rotational effects RE1 of Sec. V B). The green random trajectory of already shown in Fig. 1 is transformed into the red ones by (a) the sole geometrical restraints of the plates (reference situation), (b) the same plate restraints and a rather modest local slowdown in the stegosaurus holes with , (c) the same plate restraints and a stronger local slowdown with The meaning of the other symbols is as in Fig. 1. The random displacements of with time can be viewed by playing movies_Fig5(a).gif, movies_Fig5(b).gif, and movies_Fig5(c).gif (Ref. 58).

(a) MC hat OS-DTCF and (b) associated star relaxivity of the relative translational motion of a centered spin on the molecule with respect to centered (c) and eccentric (e) spins of on a molecule assumed to be the HS of radius without plates and without retarding solvation layers, the stegosaurus s5A of Fig. 5(a) without retarding layers, the stegosaurus s5C of Fig. 5(c) with retarding layers in its holes where has a slower relative translational diffusion such as . The slow rotation of with does not change the position of (indirect rotational effects RE1 of Sec. V B). The position of the eccentric spin is in the molecular frame . Note that the statistical errors on are larger for the eccentric spin because of the wider scattering of the values of in Eq. (18).

(a) MC hat OS-DTCF and (b) associated star relaxivity of the relative translational motion of a centered spin on the molecule with respect to centered (c) and eccentric (e) spins of on a molecule assumed to be the HS of radius without plates and without retarding solvation layers, the stegosaurus s5A of Fig. 5(a) without retarding layers, the stegosaurus s5C of Fig. 5(c) with retarding layers in its holes where has a slower relative translational diffusion such as . The slow rotation of with does not change the position of (indirect rotational effects RE1 of Sec. V B). The position of the eccentric spin is in the molecular frame . Note that the statistical errors on are larger for the eccentric spin because of the wider scattering of the values of in Eq. (18).

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