Spin relaxation theoretical simulation

Figure 3 Effect of the water exchange rate, kex, and the rotational correlation time, rR, on inner-sphere proton relaxivity. The plot was simulated for a particular value of the longitudinal electron spin relaxation rate, 1/Tie — 5.28xlOss 1. The marketed contrast agents all have relaxivities around 4—5mM 1s 1 in contrast to the theoretically attainable values over lOOrnM-1 s 1, and this is mainly due to their fast rotation...

FIGURE 10.5 Contributions to in the theoretical simulation of spin relaxation in the radical pairs from DCA-POZ and DCA-PSZ evaluated under the assumption of 0 (data points). The full simulations are represented by the curves denoted / -POZ/DQtot.A 0.45 and fc-PSZ/DQtot, respectively. The contribution from the esdi mechanism (k-esdi )=6E-7) corresponds to an effective translational diffusion constant of D = 6 x 10 cm s. The curves indicated as / -POZ/DQ.A represent the contributions of the ahfi mechanism, the indicating the factor by which the theoretical anisotropy parameter A is reduced. The curve indicated hy k-PSZ/DQ.g denotes the contribution due to the g tensor anisotropy in the PSZ radical. The constant values c POZ and c PSZ represent the field-independent contributions to k. For details of the calculation cf. Ref. 23. 

Because we assume that our reader is most likely more of a theoretician, working in the area of computer simulations, rather than an NMR specialist, we will start with some background in nuclear spin relaxation. It gives us a good opportunity to discuss the relaxation models from a simulators point of view -as well as - to present the expressions to implement the method. Also, we believe that the material should be valuable to the reader from the NMR community, because it both shows how naturally the formalism is incorporated into the simulation techniques and demonstrates the benefits in employing MD simulations to evaluate the theoretical models and interpret experimental relaxation data. NMR relaxation [8,9] contains information of processes on molecular time scales, from nanoseconds to picoseconds, which perfectly coincides with the time scales of MD simulations (Figure 5). Since MD simulations are based on molecular interaction models, they can be used to elucidate and extract molecular information... 

In a theoretical treatment, it is necessary to make approximations in the derivation of the spectral densities (Appendix A.2 - equation (A7)), that is, the Fourier transforms of time correlation functions of perturbations used to express the nuclear spin relaxation times. These theories have been tested against experiments and their limitations have been examined under varying conditions. The advantage of MD simulations to evaluate the theoretical models is the realism of the description and that many approximations in the theoretical model can be tested separately. Because of the conceptual differences between theories and the arbitrariness in their parameterization, it is often not possible discriminate between... 

The studies of intermolecular quadnipolar relaxation with MD simulations (and Monte Carlo simulations) was initiated by Engstrom et al. [49] in the beginning of the eighties [50-54]. The problem then concerned the nuclear spin relaxation of ions in water. Very successful and well accepted theoretical models of the electrostatic mechanism had been developed, and with computer simulations it was possible to examine some of the assumptions of these models [37,38]. Furthermore, the performance of the electrostatic models could be compared to that of theoretical models of the collision induced mechanism [36,55]. 

Experimentally, the activation energy, Ea, of the nuclear spin relaxation has been studied systematically to evaluate different theoretical models [71]. Reproducing activation energies constitute a crucial test for MD simulations of the relaxation mechanism. It has been studied in MD simulations for both inert and ionic solutes [62,66]. For Ne, Kr, and Xe in acetonitrile [66], it was difficult to relate the Ea for the relaxation and those for individual molecular processes. This reflects the general problem of rationalizing Ea for collective processes. 

MD simulations of nuclear spin relaxation in liquids were initiated at a time when the development of theoretical models for many mechanisms was more or less stagnant. Since then simulations have been used in combination with both theory and experiment to develop new ideas, and MD simulations is becoming recognized as a vital tool for the understanding of the relaxation processes. 

Dunand et describe a global analysis of EPR, O NMR relaxation and chemical shift and proton NMRD profiles for aqueous [Gd(D0TAXH20)] and [Gd(DTPA)(H20)] The analysis is more detailed than in previous studies in that it incorporates theoretical advances in EPR relaxation theory (primarily involving the reorientation of the static zfs tensor) as well as the results of MD simulations describing the internal motions of bound water. The authors conclude that complexes in which Gd is in a symmetric chemical environment should have relatively long electron spin relaxation times, thus enhancing proton relaxivity. They also conclude that, because of permanent zfs interactions, very low frequency NMRD-data and slowly tumbling complexes cannot be analysed by their methods. 

Since the CW-ESR spectra provides structural information and dynamics at different time scales, proper account of fast and slow motion of the labeled molecules is required for correct reproduction of the spectra. While the fast motion can be derived from a fast-motional perturbative model, in the slow-motion regime the effects on the spin relaxation processes exerted by the molecular motions requires a more sophisticated theoretical approach. The calculation of rotational diffusion in solution can be tackled by solving the stochastic Liouville equation (SLE) or by longtime-scale molecular dynamics simulations [94—96]. 

Several papers have dealt with theoretical aspects of electron spin relaxation in transition metal and lanthanide complexes. The topic is of central importance for understanding the PRE. I choose to discuss these papers here even if they do not explicitly discuss the nuclear spin relaxation. Hakansson reported an X-band ESR study of Cu(ii)-porphyrin in fluid and frozen solution. The experimental data were interpreted using a simulation method based on stochastic Liouville equation (SEE). The analysis provided the porphyrin rotational correlation time (in the nanosecond range) combined with a fast local motion, as well as an... 

Computational methods have been applied to study the conformations of free and metal-complexed oxathiacrown ethers 1-4 shown in Figure 6. The results were compared to variable temperature NMR and i spin-lattice relaxation time measurements <2001JP2988>. Theoretical studies included simulated 111 NMR spectra using PERCH and molecular modeling with PM3 semi-empirical quantum-chemical calculations. The NMR and the computational data both show that Ag+ coordinates equally well to S and O atoms, Bi3+ and Sb3+ prefer O atoms, and that Ptz+ and Pdz+ prefer exo-cyclic coordination only to the S atoms in this maleonitrile macrocycle. 

We will present the topic by introducing the nuclear spins as probes of molecular information. Some basic formal NMR theory is given and connected to MD simulations via time correlation functions. A large number of examples are chosen to demonstrate different possible ways to combine MD simulations and experimental NMR relaxation studies. For a conceptual clarity, the examples of MD simulations presented and discussed in different sections, are arranged according to the specific relaxation mechanisms. At the end of each section, we will also specify some requirements of theoretical models for the different relaxation mechanisms in the light of the simulation results and in terms of which properties these models should be parameterized for conceptual simplicity and fruitful interpretation of experimental data. 

P.-O. Westlund, T. Larsson, and O. Teleman, Paramagnetic Enhanced Proton Spin-Lattice Relaxation in the Ni " " Hexa-aquo Complex. A Theoretical and Molecular Dynamics Simulation Study of the Bloembergen-Morgan Decomposition Approach, Mol. Phys., 78 (1993), 1365-1384. 

---