# NMR spectroscopy

In NMR spectroscopy, we exploit that atomic nuclei, which exhibit a non-zero nuclear spin, interact with external and internal fields. The external fields are the static magnetic flux density B0 and the time-dependent magnetic flux density B1, which are used to generate a suitable NMR signal. The nuclear magnetic moments, which are associated with the nuclear spins, interact with the magnetic fields B0 and B1. Presence of the field B0 results in Zeeman splitting, i.e., the possible orientations of the nuclear magnetic moments with respect to the field differ in energy, where the energy difference corresponds to the Larmor frequency. Application of the alternating magnetic field B1 in a direction perpendicular to B0 and in form of radio-frequency pulses leads to transitions between these Zeeman levels provided the resonance condition is fulfilled. The resonance frequencies of these transitions depend on additional interactions of the atomic nuclei with internal magnetic or electric fields, which reflect the local properties of the sample. Thus, their determination provides access to microscopic information about the studied substance.

When using NMR spectroscopy to investigate molecular dynamics, it is exploited that, due to these internal interactions, the resonance frequencies of the nuclear spins depend on the molecular positions or molecular orientations. Hence, molecular rotational or translational dynamics render the resonance frequency of a nucleus time dependent. Various NMR techniques can be used to determine this time dependence.

Very detailed insights are available when we utilize multi-pulse sequences to correlate the resonance frequency of a nucleus at several times, i.e., to measure multi-time correlation functions. Two-time correlation functions provide straightforward access to the correlation time of the molecular dynamics. For disordered materials, the correlation decays non-exponentially in the majority of the cases. Analysis of three-time correlation functions enables determination of the origin of the non-exponentiality. Often, this effect is a consequence of the existence of dynamical heterogeneities, i.e., presence of a broad distribution of correlation times. Four-time correlation functions allow us to determine how long a particle remembers its initial correlation time. Hence, we can measure the time scale of exchange processes between fast and slow particles from the distribution.

While multi-time correlation functions provide detailed insights into slow molecular dynamics in the micro- and milliseconds regimes, field-cycling relaxometry yields valuable information about faster dynamics in the pico- and nanoseconds regimes. In field-cycling experiments, the spin-lattice relaxation time T1, which describes the buildup of the magnetization in the magnetic field, is measured as a function of the Larmor frequency, providing access to the spectral density of an observed dynamical process.

The above described experiments are carried out in homogeneous magnetic fields B0. When, instead, we use inhomogeneous magnetic fields, the Larmor frequency depends on the position of the nuclear spin. Akin to clinical applications (MRI), we exploit this spatial variation of the Larmor frequency to perform spatially resolved investigations of heterogeneous materials. Moreover, we employ such field-gradient experiments to measure molecular self-diffusion coefficients.

# Molecular dynamics (MD) simulations

Performing MD simulations, the interactions between the particles of a system are described by classical potentials. For the many-particle system, Newton's equations of motion are solved per iteration. In this way, one obtains a deterministic trajectory of the system in phase space. Thus, complete microscopic information about the simulated system is available. Observables are calculated by averaging along the trajectory, using ergodicity hypothesis, which predicts equality of time and ensemble averages. The significance of MD simulations depends on the availability of a realistic interaction potential. Often, reliable potentials can be obtained from quantum chemical calculations. The time window of MD simulations is determined by computer power. Nowadays, assuming a computer time of one week, it is possible to follow the time evolution of a classical 10000 particle system for about 100 nanoseconds.

When striving for MD simulations of larger systems for longer times, it is useful to move from all-atom to coarse-grained descriptions. In such coarse-grained models, several atoms are grouped together to form a particle. While this approach allows one to strongly reduce the number of pair interactions to be calculated, it reamains an active field of research to derive coarse-grained interaction potentials from the underlying all-atom force fields in a systematic manner. We tackle this problem in the framework of a DFG collaborative research center (TRR 146).