Rotational relaxation

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Rotational relaxation refers to the decay of certain autocorrelation magnitudes related to the orientation of molecules. If a molecule has an orientation along a unit vector {\mathbf n}, its autocorrelation will be given by

c_1(t)=\langle \mathbf{n}(0)\cdot\mathbf{n}(t) \rangle.

From the time decay, or relaxation, of this function, one may extract a characteristic relaxation time (either from the long-time exponential decay, or from its total integral, see autocorrelation). This magnitude, which is readily computed in a simulation is not directly accessible experimentally, however. Rather, relaxation times of the second spherical harmonic are obtained:

c_2(t)=\langle P_2(  \mathbf{n}(0)\cdot\mathbf{n}(t) ) \rangle,

where P_2(x) is the second Legendre polynomial.

According to simple rotational diffusion theory, the relaxation time for c_1(t) would be given by \tau_1 = \frac{1}{2D_\mathrm{rot}}, and the relaxation time for c_2(t) would be \tau_2 = \frac{1}{6D_\mathrm{rot}}. Therefore, \tau_1= 3 \tau_2. This ratio is actually lower in simulations, and closer to 2; the departure from a value of 3 signals rotation processes "rougher" than what is assumed in simple rotational diffusion (Ref 1).

Water[edit]

Main article Rotational relaxation of water

Often, molecules are more complex geometrically and can not be described by a single orientation. In this case, several vectors should be considered, each with its own autocorrelation. E.g., typical choices for water molecules would be:

symbol explanation experimental value, and method
HH H-H axis \tau_2=2.0ps (H-H dipolar relaxation NMR)
OH O-H axis \tau_2=1.95ps (17O-H dipolar relaxation NMR)
\mu dipolar axis not measurable, but related to bulk dielectric relaxation
\perp normal to the molecule plane not measurable

See also[edit]

References[edit]

  1. David van der Spoel, Paul J. van Maaren, and Herman J. C. Berendsen "A systematic study of water models for molecular simulation: Derivation of water models optimized for use with a reaction field", J. Chem. Phys. 108 10220 (1998)