# Rotational relaxation

**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 **n**, its autocorrelation
will be given by

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:

where is the second Legendre polynomial.

According to simple rotational diffusion theory, the relaxation time for would be given by , and the relaxation time for would be . Therefore, . This ratio is actually lower in simulations, and closer to ; the departure from a value of 3 signals rotation processes "rougher" than what is assumed in simple rotational diffusion (Ref 1).

## 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 | ps (H-H dipolar relaxation NMR) |

OH | O-H axis | ps (O-H dipolar relaxation NMR) |

dipolar axis | not measurable, but related to bulk dielectric relaxation | |

normal to the molecule plane | not measurable |