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:

$c_1(t)=\langle P_2( \mathbf{n}(0)\cdot\mathbf{n}(t) ) \rangle,$

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 $τ 1 = 1 / 2 D rot$ , and the relaxation time for $c 2 (t)$ would be $τ 2 = 1 / 6 D rot$ . Therefore, $τ 1 = 3τ 2$ . 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).

[edit] Water

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	$τ 2 = 2.0$ ps (H-H dipolar relaxation NMR)
OH	O-H axis	$τ 2 = 1.95$ ps ( $17$ O-H dipolar relaxation NMR)
$μ$	dipolar axis	not measurable, but related to bulk dielectric relaxation
$\perp$	normal to the molecule plane	not measurable