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

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 $\tau _{1}=1/2D_{\mathrm {rot} }$ , and the relaxation time for $c_{2}(t)$ would be $\tau _{2}=1/6D_{\mathrm {rot} }$ . 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

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	$\tau _{2}=2.0$ ps (H-H dipolar relaxation NMR)
OH	O-H axis	$\tau _{2}=1.95$ ps ( $^{17}$ O-H dipolar relaxation NMR)
$\mu$	dipolar axis	not measurable, but related to bulk dielectric relaxation
$\perp$	normal to the molecule plane	not measurable