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 $\mathbf{n}$, its autocorrelation will be given by

$c_1(t)=⟨\mathbf{n}(0)\cdot \mathbf{n}(t)⟩.$

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:

$c_2(t)=⟨P_2(\mathbf{n}(0)\cdot \mathbf{n}(t))⟩,$

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 ${\tau }_1=\frac{1}{2D_{\mathrm{r}\mathrm{o}\mathrm{t}}}$, and the relaxation time for $c_2(t)$ would be ${\tau }_2=\frac{1}{6D_{\mathrm{r}\mathrm{o}\mathrm{t}}}$. 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

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	${\tau }_2=2.0$ps (H-H dipolar relaxation NMR)
OH	O-H axis	${\tau }_2=1.95$ps (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

• Rotational diffusion
• Diffusion
• Autocorrelation

References

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)