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

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

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:

$c_1(t)=⟨P_2(\mathbf{n}(0)\cdot \mathbf{n}(t))⟩,$

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

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

References

• 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)

See also

• rotational diffusion
• diffusion
• autocorrelation