Editing 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
:<math>c_1(t)=\langle \mathbf{n}(0)\cdot\mathbf{n}(t) \rangle.</math>
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 harmonics|spherical harmonic]] are obtained:
:<math>c_1(t)=\langle P_2(  \mathbf{n}(0)\cdot\mathbf{n}(t) ) \rangle,</math>
where <math>P_2(x)</math> is the second [[Legendre polynomials|Legendre polynomial]].

According to simple [[rotational diffusion]] theory, the relaxation time
for <math>c_1(t)</math> would be given by
<math>\tau_1 = 1/2D_\mathrm{rot}</math>, and the relaxation time for
<math>c_2(t)</math> would be <math>\tau_2 = 1/6D_\mathrm{rot}</math>.
Therefore, <math>\tau_1= 3 \tau_2</math>. This ratio is actually lower in simulations,
and closer to <math>2</math>; 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:

{| border="1"
|- 
| symbol || explanation || experimental value, and method
|-  
| HH   || H-H axis  ||  <math>\tau_2=2.0</math>ps  (H-H dipolar relaxation NMR)
|-  
| OH   || O-H axis  ||  <math>\tau_2=1.95</math>ps (<math>^{17}</math>O-H dipolar relaxation NMR)
|-
| <math>\mu</math> || dipolar axis || not measurable, but related to bulk dielectric relaxation
|-
| <math>\perp</math>   || normal to the molecule plane|| not measurable
|-
|}


==References==

*[http://dx.doi.org/10.1063/1.476482 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]]


[[Category: Non-equilibrium thermodynamics]]