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 \[c_1(t)=\langle \mathbf{n}(0)\cdot\mathbf{n}(t) \rangle.\] 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).
[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 | \(\tau_2=2.0\)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 |
