# Derivation of the form of the time correlation function in NMR spectroscopy

For a given protein, I know that the NMR spectrometer magnet generates a field $B_0$ and that the interactions with the spins in the local environment generates a much smaller field $B_\mathrm{loc}$ (not necessary aligned with $B_0$).

Owing to Brownian collisions between solvent molecules and the protein, the atoms associated with these spins move, hence making $B_\mathrm{loc}$ a function of time ($B_0$ is constant).

We define a time correlation function$$G(t,\tau) = \overline{B_\mathrm{loc}(t)B_\mathrm{loc}(t+\tau)},$$ where $G(t,\tau)$ is a stationary random function (see Time evolution of correlation functions (specifically Onsager's hypothesis) in time correlation link)
Hence $G(t,\tau)$ only depends on $\tau$, the delay in measuring $B_\mathrm{loc}$.

So, for simplicity we set $t = 0$ and note that

$$G(t,\tau) = \overline{B_\mathrm{loc}(0)^2} \mathrm{e}^{-\tau/\tau_\mathrm{c}},$$

where $\tau_\mathrm{c}$ is the correlation time, the time it takes for the whole molecule to rotate by 1 radian in a process called rotational diffusion.

How do I derive the second equation from the first?

• You can't prove this, simply because it is not generally true. But it still makes sense to assume this functional form because it seems to fit to many systems. – Nino Mar 24 '17 at 15:34

Your function $G(\tau)= \langle B_x(t)B_x(t+\tau) \rangle$ is an autocorrelation function. If it is only a function of $\tau$, the time delay, and not t, this is called the stationary assumption.

The fluctuating field has zero average $\langle B_x(t) \rangle =0$. The bracket implies averaging over a long time for a single spin or average over many spins at a particular time, which is the ergodic hypothesis.

The magnitude of the fluctuating fields is the mean square fluctuation $\langle B_x^2(t)\rangle \ne 0$ and is positive thus the mean square is not zero.

It is also necessary to know how rapidly the fields fluctuate and this is where the autocorrelation $G(\tau)= \langle B_x(t)B_x(t+\tau) \rangle \ne 0$ comes in. We compare the field at one point t with another point $t+\tau$ some time later. If $\tau$ is small then the value of $B_x(t)$ and $B_x (t+ \tau )$ are similar and $B_x(t)B_x(t+\tau)$ is large and positive, by comparison if $\tau$ is large the product $B_x(t)B_x(t+\tau)$ is small approaching zero as $\tau$ increases.

Thus the general shape is that $G(\tau)$ is large at small $\tau$ and decays away to zero with increase in $\tau$. Often we assume that the decay is exponential thus $$G(\tau)= \langle B_x^2 \rangle \mathrm{e}^{-|\tau|/\tau_\mathrm{c}}$$ where $\tau_\mathrm{c}$ is a correlation time. This has the correct qualitative form but is difficult to justify on the basis of proper theory.

You state that $\tau_\mathrm{c}$ is caused by rotational diffusion thus it will, in general, be small in fluid solution and large in viscous ones. Of course $\tau_\mathrm{c}$ also depends on the temperature (smaller at larger $T$) and size of the whole molecule's rotation (as a prolate or oblate ellipsoid) or of a mobile group therein, depending upon which property you are observing. Groups in a protein can undergo 'wobbling in a cone' type motion in addition to rotation of the whole protein. Note also that if the molecule is not spherical then it will have more than one rotational relaxation time so $G(\tau)$ may decay with more than one $\tau_\mathrm{c}$. Hope that this helps!

• Thanks, so no way to prove that assumption? I am assuming spheric molecule and there is an average $\tau_c$ for whole protein. – HighEnergy Jul 21 '16 at 20:04
• And if you could explain the real significance of the spectral density function I get when I Fourier Transform $G(\tau)$ that would be great. It is of the form $J(\omega) = \frac{2\tau_c}{1+\omega^2\tau_c^2}$ – HighEnergy Jul 21 '16 at 20:06
• I am trying to prove the exponential quality of $G(\tau)$, so all help would be appreciated. – HighEnergy Jul 21 '16 at 20:13
• @HighEnergy The Fourier transform of an exponential is the Lorentzian function $J(\omega)$ so its the same thing as $G$ but in the frequency domain. It may make it easier to interpret to look at it this way. You should also probably look at the Weiner-Kninchin relations as these allow you to find the power spectrum (or spectral density) of an autocorrelation. – porphyrin Jul 22 '16 at 7:47
• You gave me information that I needed (and knew somewhat since I had looked at the Weiner-Kninchin relations). I am curious for a given frequency what $J(\omega)$, the spectral density function, means. Also, could I use the fluctuation-dissipation theorem to show that because NMR relaxation (dissipation) is exponentially decaying the time correlation function of $\mathrm{B_{loc}}$ (fluctuation owing to Brownian collisions) should as well? That is my ultimate goal. – HighEnergy Jul 22 '16 at 13:44