What is Hydrogen bond Auto-correlation function Can somebody explain the Hydrogen bond Auto-correlation function in Computational Chemistry (Molecular Dynamics and Simulations) and what does Intermittent and Continuous Hydrogen bond means? 

Why the graph decays over time? and it's applications?

This is with reference to the paper: Chandra, A. Effects of Ion Atmosphere on Hydrogen-Bond Dynamics in Aqueous Electrolyte Solutions. Phys. Rev. Lett. 2000, 85 (4), 768–771. DOI: 10.1103/PhysRevLett.85.768.
In eqn 1 and 2 the correlation functions are written in the form $&lt;A B &gt;/&lt;A&gt;$ , the denominator  is the average of $A$ and the numerator the correlation function. This is the the integral product $U(t_d)=\int A(t) B(t-t_d)dt$ where $t_d$ is some time delay. When $A$ and $B$ are the same thing then this is the auto-correlation. It measure how similar the signal A is to B at different times. If $A$ and $B$ are random signals (such as white noise, or from a random number generator) then $U$ is 1 at zero time delay (both signals exactly superimposed) and zero at all other delays and this is because the correlation has no 'memory'; any point in a random signal is not related to any other. If $A$ and $B$ are constant then $U$ is one at all times, i.e perfect 'memory'. In your example the H bonds have a certain lifetime because are being broken due to molecular motion (as explained in the paragraph under eqns 1 &amp; 2) and so the correlation function decays away. A plot of the correlation gives the lifetime of the H bonds.