Time-correlation function of the flux operator

In Section 5.2, we have seen how the rate constant for a chemical reaction may be determined as a time integral of the auto-time-correlation function of the flux operator 

In order to use this formal expression in a calculation of the rate constant we need to choose a representation. In the following we will determine the coordinate representation of the correlation function. We use the coordinate representation of the flux operator as derived above. It is introduced in the expression for the time-correlation function by introducing three unit operators like 

We see that there are four terms to be evaluated, and each term is evaluated by partial integration with respect to, respectively, c/, , q, q , and r/ . They may look a little complicated, so let us take them term by term. Let us start with an evaluation of the two terms associated with [ - q,q The first term in [ -]qiq , in Eq. (F.61) gives 

Then we need to evaluate the two terms associated with the [ operator in each 

Similarly, the result in Eq. (F.63) is combined with the first term in 

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