Properties of Renormalized Interactions

We have introduced three renormalized interactions V, W, and r(z). An understanding of their properties is essential for deriving useful approximations to G. We first consider the purely static quantity V. By using (51) we have 

A general technique for dealing with quantities of the form ((Lijf(l))/(2)/(3) /(n) has been discussed elsewhere/ Using these results one can show that 

We note that the quantity Li(ll ) is the same as Li(ll ) except the bare interaction potential is replaced by the average potential —/3 Mn g(ti—ri-). This replacement involves a summation of an entire class of diagrams in a direct density expansion of the memory function. The Fourier transform of V2 over the variable f] can be written in the following form  

We next consider the static interaction W, which is similar to V, but is considerably more complicated. We see that W plays essentially the same role in the equation for G that plays in the memory function equation for C 

It is useful to note that, for moderate densities, a reasonable approximation is 

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