I haven't been able to look at the paper yet, but am I understanding correctly that the cluster probabilities are functions of the value of the correlation function? In that case, you are either using the wrong formula for the probability or the correlation function. Regardless of how you select your clusters the correlation function has to be between -1 and 1. As long as the correlation function is in that range, even if the cluster selection doesn't make sense, the probability function should have to return a value between 0 and 1.

@Tyberius thank you for the MathJax edit. Value of the correlation functions are between -1 and 1. One thing does match though, the sum of cluster probabilities for each kind of cluster (points, pairs, and tetrahedrons) does equal to 1. I rechecked the formulae a few times and they are exactly what is in this paper.

If you sum over one/two indices of the triangle/tetrahedron probabilities, due you get the corresponding pair probability?

I also noticed there is a typo in the paper for the general probability expression. For the terms related to $\xi_2$, there should be an $i_2i_3$ rather than $i_1i_3$

@Tyberius yes, I do get corresponding probabilities. I also noticed how negative and positive pair or triangle or tetrahedron probabilities on summation end in point probabilities that are always positive. This makes sense since point probabilities are atom fractions as such and they can't be negative. I am starting to think that maybe pair and higher probabilities mean something other than cluster concentrations. Should I add a sample of these calculated values to the question?

@Tyberius I don't think that is a typo. Consider n = 3, then the biggest cluster would be a triangle in which vertex 1 forms a pair with each of the other two vertices. Hence, $i_1i_2$, $i_1i_3$, and $i_2i_3$.

@Tyberius The issue with any of these terms being negative is that there are natural logs of these terms in the configurational entropy formula.

A sample of the calculated values may help, as otherwise it seems like you are doing everything right.

@HitanshuSachania You are probably right about the "typo". I misinterpreted the ... in their formula.

@Tyberius I'll add the values then. Thank you for your interest.

Added the data.

@HitanshuSachania it looks like you got a negative probability for just the pairs as well.

@HitanshuSachania at a minimum, something must be wrong with $\xi_2$. $y_{AA}$ reduces down to $\xi_2/4$