# Correlation Functions in the Cluster Variation Method [closed]

I've been referring to Dr Mohri's paper [DOI: 10.1007/s11837-013-0738-5] for the cluster variation method (CVM). I wish to calculate the configurational entropy of a binary FCC system.

Cluster probabilities are probabilities of certain clusters existing in the system. Let's consider nearest neighbour (NN) pairs as clusters. A binary FCC cell (let's say AB type) can have $$\ce{AA}$$, $$\ce{AB}$$, $$\ce{BA}$$, and $$\ce{BB}$$ pairs. For this, $$y_{\ce{A,A}}$$ would be the probability of $$\ce{AA}$$ pairs existing. If the total number of pairs were $$N$$ then the number of $$\ce{AA}$$ pairs would be $$N\cdot y_{\ce{A,A}}$$.

In the CVM, we define $$n$$-point correlation functions $$\xi_n$$, which are used to calculate cluster probabilities, as:

$$\tag{1}\xi_n=\frac{\sum_{p_1}\sum_{p_2}...\sum_{p_n}\sigma(p_1)\sigma(p_2)...\sigma(p_n)}{N(n)}\label{1}$$

In the case of NN pairs, we use a 2-point correlation function:

$$\tag{2}\xi_2=\frac{\sum_{p}\sum_{p'}\sigma(p)\sigma(p')}{N(2)}\label{2}$$

where $$N(2)$$ is the total number of pairs in the crystal (cell), $$p$$ are each of the lattice points, $$p'$$ are the NN points of $$p$$, and $$\sigma(p)$$ ($$\sigma(p')$$) are spin variables which are taken to be $$+1$$ if $$\ce{A}$$ is at $$p/p'$$ and $$-1$$ if $$\ce{B}$$ is at $$p/p'$$.

The general probability formula is: $$y_{{i_1}{i_2}...{i_n}}=\frac{1}{2^n}\bigg[1+\sum_{j=1}^nC_j(i_{k})\xi_j\bigg]\tag{3}$$ where $$C_j(i_k)$$ is all the unique j-member combinations of $$i_k$$ values.

I understand the correlation formula up to pairs. I want to use it for bigger clusters like NN triangles and NN tetrahedrons.

Let's consider NN triangles now. In the $$3$$-point correlation function formula, we'll have $$p_1$$, $$p_1$$'s NN $$p_2$$, and one of these:

1. $$p_1$$ and $$p_2$$'s NN $$p_3$$, that is the 3rd vertex of all possible NN triangular clusters containing $$p_1$$ and $$p_2$$.
2. $$p_2$$'s NN.

I assumed 1 to be correct since it makes sense for the idea of considering NN triangular clusters, but when I use it I get negative values for certain cluster probabilities.

To be specific, I used it for NN tetrahedrons which have $$2^4 = 16$$ possible clusters ($$\ce{AAAA}$$, $$\ce{BBBB}$$, $$\ce{AAAB}$$, $$\ce{AABB}$$, $$\ce{BBBA}$$, and so on). I got a negative value for the cluster probability for $$\ce{BBBB}$$, i.e. $$y_{\ce{BBBB}} < 0$$. So my query is, how come I got a negative cluster probability?

It works perfectly well as long as I work with just NN pairs.

Update: The following are values I got for an $$AB_3$$ type FCC cell:

Point correlation function: -0.5
Pair correlation function: -0.0658682634730539
Tetrahedron correlation function: 0.034482758620689655
Triangle correlation function: 0.2328767123287671
xA = 0.25
xB = 0.75
yA,A = -0.016467065868263474
yA,B = 0.26646706586826346
yB,A = 0.26646706586826346
yB,B = 0.48353293413173654
wA,A,A,A = -0.07211067944413488
wA,A,A,B = -0.07654919259261354
wA,A,B,A = -0.07654919259261354
wA,A,B,B = 0.02760427421020029
wA,B,A,A = -0.08478272552674529
wA,B,A,B = 0.01937074127606855
wA,B,B,A = 0.01937074127606855
wA,B,B,B = 0.09843645255544649
wB,A,A,A = 0.014019669682835561
wB,A,A,B = 0.11817313648564941
wB,A,B,A = 0.11817313648564941
wB,A,B,B = 0.19723884776502737
wB,B,A,A = 0.12640666941978113
wB,B,A,B = 0.2054723806991591
wB,B,B,A = 0.2054723806991591
wB,B,B,B = 0.1602533596010624


The cell in POSCAR (VASP) format:

POSCAR
3.5639592128110129
0.00000000 -0.50000000 -1.50000000
2.00000000  1.00000000 -1.00000000
2.00000000 -2.50000000  0.50000000
A   B
12  36
Direct
1.00000000  1.00000000  1.00000000
0.25000000  0.25000000  0.25000000
0.29166667  0.29166667  0.45833333
0.45833333  0.45833333  0.29166667
0.08333333  0.58333333  0.41666667
0.12500000  0.62500000  0.62500000
0.62500000  0.62500000  0.12500000
0.66666667  0.66666667  0.33333333
0.70833333  0.70833333  0.54166667
0.37500000  0.87500000  0.87500000
0.45833333  0.95833333  0.29166667
0.91666667  0.91666667  0.58333333
0.04166667  0.04166667  0.20833333
0.50000000  1.00000000  0.50000000
0.08333333  0.08333333  0.41666667
0.54166667  0.04166667  0.70833333
0.12500000  0.12500000  0.62500000
0.58333333  0.08333333  0.91666667
0.16666667  0.16666667  0.83333333
0.62500000  0.12500000  0.12500000
0.20833333  0.20833333  0.04166667
0.66666667  0.16666667  0.33333333
0.70833333  0.20833333  0.54166667
0.75000000  0.25000000  0.75000000
0.33333333  0.33333333  0.66666667
0.79166667  0.29166667  0.95833333
0.37500000  0.37500000  0.87500000
0.83333333  0.33333333  0.16666667
0.41666667  0.41666667  0.08333333
1.00000000  0.50000000  1.00000000
0.87500000  0.37500000  0.37500000
0.04166667  0.54166667  0.20833333
0.91666667  0.41666667  0.58333333
0.50000000  0.50000000  0.50000000
0.95833333  0.45833333  0.79166667
0.54166667  0.54166667  0.70833333
0.58333333  0.58333333  0.91666667
0.16666667  0.66666667  0.83333333
0.20833333  0.70833333  0.04166667
0.25000000  0.75000000  0.25000000
0.29166667  0.79166667  0.45833333
0.75000000  0.75000000  0.75000000
0.33333333  0.83333333  0.66666667
0.79166667  0.79166667  0.95833333
0.83333333  0.83333333  0.16666667
0.41666667  0.91666667  0.08333333
0.87500000  0.87500000  0.37500000
0.95833333  0.95833333  0.79166667

• +1. But are there better tags you can use? You've added two brand new tags that are not yet being "watched" by any people! Jun 27 '20 at 21:44
• Cluster-variation-method could be clubbed with cluster-expansion but that isn't being watched either. I hope more questions with these tags pop up since these are interesting techniques in the study of alloys, especially disordered alloys. I was sceptical about cluster-correlation-functions, and I guess it doesn't make sense to add it. I'll remove that one. Jun 27 '20 at 21:50
• 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
Jun 28 '20 at 15:29
• @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. Jun 28 '20 at 17:26
• If you sum over one/two indices of the triangle/tetrahedron probabilities, due you get the corresponding pair probability?
– Tyberius
Jun 28 '20 at 18:34