Convergence of Coupling-Parameter Expansion-Based Solutions to Ornstein–Zernike Equation in Liquid State Theory

Round 1

Reviewer 1 Report

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Author Response

Reviewer-1

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Reviewer 2 Report

This manuscript is devoted to theoretical study of thermodynamic properties of liquids using the Ornstein–Zernike equation and different pair potentials. The manuscript contains new results and will be of interest to Condensed Matter readers.

The manuscript is very well written. However, I would like to make a few comments to make this work excellent.

It seems that the manuscript should be recommended for publication in Condensed Matter after minor improvement as listed in the attached PDF file.

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Reviewer-2

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Reviewer 3 Report

This paper analyzes the convergence of the coupling-parameter expansion (CPE) of the solutions of approximate integral equations in liquid state theory. The paper is technically well written and the author is an expert in the field. On the other hand, I cannot recommend acceptance because, in my opinion, the work does not represent a sufficiently important contribution to the field. 
In conventional perturbation theories of liquids (e.g., Barker-Henderson's), the statistical-mechanical properties of the reference system are assumed to be known (for instance, from the solution of an integral equation) but the subsequent perturbation terms are obtained by well defined (and formally exact) steps derived from  statistical mechanics.
In this work, however, the CPE is employed to solve approximately an approximate integral equation. What is the advantage of numerically solving the Sarkisov integral equation for Lennard-Jones fluids by means of the CLE instead of just solving the full problem altogether? What do we learn by replacing the well-known full analytical solutions of the Percus-Yevick and MSA equations for the adhesive hard-sphere and Yukawa models, respectively, by the approximate CLE partial solutions? If the answer is that we learn that the CLE converges whenever a real solution exists and does not converge otherwise, this seems to be a rather trivial conclusion.
Apart from this main criticism, other minor points are:
o p. 2 and Ref. 6: Yevic->Yevick
o p. 2: Anderson->Andersen
o p. 3: the the->the; repulsive repulsive->repulsive
o On p. 3 one reads "The latter is the appropriate function to be used in the OZE as it tends to zero for large $r$, just like the pair-potential." But this sentence applies to the direct correlation function $c(r)$ much more accurately than to the total correlation function $h(r)$.
o Is Eq. (5) entirely correct? It seems to me that the quantity $\rho$ should not be there. Otherwise, Eq. (5) is not dimensionally correct. The same applies to Eq. (11).
o In the square-well potential, setting the energy depth $u_0$ equal to 0 makes the potential become the hard-sphere one, regardless of the value of the width $\Delta$. In the CLE, this is equivalent to $\lambda=0$. What is then the physical meaning of the parameters $\tau_0$ and $\Gamma_0^*$ in Eqs. (20)-(22)?
o When referring to the HNC closure and its lack of real solutions (p. 11), it should be pointed out that HNC lacks a true critical point [see Brey and Santos, Mol. Phys. 57, 149 (1986) and references in Caccamo, Phys. Rep. 274, 1 (1996)].
o pp. 11 and 13: insert->inset
o p. 13: Eq.(??) is lacking the equation number
o p. 15: ppssible->possible; There results->The results
Ref. 28: Chyandler->Chandler

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Reviewer-3

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Round 2

Reviewer 1 Report

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Author Response

Reviewer#1-Round-2

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Reviewer 3 Report

Unfortunately, the revised manuscript does not change the substantial criticism of my first report: "... the work does not represent a sufficiently important contribution to the field" and "... the CPE is employed to solve approximately an approximate integral equation." Moreover, the new sentence "The parameters $\tau_0$ and $\Gamma_0^*$ correspond to the AHS model for depth $u_0 = 0$, but finite and small $\epsion$. The hard sphere limit is obtained only when $\epsilon\to 0$" is entirely wrong. Starting from the SW potential, the HS one is recovered by taking either $u_0=0$ for arbitrary $\epsilon$ or $\epsilon=0$ for arbitrary (finite) $u_0$. It is not necessary to take both limits simultaneously. 

Author Response

Reviewer-3-Round-2

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