VOLUME 62, NUMBER 25 PHYSICAL REVIEW LETTERS 19 JUXE 1989 Comprehensive Theory of Simple Fluids, Critical Point Included A. Parola Scuola Internazionale Superiore di Studi Avanzati, Strada Costiera I I, Trieste, Italy A. Meroni and L. Reatto Dipartimento di Fisica dell'Universita, Melano, Italy (Received 27 March 1989) We present a comprehensive theory of fluids which has the typical accuracy of a good liquid-state theory in the dense regime but in addition has a genuine nonclassical critical behavior. The theory is based on the hierarchical reference theory of fluids decoupled with an approximation inspired by the op- timized random-phase approximation. The Lennard-Jones interaction is studied in detail above the criti- cal temperature. PACS numbers: 61. 20.Gy, 05. 70. — a We do not have yet a liquid-state theory which is able to treat correctly also the region of the critical point of the liquid-vapor phase transition. Various liquid-state theories can be very accurate elsewhere in the phase dia- gram but none of them is correct even qualitatively close to the critical point in the sense of having scaling satisfied with reasonable critical exponents. Renormal- ization-group theory has not yet produced a practical scheme to compute nonuniversal quantities starting from a realistic model of a fluid. A few years back two of the authors' introduced a new scheme, the hierarchical reference theory (HRT), which held promise to fulfill this goal. This approach, however, leads to an infinite hierarchy of equations and it was not guaranteed that it would lead to a practical and accurate scheme of compu- tation. We have performed a decisive step to solve this prob- lem. In fact we show in this Letter how a well known and accurate liquid-state theory, the optimized random- phase approximation (ORPA) which has a rather trivial critical-point behavior, can be transformed into a theory with a nontrivial critical behavior. We can consider the new theory either as a sophisticated generalization of ORPA or as the lowest-order approximation of HRT which satisfies the core condition, i.e. , the vanishing of the radial distribution function g(r) inside the core. In ORPA the interatomic pair potential v(r) is de- composed into a repulsive part vR(r) and an attractive tail w(r) =v(r) — vR(r), and the eKect of w(r) on the properties of the reference system, i.e. , the system with interaction vR(r), is determined. However, following the HRT approach we turn on w(r) not just in one step but selectively in wave-vector space: We consider a family of partially coupled Q systems with potential vg(r) =v~(r)+wg(r), where the Fourier transform of wg(r) is wg(q), When Q =~, vg is just the reference system but when Q=0 the fully interacting system is recovered. This pro- cedure allows a gradual turning on of the critical fluctua- tions on diA'erent length scales because density fluctua- tions with k & Q are strongly depressed in the Q system. In fact in RPA the S(k) of the Q system for k & Q is simply the structure factor of the reference system, and only for k & Q do critical fluctuations show up. Notice, however, that the Q system is treated over all of its length scales and we do not trace out degrees of freedom as in renormalization-group calculations. The flow of the excess Helmholtz free energy Ag" of the Q system is given in three dimensions by the exact equation Q', „„p@g) dg 4 2 1 — pC g(g) (3) where A'" differs from A'" by analytic terms (see I), p(k) =— w(k)/k8T, and C g is related to the direct correlation function cg of the Q system by C g (k)— : cg (k) + p(k) — pg (k) (4) evaluated on the shell k =Q. The flow of C g is deter- mined by an equation which involves the three- and four-body direct correlation functions, but we truncate the problem at the first equation (3) of the hierarchy by an Ansatz for Pg(k). Precisely, for r &d we write C g(r) =cR(r)+age(r), (5) and C g(r) for r &d is determined by the condition of vanishing radial distribution function of the Q system: cg(k) gg (r) — = 1+ e'"' ' =0, for r & d . (6) (2') 3 1 — pcg(k) cR(r) is the direct correlation function of the reference system, d is its hard-core diameter, and kg is determined by the compressibility sum rule which reads w(q), for q & Q, 0, for q &Q. (2) Cg(k =0) =8 A"/Bp If lg =1, Eqs. (4)-(6) are simply the ORPA for the Q 1989 The American Physical Society 2981