Blip function theory

The blip function theory was originally developed as a way of calculating the thermodynamic properties and the pair correlation function of a fluid whose potential is continuous, positive, and repulsive. For an atomic liquid. 

The starting point was the knowledge of the thermodynamic properties of a hard-sphere fluid, one whose potential uair) is 

The difference u (r) - Ua (r) can be very large and infinite for particular values of r, so it does not make sense to perform an expansion in powers of this difference. The corresponding Mayer / functions 

We shall give a brief graphical derivation of a general form of the theory. Suppose we are interested in calculating jd and g for a fluid with some potential u (xi, X2). Suppose we know sd and g for another fluid with some other potential Uo(xi, X2) at the same temperature and density. This other fluid will be called the reference fluid, sdo and go denote sd and g for the reference fluid. The Mayer / functions for the fluid of interest and the reference fluid are 

The cluster series for sd and g are given in Eqs. (16) and (17). The series for sio and go are of exactly the same form, and we include them here for convenience  

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Calculation of the second-order term in Eq. (3.5.4) and the first-order term in Eq. (3.5.5) requires knowledge of the triplet distribution function in the reference fluid which is usually replaced by the Kirkwood superposition approximation. Following Smith, we will refer to the approach as a whole as the reference averaged Mayer (RAM) function theory. Another choice of reference system based upon a division of the Mayer function is that of hard spheres with a diameter chosen so that the first-order term in the free energy vanishes. This gives rise to the so called blip function theory. ... 

The Golden Rule formula (9.5) for the mean rate constant assumes the Unear response regime of solvent polarization and is completely equivalent in this sense to the result predicted by the spin-boson model, where a two-state electronic system is coupled to a thermal bath of harmonic oscillators with the spectral density of relaxation J(o)) [38,71]. One should keep in mind that the actual coordinates of the solvent are not necessarily harmonic, but if the collective solvent polarization foUows the Unear response, the system can be effectively represented by a set of harmonic oscillators with the spectral density derived from the linear response function [39,182]. Another important point we would like to mention is that the Golden Rule expression is in fact equivalent [183] to the so-called noninteracting blip approximation [71] often used in the context of the spin-boson model. The perturbation theory can be readily applied to... 

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