Elements of the Scaled-Particle Theory

In the preceding section we discussd the work required to create a cavity in the liquid. This concept is fundamental in the study of the solvation of solutes in any solvent. The simplest solute is a hard-sphere (HS) particle, and the simple solvent also consists of HS particles. We shall see in section 6.14 that the solvation of any solute in any solvent can always be decomposed into two parts, creating a suitable cavity and then turning on the other parts of the solute-solvent interaction. The scaled-particle theory (SPT) provides an approximate procedure to compute the work required to create a cavity. 

Originally, the SPT was devised and used for the study of a hard-sphere (HS) fluid. It was later also found useful and successful for simple fluids such as inert gases in the liquid state. More recently the SPT has also been applied to aqueous solutions. The basic ingredients of the SPT and the nature of the approximation involved are quite simple. We shall present here only a brief outline of the theory, skipping some of the more complicated details. 

In a fluid of HS particles, the sole molecular parameter that fully describes the particles is the diameter a.f It is important to bear this fact in mind when the theory is applied to real fluids, in which case one needs at least two molecular parameters to describe the molecules, and more than two parameters for complex molecules such as water. It is a unique feature of the HS fluid that only one molecular parameter is sufficient for its characterization. 

The fundamental distribution function in the SPT is Po(r), the probability that no molecule has its center within the spherical region of radius r centered at some fixed point Ro in the fluid. Let Po r + dr) be the probability that a cavity of radius r dr empty. (In all the following, a cavity is always assumed to be centered at some fixed point Ro, but this will not be mentioned explicitly.) This probability may be written as 

Clearly, since P idrjr) is the conditional probability of finding the spherical shell empty, given that the sphere of radius r is empty, the rhs of (5.11.3) is the conditional probability of finding the center of at least one particle in this spherical shell, given that the sphere of radius r is empty. 

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Potential 1 is extremely simple the only parameter is the radius of the sphere. In spite of this simplicity, an impressive number of physical results have been obtained using the HS potential on the whole range of densities and aggregations. Also, mixtures of liquids have been successfully treated, introducing in the computational machinery the desired number of spheres with appropriate radii. The HS model is at the basis of the Scaled Particle Theory (SPT), which still constitutes a basic element of modem solvation methods (see later for more details). 

The first attempt to explain the mechanism of the CM process was made in [1-2]. An example with the creation of two electron-hole pairs as a result of one photon absorption, was considered in the second order of the perturbation theory. In its first step the Hamiltonian of the electron-photon interaction, giving rise to band-to-band transition, was used. The virtual state of the electron-hole (e-h) pair. lr ) was situated on the energy scale not far from the final state Ixr). The second virtual transition between the states lr ) and Irx) was calculated using the matrix element of the Coulomb electron-electron interaction, which describes the scattering of one particle with the simultaneous creation of an e-h pair. This matrix element is much smaller than the diagonal one. Nevertheless, the general enhancement of the Coulomb interaction introduced by the size confinement could favor to the realization of this mechanism. 

The transformation of the time-dependent function Pj into a momentum operator is consistent with Einstein s description of light in terms of particles (photons), each of which has momentum hv (Sect. 1.6 and Box 2.3). We can interpret the quantum number rij in Eq. (3.50) either as the particular excited state occupied by oscillator j, or as the number of photons with frequency Vj. The oscillating electric and magnetic fields associated with a photon can stiU be described by Eqs. (3.44) and (3.45) if the amplitude factor is scaled appropriately. However, we will be less concerned with the spatial properties of photon wavefunctions themselves than with the matrix elements of the position operator Q. These matrix elements play a central role in the quantum theory of absorption and emission, as we ll discuss in Chap. 5. 

Now that we have described how to reduce the scaling behavior for the construction of the Coulomb part in the Fock matrix (Eq. [9]), the remaining part within HF theory, which is as well required in hybrid DFT, is the exchange part. The exchange matrix is formed by contracting the two-electron integrals with the one-particle density matrix P, where the density matrix elements couple the two sides of the integral ... 

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