Reduced density-functions discussion

J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, The Molecular Theory of Gases and Liquids, Wiley, New York, 1960. [A classic text on which most of the discussion of reduced density functions given here was based.]... 

However, billiard balls are a pretty bad model for electrons. First of all, as discussed above, electrons are fermions and therefore have an antisymmetric wave function. Second, they are charged particles and interact through the Coulomb repulsion they try to stay away from each other as much as possible. Both of these properties heavily influence the pair density and we will now enter an in-depth discussion of these effects. Let us begin with an exposition of the consequences of the antisymmetry of the wave function. This is most easily done if we introduce the concept of the reduced density matrix for two electrons, which we call y2. This is a simple generalization of p2(x1 x2) given above according to... 

From the discussion so far, it is clear that the mapping to a system of noninteracting particles under the action of suitable effective potentials provides an efficient means for the calculation of the density and current density variables of the actual system of interacting electrons. The question that often arises is whether there are effective ways to obtain other properties of the interacting system from the calculation of the noninteracting model system. Examples of such properties are the one-particle reduced density matrix, response functions, etc. An excellent overview of response theory within TDDFT has been provided by Casida [15] and also more recently by van Leeuwen [17]. A recent formulation of density matrix-based TD density functional response theory has been provided by Furche [22]. 

For simplicity, we shall commonly refer to the Q-electron distribution function as the 2-density and the 2-electron reduced density matrix as the 2-ntatrix. In position-space discussions, the diagonal elements of the 2-ntatrix are commonly referred to as the 2-density. In this chapter, we will also refer to the diagonal element of orbital-space representation of the Q-vaatnx as the 2-density. 

In addition to density functional models, MP2 models provide a good account of activation energies for organic reactions (see discussion in Chapter 9). Unfortunately, computer time and even more importantly, memory and disk requirements, seriously limit their application. One potential savings is to base the MP2 calculation on Hartree-Fock orbitals which have been localized. This has a relatively modest effect on overall cost, but dramatically reduces memory and disk requirements, and allows the range of MP2 models to be extended. 

Figure 9 shows the temperature dependence of the recovered kinetic rate coefficients for the formation (k bimolecular) and dissociation (k unimolecular) of pyrene excimers in supercritical CO2 at a reduced density of 1.17. Also, shown is the bimolecular rate coefficient expected based on a simple diffusion-controlled argument (11). The value for the theoretical rate constant was obtained through use of the Smoluchowski equation (26). As previously mentioned, the viscosities utilized in the equation were calculated using the Lucas and Reichenberg formulations (16). From these experiments we obtain two key results. First, the reverse rate, k, is very temperature sensitive and increases with temperature. Second, the forward rate, kDM, 1S diffusion controlled. Further discussion will be deferred until further experiments are performed nearer the critical point where we will investigate the rate parameters as a function of density. 

This chapter mainly focuses on the reactivity of 02 and its partially reduced forms. Over the past 5 years, oxygen isotope fractionation has been applied to a number of mechanistic problems. The experimental and computational methods developed to examine the relevant oxidation/reduction reactions are initially discussed. The use of oxygen equilibrium isotope effects as structural probes of transition metal 02 adducts will then be presented followed by a discussion of density function theory (DFT) calculations, which have been vital to their interpretation. Following this, studies of kinetic isotope effects upon defined outer-sphere and inner-sphere reactions will be described in the context of an electron transfer theory framework. The final sections will concentrate on implications for the reaction mechanisms of metalloenzymes that react with 02, 02 -, and H202 in order to illustrate the generality of the competitive isotope fractionation method. 

There are several problems in the physics of quantum systems whose importance is attested to by the time and effort that have been expended in search of their solutions. A class of such problems involves the treatment of interparticle correlations with the electron gas in an atom, a molecule (cluster) or a solid having attracted significant attention by quantum chemists and solid-state physicists. This has led to the development of a large number of theoretical frameworks with associated computational procedures for the study of this problem. Among others, one can mention the local-density approximation (LDA) to density functional theory (DFT) [1, 2, 3, 4, 5], the various forms of the Hartree-Fock (HF) approximation, 2, 6, 7], the so-called GW approximation, 9, 10], and methods based on the direct study of two-particle quantities[ll, 12, 13], such as two-particle reduced density matrices[14, 15, 16, 17, 18], and the closely related theory of geminals[17, 18, 19, 20], and configuration interactions (Cl s)[21]. These methods, and many of their generalizations and improvements[22, 23, 24] have been discussed in a number of review articles and textbooks[2, 3, 25, 26]. 

Jossi et al. (1962) presented a generalized correlation for the viscosity of high density fluids as a function of the reduced density via a corresponding states method. This method was discussed earlier. Among the gases that Jossi et al. (1962) used to build their correlation were carbon dioxide, methane, ethane, and propane. This gives us some confidence that this approach should be satisfactory for our acid gas mixtures. 

Ho et al. (1990) have presented an approach to the description of independent kinetics that makes use of the method of coordinate transformation (Chou and Ho 1988), and which appears to overcome the paradox discussed in the previous paragraph. An alternate way of disposing of the difficulties associated with independent kinetics is intrinsic in the two-label formalism introduced by Aris (1989, 1991b), which has some more than purely formal basis (Prasad et al, 1986). The method of coordinate transformation can (perhaps in general) be reduced to the double-label formalism (Aris and Astarita, 1989a). Finally, the coordinate transformation approach is related to the concept of a number density function s(x), which is discussed in Section IV,B.5. 

Electron correlation in atoms and molecules finds a very natural expression in terms of the reduced density matrices within the formalism of second quantization. The fundamental relations of this theory lead here to a discussion of the correlation effects and their connexion with two properties of the group functions frequently studied by Prof. Ede Kapuy strong orthogonality and the concept of independence. The other aim of this paper is to examine the direct expression of the correlation effects in terms of reduced density matrices. 

The Sanchez-Lacombe equation-of-state provides a good example to help clarify the rather abstract discussion given above. It will now be discussed further. It is given by Equation 3.26 for a pure molecular liquid or gas. The variable r is defined by Equation 3.27, where M is the molecular weight and R is the gas constant. If T, p and p are known, Equation 3.26 can be solved iteratively to estimate the density as a function of temperature and pressure. Since the reduced density p depends on M through the variable r defined by Equation 3.27, it is not equal for all molecules at the same combination of T and p values. Consequently, for ordinary molecules, the Sanchez-Lacombe equation-of-state is not a corresponding states theory. 

An alternative approach starts from the HF method and adds a gradient-corrected density functional for the correlation energy[96,97], which reduces the severity of the HF underbinding of molecules. Fuentealba et al.[96] have found that the Wilson-Levy correlation functional provides a more realistic correction to HF dissociation energies than does even PW91. However, results still closer to experiment (especially for molecules like C2 and 02) are found by taking both exchange and correlation within GGA, as in Table 5, presumably for the reasons discussed at the end of section 5.1. 

In the present section we first review some of the standing difficulties appearing in density matrix theory, such as the IV-representability of the the reduced 2-matrix. We also discuss the nature of the functional" v- and iV-representability problems in the Hohenberg-Kohn versions of density functional theory. We show how the neglect of functional iV-representability leads to an ill-posed variational problem. We then indicate how these difficulties can be removed from density functional theory by a reformulation based on local-scaling transformations, 21, 22]. 

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