Density function definition

This is a central quantity in chemistry [14], which was introduced by Pauling as the power of an atom in a molecule to attract electrons to itself and was quantified originally by thermochemical data [15]. The density functional definition of this quantity can be viewed as a generalization of the Mulliken definition [16] and is in accordance with earlier work of Iczkowski and Margrave [17]. 

We do not distinguish here this density functional definition of exchange energy from that of Hartree-Fock (HF). This simplification is well-justified, if the HF electron density and the exact electron density differ only slightly [40]. Similarly, the coupling-constant averaged exchange-correlation hole is the usual... 

All this is in contradiction to the density functional definition of electronegativity as presented originally by Parr[36], where it is also inferred that the electronegativity for all the (natural) orbitals of the system should be the same. The discrepancy between these two different derivations and points of view[29,36] is still to be resolved. In addition, the global definition of electronegativity as promulgated within density functional theory, does not appear to be particularly useful, as it yields the same... 

But now, in analogy with the probability function or density function definite state, as defined above (p. 131), we can... 

By exchange, we mean the density functional definition of exchange, in which the wavefunction is a Slater determinant whose density is the exact density of the interacting system, and which minimizes the energy of the non-interacting system in the Kohn-Sham external potential, iv,a=o- Another useful concept is the pair distribution function, defined as[14]... 

The density functional definition of the hardness parameter can in fact be obtained by introducing the concept of local hardness, n(r), through... 

Using the reduced density function definitions from Chap. 3... 

Besides the already mentioned Fukui function, there are a couple of other commonly used concepts which can be connected with Density Functional Theory (Chapter 6). The electronic chemical potential p is given as the first derivative of the energy with respect to the number of electrons, which in a finite difference version is given as half the sum of the ionization potential and the electron affinity. Except for a difference in sign, this is exactly the Mulliken definition of electronegativity. ... 

Equation (3-104) (sometimes called the stationarity property of a probability density function) follows from the definition of the joint distribution function upon making the change of variable t = t + r... 

We must now verify that the definition (3-219) does indeed generate a possible set of probability density functions. First of all, pXtfn, as defined by Eq. (3-219) is everywhere non-negative, and, moreover,... 

Kozlowski, P. M., Rauhut, G., Pulay, P., 1995, Potential Symmetry Breaking, Structure and Definite Vibrational Assignment for Azulene Multiconfigurational and Density Functional Results , J. Chem. Phys., 103, 5650. 

The statistical fundamentals of the definition of CV and LD are illustrated by Fig. 7.8 showing a quasi-three-dimensional representation of the relationship between measured values and analytical values which is characterized by a calibration straight line y = a + bx and their two-sided confidence limits and, in addition (in z-direction) the probability density function the measured values. 

A demand density is by definition a continuous density function defined over the set of all nonnegative numbers, thus demands are always positive or zero. 

Each of these columns of this symmetrical matrix may be seen as representing a molecule in the subspace formed by the density functions of the N molecules that constitute the set. Such a vector may also be seen as a molecular descriptor, where the infinite dimensionality of the electron density has been reduced to just N scalars that are real and positive definite. Furthermore, once chosen a certain operator in the MQSM, the descriptor is unbiased. A different way of looking at Z is to consider it as an iV-dimensional representation of the operator within a set of density functions. Every molecule then corresponds to a point in this /V-dimensional space. For the collection of all points, one can construct the so-called point clouds, which allow one to graphically represent the similarity between molecules and to investigate possible relations between molecules and their properties [23-28]. 

The aim of this chapter is to discuss chemical reactivity and its application in the real world. Chemical reactivity is an established methodology within the realm of density functional theory (DFT). It is an activity index to propose intra- and intermolecular reactivities in materials using DFT within the domain of hard soft acid base (HS AB) principle. This chapter will address the key features of reactivity index, the definition, a short background followed by the aspects, which were developed within the reactivity domain. Finally, some examples mainly to design new materials related to key industrial issues using chemical reactivity index will be described. I wish to show that a simple theory can be state of the art to design new futuristic materials of interest to satisfy industrial needs. 

In this book, an alternative description based on the joint probability density function (PDF) of the species concentrations will be developed. (Exact definitions of the joint PDF and related quantities are given in Chapter 3.) The RTD function is in fact the PDF of the fluid-element ages as they leave the reactor. The relationship between the PDF description and the RTD function can be made transparent by defining a fictitious chemical species... 

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