Basic principles of density functional theory

The Basic Principles of Density Functional Theory 7.2.1 Preliminaries... 

In the last years, it has been found that the intuitive concepts of electronegativity, hardness, and softness are closely related to basic variables of density functional theory [5]. Thus, through the use of this theory it has been possible to establish expressions that allow one to quantify these concepts, and it has also been possible to demonstrate the principles associated with them. The purpose of the present section is to review some fundamental aspects of density functional theory, and to establish the expressions for several reactivity parameters. 

The purpose of this work is to start from the basic equations of density functional theory to describe the changes in the energy associated with the transition from one ground-state to another, in terms of different sets of variables. In this process one will find the natural definitions of the hardness and softness kernels, the local hardness, the local softness, the global hardness and the global softness [23]. Then, we will proceed to establish their relation with ionization potentials and electron affinities, in order to confirm their behavior as a measure of chemical hardness or softness [14, 24]. Finally, this theoretical framework will be used to analyze the maximum hardness and the HSAB principles. 

It is however possible to obtain a physically meaningful representation of 0(r) for cations, in the context of density functional theory. The basic expression here is the fundamental stationary principle of DFT, which relates the electronic chemical potential ju, with the electrostatic potential and the functional derivatives of the kinetic and exchange-correlation contributions [20] ... 

One branch of quantum Monte Carlo research aims at providing a quantitative first-principle description of atoms, molecules, and solids beyond the accuracy of density functional theory." " If the basic physics and chemistry of the material in question is well understood at least qualitatively, as is the case for many bulk semiconductors, for example, good trial wave functions such as the Jastrow-Slater type can be constructed. These functions can... 

In this section, we will only discuss the basic principles of kinetic theory, where for detailed derivations we refer to the classic textbook by Chapman and Cowling (1970), and a more recent book by Liboff (1998). Of central importance in the kinetic theory is the single particle distribution function /s(r, v), which can be defined as the number density of the solid particles in the 6D coordinate and velocity space. That is, /s(r, v, t) dv dr is the average number of particles to be found in a 6D volume dv dr around r, v. This means that the local density and velocity of the solid phase in the continuous description are given by... 

During the last decade, density-functional theory (DFT)-based approaches [1, 2] have advanced to prominent first-principles quantum chemical methods. As computationally affordable tools apt to treat fairly extended systems at the correlated level, they are also of special interest for applications in medicinal chemistry (as demonstrated in the chapters by Rovira, Raber et al. and Cavalli et al. in this book). Several excellent text books [3-5] and reviews [6] are available as introduction to the basic theory and to the various flavors of its practical realization (in terms of different approximations for the exchange-correlation functional). The actual performance of these different approximations for diverse chemical [7] and biological systems [8] has been evaluated in a number of contributions. 

Density functional theory (DFT) uses the electron density p(r) as the basic source of information of an atomic or molecular system instead of the many-electron wave function T [1-7]. The theory is based on the Hohenberg-Kohn theorems, which establish the one-to-one correspondence between the ground state electron density of the system and the external potential v(r) (for an isolated system, this is the potential due to the nuclei) [6]. The electron density uniquely determines the number of electrons N of the system [6]. These theorems also provide a variational principle, stating that the exact ground state electron density minimizes the exact energy functional F[p(r)]. 

However, a question arises - could similar approach be applied to chemical reactions At the first stage the general principles of the system s description in terms of the fundamental kinetic equation should be formulated, which incorporates not only macroscopic variables - particle densities, but also their fluctuational characteristics - the correlation functions. A simplified treatment of the fluctuation spectrum, done at the second stage and restricted to the joint correlation functions, leads to the closed set of non-linear integro-differential equations for the order parameter n and the set of joint functions x(r, t). To a full extent such an approach has been realized for the first time by the authors of this book starting from [28], Following an analogy with the gas-liquid systems, we would like to stress that treatment of chemical reactions do not copy that for the condensed state in statistics. The basic equations of these two theories differ considerably in their form and particular techniques used for simplified treatment of the fluctuation spectrum as a rule could not be transferred from one theory to another. 

The following is a very short outline of the basic ideas of the relevant theoretical methods and aims at giving experimental chemists an understanding of the underlying principles. For those readers who wish to learn more about present methods in computational chemistry, we recommend the textbook Introduction to Computational Chemistry by Jensen". An excellent book about the theory and application of DFT given from a chemist s point of view is A Chemist s Guide to Density Functional Theory by Koch and Holthausen". Two reviews are available which discuss the application of ECPs to heavy atom molecules . We also mention the Encyclopedia of Computational Chemistry which contains a large number of reviews written by experts about nearly all aspects of the field". 

First-principles simulations are techniques that generally employ electronic structure calculations on the fly . Since this is a very expensive task in terms of computer time, the electronic structure method is mostly chosen to be density functional theory. Apart from the possibility of propagating classical atomic nuclei on the Born-Oppenheimer potential energy surface represented by the electronic energy V (R ) = ji(R ), another technique, the Car-Parrinello method, emerged that uses a special trick, namely the extended Lagrangian technique. The basic idea... 

A quantum mechanical formulation of solute charge density can be pursued in a number of ways. The most accurate treatment is the one that uses quantum mechanical first principle or ab initio approaches. However, the ab initio calculation of the electronic structure of a macromolecule is currently prohibitively expensive due to the large number of degrees of freedom. A variety of elegant theories and algorithms have been developed in the literature to reduce the dimensionality of this many-body problem [165-172]. In earlier work from the Wei group, a density functional theory (DFT) treatment of solute electron distributions was incorporated into our DG-based solvation model [132]. In this work, we review the basic formulation and present an improved DG-DFT model for solvation... 

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