Density functional theory mathematical methods

In formulating a mathematical representation of molecules, it is necessary to define a reference system that is defined as having zero energy. This zero of energy is different from one approximation to the next. For ah initio or density functional theory (DFT) methods, which model all the electrons in a system, zero energy corresponds to having all nuclei and electrons at an infinite distance from one another. Most semiempirical methods use a valence energy that cor-... 

The density functional theory (DFT) methods [18-21] focus on finding the optimal electron density (ED). The functional comes from the fact that the energy, and all other properties, of the system are computed as a function of the ED and the ED is itself a function of position (r), p(r), and in mathematics a function of a function is called a functional . Its advantages include less demanding computational effort, less computer time, and, in some cases (particularly rf-metal complexes), better agreement with experimental values than is obtained from HE procedures. 

In [170] the authors obtain a test set of ten molecules of specific atmospheric interest in order to evaluate the performance of various Density Functional Theory (DFT) methods in (hyper)polarizability calculations as well as established ab initio methods. The authors make their choice for these molecules based on the profound change in the physics between isomeric systems, the relation between isomeric forms and the effect of the substitution. In the evaluation analysis the authors use arguments chosen from the information theory, the graph theory and the pattern recognition fields of Mathematics. The authors mentioned the remarkable good performance of the double hybrid functionals (namely B2PLYP and mPW2PLYP) which are for the first time used in calculations of electric response properties. 

Electronic strucmre methods are characterized by their various mathematical approximations to its solution, since exact solutions to the Schrddinger equation are not computationally practical. There are three classes of electronic structure methods semi-empirical methods, density functional theory (DFT) methods, and... 

The methods that will be discussed in the following are all of the ab initio type. Given the molecular field free Hamiltonian in Eq. (2.9), with the nuclear coordinates and charges and the electronic mass given as parameters, in these methods all integrals over this Hamiltonian or parts of it are evaluated ab initio, i.e. by strict application of the appropriate mathematical rules and without using further data from experiment or otherwise. The emphasis will be in particular on the SCF and on so-called correlated methods. Semi-empirical or density functional theory (DFT) methods on the other hand will not be mentioned explicitly. However, most of what will be said about SCF-based methods for the calculation of properties will also apply to semi-empirical or DFT methods, because one can consider to a certain extent the semi-empirical and DFT methods as variants of SCF, just with a slightly different Hamiltonian. 

The dynamic mean-field density functional method is similar to DPD in practice, but not in its mathematical formulation. This method is built around the density functional theory of coarse-grained systems. The actual simulation is a... 

Ab initio molecular orbital theory is concerned with predicting the properties of atomic and molecular systems. It is based upon the fundamental laws of quantum mechanics and uses a variety of mathematical transformation and approximation techniques to solve the fundamental equations. This appendix provides an introductory overview of the theory underlying ab initio electronic structure methods. The final section provides a similar overview of the theory underlying Density Functional Theory methods. 

It is a truism that in the past decade density functional theory has made its way from a peripheral position in quantum chemistry to center stage. Of course the often excellent accuracy of the DFT based methods has provided the primary driving force of this development. When one adds to this the computational economy of the calculations, the choice for DFT appears natural and practical. So DFT has conquered the rational minds of the quantum chemists and computational chemists, but has it also won their hearts To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations. There has been misunderstanding concerning the status of the one-determinantal approach of Kohn and Sham, which superficially appeared to preclude the incorporation of correlation effects. There has been uneasiness about the molecular orbitals of the Kohn-Sham model, which chemists used qualitatively as they always have used orbitals but which in the physics literature were sometimes denoted as mathematical constructs devoid of physical (let alone chemical) meaning. 

Density functional calculations (DFT calculations, density functional theory) are, like ab initio and semiempirical calculations, based on the Schrodinger equation However, unlike the other two methods, DFT does not calculate a conventional wavefunction, but rather derives the electron distribution (electron density function) directly. Afunctional is a mathematical entity related to a function. 

C. Density functional theory Density functional theory (DFT) is the third alternative quantum mechanics method for obtaining chemical structures and their associated energies.Unlike the other two approaches, however, DFT avoids working with the many-electron wavefunction. DFT focuses on the direct use of electron densities P(r), which are included in the fundamental mathematical formulations, the Kohn-Sham equations, which define the basis for this method. Unlike Hartree-Fock methods of ab initio theory, DFT explicitly takes electron correlation into account. This means that DFT should give results comparable to the standard ab initio correlation models, such as second order M(j)ller-Plesset (MP2) theory. 

For further reading, see Density Functionals Theory and Application s. D. Joubert, Ed.. Springer 1998. G. Arfken, Mathematical Methods for Physicists, Acad. Press. 3rd ed. (1985). For a very readable Introduction by Richard Feynman, see The Principle of Least Action, in. R.P. Feynman. R.B. Leighton emd M. Sands. The Feynman Lectures on Physics. Addison-Wesley (1966). Vol. n chapter 19. The book by H.T. Davis. Statistical Mechanics of Phases. Interfaces and Thin FUms. Wiley (1996). contains a chapter (9) on this matter. 

In this section we provide a unified point of view of the different theoretical methods used in the study of electronic structure. This includes two rather different families of methods that nevertheless arise from the principles of quantum mechanics. On the one hand, one has the ab initio methods of computation of electronic wave functions and, on the other hand, one has the methods based in the modern density functional theory. In the forthcoming discussion we attempt to focus mainly on the physical significance rather than on mathematical foundation and technical aspects of computer implementation. Details of the methods outlined in this section can be found in specialized references, monographs [70], and textbooks [71]. 

As an alternative to ab initio methods, the semi-empirical quantum-chemical methods are fast and applicable for the calculation of molecular descriptors of long series of structurally complex and large molecules. Most of these methods have been developed within the mathematical framework of the molecular orbital theory (SCF MO), but use a number of simplifications and approximations in the computational procedure that reduce dramatically the computer time [6]. The most popular semi-empirical methods are Austin Model 1 (AMI) [7] and Parametric Model 3 (PM3) [8]. The results produced by different semi-empirical methods are generally not comparable, but they often do reproduce similar trends. For example, the electronic net charges calculated by the AMI, MNDO (modified neglect of diatomic overlap), and INDO (intermediate neglect of diatomic overlap) methods were found to be quite different in their absolute values, but were consistent in their trends. Intermediate between the ab initio and semi-empirical methods in terms of the demand in computational resources are algorithms based on density functional theory (DFT) [9]. 

Electronic structure methods or quantum mechanics (QM) methods use the laws of quantum mechanics rather than classical physics as the basis for their computations. QM states that the energy and related properties of a molecule may be obtained by solving the Schrddinger equation H- r = Eiji. For any but the smallest systems, however, exact solutions to the Schrddinger equation are not computationally practical. QM methods are characterized by various mathematical approximations to its solution. There are three classes of QM methods semiempirical, ab initio, and density functional theory (DFT). 

Density Functional Theory Methods In parallel to the development of ab initio theory, it was theorized that all molecular properties could be described as a function of the electron density [24]. By using mathematical functions, called functionals, to describe the electron density, a new theoretical approach, density functional theory (DFT), was developed. In 1965, Kohn and Sham [25] produced a set of equations that demonstrated how to determine a self-consistent density from DFT decomposition of the Schrodinger equation ... 

The generator coordinate method (GCM), as initially formulated in nuclear physics, is briefly described. Emphasis is then given to mathematical aspects and applications to atomic systems. The hydrogen atom Schrodinger equation with a Gaussian trial function is used as a model for former and new analytical, formal and numerical derivations. The discretization technique for the solution of the Hill-Wheeler equation is presented and the generator coordinate Hartree-Fock method and its applications for atoms, molecules, natural orbitals and universal basis sets are reviewed. A connection between the GCM and density functional theory is commented and some initial applications are presented. 

Density Functional Theory (DFT) [40, 41], DFT calculations yield results comparable to those obtained from ab initio methods but at lower computational costs. DFT methods compute electron correlation via general functionals of the electron density (a functional is defined in mathematics as a function of a function). 

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