Overview of Density Functionals

The main task in the Kohn-Sham formulation of DFT is finding good approximations to the exchange-correlation functional contrast to wave function 

Local Density Approximation (LDA) This approach (usually used as local spin density approximation, LSDA) only considers the electron density p(r) or p (r) and (r) and consists in applying the exact results for the uniform electron gas to a real system. For the exchange part, this involves a p term. Since this has been used in the method of Slater, the name Slater is often used as synonym for the LSDA exchange fimctional For the correlation part analytic forms are 

Meta-CCA Functionals These contain in addition higher derivatives of the density, for example, the Laplacian V p(r), or essentially equivalently, the orbital kinetic energy density r (hence the prefix T in some DF acronyms). Calculation of the kinetic energy density does not require much additional computational efforts but a finer than usual grid is recommended. Examples of meta-GGAs are t-HGTH [43], VSXC [44], the TPSS family [45] and the Minnesota functionals M06-L and the dual-range Mll-L [46]. 

Hybrid Functionals The functionals mentioned so far contain the density, its derivative, and the kinetic energy density (in case of meta-GGAs) at some given point and, hence, may be termed local. One possibility (and the most frequently used so far) to include nonlocality is to incorporate HE exchange [47]  

The first such hybrid GGA, termed half-and-half (a = 0.5), has been proposed by Becke, followed by a more empirical three-parameter hybrid functional of the form [48] 

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This paper gives a short overview of density functional calculations mainly based on the DV-Xa approach organized as follows. A short overview of Density Functional Theory, DFT, and Kohn-Sham equations is given in section II followed by a summary of different ways of solution of the Kohn-Sham equations in Sec. III. Comparisons of results from some old and some up-to-date density functional electronic structure calculations made by our group to show applications to clusters, surfaces, adsorbates on surfaces and Ceo are given in Sec. IV. Conclusions and outlook are summarized in Sec. V. 

To end this brief overview of density-functional-based computations of molecular conduction, we should note that this approach suffers in principle from problems similar to those encountered in using the HF approximation, that is, the inherent inaccuracy of the computed LUMO energy and wave functions. The errors are different, for example HF overestimates the HOMO-LUMO gap (since the HF LUMO energy is too high [143-146, 174-176]) while density functional theory (DFT) underestimates it [167, 177]. Common to both approaches is the observation that processes dominated by the HOMO level will be described considerably better by these approaches than by processes controlled by coupling to the LUMO [137,178]. 

The objective of this article is to expose the chemical engineering community to Car-Parrinello methods, what they have accomplished, and what their potential is for chemical engineering. Consistent with this objective, in Section IV, I give an overview of the most widely used quantum mechanical method for solving the many-body electronic problem, density-functional theory, but describe other methods only cursorily. I also describe the practical solution of the equations of density-functional theory for molecular and extended systems via the plane-wave pseudopotential method, mentioning other methods only cursorily. Finally, I end this section with a description of the Car-Parrinello method itself. 

In this work we have given on overview of the mathematical foundations of stationary density functional theory. We discussed in great detail the question of differentiability of the functionals and showed that the Kohn-Sham theory can be put on a solid basis for all practical purposes, since the set of noninteracting E-V-densities is dense in the set of interacting E-V-densities. The question whether these two sets are in fact identical is still an open question. We further discussed two systematic approaches for the construction of the exchange-correlation functional and potential. What can we say about future developments within density functional theory There have been many extensions of density functional theory involving... 

In spite of the advent of density functional theory (see Section 21.2), the number of publications with semiempirical calculations remains high. In the Science Citation Index, one finds for each of the past ten years more than 1000 such papers under the topic semiempirical/MNDO/AMl/PM3 , the actual numbers fluctuating between 1100 and 1500 (1994-2003) this should be regarded as a lower limit of the actual usage, for obvious reasons. Since DFT calculations have replaced semiempirical calculations in many studies on medium-sized molecules, the latter must have found new areas of application. In this section, we attempt to identify such areas from a survey of the recent literature. Given the diverse activities in this field, it is clear this cannot be a comprehensive overview and that the selection of topics will necessarily be subjective. 

From the above brief overview one can conclude that the development of density functional theory and the demonstration of the tractability and accuracy of the local density approximation to it constitutes an important milestone in soUd-state physics and chemistry. 

The hrst hve chapters (Part 1) present an overview of some methods that have been used in the recent hterature to calculate rate constants and the associated case studies. The main topics covered in this part include thermochemistry and kinetics, computational chemistry and kinetics, quantum instanton, kinetic calculations in liquid solutions, and new applications of density functional theory in kinetic calculations. The remaining hve chapters (Part II) are focused on apphcations even though methodologies are discussed. The topics in the second part include the kinetics of molecules relevant to combustion processes, intermolecular electron transfer reactivity of organic compounds, lignin model compounds, and coal model compounds in addition to free radical polymerization. 

It is the task of the new generation to continue the past and present efforts in this exciting field. We hope with this primer in density functional theory to provide students, and even established researchers, an overview of the present state and prospects of density functional methods. 

Results on the investigation of atomic density functions are reviewed. First, ways for calculating the density of atoms in a well-defined state are discussed, with particular attention for the spherical symmetry. It follows that the density function of an arbitrary open shell atom is not a priori spherically symmetric. A workable definition for density functions within the multi-configuration Hartree-Fock framework is established. By evaluating the obtained definition, particular influences on the density function are illustrated. A brief overview of the calculation of density functions within the relativistic Dirac-Hartree-Fock scheme is given as well. 

Parr B 2000 webpage http //net.chem.unc.edu/facultv/rap/cfrap01. html Professor Parr was among the first to push the density functional theory of Hohenberg and Kohn to bring it into the mainstream of electronic structure theory. For a good overview, see the book ... 

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. 

We conclude this chapter with an overview of how modem density functional theory deals with electronic excitation energies. From the very beginning, electronically excited states... 

The general theory of the quantum mechanical treatment of magnetic properties is far beyond the scope of this book. For details of the fundamental theory as well as on many technical aspects regarding the calculation of NMR parameters in the context of various quantum chemical techniques we refer the interested reader to the clear and competent discussion in the recent review by Helgaker, Jaszunski, and Ruud, 1999. These authors focus mainly on the Hartree-Fock and related correlated methods but briefly touch also on density functional theory. A more introductory exposition of the general aspects can be found in standard text books such as McWeeny, 1992, or Atkins and Friedman, 1997. As mentioned above we will in the following provide just a very general overview of this... 

The identification of unknown chemical compounds isolated in inert gas matrices is nowadays facilitated by comparison of the measured IR spectra with those computed at reliable levels of ab initio or density functional theory (DFT). Furthermore, the observed reactivity of matrix isolated species can in some instances be explained with the help of computed reaction energies and barriers for intramolecular rearrangements. Hence, electronic structure methods developed into a useful tool for the matrix isolation community. In this chapter, we will give an overview of the various theoretical methods and their limitations when employed in carbene chemistry. For a more detailed qualitative description of the merits and drawbacks of commonly used electronic structure methods, especially for open-shell systems, the reader is referred to the introductory guide of Bally and Borden.29... 

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]. 

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