Density-Functional Theory for Electrons

The mean-field approximation (MFA) is the basis of what we use to analyze the molecular properties created by the complex correlations among the electrons. The aim of this chapter is to provide the details of the conventional methods for constructing the mean field in the wave-function theory and in the density-functional theory (DFT). Section 1.1 describes an overview of the MFAs in these theories, which identifies the problems of the approximations in each of the theories. A novel approach developed to solve the problem in the DFT, which constitutes the major issue of this book, is also reviewed in this section. In Sect. 1.2, the MFA termed Hartree-Fock (HF) method of the wave-function theory is formulated in detail for later discussion. In the framework of the Kohn-Sham DFT (KS-DFT), the mean field is directly given by the functional derivative of the exchange-correlation functional \(E_{xc}[n]\) with respect to the electron density n. The theoretical foundations underlying the KS-DFT are provided in Sect. 1.3. The guiding principles for the developments of the approximate functionals used in the KS-DFT are also shown with emphasis on their physical meanings. An approach referred to as optimized-effective potential (OEP) offers a method to obtain the KS effective potential which minimizes an implicit energy functional of the density n. The method to implement the HF-OEP will be discussed in the last section. The inverse Kohn-Sham (inv-KS) method is another approach to construct the mean field from a given electron density. The relationship between the HF-OEP and the inv-KS for the HF density will also be discussed. In this chapter, the atomic units (a.u.) are used unless otherwise stated.

Static correlation (SC) is essential in describing the dissociation of a covalent bond. It is one of the main issues in this book to develop a density functional which allows to describe the SC in the bond dissociations, and the aim of this chapter is to review the methods for describing the SC in the wave-function theory and in the density-functional theory (DFT). The first section is devoted to illustrate an a priori approach to construct the wave function that realizes the SC for the simplest chemical bond at the dissociation limit. Based on the result, in Sect. 2.2, a general approach in the wave-function theory, termed multi-configuration self-consistent field (MCSCF) method, is briefly reviewed. It is well known that the realization of the SC in the DFT is quite difficult since the approximate functionals refer only to the local and semi-local properties of the system, i.e. the density and its derivatives. In Sect. 2.3, we first discuss the source of the difficulty and then provide an outline of an approach in the DFT to incorporate the SC into an explicit exchange functional. An implicit method to represent the SC within the framework of the Kohn-Sham DFT is reviewed in Sect. 2.4, where the optimized effective potential (OEP) method will be adapted to the MCSCF wave function. In this chapter, the atomic units (a.u.) are used unless otherwise stated.

A detailed review of a novel density-functional theory (DFT) [1] is described in this chapter, where the electron density \(n^e(\epsilon )\) on the energy coordinate \(\epsilon \) serves as the fundamental variable for the DFT. The definition of the energy electron density \(n^e(\epsilon )\) is provided in the first Section. The exchange functional \(E_x^{e,\text {LDA}}[n^e]\) based on the local density approximation (LDA) for the variable \(n^e(\epsilon )\) is also introduced. In the Sects. 3.2 and 3.3, the Hohenberg-Kohn (HK) functional \(F_\text {HK}^e[n^e]\) is formulated for the variable \(n^e(\epsilon )\) in parallel to the original HK functional. It is shown in Sect. 3.4 that the domain of definition of the functional can be extended to the set of N-rep. energy electron densities by means of Levy’s constraint search method. In Sect. 3.5, it is shown that the exchange-correlation functional \(E_{xc}^e[n^e]\) of the density \(n^e(\epsilon )\) can be incorporated into the equation for the Kohn-Sham DFT. In the last section, the correction based on the generalized gradient approximation (GGA) is applied to the LDA functional \(E_{xc}^{e,\text {LDA}}[n^e]\) by introducing an inhomogeneity parameter defined on the energy coordinate. The functional is employed to calculate the potential energies of the chemical bonds. In this Chapter, the atomic units (a.u.) are used unless otherwise stated.

As described in Chap. 2, the static correlation (SC) plays an essential role in describing theoretically the formations and the dissociations of chemical bonds. On the basis of the novel density-functional theory (DFT) formulated in Chap. 3, we develop a static-correlation functional as a main subject in this chapter. In Sect. 4.1, we first discuss the characteristic property of the energy electron density \(n^e(\epsilon )\) on the energy coordinate \(\epsilon \), which plays a central role in the construction of the SC functional. The advantage of using the density \(n^e(\epsilon )\) will be illustrated for the dissociating H\(_2\) molecule. At the dissociation limit of the molecule, the distribution \(n_1^e(\epsilon )\) for the spin-symmetry adapted density becomes identical to the distribution \(n_0^e(\epsilon )\) for the symmetry broken density, that is, \(n_1^e(\epsilon ) = n_0^e(\epsilon )\). On the basis of this property, in Sect. 4.2, a simple exchange-correlation functional which includes the SC energy is developed using the energy distribution \(n_1^e(\epsilon )\) as an argument. In this chapter, the atomic units (a.u.) are used unless otherwise stated.

The development of the kinetic-energy density functional (KEDF) is known as the most difficult task in the density-functional theory (DFT). Given that the history of the study of the KEDF began with the Thomas-Fermi (TF) model in 1927 (Thomas, Proc Camb Phil Soc 23:541, 1927, [1]), almost a century has been devoted to the functional development. The aim of this chapter is to present an overview of the history of the KEDF and also to provide the details of the recent developments utilizing the energy electron density \(n^e\) based on the novel DFT framework introduced in Chap. 3. The TF model and the Weizäcker correction is reviewed in Sect. 5.1, where an approach by Wang and Teter (Phys Rev B 45(23):13196, 1992, [2]) to incorporate a nonlocal kernel into the KEDF is also introduced. In Sect. 5.2, the WT nonlocal functional is reconsidered in the formalism of the Taylor expansion of the KEDF up to the 2nd-functional derivative to generalize their approach. Then, in Sect. 5.3, a nonlocal KEDF is developed for the distribution \(n^e(\epsilon )\) on the energy coordinate \(\epsilon \). The important feature of the method is that the nonlocal kernel employs the response function for the molecule of interest and not for the homogeneous electron gas, in contrast to the WT functional. The fatal problem with the TF model is that it does not provide a sound potential energy profile of a dissociating chemical bond. In Sect. 5.4, the mechanism underlying the failure is elucidated, and a functional that resolves the problem is developed in parallel to the static correlation functional in Chap. 4. In this chapter, the atomic units (a.u.) are used unless otherwise stated.