Nonrelativistic density functional theory

A Brief Review of Nonrelativistic Density Functional Theory... 

Consider the total nonrelativistic fitted electronic energy [26] functional, , which in Kohn-Sham density functional theory takes the form (in atomic units)... 

For the case of a purely electrostatic external potential, P = (F , 0), the complete proof of the relativistic HK-theorem can be repeated using just the zeroth component f (x) of the four current (in the following often denoted by the more familiar n x)), i.e. the structure of the external potential determines the minimum set of basic variables for a DFT approach. As a consequence the ground state and all observables, in this case, can be understood as unique functionals of the density n only. This does, however, not imply that the spatial components of the current vanish, but rather that j(jc) = < o[w]liWI oM) has to be interpreted as a functional of n(x). Thus for standard electronic structure problems one can choose between a four current DFT description and a formulation solely in terms of n x), although one might expect the former approach to be more useful in applications to systems with j x) 0 as soon as approximations are involved. This situation is similar to the nonrelativistic case where for a spin-polarised system not subject to an external magnetic field B both the 0 limit of spin-density functional theory as well as the original pure density functional theory can be used. While the former leads in practice to more accurate results for actual spin-polarised systems (as one additional symmetry of the system is take into account explicitly), both approaches coincide for unpolarized systems. 

It is directly possible to prove a HK-theorem for the form (3.55) using the density n and the gauge-dependent current jp — (c/e)V x m as basic DFT variables, but not for the form (3.54) which would suggest to use n and the full current j. One is thus led to the statement that the first set of variables can legitimately be used to set up nonrelativistic current density functional theory, indicating at first glance a conflict with the fully relativistic DFT approach. 

Equations (4.34)-(4.37) are immediately identified as the relativistic extension of the standard form of nonrelativistic spin-density functional theory. 

To get a first idea of what density-functional theory is about, it is useful to take a step back and recall some elementary quantum mechanics. In quantum mechanics we learn that all information we can possibly have about a given system is contained in the system s wave function, T. Here we will exclusively be concerned with the electronic structure of atoms, molecules and solids. The nuclear degrees of freedom (e.g., the crystal lattice in a solid) appear only in the form of a potential u(r) acting on the electrons, so that the wave function depends only on the electronic coordinates.2 Nonrelativistically, this wave function is calculated from Schrodinger s equation, which for a single electron moving in a potential v(r) reads... 

In spite of the impressive progress which has been achieved with conventional ab-initio methods as the Configuration-Interaction or Coupled-Cluster schemes in recent years density functional theory (DFT) still represents the method of choice for the study of complex many-electron systems (for an overview of DFT see [1]). Today DFT covers an enormous variety of fields, ranging from atomic [2,3], cluster [4,5] and surface physics [6,7] to the material sciences [8-10]. and theoretical biophysics [11-13]. Moreover, since the introduction of the generalized gradient approximation DFT has become an accepted method also for standard quantum chemical applications [14,15]. Given this tremendous success of nonrelativistic DFT the question for a relativistic extension (RDFT) arises quite naturally in view of the large number of problems in which relativistic effects play an important role (see e.g. Refs.[16,17]). 

After insertion of (103) Eqs.(99)-(102) are immediately identified as the direct relativistic extension of the standard two-component form of nonrelativistic spin-density functional theory. This suggests the application of nonrelativistic spin-density functionals Exc[npn with replaced by in Eq.(102), thus neglecting the relativistic contributions to the dependence of Exc[n, n ] on With this approximation Eqs.(99)-(102) represent the standard RDFT approach to magnetic systems. 

In the nonrelativistic context current-density functional theory is based on the nonrelativistic limits of the paramagnetic current (87) and/or the magnetization density (89) [128,129]. In the relativistic situation, however, a density functional approach relying on jp or m can only be considered an approximation, as long as the external magnetic field does not vanish. In order to clarify the relation between these two points of view the weakly relativistic limit of RDFT has to be analyzed. The weakly relativistic limit of the Hamiltonian (23) can be derived either by a direct expansion in 1/c or by a low order Foldy-Wouthuysen transformation,... 

Eqs. (l)-(3), (13), and (19) define the spin-free CGWB-AIMP relativistic Hamiltonian of a molecule. It can be utilised in any standard wavefunction based or Density Functional Theory based method of nonrelativistic Quantum Chemistry. It would work with all-electron basis sets, but it is expected to be used with valence-only basis sets, which are the last ingredient of practical CGWB-AIMP calculations. The valence basis sets are obtained in atomic CGWB-AIMP calculations, via variational principle, by minimisation of the total valence energy, usually in open-shell restricted Hartree-Fock calculations. In this way, optimisation of valence basis sets is the same problem as optimisation of all-electron basis sets, it faces the same difficulties and all the experience already gathered in the latter is applicable to the former. 

The thirty three papers in the proceedings of QSCP-Xni are divided between the present two volumes in the following manner. The first volume, with the subtitle Conceptual and Computational Advances in Quantum Chemistry, contains twenty papers and is divided into six parts. The first part focuses on historical overviews of significance to the QSCP workshop series and quantum chemistry. The remaining five parts, entitled High-Precision Quantum Chemistry, Beyond Nonrelativistic Theory Relativity and QED, Advances in Wave Function Methods, Advances in Density Functional Theory, and Advances in Concepts and Models, address different aspects of quantum mechanics as applied to electronic structure theory and its foundations. The second volume, with the subtitle Dynamics, Spectroscopy, Clusters, and Nanostructures, contains the remaining thirteen papers and is divided into three parts Quantum Dynamics and Spectroscopy, Complexes and Clusters, and Nanostructures and Complex Systems. ... 

Nowadays, many electronic structure codes include efficient implementations [37—41] of the Ramsey equations [42] for the calciflations of nonrelativistic spin—spin coupling constants. A vast number of publications devoted to the calculation of/-couplings can be found in the Hterature, covering different aspects such as the basis set effects [43-55], the comparison of wave function versus density functional theory (DFT) methods [56-60], or the choice of exchange-correlation functional in DFT approaches [61-68]. Excellent recent reviews of Contreras [69] andHelgaker [70] cover these particular aspects. 

An interesting approach to the quantum mechanical description of many-electron systems such as atoms, molecules, and solids is based on the idea that it should be possible to find a quantum theory that refers solely to observable quantities. Instead of relying on a wave function, such a theory should be based on the electron density. In this section, we introduce the basic concepts of this density functional theory (DFT) from fundamental relativistic principles. The equations that need to be solved within DFT are similar in structure to the SCF one-electron equations. For this reason, the focus here is on selected conceptual issues of relativistic DFT. From a practical and algorithmic point of view, most contemporary DFT variants can be considered as an improved model compared to the Hartree-Fock method, which is the reason why this section is very brief on solution and implementation aspects for the underlying one-electron equations. For elaborate accounts on nonrelativistic DFT that also address the many formal difficulties arising in the context of DFT, we therefore refer the reader to excellent monographs devoted to the subject [383-385]. 

We have updated the material considering the latest developments in the field over the past five years. These developments comprise both computational and more fundamental advances such as exact two-component approaches and the study of explicitly correlated two-electron wave functions in the context of the Brown-Ravenhall disease, respectively. Other topics, such as relativistic density functional theory and its relation to nonrelativistic spin-... 

First of all, a few words on the scope of this review seem to be appropriate. For simplicity, all explicit formulae in this chapter will be given for spin-saturated systems only. Of course, the complete formalism can be extended to spin-density functional theory (SDFT) and all numerical results for spin-polarized systems given in this paper were obtained by SDFT calculations. In addition, the discussion is restricted to the nonrelativistic formalism - for its relativistic form see Chap. 3. The concept of implicit functionals has also... 

The scope of this chapter is to provide a rudimentary imderstanding of response theory as implemented in a number of molecular electronic structure packages based on wave function mo dels or density functional theory. Only the general structure of response theory and its computer implementation are discussed, leaving out the often comphcated details of advanced wave function and density functional models. For these details the reader is referred to the literature mentioned in the last section and references therein. Although the discussion of this chapter is restricted to nonrelativistic theory, it is the same line of reasoning that is applied in relativistic response theory. In conjunction with the chapter on applications of response theory, the reader should become sufficiently familiar with the concepts and practices of response theory to allow educated use of quantum chemistry software packages. 

The treatment of the many-body system of nuclei and electrons of which solids and metals consist, makes it necessary to introduce approximations. Traditional approximations in electronic structure theory are those due to Hartree and Fock. We shall here only briefly give the main equations relevant for the discussion of the free-electron gas results (Chapter 9) and the density functional theory (Chapter 10). Consider, for instance, a metal of N atoms and each atom having Z electrons, where the number of atoms is of the order Avogadros number. The number of electrons to be considered for each atom can be lowered from the actual number to the number of valence electrons by introducing effective core potentials. The nonrelativistic Schrddinger equation for the electronic part of the problem is then... 

Some areas of computational electronic-structure theory are not treated in this book. All methods discussed are strictly ab initio. Semi-empirical methods are not treated nor is density-functional theory discussed all techniques discussed involve directly or indirectly the calculation of a wave function. Energy derivatives are not covered, even though these play a prominent role in the evaluation of molecular properties and in the optimization of geometries. Relativistic theory is likewise not treated. In short, the focus is on techniques for solving the nonrelativistic molecular... 

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