Basics of Electronic Structure Theory

In the previous section, we have seen that in a vast majority of cases, the quantum mechanical description of a molecular system can be greatly simplified if the nuclear and electronic motions are separated. In this case, the electronic problem can be treated for fixed nuclei by solving the clamped nucleus or electronic TISE of Eq. (2.5). Finding accurate and efficient numerical procedures to solve the electronic TISE has been a major goal of theoretical chemistry since the beginning of the second part of the previous century [10, 11]. 

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Basics of Electronic Structure Theory The energy now reads... 

First, we introduce the two basic frameworks of electronic structure theory, molecular orbital (MO) theory and band theory. Electronic structure theory can provide calculation of the total energy of a system. In addition, MO and band theories give one-electron states, which are often used to represent electron (hole) dynamics. 

In Chapters 4 and 5, we turn our attention away from the tools of second quantization towards wave functions. First, in Chapter 4, we discuss important general characteristics of the exact electronic wave function such as antisymmetry, size-extensivity, stationarity, and the cusps that arise from the singularities of the Hamiltonian. Ideally, we would like our approximate wave functions to inherit most of these properties in practice, compromises must be made between what is desirable and what is practical as illustrated in Chapter 5, which introduces the standard models of electronic-structure theory, presenting the basic theory and employing numerical examples to illustrate the usefulness and shortcomings of the different methods. 

We recently proposed a new method referred to as RISM-SCF/MCSCF based on the ab initio electronic structure theory and the integral equation theory of molecular liquids (RISM). Ten-no et al. [12,13] proposed the original RISM-SCF method in 1993. The basic idea of the method is to replace the reaction field in the continuum models with a microscopic expression in terms of the site-site radial distribution functions between solute and solvent, which can be calculated from the RISM theory. Exploiting the microscopic reaction field, the Fock operator of a molecule in solution can be expressed by... 

There are two broad areas within computational chemistry devoted to the structure of molecules and their reactivity molecular mechanics and electronic structure theory. They both perform the same basic types of calculations ... 

A different approach is adopted here. Within the LMTO-ASA method, it is possible to vary the atomic radii in such a way that the net charges are non-random while preserving the total volume of the system . The basic assumption of a single-site theory of electronic structure of disordered alloys, namely that the potential at any site R depends only on the occupation of this site by atom A or B, and is completely independent of the occupation of other sites, is fulfilled, if the net charges... 

The problem of minimizing functions is ubiquitous in many different branches of science. It arises very naturally and rather directly in the electronic structure theory when the strategy adopted is variational for, the basic task in the variational approximation boils down to finding out values of a set of parameters present in the trial wavefunction (assuming expansion in terms of finite dimensional analytic... 

The purpose of this chapter will be to review the fundamentals of ab initio MD. We will consider here Density Functional Theory based ab initio MD, in particular in its Car-Parrinello version. We will start by introducing the basics of Density Functional Theory and the Kohn-Sham method, as the method chosen to perform electronic structure calculation. This will be followed by a rapid discussion on plane wave basis sets to solve the Kohn-Sham equations, including pseudopotentials for the core electrons. Then we will discuss the critical point of ab initio MD, i.e. coupling the electronic structure calculation to the ionic dynamics, using either the Born-Oppenheimer or the Car-Parrinello schemes. Finally, we will extend this presentation to the calculation of some electronic properties, in particular polarization through the modern theory of polarization in periodic systems. 

This new perspective has enriched the theory of electronic structure and chemical reactivity by both rationalizing and quantifying basic, classical ideas and rules of chemistry, e.g., the electronegativity/chemical potential equalization of Sanderson [104] or the hard-soft acids and bases (HSAB) principle of Pearson [95,105], bringing about a deeper understanding of the nature of chemical bonds and variety of reactivity preferences [3-5,11,117]. 

The Hartree-Fock model is the simplest, most basic model in ab initio electronic structure theory [28], In this model, the wave function is approximated by a single Slater determinant constructed from a set of orthonormal spin orbitals ... 

Under the second topic of Ligand-field Theory and its Extensions we describe the basic concepts behind the various versions of LFT - the angular-overlap model (AOM) and its extensions. In the section named The physical background conditions for the applicability of the ligand-field approach we sketch briefly the theoretical foundation and limits of applicability of the effective-hamiltonian approach with special attention to electronic multiplets. In the theory section, we describe various approaches in current calculations of electronic structure, such as LFDFT, SORCI and TDDFT, with the various applications detailed in the following section, before an outlook for further developments. 

Interpretations of the anomeric effect depend on the theory that is applied, and only within a given theory can their correctness be checked. Hence, the number of explanations of the anomeric effect is quite large. They can be divided (23) into two basic groups classical structural theory supplemented with electronic effects (Section III. A) and quantum chemistry (Section III.B). 

An overview of relativistic state-of-the-art calculations on electric field gradients (EFG) in atoms and molecules neccessary for the determination of nuclear quadrupole moments (NQM) is presented. Especially for heavy elements four-component calculations are the method of choice due to the strong weighting of the core region by the EFG operator and the concomitant importance of relativity. Accurate nuclear data are required for testing and verification of the various nuclear models in theoretical nuclear physics and this field represents an illustrative example of how electronic structure theory and theoretical physics can fruitfully interplay. Basic atomic and molecular experimental techniques for the determination of the magnetic and electric hyperfine constants A and B axe briefly discussed in order to provide the reader with some background information in this field. 

In the next section, the different aspects and methods of ab initio quantum chemistry will be discussed. This will include a brief and incomplete discussion of the basics of solving a quantum-chemical problem, i. e. an electronic structure calculation. The practical calculation may proceed using either wave-function-based methods or density functional theory-based methods, and my focus will be on the latter as at the present time these methods are by far the most popular, giving the best accuracy for large systems in a limited amount of computer time. Different methods of analyzing the outcome of an ab initio calculation will also be discussed briefly. Finally, a development will be discussed which has become very important for modeling processes in condensed phases, namely the combination of electronic structure calculations (which are usually static and apply to a temperature of 0... 

There have been tremendous interests in the literature to apply information theory to the electronic structure theory of atoms and molecules [1, 2]. The concepts of uncertainty, randomness, disorder, or delocalization, are basic ingredients in the study, within an information theoretic framework, of relevant structural properties for many different probability distributions appearing as descriptors of several chemical and physical systems and/or processes. 

Since DFT uses the electron density as the basic variable, instead of the wave function in the conventional quantum theory of electronic structure, not only would it be advantageous to have the energy functional and derivatives defined for densities with fractional number of electrons, but it is also necessary to treat systems with a fractional number of electrons. To show this necessity, one only needs to consider the dissociation of Hj. In the exact quantum-mechanical theory, at the dissociation limit, H(a)-H(b)+ and its nuclear permutation H(a) — H(b) are two degenerate states. Any linear combination of these two states is also a ground state of the dissociation, including the state j o.5+ limit, we have two independent... 

The molecular orbital (MO) is the basic concept in contemporary quantum chemistry. " It is used to describe the electronic structure of molecular systems in almost all models, ranging from simple Hiickel theory to the most advanced multiconfigurational treatments. Only in valence bond (VB) theory is it not used. Here, polarized atomic orbitals are instead the basic feature. One might ask why MOs have become the key concept in molecular electronic structure theory. There are several reasons, but the most important is most likely the computational advantages of MO theory compared to the alternative VB approach. The first quantum mechanical calculation on a molecule was the Heitler-London study of H2 and this was the start of VB theory. It was found, however, that this approach led to complex structures of the wave funetion when applied to many-electron systems and the mainstream of quantum ehemistry was to take another route, based on the success of the central-field model for atoms introduced by by Hartree in 1928 and developed into what we today know as the Hartree-Foek (HF) method, by Fock, Slater, and co-workers (see Ref. 5 for a review of the HF method for atoms). It was found in these calculations of atomic orbitals that a surprisingly accurate description of the electronic structure could be achieved by assuming that the electrons move independently of each other in the mean field created by the electron cloud. Some correlation was introduced between electrons with... 

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