Basic electronic structure methods

The main purpose of this chapter is to present the basics of ab initio molecular dynamics, focusing on the practical aspects of the simulations, and in particular, on modeling chemical reactions. Although CP-MD is a general molecular dynamics scheme which potentially can be applied in combination with any electronic structure method, the Car-Parinello MD is usually implemented within the framework of density functional theory with plane-waves as the basis set. Such an approach is conceptually quite distant from the commonly applied static approaches of quantum-chemistry with atom-centered basis sets. Therefore, a main... 

Use of the plane wave based electronic structure methods introduces two basic parameters the kinetic energy cutoff value, controlling the basis set quality, and the periodic unit-cell (supercell) size, present due to periodic nature of these approaches. Both of these parameters should be large enough to guarantee the convergence in the total energy and in all the physical quantities that are supposed to be determined from the simulation. 

When compared to developments associated with the structure and synthesis of chiral structures, less attention has been focused on the electronic and magnetic properties of chiral molecules. Circular dichroism and related optical probes of chirality have been developed mainly as analytical tools and, indeed, they are applied routinely. However, as demonstrated in this volume, the basic physical underpinnings that link structure and chiral properties, including chiro-optical properties, continue to emerge and require the development of physical models and improvement of electronic structure methods before they are fully elucidated. 

The electronic structure methods are based primarily on two basic approximations (1) Born-Oppenheimer approximation that separates the nuclear motion from the electronic motion, and (2) Independent Particle approximation that allows one to describe the total electronic wavefunction in the form of one electron wavefunc-tions i.e. a Slater determinant [26], Together with electron spin, this is known as the Hartree-Fock (HF) approximation. The HF method can be of three types restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and restricted open Hartree-Fock (ROHF). In the RHF method, which is used for the singlet spin system, the same orbital spatial function is used for both electronic spins (a and (3). In the UHF method, electrons with a and (3 spins have different orbital spatial functions. However, this kind of wavefunction treatment yields an error known as spin contamination. In the case of ROHF method, for an open shell system paired electron spins have the same orbital spatial function. One of the shortcomings of the HF method is neglect of explicit electron correlation. Electron correlation is mainly caused by the instantaneous interaction between electrons which is not treated in an explicit way in the HF method. Therefore, several physical phenomena can not be explained using the HF method, for example, the dissociation of molecules. The deficiency of the HF method (RHF) at the dissociation limit of molecules can be partly overcome in the UHF method. However, for a satisfactory result, a method with electron correlation is necessary. 

An additional major problem with natural membrane systems is that the composition of the one-dimensional unit cell may be unknown because of the presence of proteins of unknown structure and composition. Nevertheless, it is possible to model the electron density profile of a multilamellar system to arrive at a crude model for the distribution of lipids and proteins. The models improve considerably if the lipid composition and lipid stmctures are known. In any case, the basic crystallographic structural method is used The model is refined on the basis of comparisons between the calculated and the experimentally determined intensities through an iterative process. 

First-principles simulations are techniques that generally employ electronic structure calculations on the fly . Since this is a very expensive task in terms of computer time, the electronic structure method is mostly chosen to be density functional theory. Apart from the possibility of propagating classical atomic nuclei on the Born-Oppenheimer potential energy surface represented by the electronic energy V (R ) = ji(R ), another technique, the Car-Parrinello method, emerged that uses a special trick, namely the extended Lagrangian technique. The basic idea... 

We have have now paved the way for computational strategies to take form. Immediately Eq. (94) invites to the basic and straightforward idea of numerical differentiation, which is known in the literature as the finite-field technique. For atomic calculations, this idea is easily implemented in an existing code for any electronic structure method, as it only involves the response of the electronic density to the external static field in accordance with Fig. 6. In the absence of the perturbation, the dipole moment is zero due to the spherical symmetry of the atom, but in the presence of the external field the Coulomb force acts on the electrons as well as the atomic nucleus thereby displacing them in opposite directions the nucleus tends to move along with and tire electrons opposed to the electric field E with an induced dipole moment pc as a result. 

In this chapter, we will not be concerned with the detailed expressions of the response functions that we find for the standard electronic structure methods in theoretical chemistry. However, we will briefly outline the basic elements in two alternative formulations of response tlieory, namely the polarization propagator and the quasi-energy derivative approaches. 

Electronic structure methods that use the electron density as the basic variable trace their origin to the Thomas-Permi [2], Thomas-Permi-Dirac [3], and related models [4-7] developed in the early years of quanmm mechanics. Many similarities with the present day DPT can be also found in Caspar s exchange potential [8] and Slater s Xa-methods [9-11]. By the 1960s, these precursors of DPT were fuUy developed and used extensively for the calculations of atoms and solids, but their impact on molecular quanmm chemistry remained insignificant. The Thomas-Permi and Thomas-Permi-Dirac models proved to be of little use in chemistry because they can never yield a lower... 

When spin-orbit interactions are not included in the RECP approximation, the formalism is basically identical for both the relativistic and nonrelativistic case. Even when spin-orbit terms are included in the RECP, the formalism is straightforward and similar to conventional electronic structure methods. We will examine some specific aspects of the two-component HF approach in the following. 

In this chapter we restrict our review to research that has been done on the stability of fully aprotic liquid electrolytes for Li-air batteries. The review includes both experimental and computational aspects of this work, although the emphasis is on theoretical aspects. In the second section we review the basics of Li-air batteries and the electrochonical reactions involved in their operation and how they relate to the electrolyte. In the third section we review electronic structure methods for investigations of electrolytes. In the fourth section we discuss some of the initial Li-air... 

As noted in section 6.1, the basic electronic structures of the metal hydrides are rather well delineated by the various photoemission studies. The abilities of various calcula-tional methods to reproduce experimental data or even now to predict new results are remarkably good. One puzzlement has been at least partially removed by the realization that some artifical band-broadening must be introduced to better simulate experimental data, either because of intrinsic (i.e., real) electronic effects or because instruments have limited resolution. 

Hiickel theory is the electronic structure method that most chemists have as the basis of their understanding of molecular orbitals. It is the simplest of all the methods, but can be amazingly insightful. In the present context, Hiickel theory can be summarized as follows only valence tt electrons are considered only nearest neighbor interactions are included the orbital overlap S is set equal to zero and electron-electron repulsion is neglected. Basically,... 

The basic quantity of WF-based electronic structure methods, namely the wave function, can be obtained as solution of the time-independent, nonrela-tivistic SchrSdinger equation. In addition, the vahdity of the Born-Oppenheimer approximation (separabiUty of nuclear and electronic motion) is assumed. The electronic SchrSdinger equation is then solved for fixed nuclei in other words, the nuclear coordinates are parameters rather than variables in the wave function. The electronic Hamiltonian contains pairwise electron-electron interaction energies, meaning that the motion of the individual electrons is not independent of each other but is correlated. ... 

Molecular orbitals were one of the first molecular features that could be visualized with simple graphical hardware. The reason for this early representation is found in the complex theory of quantum chemistry. Basically, a structure is more attractive and easier to understand when orbitals are displayed, rather than numerical orbital coefficients. The molecular orbitals, calculated by semi-empirical or ab initio quantum mechanical methods, are represented by isosurfaces, corresponding to the electron density surfeces Figure 2-125a). 

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... 

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... 

Valence band spectra provide information about the electronic and chemical structure of the system, since many of the valence electrons participate directly in chemical bonding. One way to evaluate experimental UPS spectra is by using a fingerprint method, i.e., a comparison with known standards. Another important approach is to utilize comparison with the results of appropriate model quantum-chemical calculations 4. The combination with quantum-chcmica) calculations allow for an assignment of the different features in the electronic structure in terms of atomic or molecular orbitals or in terms of band structure. The experimental valence band spectra in some of the examples included in this chapter arc inteqneted with the help of quantum-chemical calculations. A brief outline and some basic considerations on theoretical approaches are outlined in the next section. 

It should be emphasized that whereas the theoretical modelling of An3+ spectra in the condensed phase has reached a high degree of sophistication, the type of modelling of electronic structure of the (IV) and higher-valent actinides discussed here is restricted to very basic interactions and is in an initial state of development. The use of independent experimental methods for establishing the symmetry character of observed transitions is essential to further theoretical interpretation just as it was in the trivalent ion case. 

Martin, R.M. (2004) Electronic Structure. Basic Theory and Practical Methods, Cambridge University Press, Cambridge. 

It should be noted that the above classification system of technetium cluster compounds is not the only possible one. In section 4 another classification is described, which is based on thermal stability and the mechanism of thermal decomposition. Section 2.2 is concerned with the classification based on methods of synthesizing cluster compounds. The classifications based on specific properties of clusters do not at all belittle the advantages of the basic structural classification they broaden the field of application of the latter, because for a better understanding and explanation of any chemical, physico-chemical and physical properties it is necessary to deal directly or indirectly with the molecular and/or electronic structures of the clusters. 

The description theoretical study of defects frequently refers to some computation of defect electronic structure i.e., a solution of the Schrodin-ger equation (Pantelides, 1978 Bachelet, 1986). The goal of such calculations is normally to complement or guide the corresponding experimental study so that the defect is either properly identified or otherwise better understood. Frequently, the experimental study suffices to identify the basic structure of the defect this is particularly true when the system is EPR (electron paramagnetic resonance) active. However, if the computational method properly simulates the defect, we are provided with a wealth of additional information that can be used to reveal some of the more basic and general features of many-electron defect systems and defect reactions. 

The basic idea underlying AIMD is to compute the forces acting on the nuclei by use of quantum mechanical DFT-based calculations. In the Car-Parrinello method [10], the electronic degrees of freedom (as described by the Kohn-Sham orbitals y/i(r)) are treated as dynamic classical variables. In this way, electronic-structure calculations are performed on-the-fly as the molecular dynamics trajectory is generated. Car and Parrinello specified system dynamics by postulating a classical Lagrangian ... 

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