Basic theoretical method

For simplicity, we shall illustrate the basic theoretical methods employed in model calculations by an idealized case of the interaction between a substituted benzene, C6H5X, and a transition-metal atom such as chromium. This treatment is then readily extended to either a diarene complex or a monoarene complex such as a tricarbonylarene metal. 

The organization of this review is as follows In Section II we describe the theoretical and experimental background of the field. Section HI reviews experimental work on the criticality of ionic fluids. Section IV presents the basic theoretical methods for describing ionic phase transitions at the mean-field level. Results obtained by these techniques are reviewed in Section V. Section VI reviews the theoretical work concerned with the nature of the critical point. The review closes in Section VII with a brief summary and outlook. 

In this section DFT treatments of the iron-molybdenum cofactor and the activation, reduction and protonation of N2 proceeding at this cluster are presented. The earliest of these calculations appeared after publication of the first crystal structures of nitrogenase, and this was well before the discovery of the central atom X. After the discovery of X and its identification as carbon these treatments were, in part, updated. Here we will focus both on the basic theoretical methods to treat the electronic structure of the FeMoco and the reduction of N2 mediated by this cluster. 

Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

A variety of theoretical methods have been developed which include some effects of electron correlation. Traditionally, such methods are referred to as post-SCF methods because they add correlation corrections to the basic Hartree-Fock model. As of this writing, there are many correlation methods available in Gaussian, including the following ... 

The use of theoretical methods in the study of bicyclic systems with P-, As-, Sb-, or Bi- bridgehead atoms has contributed to an increased understanding of the geometry, stability, and ring-strain effects of these systems. In addition, important data relating to basicity and the interpretation of nuclear magnetic resonance (NMR) and X-ray data have been generated. A vast majority of the work done has focused on P. 

There are many varieties of density functional theories depending on the choice of ideal systems and approximations for the excess free energy functional. In the study of non-uniform polymers, density functional theories have been more popular than integral equations for a variety of reasons. A survey of various theories can be found in the proceedings of a symposium on chemical applications of density functional methods [102]. This section reviews the basic concepts and tools in these theoretical methods including techniques for numerical implementation. 

The orbital coefficients obtained from Hiickel calculations predict the terminal position to be the most reactive one, while the AMI model predicts the Cl and C3 positions to be competitive. In polyenes, this is true for the addition of nucleophilic as well as electrophilic radicals, as HOMO and LUMO coefficients are basically identical. Both theoretical methods agree, however, in predicting the Cl position to be considerably more reactive as compared to the C2 position. It must be remembered in this context that FMO-based reactivity predictions are only relevant in kinetically controlled reactions. Under thermodynamic control, the most stable adduct will be formed which, for the case of polyenyl radicals, will most likely be the radical obtained by addition to the C1 position. 

As with the methylbenzenes the methyl derivatives of condensed aromatic substances show an increase in basicity compared to the unsubstituted compound. Thus, the methyl groups exert a very profound influence on the electron distribution. The extension of these considerations by theoretical methods will be discussed in Section V. 

When an electron is injected into a polar solvent such as water or alcohols, the electron is solvated and forms so-called the solvated electron. This solvated electron is considered the most basic anionic species in solutions and it has been extensively studied by variety of experimental and theoretical methods. Especially, the solvated electron in water (the hydrated electron) has been attracting much interest in wide fields because of its fundamental importance. It is well-known that the solvated electron in water exhibits a very broad absorption band peaked around 720 nm. This broad absorption is mainly attributed to the s- p transition of the electron in a solvent cavity. Recently, we measured picosecond time-resolved Raman scattering from water under the resonance condition with the s- p transition of the solvated electron, and found that strong transient Raman bands appeared in accordance with the generation of the solvated electron [1]. It was concluded that the observed transient Raman scattering was due to the water molecules that directly interact with the electron in the first solvation shell. Similar results were also obtained by a nanosecond Raman study [2]. This finding implies that we are now able to study the solvated electron by using vibrational spectroscopy. In this paper, we describe new information about the ultrafast dynamics of the solvated electron in water, which are obtained by time-resolved resonance Raman spectroscopy. 

Theoretical and Experimental NMR techniques provide powerful tools for the investigation of heterogeneous catalysis. Recent advances in in situ NMR techniques are summarized, as are advances in theoretical methods. The utility of our combined theoretical/experimental approach is illustrated by studies of the pentamethylbenzenium cation and the 1,3-dimethylcyclopentenyl cation in zeolite HZSM-5, acetylene adsorption on MgO, and the isopropyl cation on frozen SbF5. We also discuss the role of the basicity of adsorbates in the formation of stable carbenium ions on zeolites. 

Applying computational techniques to chemical problems first requires a careful choice of the theoretical method. Basic knowledge of the capabilities and the drawbacks of the various methods is an absolute necessity. However, as no practical chemist can be expected to be well versed in the language and fine details of computational theory, we approach the subject by briefly (and by no means completely) reminding the reader of the underlying concepts of particular methods and of their often less well-documented limitations. 

The main goal of this chapter is to present the theoretical background of some basic chemometric methods as a tool for the assessment of surface water quality described by numerous chemical and physicochemical parameters. As a case study, long-term monitoring results from the watershed of the Struma River, Bulgaria, are used to illustrate the options offered by multivariate statistical methods such as CA, principal components analysis, principal components regression (models of source apportionment), and Kohonen s SOMs. 

The remainder of this paper is organized as follows In Sect. 5.2, we present the basic theory of the present control scheme. The validity of the theoretical method and the choice of optimal pulse parameters are discussed in Sect. 5.3. In Sect. 5.4 we provide several numerical examples i) complete electronic excitation of the wavepacket from a nonequilibrium displaced position, taking LiH and NaK as examples ii) pump-dump and creation of localized target wavepackets on the ground electronic state potential, using NaK as an example, and iii) bond-selective photodissociation in the two-dimensional model of H2O. A localized wavepacket is made to jump to the excited-state potential in a desirable force-selective region so that it can be dissociated into the desirable channel. Future perspectives from the author s point of view are summarized in Sect. 5.5. 

In this section we present the results of the calculation of the basic positronium properties using theoretical methods described above. We begin with positronium spectroscopy and then discuss the calculation of the second order corrections to the parapositronium decay rate. 

Let us discuss now a usual theoretical method employed in the calculation of the radial distribution function. The basic equation obeyed by g(r) is the integral equation, introduced by Ornstein and Zernike in 1914 [25,32]... 

During the past two decades, much attention has been drawn in this area and advances have been made in theoretical analysis concerning the applicability of Eq. (1) in a variety of systems. This chapter presents the state of understanding of the electrophoretic motion of colloidal particles under various conditions. We first introduce the basic concept and fundamental electrokinetic equations for electrophoretic motion. Then, we review some recent studies on the mobility of a single particle, the boundary effects and the particle interactions in electrophoresis. In addition, a few theoretical methods, which have been used to investigate the boundary effects and particle interactions, will be highlighted and demonstrated in the context. 

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