Molecular orbital theory LCAO method linear combination

Various theoretical methods and approaches have been used to model properties and reactivities of metalloporphyrins. They range from the early use of qualitative molecular orbital diagrams (24,25), linear combination of atomic orbitals to yield molecular orbitals (LCAO-MO) calculations (26-30), molecular mechanics (31,32) and semi-empirical methods (33-35), and self-consistent field method (SCF) calculations (36-43) to the methods commonly used nowadays (molecular dynamic simulations (31,44,45), density functional theory (DFT) (35,46-49), Moller-Plesset perturbation theory ( ) (50-53), configuration interaction (Cl) (35,42,54-56), coupled cluster (CC) (57,58), and CASSCF/CASPT2 (59-63)). 

Hiickel theory separates the tt system from the underlying a framework and constructs molecular orbitals into which the tt electrons are then fed in the usual way according to the Aufbau principle. The tt electrons are thus considered to be moving in a field created by the nuclei and the core of a electrons. The molecular orbitals are constructed from linear combinations of atomic orbitals and so the theory is an LCAO method. For our purposes it is most appropriate to consider Hiickel theory in terms of the CNDO approximation (in fact, Hiickel theory was the first ZDO molecular orbital theory to be developed). Let us examine the three types of Fock matrix element in Equations (2.252)-(2.254). First, In... 

When the Hartree-Fock method is applied to molecules, molecular orbitals are used instead of atomic orbitals. To construct the molecular orbitals, one widely used approximation is LCAO (linear combinations of atomic orbitals). According to molecular orbital theory, the total wave function of the system is written as a combination of molecular orbitals, spin functions describing electrons in terms of spin j(a) or — j p). 

In the molecular-orbital theory, each successive valence electron is considered as entering a field of positive electric charge furnished by the nuclei. One mathematical approach is quite extensively used as an aid in setting up a description of electron probability densities in the vicinity of more than one nucleus this approach is called the method of linear combinations of atomic orbitals (coveniently abbreviated LCAO). 

Linear combination of atomic orbitals (LCAO theory) A method for combining atomic orbitals to approximately compute molecular orbitals. 

Molecular orbital theory originated from the theoretical work of German physicist Friederich Hund (1896-1997) and its apphcation to the interpretation of the spectra of diatomic molecules by American physical chemist Robert S. MuUiken (1896-1986) (Hund, 1926, 1927a, b Mulliken, 1926, 1928a, b, 1932). Inspired by the success of Heitler and London s approach, Finklestein and Horowitz introduced the linear combination of atomic orbitals (LCAO) method for approximating the MOs (Finkelstein and Horowitz, 1928). The British physicist John Edward Lennard-Jones (1894-1954) later suggested that only valence electrons need be treated as delocalized inner electrons could be considered as remaining in atomic orbitals (Lennard-Jones, 1929). 

The theory of the chemical bond is one of the clearest and most informative examples of an explanatory phenomenon that probably occurs in some form or other in many sciences (psychology comes to mind) the semiautonomous, nonfundamental, fundamentally based, approximate theory (S ANFFBAT for short). Chemical bonding is fundamentally a quantum mechanical phenomenon, yet for all but the simplest chemical systems, a purely quantum mechanical treatment of the molecule is infeasible especially prior to recent computational developments, one could not write down the correct Hamiltonian and solve the Schrodinger equation, even with numerical methods. Immediately after the introduction of the quantum theory, systems of approximation began to appear. The Born Oppenheimer approximation assumed that nuclei are fixed in position the LCAO method assumed that the position wave functions for electrons in molecules are linear combinations of electronic wave functions for the component atoms in isolation. Molecular orbital theory assumed a characteristic set of position wave functions for the several electrons in a molecule, systematically related to corresponding atomic wave functions. 

In contrast to force-field calculations in which electrons are not explicitly addressed, molecular orbital calculations, use the methods of quantum mechanics to generate the electronic structure of molecules. Fundamental to the quantum mechanical calculations that are to be performed is the solution of the Schrodinger equation to provide energetic and electronic information on the molecular system. The Schrodinger equation cannot, however, be exactly solved for systems with more than two particles. Since any molecule of interest will have more than one electron, approximations must be used for the solution of the Schrodinger equation. The level of approximation is of critical importance in the quality and time required for the completion of the calculations. Among the most commonly invoked simplifications in molecular orbital theory is the Bom-Oppenheimer [13] approximation, by which the motions of atomic nuclei and electrons can be considered separately, since the former are so much heavier and therefore slower moving. Another of the fundamental assumptions made in the performance of electronic structure calculations is that molecular orbitals are composed of a linear combination of atomic orbitals (LCAO). 

This agrees quite well with simple molecular orbital theory. We assume that the molecule is planar, and that the atoms are joined by a-bonds in sp hybridization. Ten electrons are used up for these bonds, and additionally four are at inner Is shells of the two nitrogen atoms. There are totally seventeen electrons. The remaining three are placed in jr-orbitals. These are constmcted as a linear combination of atomic orbitals according to the LCAO method ... 

The fundamental assumption of HMO theory is that we may calculate molecular orbitals through a process known as LCAO the linear combination of atomic orbitals. That is, we use some combination of the wave functions of the atomic orbitals to produce a set of molecular orbitals. In the Huckel method, we combine a set of atomic p orbitals to produce a set of n molecular orbitals. For a set of n parallel p orbitals, the Huckel molecular orbitals have the form shown in equation 4.1. In this equation is the wave function... 

LzJ The linear combination of atomic I—I orbitals (LCAO) method is the most straightforward approach to molecular orbital theory. 

MOI.ECULAR-ORBITAL THEORY According to molecular-orbital theory, electrons in molecules are in orbitals that may be associated with several nuclei. Molecular orbitals in their simplest approximate form are considered to be linear combinations of atomic orbitals. We assume that when an electron in a molecule is near one particular nucleus, the molecular wave function is approximately an atomic orbital centered at that nucleus. This means that we can form molecular orbitals by simply adding and subtracting appropriate atomic orbitals. The method is usually abbreviated LCAO-MO, which stands for linear combination of atomic... 

Molecular orbital theory uses a mathematical method called the linear combination of atomic orbitals (LCAO) to form molecular orbitals. Each molecular orbital is associated with the entire molecule, rather than just two atoms. 

The molecular orbital theory of chemical bonding rests on the notion that, as electrons in atoms occupy atomic orbitals, electrons in molecules occupy molecular orbitals. Just as our first task in writing the electron configuration of an atom is to identify the atomic orbitals that are available to it, so too must we first describe the orbitals available to a molecule. In the molecular orbital method this is done by representing molecular orbitals as combinations of atomic orbitals, the linear combination of atomic orbitals-molecular orbital (LCAO-MO) method. 

This theory endeavors to describe the molecule by a method intrinsically similar to that used for obtaining atomic orbitals but considering multicenter wave functions. Thus this approach consists of finding the best functions for describing the state of one electron in a field formed by the totality of the nuclei placed in their equilibrium positions. These monoelectronic molecular wave functions may be obtained according to the MO-theory by a linear combination of atomic orbitals (LCAO). 

Linear combination of atomic orbitals (LCAO) method [27] has been utilized as a common technique to represent one-electron wave functions that work as building blocks in density functional theories as well as in molecular orbital theory. 

Most of the commonly used electronic-structure methods are based upon Hartree-Fock theory, with electron correlation sometimes included in various ways (Slater, 1974). Typically one begins with a many-electron wave function comprised of one or several Slater determinants and takes the one-electron wave functions to be molecular orbitals (MO s) in the form of linear combinations of atomic orbitals (LCAO s) (An alternative approach, the generalized valence-bond method (see, for example, Schultz and Messmer, 1986), has been used in a few cases but has not been widely applied to defect problems.)... 

These abbreviated answers will be of little use to those who are not already familiar with the subject. They will be expanded in Chapters 4-7. Before considering them in detail, we will look at the concepts and methods which are needed to use frontier orbitals efficiently. Since the molecular orbitals employed in perturbation theory are generally expressed as linear combinations of atomic orbitals (LCAOs), Chapter 2 will review atomic orbitals (AOs), outline molecular orbitals (MOs) and describe the Hiickel method for calculating them. Chapter 3 will set out perturbation methods in a practical fashion, putting more emphasis on applications and physical interpretation than upon mathematical derivation. 

The aim of this chapter is to review the current status of the quantum-mechanical calculation of electric and magnetic properties of isolated atoms and molecules. In view of the rapid advances made during the past decade in the calculation of ab initio molecular wavefunctions, we will clearly concentrate for the most part on the calculation of such properties using standard ab initio methods such as gaussian orbital LCAO-MO-SCF (linear combination of atomic orbital-miolecular orbital-self-consistent field), configuration interaction (Cl), coupled Hartree-Fock, and the like, but will also review similar calculations at the semi-empirical and empirical level where appropriate. For readers unfamiliar with the theory of electric and magnetic properties, the books by Davies and by Atkins review the subject thoroughly, whilst the more technical details of quantum-mechanical calculations on atoms and molecules have been described in many other places. ... 

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