Molecular orbital theory Schrodinger equation

The simplest approximation to the Schrodinger equation is an independent-electron approximation, such as the Hiickel method for Jt-electron systems, developed by E. Hiickel. Later, others, principally Roald Hoffmann of Cornell University, extended the Hiickel approximations to arbitrary systems having both n and a electrons—the Extended Hiickel Theory (EHT) approximation. This chapter describes some of the basics of molecular orbital theory with a view to later explaining the specifics of HyperChem EHT calculations. 

We know that not all solids conduct electricity, and the simple free electron model discussed previously does not explain this. To understand semiconductors and insulators, we turn to another description of solids, molecular orbital theory. In the molecular orbital approach to bonding in solids, we regard solids as a very large collection of atoms bonded together and try to solve the Schrodinger equation for a periodically repeating system. For chemists, this has the advantage that solids are not treated as very different species from small molecules. 

In molecular orbital theory, there is a clear and well defined path to the exact solution of the Schrodinger equation. All we need do is express our wave function as a Unear combination of all possible configurations (full CI) and choose a basis set that is infinite in size, and we have arrived. While such a goal is essentially never practicable, at least the path to it can be followed unambiguously until computational resources fail. 

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). 

Although a separation of electronic and nuclear motion provides an important simplification and appealing qualitative model for chemistry, the electronic Schrodinger equation is still formidable. Efforts to solve it approximately and apply these solutions to the study of spectroscopy, stmcture and chemical reactions form the subject of what is usually called electronic structure theory or quantum chemistry. The starting point for most calculations and the foundation of molecular orbital theory is the independent-particle approximation. 

It is straightforward to write down and solve the many-electron Schrodinger equation if it is assumed that the electrons do not interact, or interact only to a very small extent. Indeed, it is on this premise that the fabric of modem qualitative molecular orbital theory is based. For the two electrons in a helium atom [Z = 2] for example, this independent particle model Schrodinger equation is simply... 

The first target of quantum chemistry was how to solve the SchrOdinger equation for electronic motions in molecules. To address this challenge, the Hartree-Fock method (Hartree 1928) and its variational method (Slater 1928), molecular orbital theory (Hund 1926 Mulliken 1927), and the Slater determinant (Slater 1929) were developed, resulting in the Hartree-Fock method (Fock 1930 Slater 1930), which is accepted as the precursor of quantum chemistry. Soon afterward, the configuration interaction (Cl) method (Condon 1930), M0ller-Plesset perturbation method (Mpller and Plesset 1934), and multiconflgurational SCF method (Frenkel... 

The second broad approach to the description of molecular structure that is of importance in organic chemistry is molecular orbital theory. Molecular orbital (MO) theory discards the idea that bonding electron pairs are localized between specific atoms in a molecule and instead pictures electrons as being distributed among a set of molecular orbitals of discrete energies. In contrast to the orbitals described by valence bond theory, which are usually concentrated between two specific atoms, these orbitals can extend over the entire molecule. Molecular orbital theory is based on the Schrodinger equation,... 

Ab initio molecular orbital theory is based on the laws of quantum mechanics, under which the energy (E) and wave function ( 1 ) for some arrangement of atoms can be obtained by solving the Schrodinger equation 1 (17). 

Under ab initio molecular orbital theory, the exact solution to the Schrodinger equation could be obtained using the Full Cl method in conjimction with an infinite basis set. Since this is impractical, computational methods must place restrictions... 

Molecular orbital (MO) theory begins with the hypothesis that electrons in atoms exist in atomic orbitals and assumes that electrons in molecules exist in molecular orbitals. Just as the Schrbdinger equation can be used to calculate the energies and shapes of atomic orbitals, molecular orbital theory assumes that the Schrodinger equation can also be used to calculate the energies and shapes of molecular orbitals. Following is a summary of the rules used in applying molecular orbital theory to the formation of covalent bonds. 

According to the ab initio molecular orbital theory methodology, atomic orbitals (set of functions, also called basis sets) combine in a way to form molecnlar orbitals that snrronnd the molecule. The molecular orbital theory considers the molecnlar wave function as an antisymmetiized product of orthonormal spatial molecular orbitals. Then they are constructed as a Slater determinant [56], Essentially, the calculations initially use a basis set, atomic wave functions [57, 58], to constract the molecular orbitals. The first and basic ab initio molecular orbital theory approach to solve the Schrodinger equation is the Hartree-Fock (HF) method [59, 60], Almost all the ab initio methodologies have the same basic numerical approach but they differ in mathematical approximations. As it is clear that finding the exact solution for the Schrodinger equation, for a molecular system, is not possible, various approaches and approximations are used to find the reliable to close-to-accurate solutions [61-68]. 

The term "ab initio means "from first principles" it does not mean "exact" or "true". In ab initio molecular orbital theory, we develop a series of well-defined approximations that allow an approximate solution to the Schrodinger equation. We calculate a total wavefunc-tion and individual molecular orbitals and their respective energies, without any empirical parameters. Below, we outline the necessary approximations and some of the elements and principles of quantum mechanics that we must use in our calculations, and then provide a summary of the entire process. Along with defining an important computational protocol, this approach will allow us to develop certain concepts that will be useful in later chapters, such as spin and the Born-Oppenheimer approximation. 

The use oE self-consistent field molecular orbital theory (SCF-MO) implies a number of approximations which are necessary, since the Schrodinger equation is generally unsolvable for three or more bodies. Thus SCF MO s represent an optimum approximation to the many-electron wave function. 

There have been a number of instances where EAs have been used to obtain approximate solutions to the Schrodinger equation. Zeiri et al. use a real-value encoding to aid in the calculation of bound states in a double well potential and in the non-linear density functional calculation. Rossi and Truhlar have devised a GA to fit a set of energy differences obtained by NDDO semiempirical molecular orbital theory to reference ab initio data in order to yield specific reaction parameters. The technique was applied to the reaction Cl -I- (THa. In a third example, Rodriguez et al. apply a GA to diagonalization of the... 

Indeed, chemists think of atoms as the building blocks of molecules (and their assemblies), whereas the physically rooted Schrodinger equation thinks of molecules in terms of electrons and nuclei. Another example of such dislocation is the computationally convenient molecular orbital theory versus the chemically more intuitive valence bond theory. In this chapter we will introduce QCT, starting with QTAIM [2, 17, 18]. This theory will serve as a tool to bridge the gap between the numerical emptiness of modem wave functions and the wealth of chemical concepts. In an ideal world, chemical insight can indeed be safely extracted from modem wave functions. If this extraction persistently fails for a chemical concept such as aromaticity, for example, then the concept should be modified or abandoned. 

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