Molecular orbital theory: LCAOs

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). 

The half-wave reduction potentials (HWP) of dibenzothiophene and some of its derivatives have been measured for comparison with those of dibenzofuran and dibenzoselenophene. A shift to more negative HWP was observed for all of the methyl derivatives studied, the magnitude of which depended on the position of substitution. These shifts are in accord with LCAO molecular orbital theory predictions if the sulfur d orbitals are excluded from the calculations (Section III, A). ... 

The basic concept of molecular orbital theory is that molecular orbitals may be constructed from a set of contributing atomic orbitals such that the molecular wave functions consist of linear combinations of atomic orbitals (LCAO).. in the case of the combination of two Is hydrogen atomic orbitals to give two molecular orbitals, the two linear combinations are written below so that atomic wave functions are represented by j/ and molecular wave functions by <)> ... 

Molecular orbital theory was introduced by the use of linear combinations of atomic orbitals (LCAO). 

To further illuminate the LCAO variational process, we will carry out the steps outlined above for a specific example. To keep things simple (and conceptual), we consider a flavor of molecular orbital theory developed in the 1930s by Erich Huckel to explain some of the unique properties of unsaluralcd and aromatic hydrocarbons (Huckel 1931 for historical... 

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). 

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). 

Orbitals) (CO) are expressed as a linear combination of atomic orbitals (LCAO) or similar local functions. There is an obvious parallelism with the traditional Molecular Orbital Theory see Molecular Orbital Theory) developed by chemists, in which the wavefimctions describing the motion of an electron in the molecule (i.e. the Molecular Orbitals) (MO) are also expressed as a LCAOs. As a matter of fact, the first orbital describing the motion of an electron in a polyatomic system was written by Bloch in 1928 and it was a CO. Hence, MO theory finds its roots in solid-state physics. 

Both rigorous and approximate levels of theory express molecular orbitals as LCAOs, thus a short description of the AOs is commonly used. The hydrogen-like AOs that are used in the MO and VB approaches are expressed in the form... 

The molecular integrals [Eqs. (4.2)-(4.4)] are calculated by these molecular orbitals of LCAO approximation. In Roothaan molecular orbital theory, the developing coefficients are determined as the electronic energy (E) becomes minimal (51MI1). Then the developing coefficients (Crj) and the molecular orbital energies (e,) are obtained by solving the Fock equa-... 

These approximations are called the Hiickel approximation. The molecular orbital theory using LCAO and Hiickel approximations is called the Hiickel molecular orbital theory (HMO). HMO gives nearly correct and reasonable results when the distribution of the charge density does not largely deviate from homogeneity. However, HMO calculations cannot... 

Almost all applications of molecular-orbital theory are based on the LCAO... 

HUckel (p. 328) was a pioneer in the field of molecular orbital theory. He developed the LCAO method in its simplest form, yet Htickel molecular orbitals have proved enormously successful in dealing with organic molecules. Htickel proposed the 4/i + 2 rule in 1931. It has been tested in many ways since then, and it works. Now, what is the theoretical basis for this rule ... 

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). 

In the next section we will describe all of the semiempirical molecular orbital theories in common use by employing the ab initio expression for the Fock matrix, Eq. [7b], and the LCAO-MO SCF steps just described above. 

In this book we shall adopt the Linear-Combination-of-Atomic-Orbitals (LCAO) form of molecular-orbital theory. That is to say, we suppose that each of the 71-electrons has a description which allows it to move over the whole framework of the conjugated system. This description allows a molecular orbital, , to be expressed... 

Single determinant ab initio LCAO-SCF molecular orbital theory is used throughout this study (6,7), Molecular geometries were optimized with the minimal STO-3G basis set (8,9), and where possible the energy of the final structure was recalculated with the extended 4-31G basis set (9,10), Such a procedure has been shown to provide a reasonable description of the structures and energies of neutral organic molecules (11) and carbocations (12), The potential surface for the concerted elimination of hydrogen chloride from ethyl chloride also has been studied successfully (13) with this technique. 

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... 

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