Valence bond theory using orthogonalized orbitals

Valence Bond theory using localized orthogonal orbitals For small couplings, i.e. nearly degenerate 5 = 0 and 5=1 states. Valence Bond (VB) theory provides a more intuitive starting point than the previous molecular orbital reasoning. For the Ms = 0 wave functions we make use of the local orthonormal orbitals and tjrb as defined in Eq. 3.10a and use them to construct two neutral determinants and... 

One widely used valence bond theory is the generalised valence bond (GVB) method of Goddard and co-workers [Bobrowicz and Goddard 1977]. In the simple Heitler-London treatment of the hydrogen molecule the two orbitals are the non-orthogonal atomic orbitals on the two hydrogen atoms. In the GVB theory the analogous wavefunction is written ... 

The Hy-CI function used for molecular systems is based on the MO theory, in which molecular orbitals are many-center linear combinations of one-center Cartesian Gaussians. These combinations are the solutions of Hartree-Fock equations. An alternative way is to employ directly in Cl and Hylleraas-CI expansions simple one-center basis functions instead of producing first the molecular orbitals. This is a subject of the valence bond theory (VB). This type of approach, called Hy-CIVB, has been proposed by Cencek et al. (Cencek et.al. 1991). In the full-CI or full-Hy-CI limit (all possible CSF-s generated from the given one-center basis set), MO and VB wave functions become identical each term in a MO-expansion is simply a linear combination of all terms from a VB-expansion. Due to the non-orthogonality of one-center functions the mathematical formalism of the VB theory for many-electron systems is rather cumbersome. However, for two-electron systems this drawback is not important and, moreover, the VB function seems in this case more natural. 

The hybridization scheme in valence bond theory is a very useful concept for chemists since it permits a localized view of the bonding. The most general method for generating hybridized orbitals is based on defining a bond wavefunction (a linear combination of atomic orbitals) in a specific bond direction (usually the z-axis direction). Then the second and subsequent hybrids are obtained by a rotation transformation. Orthogonality conditions are then used to evaluate the hybrid coefficients. These bond wavefunctions are defined as equivalent because they differ from one another only by a rotation. Generally, the first bond wavefunction is... 

Most organic chemists are familiar with two very different and conflicting descriptions of the 7r-electron system in benzene molecular orbital (MO) theory with delocalized orthogonal orbitals and valence bond (VB) theory with resonance between various canonical structures. An attitude fostered by many text books, especially at the undergraduate level, is that the VB description is much easier to understand and simpler to use, but that MO theory is in some sense more fundamental . 

Abstract The wave function of Coulson and Fischer is examined within the context of recent developments in quantum chemistry. It is argued that the Coulson-Fischer ansatz establishes a third way in quantum chemistry, which should not be confused with the traditional molecular orbital and valence bond formalisms. The Coulson-Fischer theory is compared with modern valence bond approaches and also modern multireference correlation methods. Because of the non-orthogonality problem which arises when wave functions are constructed from arbitrary orbital products, the application of the Coulson-Fischer method to larger molecules necessitates the introduction of approximation schemes. It is shown that the use of hierarchical orthogonality restrictions has advantages, combining a picture of molecular electronic structure which is an accord with simple, but nevertheless empirical, ideas and concepts, with a level of computational complexity which renders praetieal applications to larger molecules tractable. An open collaborative virtual environment is proposed to foster further development. 

The generalised valence bond (GVB) method, developed by Goddard in 1970, is one of the simplest and oldest valence bond methods that use flexible orbitals in a general way. The generalised Coulson-Fischer theory for the hydrogen molecule mentioned above is used to describe every electron pair in a molecule. The orbitals for each electron pair are expanded in terms of the full basis set and are non-orthogonal. Orbitals from different pairs are forced to be orthogonal. This condition simplifies the calculations but may lead to some difficulties [160,161],... 

The fascinating thing about the actual computational developments in this period, however, was that almost none used the so-called valence bond (VB) approach that Pauling had proposed as the foundation of the theory of the bond. The technical reasons for this are well enough known. The non-orthogonality between the hybrid orbitals, a feature essential for the justification of Pauling s approach, made formulating the equations for calculation just too complicated and difficult and. 

This use of two non-orthogonal jr-orbitals forms the basis of Paoloni s theory" of the quinquevalent nitrogen atom. Paoloni was not concerned with the construction of a wave-function for the quinquevalent nitrogen atom in a molecule, but only with indicating by means of the valence-bond structure that the two ji-electrons of the pyrrolic nitrogen (tr, tr, tr, ) configuration (tr = sp ) are involved in bonding to the adjacent carbon atoms. 

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