LCAO-MO description

In order to contrast the MO-type and VB-type descriptions, we may say that the former attempts to describe the MOs (or NOs) in terms of general mixings of AO basis functions (LCAO-MO description),... 

The Huckel approximation is defined by a set of simplicafions to the form of the Hamiltonian in the LCAO-MO description of planar conjugated molecules. Although the Huckel approximations are quite severe, nevertheless they produce results that rationafize qualitatively the resonance energies and spectra of these molecules. 

We have assumed a classical model for molecules in which the majority of the electrons are localized on definite atoms or in definite two center bonds. Although this assumption works well in practice, it certainly needs justification, for in a correct LCAO MO description of a typical molecule, each MO is composed of AO s contributed by all the atoms present and there is nothing in this description that corresponds even remotely to the classical idea of localized bonds. 

Among the countless concepts that Linus Pauling introduced from Quantum Mechanics into chemistry[l,2], and that became standard principles of the trade, there is the idea of hybridization . In the framework of the valence-bond description of a system, it is useful to mix atomic orbitals of the same n-quantum number , or of similar spatial extent, to construct directed, asymmetric atomic contributions. Although hybrids are not needed in an LCAO-MO description of the system, they have so much become part of the language of both organic and inorganic chemistry, that people will go out of their way to arrive at descriptions that are compatible with them. 

In the LCAO MO description, the H2 molecnle in its ground state has a pair of electrons in a bonding MO, and thus a single bond (that is, its bond order is 1). Later in this chapter, as we describe more complex diatomic molecules in the LCAO approximation, bond orders greater than 1 are discussed. This quantum mechanical definition of bond order generalizes the concept first developed in the Lewis theory of chemical bonding—a shared pair of electrons corresponds to a single bond, two shared pairs to a double bond, and so forth. 

Obviously, the distinction between a and it orbitals cannot break down completely if there is a slight deviation from planarity, e.g. for a planar molecule in the course of an out-of-plane vibration. Thus, it can be useful to distinguish quasi-linear combinations of 2p atomic... 

Some alternative approaches have been taken to study the positron annihilation characteristics without explicit band-structure calculations. As far as YBa2Cu307 is concerned, the LCAO-MO approach developed by Chiba (1976) for oxides has been applied by Chiba (1992), Turchi et al. (1988, 1990) and Saul and Weissmann (1990). Although the Fermi surface is not a concept in the LCAO-MO description, this method is quite appropriate for the direct physical insights it provides for the analysis of the anisotropies A(r,d) of the 2D-ACAR distribution, eq. (15). 

Later there was an attempt by ab initio calculation to fit the electron structure of diazirine into the Walsh model of cyclopropane (69MI50800). According to these SCF-LCAO-MO calculations three MOs add to the description of the lone electron pairs, all of which also contribute to some extent to ring bonding. As to strain, 7r-character and conjugative effect, the term pseudo-rr-character was used. 

The origin of cyclopropenone chemistry goes back to the successful preparation of stable derivatives of the cyclopropenium cation <5 3), the first member of a series of Huckel-aromatic monocyclic carbo-cations possessing a delocalized system of (4n + 2)-7r-electrons. This experimental confirmation of LCAO-MO theory stimulated efforts to prepare other species formally related to cyclopropenium cation by a simple resonance description of electron distribution, namely cyclopropenone 7 and methylene cyclopropene (triafulvene) 8 ... 

In general, the orbitals in this method are expanded as linear combinations of a basis set of Slater functions in the same spirit as in the LCAO-MO-SCF method. However, in the present case the orbitals are essentially localized, and a description such as equation (67) is clearly equivalent to a linear combination of a great many VB structures, both covalent and ionic. Thus, in the case of methane this should provide a very good description of the ground state, particularly of the potential surfaces for such processes as... 

Since the discussion of molecular orbitals for the intermediate species involved in collisions will be used extensively in this chapter, it seems appropriate to describe now the various LCAO-MO methods in most common usage for calculating molecular orbitals. This presentation will be general and is equally applicable to diatomic and polyatomic molecules. This description of MO methods is not intended as a detailed comprehensive review of quantum chemical calculational techniques, but rather as an indication of the different methods available. 

The reality, of course, is that any complete set of square integrable functions, such as all the atomic orbitals of any atom, provide for an exact description of the eigenfunction solutions to the Schrodinger equation for a molecule. However, since such an approach is not practicable even on the largest modem computers, we settle for an approximation, which leads to good results when comparisons with experimental data are made. This is the essence of the LCAO-MO approximation and it would be normal to extend equation 6.2 and take linear combinations of valence atomic orbitals chosen from all... 

Note that such calculations assume the total wave function as a single Slater determinant, while the resultant molecular orbital is described as a linear combination of the atomic orbital basis functions (MO-LCAO). Multiple Slater determinants in MO description project the configurational and post-HF methods and will not be discussed here. 

The LCAO-MO model is the most popular one in the description of covalent bonding in atomic lattices of metals, semiconductors, and insulators. As in the case of the MO model for molecules, the atomic orbitals on the atoms in a solid can be combined into molecular orbitals by linear combination. As many molecular orbitals can be made out of atomic orbitals as there are atomic orbitals for them. In solids that number is very high and the many molecular orbitals made from one atomic orbital on each atom form continuous bands. The number of nodal planes in the molecular orbitals increases with their energy. 

In MO theory, the network of chemical bonds is determined by the occupied MO in the system ground state. For reasons of simplicity, assume the closed-shell (cs) configuration of N = 2n electrons in the standard spin-restricted HF (RHF) description, which involves the n lowest (doubly occupied, orthonormal) MO. In the familiar LCAO MO approach, they are expanded as linear combinations (LC) of the (Lbwdin) orthogonalized AO x = iXoXi - -XJ = iXi) contributed by the system constituent atoms (x x)= 8, si, = ( 2,... ) = J = xC, where the rectangular matrix C = = (x ) groups the relevant expansion coefficients of MO (i.e., LC... 

The individual bonds, or if we prefer a more realistic MO description, also the molecular orbitals, will be expressed in the form of the usual LCAO approximation in the basis of atomic orbitals. In our case, if we confine ourselves to a simpler localized description, the conesponding orbitals (bonds) are given by eq. (12). 

Consider what happens when a molecule is formed Two (or more) atoms combine to make a molecular system. The individual orbitals of the separate atoms combine to make orbitals that span the entire molecule. Why not use this description as a basis for defining molecular orbitals This is exactly what is done. By using linear variation theory, one can take linear combinations of occupied atomic orbitals and mathematically construct molecular orbitals. This defines the linear combination of atomic orbitals—molecular orbitals (LCAO-MO) theory, sometimes referred to simply as molecular orbital theory. 

According to MO-theory, the wave function for the system H2 may be formulated as a linear combination of two hydrogen atomic orbitals Is. The meaning of these two LCAO-MOs can be appreciated in both analytical and pictorial descriptions illustrated in Fig. 1.10. The electron density distribution associated to the orbital i/ ab shows that the charge concentration between the nuclei is indeed greater than the simple sum of contributions of two separate orbitals since (i/ a + + b- That results in binding the nuclei together. 

Some more rigorous, non-parametric calculations also confirm the validity of the isolobal principle. Fenske-Hall LCAO-MO-SCF calculations of the structure of B5H9 and some of its ferroborane derivatives, i.e. l-Fe(CO)3B4Hs, 2-Fe(CO)3B4Hs and l,2-[Fe(CO)3]2B3H3 have been completed. A summary of a qualitative description of the properties of frontier molecular orbitals in... 

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