The LCAO-Molecular Orbital Model

In Chapter 3,1 discussed the construction of simple LCAO-MOs for the hydrogen molecule-ion, starting from Is atomic orbitals on the hydrogen centres. Thus, we constructed LCAO-MO approximations to the two lowest energy molecular orbitals as 

A careful analysis shows that the dissociation process corresponds to 

This is a near-universal failure of molecular orbital calculations. 

Spatial function Spin function Electronic state Symmetry Configuration  

Once the calculation is done, we get a better prediction of the equilibrium bond length and dissociation energy, but most important of all we recover the correct behaviour for large 7 Ab- 

Spatial function Spin function Symmetry Configuration 

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Diatomic Molecules.—We shall begin by discussing diatomic molecules, which bear some relation to the models discussed in the previous section. To begin With, the hydrogen molecule has two electrons which both occupy the lowest molecular orbital whose LCAO form is... 

The cluster model of HAp/methyl acetate interface was shown in Fig.2 overlap population analysis was applied to this model. Using Monte Carlo method, 300 sampling points were put around each atom in the cluster. Molecular orbitals in the cluster were constructed by a linear combination of atomic orbitals (LCAO). Atomic orbitals used in this model were ls-2p for C, ls-2p for O, Is for H, ls-3d for P and ls-4p for Ca, which were numerically calculated for atomic Hartree-Fock method. Overlap population was evaluated by Mulliken s population analysis. 

He(I) and He(II) photoelectron spectra of M(BH4)4 (M = Zr or Hf) have been reported. Assignments were made on the basis of qualitative molecular-orbital models and an LCAO-HFS(Xo ) calculation on Zr(BH4)4- ... 

In 1962, Sugano showed that the Seitz model (115) could be interpreted as a molecular orbital model (123), an interpretation that clarifies analysis of these systems. In this interpretation, the absorption bands observed in the TI(I) doped alkali halide system come from the electronic transition aigf a g) hu), but the excited states are still calculated assuming an ionic interaction between the metal and the hgand. Since the thallium-chlorine bond is actually largely covalent, Bramanti et al. (118) modified the approach and used a semiempirical molecular orbital (MO) calculation to describe the energy levels of T1(I) doped KCl. Molecular orbitals were constructed by the linear combination of atomic orbitals (LCAO) method from the 6s and 6p metal orbitals and the 3p chlorine orbitals. Initial calculations were conducted with the one-electron approximation the method was then expanded to include Coulomb and spin-orbit interactions. The results of Bramanti et al. were consistent with experimental... 

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. 

Figure 3.6 shows the LCAO method for generating molecular orbitals of diatomic molecules such as H2. In real molecules, the atomic orbitals of elemental carbon are not really transformed into the molecular orbitals found in methane (CH4). Figure 3.6 represents a mathematical model that mixes atomic orbitals to predict molecular orbitals. Molecular orbitals exist in real molecules and the LCAO model attempts to use known atomic orbitals for atoms to predict the orbitals in the molecule. Molecular orbitals and atomic orbitals are very different in shape and energy, so it is not surprising that the model used for diatomic hydrogen fails for molecules containing other than s-orbitals. 

The Extended Hiickel model treats all valence electrons within the spirit of the TT-electron model. Each molecular orbital is written as an LCAO expansion of the valence orbitals, which can be thought of as being Slater-type orbitals (to look ahead to Chapter 9). Slater-type orbitals are very similar to hydrogenic ones except that they do not have radial nodes. Once again we can understand the model best by considering the HF-LCAO equations... 

Exact solutions to the electronic Schrodinger equation are not possible for many-electron atoms, but atomic HF calculations have been done both numerically and within the LCAO model. In approximate work, and for molecular applications, it is desirable to use basis functions that are simple in form. A polyelectron atom is quite different from a one-electron atom because of the phenomenon of shielding", for a particular electron, the other electrons partially screen the effect of the positively charged nucleus. Both Zener (1930) and Slater (1930) used very simple hydrogen-like orbitals of the form... 

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)). 

The obvious deficiency of crystal-field theory is that it does not properly take into account the effect of the ligand electrons. To do this a molecular-orbital (MO) model is used in which the individual electron orbitals become a linear combination of the atomic orbitals (LCAO) belonging to the various atoms. Before going into the general problem, it is instructive to consider the simple three-electron example in which a metal atom with one ligand atom whose orbital contains two electrons. Two MO s are formed from the two atomic orbitals... 

This highly successful qualitative model parallels the most convenient quantum mechanical approach to molecular orbitals the method of linear combination of atomic orbitals (LCAO). We have assumed that the shapes and dispositions of bond orbitals are related, in a simple way to the shapes and dispositions of atomic orbitals. The LCAO method makes the same assumption mathematically to... 

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