The LCAO Model

The exact solution of the electronic Schrddinger equation is no mean feat, even for sueh a simple molecule as H2 . The electronic Schrddinger equation is 

Let s examine the limiting behaviour when the electron is in the vicinity of nucleus A but far away from nucleus B. In this case the electronic Schrddinger equation is 

From simple symmetry arguments concerning the electron density, we can deduce that ca = cb and we label the two molecular orbitals by symmetry  

In order to test the accuracy of the LCAO approximations, we use the variation principle if V lcao is an approximate solution then the variational integral 

From simple symmetry arguments concerning the electron density, we can deduce that cA = cB and we label the two molecular orbitals by symmetry lcrg = lsA + lsB and lo-u = lsA — lsB. Neither is a solution of the electronic Schrodinger equation, but each has the correct boundary conditions and so they are possible approximate solutions. 

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In later chapters we will be concerned with the LCAO model. Suppose we have a set of n atomic orbitals > i(r), X2( )> > d a normalized LCAO orbital... 

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... 

In conclusion, let us say that even though the method becomes rather complicated here because the factor in polynomial form is replaced by a Fermi function, we consider that the method nevertheless presents a certain interest because at the same time it gives a solid base for comparison between the LCAO model and the metallic model. 

The LCAO Model of tt-MOs of Ethene, Acetylene, and Butadiene Frontier Orbitals... 

The characteristics of most chemical bonds (bond length, bond energy, polarity, and so forth) do not differ significantly from molecule to molecule (see Section 3.7). If the bonding electrons are spread out over the entire molecule, as described by the LCAO model, then why should the properties of a bond be nearly independent of the nature of the rest of the molecule Would some other model be more suitable to describe chemical bonds ... 

This also puts in doubt the nomic character of the ab initio procedure. If, as it were, the final model that is used in a ah initio calculation, i.e., the Hartree Fock operator applied to the LCAO model is not deducible from the primordial model, i.e., the Schrodinger-equation with the original Hamiltonian, then it seems that the law that is de facto operating in the ab initio derivation is contained in the Hartree Fock model. But this simplified model is based on assumptions that in part are contradictory to the assumptions of the original model. These assumptions include ... 

Draw the LCAO model of ethylene in the bonding and antibonding orbitals. Distinguish the ground state of ethylene from its excited state. Distinguish from ir 7T. 

The LCAO models may readily be understood by viewing each orbital as a wave function, whereby like phases will add, and opposite phases will tend to cancel each other. 

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. 

When the orbitals for two carbon atoms are mixed to form a C-C bond, the core electrons in the Is-orbital are omitted because they are not involved in bonding. Two electrons in the 2s-orbital remain, but they are found in only two of the three 2p-orbitals. When these atomic orbitals are mixed using the LCAO model, the valence orbitals are derived from the 2s- and 2p-orbitals, generating two 2s-type molecular orbitals and six 2p-type molecular orbitals. This model predicts unshared electrons in the valence shell (atoms with unshared electrons of this type are called radicals see Chapter 7, Section 7.4.3). 

This solution is not correct, as is apparent when the chemical bonds in many real organic molecules are examined. The four C-H bonds in methane (CH4) are identical, for example, and the four bond lengths and bond strengths are identical. The LCAO model did not make a correct prediction to obtain a correct solution, the mathematical model must be modified. To form a covalent bond, an atom must share electrons with another atom. This means that the position of the electrons relative to the nucleus is different from their positions in the elemental atom they are located in a molecular orbital. 

The importance of relativity to chemistry was first noticed on the basis of atomic calculations [1008,1009], because effects on bonding may be qualitatively anticipated in terms of the LCAO model. Moreover, atoms and their ions can be studied with the utmost accuracy (in the case of one-electron ions even exactly), and we are therefore advised to start with a comparison of Schrodinger and Dirac atoms. Such comparisons have been made several times throughout this book on a formal basis. Now, we recall some typical effects on one-electron states and orbital energies. 

Another, different way of utilizing the atomic hypothesis was realized in the Chemical Hamiltonian Approach [46] which exploits the LCAO model of quantum chemistry. It is important to stress that the LCAO (linear combination of atomic orbitals) expansion of the molecular wave functions is itself, in a certain sense, based on the concept of building blocks . In effect, the idea behind the LCAO method is the chemical experience that the basic building blocks of the molecules are the atoms. Thus the atomic orbitals obtained from atomic wave functions could be suitable for the expansion of molecular orbitals. 

If the transfer integral magnitude is dominated by ligand-ligand contact as assumed above, the "LCAO" model yields... 

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