Linear combinations of the atomic orbitals

LCAO method A method of calculation of molecular orbitals based upon the concept that the molecular orbital can be expressed as a linear combination of the atomic orbitals. 

We wish to construct linear combinations of the atomic orbitals such that the overall wavefunction meets the Bloch requirement. Suppose the s orbitals in our lattice are labelled X , where the wth orbital is located at position x = na. An acceptable linear combination of these orbitals that satisfies the Bloch requirements is ... 

VVe now need to consider how the form of the wavefunction varies with k. The first situation we consider corresponds to fc = 0, where the exponential terms are all equal to 1 and the overall wavefunction becomes a simple additive linear combination of the atomic orbitals ... 

In die HMO approximation, the n-electron wave function is expressed as a linear combination of the atomic orbitals (for the case in which the plane of the molecule coincides with the x-y plane). Minimizing the total rt-electron energy with respect to the coefficients leads to a series of equations from which the atomic coefficients can be extracted. Although the mathematical operations involved in solving the equation are not... 

In order to obtain an approximate solution to eq. (1.9) we can take advantage of the fact that for large R and small rA, one basically deals with a hydrogen atom perturbed by a bare nucleus. This situation can be described by the hydrogen-like atomic orbital y100 located on atom A. Similarly, the case with large R and small rB can be described by y100 on atom B. Thus it is reasonable to choose a linear combination of the atomic orbitals f00 and f00 as our approximate wave function. Such a combination is called a molecular orbital (MO) and is written as... 

In MO calculations, a wave function is formulated that is a linear combination of the atomic orbitals that have overlapped (this method is often called the linear combination of atomic orbitals, or LCAO). Addition of the atomic orbitals gives the bonding MO ... 

Consider the trans isomer of butadiene. Both the symmetry operations that define the group < 2h and the characters of the representation r are given in Table 3. The reduction of this representation leads to Tn =2Bg 2Aa. Thus, two linear combinations of the atomic orbitals can be constructed of symmetry Bg and two others of symmetry A. Their use will factor the secular determinant into two 2x2 blocks, as described in the following paragraph. 

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... 

We will explain how to do this by taking the specific example of methane. Methane has a central carbon atom which is a-bonded to four hydrogen atoms with each a-bond pointing to one of the comers of a tetrahedron. We therefore require four hybrid orbitals on the carbon atom which similarly point to the comers of a tetrahedron. Since the four bonds are indistinguishable, the four hybrids must be equivalent, that is to say they must be identical in all respects except for their orientation. For the reasons given in 11-2, they will be taken to be linear combinations of the atomic orbitals of carbon, which are... 

The incorporation of the d orbitals complicates the group theoretic procedure to some extent. Table 7.1.7 gives some useful steps in the derivation of the linear combinations of six atomic orbitals. With the results in Table 7.1.7, the linear combination of the atomic orbitals can be readily derived ... 

Various LCAO-MO Methods. In order to render tractable the problem of determining the molecular electronic eigenfunction, Fe, it is customary to assume the individual molecular orbitals to be functions of the atomic electron eigenfunctions, xr> centered on each atom. The molecular orbitals (MO s), it are taken to be linear combinations of the atomic orbitals (LCAO s), Xr-... 

Crystal orbitals are built by combining different Bloch orbitals (which we will henceforth refer to as Bloch sums), which themselves are linear combinations of the atomic orbitals. There is one Bloch sum for every type of valence atomic orbital contributed by each atom in the basis. Thus, the two-carbon atom basis in diamond will produce eight Bloch sums - one for each of the s- and p-atomic orbitals. From these eight Bloch sums, eight COs are obtained, four bonding and four antibonding. For example, a Bloch sum of s atomic orbitals at every site on one of the interlocking FCC sublattices in the diamond structure can combine in a symmetric or antisymmetric fashion with the Bloch sum of s atomic orbitals at every site of the other FCC sublattice. 

I and there are two possible linear combinations of the atomic orbitals to be considered viz,... 

As in the case of atomic orbitals, Schrodinger equations can be written for electrons in molecules. Approximate solutions to these molecular Schrodinger equations can be constructed from linear combinations of the atomic orbitals (LCAO), the suras and differences of the atomic wave functions. For diatomic molecules such as H2, such wave functions have the form... 

We will consider first the combination of two s orbitals, as in H2. For convenience, we label the atoms of a diatomic molecule a and b, so the atomic orbital wave functions are i(i( Ir ) and il<( li ). We can visualize the two atoms moving closer to each other until the electron clouds overlap and merge into larger molecular electron clouds. The resulting molecular orbitals are linear combinations of the atomic orbitals, the sum of the two orbitals and the difference between them ... 

Rather than trying to find those fimctions which satisfy the Schrodinger equation written above, we adopt a simpler variational approach in which the search for imknown fimctions is replaced by the search for unknown coefficients. The basis for this strategy is a variational argument. We represent the wave function for the molecule as a linear combination of the atomic orbitals centered on the two nuclei, that is. 

When the molecular orbitals are taken as a linear combination of the atomic orbitals only, the form taken by the equations amounts to solving a determinantal equation ... 

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. 

Exclusion Principle. The orbitals are calculated in a self-consistent fashion in a manner analogous to those developed previously for atomic orbitals and are based on linear combination of the atomic orbitals of the individual atoms. The number of molecular orbitals equals the number of atomic orbitals in the atoms being combined to form the molecule. A molecular orbital describes the behavior of one electron in the electric field generated by the nuclei and some average distribution of the other electrons. This approximation proved to be more amenable to computer programming than the valence bond model and was widely developed and used in increasingly less approximate forms from 1960 to 1990. 

Linear Combination of the Atomic Orbitals (LCAO) Method... 

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