Linear combinations of the atomic

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... 

Consider now the rr-system in benzene. The MO approach will generate linear combinations of the atomic p-orbitals, producing six rr-orbitals delocalized over the whole molecule with four different orbital energies (two sets of degenerate orbitals). Figure 7.3. The stability of benzene can be attributed to the large gap between the HOMO and LUMO orbitals. 

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 information obtainable upon solution of the eigenvalue problem includes the orbital energies eK and the corresponding wave function as a linear combination of the atomic basis set xi- The wave functions can then be subjected to a Mulliken population analysis<88) to provide the overlap populations Ptj ... 

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... 

Normal mode coordinates are linear combinations of the atomic displacements (x, yt z,, which are the components of a set of vectors Q in a 3/V-dimcnsional vector space called... 

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

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 ... 

Up to this point nothing changes. The next assumption extends the above treatment of atoms to molecules. Within it the molecular orbitals - linear combinations of the atomic core orbitals with zero overlap - are taken to be the molecular core orbitals and are assumed to be filled. This allows one to write... 

Fig. 14. Linear combination of the atomic valence orbitals s,px,py to form sp and sp2 orbitals in the (x,y)-plane. The resulting orbitals are symmetric perpendicular to the (x,y)-plane and build the structure framework in the (x,y)-plane with the orbitals of the neighboring atoms. The remaining antisymmetric pz orbital is orthogonal to the orbitals in the (x,y)-plane and is used for the delocalized electron orbitals...

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... 

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