Atomic orbitals, linear combination

The theoretical calculation of the surface states is considered by the linear combination atomic orbital (LCAO) method however, in some cases the relation between the electrode potential and the distance from the electrode surface is used. These electric potentials obey the Schrodinger equation and the surface states are obtained as eigenvalues. Inside the crystal lattice, the equation is reduced to the Mathieu differential equation with known solutions. Using the LCAO method, the interaction between the surface states and the electron is avoided. In this case, the proposed wave function is of the form ... 

One important feature of these exact wavefunctions for a simple molecule is that they resemble Is orbitals on the individual hydrogen atoms that were either added (the g state) or subtracted (the u state). This observation is one of the justifications for linearly combining atomic orbitals on atoms to create molecular orbitals. 

A special aspect of this description appears if one starts the orbital optimisation process with orbitals obtained by linear combinations ofRHF orbitals of the isolated atoms (LCAO approximation s.str.). Let Pn.opt and be the starting and final orbitals of such a calculation. Then the difference between c n.opi and Papt in the vicinity of each atom merely consists in a distortion of the atomic orbitals of each atom. This distortion just compensates the contribution of the orbitals of the other atoms to Pn.ctpt in order to restore the proportionality between the partial waves of ipopi and the appropriate atomic orbital. 

An approach based on orbital radii of atoms effectively rationalizes the structures of 565 AB solids (Zunger, 1981). The orbital radii derived from hard-core pseudopotentials provide a measure of the effective size of atomic cores as felt by the valence electrons. Linear combinations of orbital radii, which correspond to the Phillips structural indices and have been used as coordinates in constructing structure maps for AB solids. 

When we linearly combine the orbitals on Ha and II, two (un-normalized) combinations are obtained lsa + Is (Aj symmetry), sa — Is (Aj). On the other hand, the combinations for the orbitals on the equatorial hydrogens are lsc + lSd + lse (Aj) 2(lsc) - lSd - lse and Is a - lsg ( symmetry). If we assume central atom A contributes ns and np orbitals to bonding, we can easily arrive at the results summarized in Table 7.1.4. Note that we can further combine the two ligand linear combinations with Aj symmetry by taking their sum and difference. 

Linear combination of atomic orbitals (LCAO theory) A method for combining atomic orbitals to approximately compute molecular orbitals. 

In theoretical terms, the total electron density in a molecule is easily expressed in terms of the occupied molecular orbitals. Additional information is gained from the m.o. approach especially regarding the electronic energy for ground and excited states and the detailed features (e.g. phase) of individual m.o.s. Molecular orbitals are mathematical functions that can be constructed as linear combinations of orbitals of the contributing atoms, in a process where the atoms lose their individuality, except for the respective nuclei and, perhaps, the core electrons. The valence electrons are described by functions which, in general, extend to several atoms or even to the whole molecule. 

Initially it would appear that, of the four bonds formed by a carbon atom, one would involve the 2s orbital and the other three the three 2p orbitals. This, however, implies that one bond is different from the other three, whereas experimentally the CH4 molecule is perfectly symmetrical—all four bonds being identical. The solution to this dilemma was given by Pauling who suggested that we should use a linear combination of orbitals instead of the pure 2s and 2p orbitals of the carbon atom. On the basis of a simple quantum-mechanical treatment, he concluded that from the one 2s and three 2p orbitals one can construct four hybridized orbitals as follows ... 

TERM SYMBOLS FOR LINEAR MOLECULES Electronic states of a linear molecule may be classified conveniently in terms of angular momentum and spin, analogous to the Russell-Saunders term-symbol scheme for atoms. The unique molecular axis in linear molecules is labeled the axis. The combining atomic orbitals in any given molecular orbital have the same mi value. Thus an mi quantum number is assigned to each different type of MO, as indicated in Table 2-3. The term designations are of the form... 

In most cases it is convenient to have a normalized linear combination of orbitals to bond with a central atom. For example, the combination appropriate for 2s in a trigonal-planar molecule is... 

FIGURE 13.20 The valence atomic orbitals used to construct symmetry-adapted linear combination molecular orbitals of H2O. Although the atomic orbitals do not have Cjv symmetry, the proper combinations of atomic orbitals wilt See Example 13.13. 

In the 1930s, the development of valence bond theory (most notably by Linus Pauling) was extended to include linear combinations of the valence orbitals themselves. Such linear combinations are called hybrid orbitals. Specifically, the combination of a certain number of atomic orbitals provides linear combination hybrid orbitals that collectively have the proper symmetry. This single fact is what makes valence bond theory and hybrid orbitals such a usefijl interpretational tool in chemistry (whether or not such orbitals actually exist). 

The description of 3c-4e bonds, the sort which Musher (35) has called hypervalent, in terms of linear combinations of orbitals of the three colinear atomic centers is pictured for the product sulfurane (2i ) at the right below. The 5c-6e bond i stu-lated for transition state 23 is illustrated in similar terms in the MO diagram to the left. Arduengo and Burgess (36) have discussed related multicenter bonds. 

Imagine next that we repeat this exercise we take another atom with the same linear combinations of orbitals as A, which we will call A, and place it in the direction of the vector relative to the position of atom B, and at the... 

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. 

B3.1.5.2 THE LINEAR COMBINATIONS OF ATOMIC ORBITALS TO FORM MOLECULAR ORBITALS EXPANSION OF THE SPIN ORBITALS... 

HMO theory is named after its developer, Erich Huckel (1896-1980), who published his theory in 1930 [9] partly in order to explain the unusual stability of benzene and other aromatic compounds. Given that digital computers had not yet been invented and that all Hiickel s calculations had to be done by hand, HMO theory necessarily includes many approximations. The first is that only the jr-molecular orbitals of the molecule are considered. This implies that the entire molecular structure is planar (because then a plane of symmetry separates the r-orbitals, which are antisymmetric with respect to this plane, from all others). It also means that only one atomic orbital must be considered for each atom in the r-system (the p-orbital that is antisymmetric with respect to the plane of the molecule) and none at all for atoms (such as hydrogen) that are not involved in the r-system. Huckel then used the technique known as linear combination of atomic orbitals (LCAO) to build these atomic orbitals up into molecular orbitals. This is illustrated in Figure 7-18 for ethylene. 

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