This method is able to explain chemical bonding and rationalize it in many cases where other methods fail to explain the same phenomenon. For example, the molecule of oxygen or O2 where there is a triplet state of two unshared electrons. This paramagnetic structure of the oxygen molecule could not be explained by other chemical theories but the molecular orbitals theory succeeded to explain the existence of the triplet state of oxygen based on the construction of a molecular orbitals diagram that shows that there are two degenerate orbitals in the molecule that each accomodates one electron with the same spin of the other electron. This explanation was a triumph for the theory of molecular orbitals theory that succeeded in this case where other theories failed.
The method of constructing molecular orbitals from linear combination of atomic orbitals is a feature of a calculation method in theoretical chemistry that is called the Hartree-Fock method or the self consistent field. By choosing a basic set of atomic orbitals, and making a linear combination of them as a potential molecular orbital function, this function is then used to find the optimal energies and wave functions that are called molecular orbitals. This is done by using the variation method in order to minimize the coefficients in the linear combination.
This method of calculation (the Hartree-Fock method) is considered now an obsolete method and has been replaced by more accurate methods that take into consideration electron-electron interaction which is not considered in the Hartree-Fock method. In the Hartree-Fock method the obtained orbitals are obtained from the solution of the Schroedinger equation of multi-electronic atoms by assuming an averaged electronic potential of all the electrons in the atom. This approximation is the main deficiency of this method that can affect the energy values of the orbital wave functions and give inaccurate results for them. This deficiency is corrected or is taken into account in relatively new calculation methods such as the Moeller-Plesset method which takes into consideration electron-electron repulsion in the calculated Hamiltonian of the system.
Thus its results are more accurate than that of the Hartree-Fock method. Other calculation methods that give better results than the Hartree-Fock method include the coupled clusters method which is a potential use for calculations of transition states of chemical reactions. Yet another important method that is being used extensively nowadays by many theoreticians of chemistry is the density functional theory method which is less time consuming than other methods due to its use of a density functional instead of a normal probability functions which its solution can be lengthy and time consuming.
An orbital is a wave function that can be atomic or molecular. Its square value gives the probability of finding the electron in the space of the atom or molecule. It is basically obtained from the solution of the Schroedinger equation for the hydrogen atom which is a reference system for all many electron atoms and molecules due to the exact solution that is obtained when applying the Schroedinger equation for the hydrogen atom. The solution of the orbital functions of the many electron atoms uses orbitals that are similar to the orbitals of the hydrogen atom. This is due to the possibility to obtain an analytical or exact solution of the hydrogen atom wave functions. This is not possible for other atoms due to the existence of electron-electron interactions and the spin-orbit coupling which both add additional terms to the Hamiltonian of the atom the thing that makes the solution of the appropriate Schroedinger equation not possible analytically or exactly. A numerical solution is possible in this case to this problem and all other problems that involve many electron atoms and molecules.
The solution of the Schroedinger equation for the hydrogen atom, although it is accurate, is a lengthy and fairly complicated process that requires knowledge of differential equations. The solution to the hydrogen atom involves the use of Laplacian in spherical coordinates and the separation of differential equations into a radial part and an angular part. The solution of both equations leads to much information about the shape and energies of the hydrogen atomic orbitals. Also there arises in this solution new quantum numbers that have importance for the specification of the energy of the orbitals and their shape and their degeneracy or the amount of orbitals with the same number of orbitals.
Molecular orbitals that are formed from the combination of two atomic orbitals can be mostly of two types. These are sigma type orbitals and pi-type orbitals. The sigma type molecular orbital has symmetry that is along the line that connects the bonded atoms. Pi-type orbitals on the other hand are less symmetrical and use p orbitals in order to make the bond. Sigma molecular orbitals are usually energetically lower than the pi-orbitals. In addition sigma bonds are in most cases localized bonds that are located between the two nuclei of the bonded atoms. Pi-bonds on the other hand can be delocalized. This means that the electron cloud in a pi-bond can be spread over several nuclei. This fact confers extra-stability to a delocalized bond that is manifested in conjugated systems. An example of a molecule which has extra-stability due to the existence of delocalization effect is the benzene molecule. This molecule has six bonded pi electrons that if we compare them with three ethylene molecules would not give the same energies. The difference arises due to the existence of the delocalization of the pi-electrons along the six carbon atoms of the benzene ring.