Computational chemistry is a branch of chemistry that deals with mathematical calculations that lead eventually to the determination of the energy and geometry of particular chemical compounds of interest to the chemist. For the most part, the interesting information that a chemist seeks are electronic energies of atoms and molecules. The calculation can also predict if the compound is stable enough to be existent.

The most important electrons in the atom or the molecule are those that are located in the highest electronic shell or in the language of orbitals the highest occupied (HOMO) and unocupied (LUMO) molecular orbitals. These orbitals are responsible for the chemical characteristics of the molecules, namely whether they will react with other molecules or not and by which symmetry.

Chemical calculations on organic compounds are part of the computational chemistry scheme. They are difficult to calculate accurately due to the involvement of many atoms in their molecular structure. However, organic compounds which have small number of atoms are relatively easy to calculate due to the low electron count in them.

Chemical compounds reactivity is governed by the HOMO orbital and the LUMO or the frontier molecular orbitals. In particular, the interaction between the LUMO of one compound and the HOMO of the other reactant are important. The difference in energetics between these two orbitals as well as the matching of the symmetry will dictate if there will be a reaction between these two compounds. Therefore, it is important to perform calculations on the energetics of these orbitals and not on the inner core orbitals in order to verify if there will be a reaction between these two molecules or not.

The basic equation that is used to determine the energetics of chemical compounds is the Schroedinger equation. This equation is nonrelativistic and does not account for relativistic effects such as the phenomenon of the spin. The quantum mechanical variant of this equation that involves relativistic effects on atoms and molecules is called the Dirac equation.

The Schroedinger equation is a linear differential equation that is mathematically solvable in exact manner only for simple systems such as the particle in a box and the energetics of the hydrogen atom. For atomic systems it is solved only for the hydrogen atom in an exact manner. In this case also the solution involves formidable mathematical equations that involve laplacian in spherical coordinates. In addition, it involves also differential equations that are not straightforward to solve for the practitioner chemist.

For atoms other than hydrogen, solutions of the Schroedinger equation cannot be obtained mathematically in an accurate fashion due to the presence in the Hamiltonian of the Schroedinger equation a potential energy term that makes the analytical solution of the Schroedinger equation impossible. For this reason many efforts were done in order to overcome this difficulty by deriving several approximation methods that are called perturbative methods. They involve solving the the energy term of the perturbation itself and then adding it to the general energy of the unperturbed Hamiltonian.

Most calculations in chemistry involve this type of perturbation theory. This is especially important for many electron atoms and molecules. In many electron atoms there are two main perturbations that prevent from solving the Schroedinger equation in an exact manner. These are: the correlation between electrons or the repulsion forces between the various electrons which add a term to the Hamiltonian that makes the mathematics impossible for solving the problem. This electronic correlation is manifested not only in many electrons atoms but also in all molecules.

The other part of the Hamiltonian that is difficult to solve exactly is due to the spin-orbit coupling which is a relativistic effect. It originates due to the relative motion of the nucleus around the electron itself. The movement of the nucleus around the electron generates a magnetic field that can interact with the spin and angular magnetic moment of the electron, thus giving rise to an additional term in the atomic Hamiltonian that impedes an accurate solution of the Schroedinger equation.

In the 1990th the most well known calculation method for the chemist was the Hartree-Fock method or the self consistent field. Nowadays there are many perurbative methods such as the Muller plesset method which includes correlation between electrons in the Hamiltonian. In addition a popular method that is being used more and more in computational chemistry is the density functional theory which has the advantage of being less time consuming than the other methods. Additional computational methods especially useful for transition state calculations include the coupled cluster methods. for thermodynamic calculations a very efficient method is the semi-empirical method especially the PM3 method.