Quantum chemistry applies quantum mechanics[2,3] to the problems in chemistry. Experimental evidence shows that classical mechanics fails when is applied to particles as small as electrons [5-8]. In context of such objects one should take care on the wave-particle duality proposed in the de Broglie relation[9-12]. This concept plays a fundamental role in quantum mechanics.
Each physically measurable quantity has a corresponding operator. The eigenvalues of the operator tell the values of the corresponding physical property that can be observed (observable)[13-15]. The eigenfunctions of a quantum mechanical operator depend on the coordinates upon which the operator acts. These functions are called wavefunctions[15,16].The Schrödinger equation is an eigenvalue equation for the energy or Hamiltonian operator - its eigenvalues provide the energy levels of the system [15,16]. It is a principal tenet of quantum mechanics that the wavefunction, which is a solution of the Schrödinger equation, contains all the dynamical information about the system it describes. The interpretation of the wavefunction in terms of the location of the particle is based on the Born interpretation(or Copenhagen interpretation)[17,18]. To use any wavefunction in this context one should remember that it has to be normalized[16,19]. Moreover from physical point of view an acceptable wavefunction must be continuous, have a continuous first derivative and be single-valued. Other very important postulate of the quantum mechanics is the Heisenberg uncertainty principle.
To find the properties of systems according to quantum mechanics one need to solve the appropriate Schrödinger equation . To see the basics of the machinery used in the solutions it is useful to start with simple model of a particle in a box[16,22,23]. Even so simple model can be suitable for the description of the electronic absorption spectra of linear polyenes. Going further an explanation of the tunneling phenomena  would be helpful in understanding some microscopy techniques like SPM or STM.
Although the exact solution of the Schrödinger equation for the real chemical systems is almost impossible in nearly all cases, there are few exceptions . One of it is the hydrogen atom[28,29]. Analysis of this exact, analytical solution leads to very useful concepts like quantum numbers[29-32] or orbitals[33-34] which applies not only to the hydrogen-like atoms. In order to gain consistency with reality and to get full description a quantum state of a particle a property called spin should be introduced . This statement is a part of the postulates in quantum mechanics based on Schrödinger equation but comes naturally in context of the Dirac equation.
There exist a variety of approximated methods which can be applied to the real chemical problems (many-body systems). Many of them use the variational principle,adiabatic approximation[389 or Born-Oppenheimer approximation. Most widely used ab-initio methods are Hartree-Fock(HF) method and Møller–Plesset perturbation theory(MP). Moreover the density functional theory(DFT)is now a leading method for electronic structure calculations of especially large molecules [41,42].