Nuclei are thousands of times heavier than electrons. While a nucleus moves a little, an electron travels many times through the molecule. A lot can be simplified when assuming electronic motion in a field created by immobile nuclei. This concept is behind what is called adiabatic approximation. We obtain two coupled equations: one for the motion of the electrons in the field of fixed nuclei, and the other for the motion of the nuclei in the potential averaged over the electronic positions. Only after this approximation is introduced can we obtain the fundamental concept of chemistry: the molecular structure in three-dimensional space. Also, for each electronic state we obtain the concept of potential energy surface (PES) for the motion of the nuclei. These surfaces may intersect, which leads to the idea of the conical intersections, crucial for chemical reactions.

The greater the velocity of an object, the greater the errors in Newton dynamics. Electrons have greater velocity when close to nuclei of large electric charge. This is why relativistic corrections may turn out to be important for heavy elements. The Schrödinger equation is incompatible with special relativity theory. This has to be corrected somehow. This is far from being solved, but progress so far shows the origin of the Schrödinger equation, of the spin of a particle, etc., in a new light.

Interaction energy of molecules may be calculated by subtracting the subsystems' energies from the total system energy (supermolecular method). The method works independently of the interaction strength. There are some disadvantages though: a loss of accuracy and a lack of information about physical components of the interaction energy. The alternative perturbational method requires long- or medium-range intermolecular distances. Its most important contributions, electrostatic, valence repulsion, induction, and dispersion, lead to a richness of supramolecular structures. The hydrogen bond X–H⋯Y represents an example of domination of the electrostatic interaction. The all important valence repulsion controls how the interacting molecules fit together in space. In aqueous solutions the solvent structure contributes very strongly, leading to what is known as the hydrophobic effect – expulsion of the nonpolar subsystems from the bulk, which looks like their attraction. A precise molecular recognition may be planned by chemists helping to build complex molecular architectures by molecular self-organization.

What electrons are doing in molecules (at fixed positions of the nuclei) is described by their wave function – a solution to the Schrödinger equation. This equation however cannot be solved analytically. To obtain an approximate wave function the variational method is most useful. In this chapter a special kind of the variational method is presented, which restricts the form of the trial function to a single Slater determinant composed of molecular orbitals. In this way the variational method returns the best possible single Slater determinant. Each molecular orbital represents a kind of a single-electron wave function – a “home” for two electrons of opposite spin. We will now learn how to get the optimum molecular orbitals (Hartree–Fock method). Not only does this orbital picture give reasonable results, at the same time it also provides a kind of universal language of theoretical chemistry.

The main theoretical concept of density functional theory is the electron density ρ. The ground-state electronic density distribution (ρ0) contains the same information as the ground-state wave function. A total energy functional of ρ that attains its minimum at ρ=ρ0exists, but its mathematical form is unknown. In the local density approximation this form is approximated using the exact results for the uniform electron gas model. There are also DFT functionals that go beyond the local approximation. There is a possibility in DFT to calculate the excited states' energies. A good quality of results and low cost made from DFT the main computational tool in quantum chemistry.

We have to know how to compute wave functions. The exact wave function is definitely out of reach; therefore in this chapter we will learn how to calculate its approximations.

The Schrödinger equation is nowadays quite easy to solve with a desired accuracy for many systems. There are only a few systems for which exact solutions are possible. These problems and solutions play an extremely important role in physics, since they represent beacons for our navigation in science, where as a rule we deal with complex systems. Real systems may most often be approximated by those for which exact solutions exist. Thus, from the beginning we know the (idealized) solution.

A molecule changes its properties even if placed in a uniform and weak electric field. These changes can be calculated knowing the field intensity and some quantities characterizing the isolated molecule: its electric moments and its polarizabilities and hyperpolarizabilities, the latter being our target. In the LCAO MO approximation, the dipole moment of the molecule can be decomposed into the sum of the atomic dipole moments and the dipole moments of the atomic pairs. The dipole polarizability may be computed by using the perturbative sum over states method, or within the finite field method (usually a variational approach). In laser fields we may obtain a series of nonlinear effects, including the doubling and tripling of the incident light frequency. For electromagnetic fields applied to a molecule one is able to calculate the energy states of a system of nuclei (detectable in NMR spectroscopy). This makes it possible to compute in an ab initio way both the shielding (σA) and the coupling (JAB) constants, the main experimental information.

A crystal is often approximated by a periodic superposition of identical unit cells with some molecular motif. For this reason the crystal electronic orbitals transform according to an irreducible representation of the translational group labeled by the wave vector k. Knowledge of the crystal lattice basis vectors allows for the formation of the inverse crystal lattice with the first Brillouin zone (FBZ) as the unit cell. The crystal orbital (CO) theory is derived in full analogy with the molecular Hartree–Fock theory. The vectors k inside the FBZ label all possible nonequivalent crystal orbitals, their energies forming electronic energy bands – occupied or empty. The calculated band structure may be traced back to the subsystem orbitals. The energy gap between the valence and conduction bands decides about electrical conductivity: zero gives a metal, a large gap means an insulator, and a small gap corresponds to a semiconductor. Finally it is shown that the choice of unit cell motif is irrelevant only after the long-range interactions are properly taken into account.

A chemical reaction is described as a system's passing through a characteristic reaction “drain-pipe,” from the entrance channel (reactants) to the exit channel (products) as a function of the nuclear coordinates, on the electronic energy map. This usually requires overcoming a reaction barrier. The electronic energy is expensive in massive calculations; this is practically possible for three-atomic reactions. When focusing on the immediate neighborhood of the “drain-pipe” bottom (reaction path), the chemical reaction details may be obtained by the reaction path Hamiltonian method. It describes the interplay of the system's motion along the reaction path and the orthogonal vibrations, as well as the vibration–vibration interactions. A reaction barrier represents a consequence of the conical intersection of the diabatic reactants-like and product-like energy hypersurfaces. Usually the two intersecting diabatic hypersurfaces, at the reactant configuration, represent the electronic donor–acceptor ground state DA and the charge transfer state D+A−, the latter resembling already the electronic charge distribution of the products. The barrier appears therefore as a cost of opening the closed shells in such a way as to prepare the reactants for the formation of new bond(s). Within this picture, in Marcus electron transfer theory, the barrier height depends mainly on the (solvent and reactant) reorganization energy.