level of complexity, starting with the simplest atoms, through molecules and
clusters, to amorphous and periodic solids. In the past, experimental studies were
the main and, in some cases, practically the only source of knowledge about the
properties of materials. The development of computer technology and theoretical
methods leads gradually to the increasing use of computer calculations and simulations (the so-called in silico experiments) in materials research, which in many
cases becomes an equivalent (and sometimes also the first, preliminary) tool
allowing understanding of the properties of materials and their modification and
functionalization.
Among the theoretical methods used in material research, the so-called ab initio
methods (methods from the first principles), whose basic idea is to treat
multi-atomic systems as systems of many bodies composed of atomic nuclei and
electrons and to analyze them employing quantum mechanics—the arguably most
important contemporary physical theory, without using any empirical parameters.
In the following few sections, the most important ingredients of quantum
mechanics formalism as well as the density functional theory, rooted in this
formalism (and nowadays the dominant approximation used in theoretical calculations), required for their practical application in modeling of the properties of
complex many-body systems will be presented. Apart from the above-mentioned
“static” methods, i.e., methods based on the solutions of appropriate equations
formulated for stationary states, there are also other, computationally much more
complex and demanding, dynamic (ab initio molecular dynamics allowing to
study the changes of system state with time) and statistical approaches (ab initio
quantum Monte Carlo, allowing going beyond the mean field theory and
independent electrons approximations and direct treatment and description of
complex interactions in many-body systems). However, due to the limited volume
of the chapter and still relatively rare use of these methods, they will not be
presented here and interested readers are referred to the rich literature devoted to
this topic [e.g., 17–21].
1.2.1 Ab Initio Methods
1.2.1.1 Schrödinger Equation
The physical theory describing the phenomena occurring in the micro-world, and
therefore also the processes studied by various spectroscopic methods, is quantum
mechanics. The most popular formulation of quantum theory is the formalism of
wave mechanics proposed by Erwin Schrödinger [22–27]. The starting point in
Schrödinger’s formulation is a number of postulates regarding the wave function of
complex values (representing the state of the system), linear Hermitian operators
and their eigenvalues (mathematical representations of observable quantities, the
so-called observables), a recipe enabling finding an operator associated with a given
observable, a description of the measurement process and a way of predicting its
6
A. Koleżyński
clusters, to amorphous and periodic solids. In the past, experimental studies were
the main and, in some cases, practically the only source of knowledge about the
properties of materials. The development of computer technology and theoretical
methods leads gradually to the increasing use of computer calculations and simulations (the so-called in silico experiments) in materials research, which in many
cases becomes an equivalent (and sometimes also the first, preliminary) tool
allowing understanding of the properties of materials and their modification and
functionalization.
Among the theoretical methods used in material research, the so-called ab initio
methods (methods from the first principles), whose basic idea is to treat
multi-atomic systems as systems of many bodies composed of atomic nuclei and
electrons and to analyze them employing quantum mechanics—the arguably most
important contemporary physical theory, without using any empirical parameters.
In the following few sections, the most important ingredients of quantum
mechanics formalism as well as the density functional theory, rooted in this
formalism (and nowadays the dominant approximation used in theoretical calculations), required for their practical application in modeling of the properties of
complex many-body systems will be presented. Apart from the above-mentioned
“static” methods, i.e., methods based on the solutions of appropriate equations
formulated for stationary states, there are also other, computationally much more
complex and demanding, dynamic (ab initio molecular dynamics allowing to
study the changes of system state with time) and statistical approaches (ab initio
quantum Monte Carlo, allowing going beyond the mean field theory and
independent electrons approximations and direct treatment and description of
complex interactions in many-body systems). However, due to the limited volume
of the chapter and still relatively rare use of these methods, they will not be
presented here and interested readers are referred to the rich literature devoted to
this topic [e.g., 17–21].
1.2.1 Ab Initio Methods
1.2.1.1 Schrödinger Equation
The physical theory describing the phenomena occurring in the micro-world, and
therefore also the processes studied by various spectroscopic methods, is quantum
mechanics. The most popular formulation of quantum theory is the formalism of
wave mechanics proposed by Erwin Schrödinger [22–27]. The starting point in
Schrödinger’s formulation is a number of postulates regarding the wave function of
complex values (representing the state of the system), linear Hermitian operators
and their eigenvalues (mathematical representations of observable quantities, the
so-called observables), a recipe enabling finding an operator associated with a given
observable, a description of the measurement process and a way of predicting its
6
A. Koleżyński