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W. Piskorz and F. Zasada
there exists a unique mapping between the time-dependent external potential of a
system and its time-dependent density. Hence, instead of the wavefunction of 3N
variables (excluding spin), the three-dimensional electron density can be concerned
as a fabric for the determination of all observables of the system. Based on the linear
response theory, the spectra and other information on the excited states to be derived
from TD-DFT [33].
The other computationally demanding effect, the London dispersion interactions,
can be parameterised (based on either experimental or very accurate quantumchemical calculations) and form an additive term, essentially force field-like, to
the Hamiltonian. Such semi-empirical approach is coded as DFT+D and is computationally as cheap as plain DFT [34], while the accuracy is close to the high quality
of very costly benchmark CCSD(T) calculations.
For the Grimme’s D2 [35, 36] and D3 [37–39] versions of DFT+D, the dispersion
term has the simple and computationally fast form of single-atomic parameters, what
implies, besides their accuracy, the strength of these methods. Hence, the calculation
of the dispersion energy is independent of the electronic structure and its computational cost is limited to the iteration of the atomic pairs. This DFT+D approach
is possible due to the mainly two-atomic nature of London dispersion interactions,
namely the potential of interaction between two atoms of given elements is a function
of that elements only and is independent of the chemical environment of that atoms.
Moreover, such diatomic term is a simple function of the atomic scalar terms. Once
the atomic terms are parameterised, the atom–atom contribution can be calculated
instantly. Alternative scheme, e.g., by Dion et al. [40–42], called the van der Waals
density functional (vdW-DF) method, is based directly on the electron density. It
comprises the non-local term accounting in an approximate way for the non-local
electron correlation effects. Comparing to the local functional, it is an improvement
even though it is obtained quite simply, using the double space integration. Its accuracy is improved for the systems with pronounced dispersion effects, it tends to be,
however, inferior than the GGA functionals for the systems with hydrogen bonds
[43, 44].
2.2 The Classical Mechanics—Force Fields
The classical force-field (FF) dynamics is a versatile tool for the description of physical processes like diffusion, sorption, solution, or conformational analysis. Due
to little computational cost, FF is capable to describe the systems with millions
of atoms for the timescale as long as nanoseconds. In its classical formulation,
it cannot, however, describe the making or breaking of chemical bonds. The step
forward in the improvement of description of “chemical” properties is the reactive force-field (ReaxFF) level of theory [45], where the possibility of bond breaking and making is based on the “bond order” analysis which in turn depends on
the interatomic distances. The interatomic potential parameters are obtained either
semi-empirically, or by means of quantum-chemical electronic structure calculations
W. Piskorz and F. Zasada
there exists a unique mapping between the time-dependent external potential of a
system and its time-dependent density. Hence, instead of the wavefunction of 3N
variables (excluding spin), the three-dimensional electron density can be concerned
as a fabric for the determination of all observables of the system. Based on the linear
response theory, the spectra and other information on the excited states to be derived
from TD-DFT [33].
The other computationally demanding effect, the London dispersion interactions,
can be parameterised (based on either experimental or very accurate quantumchemical calculations) and form an additive term, essentially force field-like, to
the Hamiltonian. Such semi-empirical approach is coded as DFT+D and is computationally as cheap as plain DFT [34], while the accuracy is close to the high quality
of very costly benchmark CCSD(T) calculations.
For the Grimme’s D2 [35, 36] and D3 [37–39] versions of DFT+D, the dispersion
term has the simple and computationally fast form of single-atomic parameters, what
implies, besides their accuracy, the strength of these methods. Hence, the calculation
of the dispersion energy is independent of the electronic structure and its computational cost is limited to the iteration of the atomic pairs. This DFT+D approach
is possible due to the mainly two-atomic nature of London dispersion interactions,
namely the potential of interaction between two atoms of given elements is a function
of that elements only and is independent of the chemical environment of that atoms.
Moreover, such diatomic term is a simple function of the atomic scalar terms. Once
the atomic terms are parameterised, the atom–atom contribution can be calculated
instantly. Alternative scheme, e.g., by Dion et al. [40–42], called the van der Waals
density functional (vdW-DF) method, is based directly on the electron density. It
comprises the non-local term accounting in an approximate way for the non-local
electron correlation effects. Comparing to the local functional, it is an improvement
even though it is obtained quite simply, using the double space integration. Its accuracy is improved for the systems with pronounced dispersion effects, it tends to be,
however, inferior than the GGA functionals for the systems with hydrogen bonds
[43, 44].
2.2 The Classical Mechanics—Force Fields
The classical force-field (FF) dynamics is a versatile tool for the description of physical processes like diffusion, sorption, solution, or conformational analysis. Due
to little computational cost, FF is capable to describe the systems with millions
of atoms for the timescale as long as nanoseconds. In its classical formulation,
it cannot, however, describe the making or breaking of chemical bonds. The step
forward in the improvement of description of “chemical” properties is the reactive force-field (ReaxFF) level of theory [45], where the possibility of bond breaking and making is based on the “bond order” analysis which in turn depends on
the interatomic distances. The interatomic potential parameters are obtained either
semi-empirically, or by means of quantum-chemical electronic structure calculations