of normal modes of vibration in a given system and thanks to the ability to easily
visualize these vibrations, allowing for a simpler and much more accurate analysis
and interpretation of the vibrational spectrum.
The fundamental problem in theoretical calculations carried out for zeolite
structures is the necessity to apply far-reaching simplifications when creating theoretical models of such structures, resulting from the structural complexity of real
systems. Since experimental studies show the existence of well-defined structurally
hierarchical fragments of the zeolite framework, the first approaches using quantum
mechanics methods focused on the analysis of the smallest (and therefore simplest)
structure fragments, i.e., tetrahedra [TO 4 ] and simple rings, for which geometry
optimization and vibrational spectra calculations were possible in an acceptable
time using available computer resources. Such calculations were usually performed
for simple cluster models terminated with protons, mono- or divalent cations, or
simple functional groups—such termination was necessary to stabilize the geometry
of the clusters, naturally unstable due to the excess negative charge of such fragment “cut out” from the larger structure of the zeolitic framework. Such calculations
were made using available programs, initially employing Hartree–Fock or post-HF,
and later on more and more often also the density functional theory formalism, with
molecular orbitals defined in terms of localized atomic orbitals which in turn were
defined in terms of basis set functions using analytical Gaussian or Slater functions.
This allowed for the first, important results, especially for the mid-infrared range,
but nonetheless caused a number of problems related to the omission of significant
interactions with the rest of zeolite framework (and hence the lack of the possibility
to analyze far-infrared region with characteristic lattice vibrations) and artificial
impact of terminal groups on the properties of vibrational spectrum of given
structural units.
Only in recent years, thanks to both, the dynamic development of numerical
methods and an increase of available computing power, it has been possible to
perform calculations for more complex structural models, including translational
symmetry of real zeolite structures. As a result, a detailed analysis of the far-infrared
range, and thus the analysis of the influence of the type, amount and position of
non-tetrahedral cations on the structure and vibrational spectra of zeolites became
also possible. Moreover, periodic models allow the analysis of the impact on the
vibrational spectra of zeolites of the increasing complexity of structural units
(starting from the isolated tetrahedron, through double tetrahedra [T 2 O 7 ] clusters,
SBU and CBU, up to the full frameworks), allowing verification of the degree of
correctness of the earlier analyses based on simple cluster models.
In the case of periodic models, classical quantum mechanics programs for solids
routinely used by physicists are normally used for vibrational properties’ calculations. Available programs allow performing calculations using Hartree–Fock,
post-HF, or DFT formalism (the latter for different exchange–correlation potentials). Different programs use different types of crystal orbitals, based on classical,
localized atomic orbitals defined by means of analytic Gaussian (e.g., Crystal14) or
Slater (ADF) functions or in the form of numerical atomic orbitals (SIESTA,
DMol3), completely delocalized crystalline orbitals defined by means of plane
304
M. Król et al.
visualize these vibrations, allowing for a simpler and much more accurate analysis
and interpretation of the vibrational spectrum.
The fundamental problem in theoretical calculations carried out for zeolite
structures is the necessity to apply far-reaching simplifications when creating theoretical models of such structures, resulting from the structural complexity of real
systems. Since experimental studies show the existence of well-defined structurally
hierarchical fragments of the zeolite framework, the first approaches using quantum
mechanics methods focused on the analysis of the smallest (and therefore simplest)
structure fragments, i.e., tetrahedra [TO 4 ] and simple rings, for which geometry
optimization and vibrational spectra calculations were possible in an acceptable
time using available computer resources. Such calculations were usually performed
for simple cluster models terminated with protons, mono- or divalent cations, or
simple functional groups—such termination was necessary to stabilize the geometry
of the clusters, naturally unstable due to the excess negative charge of such fragment “cut out” from the larger structure of the zeolitic framework. Such calculations
were made using available programs, initially employing Hartree–Fock or post-HF,
and later on more and more often also the density functional theory formalism, with
molecular orbitals defined in terms of localized atomic orbitals which in turn were
defined in terms of basis set functions using analytical Gaussian or Slater functions.
This allowed for the first, important results, especially for the mid-infrared range,
but nonetheless caused a number of problems related to the omission of significant
interactions with the rest of zeolite framework (and hence the lack of the possibility
to analyze far-infrared region with characteristic lattice vibrations) and artificial
impact of terminal groups on the properties of vibrational spectrum of given
structural units.
Only in recent years, thanks to both, the dynamic development of numerical
methods and an increase of available computing power, it has been possible to
perform calculations for more complex structural models, including translational
symmetry of real zeolite structures. As a result, a detailed analysis of the far-infrared
range, and thus the analysis of the influence of the type, amount and position of
non-tetrahedral cations on the structure and vibrational spectra of zeolites became
also possible. Moreover, periodic models allow the analysis of the impact on the
vibrational spectra of zeolites of the increasing complexity of structural units
(starting from the isolated tetrahedron, through double tetrahedra [T 2 O 7 ] clusters,
SBU and CBU, up to the full frameworks), allowing verification of the degree of
correctness of the earlier analyses based on simple cluster models.
In the case of periodic models, classical quantum mechanics programs for solids
routinely used by physicists are normally used for vibrational properties’ calculations. Available programs allow performing calculations using Hartree–Fock,
post-HF, or DFT formalism (the latter for different exchange–correlation potentials). Different programs use different types of crystal orbitals, based on classical,
localized atomic orbitals defined by means of analytic Gaussian (e.g., Crystal14) or
Slater (ADF) functions or in the form of numerical atomic orbitals (SIESTA,
DMol3), completely delocalized crystalline orbitals defined by means of plane
304
M. Król et al.