of atoms in the cell and therefore simplifications are necessary: If occupation of a
given position is close to zero, it is often possible to omit such an atom altogether,
whereas if occupation is close to 1, then full occupation can be assumed.
The problem arises when a given position is occupied by considerable fraction
that its omission is not possible without significant undesirable influence on the
results.
One can then use one of two approaches: The first is to reduce the symmetry of
the system with full occupation of only part of the original incomplete Wyckoff’s
positions and leaving the remaining sites empty. This is, however, connected with
the additional problem of which position to choose as being occupied and the
influence of the introduced in the process artificial local order, on the structural and
physicochemical properties of the system of interest—for example, if the original
Wyckoff’s position had a multiplicity of 4, and its occupation number was about
0.25, then after symmetry reduction one can select just one of new symmetrically
non-equivalent sites, leaving remaining three empty. If, however, the multiplicity of
such a position is so large that for a given occupation, after reducing the symmetry,
it will be necessary to fill more than one new symmetrically non-equivalent position; the problem is additionally complicated, because there is more than one
possible configuration of occupied positions (e.g., for multiplicity of 8 and fractional occupation of 0.25, there are eight new positions to be filled with just two
atoms). This means that a series of calculations must be carried out for all different
local arrangements in order to find energetically optimal structure and reproduce the
experimental results best. Again, an example of such a complex structure may be a
natural zeolite A, whose unit cell in a hydrated form (Fig. 1.3a) [125] with the
chemical formula |Na 92.7 (H 2 O) 6.95 | [Si 96.96 Al 95.04 O 384 ] contains 1840 atomic sites
(not counting hydrogen atoms from water molecules!), including part of the Na and
O sites partially occupied, and in the dehydrated form (Fig. 1.3b) [126] with the
chemical formula |Na 91.78 | [Si 96 Al 96 O 384 ] 1072 atomic sites (in the case of sodium
partially occupied). In the latter case, sodium occupies three different Wyckoff’s
positions: 64 g, occ. 0.972; 96i, occ. 0.242; and 96 h, occ. 0.066. Therefore, we can
simplify the model structure considerably, assuming full occupation of the 64 g
position, and 0.25 partial occupation in position 96i (neglecting completely hardly
occupied position 96 h). However, the problem of a very large number of possible
configurations of the position 96i filling remains (96 sites to be filled with four
times smaller number of sodium atoms), which is connected with the necessity of
creating many model structures for all possible configurations and performing
calculations to find the best model structure. It is worth mentioning here that even if
such extensive calculations can be made, the resultant best structure will still be too
complex to be able to carry out even relatively simple calculations of energy
Hessian matrix and the vibrational spectrum simulation; therefore, further simplifications are still needed, such as the reduction of the unit cell size discussed earlier,
at the cost of a small change in the structure (in mentioned case the appearance of
homonuclear aluminum and silicon bonds).
The second, much more computationally complex approach involves the use of
superstructures (creation of bigger unit cell built of many original unit cells) and
1 Computational Methods in Spectroscopy
35
given position is close to zero, it is often possible to omit such an atom altogether,
whereas if occupation is close to 1, then full occupation can be assumed.
The problem arises when a given position is occupied by considerable fraction
that its omission is not possible without significant undesirable influence on the
results.
One can then use one of two approaches: The first is to reduce the symmetry of
the system with full occupation of only part of the original incomplete Wyckoff’s
positions and leaving the remaining sites empty. This is, however, connected with
the additional problem of which position to choose as being occupied and the
influence of the introduced in the process artificial local order, on the structural and
physicochemical properties of the system of interest—for example, if the original
Wyckoff’s position had a multiplicity of 4, and its occupation number was about
0.25, then after symmetry reduction one can select just one of new symmetrically
non-equivalent sites, leaving remaining three empty. If, however, the multiplicity of
such a position is so large that for a given occupation, after reducing the symmetry,
it will be necessary to fill more than one new symmetrically non-equivalent position; the problem is additionally complicated, because there is more than one
possible configuration of occupied positions (e.g., for multiplicity of 8 and fractional occupation of 0.25, there are eight new positions to be filled with just two
atoms). This means that a series of calculations must be carried out for all different
local arrangements in order to find energetically optimal structure and reproduce the
experimental results best. Again, an example of such a complex structure may be a
natural zeolite A, whose unit cell in a hydrated form (Fig. 1.3a) [125] with the
chemical formula |Na 92.7 (H 2 O) 6.95 | [Si 96.96 Al 95.04 O 384 ] contains 1840 atomic sites
(not counting hydrogen atoms from water molecules!), including part of the Na and
O sites partially occupied, and in the dehydrated form (Fig. 1.3b) [126] with the
chemical formula |Na 91.78 | [Si 96 Al 96 O 384 ] 1072 atomic sites (in the case of sodium
partially occupied). In the latter case, sodium occupies three different Wyckoff’s
positions: 64 g, occ. 0.972; 96i, occ. 0.242; and 96 h, occ. 0.066. Therefore, we can
simplify the model structure considerably, assuming full occupation of the 64 g
position, and 0.25 partial occupation in position 96i (neglecting completely hardly
occupied position 96 h). However, the problem of a very large number of possible
configurations of the position 96i filling remains (96 sites to be filled with four
times smaller number of sodium atoms), which is connected with the necessity of
creating many model structures for all possible configurations and performing
calculations to find the best model structure. It is worth mentioning here that even if
such extensive calculations can be made, the resultant best structure will still be too
complex to be able to carry out even relatively simple calculations of energy
Hessian matrix and the vibrational spectrum simulation; therefore, further simplifications are still needed, such as the reduction of the unit cell size discussed earlier,
at the cost of a small change in the structure (in mentioned case the appearance of
homonuclear aluminum and silicon bonds).
The second, much more computationally complex approach involves the use of
superstructures (creation of bigger unit cell built of many original unit cells) and
1 Computational Methods in Spectroscopy
35