14
1 A Historical Review of the Structures of Water and Ice
where x i is the atomic fraction of atoms of type i, r i j are the intramolecular distances
between the centers of nuclei.
At this level of approximation, the geometric structure of the water molecule is
assumed to be a rigid sphere, and individual atomic scattering factors are calculated
for isolated unbound particles. In fact, covalent bonding and the mutual interaction
of molecules change the charge distribution and, therefore, the atomic form factors
defined by (1.4).
The structural factor S(Q) is determined from the molecular form factor by
S(Q) =
I (Q)
F(Q)
2
.
(1.5)
The contribution of intermolecular correlations to the total intensity (see the last term
in (1.3)) is given by [33]
I inter (Q) =
i≤ j
(2 − δ i j )x i x j f i (Q) f j (Q)S i j (Q),
(1.6)
where the correlation factor S i j (Q) between atoms of type i and j is directly related
to the RDF by
S i j (Q) = 4πρ
∞
0
r
2
(g i j (r ) − 1)
sin Qr
Qr
dr,
(1.7)
where ρ is the atomic density.
Therefore, the structure factor S(Q) and the RDF itself can be calculated directly
from (1.7). However, a preliminary structural model is needed to weigh the scattering
correlations between the oxygen and hydrogen atoms. For this purpose, the diffraction pattern is analyzed by the empirical potential structure refinement [40]. Using
molecular dynamics and ab initio methods, interatomic potential energy functions are
consistently tuned to reconcile the simulated and measured structure factors S(Q).
The aim of the simulation is to find realistic constraints on the forces between atoms,
inside and between molecules, which will fit the experimentally observed RDF.
Since this problem generally has an unlimited number of solutions, the result always
depends on the model used. This is why there are still some differences between
the RDFs obtained by different authors depending on the initial assumptions and the
potentials that were applied.
The first RDF for water was reported almost a century ago [41]. Figure 1.9 shows
the modern RDFs for water and ice. The g OO (r ) function for water shows well-defined
maxima at 2.9, 4.5, 6.7 Å and minima at 3.4, 5.5 Å. The positions of the peaks have
a ratio close to the 1.63, which is expected for a tetrahedral water structure [42].
However, they are broad and overlapping, showing a wider range of O–O–O angles
than those expected for the ideal tetrahedral arrangement introduced by Bernal and
Fowler. The exact degree of water tetrahedrality is still debated and the structural
1 A Historical Review of the Structures of Water and Ice
where x i is the atomic fraction of atoms of type i, r i j are the intramolecular distances
between the centers of nuclei.
At this level of approximation, the geometric structure of the water molecule is
assumed to be a rigid sphere, and individual atomic scattering factors are calculated
for isolated unbound particles. In fact, covalent bonding and the mutual interaction
of molecules change the charge distribution and, therefore, the atomic form factors
defined by (1.4).
The structural factor S(Q) is determined from the molecular form factor by
S(Q) =
I (Q)
F(Q)
2
.
(1.5)
The contribution of intermolecular correlations to the total intensity (see the last term
in (1.3)) is given by [33]
I inter (Q) =
i≤ j
(2 − δ i j )x i x j f i (Q) f j (Q)S i j (Q),
(1.6)
where the correlation factor S i j (Q) between atoms of type i and j is directly related
to the RDF by
S i j (Q) = 4πρ
∞
0
r
2
(g i j (r ) − 1)
sin Qr
Qr
dr,
(1.7)
where ρ is the atomic density.
Therefore, the structure factor S(Q) and the RDF itself can be calculated directly
from (1.7). However, a preliminary structural model is needed to weigh the scattering
correlations between the oxygen and hydrogen atoms. For this purpose, the diffraction pattern is analyzed by the empirical potential structure refinement [40]. Using
molecular dynamics and ab initio methods, interatomic potential energy functions are
consistently tuned to reconcile the simulated and measured structure factors S(Q).
The aim of the simulation is to find realistic constraints on the forces between atoms,
inside and between molecules, which will fit the experimentally observed RDF.
Since this problem generally has an unlimited number of solutions, the result always
depends on the model used. This is why there are still some differences between
the RDFs obtained by different authors depending on the initial assumptions and the
potentials that were applied.
The first RDF for water was reported almost a century ago [41]. Figure 1.9 shows
the modern RDFs for water and ice. The g OO (r ) function for water shows well-defined
maxima at 2.9, 4.5, 6.7 Å and minima at 3.4, 5.5 Å. The positions of the peaks have
a ratio close to the 1.63, which is expected for a tetrahedral water structure [42].
However, they are broad and overlapping, showing a wider range of O–O–O angles
than those expected for the ideal tetrahedral arrangement introduced by Bernal and
Fowler. The exact degree of water tetrahedrality is still debated and the structural