Theor Chem Acc (2015) 134:115
1 3
it is necessary to bear in mind that it comes with a significant increase in the number of graph invariants. For
example, in the case of INT 4
4 the number of independent
invariants increases from 21 to 57 and R 2 changes from
0.71627 to 0.74785 when the second-order graph invariants for H-bonds are replaced by third-order ones. The
increase in R 2 is smaller when invariants for dangling
O–H bonds are considered (for example, R 2 changes
from 0.26268 to 0.27331 when the second-order invariants are replaced by fourth-order invariants for the same
ice nanotube INT 4
4 ). Even a significant increase in the
number of graph invariants doesn’t necessarily bring
serious improvement of the approximation. For example, in the case of graph invariants defined for dangling
O–H bonds of water cage (1), the number of invariants
increases from 5 to 58 when the order of invariants
changes from 2 to 4, but the value of R 2 is improved
only marginally (from 0.96378 to 0.96907).
Some systems allow for specifi c approaches. In the
case of mixed cluster (4), it is possible to separately consider the confi gurations with a dangling F–H bond and
those with the H-bond F–H · · · O . It turns out that the
value of R 2 is smaller for the fi rst set (0.98550) than for
the other (0.99435). Approaches employing graph invariants based on bond functions b and d are different in the
case of clusters with broken H-bonds. Our calculations
show that the change of the defi nition of the graph invariants does not bring signifi cant improvements in terms
of the quality of the approximation: the values of R 2 for
two schemes are relatively close (larger for that based on
bond functions d ) and are equal to 0.97290 and 0.97763
for cluster (2) and 0.93086 and 0.94077 for cluster (3),
respectively.
Finally, we provide a few thoughts about the general
idea of graph invariants as given by totally symmetric
projectors of the products of bond variables. Sometimes
other representations of the symmetry group may be useful. An example is provided by ice nanotubes. They are
thought to possess ferroelectric order with a net polarization when n is odd and to be “anti-ferroelectric” when
n is even [ 26 , 27 ]. This prediction is based on a simple
model discarding most of the H-bond confi gurations.
Alternatively, we can consider an idealized structure of
ice nanotubes. In this case, the polarization parallel to
the INT axis is proportional to the difference of the numbers of dangling O–H bonds on the INT ends, thereby
confi rming the original predictions. This parameter
should also play a signifi cant role for relaxed structures,
but it corresponds to the representation A
1 for INT m
n with
an odd n and to the representation A 1u for INT m
n with an
even n . Therefore, analysis of properties like polarization may require specialized graph descriptors based on
the symmetry of the system.
4 Conclusion
In summary, a very large database of the energies of
H-bond networks in water clusters is constructed with
the help of an ultra-fast semiempirical method based
on strictly local electron groups. It includes water clusters of different symmetry and morphology, clusters
with ideal and defected structures, systems where one
water molecule is replaced with hydrogen fl uoride.
Two varieties of graph invariant descriptors are benchmarked against quantum-chemical calculations, one of
them being proposed in the present paper. Both types
of descriptors lead to identical results when applied to
ideal networks satisfying ice rules but differ for structures with defects. We established that the use of highorder invariants is not justifi ed unless H-bond networks
signifi cantly deviating from ice rules are taken into consideration. Approximation of the water cluster energies
by linear combinations of second-order graph invariants
shows that this approach gives surprisingly good results
for clusters without signifi cant strains (including those
with defects) but fails when four-membered water rings
are present in the structure.
Acknowledgments This work is partially supported by National
Research Center “Kurchatov Institute”, Russian Science Foundation
(Grant 14-19-00662), and Russian Foundation for Basic Research
(Grants 13-07-00095 and 14-03-00867).
References
1. Hoepfner V, Deringer VL, Dronskowski R (2012) Accurate
hydrogen positions in organic crystals: assessing a quantumchemical aide. Cryst Growth Des 12:1014–1021
2. Hoepfner V, Deringer VL, Dronskowski R (2012) Hydrogenbonding networks from fi rst principles: exploring the guanidine
crystal. J Phys Chem A 116:4551–4559
3. Deringer VL, Englert U, Dronskowski R (2014) Covalency of hydrogen bonds in solids revisited. Chem Commun
50:11547–11549
4. Bernal JD, Fowler RH (1933) A theory of water and ionic solution, with particular reference to hydrogen and hydroxyl ions. J
Chem Phys 1:515–548
5. Yoo S, Kirov MV, Xantheas SS (2009) Low-energy networks of
the T-cage ( H 2 O) 24 cluster and their use in constructing periodic unit cells of the structure I (sI) hydrate lattice. J Am Chem
Soc 131:7564–7566
6. Chihaia V, Adams S, Kuhs WF (2004) Infl uence of water molecules arrangement on structure and stability of 5 12 and 5 12
6 2 buckyball water clusters. A theoretical study. Chem Phys
297:271–287
7. Kirov MV, Fanourgakis GS, Xantheas SS (2008) Identifying the
most stable networks in polyhedral water clusters. Chem Phys
Lett 461:180–188
8. Anick DJ (2003) Application of database methods to the
prediction of B3LYP-optimized polyhedral water cluster geometries and electronic energies. J Chem Phys
119:12442–12456
163
Reprinted from the journal
1 3
it is necessary to bear in mind that it comes with a significant increase in the number of graph invariants. For
example, in the case of INT 4
4 the number of independent
invariants increases from 21 to 57 and R 2 changes from
0.71627 to 0.74785 when the second-order graph invariants for H-bonds are replaced by third-order ones. The
increase in R 2 is smaller when invariants for dangling
O–H bonds are considered (for example, R 2 changes
from 0.26268 to 0.27331 when the second-order invariants are replaced by fourth-order invariants for the same
ice nanotube INT 4
4 ). Even a significant increase in the
number of graph invariants doesn’t necessarily bring
serious improvement of the approximation. For example, in the case of graph invariants defined for dangling
O–H bonds of water cage (1), the number of invariants
increases from 5 to 58 when the order of invariants
changes from 2 to 4, but the value of R 2 is improved
only marginally (from 0.96378 to 0.96907).
Some systems allow for specifi c approaches. In the
case of mixed cluster (4), it is possible to separately consider the confi gurations with a dangling F–H bond and
those with the H-bond F–H · · · O . It turns out that the
value of R 2 is smaller for the fi rst set (0.98550) than for
the other (0.99435). Approaches employing graph invariants based on bond functions b and d are different in the
case of clusters with broken H-bonds. Our calculations
show that the change of the defi nition of the graph invariants does not bring signifi cant improvements in terms
of the quality of the approximation: the values of R 2 for
two schemes are relatively close (larger for that based on
bond functions d ) and are equal to 0.97290 and 0.97763
for cluster (2) and 0.93086 and 0.94077 for cluster (3),
respectively.
Finally, we provide a few thoughts about the general
idea of graph invariants as given by totally symmetric
projectors of the products of bond variables. Sometimes
other representations of the symmetry group may be useful. An example is provided by ice nanotubes. They are
thought to possess ferroelectric order with a net polarization when n is odd and to be “anti-ferroelectric” when
n is even [ 26 , 27 ]. This prediction is based on a simple
model discarding most of the H-bond confi gurations.
Alternatively, we can consider an idealized structure of
ice nanotubes. In this case, the polarization parallel to
the INT axis is proportional to the difference of the numbers of dangling O–H bonds on the INT ends, thereby
confi rming the original predictions. This parameter
should also play a signifi cant role for relaxed structures,
but it corresponds to the representation A
1 for INT m
n with
an odd n and to the representation A 1u for INT m
n with an
even n . Therefore, analysis of properties like polarization may require specialized graph descriptors based on
the symmetry of the system.
4 Conclusion
In summary, a very large database of the energies of
H-bond networks in water clusters is constructed with
the help of an ultra-fast semiempirical method based
on strictly local electron groups. It includes water clusters of different symmetry and morphology, clusters
with ideal and defected structures, systems where one
water molecule is replaced with hydrogen fl uoride.
Two varieties of graph invariant descriptors are benchmarked against quantum-chemical calculations, one of
them being proposed in the present paper. Both types
of descriptors lead to identical results when applied to
ideal networks satisfying ice rules but differ for structures with defects. We established that the use of highorder invariants is not justifi ed unless H-bond networks
signifi cantly deviating from ice rules are taken into consideration. Approximation of the water cluster energies
by linear combinations of second-order graph invariants
shows that this approach gives surprisingly good results
for clusters without signifi cant strains (including those
with defects) but fails when four-membered water rings
are present in the structure.
Acknowledgments This work is partially supported by National
Research Center “Kurchatov Institute”, Russian Science Foundation
(Grant 14-19-00662), and Russian Foundation for Basic Research
(Grants 13-07-00095 and 14-03-00867).
References
1. Hoepfner V, Deringer VL, Dronskowski R (2012) Accurate
hydrogen positions in organic crystals: assessing a quantumchemical aide. Cryst Growth Des 12:1014–1021
2. Hoepfner V, Deringer VL, Dronskowski R (2012) Hydrogenbonding networks from fi rst principles: exploring the guanidine
crystal. J Phys Chem A 116:4551–4559
3. Deringer VL, Englert U, Dronskowski R (2014) Covalency of hydrogen bonds in solids revisited. Chem Commun
50:11547–11549
4. Bernal JD, Fowler RH (1933) A theory of water and ionic solution, with particular reference to hydrogen and hydroxyl ions. J
Chem Phys 1:515–548
5. Yoo S, Kirov MV, Xantheas SS (2009) Low-energy networks of
the T-cage ( H 2 O) 24 cluster and their use in constructing periodic unit cells of the structure I (sI) hydrate lattice. J Am Chem
Soc 131:7564–7566
6. Chihaia V, Adams S, Kuhs WF (2004) Infl uence of water molecules arrangement on structure and stability of 5 12 and 5 12
6 2 buckyball water clusters. A theoretical study. Chem Phys
297:271–287
7. Kirov MV, Fanourgakis GS, Xantheas SS (2008) Identifying the
most stable networks in polyhedral water clusters. Chem Phys
Lett 461:180–188
8. Anick DJ (2003) Application of database methods to the
prediction of B3LYP-optimized polyhedral water cluster geometries and electronic energies. J Chem Phys
119:12442–12456
163
Reprinted from the journal