1 3
Theor Chem Acc (2015) 134:115
DOI 10.1007/s00214-015-1720-9
REGULAR ARTICLE
Benchmarks of graph invariants for hydrogen-bond networks
in water clusters of different topology
Andrey M. Tokmachev
1 · Andrei L. Tchougréeff
2,3,4 · Richard Dronskowski
4
Received: 18 July 2015 / Accepted: 1 September 2015 / Published online: 11 September 2015
© Springer-Verlag Berlin Heidelberg 2015
Keywords Water clusters · Hydrogen bonds · Graphs ·
Invariants · APSG
1 Introduction
Water in all its forms constitutes the basis of our existence.
The properties of water are unique and its beauty is limitless. Aggregates of water molecules are among the most
studied chemical systems, but despite the intensive concerted efforts, water is still far from being understood. The
reason behind is the set of hydrogen (H–) bonds keeping
water molecules together. Networks of hydrogen bonds
exhibit complex behaviour manifested as cooperative contributions to properties of water systems. The enormity
of these contributions gives impetus to studies of the collective structure of H-bond networks. Among the most
recent successes in electronic-structure theory devoted to
H-bonded systems, we mention density-functional studies
on carefully quantifying H-bonds in molecular crystals [ 1 ,
2 ] and in assessing the amount of covalency involved [ 3 ].
Water systems allow for numerous H-bond confi gurations even when the morphology of the system (adjacency
matrix) is fi xed and the restrictions of the ice rules (basically requiring the absence of ionized individual water molecules in a water system) [ 4 ] are imposed. The number of
these confi gurations grows exponentially with the size of
the system. Simple gas hydrate shells are a perfect illustration: the 5 12 D-cage (H 2 O) 20 allows for 30,026 symmetry
independent confi gurations, the numbers for the larger
5 12 6 2 T-cage (H 2 O) 24 and the 5 12 6 4 H-cage (H 2 O) 28 are
3,043,836 and 61,753,344, respectively [ 5 ]. Clearly, not
all of them are important, but information extracted from
studies of the minimal energy confi guration (which is
commonly the only subject of the investigation) is often
Abstract The diversity of the various forms of water
stems from systems of hydrogen bonds. Cooperative
behaviour of hydrogen-bond networks gives rise to unique
properties of water systems. A number of approaches to
understand and model the collective behaviour of hydrogen
bonds and predict their properties on the basis of a small
number of calculations have been put forward. Among
them, the concept of graph invariants provides most general descriptors for hydrogen-bond networks, which are
routinely used to predict properties of water systems. In the
present work, we examine the formalism of graph invariants and propose its modifi cation which may be benefi -
cial for water structures with defects. To benchmark graph
invariants, we carried out quantum-chemical calculations
of more than 10 7 water clusters with different hydrogenbond confi gurations. The quality of the approximation is
studied as a function of the type of graph invariant and its
order. The results demonstrate that the method is applicable
only to cage-like structures without signifi cant strains.
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Andrei L. Tchougréeff
andrei.tchougreeff@ac.rwth-aachen.de
1
NBICS Centre , NRC Kurchatov Institute , Kurchatov Sq. 1 ,
Moscow 123182 , Russia
2
Moscow Center for Continuous Mathematical Education ,
Bol’shoy Vlas’evskiy 11 , Moscow 119002 , Russia
3
Department of Chemistry , Moscow State University ,
Vorob’evy Gory 1, Build. 3 , Moscow 119992 , Russia
4
Institut für Anorganische Chemie , RWTH Aachen University ,
Landoltweg 1 , 52056 Aachen , Germany
157
Reprinted from the journal
Theor Chem Acc (2015) 134:115
DOI 10.1007/s00214-015-1720-9
REGULAR ARTICLE
Benchmarks of graph invariants for hydrogen-bond networks
in water clusters of different topology
Andrey M. Tokmachev
1 · Andrei L. Tchougréeff
2,3,4 · Richard Dronskowski
4
Received: 18 July 2015 / Accepted: 1 September 2015 / Published online: 11 September 2015
© Springer-Verlag Berlin Heidelberg 2015
Keywords Water clusters · Hydrogen bonds · Graphs ·
Invariants · APSG
1 Introduction
Water in all its forms constitutes the basis of our existence.
The properties of water are unique and its beauty is limitless. Aggregates of water molecules are among the most
studied chemical systems, but despite the intensive concerted efforts, water is still far from being understood. The
reason behind is the set of hydrogen (H–) bonds keeping
water molecules together. Networks of hydrogen bonds
exhibit complex behaviour manifested as cooperative contributions to properties of water systems. The enormity
of these contributions gives impetus to studies of the collective structure of H-bond networks. Among the most
recent successes in electronic-structure theory devoted to
H-bonded systems, we mention density-functional studies
on carefully quantifying H-bonds in molecular crystals [ 1 ,
2 ] and in assessing the amount of covalency involved [ 3 ].
Water systems allow for numerous H-bond confi gurations even when the morphology of the system (adjacency
matrix) is fi xed and the restrictions of the ice rules (basically requiring the absence of ionized individual water molecules in a water system) [ 4 ] are imposed. The number of
these confi gurations grows exponentially with the size of
the system. Simple gas hydrate shells are a perfect illustration: the 5 12 D-cage (H 2 O) 20 allows for 30,026 symmetry
independent confi gurations, the numbers for the larger
5 12 6 2 T-cage (H 2 O) 24 and the 5 12 6 4 H-cage (H 2 O) 28 are
3,043,836 and 61,753,344, respectively [ 5 ]. Clearly, not
all of them are important, but information extracted from
studies of the minimal energy confi guration (which is
commonly the only subject of the investigation) is often
Abstract The diversity of the various forms of water
stems from systems of hydrogen bonds. Cooperative
behaviour of hydrogen-bond networks gives rise to unique
properties of water systems. A number of approaches to
understand and model the collective behaviour of hydrogen
bonds and predict their properties on the basis of a small
number of calculations have been put forward. Among
them, the concept of graph invariants provides most general descriptors for hydrogen-bond networks, which are
routinely used to predict properties of water systems. In the
present work, we examine the formalism of graph invariants and propose its modifi cation which may be benefi -
cial for water structures with defects. To benchmark graph
invariants, we carried out quantum-chemical calculations
of more than 10 7 water clusters with different hydrogenbond confi gurations. The quality of the approximation is
studied as a function of the type of graph invariant and its
order. The results demonstrate that the method is applicable
only to cage-like structures without signifi cant strains.
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Andrei L. Tchougréeff
andrei.tchougreeff@ac.rwth-aachen.de
1
NBICS Centre , NRC Kurchatov Institute , Kurchatov Sq. 1 ,
Moscow 123182 , Russia
2
Moscow Center for Continuous Mathematical Education ,
Bol’shoy Vlas’evskiy 11 , Moscow 119002 , Russia
3
Department of Chemistry , Moscow State University ,
Vorob’evy Gory 1, Build. 3 , Moscow 119992 , Russia
4
Institut für Anorganische Chemie , RWTH Aachen University ,
Landoltweg 1 , 52056 Aachen , Germany
157
Reprinted from the journal