1 3
Theor Chem Acc (2015) 134:134
DOI 10.1007/s00214-015-1728-1
REGULAR ARTICLE
Relations between real molecules through abstract molecules:
the reference cluster approach
Paul G. Mezey
1,2,3
Received: 15 June 2015 / Accepted: 13 September 2015 / Published online: 16 October 2015
© Springer-Verlag Berlin Heidelberg 2015
Keywords Nuclear charge space · Electronic energy
inequalities · Reference cluster · Universal molecule model
1 Introduction
Originally motivated by the United Atom studies of Thirring, Narnhofer, and Lieb [ 1 – 3 ], the idea of exploiting
nuclear charge convexity relations between electronic
energies of molecules, without involving the extreme case
of uniting all nuclei of the molecule into a single, formal
nucleus of the United Atom, has lead to a variety of rigorous quantum chemical energy inequalities for various sets
of molecules, starting with simple diatomic cases [ 4 , 5 ].
After the very fi rst examples of such electronic energy
inequalities [ 4 , 5 ], including one of the simplest cases, the
N 2 and CO pair, the concept of the nuclear charge space
was elaborated [ 6 ] and the level set topology approach was
introduced [ 7 ]. A generalization to other linear parameters
besides the nuclear charges was described [ 8 ], followed by
several additional applications [ 9 – 15 ]. As one important
generalization, geometry-independent relations were also
derived [ 9 , 14 ]. Analogous relations for isomerisation energies [ 10 ], as well as for potential surfaces for higher multiplicities, were derived [ 11 ], and applications to clusters
[ 13 ], an early step toward the more recent reference cluster
model, relevant to this study, were also described. Exploiting the quadratic dependence of the nuclear repulsion on
the nuclear charges, in some cases, for example, in the
prototype case of N 2 and CO, analogous inequalities were
derived for molecular total energies [ 12 , 15 ]. Special applications, for example, to polymers of different unit cell sizes
[ 16 ], and to special problems of vibrational spectra [ 17 ],
have also been described.
Abstract Replacing integer atomic numbers representing
nuclear charges by continuous variables has already provided some rigorous quantum chemical relations between
real molecules, using a formal interpolation through nonphysical abstract molecules of continuously varying nuclear
charges. Extending this approach to the more general “universal molecule” model, where all discrete parameters of
molecules are generalized and replaced by continuous variables, provides further relations, actually interconnecting
all real molecules through abstract, non-physical “molecules,” involving, for example, non-integer number of electrons. One simplifying idea of this model is the so-called
“reference cluster,” originally defi ned for isoelectronic systems of a fi xed number N of nuclei, where each nucleus is
replaced by a possibly fi ctive nucleus with a nuclear charge
equal to the average of the N nuclear charges. Based on the
earlier results, some new relations are derived interrelating
energies and some other properties of some real molecules,
and also providing a unifying framework for the utilization of both symmetry and energy relations of the universal
molecule model.
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Paul G. Mezey
paul.mezey@gmail.com
1
Scientifi c Modeling and Simulation Laboratory (SMSL),
Department of Chemistry and Department of Physics
and Physical Oceanography , Memorial University
of Newfoundland , 283 Prince Philip Dr. , St. John’s ,
NF A1B 3X7 , Canada
2
Institute of Chemistry , Eötvös University , Budapest , Hungary
3
Institute of Chemistry , Babes-Bolyai University ,
Cluj-Napoca , Romania
25
Reprinted from the journal
Theor Chem Acc (2015) 134:134
DOI 10.1007/s00214-015-1728-1
REGULAR ARTICLE
Relations between real molecules through abstract molecules:
the reference cluster approach
Paul G. Mezey
1,2,3
Received: 15 June 2015 / Accepted: 13 September 2015 / Published online: 16 October 2015
© Springer-Verlag Berlin Heidelberg 2015
Keywords Nuclear charge space · Electronic energy
inequalities · Reference cluster · Universal molecule model
1 Introduction
Originally motivated by the United Atom studies of Thirring, Narnhofer, and Lieb [ 1 – 3 ], the idea of exploiting
nuclear charge convexity relations between electronic
energies of molecules, without involving the extreme case
of uniting all nuclei of the molecule into a single, formal
nucleus of the United Atom, has lead to a variety of rigorous quantum chemical energy inequalities for various sets
of molecules, starting with simple diatomic cases [ 4 , 5 ].
After the very fi rst examples of such electronic energy
inequalities [ 4 , 5 ], including one of the simplest cases, the
N 2 and CO pair, the concept of the nuclear charge space
was elaborated [ 6 ] and the level set topology approach was
introduced [ 7 ]. A generalization to other linear parameters
besides the nuclear charges was described [ 8 ], followed by
several additional applications [ 9 – 15 ]. As one important
generalization, geometry-independent relations were also
derived [ 9 , 14 ]. Analogous relations for isomerisation energies [ 10 ], as well as for potential surfaces for higher multiplicities, were derived [ 11 ], and applications to clusters
[ 13 ], an early step toward the more recent reference cluster
model, relevant to this study, were also described. Exploiting the quadratic dependence of the nuclear repulsion on
the nuclear charges, in some cases, for example, in the
prototype case of N 2 and CO, analogous inequalities were
derived for molecular total energies [ 12 , 15 ]. Special applications, for example, to polymers of different unit cell sizes
[ 16 ], and to special problems of vibrational spectra [ 17 ],
have also been described.
Abstract Replacing integer atomic numbers representing
nuclear charges by continuous variables has already provided some rigorous quantum chemical relations between
real molecules, using a formal interpolation through nonphysical abstract molecules of continuously varying nuclear
charges. Extending this approach to the more general “universal molecule” model, where all discrete parameters of
molecules are generalized and replaced by continuous variables, provides further relations, actually interconnecting
all real molecules through abstract, non-physical “molecules,” involving, for example, non-integer number of electrons. One simplifying idea of this model is the so-called
“reference cluster,” originally defi ned for isoelectronic systems of a fi xed number N of nuclei, where each nucleus is
replaced by a possibly fi ctive nucleus with a nuclear charge
equal to the average of the N nuclear charges. Based on the
earlier results, some new relations are derived interrelating
energies and some other properties of some real molecules,
and also providing a unifying framework for the utilization of both symmetry and energy relations of the universal
molecule model.
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Paul G. Mezey
paul.mezey@gmail.com
1
Scientifi c Modeling and Simulation Laboratory (SMSL),
Department of Chemistry and Department of Physics
and Physical Oceanography , Memorial University
of Newfoundland , 283 Prince Philip Dr. , St. John’s ,
NF A1B 3X7 , Canada
2
Institute of Chemistry , Eötvös University , Budapest , Hungary
3
Institute of Chemistry , Babes-Bolyai University ,
Cluj-Napoca , Romania
25
Reprinted from the journal