Who is who in Mathematical Chemistry
(Selected and very subjective alphabetic
listing of some individuals and groups involved in the various topics of
mathematical chemistry, mainly in chemical graph theory. With some bio/biblio/graphic
links.) In progress.
Links to mathematical
chemistry papers published by Eugene Babaev
Intuitive Chemical Topology Concepts
Consider a molecular "ballandstick" model made from rubber tubes.
What happens if we "inflat" such a model?
We shall get a surface (a manifold) with intriguing properties. One
particular feature of this approach is that the Euler characteristic of
such a manifold will be preserved in chemical reactions...
Abstracts:
In the present paper we treat chemical similarity in
terms of the homeomorphism concept and electron count rules. We suggest
a novel interrelationship between molecular graphs and molecular 2D surfaces
by direct mapping of chemical structures (like the Lewis dot formulas)
to the specific 2D manifolds and pseudomanifolds. We define this mapping
in such a way that the lone pairs, free radical centers, and multiple and
multicentered bonds serve as the intrinsic topological invariants of the
2D models. This approach allows us to apply the homeomorphism concept to
chemical problems in a new way, classifying molecular structures and reactions
from the viewpoint of the topology of surfaces.
The structure of the paper is the following. Section
2 recalls several concepts of the graph theory and topology of surfaces,
necessary for further discussion, and Section 3 is the overview of the
common types of molecular graphs and molecular surfaces used in chemistry.
In Section 4 we treat the concepts of free radicals, lone pairs, and multiple
bonds as intuitively topological concepts, whereas Section 5 provides an
explicit definition and visualization of their topology on the graphtheoretical
level. In Section 6 we suggest an explicit mathematical concept of molecular
topoid, which visualizes the lone pairs, free radicals, and multiple bonds
as sorts of “holes” in an appropriate 2D surface. The topoid (a “rubber
2D molecule” without geometry) is a novel combinatorial 2D image of molecule,
intermediate between ordinary surfaces and graphs. Operations on topoids
(resembling cutandpaste operations of a topologist with imaginary 2D
manifolds) reflect the key types of formation and cleavage of chemical
bonds. In Section 7 we suggest a novel conservation law, the invariance
of Euler characteristic of molecular topoids, and use it for classifying
the chemical reactions. Section 8 has the goal to illustrate, how the homeomorphism
of topoids brings together diverse molecular similarity types in a unique
manner. In Section 9 we prove that the explicit 2D image of a graph should
be a surface with embedded Jordan curves, and in Section 10 the nonequivalent
types of embedding are used to expand the common principles of 2D modeling
in chemistry. In Section 11 we use the generalized concept of hypertopoids
to classify the structure and reactivity of molecules with multicentered
and delocalized bonds on the 2D level. Finally, in Section 12 we investigate
the possibility of 2D modeling of the excited states of molecules with
several unpaired electrons by using nonorientable surfaces.
Table of Contents. Full
text
The Alternation Rule: An Old Heuristic Principle or the New Conservation
Law?
How many conservation laws exist in chemistry? Not so much: conservation
of mass, charge, energy, orbital symmetry... Above we discussed the conservation
of the Euler characteristics. Anything else? Let us examine one more: conservation
of "chemical parity"
Consider a set of objects closed in relation to an operation of their
change. A group, an algebra? No. The objects are some specific molecules
with bipartite graphs (which follow socalled "alternation rule", and the
operation is a polar chemical reaction.
From Introduction:
We will attempt in this paper to find an example of a closed set where
it was earlier not looked for or was not noted. We will show that among
structures, habitual to the glance of organic chemists, a class of molecules
can be mathematically rigidly separated, possessing certain "rhythmic"
similarity of electronic structure, associated with alternation of polar
(donor and acceptor) centers. As an example of the operation of changes
of such objects (called superconsonant below), we will examine general
polar reactions, including stages of heterolytic formation and/or cleavage
of bonds. The main idea of this approach is that the indicated set with
certain assumptions can in fact be considered "almost closed" in relation
to the indicated operation of changes. In its turn, such an assertion is
equivalent to a new principle, nontrivial for organic chemistry, of retention
of the topological property of polar bipartition (charge alternation) in
polar processes. The formulation of the model is preceded by a review of
early papers, devoted to the principle of alternation and the problem of
consonance, in no way finding reflection to date in the domestic literature.
Abstracts:
Early concepts and models of polarity alternation along the chains
with polar groups are critically reviewed. The proposed new model describes
alternation along the chains as the result of specific mutual disposition
of Lewis' basic and acidic centres. Identical results can be obtained
from both the models of polar bipartite multigraphs and the model of "ions
in molecule". This gives rise to separate the specific class of molecules
and ions (called "superconsonant"), whose polar bipartite structure strictly
determine possibility of their presentation as ion assemblage. The author's
quantum chemical AM1 calculations proved charge alternation even at the
structures of nonpolar consonant molecules and/or their ions. The hypothesis
of "alternation conservation" in ionic processes is claimed, that permits
one to consider the consonant set of molecules to be closed (for the pelements
of 2row) in relation to polar reactions. Wide applications of this
hypothesis and rare counterexamples are discussed. Simple genetical interrelationship
among the structures of the super consonant series is proposed as the model
of computer generation of nontrivial synthetic equivalence types.
Full text
Concept of Chemical Periodicity: from Mendeleev Table to Molecular
HyperPeriodicity Patterns (with R. Hefferlin)
Can we built a Periodic Table of Molecules? How may it look like?
Is it multidimensional?
Trying to answer these questions we should recall that classical language
of chemistry is more deeply related to mathematics than it is commonly
considered. Indeed, a chemical formula is nothing else but a partition
of a natural number, and the Lewis dot diagram from mathematical viewpoint
is only a sort of a pseudograph. Let us order these mathematical objects
(partitions or graphs) on the plane and analyse the chemical trends at
this flat "hyperperiodic" table...
Table of Contents:
I. ORIGIN OF PERIODICITY
II. MENDELEEV PERIODIC TABLE
II.1. What chemists use it for
II.2. How physicists "explain" it
III. PERIODIC SYSTEMS IN OTHER SCIENCES
III.1. Some known criteria for natural
systems
III.2. Criteria for periodic systems
III.3. Periodic systems of objects,
smaller than atoms
IV. MOLECULAR PERIODICITY
IV.1. How to talk about molecular
periodicity
IV.1.A. Local
models: examples of diversity
IV.1.B. Early
attempts of global classification
IV.1.C. Global
models: what to classify and why?
IV.1.D. Atomic
periodicity versus molecular?
IV.2. "Nightmares" of global classification
and how to avoid them
IV.3. Choice of global similarity
parameters: importance of the electron count
V. THE ART AND THE LOGIC OF EQUALIZATION: Classification
of isosteric ensembles
V.1. Regularities in the polymorphism
of isosteric ensembles
V.2. Chemical trends: the rule of
two poles
V.3. Distinguishing between molecules
in the PIE
V.4. Topological trends in the PIE
V.4.A. The
point on the PIE as a set of molecular pseudographs
V.4.B. Counting
of cycles and components from electrons and atoms
V.4.C. Cyclomatic
number of pseudograph and homeomorphism of structures
V.4.D. Criteria
of connectedness for molecular pseudographs
V.5. Molecular disconnectedness as
a hyperperiodic function in the PIE
VI. THE HYPERPERIODICITY PATTERN: Classification of
isovalent ensembles
VII. SPECIAL TYPES OF CHARTS: Diatomic molecules
VIII. CONCLUSION
Acknowledgements
REFERENCES
From Conclusion:
We may ask, to which branch of exact science we should
attribute the art and logic of natural and periodic molecular classifications?
Molecules are objects of chemistry and physics, and their classification
(as we have seen) requires rather delicate mathematical models. In order
to define the appropriate place of such an activity, we may arrange the
exact sciences in some sort of a "periodic table":

Physics 
Chemistry 
Mathematics 
Physical 
Pure&Applied Physics 
Physical
Chemistry 
Physical
Mathematics 
Chemical 
Chemical
Physics 
Pure&Applied Chemistry 
Chemical
Mathematics 
Mathematical 
Mathematical
Physics 
Mathematical Chemistry 
Pure&Applied Mathematics 
The vertical "groups" are pure sciences, while the "rows"
are their applications to other sciences. It is easy to see familiar sciences
like chemical physics, and physical chemistry, and the branches of applied
mathematics. (These terms may be observed, say, as the names of scientific
journals, that "periodically" appear in libraries.) We can "predict" two
new (still littleknown) sciences in the upperright corner of the table,
namely physical and especially chemical mathematics. We may remember that
there are relatively new disciplines called chemical topology and chemical
graph theory (not with reversed word order), and that these fields are
most closely related to the problems discussed in this chapter. It seems
that global molecular classifications may be related specifically to chemical
mathematics. Let us explain why.
We mean that mathematicians often develop ideal objects
and forms without any idea how to apply them to real physical and chemical
objects of the Universe. Vice versa, chemists often propose pragmatic,
empirical, generalizations about real objects (e.g., homology, isovalency,
isomerism, aromaticity, degree of saturation, the octet rule, the repulsion
of electron pairs, and so on) without any idea how these concepts relate
to one another in a mathematical sense. As we have seen, the interrelation
between such archetypal chemical concepts is clearly displayed and clarified
in global molecular classifications. Surprisingly, such interrelations
appear to have the same nature and the same beauty that exists in rather
abstract mathematical objects (homeomorphism of surfaces, connectedness
of graphs, properties of partitions, etc.). We can say that we are applying
chemistry to mathematics and finding ideal mathematical forms inside chemistry,
rather than bringing a mathematical model to chemistry.
Full text
More... (to be appeared soon)