Concept of Chemical Periodicity: from Mendeleev Table
The paper of Eugene Babaev and Ray Hefferlin&
"The Concepts of Periodicity and Hyper-Periodicity:
from Atoms to Molecules"&
was published in the book: Concepts in Chemistry:
a Contemporary Challenge. (Ed. D.Rouvray).&
Research Studies Press, London, 1996, pp. 24-81.
Below is the original text of this paper (copyright
by the authors).&
Before editorial revision the initial manuscript had
another title:
Concept of Chemical Periodicity:
from Mendeleev Table to Molecular Hyper-Periodicity
E. V. Babaev* and Ray Hefferlin+
*Chemistry Department, Moscow State University, Moscow,
119899, Russia
+Physics Department, Southern College, P.O. Box 370,
Collegedale, TN 37315, USA
I. ORIGIN OF PERIODICITY
II. THE 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. Problems of global classification and their avoidance
IV.3. Choice of global similarity parameters: importance of the
electron count
V. THE ART AND 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 plane of isoteric ensembles
V.4. Topological trends in the plane of isoteric ensembles
V.4.A. The point on the plane of isoteric ensembles 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 plane of isoteric ensembles
VI. THE HYPER-PERIODICITY PATTERN: Classification of isovalent ensembles
VII. SPECIAL TYPES OF CHARTS: Diatomic molecules
VIII. CONCLUSION
Acknowledgements
REFERENCES
I. ORIGIN OF PERIODICITY
We meet with periodicity when something is repeated in time or in space.
Periodic processes -- like the rotations of planets, changes of seasons,
high and low tides -- are examples of global (all-pervading) periodicity
in our solar system and in nature. Ancient astronomers and astrologers
used these observations to regulate the activities of individuals or even
of nations. Periodicity is an essential part of our life, as evidenced
by the beating of the heart and our eating, sleeping, and waking patterns.
Geometrical or physical objects that are repeated in space (such as infinite
mosaics or the atoms in crystals) express another familiar example of periodicity.
Because periodicity is such an important phenomenon, scientists have
been interested in it for several centuries. Mathematicians describe periodicity
in terms of periodic functions such as F(X) = F(X+nT), where X is a time
or space coordinate, F(X) is a function describing something variable that
repeats its value after every addition of the constant value T (the period)
to the argument X, and n is an integer. The classical periodic functions
Y=sinX and Y=cosX behave in this way, with clear alternations of maxima
and minima. It is easy to imagine regular processes that are "almost periodic"
with a variable period T (for instance, if one eats or sleeps irregularly)
or with changes in the amplitude of F(X), as in the damped saw-tooth curves
shown in Fig. 1.
Figure 1. Example
of an "almost periodic" function
(ionization potentials
of atoms against their atomic numbers).
In a case such as that in Fig. 1, a strict mathematical definition
of the periodic function would be somewhat difficult. Nevertheless, it
is only common sense that such a function could be treated as a periodic
one. Such a qualitative, approximate understanding of periodicity and periodic
functions has been widely adopted by the chemical community since the classical
work of the end of the last century leading up to the periodic table.
II. THE PERIODIC TABLE
By the middle of the last century chemists generally understood
that chemical elements can be grouped together in separate classes according
to obvious similarities or dissimilarities in their properties. Thus, flammable
alkali metals (that form stable cations) can be naturally separated from
poisonous halogens (that prefer to form anions). It had also been demonstrated
by Doebereiner that some elements may be grouped into triads so that the
middle element's properties can be approximated as the average of the properties
of its neighbours. Clarification of the concept of atomic weight by Cannizzaro
in 1858 stimulated attempts to find rational classification of elements.
In the s several workers combined the principle of triads and
chemical similarity and used the atomic weights (really masses) of the
elements to formulate the periodic law, see Table 1.
Table 1. Some milestones of discovery of the periodic
Contributor
Contribution
Dobereiner
Classification of elements into triads [1]
Development of rational classifications
of elements and regularities in their atomic weights [2-4]
Pettenkofer
Table of 43 elements arranged in 13 groups
De Chancourtois
Classification of elements (spiral around
cylinder) with increase of their atomic weights [6]
Arranging of similar elements in groups,
periodicity of atomic volumes [7]
Octaves' Law [8]
P prediction of new elements and
changes/improvements of known atomic weights [9,10]
The idea of periodicity is expressed in Mendeleev?s periodic
chart, Fig. 2. In this table the elements are arranged in rows (according
to increasing atomic mass) and columns (according to chemically similar
behaviour) like in other early tables. However, only Mendeleev used this
chart to predict previously unknown elements and to improve known atomic
weights of some elements (see part II.1). For simplicity, the elements
of the table can be numbered in order of increasing atomic mass, though
the physical meaning of such numbers was explained only a few decades later.
What sort of periodicity is expressed by this chart? Qualitatively
speaking, the periodicity displayed here is simply the regular appearance
of element-analogs with increasing atomic mass. Thus, the elements with
numbers 3, 11, 19, 37, 55 are alkali metals, and those with numbers 9,
17, 35, 53 are halogens. It can be seen that the lengths of the rows (periods)
in the periodic table are different and equal to 2, 8, 8, 18, 18, and 32.
A mathematician would claim that there is no exact periodicity, since the
period itself is not of constant value [11]. From the chemical viewpoint,
however, it is a very important matter to place each element (especially
each heavy element) such that its chemical nature resembles that of lighter
elements. As a result, a lot of attempts have been undertaken by generations
of chemists to express this non-exact periodicity in different graphic
forms of the periodic table. Many examples have been collected in books
[2, 3, 12, 13], and we refer here only to few specially interesting cases
(Figs. 3a-c), such as the spiral, helical, and "dumb-bell" forms.
Figure 3. Examples
of different non-traditional forms of the periodic chart for the elements:
(a) the spiral form
due to Baumgauer [14],
(b) the helical form
due to Bilecki [15] and
(c) the 'dumb-bell' form
due to Basset [16] (Adapted from ref. [13].)
II.1. What chemists use it for
The law of periodicity has stimulated better understanding of interrelationship
between elements, design of new classes of compounds, and search and discovery
of new elements. The topic is extensively reviewed in many books [2-4,
10-12, 17-20], and according to the comrehensive review [4] the total number
of references related to the chemical periodicity up to 1969 had reached
The heuristic role of the periodic table was been recognized
most clearly at the end of the last century. At that time it was been proved
that many macroscopic properties of elements (such as density, atomic volumes,
melting points etc.) could be treated as periodic-like functions. A method
for the quantitative calculation of macroscopic properties for new (or
even unknown!) elements was developed (Table 2). This method (most consistently
and successfully used by Mendeleev) included comparison of the properties
of all neighbours of a given element in the periodic table. An element
is surrounded by a maximum of eight neighbours, thus belonging to four
triads (horizontal, vertical and two diagonal triads). An analysis of the
trends in properties in each triad provides a way to estimate the desired
unknown property. This methodology, equally applicable to elements and
their compounds, opened up the possibility of estimating desired properties
with high accuracy, see Table 2.
Table 2. Use of the periodic table for prediction of
elements and their properties in XIX-th century
Prediction of element/propertyb)
principal researcher, date
Experimental discovery
contributor, date
Eka-Aluminium (Ea)
M=68, d=5.9, low m.p.
Mendeleev, ,10]
M=69.72, d=5.904, m.p.=30
Lecoq de Boisbaudran, ]
Eka-Boron (Eb)
M=44, d (Eb2O3)=3.5
Mendeleev, ,10]
M=45.1, d (Sc2O3)=3.864
Eka-Silicium (Es), M=72, d=5.5
d (EsO2)=4.7; b.p.(EsCl4)=90
Mendeleev, ,10]
Germanium, M=45.1 d=5.35
d (GeO2)=4.703; b.p.(GeCl4)=86
Winkler, 1886 [23]
New gaseous element, M=20
Elements analogous to He and Ar
Neon, M=19.96
Krypton, Xenon
Ramsay, Travers, ]
Decrease of atomic weight (1.5
Mendeleev, ];&
Brauner, 1878 [26]
Nilson, Petersen, 1884 [27]
Increase of atomic weight (1.5
Mendeleev, ,10];&
Meyer, 1870 [7]
Bunzen, Mendeleev, 1871 [28]
Increase of atomic weights (1.5 times)&
Brauner, 1881; Cleve, 1875;
Increase of atomic weights (2
Mendeleev, ,10]
Roscoe, 1874;&
Rammelsberg, 1872 [29]
a)Actually predicted by use of the periodic table and discovered
b)M - d - density (g/cm3); m.p./b.p. melting/boiling
point (oC); changes of atomic weights from those accepted in
However, this general method of comparative calculation of elemental
properties is rather rarely used in practice. Instead, nowadays, chemists
use the Periodic Law (and the periodic chart) to analyse qualitative (but
not quantitative) trends in different properties of elements. This opportunity
to see general chemical trends directly from the periodic chart is of great
importance in modern chemical education. Indeed, it is easy to distinguish
(say) metals from non-metals at first glance in the "school" periodic chart,
by locating them respectively at the lower left and upper right corners
of the table.
II.2. How physicists "explain" it
The modern interpretation of the periodic table appeared after
the development of the physical model of the atom as a positively charged
nucleus surrounded by negatively charged electrons. Atoms of different
elements differ by the charge (number of protons) of their nuclei. This
hypothesis, which was proved experimentally [30], explained the formal
numbers for elements in the periodic table as the set of natural physical
invariants for atoms.
Moreover, an explanation has been given for the irregular lengths
of the rows in the periodic table as well as for the origin of groups with
chemically similar elements. The electrons of every atom can be subdivided
into shells, and these shells can hold different numbers of electrons.
The shells are concentric about the nucleus. Atoms with a fixed number
of electrons in the inner shells correspond to a row of elements in the
periodic table, while atoms with a constant number of electrons in the
outer shell correspond to a group of elements. The electrons in the outer
shell are usually called valence electrons, since they are responsible
for chemical bonding and their number is connected with the chemical valency
of the atom. Thus, the periodic system is constructed (and can be explained)
on the basis of the isoelectronic principle: atoms are arranged as isovalent
families (vertical groups) and "iso-inner-electronic" families (horizontal
rows). The different numbers of electrons that can be held by the inner
shells reflect the different lengths of the periods observed in the periodic
It is significant that not only macroscopic properties of elements,
but also many microscopic properties of atoms demonstrate pronounced periodicity.
Examples of such properties are atomic radius and volume, ionization potential,
electron affinity, electronegativity, and some optical and magnetic properties
[2, 4, 17-20]. The function in Fig. 1 is actually the first ionization
energy of atoms plotted against their atomic number.
III. PERIODIC SYSTEMS IN OTHER SCIENCES
The beauty of the periodic system of the elements exerted a
strong influence on other sciences. Indeed, there are a lot of objects
that can be generally considered as "elements." One can consider ions,
atomic nuclei (isotopes of elements), elementary particles, and quarks
as examples of such objects in physics. In addition, in many descriptive
disciplines (mineralogy, organic chemistry, botany, zoology) there are
enormous numbers of closely related objects that require natural classifications.
In many cases scientists have developed classifications, with the periodic
table in mind as the standard of excellence. Some attempts have been made
to generalise the meaning of the terms "natural" and "periodic" systems
in order to develop a general methodology of classification in science
and to consider the system of chemical elements as an important special
III.1. Some known criteria for natural systems
An early attempt to give general criteria for natural systems
was made by entomologist and philosopher Lyubischev. It was his dream (first
conceived in the 1920s and later developed in the 1970s) to construct a
periodic system of living organisms [31]. Although he did not achieve his
general goal, Lyubischev formulated many stimulating ideas about natural
his papers on general taxonomy and classification [32], as well
as his philosophical works, pointed out their true value [33]. According
to one of his definitions, "a system should be considered as natural if
the position of an element in it reflects the maximum number of the elemental
properties." In spite of some vagueness in this definition, it opens the
possibility of developing new systems and improving existing ones. It is
easy to see that the periodic table is an excellent example of the validity
of th other examples are also discussed below.
Another important criterion for natural systems was developed
in the 1970s by Urmantsev, another Russian biologist. He discussed some
original parallels between isomerism in chemistry and in biology (e.g.,
the structural peculiarities of non-identical leaves on the same tree or
the rather intriguing "periodic system of flowers"). [34].Urmantsev seems
to be the first to have considered the special case of "systems of objects
of a given genus," and he tried to formulate some their general properties
[34,35]. One important principle is that any element of such a system should
possess the property of polymorphism. (Originally, the phenomenon of polymorhism
was ob it is the property of a mineral compound to
exist in more than one crystalline form, for example, octahedral and cubic.)
Considered globally, this principle of polymorphism reflects
the very simple idea that any kind of element (that is claimed to be a
simple one) any "simple element" may actually consist of
simpler parts. A triad of concepts -- the nature, number, and interrelations
of these smaller parts -- should be taken into account, otherwise the polymorphism
will appear as an inexplicable property. Indeed, the elemental parts may
be related in different ways leading to the property of polymorphism. Urmantsev
illustrated this principle in his table of flowers, where the number of
petals is fixed, but the petals can be arranged in a geometrically different
manner [34]. Chemical isomerism [36,37] is as a good example of such global
the term "isomerism" (in such a wide sense) has also been
accepted in biology [38] and nuclear physics [39]. An example of polymorphism
for the elements in the periodic table is the well-known phenomenon of
allotropy [40] (e.g., the existence of the one element carbon as
diamond or graphite). Yet again, isotopes, ions, and excited states of
atoms also illustrate possibilities of different polymorphism types for
III.2. Criteria for periodic systems
Dias has attempted to formulate general criteria and properties
for a periodic table set [41]. In attempting to classify polyaromatic benzenoids
and their analogs (see part IV.1.A), Dias came up with a periodic system
of aromatic hydrocarbons. Making parallels of his tables with the periodic
chart, Dias proposed the following criteria for a periodic table set:
is a partially ordered set (i.e., it obeys reflexivity, antisymmetry, transitivity);
2. It is two-dimensional
3. It complies with the triad principle (any central element has a
metric property that is the arithmetic mean of two oppositely adjacent
elements).
Also among the properties of a periodic table set mentioned by Dias
(1) hierarchical ordering,
(2) periodicity of at least one invariant,
(3) edge effects (i.e., elements on at least one two-dimensional edge
have values for various properties which are more extreme than do elements
chosen at random),
(4) the greatest difference is frequently to be found in the properties
of the smallest element.
III.3. Periodic systems of objects smaller than atoms.
Elementary particles. High-energy physics has
resulted in the discovery of great number of "elementary" particles. There
are three leptons, each w each of these six objects
has its own neutrino. Then there are mesons, which were originally defined
as having masses between that of the electron and the proton but are now
defined on the basis of their having integer spin. These mesons come in
groups of one, or two, or three with almost one mass value. As particles
having larger and larger masses are studied, some of them appear to be
very similar to le these particles are excited forms
of the least massive particles. Thus, there are hundreds of mesons but
only a few are in their ground states. Exactly the same situation pertains
to baryons, which are particles with masses equal to (or greater than)
that of the proton and with half-integer spin. Some of their groups even
have four particles. There are many more groups of excited baryons with
larger masses. Two important invariants -- the charge and the isotopic
spin -- can be used [42, 43] to arrange multiplets of the elementary particles
in a periodic-like table. In Fig.4, such a grouping (that pertains to a
very early version of the standard model of fundamental particle physics)
is shown for mesons (a) and baryons (b) in their ground states.
Figure 4. Left: Two JP = 0- multiplets of mesons. Right:
one JP = 1/2+ multiplet of baryons. I3 is the isotopic spin and Y is the
both of these quantities are related to the charges of particles
and of groups of particles. The Greek letters p
, h , S , L
, and X refer to various elementary particles
aside from the neutron and proton. The signs indicate whether the particles
have positive, neutral, or negative charge with magnitudes equal to that
a bar over the particle symbol indicates that it is an
antiparticle. (Taken with permission and adapted from Ref. 43.)
The principle of periodicity here is that such diagrams are repeated
at higher masses for excited particles. In theoretical physics such diagrams
can be constructed from first principles, namely by combination of analogous
diagrams for quarks which are hypothetical sub-elementary particles.
Isotopes. The classification of isotopes is another well-developed
field [11, 12, 44, 45]. The number of known isotopes is over 1700 and exceeds
the number of known atoms by a large factor. It was mentioned early on
that the stability of isotopes is connected with "magic numbers" of nucleons:
protons, neutrons, and/or their sum. In this sense, there is a parallelism
between stable nucleonic configurations (the isotopes with the magic numbers
2, 8, 20, 50, etc.) and atoms with closed shells (magic numbers 2, 10,
18, etc., for atoms of the noble gases). The periodic system of isotopes
can be expressed in several alternative ways, one of which is presented
Figure 5. Possible
periodic ta vertical lines separate the periods in
the Valley of Stability.
(N - number of nucleons, Z - number of protons).
One such table proposed in the 1960s by Selinov [45] and designed as
a seven-coloured wall-chart is shown in Fig. 6. The bold line on this table
that connects black squares (stable isotopes) reflects the idea of periodicity.
Figure 6. The
periodic table of isotopes {the wall-chart of h Selinov). The running
axis (1 to 104) indicates the element.
The perpendicular axis
indicates the excess of neutrons over protons. The black triangles indicate
the magic numbers of protons.
Ions. One intriguing problem that arises from with
the periodic table of atoms is the possibility of constructing periodic
systems of ions [46]. An atom can be completely or partially ionized to
a cation by removing electrons or transformed into an anion by the addition
of new electrons. In Fig.7 the energy required for a few consecutive ionisations
of atoms is plotted against the atomic number. One can see that the curves
are periodic, and hence it is possible to construct periodic tables for
mono-, di-, and multi- charged cations. If we look at the dispositions
of the maxima and minima of the curves and compare them with those for
atoms, it becomes evident that the magic numbers of electrons for ions
are the same as for neutral atoms. Therefore, the number of electrons (but
not the charge of the nucleus) is responsible for the periodicity of ions.
Figure 7. Successive
ionization potentials of atoms plotted against atomic numbers can be used
to construct a period table of ions.
IV. MOLECULAR PERIODICITY
Progress in periodic classifications of objects smaller than
atoms leads to the fascinating question: is it possible to apply the idea
of periodicity to molecules? Can we construct, analogously to the chart
for atoms, a periodic chart for molecules -- organic, inorganic, and organometallic
molecules? Can we imagine a natural system where the role of an
"element" can be played by a molecule, so that the position of an object
in such a system reflects the behaviors of data for a maximum number of
the molecular properties?
Such ideas have been around for a very long time and are related
to the general problem of classifications o they have
in fact stirred the thoughts of many scientists since the middle of the
last century. The French chemist Gerhardt in the 1840s proposed playing
a game of chemical "patience" with cards that would place heterological
and homological series of molecules in a natural arrangement [47]. He then
added smaller and smaller pieces of paper (isological series) on every
card of his patience game to classify compounds in formal three-dimensional
space. The Russian chemist Butlerov was seriously interested in Gerhardt?s
idea, but failed to integrate isomers into such a three-dimensional system,
and so finally criticized and rejected it [48].
Many contributors to the periodic law (in particular, Dumas,
Pettenkofer, and Newlands [49]) have described parallels between triads
of the elements and members of homological series of hydrocarbons differed
by CH2-group (see also Section IV.1.A). In 1862, Newlands presented
two of many tables exhibiting the composition and mutual relations of organic
substances and serving as "a map of organic chemistry" (see Ref. [49c]).
The tables show, along vertical, horizontal, upper-left/lower right, and
upper-right/lower left diagonals, bodies differing by H2, O,
and CO2 (among others) or bodies in which one such symbol is
replaced by another. Later Pelopidas mentioned parallelism between the
series of organic radicals with decreasing saturation degree and the elements
in rows (see Ref. [49d]). Thus, the saturated methylammonium radical CH3NH3
is analogous to an alkali metal, while the unsaturated radical CN behaves
similar to a halogen.
After the discovery of the Periodic Law, many attempts -- some
intuitive and others based on a solid mathematical and physical background
-- have been made to study the problem of molecular periodicity. The present
chapter sets itself the goal of reviewing the general methodology, the
history, and recent progress in this interesting field. Early results on
this topic can be found in the authors' publications [43, 50-52] and in
the monograph of Gorski [53].
IV.1. How to talk about molecular periodicity
Let us first distinguish the local and the global
approaches to molecular periodicity. The local approach seeks to classify
a separate (finite or infinite) the global approach
seeks to classify all sets of possible molecules. The local models of molecular
classification are more widespread, are characterised by a great diversity
of viewpoints, and in some cases their authors refer to such models as
"periodic tables." Let us consider three impressive examples of such local
tables: two from organic chemistry and one from inorganic chemistry.
IV.1.A. Local models: examples of diversity
Periodic Table of Hydrocarbons. As we mentioned
above, a lot of parallels between the elements and hydrocarbons have been
discussed in XIX-th century [49]. One most elegant attempt to express this
analogy in graphical form belongs to the Russian Morozov [54]. In his table
of hydrocarbon radicals (Fig.8) the homological hydrocarbons are arranged
in columns in the same way as are similar elements in the periodic chart.
Moreover, he drew many curious analogies between the organic radicals and
corresponding elements (the parallelism in atomic and molecular mass, maximal
valency, and acidic and basic properties of corresponding hydroxides).
It is less known that in the first version of his table (distributed in
1885) Morozov has predicted the existence of a whole column of "nonvalent"
elements (appearing as analogs of the paraffins), calculated their atomic
weights (namely, 4, 20, 36, and 82), and arrangement in the periodic table
[55] (see also ref. 10c, p.135 and ref. 12, p.51). Although the idea was
criticized and rejected by contemporaries, nevertheless, decade later [24,25]
these elements were found: the noble gases! It is worth noting that Morozov
wrote his giant book in the Schlisselburg Fortress during his 25 years
of incarceration for his pamphlets against the Russian Czar's family. The
book with the table, therefore, finally appeared in 1907 (far after discovery
of the noble gases), and when Mendeleev read it, he was so impressed with
its content that he immediately had Morozov appointed to a full professorship:
the silver lining in the cloud of his imprisonment!
Figure 8. Formal
analogy between the periodic tables of elements and hydrocarbon radicals
due to Morozov [54]. The title: "Two periodic systems"; subtitles:
I - The system of normal
aliphatic hydrocarbons (key to the system - hydrogen);
II - The system of modern
mineral elements "archeohelides" (key to the system - helium). Bottom numbers:
valency in respect of halogens and metals.
Another model called "Natural System of Hydrocarbon Compounds"
has been proposed in 1922 by Decker [56]. The system was simply a two-dimensional
plane with the numbers of the carbon and hydrogen atoms on the axes. Such
oversimplified classification, however, has appeared useful in comparison
of known types of organic homology (vinylogy, phenylogy, benzannelation,
etc.) and design of novel types of homological series.
Periodic Table of Functional Groups (paraelements). In
the 1980s Haas, in trying to explain the halogen-like properties of the
pseudohalogen groups (e.g., CF3, SCF3, CN,
NCO, N3, etc.) proposed the so-called "element displacement
principle" [57]. In his model certain radicals with element-like behaviour
(the paraelements) can be designed by formal addition of x ligands (atoms
or other paraelements) to prototype p-elements with a corresponding shift
of x places to the right within the period of the periodic system. The
paraelements, therefore, can be arranged into a set of periodic tables
(Fig.9). Haas proved analogies between the elements and paraelements by
means of extensive physical data (NMR shifts, dissociation enthalpies,
electronegativities), structural features (interchangeability in known
structures), and examples of chemical similarity. He is also currently
applying this concept in the practical design of novel inorganic molecules
and functional groups.
Figure 9. Haas'
periodic system of functional groups. The first and second diagrams show
how Haas generalized Grimm's hydrogen displacement principle by the substitution
of f he names these species "paraelements". The third
and fourth diagrams show how an entry in the first diagram, CF3,
can be substituted for the fluorine in the first two diagrams. Haas designates
the resulting species "first-order derivative paraelements". The fifth
and sixth diagrams show how an entry in the fourth diagram, CF3S
can be substituted for the CF3 in the third and fourth diagrams.
Haas designates the resulting species "second-order derivative paraelements".
The seventh diagram shows what happens if ligands like oxygen, sulfur,
displacements of two or three groups must take place.
The last diagram shows what happens if various of the paraelements of the
seventh diagram are substituted with the appropriate displacements. The
diagrams are from ref. [57]; used with permission.
Periodic Table of Polyaromatic Hydrocarbons. Benzenoid
hydrocarbons are an interesting series of molecules wi
it is easy to imagine such structures if one looks at a honeycomb with
some cells full of honey. Dias has recently discussed [41] many regularities
within this class in terms of their periodic table (see part III.2). The
parameters used to construct such a table are the internal structural features
of joined polygons (Fig.10), and such parameters turn out to reflect many
aspects of similarity between the polyaromatic hydrocarbons.
Figure 10.
Formula periodic table for benzenoid polycyclic aromatic hydrocarbons
due to Dias [41]. Nc
and NH are the numbers of carbon and hydrogen atoms,
ds is the net number of disconnections among the internal edges
(see ref. 41); NIc is the number of internal carbon vertices
in the structures.
Bottom: examples of hydrocarbon - analogs (Used
with permission.)
IV.1.B. Early attempts of global classification
The periodic tables just described are elegant examples of
local periodic tables. We must now discuss the equally interesting approaches
to a global periodic system of molecules. Two names should be mentioned
in relation to the history of global molecular classifications. The first
is the Russian Shemyakin who published in the 1930s a series of papers
under the title "Natural classification of chemical compounds" [58]. The
other is the Pole Gorsky who developed in the 1970s the "morphological"
classification of compounds [53,59].
Shemyakin proposed the "molecular number" -- that is,
the sum of atomic numbers of all atoms in a molecule (actually the total
number of electrons) -- as the basis of his classification. Evidently,
such an approach immediately runs up against polymorphism: there is vast
growth in the number of isoelectronic molecules with increase of molecular
number. As a result, Shemyakin had to introduce additional parameters (such
as the number of hydrogen atoms and the"characteristic structural number"
for isomers) and used them to draw three-dimensional structure-property
correlation charts for both organic and inorganic molecules (as in Fig.11a).
Unfortunately, such classification tables consist of many senseless combinations
of atoms and, therefore, it is somewhat difficult to use them in practice.
Nevertheless, Shemyakin discussed at least nine classes of tables (isostructural
and isoelectronic, isoelectronic but not isostructural, etc.) that he declared
to be the projections of a unique table of all molecules. (An example of
a building block of such a table is presented on Fig.11b.) One result of
his classification was his formulas to calculate boiling points of alkanes
from the boiling points of the noble gases.
Figure 11 Early
attempts at global classification by Shemyakin, (a) and (b) [58] and Gorsky,
(c) and (d) [59].
(a) three-dimensional
graph with the number of hydrogen atoms and molecular number on the horizontal
axes and boiling point on the vertical axis, (b) and (c) examples of building
blocks for more complete classifications, (d) three-dimensional graph with
parameters ev and ez (see text) on the horizontal
axes and enthalpy on the vertical axis.
In the approach of Gorsky, inorganic molecules are classified by
two parameters, ev and ez, that reflect changes of
molecules in redox and acid-base reactions, respectively. The first number
is the so-called "normalised" oxidation number of the central atom (or
of a few atoms), and the second number is the total charge of the ligands
around the central core(s). Having used these parameters as the coordinates,
Gorsky constructed many classification tables for different inorganic molecules
(e.g, for oxy-acids of phosphorus and sulfur), drawing either concrete
formulas or general structures, as in Fig.11c. He also illustrated the
usefulness of such tables by constructing three-dimensional charts capable
of predicting the thermodynamic properties of molecules, as shown in Fig.11d.
All of the above-mentioned global classifications have neither
been adopted nor used by the general chemical community. Let us now try
to analyse the underlying problem here.
IV.1.C. Global models: what to classify and why?
In order to clarify the problem of global molecular periodicity,
we first attempt to answer some simpler questions: what are the objects
and the goals of such a classification? In other words: what should we
classify and why?
What to classify? In the case of elements we have objects
(atoms or their condensed phases) and measurable data (microscopic
and macroscopic properties) that form the basis for their periodic classification.
In the case of molecules -- if we consider them as the "elements"
of an alternative classification -- problems may appear both in
relation to the objects themselves (the existence and stability of an isolated
molecule), their relation to the condensed phase, and even the possibility
of measuring molecular properties.
Suppose we decide to compare only isolated molecules. In such
a case we would still need to define clearly the meaning of the term "molecule"
in relation to the condensed phases. We usually define a molecule as an
aggregate of chemically connected atoms (with given atomic numbers), that
can be separated as an entity from surrounding media because of a greater
stability -- a lower energy of the internal (intramolecular) bonds compared
to the external (intermolecular) ones. Of course, the difference between
such intra- and intermolecular forces depends on the nature of the atoms.
On the other hand, in many cases this difference may be inessential (say,
for ionic crystals, alloys, or atomic lattices such as diamond), and it
is somewhat difficult to identify a "molecule" in the condensed phase.
The problem can be illustrated by use of the well-known "Grimm
Tetrahedron" (Fig.12) [60]. This diagram symbolically reflects with the
four vertices of a tetrahedron the four main types of bonding in solid
chemical compounds
-- i.e., metallic, ionic, atomic, and van der
Waals. The six edges between these vertices correspond to the intermediate
types of bonds. It is clear that the idea of isolated molecules can be
most naturally applied only to one vertex of this diagram (the central
one in Fig.12, where the intermolecular interactions are the weak van der
Waals forces).
Figure 12.
Different types of intermolecular bonding represented by chemical compounds
on the vertices and edges of the "Grimm tetrahedron"(see text).
Such compounds correspond to typical organic and (simple) inorganic
molecules with covalent bonds. For the remaining types of bonding, the
corresponding prototype molecules may be unstable to the processes of self-association,
and so lose their identities in the condensed phase under usual conditions.
In such cases, special efforts must be made to vaporise the compound and
investigate the structure and properties of its isolated molecules.
Among the set of theoretically possible molecules (considered
under typical "wet chemistry" conditions) molecules with covalent bonds
predominate over those with the other three types of bonds. This suggests
that (i) a global molecular classification should be first addressed to
such separate classes of molecules, and that (ii) the problem of "empty
places" may turn out to be unavoidable for a global system until more representative
data can be collected. This pronounced asymmetry between the ideal objects
of classification and real molecules for many years hindered significant
development of the entire area of molecular periodicity. Fortunately, in
the past few decades short-lived species and especially clusters have become
more available for experimentalists. Major progress in spectroscopy and
mass spectrometry, and the use of lasers and the matrix isolation technique
open up the possibility of studying their properties systematically. The
development of quantum-chemical calculations provides an alternative route
to the estimation of properties of unstable molecules and clusters of almost
any imaginable combination of atoms.
Why classify? There are several reasons. (1)
A pragmatic aim: to arrange molecules in naturally related series in order
to calculate any desired molecular property by interpolation (or extrapolation)
from the known properties of their neighbours. (2) An educational goal:
the possibility of discovering qualitative chemical trends of molecules
directly from a molecular chart. (3) A philosophical question as to whether
the whole is larger than the sum of its parts. A molecule is not the arithmetic
although a large number of molecular properties can be
treated as additive, many are not. (4) A psychological reason based on
scientific and human curiosity: why not try?
IV.1.D. Atomic
periodicity versus molecular
Another related question is whether it is necessary at all
to separate molecular periodicity from atomic periodicity. Although the
periodic law was formulated to be equally valid for both elements and
their compounds, the proof was actually limited to the series of binary
compounds (e.g.,the oxides, hydrides, halides, etc.) of a given
row or column. In such simple cases molecular periodicity can definitely
be reduced to the atomic case, a viewpoint often reflected in chemistry
textbooks. This approach, however, immediately fails if one looks at the
relation between different classes of binary compounds (e.g., NaCl
and CaO), or between binary and ternary compounds (e.g., NaCl and
K2SO4), or between an inorganic and an organic substance.
For such cases molecular periodicity may only indirectly be connected with
atomic periodicity (or not follow from it at all). Hence, an alternative
methodology, capable of comparing any pair of molecules, is required.
IV.2. Problems with global classification and their avoidance
The infinity problem: The first serious problem
directly associated with the idea of molecular periodicity is the infinity
problem. Indeed, in contrast to the finite number of atoms (about 100),
the number of possible molecules constructed from them is actually infinite
and hence we can never construct a wall-chart for all molecules, since
there is no such an infinite wall. (Of course, this assertion assumes
that there is no limit to the numbers of atoms which we can imagine to
be in molecules!)
How can the problem of infinity be resolved? We see two possible
solutions: to separate the infinity into finite sets (of which there will
be an infinite number) or to separate the infinity into "smaller" infinities
(of which there will be a finite or infinite number).
(1) Infinite number of finite sets. The number of all
imaginable molecules is infinite, but the number of diatomic or triatomic
molecules is finite. The same is true for any case when the number of atoms
in the molecules is fixed. So, a global classification can be considered
as a step-by-step constructing of the local systems of N-atomic molecules
and some comparisons of such systems. Analogously, we can separate infinite
sets of molecules into (say) finite subsets of isovalence-electronic or
iso-inner-electronic molecules.
(2) "Smaller infinities". Let us take a familiar example
from mathematics: the infinite set of natural numbers can be separated
to subsets of odd and even numbers. In such a case one infinity breaks
down into two infinities, each with special peculiarities. Similarly, we
can develop some finite or infinite number of important features to separate
one infinite class of molecules from another. A good example from chemistry
is the separation of the hydrocarbons CnH2n+x according
to the saturation degree x (as it is in the Beilstein handbook). Another
example is the classification of planar cyclic delocalized systems according
to the Huckel rule [61] (4n or 4n+2 p-electrons).
Analogously, the boron hydrides (boranes) and carboranes can be arranged
according to Wade?s electron counting rules of (n+k)-electrons [62]. Of
course, these rules are valid only for certain l analogous
rules for global classifications still need to be developed.
The multidimensional problem. Theoretically we
can imagine one classification of molecules as a hyperspace in which the
number of axes is equal to the number of elements, and where the calibrations
on the axes will be the numbers of the appropriate atoms in the molecule.
(Thus, the system of Decker for hydrocarbons [56] will be a plane in this
hyperspace.) However, this classification results in a complete loss of
visual clarity and is therefore unacceptable. The human eye is able to
perceive only two-dimensional or three-dimensional pictures and, therefore,
any attempt to compare molecules by more familiar planar charts may lead
to degeneracies. In such a situation the requirement for any molecular
"chart" has to be as expressive as possible. Development of computer software
opens new horizons for perceiving and manipulating objects in the multidimensional
spaces, and this may turn out to be the most promising way to resolve the
problem of many dimensions.
The polymorphism problem. Seeking for a natural
system of molecules can be considered as a search for a finite set of natural
descriptors, capable of distinguishing between any pair of molecules. Evidently,
such descriptors should be capable not only of distinguishing the molecules
of different constitution, but also of discriminating among the isomers
of a given formula. It should be mentioned, however, that the isomerism
phenomenon in chemistry is the worst case of polymorphism of natural objects.
Indeed, the vast growth in the number of isomers with an increase of number
of atoms in organic molecules is well-known (combinatorial explosion) [36,
63]. Thus, the hydrocarbon C30H62 has 4 111 846 763
isomers (see e.g., Ref. [36b]). Another aspect of this problem is
the polymorphism of the isomerism types [36, 37]. Examples of such types
are alternative arrangements of atoms in the skeleton (as in HCNO and HNCO),
topological differences of structures (as in the cyclic and acyclic structures
of O3 and as in the differently branched skeletons of butane
and iso-butane), differences in geometry (as in the cis- and trans-, and
syn- and anti-isomers), conformation (as in rotamers), and chirality (as
in the enantiomers).
In spite of some pessimism caused by such a gloomy succession
of polymorphism problems, modern chemistry has developed various approaches
and techniques in order to distinguish isomers of different types. Let
us consider only one illustrative example: how to distinguish by natural
numbers two isomers with a different "degree of branching" -- butane and
isobutane. A very crude image of a molecule is its presentation as a set
of atoms and "rubber" bonds. More strictly speaking, skeletal structures
of molecules can be represented by graphs -- mathematical objects
that are sets of points (vertices) connected together by lines (edges)
[63]. Atoms of a molecule can be naturally associated with the vertices
and the bonds with the edges of a graph. In mathematical chemistry, molecular
structures are often presented without terminal hydrogen atoms by so-called
"hydrogen-suppressed graphs". Such graphs for butane and isobutane structures
are presented on Fig. 13a,b.
Figure 13.
Distinguishing between butane isomers. Initial structures (a) are changed
to their hydrogen-suppressed graphs (b). The latter are compared with the
graph of propane in (c). The number of propane fragments found in the isomers
is different, (d).
These two structures can easily be distinguished from one another by comparing
them with the graph of propane -- the previous member of the homologous
series (Fig. 13c). One can conclude that the structure of propane can be
found two times in the structure of butane, and three times in the structure
of isobutane (Fig. 13d). These numbers -- 2 and 3 -- can
be considered as the simplest numerical parameters (topological indices)
that distinguish the two isomers. We can apply the same operation (i.e.,
comparison of a given structure with propane) to higher members of the
alkane family to distinguish between isomeric pentanes, hexanes, etc.
Such topological indices can be defined in quite different
they may then have more or less discriminative power, and they may
(or may not) correlate with different physical properties of molecules
[36, 63]. At least, various combinations of different indices can be used.
A discipline that studies this area -- the Quantitative Structure-Property
Relationship (or QSPR) approach -- is now a rapidly developing branch
of chemistry [64, 65]. Methods have recently been put forward in this field
that are directed at distinguishing between geometrical isomers. We have
given special attention to this aspect of the isomerism in order to underline
that at least one of the above mentioned problems of global polymorphism
may be avoided.
Finally, we conclude by observing that the model of global
periodicity is not entirely hopeless. The history of chemistry is full
of intriguing empirical observations and generalizations that may help
us to avoid the problems discussed above. Moreover, as we shall show later,
chemists do have tools not only for natural distinguishing of dissimilar
structures, but also for the natural ordering of chemically similar molecules.
IV.3. Choice of global similarity parameters: importance
of the electron count
Any classification implies an arrangement of objects into classes
according to their similarities and dissimilarities. The problem of similarity
in chemistry has attracted special attention in the past decade, and a
classic book [65] discusses different aspects of this problem. Of course,
any qualitative conclusion on the similarity or dissimilarity of two molecules
might be arbitrary, though chemists have discovered some parameters that
are of major importance for molecular similarity studies.
In an effort to determine useful parameters for molecular classification,
it might seem reasonable to assume that the origin of molecular similarity
lies in atomic similarity. As we discussed above (Part I.2.B), any atom
has two invariants responsible for its similarity with other atoms. These
are its column (group) number and its row (period) number, and these two
numbers reflect the numbers of valence and inner electrons in the atom.
Modern molecular orbital theory treats molecules somewhat analogously to
it also separates the electrons in a molecule into valence and core
electrons. These two key atomic invariants -- the number of valence
and of inner electrons, therefore, can be considered as transferable parameters
applicable both to atoms and molecules. Consequently, the total number
of valence electrons (Zv) and the total of inner electrons (Zi)
of molecules could serve as promising parameters to establish molecular
kinship. The number of atoms (N) in the molecule is another important and
simple parameter.
The isovalency of atoms of the same group often causes structural and
chemical similarity in a series of related molecules formed from them.
Thus the series of the halides in group five -- binary compounds (such
as NF3, NCl3, PF3, PCl3) or
mixed derivatives (such as NF2Cl, PFCl2, PFClBr)
-- are isovalent and differ only in their inner shells. In such cases the
entire family of molecules can be represented by its isovalence type
by simply changing the symbols of atoms to the appropriate number for the
group of elements in the periodic table. In the examples of fifth-group
halides which were just discussed, Zv=26=5+7+7+7 and the isovalence
type (which is actually the "chemical formula" of entire class) is 5777
The isovalency of molecules may arise not only from the "vertical
isovalency" of atoms in groups of the periodic chart, but also from the
"horizontal isovalency" of atoms and corresponding ions from the same row.
The general rule is that similar substitutions in the isovalent series
provide isostructural molecules. Thus, we can take the series of ions Be2+,
B3+, C4+, N5+ (isoelectronic with the
helium atom) and add a fixed number of ligands such as H- or
F-. In this manner the structurally similar families of tetrahedral
hydrides (BeH42-, BH4-, CH4,
NH4+) or fluorides (BeF2, BF4-,
CF4, NF4+) may be obtained. Analogously,
we could use different ligands (e.g., nitrogen, as in the linear
anions NBN3-, NNN-, NCN2-, or oxygen,
as in the triangular anions BO33-, CO32-,
NO3-). We can also start from another series of ions
(e.g., ions C4-, N3-, O2-, that
are isoelectronic with Ne and give pyramidal structures CH3-,
NH3, H3O+ with proton as the ligand) to
yield ever more new series of isostructural molecules.
Comparing the structures of isoelectronic molecules obtained
from "horizontally isoelectronic" ions (as in the examples above) one can
conclude that their difference can be alternatively considered as the result
of an imaginary theoretical operation of addition (or removal) of a proton
from the nucleus of the central atom of molecule. Thus, we can view the
series BO2-, CO2, NO2+
as the result of proton shifts to (from) the central carbon atom of CO2
rather than as the result of addition of two O2- ligands to
the isoelectronic ions B3+, C4+, N5+.
(Long ago Bent, having reviewed many such series, proposed the apt term
"alchemical" for such theoretical proton shifts [66].) This idea can be
generalized and applied also to the ligands or, broadly speaking, to any
heavy (non-hydrogen) atoms of more complicated structures. Thus, we can
get linear ions ONO+ and NNN- by such addition (removal)
of a proton to (from) terminal ligands of the linear molecule NNO or obtain
the hexagonal pyridinium cation from benzene.
Perhaps the most intriguing and surprising result is that molecular
similarity appears not only via imaginary additions or removals of protons,
but also in the mental "internuclear rearrangements" of protons between
the heavy nuclei of the same molecule. Thus, a consistent shift of protons
among the heavy atoms of NNO leads to the isostructural family of linear
OCO, FCN, FBO, FBeF molecules. This phenomenon -- called isosterism
was first described in the 1920s by Langmuir [67], who observed pronounced
similarity in the macroscopic properties of CO and N2 or NNO
and CO2; another frequently cited pair of isosters is benzene
and borazene, again with very close physical properties. The isosterism
principle has proved its usefulness both in chemical education and in the
practical search for novel classes of molecules [66-68]. Isosterism is
considered to be responsible for the isomorphism of crystals, the constitution
of alloys (Hume-Rothery's phases) and the similarity in spectra of isoelectronic
molecules.
The cited examples underline the unique role of the proton
in deciding on the similarity between molecules of different constitution.
Unlike other particles, the proton has two symbols (p or H+),
indicating that it is simultaneously an elementary particle of physics
(symbol p) and an important "molecule" of chemistry (symbol H+),
responsible for the usual acid-base properties of chemical compounds. A
well-known chemical characteristic of the proton is that its addition or
removal as a ligand (as in acid-base processes) again only slightly perturbs
the initial molecular structure. Indeed, the molecule of ammonia NH3
(which is almost tetrahedral with one vertex occupied by a lone pair) can
be protonated (to form tetrahedral NH4+) or deprotonated
(to form the amide anion NH2- as a tetrahedron with
two vertices occupied by lone pairs) with approximate conservation of initial
structure. Exceptions appear only if there is another driving force (such
as delocalization or aromaticity) that tends to change the geometry of
charged species. Analogously, tautomerism -- the chemical shift of a proton
from one heavy atom to another (as in HNCO and HOCN or in the C- and O-forms
of the acetoacetic ester) -- in most cases conserves the initial skeletal
geometry of the molecules.
Invariance of the structures of isoelectronic molecules toward
both theoretical and actual proton shifts can be also illustrated by the
superposition of such shifts. Grimm in the 1920s first drew attention to
the fact that structural similarity of molecules exists in the special
isoelectronic series that have hydrogen as the variable atom [69]. Grimm
illustrated his "hydrogen displacement principle" by considering both neutral
molecules (e.g., Ne, HF, H2O, NH3, CH4)
and radicals or functional groups (e.g., Cl, SH, PH2,
SiH3). (It should be mentioned that in the 1930s there was a
sharp controversy (see Refs. [58b, 58c]) between Shemyakin and Grimm over
credit for the discovery of this "hydrogen displacement"; Shemyakin considered
this principle as a particular case of his nine tables, see above.)
Evidently the relationship in Grimm series is the consequence
(or superposition) of theoretical ("alchemical") and chemical proton shifts.
For example, in the simplest pair (Ne &
HF), the proton is first taken from the nucleus of a heavy atom as the
"physical" particle (Ne & F- + p) and is
then reconsidered as the "chemical" particle (p = H+) retained as a ligand
by formal protonation of the resulting anion F-. This procedure is equally
applicable to molecules with one or more heavy atoms (cf. the relationship
between the structures O=C=O, HN=C=O, H2C=C=O, and H2C=C=CH2
or between isobutene and F2C=O).
It should be emphasized that the local environment of a heavy
atom in a molecule appears to be almost insensitive to the chemical and
theoretical addition or removal of protons. As a result, the entire molecular
skeleton of heavy atoms also remains almost unchanged toward such proton
shifts, as well as toward internuclear (isosterism) or intramolecular (tautomerism)
rearrangements of protons. Since the proton is a unique chemical particle
bearing no electrons, the invariance of the electron number may be considered
as the parameter responsible for the skeletal similarity of isovalent families.
These early observations, later intensively and extensively studied in
different fields of inorganic and organic chemistry, have been met with
a rather limited number of counterexamples. For the quantum-chemical aspect
of the problem and analysis of the known exceptions see, e.g., review
in Ref. [70]. Isovalency, therefore, can be treated as a generalization
of elements' similarity in groups that is quite naturally applicable to
molecules.
V. THE ART AND THE LOGIC OF EQUALIZATION:
Classification of isosteric ensembles.
General remarks. Surprisingly enough, the isovalency
principle has usually been applied only to local classifi
let us consider its possible role in the area of global classification.
How can the principle be used in respect to global molecular periodicity?
The early Renaissance philosopher Nicolaus von Kues (Cusanus) said [71]
"equality foregoes inequality." Let us take this expression as our motto
and declare molecules with equal numbers of valence electrons Zv
to be equal objects. Of course, we should ignore any differences
between the isomers, isosters, and molecules differing by the chemical
and "alchemical" (see Section IV.3) shifts of protons. As we mentioned
above, in many cases there are very good chemical reasons why we are able
to neglect such differences.
We should also neglect any difference in the inner shells and
consider as equals the members of such sets as Li2O and Cs2S,
and F2, I2, and BrCl. This procedure is somewhat
similar to approximating the entire periodic chart of elements by only
one of its periods, say that one which contains the elements from Li to
Ne. This approximation makes sense if only elements from the main groups
are considered, hence, let us also limit ourselves to molecules constructed
from atoms of the main groups. Having arranged all molecules into isovalent
series, we may order the series by increasing Zv value. Actually,
in this way, we project a multidimensional space of all molecules to one-dimensional
space or a line.
Of course, the number of valence electrons has rather poor
discriminative power. The degree of degeneracy (or polymorphism), that
is the number of molecules with the same Zv, quickly increases
with increasing of Zv. However, we may raise the question: what
is the extent of this degeneracy, and how many neutral molecules are theoretically
possible for a given Zv? To avoid the complexity of isomerism,
let us forget it and simplify the question thus: how many different chemical
formulas are possible for a given Zv? The question is still
ill-defined, since the total of valence electrons does not reflect the
exact formulas but rather symbolic "valence formulas", that we called isoelectronic
types. So, the only question we can ask is: how many isoelectronic types
exist for given Zv?
The answer exists and follows from the mathematical theory
of numbers: the number of isoelectronic types associated with any given
Zv electrons is equal to the number of partitions of the number
Zv. The partition of a number Zv is any unordered
sequence of numbers whose sum is equal to Zv; thus (2,2) is
partition of 4 since 2+2=4, and (3,1) or (1,3) are also partitions of 4
for the same reason. There are only 5 possible partitions of Zv=4,
and these partitions are (1,1,1,1), (2,1,1), (3,1), (2,2), and (4). Therefore,
there are exactly 5 isoelectronic types of neutral molecules with 4 valence
electrons. Since we identify the size of any part of the partition as the
group number of a main-group atom, it follows that the parts of a partition
have limited size, up to eight.
The numbers of partitions (i.e., isoelectronic types) for first
seven Zv are as follows:
No. of partitions:
General formulas to calculate the partitions' number are rather
complicated, but they do exist [72]; for specific cases one may calculate
by hand or use a simple computer program. In any case, the very existence
of the possibility to calculate the number of isoelectronic types (partitions)
as a function of Zv is very important. Its importance lies in
that now the problem of polymorphism (at least in respect to one of its
possible types) can be treated in purely combinatorial sense. (Earlier,
when we deliberately admitted polymorphism into our classification, we
had absolutely no idea about its nature and degree.)
Let us consider the additional and quite natural parameter
-- the number of atoms N in the molecule. We can use it as the second axis,
making our global classification into a two-dimensional projection of a
system of all molecules on the coordinate plane (Zv,N). Every
point on this plane contains a certain finite set of isovalent molecules
with a fixed number of atoms, i.e., isosters. Let us call such sets isosteric
ensembles, and call the entire plane the Plane of Isosteric Ensembles.
The coordinates of discrete points in the Plane of Isosteric
Ensembles lie within boundaries. Thus, when increasing the number N along
isovalent series with a given Zv we "crush" larger parts into
smaller parts. The maximum extent of this crushing is an association of
N protons (or alkali metals each with one valence electron) and hence the
maximum N for given Zv is N=Zv. Analogously, being
limited by a maximum of eight valence electrons of any (main group) atom
in the molecule, we can not have in an N-atomic molecule more than 8N electrons
(as in an association of N noble gases atoms), so the maximum Zv
for a given N is 8N. Consequently, all points of the (Zv,N)
plane fall inside the sector bounded by lines N = Zv and N =
1/8 Zv. Evidently, any line parallel to one of the coordinate
axes (i.e., isovalent and isoatomic families) should cross the region between
these boundary lines (Fig. 14). The Mendeleev table on this chart is simply
the line N=1 with eight points.
Figure 14. A possible arrangement of isosteric ensembles
on the Plane of Isosteric Ensembles (shown by the gray area).
Since chemists consider the grouping of molecules into isosteric families
to be quite reasonable, the question arises as to the similarity and dissimilarity
relationships between isosteric ensembles. The Plane of Isosteric
Ensembles (first introduced by one of us in the 1980s [50, 51]) seems to
be just the needed answer to this question. As we shall prove below, the
Plane of Isosteric Ensembles is an important pattern for molecular classification,
(1) possesses a unique symmetrical structure with respect to
polymorphism,
(2) may be used for qualitative dichotomy of molecular chemical types,
(3) reflects key topological trends of molecular structures, and
4) obeys a kind of periodic law but in a sense quite different from
that of atoms: we shall call it the law of "hyper-periodicity".
In other words, the Plane of Isosteric Ensembles (in spite of its seeming
simplicity) may serve as a source of novel knowledge about molecular
periodicity.
V.1. Regularities in the polymorphism of isosteric ensembles
The first problem appearing in the model based on the Plane
of Isosteric Ensembles is the polymorphism of isosteric types. Every isosteric
ensemble has a different "capacity" due to the different number of theoretically
possible isosteric types (i.e., partitions). Let us estimate, at least
qualitatively, how this capacity varies, say along the N-atomic series.
In diatomic series, for instance, with an increase of Zv this
capacity first increases and then decreases. Thus, for Zv=2,
4, 6, 8, 10, 12, 14, and 16, the capacities (i.e., number of partitions
into two parts each smaller than 8) are 1, 2, 3, 4, 4, 3, 2, and 1, respectively.
These numbers of partitions are plotted perpendicular to the Zv,N
plane in Fig. 15.
Figure 15.
Regularities in the capacity of the isosteric ensembles along N-atomic
series. (The number of partitions is plotted against the (ZV,N)
The result is remarkable: the capacity along any isoatomic series
is symmetrical and is represented by a Gaussian-like curve. This peculiarity
follows from the combinatorial properties of partitions with parts of a
limited size. (The proof can be found in any elementary course in partition
theory (e.g., Ref. [72]).
This observation gives us at least two advantages. The first
is that the degree of polymorphism is small (and may even be neglected)
for molecules at the beginning and at the end of
thus the discriminative power of Zv and N is enough for such
particular cases. The second is that polymorphism itself (at least, in
the number of isosters) appears to be a periodic-like phenomenon.
V.2. Chemical trends: the rule of two poles
Opposites in chemistry. Since the time of Plato
and Aristotle scientists have spoken about the objects of the Universe
in terms of binary opposites. In the chemical tradition opposites are also
widely used, and one can easily recall such archetypal opposites such as:
(a) metals and non-metals,
(b) reductants and oxidants,
(c) acids and bases,
(d) electrophiles and nucleophiles, and even
(e) organic and inorganic compounds.
The first opposite (a) is related to the elements, while the other opposites
(b), (c), (d), and (e) are, of course, related to molecules and compounds
rather than atoms and elements. The opposite "metals -- non-metals" for
elements is connected with their disposition in the periodic chart (locations
in the lower-left and upper-right corners of the table, respectively).
Is it possible to treat opposites from (b) to (e) among molecules in relation
to the dispositions of their isosteric ensembles in the Plane of Isosteric
Ensembles? Let us look at the Plane in a new way by placing on it concrete
examples of chemical formulas instead of partition symbols (Fig. 16).
Figure 16.
Arrangement of concrete molecul