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New Applications of Computers in Chemistry.

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New Applications of Computers in Chemistry
By Ivar Ugi, Johannes Bauer, Josef Brandt, Josef Friedrich, Johann Gasteiger,
Clemens Jochum, and Wolfgang Schuhert[*]
The mathematical model of constitutional chemistry described here is based on a concept
of isomerism which has been extended from molecules to ensembles of molecules. A chemical
reaction is the conversion of an ensemble of molecules into an isomeric ensemble. An ensemble
of molecules is representable by an atomic vector and an associated bond and electron (BE)matrix, and a reaction by a reaction (R)-matrix. These BE- and R-matrices serve as a basis
for computer programs for the deductive solution of chemical problems. We present here
algorithms and computer programs based on the theory of BE- and R-matrices. They enable
the classification and documentation of structures, substructures and reactions, the prognosis
of reaction products, the design of syntheses, the construction of networks of mechanistic
and synthetic pathways and the prediction of chemical reactions.
1. Approaches to the Deductive Solution of Chemical
In chemistry the use of computers has been customary
for a long time. Nevertheless, only a modest part of the inherent
capabilities of modern computers is utilized for the solving
of chemical problems. Numerical problems are solved, such
as the ones encountered in quantum chemistry, and in the
collection and evaluation of experimental data, or large sets
of data are subjected to storage and retrieval operations.
The challenge to solve chemical problems by algorithms which
simulate human intelligence in the sense of decision processes
and deductive thought was felt at a rather early stage. It
led to studies in a direction which is now associated with
the term “artificial intelligence”.
This development began more than ten years ago with
computer programs for the elucidation of molecular structures
from measured physicochemical data“] (e.g. mass spectra),
and also with retrieval oriented computer assisted synthesis
design‘*].In the latter type synthesis design programs, stored
information on chemical reactions is used to generate the
precursors of a given synthetic target molecule. Here the structural features of the target molecule are perceived and analyzed
in order to determine which of the reactions in a libraly
of stored known reactions may lead to the target. For such
reactions the corresponding precursors of the target are
A theory of constitutional chemistry is needed for the development of synthesis design programs which are also able
to propose -without detailed knowledge of chemical reactions
-synthetic pathways containing unprecedented steps. This
theory should be the basis for finding the molecular systems
from which a given system may be formed by chemical reactions. or into which it may be converted.
Such a theory would not only serve as a suitable foundation
for synthesis design programs, but it would also enable the
solution of a great variety of other chemical problems.
Deductive computer programs which are based on mathematical representations and models of logical structures will
[*] Prof. Dr. 1. Ugi, Dip].-Inform. J. Bauer, Dr. I . Brandt, Dip].-Inform.
J . Briedrich. Dr. J. Gasteiger, Dr. C. Jochum, Dr. W. Schubert
Organisch-chemisches Institut der Technischen Universitat Miinchen
Lichtmhergstr. 4. D-8046 Garching (Germany)
( ~ l l < ~ l l lI. l l 1
Ell<//.I S .
111-123 (1’179)
play an essential role in future chemistry. With this approach,
one is not restricted to the retrieval and manipulation of
stored empirical data. Instead, the whole set of concrivuhle
solutions of a chemical problem can be taken into account
by deduction based on a suitable theory. There is no need
to refrain from making use of relevant empirical data; it
is possible and meaningful to condense these into the form
of general selection rules which are applied in the analysis
and evaluation of the intermediate and final results.
In this article a mathematical model of constitutional
chemistry is described, and its use as a theoretical basis of
deductive computer programs is demonstrated by some
examples. This mathenzuricul model of constitutionul chenzistry[31is based on the consequent use of the equivalency concept
in the representation and classification of molecular systems.
The basis for the additional treatment of stereochemical
aspects has been discussed beforeL4].
2. A Mathematical Model of Constitutional Chemistry
As an introduction to representing the logical structure
of constitutional chemistry by a mathematical model[31,we
observe that: An ensemble of molecules (EM)”. 61 consists
of molecules which may be chemically different. or indistinguishable. Just like a molecule, an EM has an empirical
formula. It is the union of the empirical formulas of the
molecules belonging to the ensemble.
2.1. The Extended Concept of Isomerism
The above statement leads directly to an extension of the
concept of isomerism to ensembles of molecules:
Ensembles of molecules ( E M ) are isomeric, if they have
the same empirical formula. The empirical formula describes
the collection A of atoms which the E M contains. All EM’S
which can be formed from A have the same ensemble empirical formula (A).
Accordingly, an E M ( A ) consists of one o r more than one
molecule that can be obtained from A by using each atom
belonging to A exactly once.
An F Z E M ( A ) ,the junii/j of the isomeric ensembles o/ molecules which contains the atoms of A, is the collection of all
E M ( A ) . An F I E M is simply determined by the empirical
formula (A) of the collection A of atoms.
A chemical reaction, or a sequence of chemical reactions,
is the conversion of an E M into an isomeric E M . An F I E M
contains all EM’s which are chemically interconvertible, as
far as stoichiometry is concerned. Thus the F I E M ( A )contains
the whole chemistry of the collection A of atoms. Since any
collection of atoms may be chosen here, a theory of the
F I E M is a theory of all of chemistry.
The left-hand sides and the right-hand sides of the “chemical
equations”(1) to (4)arepairs of isomeric EM’s whose empirical
formulas are given in brackets. The addition of eqs. (1) to (3)
results in eq. (4) for the overall reaction.
The EM’s which participate in the reactions (1) to (4) can
be imbedded into the FIEM (CH6Cl2N20) when the ensembles of the reacting molecules are supplemented by those
molecules that d o not participate directly in the reactions
[eq. ( 5 ~ .
HCN + Clz + NH3 + HzO + ClCN + HCI + NH3 + HZO +
The adjacency matrix J(HCN) (6) and the connectivity
matrix C(HCN) (7) of hydrogen cyanide may serve as an
The BE-matrix B of an E M ( A ) with a set of n arbitrarily
indexed atoms A = {Al,....An] is an n x n matrix whose i-th
row and column belong to the i-th atomic core A,. The off-diagonal entry bij of the i-th row and j-th column is the formal
bond order of the covalent bond from Ai to Aj. Since this
corresponds to a bond with the same formal order from
A j to Ai, we have bij=bji. Thus the BE-matrices must be
symmetric. The i-th diagonal entry bii indicates the number
of free valence electrons of the atom A;. For EM’S with closed
valence shells and spin paired valence electrons all diagonal
entries are even.
The BE-matrices B(HCN) and B(HNC) correspond to the
formulas of hydrogen cyanide and hydrogen isocyanide.
2.2. BE-Matrices
Seen chemically, atoms consist of a core (atomic nucleus
and the electrons of the inner orbitals) and the valence electrons
(electrons of the outer orbitals]. In molecules the cores are
held together by the valence electrons. The chemical constitution of a molecular system is described in terms of its coculentlj
bonded pairs of cores. The distribution of the “free” valence
electrons which do not belong to covalent bonds can also
be considered as part of the constitution. Customarily, the
chemical constitution is described by constitutional formulas. In these the atomic cores are represented by element
symbols, and covalent bonds by lines connecting the element
symbols. The placement of the free valence electrons is generally indicated by dots next to the element symbols.
In our model, the chemical constitution of an E M is represented by its atomic vector, which corresponds to the atoms
in A, and by the associated bond and electron (BE)-matrix,
whose rows and columns belong to the individual atomic
cores. The off-diagonal entries are the formal covalent bond
orders, and the diagonal entries are the numbers of the free
- ’I.
In the past, some other matrix representations of the chemical constitution have been introduced: The adjacency matrix
J, which originates in graph theory, indicates only, which
atoms are bonded. Spialter[141defined the connectivity matrix C, whose off-diagonal entries are the formal covalent
bond orders; the diagonal entries contain the atomic vector.
E . Meyer“ ’1 used topological matrices for documentation
purposes. Yonedd’ ‘ 1 uses a matrix representation for the treatment of heterogenous catalysis. The reactive centers of the
reactants and the catalysts are accounted for on the diagonal.
In B(HCN) the entries b 1 2= bZ1= 1 stand for the H-C bond,
the entries bz3= b32= 3 for the C-N triple bond, and b33= 2
for the pair of free electrons on the nitrogen atom.
As can be seen from the example, a BE-matrix is a representation of a resonance structure in the sense of the valence
bond theory.
The number of valence electrons si which belong to the
atom Ai is the sum over the entries of the i-th row or column
of a BE-matrix:
The formal electrical charge of the atom A, is the core charge
minus si. For the second row or column of B(HNC) we have
s2=0+2+3=5. It follows that the C-atom of HNC with
a core charge of + 4 has a formal electrical charge of - 1.
The cross sum
I,= 2si - bi,
over a diagonal entry b,, comprises the entries in the 1-th
row and column, i.e. all entries b,, and b,, whose indices
include an i. This is twice the row/column sum s,, minus
the diagonal entry b,,. The cross sum 3, IS the number of
Angew. Chem. Int. Ed. Engl. 18. 1 1 1 ~
123 (1979)
electrons in the valence orbitals of the atom Ai. For the
C-atom o f HNC we find
+R =E
Since B and E have the same sum S of entries:
S = Chi, = c e ,
i. j
i. j
i. 6'. this atom fulfils the octet rule.
The row of a BE-matrix describes the distribution of valence
electrons at the respective atomic core. For a given core,
only a limited number of such distributions (valence schemes)
is permissible. These may be collected in a list of acceptable
valence schemes['31.
The sum
C S=~ C b , ,
over all entries of a BE-matrix is the number of valence
electrons which the E M contains. This sum S has the same
value for all EM'S belonging to the same F I E M .
The BE-matrices of all relevant resonance structures, or
BE-matrices with fractional formal bond orders, may be used
for the description of the constitution of chemical compounds
which are not representable in terms of localized bonds, such
as systems capable of resonance, or systems with multicenter
Then atoms of an EM can be numbered in up to n ! different
ways, thus leading to up to n! distinguishable but equivalent
BE-matrices. By appropriate rules one of these numberings
can be declared canonical[' 'I. This numbering is important
when dealing with stereochemical aspects.
A chemical reaction is the conversion of an E M into an
isomeric E M by the redistribution of valence electrons. Here
the following invariances hold which are based on the conservation laws of matter and charge:
1. The atomic cores of an E M are preserved.
2. The total number of the valence electrons of an EM
is preserved.
From 2 it follows that the conversion of the starting
materials EM(B) into the final products EM(E) through a
chemical reaction
is representable by those transformations B-E of BE-matrices
in which the sum over all entries (S) [see eq. (12)] is maintained.
2.3.1. R-Matrices
We define the R-matrix (reaction matrix) R through the
transformation :
The addition of an R-matrix to a BE-matrix may be interpreted
as the action of an operator R,, on B:
Angew. Chem. lnt. Ed. Eiigl. 18, 111-123 ( 1 9 7 9 )
Chi, + C r t j
i, j
the sum of the entries rjj=ejj-bij of the matrix R = E - B
must be zero:
i. i
The matrix must be symmetric, because E = B + R is a BEmatrix, and must thus by definition be symmetric, i. e. from
b 'i.-- bji and elj=eji follows rii= rji.
The off-diagonal negative entries rzj=rji= - 1 reflect the
breaking of a covalent bond A,-A,; a negative diagonal
entry rii indicates the number of free valence electrons that
the atom A, loses by a chemical reaction. Correspondingly.
positive off-diagonal entries show how many bonds between
Aj and Aj are made, and a positive diagonal entry shows how
many free valence electrons the atom Ai gains during the
When R represents the reaction EM(B)+EM(E) according
to B + R = E , then its inverse R describes the vefroreurfion
EM (E)-t EM (B) according to :
(1 8)
The entries of R are thus given by:
tJ-J -- - r
2.3. R-Transformations
For example, the reaction
is represented by:
The inverse matrix R = - R = B(HCN) - B(HNC) corresponds
to the reaction HNC-HCN.
The R-matrices of reactions belonging to the chemistry
of valence shells with paired electrons have in the diagonal
zeros, or even numbers only. The R-matrices belonging to the
octet chemistry of molecules without formally charged atoms
have only zero entries in the diagonal. Such R-matrices exist
only from n 2 4 onwards. From this it follows, that one cannot
obtain from any collection of three atoms more than one
molecule or EM which belongs to the octet chemistry, and
that no product of a reaction of this E M can also belong
to octet ~hemistry[~J.
2.3.2. The Basis Elements of R-Matrices
An R-matrix of the type
2.3.3. Fitting Conditions for R-Matrices
with all entries=0, except u:{= + i and u$= -1, describes
an elementary redox reaction
by which an electron is transferred from Aj to A'. The elementary homoupsis according to
The transformation of a BE-matrix B by the addition of
an R-matrix R according to B + R = E corresponds to a chemical reaction only, if we have for all entries:
since, by definition, a BE-matrix cannot contain any negative
entries. Thus the negative entries of an R-matrix must be
chosen to satisfy
is represented by the R-matrix
in which v$ = v;{ = + 1 and v:i = vjj = - 1 are the only non-zero
The R-matrix -Vij describes the retroreaction, i.e. the elementary homolysis
An R-matrix with the entries rii and the i-th row sum
r, = Crij
is uniquely represented (up to sequence) as a sum of the
R-matrices U'" and V'j.
1 r,jVij + 2 r, U'"
The set of the U'"and the v" forms a basis for the R-matrices.
Since the R-matrices may be interpreted as vectors belonging
to IR 'Ii, the U'" and V'j are also called the busis oectors.
The vector basis of the R-matrices of reactions within the
chemistry of valence shells with paired electrons only (with
rii=0, k2,..,) consists of R-matrices of the type (U"+V")
and 2U'". The basis vector (U"+ V'J) represents reactions
of the types (28) and (29):
A, + : A +A?-A?
+ :A?+A,
i, j
selects those R-matrices which are acceptable with respect to
the valence chemical boundary conditions for E. The transformation properties of the BE-matrices, and the known
valence chemical properties of the chemical elements can thus
be used to predict all the conceivable reactions and their
products for a given E M . The action of all fitting R-matrices
on a single BE-matrix leads to the BE-matrices of the whole
family of isomeric ensembles of molecules ( F I E M ) .
On investigating the conceivable reactions of HCN through
the transformation properties of the BE-matrix B(HCN), one
finds, e. g., a fitting R-matrix with r I 2 = r z 1= - 1 and r33= -2.
Taking into account the structure of the R-matrices, as defined
above, and the valence chemical properties of the elements
H, C, and N, one finds that r l 3 = r j 1= + I and r Z 2 =+ 2
are suitable as the complementary positive entries. The result
is the R-matrix of eq. (20).
In general, there may be several BE-matrices having the
required mathematical and valence chemical properties to
fit a given R-matrix. Therefore an R-matrix represents a whole
category of chemical reactions which have in common the
"electron redistribution pattern" as given by the R-matrix and
some features of the participating bond systems.
The R-matrix from eq. (20) may serve to illustrate the decomposition of R-matrices into their basis vectors.
Further, the entries eij of the resulting matrix E must have
values which agree with the acceptable valence chemical
schemes of the respective chemical elements.
With a given BE-matrix B, one can thus use its positive
entries bij to determine the negative entries rij of the mnthematically fitting R-matrices. The positive entries are then
chosen to yield R-matrices with Crij=0. Among these, one
The reaction scheme (34) describes a Diels-Alder diene synthesis, while scheme (35) describes a Cope rearrangement.
( 3 5)
Accordingly, all resonance structures of the diatomic hydrogen
fluoride are already representable in three dimensional space
IR3 (Fig. 1).
The addition of an R-matrix of the form
2.4. Chemical Metric
BE-matrix B can also be represented as a vector
‘ I
to a BE-matrix corresponds to a Diels-Alder synthesis, or a
Cope rearrangement, depending on the chemical constitution
which this BE-matrix represents [cf. reactions (34) and (35)].
Any two R-matrices R and R’belong to the same R-category
and represent similar reactions if there exists an R-matrix
R” which can be transformed into R by row/column permutations, and which is convertible into R‘by removal, or addition
of rows and columns of zeros. Any two chemical reactions
belong to the same R-category if they are represented by
BE-matrix transformations of the same R-category.
In those reactions that are important for organic syntheses,
generally a redistribution of electrons takes place in which
up to six atoms participate and through which up to three
covalent bonds are broken or made. The formal charge
may increase by one unit at one of the atoms while decreasing by one unit at another atom, through participation
of free valence electrons. Such reactions correspond to Rmatrices with up to three pairs of negative and positive off-diagonal entries rij= rji = - 1 and rkl= rtk= + I , respectively. The
non-zero diagonal entries rii= + 2 are chosen to make all
row,kolumn sums of the R-matrix zero, except one pair of
rows/columns with r, = Crij= 1.
b with
Fig. I . The resonance structures of hydrogen fluoride lie on a plane which
intersects the coordinate axes at 8 ( = t h e number of valence electrons).
Similarly an R-matrix R corresponds to a vector r in IR”’.
A chemical reaction as given by B + R = E , is represented
by the vector r from P(B) to P(E). The sum
over the absolute values of the entries of R is twice the number
of valence electrons which are redistributed during the reaction, since these are taken into account by the negative, as well
as the positive entries of R.
We call D(B,E) the chemical r l i s t a n ~ 4 ~between
B and
E, and also between EM(B) and EM(E). By D(B,E) a metric
is defined in the space IR”’ which is equivalent to the euclidean
metric. The chemical distance is thus a genuine distance function for the BE-matrices of an F I E M . The origin of the coordinate system i n IR”’ corresponds to the JZ x n zero matrix, 0.
The sum over the entries S = I b i j is the number of valence
i. I
This results in an imbedding of the BE-matrices B of an
F I E M into IR“’,i.e. an n2-dimensional metric space over the
field of real numbers. The entries b,jof B can also be interpreted
as the Cartesian coordinates of a point P(B) in IR”’,or as
the components of a vector in IR”’. We call P(B) the BE-poinf
of the BE-matrix B.
Since the BE-matrices are symmetric, any BE-matrix B can
. a space
further be associated with a BE-point b in IR“‘”+”~2
with fewer dimensions, by writing the upper triangle of B
as a vector. In order to account for the constancy of the
overall number of valence electrons, the lower triangle must
be added to the upper one:
Angrw. Chon. I i i l . Ed. Enyl 18. 1 1 1-1 23
( IY79)
electrons, and has the same value for all BE-matrices of an
F l E M . Therefore an F I E M can be mapped into a lattice of
points with integral positive coordinates. on a surface with
a “radius”S=D(B,O). Note that the chemical distance D(B, E)
refers directly to the representation space IR“’.
3. Algorithms and Computer Programs Based on
BE- and R-Matrices
A BE-matrix contains the complete constitutional information about a molecular system, including the placement of the
free valence electrons. An R-matrix represents the “electron
redistribution pattern” underlying a chemical reaction.
Documentation qf structures and reactions: The BE-matrices
are most suitable for the documentation of molecular structures and substructures (see Section 3.1.1).
A hierarchic order and documentation of chemical reactions
is for the first time feasible, through the R-matrices which
describe the reactions directly, in contrast to the customary
methods by which the reactions are described in terms of the
reactants only (see Section 3.1.2).
The transformation of BE-matrices by the addition of Rmatrices leads to novel deductive methods for the solution
of chemical problems with computers. One needs a universal
model of the logical structure of chemistry, as described here,
for the deductive approach by which the solutions of the
individual problems are derived from general principles. Our
model affords the following general ways to treat the dynamic
aspect of chemistry:
One BE-mutrix is given: It is possible to start from a given
ensemble of molecules ( E M ) with the BE-matrix B, EM(B),
and to generate the R-matrices which fit B mathematically
within the valence chemical boundary conditions. This is one
basis for programming the design of syntheses and the prediction of products which may be obtained from given starting
materials (see Section 3.2).
?iw BE-matrices are gicenr With two given BE-matrices,
B and E, belonging to the same family of isomeric ensembles
of molecules (FJEM),the expansion of the R-matrix R = E - B
R = R I + R 2 + ” . R , yields information on the chemical pathways which lead from EM(B) to EM(E). Computer programs
on this basis generate networks of reaction pathways which
may consist of conceivable reaction mechanisms, or syntheses
(see Section 3.3).
A H R-mutrix is given: The determination of all pairs (B,E)
of BE-matrices which mathematically fit a given R-matrix,
under the valence chemical boundary conditions, enables us
to predict chemical reactions in a systematic manner (see
Section 3.4).
3.1. Classification and Documentation
3.1 . l . Structures and Substructures
In chemical documentation systems molecular structure is
only one of many keys of access to further information, such
as the contents of publications, and patents, as well as data
on the properties of compounds. It is typical for chemical
documentation systems that the key information is structured
in a characteristic manner.
On the one hand, the structure of a chemical compound is
subject to a variety of rules, in particular to valence chemical
rules. On the other hand, it always contains substructures
(functional groups, ring systems etc.), which may also serve
as keys of access in their own right. This holds for applications
such as
searching files for molecules with a given substructure (pure
correlation of substructures with molecular properties (subs truct ure/activity correlation)
analogy search based on incomplete structural information
(search for “similar” structures)
recognition of substructures which target molecules of
syntheses have in common with starting compounds
documentation and analysis of spectroscopic data.
A documentation system capable of doing this must have
the capacity for complete storage and retrieval of structural
information. It must, however, also perform special manipulations, such as fragmentation, valence chemical plausibility
checks, and afford access to substructures.
The representation of constitutional properties on the basis
of a mathematical model is a prerequisite for the performance
of such operations (see Section 2).
The treatment of chemical reactions is thus possible within
the framework of a unified theory. It is described in more
detail in the following sections. Firstly, the manipulation of
chemical data on the basis of BE-matrices, which are defined
as part of the mathematical model, is discussed.
Matrix representations of chemical constitution, e. g. the
adjacency and connectivity r n a t r i c e ~ “ I~’I,. have been used
in various documentation systems. Such matrices afford the
basic services of a documentation system, i.e. storage and
retrieval of data, but generally speaking they are not directly
usable for valence chemical checks and the manipulation of
data in the sense of chemical conversions.
The entries in any row of a BE-matrix must conform to
at least one of the valence chemically acceptable electron
distribution schemes of the respective atomic core. The permissible valence schemes for any given atomic core can be
listed, and they determine the range of chemistry which is
admitted for a particular purpose for the chemical element
under consideration. The valence schemes afford effective
plausibility checks on input data, or on recoding of foreign
data. They are particularly useful when older data banks that
represent a considerable investment are supplemented or
tested for conbistency. In such collections of data some o f the
structural information has been suppressed by the form of
coding used. The information that is lacking must sometimes
be reconstructed. Often the information on free electrons and
formal electrical charges is missing. A comparison with the
allowable valence schemes yields one or more proposals for
the required supplementary data. These are further processed according to additional information, or, in exceptional
cases, subjected to a manual selection procedure. In any case,
this balance of the valence electrons ensures that only acceptable and complete constitutional information enters the collection of data. For documentation systems such perfect representation by BE-matrices has the unique advantage that the
types of problems which may be of interest in the future
d o not have to be known when the system is implemented.
Hitherto, documentation systems usually have been so
designed as to cope with the solution of particular problems.
This approach frequently leads to some constitutional information being neglected. This is especially true for the widely
used fragment codes.
On the basis of such a complete representation of chemical
constitution, an interactive system for the filing and retrieval
of chemical structures and substructures has been implemented.
The methods for plausibility checks, of fragmentation,
and of substructure access have been realized in a FORTRAN
program study. A more advanced system is being implemented
Anyiw. Chenr. lnt. Ed. Engl. I S . 111-123 (19’9)
in PL/I, It offers higher performance in substructure retrieval
and in substructure/activity correlation.
In order to cope with the conflicting requirements of
semantic assistance (for example in synthesis design : recognition of functional groups and common substructures), minimal
memory space, and minimal access time, the retrieval of
substructures has been segmented into a time consuming fragmentation phase, and a fast retrieval phase.
By fragmentation (phase one) we mean the decomposition
of molecular structures into partial structures (fragments)
which are generated by breaking one bond, or lowering a
bond order. Each particle which is thus obtained is associated
with a reference to its immediate precursor Uuther reference),
and rice versa: The molecule is tagged with references to the
partial structures directly generated from it (son reference).
This process of direct fragmentation is recursively applied
to the partial structures, until single atoms are encountered as fragments. Thus one obtains a network in which
the molecules and their fragments are the nodes, and the
references are the oriented lines.
When the next molecule is fragmented, usually only rather
large new substructures are found, while the smaller substructures are recognized as already being documented. They do
not need further dissection.
Thus, the production of fragments leads to saturation of
the substructure file (Fig. 2).
3.1.2. Reactions
The systematics and nomenclature of chemical reactions
are far less elaborate than those of chemical structures.
It is symptomatic that the customary description of chemical
reactions is still largely based on trivial names.
Hitherto, chemical reactions have been represented by their
educts and products, wholly in the sense of a statically oriented
approach based on structures. Traditional documentation
systems for chemical reactions have been designed on this
In terms of the mathematical model of chemistry, this means
that two points in the vector space of an FIEM are used
to describe a reaction. A description by the vector between
these points was lacking.
In the description of a chemical reaction we begin with
the underlying scheme of electron redistribution. This is represented by an R-matrix. Knowing that a chemical reaction
can be represented by matrices of a certain kind (the Rmatrices). and that this rcpresentation is in itself structured,
affords the required generalization. The steps of the generalization are (for chemical reactions): irreducible R-matrices
(nucleus of the reaction; see below) 3 nucleus of the reaction
with associated atomic vector 3 BE-matrix of the nucleus
3 atomic vectors and BE-matrices of the spheres adjacent
to the nucleus.
A given electron redistribution pattern may describe many
different reactions which belong to the same R-category (see
Section 2.3.2). A collection of chemical reactions grouped in
this manner has been published recently (cf: Fig. 3)[”11.
Y1 F:
Fig. 2 Fragmentation of formaldehyde and hydrogen cyanide with son references.
A typical request for retrieval of a substructure is treated
in the retrieval phase (phase two) as follows: The fragment
to be retrieved (e.g. C-H) is brought into canonical form.
Its memory address is computed from its canonical representation. This enables one to enter the network of references
and follow the father references up to the molecule level.
Because of the design of the reference network from phase one,
the molecules arrived at contain the query fragment as a
partial structure.
In the retrieval phase of an analysis of substructure/activity
relations one is interested in the substructures common to
those molecules which have a certain activity (e.g. antioxidant,
fungicidal, bacteriostatic). Here, the program tracks the son
references which originate from molecules with the considered
activity, until, r i u many generations of sons, substructures
are found which occur, with statistical significance, more often
in the active molecules than in the other
E d . Engl. 18, 111 -123 ( 1 9 7 9 )
+ 1%-0-11
+ kkl
Angen Chem. I n r .
Fig. 3. Four reactions which belong to the same R-category. (a) differs from
(bj, (c), and id) by different entries in the atomic vector at the nuclcus
of the reaction: 0, C, H, N in (a), and CI, C , H, C in (b), (c), (d) (same “RAtype”); (h) differs from (c) and (dj by differences in the BE-matrix at the
nucleus of the reaction: bZ4=2 in (b), b24 = I in (cj and (d) (same “RE-type”):
(cj and (dj differ only with respect to the vicinity of the nucleus. The
R-matrix of this R-category is
It follows from the mathematical model that only a certain
number of irreducible R-matrices can exist within given limits.
Each one of these irreducible R-matrices defines an R-category, and any chemical reaction belongs to one of these R-categories. Some limited number of valence schemes may be associated with an R-category. Thcse determine atomic vectors that
fit an R-matrix. Here, only those atoms are taken into account
which belong to the nucleus of the reaction, i.e. the atoms
which d o not correspond to any one of the all-zero rows/columns of the R-matrix. The distinct atomic vectors partition
an R-category into reaction types which we call the RA-types.
These are, in turn, partitioned according to differences in
bonding between the atoms of the reacting nucleus, i.e. by
the differences in the entries in that block of the BE-matrix
which belongs to (the atomic vector of) the reacting nucleus.
Further specific classifications may be obtained by considering
some additional neighboring spheres of the reacting nucleus.
With this formalism, reactions may be coded in a general
form. This opens new possibilities in the retrieval of reactions
having unknown features.
3.2. Prediction of Reaction Products, and the Design of Syntheses
Designing syntheses usually involves analysis of the target
molecule of the synthesis for those structural moieties which
can be formed through known reactions, and one uses the
retroreactions of these for the construction of the synthetic
precursors. The realization of such an approach in a computer
program would require a data bank of individual chemical
reactions. Progressive improvement of such synthesis design
programs would necessitate continuous maintenance, updating and extension of the reaction library-and
yet a
program of this kind could only reproduce known reactions.
Our mathematical model affords an entirely different
approach in the design of syntheses[8,20-22J
. Th e application
of R-matrices to BE-matrices of given molecules and EM's
leads to the BE-matrices of the reaction products into which
they are convertible. The R-matrices may, however, not only
be used to describe the conversion of some reactants into
products, but they may also be interpreted as retroreactions
which lead from the products to the educts. Beginning with
the target molecule of a synthesis, one can thus infer the
precursors which lead to that target molecule.
has shown that here mostly small molecules form the coproducts. One encounters again and again some low-energy species, like H 2 0 ,HCI, NaCI, NH3, C 0 2 , N,. In a retrosynthetic
approach, molecules of this kind are combined with the target
into an E M . Thus those synthetic reactions are accounted for
in which these molecules are the coproducts.
Since the mathematical model affords the elaboration of
all conceivable reactions, the problem now is their meaningful
evaluation from the standpoint of the given problem. A substantial part of the evaluation can be accomplished by the
program itself. Physicochemical and heuristic criteria are used
for this.
Multiple bonds and bonds involving heteroatoms, as well
as their neighboring bonds are broken with preference. Parameters for 1,2- and 1,3-interactions are obtained from thermochemical data, and are used to estimate the enthalpy of the
Several models for estimating the activation parameters
of reactions are being tested. The effect of steric influences
of a reaction can be determined by looking at the bond
lists of the participating atoms. Certain unfavorable combinations of atoms may be recognized through electronegativity
Since syntheses generally involve many steps, the problem
of synthesis design cannot be solved by merely generatingfrom a target EM through R-matrices -the precursors of
the first level, and subsequently evaluating these. Instead,
the precursors of the first level are reintroduced into the
program and used to generate precursors of a further level.
Repetition of this procedure leads to a "synthesis tree''
whose root is the target, and whose nodes are reactions,
or molecules, respectively (see Fig. 4).
3.2.1. Operations with R-Matrices
The program EROS (Elaboration of Reactions for Organic
Syntheses)[2o1generates, by the action of R-matrices, from the
BE-matrix of an ensemble of molecules ( E M )the BE-matrices
of further EM's Depending on the direction in which the
reactionsareconsidered, forward or reverse, -this is a matter of
formulation-EROS can beused to predict the products which
are formed from given compounds, or it may serve in the
design of syntheses.
If an E M consists of one molecule only, the program generates the products of rearrangements and fragmentations.
Further molecules must be added if other types of reactions
are to be included; the choice of the supplementary reactants
depends on the problem at hand.
An investigation of the products which may be obtained
from a given molecule requires the addition of its potential
reaction partners: In a study of the products which may
be formed from a chemical (e.g. an agricultural chemical)
in the environment, these are HzO, 02,03,C 0 2 efc. When
searching for the potential uses of an industrial by-product
some readily available chemical can be added as the supplement.
In the design of syntheses, the molecules by which the
target molecule is to be augmented cannot be easily determined. A detailed study of the synthetically relevant reactions
Fig. 4. "synthesis trcc";
A reaction may involve more than one precursor. The development of a synthetic pathway is complete when all required
precursors can be identified as available starting materials.
In the evaluation of syntheses, it does not suffice to look
at the individual steps, i. e. the individual synthetic reactions.
Each branch of the synthesis tree must be considered as a
whole, because success in a synthesis depends on the quality
of all steps and the availability of all required compounds.
The program EROS has been used to study the formation
of products from given EM's, as well as the design of
The synthetic pathway A encompasses the last two steps of the
biosynthesis. Pathway B corresponds to the first two steps
of Traube's synthesis of uric acid; its further development
involving dicyanamide and glycine ester is novel. It has not
syntheses, using examples from the fields of organic synthesis,
industrial chemistry, and biochemistry. With such examples
the flexibility and prediction capability of EROS could be
Fig. 5. Selected synthetic pathways which were generated by the program EROS for guanine.
A study of the syntheses of guanine may serve as an illustration. Figure 5 contains a selection of the proposed syntheses.
been checked in the literature whether there is precedence
for pathways C and D. In order to influence the direction of
C2H4 [H@]
[C zH SO']
/ i)
Fig. 6. Synthetic pathways from ethyl acetate which were generated with the basis elements of R-matrices by program
system ASSOR. (a) (el. esterification of acetic acid with ethanol; (b): dimerizatlon of acetaldehyde ( T b c h r \ t h ~ n k o ) ;
(c): addition of water to acetylene to form acetaldehyde: (a)+(d) or (h)+(i): addition of ethanol to ketene (Wh(!,er);
(f)+(g): addition of water to ethylene to form ethanol.
the syntheses favorably, it may be necessary to activate some
of the reactive sites, and to protect others during certain
synthetic steps.
B l
+ H-O-H
3.2.2. Operations with Basis Elements of R-Matrices
Presently we are developing a program ASSOR’’’] which
does not use R-matrices as linear combinations of basis elements (see Section 2.3.1), but operates with the basis elements
themselves in the simulation of chemical reactions. These
basis elements form the simplest complete set of R-matrices
by which ull conceivable reactions can be represented. Since
a reaction is simulated by the basis elements in terms of
its elementary mechanistic steps, it is possible to decide through
plausibility checks after any one of these steps whether or
not the pathway should be pursued any further. This yields
a network of reactions which consists of mechanistic steps.
Such a network is free of mechanistically unrealizable
branches. Besides this selection, which is associated with the
generation of the pathways, further evaluation is still necessary
in order to avoid too many proposals.
The result of a trial run with ethyl acetate as the target
molecule and water as the coproduct is shown in Figure 6.
3.3. Networks of Reaction Mechanisms and Synthetic Pathways
Reaction mechanisms and synthetic pathways are equally
well represented by the mathematical model of constitutional
chemistry described above. In the investigation of a reaction
mechanism the equation
describes the overall reaction of the educts to the products.
In the treatment of syntheses, eq. (14) refers to the union of
all starting materials (B), and all products formed (E). Thus,
in one case R indicates the bonds to be broken and made by
a single reaction, whereas in the other case it represents all the
bonds to be broken and made by the reactions of a whole
pathway. Therefore, reaction mechanisms and synthetic pathways can mathematically be treated in rather a similar way.
Only when the overall reaction matrix R is expanded into the
matrices of individual steps, are a reaction mechanism or a
synthesis treated differently (see below).
? I
Fig 7. l o illustrate the postulate of ininimal redistribution of electrons.
redistribution ofa minimal number ofelectrons. This is equivalent to minimizing the chemical distance (see Section 2.4).
Mathematically, this means a permutation matrix P must
be found for which the function
F(P)= D(B, PEPT)
has a minimal value.
Ansatz: Even with the use of modern computers and
problems of moderate size performing all permutations is
too time consuming. In the case of educts and products
with 10 C-atoms more than three million permutations
would have to be carried out.
We are therefore presently studying the applicability of
some methods of integer optimization, namely quadratic
2i1. to this problem (see Fig. 8).
In order to avoid excessive computation time, these methods need to start with a permutation P for which F ( P )
is already close to the minimum. We have developed heuristic
methods for finding such an approximate solution P . These
are based on a comparison of the vicinities U(Ai,Aj) of the
atoms Ai in E M ( B ) with A, in EM(E). Here the attempt
D = 36 (a)
3.3.1. Shortest Reaction Pathways
We know by experience that chemical reactions tend to
proceed with the redistribution ofa minimal number of valence
electrons. We therefore postulate: When a reaction is given
in terms of its educts and products, the associated redistribution of electrons corresponds to an overall R-matrix which is
close to the minimal chemical distance (D)I2”.
If the atoms in the educts and products, E M (B) and E M ( E )
of a chemical reaction are indexed in an arbitrary manner, the
mutual assigment of the atoms by their respective indices may
falsely indicate a more complex reorganization of bonds and
electrons (b) than is actually taking place (a)(Fig. 7).
The aim is to isolate the actual redistribution of bonds and
electrons from such formal processes: in E M ( E )the indexing of
the atoms of the same chemical element must be permuted in
such a manner that the reaction is formally achieved with
D = 44
+- 21 / 0,
2 0,
D = 60
Fig. 8. The formation of proztagl.iiidli I lroiu 5.X.12.1l-eicoaatclraznic
acid and oxygen involving rsactionr which differ i n chemical dihtance:
(a) t h o a s the correlation of Ihc :iltiiii> ‘it minimal chemical distance,
(b) to ( d ) are examples for thc correlations near mininial chemical
is made to map the atoms of E M (B) onto the atoms of E M (E),
such that the vicinities "match" as well as possible.
Once the minimal chemical distance has been determined,
we obtain a minimal reaction matrix Ruin by the formula
the actual reactions proceed presumably close to a minimal
pathway, and on the other hand, synthetic pathways must
be rejected as wasteful if they are too remote from a minimal
RMinz= PEPT- B
3.4. Prediction of Chemical Reactions
A suitable expansion of RMi, into components R' with
RMin= C R '
leads to a description of the shortest reaction pathways.
3.3.2. Construction of Networks
Let EM(B) and EM(E) be some given initial and final
E M . By minimization of the chemical distance D(B,E) (see
above) we obtain a minimal reaction matrix RMi, which satisfies
the equation
In the preceding section we discussed the construction of
R-matrices which are acceptable when pairs of BE-matrices
are given. Now the opposite approach is described, namely
the generation of all reactions which "fit chemically" a given
electron redistribution scheme. This means the search for pairs
of BE-matrices which "fit" a given R-matrix, The creative
aspect of this approach lies in the fact that it leads to a
complete set which also contains the reactions without precedence.
To generate ull cornhinutnriully possible reactions is neither
chemically meaningful, nor feasible with present computer
technology. Instead, one must restrict the solution space to
some domain through suitable boundary conditions. Within
G i w n r e ~ i c r i .siliemc
:A + B- C
+ :C
In order to find the reaction mechanisms or synthetic pathways
by which EM(B)is convertible into EM(E), the overall reaction
matrix is expanded into sums SAof components:
This corresponds to an expansion of an overall reaction
into partial reactions. In the elucidation of reaction
mechanisms rather small partial reactions may be chosen
(e. 9. a basis vector, or a sum of a few basis vectors), because
as many as possible intermediates should be considered, the
unstable ones, too. When synthetic pathways are determined
the kR' are chosen to yield intermediates only. For both
types of problems those R-matrix components are also
admissible whose entries cancel in the overall matrix, i. e. along
some reaction pathway certain bonds are broken and remade,
and appear as unchanged by the overall reaction.
1 2
2 +
- ..
Fig. Y. Construction of a network of sqnthetic pathuays bet*een the educts
and product\ B and E.
A netuork ofintermediates kZ' is constructed from such components of R-matrices (Fig. 9). The components of the reaction
matrices must, however, be chosen to yield intermediates that
lie close to a shortest pathway within a network of rcaction
or synthctic pathways. This is so, because, on the one hand,
I I,lrl\
CYI O l t I
+ B -C
+ -1'
+ ti-CI
+ Cl'
+ 0-13''
H- 0 - - H + CI"
Fig. 10. Simplified example for the representation of reactions belonging
to the same R-category. The atomic vectors which are compatible with
a list of valence schemes (here appreciably abbreviated) describe thc whole
set of the conceivablc reactions. regardless wlicther or not thcq are raiilirable.
this domain exhaustive treatment of the problem is feasible,
and desirable. For the R-matrices this can be achieved through
suitable restrictions. The BE-matrices can then be generated
according to the valence schemes. Such valence schemes are
mappings of the individual rows in the BE-matrices. Suitable
BE-matrices are constructed from these row by row. The
addition of a corresponding row of an R-matrix yields a new
valence scheme. Each chemical element is assigned to one or
several valence schemes. Each row of the initial, and of the
final BE-matrix results in a pair of valence schemes. If such
a pair is contained in the list of valence schemes of some
chemical element, this element may occupy the site in the
atomic vector which corresponds to the row in question. As
a result one obtains for each site of the atomic vector a list of
all allowable element assignments (see Fig. 10). These
combinations of element assignments are further restricted
by additional valence chemical, physicochemical and heuristic
4. Perspectives
In chemistry, progress is achieved through the results of
experiments which have been initiated by heuristically valuable
hypotheses. The need for a universal theory with predictive
power has not yet been satisfied. As a theoretical foundation
of chemistry, quantum theory, too, has not attained the impact which had been expected because of the comprehensive
nature of its approach. Up to now, inductive inference and
reasoning by analogy have dominated in the interpretation
and planning of chemical experiments. When proceeding in
this way progress depends very much on chance. In order
to open up new possibilities in a systematic manner, it is
necessary to supplement the known approaches by deductive
procedures. For this the prerequisite is a theory which permits
one to take into account all of the conceivable solutions
for a wide variety of problems[281.
The theory of a family of isomeric ensembles of molecules,
as presented in this article, satisfies these requirements. In
the mathematical formulation of this theory BE-matrices are
used to represent the chemical constitution. The electron redistribution schemes of the chemical reactions are expressed
by R-matrices. A chemical reaction is then described by the
addition of an R-matrix to a given BE-matrix, according
to the master equation
of the model.
In this equation B stands for the ensemble of molecules at
the beginning of the reaction, and E for the ensemble on
termination of the reaction.
It is essential for the applications of the master equation
that it can be solved when one of the matrices is known.
This is possible, using our theory. In order to select from
the set of all conceivable solutions of the master equation
those which are likely to be relevant to the issue, without
discarding valuable information, one needs selection rules.
These rules must be based on chemical experience. Thus empirical data are used to restrict, and not to generate, the solutions
of the problernslz8].
For the deductive chemical computer programs chemical
literatureis not only a source of general selection rules; in the
form of suitably organized files on chemical compounds and
reactions, chemical literature can also be used to determine
whether a computer generated solution of a problem is already
known. For future developments this approach is particularly
promising, because it comprises access to the literature, and it
enjoys theadvantages of the mathematical model which enables
one to take into account all of the conceivable possibilities. As
a computer operation, matrix addition is much more effective
than the manipulation of data based on published reactions.
The programs which have hitherto been implemented, and
the results obtained with them, have demonstrated that computer programs for the deductive solution of chemical problems in combination with suitably organized documentation,
will relieve the research chemist in his routine information
and planning work, and will call to his attention possibilities
which he might otherwise have overlooked. This will never
replace the chemist; with the computer and its new
applications, however, he has a tool which will help him to
perform his tasks more effectively.
We acknowledge gratefully the financial support of our work
by Stifung Volkswagenwerk e. I! (development and testing of
mathematical models, in purticular in stereochemistry), the Deutsrhe Focschungsgemeinschaftft (synthesis design), the European
Communities (Contract Nos. 115-74-10-ENVD and 310-77-5E N V D , documentation and ecology) and the Bundesministerium fur Forschung und Technologie, project administrator:
Gesellschaftfur Information und Dokumentation (cla.ssi$cation
of reactions). We bsish to thank Pros. J . Dugundji, Los Anyeles,
and R. E. Burkurd, K d n , f o r helpful discussions und .stirnulnting
suggestions. The Institut fur Plasmaph~sik(Garching), and the
Leibniz-Rechenzentrum kindly procided us with computer time.
Received: December 27. 1977 [A 256 IE]
Supplemented: November 9, 1978
German version: Angew. Chem. 91, 99 (1979)
[ I ] J . Lederbery. G. L. Siithcrlurrrl. B. G. Burhairan, E . A. Frigenbaum,
A. I: Robertson, A . Al. Drrffirld. C . Djermsi. .I.Am. Chern. SOC.91,
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Activation Analysis-Its Present State of Development and Its Importance as an Analytical Tool
By Viliam KrivanL"]
Activation analysis is one of the most important methods available for the determination
of traces of elements and of isotopes. One special advantage of activation analysis is its
ability to cope with the simultaneous determination of many elements with very high detection
sensitivity-for many elements lying below the ppb level. In some cases as little as 10- l4-lO- l 5 g
of an element can be detected. A number of sources of systematic errors common to other
analytical procedures, mainly those caused by the blank, are eliminated in activation methods.
The disadvantages may be seen as the inconvenience of handling radioactive materials, the
dependence on large irradiation facilities, and sometimes the unavoidably long irradiation
and cooling periods. O n account of its unique capabilities activation analysis finds applications
in all fields of endeavor where minute amounts of elements are of significance, from materials
research through medicine to archaeology. Further improvements in .performance and increases
in scope are seen to lie in the development of new radiation sources, activation techniques,
measurement systems, and separation procedures.
1. Introduction
A knowledge of the occurrence and content of trace elements
in various materials is often absolutely necessary for the success
of the actual investigations in fields such as medicine, environmental studies, and life, earth, and materials sciences" -51.
In many branches of industry trace analysis of elements has
already become a routine procedure.
Activation analysis has always played an important role
in the determination of trace elements. Following its tempestuous development in the post-war years, the attraction and
significance of activation analysis lay primarily in the uniquely
high sensitivity which it offered for the detection of many
elements. The subsequent introduction of high-resolution
[*] Prof. Dr. V. Krivan
Sektion Analytik und Hochstreinigung der Universitat Ulm
Oberer Eselsberg N-26
D-7900 Ulm (Germany)
Ge(Li) detectors in the sixties opened the way for instrumental
multi-element analysis.
In the sixties and seventies, some powerful methods based
on atomic spectroscopy and other techniques were developed
in addition to activation analysis, and sometimes with comparable sensitivities. It is only since analysts have had access
to several high-performance techniques, that they have been
able to assess the reliability and accuracy of the individual
methods. Comparisons of analytical results obtained by various techniques and in particular of interlaboratory results
obtained from collaborative investigations-so-called roundrobins--have revealed the real limitations of all these
methods['-']: the lower the levels of the elements being determined, the greater the deviations of the results obtained by
individual laboratories from the mean experimental value, or
from the "true" value. At the ppb level and below, deviations
of several orders of magnitude are by no means uncommon.
The main problem in element trace analysis is undoubtedly
that of the accuracy of the results, i. e. the problem of systematic
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