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 Problems 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 generated. 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) 411~/1~ll. ( ~ l l < ~ l l lI. l l 1 Ed 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 111 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 + HzNCN + HCI + HC1+ HzO + HZNCONHz+ HCI + HCI (5) The adjacency matrix J(HCN) (6) and the connectivity matrix C(HCN) (7) of hydrogen cyanide may serve as an illustration. H-C=N: J(HCN) = C(HCN) = [#) (?#J (7) 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 valence - ’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. ’ 112 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,,(B) =B +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 S= C S=~ C b , , i iJ 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 bonds. 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 : B+R=E (14) 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 I,J the sum of the entries rjj=ejj-bij of the matrix R = E - B must be zero: CrEj=O 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 reaction. 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 : E+R=E-R=B (1 8) The entries of R are thus given by: tJ-J -- - r 2.3. R-Transformations = 'I For example, the reaction HCN + HNC is represented by: B(HCN) + R + B(HNC) 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. 113 2.3.2. The Basis Elements of R-Matrices (30) An R-matrix of the type i u" 1 = 2.3.3. Fitting Conditions for R-Matrices with all entries=0, except u:{= + i and u$= -1, describes an elementary redox reaction by which an electron is transferred from Aj to A'. The elementary homoupsis according to 'Ai+'AJ + (23) A,-Aj The transformation of a BE-matrix B by the addition of an R-matrix R according to B + R = E corresponds to a chemical reaction only, if we have for all entries: since, by definition, a BE-matrix cannot contain any negative entries. Thus the negative entries of an R-matrix must be chosen to satisfy is represented by the R-matrix i~ . .. in which v$ = v;{ = + 1 and v:i = vjj = - 1 are the only non-zero entries. 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 J is uniquely represented (up to sequence) as a sum of the R-matrices U'" and V'j. R= 1 r,jVij + 2 r, U'" i<J i<n 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?+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. (34) -A, The R-matrix from eq. (20) may serve to illustrate the decomposition of R-matrices into their basis vectors. 114 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 j k I . -1 . . +1 . . +1 . . -1 . +1 . . . . +1 . -1 i -1 R = . . tl 111 II , +I . . . , -1 . . -1 i 2.4. Chemical Metric An BE-matrix B can also be represented as a vector components JI x JZ ti2 ‘ 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 I 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. 115 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 into its components R1,R2,...,R, according to 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 retrieval) correlation of substructures with molecular properties (subs truct ure/activity correlation) analogy search based on incomplete structural information (search for “similar” structures) 116 - ~ 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). N S C-H 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 basis. 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. Hh I O=C-H N=C-H 1\ 0-C-H O=C N-C n i N-C-H $ CH,-$H-y-CH, 61 H I I CH3-$=Y-C,H, Y1 F: H-$H-$H-C,H, 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 ) -----t 2 4 2 4 2 4 CH3-CH=N-CH3 CH3-C-C-C,H, I + 1%-0-11 + 1 ? , (a) HC1 (b) + kkl (d) N=C C-H Angen Chem. I n r . - H-CH=CH-C3H, 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 117 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 reactions[23! 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 considerations[24! 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 218 Fig. 4. "synthesis trcc"; 3 inolecules: 7 reactions. 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 demonstrated. 52J. RO-C-CHZ-NHZ + B RO-C, HZN-C-NHZ NII H "'p\ ,C-H ,,N + >c-H H&-N HzN-C-NH-COZR HOxC-y .1 1 H,N-C-NH-CN .1 ROZC-CH-NH-CH I + AH I C H 1 I IH KC,N\ 6 II RO"*O HzN-C-NH-C=N AH H' NH -z ,C-H N ' I H 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 :0: 11 H,C-C-~-CH~-CH, 1 +Ha H,C-COOH J A C2H4 [H@] A [C zH SO'] H\ .. H,C=C=O / i) l+H" [H3C-E<H] 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. H I H-C2-II B l ~-73-q-ci + H-O-H D(a)=8 H I H-C4-H 3.2.2. Operations with Basis Elements of R-Matrices I 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 B+R=E (14) 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). H I I H H-C’-H ? I H-C3-C2-C1 I I + H H-0-H D(b)=28 H I H-q4-H il 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) (39) 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 assignment 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) SI, 120 H-C1 I 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 + H-C3-&-OH ,6 17 IY OH D = 44 +- 21 / 0, 2 0, (b) \ 0 10 I2 OH D = 60 (d) 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 m.in) correlations near mininial chemical distance. 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 pathway. RMinz= PEPT- B 3.4. Prediction of Chemical Reactions (40) 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 i~~ :A + B- C + + :C A-B 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 + -0- -. . ' a - 0 /-O - .. .. 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\ A CYI O l t I + B -C erulilyl<,\ + + -1' 11-0" -4 CI" + H 0" A-B 1 -0-H + ti-CI +C CI + H-0-1.- + CI-~C' I + I + Cl' + 0-13'' H- 0 - - H + CI" etc. 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. 121 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 rules. 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 B+R=E (14) 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]. 122 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, 2973 ( I 969). [2] a) 1. Cyi. G. KauJinld (unpublished 1966-67) described in [5] and I . 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C y i in W T WipLe, S.Hel/tv,R. Fddmariii, E . Hyde: Computer Representationand Manipulation of Chemical Inlormation. Wiley, New York 1974. p. 129. IBM-Nachr. 24, 180 rlV74). J . C~i.\reiyt~v, J . Branrlr, J . Brunnert, W Schuherf, IBM-Nachr. 24. 1x5 (1974). J . Branrlr, J . Fririlrich, J . Gasteiyer. C. Jochum. W Schuherf, 1. Uyi in E. !/ LudeEu, N . H . Subelli, A . C. Wuhl: Computers in Chemical Education and Research. Plenum, New York 1977. J. Brandf, J . Frirdrrch. .I. C o r e r y e r . C .Jothum, W Schuhrrt. 1. C y i : Computer-Assisted Organic Synthesis. ACS Syinposium Ser. 61. Aiigew. C l i e m 111r.Ed. Enyl. 18. / 1 / - - / 2 3( 1 9 7 9 ) [ I 3 ) J . Brand/, J . Frredrich, J . Custetger, C . Jocltum, W Schuberf, P . Lemmen, 1. Ugi, Pure Appl. Chern 50, I303 (1978). [I41 L. Spialter, J. Chem. Doc. 4. 261, 269(1964). [I51 E . Meyrr, Angew. Chcm. 82, 605 (1970): Angew. Chem. Int. Ed. Engl. 9. 583 (1970). [I 61 a ) Y Yoneda: Saikin no Kagakukogaku (Progress of Chemical Engineerb) E Yoneda: Kemoguramu 1 -Keiing), Maruzen, Tokio 1 9 7 0 , ~145ff.; . sanki Kogyo Kagaku (Chemogram 1 -Computer Industrial Chemistry). Band 1. Maruzen, Tokio 1972, p. 3098. 1171 a ) W Schuhert, I , Ugi, J. Am. Chem. SOC.100, 37 (1978); b ) C. lorhum, J . Gasteiger, J. Chem. Inf. Comput. Sci. 17, 1 13 ( 1 977). 1181 J . friedrich, 1. Ugi, Informal Commun. Math. Chem., in press. 1191 J . C. J . Bart. E . Garugnuni, Z . Naturforsch. B31, 1646 (1976); B32, 455, 465, 678 (1 977). [20] a) J . Gusteiger, C . Jochum, Top. Curr. Chem. 74, 93 (1978); b) J . Gasteigrr, C . Jochum, M . Marsili, J . Thorna, Informal Commun. Math. Chem., in press. [21] a) W Schubert, I. Ugi, Chimia, in press; b) W Schubert, Informal Commun. Math. Chem., in press. [22] A . W i s e , 2. Chem. 15, 33 (1975). [23] J . Gasreiger, Comput. Chem. 2,85 (1978). [24] J . Gasteiger, iM.Mlirsili, Tetrahedron Lett. 1978, 3184. [25] C . Jochum, J . Gnstcqer, 1. U g i , unpublished. [26] a) K . H. Strarmann, Diplomarbeit, Universitat Koln 1976; b) R. E. Burkard, K . H . Strutmann, Naval Res. Logistics Quart., in press. 1271 R. E. Burkard' Methoden der Ganzzahligen Optimierung. Springer, Wien 1972. [28] I. Ugi, Giessener Universitatsbl. 11, 68 (1978). 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) .4Itp'li. 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 123 Cllcw. 1 1 1 1 . Ed. Enql. 1 8 . 123-147 ( 1 9 7 9 ) 0 I.i.dag Ckrmie. GmhH. 6Y40 W e i ~ ~ l t ~ ~I Y79 uir, 0570-0833, 7910202-0123 S 02.5010

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