Computer Programs for the Deductive Solution of Chemical Problems

2 Computer Programs for the Deductive Solution of Chemical Problems on the Basis of a Mathematical

Downloaded by UNIV OF ARIZONA on January 17, 2013 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0061.ch002

Model of Chemistry JOSEF BRANDT, JOSEF FRIEDRICH, JOHANN GASTEIGER, CLEMENS JOCHUM, WOLFGANG SCHUBERT, and IVAR UGI Organisch-Chemisches Institut, Technische Universität München, Arcisstr. 21, D-8 München-2, Germany

1.

Introduction

1.1

Computers and P r o g r e s s in Chemistry

In t h e p a s t decades t h e r e has been d r a m a t i c p r o g r e s s in c h e m i s t r y . The c o n c e p t u a l and theoretical approaches t o c h e m i s t r y have been decisively shaped by quantum mechanics. The r e s e a r c h t o p i c s and t e c h n i q u e s o f c h e m i s t r y have been changed in a p r o f o u n d manner by the advent o f modern s e p a r a t i o n and purification methods as w e l l as advances in structural d e t e r m i n a t i o n by t h e v a r i o u s types o f s p e c t r o s c o p y and X-ray crystallography. The use o f computers in c h e m i s t r y has a l s o p l a y e d an essential r o l e in this c o n t e x t . However, it is s a f e t o p r e d i c t t h a t t h e major c o n t r i b u t i o n s o f computers t o c h e m i s t r y a r e still t o be expected i n the future. Computers a r e a l r e a d y an important t o o l o f c h e m i s t r y . C o m p u t e r - a s s i s t e d documentation, the collection and e v a l u a t i o n o f e x p e r i m e n t a l d a t a , and quantum m e c h a n i c a l c a l c u l a t i o n s o f m o l e c u l a r p r o p e r t i e s predominate. The importance o f t h e solution o f c h e m i c a l problems such as the d e s i g n o f syntheses w i t h t h e aid o f computers is not y e t w i d e l y r e c o g n i z e d , but will have f a r - r e a c h i n g consequences and may w e l l l e a d t o r a t h e r fundamental changes in the activity o f c h e m i s t s . In

33 In Computer-Assisted Organic Synthesis; Wipke, W., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

34

COMPUTER-ASSISTED

ORGANIC SYNTHESIS

p a r t i c u l a r , the computer programs f o r the d e d u c t i v e s o l u t i o n o f c h e m i c a l p r o b l e m s on t h e b a s i s o f l o g i c a l s t r u c t u r e models and m a t h e m a t i c a l r e p r e s e n t a t i o n o f c h e m i s t r y have g r e a t p o t e n t i a l f o r the f u t u r e .


1.2

The S o l u t i o n o f C h e m i c a l P r o b l e m s on t h e o f T h e o r e t i c a l P h y s i c s and M a t h e m a t i c s

Basis

The m o l e c u l a r s y s t e m s , t h e s u b j e c t o f c h e m i s t r y , consist o f a r a t h e r s m a l l number o f b u i l d i n g b l o c k s , n a m e l y t h e c a . 100 c h e m i c a l e l e m e n t s w h i c h a r e c o m b i n e d a c c o r d i n g to r u l e s d e r i v e d from m a t h e m a t i c a l l y s t a t e d p h y s i c a l p r i n c i p l e s . The r e p r e s e n t a t i o n o f chemistry i n m a t h e m a t i c a l t e r m s seems t h e r e f o r e q u i t e n a t u r a l . T h i s has not yet been u t i l i z e d to the f u l l c o n c e i v a b l e extent. The e n e r g y h y p e r s u r f a c e d e s c r i b e s t h e e n e r g y o f c h e m i c a l s y s t e m s w i t h a g i v e n s e t o f atoms as a f u n c t i o n o f t h e s p a t i a l d i s t r i b u t i o n o f t h e a t o m i c n u c l e i . I f one c o u l d compute t h e c o m p l e t e e n e r g y h y p e r s u r f a c e f o r any s e t o f atoms and a n a l y s e t h e c o r r e s p o n d i n g d a t a i n a s u i t a b l e m a n n e r , one c o u l d p r e d i c t m o s t o f t h e relevant p r o p e r t i e s o f m o l e c u l a r systems w i t h the methods o f t h e o r e t i c a l p h y s i c s and m a t h e m a t i c s . T h i s , however, i s not f e a s i b l e i n f u t u r e , because the computational extremely l a r g e , even i n the case molecular systems.

the f o r e s e e a b l e e f f o r t w o u l d be of rather small

F o r t h e s o l u t i o n o f many c h e m i c a l p r o b l e m s t h e t h e o r e t i c a l t r e a t m e n t o f a few s e l e c t e d p o i n t s a n d t h e i r vicinity on t h e e n e r g y h y p e r s u r f a c e w o u l d s u f f i c e . Y e t , i n many c a s e s , o n e d o e s n o t know w h i c h p o i n t s a r e t h e r e l e v a n t o n e s . F o r c h e m i s t r y i t w o u l d be r a t h e r u s e f u l t o have a method f o r p r o v i d i n g a s u r v e y o f t h o s e p o i n t s and pathways on an e n e r g y h y p e r s u r f a c e which are e s s e n t i a l f o r the s o l u t i o n of a g i v e n problem, w i t h o u t t h e n e c e s s i t y o f a quantum m e c h a n i c a l t r e a t ment o f w i d e a r e a s o f a n e n e r g y h y p e r s u r f a c e . In t h e p r e s e n t p a p e r a s i m p l e m a t h e m a t i c a l model o f t h e c h e m i s t r y o f a g i v e n s e t o f atoms i s p r e s e n t e d w h i c h a f f o r d s p r e c i s e l y t h e l a t t e r , a n d may s e r v e a s a b a s i s o f computer programs f o r the d e d u c t i v e solution of a great v a r i e t y of chemical problems.

In Computer-Assisted Organic Synthesis; Wipke, W., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.


2.

BRANDT

E T

AL.

Computer

Programs

2·

The C h e m i s t r y

2.1

Chemical Equivalence

for Chemical

of a Fixed

Set o f

Problems

35

Atoms

Classes

Any p r o g r e s s i n c h e m i s t r y may b e i n t e r p r e t e d a s t h e r e c o g n i t i o n o f new e q u i v a l e n c e r e l a t i o n s a n d c l a s s e s . U n t i l r e c e n t l y , a u n i v e r s a l model o f c h e m i s t r y could n o t b e d e f i n e d , b e c a u s e we l a c k e d t h e c o n c e p t s f o r a mathematical treatment of the c o n s t i t u t i o n a l aspect of c h e m i s t r y . T h e r e f o r e i t i s n e c e s s a r y t o c o n s i d e r any c o n s t i t u t i o n a l l y d i f f e r e n t m o l e c u l a r systems as e q u i v a l e n t r e g a r d l e s s o f t h e number o f m o l e c u l e s w h i c h they c o n t a i n , i f , i n p r i n c i p l e they are i n t e r c o n v e r t i b l e through chemical r e a c t i o n s .

2.2

Isomeric

Ensembles

of Molecules

The s e t o f a l l m o l e c u l e s c a n be p a r t i t i o n e d into e q u i v a l e n c e c l a s s e s w h o s e members h a v e a l l t h e same molecular formula, i . e . the equivalence classes of isomers. Isomerism i s here the r e l e v a n t equivalence relation. Extending the concept o f isomerism from molecules t o e n s e m b l e s o f m o l e c u l e s (EM) l e a d s t o a m a t h e m a t i c a l m o d e l o f c o n s t i t u t i o n a l c h e m i s t r y ^ . A n EM i s d e s c r i b e d by i t s l i s t o f t h e m o l e c u l a r s p e c i e s i n t h e f o r m o f s u i t a b l e c h e m i c a l f o r m u l a s . F o r a n EM o n e c a n d e f i n e two t y p e s o f e m p i r i c a l e l e m e n t a r y f o r m u l a s , o n o n e hand t h e e n s e m b l e f o r m u l a , on t h e o t h e r hand a p a r t i t i o n e d e m p i r i c a l ensemble formula which c o n s i s t s o f the m o l e c u l a r formulas o f t h e m o l e c u l e s i n t h e EM. 1

L e t A b e a s e t o f a t o m s . T h e n a l l E M ( A ) , i . e . a l l EM w h i c h c a n b e made f r o m A , h a v e t h e same e n s e m b l e f o r m u l a . T h e p a r t i t i o n e d e m p i r i c a l f o r m u l a o f a n EM i s a p a r t i t i o n i n g { < A > , . . . < A >} o f A i n w h i c h e a c h i s t h e f o r m u l a o f a m o l e c u l e . Accordingly, a n E M ( A ; c o n s i s t s o f o n e o r more m o l e c u l e s , w h i c h a r e o b t a i n e d f r o m A , i f e a c h atom o f A i s u s e d e x a c t l y o n c e . t

S i n c e i s o m e r i c m o l e c u l e s may d i f f e r constitutionally and s t e r e o c h e m i c a l ^ , a p a r t i t i o n e d e n s e m b l e f o r m u l a g e n e r a l l y c o r r e s p o n d s t o more t h a n o n e E M ( A ) . T h e c o n s t i t u t i o n a l f o r m u l a o f a n EM(A) c o n t a i n s t h e c o n s t i t u t i o n a l formulas of a l l molecules i n that EM(A). An F I E M ( A ) , t h e f a m i l y o f i s o m e r i c e n s e m b l e s o f m o l e c u l e s o f an atom s e t A i s t h e c o l l e c t i o n o f a l l


COMPUTER-ASSISTED

36

E M ( A ) . An F I E M i s underlying set of

g i v e n by t h e atoms A .

empirical

ORGANIC SYNTHESIS

formula

of

the


A c h e m i c a l r e a c t i o n or a sequence of c h e m i c a l r e a c t i o n s , i s t h e t r a n s f o r m a t i o n o f a n FM i n t o a n i s o m e r i c E M . T h u s t h e w h o l e c h e m i s t r y o f a s e t o f atoms A i s g i v e n by t h e EM(A) and t h e i r i n t e r c o n v e r s i o n s w i t h i n t h e FIEM(A). An e n e r g y h y p e r s u r f a c e d e s c r i b e s a l l c h e m i c a l s y s t e m s w h i c h c o n t a i n a g i v e n s e t o f a t o m s . An F I E M ( A ) o f s t a b l e EM(A) c o r r e s p o n d s t o a f a m i l y o f e n e r g y m i n i m a on t h e e n e r g y h y p e r s u r f a c e o f t h e a t o m s e t A . S e q u e n c e s o f EM(A) w h i c h a r e c h e m i c a l l y i n t e r c o n v e r t e d correspond t o pathways on t h e e n e r g y h y p e r s u r f a c e o f A .

3.

BE-Matrices

In m o l e c u l a r systems t h e a t o m i c c o r e s a r e h e l d t o g e t h e r by v a l e n c e e l e c t r o n s w h i c h o c c u p y m o l e c u l a r o r b i t a l s i n v o l v i n g t w o , o r more a t o m s . A c o v a l e n t bond i s a p a i r o f v a l e n c e e l e c t r o n s i n a m o l e c u l a r o r b i t a l a b o u t two c o r e s . G e n e r a l l y , the chemical c o n s t i t u t i o n of a m o l e c u l a r system i s d e s c r i b e d by i t s c o v a l e n t l y b o u n d p a i r s o f a t o m i c c o r e s . The d i s t r i b u t i o n o f t h o s e v a l e n c e e l e c t r o n s w h i c h a r e n o t c o n t a i n e d i n c o v a l e n t c h e m i c a l b o n d s c a n be i n c l u d e d i n the d e s c r i p t i o n of a chemical c o n s t i t u t i o n . A c h e m i c a l c o n s t i t u t i o n i s u s u a l l y r e p r e s e n t e d by a c o n s t i t u t i o n a l f o r m u l a i n which the c h e m i c a l element symbols c o r r e s p o n d to the r e s p e c t i v e atomic c o r e s , the c o v a l e n t b o n d s a r e i n d i c a t e d by l i n e s b e t w e e n t h e e l e m e n t s y m b o l s , a n d t h e f r e e v a l e n c e e l e c t r o n s by d o t s at the atomic symbols. The c h e m i c a l c o n s t i t u t i o n o f m o l e c u l e s h a s a l s o b e e n r e p r e s e n t e d by v a r i o u s t y p e s o f m a t r i c e s . F o r t h e present purpose the BE-matrices are p a r t i c u l a r l y suitable. An η χ η

BE-matrix

Β =

\

(13)


2.

BRANDT

ET AL.

Computer

Programs

for Chemical

Problems

37

r e f e r s t o a m o l e c u l e o r EM w h i c h c o n t a i n s a n a t o m s e t A = { A ^ , . . . , A } w i t h η i n d e x e d a t o m s . T h e i - t h row a n d i - t h column of A.

of a BE-Matrix

An o f f - d i a g o n a l

entry

b..

belongs

to the i - t h

atom

i n t h e i - t h row a n d j - t h

-J-J

column

is

the formal

covalent

bond

order

between

the

atoms

A. and Α . . S i n c e t h i s i m p l i e s a l s o a bond from ι J A. to A . we h a v e b . . = b . . . T h u s B E - m a t r i c e s a r e J ι ^-J J ι symmetric.


3

T h e i - t h d i a g o n a l e n t r y o f a B E - m a t r i x i s t h e number f r e e valence e l e c t r o n s which belong to A^. T h e number a t o m A^ i s

of valence e l e c t r o n s which belong g i v e n b y t h e r o w - c o l u m n sum

to the

T h e c r o s s sum b . o v e r a d i a g o n a l e n t r y c o m p r i z e s e n t r i e s i n t h e i - t h row a n d c o l u m n s a n d i s e q u a l b.

=

2

b

i

- b

i

±

of

all to

.

T h e c r o s s sum b . i s t h e number o f v a l e n c e which occupy the v a l e n c e o r b i t a l s o f A^.

electrons

A t a b l e which c o n t a i n s f o r a l l ^ c h e m i c a l elements the a l l o w a b l e c o m b i n a t i o n s o f b^, b^, c o o r d i n a t i o n numbers and b o n d o r d e r s a f f o r d s a q u i c k c h e c k w h e t h e r a B E matrix represents a valence chemically stable molecular system. The

sum

S

=

£

b

i j

o v e r a l l e n t r i e s o f a B E - m a t r i x i s t h e t o t a l number v a l e n c e e l e c t r o n s i n t h e E M . F o r a l l EM o f a n F I E M t h i s number i s t h e s a m e .


of

COMPUTER-ASSISTED

38

Unless the some r u l e , represented equivalent the atomic indices.

ORGANIC SYNTHESIS

i n d e x i n g o f t h e atoms i s f i x e d a c c o r d i n g t o a g i v e n chemical c o n s t i t u t i o n i s not only by one B E - m a t r i x , b u t t h e r e a r e up t o η ! B E - m a t r i c e s w h i c h d i f f e r by p e r m u t a t i o n s of i n d i c e s and t h e r e s p e c t i v e row/column

Any B E - m a t r i c e s Β a n d B w h i c h d i f f e r o n l y b y row/ c o l u m n p e r m u t a t i o n s d e s c r i b e t h e same E M . T h e t r a n s f o r m a t i o n o f a n η χ η B E - m a t r i x Β by row/column p e r m u t a t i o n i n t o an e q u i v a l e n t B E - m a t r i x B c a n be a c h i e v e d by a n η χ η p e r m u t a t i o n m a t r i x Ρ a n d i t s i n v e r s e , P^, a c c o r d i n g t o f


f

B

t

_

p

t

.

. ρ

B

F o r v a r i o u s p u r p o s e s , s u c h a s d o c u m e n t a t i o n , one n e e d s an u n a m b i g u o u s c o r r e s p o n d e n c e b e t w e e n a n EM a n d i t s B E - m a t r i x . T h e r e f o r e , we h a v e d e v e l o p e d p r o c e d u r e s for t h e a s s i g n m e n t o f a t o m i c i n d i c e s w h i c h l e a d t o an canonical BE-matrix. Among t h e e q u i v a l e n t of several molecules

B E - m a t r i x o f a n EM w h i c h c o n s i s t s i>**«> > there exist BE-matrices

M

M

m

i n b l o c k f o r m , s u c h t h a t e a c h b l o c k r e p r e s e n t s one c o n t i g u o u s m o l e c u l e . The b l o c k f o r m o f B E - m a t r i c e s u s e f u l f o r t h e s e p a r a t i o n o f m o l e c u l e s o f EM w h o s e B E - m a t r i c e s have b e e n g e n e r a t e d by a c o m p u t e r .

4.

The R e p r e s e n t a t i o n R-Matrices

of

Electron

Relocation

is

by

A c h e m i c a l r e a c t i o n i s t h e c o n v e r s i o n o f a n EM i n t o i s o m e r i c EM by r e l o c a t i o n o f v a l e n c e e l e c t r o n s .

an

S i n c e t h e t o t a l number o f v a l e n c e e l e c t r o n s S d o e s n o t c h a n g e i n a c h e m i c a l r e a c t i o n , i t must be t h e same f o r t h e i n i t i a l EM(B) and t h e f i n a l EM(E) o f a c h e m i c a l r e a c t i o n . I t f o l l o w s t h a t t h e sum o f e n t r i e s o f B E m a t r i c e s must b e i n v a r i a n t u n d e r t h o s e B E - m a t r i x t r a n s f o r m a t i o n s Β -> Ε w h i c h r e p r e s e n t c h e m i c a l r e a c t i o n s EM(B) + EM(E). L e t Β and Ε be t h e B E - m a t r i x o f t h e s t a r t i n g EM(B) and t h e f i n a l E M ( E ) o f a c h e m i c a l r e a c t i o n . T h e n a n Rm a t r i x ( r e a c t i o n m a t r i x ) i s d e f i n e d by t h e transfor-


2.

Computer

BRANDT E T A L .

The

sum o f

the

entries

Programs

for Chemical

r..

an R - m a t r i x

of

Problems

39

is

because


s

Σ

e..

=

Σ .. b

r

Σ b..

T h e m a t r i x R = Ε - B must b e s y m m e t r i c b e c a u s e m a t r i c e s Β a n d Ε a r e s y m m e t r i c by d e f i n i t i o n .

the

BE-

T h e m a t r i x R = -R i s t h e i n v e r s e o f R. T h e t r a n s f o r m a t i o n Ε + R = Β r e p r e s e n t s t h e r e a c t i o n E M ( Ε ) -> E M ( Β ) , i . e . t h e r e t r o - r e a c t i o n o f EM(B) EM(E) . An R - t r a n s f o r m a t i o n Β + R = Ε r e p r e s e n t s a c h e m i c a l r e a c t i o n i f , and o n l y i f i t obeys t h e m a t h e m a t i c a l f i t t i n g condition e.. = b.. + r.. > 0 IJ IJ i J f o r a l l e n t r i e s b e c a u s e , by d e f i n i t i o n , a B E - m a t r i x rrust have n o n - n e g a t i v e e n t r i e s . F u r t h e r m o r e , the r e s u l t Ε o f an R - t r a n s f o r m a t i o n o f a B E - m a t r i x Β o f a s t a b l e EM(B) w i l l represent a s t a b l e EM(E), i f the e n t r i e s e . . of Ε have v a l u e s t h a t are p e r m i s s i b l e f o r atoms A . and A . ( " c h e m i c a l f i t t i n g " ) .

the

respective

A c c o r d i n g l y , the p o s i t i v e e n t r i e s b . . of a given BE m a t r i x Β may b e u s e d t o d e t e r m i n e the negative e n t r i e s of mathematically f i t t i n g R-matrices. The p o s i t i v e e n t r i e s must be s e l e c t e d t o y i e l d a n R - m a t r i x 1

with can of

. .

r

i j

= 0.

Moreover,

the

J

entries

of

an

be s e l e c t e d t o meet t h e v a l e n c e c h e m i c a l t h e c h e m i c a l e l e m e n t s i n Ε = Β + R.

R-matrix restrictions

T h u s i t i s p o s s i b l e t o f i n d f o r a g i v e n EM a l l o f i t s c h e m i c a l r e a c t i o n s a n d t h e i r p r o d u c t s by t h e t r a n s f o r m a t i o n p r o p e r t i e s o f the B E - m a t r i c e s , and t h e v a l e n c e c h e m i c a l c o n s t r a i n t s o f t h e l a t t e r . The a p p l i c a t i o n o f a l l m a t h e m a t i c a l l y and c h e m i c a l l y f i t t i n g R-matrices to a BE-matrix generates the BE-matrices o f the whole FIEM.


40

COMPUTER-ASSISTED

4.2

R-Categories

The t r a n s f o r m a t i o n o f a B E - m a t r i x by corresponds to a chemical r e a c t i o n . The

off-diagonal

the

number

and

a negative

electrons Downloaded by UNIV OF ARIZONA on January 17, 2013 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0061.ch002

ORGANIC SYNTHESIS

entries

the

r..

broken

atoms

bonds

entries

covalent

diagonal

= r..

-LJ

covalent i i numbers

of

negative

A^

are

entry lose.

the

J -L

Α.-Α.,

r..

bonds r^ The

= r..

how

positive of

the

positive

R-matrix

indicate

between

tells

numbers

and the

a fitting

A.

and

many

A.,

free

off-diagonal newly

made

diagonal

entries

to the i n c r e a s e s i n f r e e e l e c t r o n atoms Α . . ι G e n e r a l l y , a n R - m a t r i x d o e s n o t o n l y f i t o n e , b u t many B E - m a t r i c e s . A c c o r d i n g l y , an R - m a t r i x does not o n l y r e p r e s e n t an i n d i v i d u a l c h e m i c a l r e a c t i o n , b u t a whole c a t e g o r y o f r e a c t i o n s w h i c h h a v e i n common t h e e l e c t r o n r e l o c a t i o n p a t t e r n r e p r e s e n t e d by t h e R - m a t r i x ) . An R - c a t e g o r y i s an e q u i v a l e n c e c l a s s o f c h e m i c a l r e a c t i o n s w h i c h h a v e i n common t h e same e l e c t r o n r e l o c a t i o n p a t t e r n and c e r t a i n f e a t u r e s o f t h e p a r t i c i p a t i n g bond s y s t e m s . The row/column p e r m u t a t i o n e q u i v a l e n c e o f B E - m a t r i c e s i m p l i e s t h a t R - m a t r i c e s r e p r e s e n t t h e same r e a c t i o n s when t h e y a r e i n t e r c o n v e r t e d b y row/column r

c

o

r

r

e

s

P ° ^ at the n

CH^-CH^l

+ CH -CH=CH

3 T

-> C H = C H 2

CH^-CH=CC1-CH

3

0

C

\

T

3

H

0

+

Ho0

2

+ HC1

2

-> CH - C = C - C H . , + HC1

3

OH H I I C H — C — N-CH

d. ο

T

2

3

3

+ C H -C=N-CH n

c

2 5 I

H

T

+

3

Ho0

2

In o r d e r t o c l a s s i f y c h e m i c a l r e a c t i o n s i n a c o n s i s t a n t m a n n e r w h i c h i s s u i t a b l e f o r c h e m i c a l p u r p o s e s , we d e f i n e the f o l l o w i n g h i e r a r c h y of equivalent classes: R - c a t e g o r y ^ R A - t y p e ^ R B - t y p e r > R l - t y p e z> R 2 - t y p e 1)

Any

two

R-matrices

R-category

if

their

R

1

and R

2

belong

irreducible

to

forms

the

. . . etc .

same

R ° i and

R

0

2

d i f f e r o n l y by r o w / c o l u m n p e r m u t a t i o n s . 2) Any two r e a c t i o n s b e l o n g t o t h e same R A - t y p e , i f t h e y b e l o n g t o t h e same R - c a t e g o r y , a n d i f t h e i r n o n - z e r o e n t r i e s r e f e r t o t h e same c o m p o n e n t s o f a n a s s o c i a t e d v e c t o r of atoms. 3) Any two r e a c t i o n s o f t h e same R A - t y p e b e l o n g t o t h e same R B - t y p e , i f t h e y h a v e t h e same p o s i t i v e entries


2.

BRANDT

ET

Computer

AL.

Programs

for Chemical

Problems

45


i n the a f f e c t e d B E - m a t r i c e s , where t h e R - m a t r i c e s have n o n - z e r o entries. 4) Any two r e a c t i o n s o f t h e same R B - t y p e b e l o n g t o t h e same R l - t y p e , i f t h o s e e n t r i e s o f t h e B E - m a t r i c e s whose e n t r i e s a r e a f f e c t e d by t h e R - m a t r i x r e f e r t o t h e same c o m p o n e n t s o f t h e a t o m v e c t o r . 5) R 2 - , R 3 - j . . . t y p e s may b e d e f i n e d i n a n a l o g y t o t h e R l - t y p e s by f u r t h e r c l a s s i f y i n g a c c o r d i n g t o t h e second, t h i r d e t c . sphere of n e i g h b o r i n g atoms. T h i s h i e r a r c h i c c l a s s i f i c a t i o n o f c h e m i c a l r e a c t i o n s by t h e i r R- a n d B E - m a t r i c e s may n o t o n l y s e r v e a s a means o f f o r m a l o r d e r i n g o f r e a c t i o n s and as a b a s i s o f d o c u m e n t a t i o n s y s t e m s , b u t can a l s o s e r v e as a d e v i c e in the systematic c o m p u t e r - a s s i s t e d deductive search f o r new c h e m i c a l r e a c t i o n s , by a n a l g o r i t h m w h i c h f i n d s all o f t h e m a t h e m a t i c a l l y and c h e m i c a l l y f i t t i n g p a i r s ( Β , E) o f B E - m a t r i c e s f o r a r e p r e s e n t a t i o n R - m a t r i x o f an R - c a t e g o r y .

G e o m e t r i c and G r o u p - T h e o r e t i c a l t u t i o n a l Chemistry

5.

Aspects

of

Consti-

The g e o m e t r i c and g r o u p t h e o r e t i c a l a s p e c t s o f t h e B E and R - m a t r i c e s a r e i m p o r t a n t f o r t h e s o l u t i o n o f c h e m i c a l problems. 1)

. in will

S i n c e these a s p e c t s have been d i s c u s s e d elsewhere d e t a i l , a b r i e f o u t l i n e of the e s s e n t i a l f e a t u r e s suffice here.

The B E - a n d t h e R - m a t r i c e s b e l o n g t o S ( n ) , t h e additive group of a l l η χ η symmetric m a t r i c e s w i t h i n t e g r a l e n t r i e s . L e t P . . be a n η χ η m a t r i x i n w h i c h a l l entries

are

zero,

except

a

"one"

in

the

υ ί

ϋ Ι ί

= 1,

S(n)

whose r a n k

(i,

j)-position.

Then {

is

P

ij

+

1

V' "^'

a basis

of

n(n

the -

η

}

group

1)

ρ

+ η = n(n

2

is

+ 1) 2

We d e n o t e

by

points

an e u c l i d e a n

of

..., η}

Z

s

the

group

whose

elements

s-dimensional

are

the

space R , s

lattice

the


46

COMPUTER-ASSISTED

ORGANIC SYNTHESIS

group o p e r a t i o n b e i n g v e c t o r a d d i t i o n . Then, Z , i s a f r e e a b e l i a n group of rank s w i t h ( 1 , 0 . . . 0 ) , ( 0 , 1 , 0 . . . 0 ) ( 0 , . . . , 0 , 1 ) a s a b a s i s . Two f r e e a b e l i a n g r o u p s a r e i s o m o r p h i c i f , a n d o n l y i f , t h e y h a v e t h e same rank. Therefore, Z a n d S ( n ) a r e i s o m o r p h i c . The i m b e d d i n g o f zs i n R i s a geometric i n t e r p r e t a t i o n of S(n). s

s


s

The B E - m a t r i c e s a r e c o n t a i n e d i n B ( n ) < S ( n ) , t h e s e t o f a l l η χ η symmetric m a t r i c e s w i t h n o n - n e g a t i v e integral e n t r i e s . The R - m a t r i c e s b e l o n g t o R ( n ) S ( n ) , t h e g r o u p o f a l l η χ η s y m m e t r i c i n t e g r a l m a t r i c e s whose sum o f entries is zero. In the mapping π : Β ( η ) R , π | B(n) | i s v i s u a l i z e d as a cone i n R w i t h t h e v e r t e x a t t h e o r i g i n , and π | R ( η ) | l i e s on a l i n e a r s u b s p a c e g o i n g t h r o u g h t h e o r i g i n and h a v i n g no o t h e r p o i n t i n common w i t h π | Β ( η ) | . s

s

The

mapping P:B(n) b

matrices dean The of

)}

2 1

...,

yields

an FIEM i n

R

n 2

b

l

n

;

b

2 1

,

t> , n l

an i m b e d d i n g o f ,

the

BE-

an n - d i m e n s i o n a l

eucli-

2

space. entries

a vector

nates of

Β of

-> { ( b

of

the

b..

of

Β c a n be

in

R

, or

a point

BE-matrix

n

P(B)

c o n s i d e r e d as t h e

also

in

R

n 2

as t h e

.

cartesian

We c a l l

P(B)

the

component? coordi BE-point

B. 2

n S i m i l a r l y , an R - m a t r i x R r e p r e s e n t s a v e c t o r i n R . A c h e m i c a l r e a c t i o n w h i c h i s r e p r e s e n t e d by t h e t r a n s f o r m a t i o n Β + R = Ε c a n be g e o m e t r i c a l l y interpreted by t h e v e c t o r R f r o m t h e B E - p o i n t P(B) t o t h e B E - p o i n t P(E). The

sum o f D(B,E)

the = g

absolute |b±J

-

values e..|

=

of

the

entries

of

R,

£.|r..|

i s e q u a l t o t w i c e t h e number o f v a l e n c e e l e c t r o n s t h a t p a r t i c i p a t e i n t h e r e a c t i o n EM(B) E M ( E ) . We c a l l D ( B , E ) t h e c h e m i c a l d i s t a n c e between P(B) and P ( E ) , o r EM(B) and E M ( Ε ) , r e s p e c t i v e l y . Note t h a t c h e m i c a l distances r e f e r to R , and t o R . n

s

The o r i g i n o f t h e c o o r d i n a t e s y s t e m i n R corresponds t o t h e z e r o m a t r i x . S i n c e t h e sum o f e n t r i e s S = r i j i j > n

b


2.

BRANDT E T A L .

Computer

Programs

for Chemical

Problems

47

t h e number o f v a l e n c e e l e c t r o n s , i s t h e same f o r a l l B E - m a t r i c e s o f an F I E M , a n d F I E M i s mapped o n t o a l a t t i c e of points with non-negative i n t e g r a l coordinates in R , l y i n g on a s e g m e n t o f a " h y p e r s h e r e s u r f a c e " whose r a d i u s i s S = D ( B , 0 ) . n


T h i s r e m a r k a b l e t o p o l o g i c a l o r d e r o f t h e FIEM i s not only t h e o r e t i c a l l y a p p e a l i n g , but a l s o d i d a c t i c a l l y u s e f u l , because t h i s order r e v e a l s w e l l - d e f i n e d u n i v e r s a l l o g i c a l s t r u c t u r e s i n t h e immense w e a l t h o f individual facts. In p a r t i c u l a r , the o r d e r p r o v i d e s a b a s i s f o r t h e computer-oriented r e p r e s e n t a t i o n of molecular systems, and a l l o w s t o f o r m u l a t e c h e m i c a l f a c t s and p r o b l e m s i n a manner w h i c h i s w e l l - s u i t e d f o r t h e i r m a n i p u l a t i o n by c o m p u t e r s . 6.

MATCHEM

The m a t h e m a t i c a l m o d e l o f c o n s t i t u t i o n a l c h e m i s t r y which has been d e s c r i b e d i n the p r e c e e d i n g s e c t i o n s can be u s e d as a b a s i s f o r a m o d u l a r s y s t e m o f c o m p u t e r programs f o r the d e d u c t i v e s o l u t i o n of c h e m i c a l problems. I n i t i a l l y , t h e m a t h e m a t i c a l m o d e l was u t i l i z e d f o r C I C L O P S ^ , a p i l o t program f o r s y n t h e t i c d e s i g n . The i n s i g h t t h a t t h e m a t h e m a t i c a l m o d e l may s e r v e f o r the c o m p u t e r - a s s i s t e d s o l u t i o n o f a wide v a r i e t y of c h e m i c a l p r o b l e m s l e d t o t h e d e s i g n o f MATCHEM, a m o d u l a r s y s t e m o f c o m p u t e r p r o g r a m s , whose i n d i v i d u a l p a r t s may b e c o m b i n e d i n d i f f e r e n t ways t o s u i t d i f f e r e n t p u r p o s e s . The o r i g i n a l s y n t h e t i c d e s i g n p r o g r a m C I C L O P S was m o d i f i e d v i a t h e i n t e r m e d i a t e s t a g e MATSYN t o y i e l d t h e p r e s e n t s y n t h e t i c d e s i g n p r o g r a m EROS ( E l a b o r a t i o n o f R e a c t i o n s f o r O r g a n i c S y n t h e s i s ) w h i c h s e r v e s a s i m i l a r p u r p o s e as SECS ) and t h e o t h e r r e a c t i o n l i b r a r y o r i e n t e d s y n t h e t i c d e s i g n programs LHASA \ SYNC HEM °) , e t c . I n c o n t r a s t t o t h e p i l o t p r o g r a m C I C L O P S , i n EROS t h e m o d u l a r t y p e s t r u c t u r e i s emphasized more, i n o r d e r to enable the c o m b i n a t i o n w i t h o t h e r p a r t s o f t h e MATCHEM s y s t e m . 6

7

T h e s y n t h e t i c d e s i g n p r o g r a m EROS a n d i t s p r e d e c e s s o r s g e n e r a t e a t r e e o f B E - m a t r i c e s s t a r t i n g f r o m one B E matrix which r e f e r s to the s y n t h e t i c t a r g e t . This t r e e may b e i n t e r p r e t e d a s a t r e e o f s y n t h e t i c p a t h w a y s . If, however, the i n i t i a l m a t r i x does not r e p r e s e n t a

American Chemfeaf Society Library 1155 16th St. N. W. In Computer-Assisted Organic Synthesis; Wipke, W., et al.; Washington, D. C. 20031 ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

COMPUTER-ASSISTED

48

ORGANIC SYNTHESIS


s y n t h e t i c t a r g e t and i t s b y - p r o d u c t s , but a g i v e n c h e m i c a l compound i n c o m b i n a t i o n w i t h a l i s t o f p o t e n t i a l r e a c t a n t s , the generated t r e e of B E - m a t r i c e s can be i n t e r p r e t e d i n t e r m s o f s e q u e n c e s o f p r o d u c t s o b t a i n a b l e f r o m t h e i n i t i a l r e a c t a n t s . In t h i s f o r m t h e p r o g r a m f i n d s new u s e s f o r c h e m i c a l c o m p o u n d s , e . g . industrial by-products, or permits the p r e d i c t i o n of t h e c o n c e i v a b l e f a t e o f a c h e m i c a l compound i n t h e environment. The m a t h e m a t i c a l m o d e l a f f o r d s a l s o e n t i r e l y different a p p r o a c h e s t o c h e m i c a l p r o b l e m s . I f t h e i n i t i a l and f i n a l e n s e m b l e o f m o l e c u l e s , EM(B) a n d E M ( Ε ) , o f a chemical r e a c t i o n , or a sequence of r e a c t i o n s are k n o w n , o r a t t a i n e d by a n e d u c a t e d g u e s s , s u c h a s a comparison of substructures ( s e e s e c t i o n 5.1.3) t h e d i f f e r e n c e of the c o r r e s p o n d i n g B E - m a t r i c e s E-B = R c a n be a n a l y z e d t o y i e l d t h e p a t h w a y s o f individual s t e p s w h i c h l e a d f r o m EM(B) t o E M ( E ) . T h e s e p a t h w a y s may be s e q u e n c e s o f i n t e r m e d i a t e s i n a r e a c t i o n mechanism, or a l s o sequences of s y n t h e t i c inter m e d i a t e s . The p r e c o n d i t i o n f o r t h i s a p p r o a c h i s t h a t t h e i n d i c e s o f t h e a t o m s i n EM(B) a n d E M ( Ε ) a r e a p p r o p r i a t e l y a s s i g n e d . The r e q u i r e d a s s i g n m e n t o f a t o m i c i n d i c e s i s d i s c u s s e d i n s e c t i o n 5.3.1). A l l o f t h e s e a p p l i c a t i o n s o f the m a t h e m a t i c a l m o d e l , as w e l l as t h e s e a r c h o f B E - m a t r i x p a i r s ( Β , E) w h i c h f i t a g i v e n R - m a t r i x w i l l be c o n t a i n e d i n MATCKEM. I n t h e now f o l l o w i n g s e c t i o n s some r e c e n t l y p a r t s o f MATCHEM w i l l b e p r e s e n t e d .

6.1

Manipulation

6.1.1 C a n o n i c a l

of

implemented

BE-Matrices

Ordering

A p r e c o n d i t i o n f o r an e f f i c i e n t m a n i p u l a t i o n o f B E m a t r i c e s i n the computer i s a c a n o n i c a l i n d e x i n g o f t h e atoms i n a m o l e c u l e . In o r d e r t o g e n e r a t e a u n i q u e n u m b e r i n g we u s e t h e c o n n e c t i v i t y m a t r i x o f t h e m o l e c u l a r g r a p h and t h e l a b e l s a l r e a d y a s s i g n e d t o its v e r t i c e s , i . e . the c h e m i c a l symbols. An atom k i s c o n s i d e r e d t h e j - t h n e i g h b o r o f i i f j i s t h e m i n i m a l number o f b o n d s w h i c h s e p a r a t e i a n d k. All a t o m s k w h i c h meet t h i s c o n d i t i o n a r e c a l l e d t h e j - t h n e i g h b o r h o o d o f i . The z e r o t h n e i g h b o r h o o d o f i i s a t o m i i t s e l f . We d e f i n e a n a t o m i c d e s c r i p t o r


2.

as

BRANDT E T A L .

Computer

for Chemical

Problems

49

follows

11 a 1 2 a., is J neighborhood

where

descending


Programs

the

...|a22..•a31l 32 a

atomic

and a l l

order.

The

a.,

number for

of

the

operator

11

atom k i n same j

are

[" means

the

j-th

put

in

concatenation.

The d . a r e c o n s t r u c t e d i n s u c h a manner t h a t a l l j - t h n e i g h b o r h o o d s a r e a l i g n e d . T h i s i s a c h i e v e d by i n s e r t i n g dummy n e i g h b o r s w i t h a t o m i c number o f z e r o i f necessary. T h e s e q u e n c i n g o f a t o m s i s d o n e by p u t t i n g t h e a t o m i c d e s c r i p t o r s i n d e s c e n d i n g o r d e r . Atoms w h i c h c a n n o t b e a s s i g n e d a u n i q u e number by t h i s p r o c e d u r e a r e c o n s t i t u t i o n a l l y e q u i v a l e n t . A f i n a l numbering of the a t o m s i s r e a c h e d by i n t r o d u c i n g a r b i t r a r y information. One a t o m i s p i c k e d o u t o f t h e h i g h e s t r a n k i n g g r o u p o f c o n s t i t u t i o n a l l y e q u i v a l e n t atoms and i t s a t o m i c number s e t h i g h e r t h a n t h o s e o f t h e o t h e r a t o m s . The a t o m i c d e s c r i p t o r s a r e m o d i f i e d a c c o r d i n g l y and s o r t e d a g a i n . The g r o u p o f c o n s t i t u t i o n a l l y e q u i v a l e n t atoms i s thus s p l i t up. If a unique numbering i s s t i l l not p o s s i b l e the m o d i f i c a t i o n of the atomic d e s c r i p t o r s i s r e v e r s e d a n d a n o t h e r a t o m o u t o f t h e now h i g h e s t r a n k i n g g r o u p o f e q u i v a l e n t atoms i s p i c k e d , and t h e p r o c e d u r e j u s t d e s c r i b e d i s r e p e a t e d u n t i l a l l atoms a r e a s s i g n e d a u n i q u e number. I t s h o u l d be n o t e d t h a t i t d o e s n o t m a t t e r w h i c h a t o m i s p i c k e d o u t o f a g r o u p o f a t o m s w h i c h c a n n o t be a s s i g n e d a u n i q u e number i n t h e f i r s t p l a c e b e c a u s e t h e s e atoms a r e i n f a c t c o n s t i t u t i o n a l l y equivalent. I n t h i s a s p e c t o u r new i n d e x i n g r o u t i n e d i f f e r s f r o m o u r p r e v i o u s a p p r o a c h ^ . Our new p r o c e d u r e d o e s n o t have t h e drawbacks o f o t h e r methods ) known, like i n s t a b i l i t y of the c a l c u l a t i o n s , o s c i l l a t o r y behaviour, n o n c o n v e r g e n c e and i n d e t e r m i n a n c y . 9

6.1.2

Direct

Access

to

Structures

and

Substructures

The B E - m a t r i x , t o g e t h e r w i t h i t s a s s o c i a t e d v e c t o r s (atoms, s t e r e o c h e m i c a l p a r i t y b i t s ) c o n t a i n s a l l s t r u c t u r a l i n f o r m a t i o n about a m o l e c u l e . In a v a r i e t y o f a p p l i c a t i o n s , s u c h i n f o r m a t i o n has t o be s t o r e d and


50

COMPUTER-ASSISTED

a c c e s s e d r a t h e r f r e q u e n t l y . Examples c a t i o n s are documentation systems or r e a c t i o n graphs (trees or networks).


6.1.2.1

Inter-Machine

ORGANIC SYNTHESIS

of such a p p l i the a n a l y s i s of

Manipulation

T h e s t o r a g e r e q u i r e m e n t s o f t h e f u l l B E - m a t r i x make i t r a t h e r u n s u i t a b l e f o r d i r e c t s t o r a g e and r e t r i e v a l . C o m p r e s s i o n c a n b e a c h i e v e d i n two s t a g e s , d e p e n d i n g upon t h e a p p l i c a t i o n . F o r d a t a exchange between d i f f e r e n t computer systems (with p o s s i b l y different c o d e s a n d f i l e s t r u c t u r e s ) , c o m p r e s s i o n down t o t h e l e v e l of characters (purely numerical) i s a useful c o m p r o m i s e . The c o m p r e s s i o n m e t h o d i s straightforward i n t h a t i t f i r s t a s s i g n s e a c h s t r u c t u r e ( i . e . an i n t a c t m o l e c u l e or a m o l e c u l a r fragment) a unique number, w h i c h may b e d e r i v e d f r o m some s o r t o f r e g i s t r y number, o r may b e g e n e r a t e d by t h e s y s t e m i t s e l f , i n w h i c h c a s e i t u s u a l l y w i l l c o n t a i n some g r a p h i n f o r m a t i o n , s u c h a s n o d e number o r l i n k p o i n t e r . The i n f o r m a t i o n t h a t i s c o n t a i n e d i n t h e B E - m a t r i x and its a s s o c i a t e d atom v e c t o r i s c o m p r e s s e d t o t h e mininum number o f b y t e s w h i c h i s n e c e s s a r y f o r t h e e x p e c t e d range of values f o r each of these items. Such a d a t a format a l l o w s r a t h e r e c o n o m i c a l machine i n d e p e n d e n t p a c k i n g and u n p a c k i n g o f B E - m a t r i c e s and i s u s e d f o r d a t a t r a n s f e r on s e q u e n t i a l l y o r g a n i z e d , byte oriented media.

6.1.2.2

Intra-Machine

Manipulation

I n t e r n a l h a n d l i n g o f l a r g e amounts o f s t r u c t u r a l d a t a p u t s more s e v e r e r e q u i r e m e n t s o n d a t a f o r m a t a n d o n d a t a o r g a n i s a t i o n ( i n t h i s p a r a g r a p h , use o f random a c c e s s m e d i a , s u c h a s c o r e memory b a n k s - p o s s i b l y w i t h page o r b l o c k o r g a n i s a t i o n - o r m u l t i - t r a c k d i s c s is assumed). S t r u c t u r a l i n f o r m a t i o n c a n be - a n d i n f a c t i s f u r t h e r c o m p r e s s e d by a . ) r e p l a c i n g s e q u e n c e s o f z e r o e n t r i e s by p o i n t e r s , b . ) r e d u c i n g t h e s t o r a g e s p a c e f o r t h e r e m a i n i n g e n t r i e s (atom s y m b o l : 7 b i t , d i a g o n a l elements: 4 b i t , o f f d i a g o n a l upper t r i a n g l e : 2 b i t ) .


2.

BRANDT E T A L .

Computer

Programs

for Chemical

51

Problems


F u r t h e r , the f i l e s c o n t a i n r e f e r e n c e s of d i f f e r e n t o r i g i n t o o t h e r e n t r i e s i n t h a t f i l e . These appear i n the form of n u m e r i c a l p o i n t e r s and, i f improperly o r g a n i z e d , c a n u s e up l a r g e s t o r a g e s p a c e . We d e v e l o p e d a f i l e o r g a n i z a t i o n to minimize storage requirement w h i c h makes u s e o f r e l a t i v e p o i n t e r s a n d p o i n t e r length c l a s s e s . Furthermore, short access paths from p o i n t i n g element (son) to p o i n t e d - a t element ( f a t h e r ) , are p o s t u l a t e d , i . e . minimization of core page-switching, o r d i s k h e a d movement respectively. I t h a s b e e n f o u n d t h a t t h e a b o v e two p o s t u l a t e s l e a d t o a d a t a o r g a n i z a t i o n , whereby sons a r e a s s i g n e d s t o r a g e i n t h e v i c i n i t y o f t h e i r f a t h e r . The a d d r e s s t o be a s s i g n e d i s d e r i v e d f r o m t h e d a t a t h e m s e l v e s by a s u i t a b l e h a s h - f u n c t i o n , which a s s i g n s linearly i n c r e a s i n g d i s t a n c e s f o r cases without c o n f l i c t , (addressed storage n o n - o c c u p i e d ) , but switches to f u n c t i o n s o f s e c o n d d e g r e e when c o n f l i c t s ( o v e r f l o w s ) a r e t o be h a n d l e d . S i n c e t h e r e a r e o n l y m o d e r a t e c o m p u t i n g r e q u i r e m e n t s , and s i n c e a c c e s s t i m e s on t h e storage media determines p r o c e s s i n g speed, a r a t h e r moderate computer i s needed f o r h a n d l i n g f a i r l y large a m o u n t s o f d a t a . We a r e a t p r e s e n t i n v e s t i g a t i n g t h e behaviour of a system with s e v e r a l thousand s t r u c t u r e s and t h e i r s u b s t r u c t u r e s ( s e e 5·1-3) on a " m i n i " c o m p u t e r o f t h e w e l l - k n o w n PDP11 f a m i l y t h a t i s equipped w i t h moving head d i s k packs t h a t have a s t o r a g e c a p a c i t y o f a b o u t two mega b y t e e a c h .

6.1.3

Hierarchically

Organized

Substructure

Files

I n a g r e a t many a p p l i c a t i o n s , a g i v e n s t r u c t u r e i s t r e a t e d as a combine o f i t s s u b s t r u c t u r e s . D e p e n d i n g on t h e c o n t e x t , s u b s t r u c t u r e s a p p e a r u n d e r t h e name of " f u n c t i o n a l group", "fragment", " r e a c t i o n i n v a r i a n t " "building block" etc. T y p i c a l a p p l i c a t i o n s are optimization strategies in synthesis planning, avoidance o f p r o h i b i t e d c o m b i n a t i o n s o f f u n c t i o n a l groups in generating chemical r e a c t i o n s , structure-activity correlations etc. Computer g e n e r a t i o n , s t o r a g e and r e t r i e v a l o f s u b s t r u c t u r e s i s g r e a t l y f a c i l i t a t e d , when t h e m a t h e m a t i c a l model of s t r u c t u r a l c h e m i s t r y i s employed to g e n e r a t e a h i e r a r c h i c a l l y organized substructure f i l e in a s y s t e m a t i c f a s h i o n . Such h i e r a r c h i c a l o r d e r i n g i s not only a p r e r e q u i s i t e i n avoiding d u p l i c a t i o n of


52

COMPUTER-ASSISTED


i n f o r m a t i o n and m i s s i n g l i n k s , i t f o r s e t t i n g up d a t a o r g a n i s a t i o n s previous chapter (5.1.2).

is as

ORGANIC SYNTHESIS

also indispensable d e s c r i b e d i n the

The a p p r o a c h t a k e n h e r e t r e a t s t h e o f f d i a g o n a l e n t r i e s o f a B E - m a t r i x as an o r d e r e d s e t o f numbers. E a c h e l e m e n t i n t u r n i s l o w e r e d by o n e , t h e r e s u l t i n g m a t r i x s e p a r a t e d i n t o b l o c k s and t h e d a t a f i l e i s t e s t e d f o r the presence of each r e s u l t i n g b l o c k . If the t e s t i s p o s i t i v e , then only a p o i n t e r to the g e n e r a t i n g s t r u c t u r e ( " f a t h e r " ) i s a d d e d t o t h e f i l e , a n d no f u r t h e r f r a g m e n t a t i o n i s n e e d e d , s i n c e a l l s u c c e s s o r s must b e i n the f i l e . If the t e s t i s n e g a t i v e , the son i s e n t e r e d i n t o t h e f i l e and i t s f a t h e r (and b r o t h e r if a n y ) a r e p u s h e d on a s t a c k f o r f u r t h e r t r e a t m e n t . E a c h e l e m e n t o f t h e s t a c k i s t r e a t e d i n t h e same way (thus becoming the f a t h e r of the next g e n e r a t i o n ) a f t e r the f i r s t son i s p r o c e s s e d . T h i s d e p t h - f i r s t fragmentation y i e l d s d a t a - f i l e s that f u l f i l the requirement of the p r e v i o u s c h a p t e r , i n t h a t sons a p p e a r as c l o s e as p o s s i b l e to t h e i r f a t h e r i n the g e n e r a t i o n p r o c e s s , thereby l e a d i n g to minimal pointer lengths. With such a f i l e s t r u c t u r e q u e r i e s take the form p a t h s w i t h i n a d i r e c t e d g r a p h and t h e r e f o r e a r e p r o c e s s e d w i t h a minimum o f c o m p u t i n g e f f o r t a n d extremely short response time.

6.2

Operating

6.2.1

The

with

Synthetic

of

R-Matrices Design

Program

EROS

Our p r e v i o u s s t u d i e s i n d e v e l o p i n g C I C L O P S , t h e p r o t o t y p e o f a s y n t h e t i c d e s i g n p r o g r a m , h a v e now l e d t o a new s t a g e . We a r e now p r e s e n t i n g EROS ( E l a b o r a t i o n o f R e a c t i o n s f o r O r g a n i c S y n t h e s i s ) which has matured to the p o i n t of becoming a r o u t i n e t o o l f o r the s y n t h e t i c chemist. A number o f o p t i o n s a l l o w s t h e u s e r t o d i r e c t t h e p r o g r a m t o meet h i s s p e c i f i c p r o b l e m s a n d n e e d s . Rather than t a k i n g the complete set of a l l mathematically possible r e a c t i o n matrices a s e l e c t i o n of only three R - m a t r i c e s i s c o n t a i n e d i n the program. These t h r e e R - c a t e g o r i e s i n c l u d e t h e m a j o r i t y o f a l l known s y n t h e t i c r e a c t i o n s and i t i s b e l i e v e d t h a t most o f t h e r e a c t i o n s w h i c h w i l l be d i s c o v e r e d i n t h e f u t u r e w i l l a l s o f a l l i n t h e s e t h r e e R - c a t e g o r i e s . So t h e a d v a n t a g e s o f t h e


2.

BRANDT

E T

Computer

AL.

Programs

for Chemical

53

Problems

m a t h e m a t i c a l model o f p r o v i d i n g a l s o u n p r e c e d e n t e d r e a c t i o n s i s l a r g e l y r e t a i n e d . On t h e o t h e r h a n d s u b s t a n t i a l r e d u c t i o n of the output of u n l i k e l y intermediates i s achieved.


EROS c a n b e a p p l i e d t o t h e s t u d y o f c h e m i c a l r e a c t i o n s i n two b a s i c a l l y d i f f e r e n t w a y s : One c a n i n p u t several m o l e c u l e s and g e n e r a t e t h e p r o d u c t s t o be e x p e c t e d i n t h e r e a c t i o n s o f t h e s e m o l e c u l e s . Or o n e c a n l o o k f o r all s y n t h e t i c r e a c t i o n s l e a d t o a t a r g e t compound w h i c h is input. R e a c t i o n s mechanism c a n be s t u d i e d by a l l o w i n g e l e c t r o n i c c o n f i g u r a t i o n s f o r d i s t i n c t atoms.

certain

The r e a c t i o n s i t e i s d e t e r m i n e d by s t a t i n g w h i c h b o n d s a r e b r e a k a b l e . T h e s e c a n be f o u n d by a s t a n d a r d r o u t i n e o r p r e s e l e c t e d b y t h e u s e r . F o r e a c h r e a c t i o n t h e energy i s c a l c u l a t e d u s i n g parameters obtained from thermoc h e m i c a l d a t a - ' ) . By d e f i n i n g e n e r g y l i m i t s , r e a c t i o n s can be r e j e c t e d on thermodynamic grounds. 1

6.2.2

0

Predictor

1 1

Systems

F r o m t h e p r e v i o u s c h a p t e r i t b e c o m e s o b v i o u s t h a t Rm a t r i c e s , when c o m b i n e d w i t h s u i t a b l e s e l e c t i o n r u l e s , are a general t o o l f o r p r e d i c t i n g chemical r e a c t i o n products. I n some a p p l i c a t i o n s , c h e m i c a l s e l e c t i o n r u l e s may b e l e s s s t r i n g e n t b e c a u s e some o t h e r e x t r a n e o u s selection a l l o w s f o r f u r t h e r r e d u c t i o n o f output to manageable size. One g r o u p o f a p p l i c a t i o n s a r e p r e d i c t o r s y s t e m s . S u c h systems have, i n g e n e r a l , a c c e s s t o s t r u c t u r e f i l e s , i . e . f i l e s of i n t a c t molecules or molecular fragments o f t h e n a t u r e d e s c r i b e d i n c h a p t e r s 5.1.2) and 5.1.3) whose e n t r i e s a r e s e l e c t e d u n d e r a g i v e n a s p e c t . A s p e c t s may b e t o x i c i t y , p h a r m a c e u t i c a l a c t i v i t y , environm e n t a l i m p a c t , a v a i l a b i l i t y as an unwanted by-product i n an i n d u s t r i a l environment e t c . In such cases query e n t r i e s a r e p r o c e s s e d through r e a c t i o n g e n e r a t o r s . T h e s e may b e t h e c o m p l e t e s e t o f R - c a t e g o r i e s , b u t o f c o u r s e a s u b s e t o f t h o s e c a n be s e l e c t e d i f t h e a p p l i c a t i o n j u s t i f i e s i t . Output o f t h e r e a c t i o n g e n e r a t o r s i s t h e n c h e c k e d a g a i n s t structure f i l e s and matches a r e o u t p u t t o t h e u s e r .


54

COMPUTER-ASSISTED

ORGANIC SYNTHESIS


One s u c h s y s t e m i s a t p r e s e n t b e i n g i m p l e m e n t e d u n d e r a r e s e a r c h c o n t r a c t w i t h a p u b l i c agency w i t h the purpose of d e t e c t i n g sources of environmentally active c h e m i c a l s among c h e m i c a l s r e g a r d e d a s h a r m l e s s , themselves. A l t h o u g h an e s s e n t i a l p a r t o f s u c h s y s t e m s , the r e a c t i o n g e n e r a t o r s , due t o t h e i r a l g e b r a i c n a t u r e , t a k e up o n l y a n e g l i g i b l e amount o f t h e t o t a l c o m p u t i n g e f f o r t . It has t h e r e f o r e been r e m a r k e d , not w i t h o u t j u s t i f i c a t i o n , t h a t i n many a p p l i c a t i o n s p r e d i c t o r s y s t e m s a p p e a r t o t h e u s e r a s a mere e n h a n c e m e n t o f the c a p a b i l i t i e s of a documentation system. It should be s t r e s s e d , h o w e v e r , t h a t d o c u m e n t a t i o n s y s t e m s w i l l be a b l e t o s u p p o r t p r e d i c t o r m o d u l e s o n l y i f t h e y m a i n t a i n the complete s t r u c t u r a l i n f o r m a t i o n i n a form t h a t i s c o m p a t i b l e w i t h , o r c o n v e r t i b l e t o an a l g e b r a i c a l l y d e f i n e d B E - m a t r i x . D e s i g n , or r e d e s i g n of systems should take t h i s f a c t i n t o a c c o u n t .

6·3

Generation

and C l a s s i f i c a t i o n o f

6.3.1

Determination

of

the

R-Matrices

Minimal Chemical

Distance

L e t EM(B) and EM(E) be t h e i n i t i a l and f i n a l e n s e m b l e of molecules of a c h e m i c a l r e a c t i o n or sequence of r e a c t i o n . This chemical conversion determines i n a c h a r a c t e r i s t i c manner a c o r r e l a t i o n o f t h e atoms i n EM(E) a n d E M ( B ) . We c o n j e c t u r e t h a t c h e m i c a l r e a c t i o n s p r o c e e d i n s u c h a m a n n e r t h a t a minimum o f v a l e n c e e l e c t r o n s i s r e l o c a t e d . This i s c r i t e r i o n f o r the c o r r e l a t i o n o f t h e atoms o f EM(E) t o t h e atoms o f E M ( B ) . We c a l l t h e number o f v a l e n c e e l e c t r o n s w h i c h must b e r e l o c a t e d d u r i n g t h e t r a n s f o r m a t i o n f r o m E M ( B ) t o EM(E) w i t h a g i v e n c o r r e l a t i o n o f atoms t h e c h e m i c a l d i s t a n c e D ( B , E) between t h e s e EM. M a t h e m a t i c a l l y the chemical d i s t a n c e D(B, f o r m u l a t e d i n t h e f o l l o w i n g way Let

Then

(b..)

be

the

BE-matrix

of

A

(e..)

be

the

BE-matrix

of

Z.

E)

c a n be

and

i> j = l


2.

BRANDT

ET

Computer

AL.

Programs

for Chemical

Problems

55

F o r t h e c h e m i c a l l y m e a n i n g f u l r e p r e s e n t a t i o n E M ( B ) •> EM(E) t h e a t o m s o f E M ( B ) must b e a s s i g n e d t o a t o m s o f EM(E) i n s u c h a manner t h a t t h e c h e m i c a l d i s t a n c e D has i t s minimal value. This


1)

problem

c a n be a t t a c k e d i n

several

ways

E x h a u s t i v e e n u m e r a t i o n : a l l a t o m s w i t h t h e same a t o m i c number a r e p e r m u t e d a n d t h e p e r m u t a t i o n corresponding to the minimal chemical d i s t a n c e i s taken. Let n

EM(E) atoms

2

atomic (n !)

number

0^.

k

method

of

atomic

,..(n !)

2

this

consist of

is

n^ a t o m s

number Then

of

0^,

atomic

. . .,

one h a s

to

carry

permutations.

Since

suitable

for

only

number

n^. a t o m s

10!

very

out

0^,

of (n^!)«

= 3 628

800

small

molecules. 2)

The c r i t e r i o n f o r m i n i m i z a t i o n o f t h e c h e m i c a l d i s t a n c e c a n be p a r t i t i o n e d i n t o t h e t h r e e c o m p l e mentary c r i t e r i a ( i ) , ( i i ) and ( i i i ) . (i) Atoms w i t h t h e same a t o m i c number n u m b e r s f r o m t h e same s e t o f n u m b e r s .

Example

In

I

(i)

I:

H^-C^N:.

^2

suffices

1

to

+

are

given

the

H-N^C:

3 12

minimize

the

chemical distance.

(ii) The atoms a r e a s s i g n e d i n s u c h a manner t h a t , f o r a t o m s w h i c h a r e a s s i g n e d t o e a c h o t h e r , a s many spheres o f n e i g h b o r s as p o s s i b l e c o r r e s p o n d . T h i s c r i t e r i o n f o l l o w s from the assumption, t h a t r e a c t i o n e a c h a t o m w i l l t r y t o m a i n t a i n a s many s p h e r e s o f n e i g h b o r s as p o s s i b l e .


in

a

56

COMPUTER-ASSISTED

Example

ORGANIC SYNTHESIS

II:

H—

H—CjpH H.

- C l . + H - 0 H -* H - ^ C — C ^ — 0 - H H-^C 1 2 H / 5 |3 2 |3 H• ^ 5 H àr H H - Ç - H o

C

o

+ H-Cl.

1

6

H


H

In II ( i ) s u f f i c e s t o a s s i g n 0 and C l ; but f o r t h e assignment of the carbon-atoms ( i i ) i s necessary. (iii)

If there are c o n s t i t u t i o n a l l y symmetric molecules i n EM(B) a n d ( o r E M ( Ε ) ) t h e n a n a t o m A i n E M ( B ) c a n be a s s i g n e d t o more t h a n o n e a t o m Z ^ , · · · * Zk i n E M ( E ) e v e n i f one t a k e s ( i i ) i n t o a c c o u n t . T h e n t h e r e e x i s t two p o s s i b i l i t i e s : l 2 )

1) to to

A i s t h e f i r s t atom i n EM(B) w h i c h s h a l l be a s s i g n e d an atom i n E M ( E ) . Then A can be a s s i g n e d a r b i t r a r i l y one o f t h e a t o m s Ζ ^ , . . . Ζ ^ . .

2) One h a s a l r e a d y a s s i g n e d a t o m s Aj_ -> Z±9...9 A -> T h e n A c a n b e a s s i g n e d t o Z^ i n s u c h a way t h a t t h e

Z .

e

atoms to

Α ^ , - . , , Α

A as

the

are

atoms

Z,

in

the ,Z

same s p h e r e s e

to

In by

neighbors

Z..

T h i s m e t h o d w i l l be i t e r a t e d u n t i l neighbors correspond to each other Example

of

e

a l l spheres of as good as p o s s i b l e ,

III:

example III one c a n f i n d criterion (iii).

the

optimal

assignment

only

I n t h e c o m p u t e r p r o g r a m w h i c h we h a v e d e v e l o p e d f o r t h e m i n i m i z a t i o n o f t h e c h e m i c a l d i s t a n c e we u s e a l g o r i t h m s f r o m t h e f i e l d o f o p e r a t i o n s r e s e a r c h known as " o p t i m a l a s s i g n m e n t " - m e t h o d s . ^ 13


2.

BRANDT


6.3.2

ET

AL.

Computer

Documentation

Programs

Systems

for Chemical

for

Problems

Chemical

57

Reactions

E n c o d i n g and r e t r i e v a l p r o c e d u r e s f o r c h e m i c a l s t r u c t u r e s h a v e b e e n d e v i s e d a n d u s e d s i n c e t h e t i m e s when s t r u c t u r a l o r g a n i c c h e m i s t r y became known. Although t h e y had t o be t a i l o r e d t o t h e t e c h n i c a l n a t u r e o f t h e d a t a c a r r y i n g m e d i a - be i t s p o k e n o r p r i n t e d w o r d s , manually processed card f i l e s , s e q u e n t i a l l y processed punch h o l e cards or magnetic t a p e s , - they a l l f u l f i l l e d t h e i r p u r p o s e a t t h e i r t i m e . The s i t u a t i o n i s r a t h e r d i f f e r e n t f o r c h e m i c a l r e a c t i o n s . We s t i l l r e l y o n s u c h a l c h e m i s t i c means a s n a m i n g - b y - a u t h o r or citing-by m a j o r - p r o d u c t e t c . T h i s i s symptomatic f o r our s t a t e of k n o w l e d g e . A p p a r e n t l y we h a v e no s y s t e m a t i c way o f d e s c r i b i n g r e a c t i o n s u n t i l now. (The n e e d , o f c o u r s e , was l e s s p r e s s i n g , s i n c e t h e number o f r e a c t i o n s i s much s m a l l e r t h a n t h e number o f s t r u c t u r e s . ) A p p r o a c h e s t h a t use s t a r t i n g and/or f i n a l p r o d u c t s as a v e h i c l e f o r d e s c r i b i n g a n d c l a s s i f y i n g r e a c t i o n s must b e regarded u n s u i t a b l e i n the l i g h t of our mathematical m o d e l : A r e a c t i o n i s a p a t h b e t w e e n p o i n t s on t h e s u b s p a c e , t h a t i s d e f i n e d by a n F I E M . A l t h o u g h a n i n d i v i d u a l p a t h may be d e s c r i b e d by i t s s t a r t i n g , f i n a l ( a n d p o s s i b l y some i n t e r m e d i a t e p o i n t s ) e q u i v a l e n c e c l a s s e s o f s u c h p a t h s c a n n o t be a d e q u a t e l y d e s c r i b e d and c l a s s i f i e d t h a t way. What we n e e d i s a d e s c r i p t i o n o f the path i t s e l f . An R - m a t r i x i s s u c h a d e s c r i p t i o n . I n i t s most g e n e r a l f o r m i t c o n t a i n s no r e f e r e n c e t o a p a r t i c u l a r F I E M . Therefore ( a s e x p l a i n e d i n s e c t i o n 4.3) e a c h i r r e d u c i b l e R - m a t r i x d e f i n e s a n R - c a t e g o r y . The m i n i m a l r e f e r e n c e to a s p e c i f i c FIEM, t h e n , i s through those elements of t h e atom v e c t o r , t h a t c o r r e s p o n d t o t h e rows and columns i n t h e i r r e d u c i b l e R - m a t r i x , i n o t h e r words to the nonzero e n t r i e s i n the f u l l R - m a t r i x . Increasing d e g r e e s o f s p e c i f i c i t y a r e o b t a i n e d by a r e f e r e n c e f r o m t h i s f i r s t s e t o f atoms ("the r e a c t i o n c o r e " ) t o t h e i r corresponding elements i n the BE-matrix, that i n t u r n e n l a r g e t h e atom v e c t o r ( " f i r s t s p h e r e " ) e t c . A h i e r a r c h y i s thereby a c h i e v e d , that w i l l lead to a r e a c t i o n f i l e s t r u c t u r e , where e a c h e n t r y i s r e f e r e n c e d t h r o u g h t h e v a r i o u s l e v e l s o f s p e c i f i c i t y up t o t h e R - c a t e g o r i e s . Q u e r i e s c a n be e n t e r e d a t any d e g r e e o f generality.


COMPUTER-ASSISTED ORGANIC SYNTHESIS


58

In a p r a c t i c a l system, a c a n o n i c a l o r d e r i n g has t o be imposed onto R - m a t r i c e s , which i s most e a s i l y a c h i e v e d by s e l e c t i n g t h e l e x i c o g r a p h i c a l l y s m a l l e s t i r r e d u c i b l e R-matrix among t h e n! e q u i v a l e n t ones. L e x i c o g r a p h i c a l p r i o r i t y i s given to the d i a g o n a l elements. T h i s , a c t u a l l y , i s an a r b i t r a r y measure, but i t i s j u s t i f i e d on t h e ground, t h a t non-zero d i a g o n a l elements cause changes i n v a l e n c e s t a t e o f the a f f e c t e d atom, thereby earmarking i t as a key atom i n t h e r e a c t i o n c o r e . Remaining symmetries i n R - m a t r i c e s , t h e n , a r e r e s o l v e d on t h e f i r s t occurence o f d i f f e r e n t i a t i o n i n the course of i n c r e a s i n g s p e c i f i c i t y . I t i s worth remarking t h a t the d i f f e r e n t R - c a t e g o r i e s govern w i d e l y d i f f e r e n t p o p u l a t i o n s o f known r e a c t i o n s . One obvious example i s t h e simple f o u r c e n t e r r e a c t i o n of t h e k i n d A-B + C-D = A-C + B-D, which i s i n c i d e n t a l l y r e p r e s e n t e d by t h e s i m p l e s t R-matrix i n c l o s e d s h e l l c h e m i s t r y t h a t has a zero d i a g o n a l . We have s t a r t e d t h e d e s i g n o f a model program t o e x t r a c t i r r e d u c i b l e R-matrices from known r e a c t i o n s , t o put them i n t o c a n o n i c a l o r d e r , and t o s e t t h e sequence o f references i n a h i e r a r c h i c a l f i l e , that i s organized i n a s i m i l a r way as s u b s t r u c t u r e f i l e s d e s c r i b e d i n (5.1.3).

References 1.

J . D u g u n d j i , I . U g i , T o p i c s C u r r . Chem. 39, 19 (1973) see a l s o : I . U g i , P. Gillespie, Angew. Chem. 83, 980, 982 (1971), I b i d . I n t e r n a t . Edit. 10, 914, 915 (1971).

2.

I . U g i , P.D. Gillespie, C. Acad. Sci. 34, 4 l 6 (1972).

3.

J . Blair, J . Gasteiger, C. Gillespie, P. Gillespie, I . U g i in "Computer R e p r e s e n t a t i o n and M a n i p u l a t i o n o f Chemical I n f o r m a t i o n " , Ed. W.T. Wipke, S. H e l l e r R. Feldmann, E. Hyde, W i l e y , N.Y. (1974).

4.

J . B l a i r , J . Gasteiger, C. Gillespie, P.D. G i l l e s p i e I . U g i , T e t r a h e d r o n 30, 1845 (1974).

Gillespie,

Trans.


N.Y.


2. BRANT ET AL. Computer

Programs

for Chemical

Problems

59

5.

I . U g i , J. G a s t e i g e r , J. B r a n d t , J . F . B r u n n e r t , W. S c h u b e r t , IBM-Nachr. 24, 185 (1974) I. U g i , IBM-Nachr. 24, 180 (1974).

6.

W.T. Wipke, T.M. D y o t t , J. Amer. Chem. Soc. 96, 4825 (1974), and earlier r e f e r e n c e s .

7.

E . J . Corey, W.J. Howe, H.W. O r f , D.A. Pensak, G. P e t e r s s o n , J. Amer. Chem. Soc. 97, 6116 (1975), and earlier references.

8.

H. G e l e r n t e r , N.S. S r i d h a r a n , H.J. H a r t , S.C. Yen, F.W. F o w l e r , H.J. Shue, T o p i c s C u r r . Chem. 41, 113 (1973).

9.

M. R a n d i ć , J o u r n . o f Chem. Inform. and Comp. Sci., V o l . 15, No. 2, 1975 (105).

10. S.W. Benson, Thermochemical Kinetics, W i l e y , N.Y. (1968), S.W. Benson e t al. Chem. Rev. 69, 279 (1969). 11. T.L. Allen, J . Chem. Phys. 31, 1039 (1959), A . J . K a l b , A.L.M. Chung, T.L. Allen, J . Amer. Chem. Soc. 88, 2938 (1966). 12. J . G a s t e i g e r , P. Gillespie, D. Marquarding, T o p i c s C u r r . Chem. 48, 1 (1974).

I. Ugi,

13. R.E. B u r k a r d , Methoden d e r g a n z z a h l i g e n Optimierung, S p r i n g e r - V e r l a g , Wien-New York (1972).

Acknowledgements

We acknowledge g r a t e f u l l y or work by Deutsche Fonds d e r Chemischen

the f i n a n c i a l

support o f

F o r s c h u n g s g e m e i n s c h a f t and Industrie.


Computer Programs for the Deductive Solution of Chemical Problems

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