Bull. SOC. Chim. Belg. vol. 84/n05/1975

ON THE RELATION BETWEEN ISOMERIZATION MODES

FOR IDEALIZED AND DISTORTED SKELETA

by J. Brocas, R. Willem', D. Fastenakel** and J. Buschen***

Collectif de Chimie Organique Physique, Faculte des Sciences

Rece ived 2 1 / 2 / 7 5 -Accepted 2 0 / 5 / 7 5 .

Universite Libre de Bruxelles, av. F.D. Roosevelt,50 B-1050 Brussels - Belgium

Summary : Permutational isomerization reactions [11[*1 may occur in two extreme situations. If the n ligands do not distort the symmetry of the

and will distort the ske+etal framework. In this case, the allowed permutations belong to S , a subgroup of Sn and have to be classified [2] according to G' , the syndetry group of the distorted skeleton. In this paper, the relation between these two descriptions is analysed and illustrated on examples of chemical interest.

I. INTRODUCTION Permutational isomerization reactions may be described by using two permutation groups [1], [3] i) The symmetric permutation group of the n ligands Sn ii) The point group G expressing the symmetry of the skeleton and related to the group of proper rotations A through G=A+aA, where a is any improper symmetry operation. G is a subgroup of Sn. If the differences in chemical nature between the n ligands may be considered as vanishingly small, then the geometry of the skeleton is independent of the distributions of the ligands on the skeleton : we will say that such ligands do not distort the idealized geometry of the skeleton[4! When the above conditions are fullfilled, the symmetry operations G provide a classification principle for the permutations of Sn : an isomerization mode [5] M(xi) has been defined as the set of permutations of Sn which are symmetry and/or rotationaly equivalent to a permutation x of Sn [3] :

M(xi) = (AxiA) U (Aaxio-lA) (1)

where BUC means union of the complexes B and C. In this way the group Sn is partitioned into an union of disjoint modes M(xi).

* Also at the Vrije Universiteit Brussel Dienst voor Algemene en Organische Scheikunde Fakulteit der Toegepaste Wetenschappen

** Aspirant at the Fonds National de la Recherche Scientifique *** Boursier de Specialisation at the Institut pour 1'Encouragement de la

Recherche Scientifique dans 1'Industrie et 1'Agriculture.

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However as soon as the differences in chemical nature of the liqands increase, significant departure from the idealized geometry is to be expected : each ligand distribution within a given ligand partition will induce a distorted geometry whose symmetry is lower than the idealized one and characteristic of the isomer corresponding to this ligand distribution. For this reason, various distorted symmetries G' will correspond to any idealized G. Moreover, in many cases of chemical interest, some ligands will occupy specific skeletal positions or specific relative positions on the skeleton. If the differences in the ligand chemical natures are sufficiently strong, only the ligand distribution leading to the most stable molecule will be realized : we will speak of blocked molecules. In this case, the group of the allowed ligand permutations[21is no longer Sn but SA , a subgroup of Sn (see 111. Examples). If one expresses the different groups Sn,SA, G and G o as permutations of the liqands on the labelled skeleton sites", then it is possible to the point group G' of the distorted symmetry as :

[ 4 ]

G ' = S ; ~ G (2)

where A n B means intersection of A and B. In general G' is a subgroup of both SA and G. We will comment later (see Examples - Octahedra) on the limitations of this formula. In equation (2) either (a) G is a subgroup of SA or (b) it is not. Two different situations result :

(a) G' = S'nG = G

(b) G ' is a proper subgroup of G. ( 3 )

11. THEORETICAL DESCRIPTION Let us now discuss the relations between modes for idealized and distorted skeleta in both situations (a) and (b). We do it on a classification of isomerizations which corresponds to Klemperer's ['I definition of "a set of non differentiable permutational isomerization reactions in an achiral environment" (or "racemic modes" [3i ) . This classification coincides also with the equivalent basic permutational sets of reference "1 for example :

E(xi) = G xi G ( 4 )

The same discussion could be performed starting from formula (1) but it is expected to be more complicated. Situation (a) is the simplest one. Indeed, since G is a subgroup of SA and of Sn, these two groups may be partionned into double cosets GxiG :

sn = ui A(Xi) + Uk B(x,) ~

* For instance, the cyclic permutation ( 1 2 3 ) will mean : ligand on site 1 replaces ligand on site 2, ligand on site 2 replaces ligand on site 3, ligand on site 3 replaces liqand on site 1.

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These two equations mean that any set A(xi) contained in SA is also found in Sn. Some sets E(xk) of Sn are however missing in SA : if one restricts Sn to the allowed permutations SA one suppresses completely (in situation (a)) some isomerization modes A(xk) without affecting the others R(xi). The connectivity of mode z(xi) defined by ( 4 ) and for the idealized skeleton is given by [8] :

(6) ]GI 6i =

]xi1 G xinGI

For the distorted skeleton it has the same value since G'=G. Therefore, with both idealized and distorted skeleta, the same set of enantiomeric pairs will be reached from a given one through one step of mode R(xi). Situation (b) is more difficult to analyse. First of all (5a) is no longer true since G is not a subgroup of SA. However, because it is a subgroup of Sn, one has

Moreover

with the definition

w'(x') = G' x' G' q 9

Equation (8) holds since G' I s a subgroup of G. We have now to restrict again Sn to the allowed permutations SA. Equation (7) may be written

= u s;n A(x.1 j [ 3 1

Since G' is a subgroup of SI, and G, one has (see ( 8 ) ) :

It should be stressed that in situation (b) , the number of sets A' (x') in (11) is smaller or equal to the number of sets R'(x') in ( 8 ) . This means

9 that the parent mode H(x.) splits into submodes R'(x') due to a symmetry

3 9 lowering from G to G' and that some of these submodes could vanish (see (11)) when the allowed permutations S; only are taken into account. This time a given mode M(x.) could be partially, totally or not affected by the exclusion of not allowed permutations (in contrast to (Sa)). As a consequence of (ll),,it is possible to show that

P

I

6 a L 6' (12) j P p

an equation meaning that the connectivity of the parent mode 6 j (idealized symmetry) is bigger or equal to the sum of the connectivities of the

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submodes into which it splits. This is done in Appendix A.

111. EXAMPLES The above statements are now illustrated on some examples of chemical interest. 1) TEIgs98LLieyz8mids This idealized skeletal geometry G = D3h is common for non-rigid

Complexes of MA5 type and Dgh symmetry have been studied by NMR - line shape analysis[lq . The experimental spectra are only compatible with mechanisms whose permutational result belongs to G(24) (35)G (see Table I, first column). In the ligand partition MA4B (SA = S4S1), two situations are possible : B prefers the axial position or the equatorial one according to its electronegativity difference relative to A 1111 . The experimental study of non-rigidity has been performed for molecules of both types and using again NMR-line shape analysis . B equatorial : (CH3) 2NPF4 c12'1iJ ; C1PF4 [13] bearing C2v symmetry.

B axial : HM(PF314 (M=Co, Rh, Ir); HM(PF3)i (M=Fe, R u , 0 s ) ;

entacovalent MA molecules (M = P,As) and has also been pointed out for M = Sb 791 .

5

HM P(0C2H5I3l4 (M=Co, Rh) [141 L The molecules with axial B do not lead to mechanistic information, since, as quoted in Ref [14] every mechanism of non rigidity leads to permutations of G'(24)G'. (see Table I, last column) The molecules with equatorial B show a line shape only compatible with permutations of G'(24)(35)G' (see Table I, second column). It should be noted that C2v symmetry of equatorial situation and the observed point group[14]of the axial structure (H on a pseudotetrahedral face of the MA4 framework) may both be described by formula ( 2 ) which leads to G' groups respectively isomorphic to CaV and Cjv (see Table I below). In the ligand partition MA3B2 (S; = S3S2) the B ligands can be diaxial, diequatorial or axial-equatorial. From the experimental point of view, the diaxial situation is not interesting because all the permutations of S '

belong to the same G'xG' mode. Indeed, the condition characteristic of case (a) situation has bean analysed recently for PF3H2 b5i, a molecule of C2v symmetry. Its line shape analysis is compatible with a mechanism of G'(14)G' (Table 11, third column). The basic permutational sets for the diaxial, diequatorial or axial-equatorial species of Cs symmetry have been given in Ref [14] .

(G is a subgroup of SA) reduces here to G = SA . The diequatorial

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a

J? m

w W v

W 4

II '0

W rl

II '0

W .-I

w II

II w

:: W

- - m - - m m - 4 A m 4 * V 2.2. II

4 W - m

m II m -

d

m

II W

W

!I

W

II

W W - w N - W

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w

rl

E II

:: - L o

W

w

4

II Lo

E R u

I( u

I I - I m I . $ I N I - I . I - I d I m

N I - - I - - 1 - LD I m 3 I - - I .

;; ; c !

I - w ; u

m

u

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In Tables I and 11, the various distorted skeleta corresponding to the trigonal bipyramidal idealized symmetry and for two ligand partitions have been given. The G' group for each case derives from (2) with

G = I, (123), (132), (12) (45), (13) (45), (23) (45) (45) t(123) (45)t (132) (45) t (12) r (13) 8 (23)

The sets GIG, G(24)(35)G,G(24)G of the first column contain 12, 36 and 72 permutations of Sn respectively. They correspond to E+PgI P1+P4, P2+P3 [16,17] and to the basic permutational sets E, A , B of Ref . AS

indicated at the top of each column, the only situation where G is a subgroup of SA (Equation 3(a)) is the diaxial geometry of Table 11. It is also the only case where the connectivity of each mode of the distorted geometry is either zero or equal to the connectivity of ttie corresponding mode of the idealized geometry. The four other situations correspond to Equation 3(b). For each of them, Table I and I1 provide a list of permutations (Gx.G)nSA common to the Gx.G complexes of the left column and the relevant SI;. For example, for axial B , Table I gives the permutations of GIG, G(24) (35)G and G(24)G contained in S4(1234)' S1(5). (see [2]) For equatorial B I (24) (351, (25) (34) I (2435) and (2534) are the permutations of G(24) (35)G contained in S4(2345) S1(l) The set (Gx.G)nSA have moreover to be classified into G'x G' submodes

3 q (see Equation ( 8 ) ) . It happens that a given set (Gx.G)nSA splits into much submodes. For instance, in Table 11, (G(24)G)nS3(234) S2(15) splits into G' (24)G' and G' (15)G'. In any case it is easy to verify, for each distorted geometry, that the connectivity of a Gx.G mode is bigger or equal to the sum of the connectivities of the submodes into which it splits. Finally, the capital symbols E , A , B , C refer to the basic set numbering in Ref [14] . Unfortunately, they do not reflect the relation between parent modes and submodes.

[lo]

3 ?

(see Table I, equatorial B).

3

3

2) Q&&&g

The non-rigidity of hexacoordinate molecules has been studied experimentaly on H ML complexes (M=Fe,Ru and L represent various phosphorus ligands)

2 - 4 [18,19J. Table I11 is constructed in the same way as I and 11. The first column refers to idealized octahedra (symmetry group G = Oh whose permutational expression is easy to obtain). The double cosets GIG, G(13) (24)G, G(34)G correspond respectively to E+P8,P6+Pg,P, of Ref[16] differentiable permutational isomerization reactions of Ref [l] . We do not consider the ligand partition MA5B. It should lead to the same distorted symmetry group and mode classification as for the undistorted tetragonal pyramid. (Table IV, first column). The second and third columns correspond to cis and trans isomers. The dihydride systems where the cis isomer is the only present are rather commun but dihydrides with only trans isomers are exceptional [18,19] . We give the G' groups for both situations. They are derived from formula(2). For the cis dihydrides a pseudotetrahedral array of the ML4 framework of

and to the three

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Cav symmetry has been proposed [18, 191. This is consistent with formula ( 2 ) .

For the trans situation, formula (2) leads to a group isomorphic to Dqh. In fact an additional distortion giving rise to the lower D2d geometry has been suggested 1193. This illustrates the fact that formula (2) accounts for the minimum distortion due to the ligand influence on the idealized skeleton. In specific cases, the actual distorsion may be more important. For the trans situation the group S; splits into two double cosets G'IG' and G'(34)G' whose connectivity is lower or equal to the connectivity of the parent double cosets GIG and G(34)G in the idealized geometry, whereas G(13) (24)G vanishes for this case. This situation corresponds to equation (3b). The cis geometry also obeys (3b). It is seen that G(13) (24)G splits into two submodes G' (13) (24)G' and G' (13) (56)G' containing the basic permutation sets I and I11 G' (13)G' and G' (34)G' which contain IV and I1 of the parent mode is bigger (or equal) to the connectivity of the submodes into which it splits

1191 respectively. In the same way G(34)G splits into [19] . Again the connectivity

6E = 1

*I + *I11 < 8

Tetragonal pyramidal geometry has been suggested to be the structure of pentaphenyantimony in the solution state[20]. In the ligand partition MA4B, B will be axial (or basal) if it is more electropositive (or more electronegative) than A The first column refers to the undistorted skeleton whose labelling and symmetry group are given (apical view of the skeleton). The double cosets GIG,G(25)G,G(23)G and G(23) (451G correspond respectively to E+A, C+D, B and F+G of Ref [16]. The next column describes a skeleton with apical B. Its G' and S;I groups satisfy condition (3a). Therefore double cosets either disappear in this case (like G(25)G and G(23) (45)G) or remain untouched (like GIG and G(23)G). Such a situation has been studied theoretically by Faller and AndersonL221and experiments have been performed on related systems. [22,23] Brunner and Hermann [23] as well as Musher and Agosta [24] have interpreted these results in terms of G'(23)G' modes or of the alternative three stepreverse Berry pseudo-rotations. Finally the column at the extreme right describes a skeleton with basal B, corresponding to equation (3b). Except GIG and G(23)G the modes for idealized skeleton split into submodes, the sum of their connectivities being smaller or equal to the connectivity of the parent mode. It should be stressed that among the three modes G' (23) (45)G', G' (235)G' and G' (253)G' issued from G(23) (45)G, only G' (23) (45)G' is self-inverse whereas the two others are not. Some comments relative to this situation are given in the Appendix B.

[2l] . These two situations are described in Table IV.

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1

.~

~~

.

G' (

13

)G1

=(1

3),

(23

1,

(14

1,

(24

) (1

23

4)

(56

), (1

43

2)

(56

1,

(13

42

) (5

61

,

4j1;:

2

G'

(34

)G1

=(3

45

6),

(36

54

1,

(35

46

), (

36

45

) (3

45

), (

34

61

, (3

56

1,

(45

6)

Sn =

S

6(1

23

45

6)

G=

O h

GIG

=

G

6=

1

~~ G

(13

) (2

4)G

6=

8

G(3

4)G

6=

6

(111)

I

Ta

ble

I11 : I

de

ali

ze

d s

ke

leto

n : o

cta

he

dro

n.

Dis

tort

ed

sk

ele

ta for M

A4B

2

I +b

9

N

I

14

'E

l3

GIG

= G

6=

1

G(2

5)G

6

=4

G(2

3)G

6=

2

G(2

3)

(45

)G

6=

8

S; =

S4

(12

34

)S1

(5)

G'

= G

G'I

G'

= G

6=

1

G'

(23

)G'

=

(23

), (

12

), (

14

1,

(34

)

(14

21

, (2

43

1,

(13

21

, (1

43

)

(12

4)t

(2

34

) r (1

23

)I (1

34

) (1

32

4),

(1

42

31

, (1

34

2),

(1

24

3)

= G

(23

)G

6=

2

AmA

EA

S; =

S4

(2

34

5) S

1 (1

)

G'

=

I,(2

4)

G'I

G'

= G

'

6=

1

G'

(23

)G'

=

(23

1,

(34

), (

23

4),

(2

43

)

6=

2

6=

2

QH

r

a

rt

pl

O

F

MI-

rt

N

mm

a

a

am

r a

Y

W a F!

IV. CONCLUSIONS According to the differences in chemical nature existing between the ligands, two different descriptions of permutational isomerization reactions have been given. The first one uses the full permutation group of the n ligands Sn and the symmetry group G of the idealized skeleton[l,3] . The second one uses an appropriate subgroup S,', of Sn (allowed permutations) and the symmetry group G' of the distorted skeleton (a subgroup of G) [Z] . We have shown that the second description may be obtained from the first one through simple group theoretical arguments. This establishes a correspon- dence between isomer counting and classification of permutational isomeriza- tion reactions in both langages. In fact, if equation (3a) is satisfied, isomerization modes for the idealized skeleton either disappear completely or coYncide with isomerization modes of the distorted skeleton and have, in this case, the same connectivity. If (3b) is satisfied, each parent mode of the idealized skeleton splits into submodes of the distorted skeleton. We have shown, in Appendix A, that the sum of the connectivities of the submodes is smaller or equal to the connectivity of the parent mode. We think that these considerations show some of the relations existing between two apparently different classification principles for permutational isomerization reactions.

APPENDIX A : The relation between idealized and distorted skeleta Any double coset $(x.) or 8'(x') may be expressed as a union of cosets : 1 q

M'(x') = U G' y' g I 3 B

(A. 2 )

each coset GYa (or G'y;O representing a pair of enantiomers of the idealized (or distorted) geometry. The connectivity given by (6) is the number of enantiomeric pairs reached in one step of a given mode, i.e. the number 6. (or 6 ' ) of Gy, (or G'y;) cosets in the double coset 8(x.) (or R'(x'f). 1 q 1 4 We start from equation (10) and we want to analyse the properties of s;lnM(x.). Using (~.1) :

3

A given intersection S' n Gy, may be empty. This means that, in the blocked molecule, described by the group of allowed permutations S,',, the pairs of enantiomers corresponding to the coset Gy, is missing.(For example, in the ligand partition MA4B of the trigonal bipyramid, if one uses a S,', group expressing that B is apical, then the intersection SAnGy, is empty each time Gy, corresponds to an isomer with equatorial B. See examples - Trigonal Bipyramids) . If Sl;n Gya is not empty, let pr be one of its elements. One has :

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Let now g; be an element of G' = G n S,!, . From (A.4) it follows that grl;pr belongs to both S,', and GY . Therefore, the intersection St;n Gyk contains at least IG'I distinct elements if it is not empty. Let us show that it contains exactly IG'I distinct elements or, in other words, that if pr and p, both belong to st;nGYk they are necessarily related to each other by

(g'tG') Pr = 4;1Ps 9 Indeed, starting from (A.4) and from

= s; ; p, = gvYa

-1 -1

(g$G; S ~ E Sl;) PS

PrPs 6 SI; and PrPs E G one obtains

or -1

a relation equivalent to (A.5)

PrPs t G'

( A . 5)

(A.6)

( A . 7 )

Conclusion : a coset GYa (pair of enantiomers in the idealized symmetry) limited to the "allowed permutations" of St; may either disappear or generate a coset G'Y'~ (see (A.5)) corresponding to a pair of enantiomers in the distorted geometry. As a consequence, Sl;nfi(x.) contains a number of G'y' enantiomers of distorted geometry) which is smaller or equal to the number of Gya cosets of g(x.1 (pairs of enantiomers of idealized geometry). This statement is shown by equations (11) and (A.2) and is equivalent to (12). The above "Conclusion" provides a bridge between the group theoretical treatment of stereoisomerizations and physical intuition leading to the suppression of isomers which has been proposed earlier [25] to describe systems of distorted geometry (blocked ligands).

cosets (pairs of 1 6

I

APPENDIX B : Non self-inverse modes In the case of tetragonal pyramids, the mode G(23) (45)G for the idealized geometry CqV is splitted into three submodes G'(23)(45)G', G'(235)G' and G'(253)G' for the distorted geometry Cs (basal blocking). In G'(23) (45)G' the inverse of each of its elements is present; it is self- inverse. However G'(235)G' and G'(253)G' do not possess this property :

to each element of one of these double cosets corresponds an inverse in the other one.(see Table IV). Thus, permutations such as (354) and (345) are considered as non equivalent in this classification. Let u s imagine a mechanism consistent with (345) : it is completely defined if we know any internal distance or angle as a function of the time t. (See Fig I, t varies from 0 to T). Of course, the ~ a m e mechanism is consistent with its inverse (354) but now t varies from T to 0.

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t E n e r g y

Symmetry

AxA

GXG

( A ~ A ) u (~oxu-l~)

Fig.1. Energetic profile for a mechanism consistent with (345) and (354)

Ref Extended Ref

111 ( A ~ A ) u (AX-~A) P 63 [i ,3] (GXG) u (GX-~G) [3 1 ( A ~ A ) u (A~-'A) [4,6,27]

u (~ox0-l~)

u (~ox-lu-l~)

Even if the extended principles are well adapted to describe specific experimental situations, it is true nevertheless that the symmetry classification describes more detailed properties. Indeed, there exist no symmetry reason imposing that the maximum of the energy profile (Fig.1) should happen for t = T/2. Therefore, to the same mechanism, there correspond two ways-(345) and (354)-to trace this profile. They are not equivalent, according to the symmetry classification but equivalent as far as rate constants are concerned. It should be noticed that, in the case discussed by Klemperer[6], non self-inverse AxA double cosets give rise to self-inverse complexes by considering either (AxA)U (Ax-lA) even GxG complexes are not self-inverse and (GxG)U(Gx-lG) has to be constructed.

GxG double cosets. In this case,

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ACKNOWLEDGMENTS Two of us (D. Fastenakel and J. Buschen) want to thank the F.N.R.S. and I.R.S.I.A. respectively for financial support. We thank Professor A.H. Cowley for having sent us a preprint of his paper. We also acknowledge discussions with Professors M. Gielen and J. Nasielski.

REFERENCES [l] W.G. Klemperer : Journal of Chem.Phys. 56, 5478 (1972) 123 W.G. Klemperer : 1norg.Chem. 11, 2668 (1972) [3] E. Ruch and W. Hasselbarth : Theor.Chim.Acta 2, 259 (1973) [4] W.G. Klemperer : J.Am.Chem.Soc. 94, 6940 (1972) [5] J.I. Musher : J.Am.Chem.Soc. 94, 5662 (1972) [6] D.J. K1einandA.H. Cowley : J.Am.Chem.Soc. (to appear, 1975) [7] J.P. Jesson and P. Meakin : Acc.Chem.Res. 6 , 269 [8: W.G. Klemperer : J.Am.Chem.SOc., 94, 8360 (1972) L9j

(1973)

R. Luckenbach : Dynamic stereochemistry of Pentacoordinated Phosphorus and Related Elements, Georg.Thieme Publishers, Stuttgart 1973.

[lo] P. Meakin and J.P. Jesson : J.Am.Chem.Soc. 95, 7272 (1973) [ll] H.A. Bent : Chem.Rev. 61, 275 (1961) ;12] [13] E. Eisenhut, H.L. Mitchell, D.D. Traficante,

G.M. Whitesides and H.L. Mitchell : J.Am.Chem.Soc. 91, 5384 (1969)

R.J. Kaufman, J.M. Deutch and G.M. Whitesides : J.Am.Chem.Soc. 96, 5385 (1974) P. Meakin, E.L. Muetterties and J.P. Jesson : J.Am.Chem.Soc. 94, 5271 '141

' (1972) 1151 J.W. Gilje, R.W. Braun, A.H. Cowley : J.C.S. Chem.Comm. 15 (1974)

116; M. Gielen andN. Van Lautem : Bull.Soc.Chim.Be1ges 79, 679 (1970) .17. J. Brocas and R. Willem : Bull.Soc.Chim.Be1ges E, 469 (1973) '181 P. Meakin, E . L . Muetterties, F.N. Tebbe and J.P. Jesson : J.Am.Chem.Soc.

93, 4701 (1971) - [IS]

(1973) :20] I.R. Beattie, K.M.S. Livingston, G.A. O z i n and R. Saline : J.Chem.Soc.

(London) Dalton Trans. 784 (1972) 121: R. Hoffmann, J.M. Howell, E.L. Muetterties : J.Am.Chem.Soc. 94, 3047

(1972)

[22] J.W. Faller and A . S . Anderson : J.Am.Chem.Soc. 92, 5853 (1970) [231 H. Brunner and W.A. Herrmann : Chem.Ber. 106, 632 (1973) :24] J.I. Musher and W.C. Agosta : J.Am.Chem.Soc. 96, 1320 (1974) r25] M. Gielen, Med.Vl.Chem.Veren z, 185,201 (1969) '26: G.M. Whitesides, M. Eisenhut and W.M. Bunting : J.Am.Chem.Soc. 96,

'27; J. Buschen, MBmoire de Licence U.L.B. (octobre 1974).

P. Meakin, E.L. Muetterties and J.P. Jesson : J.Am.Chem.Soc. 95, 75

5398 (1974)

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