Structural aspects of the allotropy of sulfur and the other divalent elements

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STRUCTURAL ASPECTS OF THE

ALLOTROPY OF

SULFUR AND THE OTHER

DIVALENT ELEMENTS

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STRUCTURAL ASPECTS OF THE

ALLOTROPY OF

SULFUR AND THE OTHER

DIVALENT ELEMENTS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP-PEN AAN DE TECHNISCHE HOGESCHOOL DELFT OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. C. J. D. M. VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECH-NISCHE NATUURKUNDE, VOOR EEN COM-MISSIE U I T DE SENAAT TE VERDEDIGEN OP WOENSDAG 22 NOVEMBER 1967 TE 14 U U R

DOOR

FOKKE TUINSTRA

natuurkundig ingenieur

geboren te Assen

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DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR: PROF. DR. J. A. PRINS

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Yn neitins oan üs heit Foar Pieter

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PART I A-PRIORISTIG CONSIDERATIONS

CHAPTER 1

G E N E R A L I N T R O D U C T I O N

1.1 The polymorphism of sulfur and its homologues

Compared with the other elements sulfur shows a large number of polymorphs. A recent compilation (Meyer, 1965) includes some thirty-odd modifications occurring in scientific literature. Quite a few of these may not bear a critical re-examination, but on the other hand, at least three new ones have recently been added to the list. Even the following conservative estimate (table I) shows how sulfur stands out among the elements surrounding it in the periodic system.

Table I. Number of crystalline modifications of the elements in the right-hand portion of the periodic system

Atomic numbers 6 7 8 9 14 15 16 17 32 33 34 35 50 51 52 53 82 83 84 10 18 36 54 Elements C N O Si P S Ge As Se Sn Sb Te Pb Bi Po F Ne CI Ar Br Kr I Xe Number of modifications 2 2 4 1 1 2 4 8 1 1 1 2 3 1 1 2 2 1 1 1 1 1 2

In table II more detailed data on the distinct modifications are presented, elucidating particularly the atomic neighbourhood relations in the correspon-ding structures. Table II shows clearly the correlation between the large number of modifications and the large number of different molecular forms for the elemental sulfur as compared with the surrounding elements. The large number of molecular forms evidently is one of the main reasons for the poly-morphism of sulfur.

In the present treatise this structural aspect of the polymorphism is examined in detail for the elements in the right-hand side of the periodic table. In this chapter the reasons leading to the different numbers of molecular forms and their connection with the number of the crystalline arrangements will be dis-cussed briefly. In the subsequent four chapters of the book these ideas are elaborated, using geometrical models and knowledge of the chemical bonds. In the second part of this treatise these considerations are compared with ex-perimental findings.

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Table II. Further particulars on molecules, atomic coordinations and lattices of the elements of table I *) Column Valency Molecules Coordination number - . f number L^'»"" itype Molecules Coordination number - . ƒ number L»"''^^ {type Molecules Coordination number T ,,. f number L»"'^^ ( t y p e Molecules Coordination number T . f number Lattice . _ Itype Molecules Coordination number ^ . t number L^«'" {type IV Tetravalent

c„-3 A 9 Si, •? 1 H G e „ . 4 1 Sn„> 4 1 A 4 P b „ ' 12 1 Co,' 4 A 4 S i „ ' 4 A 4 Sn • 00 (4+2) A 5 V Trivalent N , 1 2 M P4 3 2 M A s j 3 I M Sb, 3 1 M Bi • 3 1 P » ' Po.' 00 CD 3 3 1 I A A 7 17 As • CO 3 A 7 Sb • OD 3 1 VI Divalent 0 , 1 3 M

s.

2 1 M Se, 2 2 M Tc • OD 2 1 A 8 P o „ ' 6 1 R 0 , 4/3 1 M Sg S,j S„> 2 2 2 3 1 3 M M M Se„' 2 A 8 T e „ . 6 1 R POoo' 6 1 K VII Mono-valent F , 1 1 M CI, 1 1 M Br, 1 1 M h 1 1 M VIII Zero-valent Ne 12 A Ar 12 1 A 1 Kr 12 1 A Xe 12

K

*) The symbols 0 0 ' , 0 0 ' and 0 0 ' refer to "molecules" extending throughout the total structure in one, two or three dimensions, respectively; for the corresponding lattice types the notation of the "Strukturberichte" of Ewald and Hermann has been used. M means a molecular lattice; R and K indicate simple rhombohedral and simple cubic, respectively; H means hexagonal. The data on the elements in the column V I will be dealt with in subsequent chapters; the other data are taken from:

Gmelins Handbuch, achte Auflage, Verlag Chemie, Weinheim.

Pascal, P., "Nouveau traite de la chimie minerale" Masson et Cie, Paris, 1956. Wyckoff, R. W. G., "Crystal Structures" Vol. I, Interscience Publ., New York, 1965.

T h e elements contained in a single column of table I I (or I) have similar configurations of the outer electronic shell. T h e elements of column V I , for instance, have electronic shells complete but for two electrons. In the ground state the term symbol for these elements is ^PE , indicating that the two unpaired electrons are situated in two different p-orbitals. This explains why these ele-ments are called divalent: the atoms will preferably form only two covalent bonds with adjacent atoms. In table I I this preference is expressed by the atomic coordination number. I n an analogous way the other atomic coordination numbers 0, 1, 3 a n d 4 correspond in table I I to the other valences.

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Let us consider briefly the a priori possible molecular conformations for each of the elements in table I.

The only molecular forms generated by the z e r o v a l e n t and m o n o v a l e n t elements are the single atom (inert gas) and the diatomic molecules (halogens), respectively; both "molecules" are objects which are highly symmetrical.

For the d i v a l e n t e l e m e n t s the number oï a priori possible molecular forms seems to be much higher. These elements, sometimes called the c h a l c o g e n s , can in principle form linear molecules of arbitrary extent, which may close to form ring molecules; otherwise they become infinitely long chains. A great many of these hypothetical molecular conformations will be ruled out on ste-reochemical grounds; others do not easily fit into a crystalline packing. In principle, however, all ring molecules can be packed in a crystallographic way without penetrating each other. Of the linear chains, only the ones with straight axes (which are linear in the c r y s t a l l o g r a p h i c s e n s e of t h e w o r d ) can be arranged in a crystalline packing. Therefore, with the exception of the "non-straight" types of chains, molecular lattices can be formed with any of the molecular types. Since the orientation in space of the covalent bond depends on the orientation of the next nearest bond in the chain, the permissible mol-ecules will have a staggered shape. In other words, the expected molecular conformations have a low degree of symmetry.

In an analogous way it can be seen that only two-dimensional structures can be formed if to each atom t h r e e valencies are available. These "surfaces" can be extended to infinity in one or two directions, or they can form closed "cages". If saturation of all valencies is postulated, finiteness of a non-closed structure is incompatible with the equality of the atomic positions in the structure. T h e infinite (non-closed) structures can be arranged in a crystal structure only when they are f l a t i n t h e c r y s t a l l o g r a p h i c s e n s e of the word; in the latter case, they must be oriented parallel. Of the cage structures only the smaller ones can effectively fill space. These small cages are highly symmetrical as compared with the corresponding ones of the divalent elements.

For the t e t r a v a l e n t elements only three-dimensional space-filling structures may be expected. T h e molecular lattices of the preceding elements are replaced here by covalent c o o r d i n a t i o n l a t t i c e s (graphite is not tetravalently bond-ed).

From the afore mentioned reasoning it may be expected that the elements included in column V I will have a large number of possible molecular con-formations.

Let us now review the possible crystalline arrangements of the above-mentioned molecular forms, in order to obtain a rough estimate of the polymorphism.

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by means of the London dispersion forces only. These forces will be taken to be central forces between a t o m s . The dispersion forces increase from element to element if we go down along a column of table I I i ) ; finally they become metallic forces. For reason of these central forces, any atom will prefer to be surrounded by the greatest possible number of non-covalently bonded neigh-bours.

This simple argumentation would result in the closest packings of spheres, when applied to the zerovalent elements. Due to the symmetry of the diatomic molecules, the possible number of packings for the monovalent elements is small. The same reason limits the number of possible packings of the trivalent elements. Due to the staggered shape of the molecules of the divalent elements, however, each of the possible molecular types can be packed in various man-ners, slightly different from each other and corresponding to slightly different lattice energies ^). If, for each of the elements of the sixth column, similar (hypothetical) molecules are packed in a similar crystalline manner, the lattice energies will be proportional to the dispersion energies, as are the energy differences between the slightly different packings. Thus, going down from oxygen to polonium a decreasing polymorphism may be expected. This is indeed observed in the tables I and I I , with the exception of oxygen. With the other elements in the first row of the periodic table, oxygen has in common its preference for a m u l t i p l e b o n d . In this treatise we will accept this feature as a fact, without attempting to explain it. Taking into account this feature, t h e e l e m e n t s u l f u r w o u l d a p p e a r t o s h o w m o r e p o l y m o r h p i s m t h a n t h e s u r r o u n d i n g e l e m e n t s i n t h e p e r i o d i c t a b l e .

In chapter 2 a mathematical analysis is presented of the symmetry of geomet-rical molecular models for the divalent elements as well as for the trivalent elements. In chapter 3 and 4 conformational calculations are given for the divalent elements taking into account bond parameters selected in chapter 2. In chapter 5 and 6 the conformations of the preceding chapters are com-pared with existing structures. New experimental data are used in chapter 6 to fill the gap in our knowledge about the configurations of the elemental sulfur polymers.

1.2 M a i n p o i n t s

(i) For the divalent elements, which are included in the sixth column of the periodic table, the number of a priori possible molecular

conforma-') The effect of the increasing dispersion forces is counteracted by the increase of the atomic radii to only a small extent.

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tions which can properly form a crystal seems to be h i g h e r than for the elements in neighbouring columns.

(ii) As compared with the elements in the neighbouring columns the sym-metry of the molecular conformations of the divalent elements is pre-dicted to be low.

(iii) The differences in lattice energy between analogous atomic arrange-ments of the successive elearrange-ments in a single column increase with the atomic number.

(iv) The elements in the first period do not fit into this scheme because of their preference for "contracted" multiple bonds.

(v) Accepting the above four points as postulates, the number of polymorphs may be expected to be large for sulfur as compared with the surrounding elements in the periodic table.

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CHAPTER 2

P O S S I B L E M O L E C U L A R F O R M S

2.1 Introduction

In the present chapter we shall examine the premises (i) and (ii) of section 1.2. In section 2.2 I shall give a survey of all possible molecular conformations of the divalent elements. T h e complete collection of possible conformations together with their symmetry can be enumerated if each configuration is represented by a distinct succession of two types of operations i ) .

T h e collection of possible molecular conformations of the trivalent elements can be enumerated with the help of the mathematical theory of polyhedra; this is done in section 2.3. For the monovalent, zerovalent and tetravalent ele-ments every time one conformation is possible as is shown in chapter 1.

In section 2.4 values for the bond parameters are selected, taking into account experimental data together with theoretical considerations.

2.2 D i v a l e n t e l e m e n t s

Introduction

I n the present section we shall introduce a formalism with which the possible molecular forms of the divalent elements can be classified. For this purpose we cannot do without some knowledge of the stereochemistry of the elements under consideration.

T h e angle between the directions of the two single covalent bonds of a diva-lent atom will be called the bond angle, /3. For each of these elements the value of this bond angle is practically independent of the kind of atoms with which the element is connected. For all chalcogens the value of the bond angle /3 is near to 100°.

For the ring molecules a constant value of /3 in itself is in general sufficient to enforce a non-planar configuration. From structural data about the chain molecules it is moreover known that the rotation around the single bonds of the chalcogens is hindered. Thus any sequence of four atoms of which the middle two are chalcogenic will be non-planar, i.e. the configurations are skew. T h e angle between two planes containing two successive triads of atoms will be called the dihedral angle y; its definition is illustrated in figure 1. For all the chalcogens the dihedral angle is about 90°, corresponding to two isomeric states.

1) The present treatment is presented in a slightly different form in an article (Tuinstra, 1967a).

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Fig. 1 Definition of the dihedral angle y;

the two possible configurations for four atoms corresponding to one value of y; the confi-gurations initiate a left-handed and a right-handed helical molecule, respectively. Restrictions

A systematic synopsis of all a priori possible molecular conformations of the divalent elements can only be given if we restrict ourselves slightly; some natural restrictions will now be given.

Excluding all excited and all ionized states for the molecules under conside-ration, it is reasonable to expect all bonds to be equal, that is: for all atoms the values of the bond distance d, and of the angles ^ and y are the same.

If now our treatment is confined to these cases, the number of conformations for a distinct number of atoms still remains large. With respect to its three predecessors any atom in a chain can make a choice out of two different positions (see fig. 1):

a) the fourth atom initiates with its three predecessors a right-handed screw; b) the fourth atom initiates with its three predecessors a left-handed screw. If n is the number of atoms forming a molecule, the number of possible different molecular conformations is now 2<"~3) (« > 2). Since we are interested here in crystalline modifications only, we can exclude all configurations in which the successive choices (a) and (b) are distributed at random. We will confine ourselves to those configurations in which a distinct sequence of choices (a) and (b) is indefinitely repeated i).

Because of the non-penetration of non-bonded atoms, a geometrical model

') Note that the treatment now also comprises all ring molecules; after a number of repetitions the "skeletons" may close in themselves.

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has much more effective space to its disposal than the real molecules have. This limitation of the effective space makes an elegant mathematical formula-tion impossible. We therefore will not take into account this "volume effect" until we enter into the ultimate calculations.

The motif

The repeating "unit" is a distinct sequence of choices out of (a) and (b); in the molecule this sequence corresponds to a row of successive atoms, the rela-tive positions of which are fixed if fi and y are kept constant. Such a repeating sequence of atoms will be called the motif or module. In general a motif will coincide with its successor in the molecular string after a translation combined with a rotation in space. This implies that in general these molecular confor-mations are h e l i c a l ones. In this argumentation r i n g molecules can be con-sidered as special helical molecules with pitch zero.

The molecular skeleton can be represented as a chain of vectors pointing from each atom towards its successor; these vectors will be called the bond vectors. Without loss of generality the length of these vectors can be thought to be unity.

Leaving their directions unaltered the successive vectors can be shifted to a common origin, which is taken to be the centre of a unit sphere. Three succes-sive bond vectors now form either a right-handed set or a left-handed set, ac-cording to the corresponding choices (a) or (b). The motif is now represented by a number of successive bond vectors. On the surface of the unit sphere the motif can be represented by a figure which appears when the end points of the successive vectors are connected.

Now since the vectors all start from one single point, in the present picture, the set of vectors representing the motif only needs to be rotated in space to make it coincide with its successor. Thus the entire molecule can now be re-presented by a number of successive rotations alone. If after a number p of these rotations the "head" of the figure coincides with the "tail", this means that in the real molecule the motif has returned to its original o r i e n t a t i o n in space. This corresponds to a regular translatory repetition length along the helical axis of the molecule. The directions of the screw axis and of the rotation axis of the "spherical" representation are the same.

The rotation resulting in the coincidence of the set of bond vectors, repre-senting the motif, with its successor will be called operation M. Closure of the figure on the sphere after p rotations then can be written as:

{M)P = E,

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For our treatment we will select only those conformations which can be represented by this equation.

This is, however, not a real restriction since if the rotation Af has been applied

p times, the total rotation may have gone q times through the angle 2v:. We

will call the number k = pjq the order of the screw axis of the helical molecule. The operation M transforms any vector in the motif into its image in the subsequent motif. Now the operation M can be factorized into the successive operations which transform the b o n d v e c t o r s into t h e i r s u c c e s s o r s . Ac-cording to the choices (a) and (b) we must distinguish now between two types of elementary operations transforming the three bond vectors 1, 2 and 3 into the triad 2, 3 and 4 respectively:

(i) the operation A, which does not reverse the handedness of the initial triad; this means that choice (a) is transformed into (a) and choice (b) into (b).

(ii) the operation A, which reverses the handedness of the triad, so A trans-forms choice (a) into (b) and choice (b) into (a).

T h e operation A is a.proper rotation, whereas the operation Ais a. rotary reflection, that is a combination of a rotation with a reflection in a plane normal to the axis of rotation. After a distinct sequence of these operations the original vector is transformed into its image in the succeeding motif. M is the product of these successive operations of type A and A. Figure 2 illustrates that the motif can be represented by this succession of operations as well as by the succession of the choices (a) and (b).

Now Af is a proper rotation; proper rotations can never transform a left-handed set of vectors into a right-left-handed set. Thus, the product of elementary operations, representing M, must contain the operation A an even number of times.

M

^ 1

Fig. 2 Correspondence between two representations of a motif; M = AAAAAA, represented schematically:

(i) as a sequence of left-handed and right-handed choices, (ii) as a sequence of operations of the types A and A

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We are able now to classify the motifs with respect to the corresponding number of elementary operations and the succession of the two types of them: A;AA; AAA{ - AAA = AAA); AAAA{ = AAAA = AAAA = AAAA ^ AAAA) ; etci).

Symmetry aspects

The symmetry of the molecular conformation will be reflected in the symmetry of all its representations; for instance, in the symmetry of the figures on and in the unit sphere and in the symmetry of the succession of the operations A and A. The symmetry of the latter is determined by the order k and the suc-cession within the motif. To visualize the symmetry of the sucsuc-cession of opera-tions, within the motif, we should arrange the successive operations in a circle. The symmetry elements of such an arrangement of the operations are:

(i) A twofold axis. The sequence of operations consists of two identical parts. This case is illustrated in figure 3. Both parts must contain the operation of type A an odd number of times. This requires both parts of M to be rotary reflections, we will denote these operations here by the symbol E, with (5)^ = M. Since the reflection and the rotation contained in B may commute it is easily seen that E and M have their axes of rotation in common. The planes of reflection corresponding to the operations B are normal to this axis. This implies that the projection on the rotation axis of the sum vector, corresponding to B, is

balan-Fig. 3 Circular representation of the motii AAAAAA with its symmetry elements: two mirror planes and a twofold axis.

') Note that two motifs are identical if their sequences of operations can be made to coincide after only cyclic transpositions.

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eed by the projected sum vector of the second E; so the pitch of the helix is zero.

Thus: the pitch of a "helical" molecule having a motif of the here mentioned type is zero, independent of the i n t e r n a l d e t a i l s of the motif.

For the corresponding figure on the sphere one half of the motif is coinciding with the second half, after a rotation followed by a reflection in the equatorial plane of the rotation axis. As a consequence r i n g molecules are o b t a i n e d when and only when ^ i s a n i n t e g e r . In figure 4 this is illustrated on the unit sphere.

Fig. 4 A proper ring represented on the bond sphere; the motif corresponds to two successive commas, the second of which is generated by a rotary reflection of the first one; z is the

rotation axis.

Any motif having a symmetry of the sequence of operations such that the pitch is zero, independent of the internal details of the operations of the types A or A, will be called a ring motif.

All other motifs are essentially helical motifs. Due to the internal details of the operations A and A the pitch of the corresponding helix may a c c i d e n t a l l y be zero.

A ring formed by a helical motif will be called an improper ring, in contra-distinction to the proper rings, which are only generated by the ring motifs.

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(ii) A p l a n e of r e f l e c t i o n c o n t a i n i n g A. This case is also illustrated in figure 3. T h e corresponding figure on the unit sphere has a twofold axis. This twofold axis must go through the centre of the sphere; on the opposite side of the sphere the axis is again incident with a point of the figure. We now choose the set of vectors representing the motif so that the twofold axis transforms this set into itself. With this choice the sum vector of the motif is pointing along the twofold axis. As soon as the order k of the principal axis is an integer greater t h a n unity, the total sum vector over all bond vectors is zero; the result is a ring.

Thus, a motif having a plane of reflection containing A is a r i n g m o t i f ; when and only when k is an integer greater than unity the resulting molecule is a p r o p e r r i n g .

For the real molecular conformations the corresponding symmetry element is a plane of reflection containing the principal axis.

(iii) A p l a n e of r e f l e c t i o n n o t c o n t a i n i n g A (see fig. 3). This corre-sponds to a plane of reflection for the figure on the sphere. For the real mole-cular conformation this results in a twofold axis perpendimole-cular to the molemole-cular principal axis.

For both distinct types of molecular forms, the closed rings and the open chains, only the seven axial point groups come into consideration; for both types the principal axis is pointing along the rotation axis of M. Since we are only interested in single chains of atoms of the divalent elements, the number of point groups can be limited further.

For the ring molecules those point groups which contain a "horizontal" plane of reflection are unimportant as they correspond only to planar configu-rations. For all atoms must be situated in the plane of reflection, since other-wise branches would appear, which is in conflict with the divalent character.

For the chain molecules, all point groups containing a principal r o t a t i o n axis can be discarded as they would result in twisted multiple chain molecules which are improbable for these elements. Further only those point groups will come into consideration which appear in two enantiomorphic forms; all other possibilities are very special cases, which can be considered as limiting cases of ring molecules.

T h e horizontal planes of reflection which we have not taken into account, are exactly those symmetry elements which are absent in the presentations by means of the sequences of operations. There is a one to one correspondence between the symmetry of the sequence of operations and the symmetry of the molecular skeletons considered here.

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Table III presents the correspondence between the "structure" of the ope-ration M and the symmetry of the molecular conformation. T h e notation for the axial point groups is due to Schoenflies; the molecular groups for the helical molecules are given in accordance with Vainshtein (1966), S represents a screw axis, the index its order.

Table III. Correspondence between the symmetry of a molecule and the symmetry of its motif represented by a sequence of operations *).

Symmetry of the motif expressed in A and A

No symmetry

Mirror plane not through A Twofold axis, M = BB Mirror plane through A Mirror plane through A and mirror plane not through A

Symmetry of the ring

c*

D*

S2*

Cfcv Dfcd

Symmetry of the helix

s*

Sife/2

-*) The symbol Sj, where q is an arbitrary integer, appearing in the second and in the third column has different functions. For the rings, S^ means a g-fold axis of rotary reflection; for the helices it means a j-fold screw axis. Later on, the same symbol is used for a sulfur molecule containing q atoms; the meanings will always be clear from the context.

In table III the last three rows correspond to proper rings. T h e symmetry Did in the last row includes the symmetry elements of S21: and C^v. T h e treat-ment of Pauling (1949) is restricted to the cases corresponding to the last row, and taking into account only the motif of type AA. Pauling called this limited group of molecules "staggered rings". It will be clear after our general treatment that many more molecules have claim to this n a m e ; especially, all other mole-cules having symmetry Dfca, as well as all proper rings with symmetry Sa*.

For pictures of some of these molecules the reader is referred to figure 7.

2.3 Trivalent e l e m e n t s

Molecular forms

For all trivalent elements the bonds are mainly governed by the p-orbitals, one from each of the two bonding atoms. This causes the three bond directions in one atom to be non-planar.

Now as we consider the geometrical bond vectors to be straight, the theory of polyhedra can easily be applied to the geometry of the molecules. The central theorem for polyhedra is Euler's, which relates the numbers of edges, vertices and planes. This theorem can be formulated more generally for nets on closed surfaces in space. These nets consist of e lines and c points which

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together divide the surface into a areas. To these nets in general the theorem of Euler can be applied in the form i ) :

c—e^a = 2.

If now the points are the atoms and the lines are the bonds, a net represents a molecular conformation. For the trivalent elements we have to restrict the number of lines meeting in a point to three. Moreover we suppose all points to have equivalent positions, which means that all the areas are polygons (mostly skew), all having the same number of sides. T h e number of points now will be the number of atoms, n, in the molecule. T h e number of bonds will be

b = e = 3re/2. If we introduce the above restriction, of the polygons all

having m sides, then a = 3«/m. If m goes through all positive integers we obtain all possible conformations. Applying Euler's theorem yields:

4TO

o—m

From this expression we can see that the number of possible configurations is confined to five: « = 2, 4, 8, 20 or oo.

So far nothing has been said about the bond lengths or the bond angles being equal. Only the number of vertices per area was assumed to be the same for all areas.

If, however, the bonds are "equalized" the only possible conformations are the regular polyhedra: the "dumb-bell", the regular tetrahedron, the cube and the regular dodecahedron, corresponding to « = 2, 4, 8 and 20 respectively. T h e case corresponding to n = oo is a two-dimensional puckered layer.

O n e of the cases in which the theorem of Euler is not applicable is that of a closed surface enclosing a second independent "smaller" surface. These " c a g e " structures in which a small molecule is enclosed in a larger one seem to be highly improbable.

So we see only a small amount of a priori possible conformations can be formed for the trivalent elements. These conformations are highly symmetrical.

2.4 T h e b o n d p a r a m e t e r s

The relative orientations of the bonds

T h e description of the molecular conformations using the two types of opera-tions A and A, informs us only about the symmetry of the molecules. For the

') See for instance:

Alexandrow, A. D., "Konvexe Polyeder", Akademieverlag, Berlin, 1958; or Steinitz, E. and Rademacher, H., "Vorlesungen iiber die Theorie der Polyeder unter Einschluss der Elemente der Topologie", Julius Springer, Berlin, 1934.

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ultimate conformational calculations we need correct values for the bond angle /?, the dihedral angle y and the distance d, so as to define the operators corre-sponding to A and A explicitly.

In table I V the experimentally observed values of ^, y and d are given, together with the corresponding molecules. For oxygen, for which a single bonded O - O - O chain so far has not been observed, we have to rely on triads containing at least one heterogeneous atom. Only in a very few cases m o l e c u l a r chains of more than two atoms Se-Se-Se or Te-Te-Te have been reported.

Table IV. Observed bond parameters from literature *). Molecule H , 0 , S.F, S,C1, S,Br, CjHjSj C,F.S, CjGl.S, CiHgljSj

(s.)( S . O . ) ( S 5 0 . )

-

(s.)-s,

_{(S.O.)-(<rw-m)} (S.O,) —(franj-franj)

s,

s»

Soo CijHgCljSej (C.H.),Se, Se(SeCN), Se. Se„ T e „ rfin A 1,49 1,89 1,98 1,98 2,04 2,06 2,03 2,05 2,07 2,02 2,04 2,02 2,06 2,04 2,05 2,06 2,05 2,07 2,33 2,29 2,33 2,34 2,32 2,86 /? in degrees (100) _ -103,8 104 106 113 104,5 103,8 105 107,8 102,2 111 107,4 108 106,5 106 101 106 101 105,5 105 102 y in degrees 101 88 82,5 83,5 93 103,5 94 82 75,5 90,4 108 82 74,5 98; 106,5 85,2; 73,2 98,9 (87,5?) 84,2 74,5 82 94 102 102 100

*) The data are mainly taken from: Sutton, L. E., ed., "Tables of Interatomic Distances and Configurations in Molecules and Ions", spec. publ. of The Chemical Society, London,

1957 and its Supplement, London, 1965; others are from more recent literature.

From table I V the angle /? is seen to have a value somewhat greater than 100°, the normal value for divalently bonded atoms. This angle is produced by atomic p-orbitals occurring in each of the bonds. Pure p-orbitals would give rise to a bond angle of 90°. T h e experimental values of ^ might point to s p -hybridization of the orbitals, causing the bond angles to exceed 90°. Using a simple argumentation in which the amount of the s-p-promotion energy, needed for the hybridization, is compared with the increase of the bond energy,

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due to the better bonding by the hybridized orbitals, Pauling (1949) concluded that the amount of s-character in the bonding orbitals was less than 1 percent. (The bond angles of H2S, H2Se and H2Te, being 92,2°, 91,0° and 89,5°, respec-tively, are in perfect agreement with this picture).

In view of this argument the high value of /? must be caused by an other effect. An explanation by steric hindrance may be given assuming atomic Pauli repulsion of next nearest neighbours in the molecule. The atoms are assumed to be hard spheres as far as they extend out of the spheres of their bonded nearest neighbours. In this treatment we will take the van der Waals radii, selected by Bondi (1964) in a recent compilation of experimental data, to be the radii of the "hard spheres". T h e lowest value ^ can attain in this model will be called /5min. T h e resulting values of ;ömin for the chalcogens are presented in table V, together with some experimental data.

Table V. Data involved in the "hard sphere" model *).

Element 0 S Se Te Po Shortest inter-molecular radius, rg, 1,47 1,70 1,73 1,74 1,68 atomic in A

Van der Waals radius, 1,50 1,80 1,90 2,06 -''w, in A Covalent distance, 1,48 2,06 2,34 2,74 -bond d, in A /?mlii in degrees 121 109 104 97 -*) The data are taken from the following references:

Bondi, 1964: ra(0) and all r»; DeSando and Lange, 1966: ra(Po);

Donohue in "Elemental Sulfur," Meyer, 1965: rg(S);

Pauling, 1960: all d, the value 2, 06 A for sulfur seems to be in better agreement with present data than 2,08 A;

Wyckoff, 1965: rs(Se) and rg(Te) (recalculated). For a "hard sphere" model see also Bartell, 1960.

The first column of table V presents the shortest intermolecular atomic radius, which is de-fined as half of the shortest distance observed between atoms in adjacent molecules. The values given for selenium and for tellurium are observed in the trigonal crystals of these elements. The value for sulfur is observed in one of its crystalline allotropes. The table suggests, at least for selenium and for tellurium, the value of this intermolecular radius to be significantly smaller than the corresponding van der Waals radius. This phenomenon was noted by de Boer (1948). The difference between the intermolecular radius and the van der Waals radius is related to the amount of "metallic character" of the elements, which increases as we go down the columns of table V.

T h e values of /Smm may be compared with those of P represented in table I V . T h e values of both parameters are of the same magnitude and both decrease from sulfur to tellurium. T h e table suggests that for tellurium the atomic repulsion of the next nearest neighbours does not affect the value of /3. For

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selenium this seems to be doubtful, as the smaller value from the first column of table V would result in /9min(Se) = 95,2°. For sulfur the repulsion surely will be one of the effects determining the value of ^. If the value rw = 1,75 A i) is apphed /9min would be 108°, whereas rw = 1,70 A would correspond to ^mm = 105,5°. (Pauhng (1960) even states r„(S) to be 1,85 A ) .

For oxygen the value of /3min of table V could offer a new "explanation" for the absence of O - O - O chains, or for the preference of oxygen for the double bond. Abrahams (1956) found the bond angle /3 for oxygen to be smaller than 109,5°, as long as one of the bonds is not occupied by an aromatic group; the only exception to this rule is /?ozone = 117°, in which, however, the bonds are not single.

Table IV shows the angle y to be near 90° in all cases. Recently Hordvik (1966), surveying experimental data of sulfur, showed a correlation to be present between the dihedral angle, y, and the bond distance, d. T h e shorter values of </ correspond to values of y close to 90°; the d changes 0,07 A as y is varied from 0° to 90°.

T h e value of 90° for y and the above-mentioned correlation between d and y can be explained by the repulsion of the p-Ti-electrons 2) of nearest neighbours. A treatment according to this picture has been given by Pauling (1949):

I n the valence bond theory, the exchange integrals concerning "non-paired" electrons represent repulsive terms in the expression for the energy. T h e total "exchange repulsion" is for the planar configuration (y = 0° or 180°) 5/4 times as large as for the skew configuration corresponding to y = 90°. For the ex-change integrals occur in the expression for the exex-change repulsion only as far

c f e

b /

plane

Fig. 5 Interaction of the p-orbitals, on adjacent atoms, for the planar {y — 0) and the skew (y = 90°) configuration, respectively. The expressions below the figures represent the values of the total exchange integrals which are supposed to be proportional to the repulsion energy

(Pauling, 1949).

*) This value is the shortest intermolecular radius mentioned by Bondi (1964).

*) p-7r-electrons are those electrons which occupy p-7t-orbitals. These orbitals are character-ized by a plane of "antireflection" which contains the bond directions.

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as they refer to electrons the spins of which are not mutually paired; the ex-change repulsion is represented by half the sum of these exex-change integrals. From figure 5 we can see that Jet = Jbe = Jc'e' = Jb'f, where J presents the exchange integral corresponding to the indicated orbitals. If this result is com-bined with the fact that a bonding orbital contains only one electron per atom the resulting repulsion term is the one given below the pictures in figure 5.

A semi-empirical "molecular orbital" treatment was carried out by Bergson (1960 and 1961) for sulfur. His results confirm Pauling's (1949) result: the skew configuration corresponding to y = 90° is the stable one. In Bergsons treatment the skew configuration is also caused by the mutual repulsion of the p-7t-electrons of nearest neighbours.

Experimental findings show that the height of the restricting rotational barrier of the sulfur-sulfur bond (and of the Se-Se bond) is considerable greater than kT (Pauling, 1949 and 1960; Semlyen, 1967).

Repulsion of the next nearest neighbours

Since the reason for the skew configuration and for the value of y was found in the repulsion of non-bonded electrons in adjacent atoms, it is also reasonable to examine the orbitals of next nearest neighbours. T h e presentation of the orbitals by the vectors in figure 6 shows us that p-Tt-electrons of next nearest neighbours come very close to each other for the normal values of y (90°) as long as /9 is not too large. T h e corresponding " h i n d r a n c e " cannot easily be accounted for numerically. The figure shows, however, that if the repulsion of next nearest neighbours causes the angle /3 to increase, a small deviation from 90° for the angle y should be expected to be energetically profitable. This repulsion, which in a previous section was treated in terms of the hard sphere model, should be calculated with exchange integrals too, as was done for the

O

I

O"

Ó

Fig. 6 Interaction of p-orbitals on next nearest neighbours. In the skew configuration, corresponding to y = 90°, p-orbitals of the atoms b and d will interact. The configurations with the atoms e' and e" will be called the cis- and the tranj-configuration, respectively.

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repulsive terms of nearest neighbours. Unfortunately these integrals cannot easily be calculated and compared with the ones corresponding to nearest neighbours.

One can, however, roughly estimate the repulsion terms in the expression for the energy using overlap integrals of the corresponding orbitals. Mulliken (1950; 1952) states that, as a first approximation, the repulsion terms in the expression for the energy can be expressed as a function of the overlap integrals. The overlap integrals, in turn, can be calculated for orbitals of the Zener-Slater type ') using tables and formulas compiled by Mulliken et al. (1949). For one kind of sulfur molecule such a treatment will be carried out in chapter 5.

The volume effect

The same argumentation as before applies to the more "distant" neighbours; however, the effect is now called the "volume effect". As has been mentioned already, this effect can only be accounted for in a rather rough manner.

In figure 6 a chain is shown in the m-configuration (e') as well as in the /ran^-configuration (e"). For the /ranj-configuration the interaction between the atoms (a) and (e") will be negligible. For the cü-configuration small values of /3 are impossible. For a number of important configurations containing even more atoms, the relation between the minimal values of /3 and y will be cal-culated using the hard sphere model. It must be noted that a value ymin is meaningless when atom (a) and atom (e') have a neighbour (f) in common.

A choice for the values of the bond parameters

For a reasonable approximation of the bond angle /? for the undistorted S S S -configuration, we will select a number of molecules from table IV, which conform to the following requirements:

(i) at least a sequence of three bonded sulfur atoms should be present; (ii) the two outermost atoms of the triad should be divalently bonded;

electrical charges on them should be absent;

(iii) the molecules should not be ring molecules, as the distortion caused by the ring closure is not known;

(iv) steric hindrance of different parts of the molecule should be absent. These requirements are met by only three of the molecules presented in table IV: (Se) , (SeOg)—(trans-trans) and S„. The other molecules have mostly the cü-m-configuration, for which mutual hindrance is present between the outermost atomic groups. The resulting undistorted value of /3 for sulfur is 107°. (If all values of table IV are taken into account a mean value of 106° is obtained).

Selenium and tellurium should have values of 105° and 102° respectively for /S; however, these values are based only on a few observed data.

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In search for the best values of the dihedral angle y for sulfur, we should at first sight have limited the suitable configurations to the ones containing a string of four sulfur atoms. T h e theoretical treatment, however, has shown us that the skew configuration is caused by the interaction of nearest neighbours, therefore molecules containing only two divalently bonded sulfur atoms can be included in our considerations. For the best choice of the undistorted value of y we will compare the values of: (Se)—, (SeOe) {trans-trans), (S4) and S „ , whereas the molecules S2Br2, S2CI2 and S2F2 should have values near to the selected one. For these molecules, for which the distortion seems to be as small as possible, the value of y is below 90°.

For the angle y, in the case of sulfur, we choose the value of 80°, which is the average value.

It is a much more difficult problem to obtain a reasonable value of y for selenium and still more for tellurium. T h e number of suitable molecules is very small, whereas the values for Se„ and Te„ are mainly governed by the intermolecular forces in the trigonal lattices. From table IV the value of y for selenium seems to be smaller than 90°, as was the case for sulfur.

T h e values for the bond distance d will be chosen according to the values presented in table V.

2.5 M a i n p o i n t s

(i) For the divalent elements the molecular forms can be classified by means of a repeating unit, the motif, by repetition of which the whole molecule is formed.

(ii) Looking at the structure of the motifs, the molecules may be divided into

proper rings and helical molecules (including improper rings).

(iii) T h e symmetry of all these geometrical molecular models of the divalent elements is complex.

(iv) Of the trivalent elements only five molecular forms are possible; these forms are highly symmetrical as compared with the corresponding mo-lecular forms of the divalent elements.

(v) In view of theoretical considerations, together with experimental data, the following set of bond parameters is selected from the experimental values, preferring those for free (or quasi-free) molecules:

)3(S) = 107° y(S) = 80° d{S) = 2,06 A /3(Se) = 1 0 5 ° y(Se) ^ 9 0 ° d{Se) = 2,34 A /9(Te) = 102° y(Te) (^ 90° d{Te) = 2,74 A

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CHAPTER 3

T H E O R E T I C A L C O N F O R M A T I O N S O F R I N G M O L E C U L E S A N D T H E I R

C R Y S T A L L I N E P A C K I N G

3.1 Introduction

With the aid of the results of chapter 2 we are able to carry out the exact conformational calculations for the divalent elements.

The operations introduced in chapter 2 enable us to treat all possible ring molecules. For this purpose the abstract operations are replaced by matrices, which are functions of the bond parameters. These matrices are similar to the well-known "Eyring matrix" (Eyring, 1932). In this way the treatment results in purely geometric ring conformations i). The conformational calculations are carried out in section 3.2.

Most of the geometrical forms, however, will be ruled out by physical reasons. In the first place the volume effect should be taken into account and secondly the expected stability of the conformations in view of the selected bond para-meters. The stability of the diflerent forms can now be estimated, as is done in section 3.3.

In section 3.4 a survey of all possible crystalline arrangements of rings is attempted. Such a survey gives an insight into the polymorphism, which is a subject of special interest to us.

3.2 Geometrical molecular models Classification

The molecular conformations can be classified according to their motifs. As has been pointed out in chapter 2 for this purpose the motifs should be represented as a sequence of operations of types A and A. In the present section we shall work out this classification for rings. For this purpose we first distinguish between proper and improper rings. For either of these two classes of conformations we can arrange the motifs in a one-dimensional array, considering first its number of elementary operations; if these numbers are equal, motifs can be arranged according to the number of operations of type A they contain; these operations A can be permuted in a systematic way over all available places in the motif 2). Two motifs are provisionally taken to be identical, if the se-•) The sections 3.2 and 3.3 are extensions of a treatment published elsewhere (Tuinstra,

1967a).

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quences of operations can be made identical by complete reversal of one of the two sequences.

As we are now able to enumerate all motifs in a systematic way all possible molecular conformations are obtained by adding to each motif the order k of

r

Al MMl. AAAAKTI. AAAAAA n-11

w

Fig. 7 Classification of the proper rings, showing in topview and sideview the conforma-tions. The motifs and the values of k are indicated as horizontal and vertical coordinates.

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the principal molecular axis - i.e. the number of times the motif must be re-peated so as to obtain the whole molecule - , k passing through all natural numbers.

For proper rings the classification is illustrated in figure 7, which contains only a few simple cases. In figure 7 from left to right the magnitude of the motif and the number of "A's" increases; downward the number k increases. T h e presentation is confined to molecules containing less than 13 atoms.

The method of calculation

In a given conformation only one value for p, only one value for y and only one value for d will occur. The quantities p, y and d in our calculations will be used as parameters.

For explicit calculations we need a set of axes to be fixed in the "bond vector space". We shall fix the "first three bond vectors" with respect to these axes; these "first three bond vectors" are those successive bond vectors in the con-formation which we will consider (quite arbitrarily) as the initial vectors. Details of the exact relation between the axes and these bond vectors are dealt with in appendix I.

Now the axes having been defined, the operations A and A can be expressed in m a t r i x form; the m a t r i c e s A and A transform the initial three vectors into the "second triad". The elements of these matrices can be calculated applying spherical trigonometry to the unit sphere. The elements of A and A are trigonometrical functions of the parameters /3 and y. T h e matrices A and

A are written down explicitly in appendix I. The matrix corresponding to M

is the product of the corresponding sequence of matrices A and A; since A and A are orthogonal, M will also be so; the elements of M are again trigono-metrical functions of ^ and y.

T h e parameter d is absent in all these calculations, its value can be chosen afterwards. It acts as a scale factor only.

From the theory of matrices it is well known that the trace of a rotation matrix is independent of the choice of the axes in space. T h e trace depends only on the rotation angle •&:

tr(M) = + l + 2 c o s i ? . (1) T h e term + 1 must be replaced by — 1 if the proper rotation is replaced by a

rotary reflection.

T h e order of the principal molecular axis must be an integer to achieve ring closure: •& = 27t/A;, where A: is a natural number. Thus, if A: is fixed the right-hand part of the relation (1) is fixed.

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only. This function is determined by the elements of A and ^ and by the sequence of the elementary operations in the motif. T h u s :

l+2cos27i:/A:= + 1 + 2 c o s ^ = tr(M) == tr(prod. of elem. op.) =f{p,y). (2) A definite choice for the motif and the order k, defines a relation between /3 and y.

T h e restriction of the values of k to integers is necessary but not sufficient for ring closure. For the ring closure it is necessary for the pitch of the molecular helix to be zero. T h e pitch is k times the projection, on the principal molecular axis, of the sum of the vectors of one motif. Thus, the pitch will in general be a function of k, /5 and y.

For p r o p e r rings the pitch is zero, independent of the value oï k, /? or y. Hence, if for proper rings the (integer) value of k is chosen, a relation between

fi and y is generally defined.

For the i m p r o p e r rings the extra requirement for closure must be fulfilled. Improper rings are essentially helices for which the parameters (i and y are chosen so as to make the pitch zero. T h e extra relation between fi and y in-duced by this requirement is independent of the relation presented in (2). As a result we may conclude that for improper rings ring closure is possible only for some discrete combinations of /3 and y; in many cases only the planar con-formations will fulfil these requirements.

Proper rings

From table I I I we know that the symmetry of proper rings corresponds to one of the three point groups: 82*, Cj;v or Djd; the symmetry elements of the first two are included in the last one.

a) Rings having the symmetry elements corresponding to 82*. This class in-cludes all rings with point groups 82* and D^d; they have an axis of rotary reflection which coincides with the principal molecular axis. T h e method of calculation derived in the preceding section can be simplified since M can split up here in two equal parts, B, which are rotary reflections, having the rotation angle ^ / 2 :

-l+2(cos27t/2A;) = tr(B) =f{P,y).

For the simplest motif of this class, AA, the method results in the equation:

+ 1 - cos /3 — 2 COS(27C/K)

cos y = ; . (3)

'^ 1 + cos j3 ^ ^

These are the only conformations taken into account by Pauling (1949). T h e quantity n is the number of atoms in the molecule: n = 2k.

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The next motif of our classification is AAAA. The corresponding relation between /3 and y is found to be:

- s i n / 3 + 2 sin(Ti;/2A:) sin p

For larger motifs the method produces equations of higher degree, which in general can only be solved numerically.

b) Rings having the symmetry elements of C t v Since the rings with symmetry D M have already been treated under point (a), we can restrict ourselves to rings having only the symmetry C^v.

For these rings the smallest possible motif is found to be AAAAAA. T O obtain ring closure, the motif must be applied at least two times, so that the smallest proper ring having symmetry Ctv is a "12-ring". The motif is too large for application of equation (2); it is impossible to express y as a function of /3 using relation (2).

T h e next motif of this type is AAAAAAAA, the corresponding smallest ring is a "16-ring".

T h e relations between ,8 and y are graphically represented in figure 8 for all proper rings in which the number of atoms is less than 14.

Improper rings

As long as horizontal planes of reflection are absent, the improper rings can only have a symmetry according to C* or D*, where k is a positive integer; the presence of a horizontal plane of reflection introduces planar conformations only. For the improper rings equation (2) can only be applied with the use of an electronic computer, as the method for the proper rings fails here because of the large number of A's and A's in the motifs.

Moreover the extra requirement for the ring closure must be fulfilled here. For improper rings containing not too many atoms, rather simple geometrical considerations enable us to decide whether a conformation is planar or not.

Using the matrix method we always start with three successive initial bond vectors, forming either a left-handed or a right-handed set. From these the elementary operations corresponding to the motif generate two initial vectorial sets, two molecular conformations which are mutually mirror images; as al-ready mentioned, two conformations which are related in this way will be taken to be identical for the time being.

The elementary operations of the motif will generate the successive bond vectors; to get all bond vectors, only (ffz-3) operations are needed as the initial

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set is already present; m is here the number of atoms in the motif. Hence, three successive elementary matrices are not needed in producing all the vectors of a motif; from the row of elementary matrices we may choose for these initial matrices any triad of successive matrices from the motif (in circular representa-tion).

T h e result of this argumentation is that if we can find in M a sequence of

(OT-3) operations of type A, then the whole structure of bond vectors can be found using operations of type A alone; they will form a right- or a left-handed helix according to the handedness of the initial triad of vectors.

Now a single helix can be closed only if its pitch is zero, that is y = 0. Therefore if such a motif is applied once in a molecule the conformation is planar.

If the ring is improper an analogous argument can be used if all but three successive operations are of the type A. All conformations of rings containing less than six atoms can easily be surveyed using these two rules.

This treatment applies also to larger molecules. However, the larger the mole-cules, the smaller the fraction of them suitable for appUcation of the rules. For these larger molecules much more detailed reasonings can be used, applying elementary spherical geometry; these arguments are, however, very tedious. T h e motif can have been applied only once in molecules containing 3, 5, 7, 11 or 13 atoms, since these numbers are prime numbers. For proper rings, however, the number of atoms must be even, so the prime-numbered rings are all improper ones.

3.3 P h y s i c a l m o l e c u l a r m o d e l s

Possible rings

If the calculations of the preceding sections are applied, we can survey the collection of all rings which are geometrically possible. For a number of confor-mations, which correspond to a ring motif which is applied only once in the molecule, the method for the improper rings shows them to be planar. In cases of large motifs we will eventually have to use numerical calculations.

T h e final results are as follows: 3-rings: the 3-rings are all planar.

4-rings: the ring corresponding to AAAA is a proper ring, which is, however, impossible for values of ;S > 90° (see formula (3)); if the ring is not planar, then ^ < 90°; the ring characterized by AAAA should be planar as formula (4) shows;

the rings corresponding to AAAA and to AAAA are both improper rings which can only be planar.

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1 - j ^ ( S ) •-/J (Se) -j3Cre) -~-_au ..^ 1 ' ' < ^ 1 ' " \ ' ' \ ' \ \ ' ^S^. ^ \ V \ \ \ X > ^ ^ \ N\ \ \ \ ~---.^{S,-) N>s(S.'«) \(0.'«) \S,MA) \ \ \ \ ^ (6,lK \ \(8. iK\ (10, i\(n, l)\ ^\ \ \ ^ \ ^ \ \

^^A

\

^A\

\ ^

7(5)^^"^^ \ \ \ \ ' 1 ' \ W, 1) \ \ \ , , , _ -y in degrees y in degrees

Fig. 8 ^-y-relations for rings.

curves for n-membered proper rings labelled by {n, 2"), where Tis the type of the motif:

I = AA, II = AAAA, III = AAAAAA, and IV = AAAAAA;

the volume effect does not discard the conformations along the heavier parts of the curves.

limiting (minimum) curves, smaller values of ^ and y being ruled out by the volume effect, indicated by the symbol (element, configuration); - means a sequence of four atoms, ^ is a iranj-configuration, A is a m-configuration (for their definitions see fig. 6).

o represents an n-membered plane ring, labelled («, plane). 5-rings: all 5-rings are planar.

6-rings: the ring AAAAAA is a proper one; all other 6-rings are planar, even the proper ring, AAAAAA.

7-rings: all 7-rings are planar.

8-rings: all 8-rings are planar except two, both of which are geometrically possible, corresponding to the motifs AA and AAAA.

9-rings: there are only two motifs which, when applied more than once, result in 9-rings: A and AAA, both giving planar rings;

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all other motifs can be applied only once, they all result in planar conformations too.

10-rings: the 10-ring corresponding to the motif Z^T is a proper ring;

all other ring motifs must be applied only once; the corresponding rings are planar;

all improper rings are planar.

11-rings: all 11-rings are improper rings, they all are planar.

12-rings: 12 is divisible by 2, 3, 4 and 6; ring motifs which can be applied more than once are: AA, AAAA, AAAAAA, and AAAAAA;

other ring motifs can be applied only once, they result in planar rings only;

all improper rings are planar. 13-rings: all 13-rings are planar.

14-rings: the only ring motif applicable more than once is AA; all other pos-sible motifs result in planar conformations.

In figure 7 the possible conformations are illustrated; figure 8 gives the cor-responding ^-y-relations.

An analogous treatment can be given for still larger rings. From the above reasonings it can be seen that for molecules containing n atoms, the number of conformations is large if n has many even divisors. We therefore may expect relatively large numbers of possible conformations for the 16-rings and for the 24-rings. We will, however, confine ourselves to rings containing less than 15 atoms.

The volume effect

I n the present section, the geometrical model will be subjected to the require-ments of the volume effect, which takes into account the volumes occupied by the atoms. T h e atoms are considered to be hard spheres (radius r») in the sense mentioned in chapter 2.

For a distinct succession of ^ ' s and ^ ' s in the motif, the volume effect res-tricts the values of ji and y to distinct intervals. Applying elementary spherical trigonometry the coupled limiting values of /3 and y can be calculated for rather short strings of atoms, taking into account the values of rw (table V ) .

To a certain sequence of elementary operations corresponds a section of the /3,y-plane, in which the points refer to combinations of /? and y which meet the requirements of the volume effect. T h e section will vary a little from element to element, due to the differences in rw and d (table V ) . In figure 8 the dashed lines represent, for a number of cases, the relations between fim\n and ymm; the other curves give the relation between fi and y for the proper rings. T h e improper rings are presented in the diagram by the corresponding d i s c r e t e

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points. T h e diagram is confined to the intervals 87^/2°-120° and 0°-180° for /9 and y respectively.

Stability of the rings for O, S, Se and Te

T h e stability of the different rings can now be estimated if we take account of: (i) normal values of /5 and y,

(ii) the relation between p and y for the corresponding rings, (iii) the volume effect for the configurations in the ring.

It should be noted that this kind ofstability is the stability ofthe "free molecule", leaving aside intermolecular forces, i)

For oxygen the "volume effect" between next nearest neighbours would al-ready result in values for p and y which are far out of the region of their normal values; hence, since even the formation of - O - O - O - s t r i n g s is improbable, the ring molecules would be very unstable indeed.

For sulfur, selenium and tellurium we may expect decreasing stability in the following order: 12-ring, AAAA; 8-ring, AA; 6-ring, AA; 10-ring, another 12-ring and a 14-ring, all three having motiS AA. T h e last three con-formations will be very unstable, as the combinations of fi and y are far out of the region of the normal values. T h e "smaller" volume effect for tellurium might make possible two extra rings for this element, namely a 12-ring and an 8-ring having motifs AAAAAA and AAAA, respectively.

3.4 Crystalline p a c k i n g

A ring molecule occupies only a limited volume in space. For this reason the number of different ways of packing these molecules in a three-dimensional array is innumerable. A systematic a priori classification of these different arrangements is impossible, especially because of the complex symmetry cha-racter of the rings.

If we still want to say something about their possible three-dimensional ar-rangements we must simplify the whole picture drastically. For instance, we can disregard the staggered form ofthe rings and treat them as if they were "disks". Now by first treating crystalline, one-dimensional arrays of these disks, using close packing principles, we may expect "sheared penny rolls" or "upright penny rolls" (piled one right on top o f t h e other). If, however, sheared penny rolls are expected, then the zigzag formation should also be expected; this form is still free in chosing its „amplitude". T o be complete helical arrangements should also be expected.

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All these one-dimensional arrays in turn can be packed according to the packing principles of the chain molecules, which will be derived in the follow-ing chapter.

It will now be clear that a systematic approach to this problem is impossible, since even oversimplified models lead to innumerable possibilities.

3.5 M a i n p o i n t s

i) There are two requirements for ring closure:

a) T h e pitch of the molecular helix should be zero.

b) T h e configuration should be closed in itself (bottom page 16). ii) T h e requirement (b) can be treated with a matrix method resulting in a

relation between /S and y, which must be satisfied.

iii) For the proper rings the requirement (a) is automatically satisfied; these rings retain one degree of freedom; either jS or y may be chosen arbitrarily. iv) For the improper rings these two requirements (a) and (b) result in two independent relations between fi and y, both of which must be satisfied; these rings have no freedom left.

v) With the aid of elementary geometry it can be shown that all improper rings containing less than 15 atoms are planar.

vi) T h e stability of the non-planar proper rings can be estimated by taking into account the values of the normal bond parameters, the relation between y and /3 necessary for ring closure and the volume effect. vii) T h e possible crystalline packings can be treated in a general manner

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CHAPTER 4

T H E O R E T I C A L C O N F O R M A T I O N S O F T H E C H A I N M O L E C U L E S A N D T H E I R

C R Y S T A L L I N E P A C K I N G

4 . 1 I n t r o d u c t i o n

T h e chain molecules can now be treated in a similar way as was done for the ring molecules. The conformational calculations are carried out in section 4.2 with the use of the matrix method. T h e stability of the conformations is esti-mated, taking into account the volume effect and the selected bond parame-ters. As intramolecular forces we only consider the covalent bond forces and the atomic repulsions. The effect of intramolecular dispersion forces has not been taken into consideration as their energy is small in comparison with the height o f t h e rotation barrier o f t h e sulfur-sulfur bond (Semlyen, 1967). More generally we remark that in the condensed state it does not matter whether the dispersion energy is due to inter- or intramolecular neighbours; therefore the effect of intramolecular dispersion forces on the molecular con-formations is practically irrelevant.

The number of possible a r r a n g e m e n t s of chain molecules can be surveyed taking into account close packing principles and prohibited interpenetration. T h e systematic treatment given here for helical molecules has points of contact with a more general treatment by Kitaigorodskii (1961), with the papers of Huggins (1945; 1966) and with the work on the isotactic organic polymers by Natta and Corradini (1960).

4.2 G e o m e t r i c a l m o l e c u l a r m o d e l s

Classification

For the chain molecules a classification of the possible conformations can be made in the same way as was done for the rings; the starting point is the se-quence of operations A and A in the motif. The chain molecules have been shown to have heUcal conformations (zigzags included, section 2.2).

For the rings the two requirements for closure involved relations between the parameters k, fi and y. For the helical molecules we want the pitch of the helix to be different from zero, as otherwise distinct parts of the molecule would interpenetrate. As a second requirement for ring closure k had to be an integer; for the helical molecules this requirement would restrict the number of motifs on one turn of the helix to integers. For the helical molecules with saturated bonds there can, however, only be external reasons requiring k to be an integer

Structural aspects of the allotropy of sulfur and the other divalent elements

ALLOTROPY OF

DIVALENT ELEMENTS

STRUCTURAL ASPECTS OF THE

ALLOTROPY OF

SULFUR AND THE OTHER

DIVALENT ELEMENTS

FOKKE TUINSTRA

CONTENTS

PART I A-PRIORISTIG CONSIDERATIONS

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