Properties of the single and double D4d groups and their isomorphisms with D8 and C8v groups A. Le Paillier-Malécot, L. Couture

To cite this version:

A. Le Paillier-Malécot, L. Couture. Properties of the single and double D4d groups and their isomorphisms with D8 and C8v groups. Journal de Physique, 1981, 42 (11), pp.1545-1552. ￿10.1051/jphys:0198100420110154500￿. ￿jpa-00209347￿

HAL Id: jpa-00209347 https://hal.archives-ouvertes.fr/jpa-00209347 Submitted on 1 Jan 1981

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. J. Physique 42 (1981) 1545-1552 NOVEMBRE 1981, 1545

Classification Physics Abstracts 75.10D

Properties of the single and double D4d groups and their isomorphisms with D8 and C8v groups

A. Le Paillier-Malécot

Laboratoire de Magnétisme et d’Optique des Solides, 1, place A.-Briand, 92190 Meudon Bellevue, France and Université de Paris-Sud XI, Centre d’Orsay, 91405 Orsay Cedex, France

and L. Couture

Laboratoire Aimé-Cotton (*), C.N.R.S., Campus d’Orsay, 91405 Orsay Cedex, France and Laboratoire d’Optique et de Spectroscopie Cristalline, Université Pierre-et-Marie-Curie, Paris VI, 75230 Paris Cedex 05, France

(Reçu le 15 mai 1981, accepté le 9 juillet 1981)

Résumé. 2014 Le groupe de symétrie D4d est intéressant car certains ions dans les cristaux peuvent présenter cette symétrie de façon approchée. Nous donnons la table de caractères pour les trois groupes isomorphes D8, C8v et D4d, simples et doubles. Le type d’isomorphisme choisi établit une correspondance entre les rotations inverses notées IC8 de D4d et les rotations directes notées C8 de D8 et de C8v. Le groupe D4d ayant seul un intérêt spectroscopique, nous présentons, uniquement pour ce groupe, la table de décomposition des représentations du groupe des rotations, la table de décomposition des représentations de D4d pour les différents sous-groupes cristallographiques, ainsi qu’une table relative aux règles de sélection. Enfin nous discutons les avantages du type d’isomorphisme choisi.

Abstract. 2014 D4d group is of interest because it may be an approximate symmetry for ions in crystals. We present here the character table for the three isomorphic D8, C8v and D4d single and double groups. The chosen type of isomorphism establishes a correspondence between the inverse rotation IC8 of D4d and the direct rotation C8 of D8 and C8v. Since only D4d is of spectroscopic interest we give, for this group only, the full rotation group compatibility table, the subgroup compatibility table and the spectroscopic selection rules. The advantages of the chosen type of isomorphism are discussed.

1. Introduction. - D4d symmetry group which approximate higher symmetry if there is one. For presents an inverse eightfold axis is not one of the example, using the approximate icosahedral sym- thirty-two point groups which are allowed in crystal- metry of rare earth ions in double nitrates, Judd lography. However, in crystals, an approximate D4d was able to explain many spectral features of these symmetry may occur in the surroundings of an ion ions [2]. when the coordination number of this ion is eight In a study of the Zeeman effect of ytterbium ions in and when the coordination polyhedron is a square monoclinic YbCl3, 6 H20, Dieke and Crosswhite [3] or tetragonal Archimedean antiprism (a trigonal first observed that the spectra exhibited an approxi- antiprism is an octahedron). mate symmetry higher than monoclinic. After that, As an example of an approximate D4d site sym- many studies of Zeeman effect or paramagnetic metry in crystals we may mention the site of a tan- resonance of rare earth hexahydrated chlorides reveal- talum Ta+ 5 ion in the monoclinic crystal Na3(TaFg) [1]. ed a similar symmetry, which was sometimes inter- To understand the spectroscopic properties of preted as being a sixfold approximate symmetry. In paramagnetic ions in crystals, it is important to know a forthcoming article Couture and Le Paillier-Malécot the true symmetry of the crystal field, but also the will show that a sixfold axis cannot be found in the surroundings of rare earth ions in such crystals; the coordination polyhedron of rare earth ions will (*) Laboratoire associé à l’Université Paris-Sud. be shown to have the form of an approximate square

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198100420110154500 1546 antiprism made of six water molecules and two (rotation of 2 03C0/8 around z axis) and C’ (rotation of chloride ions (a preliminary presentation is given 03C0 around x axis perpendicular to z). in [4]). The observations can then be interpreted by The symmetry elements of the Os group are an an approximate D4d symmetry. eightfold axis taken as the z direction and eight The surroundings of paramagnetic ions with the twofold axes perpendicular to it. The four twofold shape of a square antiprism are very interesting for axes related to C2 operations are taken as the x and spectroscopic and magnetic applications because they y axes, and the first and second bisectors of the (x, y) have the following three properties : angle ; the four twofold axes related to C2 operations bisect the of the four axes. (i) an approximate axial symmetry of high order, angles C2 The order of this is sixteen. (ii) no centre of symmetry, even approximate, and group (iii) a high anisotropy. C8, group For these reasons we think that the is D4d group All the operations of the C., group may be obtained worth in a similar that the studying way crystal- as products of the two operations C’ and a’ (reflec- have been studied Koster, lographic point groups by tion in a plane containing z and perpendicular to Wheeler and Statz and Prather Dimmock, [5] [6]. x axis which is perpendicular to z). In contrast to ions in molecules crystalline sites, The symmetry elements of the C8v group are an exhibit true and this has may D4d symmetry, group eightfold axis taken as the z direction and eight been studied by molecular spectroscopists [7-9]. The reflection planes containing this axis. The four exhibits such a puckered ring sulfur molecule, S8, reflection planes related to av operations are taken symmetry [8]. in the planes yz and xz and the first and second The D4d symmetry group is isomorphic with D8 bisecting planes of the (yz, xz) angle ; the four reflec- and and we shall the character table and C8, give tion planes related to Oj operations bisect the angles the which are for the multiplication table, common, between the planes (J" v’ three groups. But as D4d group is the only one of interest for spectroscopists we shall present, for this D4d group group only, the full rotation group compatibility All the operations of the group may be obtained table, the subgroup compatibility table and the as products of the operation IC’ and the opera- spectroscopic selection rules. tion C’ (with x axis perpendicular to the z axis). This study will be treated in part 2 and will use The symmetry elements of the D4d group are an inverse rotations as operations of the second kind inverse eightfold axis in the z direction, four twofold of D4d group. The chosen type of isomorphism will axes perpendicular to z, two of them being taken be what we call inversion isomorphism. In part 3 along the x and y axes (Cz operations), and four we shall discuss another type of isomorphism (Tisza’s reflection planes bisecting the angles between these isomorphism) and shall consider improper rotations, binary axes (ad operations). Sn, according to Schoenflies notations. 2.3 GEOMETRICAL REPRESENTATIONS OF THE D8, C8v AND D4d SYMMETRY GROUPS. - We present in 2. of and and double Properties D8, C8, D4d single 1, for each of the three and inverse rotations and inversion isomor- figure D8, C8v D4d groups groups using and with the same conventions as in the Interna- phism. - 2.1 NOTATIONS FOR OPERATIONS OF THE tional Tables for X-ray Crystallography [10], stereo- GROUPS. - The following notations are used : grams of poles of general equivalent directions and E : identity, the symmetry elements of the groups. We have in addition shown the directions of the x axes. Cn : rotation of + 2 03C0/n around an axis and y (n integral), I : inversion, 2.4 CHOICE OF ISOMORPHISM OF THE THREE SINGLE 2 around an rotation of + axis, - 1Cn : 03C0/n GROUPS OS, C8v AND D4d The choice of the corres- followed by an inversion (the inverse ponding operations in the isomorphism of the D. rotation must be considered as a IC,, and C8v groups is obvious, namely C8(D8) H C8(C8v) single operation), and C2(Dg) H u’(C8v); we must nevertheless remark Q = reflection in a to 1C2 : plane perpendicular that the choice of isomorphism is not unique. the axis of rotation of C2. In the case of the D. and D4d group isomorphism, the choice of the corresponding operations which we 2.2 DEFINITIONS OF THE THREE SINGLE POINT GROUPS have made is the following : D8, Csv AND D4d. D8 group All the operations of the D8 group may be obtained We shall call this correspondence inversion iso- as products of the two elementary operations C8z morphism. 1547

the spinor : this is the operation E. An additional rotation of 2 n, thus a total rotation of 4 03C0 brings us back to the identity E. The multiplication table for rotations around the z axis can then be visualized in a diagram where a rotation of 2 n represents a physical rotation of 4 03C0 [5]. Such a diagram for the direct rotations of the studied groups is shown in figure 2 ; it introduces also the notations used by Koster, Dimmock, Wheeler and Statz [5].

Fig. 2. - Schematic multiplication diagram for double group direct rotation in use for and The notations Fig. 1. - a) Stereograms of poles of general equivalent directions, operators D. C,,,. indicated are those of reference b) symmetry elements, for the three point groups D., C,,, and D4d. [5].

We present in table 1 the correspondence of ele- The choice of isomorphisms for the double groups ments and classes for the three single groups D8, is based on the same correspondence as in the single C8v and D4d with the chosen isomorphisms. The groups, namely C,(D8) H C,,(C,,) and sixteen elements of each group separate into seven classes.

Table I. - Correspondence of elements and classes 2.6 CHARACTER TABLE AND BASIS FUNCTIONS FOR fôr the three .single groups D8, C8, and D4d with the THE DOUBLE AND SINGLE GROUPS Dg, C8v AND D4d. - chosen isomorphisms C8(D8) H C8(C8,) H IC8(D41)- Table II is the character table for the even irreducible representations of the three isomorphic double groups. According to the rules given by Opechowski [11] unbarred operations of the first four classes of table 1 and related barred elements are in distinct classes, whereas, for the three following classes, the unbarred 2.5 DOUBLE GROUPS ASSOCIATED WITH THE POINT and related barred elements are in the same classes.

GROUPS D,, C8, AND D4d AND THEIR ISOMORPHISMS. - The elements of the double groups therefore divide In problems involving the spin of the electron, we into eleven classes as is shown in the first three lines have to consider the way a spinor is changed under of table II. direct or inverse rotations. Then corresponding to The eleven representations are named according the identity E of the single group there will be in the to Bethe’s notations ri (i = 1, 2, ..., II) [12]. For double group two operations E and E ; the operator E D8 and C., we have also written Herzberg’s nota- changes only the sign of the spinor. Similarly each tions [8]. For each irreducible representation, we give operation R gives rise to the two operations R and in the last four columns some basis functions common R = ER. The order of the double groups is there- to all groups or, if they are different, corresponding fore thirty-two. By a rotation through 2 03C0 we do not to each group. The character, for each class, is found return to identity but obtain a change in the sign of from the transformation matrices of the basis func- 1548

Table II. - Character table and basisjunctionsjor the groups D8, C8v and 04d with the inversion type ojîsomorphism.

Table III. - Multiplication table jor the irreducible representations oj’ Os, Csv and 04d group.s. 1549 tions under the operations of the class. The notations (c) means that Tx(R) is complex and equivalent for the basis functions are the following : to F,,(R)*. - a means a function transforming into itself We finally indicate that the table of characters for under all operations of the groups, e.g. x2 + y2 + z 2 the single groups may be obtained from that for the - x, y, z are the components of a vector on the double groups (Table II) by neglecting the barred axes defined in paragraph 2, elements R and considering only the first set of - sx, Sy,, Sz are the components of an axial vector representations. (or second order anti-symmetric tensor) in the x, y and z directions ; they transform as x, y and z in a 2.7 MULTIPLICATION TABLE. - For completeness direct rotation but do not change sign under inver- we give in table III the multiplication table for the sion, irreducible representations of the double groups - aij (i, j = x, y, z) are the components of a associated with D8, C,,, and D4d, which has already second order symmetric tensor, been given by Herzberg III, p. 572 [8]. - the basis functions rp(J, m) transform like eigen- states with total angular momentum J, and z compo- 2. 8 FULL ROTATION GROUP COMPATIBILITY TABLE nent m ; under a direct rotation these functions FOR D4d’ - The irreducible representations of the transform according to the matrices Dj(a, fi, y) as full spherical group D± break up into irreducible defined by Wigner [13]; under the inversion these representations of the D4d group as is given in the functions transform into themselves. full rotation group compatibility table (Table IV). The ± indices show whether the representation is The which form a first representations r 1 to r 7 even or odd under the inversion. For J integral or set above the horizontal broken line in table II are half integral, D 4d representations of the first and used for an even number of electrons (J integral), second set are respectively obtained. we = have for the characters x(R) x(R) ; T 8 to T 111 which form a second set are used representations - 2.9 SUBGROUP COMPATIBILITY TABLE FOR D4d. for an odd number (J half and of electrons integral), Figure 3 shows how the different subgroups of the we have for them x(R) = - x(R). D4a group are related to it and between themselves. For each irreducible representation we indicate in We have considered only subgroups which are crys- the column labelled « time inv. » (time inversion) how groups. The way each represen- the is related to tallographic point complex conjugate representation tation of into irreducible the original D4d decomposes repre- representation : sentations of these subgroups is given in table V. (a) means that Tx(R) = Tx(R)*, i.e. the represen- A similar table has been given for the D4d single tation is real, group [9].

Table IV. - Full rotation group compatibility table for D4d. 1550

Table V. - Subgroup compatibility table oj’ D4d (*).

(*) The Fi notations for representations are those of reference [5].

characterized by the correspondence Cn --> Sn, Sn being an improper rotation, whose definition in single groups is Sn = Uh Cn (uh being a reflection in a plane perpendicular to the axis of the rotation Cn). The character table of the D4d single group has been given in references [8] and [9]. The character table of D., C., and D4d double groups has been studied with Tisza’s isomorphism by Herzberg [8], using Schoenflies notations involving improper rotations which are familiar to spectroscopists. A new study of this character table [14] has corrected some errors in the previous work [8]. We want now to present the advantages of the use of the inversion isomorphism and the inverse rota- tions. One advantage of inversion isomorphism is that it can be quite general. It is the only type of isomorphism used Wheeler and Statz - Koster, Dimmock, [5], Fig. 3. Subgroup decomposition of D4d group. by and by Prather [6] in their studies of the crystallo- graphic point groups. On the other hand Tisza iso- 2. lo SELECTION RULES TABLE FOR -- D4d. Selection morphism cannot be used in every case. For instance, rules for electric and dipolar magnetic dipolar tran- in the isomorphism of D6 and D3h groups the cor- sitions for are in table VI. The = D4d symmetry given respondence can be : C6(D6) H IC6 S3 1 (D3h), but electric transforms as for the z dipolar operator r 4 it cannot be C6 +--> S6 as S6 is not an operation of for the x and (or 03C0) component and r 7 y (or Q) compo- the D3h group. So, for these groups, all authors use nents ; the magnetic dipolar operator transforms as inversion isomorphism. r2 for the Sz (or 03C0) component and as rs for the Sx Another advantage is that the transformation rules and Sy, (or 03C3) components. Selection rules are given of the basis functions ~(J, m) and qJ(J, - m) under for an even number of electrons and separately (J operations of the groups are easier to apply with m first set of and for an integral, representations) odd inverse rotations, as the inversion leaves them inva- number of electrons and m half (J integral, second riant. A consequence is that with the inversion type set of representations). of isomorphism all these basis functions with the same value of m transform according to the same 3. Discussion on Tisza isomorphism and Schoenflies representation for all isomorphic groups.

-- notations. 3.1 TYPES OF ISOMORPHISM. - The Finally, the definition of I Cn is not ambiguous, type of isomorphism that we have chosen in this whereas the definition of Sn in the double groups is study, inversion isomorphism, establishes corres- not unique. pondence between a rotation Cn of a group and the related inverse rotation 1Cn of the other group ; it is 3 . 2 DEFINITIONS OF IMPROPER ROTATIONS IN DOUBLE then easier to use, as we have done, inverse rotations GROUP D4d. -- As the improper rotations are widely as operations of this group. used by spectroscopists, it is useful to grve the cor- Tisza [7] has proposed another type of isomorphism respondence between inverse and improper rotations. 1551

Table VI. - Selection and polarization rule.s for D41 .symmetry,

The table VII presents the successive powers of So we obtain, for the particular following inverse the /Cg inverse rotation in the double group D4d rotations, the relations : and gives them below with Herzberg’s notations for rotations [8] and with Koster, Dimmock, Wheeler and Statz’s notations [5] (extending to inverse rota- and tions the conventions given for proper rotations in 2). We find then the two following options for the defi- Fig. nition Since in the single group the improper rotations of Sg : are defined by Sn = Uh Cn with Uh = IC2 we have to - first option : Ici = Ies-3 = 6h C8 = S8 which consider the inverse rotations as being the 03C3h reflec- is the definition of the improper rotation S8 in the tion times some proper rotation, which is given by single group D4d used by Herzberg [8] and Prather [6] using the relation IC,, = I C2(C2 CJ = ah(e2-1 1 Cn), and which is considered still valid in the double and by consulting the figure 2. group ;

Table V". - IC8 inverse rotation powers given with Herzberg’,s notation,s [8] and Ko.ster, Dimmock, Wheeler and Statz’,s ones [5] and their corresponding ilnproper rotation,s with the two option,s jor S8 definition. 1552

- ,second option : 1C8 3 = 6h C8 E S8 which As D4d is isomorphic with D8 and C8, groups we is the improper rotation definition proposed by have presented the character table common to all Koster, Dimmock, Wheeler and Statz [5] for double groups. The chosen type of isomorphism is here the groups. inversion isomorphism with corresponding elements In the table VII, the last two lines give the cor- responding improper rotations for the first and the second with notations [8]. option only Herzberg’s and we have used as operations of D4d direct and Indeed, if we want to use Koster, Dimmock, inverse rotations. Wheeler and Statz’s notations a [5], difficulty appears The character table of these three groups has been for the double In their of the D4d group. study thirty- presented before [8, 14] with Tisza’s type of isomor- two point groups these authors based their definition phism [7], C8(D8) H S8(D4d)’ and using as operations of rotations on the improper following relations, of D4d group proper (or direct) and improper rota- labelled here convention and (1) : IRi=Sj IRi=8;, tions, according to Schoenflies notations. We have is a an rotation. where Ri proper and Sj improper shown that for groups which have improper rotations But in the double group both and appear D4d’ Sg S8 S8 (or Sn with Il > 6) one cannot apply Koster, as fundamental operations which cannot together Dimmock, Wheeler and Statz’s rules simultaneously obey convention (1). In fact if we apply this condition for the definition of all improper rotations. The to the definition we find = = Of S8, IC8 3 03C3h C8E Sg choice of Sg definition is then left open here and we as in the second but then which option ; S8 = les-l, think it preferable to use Herzberg’s notations for does not follow this condition. The reverse situation successive powers of S8- arises for the first option where Sg = IC 8-3, in con- We have discussed the many that the 1 advantages tradiction with convention (1); but S8 = 1C8 1 is in inversion isomorphism presents, together with inverse accord with it. rotations as operations, particularly for the study of If we wanted to keep convention (1), we should be double groups. obliged to designate in second option S8 by S8-5 and Crystallographers have introduced international in the first option Sg by S9. These complications conventions for the symmetry groups (D8 : 8 2 2, remove the interest of convention (1) for the defini- C8, : 8 m m, D4d : 8 2 m) and are using for the sym- tion Of S8, and in this article we do not make a choice metry operations of the second kind only inverse between the two options of the table VII. For the rotations, and not improper rotations ; the interna- same reason we use preferably Herzberg’s notations tional notations are Hermann-Mauguin notations. for the powers of Sg operations, which do not present We regret that most spectroscopists have not adopted any difficulty. these notations, since the use of inverse rotations In their study of the character table of D8, C8v makes the various studies much easier and also and D4d double groups, Couture and Le Paillier- coordination with crystallography is more straight- Malécot [14] have chosen the first option for the forward. definition of S8 and they use Herzberg’s notations for the successive powers of this operation. Acknowledgments. - We wish to express our 4. Conclusion. - We have studied the properties of gratefulness to Professor G. Herzberg for helpful D4d group which can be of interest for spectroscopists. correspondence. D4d may be an approximate symmetry for an ion in We want to thank Doctor D. A. Ramsay who a crystal when it has a coordinate polyhedron in the discussed the manuscript and Professor S. Feneuille shape of an Archimedean square antiprism. for reviewing all the work.

References

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