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

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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�

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.}

Classification Physics ^Abstracts

75.10D

1. Introduction. ^- D4d symmetry group which presents ^{an inverse}

eightfold

^{axis is not one of the}

thirty-two point

groups which are allowed in

crystal- lography.

However, ⁱⁿ

crystals,

^an

approximate D4d

symmetry may ^occur in the

surroundings

^{of an ion}

when the coordination number of this ion is

eight

and when the coordination

polyhedron

^{is a} square

or

tetragonal

Archimedean

antiprism

^(a

trigonal antiprism

^{is an} octahedron).

As an

example

^{of an}

approximate D4d

^site sym- metry ⁱⁿ

crystals

^we may mention the site of a tan- talum Ta+ 5 ion in the monoclinic

crystal Na3(TaFg) [1].

To understand the

spectroscopic properties

^of

paramagnetic

^{ions in}

crystals,

^{it is}

important

^{to know}

the true symmetry ^{of the} crystal field, but also the

(*) Laboratoire associé à l’Université Paris-Sud.

approximate higher

symmetry if there is one. For

example, using

^the

approximate

icosahedral _sym- metry ^{of rare} earth ions in double nitrates, ^Judd

was able to

explain

many

spectral

^{features of these}

ions

[2].

In a

study

of the Zeeman effect of

ytterbium

^{ions in}

monoclinic

YbCl3,

⁶

H20,

^{Dieke and} Crosswhite

[3]

first observed that the spectra ^{exhibited an}

approxi-

mate symmetry

higher

than monoclinic. After that,

many studies of Zeeman effect or

paramagnetic

resonance of rare earth

hexahydrated

chlorides reveal- ed a similar symmetry, ^{which was} sometimes inter-

preted

^as

being

^{a sixfold}

approximate

^{symmetry. In}

a

forthcoming

article Couture and Le Paillier-Malécot 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 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 chloride ions

(a preliminary presentation

^is

given

in

[4]).

^The observations can then be

interpreted by

an

approximate

D4d symmetry.

The surroundings ^of

paramagnetic

ions with the

shape

^{of a} square

antiprism

^{are very}

interesting

^for

spectroscopic

^and

magnetic applications

^because

they

have the

following

^three

properties :

(i) ^an

approximate

^{axial symmetry of}

high

order, (ii) ^{no centre of} symmetry, ^even

approximate,

^and (iii) ^a

high

anisotropy.

For these reasons we think that the D4d group is worth

studying

^{in a similar} way that the

crystal- lographic point

^groups have been studied

by

Koster, Dimmock, Wheeler and Statz

[5]

and Prather [6].

In contrast to ions in

crystalline

sites, molecules may exhibit true D4d symmetry, ^{and this} group has been studied

by

^molecular

spectroscopists

[7-9]. ^The

puckered ring

^sulfur molecule,

S8,

exhibits such a

symmetry [8].

The

D4d

symmetry group is

isomorphic

^with

D8

and

C8,

^{and we shall}

give

the character table and the

multiplication

table, ^{which are} common, for the three groups. But ^as D4d group is the only ^{one of}

interest for

spectroscopists

^{we shall} present, ^{for this} group

only,

the full rotation _group

compatibility

table, ^the

subgroup compatibility

table and the

spectroscopic

selection rules.

This

study

will be treated in part ^{2 and will use} inverse rotations as

operations

of the second kind of D4d group. The chosen type ^of

isomorphism

^will

be what we call inversion

isomorphism.

^In part ³

we shall discuss another type ^of

isomorphism

(Tisza’s

isomorphism)

and shall consider

improper

rotations, Sn,

according

^to Schoenflies notations.

2.

Properties

^of

D8, C8,

^and

D4d

single ^{and double}

groups using inverse rotations and inversion isomor-

phism. ^{- 2.1 NOTATIONS} FOR OPERATIONS OF THE GROUPS. ^- The

following

^{notations are used :}

E :

identity,

Cn : rotation of + 2 03C0/n ^{around an axis} (n

integral),

I : inversion,

1Cn : rotation of + 2 03C0/n ^{around an} axis, followed

by

^{an inversion (the inverse}

rotation IC,, ^must be considered as a

single operation),

Q ⁼ 1C2 : reflection in a

plane perpendicular

^to

the axis of rotation of C2.

2.2 DEFINITIONS OF THE THREE SINGLE POINT GROUPS

D8, Csv

^AND D4d.

D8 group

All the

operations

^{of the}

D8

group may be obtained

as

products

^{of the two} elementary

operations

C8z

(rotation ^{of 2} 03C0/8 ^{around z} axis) ^and C’ (rotation ^of

03C0 around x axis

perpendicular

^to z).

The symmetry elements of the Os group ^{are an}

eightfold

axis taken as the z direction and

eight

twofold axes

perpendicular

^to it. The four twofold

axes related to C2

operations

^{are taken as the x and}

y axes, and the first and second bisectors of the (x, y)

angle ;

the four twofold axes related to

C2 operations

bisect the

angles

of the four

C2

^axes.

The order of this _group is sixteen.

C8,

group

All the

operations

^{of the}

C.,

^{group may} be obtained

as

products

^{of the two}

operations C’

^and a’ (reflec-

tion in a

plane containing

^{z and} perpendicular ^to

x axis which is

perpendicular

^to z).

The symmetry elements of the

C8v

group ^{are an}

eightfold

axis taken as the z direction and

eight

reflection

planes containing

^this axis. The four reflection

planes

^related to av

operations

^{are taken}

in the

planes

^{yz and xz} and the first and second

bisecting planes

^{of the} (yz, xz)

angle ;

the four reflec- tion

planes

^{related to} Oj

operations

bisect the

angles

between the

planes

(J" v’

D4d group

All the

operations

^{of the} group may be obtained

as

products

^{of the}

operation IC’

^{and the} opera- tion C’ (with ^{x axis}

perpendicular

^{to the z axis).}

The symmetry elements of the D4d group ^{are an} inverse

eightfold

axis in the z direction, four twofold

axes

perpendicular

^{to z, two of them}

being

^taken

along

^{the x}

and y

^axes

(Cz operations),

^{and four}

reflection

planes bisecting

^the

angles

between these

binary

^axes (ad

operations).

2.3 GEOMETRICAL REPRESENTATIONS OF THE D8,

C8v

^AND D4d SYMMETRY GROUPS. ^- We present ⁱⁿ

figure

1, for each of the three

D8,

C8v ^and D4d groups and with the same conventions as in the Interna- tional Tables for X-ray

Crystallography

[10], ^stereo-

grams of

poles

^of

general equivalent

directions and the symmetry ^elements of the groups. We have in addition shown the directions of the x and y ^axes.

2.4 CHOICE OF ISOMORPHISM OF THE THREE SINGLE GROUPS

OS, C8v

^AND D4d ^{- The choice of the corres-}

ponding operations

^{in the}

isomorphism

^{of the} D.

and

C8v

groups is obvious,

namely C8(D8)

^H

C8(C8v)

and

C2(Dg) H

u’(C8v); ^{we must} nevertheless remark that the choice of

isomorphism

^{is not}

unique.

In the case of the

D.

^and D4d group

isomorphism,

the choice of the

corresponding operations

^{which we}

have made is the

following :

We shall call this

correspondence

inversion iso-

morphism.

Fig. ^{1. -} a) Stereograms ^of poles ^of general equivalent directions, b) symmetry elements, ^{for the three} point groups D., C,,, ^and D4d.

We present ^{in table 1 the}

correspondence

^{of ele-}

ments and classes for the three

single

groups D8,

C8v

^and

D4d

with the chosen

isomorphisms.

^The

sixteen elements of each _group separate ^{into seven} classes.

Table I. ^-

Correspondence

of elements and classes

fôr

^{the three}

.single

groups D8,

C8,

^and

D4d

^{with the}

chosen

isomorphisms

C8(D8) ^H

C8(C8,)

^H IC8(D41)-

2.5 DOUBLE GROUPS ASSOCIATED WITH THE POINT GROUPS D,,

C8,

^AND

D4d

^{AND THEIR} ISOMORPHISMS. ^- In problems

involving

^the

spin

^{of the} electron, ^we

have to consider the _way a

spinor

^is

changed

^under

direct or inverse rotations. Then

corresponding

^to

the

identity

^{E of the}

single

group there will be in

the

double _group two operations ^{E and}

E ;

^the operator

E changes

only ^the

sign

^{of the}

spinor. Similarly

^each

operation R

^{gives rise to the two} operations R ^and R ⁼

ER.

The order of the double _groups is there- fore

thirty-two.

By ^{a rotation}

through

^{2 03C0 we do not}

return to

identity

but obtain a

change

^{in the} sign ^of

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 operators ^{in use for} D. ^and C,,,. The notations indicated are those of reference [5].

The choice of

isomorphisms

for the double _groups is based on the same

correspondence

^{as in the}

single

groups,

namely C,(D8)

^H

C,,(C,,)

^and

2.6 CHARACTER TABLE AND BASIS FUNCTIONS FOR THE DOUBLE AND SINGLE GROUPS

Dg, C8v

^AND D4d. ^-

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 and related barred elements are in the same classes.

The elements of the double _groups therefore divide into eleven classes as is shown in the first three lines of table II.

The eleven

representations

^{are named}

according

to Bethe’s notations ri (i = 1, 2, ..., II) [12]. ^For D8 ^and C., ^we have also written Herzberg’s ^nota-

tions [8]. ^For each irreducible

representation,

^we give

in the last four columns some basis functions common to all _{groups or,} if

they

^are different,

corresponding

to each _group. The character, ^{for each} class, ^{is found} 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.

tions under the

operations

of the class. The notations for the basis functions are the

following :

- a means a function

transforming

^{into itself}

under all

operations

of the groups, e.g. x2

+ y2

⁺

z 2

- x, y, ^{z are} the components ^{of a vector on the}

axes defined in

paragraph

2,

- sx,

Sy,,

^{Sz are the} components ^{of an axial vector} (or second order

anti-symmetric

tensor) ^{in the} x, y and z directions ;

they

^{transform as x, y and z in a}

direct rotation but do not

change sign

under inver-

sion,

- aij

(i, j

= x, y, z) ^{are the} components ^{of a} second order

symmetric

tensor,

- the basis functions rp(J, m) transform like

eigen-

states with total

angular

^momentum J, ^{and z} compo- nent m ; under a direct rotation these functions transform

according

^{to the matrices}

Dj(a, fi,

y) ^as

defined

by Wigner [13];

under the inversion these functions transform into themselves.

The

representations

r 1 to r 7 which form a first set above the horizontal broken line in table II are

used for an even number of

electrons

^(J

integral),

we have for the characters

x(R)

⁼

x(R) ; T 8

^to T 11 1

representations

which form a second set are used for an odd number

of electrons

^{(J half}

integral),

^and

we have for them

x(R) = -

x(R).

For each irreducible

representation

^we indicate in the column labelled ^« time inv. » (time

inversion)

^how

the

complex conjugate representation

is related to the

original representation :

(a) ^{means that}

Tx(R)

⁼

Tx(R)*,

^i.e. the represen- tation is real,

(c) ^{means that}

Tx(R)

^is

complex

^and

equivalent

to

F,,(R)*.

We

finally

indicate that the table of characters for the

single

groups may be obtained from that for the double _groups

(Table

II)

by neglecting

^{the barred}

elements

R

^and

considering only

the first set of

representations.

2.7 MULTIPLICATION ^{TABLE. -} For

completeness

we

give

ⁱⁿ table III the

multiplication

table for the irreducible

representations

of the double _groups associated with D8,

C,,,

^and

D4d,

^{which has}

already

been

given by Herzberg

^III, p. 572

[8].

2. 8 FULL ^ROTATION GROUP COMPATIBILITY TABLE FOR D4d’ ^- The irreducible

representations

^{of the}

full

spherical

group

D±

^break up into irreducible

representations

^{of the}

D4d

group ^as is

given

^{in the}

full rotation _group

compatibility

^table (Table IV).

The ± indices show whether the

representation

^is

even or odd under the inversion. For J

integral

^or

half

integral, D 4d representations

of the first and second set are

respectively

^obtained.

2.9 SUBGROUP COMPATIBILITY TABLE FOR D4d. ^-

Figure

3 shows how the different

subgroups

^{of the}

D4a

group ^are related to it and between themselves.

We have considered

only subgroups

^{which are} crys-

tallographic point

groups. The way each represen- tation of D4d

decomposes

into irreducible _repre- sentations of these

subgroups

^is

given

in table V.

A similar table has been

given

^{for the}

D4d single

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].

Fig. ^{3. -} Subgroup decomposition ^of D4d group.

2. lo SELECTION RULES TABLE FOR D4d. ^{-- Selection}

rules for electric

dipolar

^and

magnetic dipolar

^tran-

sitions for D4d symmetry ^are

given

in table VI. The electric

dipolar

operator transforms as r 4 ^{for the z} (or 03C0) component and r 7 ^{for the x and} y (or Q) compo- nents ; the

magnetic dipolar

operator transforms as

r2 ^for

the Sz

(or 03C0) component ^{and as} rs ^{for the} Sx

and

Sy,

(or 03C3) components. Selection rules are

given separately

^{for an even} number of electrons (J ^and

m

integral,

^{first set of}

representations)

^{and for an odd}

number of electrons (J and m half

integral,

^second

set of

representations).

3. Discussion on Tisza isomorphism ^and Schoenflies notations. ^-- 3.1 TYPES OF ISOMORPHISM. ^- The type ^of

isomorphism

^{that we} have chosen in this

study, ^inversion

isomorphism,

establishes corres-

pondence

^{between a rotation} Cn ^{of a} group and the related inverse rotation 1Cn of the other group ; it is then easier _{to use,} as we have done, inverse rotations

as

operations

^{of this} group.

Tisza [7] ^has proposed ^another type ^of

isomorphism

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

by

Koster, Dimmock, Wheeler and Statz [5],

and

by

^Prather [6] ⁱⁿ their studies of the

crystallo- graphic point

^{groups. On} the other hand Tisza iso-

morphism

^cannot be used in _every case. For instance,

in the

isomorphism

^of

D6

^and

D3h

groups the cor-

respondence

^{can be :}

C6(D6) H

IC6 ⁼

S3 1 (D3h),

^but

it cannot be C6 +--> S6 ^as

S6

^{is not an}

operation

^of

the

D3h

group. So, for these _groups, all authors use

inversion

isomorphism.

Another

advantage

is that the transformation rules of the basis functions ~(J, m) ^and qJ(J, - m) ^under

operations

^{of the groups are easier to}

apply

^with

inverse rotations, ^as the inversion leaves them inva- riant. A consequence is that with the inversion type of

isomorphism

all these basis functions with the

same value of m transform

according

^{to the same}

representation

^{for all}

isomorphic

groups.

Finally,

the definition of I Cn ^{is not}

ambiguous,

whereas the definition

of Sn

ⁱⁿ the double _groups is not

unique.

3 . 2 DEFINITIONS OF IMPROPER ROTATIONS IN DOUBLE GROUP D4d. ^{-- As the}

improper

^{rotations are widely}

used

by spectroscopists,

^{it is useful to}

grve

^{the cor-}

respondence

between inverse and

improper

^rotations.

Table VI. ^- Selection and

polarization

^rule.s for

D41

.symmetry,

The table VII presents the successive _powers of the /Cg inverse rotation in the double _group

D4d

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-}

tions the conventions

given

^for proper rotations in

Fig.

2).

Since in the

single

group the

improper

^rotations

are defined

by Sn

^{= Uh} Cn ^with Uh ⁼ IC2 ^{we have to}

consider the inverse rotations as

being

^the 03C3h reflec- tion times some proper rotation, ^{which is}

given by using

the relation IC,, ⁼

I C2(C2 CJ = ah(e2-1 1 Cn),

and

by consulting

^the

figure

^2.

So we obtain, ^{for the}

particular following

^inverse rotations, the relations :

and

We find then the two

following options

for the defi- nition

of Sg :

-

first

option :

Ici

⁼

Ies-3

⁼ 6h C8 ⁼ S8 ^which

is the definition of the

improper

^rotation S8 ^{in the}

single

group D4d ^used

by Herzberg

[8] and Prather

[6]

and which is considered still valid in the double 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

is the

improper

rotation definition proposed

by

Koster, Dimmock, Wheeler and Statz [5] ^{for double}

groups.

In the table VII, ^{the last two lines}

give

^{the cor-}

responding improper

rotations for the first and the second

option

^with

only Herzberg’s

^notations

[8].

Indeed, ^{if we want to use} Koster, Dimmock,

Wheeler and Statz’s notations

[5],

^a

difficulty

appears for the D4d ^double group. In their

study

^{of the}

thirty-

two

point

groups these authors based their definition of

improper

^{rotations on the}

following relations,

labelled here convention (1) :

IRi=Sj

^and

IRi=8;, where Ri

^{is a} proper

and Sj

^an

^improper

^rotation.

But in the double group

D4d’

^both Sg ^and

S8

appear

as fundamental

operations

^{which cannot}

together obey

convention (1). In fact if we

apply

^this

condition

to the definition

Of S8,

^{we find}

IC8 3

⁼

_{03C3h C8E}

⁼

Sg

as in the second

option ;

^{but then}

S8 = les-l,

^which

does not follow this condition. The reverse situation arises for the first

option

^where

Sg

⁼

IC 8-3,

^{in con-}

tradiction with convention (1); ^but

S8

⁼

1C8 1

^{1 is in}

accord with it.

If we wanted to

keep

convention (1), ^{we should be}

obliged

^to

designate

ⁱⁿ

second ^option S8 by S8-5

^and

in the first

option

Sg

by S9.

^These

complications

remove the interest of convention (1) for the defini- tion

Of S8,

and in this article we do not make a choice between the two

options

of the table VII. For the

same reason we use

preferably Herzberg’s

^notations

for the _powers of

Sg operations,

^{which do not} present any

difficulty.

In their

study

of the character table of

D8, C8v

and

D4d

^double groups, Couture and Le Paillier- Malécot [14] have chosen the first

option

^{for the}

definition of

S8 and they

^use

Herzberg’s

^notations

for the successive _powers of this

operation.

4. Conclusion. ^- We have studied the

properties

^of

D4d

group which can be of interest for

spectroscopists.

D4d

may be an

approximate

symmetry ^{for an ion in}

a

crystal

^{when it has a} coordinate

polyhedron

^{in the}

shape

^{of an} Archimedean _square

antiprism.

As D4d ^is

isomorphic

^with

D8

^and

C8,

groups ^we have

presented

the character table common to all groups. The chosen type ^of

isomorphism

is here the inversion

isomorphism

^with

corresponding

^elements

and we have used as

operations

^of D4d direct and inverse rotations.

The character table of these three _groups has been

presented

^before

[8, 14]

with Tisza’s type ^{of isomor-}

phism

[7],

C8(D8)

^H

S8(D4d)’

^and

using

^as

operations

of D4d group proper (or direct) ^and

improper

^rota- tions,

according

^to Schoenflies notations. We have shown that for _groups which have

improper

^rotations

S8

(or Sn ^{with Il} > 6) ^{one cannot}

apply

^Koster, Dimmock, Wheeler and Statz’s rules

simultaneously

for the definition of all

improper

rotations. The choice of Sg definition is then left _open here and we

think it

preferable

^{to use}

Herzberg’s

notations for successive _powers of

S8-

We have discussed the _many

advantages

^{that the}

inversion

isomorphism

presents,

together

with inverse rotations as

operations, particularly

^{for the}

study

^of

double _groups.

Crystallographers

^have introduced international conventions for

the

^{symmetry groups}

^{(D8 :}

^{8 2 2,}

C8, :

^{8 m} m, D4d :

8

2 m) ^{and are}

using

^{for the sym-}

metry

operations

of the second kind

only

^inverse rotations, ^{and not}

improper

rotations ; the interna- tional notations are

Hermann-Mauguin

^notations.

We regret ^{that most}

spectroscopists

^{have not}

adopted

these notations, ^{since the use} of inverse rotations makes the various studies much easier and also coordination with

crystallography

^{is more}

straight-

forward.

Acknowledgments.

^{- We wish to} express ^our

gratefulness

^to Professor G.

Herzberg

^for

helpful correspondence.

We want to thank Doctor D. A. Ramsay ^who

discussed the

manuscript

and Professor S. Feneuille for

reviewing

all the work.

References

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[2] JUDD, B. R., ^{Proc. R. Soc.} (London) ^{A 241} (1957) ^122.

[3] DIEKE, ^G. H. and CROSSWHITE, ^H. M., ^J. Opt. ^{Soc. Am. 46} (1956) 885.

[4] COUTURE, L., ^{J. Lumin.} 18/19 (1979) ^891.

[5] KOSTER, G. F., DIMMOCK, ^J. O., WHEELER, R. G. and STATZ, H., Properties of ^the thirty-two point groups (M.I.T. Press, Cambridge, Massachusetts, U.S.A.) ^1963.

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Bureau of Standards, Monograph ¹⁹ (1961) ^U.S.A.

[7] TISZA, L., ^Z. Phys. ⁸² (1933) ^48.

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[9] ^{WILSON Jr., E.} B., DECIUS, J. C. and CROSS, P. C., ^Molecular Vibrations (McGraw-Hill ^Book Company, ^New York, U.S.A.) ^1955.

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[12] BETHE, H., ^Ann. Phys., Lpz. ³ (1929) ^133.

[13] WIGNER, ^E. P., Group Theory (Acad. Press., Inc., ^New York, U.S.A.) ^1959.

[14] COUTURE, ^{L. and LE} PAILLIER-MALÉCOT, A., ^{submitted to} Molec. Phys.