Stark effect and isometric groups of nonrigid molecules.

..‘, ‘thhiti~~ PJ~ytics X.(J978),4331463

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-.ONorth-Holland Publishing Company. ; >- .:.-...i:.: .; ,,.; _~ . . . . . . ._:

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: STARR EEEECT AND ISOMETRIC GROUPS OF NONRIGID MOLECULES.

i. &neral.&eory aCd d~terminatkiof the relative signs of components of the $ectric dipole moment of semirigid moleciiles with one atid tvio symmetric rotors W. BOSSERT, J. EKKERS, A. BAUDER and Hs.H. GmTHARD Loboratory for Physical Chemistry, Swiss Federal Institute of Technology, CH-8092 Zurich, Switzerland Received 25 July 1977

A study of the St&k effect of nonrigid molecules is presented. First the group theoretical aspects of the hamiltonian of the rotation large amplitude internal motion problem is discussed, based on the concept of isometric groups of semirigid mode& (SRhfs). General group theoretical aspects of the splitting of electric dipole transitions and general expressions for .the Wigner-Eckart theorem of SRMs in electric fields are derived. The general theory is applied to a rather wide fan& of SRI$s with one and two symmetric tops yielding a complete survey of the behavior of such systems in Stark fields. It further is used to formulate a method for the experimental determination of the orientation of the electric dipole moment from @adratic Stark effect measurements in molecules of this family-The method is exemplified by the analysis of the Stark effect of some transitions of the acetaldehyde molecule. The analysis is based on a solution of the rotation-internal rotation problem by infmite mat* diagonal&ion for one and two top molecules with iocal frame symmetry Cs, for which computer progmms have been developed. The possibility of detection of electric dipole transitions of molecules with vanishing

constant part of the electric dipole moment is considered, basing on modulation effects by internal rotation.

1. Introduction In this paper we wish to present a study on the symmetry groups of nonrigid molecules subject to a homogeneous electric field. The Stark effect is of considerable practical importance for the analysis of spectra, particularly rotational spectra, for the determination of the electrical dipole moment, deflection by inhomogeneous fields, etc. For quasirigid molecules the Stark effect has been treated extensively [1,2, p-244; 3,4]. Although Stark spectroscopy of nonrigid molecules is commonly employed, in particular for molecules with symmetric internal rotors,

no systematic treatment of the symmetry problem of semirigid systems in homogeneous or inhomogeneous electric fields has been presented to our knowledge. The specific case of deflection of molecules with inversion or pseudo-rotation has been treated by Bernstein and Pitzer [5] within the framework of rigid symmetric top perturbation theory. In this paper the symmetry problem of semirigid models (SRMs), its application to a wide family of such systems and a method for the determination of the orientation of the electric dipole moment vector will be considered. In section 2 a general derivation of isometric groups and full symmetry groups of semirigid models in a homogeneous electric field (Stark effect) is given and a number of relations between the symmetry groups of the hamiltonians with and without Stark field will be derived. This treatment is based on the theory of irreducible tensors and on the selection rules of SRMs recently published by Frei et al. [6]. For particular cases these relations will be discussed in terms of the homomorphisms existing between various representations of the isometric group. Application of the general results of section 2 to a rather wide family of semirigid models consisting of a rigid frame withcoveringsymmetry CI, Cs, C2, Czv, Ci, C,h to which one or two rigid tops of Csv symmetry are attached, wiIl be put forward in section 3, including tables of all symmetries pertinent to the Stark effect of such systems; IrLsection 4 a practical method for the determination of the-orientation of the molecular electric dipole

434

W. Bosert et al.lStark effect and isometric groups of non&id m@e&!e$

-. :. ,: :_ ..~. I : .~_. .. ., : : ,: :. vector with respect to (w.r.t.) the principal axis system will be discussed. It is based on.quadratic S&k effect measurements of rotation-internal rotation transitions of molecules of the types.considered in section 3. An exact solution of the rotation-internal rotation problem by infinite matrix diagonalization and 2nd_orderperturbation treatment of the Stark effect wi!l be used. An application of this new technique to deteimine the orientation of the electric dipole moment of acetaidehyde is presented in section 5. Practical requirements and limitations of the method will be discussed. The general results reported in this paper also form the basis for the problem.pf focusing molecular beams of semirigid molecules and of the behavior of such systems colliding with a cold matrix. These topics wiIl be treated in forthcoming papers.

2. Isometric group of SRMs in electric fields 2.1. Framework of the treatment The following investigation will be based on the isometric group concept for semirigid models (SRMs) published by Bauder et al. [7] and Frei et al. [6]. The reader is referred to these papers,for details and notation. As has been shown earlier the isometric group is the natural generalization of the conventional concept of covering symmetry from rigid molecules to SRMs [6,8]. This allows the formu!ation of selection.rules and irreducible tensors in a strictly analogous manner to the case of quasirigid molecules. These have been derived earlier, both for electric and magnetic dipole transitions for a considerable number.of SRMs. Although the latter result can be used directly for the calculation of the Stark energy matrix elements, no discussion of the symmetry groups of SRMSin homogeneous electric fields has yet been given. Therefore this aspect of the Stark effect (SE) will be considered

first.

2.2. Exergv eigenvalue problem and Stark effect (SE) operator The following discussion will be based on the assumption that the classical energy function of the force free SRM may be written in the form [7] (2-1 j

where $ and $ denote the finite internal coordinates (h, & , .... tf) and their time derivatives (velocities) respectively_ The symmetry group of the hamiltonian do associated with (2-l) may be written as $?{riOI = SO(3)1@ &riO).

(2-2)

The isometric group 9f(5_)of the SRM is either a proper or improper subgroup of g{$}:

969

c sfJ+“l;

(2-3)

more exactly the group r[BI} represented by the substitutions of the dynamical variables (eulerian angles e, finite internal coordinates .$j):

is a (proper or improper) subgroup of the group g.{k”> of the hamiitonian.

Below it will be assume&as a rule

._

W:%osserret aL/Stttrk eflecr aitd ismetnk groups of nonn’gidntoiectdes

435

that g{@) 2 I’(GT]; cases in ;rJhch I’{%] < g{&O> wif.fbe mentioned separately; However, it is obvious that in jhe latter &se the group g{f?O], i.e. an appropriate extension of I’{%), has to be used. The quantum mechanicaf energy eigenvalue problem associated with (2-1)

(2-S) will be asstim’edto be solved in an appropriateIy chosen zeroth order basis symmetrized w.r.t. the irreducibfe representations L](j) 8 I’@) of the group (2-2). Accord~n~y f9J * (2-6’)

The Stark effect operator wifl be taken in the form riSE = - (&ji: * D”‘(e)+ ?‘ttl’($) ,

(2-7)

where the external electric field _I?’and the molecufar efectric dipole moment il$(t) will be expressed by sphericaf and cartes&n coordinates, respectively. Tis the matrix which transforms from cartesian to spherical coordinates [P, p. 1691. ft shoufd be pointed out that the expression (2-7) is to be considered as an operator expressed in the dynamical variables e, & whereas (&) plays the role of an external parameter. The operator of the total energy is &z&O +jjSE,

(2-Q

and its symmetry group is given by !a&

= 8 EfiOlI fl $wEl,

P-9)

whe;e iSj{~?~~) stands for the symmetry group of the operator (2-7). Generally ~{II@~) < 9 @?I, 9{H) = 9 {GSE).

hence

2.3. Symmetry group of rke SE operator In strict analogy to the determ~ation of the symmetry of the rotation-internal motion problem the operator (2-8) admits two types of operations: (i} orthogonal transformations of the laboratory system, expressed by Ff’ = ZrD(&, II@!} G O(3), (.~)z’= j@)(X)P’ .-1 (x) P’, to which the operators fiRl are associated, .- R (ii) t~~formations

of the frame system

T’f’ = ;i;f@ef) induced by isometric substitutions {’ = fl_$) E F(t) and mappings of the covering group$#); tors PH are associated.

* The operator

techniqueof ref. 191wiUbe usedthroughut in this paper.

to these the opera-

,

..

-..

_ __.

C, F{IR tRESO(3);R=R{e~,~~,O~~<2p)

_.... @-12):

formsthe one-parametric rotation group C, of all rotations with rotation axis eE =E~/lE!l{and represents the 1 (properly) orthogonal part of the symmetry group of fisE w.r.t. rotations of the laboratory System. 2 Owing to the fact that (Zl = --l(3)) f; fifiSE~-1 =&SE (2-13) zl 2 > the fuU symmetry group of tiSE w.r.t. operation on the laboratory system is the extension C-i rfRfeE,@j,

1cPE(O
-RfeE,@3

(2-14) ._ C-1 is an abelian group with two parameters; a complete system of irreducible represerkations is &en in appeklix A. The eigenfunctions of both energy operators (2-7) and (Z-8) belong exclusively to the even (gerade) representations X”“*‘EC,iIA brief comment concerning this result wiii be given in section 2.4.1. 2.3.2. isonzetric group off$‘E Application of any operator& to fi SE is based ORthe general transfo~at~on formula of the irreducible representations of the rotation group SO(3) w.r.t. transformations of the frame system * &fO(~.+)

*

= D(n(R f)M&)

(2.15)

Accordingly

= -(E’)~o(‘)(~)~Tnif(Ulr(3)CH)f D

*

ffkH is required to be a symmetry ofi”jSEz I~HII?‘~&I lr-Q.fJl=

{2-I@ =fiSE, one obvibusly must request

’ (2-R)

I ,

i.e. thk rotational part of the element [6,7] I’(NC)(H) = lI(H) QO I’t3)(H) must be properly orthogonal fi3)(H) = R(B) E Scl(33).

(2-l 7’)

We now define the set PSE~~[~])

:= {rl3)(L)

JL EL?[gl], Irc3’(H)J = 13,

G-f@

i.e. the set of all elements of the group I’(3) @[5X] 1, which leave &SE symmetric. For this set the theoKern holds: either r(3)SE(L![5XKf} ~JI?[$X!] or

@)sE{J$GX]}

fP(R],

If?+I=~lL?i.

(2-18’)

The alternatives correspond to the two cases (a) and (b) defined in ref. [6j, where in case (b)et’[st] GQ[sl] is a normal sublifroup of index 2 of %. Next consider the relation of I’(j)SE {.@[a]) to the linear transformation group fiNC) {5X)_-Earlier it has been proved that there exists a homomorphism . ’ lfie transformation formula given in ref. [6] is not correct and has to be replaoed by eq. (2-15j,~Adi&xsion of this point will _.. ._ ‘_,. be published shortly.

437

W.Eossqt etal&ark effect and isometricgroupsof nonrigidmolecules &tiNC’~@]+fi3&[sX]],

ij(ll(H) @r’qff))

= W(H)

)

(2_” :

-According to well known theorems we may state that there exists a subgroup PC)sE

c PC)

(BI}

The group J?c3jSE{e[GY]} is therefore isomorphic to one of the properly orthogonal covering groups C,, C,, D,, nZ=2,T,O,I. The set fi3jsE {.@[%I) defines a subgroup l?fSE) { 81) C I’{ a} by virtue of eq. (24, which assigns to any element r(3)SE(L) f fi3jSE {.@[a]} (at least) one element I’fSE)(L) E I’{ 5X). This leads to the theorem

ifsE){5xlcr(a),

(2-20)

flsE) { 9) is the substitution group of the dynamical variables which leaves @E symmetric. 2.3.2.1. Speciul cases in which G(E) 9 C, As a special case in which the structure of the isometric group is particularly simple, consider SRMs with S(.$) = C,.This case is realized for the systenis treated in section 3 and is characterized by the relations SK(E)k y(g) k rM){9}

!? r(NC) {?I !! r(y)

.

For all three cases(a), (bl) and (b2) discussedearlier [6] the followingequations hold

r(3)SEC~[9]jCr(3)(~+[~]), flSE)Is) s r-I-3). In many cases, when l? SE) {s} is a proper subgroup of I’{q,

fiSE)(s}:r{Fl,

r{ 9}/r(SE) {7) g W 2.

(2-2 1)

one finds (i : denotes normal subgroup) (2-21’)

2.4. Wigner-Eckurt theorem for fisE The Wigner-E&art theorem (WET) for the matrix elements off@ w.r.t. the eigenbasis ofko reads (if the external electric field is taken along the laboratory z axis) -_ (JMr(k)~~I~sEIJMr(k)KN) =-E&2J t 1)~IS JM jMIos~M~(J,7, ij (~r(k)itlvllnjllJr(k)KN). (2-22) This follows directly from the formulae given by Frei et al. [6] for the selection rules of transitions induced by an oscihating field along the laboratory z axis. The double bar element contains the matrix (K’, K being dummy indices) (2.22’) which expresses the selection rules w.r.t. the isometric group l?{ 5X). The selection rules w.r.t. SO(3)l are given by the Wigner coefficient AJ=O,+l,

SJM,~Mlo and read as J=O+J=O,

usual

AM=o.

(2-22”)

438~

W. Bosenet

&SYark.effect and .isometnk &ups-qf hu&id mo&ules -.

‘.

’

‘-.

2_.$[1.. &~nm& &nfp~&gq. fz_14) _. :.~.: ?- -- 1. 1 :__ .,. ..__-~ .-J.I_._;-_. -)‘.e-~FI’--‘. ‘Y ., ...-.. Bqth&ups. (2, and Cii are abelian grcqx and therefore ha~e-one-dimenaional.i~~ducjbie re@&enta$ns : only. This could lead to the conclusion that the degeneracy of-the levels EJ&&$&~ ~J~&&~~.&&l.d be. : removed by the Stark field, in contradiction to Kramer’s theorem [iO; 1.1,p_ 6751 and generW5ccepted sym- ._ metries for the Stark effect [9, p. 204; 11, p. 6621. In appendix A, a coinplete system ofirreducible representations ofC,i is given. _&y pair’&f &r&entat& r(+Mp), r(-Mp), ME IN+ is complex conjugate and.is equivalent to the.two-dimensional repiesentaiion’ DVMp). (irreducible in IR)).From eq. (2-22) it firstly follows that the Stark energy matrix is diag&$ti~r.t.N, owing to the factor 5~~~. Furthermore, owing to $_,,7-M10

=(-l)J+l-J~J~.~&~,,

>

the two matrices $E belonging to the two rows -M and +M (of DcJ)) differ at the most by the sigri. Consideiing the Stark splittings separately in-the blocks -M, +M, closer inspection shows that a.paii of corrGpotiding levels : EJnrr(k),N, EJ_Afr(k)KN remains degenerate for an arbitrary field. The same arguments apply if the niatr&s of A0 and Hare calculated in a zeroth order basis symmetrized w.r.t. I’(SE) {a}. rn this.ca%ethe equivalence of the pair of energy matrices belonging to -Mand +M has to be verified by actual calculation: Since by the presence of the Stark field the isometric group l?{Q) goes over into I’(SE){g) , the behavior of the eigenstates of$ under the influence of the Stark field is described by the-irreducibIe rearesentations of r(SE) {Q}. In case (a) in which 03)S.E {_!?[%I) 2 I’(3) {f?[ 811) and therefore rcSE) {%} g r{%};. the eigenstates of fro and Bare classified according to the irreducible representations of r { %}, i.e. the Stark’field cannot cause any splittings of the eigenstates w.r.t. the isometric group. In principle in case (b) there exist two possibilities according to whether or not I’cSE) {%} is a subgroup of index 2 of I’(%} or is isomorphic to r{sr). If r(SE)(a)-‘=sr{sl) no splittings w.r.t. the isomorphic groups will be produced by the Stark effect. If II’@? { g}l.= iI I’{%}1 one has to distinguish the following two cases [12]; (i) Levels belonging to a self associated irreducible representation I’(i) {tirl) of,I’f_81} til belong to two conjugate irreducible representations subduced from I’(i) { %} by J?(SE) {5X). Stark splittings may now occur if the pair of conjugate representations is not conjugate complex. (ii) States belonging to a pair of associated representationi I’(i+), l?-) of I’{n) will belong to the irreducible representation I’v) of WE) {SY},and therefore will not experience any splitting by the Stark field. For particular cases this relation may be established in a straightforward manner, e.g. for the SEWS treated in section 3.

3. SRMs with on!: or two symmetric rotors 3. I. Families of SRMs Fig. .l depicts in a symbolic way a model of the systems treated in this section. The semirigid model-will be assumed to consist of a rigid frame F with local covering symmetry group QF, to which one or two tops T, Tof .. .. local symmetry C,, and rotation angles r. and i-l, _resp ectively, are attached. For the sake of brevity the self explanatory notation QFF CsVT and BFF Cs.T CsvT will be used. Systems of this Sort have previousiy been studied by several authors, e.g. by Dreizler [13,14] and more recently by Durig eVaI. [15] tid.by Gioner and Durig [16] w.r.t. symmetry, infrared and Raman‘ transitions, The following treatment therefore concentrates on.’ .. -: aspects not covered earlier, e.g. Stark effect. : In table 1 the systems treated in this work are collected, together.with representatbe molecules..Both’the one and two top systems each form a family of SRMs, which mai be.inieirelated by raising or lqwering the symmetry of the frame. Systems with a center of symmetry in the frame covering group have beeh included, alth&b strictly

.:

W.Bossirr et k/Sterk effect and

.

.

:.

.

isbmetric groups

ojnontigf2

molecdes

439

_._:.

.

.

;.

-

a

Fig. 1. Schematic representation of SW

with one or two symmetric internal rotors.

Tsbfe t

SRMs of type $&F C3vT and $&F C-9 C3vT Notation

Local frame covering symmetry %

CI F C3vT

Number of tops

Typica! mclecule

1

CHFCL- CH3 HCO *CH3

CjF C3J C2F C3vT C2vF C3vT ._. CIF C3vT c3vT

2

H3SiCHF - 7H3

GF C3vT C&T

2

CH3CN:N-Cl-83

%F C3-J C3vT

Z(conxial)

%P

2(coaxial)

C3vT C3vT

2

G c2

* 0(=3)2

c2v ci

‘=3

2

C2h D2b

l(coauiaIj

Cf& .e

CH3

semir@i systems with a centros~mmet~c frame do not exhibit an electric dipole spectrum. It should be pointed ‘out that SRMS including structural relaxation should exhibit electric dipole transitions, even if the constant part of the ~~anen~ electric dipole moment vanishes, cf. section 4.

3.2. Isometrii groupsofsem~rigfdmode& with symmeftic rotors in Stark fieids The isometric groups of the systems listed in table 1 are collected in tables 2 and 3. The information given in

a) Internalisometric goup.

C) FulI isometric group.

8) Covering group (~cb~nflies symbols).

d) Group of internaI isometric sub$tutions

4 I33 {P [a 1) is given as covering group, generating eIemen& are f~34&j

=

f) Symmetry group of rotation-internal rotation hamlItonian; the potential fiinction is assufned to not vanish identically. The operator Q, which is an additionaI symmetry of the hamiltonian is represented by

!Z)c?(n) cyclic group of ordern. a,, dihedmI group of index n, SO(3) special orthogonal group ove$!s.

these tabIes may be derived in a straightforward manner. Among the properties of isometric groups the foIIowi$ features wiII be relevant in this work: (i} AII systems Iisted in tables 2 and 3 b&rig to the special case

Q(9=C1

9

9x7) = s?!(r) ;

$?(Q*$ = Cl

I)

~hJ,q~=~(q)lq)-

The relation between 9(‘(8 and $@j is aIso shown in tables 2 and 3. For some of the systems ZQ) is a proper subgroup of @(&j, 9@} C &$k for ail other SRMs of the famiIy in tables 2 and 3_9(_$)2 @j$j. (ii) In table 4 information about the abstract structure of the isometric groups is colhcted, including group’ relations, order and number of classes, the commutator groups fi7, p.431 and at least one composition seiies [17, p-381. From these data it is a straighlforward matter to construct complete &terns of nonequivalent irreducible representations (IRREPS) of the isometric groups involved. Such systems are given in appendix B; the no& tion of the IRREPS is used later in the fo~uia~on of the nonvanis~~g matrix elements ofHSE. (iii) fn tables 2 and 3 the extension from Y(E) to && is included; it is induced by&e element b E $&& represented by a substitution l?(Q) in the dynamical coordinates e alone. ThIs’substltution expresses the syinmefry of H w-r-t. rotation of the frame system by t around the ef axis. These s$stems are examples for a generai theorem on SRMs recentIy reported by Frei et al. [6]; which states conditions under whi$ the isometric group is : .. ._ a proper subgroup of g{ri). (iv) The kernel of the homomo~~sm i : @NC){Y) g’?(E) -+ ~(3~~~~~-]3.~d the g&ups I’{%],-. :. fi3jSE {f?) and l?@E) {F) are given explicitiy in tabIes 5-7. From fi3jSE {.@I the group l’@JC)8E-{F] may I. easiIy be reconstructed, but since it is not required subsequently it w.iIInot ,be given.: lnsl%?io~ of tables $-7. I .. reveals that both cases (2-18’) occur. This fact is important for the explicit form of the Wigner-Eckart *e&em &Ed for the mat@ &rie&s of: : _ ”

_..

:,.. . ., -’-_ .’I

:..._

: .: _.

&SE. The latter is expressedby eq. (2-223, the quantity 0 now takes the explicit for& case (a)

0=flfri)S~i6;;,S;,1(3)

case(bI)

Q = $ [l - (~(~~~{l/~+~~~~~~~Sifl l(3) ,

) cw

which determines the selection rules for electric dipole transitions and Stark energy matrix elements w.r.t. the isometric group. Table 8 givesa complete list of the nonvan~h~g matrix eIemenrsof$n w.r.t. F(f) for the SRMsIisted in. table I. The nonvanishingmatrix elementsw-r-t. the group O(3) are determined by the function a(?, J, I) as usual- The notation used in table 8 is based on the notation of irreduciblerepresenta~onsof F(t) givenin appendix 3. The matrix efements offisE may in generalbe calculated directly in a complete orthonorm~ zeroth or first order b&issymmetrized w.r.t. Sof3) X LJ(w.These matrix elements follow the same WET,but most often may be structured in more detail Exampleswill be givenin section 4.

.. .-

C3=E

Cl F C3vT

GF%vT

~3,y2=~

C2=3vT

e(3)

3

‘2

3

33

6

3

‘2

@(3,2)

6

6

6

5

2

iC=CV

G$=C3vT

..

CzF C3J

C3J

c-x3,31

9

9

9

$$1”,9’

18

6

2

@(3,3,2) 18

18

18

a3,31 ‘T et3i

co,321

5 ec3,3>

%e~~)

%F(c3yT)2

36

12

18

9

18

9

I

6

‘.

‘: . .

443

W.Bossert et &/Stark effect and isometric groups of nonrigid molecules

.Table4 (continuid)

-. S&

‘I-

‘. .: &up relationsa)

Isometric group b)

C&F(C3&

C,=C:=T2=U2=E

.. 1. Cztp(C3vT)2

C&l = ClCO,

%I

ISWI

$?$$

36

Commutator

sCX91 9

[X91

group o)

1[%9ll

@(3,3)

Composition

series d)

iW1)

9

4

ss’d ‘9 5% im J ea.n 9.e

(3)

~C~C:'=C'$~T

Bs"d'!?91"8"

u&c~=&;'%I

im 3 @(3,3) ‘L%(3)

TU=CJT=S a)For&initions

ofgroupelementsseetables2 and3.

b)tGtands@}de no t e order and number of equivalence classes of the group 9. ‘) [$.?I and i@?I 1 d enote the conxmutator group of the group $7 and its index under 9, respectiveIy. d) The symbol

$$@a

means that g is a maximal invariant subgroup of % [17].

Table 5 SRMs with one symmetric

10) {.fi?[~~ )a)

SRM

between r( NC) and ~(3) and symmetry groups of Stark operator

internal top. Homomorphism

r {‘SC} b)

Ker : C)

r(3)SE{.Q[Yj

) d)

fiSE

r(SQ {S} e)

:9-r(3){.@} l(3)

C,FC,J

W

f-(c)

l(3), I-w;,)

r(O.

C2FC3vT.

l(3),r(3)(c2&))

r(c),r(Y)

CzvFC3vT

P! r(3)(c2cx4,>, WJ, rG9,

C3J

r(34s[3),fi3)(S~33)

1 I

=

-1

1

l(3)

{r(&lk=O,l,2}%?(3)

e(3)

l(3)

e(3)

e(3)

lc3), r(3'(c2(x$)

{r(ck),r(vck)I

t?(3)

I@! r(3)(C2(x$,

>

’ -1 r(3)(S:3)

=

I

1I

b, Representation of the isometric group by linear substitutions angles), cf. eq. (2-4) (x = 2~/3) ‘1 r(C)=

. _ . .’ 1 . .l..)*

-1

c) Kernel of the homomorphism

dJ%&oup

r(3)SE{E[Y]}c

e) Symmetry

. -1

rm =

. .

. .

-1.

-1

lx1,

27’ x I, . 1,

5

ke(3,z)

@(3,2)

rwhrwn

,

-1

a) r(3)(C2(&

r(s)

e(3)

-I 1 d3)(S,f3)= [

_ 1I

of the dynamical variables (eulerian angles and internal rotation -1

. . _. 1.

r(y)=

1 -1,. 1X

,

1.I

: r (NC){7)+ r(3){i.i?[T]}, cf. eq. (2-19). Notation (?(nt, n2...): abelian group of type ttt,

~3){~[~]),cf.eq.(2-19).

group of riSR represented

by substitutions

in the dynamical variables E, 6.

a) c) d) o) See key for table 5. b) Representation of the isometric group by linear substitutions of the dynamical variables &deria~ an&s, i&em&

rotation

an&s) cf. eq. (24) ‘(x = 2d3) 1.. *. . 1. . - .

11 1..

-1 S-(T)=

I n

-,

. -1

. . .

.

i ? .

n

1 . I

4. Method for the determination of the orientation of the electric dipole moment vedtor In this section a method will be presented which under certain practical cbnditions’&ows the determination of the relative signs of the comporients of the constant part of the electric dipole moment.w.r.t:the~fr&e &em or the p~c~pal axis system for SRMs of the type considered in section 3. The met&d is based dn the me~umment ” -.: of the {quadratic) Stark effect of rotation-internal rotation transitions l&f Jqjg -sJ jpf ~R&f

_. It pulpily makes use of the fact that inter&l rotation axes define’ ~st~~i~~e~ &r&ions-y.r.t. principal axis) system. In the case of the asymmetric rigid rotor without hyperfine interactions qo such ~s~~~n~shed .. ,: . .’ .I:

::. ._.,_ :._t& frame-(ir , diiection exists, . .‘.

.

.:

..

..‘..

445

W. ‘Bossen et oL/Stbrk effect and isomeiric gmups of nonrigid molecules

:

‘..

-.

k,ble,.7

SRI&with two &ninetric tops and centrosymmetric frame. Homomorphism between r( NC) and I’@) and symmetry groups of .. .: Siar~operator-

_.&.;

r-w{&])?)

Km;: C)

r(9) b)

74

CzhF(Cav-%

1(3),r(3)(c~(*:,,,

rcc~c~,.r
d3)SE{L’[Y]

IT=)

} d)

rC3k2k5,r%+ P(C*(x$

’

c r {Y}

d

{r(cboc:‘),r(TC$&

P),rqc&))

e(3,3)

}

is

=sg

rc&c:l), r(u), r(s),rm,

l(3), _&3!

{9}

(2)

r(T), rwl

r(3)(S;3),-1(3).

%hF(C3&

r(3)

ec3,3>

1(3),r(3)(C2(x:)),

{r(C~°C:lUVC$%$),

r(3k_2(x$)

rw~ooc~l)}

rCv,

a) c) d) e) See key for tables 5 and 6 (x = 2x/3). h) Only generators are given ‘1 .

1.

Ill 1 . w,

rQ

’

q

1 Xl 1 I

1 1 .

-1

. 1 .

. . . . 1 . .-1 -1 _

. .

. . .

. n . x -1 . . , -1 . . -1 .

‘1 r(T) =

1J

.’ . . . ’ . 1,

‘1..

rm =

. -1

\

. . .

. n . .

1 .

1 . 1 _ 1

. . . 1.. II 1 . n 1* .

1..

. _ . -1

.

1

Therefore, no information concerning relative signs of the components of the electric dipole vector w.r.t. the principal axes system may be derived from quadratic Stark effect measurements. Use of isotopic substitution leading to small changes in the components of the electric dipole vector, use of bond moment models [3] and of quantum chemical calculations to obtain the first moments of the electric charge distribution [ 18,191 have been

used to derive information about the direction of the electric dipole moment vector. The following arguments will be formulated for SRhIs with structural relaxation, i.e. semirigid molecular models, whose structural parameters vary with the finite coordinates in such a way that the isometric substitutions of the SRM remain isometric substitutions of the SRM with relaxation (SRMR). Effects of structural relaxation in rotational spectra have been recently investigated [20-231. 4.1. Functionalfond of the electric dipole moment of SRMs For SRMSwith structural relaxation the electric dipole moment will always be a Function of the internal coordinates; its general form is determined by the transformation formula p;: nif(g)&-l

= ?3)(F)njf(g)

= kf@-‘(g))

(4-l)

.

In the case of the SRI& listed in table 1 this formula requires, respectively (VFE F(g)) ?3)(J’)nif(7>

=.ni’(F-‘(T))

,

~c3’(F)nif(ro, q) =tif(F-l(ro,

q)) .

(4-l’)

CIF C3vT C3”T GF C3JC3vT C2F C3”T C3v7 CzvFC3JC3vT

+wl)

a b2

@OWP) cTr(mo%P)

1

rW-d

b2

c2vF(c3v%

+f r(m”q),

r(mn) * r(rnn),

m0, ml = 0,1,2, i =.+i,‘-

(mn) = (01),(10),(11),(12)

(mn) =(00),(12),(21)

(rm) = (01),:(11),(02)

r(mW) _ dmnP),

a

.

++ r(OO+S), q = f, _

rim&) - dmn-),

b2

CiF(C3vT)2 I CZF(C3vT)Z

, qhrnl = OJ.2 ‘r(Yu) ct firnn), c&r) = (01),(10)~(11),(12)

#Jo+) cf rtoo:~,

dmw) csF(c3vn2

* @06)

On, n) = (OO), (1 i), (22)

rtmn) e rJmn), (m, i2)= (011, (02), (12) r”+@ rot e r-o+@ye-, yu-@rot u yu-@yob2

C2hF(c3v'D2 I

t

r(OP)~r(l),r(OP)~r(l),

~(l)~rr(O+),~(~)~~(O-_)

~O)@rO)-r(~)~rO)

a) The electric field is taken along the laboratory

z axis, (E$ =

0 1 00

IE I.

b, For discrimination of cases cf. ref. [6] _ ‘) For selection rules w&t. SO(3) cf. (2-22) and for notation of irreducible representations hfatrix elements with all other combination

of irreducibIe.represenhtion

of the isometric group cf. appendix B.

of y(.$) vanish.

Using the information given in tables 2-7 one may derive the formulae given in table 9 and appendix C in a straightforward manner. Each dipole moment component n;i,f is expressed as a Fourier series obeying the sym-. metry requirements (4-1). The following comments concerning table-9 should be made: (i) All SRMR considered in this paragraph will in general exhibit dipole moment .components it?; modulated by the internal rotation even if the constant term of the Fourier series vanishes. The. most extreme-examples in this respect are SRMs with centrosymmetric frames (with reiaxation), where the constant terk of all components vanishes. Such systems may nevertheless have an electric dipole absorption.spectrum due.to the higher Fourier coefficients. _. -: .. (ii) To our knowledge no information about the magnitude of the Fourier coefficients is available. Ab initio calculations should in principle give estimates of these molecular constants and also might serve to design experiments for observation of rotatiorral spectra of say, molecules vvith centrosymmetric frame andtwo equivalent symmetric tops. Ab initio studies of this type are underway in our laboratory. (ii) The transformation properties of ML discussed in this paper v&l prove useful for the formulations of a ., theorem concerning determination of its orientation w.r.t. the frame system, cf. section 4;3.. : ..

W.:k&sert et &/Stark effectand isometricgroupsof now&id molecules

_I

447

- .. .Table$J ' Gerieial fomi of.electricdip?le.momkntdependenceon internal rotation anglesof two top SRMs

-CIFC~~T

f(T)

f(d

f(T)

W

C3J

fcs

f-s.

fcs

‘3

C3vT

f-v

f-v

f+v

f+s-v

f-s-v

f+s+v

fhJ,q)

f(70,q)

fbo, 11)

CzvF

C3vT

GFC3vT

-

C3vT

C3vTC3vT C2F C&T C3”T

f+s

fs

f+s

f-v

f-v

f+v

C2vFC3vTC3vT

f+s-v

f-s-v

f+s+v

6W3vTh

f+iY

f+u

f-u

W%vT)2

f+T

f-T

f-T

cZvF(c3vT)2

f+S+T

f-S-T

f+S-T

CiF(C3vTI2

f-u

f-u

f-u

CmF(C3vT)z

f-S+T

f+S+T

f+S-T

DzhF(C3v~2

f-S+T-V

f+S+T-V

f+S-T+V

W

4.2. Dipole moment matrix elements

In this section explicit expressions for electric dipole moment matrix elements will be given within the following framework: (i) The eigenbasis of the rotation-internal rotation hamiltonian

+ vro,71) ,

(4-2)

is constructed (by infinite matrix diagonalization) from the eigenbasis functions iVr~~in(~o, I-~) of the internal

motion problem arising from the operator (4-2’)

and the zeroth order symmetrized rotational basis function dJKMr(~~)(QPr) (actually obtained from the rotation group coefficients DK(J(c$y) [9, p. 164;24]). The index IWO denotes the irreducible representations of the isometric group to which the rotational basis functions belong. A detailed discussion of the symmetry properties of the (g,,), (g”“) matrices and the potential energy functions of the family of SRMs listed in table 1 has recently been published by Croner et al. [16] _ (ii) The dipole moment operator is assumed to have the most general form for the SRM C,F C3,T C3,T. According to table 9 and appendix C it may be expressed as =a Iji,f(.,,, 71) ‘n&o

Mfbpp sin 3pro sin 34rl , cos 3pq cos 3471 + c P, q=l

(4-3)

.-

i:~imagirWy unit}

(A; Q-type transition

_’

Eqs. (4.4) allow a more detailed verification of the &mew degeneracy beyond the generaf arguments discussed in section 2.4.1. The structure of the matrix of @ + fiSE (for any symmetry species D(fi, I’(k) and the corresponding rowsMand K, respectiveIy) may accordingly be pictured schematically by (4-5) provided a real symmetrized basis is used for calculation of the matrix of@. The phases of the basis may furthermorebe chosen_to yield a real (symmetric) matrix H* (blocks A and B) and real Stark matrices (Cand 0) [23]. The states withM Q 0 yield the matrix

:_ : _I which has the same eigenvalues as the matrix (4-S) and corn~~e~.conj~~a’~eeigenvectok forking _ a_:~~~~~-~bi~~~~ gc3kil W-r-t. that of the matrix (4-5). ‘. : __

..

._:-4..!?. determination

of the relative signs of electriC dipole vector components

;..

I

-

Tfie rest&s obt~~ed before~nd allow one to formulate a theorem for the detktiination of.the r&a&e signs _: of the components of the constant part of kf ih the case of the SRMs Ci e CivT,‘C$ C&,‘&C,i: C&T C&T,‘ CsF C3vTC,vT, C,F(C3,,T)l. According to eq. (4-l) ki is subject to the condition ‘: _’ &g = i;(3>(F)&f

“ia

: Furthermore, by eq. (4.4), the matrix of&$ in the &enbasis of I@ is g linear combinatidn mat&& whose scalar coefficients are the componentsM&00 (s,= l&3) w.r.t. the frame axes. This is made evident as follows; Denoting the transformation from the first order basis of I? to its eigenbasis by

df

(4-7) where the transformation matrix is independent of the row numbers M and K, one may write (cf. eqs. (4-3) and @-%D

(4-s) For the calculation of second order Starkshifts of the energy levels by RayIeigh-Schr&linger perturbation theory one has to use

a

_

4

:

... ..;

I;‘ ; ;..:.<

.I _.-.. .__: .1x.. _’ Fig. 12~ Dete=nination of relativetign of constantpart of eIectic dipole moment ‘~ornponen;ts..~~~:.%&i&Lvitti C, f&e. _

.. .

.:

-:::

T

:

_’

:;.

:_

.:

:

-.

451

N. Bosiert et &./Stark effect and isometricgroups of nonrigid molecules

_. .. .. .Th&&@tio~

shows that the second order Stark shifts depend on the relative sign of the quantities @a,,. For the SRMslisted above the last equation therefore allows the determination of the relative signs of the dipole moment .components Mfsao,-,by measurements of Stark shifts of either P, Q or R transitions provided the involved levels experience a predominantly second order Stark effect. This theorem may be illustrated for the SRM classes .&th’C, frame symmetry by fig. 2.

5. Determination of the orien@ion of the electric dipole in the acetaldehyde molecule 5.1. Framework of the treatment

The acetaldehyde molecule has been subject to several microwave studies [25,22]. In this section the analysis of the Stark effect of a set of transitions is reported, from which the relative sign of the components of the constant part of the electric dipole moment is derived. The analysis is carried out as follows: (i) The data will be treated by infinite matrix diagonalization of the rotation-internal rotation hamiltonian of a SEW of the type C,F C&T without relaxation, essentially by the PAM approach [26; 2, P_158; 231. The (pfl)

matrix then takes the form I gtt 0 0 g14 ’ (gm”) =

0

$20

0

0

0 g33

$4

*

G-1)

Zeroth order basis functions symmetrized w.r.t. S(T) $ $‘{I?} 5 CD, h ave recently been given by Bauder and Ciinthard [22]; the information given by these authors should be completed by the formulae collected in table 10. In the first step of the energy eigenvalue problem the internal problem is solved (J = 0 states) lY4 = f&;

f V(T) )

(5-2)

by infinite matrix diagonalization, providing thereby first order basis functions for the solution of the rotationinternal rotation problem (J> 0):

Diagonalization-of the resulting infinite matrix is then carried out observing the following rules: - if II E lN is the number of eigenvalues of the internal problem to be calculated to kHz accuracy, matrices of dimension 3rz X 312 (ru+, ~--blocks) and 6n X 6n @‘I-blocks) are to be diagonahzed -consider the maximum rotational energy level associated with the energy operator of the rigid rotor belonging to a previously chosen J-value E, = max E,(J, K_, K+) . Then if sup I?&,, sup E$N W

is the highest internal rotation energy eigenvalue of a species I’(j) fulfilling the relation < max E,(J, K_, K+) ,

one has to diagonaliie a matrix of dimension 3N(2J+l) and 6N(2Jtl) respectively. As a rule the lowest N(2J+l) .eigenvalues will have kHz accuracy. (ii) Values for rotational constants, structural data and barrier to internal rotation will be taken from ref. [22]. The molecularmodel is depicted in fig. 3..

dJoMr* d

JKMro-

d

JOMi’*

a)Internalrotation

problem.

b, D~(@?yI denotes rotation group coefficient cf. ref. 19, p. 1641.

..

c3 Pairs in g&&bases denote basis functions belonging to the 1st and 2nd row of ft, respectively.

d, Rotational basis functions. e)U.rb-)Ndenote ~i~e~~ctions of the internal probiem.

‘.

:

.:.

.i.,‘.

,_

:_;

:

_._

._

.’

.

‘...1.

Fig- 3. Acetaldebyde molecular model and orientation of electric dipole &om&tY ‘_‘.’ .z ‘( 1, .-

:

. ‘..,

.-.:

.:

: W.B&ert &ral. fSttirkeffe$thndisometricgroupsof nonrigidmoleciles

453

(iii> Ex@&ons for the dipole matrix elements may be taken from eq. (44), which s.pecifically.for the _C,F C$,T.case read in the zeroth order basis (constant part of the dipole moment only): @type ilemen ts. -.. _-i

: UKML

l?%i,~;,JiML

I%m)

.. -= (-i~~~(J+l)~~~~(-l)J+r.

F(K,K){~~~10[A(J,-K)~~,K+1+A(J.K)6~,K_1]

Calculation of Stark shifts is effected by 2nd order Rayleigh-Schriidinger perturbation theory and by numerical diagonalization. For the perturbatiorrapproach Stark shifts are calculated in the form (cf. section 4.3)

During numerical diagonshzation of the infmite matrix of fro + k SE, for any given value IMI a block composed of at least al! matrix elements included in the 2nd order perturbation sums is directly diagonalized. For the levels considered in this work, blocks up to dimension 100 X 100 have to be diagonalized. Comparison between experimentally observed shifts and analysis by perturbation treatmentand direct diagonalization reveals that shifts not obeying a square law dependence on the Stark field strength show as a rule large discrepancies between perturbation treatment and direct diagonalization. 12. Experimental All experiments were carried out with a computer controlled microwave spectrometer equipped with conventional 4 m and cj m X-bandStark cells. 30 kHz Stark modulation was used and the Stark voltageswere measured with a digital voltmeter up to 4 digits. The exact electric field in the wave guide was determined by comparing the theoretical and experimental Stark shifts for OCS @= 0.71521 Debye [27]) in both Stark cells. For acetaldehyde the line shifts due to the Stark effect were measured over a range of 1 to 20 MHz.In addition, the observed spectra were smtiothed’tith a 7 point parabolic fit [28]. All measurements were carried out at room temperature and pressures of apprtiximately 1 mtorr. The line centers could be determined with an accuracy of approximately 10 kHz (singlc~standard deviation).

-454. :

:

IV.Bossertet al./.Starkeffebtand isometric groups of nonrigid~olecriles~

..

-Table.lf ._ Stark Shifts for acetafdehydc

-

.Transition Jti, Ic;r c JK_r K,r

lo1 + 000

202

-

101

Stark shift (Hz V” cm*) WJ

OA IA 2A OE IE 2E OA

M

0 0 0 0

0 0 0

1 2A OE 2E

303-212 _

-.

OA

0

1 0 1 0 I 0 1 2

obs.

45.7632 45.8486 41.9626 45.7620 45.1023 39.7950 -15.2550 9.7025 -12.1290 10.0609 -15.1289 9.6467 -11.5961 9.2503 3.2759 17.7586 60.6164

..

P#b>O

-~#~
cdc..

talc.-obs.

talc:-

45.1481 46.0983 42.6874 45.1886 45.3954 40.4355 -14.9006 9.5925 -12.3061 10.1219 -15.0633 9.4932 -11.4697 9.4323 3.2904

-0.6151 0.2497 0.7248 -0.5734 0.2931 0.6405 0.3544 -0.1100 -0.1771 0.0610 0.0656 -0.1535 0.1264 0.1820 0.0145 -0.0660 0.2826

4312741 42.9304 44.0034 43.2604 43.1175 44.5672 714.3887 9.1471 -12.6458 lo.3577 -14.$567 9.0245 -12.7301 10.3474 3.3398 17.1088 58.4158

17.6926 60.8990

-.’

c&c.-ohs. 2.4891 -2.9182 2.0408 -2.5016 -1.9848 4.7722 0.8663 -0.5544 -0.5168 0.2968 0.5722 -0.6222 -1.1340 1.0971 0.0639 -0.6498 -2.2006

In table 11 the set of low J transitions investigated in this work is collected, together with the Stark shifts measured. For each of the transitions the line shifts were measured for 8 to 10 different voltages and the data were fitted to a pure quadratic function of the Stark field for eachM-component. For the determination of the dipole moment components only transitions were used for which no significant deviation of the square law dependence could be found. For the lo, + Ooo transition the ground state, and the fmt and second torsionally excited states (U = OJ ,2) for both symmetry species A and E could be used, whereas for the 202 + loI transition the first excited state had to be discarded, since the Stark shift contains a strong iinear component in addition to ‘he second order effect.

5.3. Analysis of results The.experimental data collected in table 11 were analyzed as follows: To fit the calculated Stark shiftvalues to. the measurements a least squares method was used for the two possible signs ofIV~01.M,f03 hereafter denoted by &fib_ In version 1 the sign ,u&, > 0 was chosen and as a starting value for the least squares fit I&] .w 2.55 D and I&,] = 0.85 D 1251 was selected. In table 11 the calculated Stark shifts resulting from the choice &pb > 0 are listed for the transitions investigated. In version 2 &c(b was chosen negative, and the fitting process started with the same zeroth order values for ]q] and 1~~1as in version 1. Comparison of observed and calculated Stark shifts for this version is also given in table 11. In table 12 the values of the dipole moment components of the two versions and the residual sum of squares of the differences of observed and calculated Stark shifts are contrasted. :

5.4. Discussion The foregoing results deserve a number of comments:

:

:-

-

W. Bmsert et al. f&ark effeer and isometric groups Table i2 Acetatdehyde:

components

and orientation

;&ye) f$Pb>‘o

Kilb et at. [X5]

2.5160(43) -2460(22) 2.55

0

mo!emIes

4.55

of electric dipote moment

:_ -Version

&Pb~O

of nonrigid

1

pb

M

E(cidc.-obsJ2


(debye)

(Hz V-* cmz)2

1.0700~65!

2.7341(65)

1.079(33)

2.686(33)

0.85

2.69

2

0

*

2.125 61.735

2

Fig. 4. Acetaldehyde. Determination of relative sign of electric dipole moment components from secondorder Stark shifts.Key: measured and caIculated Stark shifts Au (Hz V-’ cm2) for the lot - Ooo rotational transition. (a) A *A transitions;@) E +fE

transitions. l measuredStark shifts, l calculatedwith pa = 2.516 D, &&= 1.070 D, Acalculatedwith P~ = -2.460 D, PB= 1.079 D. (i) The two versions pa/+, >O and&lug < 0 Iead to the same values of 1gJ and l&l within their standard deviations. (ii) In fig. 4 the Stark shifts Gf the lot +Ooo transitions for A(I? J+ I’O-) and E(@ * f*) species are plotted, respectively. The plot clearly shows that the dependence of the calculated Stark shifts upon the torsional quantum number is qualitatively quite different for the two versions. (iii) Fig. 4 furthermore demonstrates that only version 1, g&b > 0 reproduces the observed Stark shifts satisfactoriiy whereas version 2 shows systematic discrepancies.

5.5. Conclusions

The discussion presented in the foregoing section shows that the appiicabihty of this method for the determination of the orientation of the dipole moment in molecules rests on a number of practical requirements: (i) In the case of the SKM C,F C&T Stark shifts of several A(@- * Pi) and E(I’l e I?) torsionally excited states with low J values must be measurable. This impbes relatively high resolving power of Stark measurements on one hand and sufficiently Iarge A-E splittings, i.e. sufficiently Iow barrier to intemai rotation on the other hand. (ii) The direction of the constant part of the molecular dipole moment vector and of the frame axes should deviate sufficiently, thus allowing dete~ination of the moduli l&l, l&,1 with acceptable precision. (ii) By the new procedure it is not possible to extend the information about zhe orientation of the constant part of the electric dipole moment of SRhfs of the type discussed in section 4 beyond the theorem given there, Le. to discriminate between forward and backward directions indicated in figs. 2 and 3. Such a discrimination would require further appropriate terms in the hamiltonian.

For some purpdses it may be favorable to f~rrn~ate a WET for the oper&orfiSB w.r.t. the eigenbasis of@, .’ but restricted to the symmetry group of I? BE, the latter being generated by C_r C O(3) and l?SB) El?{@). The energy eigenf~nc~o~ of go belong to irreducibJe or reducible repres’entations of c&i @flSE). According to [ _, the assumptionsmade in section 22, the eigenfunctions ~~~~~~)~~(~,~) belong (i) to the irreducible representation flMi) of C,, {ii) to irreducibkor reducible represent&ions of I?{%!) depending whether or not the representation of .’ . fisE) ~~~‘subd~ced from I’(@Csir;) is irreducible or not1 in any case the WET reads :

.: : If the representatrous A~{BISE] subduced from I’(+(%) are known, theJast coefficient of the doublibar element inay be evahrated by means of the usual orthogonal&y relatiok for [email protected]&n the matrix oP$B is:. &gonai w-&M, but to show. that the pair of matrices-for +Mand -Mstates are e~~v~e~~requ~s a-more de-. :. ‘. . ‘.I’ tailed investigation of the double bar element or application of time reversd syinmetry;_ -1; .:.. ~..~..~~‘__.,... ~. ..‘. .,.

‘.

Append& B: frreduciMe representations of isometric gr&ps of SBMs with sjmmetric inteti! .: .-

z

_:I

r+i$ -I.. -. _ ._I-__

: . . :..

.’ IrreducibIe representations if groups related to the SRMs listed in table l’havo b&n @ported on:. va&rsocca-‘j.’ -. _. .‘... ‘. .: _ :

._

:.a .’

‘,

.. :

_ .:

.,

-_.

.:_ ._i:.

._

. .

..:

I .,..

:

:

--

_.

c-..-

: :

,.‘..._

__:._. -. :

::

-.

.-.

. .

,LB&&

e&df&rk

effeciand isomer& &I~E of non&M malecules-

67

:‘s$&t goti, for ihe sak&of co&pI&eness and tiompactness of the formulatiot. of seIection rules and Stark energy :. : n$i$x elements we’give com$ete n&equivalent systems of irreducible representations of all the isometric groups of t&especific systems co&de&d in this paper. The notation of irreducible representatidns has beeo chosen in a My which allows convenient f&molation bf Clebsch-Gordan type theorems for direct products and Kronecker pdwe&. Con~eniional notations are given where available. (a = 2n/3, E = exp(ior), q 4 exp(iol/2), p = O,+; I,--) :

($3) .:Y {Ck 1k E Z, C3 =E] r(m).

ck

.

fro)(A)

1

Ek e-k

cd3:= {Ck,SCk 1k E Z, C3 = S2 = E, SC= C-%3]

I’@‘)(Ai) X@-‘(Az) T’(t)(E)

*I

r(m) e,S f@n(mod

3)) e,S y13-ml,

r(m)

0

r(n) ezS r
(3) Q6 := {Ck, sCk 1k E ?I, t? = S2 = E, Se= ?--IS)

.

:.-

:

r(w r(v)

ck

._ _

l-(O)l-o+ (A)

r(S) fil-> r(3)

r@-‘>

fl2,

r(l+)

r(4)

:

1

~fil, r(2-1

._.

.:

.

:-

: :.

.

+ 1

r(2+)

(El) (B)

o5F .(_l)k‘,

g3k

q2k (E2)

,,-+k

I-

(5) e(3,3),

e(3,3,2)

,.

.:

._

-- :

+WS)

vsd$q

po++1

1

r(oo+-)

-I

r(OO-4

-1

r(OO~y--1

1

JyWW

.m=

10,01,11,12;

ca&M+lllV)

t-vq -saQoM standard

+ l&

form:flMNs) = r@f(mod~)N(mod3)q).

TCW 0

-I 1

1

-1 &Clo+W -eio(lO+~~)

e2ia(lo+Zl)

&(IoM+llN) --m(loM + l$V)

I

iiir : .: :

e2iatl

0

,liaZo

i0

eia(t0+211)

I

I: 0

1

eia(210+Z~)

0

.o-

i

..

eia(Ljf2Zlj~

eia(lftO+Zl) 0.

:

1,

(8) 0% $ ‘9, X 93, a)

I

ref. 113:

ref.[15]

&AI

r(00++)

P’x

AI.%

r(oo+-)

1

1

1

r”+ x r”-

1

I

-1

#Jo+)

i-o-x r-0’

1

-1

-1

+1

A2A2

r(oo--)

l-0-x l-O-

1

-1

+i

-1

AIE

r(11+)

9+x

r(l)(@G)I

r(l){C!(3) I

r(l){S@Q)l

r(l){se(3)1

fill-)

rowxr'

rqC(3))

-r('){&3))

-r(l) {S&!&j

rclqsew

f=l

r(12+)

rlxro+

d1)IC(3))

d1){SC(3)}

&en)

r(‘W(3)l

E‘42

l-W-1

lqSsc(3))

-r(F){Se(3)}

-r(l){CCS}

EE

r(lO)

rcljfsc(3)~

&&?(3,)

dt~{ecsi}

AA

A2E

rQ’

r1

rlx row

r(x){E?(3)}

l-lXl-1'

rc’) {e(3)> x dqe(3)I

a) The isomorphism

xdW3))

i: G?$ + ‘03 X Q3 may be chosen as

1 1

x ril) i&3)}

... is subject to symmetry requirements of the type expressed by eq. (4-l) conditions between the coefficients arise ... which may be d&%rnine,d ma-straightforward manner. Writing the resulting series in_a real form containing only independent coefficients yields the.following expressions. For ah cases the operator P , I leads to series whose tfrequencies are all~mui.tiplesof.three. &m-&trized Fourier.series in one variable may%%ived from those in two:va&bles ina simple way. M(iO,

i1) = Moo

+ ;g0

Map4

COS-3pr0

cos 34T1

+ Pzl

&&Pq

Sin 3pT0

Sin 34Tl

‘.

c

t

p=O,q=l

MCpq cos 3pro sin

c

3qT1 t

p=l,q=O

Mdpq sin 3p~~ cos

3qrl

,

-1

M+&), 71)‘Mob + .

c Map4 cos 3pro cos 3qrl ZW’O

=pj$zl

MCp4cos 3p.ro sin

Jf-s(To,3’1)

3qT,

t

+

t

M&

c

Mdp4 sin 3pr0 cos 3qrl

c p=I,q=l

71) =M,,

t

,

m

DD M+l/(To>

sin 3pro sin 34~~ ,

p47=1

c PA=0

Mapq gcos 3pro cos 3qrr +

,$zl MbPq gsin 3pro sin 34SI

P

m

tc MEpq gcos 3PTos’n %+ p=O,q=l M_.y(~o,

T1) =P$Fo

tii

p=O,q=l

McPqu

M+u(~o> 71)=Mo,-,+ iFI -.

Map4

u cos 3p.ro

& p=l,q=o

cos

34~~ +P$=l

COS 3p~o Sin3qTl +

Mapp

MdPq

c p=l,q=O

.p=o,~=l

7pq

u

Maps

cos 3pT,-, co; 3~7~ + c

sin 3pro sin

(COS ipTo

,

34~~

34~~

,

CDS 3qTl t COS @To

P&=1.

P=l

(cos 3pTo

MbPq

Cos 34Tl

Mdpq II sin 3pro cos

MBpp sin 3pro Sin 3prf + p$zl Mep4 (sin

+- .ii:.

3pro

gsin

sin 3471

-Sin

34To

3pT,-, Sin 34Tl+

cos 3pT1!

’

Sin

34~ sin 3Prl)

COS 3pTl)

+

g=$[l+(-l)P’4],

22

.%

p=(),q=l

m

(cos3pTO sin 3q7i -siR3q?Oco~3pT~).

u=$,[l

._.’

-(-l)P*q],

‘=excludesthetermwitllp=~=O..

:

-

_’

(’

:. :

..

References

. .I

;;rk,

[I] C.H. Tow& and A-L. Schawlow, Microwaye speftroscbpy (McGr&& New r9&) p.248: t . [2j 3-E. W+!_r& Rotational spectra and malecuiar structure CAcedemic Press, New.York, 1967)~p?244., 1. . f3f W.CordYand RL. Cook, Microwave molecufar spectra (~tematio~ P&Iish~rs, New Yo+ 7970) p. 303; : [4] H-W_ Kroto, Mplecu&rotation Spectra (j. WjIey tid Sons; yew York, 1975)~. 164. y : c -; I _’ [SJ L.S..Bernsteiriand K-S. Piker; J. C&em-Phys. 62 (1975) 2530. f6] H. Freii R. Meyer, A. Bander and Hs.H. Giinthard, Moi. Phys. 32 (I&@ ‘4;; $4 [19i7) 1195 ’ ._ _. -, [7f A. Bander, R. MeyerandHs.H. G~n~d;.MoI. Phys. 28 (1974) 1335. ;.. [8],NtFreiand~.H;G~nthard,Chem.Phys..1S(197P)’155. ,. : --’ 1 ;:.:-,_--. -_ -,_\-- -: [9] E;P. Wignei, Group theory (Academic Press, New York,i959). . [IO] H.A. IGangs, Proc. Amsterdam Aizad.%I;‘33 (193oj 959. 111J AiMessiah;Quantum mechanics, 901.2 (North-Ho&and, Am&dam, 19#. _ _-._ : _,[! .,~r__ _, .-._; 1:: : ‘_ [12J H. B6n-g D&teIiun&en Van &&p& (Spring&, Berlin, i9.55) $83. ‘. [f3] H. DreizIer, in: Fortschritte der chemischen Forscbng, Vol. 10, Spekt&.&d M~I~k~~~ fC&&&, B&g, &3) $,59: 1141 H. Dreizler, in: MoIecuIarspectroscopy modern reech, eds. K-N.- f&and C.W.Ma$+ (Acadert& Press;Ne&York, .. _: 1972) p. 59. : _._ [15] J.R. LIttrig,Y.S.~abdP_Groner,J.Mol. &ectqy. 62 (1976) l&i ‘. -’ 1‘; ‘-.‘I :‘y ?-f.- ~:I: _. 2, .. -: [16J P. Gronerand J-R. Dmig, J. Chem. Phyi. 66 (1977) 1856.

._ ._-.

:.

_. ‘._.r__. .- ‘,.., ,., .. -.- ‘_:_ : :_.‘.,__” ‘., _ ~_(..~ .-.;-, . . ; _.,I__.. ,. _:. : : :...

.-.’ izsl .&yc Iwb, c.C.-wand E.B. W&on Jr., J. Chem. Phys. 26 (1951) 1695. .: f26j C.C.I&I and J-D: S&den, F&v.Mod. Phys..31(19!9) 841. [27]-$. Miibiei, J. hein; Phyi. 48.(1968) 4544. [281J.P. Porchet and ID.& Giinthard, J. Phys. E: Sci. Instr. 3 (1970) 261. . .

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._

.,

.

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