Infinite matrix solution of the rotation—internal rotation problem of glyoxal type molecules. II

&emical Physics 310978) 433-466 0 North-Holland Publishing Company

INF~ITEMATRIXSOLUTION

OFTIIE ROTATION-INTERNALR~TATIONPROBLEM

OF GLYOXAL ‘I-WE MOLECULES. II M. GUT, R. MEYER, A. BAUDER and Hs.H. GmTHARD Laboratory for Physical Chemistry, 8092 Zurich, Switzerland

Swiss Federal Institute of Technology.

Received 16 December 1977

Symmetry group and the rotation-internal rotation problem of the semirigid model representing molecules like H202, glyoxal, CHzF-CH2F etc. are reconsidered. For three models differing in the choice of the moving frame, the interrelation between the dynamical coordinates (eulerian angles and internal rotation angle), isometric transformation groups, kinematics and kinetic energy functions is first discussed. It is shown that consideration of the periodicity isometric transformation leads to a natural extension of the isometric group, and to a straightforward defiition of the function space, in which the quantum mechanical energy eigenvalue probIem has to be solved. A new solution of the latter based on infinite . matrix diagonalization is presented and two computer programs based on two models are discussed. By means of the latter representative calculations for three molecules (glyoxal, H202, FHzC-C = C-CH2F) have been carried out including Frequencies and intensities of electric dipole transitions between states with 0 a J 5 3..

1.&ctduction Recently we have presented a solution to the rotation-internal rotation (RIR) problem of giyoxal type molecules by infinite matrix diagonalization [ l]_ This work, hereafter called I, is characterized by a symmetric choice of the frame system and analytic calculation of the energy matrix elements in a basis constructed from rotation group coefficients and Fourier functions, symmetrized with respect to the isometric group of a semirigid model (SRM) consisting of a frame with D,, and two coaxial tops with C, local symmetry (D,hF(CsT)2-system)_ A number of group theoretical aspects related to primitive period and domain of the internal rotation angles and of the transformation of the dynamic problem induced by changes of the frame system were considered in some detail. In the meantime a computer program based on the analytical approach given in I has been worked out. In the course of the development and test of the program we found that the interna problem (J= 0 states) was correctly solved, that however, the computation of energy eigenvalues and eigenvectors for J> 1 states.was seriously hindered by the slow convergence of those parts of the energy matrix elements which involved the g14 and g23 coefficients of the kinetic energy matrix (g”“(r)). These phenomena were found only with the approach based on the symmetric choice of the frame system, but did not appear with the frame system attached to orie of the two equivalent tops. Difficulties of a similar kind were found in the analytical treatment of other SRM’s, where a symmetrical frame system could be chosen and prompted reinvestigation of the DmhF(CsT)2 system in particular. The results of this study, which appear relevant for a large class of dynamical problems associated with nonrigid molecules will be reported in this paper. In section 2 the two models used in I are completed by a further model consisting of two tops, whose dynamics is described dire&y from the laboratory system, without reference to a frame system. The interrelation between all three models is discussed and used for the solution of the problem of uniqueness and primitive periods of the dynamical coordinates. Implications for the isometric group and for the construction of a symmetrized complete ortbonormal basis for the solution of the energy eigenvalue problem are

434

MI Gut et aL/Rotation-internal rotation in gZyoxaZtype molecdes. II

considered in section 3. Analytical expressions for the matrix elements of the hamiitonian of one of the models are-given in section 4, which avoid the problems of slow convergence observed in the approach I. Subsequently, we give a set of numerical examples for spectra of typical molecules obtained by a computer program based on the new approach. The new approach also implies reconsideration of the electric dipole matrices, which is given in section’5

Z..Dynamics of semirigid models of the D,,F(C,T)z

system

2.1. Models andnotation

For the following treatment of the dynamics of semirigid models (SRM’s) of the D_I,F(CsT)2 system three models will be used, which are all based on the schematic model fig_ 1. The frame of the SRM is assumed to have a rigid linear structure of D_h covering symmetry. To this frame two congruent rigid tops of local C, symmetry are attached, which may be rotated against each other around the C,- axis of the frame. Without loss of generality the frame may be assumed to consist of 2 anchor nuclei and at the most one nucleus iu the center of the frame (site m and site 0, respectively in the notation of Herzberg [2])_ The nuclei of the tops are positioned either on thelocal symmetry plane or in general site (site notation tr and tZ, respectively). Any further nuclei on the C, axis are accounted for by nuclei in site to belonging to the tops. The three models used subsequently in this paper may be characterized as follows (lig. 2): (i) TwoItop model, consisting of two congruent rigid tops of local symmetry C, each including one of the frame nuclei. The two tops have one axis in common, but otherwise rotate independently with respect to (w.r.t.) the laboratory axes. (ii) Symmetric frame system model (ef-model). In this model, the relative nuclear configuration is defmed by the internal rotation angle (dihedral angle) 2~. Most of the considerations of I were based on this model, where the frame system is at the same time one of the crystallographic bases of the covering group G(r) = C2 of the lnstantaneous nuclear configuration.

I

Fig. 1. The SRM D,bF(CsT)a: laboratory basis vectors.

schematic model. ei denote

M. f&t et a~/liotahbn-intemat

rotation in g&oxal type moledes.

Ii

a Fig. 2. The SRnl! D,hF(C,T),: eulerian angle 7 and intern& rotation angle r of three kinematic models. (a) two-top modef; (II) ef-model; (c) if-model. (iii) Asymmetric frame system model (if-model), where the frame system is attached rigidly to the partial nuclear configuration consisting of frame and top 0. In this choice the relative nuclear configuration also depends on the internal rotation angle. The tr~sfo~atio~ theory between the et- and the’if-system has been given in I. The three kinematic models are depicted in fig, 2, where also the dynakrical coordinates are indicated. Furthermore, in tables ! and 2 dynamical coordinates and expressions for the nuclear coordinate vectors referred to the laboratory system of typical sets of equivalent nuclei are collected. In all cases tr~R~ation has been omitted. The notation in these tables is as follows: - eulerian angles are used in the deftition by Edmonds [3], i.e. interrelation of laboratory and frame bases is given by z+= PD(or, a, r) ,

- nuclear coordinate vectors for an arbitrary nuclear configuration (NC) are generated by mapping of a reference

Table 1 Notation for eulerian and internal rotation angks -Model

TabIe 2 D_bF(CsT&

SRM, nucIear coordinate vectors

two-top

ancIlor

--fnter1eIation

Notation a)

(Cf. fig. 2) a)

‘xf

a3

1

-3

b)

t1 (91) t2 (92) cbf

=)

w

ef

anchor

t1 t2 CM

if

anchor t1 t2

CM

a) Upper indices P, f, t indicate laboratory, frame and local top coordinate system, respectively. b) It should be noted that .A& does not depend OR yk, y and =j, respectively. c) CM denotes center of mass ofthe.wbole SRM, the quantities occurring in these coordinates tie det%ed in eq. (2%

M. Gut et d/Rotation-intend

rotation in glyoxd type molecules. II

437

contiguration def;ned in the laboratory system. The latter is in all cases generated by a two-fold symmetry axis in order to secure congruence of the two haives (tops) of the SI@l; - nuclear coonhnates of anchor and top system in the reference NC are chosen identically for alI three models as follows (cf. fig, 2). Anchor nuclei are characterized by the one coordinate vector component Xi3, denoting the distance of.the representative nucleus from the center of the frame. Top coordinate vectors Xtl , Xtz are always referred to the local coordinate system of the representative top (top 0), whose origin is located in the representative anchor atom. Accordingly

at least component two nonzero,

,

at least one of the fust or third component nonzqo;:

- letd be any matrix, then A.*, Ak. denote the Zth column and the kth row ofA, respectively. In particular, I.1 denotes the column i of the three by three unit matrix l(3); - otherwise notation is kept as close as possible to the one used in I; besides obvious adaption of the indices, the only important difference is the internal rotation angle for the if-system, which is in the present work defined as 5= 21 (as compared to 7 = -27 used in 1). Table 2 should be commented upon as follows: (i) The center of mass coordinates X& contain for all three models the same quantity

where M stands for the total mass of the SF%& (ii) The covering symmetry operation (?2 may be expressed as mappings of the X$ vectors by the matrices: two-top model -lC31 + 2D(o, P, 0) D.1(0,0,

$(Yu + r&,(0,0,

&-J + r,))&%

P, 0) I

ef-model -l@f f 2D(o, P, 0) D-1(0,0,

0, 0) I

r) Zi5,,(0,0, &x%

(2.2’)

if-model -l(3) -I-2D(ar, a, 0) D.,(O, 0,5 - $7) 5. I@, 0, r - :3&x

P,. 0) *

For derivation of the internal isometric substitutions internuclear distances for at least one set of equivalent nuclei are required. From the information given in table 2 one derives the following formula for t2 type sets (X,ll=O,l): twp-top model 2 d&L,xfi* = 2t1- (_l~~~~]X~~(X~~ + 2x4 + zz&x,, [-(-l)h+p + (-l)*“c’] sirIt& - yh;) 0 [-(-l)~+fi

+ (-l)XfU’]

Siri(-/^ - Th’)

2(_l)hf~+r*$Cos(~~ 0

-

T*,)

O-

Xt2 *

2(-l)&’ .h’i (2.3’)

438

M. Gut et al.jRotation-internal rotation in glyoxal type molecules. ZZ

:

et-model d2 , , = 2[1 - (-l)“+c =yJhP 2 [co& + (-l)h’ti -Xt2

-Cl -(-l)=z]

lXaf(Xis

f 2X,,,)

sir&] [(-l)P

+ 2Zt;,X,, -[I - (-l)“*A’]

f (-l)fi’]COST SinT

i 0

2(-l)p+p’

[(-l)P

+(-l)“]

[sin% + (--1)‘c+C Co&]

0

cos7 sinr

0 0

\ Xt

2(-l)h+hY (2.33

2.2. Kinematics of models 2.2-l. Problem of uniqueness and primitiveperiods Using the relations between eulerian angles and internal rotation angles collected in table 1, one may transform the coordinate vectors of the three models (cf. table 2) into each other. The coordinate expressions will now be used for discussion of the primitive periods and fundamental domains of the dynamical coordinates. Starting from the set of four coordinate vectors X&(h,~_r = 0, 1) of a top set in general site (t2 set) one first finds by studying the equations X&)

= x;,
x$e,

T) = X&(i,

r’),

x;,(?,

7) = x$2,

7) ,

(2.4)

in a straightforward manner the solutions collected in table 3. Besides the primitive periods given in table 3 one has for all cases X&(CY,p, __.)=X&(CY-k-27T,p, J,

x;&I,

8, .__)= X~JOL, p + ‘II,_..)-

It should be pointed out, that more general equations of the type X&&)

= x&

($)

(2.4’)

,

will be considered later in the context of the isometric group. In fig. 3 a symbolic representation of the (ru,yl) and (r, T) plane of the three models is shown. It should be noted, that in the case of the ef-model the periodicity conditions imply x&co, P. 7, r) =x&@,

P, Y, r + 27r) = X&(or, P, Y + 27r, 7) -

(2.4”)

From the interrelation of the dynamical coordinates listed in table 1 one may verify that the three sets of primitive periods given in table 3 are consistent with each other and directly give the mappings of the primitive period parallelograms onto each other as shown in fig. 3. 2.2_2_Expressiom for velocities From the nuclear coordinates listed in table 2 one may express the velocities XQ by the generalized velocities e (eulerian velocities) and i (internal angular velocity). In order to have a convenient transition to the convenTable 3 primitive periods of dynamical coordinates (eolerian angles and internal rotation angle) Model

Primitive periods of X& a)

a) ‘Ihe primitive periods of cxand p in all three oses are 271and P respectively.

M. Gut et al./Rotation-internaI

rotation in glyoxal type molecules. II

Fig. 3. Primitive periodes para!leiogams of D,bF(C& tern: (a) two-top model; (b) e*-model; (c) ifmodel.

439

sys-

tional form of the kinetic energy, we first note that the velocities for the three models may be expressed as follows

(t2 set): two-top model

L

1 Xi,(T)

=

-1 i

-1

) h

(2.5)

440

M Gur.er al./Rorarion-inremal

rotation inglyoxal.~~e.molecules.

h

if-system ~’

(2.5’)

where

Then we note the following rules concerning manipulations (i)foranyX,YE%$,ff;flEQ:

with the symbol fi [X] :

n[CYx+flY] =aS2[XJ +p!LI[Y]; (ii) for any X E 32, ,

and more generally for any D(0, 0, T) E SO(3), D(0, 0, T) a[X] @O, 0, r) = Q[D(O, 0, -r)X] ; (ii) E[DX] s2[DX] =Dsi[X] s2[X]ix Furthermore, since for calculation of the kinetic energy the nuclear coordinates have to be referred to a center of mass system, we note (cf. table 2): two-top model

e f-model

(2.7)

~~*~=D(E){~[x~~~l_I]_E(E)~COS~-x~ll.lsinri}, if-model

2.3. Kinetic energy fuwtions Assuming the SRhI to consist of a set of two anchor atoms (mass ma) and arbitrary numbers of sets of type tl (mass mtIk) and type t2 (mass mtzk) the kinetic energy function is given by T=; [

m C$CMXECM ak a a

+ C, k

tlk

cx?,“,‘,“,*;“k”, L

+ z&k k

c ZECM PCM t2kU > t2khp

x cr

1 -

cw

By means of the coordinate and velocity expressions collected in table 2, eq. (2.5) and eq. (2:7) one derives ina lengthy calculation first for the two-top model

G!Il3 =(2/Jo

[2 pT2R(XfL3

+xt2k3)

+ Frn,lJIXi3

+x,&f

m,xfg] *

(2.9’)

The kinetic energy function for the ef- and if-models have been given in I and also the transformation of the two functions into each other. The expression for the ef-model is written here for the sake of convenience in a form which allows a more direct comparison with eq. (2.9). According to eq. (2.5) the velocity equations now have the form k;=D(~){fi[X,$

E(++(ax,P/ar)i),

which, if inserted in eq. (2.8), yield (m, n = 1,2,3) T = f{=(e)

(g,,

(r)) E(e)i - g14 (r) [:E(e) 1. 1 + x 1 E(e)P] cos r i + ga4 (T) i2} ,

(2.10)

where(h=O,l,y=O,l)and

(2.10’)

442

:

M. Gut et al./Rotation-internal rotation in glyoxal type molecules. II ;

Explicit expressions .for the quantities a[X]Q[X]

are collected in appendix A. Furthermore,

(2.10”) In order to verify that the kinetic energy expressions are related by the variable transforms collected in table 1, it is convenient to note first the relations collected in appendix B. Then, inserting the results of appendix A and B, the formulae collected in table 2, eq. (ZS), (2.6) and the relation (X = 0,l)

the energy functions of the two-top model may be shown to go over by a straightforward calculation into the kinetic energy expression (2.10) of the ef-model.

3. Isometric groups In I the isometric groups of both the ef- and if-model had been considered under the assumption, that the prim& tive period of the internal rotation angle r equals ‘IT.This is derived from the fact, that by the substitution V,, i=T+Sr,

(3.1)

the set of distances is mapped identically onto itself [4], e.g. d t2kXy,X+lp(7 + 4 = dt2~Xa~h+~&~~ A, k, E= n(mod 2). It will now be shown that inclusion of this substitution into the isometric group leads to an extension SC(r) of the isometric group 9!(r), tihose operators are not only symmetries of the rotation-internal rotation hamiltonian, but also defme the periodicity conditions (boundary conditions) to which its eigenfunctions have to be subject. 3.1. ef-system

Using the results of I and the substitution (3.1) one may derive, following the procedures given by Bauder et al. [4] and Frei et al. [5] the groups collected in table 4.

A number of comments should be made concerning these groups, beyond those discussed in I: (i) As a consequence of the fact that the covering symmetry group of the SRM D_lrF(CsT)2 9(r)=C,,

ig(r))r =2,

there exist for every FE B(r)_two representation matrices r( NC0 Q for F = V3, V, as verified explicitely in appendix C. In tabIe 4 only two of the possible choices are listed, namely those generated by r(NCf)(v*)

= II

@ I?@)(Vs) = l(4) 8

(3.2)

M. Gut et al.jRotation-internal rotationin glyoxal type molecules.II

-7

----Y-z . .

::..

.Tyl

‘7

* f-----2 -7 ‘; - - . .

-

-=L--

443

444

M. Gut et Q~fRotation-interns

rotation in g&oxaI ?yp< molecules. if

The first choice will be further used in section 4 for the solution of the energy eigenvalue problem. It should & pointed out that likewise solution (C.5’) could be used instead of (C.5). (ii). In I the isometric group of the SRM DoDhF(CsT)2 has been taken to be generated by the operators V2, V3,i.e.

since the operator Vs has not been considered. fn this paper the operator Vs will be taken as a further generating element. Owing to the fact that the commutators [Vj,Vj‘]=E, the extended

k=2,3,4,

isometric group si (r) may be defmed as

G&j= ~(~)uv~~(~~=~~u~~~~ :=CV~,V~~~lk,k’=2,3,4;V~=V~=E;VkVk.=V~~Vk;V~Vg=V~V~)~sQ(2,2,2).

(3.3)

A complete set of inequivalent irreducible representations and the interrelation of these with the irreducible representations of sy1(7)z 9Q is given in a pendix D_ r(NcfQ){@>, rj@3 and I’(“>;){@) the representation (iii) Besides the representations I’( b@t2){81), fiNcat*) is listed_ This representation is generated by application of the operators&,-, V’E @ (T),to the substrate (e, T), i.e. the coordinate vectors of a typical set t2 referred to the laboratory frame. One of the many isox’p”“Ki morp c forms, which may be chosen for I?(NCQt2) is given. This representation is related to the Longuet-Higgins approach often used for setting up symmetry groups of nonrigid molecules [6-81. It is of particular interest to note the relation of the representation matrices I’fNCf)(V5) and K’(NCf)(V5) to

the periodicity conditions collected in table 3. Obviously the equations X~~(cr,Bor,7)=X~~(ff,p,y+sr,Tf1T);

p(NCnt2)(V5)

= l(4) 8 ~(31,

r(v,)=

u=&4a

(3.4)

imply each other. Therefore, the operator expressing the primitive period of the distance set {dtZhp,rG(r)} generates the periodicity condition of any one function of the nuclear coordinate vectors referred to the laboratory system in a natural way. Ihe second solution (C.5’) is related to the covering sy~et~ group $?fr) = C,, expressed by the transformation given by eq_ (Z-2). The operator & induces on the substrate _TS$(T) the permutation (S~k-,,,Sis,) and the rotational factor (l -1 _1), which then lead from (C.5) to (CS’). As is seen from table 4 this choice of I’tNcf3(V5) leads to an outer automorphism of the group JY{$?)

h : F{si)

+r’{Q) ,

-

(3.5)

where &(I’( v;>) = E’(V6). Therefore, either choice Ieads to the same periodicity condition eq. (3.4) and the same statement holds for any other possible choice out of the set of (outer) automorphisms ori~at~2 from the two vduedness d;f rcNCn(H), H E si(~). (iv) The representations I?{ %} and r( w*G){R) will be used in section 4 in connection with the symmetj group of the rotation-internal rotation hamiltonian of the ef-mo_del 3.2. Two-top model In the two-top model the coordinate vectors bf the nuclei are expressed directly w-r-t. the ~bomto~ basis ep.

M. Gut et al.IRotation-internal Table 5 Two-top

rotation in glyoxd

type molecdes.

II

44.5

model, gtierators of isometric substitutions Generator

Transformation

As a consequence no internal isometric substitutions may directly be derived from the distance formula eq. (2.3’) though the tatter is symmetric w.r.t, arbitrary transformations of the laboratory system

Eq. (2.33 admits isometric ~~s~o~ations which are arbitrary table 5 w.r.t. yo, yr. Furthermore, see eq. (2.3’) and table 3,

in or, @

and subject to the conditions collected in

express the ~&nitive periods of the anglesr. and 71 ; for neither of these angles ‘IT is a primitive period. One immediately verities from table 5 (for arbitrary 6 E [0,27r): F&

= F&I,

F2F4 = F4F2,

Hence, the isometric group

F3F4.= F4F3 .

13.71

A IS) is isomorphic to the group9, X 192. The relation of this transformation group to the isometric group of the &model may be obtained as fohows. From the derivation of the distance formula it is obvious that the difference;% = 70 - +yris an internal coordinate; this is verified by substituting 27 = rr, - yr into eq. (2.33, which thereby goes over into eq. (2.3”). The representation given in table 5 may be subject to the

(3-S)

which brings it into the second form listed in table 5. As a consequence the pe~o~c~~y conditions (3.6) transform into

i.e. into the primitive period isometric transformation ten in either form

Vs and a second periodicity condition, which may be writ-

(3.9’)

By compa~son with table 4 it then f~lIows that the subgroup defmed by S = n of the group in table 5 is isomorpbic with the isometric group B(7)_ A more complete interrelation of the isometric groups of the two-top and the ef-system is therefore obtained by transforming the group I’{@] &en in table 4 by

leading to the group F(g) = LWIT’{@)L also listed in table 5. @%I) is (direct) factor of the symmetry group of the hamiltonian function of the two-top model. Since by the transfo~a~on (3.10) the matrix I’( V,) is transformed into the unit matrix (taking r and r always mod 2rr),l?(Ys) = l@ one finds F@)

!z q$ t r@i-) ,

i.e. T@? ) is homomorphic

(3s I)

(not isomorphic) to F{@3 and ~sorno~~c to the isom~t~c group of the ef-model if in the latter the primitive period is&metric t~sformat~o~ are excluded. This result: rUustrates the fact that isometric groups of SR.M’sare derived most siniply via the representation T’tNcflf%‘). For the solution of the energy eigenvafue problem such homomo~~sms may lead to a less detailed s~ec~~cation w.r_t. sy~et~ species. One may further investigate the transfo~ations of the coordinate vectors Xf (f; 7) given in table 2, induced by the transformations defined by T{si). Since no use of this representation witi be made in &is work, it will not be given. However, it shoumbe mentioned that the operator V, expressing periodicity w.r.t. the interuai coordinate T = &a - yl ) is now represented by the identity, demonstrating thereby its nature again as an uniqueness condition for the nuclear configuration oftbe SFW and for the wavefunctions of the quantum mechanical energy eigenvahre problem. The results of this section should bring out the motivation to investigate the interrelations of the three modeIs

M. Gut

er al.fRotuti~n-i~~ernaI

rotarion

in g!yoxal

type molecules.

447

If

considered so far: In ~articuhtr by the actual construction of the connection between the two-top and the &-model the use of the latter for a correct explicit solution of the energy .eigenvalue problem is considerably strenghtened.

4. Solution of the energy eigenvalue problem Solution of the energy eigenvalue problem of the SRM D_+F(CaT)2 could be attempted with any one of the three models discussed in sections 2 and 3. However, comparison of the three kinetic energy expressions reveals that the classical hamiltonian function of the &-model is by far the simplest among the three. Solution of the eigenvahre problem associated with the two-top model would have to be carried out in a subspace A{&) of the Hilbert space , I denotes rotation group coefficients f9]) C=

for

which as a

basis one naturahy would chose ($,(e)

(4.1)

L9C~3:=_[D~r(~,P,~*)l,JEN,t(l’l,llrf~~f,X=0,13.

(4.1’)

However, both the construction ofsymmetrized zeroth order basis functions, the calculation of kinetic energy matrix elements and of the matrix of the potential energy

is complex and wih not be pursued further in this work. The if-model has been discussed in Eand like the two-top model leads to complicated symmetrization problems and.involved expressions for the kinetic energy matrix elements. However, a direct numerical method [ IO,1 I] based ORthis model will be used to check correctness and accuracy of the treatment presented in section 4.1.

From sections 2 and 3 it follows that the eigenvalue problem of the operator [cf. eq. (2. IO)]

(4.2) has to be solved in a Hilbert space defined by iH{k3:=

~(*~~~)10Ga<2?r,O9~wih be discussed.

.

(4.2’)

4.1.2. Symmetry group of rhe hamiltonian In I it has been _&own that the group By = {E, Yz, V3, V4] is a symmetry group S{$). We now first show that the operator Pys defined by l”(V,), cf. table 4, leaves the classica h~iltonian function (2.10) and its quantum mechanical analogue (4.2) symmetric. Using the results collected in table 4 and eqs. (2.10’) and (2.10”) for the kinetic coefficients one finds immediately

448

hf. Girt et d/Rotation-internal

rotation in glyoxal type molecules_II

-823

’

g33

0

0

844

By symmetry and periodicity requirements the potential function may be written as V(r) = ,c, :I&( =

1 - cos 2nr) + Vo ,

(44.3)

its symmetry w.f.t. 13V is trivial. Furthermore, it can easily be shown that T + V is likewise symmetric w.r.t. i vs if the second solution &!.S’) is taken for the latter. Therefore, the hamiltonian function is symmetric w.r.t. the extended group B (7): zj#i}

= 5%(r) f=”92(2,2,2).

The complete symmetry group of the rotation-internal inversion of the Iaboratory system)

(4.4) rotation hamiltonian may therefore be taken as (omitting

The implications of this result for the solution of the energy eigenvalue problem and its relation to the group used in I will now be discussed. 4.1.1.1. Relation to othergroup theoretical work The symmetry group of the RIR energy eigenvahre problem of molecules approximately describable by the SRM DWhF(CsT)2 has been discussed by several authors. Hunt et al. [ 121 used a four-group s{fi} = V4 in their treatment of the FIR spectrum of H202 and apparently did not pay attention to the periodicity problem. Dreizler [13] also studied the symmetry group $j’{H} of the RIR problem of the H202 molecule and arrived at a four-group, which included (in the notation of this paper) the transformation r’ = r f X. To the dihedral angle 27 the period 47r was ascribed, owing to the fact, that the coefficients g14(r) andg23(r) have primitive period 2~ In an investigation of the molecule CH2F C i C. CH2F Hougen et al. [14] concluded the symmetry group of the rotation-internal rotation problem to be l

Some aspects of the various approaches will be commented upon in connection with specific molecules later in section 4.2. However, the treatment of the symmetry problem given in sections 3.1 and 4.1 should give a rigorous explanation of the confhcting results mentioned above 4.1.2. Choice of firnction space According to the foregoing analysis, the eigenvalue problem of the energy operator is to be solved in the subspace A{&) C ME&, where a{&:

= r&,/3, y, r)l(tl, u) =st(*u siri0 da dfl dr dr , i u(ar,p, y, T) = z&Y.,p, y f 7f;T f 77)= u(cY,p, y f 27r, 7) = z& p, y, 7 -I-27r) -

(4-5)

hE Gut et okj~ot~tion-intent

rotation in glyoxd type ndectdes.

449

If

The

second requirement imposed on the eIements of_.&{& originate~.frok the inclusion bf the primitive period isametric transfo~ati5n YS in the isometric group %! {tT)_The main consequences of this are: (i) The parameter space is connected according to the requirements listed by eq. (3.9) and may be depicted by figs. 3 and 4. in b&h figures only the two variables y and 7 are considered. Fig. 4 results from fig. 3 by deforming the period parallelogram into the two-dimensional torus. The latter represents in a natural way the connectivity of the independent variable domain of the RIR probleF. (ii) From the requ~emen~ (solution 1, Vu E A(rr)). av5tc(cl,P,r,~)=uCa,~~~-n,r-n)=u(a,p,rtlr,7+~)=U(OI,P*Y,?-),

f4.61

one im$e$iai.fJy~oncludes that all eigenfunctions of &will be totally symmetric w.r.t.-filrs. Since the symmetry group $j {lvl= SK(r) may be decomposed according to [cf. eq. (33)] e(r)

= %@) U V$X($g

the representations

srCTY~{r)

Z(7) = {E, Yz, F3, V4) 2 V4 ,

Se(2,2,2),

of % fr) may be classified according to the two ~~re~n~at~ons = isI0,

P,, I”__of the factor group

V.5sc~793 .

(4.7)

Hence, the eigenfunctions off$belong to one of the four irreducible representations K&j = l,& 3,4 of R(T), i.e. they belong uniquely to the irreducible representations J+ j = 1,2,3,4 of the group Z(r) = W4 (cf. appendix D). This classification has been used already in I, where the requirement expressed equivalently by table 3.and eqs. (3.91, (4.5) has not been taken into account. However, this req~rement has profound effects on the structure . of ei~nfunctions. and spectrum of the aperator f4.2) as will be shown in the next subsection. The construction bf a zeroth order basis for the calculation of&he matrix of the energy operator eq. (4.2) follows closely the lines used in I. However, the requirement eq. (4.6) implies that for a complete orthonorma1 basis ofthe ekment of A{& one has to chose the set

ME~,~ikffMf
‘Pllz,

This set forms indeed a complete or~ogon~

fi

basis in A{&},

(4.81 the fact that the direct product ((Pi)

Fig- 4. djnodel: torus r~pr~ent~g the connectivity of the coordinate subspace (7, I). Dashed ties correspond to boundaries of the period paralielogmn fis. Zb, which have to be connected in forming the torus.

Table 6 ~ymme~zed I

basis functions a)

Rotational

J

K

r,_

J

odd M 0

r4+ J

even M 1

I&

add 0

Rotation-internal

Internalrotation F

r

MO

M -

n-I”

r2-

rotation basis functions

r

sin(2m + 1)~ urz~

r I+ emn -In cos 2mr

“r,N

r It

%TIN

%?* -m cos 2?m

a)C= f(2J+ l)j83?j’n,

stands for the set of the Fourier ~~nctions~ iiii: = {DT (E)) 0 {ip (T)), forms a complete or~ogon~ basis in HI{&, second by the fact that any element of the ~ornpiern~~t @\I!%is ~rthogo~~ to the elements of lB. The subset IB,

: = {Em ,-lf2

COS&tT,

+i’Sin

hT)

3

forms a complete or~o~on~~ basis for the so&ion 844 = $ rp’+ : G

J.Q$I&P%) fp^-

its

(4.9) of the internal ~robI~rn

(Li.lO)

IngIl + V(T) I

which defmes spectrum and eigenfunctions of the states with J = 0. This set has also been used in I for solution of the internal problem and for construction of a zeroth order basis. From the treatment presented in this paper it is obvious that for states withJ> 0 the subset “0 is not sufficient and has to be replaced by 18. Table 6 gives a complete compilation of symmetrized zeroth order basis f&ctions, which subsequently will be uSed in the calculation of the energy matrix coefficients. For the sake of brevity, the more accurate notation Fj,.&or irreducibie representations will be hencefo~ard replaced by lYFThe rotations factor may be taken in the form (as has been shown in i)

and the phase factor i”fAK) is chosen in a way that the energy matrix appears in real form: cr(A K) = 0, = 1,

K even, J arbitrary; K odd, J arbitrary 1

4.1.3. Energy mattir elements Calculation of the matrix elements will be carried out by the ie~que

‘-_ of Laurent series and complex integra-

451

M. Gut et aLlRotation-internal rotation in glyoxal type molecules. Ii

tion in the same way as discussed in 1. Only new aspects arising in connection with requirements (4..5), will be discussed in some detail. 4.1.3.1. Internal rotation problem (J= 0 states) The analytical calculation of the matrix elements of the operator given in I has to be modlfied only w.r.t. the integration domain; in the zeroth order basis one has to calculate (~~,ri44~,n)=~~&rI?‘$,,rdr 0

(4.11) It is a straightforward matter to show that the matrix elements may be expressed by the same integrals as used in I, yielding: (i) I’r block, GI, m > 0 = P‘Ii2~~~m{4~m[144(12m-22mI)-Ir44(2~

(~~,+2ii&Pr&J

+2m)]

+$[12ifi-2ml(14$J(12Fz-2ml)+144~2(12~-2m()) +(2fi+2m)(Wi([2fi

-2ml)+144~2(12Ffz

- 2ml))]

+&[144~11(i2E-2ml)+144J2(12E-2ml)+f44~22(l2E-2ml) +1443(2%

+&&

+ 2m) + PJ*

(2E + 2m) +144y22(2fi

n$l v2n h2iii -2m1,o +62ff&m,o

+ 2m)]}_

-%,2iii-2m1,2n

+62E+2m,2n)l

f%%

vokz,nl

-

(ii) r2 block, fi, m 2 1 ((pr,,ZFlY ~~~~2+2m)=$fi2{4ii2m[Z~(12i-2ml)+144(2~+2m)]

+~[12fi-2ml(144J(12E-2ml)+144~2(12Ei-2ml)) - (2% + 2m)(I”s1(2m

+ 2m) + 14472(2E + 2m))l

+ ~[~~,~l(l2~-2ml)Sf~~~2(l2rii-2ml)+144~22(l2~-2ml) -P4~~~(2iii

t2m)

---I 44,12(2~

+ 2m) - 1449i2(2fi

+ Zm)])

-S2fi+*m,2n)l+

VOSfi,m -

(4.12)

In appendix E explicit expressions for the integrals occurring in the eqs. (4.12) will be given. The pure internal rotation states belong to either rl or r2 only, since the sets of basis functions j~r-~~~cos(2rn + 1)rlm EIN],

{7d2

sin(2m t l)-rlmEIN} ,

(4.13)

violate the requirement (4.6) and have to be excluded for the solution of the internal problem. Therefore, the treatment_ of the latter does not experience any modification. Nevertheless, both sets (4_9), (4.13) have to be used in the calculation of energy matrix elements forJ> 1 states. Diagonalization of the infinite matrix defined by eq.

452

M. Gut et al/Rotation-intend

rotation in gIyoxa1iype molecules. II

(4.12) leads to the complete orthonorma~ set of eigenfunctions problem

and the energy

spectrum of thi intemai rotation

4.1.3.2. Rorg~ion-inre~l rotafion problem f.E= I,J [RIR} According to table 6 the calculation of the energy matrix coefficients for f> I states has to be modified considerably in comparison to the treatment I. The most profound modification originates from the fact that forJ> 1 also the set of Fourier functions (4.13) plays a role, besides those relevant for the internal rotation problem (4-Y). This phenomenon should be considered as a characteris& consequence of the choice of the crystallographic coordinate system ef_ As a consequence also matrix elements of the type (IPri_2nr+l,.~44~~~_zm~1),

i= L2,

will be required for JB 1 states. Therefore, the internal rotation problem has now to be solved also for the basis functions $7pt_2m+I

=

r%os(2m

+

qr,

$7ra_2m+L =?r-1~sin(2mi-I)T,

mEIN,

leading to the matrix elements (E, m E IN) )=3ri~{(2iiif1)(2m~l)[144(12~-2ml)-~~(2~~+m~2)]

(~~~_2iii+L,@%4m+~

f~f~2~-2m~(144~~f~2~-22m~)+144~2(~2~-2m[)) + (2% t 2m + 2)(144J(2Z

+ 2m f 2) i-W2(2fi

f 2m t 2))]

t~[~~~~~~(~2~-2m~)t1~~~2(f2iii-Zm])t~44~22(12i-ii-2ml) t~~,11(2~t+m1-2)+I~4,12~2rji+2mt3_)t+144~22(2iiif2mt 2)]} +i )$I Va, P**iii-_2m1,0-idSf2E-2m1,2n ki@pI;_2m+l)

&gFz+l~

=

*S2E+2m+2,2n~l

+- vOsiii,m

7

&@{(2fi i- 1)(2mf l)[144(12iE - 2ml)+144(2i?i + 2m t 211

t~[~2iii-2m~(194~1(/2iii-2ml)~~44=2(12~-2ml)) -(2E+2m+2)(144~~1(2r7i

t2m+2)+144~2(2Fzt2m+2))]

f &j[14qpii -ami +.144,12(12iii-2ml)+f44,22(12m-Zmf) -f~~1~(2@t2mt2)-I44~12(2%i-2mt2)-144~22(2fit2mt2)]] +;Evn=l

2~2[I5pm-2mt.O

-2 ‘(6 12E--2m1,2n

-82iii+2m+2,2n

)I + T1’OGiii,m

(4.15)

and to fust order basis f~ctio~ urjfv ~=m$o %&2m+l The coefficients

v;$Ap

i=

1,2,NEW.

(4.15) may be expressed by the same integrals as those occurring in the internal probIem.

(4.16)

M. Gut et ai./Romion-internal

rotation in glyoxal type molecules. II

453

First order basis functions for J% 1 states may now be constructed according to table 6 but with ~“fi replaced by uri and uri_ The matrix etements of&read exphcitly (K even = 0,2,4, .... ZfJfZ], odd = I, 3,5,1._, 2[J12] + 1): (iZ.Mrj~lklPJKi%iIjIV) +6,-K-,, I

= QQ-2

%*(-l)IC+lA(J;

s“~zA(J.K-l)A(~‘J,)[~~11(~N)-~G22(~,N)]

K)[(2K - I)$23(@

N) -f-+G14(@, N)]

~~~~{$W*[J(J+l)-~~][jcl1(~,N)+~G~~(iii.N)J~fti~K~G~~(~N)+~~~~H~~(N,N)l * “6~,K+1~~2(-i)KttA(&

K+ I)[-(ZK+

l),t-G23(1si, N) + $Gx4@ N)]

(4.17)

A(J.K)=[~(~+l)-(K-l)KJl~2=[(J+K)(~-K+1)]l~2.

(4.18)

and the quantities Gik$, iV) are listed in table 7. The matrix elements eq. (4.17) contain five further types of trigonometric integrah, which are evaluated analytically in appendix E. Relevant properties may be commented upon as foIlows: (i) The convergence probiems associated with the integrals Involving the kinetic coef~~i~ntsg14 and gz3 met in approach I, are eliminated as is evident from the analytic expressions for the integrals 123, 114. The asymptotics of the off diagonal elements of the hamiltonian are now completely dominated by the largest kinetic roots rn~(l~l~l~, max(jq2,#, 1~~1, /q2kf < 1. This amounts for ah integrals and therefore also for the matrix elements to the asymptotics 1(2(fi -m))L% t?&Jifi-mi,

tiii-ml

+m_

(4.19)

(ii) The violation of the self-adjointness of a observed in I, when the domain o
4.2. I. ehvdei Basing on the rest&s of section 4.2 a computer program (FORTRAN IV, CDC 64OOf6500) has been written, which allows numerical calculation of energy eigenvahres and eigenvectors ofSRWs of type DmhF(CsT)2 with high accuracy [16]*. For nnmerical diagonabzation the infiite energy matrix is truncated, and the truncation process is subject to the following rules, provided high and low barrier extremes are classified according to the inequalities, respectively *The reader should consult this work for more detailed information.

454

-M. Gut et a~~~otQ~on_in~em~~rot@on in g&ox& type molecules. II

Table 7

Matrix elements of the kinetic coefficients $nn(& J > 1 states

fiw, -e(max(v)

- rnhl(v)),

liw, > (max(v)

- mm{ v));

(i) in order to obtain @eigenvaIues of the interna problem with kHz accuracy one has to diagonaiize an Mdimensional matrix withN = 3E and M= 2g for high and low barrier, respectively. A more general rule is to choose#such that f&_< 10-‘3HlrH~p (ii) For J S=1 states one should use the lowest ‘JM first order basis functions uriN, uriN for nmerical cah~~!ation of the energy matrix of dimension D =&$.I or D = M(J + l), depending on the representation. This then yields appro~ate~y $D eigenvahses (accuracy k!Sz} for both high and low barrier cases. (iii) The accuracy of the eigenvectors should be checked by ex~ing the asymptotic behavior of the module _ of the vector coefficients. These should decrease asymptotically according to eq. (4.19).4.2.2. if-model . For comparison, the energy levels and wavefunctions for J = 0 and J = 1 were also calculated with a quite different approach. The method used corresponds to the direct numerical method published earlier [lo,1 11, and shows to include structural relaxation. Recent mo~~cations of the crtlculation, which will be FubIished elsewhere in detail, include an iteration method yielding more accurate wavefunctions with less computational effort and a more generally applicable choice of the moving reference axes. For D,hF(CsT)2 SRM’s the if axis system, cf. fig. 2c, may be chosen for the moving coordinate system. Hence, the dynamical variables in this approach are the eulerian angles and the internal rotation angle7. Furthermore, it differs from the program based on the &-model also w.r.t. the numerical technique and therefore allows an ~depeudent test of both correctness of the two theoretical models (ef- and if-models) and the numerical accuracy. It should be mentioned that the if-program allows c~c~atioR of states withid 1 with existing facifities. A comparison of results obtained by the two approaches for specific molecules is discussed in the next section. 4.2_3. Nz.unen+al exampies For ilhrstration of RIR spectra of D,hF(CsT)2

type SRM’s numerical values for glyoxal (high barrier case)

M. Cut et al. jRotat&m-internal rotation in giyoxat type molecules. II

H\

/c-c,

455

//* 9

Q/

H

hydrogen peroxide (high cis and low tram barrier) H\

H ,

o-o/

and 1,4-difluoro-2-butyne FH$-C

(assumed to be a zero barrier case)

= C-CH,F

wiU be presented. (11 G&o_xal The

structural parameters as reported by Kuchitsuet

tram: d(CC) = i.526 A, L(CC0) = 121.2”, cis: d(E) = 1.505 A, L(CC0) = 123.9”,

d(C0) = 1.212 A, L(CCH) = 112.2”; d(C0) = 1.212 A, L(CCH) = 112.2’.

al [I73 and Durig et al. [IX] were taken as

d(CH) = 1.122 A, (I(CH) = 1.122 A;

A value proposed by Currie et a.l.[19] for the difference of the eIectronic energy of the cis and tram conformation (I 125 A 100 cm-l), the known microwave [18] and infrared [20] data may be used to determine the potential function, yielding . m

V(T) = I v, I f ngl $ F&(1 - cos zm-) > VI2=-1133cm-1,

VG=l137cm-l,

VG=Ocm-l,

V’n=-56cm-r.f

From an analysis published by Durig et al. [Zl] one arrives at values Vz = -1182 cm-l and V, = ii 14 cm-l. These authors allowed for a larger deviation in the cis-tram energy difference from the observed vaiue. In fig. 5 internal rotation levels and wavefunctions (J= 0) are shown. It should illustrate the localization of trans and cis states in the region of appro~ateiy equal energy, besides typical low tram states. In table 8 the observed transition frequencies are contrasted with frequencies calculated by both the ef and if * in this paper T= 0 defines the cis conformation.

Fig. 5. Level dkgrm

for glyoxal.

456

hf. Gut er a~.~Rotat~~n-i~?em~Iratution in giyoxaf type molecules.II

Table 8

Rotation-internal

rotation levels of SRM glyoxal

Transition SRM

configuration

obs.“‘b)

asym. top

oral or,1 +-or,1 t or,2 0r,2+orz2 or,2+or,3 or,3~crra3 1~46 t Ifs6 2rr6 + 2fa6 3~~6 + 3rs6 4rr6 +4r26 5r46 + 5rs6 3ra6 -4rt6

tram trans trans

tram 110 + 101 211~ 202 312 + 303 413 t404 514 + 505 313 -404

41',6+-5r,6414+505

Transition frequency-

tmns cis cis cis cis

125.2 126.7 123.6 122.0

caJc. ef-model

126.722 125.467 123.972 122.240

CiS

125.6 21680.7 22886-86 24783.95 27472.50 31076.20

120.266 21841.06 23021.43 24876.01 27501.25 31014.47

Ci5

26667.21

26208.79

CiS

39389.97

38860.98

talc. if-model

125.467 126.722 cm-r 123.972 122240 120.266 21841.06 MHz

a) Structural data taken from mfr. [17,18]. b, Potential function

from fit of microwave data of ref. [18] and far infrared data of ref. [2b].

programs. Both programs give identical predictions with kHz accuracy and are found to reproduce the observed FIR transitions of the tram conformation and the energy difference Eur.6 - Eo~,c ofthe cis conformation estimated from microwave intensity measurements satisfactorily. If, however, the dependence of the rotational constants on the torsiona state in the electronic ground state is analyzed the observed behavior [18] is reproduced only qualitatively since the cafeulated dopes ofA, B and C amount to only half the observed ones. So far no acceptable mode of structural. relaxation concerted with internal rotation has been found, capable to explain the empirical dependence of ail three rotational constants on the torsional state. On the other hand it has been found possible to give a simple relaxation model explaining approximatively the observed difference A - C in the ground and first excited torsional state of the cis conformation: if the angle a! = L(CC0) is assumed to follow an equation of the form cY(T)=cro f&4COS4T,

(4.20)

withou =121.9°,cr4 = 2-O”, one fmds A - C(cis l?,6) = 21730.02MHz and A - C(cis i’26) ==21800.69 MHz, yieldiug a variation of approximately 70 MHz as compared to 59 MHz derived from microwave spectra 1181-Using the SRM model, the analogous calculation without relaxation leads to 6(A - c) x 300 MHz, widely deviating from the experimental data. The variation of cr expressed by eq. (4.20) is one order of magnitude larger than a quantum chemical prediction by Schaefer 1227. Further investigation into the glyoxal problem wili be reported in a separate paper. (.2}Hydrogen peroxide As a next exampIe the RIR problem of Hz02 wih be approximated by the SRM D,hF(CsT)2. This molecule, which is the simplest system of this type, has been given considerable attention both in the far infrared [12,23] and the microwave spectral region [24,25]. In this paper no attempt will be made to analyze the available data, since both from quantum chemical studies [25] and from inspection of the spectral data, it is obvious that for a satisfactory analysis the SRhJ has to be generalized by inclusion of structural relaxation and vibrational and centrifiigal distortion effects. It, however, is of some interest to compare the numerical accuracy achieved by Hunt et al. [12] with the methods used here. These authors treated the RfR problem by a rotating frame axes @AM) method and Fourier expansion of kinetic coefficients. Furthermore, the internal problem was used in the form

Table 9 Internal rotation ievels of Hz02 (J= 0 states} Notation

Energy

level &%Hz)

this work

Hunt et al. a)

EOl?, 1

01 (ess)

5 038 886.305

02 (osaf

5 038 886.631

fur&

flmuQ,zl)

this work

(UJ+ ~“ur#

03 (oas)

5 311463.099

EOFal

04 (eaaj

5 377 463.513

&r,2

11 (ess)

12681329.18

(‘tr*2*H-44 V,2)

12 (osa)

12 681331.92

(vr, 29Ijw V,2)

13 (oas)

16 156 6f5.00

EOI?a2

14 (eaa)

16 156 626.70

Eor,3

21 (ess)

22 x02 330.85

(u53. @“r23, -44 (?,3~ H %7,3)

22 (osa)

22 102 391.02

23 (oas)

28 272 478.52

Eor,3. Eox‘,4

24 (eaa)

28 272 792.05

31 (es@

34 973 531.24

(Uuf,4”fi”ufi4)

32 (osa)

34 975 131.09

J%riN-~or,l Hunt et al.

@m-‘)

this

work

Il.43

11.29

254.2

254.92

370.8

3?0*85

569.0

569.17

775.0

77.5.0_

998.6

99851

a) Ref. [12].

instead of the correct form given by eq. (4.2). This amounts essentially to neglecting the quantum potential term and the term (&&x))&_ In H202both terms are relatively small. In table 9 values for energy eigenvalues of the intema] rotation problem are tabulated and contrasted with values reported by Hunt et al. The computations were based on data given by these authors: d(O0) = 1.475 A,

d(OH) = 095 a;

L(OOH) = 94.8O;

V(T) = 2460.0 + f [--1985.0(1 - cos 27) - 1272.0(1 - cos 4~) - 88.0(1 - cos 6r)](cm-L)

.

The following comments should be made: (i) Eigenvalues of the internal problem calculated by the two methods used in this work agree within 10 kHz, whereas the deviation from the results of Hunt et al. do not exceed 0.7 cm-l. (ii) It appears from the paper of Hunt et al. that these authors considered functions of type uy N and I+ ~(7 = 2,3 in their notation) as eigenstates of the internal rotation problem. According to the theory pr&ented in &is paper, only the levels denoted by OI’$V and OI’$Vought to be considered as energy eigenvalues of the internal problem (J= 0) and no internal states of species I’3z r4 exist. However, in the immediate vicinity of each Of IN and 0K’2Nlevel there appear diagonal elements of HM built from ur. N and un N functions, which result from diagonalization of the matrix 0fH“44 cahxlated in the bases {n-l)2 cbs(2m + &j and {n--if2 sin(2nz + l)r), respectively.,

458

M. Gut et d/Rotation-internal

rotation in gtyoxal type matecules. II

13) 1.4di.uoro-2-butyne This moIecule has been the subject of a theoretical study by Hougen et al. 1141, who considered the symmetry of the RIR problem and partly also of the rotation-internal rotation-vibration problem, These authors concluded the symmetry group of the SRM to be a double group of order eight isomorphic to Dz@ @Z(2,2,2)). However, no attempt was made to solve the RIR problem. The discussion of the primitive period operator given in section 2 +ouId make clear through which arguments the double group appears: inciusion of the primitive period operator Py in the isometric group indeed leads to the eight group 5X(2,2,2). Nevertheless, all eigenstates of the RIR pro>Iem belong uniquely to the irreducible representations rj of the factor group [see eq. (4.7)]

i.e. exclusively to the even representations tive structural parameters

rj+ of the group &(2,2,2).

d(C’C) = 1.205 &

d&-C)

= 1.488 A,

L(CCH) = 11 lo,

L(CCF)= 111_7”,

d(C-H)

= 1.098 A,

L(HCH) = 107_9O,

Using standard geometries for approxima-

d(C-F)

= 1.382 A;

L(HCF) = 107.5”,

and assuming the internal rotation potential to be vanisbingly small, a set of energy eigenvalues and eigenvectors for states withJ = 0, 1,2,3 has been calculated. The results, collected in table 10 and represented graphically in

fig. 6, dispiay the following features: (i) Fig. 6 should serve a~ a survey for the level pattern of DmhF(CsT)2 SRM’s with free internal rotation. The levels Ear N and EOr N increase rapidly with N and tend to form neady degenerate pairs (f&. N, E,, N), whose energy difkerence dro& below 700 kHz already atiV= 3 (cf. tabIe IO). The sequence of levels I$, N ve’ry nearly follow a quadratic dependence on IV, even at low N_ For J> 1 the RIR levels group into clusters ok four and two levels. Table 10

Rotation-internal rotation energy eigenvalues of the SRM of FHzC-C = C-CHzF (MHz) N

r1

/=o

1

12 754.961 a)

0.000

2

55 573.183

3 4 5

220 543.083 496 137.830 881988.050

124 344 675 1 116

32 568.273

37 60 62 119

14 865.450 124 116.294 344 554.182 675 282.273 116 254.644

27 114.407 140 106.676 360 654.894 691379.518 1132 349.840

31991.798 61 784.230 115 787.731 144 819.922 227 533.863

38 122 133 151 262

31 115 144 282 365

39 218.766 68 788.401 122 751.142 133 902.030 151866.767

=3

1 2 3 4 5

27 173.265 51114.647 140 122.560 222 857.708 360 670.598

1

2 3

2 339.408 32 544.940 57 914.252

4 5

140 296.752 222 881.259

ri

I

44 069.100 67 119.339 126 888.902 151953.701 164 620.925

140 282.589 360 657.118 691379.491 1 132 349.836

1 2 3 4 5

a) Diagonal elements (u N, ii44v,.$,

1017.691

026.887 553.282 282.321 254.649

rz

r4

54 779.814 220 519.653 496 137.144 881988.023 1378 082.098 1

J=3

.l=2

J=l

see section 4.1.3.2 ff.

098.631 028.640 601.627 982.828

815.447 787.723 772.196 699.696 316.623

37 168.597 60 033.730 119 982.722 144 951.578 282 784.24@

14 43 67 69

866.742 751.101 902.028 771.103 082.002

033.791 929.263 093.508 646.173

126 889.426

AK Gut et d/Rotation-intend

rotation in gIyoxa1 type molecuk

II

459

Fig. 6. RIR energy level pattern of the SRM F
F

The chrsters of four consist of one level of each symmetry rr, lT2, r, , I’, and occur for both even and odd J. III contrast the clusters of two comprise either a i’r and a r2 level for even J or r3 and I’,+ for odd J.

(ii) l,Wifluoro-Zbutyne represents an example, where the kinetic coefficients and the kinetic determinant strongly modulated by the internal rotation, e.g.

are

g44 = 54.761 + 12.125 cos 2r(u AZ), gL = 15 895.558 - 3 872.061 cos 2~ - 183.483 cos 4r(u* A4), g* = 1s 305.101+ 3 355.109 cos 27 - 109.978 cos 4T(u2 .@). As a consequence the quantum potential terms arising from the fi Ing-terms appearing in eq. (4.2) are relevant for accurate calculation of energy levels, This is exemplified by table 11, which demonstrates that both levels and transitions are noticeably in error, if the quantum potential terms are neglected. As an example, the energy differences E or, 1_- Eorg a*dEoty - Eor,2 are shifted by 1.3 and 2.6 GHz, respectively by neglecting of the quantum potentrat. (iii) Again it should be pointed out that J= 0 states comprise only states arising from either {IT-~/~ cos 2m?) or {7r-l/2 sin 2rnT) basis functions. The basis functions {r-l/2 cos(2m + l)r} and {r-1/z sin(2m + 1)~) do not TabIe 11 Rotation-internal rotation problem of FHzC-C=C-CH2F. 6= 0 and on energy differences (MHz)

Level Or,1 Or,1 or,2 or,2 or,3 or,3 or,1

-or*1

or,1 + or,2

or,2+0r,2 or,*-or,3 or,3 *or23

With QP

Effect of quantum potential (QP) on c&ulated enwY lever for

Without QP

0.000 54 779.814

0.000 56 110 061

55 573.183 220 519.653

54 284.03 2 220 680.223

220 543.083 496 137.144 54 779.814

220 427.512 496 170.222

793.369 164 946.470

22.430 275 594.061

Difference 0.000 1 330.247 -1 289.151 160.570 -115.571 33.078

56 110.061 -1

826.029

166 396.191

1 330.247 -2

619.398 1449.721

-252.711

-275.141

275 742.710

148.649

M_ Gut et aI_/Rotation-internalrotationin glyoxal type molecuies. 11.

460

occur in J= 0 states, though they are required for J> I states. In both table 10 and fig. 6 the matrix elements N) are included. They occur in nearly degenerate pairs, which lie between contrast to the situation for H202.

5. Electric dipole transitions

and selection

rules

5.1. Selection rules In this section only those aspects of the electric dipole matrix eIement calculation and of the selection rules wiII be considered, which require modification from the treatment given in I. First it should be mentioned that the transformation properties of the electric dipole moment operator

w.r.t. the periodicity

Therefore,fif(T)

isometric

transforms

G-1)

cos(2k + I)7 ,

Ici(r)f = et kg0 A&+1

transformation

according

Pv5 is given by

to the representation

Table 12 Linestrength factors of FH2C-CS-CH2F.

rl_

of the group g(a)

Transition frequency (MHz)

Lime strength factor (D*)

EJriN - qr~@ Q-type transitions 131-141 132~142 211+221 212+222 232-241 23 l-242

-24833.858 -24569.707 24974.106 24685.599 28218.284 -78619.126

0.744 0.365 1.233 0.606 0.499 0.203

-22211.541 85502.775 -80237.064 27114.407 -28458.776 84533.493 -20096.716 29215.957 -78498.359 29476.039 83242.783 32860.465 -19946.050

0.244 0.226 0.241 0.497 0.239 0.234 0.248 0.122 0.113 0.744 0.366 0.897 0.365

R-type transitions

011*121 011+122 012+122 121+211 112+221 112-222 141+231 142~232 1.32~241 131~242

On the other hand

Transitions for Ml = 1.0 D. Values exceeding 0.01 D* are given

Transition “r,?ir + JF:IV I

021;Llll 022+111 022+-112

m B(r).

461

M Gut et aLlRotation-intemaIrotationin glyoxaItype molecules.II

0.6

LSidl

1

0.4

I ,

0.0 78

. 82

80

84

88

86

05Hz1

90

Y t

LSFd I.0 f

I

0.9 0.6 0.4 _

Fig- 7. RIR transition of FH2Cb3C-CHzF. Line strength factors (LSF) for Mf = 1.0 D only transitions with line-strength factors > 0.001 D* are considered. For identification of transition cf. table 12.

0.2 _ 0.0

I 18

20

,I

.

) I,

,

22

24

,

. I ,I

I

28

26

one obtains from general relations, &@P(e,

T&t

=I?@,

Since the energy eigenstates AJ=O,

+I,

Hence, the selection

,

,

30

) 32

Y)

, (GHz)

cf. table 4,

r) = rl+( V$“(E,

T) .

belong to the rj+ representations

J=O+J=O,

AM=O,

rules remain essentially

r1+ * rz+>

unaltered

(5.3) of @(T), the selection r3+ ++ r4+

by consideration

rules read (for z-polarization) (i.4)

-

of the extended

isometric

group.

5.2. Electric dipole matrix elements The general expressions for the electric dipole matrix elements in the first order energy basis have been given explicitely in I. Consideration of the extended isometric group classification implies a modification of the doublebar elements of&f(r) w.r.t. the internal base. Restriction of the general expression forif given by eq. (5.1) to the fast term leads to

5.3. Computerprogram

for transition intensities

For numerical calculation of the line strength factor a computer program has been developed, based on the programs discussed in section 4.2. As an example of application the intensities of some of the lowest RIR transitions of the molecule 1,4-dilluoro-2-butyne for J< 2 are listed together with frequencies in table 12 and graphically represented in fig. 7. These predictions might be useful in future investigations of this molecule.

Financial support of this work by the Swiss National Science Foundation (Projects NC. 2.303-0.75,2.5190.76) and by Messrs. Sandoz AG, Basle, is gratefullly acknowledged. Furthermore, we are grateful to Mr. J. Keller for his assistance with computationalhroblems and to one of the referees for drawing our attention to ref. [21]. Also we t$ank miss R. Zollinger for typing the manuscript.

462

M. Cut et al.fRotation-internal

rotation in glyoxal type molecules. II

Appendix A; Expressions for kinetic matrix coefficients Using the formulae given in table 2 for the ef-model one finds F%k

5

Stx:,lc,,l

QGk,,l

2 x,2kl

2 ‘x,2k2

-(x:2k,

=2FmQk

+ xx,f,

fX*2k312

- X;L2k2) cos

27

0

x2t2kl +&,&2(&

I

* Glil

\0

ma

-2X,,,,(X,f3

F E [Xl-J !2[X,fh] = 2ma(Xf3)2

Appendix B: Interrelation

0

0

iX,,,,)2-2X,,,,(?;~

- xf2k2) cos 27 ‘X&s~r

2G2kl

._ 1I 1

1 _ . 0

between velocities of two-top and ef-model

The variable kansforms given in table 1 may be written as:

and consequently

(B.1’)

Furthermore, starting from eq. (2.5”) we may write (abbreviating cos and sin by c and s, respectively),

M Gut et d/Rotation-intend

m&J

rotation in glyoxal type molecules. II

463

(; Lfj(I)_(fz:;‘./El:: 40,)

= SPX-J

=is(o,O,T)E(E)E+ l.,i

_

(B.2)

In the last step use has been made of the important equation E(P, Yl+ Y2) = E(Oo,0, Y+W,

72) >

71) =%oo, 0, r+W,

(B-3)

where the eulerian matrix&e) is defined by eq. (2.5”). Relating (B.3) expresses the fact that the eulerian matrix may be factorized according to E(l% r) = E&IO,0, %W%O)

(B-3’)

*

In analogy to (B.2) we fmd E(q)&

=D(O,O,T)E(E)E-

Appendix C: Determination

I.,?

.

(B.2’)

of the representation

rcNCf) {‘%)

According to the general procedure given in refs. [4,5] for generation of the representation rcNcf) {%} we consider as a substrate the coordinate vectorszf t2Xp(7>. Then the representation IQJCft2) of the isometric substitutions V3 and V, are obtained as follows from the coordinate expressions given in table 2 and eqs. (2.5):

=[[~:;L+~t-i’

=~[x~3i3+~t~O0

i’

-1

,i%(Wr)](’

-1

j%(o,o,r)][L

-1

-1

_,i^)

.

_j^.p”‘(V,))

(C-0

Since Xt200 is arbitrary one may conclude i’

-1

,)%(O,O,-T)(I

-1

_,i^;(’

-1

This equation may be solved by left multiplication sr, respectively), c% - (--l)~*GIQ)($)

= (-l)h’]I (0

+ (--l)P*Q?sT

-1

_Jr”w,,_

for I’(3)(V3), yielding (abbreviating cos r and sin r by cr and

-(-l)X[l (-l)hf+s%+ 0

d”a,,,,(i

t(-l)fi+~‘]G% (-I)@&%]

0 0 (_l)h’h’ 1

(C.2)

f13)(V3) must be independent of 7, furthermore x’ and D’ must be functions of h and p such that it becomes independent of h and u. These requirements lead to

464

M. Gut et al/Rotation-internal

@’= (II + l)(mad 2);

ra)(V,)

rotation in glyoxal type ndedes.

Ii

_(_l)A”h’

=

(6.3) (_l)h’ti

and finally to the two solutions X’= X(mod 2);

l-‘@)(V3) =

x* = XC 1 (mod 2);

,

I?(3)(V3) =

(C.33

(C-3”)

(C.3”‘)

(ii)Pvs{q,(T)} = @$T This leads straightforwardly

7t)) =

{qp8(r)rqv5)}

= {q_(T)}n(v,)

c3

rqV5) -

(C-4)

to

-[ST

f (-l)fl’+fi’&J

(-1)ql

-(-l)fi”+ST

-(-l)h[l

- (-l)“‘“‘]CTST

0

-(-l)h+h’

[s% i- (-l)p+@T]

0

0

,

(_l)“CX

0

from which we deduce the two solutions fiNCft*)(v,)

=(6y;hsEg)e

(C-5) -1

r@Cft2)(v5) =@r;,,,6+

.

1 i

One should note that the permutational indices of the nuclei.

(CS’)

-1 i factor of (C-5) is a unit, whereas (C.5’) features a proper permutation

Appendix D: Irreducible representations of c{b}

= B(T) 2 ~S(2,2,2)

8 ri r1+

r3c

W)

v, W)

E

V2

1 1

1 1

v, 1 -fi

v4 1 -1

vs 1 1

GV2

i 1

v5v3

v,v,

1 -1

-1

1

of

M. Gut et aL/Rotation-intend

465

rotation in giyoxal type molecules. II

Appendix D (continued) C

r3+ r4+

1

1 1 1

PIr2r3r4-

1

1

-1 -1

1 -1 1 1

-1 -1

1 -1 1 -1

-1 1 1 -1 -1 1

1 1 -1 -1 -1 -1

-1 -1 -1 -1 1 1

1

-1

1

1 -1 1 1 -1

-1 -1 -1 1

Appendix E: Analytical evaluation of trigonometric integrals The integrals occurring in the internal problem have been evaluated by complex integration in I. In order to remain consistent with notation, the interrelation with the integrals occurring m eq. (4.12) reads as follows: _ 14f%ik(2p) =~4Lik(p),

(E.1)’

where the barred integrals are given in I, appendix A-2.2, eq. (A-3). required for the Ja.1 problem are listed in table 7, The matrix elements of the kinetic coefficients g ll,g22,$3 expressed in the first order basis. They may be reduced to three trigonometric integrals in the zeroth order basis which have been evaluated in I; the refation to this paper is simply Pn(2p)

=Fmm(p)

(E.2)

)

where again the barred integrals are given analytically in I, appendix A3, eq. (A.10). The integrals originating from the coefficients g23 and g14, i.e. 123(2p + 1) =(1/2n)

f”

sin(2p f l)r[-g14,1sinr/g2(2~)]dr,

0

114(2p + 1) =(1;2~)

1:

COS(~P+ i)r[-g~4,1cos7/gl(2~)ld7,

0

may directly be expressed as 123(2p + 1) = -$g14,1 [G(2)(p) - Gc21(p + 1)] , 114(2~ + 1) = --$g14 1 [G(l)(p) f G(l) (p + l)] , where the Laurent eoefficiks G(l)(p) = G’+r)

(E-3)

are given as [cf. I, appendix A.3, eq. (A-12)]

= H1 (~)r+ P-1 1 fHl(“12)‘1&l.

G(2I(p) = G(21(-p) =H2(~21)l.r~;1

+ H2(~22)71$;1 .

p = 0,l .-. , (E-4)

Furthermore P

(2p) = 114(2p) = 0 .

(E-5)

4bb

M. Gut

et al.fRotation-internalrotationin gIyoxa1type molecules.11

References IL ] M- Gut, A Bauder and Hs.K. Gtinthard, Chem. Phys. 8 (1975) 252. [2] G. ‘Herzberg, Molecular spectra and molecular structure Vol. 2, Infrared and Raman Spectra of Polyatomic Molecules, (Van Nostrand, New York, 1945) p. J.39. [3] A-R. Edmonds, An&u momentum in quantum mechanics (Princeton University Press, 1957) p. 7. [4] A. Bauder, R. Meyer and KS-H. Giintkard, Mol. Phys. 28 (1974) 1305. [S] K. Frei, R. Meyer, A. Bauder and Hs.H_Giintiard. Moi. Phys. 32 (1976) 443. [6] M.C. Longuet-Higgins, Mol. Phys. 6 (1963) 445. [7] J.S.L. Ahmann, Proc. Roy. Sot. A298 (1967) 184; Mol. Phys. 21 (1971) 587. [8] P.R. Bunker, in: VTorational spectra end structure, VoJ. 3, ed. JR. D&g, (Dekker, New York, 1975) p- 2. f9] EP- ?Vigner, Group theory (Academic Press, New York, 1959) p. 167. [IO] R- Meyer, J. Chem. Phys. 52 (1970) 2053. [ll] R. Meyerand E.B. Wilson Jr., J. Chem. Phys. 53 (1970) 3969. 1121 R.H. Eiunt, R.A. Leacock, C.V.W. Peters and K.T. Hecht, J;Chem. Phys. 42 (1965).1931. [l3] K. Dreialer, 2. Naturforsch. Zla (1966) 1628. [14] G. Delfepiane, M. Gussoni and J.T. Kougen, J. Mot Specby. 47 (1973) 515. [15] E. Mathier, A. Attanasio, J. Keller, P. Niisberger, A. Bauder and Hs.H. Giinthard, Comp. Phys. Commun. 4 (1972) 20. [16] M. Gut, Ph.D Thesis, ETH-Zurich (1977). 1173 K. Kuckitsu, T. Fukuyama and T. MO&IO.J. Mol. Struct. i(l968) 463; 4 (1969) 41. 1181 JR. Durig, C.C.Tongand Y.S. Li, J. Chem. Phys. 67 (1972) 4425. [19] G.N. Currie and D.A. Ramsay, Can. J. Phys. 49 (1971) 317. [20] D.M. Agar, EJ. Bair, F-W_Birss, P. Borrell, P.C. Chen, G.N. Currie, A.J. McHugh, B.J. Orr, D.A. Ramsayand J.Y. Ron~in, Can. J. Phys.49 (1971) 323. [21] J.R. Durig. W.E. Bucy and A.R.H. Cole, Can. J. Phys. 53 (1975) 1832. [22] K-E Schaefer HI, C.E. Dykstra, 3. Am. Sot. 97 (1975) 7210. 1231 RH. Hunt and R.A. Leacbck, J. Chem. Phys. 45 (1966) 3141. [24] W-C. Oelfke, Ph.D Thesis, Duke University (1969). [25] W.C. Oelfke and W. Gordy, J. C&em. Phys. 51 (1969) 5336. [26] A. Veillard, Theoret. Chim. Acta 18 (1970) 21.