This work has been digitalized and published in 2013 by Verlag Zeitschrift für Naturforschung in cooperation with the Max Planck Society for the Advancement of Science under a Creative Commons Attribution4.0 International License.

Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschungin Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung derWissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht:Creative Commons Namensnennung 4.0 Lizenz.

Tentatively Standardized Symmetry Coordinates for Vibrations of Polyatomic Molecules

Part XVI. Further Six-Atomic Models

S . J . C Y V I N , G . H Ä G E N , a n d B . N . C Y V I N

Institute of Theoretical Chemistry, Technical University of Norway, Trondheim, Norway

(Z. Naturforsch. 23 a, 363—367 [1970] ; received 24 December 1969)

Twelve six-atomic models are treated as a part of the work on tentatively standardized sym-metry coordinates of molecular vibrations.

In a systematic treatment of symmetry coordina-tes for vibrations of polyatomic molecules a number of four-atomic five-atomic2, and some six-atomic 3

models have been considered. In the present work some further six-atomic models of importance in molecular structure studies are considered.

The specified symmetry coordinates are believed to be suitable as a standard reference when har-monic force constants are reported, and for other purposes when analysing molecular vibrations. It should be mentioned as a warning, however, that they are not always well suited for setting up an initial approximate force field in the normal coordi-nate analysis.

1. Some Cyclic Models

Suitable symmetry coordinates for the planar and puckered Z6 ring models have been specified else-where 3' 4. In the present work we have considered two cyclic models of the types X 3 Y 3 and X4Y.>, which apply to the skeletons of 1,3,5-trithiane5 '6

and 1,4-dithiane6 '7, respectively. The latter type applies also to the p-dioxane 8 skeleton.

Fig. 1 shows the considered X 3 Y 3 model, and con-tains a specification of valence coordinates, viz. d, a and ß. A complete set of symmetry coordinates is given in the following.

Reprints request to Prof . Dr. S. CYVIN, Institutt for Fysi kalsk Kjemi , Norges Tekniske Hegskole , N-7034 Trond-heim, Norwegen.

1 S.J.CYVIN, J. BRUNVOLL, B.N.CYVIN, I. ELVEBREDD, and G. HAGEN, Mol . Phys. 14, 43 [ 1 9 6 8 ] .

2 S. J. CYVIN, B. N. CYVIN, I. ELVEBREDD, G. HÄGEN, and J. BRUNVOLL, to be published.

3 B.VIZI and S.J.CYVIN, Acta Chem. Scand. 22, 2012 [1968] .

Fig. 1. The cycl ic X 3 Y 3 molecular mode l ; symmetry Ca v • The Y atoms are in the XY plane, while the X atoms are situated below this plane. The equilibrium X Y distance is denoted by D. Two additional parameters are needed to determine the equilibrium structure, e. g. the two angles 2 A ( Y X Y ) and

2 B ( X Y X ) .

(^i) = 6~i(d1 + d2 + d3 + dA + d5 + d6), S,(A1)=3-iD(al+a.2 + a3), S3(Al)=3-lD(ß1+ß2 + ß3); S (A2) = - d2 + d3-di + d5 - d6) ;

Sla (E) = 1 2 " » ( 2 di+2d,~ d3 -di-d5- d6), (E) = I ( - d3 + <f4 + d5 - d6),

S3a(E) = 6-iZ>(2ax -a2-a3), S4A (E) = 6 - 1 D(2ß1- ß2 - ß3); Sib(E) = i (d3 + di-d5~d^), Ssb(E) = \2-H2d1-2d2-d.i + d,-d5 + d6),

4 S. J. CYVIN, Molecular Vibrations and Mean Square Am-plitudes, Universitetsforlaget, Oslo ; and Elsevier, Amster-dam 1968.

5 M. J. HITCH and S. D. Ross , Spectrochim. Acta 25 A , 1047 [1969] .

6 P. KLABOE, Spectrochim. Acta (in press) . 7 M. J. HITCH and S. D. Ross , Spectrochim. Acta 25 A , 1041

[1969] . 8 F. E. MALHERBE and H. J. BERNSTEIN, J. Am. Chem. Soc.

7 4 , 4 4 0 8 [ 1 9 5 2 ] .

S3b(E) = 2 - l ö ( a ä - a s ) , S4b(E)=2~LD(ß2-ß3).

The degenerate coordinate pairs (5 , a , Sib) are oriented as to transform like the rigid translations (TX, TY) in accord with the chosen cartesian axes (see Fig. 1 ) .

Fig. 2. The cycl ic trans-X4Y2 molecular model ; symmetry C^h-Equilibrium distances: D ( X — Y ) , T(X — X ) . Two angle para-meters in addition, viz. 2 A ( X Y X ) and 2 B ( X X Y ) , are suf-

ficient to determine the structure.

Fig. 2 shows the considered X 4 Y 2 model, which is a trans- (chair-form) cyclic type. Symmetry coordi-nates:

StiAg) = b (d1+d2 + d3 + d4), S2(Ag) = 2 ~±{tx + t2), S3{Ag) = 2 ~$D(a1 + a2), S4(Ag) = i (DT)Hßt + ß2 + ß3 + ßi); Sx {Bg) = \ (d.-d. + dß-d,), S2(Bg)=i (DT)Hß1-ß2 + ßs-ßt); SxiAJ = \ (d1-d2~d3 + dA), S2(Au) = 2 - H t i - t o ) , S3{Au) = \ (DT)Hßx-ß2-ßs + ßi); SAB«) = h (dl + d2-d3-d,), S2(Bu) = 2-*D(ax-a2), S3(Bu)=h(DT) i(ß1+ß2-ß3-ß4).

9 B. N. CYVIN, S. J. CYVIN, G. HAGEN, I. ELVEBREDD, and J. BRUNVOLL, Z. Naturforsch. 24 a, 643 [1969] .

10 S. G. FRANKISS and F. A. MILLER, Spectrochim. Acta 21. 1235 [1965 ] .

11 S. G. FRANKISS, F. A. MILLER. H. STAMMREICH, and TH. T. SANS, Spectrochim. Acta 23 A , 543 [1967] .

12 A. YAMAGUCHI, I. ICHISHIMA. T. SHIMANOUCHI, and S. I. MIZUSHIMA, Spectrochim. Acta 16. 1471 [1960] .

2. Some X 2 Y 4 Models

In a previous part 9 the X 2 Y 4 models of Doh ? DM and Do symmetries were treated. The similar models of Cov, C-2h and Co symmetries (see Fig. 3) are treated in the present section. In these models the X — X Y 2 conformations are nonplanar, in contrast

(a)

(a) (c) (b)

Fig. 3. The X 2 Y 4 molecular models of symmetries (a) C%v , (b) C2/1 and (c) C2 . The four ßi bendings for each model per-tain to the X X Y i angles; for the sake of convenience only two of them (ßt and ß3 in the trans model) are indicated. r = — t 1 5 6 3 — t2564 is a twisting coordinate. Equilibrium struc-ture parameters: R{X-Y), D ( X - X ) , 2 A ( < £ Y X Y ) and

B ( < £ X X Y ) ; in case (c) also 2 T (the angle of rotation).

to the case of the previously treated ones9 . The P0CI4 molecule was found to have the trans (C2h) structure 10, as also was the case for P2I4 in solution and the crystalline state n . The gauche (C2) struc-ture was found for N2H4 12, P2H4 1 3-1 4 , and"N2F4 in the vapour and solid states1 5 '1 6 . Somewhat more detailed surveys of molecules with the here con-sidered structures are found elsewhere n ' 1 7 .

A suitable set of symmetry coordinates for the cis-X2Y4 model (symmetry Cov) is given below.

S j M i ) = % (ri + r2 + r3 + r 4 ) , S 2 ( A X ) = D,

S3{AX) = l (RD)i(ß1+ß2 + ß3 + ßA),

13 M. BAUDLER and L. SCHMIDT, Z. anorg. allgem. Chem. 289. 219 [ 1 9 5 7 ] .

14 E. R. NIXON, J. Phys. Chem. 6 0 . 1 0 5 4 [1956] . 15 D. R. LIDE, JR. and D. E. MANN, J. Chem. Phys. 31, 1129

[ 1 9 5 9 ] , 10 J. R . DURIG and R. C. LORD, Spectrochim. Acta 19, 1877

[ 1 9 6 3 ] . 17 H. SIEBERT, Anwendungen der Schwingungsspektroskopie

in der anorganischen Chemie, Springer-Verlag. Berlin 1966.

S4(AT) = 2 - » Ä ( a i + a f ) ; S1(A2) = \ ( r i - r 2 - r 3 + r 4 ) , S2(A2)=\(RD)Hß i - Ä - Ä + A). S3(A2) = RX; Si(B1) = \ ( r 1 - r 2 + r 3 - r 4 ) , s 2 ( 5 x ) = i ( Ä D J M A - Ä + Ä - A ) ; S i ( ß 2 ) = I (ri + r2 - r3 - r 4 ) , S2{B2)=\{RD)Hßi+ßz-ß t - ß i ) , S 3 ( ß 2 ) = 2 - 0 4 ) .

Formally the same expressions are applicable also in the cases of the two other models in question. The correlations between the symmetry species of C2h and C2v are: AG — A1, Bg — AU — A2 , Bu — B2. In the generalized case of the C2 model the correla-tions are: A(A1 + A2), B(B1 + B2) or A(AG + AU), B(Bg + Bu). It is proposed here that the symmetry coordinates for the C2 model are taken in the follow-ing sequence. A: S1(A1), S1(A2), S2(A1), S3(A1), S2 (A2), SA (A,), S3 (A2); B: S± (Bt), SX (Bt), S 2 ( B t ) , S 2 ( B 2 ) , S 3 ( B 2 ) ; where the notation pertains to the C2v model.

3. Two Planar X 2 Y 2 Z 2 Models

Disubstituted ethylenes (e. g. C2H2C12) may be of the eis-, trans- and asymmetric (CH2CC12) type 1 8 _ 2 0 . The appropriate models in the two former cases are shown in Fig. 4. The trans type is also applicable to glyoxal2 1 and oxalyl chloride22 . Symmetry co-ordinates for the cis-X2Y2Z2 (C 2 v ) model:

S M i ) - 2 - l ( r 1 + r 2 ) , St(At)" 2 - » ( 5 1 + 5 t ) , S3(A1) = D, S,(A,) = (RD/2)l(a1 + a2), S5(AT) = (SD/2)*-(ß1 + ß2); Si (A2) = [(RS^D/2]i(7i + 72), S2(A2) = RX; S(Bt)= [(RS)iD/2]i(yi-yt);

S1(B2) = 2~Hr1-r2), S2(B2)= 2 ~HS1-S2), S3(B2) = (RD/ 2 ) M « i - a 2 ) , S,(B2)= (SD/2)Hßi-ß2).

18 H.J.BERNSTEIN and D. A. RAMSEY, J. Chem. Phys. 17, 556 [1949] .

19 J. M. DOWLING, J. Chem. Phys. 25, 284 [ 1 9 5 6 ] . 20 D. E. MANN, T . SHIMANOUCHI, J. H. MEAL, and L. FANO,

J. Chem. Phys. 27, 43 [ 1 9 5 7 ] .

of-plane bendings yx and y2 , which involve the atoms (3, 1, 5, 6) and (4, 2, 6, 5 ) , respectively. Accord ing to the chosen conventions both central atoms (X ) move down (i. e. in the direction of the negative Z-axis) in the trans mode l ; in the eis model a positive moves 5 up, while y2 moves atom 6 down, (ii) The torsion r involves the atoms (3, 5, 6, 4 ) . Equi-

librium parameters: R(X-Y), S ( X - Z ) , D ( X - X ) , A ( < £ X X Y ) , ß ( < £ X X Z ) .

Formally the same expressions hold for the trans-X 2 Y 2 Z 2 (C2h) model. One only has to observe the correlations for C2h (C2v) ? viz.: AG (AT), Bg (B±), AU (A2), Bu (B2)~

4. Two X Y 2 W Z 2 Molecular Models

In asym-C2H2Cl2 the two C atoms are not sym-metrically equivalent. Fig. 5 shows (a) the appro-priate planar W X Y 2 Z 2 model along with (b) the twisted W X Y 2 Z 2 model, both with symmetry C2v . Suitable sets of symmetry coordinates are specified in the following. Firstly, for both models (a, b) :

SiiAJ- 2-i(r1 + r2), S , ( A 1 ) = 2 - 4 ( ^ + 5 , ) , S s M i ) = d, SMi)= (RD/2)i(a1 + a2), S 5 ( ^ ) = (S D/2) + S(A2)= (RS)* X.

Next we introduce the six coordinates:

(i) 2 - * ( r 1 - r t ) , (ii) 2 ~ i ( s i - s 2 ) ,

(iii) (RD/2)l(ai-0L2), (iv) (S D /2 ) (v) (RD)h,

(vi) (SD)*d.

21 K . KUCHITSU, T. FUKUYAMA, and Y . MORINO, J. Mol . Structure 1, 463 [1967 /68 ] .

22 J. S. ZIOMEK, A. G. MEISTER, F. F. CLEVELAND, and C. E. DECKER, J. Chem. Phys. 21, 90 [1953] .

Fig. 5. The (a) planar and (b) twisted XY 2 WZ 2 models; symmetry C%v . Additional valence coordinates: (i) Out-of-plane bendings y(4, 3, 6, 5) and 0 (2 , 1, 5, 6 ) . (ii) Twisting

T= — T1363 — TO564 . Equilibrium parameters: R(X —Y), 5 (W - Z ) , D(W - X ) , 2 A (<£YXY) and 2 B (<£ZWZ).

For the 2 + 4 ß 2 coordinates of the model in Fig. 5 (a ) one should use the expressions (v ) , ( v i ) ; and ( i ) , ( i i ) , ( i i i ) , ( iv ) . For the 3 BL + 3 B2 coordina-tes of the model in Fig. 5 ( b ) the appropriate ex-pressions are ( i i ) , ( iv ) , (v) ; and ( i ) , ( i i i ) , (v i ) .

5. Two X Y 2 Z U V Models

Fig. 6 shows two X Y 2 Z U V models with planar XYoZU parts and linear XZU chains; the ZUV con-

Fig. 6. (a) Planar symmetrical XY 2 ZUV model with linear XZUV chain; symmetry C^v, and (b) XY 2 ZUV model with planar XY 2 ZU part and linear XZU chain; symmetry Cs • Equilibrium structure parameters: R{X — Y ) , D(X — Z ) , J ( Z - U ) , S ( U - V ) , 2 A (<^YXY) , and in particular for mo-del (b) 0 (<^ZUV). The parenthesized symbols in (a) are out-of-plane valence coordinates. For model (b) in addition to the indicated valence coordinates, one has the combination of torsions: r = 2 — i (T12644-T1285), which is considered os one

single type.

formations are (a) linear and (b) bent in the two cases. As to the symmetry coordinates we start in both cases (a, b) with the following ones.

Sx = + r 2 ) , S2 = d, S3 = t, S4 = s, S5=(RD/2)i(ß1 + ß2).

In the case of model ( a ) , which possesses the C2v

symmetry, these coordinates represent the whole set

in species AX; in case ( b ) , where the model has the symmetry of CS, the coordinates belong to A', and one has furthermore:

S6(A') = (DT)LOY, S7(A') = (ST)I (P,

S8(A') = (RD)*Y

For model (a) :

S ^ ) - (DT)*6X, S2(B1) = (ST)*<px,

ST{BT)= (RD)H; SX(B2) = 2 - l ( r 1 - r 2 ) , S2(B2) = (RD/2)1 (ßx-ß2), S3(B2)= (DT)IOYT

S , ( B 2 ) = (ST)^cpy.

For model (b) :

S M ' ) - 2 - » ( r 1 - r 8 ) , S,(A") = (RD/ 2)*(ßi-ßs), S3(Ä')= (.DT)*0X, ST(A") = (RS)1* R.

6. The X Y 2 Z U V Model with Central Atom X

In this section the last model is treated, which is another X Y 2 Z U V model of symmetry CS, but differ-ent from the one of the previous section. In the pre-sent case (see Fig. 7) X is a central atom. The mo-del applies to the sulphonic acid molecule and some

T(X - U), S (U - ' V ) , 2 A (<£YXY) , Bx (<£UXY) , B2 (<£ZXY) , r (<^XU V) . t is the torsion for atoms 1 — 2 — 6 — 3.

of its derivatives 23 25. Symmetry coordinates: SM') = ( D R / 2 ) i ( / ? 2 1 + A>2),

SM') = SX(A') = 2 " » ( r 1 + r t ) , SM')- (ST)H-SM') = t, SM") = 2 - » ( r 1 - r t ) , SZW) = d, SM") = (TR/2)* (ßu-ßu),

SM') = Ra, SM") = (DR/2)* (ßn-ßn),

SM') = (TR/2)Hßn+ß12), SM") = {SD)* r .

23 R.J.GILLESPIE and E.A.ROBINSON, Can. J. Chem. 40, 644 25 S. J. CYVIN and I. HARGITTAL, Acta Chim. Hung. 61, 159 [1962] . [ 1 9 6 9 ] .

24 S. M. CHACKALACKAL and F. E. STAFFORD, J. A m . Chem. Soc. 8 8 , 4 8 1 5 [1966] .

Time Correlation Functions for Internal and Anisotropic Rotational Motion of Molecules

H . V E R S M O L D

Institut für Physikalische Chemie und Elektrochemie der Universität Karlsruhe, Germany

(Z. Naturforsch. 25 a, 367—372 [1970] ; received 24 December 1969)

By use of Green 's functions f o r the diffusional motion a very concise formulation and compu-tation of correlation functions is possible. For an axially symmetric overall diffusion with internal rotation about one and two axes correlation functions of a second rank spherical tensor are cal-culated. The results comprise all the solutions of the pertinent problem as given so far and allow the extension of the theory to a wider field of application.

1. Introduction

Nuclear spin relaxation has proved to be a power-ful tool for studying molecular motions in liquids. Firstly, valuable information concerning the rotatio-nal motions of the molecule to which the nucleus belongs can be obtained. Secondly, internal rotatio-nal motions about some fixed molecular axes also have a specific influence on the relaxation behaviour of the nucleus to be considered. This type of rota-tional motion occurs especially in organic liquids.

The molecular motions enter into the theory of nuclear spin relaxation through the autocorrelation functions. The influence of rotational diffusion on correlation functions has been the subject of several investigations1 - 4 . The paper of HUNTRESS 4 treats in detail the case of anisotropic rotational diffusion.

Reprints request to H. VERSMOLD, Institut für Physikali-sche Chemie und Elektrochemie der Universität Karlsruhe, D-7500 Karlsruhe, Kaiserstraße 12.

1 A. ABRAGAM, The Theory of Nuclear Magnetism, Oxford University Press, New York 1961.

2 H. SHIMIZU, J. Chem. Phys. 37, 765 [1962] . 3 H. SHIMIZU, J. Chem. Phys. 40. 754 [1964] . 4 W . T. HUNTRESS. J. Chem. Phys. 48, 3524 [ 1 9 6 8 ] .

However, it is restricted to the case of extreme narrowing and furthermore asymmetry parameters for the various possible interactions are neglected.

The case of internal rotational motions has been studied as well 5 - 8 . Recently WOESSNER 7 has given a rather general formula for magnetic dipole-dipole relaxation. W A L L A C H 8 treated internal motion in macromolecules but he did not make allowance for any anisotropic rotations of the macromolecule as a whole. For very long molecules this may be a poor approximation. Furthermore he neglected asymmetry parameters as indicated above.

The author of this article wants to show that by use of Green's functions 9 a very concise formulation of the computation and results is possible. For an overall diffusion of the molecule that is axially sym-metric this formulation comprises all the solutions

5 D. E. WOESSNER, J. Chem. Phys. 36. 1 [1962] , 6 D. E. WOESSNER, J. Chem. Phys. 37, 647 [1962] , 7 D. E. WOESSNER, B. S. SNOWDEN, and G. H. MEYER, J.

Chem. Phys. 50, 719 [1969] . 8 D. WALLACH, J. Chem. Phys. 47, 5258 [1967] . S D. FAVRO, in : R. E. BURGESS, Fluctuation Phenomena in

Solids, Academic Press, New York 1965.