~OOa~t Ol~ ~ MOL~CULAa S~CTr¢OSCOeY 25, 406--409 (1968)

The Effect of Orientation of Degenerate Coordinates on Coriolis Coupling Constants

S. G. W. G I N N AND SANDOR R E I C H M A N

Molecular Spectroscopy Laboratory, School of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455

The ambiguity in the definition of degenerate normal coordinates leads to a corresponding ambiguity in the rallies of the zeta constants coupling parallel and perpendicular vibrations. It is important to resolve this ambiguity for the determination of matrix elements which depend on these zeta constants.

The C matrix method of Meal and Polo (1) has been extensively employed in the discussion of zeta matrices. In particular, Kristiansen and Cyvin (2) have applied the method to the determination of the possible (~ matr ix elements (a = x, y, z) for planar symmetrical XYa molecules.

In the ease of the possible fx and fY elements, however, there is an indeter- minaney in the individual values of the Coriolis coefficients of which Kristiansen and Cyvin apparent ly were not aware, arising from the infinite number of ori- entations possible for doubly degenerate normal coordinates. This is apar t from the sign ambigui ty already mentioned in tlef. (2). Vectors for degenerate normal or symmet ry coordinates may be oriented arbitrari ly with respect to the xy axes (Fig. 1). For example, I~ristiansen and Cyvi i f s choice given below Fig. 1 leads to all four possible f2:~,~ components having nonzero values, because each co- ordinate has a Component along the y-axis. A more natural choice, which is al- ways possible for symmetr ic- top molecules, is to define the top component co- ordinate vectors of each degenerate vibrat ion to be parallel to the x and y axes respectively (Fig. 1). For instance in X Y:~ if the b component is made parallel to the x-axis, the two ~-2 :°,~b eoetIieients vanish. Calculations by I~ristiansen and Cyvin (2) for B**Fa have been repeated for this particular coordinate orientation (which corresponds to a clockwise rotat ion of their coordinate system by 120 °) leading to the following zeta values.

~-2Z,aa = f2vab = --0.929 f'~,4, = f2Y,4b = --0.371

~'2 x ,3bX = - - ~-2 y,3a~k = ~-ff~,4b )k : - - ~'2 y ,4a ~k = ) k 0

The C ~ matrix elements from which these zeta values were obtained are also given below:

4o6

CORIOLIS COUPLING CONSTANTS 407

%

5

Y '[ ~ X

02 4 Oi

I 2

KC a

$3a=(I /6 '/2) (2r l - r2- r3)

S3b= (I/2 v2) (r2 - r3)

S4a= (I/6 '/2) (2O 1 -e2-a s)

$4b=(! /2 '/2) (o 2 - 03)

aKristiansen and Cyvin, Ref. 2.

SYMMETRY COORDINATES

KC, rotated clockwise by 120°about z

(I/61~) (2ra- q - r 2 )

(I/21~) (rl - r2 )

(I/61~) (20 s- ol - 02)

(I/21/2) (~ - o 2)

FIG. 1. Atom and coordinate numbering convention

408 GINN AND REICHMAN

C2z.~: = - (6~12/2) (~y + 3ux) = C2v,ab

C2X,4a = - - (3/21/2) (#~ -t- 3~x) = C2Y,4b

C2%b = C2 "~,4b = -- C2 ~,3a = - C2 ~,4o = 0

i n general, the effect of ro ta t ing the basis vec tors f rom an a rb i t r a ry posi t ion is equivalent to opera t ing on the original C ~ ma t r ix b y a ma t r ix R to yield the new values for ~ ,

(= L-1RC~ -I

o r

= £G-~RC~£ -~.

The L mat r ix is independen t of the or ien ta t ion of the s y m m e t r y coordinates, the normal coordinates being a u t o m a t i c a l l y or iented parallel to the s y m m e t r y coordinates.

The R , m a y also be envisaged as t rans forming the s y m m e t r y s-vectors involved in cross products wi th S2, the out-of-plane s-vector , in the definit ion of the C a ma t r ix elements. Express ions for the fx coefficients following a ro ta t ion of Kris= t iansen and Cyv in ' s basis degenera te coordinates b y an angle a are given below.

r2CV3a = ( A L44 -- B L34)L22/I L [

~2:,3b = (C L44 -- D La4)L22/I L I

~2~,4b = ( D L33 -- C L4a)L22/l L [

w h e t A = (6~/2/12) cos c~ -- (2112/~) sin c~ B = (--6112/4) sin ~ + (21/2/4) cos a C = (61/2/12) sin a + (21/2/4) cos c~ D = (61/2/4) cos a -}- (21/2/4) sin a

and I L [ is the de te rminan t of the L ma t r ix for the degenera te coordinates. These more general expressions m a y be used to show t h a t the first sum rule der ived by Kris t iansen and Cyv in (2) [their equa t ion (13)],

(~-,'3.)2 q - (f2".4a)2 q - (~'2".3b)2 q - (f2'~'.4b) 2 = 1,

remains val id for p lanar symmet r i ca l X Y 3 molecules for a n y or ienta t ion of the degenerate coordinates. W h e n the a - t ype coordinates are oriented paral lel to the y-axis, this sum rule reduces s imply to

(f2".z.)2 q - (f2".4.) 2 = 1

The second sum rule given b y Kr is t iansen and Cyv in [their equat ion (14)] is not in general applicable.

CORIOLIS COUPLING CONSTANTS 409

The ambiguity in the orientation of the degmlerate coordinates may lead to uncertainty in the evaluation, for instance, of matr ix elements in the Hamil ton- Jan. An example of this is found in the first order Coriolis interaction between a parallel (t) and a perpendicular (s) vibrat ion which is dependent on the value of ftx~a (8). As stated by Nielsen (3), it is understood tha t f / . t b is ze ro in th i s instance, corresponding to an orientation of the b-type eoordinate vector along the y-axis. An unambiguous choice for ~"~ and fs is also impor tan t in the ealeula- tion of v ibra t ion-rota t ion interaction constants a and anharmoniei ty constants

(3). It should be pointed out that the more familiar ~ coefficients (~-~3 ,/'~4, ~4 for

XY~) are independent of the orientation of the degenerate coordinates, because the cross product vector, directed along the z-axis, of the two interacting (a and b) components in the C" matrix is constant in magnitude.

Note added in proof: After the completion of this work, it came to our attention that C. di Lauro and I. M. Mills (o r. Mol. Spectry. 9.1, 386 (1966)) also treated this problem and arrived at similar conclusions.

ACKNOWLEDGMENT

This work was supported by the U. S. Air Force 0ffiee of l~eseareh under Grant No. 570-67.

RECEIVED: September 8, 1967

REFERENCES

1. J. HAW~(INS MEaL ~tND S. R. POLO, J. Chem. Phys. 94, 1119, 1126 (1956). 2. L. KI~ISTrA.~SEN AND S. J. CYw~, J. Mol. Spectry. 11, 185 (1963). 3. H. It. NIELSEN, in "Handbuch der Physik." (S. FL~GOE, ed.), Vol. 37, p. 173. Springer,

Berlin, 1959.