Bond Angle Relationships in Some AX,Y,
Molecules
In the structural determination of a compound, often only the bond angles that uniquely define the molecular geometry are reported. Evaluation of the remaining angles, if required, is generally left to the reader. An example of such' a case is CH2F2, where both F-C-F and H-C-H angles Z u L. are known ex~erimentallvwhile F-C-H is not reoorted. In this note, bond angle relationships are derived for several ........, &Y. systeis ,th ,hasis on the symmetry c k c t i o n ' x -..A& between some apparently unrelated structures. Consider a "distorted tetrahedral" AXaYz molecule of CaYsymmetry with bond angles a(X-A-X) and,/3(Y-AY) both differing from cos-1 (-YJ), the angle in a regular tetrahedral AX4 molecule. With reference to an orthogonal coordinate svstem centered at atom A (Fie. unit vecD * A . " 1). ., tars directed along the A-X,, and A-Y, bonds have components ( d s i n ( a / 2 ) , \/%sin(a/2), cos(a/2)) and (-\/Y;sin(B/Z), v%sin(8/2), -cos(8/2)) respectively, and their scalar product yields the follawingrelation between the anglesa, 3/ and y(X-A-Y)
a
f
v-
cosy = -cos(a/2)cos(8/2)
(1)
Kquation r l ) ~i well-known but Joe, nut seem mattract the artention ofrrxrbmk wirers. If the an& (I 1s increased until it exceeds n. rhr molcrular symmerr). remains unchanged but the molerular skeleton bcromes "d-Irigunal b~pyramidal"las drpictrd in Figure 1. Here it is more convenient lo use u' = 2n - I,, and eqn. (1)takes the farm (2) cosy = cos(a'/2kodB/2) Consider now the special case a' = 0 in eqn. (2);the simplifiedform cosy = cos2(a'/2) = cos2(a/2) (3) is applicable to square pyramidal (C,,) AX,Z systems with bond anglesX-A-Y y and X-A-Z= a / 2 (Fig. 3). For an AX&?. molecule of Clu symmetry we can choose the coordinate system shown in Figures 4a and 4b. The scalar product of the vectors
-
-7
-
AX,/IAX,I = (cos(8
- s/2), 0, -sin(8 - ~ 1 2 ) =) (sir$,
0, cosdl
and
1. fi. Z,/IE,I = (-2wn~,
ems)
xXb
Y
b
c m = - - s ,1n '.p.+ m 2 p = 1 - - S l n 3~ . , 2 2 which further reduces to 2 . sin (T - p) = - sm(ai2).
fl
Equations (1)-(1), are generally applicable to all AX,Y,Z,
(4)
systems belonging to point groups Czu,
C3"and C4Y including the replacement of bonds by lone pairs. Some examples of their use are illustrated in the table. Examples of Use of Equations
Bond Angles Experimental Valuels)
Compound CH2CH2*CHa (CHd?PF*F1 (CHd3TeCI? BrFaF* HaCCI
H*-C-H* C-P-C C-TcC F-Br-F* H-C-C1
= 106.7°, C-C-C = 112.4' = 124.0'. F-P-F = 111.8" = 98". Cl-TeCI = 172' = 85' = 108.0'
Calculated Value H*-C-C F-P-C C-TeCl F-Br-F H-C-H
109.4° 90.5'
= =
= 87.4= 89.6' = 110.9'
Equation Used (1) (1) (2)
13) (4)
Cotton, F. A., and Wilkinson, G., "Advanced Inorganic Chemistry," 2nd Ed., Wiley-Interscience, New Yark, 1966, p. 404.
The Chinese University of Hong Kong Shatin, N.T. Hong Kong
Wai-Kee Li Thomas C. W. Mak
Volume 51, Number 9. September 1974
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571
