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On the symmetry of normal modes
JOURNAL OF MOLECULAR SPECTROSCOPY 6, 264-265
LETTER
(1266)
TO THE EDITORS
On the Symmetry
of Normal
Modes
It is well known that it is easy to draw a picture of any nondegenerate mode of a molecule by means of arrows representing atomic displacements from equilibrium as long as one knows the symmetry type, i.e., the (one-dimensional) irreducible representation of the equilibrium symmetry group associated with the mode. One simply draws the arrows in such a way that the resultant mode picture is invariant under all operations of the equilibrium symmetry group under which the representation is symmetric and such that the arrows are all reversed for the operations under which the given representation is antisymmetric. The purpose of this brief note is to generalize this rule in such a way that it includes not only the nondegenerate modes but certain two-fold degenerate modes as well. This makes it easier to draw mode pictures for these two-fold degenerate modes. The new statement of the rule is as follows: The symmetry group which leaves the mode picture invariant (corresponding to the irreducible representation p of the equilibrium symmetry group G) is the invariant subgroup P with the property that p is a faithful representation of the factor group G/P. Thus, for every one dimensional representation, the invariant subgroup P is the group of operations which leave the corresponding mode picture invariant and G/P is a group of order one (for the totally symmetric representation where P = G) or two (for all other one-dimensional representations). But the rule also applies to the two-fold degenerate E’ modes of a molecule with octahedral (0 or Oh), or with tetrahedral (?ld) symmetry. Since the two-fold degenerate E representation of the 0 group is a faithful representation of the factor group O/D, which is of order 6 and isomorphic to C’r,. , we can say that the mode picture of any molecule with octahedral (0) equilibrium symmetry must have DT symmetry. This must be true for each of the two degenerate modes (and hence any linear combination of them). A similar result holds for the tetrahedral group I’d which is isomorphic to 0, and which has the same invariant subgroup consisting of the identity and the entire class 3Cy (which constitute the subgroup D?). For Oh the invariant subgroup is Dzh. Thus it can be seen by inspection that the mode pictures in Herzberg (1) and Wu (9) for the octahedral XI’, molecule are incorrect as they have D&instead of DS symmetry. One cannot obtain in this way the mode pictures of irreducible representations of maximum dimensionalitv (degeneracy) of an equilibrium symmetry group G since these are faithful representations of G and cannot be faithful representat,ions of a factor group. Thus our increased generalization adds only the two-dimensional representations of the equilibrium symmetry groups 7’,i , 0, and 0,‘ , which are the only groups having irreducible representations of dimension great.er than one but less than the maximum dimensionality.’ HOWever, the number of possible types of molecules of these symmetries ext,ends beyond the simplest familiar cases worked out in t,he texts, and applicat,ions also include crystals having these
point
groups.
1Excluding the icosohedral group which has irreducible representations of dimension 1,3,3,4,5 but no invariant subgroups, hence all are faithful representations except the one-dimensional one. 26-1
LETTER
TO
THE
EDITORS
z&j
It is often possible to draw pictures for the modes of maximum degeneracy which have symmetry also. But it will be noted that although the operations for any one such degenerate mode may form a group it is not an invariant subgroup of the equilibrium symmet,ry group. In the latter case each of the degenerate mode pictures will be invariant under a different set of symmetry operations (although they may form a group of the same symmet.ry), and hence an arbitrary linear combination of such modes would have only the identity as a symmetry element in common. This is due to the fact that the invariant subgroup P for the faithful representations of G consists of t,he identity alone. An example is Its degenerate counterthe FZe mode pictured in Herzberg (I), which has D 2h symmetry. parts ohtained by rotation of t,he figure also have Dp,&symmetry hut all three degenerate mode pictures have only the identity in common, This illustrates the fact that t,he rule presented is bot,h a necessary and suflicient~ condition for det,ermining the appropriate mode picture REFERENCES and Raman Spectra of Polyatomic Molecules,” p. 122. Van 1. C. HERZBERO, “Infrared Nostrand, New York, 1945. d. T. WV, “Vibrational Spectra and Structure of Polyatomic Molecules,” 2nd ed., p. 285. Edwards, Ann Arbor, 1946. Deparlrnent of Physics ~‘niversity of California Los Angeles, California Received July 21, 1960
ROBERT A. SATTEN
LETTER
(1266)
TO THE EDITORS
On the Symmetry
of Normal
Modes
It is well known that it is easy to draw a picture of any nondegenerate mode of a molecule by means of arrows representing atomic displacements from equilibrium as long as one knows the symmetry type, i.e., the (one-dimensional) irreducible representation of the equilibrium symmetry group associated with the mode. One simply draws the arrows in such a way that the resultant mode picture is invariant under all operations of the equilibrium symmetry group under which the representation is symmetric and such that the arrows are all reversed for the operations under which the given representation is antisymmetric. The purpose of this brief note is to generalize this rule in such a way that it includes not only the nondegenerate modes but certain two-fold degenerate modes as well. This makes it easier to draw mode pictures for these two-fold degenerate modes. The new statement of the rule is as follows: The symmetry group which leaves the mode picture invariant (corresponding to the irreducible representation p of the equilibrium symmetry group G) is the invariant subgroup P with the property that p is a faithful representation of the factor group G/P. Thus, for every one dimensional representation, the invariant subgroup P is the group of operations which leave the corresponding mode picture invariant and G/P is a group of order one (for the totally symmetric representation where P = G) or two (for all other one-dimensional representations). But the rule also applies to the two-fold degenerate E’ modes of a molecule with octahedral (0 or Oh), or with tetrahedral (?ld) symmetry. Since the two-fold degenerate E representation of the 0 group is a faithful representation of the factor group O/D, which is of order 6 and isomorphic to C’r,. , we can say that the mode picture of any molecule with octahedral (0) equilibrium symmetry must have DT symmetry. This must be true for each of the two degenerate modes (and hence any linear combination of them). A similar result holds for the tetrahedral group I’d which is isomorphic to 0, and which has the same invariant subgroup consisting of the identity and the entire class 3Cy (which constitute the subgroup D?). For Oh the invariant subgroup is Dzh. Thus it can be seen by inspection that the mode pictures in Herzberg (1) and Wu (9) for the octahedral XI’, molecule are incorrect as they have D&instead of DS symmetry. One cannot obtain in this way the mode pictures of irreducible representations of maximum dimensionalitv (degeneracy) of an equilibrium symmetry group G since these are faithful representations of G and cannot be faithful representat,ions of a factor group. Thus our increased generalization adds only the two-dimensional representations of the equilibrium symmetry groups 7’,i , 0, and 0,‘ , which are the only groups having irreducible representations of dimension great.er than one but less than the maximum dimensionality.’ HOWever, the number of possible types of molecules of these symmetries ext,ends beyond the simplest familiar cases worked out in t,he texts, and applicat,ions also include crystals having these
point
groups.
1Excluding the icosohedral group which has irreducible representations of dimension 1,3,3,4,5 but no invariant subgroups, hence all are faithful representations except the one-dimensional one. 26-1
LETTER
TO
THE
EDITORS
z&j
It is often possible to draw pictures for the modes of maximum degeneracy which have symmetry also. But it will be noted that although the operations for any one such degenerate mode may form a group it is not an invariant subgroup of the equilibrium symmet,ry group. In the latter case each of the degenerate mode pictures will be invariant under a different set of symmetry operations (although they may form a group of the same symmet.ry), and hence an arbitrary linear combination of such modes would have only the identity as a symmetry element in common. This is due to the fact that the invariant subgroup P for the faithful representations of G consists of t,he identity alone. An example is Its degenerate counterthe FZe mode pictured in Herzberg (I), which has D 2h symmetry. parts ohtained by rotation of t,he figure also have Dp,&symmetry hut all three degenerate mode pictures have only the identity in common, This illustrates the fact that t,he rule presented is bot,h a necessary and suflicient~ condition for det,ermining the appropriate mode picture REFERENCES and Raman Spectra of Polyatomic Molecules,” p. 122. Van 1. C. HERZBERO, “Infrared Nostrand, New York, 1945. d. T. WV, “Vibrational Spectra and Structure of Polyatomic Molecules,” 2nd ed., p. 285. Edwards, Ann Arbor, 1946. Deparlrnent of Physics ~‘niversity of California Los Angeles, California Received July 21, 1960
ROBERT A. SATTEN