ORNL/TM-5666 3 The Use of Group Theory in the Interpretation of infrared and Raman Spectra E. Silberman H. W. Morgan OAK RIDGE NATIONAL LABORATORY BU BLANK PAGE Printed in the United States of America. Available from National Technical Information Service U.S. Department of Commerce 5285 Port Royal Road. Springfield. Virginia 22161 Price: Printed Copy $4.00: Microfiche $3.00 This report was preoared as an account of work sponsored by the United States Government Neither the United States nor the Energy Research and Development Administration/United States Nuclear Regulatory Commission, nor any of their employees, nor any of their contractors, subcontractors, or then employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness cr usefulness of any information, apparatus, product or process disclosed or represents that its use would not infringe privately owned rights 0RNL/TM-S666 Contract No. K-740S-eng-26 Physics Division THE USE OF GROUP THEORY IN THE INTERPRETATION OF INFRARED AND RAMAN SPECTRA E. Silbernan H. W. Morgan «Ut« I n* «M IH^IBl H 11H1HWI «wt« mw wl Wv l*** tow. Uvmmnwm »*i>i •» 11M llw Ml a. 1>M KM lav Ml ftmfciawm mwwmiiiiui mifciil M* * tfm. 4> .in w. III, w* «* WW Date Published: January 1977 «• IMMAiMM «Mi» «»»•*• MHWtm. ** VtfVW^f tf^M^ Prepared by the OAK RIDGE NATIONAL LABORATORY Oak Ridge, Tennessee 37830 operated by UNION CARBIDE CORPORATION for the ENERGY RESEARCH AND DEVELOPMENT ADMINISTRATION CONTENTS ABSTRACT v 1. INTRODUCTION 1 2. TOE POINT GROUPS I 3. GROUPS, REPRESENTATIONS, CHARACTER TABLES, AND THE REDUCTION FORMULA 7 4. TOE CHARACTER TABLES OF THE POINT GROUPS 8 5. GROUP THEORY AND MOLECULAR VIBRATIONS 21 6. THE PREDICTION OF THE FUNDAMENTAL VIBRATIONAL BANDS 22 7. EXAMPLES 24 7.1 Anoonia (NH3) 24 7.2 Monodeuterated Ammonia (NH2D) 26 7.3 1,2-Dichloroethylene (c£a) (e-ClHC-CClH) 27 7.4 1,2-Dichloroethylene (trans) (t-ClHC*CClH) 28 7.5 Ethylene (C2H^) 30 7.6 Phosphorus Pentafluoride (PFS) 32 7.7 Carbon Tetrachloride (CC1»,> 34 BIBLIOGRAPHY 37 iii ABSTRACT Application of the mathematical theory of groups to the symmetry of molecules is a powerful method which permits the prediction, classification, and qualitative description of many molecular properties. In the particular case of vibra- tional molecular spectroscopy, applications of group theory lead to simple methods for the prediction of the number of bands to be found in th-> infrared and Raman spectra, thei- shape and polarization, and the qualitative description of the normal modes with which they are associated. This report contains the tables necessary for the application of group theory to vibrational spectroscopy and instructions on how to use them for Molecular gases, liquids, and solutions. A brief introduction to the concepts, definitions, nomenclature, and formulae is also included. v THE USE OF GROUP THEORY IN THE INTERPRETATION OF INFRARED AND RAMAN SPECTRA 1. INTRODUCTION Application of the mathematical theory of groups to the symmetry of molecules is a powerful method which permits the prediction, classifi- cation, and qualitative description of many molecular properties. Most of this information would be unobtainable from first principles because of th<; difficulties in solving the complicated equations, and, in most cases, very long calculations would be necessary in order to obtain equivalent information by other methods. In the particular case of vibrational molecular spectroscopy, applications of group theory lead to simple methods for the prediction of the number of bands to be found in the infrared and Raman spectra, their shape and polarization, and the qualitative description of the normal modes with which they are associated. This summary will include the tables necessary for the application of group theory to vibrational spectroscopy and instructions on how to use them for molecular g&ses, liquids, and solutions. A brief intro- duction to the concepts, definitions, nomenclature, and formulae will be presented, but no theoretical justification of the methods will be given. More complete descriptions of the theory underlying such methods, as well as their extensions to molecular solids and polymers, can be found, at different levels of difficulty, in the annotated bibliography included at the end of this section. 2. THE POINT GROUPS The movement of a molecule from one position to another that is indistinguishable from the original one (because every atom has been replaced by another of the same kind or has not moved at all) is called a symmetry operation. Each symmetry operation has to be carried out with respect to a geometrical entity (point, line, or plane) which is called a symmetry element. There are a limited number of possible 1 BLANK PAGE 2 symmetry operations; they are listed in Table 1, together with their usual symbols and associated symmetry elements. The identity, a trivial symmetry which all molecules have, has been included because it has to be taken into account in the calculations that follow. The axis of symmetry of the highest order (if it exists) is called "principal axis," and it is customarily considered to be vertical; if a coordinate system is used, the principal axis should coincide with the z axis. Table 1. Symmetry operations Symbol Operation Description J Identity Rotation by 360° Cp p-Fold rotation Rotation by angle 2n/p a Reflection Reflection by a plane 5 Rotation-reflection Rotation by angle 2n/p followed by P reflection i Inversion "Reflection" of each point through the origin, or center of mass A molecule may have one or more symmetry elements. Not all com- binations of symmetry elements are possible; on the other hand, the presence of certain pairs of symmetry elements implies the existence of certain others. Each set of compatible symmetry operations is called a "point group" because (1) no symmetry operation can mov<i the point coincident with the center of mass (which is unique and cannot, therefore, be shifted to an "equivalent" position), and (2) each set must satisfy the conditions to constitute a mathematical "group." Because very few known molecules have symmetry axes of order greater than six (except «>), the number of point groups of practical interest is limited. Table 2 lists those groups and gives typical examples of molecules that have the corresponding symmetry in their equilibrium configuration. Some point groups for which examples are not known are listed because of the possibility of having to deal with molecules that belong to them in a perturbed configuration (as may happen in liquids) or in excited states. Table 2. The symmetry elements of the point groups of interest to molecular spectroscopy Point group*2 Symmetry elements b' a Examples Ci E CHFClBr, CH3'CHO c2 c2 M2O2 (nonplanar) c% d Cu Cu CJ(c. Cu) C*2(c. C6). C3(c. ^ (BS2) V trans-C1BrHC•CHBrC1 ^(C. SU) Se C3(c. s6), i a N0C1 (nonlinear), NH2D, C2H5C1 H2O, H CCl2» chlorobenzene °2' % 2 NH3, MCCI3, 0PC13 2o«(p. av), Cj(c. Ck) IF5, Fe(CO)5 3% 3o* (b. oy)/ Ci(c. C6), C3(c. CG) c , ooq Any Cp{c. CJ HC1, HCN, HCCCl to' V i Czh °h fcrane-ClHC=CHCl, C302 H B0 , C*(NH ) c3* °h 3 3 2 3 % V °h <?2(C' V' C3(C' C*)f 56(C. C6), 53(C. i Table 2 (continued) Point group Symmetry elements btc Examples C D2=V 2> (P» '"'2) C2(p. C2C2) 3C2(p. C3) 2C?2Cp• £«»» m.p.)^ 2(7^(b. CI), C*2(c. CO C6, 3C2(p. C6) 3C2(b. CZ.), C2 (c. C6), C3(c. <76) B CI , H C_H , N S C2» 2C'(p. C2, m.p.)» 5„(c. C2) 2 k* 2 3 2' U <• i C H , C H (cyclohexane), '3 d 3C2(p. C3), 30^ S6Cc. c3), 2 6 6 12 (SiH3)20 <V 4C2(P- 4ad <?!J(c. C^), S8Cc. C„) S8 °2h <SV C2' ^P* » p. CC\), o'(t. C2C2), C2Hi», p-dichlorobenzene, o(t. C'CJ) o"(t. C2(f'7)* i naphtalene 5 (c. C ), 3o '3 h C3, 3C2(p. C3), ah 3 3 y BF3, 1,3,5-C6H3C13 PtC1 2 \h Ck, 2Cz(p. CJ, <W , ' 2^'(p. C„), oh 2oy, 2(Jj, i C , S£7 (p. C ), o^ 5(c. C5), Sov Cyclopentane (plane and 5k s 2 5 symmetric), ferrocene eclipsed) D 6 h C6, 3Cz(p. C6), CgHg (benzene) S3(c. S6(c. C6); 3oy, Table 2 (continued) Point groupa Symmetry elements^'0 Examples ^ mfih Any cp[c. CJ, any o2, co2, c2h2 Sp(.c. CJ, »oy, i T 4C3, 3C2(m.p., b. C3's) ch4, wj, nhj, SO^" Td 4C3, 3C2(m.p., b. C3's), ss^c. Cz), 6orf(t. C3's) 4C 3C (m.p.), i C > Th y 2 3o(t. C2's), 4Sg(c. 3 s) 0 ^(m.p.), 4C3 3c2(C. CJ, 6C2 (m.p.), 4C3(cube 3C2(c. CJ, 6C2(t. centers SF6, Ptci62-, UF6 diagonals), i opposite cube edges), 3SJc. CJ, 4Sg(c. CJ, 3a^(p. CJ, 6ad(t. oppo- site cube edges) a C. In the Zu hV ,, and C.DV , groups, aV should (if possible) pass through more atoms than oV\ (which is o-, C C sometimes labeled a in and © v ); if the rule is not applicable, oy should intersect more bonds than Oy. Similar rules should be used to label C2, C'2, and in groups D2 and D^, and C2 and C^ in groups Dk, Dg, Dzd, D^, and D^. The element E (identity), which is contained in every point group, has been omitted. cAxes and planes pr<?c?ded by a number are equivalent (can be transformed one into the other by another symmetry operation).