The field of Molecular Spectroscopy was surveyed in order to determine a set of

conventions and symbols which are in common use in the spectroscopic literature. This

document, which is Part 2 in a series, establishes the notations and conventions used for the

description of symmetry in rigid molecules, using the Schoenflies notation. It deals firstly

with the symmetry operators of the molecular point groups (also drawing attention to the

difference between symmetry operators and elements). The conventions and notations of the

molecular point groups are then established, followed by those of the representations of these

groups as used in molecular spectroscopy. Further parts will follow, dealing inter alia with

permutation and permutation-inversion symmetry notation, vibration-rotation spectroscopy

and electronic spectroscopy

Bertie, J E

Bunker, P R

Hougen, J T

Mills, Ian

Schutte, C J H

Watson, J K G

Winnewisser, B P

English

Central Archive at the University of Reading

Pure &A@. Chem., Vol. 69, No. 8, pp. 1641-1649, 1997. Printed in Great Britain. 0 1997 IUPAC INTERNATIONAL UNION OF PURE AND APPLIED CHEMISTRY PHYSICAL CHEMISTRY DIVISION COMMISSION ON MOLECULAR STRUCTURE AND SPECTROSCOPY* SUBCOMMITTEE ON NOTATIONS AND CONVENTIONS FOR MOLECULAR SPECTROSCOPY NOTATIONS AND CONVENTIONS IN MOLECULAR SPECTROSCOPY: PART 2. SYMMETRY NOTATION (IUPAC Recommendations 1997) Prepared for publication by C. J. H. SCHUTTE', J. E. BERTIE2, P. R. BUNKER3, J. T. HOUGEN4, I. M. MILLS5, J. K. G. WATSON3 AND B. P. WINNEWISSER6 'University of South Africa, PO Box 392, Pretoria OOO1, South Africa 2Department of Chemistry, University of Alberta, Edmonton, Canada T6G 2G2 3Steacie Institute for Molecular Sciences, National Research Council of Canada, Ottawa, K1A OR6, Canada 4Molecular Physics Division, National Institute for Standards and Technology, Gaithersburg, Maryland 20899, USA 5Department of Chemistry, University of Reading, Whiteknights, Reading, RG6 2AD, UK 6Justus Liebig Universitat, Physikalisch-Chemisches Institut, Heinrich-Buff-Ring 58, D-6300 Giessen, Germany *Membership of the Commission during the preparation of this report (1985-97) was as follows: Chairman: J. R. Durig (USA; 1983-89); C. J. H. Schutte (RSA; 1989-93); J. E. Bertie (Canada; 1994-97); Secretary: H. A. Willis (UK, 1983-87); J. G. Grasselli (USA; 1987-89); J. F. Sullivan (USA; 1989-95); P. Klaeboe (Norway; 1996-97); Titular Members: A. M. Bradshaw (FRG; 1983-91); R. D. Brown (Australia; 1987-93); B. G. Derendjaev (Russia; 1983-91); R. K. Harris (UK; 1989-93); A. M. Heyns (RSA; 1994-97); E. Hirota (Japan; 1985-93); R. Janoschek (Austria; 1994-97); P. Klaeboe (Norway; 1994-97); J. F. J. Todd (UK; 1981-89); S. Tsuchiya (Japan; 1994-97); H. A. Willis (UK, 1983-87); B. P. Winnewisser (FRG; 1991-97); Associate Members: A. M. Bradshaw (FRG; 1991-95); S. M. Cabral de Menezes (Brazil; 1994-97); B. G. Derendjaev (Russia; 1991-95); P. Granger (France; 1989-93); R. K. Harris (UK; 1981-89); E. Hirota (Japan; 1994-97); J. Kowalewski (Sweden; 1996-97); S. Leach (France; 1987-91); G. J. Martin (France; 1983-89); J. P. Maier (Switzerland; 1989-93); L. N. Masalov (Russia; 1987-91); A. Oskam (Netherlands; 1991-97); W. B. Person (USA; 1983-89); C. J. H. Schutte (RSA; 1983-89); J. F. J. Todd (UK, 1989-91); S. Tsuchiya (Japan; 1987-93); C. Zhang (China; 1989-97); National Representatives: J. E. Collin (Belgium; 1985-97); S. M. Cabral de Menezes (Brazil; 1991-93); S. L. Spassov (Bulgaria; 1987-93); J. Lu (China; 1981-89); E. Hadjoudis (Greece; 1991-93); G. Varshyi (Hungary; 1985-93); M. Chowdhury (India; 1986-95); Z. Luz (Israel; 1983-93); S .  Califano (Italy; 1989-97); S. Ng (Malaysia; 1983-93); A. Oskam (Netherlands; 1989-91); B. Jezowska-Trzebiatowska (Poland; 1983-89); J. J. C. Teixeira-Dias (Portugal; 1991-97); Y. S. Lee (Korea; 1989-97); T. A. Ford (RSA; 1987-91); C. Pascual (Spain; 1987-91); D. Escolar (Spain; 1991-97); J. Kowalewski (Sweden; 1994-95); M. Cebe (Turkey; 1987-91); S. Suzer (Turkey; 1991-93, 1995-97); S. I@ (Turkey; 1993-95); T. Ast (Yugoslavia; 1989-93); R. K. Harris (UK; 1993-97). Republication or reproduction of this report or its storage and/or dissemination by electronic means is permitted without the need for formal IUPACpermission on condition that an acknowledgement, with full reference to the source along with use of the copyright symbol 0, the name IUPAC and the year of publication are prominently visible. Publication of a translation into another language is subject to the additional condition of prior approval from the relevant IUPAC National Adhering Organization. Notations and conventions in molecular spectroscopy: Part 2. Symmetry notation (IUPAC Recommendations 1997) Abstract -- The field of Molecular Spectroscopy was surveyed in order to determine a set of conventions and symbols which are in common use in the spectroscopic literature. This document, which is Part 2 in a series, establishes the notations and conventions used for the description of symmetry in rigid molecules, using the Schoenflies notation. It deals firstly with the symmetry operators of the molecular point groups (also drawing attention to the difference between symmetry operators and elements). The conventions and notations of the molecular point groups are then established, followed by those of the representations of these groups as used in molecular spectroscopy. Further parts will follow, dealing inter alia with permutation and permutation-inversion symmetry notation, vibration-rotation spectroscopy and electronic spectroscopy. PREAMBLE The sub-committee on Notations and Conventions in Molecular Spectroscopy was initiated by Commission 1.5, the Commission on Molecular Structure and Spectroscopy, to establish and recommend a set of conventions and symbols from those that are in common use in the spectroscopic literature. Part I of the series is titled General Spectroscopic Notation [l]. The present part deals with symmetry notation referring to point groups, and the recommendations follow those of Jenkins [2], Mulliken [3], IUPAP [4], and IUPAC [5] in the main. The notation of Herzberg [6] is recommended when it differs fiom that of Wilson, Decius and Cross [7]. It is pointed out that the International Union of Crystallography uses another approach [8]. Part 2. SYMMETRY NOTATION 2.1 2.2 The Molecular Point Groups 2.3 Symmetry Operators for the Molecular Point Groups Representations for Molecular Point Groups and Spectroscopy . .  01s used in this Dane r Greek Alphabet r 2.3.1; 2 .3 .3~  c, (3 2.3.3a p 2.3.2 a 2.1.2a r, Y 2.3 .3~  Oh 2.1.2b A, 6 2.3.3~ a,, 0,' 2.1.2c l-I, 71 2.3.3~ a d  5 a d '  2.1.2d; 2.1.7 0, + 2 . 3 . 3 ~  axy, a(x,y) 2.1.6 sz 2.3.11 1642 0 1997 IUPAC Symmetry notation 1643 Symbols matrices I I 1  9 upperllower case 2.2.17 2.1.7; 2.3.7 2.3.4 192 2.3.6; 2.3.9; 2.3.12a + -  2.3.13 Latin Alphabet and Numbers A a B b ci c n v  f 2.3.3a; 2.3.12 2.3.3a 2.3.3b; 2.3.12 2.3.3b 2.1.3; 2.2.1 2.1.3; 2.1.6 2.1.7 2.2.2 2.2.6 2.2.7 2.2.7 2.1.3 2.2.14 2.2.3 2.2.4 2.2.5 2.2.15 2.1.1; 2.3.3~; 2.3.12 2.1.1; 2 . 3 . 3 ~  2.3.3d; 2.3.12 2.3.3d 2.3.3e; 2.3.12 2.3.3e; 2.3.5a 2.3.3f 2.1.1; 2.3.12 2.2.13 2.1.5 2.3.8 2.3.10 2.2.1 1 2.2.12 2.1.4 2.1.4 2.1.4; 2.1.5 2.2.2 2.2.8; 2.3.3d 2.2.10 2.2.9 2.3.3d 2.3.5b 2.1.6 0 1997 IUPAC, Pure andApplied Chemisfry69, 1641-1649 1644 COMMISSION ON MOLECULAR STRUCTURE AND SPECTROSCOPY hical Conventions: It is desirable to distinguish by means of typographic conventions between the symmetry elements of a body (such as, for example, the invariant line called a rotation axis) and the symmetry operators (sometimes called symmetry operations) which bring a body into coincidence with itself and which form apoint group. Unfortunately there seems to be no established set of typographic conventions for doing so. The conventions used in this paper are stated below. It is recommended that authors who do not follow these conventions clearly define their own conventions. Symbols for symmetry operators are printed italic, with sub- and super-scripts that are upright except for the variables k and n which are replaced by numbers for a specific operator [5 ] .  Following Donnay (Actu Cryst. A28, (1972) 1 lo), the power of an operator, k, is specified even when it is one, e.g., C,' is used instead of C,; and S,' is used instead of S,. Symbols for symmetry elements are seldom used in this document. Those that are included are printed upright. Symbols for symmetry groups are printed italic with upright subscripts, following common usage. Symbols for irreducible representations of point groups, which are usually called symmetry species in spectroscopy, are printed upright. Subscripts are upright except when they are the variablesj or a which are replaced by numbers in specific cases. T i s  printed italic when it is a variable to be replaced by other symbols in specific cases. 2.1 SYMMETRY OPERATORS OF THE MOLECULAR POINT GROUPS The symbols for symmetry operators are defined below. Only the Schoenflies notation is followed here. The operator symbols approved by the International Union of Crystallography [8] for use in crystal chemistry, which include translational symmetry operators, differ from those given here. The correspondences are (E  = I ) ,  (C, = n) (i = y), (cr m), (Snl = or 3;. A table showing the alternative notations of the point groups used by crystallographers is found in Ref. 8, p. 801, as well as in Schutte [9], p. 393. 1 2.1.1 Identity operator E; e; I 2.1.2 Reflection operator for reflection across a plane (a) across a general plane cr (b) across a horizontal plane (c) across a vertical plane (d) in a plane bisecting two C,-axes which are perpendicular to the principal C, axis Od 2.1.3 n-fold rotation operator for k successive rotations through an angle of 2xln about an n-fold rotation axis, where k n = 2 ,  3, ...; k =  1,2, ..., n -  1 c, ; c,' = c, or: n = 00 ; rotation through an arbitrary angle 4 CW( $4 0 1997 IUPAC, Pure and Applied Chemistry69,1641-1649 Symmetry notation 1645 2.1.4 n-fold rotation-reflection (alternating; improper) operator for k successive rotation-reflections about an n-fold rotation- reflection axis for a rotation through an angle of 2rcln followed by a reflection in a plane perpendicular to the axis, where n = 2 , 3 , 4 ,  ...; k =  1,2, ... (n-1) or: n = co ; rotation through an arbitrary angle 4 Inversion operator through the centre of symmetry 2.1.5 k 1  s, ; s, = s, Sa( 4) i, notes2' = i 2.1.6 Specification of orientation: If it is needed to specify the orientation of the symmetry element with respect to a molecule- fixed set of Cartesian axes (x, y, z )  which is involved in the symmetry operation, an appropriate subscript or symbols in brackets may be added to the symbol of the symmetry operation, e.g., uv, +Y), c , k ~  2.1.7 Distinguishing between same kind of operators: symmetry operators of the same kind, but belonging to different group classes must be distinguished by consecutive right accent superscripts, e.g., C2, C2', C2", and 0, and 0,' in D4h 2.2 THE MOLECULAR POINT GROUPS The essential operators in each point group are listed. Other operators are present in the groups marked **, required by the presence of those listed. References 6 and 7 give a complete listing of the operators in each group. 2.2.1 Finite rotation groups having only C,k operators cn 2.2.2 Rotation-reflection groups having only S2,k operators S2,; s2 = ci 2.2.3 Group of operators of the group C, plus n-dihedral C2 operators Dn ; 4 2.2.4 Group of operators of the group D, plus the o h  operator** Dnh; D2h 2.2.5 Group of operators of the group 0, plus nud operators** Dnd; D2d 2.2.6 Group of operators of the group C, plus nov operators c n v  2.2.7 Group of operators of the group of C, plus the a h  operator** cnh; Clh cs 2.2.8 Group of rotation operators of a tetrahedron T 2.2.9 Group of operators of Tplus the inversion operator i Th 2.2.10 Full group of operators of a tetrahedron Td 2.2.1 1 Group of operators for the rotations of an octahedron (isomorphous with Td) 0 2.2.12 Full group of operators of an octahedron (Group of operators of 0 plus the inversion operator i )  Oh 0 1997 IUPAC, Pure and Applied Chemistry69,1641-1649 1646 COMMISSION ON MOLECULAR STRUCTURE AND SPECTROSCOPY 2.3 2.2.13 Full group of operators for an icosahedron I h  2.2.14 Group of operators for an asymmetric linear molecule caw 2.2.15 Group of operators for a symmetric linear molecule (the group of operators of C,, plus the inversion operator i)* * Dmh 2.2.16 Alternative definitions: Some of the point groups may be defined in an alternative fashion as direct product groups with the i-operator, e.g., for n odd C, x i = S,,,. The c,,h groups are also generated by C, x a h  = c,,h or for n even by C, x i = c,,h 2.2.17 Matrices representing symmetry operators: the matrices representing the symmetry operators of a group G are indicated by the corresponding heavy-print symbols. Definitions should include whether the operators are active (operating on the position vectors) or passive (operating on the molecule-fixed coordinate system), whether the rotations are clockwise or anticlockwise with respect to the positive direction of the rotation axis, and the handedness of the axial system. REPRESENTATIONS FOR MOLECULAR POINT GROUPS AND SPECTROSCOPY 2.3.1 Reducible representation of group G r 2.3.2 Irreducible representations of group G, where j = 1,2,  3,  ... numbers the representations sequentially and (a) = 1,2,  3 indicates the dimension of each representation alternatively, the symbols of 2.3.3 may replace Jca) 2.3.3 The dimensions of the representations are indicated by the following symbols (which may be sub- and superscripted according to 2.3.5 - 2.3.12), the choice between upper case and lower case letters being regulated by 2.3.4: (a) (b) polyatomic h A, a z B, b one-dimensional, symmetric (+1) with respect to the one-dimensional, antisymmetric (-1) with respect to the primary operator C,,' or S,' of the point group primary operator C,' or S,,' of the point group - -  (c) two-dimensional and pseudo two-dimensional E, e n, 71 4 6 0, 4 r, Y &&: The complex conjugate one-dimensional representations 4 and 4 are bracketed together in spectroscopy as a pseudo E-representation or corepresentation, because the Hamiltonian is invariant to the operator which expresses time-reversal, that is, the wavefunction and its complex conjugate yield the same energy. This occurs, for instance, in the groups C, S,,, and c,,h 0 1997 IUPAC, Pure and Applied Chernistry69, 1641-1649 Symmetry notation 1647 (d) three-dimensional (the symbol T is customarily used in the context of inorganic complexes, to avoid confusion with F for atomic states in the full rotation group) (e) four-dimensional (0 five-dimensional Symbols for symmetry species in 2.3.3 are used upper case or lower case as follows: (a) Individual normal vibration wavefunction (b) Overall vibrational state wavefunction (c) Individual one-electron (orbital) wavefunction (d) Overall electronic state wavefunction (e) Overall vibronic state wavefunction 2.3.4 2.3.5 Right subscripts on the representations of 2.3.3 with respect to the inversion operator i: (a) representations which are symmetric (+1) with respect to the inversion operator i (b) representations which are antisymmetric (-1) with respect to the inversion operator i lower case upper case lower case upper case upper case 2.3.6 Right subscripts on the representations A andor B of 2.3.3 for the groups D,, D,, and Dnd or for C,, (a) representations which are symmetric with respect to the C21 operator for a rotation around a C2' axis I to the generating axis C, of the representations which are symmetric with respect representations which are antisymmetric with respect representations which are antisymmetric with groups Dm Dnh,  and Dnd to the 0, operator in the group C,, to the C2' operator in the groups D,, Dnh, and Dnd respect to the a, operator in the groups C,, 1 (b) (c) (d) U: To resolve the ambiguity in assigning the subscript labels of the B species of the point groups C2, and DZh for planar molecules, the x axis must be chosen perpendicular to the plane of the molecule. For 4, the z axis is chosen so that it passes through the largest number of atoms or, if this is not unique, then so that it cuts the largest number of bonds [3]. The Mulliken [3] numbering of the B,, B2 and B3 species of DZh is followed. 1 2 2 2.3.7 Right superscripts on the representations of 2.3.3 with respect to the operator (a) (b) representations which are symmetric with respect to the ah-operator representations which are antisymmetric with respect to the ah-operator 0 1997 IUPAC, Pure and Applied Chemistry69, 1641-1649 1648 2.3.8 2.3.9 2.3.10 2.3.1 1 2.3.12 2.3.13 COMMISSION ON MOLECULAR STRUCTURE AND SPECTROSCOPY Right subscript I o n  the E-representation where I=  1,2,3,  ... ( f n  - 1) for n even and $ ( n  - 1) for n odd, where n is the order of the generating operator of the point group Right subscripts 1 and 2 on F-representations of the groups U: Since the use of 1 or 2 is arbitrary, the notation of Herzberg [6] and Wilson, Decius and Cross [7] is recommended Td,  0, Oh A left superscript m to indicate electron spin multiplicity m = 2 s  + 1 may be added to the species symbols of 2.3.4(d) and (e); for example, 3A2, 3Cg A right subscript n o r  P of 1.2.2.16 [l]  may be added to the species symbols of linear molecules, for example, n(3/2)", A5/2 Level designations according to the value of R for very strong spin-orbit coupling (Hund's case c): (a) diatomic and linear molecules 2 2 odd electron states* even electron states* (b) non-linear molecules odd electron states* *+ even electron states**+ Right superscripts on the C representations of the groups (a) representations which are symmetric with respect to the reflection operator ov (b) representations which are antisymmetric with respect to the reflection operator o, Cmv and Dmh m n 1 2 5  2 '  2 ,  2 , ' "  o+, 0-, 1, 2, ... *For Dmh add the g, u subscripts. **For D,, and 0, add the g, u subscripts, and for D3,, and c,h add the ' and " superscripts, omitting the 1 and 2 subscripts. +Multiply the orbital symmetry species by the spin symmetry species to obtain the appropriate level designation, keeping in mind the and " signs, as well as g and u. REFERENCES 1 C. J. H. Schutte, J. E. Bertie, P. R. Bunker, J. T. Hougen, I. M. Mills, J. K. G. Watsonand B. P. Winnewisser,(Part l), Pure Appl. Chem.,this issue (1997). 2 F. A. Jenkins, Phys. Rev. 43, (1953) 425. 3 Report on Notation for the Spectra of Polyatomic Molecules, prepared for the Joint Commission for Spectroscopy by R. S. Mulliken, J. Chem. Phys., 23, (1955) 1997. 4 IUPAP, Symbols, Units, Nomenclature and Fundamental Constants in Physics, 1987 Revision, E. R. Cohen and P. Giacomo, Document IUPAP-25 (SUNAMCO 87-l), Physica, 146A (1987) 1. 0 1997 IUPAC, Pure and Applied Chemistry69, 1641-1649 Symmetry notation 1649 5 I. Mills, T. Cvitas, K. Homann, N. Kallay and K. Kuchitsu, Quantities, Units and Symbols in Physical Chemistry, International Union of Pure and Applied Chemistry (Blackwell Scientific Publications, Oxford, 1993) Second Edition. 6 G. Herzberg, Molecular Spectra and Molecular Structure: 11 Infared and Raman Spectra of Polyatomic Molecules, D. Van Nostrand Company, Inc., Princeton 1945; 111 Electronic Spectra and Electronic Structure of Polyatomic Molecules, D. Van Nostrand Company, Inc., Princeton 1966. Reprinted by Krieger Publishing Co., Malabar, F1. (1 99 1). 7 E. B. Wilson, Jr., J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw-Hill Publishing Co., New York, 1955. 8 T. Hahn, International Tables for Crystallography, Volume A, Fourth Revised Edition, Kluwer Academic Publishers, Dordrecht, 1995. 9 C. J. H. Schutte, The Theory of Molecular Spectroscopy, Volume 1. The Quantum Mechanics and Group Theory of Vibrating and Rotating Molecules. North Holland, AmsterdadAmerican Elsevier, New York, 1976. 0 1997 IUPAC, Pure and Applied Chemistry69, 1641-1649 

Notations and conventions in molecular spectroscopy: part 2. Symmetry notation

The field of Molecular Spectroscopy was surveyed in order to determine a set of

conventions and symbols which are in common use in the spectroscopic literature. This

document, which is Part 3 in a series, deals with symmetry notation referring to groups that

involve nuclear permutations and the inversion operation. Further parts will follow,

dealing inter alia with vibration-rotation spectroscopy and electronic spectroscopy

Bunker, P. R.

Schutte, C. J. H.

Hougen, J. T.

Watson, J. K. G.

Winnewisser, B. P.

Pure & Appl. Chern., Vol. 69, No. 8, pp. 1651-1657, 1997. Printed in Great Britain. Q 1997 IUPAC INTERNATIONAL UNION OF PURE AND APPLIED CHEMISTRY PHYSICAL CHEMISTRY DIVISION COMMISSION ON MOLECULAR STRUCTURE AND SPECTROSCOPY * SUBCOMMITTEE ON NOTATIONS AND CONVENTIONS FOR MOLECULAR SPECTROSCOPY NOTATIONS AND CONVENTIONS IN MOLECULAR SPECTROSCOPY PART 3. PERMUTATION AND NOTATION PERMUTATION-INVERSION SYMMETRY (IUPAC Recommendations 1997) Prepared for publication by P. R. BUNKER', C. J. H. SCHUTTE2, J. T. HOUGEN3, I. M. MILLS4, J. K. G. WATSON' AND B. P. WI"EWISSER5 'Steacie Institute for Molecular Sciences, National Research Council of Canada, Ottawa, KIA OR6, Canada 2University of South Africa, PO Box 392, Pretoria OOO1, South Africa 3Molecular Physics Division, National Institute for Standards and Technology, Gaithersburg, Maryland 20899, USA 4Department of Chemistry, University of Reading, Whiteknights, Reading, RG6 2AD, UK 5Justus Liebig Universitat, Physikalisch-Chemisches Institut, Heinrich-Buff-Ring 58, D-6300 Giessen, Germany *Membership of the Commission during the preparation of this report (1985-97) was as follows: Chairman: J. R. Durig (USA; 1983-89); C. J. H. Schutte (RSA; 1989-93); J. E. Bertie (Canada; 1994-97); Secretary: H. A. Willis (UK; 1983-87); J. G. Grasselli (USA; 1987-89); J. F. Sullivan (USA; 1989-95); P. Klaeboe (Norway; 1996-97); Titular Members: A. M. Bradshaw (FRG; 1983-91); R. D. Brown (Australia; 1987-93); B. G. Derendjaev (Russia; 1983-91); R. K. Harris (UK; 1989-93); A. M. Heyns (RSA; 1994-97); E. Hirota (Japan; 1985-93); R. Janoschek (Austria; 1994-97); P. Klaeboe (Norway; 1994-97); J. F, J. Todd (UK; 1981-89); S. Tsuchiya (Japan; 1994-97); H. A. Willis (UK; 1983-87); B. P. Winnewisser (FRG; 1991-97); Associate Members: A. M. Bradshaw (FRG; 1991-95); S. M. Cabral de Menezes (Brazil; 1994-97); B. G. Derendjaev (Russia; 1991-95); P. Granger (France; 1989-93); R. K. Harris (UK; 1981-89); E. Hirota (Japan; 1994-97); J. Kowalewski (Sweden; 1996-97); S. Leach (France; 1987-91); G. J. Martin (France; 1983-89); J. P. Maier (Switzerland; 1989-93); L. N. Masalov (Russia; 1987-91); A. Oskam (Netherlands; 1991-97); W. B. Person (USA; 1983-89); C. J. H. Schutte (RSA; 1983-89); J. F. J. Todd (UK; 1989-91); S. Tsuchiya (Japan; 1987-93); C. Zhang (China; 1989-97); National Representatives: J. E. Collin (Belgium; 1985-97); S. M. Cabral de Menezes (Brazil; 1991-93); S. L. Spassov (Bulgaria; 1987-93); J. Lu (China; 1981-89); E. Hadjoudis (Greece; 1991-93); G. Varsfmyi (Hungary; 1985-93); M. Chowdhury (India; 1986-95); Z. Luz (Israel; 1983-93); S. Califano (Italy; 1989-97); S. Ng (Malaysia; 1983-93); A. Oskam (Netherlands; 1989-91); B. Jezowska-Trzebiatowska (Poland; 1983-89); J. J. C. Teixeira-Dias (Portugal; 1991-97); Y. S. Lee (Korea; 1989-97); T. A. Ford (RSA; 1987-91); C. Pascual (Spain; 1987-91); D. Escolar (Spain; 1991-97); J. Kowalewski (Sweden; 1994-95); M. Cebe (Turkey; 1987-91); S. Suzer (Turkey; 1991-93, 1995-97); S. IFli (Turkey; 1993-95); T. Ast (Yugoslavia; 1989-93); R. K. Harris (UK, 1993-97). Republication or  reproduction of this report or  its storage and/or dissemination by electronic means is permitted without the need for  formal IUPAC permission on condition that an acknowledgement, with full reference to the source along with use of the copyright symbol 0, the name IUPAC and the year of publication are prominently visible. Publication of a translation into another language is subject to the additional condition of prior approval from the relevant IUPAC National Adhering Organization. Notations and conventions in molecular spectroscopy: Part 3. Permutation and permutation-inversion symmetry notation (IUPAC Recommendations 1997) Abstract -- The field of Molecular Spectroscopy was surveyed in order to determine a set of conventions and symbols which are in common use in the spectroscopic literature. This document, which is Part 3 in a series, deals with symmetry notation referring to groups that involve nuclear permutations and the inversion operation. Further parts will follow, dealing inter alia with vibration-rotation spectroscopy and electronic spectroscopy. PREAMBLE The sub-committee on Notations and Conventions in Molecular Spectroscopy was initiated by Commission 1.5, the Commission on Molecular Structure and Spectroscopy, to establish and recommend a set of conventions and symbols from those that are in common use in the spectroscopic literature. Part 1 of the series is titled General Spectroscopic Notation [l], and Part 2 is titled Symmetry Notation [2]. The present part deals with symmetry notation referring to groups that involve nuclear permutations and the inversion operation. The recommendations follow the notation of Bunker [3], and in particular the symmetry groups involved will be called Molecular Symmetry Groups after the work of Longuet-Higgins [41. Part 3. PERMUTATION AND PERMUTATION-INVERSION SYMMETRY NOTATION 3.1 General Comments 3.2 Symmetry Operations for the Molecular Symmetry Groups 3.3 The Molecular Symmetry Groups 3.4 Representations for Molecular Symmetry Groups Appendix of Examples 3.1 GENERAL COMMENTS For a given molecule the group consisting of the identity and all permutations of the space and spin coordinates of identical nuclei is called its Complete Nuclear Permutation Group. The direct product of the Complete Nuclear Permutation Group with the group {E,  E*} ,  where E is the identity and E* the inversion of the space coordinates of all nuclei and electrons in the molecular centre of mass, is called its Complete Nuclear Permutation Inversion Group. The subgroup of the Complete Nuclear Permutation Inversion Group of a molecule, obtained by deleting all unfeasible elements, is called its Molecular Symmetry Group; the elements in the Molecular Symmetry Group are said to be feasible. An unfeasible element is one that interconverts nuclear-numbered equilibrium forms of the molecule when these forms are separated by an insuperable barrier in the potential energy surface; an insuperable barrier is one that does not allow observable tunnelling to occur through it at the energy resolution of the experiment being performed (i.e., no energy level shifts or splittings resulting from tunnelling are detected). Tunnelling may not be observable if we use a low resolution experiment yet it may be apparent if we use a high resolution experiment; the Molecular Symmetry Group that we use to analyze the results will then be different for the two cases since the elements associated with the tunnelling are feasible in the latter (high resolution) case but not in the former (low resolution) case. 1652 0 1997 IUPAC Permutation and permutation-inversion symmetry notation 1653 For molecules that have a non-linear equilibrium configuration, and for which there are no observable effects on the energy levels from tunnelling motions, the Molecular Symmetry Group is isomorphic to the Molecular Point Group and each element of the Molecular Symmetry Group has the same effect on the vibronic coordinates as its partner in the Molecular Point Group. In general, the Molecular Symmetry Group is not isomorphic to any point group. For linear molecules, and molecules having coaxial internal rotors and observable tunnelling effects, it is sometimes useful to use a group OK higher order than the Molecular Symmetry Group (see, Hougen [5] and Bunker and PapouSek [6]), and such a group is called an Extended Molecular Symmetry Group. 3.2 SYMMETRY OPERATIONS FOR THE MOLECULAR SYMMETRY GROUPS 3.2.1 Identity operation E 3.2.2 The pair permutation (or exchange) of the space and ( i j )  spin coordinates of identical nuclei labelled i and j 3.2.3 The cyclic permutation of the space and spin coordinates of identical nuclei labelled i, j ,  k, ..., m, n where nucleus i is replaced by nucleus j ,  j by k, ..., m by n, and n by i. After such an operation the space and spin coordinates of nucleus j will be the same as those that nucleus i had initially, those of k will be the same as those that j had initially, ..., those of n will be the same as those that m had initially, and those of i will be the same as those that n had initially. 3.2.4 The inversion of the space coordinates of all nuclei and electrons of the molecule in the molecular centre of mass The product of the permutation (ijk ... mn) and the inversion E*. The operation of rotating one internal rotor through 2n radians in a molecule with coaxial internal rotors for which it is necessary to use an Extended Molecular Symmetry Group. Operations of the Extended Molecular Symmetry Group of a linear molecule involving a change in the Euler angle x of the isomorphic Hamiltonian [3,6] 3.2.5 3.2.6 3.2.7 (ijk., .mn) E* (ijk.. .mn)* E’ 3.3 THE MOLECULAR SYMMETRY GROUPS 3.3.1 For a non-linear molecule with no observable tunnelling effects the Molecular Symmetry Group is isomorphic to the Molecular Point Group, and we recommend that the same symbol be used for it; in cases where it is necessary to distinguish the Molecular Symmetry Group from the Molecular Point Group we recommend adding (M) after the Point Group name, e.g. C2,(M) for the Molecular Symmetry Group of the water molecule. For linear molecules with no observable tunnelling effects the symbol for the Molecular Symmetry Group should be C,, (M) or Dmh(M) and the symbol for the Extended Molecular Symmetry Group should be C,, (EM) or &,(EM) as the Molecular Point Group is C,, or &, , respectively. 0 1997 IUPAC, Pure andApplied Chemistry69.1651-I657 1654 3.3.2 3.3.3 COMMISSION ON MOLECULAR STRUCTURE AND SPECTROSCOPY The Molecular Symmetry Group of a tunnelling molecule having order n. To distinguish between non-isomorphic groups having the same number of elements, no accepted general convention has yet emerged. Gn The m-fold Extended Molecular Symmetry Group of a tunnelling molecule for which the Molecular Symmetry group is G,,. G,(~)(EM) 3.4 REPRESENTATIONS FOR MOLECULAR SYMMETRY GROUPS 3.4.1 For a non-tunnelling non-linear molecule the same notation for the irreducible representations is to be used as in the isomorphic Molecular Point Group. 3.4.2 For a tunnelling molecule the conventions of 2.3.1,2.3.2, 2.3.4,2.3.10 and 2.3.12 of [2] apply. One-dimensional representations are labelled A or B, two-dimensional representations are labelled E following 2.3.3 (c) of [2], and recommendations 2.3.3 (d)-(f) of [2] also apply for the labelling of higher dimensioned representations. REFERENCES 1 C. J. H. Schutte, J. E. Bertie, P. R. Bunker, J. T. Hougen, I. M. Mills, J. K. G. Watson and B. P. Winnewisser,(Part l), Pure Appl. Chem.,this issue (1997). 2 C. J. H. Schutte, J. E. Bertie, P. R. Bunker, J. T. Hougen, I. M. Mills, J. K. G. Watsonand B. P. Winnewisser,(Part 2), Pure Appl. Chem., this issue (1997). 3 P. R. Bunker, Molecular Symmetry andSpectroscopy, Academic Press, New York, 1979. 4 H. C. Longuet-Higgins, Mol. Phys., 6 (1963) 445. 5 J. T. Hougen, Can. J.  Phys., 42 (1964) 1920. 6 P. R. Bunker and D. PapouSek, J.  Mol. Specrrosc., 32 (1969) 419. 0 1997 IUPAC, Pure and Applied Chemistry69, 1651-1657 Permutation and permutation-inversion symmetry notation 1655 APPENDIX OF EXAMPLES 1. The methane molecule The methane molecule is an example of a nonlinear molecule for which there is no observable tunnelling (i.e. no inversion splittings are seen in the spectrum). We number the protons 1 through 4 and the Complete Nuclear Permutation Group consists of all possible proton permutations; this group has 4!=24 elements. The Complete Nuclear Permutation Inversion Group consists of 4!x2=48 elements (see Table I). Because the effects of inversion tunnelling are not observed in the spectrum, elements such as E*, (12), (1423), (123)* and (14)(23)* are unfeasible; there are 24 unfeasible elements. Removing these 24 unfeasible elements from the Complete Nuclear Permutation Inversion Group gives the Molecular Symmetry Group of methane. This group of 24 elements is isomorphic to the Molecular Point Group Td and is called Td(M) (see Table 11). 2. The ethane molecule The ethane molecule is an example of a molecule for which tunnelling effects are observed. We label the carbon nuclei 7 and 8, the three protons attached to C-7 are labelled H-1, H-2 and H-3, and the three protons attached to C-8 are labelled H 4 ,  H-5 and H-6. The Complete Nuclear Permutation Inversion Group consists of 2!x6!~2=2880 elements. The molecule has a staggered equilibrium configuration but the effects of torsional tunnelling are observed. Thus, while elements such as (12), (78), (14) and E* are unfeasible, elements such as (123) and (456) are feasible. There are 36 feasible elements and these constitute the Molecular Symmetry Group of ethane; the group is called G36 (see Table 111). If we wish to classify the rotational and torsional wavefunctions separately then it is necessary to use the Extended Molecular Symmetry Group G36(EM) (see Table A-25 of ref. 3). Neither G36 nor G3,(EM) is isomorphic to a point group. 3. The carbon dioxide molecule The carbon dioxide molecule is an example of a linear molecule. Labelling the oxygen nuclei 1 and 2 the Complete Nuclear Permutation Inversion Group is the group of four elements {E,  (12), E*, (12)*}. These are all feasible elements and thus the Molecular Symmetry Group also consists of these four elements (see Table IV). This group is not isomorphic to the Molecular Point Group Dmh. If we wish to classify the rotational and vibronic wavefunction separately, it is necessary to use the Extended Molecular Symmetry Group &(EM) which is isomorphic to the Molecular Point Group D,h (see Table V). In the group D,h(EM) the operation (12); has the same effect on the vibronic variables as the point group operation i, and the operation (1 2);+& has the same effect as the point group operation S",'". 0 1997 IUPAC, Pure and Applied Chemistry69, 1651-1657 1656 COMMISSION ON MOLECULAR STRUCTURE AND SPECTROSCOPY TABLE I. The character table of the group G48, the Complete Nuclear Permutation Inversion Group for methane (CH4). E (123) (14)(23) (1423)* (23)* E* (123)* (14)(23)* (1423) (23) 1 8 3 6 6 1 8 3 6 6 A1+: 1 1 1 1 1 1 1 1 1 1 A;: 1 1 1 -1 -1 1 1 1 -1 -1 E': 2 -1 2 0 0 2 -1 2 0 0 F,': 3 0 -1 1 -1 3 0 -1 1 -1 F;: 3 0 -1 -1 1 3 0 -1 -1 1 A1-r 1 1 1 1 1 -1 -1 -1 -1 -1 A2-1 1 1 1 -1 -1 -1 -1 -1 1 1 E-:  2 -1 2 0 0 -2 1 -2 0 0 Fl-: 3 0 -1 1 -1 -3 0 1 -1 1 F2-z 3 0 -1 -1 1 -3 0 1 1 -1 TABLE 11. The character table of Td(M), the Molecular Symmetry Group for methane (CH4). A, : 1 1 1 1 1 A2 : 1 1 1 -1 -1 E: 2 -1 2 0 0 F,: 3 0 -1 1 -1 F2 3 0 -1 -1 1 0 1997 IUPAC, Pure and Applied Chernistry69, 1651-1657 Permutation and permutation-inversion symmetry notation 1657 TABLE 111. The character table of G3,, the Molecular Symmetry Group for ethane (c2H6). E (123)(456) (14)(26)(35)(78)* (123)(465) (132) (142635)(78)* (14)(25)(36)(78) (142536)(78) (23)(56)* 1 2  3 2 4 6 3 6 9 Al: 1 1 A,: 1 1 A3: 1 1 Ad: 1 1 El: 2 2 E,: 2 2 E3: 2 -1 E,: 2 -1 G: 4 -2 1 1 -1 -1 2 -2 0 0 0 1 1 1 1 1 1 1 1 -1 -1 -1 -1 2 -1 2 -1 -2 1 1 1 -1 -1 -1 1 0 0 0 1 1 -1 -1 1 1 -1 -1 0 0 0 0 2 -1 -2 1 0 0 1 -1 -1 1 0 0 0 0 0 TABLE IV. The character table Of &h(M), the Molecular Symmetry Group for carbon dioxide (C02). (+S)  cg+: 1 1 1 1 (-a) Zg-: 1 -1 -1 1 (+a) c:: 1 -1 1 -1 (-s) xu-: 1 1 -1 -1 TABLE V. The character table of &,(EM), the Extended Molecular Symmetry Group for carbon dioxide (COz). (+s) cg+: (-a) Zg-: (+a) xu+: (-s) xu-: ng: nu: Ag: Au: 1 1 1 1 1 1 1 1 2 2COS(&) 2 2COS(&) 2 2COS(2&) 2 2COS(2&) 0 1997 IUPAC, Pure and Applied Chemistry69, 1651-1657 ... 1 ... -1 ... 1 ... -1 ... 0 ... 0 ... 0 ... 0 ... 1 1 1 1 -1 -1 -1 -1 2 2COS(&) -2 -2COS(&) 2 2COS(2&) -2 -2COS(2&) ... 1 .*. -1 ... -1 ... 1 ... 0 ... 0 ... 0 ... 0 ... 

Notations and conventions in molecular spectroscopy: part 3. Permutation and permutation-inversion symmetry notation

Notations and conventions in molecular spectroscopy: part 2. Symmetry notation

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