Ohio State International Symposium June 20-24, 2011 1
Notation confusion of symmetry species for molecules with several large-amplitude
internal motions
Peter Groner
Department of Chemistry, University of Missouri - Kansas City, Kansas City, MO 64110, USA
Confusion ?
D2h G18 C3C3v A1E E4 G D3d Q F1 Cs B2g EE+ A D3d Q F1 Cs B2g EE+ A D2h G18 C3C3v A1E E4 G
Cs B2g EE+ A D2h G18 C3C3v A1E E4 G D3d Q F1
G D3d Q F1 Cs B2g EE+ A D2h G18 C3C3v A1E E4
C3C3v A1E E4 G D3d Q F1 Cs B2g EE+ A D2h G18
EE+ A D2h G18 C3C3v A1E E4 G D3d Q F1 Cs B2g
A1E E4 G D3d Q F1 Cs B2g EE+ A D2h G18 C3C3v
F1 Cs B2g EE+ A D2h G18 C3C3v A1E E4 G D3d Q
Ohio State International Symposium June 20-24, 2011 2
Confusion in everything?
Ohio State International Symposium June 20-24, 2011 3
Confusion through sin
Confusion through language
Confusion through notation
Grave sins, lesser sins, and customer serviceSin 1 No explanation or reference
1987 Research Group 1 Symmetry group C3vC3v ; use of AA, EE, AE,
EA notation.
No reference to character table, notation.
1988 Research Group 2
Use of AA, EE, AE, EA notation. No reference to symmetry group, character table, notation.
2010 Paper reviewed “Some lines ... appeared clearly as AA, AE, EE, and EE* quartets.”
No reference to symmetry group, character table, notation.
Sin 2 Selective or no citation
Lesser sins Missing information
1987 Research Group 3 Use of PI group with explanation of everything except, in the discussion,“...AE states (as opposed to AA states or EE states)...”
Connection between AE, AA, EE notation and PI group notation
Customer service I would have liked to see also
1965 J. K. G. WatsonCan. J. Phys. 43 (1965) 1996“The correlation of the symmetry classifications of states of nonrigid molecules”
Use of MS group with reference to everything. Cross-reference for irreducible representations with notation used in three earlier papers (1959, 1959, 1960), all different, two of them referenced.
Ohio State International Symposium June 20-24, 2011 4
Confusion through language
Ohio State International Symposium June 20-24, 2011 5
What do we mean when we say “It’s the same group as ...” ?
Groups are the same when they have the same order and the
* same operators (which ones?) and their notation
Are (14)(25)(36)(ab) for CH3CH3 and (14)(25)(36)(ab)(c) for CH3OCH3 the same operators?
* same partition of operators into classes
* same character table
Are C2h, C2v and D2 the same group or different groups?
* same abstract group (which means that groups are isomorphic)
* same type of large-amplitude motions
Are groups for CH3COOCH3 (2 LAMs) and CH3SiH3 (1 LAM) the same?
Confusion through notation
Ohio State International Symposium June 20-24, 2011 6
Notations for Groups and Irreducible Representations (“Symmetry species”)
Examples for molecules with two methyl rotors
Group of order 36: G36 (molecules like acetone CH3COCH3)
Groups of order 18: G18 (which one?)
* Cs(n) (Cs symmetry, non-equivalent rotors, rotor axes in symmetry plane)
(molecules like methyl silyl ether CH3OSiH3)
* Cs(e) (Cs symmetry, equivalent rotors, rotor axes symmetric to symmetry plane)
(molecules like 2-fluoropropane (CH3)2CHF
* C2(e) (C2 symmetry, equivalent rotors, rotor axes symmetric to C2 axis)
(molecules like dimethyl disulfide CH3SSCH3)
Confusion through notation G36
Ohio State International Symposium June 20-24, 2011 7
Two methyl rotors: group of order 36
C2v(e) (C2v symmetry, equivalent internal rotors with axes in one symmetry plane)
Group Reference Irreducible representations Used for
(C3vC
3v) 1959 KM
AAA
QAE
E1
EE
E3
EECH
3OCH
3
(C3vC
3v) 1959 SC A Bx
By
Bz Q E
1E
2E
3E
4CH
3COCH
3
C3v
–C3v
+ 1960 MW1961 D
A1A
1A
1A
2A
2A
1A
2A
2 EE EA1
EA2
EA1
EA2
CH3OCH
3
CH3SCH
3
(G6G
6) 1963 LH A
1A
3A
2A
4 G E3
E4
E1
E2
CH3CH
3
G36 1965 W A
1A
3A
2A
4 G E3
E4
E1
E2
CH3COCH
3
D3D
3 1976 DLG 00++ 00+– 00–+ 00- - 01 11+ 11– 12+ 12– CH3OCH
3
[33]C2v 1993 G
KM Kasai, Myers J. Chem. Phys. 30 (1959) 1096 SC Swalen, Costain J. Chem. Phys. 31 (1959) 1562MW Myers, Wilson J. Chem. Phys. 33 (1960) 186 D Dreizler Z. Naturforsch. A 16 (1961) 1354LH Longuet-Higgins Mol. Phys. 6 (1963) 445 W Watson Can. J. Phys. 43 (1965) 1996DLG Durig et al. J. Mol. Spectrosc. 62 (1976) 159 G Groner Spectrochim. Acta A 49 (1993) 1935
Confusion through notation G18 (#1)
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Two methyl rotors: groups of order 18
1. Cs(n) (Cs symmetry, non-equivalent internal rotors with axes in symmetry plane)
Group Reference Irreducible representations Used for
Gruppe 18 1961 D A B E
E
E
E
CH3S13CH
3
G18
1965 B2004 O
A1
AA
A2
AA
E1
EA
E2
AE
E3
EE
E4
EE
CH3C≡CSiH
3
CH3CONHCH
3
G18G18
1972 M1976 DLG1977 DGL
00+ 00– 01 10 11 12
CH3N=CHCH
3
CH3OCD
3
CH3CH
2SiH
3
Group 18 1974 LCA
1
AA
A2
AA
E1
AE
E2
EA
E3
EE+
E4
EE–
CH3OSiH
3
[33]Cs’ 1993 G
G18 2002 NH AA AA AE EA EE+
EE–
CH3CH
2SiH
3D Dreizler Z. Naturforsch. A 16 (1961) 477 B Bunker Mol. Phys. 9 (1965) 257O Ohashi et al. J. Mol. Spectrosc. 227 (2004) 28 M Meier et al. J. Chem. Phys. 57 (1972) 1219DLG Durig et al. J. Mol. Spectrosc. 62 (1976) 159 DGL Durig Lopata Chem. Phys. 21 (1977) 401LC Le Croix et al. J. Mol. Spectrosc. 53 (1974) 250 G Groner Spectrochim. Acta A 49 (1993) 1935NH Niide, Hayashi J. Mol. Spectrosc. 216 (2002) 61
Confusion through notation G18 (#2)
Ohio State International Symposium June 20-24, 2011 9
Two methyl rotors: groups of order 182. Cs(e) (Cs symmetry, equivalent internal rotors with axes symmetrical to symmetry plane)
Group Reference Irreducible representations Used forC
3C
3v1961 S AA
1AA
2E
0E E
0E AE E
0A
1E
0A
1E
0A
2E
aA
2dimethyl oxirane
C3
–C3v
+ 1961 D1991 MD
AA1
AA
AA2
AA
EaE
EaA
AEb
EbE
AEa
EbA
AEE
aE
a
EbE
b
EbA
1
EaE
b
EaA
1
EbE
a
EbA
2
EaE
b
EaA
2
EbE
a
(CH3)
2SO
(CH3)
2CHF
C3D
3 1977 GD 00+ 00– 01 01* 11 12+ 12+* 12– 12–* (CH3)
2NH
[33]Cs 1993 G
G18
(C3vC
3)
2002 OH A1
A2 E+ E– E A
1+ A
1– A
2+ A
2– (CH
3O)
2PCH
3
S Sage J. Chem. Phys. 35 (1961) 142 D Dreizler Z. Naturforsch. A 16 (1961) 1354MD Meyer, Dreizler J. Mol. Spectrosc. 148 (1991) 310 GD Groner, Durig J. Chem. Phys. 66 (1977) 1856G Groner Spectrochim. Acta A 49 (1993) 1935 OH Ohashi, Hougen J. Mol. Spectrosc. 211 (2002)
119
Confusion through notation G18 (#3)
Ohio State International Symposium June 20-24, 2011 10
Two methyl rotors: groups of order 18
3. C2(e) (C2 symmetry, equivalent internal rotors with axes symmetrical to C2 axis)
Group Reference Irreducible representations Used for
C3v
–C3
+ 1961 D1992 Me
A1A
AA
A2A
AA
EEa
AEb
EbA
EEb
AEa
EaA
A1E
a
EaE
a
A1E
b
EbE
b
A2E
a
EaE
a
A2E
b
EbE
b
EAE
aE
b
EbE
a
CH3SSCH
3
CH3SSCH
3
C3D
3 1977 GD 00+ 00– 01 01* 11+ 11+* 11– 11–* 12t-dimethyl
cyclopropane[33]C
2 1993 G
2004 GG
1,
2) =
(0,0) (0,1) (1,1) (1,2) CH3SeSeCH
3
D Dreizler Z. Naturforsch. A 16 (1961) 1354 Me Meyer J. Mol. Struct. 273 (1992) 99GD Groner, Durig J. Chem. Phys. 66 (1977) 1856 G Groner Spectrochim. Acta A 49 (1993) 1935GG Groner et al. J. Mol. Spectrosc. 226 (2004) 169 [ERHAM]
1. Notation confusion is real.2. Correlation to single rotor notation is essential.
Three different types of notations are used based onA group = direct product of smaller groups
convenient for groups that are direct products (most MS groups are not)correlation not obvious, sometimes counterintuitive
or B group = semidirect product of “permutation subgroup” and point groupmost MS groups contain an invariant “permutation subgroup”
C. M. Woodman, Mol. Phys. 19 (1970) 753-780correlation obvious b/c single rotor operators are in invariant subgroup
or C MS/PI group, letter indicating dimension of irreducible representationcorrelation not obvious
In my view, type B notation conveys the most information!
Diagnosis
Ohio State International Symposium June 20-24, 2011 11
AuthorsDefine everything carefullyCite origin of notation for group, irreducible representationsCross-reference notation for symmetry species to single rotor species or to reference
Referees and editorsCheck papers for compliance and insist on corrections/additionsDon’t gloss over things that you don’t understand completely
Anti-Confusion Convention (ACC)I propose to have a convention or a committee todraft recommendations about nomenclature, notation, presentation for symmetry groups of molecular systems with large-amplitude motions comparable to the Mulliken standard for point groups (Phys. Rev. 43 (1933) 279).
Anti-confusion measures
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For many MS groups, notation could be based on system B where
MS group = semidirect product of “permutation subgroup” and a point group
where “permutation subgroup” = direct product of cyclic groupsFine print: does not work ALL the time
AdvantagesThe order of the group is the product of the orders of the point group and all cyclic groups.The correlation between the irreducible representations of the MS group and the species of the individual internal rotors is very simple if an appropriate notation is used.Transformations of rotational basis functions are determined by the point group.
Next two tables: Examples for such notations
Proposal to Anti-Confusion Convention
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Notation system B
Ohio State International Symposium June 20-24, 2011 14
How could it look for symmetry groups?
MoleculeGroup order
Reference Notation long form Short 1 Short 2
CH3NO
2 12 1963 LH C3C
2vC
3C
2v[3]C
2v
CH3CH
2NO
2 12 (C3C
2)C
sC
3,2C
s[32]C
s
(H2O)
2 16 1977 Dy (C2C
2)C
2vC
2,2C
2v[22]C
2v
CH3COCH
3 36 1959 SC (C3C
3)C
2vC
3,3C
2v[33]C
2v
CH3P(OCH
3)
2 54 2002 OH (C3C
3C
3)C
sC
3,3,3C
s[333]C
s
CH3CON(CH
3)
2 54 2006 F (C3C
3C
3)C
s’ C
3,3,3C
s’ [333]C
s’
(H2O)
3 48 1993 Wl (C2C
2C
2)C
3hC
2,2,2C
3h[222]C
3h
XC(CH3)
3 162 1963 MA (C3C
3C
3)C
3vC
3,3,3C
3v[333]C
3v
B(CH3)
3 324 1963 LH (C3C
3C
3)D
3hC
3,3,3D
3h[333]D
3h
C(CH3)
4 1944 1963 LH (C3C
3C
3C
3)T
dC
3,3,3,3T
d[3333]T
d
CH3S13CH
3 18 1961 D (C3C
3)C
s’ C
3,3C
s’ [33]C
s’
CH3C≡CSiH
3 18 1965 B C3C
3vC
3C
3v[3]C
3v
Dy Dyke J. Chem. Phys. 66 (1977) 492 F Fujitake et al. J. Mol. Spectrosc. 236 (2006) 97Wl Wales J. Am. Chem. Soc. 115 (1993) 11180 MA Möller, Andresen J. Chem. Phys. 39 (1963) 17
Notation system B
Ohio State International Symposium June 20-24, 2011 15
Groups and irreducible representations for two methyl rotors
a Superscript numbers = symmetry numbers 1, 2 ; * denotes conjugate complex (separately degenerate reps)
b,c,d Superscript sign = sign of character for operations: (14)(25)(36), (14)(26)(35)*, (23)(56)*, resp.
Group Reference Group Irreducible representations a, b, c, d
C3C
3 1977 GD C3,3C
1 00 01 01* 10 10* 11 11* 12 12*
00 01 02 10 20 11 22 12 21
(rational notation) aa ae ae* ea e*a ee e*e* ee* e*e
C3D
3 1977 GD C3,3C
2 b 00+ 00– 01 01* 11+ 11+* 11– 11–* 12
C3v
–C3
+ 1992 Me AA AA AEb
AEa
EaE
aE
bE
bE
aE
aE
bE
bE
aE
b
C3D
3 1977 GD C3,3C
s c 00+ 00– 01 01* 11 12+ 12+* 12– 12–*
C3
–C3v
+ 1991 MD AA AA EaA AE
aE
aE
aE
aE
bE
bE
aE
aE
bE
bE
a
G18 1972 M C3,3C
s’ d 00+ 00– 01 10 11 12
Group 18 1974 LC AA AA AE EA EE+
EE–
G18 2004 O AA AA AE EA EE EE
D3D
3 1977 GD C3,3C
2v b,c 00++ 00+– 00–+ 00-- 01 11+ 11– 12+ 12–