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)

Ohio State International Symposium June 20-24, 2011 8

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–

Ohio State International Symposium June 20-24, 2011 16

Thank you