Transformation Properties under the Operations of the Molecular Symmetry Groups G36 and G36(EM) of Ethane H3CCH3
Abstract
:1. Introduction
2. The Structure of the G36 Group
3. Irreducible Representations of G36
4. Representation Matrices for the Irreducible Representations of G
5. Representation Matrices for the G Irreducible Representation of G36
6. Generation of a Symmetry Adapted Basis Set for Ethane
6.1. Definition of the Internal Coordinates Used for Ethane
- the C−C bond length R;
- six C−H bond lengths , k = 1, 2, …, 6;
- six bond angles ∠(H-C-C) = , k = 1, 2, …, 6;
- four dihedral-angle coordinates , , , and ,
- the torsional angle ; and
- the three rotational angles .
6.2. Transformation of the Vibrational Coordinates under G36
6.3. The Extended Molecular Symmetry Group G36(EM) and the Transformation of the Torsional Coordinate
6.4. Rigid-Symmetric-Rotor Function Representations
6.5. Symmetrisation of the Basis Set
7. Potential Energy Function of Ethane in a Symmetry Adapted Representation
8. Numerical Example
8.1. Subset 2 Symmetrisation
8.2. Subset 4 Symmetrisation
8.3. Subset 5 Symmetrisation
8.4. Torsional Basis Function Symmetries
9. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Character Table of the Isomorphic Groups C and C
1 | 2 | 3 | |
---|---|---|---|
1 | 1 | 1 | |
1 | 1 | −1 | |
E | 2 | −1 | 0 |
Appendix B. The G36 Transformation Matrices
Appendix B.1. Ei Transformation Matrices in G36
Appendix B.1.1. One-Member Class Containing E
Appendix B.1.2. Two-Member Class Containing (123)(456) and (132)(465)
Appendix B.1.3. Three-Member Class Containing (14)(26)(35)(ab)*
Appendix B.1.4. Two-Member Class Containing (123)(456) and (132)(456)
Appendix B.1.5. Four-Member Class Containing (123) and (456)
Appendix B.1.6. Six-Member Class Containing (142635)(ab)*
Appendix B.1.7. Three-Member Class Containing (14)(25)(36)(ab)
Appendix B.1.8. Six-Member Class Containing (142536)(ab)
Appendix B.1.9. Nine-Member Class Containing (12)(45)*
Appendix B.2. G Transformation Matrices in G36
Appendix B.2.1. One-Member Class Containing E
Appendix B.2.2. Two-Member Class Containing (123)(456) and (132)(465)
Appendix B.2.3. Three-Member Class Containing (14)(26)(35)(ab)*
Appendix B.2.4. Two-Member Class Containing (123)(456) and (132)(456)
Appendix B.2.5. Four-Member Class Containing (123) and (456)
Appendix B.2.6. Six-Member Class Containing (142635)(ab)*
Appendix B.2.7. Three-Member Class Containing (14)(25)(36)(ab)
Appendix B.2.8. Six-Member Class Containing (142536)(ab)
Appendix B.2.9. Nine-Member Class Containing (12)(45)*
Appendix C. Derivation of the Transformation of Internal Coordinates
Appendix C.1. Bond Lengths
Appendix C.2. Bond Angles
Appendix C.3. Dihedral Angles
Appendix D. Transformation of τ under the Generating Operations of G36
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= = E | = (132)(456) | = (123)(465) | = (14)(25)(36)() | = (16)(24)(35)() | = (15)(26)(34)() |
---|---|---|---|---|---|
= | |||||
= (132)(465) | (123) | (456) | (162435) | (152634) | (142536) |
= (14)(26)(35) | (152436) | (163425) | (23)(56) | (12)(46) | (13)(45) |
= (16)(25)(34) | (13)(46) | (23)(45) | (12)(56) | ||
= (15)(24)(36) | (162534) | (143526) | (12)(45) | (13)(56) | (23)(46) |
E | (123)(456) | (14)(26)(35)() | (123)(465) | (123) | (142635)() | (14)(25)(36)() | (142536)() | (12)(45) | ||
---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 2 | 4 | 6 | 3 | 6 | 9 | ||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
1 | 1 | 1 | 1 | 1 | 1 | −1 | −1 | −1 | ||
1 | 1 | −1 | 1 | 1 | −1 | 1 | 1 | −1 | ||
1 | 1 | −1 | 1 | 1 | −1 | −1 | −1 | 1 | ||
2 | 2 | 2 | −1 | −1 | −1 | 0 | 0 | 0 | ||
2 | 2 | −2 | −1 | −1 | 1 | 0 | 0 | 0 | ||
2 | −1 | 0 | 2 | −1 | 0 | 2 | −1 | 0 | ||
2 | −1 | 0 | 2 | −1 | 0 | −2 | 1 | 0 | ||
G | 4 | −2 | 0 | −2 | 1 | 0 | 0 | 0 | 0 | |
E | ||||||
(16)(24)(34) | ||||||
(15)(26)(34) | ||||||
(153426) | ||||||
(162435) | ||||||
(143624) | ||||||
(152634) | ||||||
(163524) | ||||||
(142536) | ||||||
(123) | (23)(56) | |||||
(13)(46) | ||||||
(12)(45) | ||||||
(152436) | (12)(46) | |||||
(142635) | (23)(45) | |||||
(162534) | (13)(56) | |||||
(163425) | (13)(45) | |||||
(153624) | (12)(56) | |||||
(143526) | (23)(46) |
Transformed | Transformed | G(EM) Generator | G Generator |
---|---|---|---|
(123)(456) | |||
(132)(456) | |||
c | (14)(26)(35)() | ||
(14)(25)(36)() | |||
K | |
---|---|
0 (J even) | |
0 (J odd) | |
(K even) | |
(K odd) | |
(K even) | |
(K odd) | |
(K even) | |
(K odd) |
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Mellor, T.M.; Yurchenko, S.N.; Mant, B.P.; Jensen, P. Transformation Properties under the Operations of the Molecular Symmetry Groups G36 and G36(EM) of Ethane H3CCH3. Symmetry 2019, 11, 862. https://doi.org/10.3390/sym11070862
Mellor TM, Yurchenko SN, Mant BP, Jensen P. Transformation Properties under the Operations of the Molecular Symmetry Groups G36 and G36(EM) of Ethane H3CCH3. Symmetry. 2019; 11(7):862. https://doi.org/10.3390/sym11070862
Chicago/Turabian StyleMellor, Thomas M., Sergei N. Yurchenko, Barry P. Mant, and Per Jensen. 2019. "Transformation Properties under the Operations of the Molecular Symmetry Groups G36 and G36(EM) of Ethane H3CCH3" Symmetry 11, no. 7: 862. https://doi.org/10.3390/sym11070862
APA StyleMellor, T. M., Yurchenko, S. N., Mant, B. P., & Jensen, P. (2019). Transformation Properties under the Operations of the Molecular Symmetry Groups G36 and G36(EM) of Ethane H3CCH3. Symmetry, 11(7), 862. https://doi.org/10.3390/sym11070862