Wavefunctions of symmetrical nanoparticles are considered making use of induced representation method. It is shown that when, at the same total symmetry, the order of local symmetry group decreases, additional quantum numbers are required for complete labelling of electron states. It is shown that the labels of irreducible representations of intermediate subgroups can be used for complete classification of states in the case of repeating IRs in symmetry adapted linear combinations. The intermediate symmetry approach is extended to singlet and triplet two-electron states making use of Mackey theorem on symmetrized squares of induced representations.
@article{bwmeta1.element.doi-10_2478_nsmmt-2013-0006,
author = {V.G. Yarzhemsky},
title = {Internal Symmetries and Additional Quantum Numbers for Nanoparticles},
journal = {Nanoscale Systems: Mathematical Modeling, Theory and Applications},
volume = {2},
year = {2013},
pages = {96-106},
zbl = {1273.81113},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_nsmmt-2013-0006}
}
V.G. Yarzhemsky. Internal Symmetries and Additional Quantum Numbers for Nanoparticles. Nanoscale Systems: Mathematical Modeling, Theory and Applications, Tome 2 (2013) pp. 96-106. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_nsmmt-2013-0006/
E. A. Lord , A. L. Mackay , S. Ranganathan. New geometries for new materials. Cambridge University Press, (2012). | Zbl 1256.52008
F.A. Cotton. Chemical application of group theory. Wiley (1990).
C.D.H. Chisholm. Group theoretical techniques in quantum chemistry. Academic Press (1976).
C.J. Ballhausen. Introduction to ligand field theory. N.Y. McGraw-Hill (1962). | Zbl 0113.23301
E.F. Kustov, V.I. Nefedov. Nanostructures: compositions, structure, and classification. Russian Journal of Inorganic Chemistry 53, pp. 2103-2170 (2008).
C.W. Curtis, I. Reiner, Representation theory of finite groups and associative algebras. American Mathematical Soc. (1962) | Zbl 0131.25601
A. Kartunen, M. Linnolahti , T.A. Pakkanen , P. Pyykko. Icosahedral Au72: a predicted chiral and spherically aromatic golden fullerene. Chem. Comm. 465, pp.465-467 (2008).
V.G. Yarzhemsky, C. Battocchio. The structure of gold nanoparticles and Au based thiol self-organized monolayers. Russian J. Inorganic Chemistry 56, pp. 2147-2159 (2011).
V.G. Yarzhemskii, E.N. Murav’ev. Orbits and induced representations in the quantum chemistry of nanostructures. Russian J. Inorganic Chemistry 54, pp. 1273-1276 (2009).
G. Racah. Group Theory and Spectroscopy. Springer Tracts in Modern Physics, 37, pp 28-84 (1965).
D.J. Klein. Semiregular induction of group representations. J. Math. Phys. 25, pp.200-203 (1984). | Zbl 0569.20014
V.P. Smirnov, R.A. Evarestov. Bulletin of Leningrad State University 22, pp. 107-111 (1981) (in Russian).
V.G. Yarzhemsky , E.N. Murav’ev. Induced representation method in the theory of molecular orbitals. Doklady USSR Academy of Sciences 278, pp. 945-948 (1984) (in Russian).
J. des Cloizeaux. Orthogonal orbitals and generalized Wannier functions. Phys. Rev. 129, pp.554-566 (1963). | Zbl 0113.23004
R.A. Evarestov, M.V. Losev. All-electron LCAO calculations of the LiF crystal phonon spectrum: influence of the basis set, the exchange correlation functional, and supercell size. J. Comput. Chem. 30, pp. 2645-2655 (2009)
R.A. Evarestov. Quantum Chemistry of Solids Springer-Verlag Berlin Heidelberg, (2007)
G. W. Mackey. Symmetric and antisymmetric Kronecker squares and intertwinning numbers of induced representations of finite groups. Amer. J. Math. 75 pp. 387-405.(1953). | Zbl 0051.25903
C. J. Bradley, B. L. Davies. Kronecker products and symmetrized squares of irreducible representations of space groups. J. Math. Phys. 11, pp. 1536-1552 (1970) | Zbl 0201.03403
V.G. Yarzhemsky. Mackey theorem and two-electron wave function of a multi-centre system. Few-Body Systems 22, pp. 27-36 (1997).
V.G. Yarzhemsky Nodal quantum numbers for two-electron states in solids. Few-Body Systems 53, pp. 499-504 (2012)
V.G. Yarzhemsky, E.N. Murav0ev Space group approach to the wavefunction of a Cooper pair J. Phys: Cond. Matter. 4, pp. 3525-3532(1992).
V.G. Yarzhemsky. Space-group approach to the nodal structure of superconducting order parameter in UPt3. Physica Status Solidi (B) 209, pp. 101-107 (1998)
T. Micklitz, M.R. Norman. Odd parity and line nodes in nonsymmorphic superconductors. Phys. Rev. B 80, 100506(R)(2009)
D.J. Newman. Symmetry properties of direct sum systems J. Phys. A: Math. Gen. 16, pp. 2375-23887 (1983)
S. L. Altman, Induced Representations for Crystals and Molecules. New York, Academic (1977).
D.J. Newman B. Ng. Intermediate symmetry, superposition model and induced representations. Molecular Physics 61, p. 1443-1453 (1987).
A.P. Cracknell. Applied group theory. Pergamon 417 p. (1968) | Zbl 0176.55101
L.D. Landau, E.M. Lifshits. Quantum Mechanics Non-Relativistic Theory (3ed), Pergamon) 691p (1991).
J.H. Macek , G.H. Duffey. Bonding in icosahedral complexes. J. Chem. Phys. 34, pp. 288-290 (1961)
V.G. Yarzhemsky, Yu.V. Norov, S.V. Murashov, C. Battocchio, I.Fratoddi, I.Venditi, G.Polzonetti. Modeling of interaction between gold nanoparticles and thiols. Inorganic Materials 46, pp.924-930. (2010)
V.G. Yarzhemsky, E. N. Murav’ev, M.A. Kazaryan,Yu. A. Dyakov. Electronic structure of gold nanoparticles. Inorganic Materials 48, pp. 1075-1077 (2013).
S. Bernadotte, F. Evers , C. R. Jacob. Plasmons in Molecules. The Journal of Physical Chemistry C 117, pp.1863- 1878 (2013).