title: Unifying the rotational and permutation symmetry of nuclear spin states: Schur-Weyl duality in molecular physics.
abstract: In modern physics and chemistry concerned with many-body systems, one of the mainstays is identical-particle-permutation symmetry. In particular, both the intra-molecular dynamics of a single molecule and the inter-molecular dynamics associated, for example, with reactive molecular collisions are strongly affected by selection rules originating in nuclear-permutation symmetry operations being applied to the total internal wavefunctions, including nuclear spin, of the molecules involved. We propose here a general tool to determine coherently the permutation symmetry and the rotational symmetry (associated with the group of arbitrary rotations of the entire molecule in space) of molecular wavefunctions, in particular the nuclear-spin functions. Thus far, these two symmetries were believed to be mutually independent and it has even been argued that under certain circumstances, it is impossible to establish a one-to-one correspondence between them. However, using the Schur-Weyl duality theorem we show that the two types of symmetry are inherently coupled. In addition, we use the ingenious representation-theory technique of Young tableaus to represent the molecular nuclear-spin degrees of freedom in terms of well-defined mathematical objects. This simplifies the symmetry classification of the nuclear wavefunction even for large molecules. Also, the application to reactive collisions is very straightforward and provides a much simplified approach to obtaining selection rules.
authors.affiliation: I. Physikalisches Institut, Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany.
authors.forename: Hanno
authors.initials: H
authors.lastname: Schmiedt
authors.affiliation: Physikalische und Theoretische Chemie, Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany.
authors.forename: Per
authors.initials: P
authors.lastname: Jensen
authors.affiliation: I. Physikalisches Institut, Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany.
authors.forename: Stephan
authors.initials: S
authors.lastname: Schlemmer