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.

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“…However, in the case of deuterium nuclei (bosons), the one-to-one correspondence between the nuclear-spin angular momentum and the permutation symmetry breaks down. A new method unifying the rotational and permutation symmetries was proposed recently (Schmiedt, Jensen & Schlemmer 2016).…”

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“…Schmiedt et al (2016) have recently shown that the spin angular momentum and the permutation symmetry are, after all, inherently coupled, and that it is possible to construct an unequivocal and practical representation for each spin angular momentum -symmetry species using the Young diagrams. In the present study, chemical species are distinguished solely by their nuclear spin symmetries.…”