Group theory and nuclear spin statistics of weakly‐bound (H2O)n, (NH3)n, (CH4)n, and NH+4(NH3)n

Nonrigid water octamer: Computations with the 8-cube

A 8-cube model of the fully nonrigid water octamer is considered within the 8-dimensional hyperoctahedral wreath product group with 10,321,920 operations and 185 irreducible representations by employing computational and mathematical techniques. For the two lowest-lying isomers of (H₂ O)₈ with D_2d and S₄ symmetries of a rigid (H₂ O)₈ , correlation tables and nuclear spin statistics are constructed for the tunneling splittings of the rotational levels are computed by a computational matrix polynomial generating function technique combined with Möbius inversion, and the relationship to the 8-cube multinomials are pointed out. Multinomial generating functions combined with the induced representation techniques are employed to compute and construct the nuclear spin species, nuclear spin statistical weights and tunneling splittings of rovibronic levels. We have also computed the spin statistical weights and tunneling splittings of the rotational levels for a semirigid water octamer within the wreath product O_h [S₂ ] consisting of 12,288 operations.

Computations of Colorings 7D-Hypercube's Hyperplanes for All Irreducible Representations

In the present study, we compute and enumerate the colorings of 7D-hypercube for all of its hyperplanes (q = 1-7) for all 110 irreducible representations (IRs) of the seventh-dimensional hyperoctahedral group consisting of 645,120 symmetry operations. The computations of colorings of the 7D-hypercube are motivated by a number of chemical and biological applications such as the 7D-hypercube representation of the periodic table, hypercube representations of water heptamer clusters, genetic regulatory networks, isomerization graphs, massively large data representations, and so forth. We have employed the Möbius inversion technique combined with generalized character cycle indices for 110 IRs to compute the generating functions for colorings of seven different types of hyperplanes of the 7D-hypercube varying from the vertices (q = 7) to hexeracts (q = 1) of the 7D-hypercube. Explicit computed tables are provided for 110 IRs from q = 1 to q = 7 for the 7D-hypercube. © 2019 Wiley Periodicals, Inc.

Infrared studies of ionic clusters: The influence of Yuan T. Lee

Beginning in the mid-1980s, a number of innovative experimental studies on ionic clusters emerged from the laboratory of Yuan T. Lee combining infrared laser spectroscopy and tandem mass spectrometry. Coupled with modern electronic structure calculations, this research explored many facets of ionic clusters including solvation, structure, and dynamics. These efforts spawned a resurgence in gas-phase cluster spectroscopy. This paper will focus on the major areas of research initiated by the Lee group and how these studies stimulated and influenced others in what is currently a vibrant and growing field.

Geodesic and re–coupling–induced limits to S 2 n group invariants of uniform ( k 1 ⋯ k 2 n ) dual tensorial sets in spin physics, or NMR: SR dynamical structure on {ℍ v } via polyhedral re–coupling and time–reversal invariance

For S n –based, uniform auxiliary dual tensorial (UDT) sets, explicitly with even maximal all–alike j i ( k i ) over v ≡ ( j 1 ⋯ j 2 n ), or ( k 1 ⋯ k 2 n ), sub–rank auxiliary labelling, the specific time–reversal invariance–dependent dual group invariants and their independent cardinalities (| SI | (2 n ) ) are fundamental to understanding simply reducible (SR) dynamical spin system structure over (Liouvillian) carrier space, a quantum physics property implicit in dual quasi–particle (QP) formalisms associated with, for example, uniform identical multiple spin NMR systems . Such UDT sets stand in total contrast to the distinct j i ( k i ) of ( j 1 ⋯ j″ n )(( k 1 ⋯ k″ n )) sub–rank orthogonal tensorial sets with their Z aa′ graph invariants which for reasons discussed in the text are of restricted validity. For UDTs as operator bases related to the (spin) interaction network–based tensors of high S n permutational NMR Hamiltonians (Liouvillians), the use of QP boson (superboson) formalisms and their dual projective maps (F. P. Temme 1993 Physica A 198 , 245–261) prove invaluable. Once the independent cardinality of the multiple invariants of these UDTs (alias the SU (2) × S 2 n group invariants) (here denoted | SI | (2 n ) ) are known in terms of democratic re–coupling (DR) and polyhedral combinatorial (PC) sampling applied to the time–reversal invariance (TRI) properties of the uniform multiple spin (sub–)system, so the SR dual carrier spaces , ℍ˜, ℍ˜ v follow directly as well–defined entities. For UDTs of general (2 n > 6)–fold ensembles, the augmented models of TRI discussed here (i.e. beyond linear re–coupled Weyl forms) are essential, due to the presence of non–Abelian degeneracy. A regular geometric automorphic limit is established for sequential (2 n )–based | SI | (2 n ) total series, specific to (higher) uniform ( j 1 ⋯ j i ), ( k 1 ⋯ k i )–based dual tensors. By treating the | SI | (2 n ) obtained from the DR–based TRIs as group measures and invoking an automorphic geodesic augmentation, certain otherwise inaccessible, additional | SI | (2 n′ ) (and hence UDTs) may be established; these are based on sporadic higher (regular) (2 n ≫ 12)–fold ensemble invariants and are obtained via finite group duality properties implicit in the concept of group measures. The PC DR–based views of general (2 n ) interacting spins of UDTs expressed here give specific insight into dynamic structure over (explicit TRI invariant–defined) Liouvillian carrier spaces. This lattice point (Erdösian) view of DR shows that UDT invariants are distinct and more contracted than predicted by the earlier linear–re–coupling–based | D 0 ( U )|(⨷ SU (2)) approach. The presence of degeneracy in general high (2 n ) UDTs as non–Abelian entities ensures that the homomorphic–based, | D 0 ( U )| enumeration (with its implied linear graphical re–coupling) is invalid for high (2 n ) uniform (NMR) multi–spin systems whose TRI properties are governed by DR re–coupling. The prime focus of the report concerns the positive conceptual value of PC lattice–point–based DR, as applied to the S 2 n –invariants which define UDTs (of NMR) and their completeness. Brief contrasts with the earlier, more restricted O (2 k + 1) (overall) invariants of ( O ( n ) ⊃ ⋯ ⊃ G )–based democratic re–coupled quantum systems are given for completeness.

Relativistic double group spinor representations of nonrigid molecules

The character theory of relativistic double group spinor representations is developed in order to represent the total rovibronic states of nonrigid molecules. It is shown that the double groups can be represented in terms of wreath products and powerful matrix cycle type generators that are used to construct their character tables. It is shown that these tables are of use when spin-orbit coupling is included in the Hamiltonian even for molecules containing lighter atoms. Applications to nonrigid molecules such as Tl2H4/Tl2H4+ are considered. It is shown that the tunneling splittings and the nuclear spin statistical weights can be obtained for such species using the character tables thus constructed. The spinor double groups of several other molecules such as hexamethyl dilead and heavy weakly bound clusters such as (PoH2)4 are also considered.