Symmetry Groups Of Non-rigid Molecules Research Articles

Character tables of n-dimensional hyperoctahedral groups and their applications

ABSTRACTWe have shown that powerful matrix-type-based multinomial generators together with combinatorial techniques can be applied to derive the conjugacy classes, irreducible representations and the character tables of n-dimensional hyperoctahedral or hypercube groups of order n! × 2n, which are generalisations of non-rigid molecular theory formulated by Longuet-Higgins. The hyperoctahedral groups appear in the symmetry groups of non-rigid molecules, non-rigid water clusters, and other weakly bound van der Waals complexes that exhibit rapid tunnelling motion among various low-lying surmountable potential energy minima. These groups also have applications in quantum chromodynamics in the classification of states of quarks. The associated combinatorics has applications in enumerations, nuclear spin statistics and NMR spectroscopy. Finally, the groups enhance understanding of chirality in biosystems, that is, the functionality of intrinsically disordered proteins. We have derived the character tables of hyperoctahedral groups of sixth dimension, and techniques have been outlined to generalise for higher dimensions in the form of matrix generators.

The symmetry groups of non-rigid molecules

The concept of molecular symmetry is extended to molecules such as ethane and hydrazine which can pass from one conformation to another. The symmetry group of such a molecule is the set of (i) all feasible permutations of the positions and spins of identical nuclei and (ii) all feasible permutation-inversions, which simultaneously invert the coordinates of all particles in the centre of mass. According to the representations of this group one can classify not only the spin states and states of motions of the nuclei, but even the electronic states of the molecule. Examples are given to illustrate the use of this concept in determining the statistical weights of individual levels and selection rules for electric dipole transitions between them.

The symmetry groups of non-rigid molecules: a Lie algebraic and group contraction approach

The description of the symmetry of non-rigid molecules is explored through a Lie algebraic approach. It is shown that the abstract process of contraction to an appropriate Lie algebra corresponds to restriction of molecular motion to regions located around the minima of a global potential. The contracted algebra may be determined by recasting the Hamiltonian in terms of variables which vanish at global potential minima, and the resultant direct or semidirect product group structure accords with that proposed by other workers. This approach is applied to two familiar problems of molecular symmetry, namely inversion through a potential barrier and hindered rotation; the results derived confirm and extend those obtained by other methods.

Generators of the character tables of generalized wreath product groups

A method based on the cycle-type matrix algebra is devised to generate the character tables of generalized wreath products which are useful in describing the symmetry groups of non-rigid molecules, NMR groups, groups of non-rigid van der Waals complexes, the groups of space types in configuration interaction calculations, etc. This newly developed method is illustrated with examples.

Computer generation of the symmetry elements of nonrigid molecules

AbstractAlgorithms and computer programs are developed for generating the symmetry elements of nonrigid molecules. These programs are based on the wreath product formalism for the symmetry groups of nonrigid molecules developed by the present author. Several examples are given to illustrate the procedures. Applications to weakly bound van der Waal complexes synthesized by supersonic beam expansion are also presented.

Symmetry coordinates of nonrigid molecules

Motivated by Altmann’s definition of symmetry groups of nonrigid molecules, Wigner’s method of obtaining the symmetry coordinates of a molecule is extended to nonrigid molecules with free internal rotations. The molecule BF2 CH3 is exemplified.

The symmetry groups of nonrigid molecules as generalized wreath products and their representations

The symmetry groups of nonrigid molecules first defined by Longuet–Higgins, in the most general cases are generalized wreath product groups. We first outline the representation theory of wreath product groups. Then the representation theory of generalized wreath product groups is developed. Several illustrative examples of NMR groups are offered. The character tables of several nonrigid molecular groups are presented. An example of a nonrigid triphenyl molecule is given to illustrate the use of the generalized wreath product in deriving optical selection rules.

The isometric group of non-rigid molecules

A new method for derivation of symmetry groups of non-rigid molecules is presented, based on the concept of isometry of nuclear configurations defined by a certain number of internal (structural) coordinates. The substitution group of the internal coordinates interrelating isometric nuclear configurations generates the internal isometric group by both permutations and intransitive imprimitive substitution groups, and together with the covering group it generates the full isometric group. Examples for these groups are given. It is shown that the rotation-internal motion hamiltonian is symmetric with respect to the isometric group. The relation between various approaches to the isometric group is investigated; the role of automorphisms is pointed out. The new approach is shown to give a self-consistent procedure for the quantum mechanical problem associated with the rotation-internal motion hamiltonian.

On the symmetry groups of non-rigid molecules

A counter-example is produced to show that Altmann's representation of the group S of a non-rigid molecule as a semi-direct product of an ‘isodynamic’ group I and a point group G in the form S = I ^ G does not always hold in the Longuet-Higgins classification scheme. Altmann's proof of this relation requires reconsideration. However, the discrepancies between the two treatments in the case of boron trimethyl are due to Altmann's use of an operation of questionable validity called a ‘flip’, denoted by Z 2 I.

The symmetry groups of non-rigid molecules as semi-direct products

The possibility of formulating the symmetry groups of non-rigid molecules (as defined by Longuet-Higgins) as products of subgroups is investigated. For the most general case of non-rigidity, no such formulation is possible. When the non-rigidity results from internal rotation about one or more axes, the molecular symmetry group may be written as a semi-direct product of an invariant torsional subgroup, which is a direct product of cyclic groups, and a ‘frame subgroup’, which is isomorphous with the point group of the ‘frame’ formed by detaching all the nuclei undergoing internal rotation. A correlation table may be drawn up linking the representations of the torsional and frame subgroups with those of the molecular symmetry group. This table may be used to classify the various factors of the molecular wave-function, without the need to perform the tedious construction of the molecular symmetry group. The method is extended to molecules containing rotating groups which in turn contain more rotating groups ...

The summary of nonrigid molecules: the Schrödinger supergroup

The symmetry groups of nonrigid molecules possess two subgroups: one is that of all the symmetry operations of the Schrödinger group of the molecule and the other is a group of a new type of operation called an isodynamic operation. The total group, which is called the Schrödinger supergroup, is a semidirect product of the Schrödinger group and the isodynamic group. Examples are given for borondifluoride methyl, ethane, dimethylacetylene and borontrimethyl. It is shown that in certain cases the double groups of the isodynamic group have to be used. The results do not agree always with those recently given by Longuet-Higgins: a careful analysis of the nature of symmetry and isodynamic operations indicates the origin of the discrepancy.

Statistics and the symmetry groups of non-rigid molecules

It is shown that the use of the symmetry group of a non-rigid molecule leads to the same results for the statistical weights and symmetry number as do more conventional methods.

The symmetry groups of non-rigid molecules

The concept of molecular symmetry is extended to molecules such as ethane and hydrazine which can pass from one conformation to another. The symmetry group of such a molecule is the set of (i) all feasible permutations of the positions and spins of identical nuclei and (ii) all feasible permutation-inversions, which simultaneously invert the coordinates of all particles in the centre of mass. According to the representations of this group one can classify not only the spin states and states of motions of the nuclei, but even the electronic states of the molecule. Examples are given to illustrate the use of this concept in determining the statistical weights of individual levels and selection rules for electric dipole transitions between them.