Internal spaces, kinematic rotations, and body frames for four-atom systems

This paper is concerned with the structural transition dynamics of the six-atom Morse cluster with zero total angular momentum, which serves as an illustrative example of the general reaction dynamics of isolated polyatomic molecules. It develops a methodology that highlights the interplay between the effects of the potential energy topography and those of the intrinsic geometry of the molecular internal space. The method focuses on the dynamics of three coarse variables, the molecular gyration radii. By using the framework of geometric mechanics and hyperspherical coordinates, the internal motions of a molecule are described in terms of these three gyration radii and hyperangular modes. The gyration radii serve as slow collective variables, while the remaining hyperangular modes serve as rapidly oscillating "bath" modes. Internal equations of motion reveal that the gyration radii are subject to two different kinds of forces: One is the ordinary force that originates from the potential energy function of the system, while the other is an internal centrifugal force. The latter originates from the dynamical coupling of the gyration radii with the hyperangular modes. The effects of these two forces often counteract each other: The potential force generally works to keep the internal mass distribution of the system compact and symmetric, while the internal centrifugal force works to inflate and elongate it. Averaged fields of these two forces are calculated numerically along a reaction path for the structural transition of the molecule in the three-dimensional space of gyration radii. By integrating the sum of these two force fields along the reaction path, an effective energy curve is deduced, which quantifies the gross work necessary for the system to change its mass distribution along the reaction path. This effective energy curve elucidates the energy-dependent switching of the structural preference between symmetric and asymmetric conformations. The present methodology should be of wide use for the systematic reduction of dimensionality as well as for the identification of kinematic barriers associated with the rearrangement of mass distribution in a variety of molecular reaction dynamics in vacuum.

Section: ͑26͒mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Gyration-radius dynamics in structural transitions of atomic clusters

Yanao

Koon

Marsden

et al. 2007

“…This paper proposes a complementary method for mode analysis based on a concise expression for kinetic energy of an n-body system. While the potential energy surfaces are believed to govern the dynamics of molecules, the kinetic energy can also play an important role via the non-Euclidean mass matrix or metric tensor on the molecular internal space [22][23][24][25] and deserves closer attention. [26][27][28] Moreover, while the potential energy surface changes from system to system, the expression of the kinetic energy is independent of the system once mass-weighted coordinates are used, indicating that the roles of kinetic energy are of rather universal nature.…”

Section: Introductionmentioning

confidence: 99%

Intramolecular energy transfer and the driving mechanisms for large-amplitude collective motions of clusters

Yanao

Koon

Marsden

2009

This paper uncovers novel and specific dynamical mechanisms that initiate large-amplitude collective motions in polyatomic molecules. These mechanisms are understood in terms of intramolecular energy transfer between modes and driving forces. Structural transition dynamics of a six-atom cluster between a symmetric and an elongated isomer is highlighted as an illustrative example of what is a general message. First, we introduce a general method of hyperspherical mode analysis to analyze the energy transfer among internal modes of polyatomic molecules. In this method, the ͑3n −6͒ internal modes of an n-atom molecule are classified generally into three coarse level gyration-radius modes, three fine level twisting modes, and ͑3n −12͒ fine level shearing modes. We show that a large amount of kinetic energy flows into the gyration-radius modes when the cluster undergoes structural transitions by changing its mass distribution. Based on this fact, we construct a reactive mode as a linear combination of the three gyration-radius modes. It is shown that before the reactive mode acquires a large amount of kinetic energy, activation or inactivation of the twisting modes, depending on the geometry of the isomer, plays crucial roles for the onset of a structural transition. Specifically, in a symmetric isomer with a spherical mass distribution, activation of specific twisting modes drives the structural transition into an elongated isomer by inducing a strong internal centrifugal force, which has the effect of elongating the mass distribution of the system. On the other hand, in an elongated isomer, inactivation of specific twisting modes initiates the structural transition into a symmetric isomer with lower potential energy by suppressing the elongation effect of the internal centrifugal force and making the effects of the potential force dominant. This driving mechanism for reactions as well as the present method of hyperspherical mode analysis should be widely applicable to molecular reactions in which a system changes its overall mass distribution in a significant way.

“…A geometrical analysis of these singularities is described in Ref. 12. The corresponding hyperspherical harmonics, which are eigenfunctions of the kinetic energy operator at fixed hyper-radius, provide a complete orthonormal basis which satisfies appropriate boundary conditions at those poles and therefore constitutes an attractive alternative to the arrangement channel hyperspherical coordinate basis on which to expand the wave function of the system.…”

Section: Introductionmentioning

confidence: 99%

Numerical generation of hyperspherical harmonics for tetra-atomic systems

Lepetit

Wang

Kuppermann

2006

A numerical generation method of hyperspherical harmonics for tetra-atomic systems, in terms of row-orthonormal hyperspherical coordinates-a hyper-radius and eight angles-is presented. The nine-dimensional coordinate space is split into three three-dimensional spaces, the physical rotation, kinematic rotation, and kinematic invariant spaces. The eight-angle principal-axes-of-inertia hyperspherical harmonics are expanded in Wigner rotation matrices for the physical and kinematic rotation angles. The remaining two-angle harmonics defined in kinematic invariant space are expanded in a basis of trigonometric functions, and the diagonalization of the kinetic energy operator in this basis provides highly accurate harmonics. This trigonometric basis is chosen to provide a mathematically exact and finite expansion for the harmonics. Individually, each basis function does not satisfy appropriate boundary conditions at the poles of the kinetic energy operator; however, the numerically generated linear combination of these functions which constitutes the harmonic does. The size of this basis is minimized using the symmetries of the system, in particular, internal symmetries, involving different sets of coordinates in nine-dimensional space corresponding to the same physical configuration.