Reactive Scattering with Row-Orthonormal Hyperspherical Coordinates. 2. Transformation Properties and Hamiltonian for Tetraatomic Systems

Computation and analysis of bound vibrational spectra of the neon tetramer using row orthonormal hyperspherical coordinates

This paper presents the first implementation of the row-orthonormal hyperspherical coordinate formalism for the computation of the vibrational spectrum of a tetratomic system. The wavefunction of Ne₄ is expanded on a large basis set of hyperspherical harmonics generated numerically. This method not only provides spectra with reasonable accuracy, but also gives physical insight into the vibrational dynamics of the system. The characteristics of the spectra are related to the symmetry and localization of the wavefunction in configuration space.

Roles of dynamical symmetry breaking in driving oblate-prolate transitions of atomic clusters

This paper explores the driving mechanisms for structural transitions of atomic clusters between oblate and prolate isomers. We employ the hyperspherical coordinates to investigate structural dynamics of a seven-atom cluster at a coarse-grained level in terms of the dynamics of three gyration radii and three principal axes, which characterize overall mass distributions of the cluster. Dynamics of gyration radii is governed by two kinds of forces. One is the potential force originating from the interactions between atoms. The other is the dynamical forces called the internal centrifugal forces, which originate from twisting and shearing motions of the system. The internal centrifugal force arising from twisting motions has an effect of breaking the symmetry between two gyration radii. As a result, in an oblate isomer, activation of the internal centrifugal force that has the effect of breaking the symmetry between the two largest gyration radii is crucial in triggering structural transitions into prolate isomers. In a prolate isomer, on the other hand, activation of the internal centrifugal force that has the effect of breaking the symmetry between the two smallest gyration radii is crucial in triggering structural transitions into oblate isomers. Activation of a twisting motion that switches the movement patterns of three principal axes is also important for the onset of structural transitions between oblate and prolate isomers. Based on these trigger mechanisms, we finally show that selective activations of specific gyration radii and twisting motions, depending on the isomer of the cluster, can effectively induce structural transitions of the cluster. The results presented here could provide further insights into the control of molecular reactions.

Full-dimensional time-dependent wave packet dynamics of H2 + D2 reaction

Collision induced dissociation (CID), four center reaction (4C), and single exchange reaction (SE) in H(2) (v(1) = high) + D(2) (v(2) = low) were studied by means of time-dependent wave packet approach within a full-dimensional model. Initial state-selected total reaction probabilities for the three competitive processes have been computed on two realistic global potential energy surfaces of Aguado-Suárez-Paniagua and Boothroyd-Martin-Keogh-Peterson (BMKP) with the total angular momentum J = 0. The role of both vibrationally excited and rotationally excited reagents was examined by varying the initial vibrational and rotational states. The vibrational excitation of the hot diatom gives an enhancement effect on the CID process, while the vibrational excitation of the cold diatom gives an inhibition effect. The rotational excitation of both reagents has a significant effect on the reaction process. The 4C and SE probabilities are at least one order of magnitude smaller than the CID probabilities over the energy range considered. Isotope substitution effects were also studied by substituting the collider D(2) by H(2) and HD on the BMKP potential energy surfaces. The CID process is most efficient for the H(2) + D(2) combination and least efficient for the H(2) + H(2) combination and is different for the 4C and SE processes.

Gyration-radius dynamics in structural transitions of atomic clusters

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.

A semiclassical theory for nonseparable rovibrational motions in curved space and its application to energy quantization of nonrigid molecules

The nonseparability of vibrational and rotational motions of a nonrigid molecule placed in the rotationally isotropic space induces several important effects on the dynamics of intramolecular energy flow and chemical reaction. However, most of these studies have been performed within the framework of classical mechanics. We present a semiclassical theory for the motions of such nonrigid molecules and apply to the energy quantization of three body atomic cluster. It is shown numerically that the semiclassical spectum given without the correct account of the rotational symmetry suffers from unnecessary broadening of the resultant spectral lines and moreover from spurious peaks.

Numerical generation of hyperspherical harmonics for tetra-atomic systems

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.

Incorporating the Geometric Phase Effect in Triatomic and Tetraatomic Hyperspherical Harmonics

Hyperspherical harmonics in the democratic row-orthonormal hyperspherical coordinates are very appropriate basis sets for performing reactive scattering calculations for triatomic and tetraatomic systems. The mathematical conditions for incorporating the geometric phase effect in these harmonics are given. These conditions are implemented for triatomic systems, and their explicit analytical expressions in terms of Jacobi polynomials, in both the absence and presence of the geometric phase effect, are given.

Dynamical and statistical effects of the intrinsic curvature of internal space of molecules

The Hamilton dynamics of a molecule in a translationally and/or rotationally symmetric field is kept rigorously constrained in its phase space. The relevant dynamical laws should therefore be extracted from these constrained motions. An internal space that is induced by a projection of such a limited phase space onto configuration space is an intrinsically curved space even for a system of zero total angular momentum. In this paper we discuss the general effects of this curvedness on dynamics and structures of molecules in such a manner that is invariant with respect to the selection of coordinates. It is shown that the regular coordinate originally defined by Riemann is particularly useful to expose the curvature correction to the dynamics and statistical properties of molecules. These effects are significant both qualitatively and quantitatively and are studied in two aspects. One is the direct effect on dynamics: A trajectory receives a Lorentz-like force from the curved space as though it was placed in a magnetic field. The well-known problem of the trapping phenomenon at the transition state is analyzed from this point of view. By showing that the trapping force is explicitly described in terms of the curvature of the internal space, we clarify that the physical origin of the trapped motion is indeed originated from the curvature of the internal space and hence is not dependent of the selection of coordinate system. The other aspect is the effect of phase space volume arising from the curvedness: We formulate a general expression of the curvature correction of the classical density of states and extract its physical significance in the molecular geometry along with reaction rate in terms of the scalar curvature and volume loss (gain) due to the curvature. The transition state theory is reformulated from this point of view and it is applied to the structural transition of linear chain molecules in the so-called dihedral angle model. It is shown that the curvature effect becomes large roughly linearly with the size of molecule.

Phase-space invariants for aggregates of particles: Hyperangular momenta and partitions of the classical kinetic energy

Rigorous definitions are presented for the kinematic angular momentum K of a system of classical particles (a concept dual to the conventional angular momentum J), the angular momentum L(xi) associated with the moments of inertia, and the contributions to the total kinetic energy of the system from various modes of the motion of the particles. Some key properties of these quantities are described-in particular, their invariance under any orthogonal coordinate transformation and the inequalities they are subject to. The main mathematical tool exploited is the singular value decomposition of rectangular matrices and its differentiation with respect to a parameter. The quantities introduced employ as ingredients particle coordinates and momenta, commonly available in classical trajectory studies of chemical reactions and in molecular dynamics simulations, and thus are of prospective use as sensitive and immediately calculated indicators of phase transitions, isomerizations, onsets of chaotic behavior, and other dynamical critical phenomena in classical microaggregates, such as nanoscale clusters.

Kinematic effects associated with molecular frames in structural isomerization dynamics of clusters

Kinematic effects associated with movements of molecular frames, which specify instantaneous orientation of molecules, is investigated in structural isomerization dynamics of a triatomic cluster whose total angular momentum is zero. The principal-axis frame is employed to introduce the so-called principal-axis hyperspherical coordinates, with which the mechanism of structural isomerization dynamics of the cluster is systematically analyzed. A force called "democratic centrifugal force" is extracted from the associated kinematics. This force arises from an intrinsic non-Euclidean metric in the internal space and has an effect of distorting the triatomic cluster to a collapsed shape and of trapping the system around collinear transition states. The latter effect is particularly important in that the kinematics effectively makes a basin at the saddle (transition state) on the potential surface. Based on this framework, we study the effect of the gauge field associated with the Eckart frame in internal space, which has not been carefully examined in the conventional reaction rate theories. Numerical comparison between the dynamics with and without the gauge field has revealed that this field has an effect to suppress the rate of isomerization reaction to a considerable amount. Thus a theory neglecting this effect will significantly overestimate the rate of isomerization. We show the physical origin of this suppressing effect.