Normal coordinates

About: Normal coordinates is a research topic. Over the lifetime, 632 publications have been published within this topic receiving 18584 citations.

Reaction path Hamiltonian for polyatomic molecules

Abstract: The reaction path on the potential energy surface of a polyatomic molecule is the steepest descent path (if mass‐weighted Cartesian coordinates are used) connecting saddle points and minima. For an N‐atom system in 3d space it is shown how the 3N‐6 internal coordinates can be chosen to be the reaction coordinate s, the arc length along the reaction path, plus (3N‐7) normal coordinates that describe vibrations orthogonal to the reaction path. The classical (and quantum) Hamiltonian is derived in terms of these coordinates and their conjugate momenta for the general case of an N atom system with a given nonzero value of the total angular momentum. One of the important facts that makes this analysis feasible (and therefore interesting) is that all the quantities necessary to construct this Hamiltonian, and thus permit dynamical studies, are obtainable from a relatively modest number of ab initio quantum chemistry calculations of the potential energy surface. As a simple example, it is shown how the effects o...

Simplification of the molecular vibration-rotation hamiltonian

Abstract: By use of the commutation relations and sum rules, the Darling-Dennison vibration-rotation hamiltonian for a non-linear molecule is rearranged to the form: The order of the factors in the first term is immaterial, on account of the relation: A simple expansion is given for the μαβ tensor in terms of the normal coordinates.

Some Studies Concerning Rotating Axes and Polyatomic Molecules

Abstract: The theory of small vibrations when the potential energy is invariant under the rotation-displacement group is developed. The results are compared with the Brester-Wigner theory of the normal coordinates, and it is shown that the use of these coordinates implies the use of a particular (normal) system of rotating axes whose construction is given. It is shown that when the motion of a normal molecule is referred to these axes, those terms of the Hamiltonian which are linear in the angular momenta will be especially small and of the same order of magnitude as the quadratic terms (Casimir's condition). When the amplitude of one or more of the normal vibrations becomes large, this is no longer true of the normal axes; this will always be the case when one of the normal frequencies is small compared to the others, as has been noted by other writers. The normal axes are not the principal axes of inertia of the instantaneous configuration of the system, and certain conclusions recently published by the author are wrong for that reason.

Some Mathematical Methods for the Study of Molecular Vibrations

Abstract: Developments which reduce the labor of calculating the vibration frequencies of complex molecules are described. In particular a vectorial scheme is given for obtaining the reciprocal of the matrix of the kinetic energy in terms of valence‐type coordinates. A general rule for writing down the coefficients of the transformation to symmetry coordinates is derived together with a method of obtaining the kinetic energy reciprocal matrix (G) in terms of symmetry coordinates with a minimum of algebra. A treatment of redundant coordinates is developed. In addition, reduction of the secular equation by the splitting out of high frequencies, a new type of isotope product rule, and the determination of normal coordinates are discussed. The molecule CH3Cl is worked out as an illustration.

Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent Space Alignment

Abstract: Nonlinear manifold learning from unorganized data points is a very challenging unsupervised learning and data visualization problem with a great variety of applications. In this paper we present a new algorithm for manifold learning and nonlinear dimension reduction. Based on a set of unorganized data points sampled with noise from the manifold, we represent the local geometry of the manifold using tangent spaces learned by fitting an affine subspace in a neighborhood of each data point. Those tangent spaces are aligned to give the internal global coordinates of the data points with respect to the underlying manifold by way of a partial eigendecomposition of the neighborhood connection matrix. We present a careful error analysis of our algorithm and show that the reconstruction errors are of second-order accuracy. We illustrate our algorithm using curves and surfaces both in 2D/3D and higher dimensional Euclidean spaces, and 64-by-64 pixel face images with various pose and lighting conditions. We also address several theoretical and algorithmic issues for further research and improvements.