Rotation of real spherical harmonics (original) (raw)
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Journal of Physics A: Mathematical and Theoretical, 2007
The angular factors of atomic orbitals are real spherical harmonics. This is independent of the choice of basis function. In the course of molecular electronic structure calculations, numerous rotations of real spherical harmonics are required in a suitably defined space-fixed co-ordinate system. The origin and axes are space-fixed and rotation matrices defined on a basis of spherical harmonics.
Computation of Rotation Matrices Making Lined-Up to the Local Cartesian Coordinates
Journal of the Chinese Chemical Society, 2007
Rotation matrices were expressed in terms of Gaunt coefficients and complex spherical harmonics. The rotation matrices were calculated using two different ways. In the first, Gaunt coefficients and normalized complex spherical harmonics were directly calculated using binomial coefficients; in the second, Gaunt coefficients and complex spherical harmonics were recursively calculated. The methods were compared with respect to accuracy and computation time (CPU) for low and very high quantum numbers.
Revue Roumaine De Chimie, 2007
In this work, based on simple algebraic manipulations, the divergence-free description of molecular rotations is revisited using the axis-rotation formula for the rigid-body system. The so-called axis-rotation formula is useful in various fields of computational chemistry, including molecular simulations, graphical rendering and group theory, allowing more convenient ways to construct and to manipulate the atomic or fragment structures of rotations. It is shown that the analytical expression of the axis-rotation operator facilitates obtaining the symmetry operator in analytical form, which is useful in the determination of group symmetries of molecules and the adaptation to the symmetry of atomic and molecular orbitals.
A general treatment of vibration-rotation coordinates for triatomic molecules
International Journal of Quantum Chemistry, 1991
An exact, within the Born-Oppenheimer approximation, body-fixed Hamiltonian for the nuclear motions of a triatomic system is presented. This Hamiltonian is expressed in terms of two arbitrarily defined internal distances and the angle between them. The body-fixed axis system is related to these coordinates in a general fashion. Problems with singularities and the domain of the Hamiltonian are discussed using specific examples of axis embedding. A number of commonly used coordinate systems including Jacobi, bond-length-bond-angle, and Radau coordinates are special cases of this Hamiltonian. Sample calculations on the HzS molecule are presented using all these and other coordinate systems. The possibility of using this Hamiltonian for reactive scattering calculations is also discussed.
A Note on the Representation of Cosserat Rotation
Continuous Media with Microstructure, 2010
This brief article provides an independent derivation of a formula given by Kafadar and Eringen (1971) connecting two distinct Cosserat spins. The first of these, the logarithmic spin represents the time rate of change of the vector defining finite Cosserat rotation, whereas the second, the instantaneous spin, gives the local angular velocity representing the infinitesimal generator of that rotation. While the formula of Kadafar and Eringen has since been identified by Iserles et al. (2000) as the differential of the Lie-group exponential, the present work provides an independent derivation based on quaternions. As such, it serves to bring together certain scattered results on quaternionic algebra, which is currently employed as a computational tool for representing rigid-body rotation in various branches of physics, structural and robotic dynamics, and computer graphics.
Alternative treatment of rotational quantum systems
Zeitschrift f�r Physik D Atoms, Molecules and Clusters, 1989
The method of the Hill determinant proves to be useful in treating purely rotating quantum systems. The rotational Stark effect in symmetric-top molecules and the internal rotation in molecules are discussed as illustrative examples. The procedure can be used either to obtain the energy eigenvalues for a given model potential or to built it from experimental data.
Some new results on three-dimensional rotations and pseudo-rotations
2013
We use a vector parameter technique to obtain the generalized Euler decompositions with respect to arbitrarily chosen axes for the three-dimensional special orthogonal group SO(3) and the three-dimensional Lorentz group SO(2, 1). Our approach, based on projecting a quaternion (respectively split quaternion) from the corresponding spin cover, has proven quite effective in various problems of geometry and physics [1, 2, 3]. In particular, we obtain explicit (generally double-valued) expressions for the three parameters in the decomposition and discuss separately the degenerate and divergent solutions, as well as decompositions with respect to two axes. There are some straightforward applications of this method in special relativity and quantum mechanics which are discussed elsewhere (see [4]).