Showing papers on "Normal coordinates published in 1987"

Conformal geodesics in general relativity

Abstract: Conformal geodesics, space-time curves which are related to conformal structures in a similar way as geodesics are related to metric structures, are discussed. \`Conformal normal coordinates', \`conformal Gauss systems' and their associated \`normal connections', \`normal frames' and `normal metrics' are introduced and used to study: (i) asymptotically simple solutions of Ric(g) = $\Lambda g$ near conformal infinity, (ii) asymptotically simple solutions of Ric(g) = 0 with a past null infinity, which can be represented as the future null cone of a point i$^-$, past time-like infinity. In the first case we define an $\infty$-parameter family of (physical) Gauss systems near conformal infinity, in the second case a ten-parameter family of (physical) Gauss systems covering a neighbourhood of i$^-$. The behaviour of physical geodesics can be analysed in a particularly simple way in these coordinate systems. Each of these systems allows an extremely simple transition from the conformal analysis to the physical description of space-time. For $\Lambda\eta_{00}$ < 0 (De-Sitter type solutions) all solutions are characterized in terms of the physical space-time by their data on past time-like infinity. For $\Lambda$ = 0 the conserved quantities of Newman and Penrose are characterized as the first non-trivial coefficient, given by the value of the rescaled Weyl tensor at i$^-$, in an expansion of the physical field in a Gauss system of the type considered before.

Extended coupled-cluster method. II. Excited states and generalized random-phase approximation

Abstract: This article gives a discussion of the application of the extended coupled-cluster method (ECCM) to the excited states of a general quantum many-body system. The direct eigenvalue equations for the excitation amplitudes of both the ket and the bra eigenstates are derived in the biorthogonal basis obtained by a double similarity transformation. The equations correspond to the diagonalization of a matrix involving second-order functional derivatives of the average-value functional for the Hamiltonian with respect to the basic ECCM amplitudes. The same excitation spectrum is obtained by considering small oscillations around the equilibrium. The problem with its associated effective Hamiltonian has the structure of a generalized random-phase approximation. By diagonalizing the effective Hamiltonian we perform a canonical or symplectomorphic coordinate transformation into normal coordinates in the symplectic ECCM phase space. In this coordinate system the exact average-value functional for the Hamiltonian has a structure analogous to that of classical lattice dynamics or the phenomenological Ginzburg-Landau theory. At all stages the method satisfies the property of quantum locality, which in real space shows up as a definite quasilocality. Due to this property the method allows, for example, the treatment of mesonlike excitations in the presence of topological objects or in other symmetry-broken equilibrium states.

Kinetic anharmonic coupling in the trihalomethanes: A mechanism for rapid intramolecular redistribution of CH stretch vibrational energy

Abstract: The coupling of CH stretching and CH bending vibrations in trisubstituted methanes is analyzed. Improved spectroscopic constants, especially the cubic anharmonic stretch–bend coupling constants, are extracted from Fermi resonances in the overtone spectra of HCF3 and HCCl3. Both harmonic oscillator and Morse oscillator basis functions are used in the analysis and the results compared. That part of the coupling which arises from the kinetic energy as expressed in curvilinear coordinates is calculated and compared to the coupling calculated using the more conventional rectilinear treatment. Use of curvilinear coordinates is found to provide significant advantages. The formalism for curvilinear normal coordinates is clarified and generalized. From these calculations and the spectral analysis, one of the cubic anharmonic constants of the potential energy surface is extracted for comparison with ab initio calculations. The curvilinear model of the CH stretch–bend interaction tested for these isolated CH chromop...

Rotational optical activity

Abstract: Optical activity generated by the rotational normal coordinates is investigated. The theoretical formulation is based on classical principles and uses the definition of the rotational normal coordinates and the changes in electric and magnetic dipole moment components along the principal axes of inertia due to rotations represented by these coordinates. It is assumed that the effective atomic charges and polarizabilities are uneffected during the rotational motions of the molecule.

Soliton energies in the standard map beyond the chaotic threshold

Abstract: The Birkhoff-Moser theorem guarantees the existence of an analytical reduction to its normal form of an area-preserving map in the neighbourhood of an unstable fixed point. The stable and unstable manifolds emanating from this point are then simply the images of the axes in the normal coordinates. For sufficiently high parameter values of the standard map, the series for the manifolds can be resummed into a closed form, providing good approximations to the first loops of the manifolds. The authors thus obtain simple approximations for a pair of homoclinic orbits for parameters where most orbits are chaotic. The Frenkel-Kontorova model consists of an infinite sequence of equal springs and masses under the action of a periodic potential. The configuration where all the masses lie at the minima of the potential corresponds to the unstable fixed point of the standard map. A soliton or discommensuration, where one of the minima is missed, is represented by a homoclinic orbit. Using the normal form they calculate the soliton energy and its pinning energy. Good agreement is found with the pinning energy of Pokrovsky (1981) obtained as a perturbation from the continuum approximation.

Experimental determination of CH‐stretching–bending–bending cubic force constants of ethane from Raman intensity analysis

Abstract: The vibrational anharmonicity of ethane has been experimentally investigated by means of a joint analysis of the gas‐phase mean polarizability Raman scattering cross sections and the wave numbers of the multiple vibrational resonances in the C–H or C–D stretching region of several isotopic derivatives. Thirty nine cubic force constants krst in the representation of normal coordinates were obtained in a first step. In a second step they were reduced to a smaller set of isotopically invariant cubic constants Fijk in the representation of curvilinear symmetry coordinates.

Raman tensors for planar and pyramidal AB3 molecules

Abstract: On the basis of bond polarizability model, expressions for second derivatives of the polarizability tensor with respect to the normal coordinates of planar and pyramidal AB3 molecules are presented Using the results obtained, the relationship between the first and the second derivatives of the mean polarizability is found

On the temperature dependence of the quadrupole splitting of 57Fe in MgCO3, CaCO3 and FeCO3

Abstract: The phonon-field contribution to the quadrupole splitting of the 57Fe2+ ion in carbonates is considered and a comparison of the experimental data with the theoretical estimates is presented The calculations have been performed in terms of the normal coordinates of the complex composed of the 57Fe2+ ion plus its nearest neighbours The present approach leads to results that are in good agreement with the experimental data and, in contrast with the results obtained by other authors, it makes clear that a Debye model can be used to explain the experimental data even at low temperatures