Showing papers on "Normal coordinates published in 2015"
TL;DR: Conformal Fermi Coordinates (CFC) as discussed by the authors is a generalization of FNC that is valid outside the sound horizon of the cosmological fluid, starting from the epoch of inflation until today.
Abstract: Fermi Normal Coordinates (FNC) are a useful frame for isolating the locally observable, physical effects of a long-wavelength spacetime perturbation. Their cosmological application, however, is hampered by the fact that they are only valid on scales much smaller than the horizon. We introduce a generalization that we call Conformal Fermi Coordinates (CFC). CFC preserve all the advantages of FNC, but in addition are valid outside the horizon. They allow us to calculate the coupling of long- and short-wavelength modes on all scales larger than the sound horizon of the cosmological fluid, starting from the epoch of inflation until today, by removing the complications of the second order Einstein equations to a large extent, and eliminating all gauge ambiguities. As an application, we present a calculation of the effect of long-wavelength tensor modes on small scale density fluctuations. We recover previous results, but clarify the physical content of the individual contributions in terms of locally measurable effects and ``projection'' terms.
65 citations
TL;DR: Conformal Fermi Coordinates (CFC) as mentioned in this paper is a generalization of FNC that is valid outside the sound horizon of the cosmological fluid, starting from the epoch of inflation until today.
Abstract: Fermi Normal Coordinates (FNC) are a useful frame for isolating the locally observable, physical effects of a long-wavelength spacetime perturbation. Their cosmological application, however, is hampered by the fact that they are only valid on scales much smaller than the horizon. We introduce a generalization that we call Conformal Fermi Coordinates (CFC). CFC preserve all the advantages of FNC, but in addition are valid outside the horizon. They allow us to calculate the coupling of long- and short-wavelength modes on all scales larger than the sound horizon of the cosmological fluid, starting from the epoch of inflation until today, by removing the complications of the second order Einstein equations to a large extent, and eliminating all gauge ambiguities. As an application, we present a calculation of the effect of long-wavelength tensor modes on small scale density fluctuations. We recover previous results, but clarify the physical content of the individual contributions in terms of locally measurable effects and "projection" terms.
39 citations
TL;DR: The results show that anharmonicity and coupling do noticeably affect the vibrational transitions, shifting them by several cm(-1).
Abstract: The anharmonic vibrational spectrum of UF6 is computed in full dimensionality directly from ab initio data, i.e., bypassing the construction of a potential energy surface (PES). The vibrational Schrodinger equation is solved by fitting parameters of an adaptable basis using a modified version of the rectangular collocation algorithm of Manzhos and Carrington (J. Chem. Phys . 2013, 139, 051101). The basis functions are products of parametrized Hermite polynomials that impose approximate nodal structure. The Schrodinger equation is solved in normal coordinates. The results show that anharmonicity and coupling do noticeably affect the vibrational transitions, shifting them by several cm(-1). Although UF6 has 15 coordinates, we compute hundreds of levels with fewer than 1000 basis functions and about 50,000 ab initio points. It is the efficiency of the basis that makes it possible to forego a PES.
31 citations
TL;DR: It is found that the introduced schemes for parameterizing correlated many-mode vibrational wave functions lead to at least as systematic and accurate calculations as those using more standard and straightforward excitation level definitions.
Abstract: We introduce new automatic procedures for parameterizing vibrational coupled cluster (VCC) and vibrational configuration interaction wave functions. Importance measures for individual mode combinations in the wave function are derived based on upper bounds to Hamiltonian matrix elements and/or the size of perturbative corrections derived in the framework of VCC. With a threshold, this enables an automatic, system-adapted way of choosing which mode–mode correlations are explicitly parameterized in the many-mode wave function. The effect of different importance measures and thresholds is investigated for zero-point energies and infrared spectra for formaldehyde and furan. Furthermore, the direct link between important mode–mode correlations and coordinates is illustrated employing water clusters as examples: Using optimized coordinates, a larger number of mode combinations can be neglected in the correlated many-mode vibrational wave function than with normal coordinates for the same accuracy. Moreover, the fraction of important mode–mode correlations compared to the total number of correlations decreases with system size. This underlines the potential gain in efficiency when using optimized coordinates in combination with a flexible scheme for choosing the mode–mode correlations included in the parameterization of the correlated many-mode vibrational wave function. All in all, it is found that the introduced schemes for parameterizing correlated many-mode vibrational wave functions lead to at least as systematic and accurate calculations as those using more standard and straightforward excitation level definitions. This new way of defining approximate calculations offers potential for future calculations on larger systems.
25 citations
TL;DR: This approach is found to lift the vibrational degeneracies arising from coordinate optimization and provides better agreement with experimental and benchmark frequencies than uncorrected vibrational self-consistent field theory without relying on traditional correlated methods.
Abstract: Carefully choosing a set of optimized coordinates for performing vibrational frequency calculations can significantly reduce the anharmonic correlation energy from the self-consistent field treatment of molecular vibrations. However, moving away from normal coordinates also introduces an additional source of correlation energy arising from mode-coupling at the harmonic level. The impact of this new component of the vibrational energy is examined for a range of molecules, and a method is proposed for correcting the resulting self-consistent field frequencies by adding the full coupling energy from connected pairs of harmonic and pseudoharmonic modes, termed vibrational self-consistent field (harmonic correlation). This approach is found to lift the vibrational degeneracies arising from coordinate optimization and provides better agreement with experimental and benchmark frequencies than uncorrected vibrational self-consistent field theory without relying on traditional correlated methods.
24 citations
TL;DR: This theoretical approach provides very good agreement between the predicted and experimental frequencies and intensities, however, the favorable result can be partly attributed to error cancellation within the B3LYP/6-31+G(d,p) QM model, as observed in earlier studies.
Abstract: Anharmonic vibrational frequencies and intensities (infrared and Raman) of an isolated free-base porphin molecule are predicted from the quantum mechanical (QM) geometry, the "semi-diagonal" quartic force field, and dipole moment and polarizability surfaces. The second-order vibrational perturbation theory plus the numerical diagonalization of the Hamiltonian matrix containing off-diagonal Fermi and Darling−Dennison resonance couplings (VPT2+WK) was used. The QM calculations were carried out with the Becke−Lee−Yang−Parr composite exchange− correlation functional (B3LYP) and with the 6-31+G(d,p) basis set. The harmonic force field for the equilibrium configuration was transformed into nonredundant local symmetry internal coordinates, and normal coordinates were defined. The semi- diagonal quartic rectilinear normal coordinate potential energy surface (PES), as well as the cubic surfaces of dipole moment (p) and polarizability (α) components, needed for the VPT2+WK calculation, were constructed by a five-point finite differentiation of Hessians (for PES) and of the values and first derivatives of p and α. They were obtained at the point of equilibrium and for 432 displaced configurations. This theoretical approach provides very good agreement between the predicted and experimental frequencies and intensities. However, the favorable result can be partly attributed to error cancellation within the B3LYP/6-31+G(d,p) QM model, as observed in earlier studies. Reassignments of some observed bands are proposed.
18 citations
TL;DR: In this paper, the authors considered a domain M with a varying and possibly anisotropic wave speed which they model as a Riemannian metric g. The inverse problem is to recover the metric g in local coordinates anywhere on a set M ⊂ M ˜ up to an isometry.
17 citations
TL;DR: The new Hamiltonian is shown to provide easy access to Eckart frame ro-vibrational Hamiltonians with exact Tˆ given in terms of any desired set of vibrational coordinates.
Abstract: A new ro-vibrational Hamiltonian operator, named gateway Hamiltonian operator, with exact kinetic energy term, Tˆ, is presented. It is in the Eckart frame and it is of the same form as Watson’s normal coordinate Hamiltonian. However, the vibrational coordinates employed are not normal coordinates. The new Hamiltonian is shown to provide easy access to Eckart frame ro-vibrational Hamiltonians with exact Tˆ given in terms of any desired set of vibrational coordinates. A general expression of the Eckart frame ro-vibrational Hamiltonian operator is given and some of its properties are discussed.
13 citations
TL;DR: A full derivation of the analytic transformation of the quadratic, cubic, and quartic force constants from normal coordinates to Cartesian coordinates is given.
Abstract: A full derivation of the analytic transformation of the quadratic, cubic, and quartic force constants from normal coordinates to Cartesian coordinates is given. Previous attempts at this transformation have resulted in non-linear transformations; however, for the first time, a simple linear transformation is presented here. Two different approaches have been formulated and implemented, one of which does not require prior knowledge of the translation-rotation eigenvectors from diagonalization of the Hessian matrix. The validity of this method is tested using two molecules H2O and c-C3H2D+.
13 citations
TL;DR: In this paper, the pseud Jahn-Teller effect (PJTE) was used to explain the twisting instability of 3,6-pyridazinedione in planar configurations.
Abstract: 3,6-pyridazinedione and two of its derivatives where oxygen atoms of the molecule are substituted by two sulfur or selenium (N2C4Y2H4) were studied with the goal of answering the following question: "Which N2C4Y2H4 compounds are unstable in their planar configuration?" Additionally, the origin of the twisting instability of 3,6-pyridazinedione planar configuration and three of its 1,2-dihalo derivatives (N2C4H2O2Z2) were rationalized by employing the pseudo Jahn–Teller effect (PJTE) to explain the difference between N2C4H2O2Z2 structures in series. Therefore, the structures of six 3,6-pyridazinediones (N2C4H2Y2Z2) were optimized in both equilibrium and planar configurations, and their vibrational frequencies were calculated. Then the adiabatic potential energy surface (APES) profiles along the a2 distortion coordinates were calculated. Based on the calculation results, N2C4S2H4 and N2C4Se2H4 compounds were stable in the planar structure; but, due to the vibronic coupling interaction between the 1A1 ground state and the first excited state 1A2, the twisting instability occurred in planar N2C4H2O2Z2 series. The (1A1 + 1A2) ⊗ a2 problem was found to be the reason of the breaking symmetry phenomena in all the four N2C4H2O2Z2 in series from unstable planar configuration (highest-symmetry C2v) to the stable twisted geometry with C2 symmetry. Finally, the vibronic coupling constants of the PJTE of the compounds in series were estimated by fitting the secular equation roots along the normal coordinates of distortion.
11 citations
TL;DR: In this article, a new approach to expand the Stuckelbergized fiducial metric in a covariant manner is developed, where the curved 4-dimensional space is considered as a codimension-one hypersurface embedded in a 5-dimensional Minkowski bulk.
Abstract: A new approach to expanding the ``St\"uckelbergized'' fiducial metric in a covariant manner is developed. The idea is to consider the curved 4-dimensional space as a codimension-one hypersurface embedded in a 5-dimensional Minkowski bulk, in which the 5-dimensional Goldstone modes can be defined as usual. After solving one constraint among the five 5-dimensional Goldstone modes and projecting to the 4-dimensional hypersurface, we are able to express the St\"uckelbergized fiducial metric in terms of the 4-dimensional Goldstone modes as well as 4-dimensional curvature quantities. We also compared the results with expressions gotten using the Riemann normal coordinates in Gao et al. [Phys. Rev. D 90, 124073 (2014)] and find that, after a simple field redefinition, results gotten in two approaches exactly coincide.
TL;DR: The authors address the drawbacks of LRML by using a composition procedure to combine the normal coordinate neighborhoods for building a suitable representational space and incorporate a polyhedral geometry framework to the LRML method to give an efficient background for the synthesis process and data analysis.
Abstract: The Local Riemannian Manifold Learning (LRML) recovers the manifold topology and geometry behind database samples through normal coordinate neighborhoods computed by the exponential map. Besides, LRML uses barycentric coordinates to go from the parameter space to the Riemannian manifold in order to perform the manifold synthesis. Despite of the advantages of LRML, the obtained parameterization cannot be used as a representational space without ambiguities. Besides, the synthesis process needs a simplicial decomposition of the lower dimensional domain to be efficiently performed, which is not considered in the LRML proposal. In this paper, the authors address these drawbacks of LRML by using a composition procedure to combine the normal coordinate neighborhoods for building a suitable representational space. Moreover, they incorporate a polyhedral geometry framework to the LRML method to give an efficient background for the synthesis process and data analysis. In the computational experiments, the authors verify the efficiency of the LRML combined with the composition and discrete geometry frameworks for dimensionality reduction, synthesis and data exploration.
TL;DR: The flat-space rotational Killing vector method for measuring the Cartesian components of a black hole spin can be derived from the surface integral of Weinberg's pseudotensor over the apparent horizon surface when using Gaussian normal coordinates in the integration.
Abstract: We show that the so-called flat-space rotational Killing vector method for measuring the Cartesian components of a black hole spin can be derived from the surface integral of Weinberg's pseudotensor over the apparent horizon surface when using Gaussian normal coordinates in the integration. Moreover, the integration of the pseudotensor in this gauge yields the Komar angular momentum integral in a foliation adapted to the axisymmetry of the spacetime. As a result, the method does not explicitly depend on the evolved lapse $\ensuremath{\alpha}$ and shift ${\ensuremath{\beta}}^{i}$ on the respective time slice, as they are fixed to Gaussian normal coordinates while leaving the coordinate labels of the spatial metric ${\ensuremath{\gamma}}_{ij}$ and the extrinsic curvature ${K}_{ij}$ unchanged. Such gauge fixing endows the method with coordinate invariance, which is not present in integral expressions using Weinberg's pseudotensor, as they normally rely on the explicit use of Cartesian coordinates.
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TL;DR: In this paper, the authors give a detailed exposition of the Jet Isomorphism Theorem of pseudo-Riemannian geometry, which states the equivalence of Taylor expansions of metrics in normal coordinates up to order k+2, abstract curvature kjets and their symmetrizations as affine vector bundles.
Abstract: We give a detailed exposition of the Jet Isomorphism Theorem of pseudo-Riemannian geometry. In its weak form this theorem states that the Taylor expansion up to order k+2 of a pseudo-Riemannian metric in normal coordinates can be reconstructed in a universal way from suitable symmetrizations of the covariant derivatives of the curvature tensor up to order k. In its full generality it states the equivalence of Taylor expansions of metrics in normal coordinates up to order k+2, abstract curvature k-jets and their symmetrizations as affine vector bundles. This theorem seems to be a cornerstone of local pseudo-Riemannian geometry which is hardly mentioned in the present literature anymore.
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TL;DR: In this article, the existence of pre-semigeodesic coordinates on manifolds with affine connection was proved in the case when the components of the affine connections are twice differentiable functions.
Abstract: In the present paper we consider the problem of the existence of pre-semigeodesic coordinates on manifolds with affine connection. We proved that pre-semigeodesic coordinates exist in the case when the components of the affine connection are twice differentiable functions.
17 Feb 2015
TL;DR: The vibrational coupled cluster method in bosonic representation is formulated to describe the molecular anharmonic vibrational spectra in this article, where the vibrational excited states are described using coupled cluster linear response theory (CCLRT).
Abstract: The vibrational coupled cluster method in bosonic representation is formulated to describe the molecular anharmonic vibrational spectra. The vibrational coupled cluster formalism is based on Watson Hamiltonian in normal coordinates. The vibrational excited states are described using coupled cluster linear response theory (CCLRT). The quality of the coupled cluster wave function is analyzed. Specifically, the mean displacement values of the normal coordinates and expectation values of the square of the normal coordinates of different vibrational states are calculated. A good agreement between the converged full CI results and coupled cluster results is found for the lower lying vibrational states.