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Electron-Vibrational Interaction and Its Manifestation in the Experimental Absorption and Emission Spectra of Impurity Ions in Crystals
Published in Mikhail G. Brik, Chong-Geng Ma, Theoretical Spectroscopy of Transition Metal and Rare Earth Ions, 2019
Mikhail G. Brik, Chong-Geng Ma
(compare with Eqs. (9.38) and (9.39)). Each normal coordinate oscillates independently; each equation of motion (which is, in fact, an equation of motion of a harmonic oscillator) can be easily solved separately. There is no energy exchange between different vibrational modes corresponding to these normal coordinates. The normal coordinates can be considered as a basis set in the 3N −6 dimensional space of nuclear displacements in a molecule. Any collective motion of atoms in the molecule can be represented as a linear combination of these normal coordinates (like any vector in the n-dimensional space is a linear combination of the n basis vectors).
MuIti Degree-of-Freedom Vibration
Published in Haym Benaroya, Mark Nagurka, Seon Han, Mechanical Vibration, 2017
Haym Benaroya, Mark Nagurka, Seon Han
In modal analysis, the symmetries of the stiffness and mass matrices are key to the orthogonality of the natural modes. The orthogonality relations are also useful as a check on the accuracy of a numerical computation of the modes. The physical meaning of the normal coordinates is discussed in Section 6.10. The normal coordinates are the principal coordinates, in the same sense that the term is used in solid mechanics. In a single degree-of-freedom system, by considering the vibration to be in only one direction, we imply that direction to be a normal or principal coordinate.
Symbols, Terminology, and Nomenclature
Published in W. M. Haynes, David R. Lide, Thomas J. Bruno, CRC Handbook of Chemistry and Physics, 2016
W. M. Haynes, David R. Lide, Thomas J. Bruno
by replacement of one or more carbon atoms of the fulvalene skeleton by a heteroatom). [5] Fulvenes - The hydrocarbon fulvene and its derivatives formed by substitution (and by extension, analogues formed by replacement of one or more carbon atoms of the fulvene skeleton by a heteroatom). [5] Fundamental vibrational frequencies* - In molecular spectroscopy, the characteristic vibrational frequencies obtained when the vibrational energy is expressed in normal coordinates. They determine the primary features of the infrared and Raman spectra of the molecule. - Name sometimes used for microgram. -rays* - Electromagnetic radiation (photons) with energy greater than about 0.1 MeV (wavelength less than about 1 pm). g-Factor of the electron* - The proportionality factor in the equation relating the magnetic moment µ of an electron to its total angular momentum quantum number J, i.e., µ = -gµBJ, where µB is the Bohr magneton. Also called Landé factor. Gal - A non-SI unit of acceleration, equal to 0.01 m/s. Also called galileo. Gallon (US) - A unit of volume equal to 3.785412 L. Gallon (UK, Imperial) - A unit of volume equal to 4.546090 L. Gauss (G) - A non-SI unit of magnetic flux density (B) equal to 10-4 T. Gaussian system of units - A hybrid system used in electromagnetic theory, which combines features of both the esu and emu systems. Gel - A colloidal system with a finite, but usually rather small, yield stress (the sheer stress at which yielding starts abruptly). [3] Genetic code* - The set of relations between each of the 64 codons of DNA and a specific amino acid (or other genetic instruction). Gibbs energy (G)* - An important function in chemical thermodynamics, defined by G = H-TS, where H is the enthalpy, S the entropy, and T the thermodynamic temperature. Sometimes called Gibbs free energy and, in older literature, simply "free energy". [2] Gibbs phase rule - The relation F = C - P + 2, where C is the number of components in a mixture, P is the number of phases, and F is the degrees of freedom, i.e., the number of intensive variables that can be changed independently without affecting the number of phases. Glass transition temperature* - The temperature at which an amorphous polymer is transformed, in a reversible way, from a viscous or rubbery condition to a hard and relatively brittle one. [10] Glow discharge mass spectroscopy (GDMS) - See Techniques for Materials Characterization, page 12-1. Gluon - A hypothetical particle postulated to take part in the binding of quarks, in analogy to the role of the photon in electromagnetic interactions. Glycerides - Esters of glycerol (propane-1,2,3-triol) with fatty acids, widely distributed in nature. They are by long-established custom subdivided into triglycerides, 1,2- or 1,3-diglycerides, and 1- or 2-monoglycerides, according to the number and positions of acyl groups. [5] Glycols - Dihydric alcohols in which two hydroxy groups are on different carbon atoms, usually but not necessarily adjacent. Also called diols. [5] Grain (gr) - A non-SI unit of mass, equal to 64.79891 mg.
Experimental and theoretical studies on structure, bonding and luminescence properties of Eu(III) and Tb(III) complexes of a new macrocyclic based 8HQ ligand
Published in Journal of Coordination Chemistry, 2019
To validate the assignments of IR frequencies, the theoretical calculations were performed by DFT at GGA (BLYP-D3) level. The vibrational spectral assignments were performed with the help of normal coordinate analysis. Theoretically calculated vibrational spectra of the complexes were made accessible to illustrate the experimental vibrational peaks by the structural animation at each peak. The theoretical calculated vibrational frequencies in contrast to the experimental values exhibited an identical trend, shown in Table 2. Somewhat higher values were observed for stretching vibrations in theoretical results. The variations in the calculated and experimental spectra are generally because of, to some extent, the fact that the experimental spectra were observed in the solid phase and the theoretical spectra were found in the gas phase. The trend was persistent with the experimental results. A correlation found between the theoretical and experimental data provided satisfactory results (R2 = 0.9938, 0.9978) for the complexes (Figure 3). This confirms the reliability of the presented band assignments.
Non-adiabatic dynamics of ring opening in cyclohexa-1,3-diene described by an ensemble density-functional theory method
Published in Molecular Physics, 2019
Michael Filatov, Seung Kyu Min, Kwang S. Kim
Let us assume that there exists a geometry for which the normal coordinates , are known (the mass-weighted coordinates are assumed here and below). This can be, e.g. the geometry of a (local) minimum on the ground or excited state PES. The normal coordinates describe all possible internal displacements of atoms in the given molecular force field. Hence, an instantaneous geometry of the molecule at a time t, can be decomposed in terms of the normal coordinates as where are time-dependent amplitudes. Due to orthogonality of the normal modes, , the amplitudes can be found as For a slew of trajectories, the amplitudes can be averaged over all the trajectories In the averaged amplitudes, the contributions of the normal modes which are out of synchronisation with one another will cancel out and only the normal modes persistently contributing to the displacements along the trajectory will have non-zero averaged amplitudes . Hence, the averaged amplitudes can serve as useful descriptors of the dynamics.