Excitation harmonic frequencies

Internal (i.e., bearing) and offset misalignment also excite the second (2x) harmonic frequency. Two high spots are created by the shaft as it turns though one complete revolution. These two high spots create the first (lx) and second harmonic (2x) components. 

Angular misalignment can take several signature forms and excites the fundamental (lx) and secondary (2x) components. It can excite the third (3x) harmonic frequency depending on the actual phase relationship of the angular misalignment. It also creates a strong axial vibration. 

In most cases, this failure mode also excites the third (3x) harmonic frequency and creates strong axial vibration. Depending on the severity of the instability and the design of the machine, process instability also can create a variety of shaft-mode shapes. In turn, this excites the lx, 2x, and 3x radial vibration components. 

Raman excitation. and I2s are the high-frequency and low-frequency components of the pump light pulse. A probe pulse of frequency 12 interacts with the coherence to present the optical response of the fundamental frequency 12 + C0fsl2. (c) Fourth-order coherent Raman scattering, the optical response of the second harmonic frequency 212 + co 2I2 is modulated by the vibrational coherence. 

Sinusoidal excitation provides only one harmonic at the modulation frequency. In contrast, pulsed light provides a large number of harmonics of the excitation repetition frequency. The harmonic content, the number of harmonics and their amplitude, is determined by the pulse width and shape.(25) For example, a train of infinitely short pulses provides an infinite number of harmonics all with equal amplitude. A square wave provides only three modulation frequencies with sufficient amplitude to be usable. Equation (9.74) gives the harmonic content of a train of rectangular pulses R(t) of D duty cycle (pulse width divided by period) and RP peak value ... 

Equilibrium Bond Distance and the Harmonic Frequency for N2 from the 2-RDM Method with 2-Positivity (DQG) Conditions Compared with Their Values from Coupled-Cluster Singles-Doubles with Perturbative Triples (CCD(T)), Multireference Second-Order Perturbation Theory (MRPT), Multireference Configuration Interaction with Single-Double Excitations (MRCI), and Full Configuration Interaction (FCI)". 

The resonance Raman spectrum of K4[Mo2C18] has been reinvestigated using 488.0 and 514.5 nm excitation. An enormous enhancement of the intensity of the Mo—Mo stretching mode relative to the intensity of other fundamentals was observed and an overtone progression in Vj to 5vj identified. From these data the harmonic frequency and anharmonicity constant X, were calculated as 347.1 + 0.5 cm -1... 

Kohn and Hattig [40] have presented a quite extensive study on the performance of the CC2 method for adiabatic excitation energies, excited state structures, and excited state harmonic frequencies. The systems studied include 7 diatomic molecules, 8 triatomic molecules, and 5 larger molecules. The aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets were used. The results in general are quite encouraging, and studies of this sort with CCSD and, to the extent that they are possible, with higher level methods would be most welcome. 

The latest developments of time-dependent methods rooted in the density functional theory, especially by the so-called range separated functionals like LC-coPBE or LC-TPSS are allowing computation of accurate electronic spectra even for quite large systems. Moreover, the recent availability of analytical gradients for TD-DFT " allows an efficient computation of geometry structures and harmonic frequencies (through the numerical differentiation of analytical gradients) also for excited electronic states. 

We point out that similar analyses and results have been performed and obtained also by other authors [33, 35, 38 0]. The spectral lines at 86meV and 123 meV excitation energy in the theoretical spectrum correspond to excitation of the modes V6 and vi, respectively. The first spacing deviates from the harmonic frequency of mode V6 in Table 3 because of the JT effect, while the second coincides with that of mode vi because of the linear coupling scheme adopted. For higher excitation energies the lines represent an intricate mixture of the various modes because of the well-know nonseparability of modes in the multi-mode dynamical JT effect. Overall, the excitation of the various modes can be characterized as moderately weak. The total JT stabilization energy amounts to 930 cm and is dominated by the contribution of mode ve- The barrier to pseudorotation is of the order of 10 cm only, consistent with the fact that the theoretical spectrum of Fig. 3 is obtained within the LVC scheme (see Sect. 2.1 above). 

Common spectroscopic techniques test small portions of the ground and/or excited state PES either around the gs minimum (IR and non-resonant Raman spectra, electronic absorption spectra.) or in the proximity of the excited state minimum (steady-state fluorescence). These spectra are then satisfactorily described in the best harmonic approximation, a local harmonic approach that approximates the PES with parabolas whose curvatures match the exact curvatures calculated at the specific position of interest [78]. Anharmonicity in this approach manifests itself with the dependence of harmonic frequencies and relaxation energies on the actual nuclear configuration [79]. Along these lines we predicted softened (hardened) vibrational frequencies for the ground (excited) state [74], amplified and p-dependent infrared and Raman intensities [68, 74], different Frank-Condon... 

For metallic surfaces, the three non vanishing and independent elements have been recast into three parameters, also named the three Rudnick and Stem parameters fl, b and d, corresponding to the three nonlinear currents induced at the harmonic frequency, respectively two surface currents perpendicular and parallel to the surface and one volume current perpendicular to the surface [28]. Theoretical expressions for these parameters are known within some approximations and for perfect surfaces [29, 30]. For anisotropic surfaces, the elements of the hyperpolarizability tensor would follow different relationships from Eq. (4) and would be taken from the usual Tables [31]. For metallic particles, the polarization field inside the particle is non negligible and the local exciting field should be taken as the superposition of the incoming and polarization fields. It is also still debatable whether the sheet of polarization is located inside or outside the particle. This problem has been discussed for planar interfaces and a similar discussion could be developed here. The problem of the exciting field will be discussed further below for metallic particles. 

Figure 12. Calculated Raman excitation profiles of / carotene. A harmonic three-mode model was used with ground state frequencies a> , = 1525, co2 = 1155, and coj = 1005 cm-1 and excited state frequencies

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