Frequency harmonic

We introduce the dimensionless bending coordinates qr = t/XrPr anti qc = tAcPc ith Xt = (kT -r) = PrOir, Xc = sJ kcPc) = Pc nc. where cor and fOc are the harmonic frequencies for pure trans- and cis-bending vibrations, respectively. After integrating over 0, we obtain the effective Hamiltonian H = Ho + H, which is employed in the perturbative handling of the R-T effect and the spin-orbit coupling. Its zeroth-order pait is of the foim... 

A vibrations calculation is the first step of a vibrational analysis. It involves the time consuming step of evaluating the Hessian matrix (the second derivatives of the energy with respect to atomic Cartesian coordinates) and diagonalizing it to determine normal modes and harmonic frequencies. For the SCFmethods the Hessian matrix is evaluated by finite difference of analytic gradients, so the time required quickly grows with system size. 

Excessive healing of magnetic cores, as a result of harmonic frequencies due to hysteresis and eddy current losses (equations (1.I2) and (1.13)). 

The rating of a capacitor unit will thus vary in a square proportion of the effective harmonic voltage and in a direct proportion to the harmonic frequency. This rise in kVAr, however, will not contribute to improvement of the system p.f. but only of the overloading of the capacitors themselves. 

For a harmonic order n and harmonic frequency n f the harmonic reactance will become... 

A filter circuit is a combination of capacitor and series reactance, tuned to a particular harmonic frequency (series resonance), to offer it the least impedance at that frequency and hence, filter it out. Say. for the fifth harmonic, =... 

These are parallel resonant L-C circuits, and are tuned to offer a high impedance to a particular harmonic frequency... 

It should be ensured that under no condition of system disturbance w ould the filter circuit become capacitive when it approaches near resonance. To achieve this, the filter circuits may be tuned to a little less than the defined harmonic frequency. Doing so will make the L and hence Xl always higher than Xc, since... 

Finally, the harmonic frequencies are obtained at the HF/6-31G level and scaled uniformly by the well-accepted factor of 0.8929. That allows for Boltzmann contributions to the vibrational modes. 

Table 11.13 H2O HF harmonic frequencies (cm ) as a function of basis set experimental values are 3943 cm 3832 cmand 1649 cm...

The MP2 treatment recovers the majority of the correlation effect, and the CCSD(T) results with the cc-pVQZ basis sets are in good agreement with the experimental values. The remaining discrepancies of 9cm , 13cm and lOcm are mainly due to basis set inadequacies, as indicated by the MP2/cc-pV5Z results. The MP2 values are in respectable agreement with the experimental harmonic frequencies, but of course still overestimate the experimental fundamental ones by the anharmonicity. For this reason, calculated MP2 harmonic frequencies are often scaled by 0.97 for comparison with experimental results. ... 

H2O CCSD(T) harmonic frequencies as a function of basis set (valence electrons only)... 

Table 11.16 H2O MP2 harmonic frequencies (cm function of basis set (all electrons) " ) as a...

For comparison with experimental frequencies (which necessarily are anharmonic), there is normally little point in improving the theoretical level beyond MP2 with a TZ(2df,2pd) type basis set unless anharmonicity constants are calculated explicitly. Although anharmonicity can be approximately accounted for by scaling the harmonic frequencies by 0.97, the remaining errors in the harmonic force constants at this level are normally smaller than the corresponding errors due to variations in anharmonicity. 

The harmonic frequencies calculated with different DFT functionals as a function of basis set are shown in Tables 11.17-11.19. The convergence as a function of basis set is similar to that observed for the HF method. The B3PW91 functional again shows the best performance. With the cc-pV5Z basis set the deviations from the experimental harmonic frequencies are only 0cm", 9 cm" and 17 cm", substantially better that the... 

Table 11.17 H2O highest harmonic frequency (cm ] DFT functionals the experimental value is 3943 cm" ...

Table 11.23 cc-pVTZ basis Harmonic frequencies for 0 3 with the... 

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. If a more elaborate treatment is required, the summation for the partition functions must be carried out explicitly. Many molecules also have internal rotations with quite small barriers, hi the above they are assumed to be described by simple harmonic vibrations, which may be a poor approximation. Calculating the energy levels for a hindered rotor is somewhat complicated, and is rarely done. If the barrier is very low, the motion may be treated as a free rotor, in which case it contributes a constant factor of RT to the enthalpy and R/2 to the entropy. 

Sub-harmonic frequencies (i.e., less than the actual shaft speed) are the primary evaluation tool for fluid-film bearings and they must be monitored closely. A narrowband window that captures the full range of vibration frequency components between electronic noise and running speed is an absolute necessity. 

Vertical Mechanical looseness in the vertical plane generates a series of harmonic and half-harmonic frequency components. Figure 44.37 is a simple example of a vertical mechanical looseness signature. 

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. 

Friedly (F4) expanded the theoretical analysis of Hart and McClure and included second-order perturbation terms. His analysis shows that the linear response of the combustion zone (i.e., the acoustic admittance) is not sign-ficantly altered by the incorporation of second-order perturbation terms. However, the second-order perturbation terms predict changes in the propellant burning rate (i.e., transition from the linear to nonlinear behavior) consistent with experimental observation. The analysis including second-order terms also shows that second-harmonic frequency oscillations of the combustion chamber can become important. 

Quasi-Harmonic Frequencies Calculated from Monte Carlo Trajectories on the Exact Potential Surface for Trans and Gauche Butane... 

Distances Ce are in A, dissociation energies in eV (calculated values are not corrected for the 2ero-point vibrational energy, harmonic frequencies Oe in cm , and adiabatic ionization potentials AEip and electron affinities AEea in eV. Experimental values are from Refs. [94, 159-162]. 

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