Rotor-vibrator functions

What can be the outcome of this inference If the RV representation is indeed somewhat better, one might guess that expansion of the exact function in a (truncated) basis set of rotor-vibrator functions would converge faster than a conventional expansion in products of one-electron, independent-particle functions. An investigation of this question is in progress now. 

In Eqn. (5), the angular brackets impley averages over the asymmetric rotor wave-function as well as the vibrational wavefunction. Rz is the component of R along the space-fixed z-axis. The final step is to relate the coupling constants in Eqn. (5) to those of the monomer. In general, the expressions depend on the complexity of the monomers and on the dimer rotational state observed. For a large number of cases, a linear type dimer in a K=0 rotational state may be assumed, and Eqn. (5) may be expressed as... 

The application ofc Tyir to the analysis of the vibrational-inversion-rotation spectra of ammonia will be discussed in detail in Sections 5.1-5.4. Here we mention only that if the interaction between the inversion, vibration and rotation states is neglected, the overall wave function pvit can be written as a product of the harmonic oscillator wave functions the inversion wave function, p), and the symmetric rotor wave function Sj/cm( > 4>) exp (i/cx) ... 

Analytic expressions are available for both the quantum harmonic vibrator partition function Q ib(T) and the classical rigid rotor partition function Q ot(T) [40] further simplifying the analysis. 

Except for floppy molecules, thermal contributions at room temperature can be quite accurately evaluated using the familiar rigid rotor-harmonic oscillator (RRHO) approach. If data at high temperatures are required, this approach is no longer sufficient, and an anharmonic force field and analysis, combined with a procedure for obtaining the rotation-vibration partition function therefrom, are required. Two practical procedures have been proposed. The first one, due to Martin and co-workers " is based on asymptotic expansions for the nonrigid rotor partition function inside an explicit loop over vibration. It yields excellent results in the medium temperature range but suffers from vibrational level series collapse above 2000 K or more. A representative application (to FNO and CINO) is found in Ref. 42. 

In this section, we will analyze the scattering wavefunction (s,p,y) in the transition-state region for total angular momentum J = 0 and a near-resonant energy of 0.36 eV (measured from the bottom of the entrance valley), with H2 starting in its ground vibration-rotation state. As mentioned in section II, we build the scattering wavefunction at each value of s from products of Morse oscillator functions times hindered-internal-rotor functions. For the calculations reported here, we used a basis of 10 vibrational functions with the rotational distribution 6, 6, 6, 4, 3, 1, 1, 1, 1, 1. This means that there are 6 even-j odd-j functions in the v = 0 manifold,... 

Vibration grade signifies the accuracy of rotor balancing, e.g. N = normal, R = reduced, S = special. This may be based on the type of installation and the accuracy of the function the motor may have to perform. 

This speed becomes critical when the frequency of excitation is equal to one of the natural frequencies of the system. In forced vibration, the system is a function of the frequencies. These frequencies can also be multiples of rotor speed excited by frequencies other than the speed frequency such as blade passing frequencies, gear mesh frequencies, and other component frequencies. Figure 5-20 shows that for forced vibration, the critical frequency remains constant at any shaft speed. The critical speeds occur at one-half, one, and two times the rotor speed. The effect of damping in forced vibration reduces the amplitude, but it does not affect the frequency at which this phenomenon occurs. 

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. 

Function of speed The high amplitudes at the rotor s natural frequency are strictly speed dependent. If the energy source, in this case speed, changes to a frequency outside the resonant zone, the abnormal vibration will disappear. 

Figure 12. Comparison of simple RRKM rate-energy curves, using three different loose activated complexes giving the same rates at the energy corresponding to about 10 s" . Calculations are shown for Eq values of 1.86 and 3.10 eV. The three transition states are (a) uniform frequency multiplier of 0.9 (—) (b) four low-frequency vibrations (—) (c) low-frequency vibration to internal rotor ( -). The corresponding values are as follows 1.86 eV, (a) = 8.2 eu, (b) = 4.9 eu, (c) = 1.6 eu 3.10 eV (a) = 6.9 eu, (b) = 4.9 eu, (c) = 2.9 eu. Also shown is the semiclassical RRK functional form...

Tables 1 and 2 show the lowest torsional energy levels of hydrogen peroxide and deuterium peroxide which have been determined variationally using as basis functions the rigid rotor solutions. Experimental data are from Camy-Peiret et al [15]. The first set of leval data are from Camy-Peiret et al [15]. The first set of levels (SET I) has been calculated without including the pseudopotential V = 0). The levels corresponding to the other sets (SET II, SET III and SET IV) were obtained including pseudopotentials calculated with different numerical and analytical algorithms. Finally, the zero point vibration energy correction was introduced in the SET V [14],...

The derivation above may be generalized to wave functions other than electronic ones. By evaluation of transition dipole matrix elements for rigid-rotor and harmonic-oscillator rotational and vibrational wave functions, respectively, one arrives at the well-known selection rules in those systems that absorptions and emissions can only occur to adjacent levels, as previously noted in Chapter 9. Of course, simplifications in the derivations lead to many forbidden transitions being observable in the laboratory as weakly allowed, both in the electronic case and in the rotational and vibrational cases. 

Theory. The theory of collision-induced absorption profiles of systems with anisotropic interaction [43, 269] is based on Arthurs and Dalgamo s close coupled rigid rotor approximation [10]. Dipole and potential functions are approximated as rigid rotor functions, thus neglecting vibrational and centrifugal stretching effects. Only the H2-He and H2-H2 systems have been considered to date, because these have relatively few channels (i.e., rotational levels of H2 to be accounted for in the calculations). The... 

The nuclear function %a(R) is usually expanded in terms of a wave function describing the vibrational motion of the nuclei, and a rotational wave function [36, 37]. Analysis of the vibrational part of the wave function usually assumes that the vibrational motion is harmonic, such that a normal mode analysis can be applied [36, 38]. The breakdown of this approximation leads to vibrational coupling, commonly termed intramolecular vibrational energy redistribution, IVR. The rotational basis is usually taken as the rigid rotor basis [36, 38 -0]. This separation between vibrational and rotational motions neglects centrifugal and Coriolis coupling of rotation and vibration [36, 38—401. Next, we will write the wave packet prepared by the pump laser in terms of the zeroth-order BO basis as... 

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