Energy levels quantization

To calculate N (E-Eq), the non-torsional transitional modes have been treated as vibrations as well as rotations [26]. The fomier approach is invalid when the transitional mode s barrier for rotation is low, while the latter is inappropriate when the transitional mode is a vibration. Hamionic frequencies for the transitional modes may be obtained from a semi-empirical model [23] or by perfomiing an appropriate nomial mode analysis as a fiinction of the reaction path for the reaction s potential energy surface [26]. Semiclassical quantization may be used to detemiine anliamionic energy levels for die transitional modes [27]. 

Detailed analyses of the above experiments suggest that the apparent steps in k E) may not arise from quantized transition state energy levels [110.111]. Transition state models used to interpret the ketene and acetaldehyde dissociation experiments are not consistent with the results of high-level ab initio calculations [110.111]. The steps observed for NO2 dissociation may originate from the opening of electronically excited dissociation chaimels [107.108]. It is also of interest that RRKM-like steps in k E) are not found from detailed quantum dynamical calculations of unimolecular dissociation [91.101.102.112]. More studies are needed of unimolecular reactions near tln-eshold to detennine whether tiiere are actual quantized transition states and steps in k E) and, if not, what is the origin of the apparent steps in the above measurements of k E). 

For discrete energy levels, the energies are specified functions the depend on quantum numbers, one for each degree offreedom that is quantized... 

Quantized Aetion Can Also be Used to Derive Energy Levels... 

These are the same quantized energy levels that arose when the wavefunetion boundary eonditions were matehed at x = 0, x = Lx and y = 0, y = Ly. In this ease, one says that the Bohr-Sommerfeld quantization eondition ... 

Substituting the integer n for a, we find that the energy levels are quantized because a is quantized (equal to n) ... 

These harmonic-oscillator solutions predict evenly spaced energy levels (i.e., no anharmonicity) that persist for all v. It is, of course, known that molecular vibrations display anharmonicity (i.e., the energy levels move closer together as one moves to higher v) and that quantized vibrational motion ceases once the bond dissociation energy is reached. 

This technique for finding a weighted average is used for ideal gas properties and quantum mechanical systems with quantized energy levels. It is not a convenient way to design computer simulations for real gas or condensed-phase... 

For reactions between atoms, the computation needs to model only the translational energy of impact. For molecular reactions, there are internal energies to be included in the calculation. These internal energies are vibrational and rotational motions, which have quantized energy levels. Even with these corrections included, rate constant calculations tend to lose accuracy as the complexity of the molecular system and reaction mechanism increases. 

Quasiclassical calculations are similar to classical trajectory calculations with the addition of terms to account for quantum effects. The inclusion of tunneling and quantized energy levels improves the accuracy of results for light atoms, such as hydrogen transfer, and lower-temperature reactions. 

Infrared Spectra for Molecules and Polyatomic Ions The energy of infrared radiation is sufficient to produce a change in the vibrational energy of a molecule or polyatomic ion (see Table 10.1). As shown in Figure 10.14, vibrational energy levels are quantized that is, a molecule may have only certain, discrete vibrational energies. The energy for allowed vibrational modes, Ey, is... 

Four types of transitions between quantized energy levels account for molecular UV/Vis spectra. The approximate wavelength ranges for these absorptions, as well as a partial list of bonds, functional groups, or molecules that give rise to these transitions is shown in Table 10.5. Of these transitions, the most important are the n and TZ —> TZ, because they involve functional groups that are characteristic... 

At low energies, the rotational and vibrational motions of molecules can be considered separately. The simplest model for rotational energy levels is the rigid dumbbell with quantized angular momentum. It has a series of rotational levels having energy... 

Translational energy, which may be directly calculated from the classical kinetic theory of gases since the spacings of these quantized energy levels are so small as to be negligible. The Maxwell-Boltzmann disuibution for die kinetic energies of molecules in a gas, which is based on die assumption diat die velocity specuum is continuous is, in differential form. 

The amount of energy a molecule contains is not continuously variable but is quantized. That is, a molecule can stretch or bend only at specific frequencies corresponding to specific energy levels. Take bond-stretching, for example. Although we usually7 speak of bond lengths as if they were fixed, the numbers... 

Because these photons are produced when an electron moves from one energy level to another, the electronic energy levels in an atom must be quantized, that is, limited to particular values. Moreover, it would seem that by measuring the spectrum of an element it should be possible to unravel its electronic energy levels. This is indeed possible, but it isn t easy. Gaseous atoms typically give off hundreds, even thousands, of spectral lines. 

We see that, as L (the length of the box) or m (the mass of the particle) increases, the separation between neighboring energy levels decreases (Fig. 1.26). That is why no one noticed that energy is quantized until they investigated very small systems such as an electron in a hydrogen atom the separation between levels is so small for ordinary particles in ordinary-sized vessels that it is completely undetectable. We can, in fact, ignore the quantization of the motion of the atoms of a gas in a typical flask. 

An electron in an atom is like a particle in a box, in the sense that it is confined within the atom by the pull of the nucleus. We can therefore expect the electron s wavefunctions to obey certain boundary conditions, like the constraints we encountered when fitting a wave between the walls of a container. As we saw for a particle in a box, these constraints result in the quantization of energy and the existence of discrete energy levels. Even at this early stage, we can expect the electron to be confined to certain energies, just as spectroscopy requires. 

On the other hand, even in particle systems the coulomb blockade (Van Bentum et al. 1988a) and the coulomb staircase (Van Bentum et al. 1988b) were observed, some nonlinear effects were observed in the current-voltage characteristics (Wilkins et al. 1989), and behavior related to the quantized energy levels inside the particles was described (Crom-mie et al. 1993, Dubois et al. 1996). 

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