Vibrational energy quanta

The more detailed question as to whether multiple loss of vibrational energy quanta is probable is much more difficult to answer decisively. The view is generally accepted that stepwise loss of vibrational energy is usual. The vibrational energy is much more rapidly equilibrated for molecules with low fundamental vibration frequencies such as iodine (co = 215)30 than it is for molecules with high frequencies such as nitrogen (co = 2360)30. 

The only way-out insists in the requirement, to take into account the whole electron-vibration-rotation-translational Hamiltonian. It means, that the total Hamiltonian in the crude representation, expressed in the second quantization formalism, has explicitly to contain not only the vibrational energy quanta, but also the rotational and translational ones, which originate from the kinetic secular matrix. 

From this derivation, it follows that the expansion characterized by Eqs. (51) and (54) is valid if the two expansions—the perturbation and Taylor series expansion—converge rapidly. This will be the case if (0) — (0) is large, typically much larger than a vibrational energy quantum cu,-, and the amplitudes Q are small, i.e., if only low vibrational quantum numbers are involved. 

In contrast, the vibrational energy quantum decreases with the atomic mass... 

These results do not agree with experimental results. At room temperature, while the translational motion of diatomic molecules may be treated classically, the rotation and vibration have quantum attributes. In addition, quantum mechanically one should also consider the electronic degrees of freedom. However, typical electronic excitation energies are very large compared to k T (they are of the order of a few electronvolts, and 1 eV corresponds to 10 000 K). Such internal degrees of freedom are considered frozen, and an electronic cloud in a diatomic molecule is assumed to be in its ground state f with degeneracy g. The two nuclei A and... 

As in classical mechanics, the outcome of time-dependent quantum dynamics and, in particular, the occurrence of IVR in polyatomic molecules, depends both on the Flamiltonian and the initial conditions, i.e. the initial quantum mechanical state I /(tQ)). We focus here on the time-dependent aspects of IVR, and in this case such initial conditions always correspond to the preparation, at a time of superposition states of molecular (spectroscopic) eigenstates involving at least two distinct vibrational energy levels. Strictly, IVR occurs if these levels involve at least two distinct... 

Weston R E and Flynn G W 1992 Relaxation of molecules with chemically significant amounts of vibrational energy the dawn of the quantum state resolved era Ann. Rev. Rhys. Chem. 43 559-89... 

J and Vrepresent the rotational angular momentum quantum number and tire velocity of tire CO2, respectively. The hot, excited CgFg donor can be produced via absorjDtion of a 248 nm excimer-laser pulse followed by rapid internal conversion of electronic energy to vibrational energy as described above. Note tliat tire result of this collision is to... 

The generalized Prony analysis can extract a great variety of information from the ENDyne dynamics, such as the vibrational energy vib arrd the frequency for each normal mode. The classical quantum connection is then made via coherent states, such that, say, each nomral vibrational mode is represented by an evolving state... 

Figure 7.18 shows sets of vibrational energy levels associated with two electronic states between which we shall assume an electronic transition is allowed. The vibrational levels of the upper and lower states are labelled by the quantum numbers v and u", respectively. We shall be discussing absorption as well as emission processes and it will be assumed, unless otherwise stated, that the lower state is the ground state. 

Vibrational energy, which is associated with the alternate extension and compression of die chemical bonds. For small displacements from the low-temperature equilibrium distance, the vibrational properties are those of simple harmonic motion, but at higher levels of vibrational energy, an anharmonic effect appears which plays an important role in the way in which atoms separate from tire molecule. The vibrational energy of a molecule is described in tire quantum theory by the equation... 

This rule conforms with the principle of equipartition of energy, first enunciated by Maxwell, that the heat capacity of an elemental solid, which reflected the vibrational energy of a tliree-dimensional solid, should be equal to 3f JK moH The anomaly that the free electron dreory of metals described a metal as having a tliree-dimensional sUmcture of ion-cores with a three-dimensional gas of free electrons required that the electron gas should add anodier (3/2)7 to the heat capacity if the electrons behaved like a normal gas as described in Maxwell s kinetic theory, whereas die quanmtii theory of free electrons shows that diese quantum particles do not contribute to the heat capacity to the classical extent, and only add a very small component to the heat capacity. 

Finally, it can be shown from dre quantum dreoiy of vibrational energy in dre solid state drat, at temperatures above dre Debye temperature 0d, dre density of phonons, p, is inversely related to 6 according to dre equation... 

It is a fundamental result of quantum mechanics that the vibrational energy levels of the bond are given by... 

First of all, the vibrational energy is quantized, and we write the single quantum number v. This quantum number can take values 0, 1, 2,... 

The square of the wavefunction is finite beyond the classical turrfing points of the motion, and this is referred to as quantum-mechanical tunnelling. There is a further point worth noticing about the quantum-mechanical solutions. The harmonic oscillator is not allowed to have zero energy. The smallest allowed value of vibrational energy is h/2jt). k /fj. 0 + j) and this is called the zero point energy. Even at a temperature of OK, molecules have this residual energy. 

We will use the harmonic oscillator approximation to derive an equation for the vibrational partition function. The quantum mechanical expression gives the vibrational energies as... 

The vibrational relaxation of simple molecular ions M+ in the M+-M collision (where M = 02, N2, and CO) is studied using the method of distorted waves with the interaction potential constructed from the inverse power and the polarization energy. For M-M collisions the calculated values of the collision number required to de-excite a quantum of vibrational energy are consistently smaller than the observed data by a factor of 5 over a wide temperature range. For M+-M collisions, the vibrational relaxation times of M+ (r+) are estimated from 300° to 3000°K. In both N2 and CO, t + s are smaller than ts by 1-2 orders of magnitude whereas in O r + is smaller than t less than 1 order of magnitude except at low temperatures. 

The time constant r, appearing in the simplest frequency equation for the velocity and absorption of sound, is related to the transition probabilities for vibrational exchanges by 1/r = Pe — Pd, where Pe is the probability of collisional excitation, and Pd is the probability of collisional de-excitation per molecule per second. Dividing Pd by the number of collisions which one molecule undergoes per second gives the transition probability per collision P, given by Equation 4 or 5. The reciprocal of this quantity is the number of collisions Z required to de-excite a quantum of vibrational energy e = hv. This number can be explicitly calculated from Equation 4 since Z = 1/P, and it can be experimentally derived from the measured relaxation times. 

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