Translational quantum numbers

The kinetic energy operator,however,is almost separable in spherical polar coordinates, and the actual method of solving the differential equation can be found in a number of textbooks. The bound solutions (negative total energy) are called orbitals and can be classified in terms of three quantum numbers, n, I and m, corresponding to the three spatial variables r, d and q>. The quantum numbers arise from the boundary conditions on the wave function, i.e. it must be periodic in the 0 and q> variables, and must decay to zero as r oo. Since the Schrodinger equation is not completely separable in spherical polar coordinates, there exist the restrictions n > /> m. The n quantum number describes the size of the orbital, the / quantum number describes the shape of the orbital, while the m quantum number describes the orientation of the orbital relative to a fixed coordinate system. The / quantum number translates into names for the orbitals ... 

According to the argument presented above, any molecule must be described by wavefunctions that are antisymmetric with respect to the exchange of any two identical particles. For a homonuclear diatomic molecule, for example, thepossibility of permutation of the two identical nuclei must be considered. Although both the translational and vibrational wavefunctions are symmetric under such a permutation, die parity of the rotational wavefunction depends on the value of 7, the rotational quantum number. It can be shown that the wave-function is symmetric if J is even and antisymmetric if J is odd The overall... 

Fig. 12. Partitionings of hydrogen fragment translational energy distribution into three components. The solid line denotes the contribution from H2S — 8H(,4 "S+ ) + H which yields a resolved structure with a rovibrational state assignment on the top. The dotted line denotes the contribution of hydrogen from the SH(442 +) —> S(3P) + H reaction, which is a reflection of the solid curve but the structure is smeared out. The corresponding rotational quantum numbers of the parent molecule SI I (A 2>l 1 ) l =0 is marked on the bottom. The remaining part of the P(E) spectrum is represented by the square-like dashed curve.

The time-of-flight spectrum of the H-atom product from the H20 photodissociation at 157 nm was measured using the HRTOF technique described above. The experimental TOF spectrum is then converted into the total product translational distribution of the photodissociation products. Figure 5 shows the total product translational energy spectrum of H20 photodissociation at 157.6 nm in the molecular beam condition (with rotational temperature 10 K or less). Five vibrational features have been observed in each of this spectrum, which can be easily assigned to the vibrationally excited OH (v = 0 to 4) products from the photodissociation of H20 at 157.6 nm. In the experiment under the molecular beam condition, rotational structures with larger N quantum numbers are partially resolved. By integrating the whole area of each vibrational manifold, the OH vibrational state distribution from the H2O sample at 10 K can be obtained. In... 

In the above relation, quantum states of phonons are characterized by the surface-parallel wave vector kg, whereas the rest of quantum numbers are indicated by a the latter account for the polarization of a quasi-particle and its motion in the surface-normal direction, and also implicitly reflect the arrangement of atoms in the crystal unit cell. A convenient representation like this allows us to immediately take advantage of the translational symmetry of the system in the surface-parallel direction so as to define an arbitrary Cartesian projection (onto the a axis) for the... 

Electronic levels are spaced more closely together at higher quantum numbers as the ionization limit is approached, vibrational levels are evenly spaced, while rotational and translational levels are spaced further apart at high energies. The classical principle assumes continuous variation of all energies. 

The last term refers to the 3N — 6 (or 3N — 5) vibrations. Corresponding to each of the terms in Equation 4.70 are sets of quantum numbers (e.g. translational quantum numbers, rotational quantum numbers, etc.) which are independent of each other. From this point it is quite straightforward to show that the partition function can be factored into a product of partition functions corresponding to translation, rotation, etc. 

Thermal motions A molecule has three translational degrees of freedom. Let us consider a system of M ideal monatomic gas molecules in a cubic box kept at a constant temperature. For a very dilute gas, where the molecules do not interact with one another, the quantum mechanical solution is a number of wave functions with three quantum numbers, nx, riy, and n, for the translational energies in three dimensions. The energy of a molecule in a cubic box with side length a is given by... 

The column vector is indicated by square brackets, a row vector by round brackets. The quantum numbers may be determined by the complete set of her-mitian operators commuting with the generator of time evolution. Invariance of the quantum state to frame rotation, origin displacement, parity and other symmetry operations determine quantum numbers for the corresponding irreducible representations. Frame related symmetry operations translate into unitary operator acting on Hilbert space (rigged), e.g. Ta. 

This section introduces the principal experimental methods used to study the dynamics of bond making/breaking at surfaces. The aim is to measure atomic/molecular adsorption, dissociation, scattering or desorption probabilities with as much experimental resolution as possible. For example, the most detailed description of dissociation of a diatomic molecule at a surface would involve measurements of the dependence of the dissociation probability (sticking coefficient) S on various experimentally controllable variables, e.g., S 0 , v, J, M, Ts). In a similar manner, detailed measurements of the associative desorption flux Df may yield Df (Ef, 6f, v, 7, M, Ts) where Ef is the produced molecular translational energy, 6f is the angle of desorption from the surface and v, J and M are the quantum numbers for the associatively desorbed molecule. Since dissociative adsorption and... 

A translational line like the one seen above in rare gas mixtures is relatively weak but discernible in pure hydrogen at low frequencies (<230 cm-1), Fig. 3.10. However, if a(v)/[l —exp (—hcv/kT)] is plotted instead of a(v), the line at zero frequency is prominent, Fig. 3.11 the 6o(l) line that corresponds to an orientational transition of ortho-H2. Other absorption lines are prominent, Fig. 3.10. Especially at low temperatures, strong but diffuse So(0) and So(l) lines appear near the rotational transition frequencies at 354 and 587 cm-1, respectively. These rotational transitions of H2 are, of course, well known from Raman studies and correspond to J = 0 -> 2 and J = 1 — 3 transitions J designates the rotational quantum number. These transitions are infrared inactive in the isolated molecule. At higher temperatures, rotational lines So(J) with J > 1 are also discernible these may be seen more clearly in mixtures of hydrogen with the heavier rare gases, see for example Fig. 3.14 below. 

Molecules generally interact with anisotropic forces. The accounting for the anisotropy of intermolecular interactions introduces substantial complexity, especially for the quantum mechanical treatment. We will, therefore, use as much as possible the isotropic interactions isotropic interaction approximation (IIA), where the Hamiltonian is given by a sum of two independent terms representing rotovibrational and translational motion. The total energy of the complex is then given by the sum of rotovibrational and translational energies. The state of the supermolecule is described by the product of rotovibrational and translational wavefunc-tions, with an associated set of quantum numbers r and t, respectively. 

For the rotovibrational spectra of molecules interacting through purely isotropic forces, the Hamiltonian may be written as the sum of two independent terms. One term describes the rotational motion of the molecules, the other the translational motion of the pair. The total energy of the system is then equal to the sum of the rotovibrational and the translational energies. At the same time, the supermolecular wavefunctions are products of rotovibrational and translational functions. Let r designate the set of the rotovibrational quantum numbers and t the set of translational quantum numbers, the equation for yo may be written [314]... 

Find the temperature required to reduce this translational quantum number to j = 105. 

Obtain a general formula for the most probable three-dimensional translational quantum number j = jmax for a gas (assume a Boltzmann distribution). Evaluate this expression for NO2 at 1000 K (assume a cubic container 0.1 m on each side). Determine the translational energy that this corresonds to (J/mole). Find the fraction of molecules having a translational energy level greater than jmax. Hint Solution to this problem will involve the error function, erf(x). 

In Section 5.1, we noted that to a good approximation the nuclear motion of a polyatomic molecule can be separated into translational, vibrational, and rotational motions. If the molecule has N nuclei, then the nuclear wave function is a function of 3/V coordinates. The translational wave function depends on the three coordinates of the molecular center of mass in a space-fixed coordinate system. For a nonlinear molecule, the rotational wave function depends on the three Eulerian angles 9, principal axes a, b, and c with respect to a nonrotating set of axes with origin at the center of mass. For a linear molecule, the rotational quantum number K must be zero, and the wave function (5.68) is a function of 6 and only only two angles are needed to specify the orientation of a linear molecule. Thus the vibrational wave function will depend on 3N — 5 or 3N — 6 coordinates, according to whether the molecule is linear or nonlinear we say there are 3N — 5 or 3N — 6 vibrational degrees of freedom. 

The function tr describes the relative translational motion of the photofragments. This motion can be described in the semi-classical approximation (except in the turning point region) by an oscillating wavefunction for which the number of oscillations increases with an increase of the relative momentum. An increase of the number of oscillations results in a decrease of the FC factor. Hence, the semiclassical behavior of the translational wavefunction makes a transition to a state with large momentum less favorable. Because of conservation of energy, the resulting state is characterized by a small vibrational quantum number. 

Excited electronic energy levels are sometimes occupied, especially when unpaired electrons are present. A set of quantum numbers describe the electronic levels, but because of the unique nature of these levels for each type of atom or molecule, it is not possible to write general expressions similar to those given earlier for translational, rotational, or vibrational levels. 

The slowest process will be the vibration-translation activation to the (v = 1) level, which will be rate-determining, and the subsequent vibration-vibration transfers will occur at increasingly fast rates with increasing vibrational quantum number. (For harmonic oscillators 1 = n(m+ l) 1 .) Shock-tube experi-... 

The OH radical is produced in a particular vibrational and rotational quantum state specified by the quantum numbers n and j. The corresponding energies are denoted by tnj. The probabilities with which the individual quantum states are populated are determined by the forces between the translational mode (the dissociation coordinate) and the internal degrees of freedom of the product molecule along the reaction path. Final vibrational and rotational state distributions essentially reflect the dynamics in the fragment channel. They are one major source of information about the dissociation process. 

The possible dissociation channels for the fragmentation of a triatomic molecule were discussed in Section 1.4. The linear ABC molecule can fragment into three chemical channels, A+B+C, A+BC(n), and AB(n )+C with the diatoms being produced in particular vibrational states denoted by quantum numbers n and n, respectively. Furthermore, each of the fragment atoms and molecules can be created in different electronic states. The total energy Ef = Ei + hu is the same in all cases and therefore the different channels are simultaneously excited by the monochromatic light pulse. The dissociation channels differ merely in the products and in the way the total energy partitions between translation and vibration. 

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