Separation of translational, rotational and vibrational motions

35) represents approximate kinetic energy. Tb obtain the corresponding Hamiltonian we have to add the potential energy for the motion of the nuclei, Uic, to this energy where k labels the electronic state. The last energy depends uniquely on the variables that describe atomic vibrations and corresponds to the electronic energy Uk(R) of eq. (6.28), except that instead of the variable R, which pertains to the oscillation, we have the components of the vectors Then, in full analogy with (6.28), we may write 

Since (after the approximations have been made) the translational, rotational and vibrational ( internal motion ) operators depend on their own variables, after separation the total wave function represents a product of three eigenfunctions (translational, rotational and vibrational) and the total eneigy is the sum of the translational, rotational and vibrational energies (fully analogous with eq. (6.29)) 

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Quasi-Rigid Model-Simplifying by Eckart Conditions Approximation Decoupling of Rotations and Vibration Spherical, Sjfminetric, and Asymmetric Tops Separation of Translational, Rotational, and Vibrational Motions... 

After the separation of translational, rotational, and vibrational modes of nuclear motion... 

These BO equations ean be reeognized as the equations for the translational, rotational, and vibrational motion of the nuelei on the potential energy surfaee Ej (R). That is, within the BO pieture, the eleetronie energies Ej(R), eonsidered as funetions of the nuelear positions R, provide the potentials on whieh the nuelei move. The eleetronie and nuelear-motion aspeets of the Sehrodinger equation are thereby separated. 

In a rigid molecule approximation (internal rotation and inversion barriers appreciably exceed kT), one may single out contributions from separate degrees of freedom of the translational, rotational, and vibrational motions to the entropy S and the heat capacity, with anharmonicity of vibrations and some other effects neglected ... 

The classical description in Chapter 2 separated molecular motions into translations, rotations, and vibrations. Each of these motions is treated differently in a quantum mechanical picture. In addition, electrons in molecules can be moved to higher energy levels, just as electrons in a hydrogen atom had multiple energy levels. We will treat each of these cases in turn. 

In case that the energy of a molecule can be represented as the sum of several terms (such as rotational, vibrational, electronic, and translational energy), the Boltzmann factor can be written as the product of individual Boltzmann factors, and the contributions of the various energy terms to the total energy of the system in thermodynamic equilibrium and to the heat capacity, entropy, and other properties can be calculated separately. To illustrate this we shall discuss the contributions of rotational and vibrational motion to the energy content, heat capacity, and entropy of hydrogen chloride gas. 

The coupling functions 1 and still depend on the molecular vibrational and rotational degrees of freedom as well as the relative molecule-perturber separation, R. Since the experiments imply that the physical origin of the collision-induced intersystem crossing resides in long-range attractive interactions, we may adopt a semiclassical approximation where the quantum-mechanical variables for the relative translation is replaced by a classical trajectory, R(l), for the relative molecule-perturber motion. The internal dynamics is then influenced by the time-dependent interactions f s[ (0] and Fj-j-fR(r)], which are still functions of molecular rotational and vibrational variables. For simplicity and for illustrative purposes we consider only the pair of coupled levels S and T and a pure triplet level T, which represents the molecular state after the collision. Note T may differ in rotational and/or vibrational quantum... 

The Bom-Oppenheimer separation of the electronic and nuclear motions is a cornerstone in computational chemistry. Once the electronic Schrodinger equation has been solved for a large number of nuclear geometries (and possibly also for several electronic states), the potential energy surface (PES) is known. The motion of the nuclei on the PES can then be solved either classically (Newton) or by quantum (Schrodinger) methods. If there are N nuclei, the dimensionality of the PES is 3N, i.e. there are 3N nuclear coordinates that define the geometry. Of these coordinates, three describe the overall translation of the molecule, and three describe the overall rotation of the molecule with respect to three axes. For a linear molecule, only two coordinates are necessary for describing the rotation. This leaves 3N - 6(5) coordinates to describe the internal movement of the nuclei, which for small displacements may be chosen as vibrational normal coordinates . 

Although it is easier to discuss rotations and vibrations of molecules separately, in reality such motions of molecules occur simultaneously. (Translations are also occurring, and translational motion accounts for a large part of the kinetic energy of a molecule in the gas phase. However, translations do not contribute directly to the topic at hand.) When a sample is in the gas phase, molecules are unhindered in their rotational and vibrational motions, and so both occur simultaneously. In the liquid phase, vibrational motions occur relatively unhindered but rotational motions may be hindered. In the solid phase, vibrations are relatively hindered, and with a few exceptions the rotations are quenched. 

In addition to translational and electronic motion, a diatomic gas has rotational and vibrational motion. To a good approximation the energy is a sum of four separate terms, as in Eq. (22.2-37) ... 

Of the 3n coordinates needed to describe an n-atom molecule, three are used for center of mass motion, three describe angular displacement (rotation, hindered rotation, or libration) (two if the molecule is linear, 0 if monatomic), the remaining 3n—6 (3n—5, if linear, 3n—3 = 0, if monatomic) describe atom-atom displacements (vibrations). In some cases it may not be possible to separate translation cleanly from rotation and vibration, but when the separation can be made it is a convenience. Elementary treatments assume... 

In more complex molecular systems, increased coupling between the translational motion and both rotational and vibrational modes occurs. It is difficult to separate these effects completely. Nevertheless, the velocity autocorrelation functions of the Lennard—Jones spheres [519] (Fig. 52) and the numerical simulation of the carbon tetrachloride (Fig. 39) are quite similar [452a]. 

For a temperature of 298.15 K, a pressure of 1 bar, and 1 mole of H2S, prepare a table of (1) the entropy (J/mol K), and separately the contributions from translation, rotation, each vibrational mode, and from electronically excited levels (2) specific heat at constant volume Cv (J/mol/K), and the separate contributions from each of the types of motions listed in (1) (3) the thermal internal energy E - Eo, and the separate contributions from each type of motion as before (4) the value of the molecular partition function q, and the separate contributions from each of the types of motions listed above (5) the specific heat at constant pressure (J/mol/K) (6) the thermal contribution to the enthalpy H-Ho (J/mol). 

Equation (1-13) or its body-fixed equivalent is of little use for Van der Waals complexes, as it discriminates one nuclear coordinate, e.g. y = 1. Specific mathematical forms of Hamiltonians describing the nuclear motions in Van der Waals dimers have been developed (7). This point will be discussed in more details in Section 12.4. Here we only want to stress that whatever the mathematical form of the Hamiltonian is used to solve the problem of nuclear motions, the results will be the same, if the Schrodinger equation is solved exactly. However, in weakly bound complexes there is a hierarchy of motions due to the strong intramolecular forces which determine the internal vibrations of the molecules, and to much weaker intermolecular forces which determine their relative translations and rotations. This hierarchy allows to make a separation between the intramolecular vibrations with high frequencies and the intermolecular modes with much lower frequencies. Such a separation of the fast intramolecular vibrations and slow rotation-vibration-tunneling motions can be performed if a suitable form of the Hamiltonian for the nuclear motions in Van der Waals molecules is used. 

Such a separation is exact for atoms. For molecules, only the translational motion of the whole system can be rigorously separated, while their kinetic energy includes all kinds of motion, vibration and rotation as well as translation. First, as in the case of atoms, the translational motion of the molecule is isolated. Then a two-step approximation can be introduced. The first is the separation of the rotation of the molecule as a whole, and thus the remaining equation describes only the internal motion of the system. The second step is the application of the Born-Oppenheimer approximation, in order to separate the electronic and the nuclear motion. Since the relatively heavy nuclei move much more slowly than the electrons, the latter can be assumed to move about a fixed nuclear arrangement. Accordingly, not only the translation and rotation of the whole molecular system but also the internal motion of the nuclei is ignored. The molecular wave function is written as a product of the nuclear and electronic wave functions. The electronic wave function depends on the positions of both nuclei and electrons but it is solved for the motion of the electrons only. 

The first term in this equation represents the molecule s center of mass translational motion and can be ignored since it is separable from the other terms. The second and third terms represent the molecule s rotational and vibrational kinetic energies, respec-... 

Given the separation of slow rotational from fast vibrational motions outlined in section 4.2, unperturbed rotations are described by the Hamiltonian Hr, and the translational-rotational coupling is V R.n = ViR. u).d). Using again the Magnus expansion one finds that the wave operator sr - where [16]... 

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