Classical vibrational/rotational motion

Improving on the semi-classical treatment of the vibration-rotation motion only slightly allows Rt to be recast in a form... 

Theoretical chemistry is the discipline that uses quantum mechanics, classical mechanics, and statistical mechanics to explain the structures and dynamics of chemical systems and to correlate, understand, and predict their thermodynamic and kinetic properties. Modern theoretical chemistry may be roughly divided into the study of chemical structure and the study of chemical dynamics. The former includes studies of (1) electronic structure, potential energy surfaces, and force fields (2) vibrational-rotational motion and (3) equilibrium properties of condensed-phase systems and macromolecules. Chemical dynamics includes (1) bimolecular kinetics and the collision theory of reactions and energy transfer (2) unimolecular rate theory and metastable states and (3) condensed-phase and macromolecular aspects of dynamics. 

The first step in a unimolecular reaction is the excitation of the reactant molecule s energy levels. Thus, a complete description of the unimolecular reaction requires an understanding of such levels. In this chapter molecular vibrational/rotational levels are considered. The chapter begins with a discussion of the Bom-Oppenheimer principle (Eyring, Walter, and Kimball, 1944), which separates electronic motion from vibrational/rotational motion. This is followed by a discussion of classical molecular Hamiltonians, Hamilton s equations of motion, and coordinate systems. Hamiltonians for vibrational, rotational, and vibrational/rotational motion are then discussed. The chapter ends with analyses of energy levels for vibrational/rotational motion. 

The separation of the reaction coordinate x from the other coordinates of the reacting system allows us to treat independently the relative translation or vibration of reactants and its non-reac-tive motions from the point of view of statistical physics, too. Classical statistics may be used in most cases for the motion along the reaction coordinate however, quantum statistics is usually necessary for the non-reactive vibration-rotation motions of reactants. 

In the classical vibrational spectroscopy, the subject of investigation is the vibrational-rotational motion of polyatomic molecules near the very bottom of their potential-energy surface in the electronic ground state. In this case, the normal-mode approximation proves quite applicable. Indeed, one can expand as a Taylor series the potential energy of the molecule near the equilibrium position (the potential-energy minimum) and write down the molecular Hamiltonian in the form... 

The determination of the good actions describing vibration-rotation motion requires the solution of the molecular Hamilton-Jacobi equation, which is a nonlinear partial differential equation in 3Na"5 variables (including rotation), where is the number of atoms. Even for = 3 (a triatomic molecule) an exact solution to this equation is extremely complex computationally, and it is not practical for collisional applications. Several approximations can be used to simplify this treatment, however, including (i) the separation of vibration from rotation (valid in the limit of an adequate vibration-rotation time scale separation), and (ii) the use of classical perturbation theory (in 2nd and 3rd order) to solve the three-dimensional vibrational Hamilton-Jacobi equation which remains after the separation of rotation. Details of both the separation procedures and the perturbation-theory solution are discussed elsewhere. For the present application, the validity of the first... 

Consider a gas of N non-interacting diatomic molecules moving in a tln-ee-dimensional system of volume V. Classically, the motion of a diatomic molecule has six degrees of freedom—tln-ee translational degrees corresponding to the centre of mass motion, two more for the rotational motion about the centre of mass and one additional degree for the vibrational motion about the centre of mass. The equipartition law gives (... 

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... 

A fundamental theorem of classical mechanics called the equipartition theorem (which we shall not derive here) states that the average energy of each degree of freedom of a molecule in a sample at a temperature T is equal to kT. In this simple expression, k is the Boltzmann constant, a fundamental constant with the value 1.380 66 X 10-21 J-K l. The Boltzmann constant is related to the gas constant by R = NAk, where NA is the Avogadro constant. The equipartition theorem is a result from classical mechanics, so we can use it for translational and rotational motion of molecules at room temperature and above, where quantization is unimportant, but we cannot use it safely for vibrational motion, except at high temperatures. The following remarks therefore apply only to translational and rotational motion. 

In this section, we shall consider how the solution of the classical equations of motion for more than two atoms may be used to find reaction probabilities and cross-sections for chemical reactions. Although the treatment is based on classical mechanics, it is termed quasi-classical because quantization of vibrational and rotational energy levels is accounted for. 

Although vibrational and some rotational motions certainly require quantum mechanics for their accurate consideration, we will treat these motions in their classical limit. Using Eq, (23) and integrating over a 2M-dimension phase space for a total of M rotations and vibrations, we get, using a normalization factor of hrm,... 

Ion-molecule association is seemingly well suited for the application of the quasiclassical trajectory (QCT) method (Porter and Raff 1976 Raff and Thompson 1985 Truhlar and Muckerman 1979). Since there is no potential barrier and the centrifugal potential is broad, quantum mechanical tunneling is typically unimportant. Energy transfer from relative translational to vibrational and/or rotational motions of the complex should be reasonably classical because of the... 

From the classical chemical aspect, isotopes of an element are expected to have same chemical properties. Therefore, it has been believed that the behaviors of isotopes of an element are essentially the same. However, a change in the mass of an atom in a molecule produces changes in vibrational, rotational, and translational motions of the molecule. In the diffusion of gases it has long been known that the velocity of molecules changes with the mass of the molecule, thus producing a physical isotope effect. 

Neglecting rotational motion, and assuming that the vibrational modes are all harmonic, the classical limit of the vibrational partition function of a molecule... 

Once a potential energy function is chosen or determined for a molecule, there are three major components to a trajectory study the selection of initial conditions for the excited molecule, the numerical integration of the classical equations of motion, and the analysis of the trajectories and their final conditions. The last item may include the time at which the trajectory decomposed to products, the nature of the trajectory s intramolecular motion, i.e., regular or irregular, and the vibrational, rotational and translational energies of the reaction products. 

We shall limit ourselves to the case where the mean energy of each molecule is equal to the sum of the energies of translation, of rotation and of vibration. The heat capacity at constant volume (c/. 10.5) will also be composed of three terms arising from these three kinds of motion. The contribution from the translational motion is f R per mole, and that from rotation is JR or fR depending upon whether the molecule is linear or not. This last statement is only exact if the rotational motion may be treated by classical, as opposed to quantum, mechanics. This is a good approximation even at low temperatures except for very light molecules such as Hg and HD. Finally the contribution from vibration of the atoms in a molecule relative to one another is the sum of the contributions from the various modes of vibration. Each mode of vibration is characterized by a fundamental frequency vj which is independent of the temperature. It is convenient to relate the fundamental frequency to a characteristic temperature (0j) defined by... 

The molecular configuration is a function of time. Molecular systems are not stationary molecules vibrate, rotate, and tumble. Force field calculations and the properties predicted by them are based on a. stationary model. What is needed is some way to predict what motions the atoms within a molecule will undergo at various temperatures. Molecular dynam-ics (MD) simulations use classical mechanics—force field methods—to study the atomic and molecular motions to predict macroscopic properties. "... 

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