Molecules classical complex rotation

Proceeding in the spirit above it seems reasonable to inquire why s is equal to the number of equivalent rotations, rather than to the total number of symmetry operations for the molecule of interest. Rotational partition functions of the diatomic molecule were discussed immediately above. It was pointed out that symmetry requirements mandate that homonuclear diatomics occupy rotational states with either even or odd values of the rotational quantum number J depending on the nuclear spin quantum number I. Heteronuclear diatomics populate both even and odd J states. Similar behaviors are expected for polyatomic molecules but the analysis of polyatomic rotational wave functions is far more complex than it is for diatomics. Moreover the spacing between polyatomic rotational energy levels is small compared to kT and classical analysis is appropriate. These factors appreciated there is little motivation to study the quantum rules applying to individual rotational states of polyatomic molecules. 

Two points should be emphasized. First, according to classical structure theory, all the equivalent positions of a given set should be occupied and moreover they should all be occupied by atoms of the same kind. In later chapters we shall note examples of crystals in which one or both of these criteria are not satisfied an obvious case is a solid solution in which atoms of different elements occupy at random one or more sets of equivalent positions. (The occupation of different sets of equivalent positions by atoms of the same kind occurs frequently and may lead to quite different environments of chemically similar atoms. Examples include the numerous crystals in which there is both tetrahedral and octahedral coordination of atoms of the same element—in the same oxidation state—as noted in Chapter 5, and crystals in which there is both coplanar and tetrahedral coordination of Cu(ii), p. 890, or Ni(ii), p. 965.) The second point for emphasis is if a molecule (or complex ion) is situated at one of the special positions it should possess the point symmetry of that position. A molecule lying on a plane of symmetry must itself possess a plane of symmetry, and one having its centre at the intersection of two planes of symmetry must itself possess two perpendicular planes of symmetry. If, therefore, it can be demonstrated that a molecule lies at such a position as, for example, would be the case if the unit cell of Fig. 2.13 contained only one molecule, (a fact deducible from the density of the crystal) this would constitute a proof of the symmetry of the molecule. Such a conclusion is not, of course, valid if there is any question of random orientation or free rotation of the molecules. Moreover, there is another reason for caution in applying this type of argument to inorganic crystals. 

Microwave studies in molecular beams are usually limited to studying the ground vibrational state of the complex. For complexes made up of two molecules (as opposed to atoms), the intennolecular vibrations are usually of relatively low amplitude (though there are some notable exceptions to this, such as the ammonia dimer). Under these circumstances, the methods of classical microwave spectroscopy can be used to detennine the stmcture of the complex. The principal quantities obtained from a microwave spectmm are the rotational constants of the complex, which are conventionally designated A, B and C in decreasing order of magnitude there is one rotational constant 5 for a linear complex, two constants (A and B or B and C) for a complex that is a symmetric top and tliree constants (A, B and C) for an... 

Halberstadt, N., Beswick, J.A., and Schinke, R. (1991). Rotational distributions in the vibrational predissociation of weakly bound complexes Quasi-classical golden rule treatment, in Half Collision Resonance Phenomena in Molecules Experimental and Theoretical Approaches, ed. M. Garcia-Sucre, G. Raseev, and S.C. Ross (American Institute of Physics, New York). 

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... 

Classical trajectory studies of the association reactions M+ + H20 and M+ + D20 with M = Li, Na, K (Hase et al. 1992 Hase and Feng 1981 Swamy and Hase 1982,1984), Li+(H20) + H20 (Swamy and Hase 1984), Li+ + (CH3)20 (Swamy and Hase 1984 Vande Linde and Hase 1988), and Cl- + CH3C1 (Vande Linde and Hase 1990a,b) are particularly relevant to cluster dynamics. In these studies, the occurrence of multiple inner turning points in the time dependence of the association radial coordinate was taken as the criterion for complex formation. A critical issue (Herbst 1982) is whether the collisions transfer enough energy from translation to internal motions to result in association. Comparison of association probabilities from various studies leads to the conclusion that softer and/or floppier ions and molecules that have low frequency vibrations typically recombine the most efficiently. Thus, it has been found that Li+ + (CH3)20 association is more likely than Li+ + H20 association, and similarly H20 association with Li(H20)+ is more likely than with the bare cation Li+. The authors found a nonmonotonic dependence of association probability on the assumed HaO bend frequency and also a dependence on the impact parameter, the rotational temperature, and the orientation of the H20 dipole during the collision. 

Ethylene has the well-known classical >2/1 structure with a barrier to rotation. The next in complexity of the simple hydrides is the methyl radical CH3. The obvious (sp2) planar arrangement can only accommodate six of the seven valence electrons. The electronic configuration of this molecule can therefore not be described in terms of either atomic wave functions or hybrid orbitals. An alternative approach is to view the structure of the methyl radical as a reduced-symmetry form, derived from the structure of methane, to be considered next. 

The "statistical formulation (67.Ill) cannot be applied to unimolecular reactions for which the classical activation energy and the reaction heat are equal (E = Q) without introducing some additio-nal assumptions which are necessary for the definition of the transition state. One usually considers the "activated complex (AB) as a rotating "diatomic molecule in which the centrifugal force is balanced by an attractive dipole-induced dipole or dispersion force /HO/. This "diatomic model implies that the angular momentum... 

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