Polyatomic nonlinear

The example given above provides a simple illustration of the use of moments of inertia and vibration frequencies to calculate equilibrium constants. The method can, of course, be extended to reactions involving more complex substances. For polyatomic, nonlinear molecules the rotational contributions to the partition functions would be given by equation (16.34), and there would be an appropriate term of the form of (16.30) for each vibrational mode. ... 

As illustrated by equation (28-62), rotational motion of diatomic and polyatomic nonlinear ideal gases contributes to the internal energy as follows ... 

The law of equipartition of energy reveals, once again, that each degree of rotational freedom contributes kT per molecule or RT per mole to the internal energy of a diatomic ideal gas, relative to its ground-state energy. This theorem also applies to polyatomic nonlinear ideal gases, because rotation and also... 

Equation (3.72) is also valid for linear molecules like CO2, where the rotation around the common axis of the atoms cannot be activated. For polyatomic nonlinear molecules, the third rotation coordinate can be activated. In this case, the rotation part of df is... 

The subscripts Vi, Vj, and so on are the typical labels used to represent the individual normal modes of vibration. The vibrational partition function for a polyatomic, nonlinear molecule is, by substituting into equation 18.9,... 

These electronic energies dependence on the positions of the atomic centres cause them to be referred to as electronic energy surfaces such as that depicted below in figure B3.T1 for a diatomic molecule. For nonlinear polyatomic molecules having atoms, the energy surfaces depend on 3N - 6 internal coordinates and thus can be very difficult to visualize. In figure B3.T2, a slice tln-oiigh such a surface is shown as a fimction of two of the 3N - 6 internal coordinates. 

This is the central Jahn-Teller [4,5] result. Three important riders should be noted. First, Fg = 0 for spin-degenerate systems, because F, x F = Fo. This is a particular example of the fact that Kramer s degeneracies, aiising from spin alone can only be broken by magnetic fields, in the presence of which H and T no longer commute. Second, a detailed study of the molecular point groups reveals that all degenerate nonlinear polyatomics, except those with Kramer s... 

To describe the orientations of a diatomic or linear polyatomic molecule requires only two angles (usually termed 0 and ([)). For any non-linear molecule, three angles (usually a, P, and y) are needed. Hence the rotational Schrodinger equation for a nonlinear molecule is a differential equation in three-dimensions. 

For a polyatomic molecule, the complex vibrational motion of the atoms can be resolved into a set of fundamental vibrations. Each fundamental vibration, called a normal mode, describes how the atoms move relative to each other. Every normal mode has its own set of energy levels that can be represented by equation (10.11). A linear molecule has (hr) - 5) such fundamental vibrations, where r) is the number of atoms in the molecule. For a nonlinear molecule, the number of fundamental vibrations is (3-q — 6). 

We have seen earlier that for a linear polyatomic molecule, the vibrational motions can be divided into (3rj — 5) fundamentals, where rj is the number of atoms. For a nonlinear molecule (3rj - 6) fundamentals are present. In either case, each fundamental vibration can be treated as a harmonic oscillator with a partition function given by equations (10.100) and (10.101). Thus. 

Nonlinear polyatomic molecules require further consideration, depending on their classification, as given in Section 9.2.2. In the classical, high-temperature limit, the rotational partition function for a nonlinear molecule is given by... 

The comparison of the values of Ee0 for reactions of ketyl radicals with peroxides with those for reactions R1R2C 0H + R2C(0) show that the reaction of the ketyl radical with peroxide occurs with sufficiently higher Ee0 (see Table 6.30). Obviously, this is the result of strong additional repulsion in the nonlinear polyatomic transition state of the ketyl radical reaction with peroxide... 

Trigonal-bipyramidal species and nucleophilic displacement reactivity The 3c/4e cu-bonding motif can also be achieved in nonlinear polyatomics by backside attack of a nucleophile X - on a polar Y—Z bond of a conventional Lewis-structure molecule,... 

Abstract. The development of modern spectroscopic techniques and efficient computational methods have allowed a detailed investigation of highly excited vibrational states of small polyatomic molecules. As excitation energy increases, molecular motion becomes chaotic and nonlinear techniques can be applied to their analysis. The corresponding spectra get also complicated, but some interesting low resolution features can be understood simply in terms of classical periodic motions. In this chapter we describe some techniques to systematically construct quantum wave functions localized on specific periodic orbits, and analyze their main characteristics. 

It is not possible to discuss highly excited states of molecules without reference to the recent progress in nonlinear dynamics.2 Indeed, the stimulation is mutual. Rovibrational spectra of polyatomic molecules provides both an ideal testing ground for the recent ideas on the manifestation of chaos in Hamiltonian systems and in turn provides many challenges for the theory. 

Figure 1. Translation, rotation, and vibration of a diatomic molecule. Every molecule has three translational degrees of freedom corresponding to motion of the center of mass of the molecule in the three Cartesian directions (left side). Diatomic and linear molecules also have two rotational degrees of freedom, about rotational axes perpendicular to the bond (center). Non-linear molecules have three rotational degrees of freedom. Vibrations involve no net momentum or angular momentum, instead corresponding to distortions of the internal structure of the molecule (right side). Diatomic molecules have one vibration, polyatomic linear molecules have 3V-5 vibrations, and nonlinear molecules have 3V-6 vibrations. Equilibrium stable isotope fractionations are driven mainly by the effects of isotopic substitution on vibrational frequencies.

If the atom or molecule has additional symmetries (e.g., full rotation symmetry for atoms, axial rotation symmetry for linear molecules and point group symmetry for nonlinear polyatomics), the trial wavefunctions should also conform to these spatial symmetries. This Chapter addresses those operators that commute with H, Pjj, S2, and Sz and among one another for atoms, linear, and non-linear molecules. 

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