Vibrational dynamics partitioning

Crystals lack some of the dynamic complexity of solutions, but are still a challenging subject for theoretical modeling. Long-range order and forces in crystals cause their spectrum of vibrational frequencies to appear more like a continuum than a series of discrete modes. Reduced partition function ratios for a continuous vibrational spectrum can be calculated using an integral, rather than the hnite product used in Equation (3) (Kieffer 1982),... 

Interface between two liquid solvents — Two liquid solvents can be miscible (e.g., water and ethanol) partially miscible (e.g., water and propylene carbonate), or immiscible (e.g., water and nitrobenzene). Mutual miscibility of the two solvents is connected with the energy of interaction between the solvent molecules, which also determines the width of the phase boundary where the composition varies (Figure) [i]. Molecular dynamic simulation [ii], neutron reflection [iii], vibrational sum frequency spectroscopy [iv], and synchrotron X-ray reflectivity [v] studies have demonstrated that the width of the boundary between two immiscible solvents comprises a contribution from thermally excited capillary waves and intrinsic interfacial structure. Computer calculations and experimental data support the view that the interface between two solvents of very low miscibility is molecularly sharp but with rough protrusions of one solvent into the other (capillary waves), while increasing solvent miscibility leads to the formation of a mixed solvent layer (Figure). In the presence of an electrolyte in both solvent phases, an electrical potential difference can be established at the interface. In the case of two electrolytes with different but constant composition and dissolved in the same solvent, a liquid junction potential is temporarily formed. Equilibrium partition of ions at the - interface between two immiscible electrolyte solutions gives rise to the ion transfer potential, or to the distribution potential, which can be described by the equivalent two-phase Nernst relationship. See also - ion transfer at liquid-liquid interfaces. 

The dynamics of the reactions of 0( P) with cyclohexane, cyclohexene, and cyclohexa-1,4-diene have been studied by measurement of the product OH(X II) internal state distributions in a molecular beam/LIF apparatus. The rotational state distributions were found to be similar for all three reactions and consistent with small (1—3%) partitioning of the available energy, indicating that H-abstraction occurs only when the O atom is collinear with the C-H bond under attack. Comparisons with model predictions suggested that some of the extra energy available in the more exoergic reactions between 0( P) and the unsaturated hydrocarbons is released into internal excitation of the hydrocarbon radical product, resulting in only a modest increase in OH vibrational excitation. 

Analytic energy derivatives are as important as the energies themselves. One needs first derivatives for geometry optimizations, reaction path following, dynamics simulations, and (if analytic second derivatives are not available) second derivatives via finite differencing. Second derivatives are necessary for the computation of vibrational frequencies and, subsequently, thermodynamic properties via the appropriate partition functions. 

The largest uncertainty in the calculation of partition function ratios for minerals is the magnitude of frequency shifts on isotope substitution. Because direct spectroscopic measurements of minerals made with the less abundant isotope are not widely available, these shifts must usually be calculated or estimated in some other way. The detailed approach to calculating frequency shifts employs the methods of lattice dynamics to determine force constants for each vibrational mode, from which the vibrational frequencies for a mineral and its isotopic derivative can be predicted (e.g. Bottinga 1968 ... 

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