Condensed Phase Effects

9 SOME SPECIAL CONSIDERATIONS 4.9.1 Condensed Phase Effects 

Ionization in the condensed phase presents a challenge due to the lack of a precise operational definition. Only in very few cases, such as the liquefied rare gases (LRG), where saturation ionization current can be obtained at relatively low fields, can a gas-phase definition be applied and a W value obtained (Takahashi et al., 1974 Thomas and Imel, 1987 Aprile et al., 1993). 

Operationally, a procedure may be based on measuring the yield of a reaction traceable to ionization, usually giving a lower limit to the ionization yield. Thus, in the radiation chemistry of hydrocarbon liquids, the product of an electron scavenging reaction (for example, C2H3- radical from the scavenger C2H5Br) 

FIGURE 4.5 Inelastic collision cross section of water vapor versus electron energy (LaVerne and Mozumder, 1992). Circles compilation of Hayashi (1989) dashed line unmodified theoretical formula (Pimblott et al., 1990) dot-dashed line theoretical formula scaled to match compilation full curve theoretical formula scaled to match experimental W values. 

For highly polar media, the yield of the solvated electron can serve as a lower limit to the ionization yield. This method needs short-time measurement and may work for liquid water and ammonia. Farhataziz et al. (1974) determined the G value—that is, the 100-eV yield—of solvated electrons in liquid NH3 to be about 3.1 at -50 ns. This corresponds to a W value of 32 eV, compared with the gas-phase value of 26.5 eV. The difference may be attributed to neutralization during the intervening time. In liquid water, it has been found that G(eh) increases at short times and has a limiting value of 4.8 (Jonah et al., 1976 Sumiyoshi et al, 1985). This corresponds to W,. = 20.8 eV compared with Wgas = 30 eV (Combecher, 1980). Considering that the yield of eh can only be a lower limit of the ionization yield, suggestions have 

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In order to segregate the theoretical issues of condensed phase effects in chemical reaction dynamics, it is usefiil to rewrite the exact classical rate constant in (A3.8.2) as [5, 6, 7, 8, 9,10 and U]... 

A3.8.2 THE ACTIVATION FREE ENERGY AND CONDENSED PHASE EFFECTS... 

Many of the condensed phase effects mentioned above have been studied computationally using the PI-QTST approach outlined in die first part of the last section. One such study 48 has focused on the model synnnetric... 

As seen in Eqs. (59)—(61), dephasing processes introduce two new time scales into the dynamics, in addition to the intermediate state lifetime that determines the structure of 8s in the isolated molecule case. One is the time scale of pure dephasing, and the other is the lifetime of the final state. Equation (64) illustrates that the Tff dependence of 8s is a condensed phase effect that vanishes in the limit of no dephasing. The more careful analysis later shows that the qualitative behavior of the channel phase is dominated by the rpd/rrr and Tpd / [ ratios, that is, by the rate of dephasing as compared to the system time scales. 

Abstract This chapter reviews the theoretical background for continuum models of solvation, recent advances in their implementation, and illustrative examples of their use. Continuum models are the most efficient way to include condensed-phase effects into quantum mechanical calculations, and this is typically accomplished by the using self-consistent reaction field (SCRF) approach for the electrostatic component. This approach does not automatically include the non-electrostatic component of solvation, and we review various approaches for including that aspect. The performance of various models is compared for a number of applications, with emphasis on heterocyclic tautomeric equilibria because they have been the subject of the widest variety of studies. For nonequilibrium applications, e.g., dynamics and spectroscopy, one must consider the various time scales of the solvation process and the dynamical process under consideration, and the final section of the review discusses these issues. 

Self-consistent reaction field (SCRF) models are the most efficient way to include condensed-phase effects into quantum mechanical calculations [8-11]. This is accomplished by using SCRF approach for the electrostatic component. By design, it considers only one physical effect accompanying the insertion of a solute in a solvent, namely, the bulk polarization of the solvent by the mean field of the solute. This approach efficiently takes into account the long range solute-solvent electrostatic interaction and effect of solvent polarization. However, by design, this model cannot describe local solute-solvent interactions. 

Computed and experimental data for the chemical shifts of heavy elements have been less extensively compared. Table 9.6 lists some results for Se that are illustrative of the wide range of chemical shifts typically possible for such nuclei (here more than 2000 ppm) as well as the degree to which the chemical phase may affect the comparisons. The calculations are gas phase, although in Chapters 11 and 12 we will discuss techniques for including condensed-phase effects in computational predictions. 

Collective coordinates, 35, 98 Collision theory, 528, 542 Comparative molecular field analysis, 308-310 Complete basis set, see Multilevel methods) Compressibility, 418, 446 Condensed-phase effects, see also Solvation... 

In terms of layout, it might be preferable from a historic sense to start with quantum theories and then develop classical theories as an approximation to the more rigorous formulation. However, I think it is more pedagogically straightforward (and far easier on the student) to begin with classical models, which are in the widest use by experimentalists and tend to feel very intuitive to the modern chemist, and move from there to increasingly more complex theories. In that same vein, early emphasis will be on single-molecule (gas-phase) calculations followed by a discussion of extensions to include condensed-phase effects. While the book focuses primarily on the calculation of equilibrium properties, excited states and reaction dynamics arc dealt with as advanced subjects in later chapters. 

Possible explanations for the discrepancy between theory and experiment are an accommodation coefficient less than unity, different configurations in the first condensate and the bulk of the liquid, different molecular species in the vapour and the condensed phase, effects of molecular rotation, etc. 

In addition, the molecules properties are changed due to the interaction with the surrounding medium. Several computational schemes have been proposed to address these effects. Tliey are essentially based on the extension of the Onsager reaction field cavity model and give effective hyperpolarizabilities, i.e. molecular hyperpolarizabilities induced by the external fields that include the modifications due to the surrounding molecules as well as local (cavity) field effects [40 2]. These condensed-phase effects have, however, not yet been included in the SFG hyperpolarizability calculations, which are therefore strictly gas-phase calculations. 

However, further analysis of the linear regression expression in A3.8.1 is required to achieve a useful expression for the rate constant both from a computational and a conceptual points of view. Such an expression was first provided by Yamamoto [6], but others have extended, validated, and expounded upon his analysis in considerable detail [7, 8]. The work of Chandler [7] in this regard is followed most closely here in order to demonstrate the places in which condensed phase effects can appear in the theory, and hence in the value of the thermal rate constant. The key mathematical step is to differentiate both sides of the linear regression formula in (A3.8.1) and then carefully analyse its expected behaviour for systems having a barrier height of at least several times k-Q T. The resulting expression for the classical forward rate constant in terms of the so-called reactive flux time correlation function is given by [6, 7 and 8]... 

A few relatively recent applications of PI-QTST are summarized in this subsection. For other applications and extensions of the theory, the reader is referred to the growing list of PI-QTST papers in such areas as electron transfer theory [102-105] and simulation [50,98-100,102,106], proton transfer theory [107] and simulation [46,77,107-111], hydrogen diffusion in [112] and on [113-116] metals, molecular diffusion [117] and adsorption [118,119] on metals, and in the theory of condensed-phase effects in quantum activated dynamics [43,63, 66, 96, 97,120-122],... 

Monte Carlo simulation techniques have been extensively used to study solvent effects on molecular properties and equilibrium points. Jorgensen has summarized theoretical work of condensed-phase effects on conformational equilibria [63]. 

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