Dynamics of condensed phases

In the sections below a brief overview of static solvent influences is given in A3.6.2, while in A3.6.3 the focus is on the effect of transport phenomena on reaction rates, i.e. diflfiision control and the influence of friction on intramolecular motion. In A3.6.4 some special topics are addressed that involve the superposition of static and transport contributions as well as some aspects of dynamic solvent effects that seem relevant to understanding the solvent influence on reaction rate coefficients observed in homologous solvent series and compressed solution. More comprehensive accounts of dynamics of condensed-phase reactions can be found in chapter A3.8. chapter A3.13. chapter B3.3. chapter C3.1. chapter C3.2 and chapter C3.5. 

The macroscopic properties of a material are related intimately to the interactions between its constituent particles, be they atoms, ions, molecules, or colloids suspended in a solvent. Such relationships are fairly well understood for cases where the particles are present in low concentration and interparticle interactions occur primarily in isolated clusters (pairs, triplets, etc.). For example, the pressure of a low-density vapor can be accurately described by the virial expansion,1 whereas its transport coefficients can be estimated from kinetic theory.2,3 On the other hand, using microscopic information to predict the properties, and in particular the dynamics, of condensed phases such as liquids and solids remains a far more challenging task. In these states... 

In many cases, in order to compute the dynamics of condensed phase systems, one invokes a basis representation for the quantum degrees of freedom in the system. Typically, one computes the dynamics of these systems in order to obtain quantities of interest, such as an average value, A(t) = Tr [Ap(t)], or a correlation function, as will be discussed below. Since such averages are basis independent one may project Eq. (8) onto any convenient basis. This is in principle a nice feature, and one that is often exploited to aid in calculations. However, it is important to note that the basis onto which one chooses to project the QCLE has important implications on how one goes about solving the resulting equations of motion. Ultimately the time-dependent average value of an observable is expressed as a trace over quantum subsystem... 

The QCL approach discussed thus far in this chapter provides a good approximation to the quantum dynamics of condensed phase systems. Most often other approximate quantum-classical methods, such as mean field and surface-hopping schemes, have been commonly employed to treat the same class of problems as the QCLE. These methods are attractive due to their computational simplicity however, many important quantum features, such as quantum coherence and correlations, are not properly handled in these approaches. In this section we discuss these methods and show that starting from the QCLE, an approximate theory in its own right, further approximations lead to these other approaches. 

G. Hanna and R. Kapral. Nonadiabatic dynamics of condensed phase rate processes. Acc. Chem. Res., 39(l) 21-27, Jan 2006. 

Owrutsky JC, Li M, Culver JP, Sarisky MJ, Yodh AG, Hochstrasser RM. Vibrational dynamics of condensed phase molecules studied by ultrafast infrared spectroscopy. In Lau A, Siebert F, Werncke W, eds. Time Resolved Vibrational Spectroscopy IV. Berlin Springer-Verlag, 1993 63-67. 

Variations on this surface hopping method that utilize Pechukas [106] formulation of mixed quantum-classical dynamics have been proposed [107,108]. Surface hopping algorithms [109-111] for non-adiabatic dynamics based on the quantum-classical Liouville equation [109,111-113] have been formulated. In these schemes the dynamics is fully prescribed by the quantum-classical Liouville operator and no additional assumptions about the nature of the classical evolution or the quantum transition probabilities are made. Quantum dynamics of condensed phase systems has also been carried out using techniques that are not based on surface hopping algorithms, in particular, centroid path integral dynamics [114] and influence functional methods [115]. 

Calculation of the quantum dynamics of condensed-phase systems is a central goal of quantum statistical mechanics. For low-dimensional problems, one can solve the Schrodinger equation for the time-dependent wavefunction of the complete system directly, by expanding in a basis set or on a numerical grid [1,2]. However, because they retain the quantum correlations between all the system coordinates, wavefunction-based methods tend to scale exponentially with the number of degrees of freedom and hence rapidly become intractable even for medium-sized gas-phase molecules. Consequently, other approaches, most of which are in some sense approximate, must be developed. 

A more severe problem might be the classical mechanics nature of MD simulations. It is nowadays established that quantum mechanics does not play a major role in the nuclear dynamics of condensed phase systems at room temperature (apart from, maybe, in proton transfer reactions), but the same is not necessarily true for thermal... 

Quantum-Classical Liouville Dynamics of Condensed Phase Quantum Processes... 

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