Dynamics of nuclear motion

We have presented here the quasi harmonic approximation epitomized by Eq.(51) to show one way to represent the dynamics of nuclear motions in a quantum mechanical scheme. A general solution for these equations cannot be obtained. However, a number of particular cases exist for which solutions have been worked out in the literature. 

In semiclassical ET theory, three parameters govern the reaction rates the electronic couphng between the donor and acceptor (%) the free-energy change for the reaction (AG°) and a parameter (X.) related to the extent of inner-shell and solvent nuclear reorganization accompanying the ET reaction [29]. Additionally, when intrinsic ET barriers are small, the dynamics of nuclear motion can limit ET rates through the frequency factor v. These parameters describe the rate of electron transfer between a donor and acceptor held at a fixed distance and orientation (Eq. 1),... 

The time scale of valence electrons is typically of the order of 10 femtoseconds, while that of the inner shell is of course much faster and becomes close to the speed of light velocity for heavy atoms. Just as the femto-scale laser technology made a great contribution to the analyses of chemical dynamics of nuclear motions, the attosecond laser is anticipated to play the similar role for electron dynamics. 

The universality of the tendency observed here may be justified by comparing this result with those in other molecular systems. In our previous works on several isolated molecules such as methane and methyl alcohol placed in a low frequency laser field [425], we have found the same features that the absence of X j can significantly underestimate the dynamics of nuclear motions. Therefore the nuclear derivative coupling strongly assists the field induced d3mamics. 

This chapter discusses the apphcation of femtosecond lasers to the study of the dynamics of molecular motion, and attempts to portray how a synergic combination of theory and experiment enables the interaction of matter with extremely short bursts of light, and the ultrafast processes that subsequently occur, to be understood in terms of fundamental quantum theory. This is illustrated through consideration of a hierarchy of laser-induced events in molecules in the gas phase and in clusters. A speculative conclusion forecasts developments in new laser techniques, highlighting how the exploitation of ever shorter laser pulses would permit the study and possible manipulation of the nuclear and electronic dynamics in molecules. 

Nuclear magnetic resonance spectroscopy of dilute polymer solutions is utilized routinely for analysis of tacticlty, of copolymer sequence distribution, and of polymerization mechanisms. The dynamics of polymer motion in dilute solution has been investigated also by protoni - and by carbon-13 NMR spectroscopy. To a lesser extent the solvent dynamics in the presence of polymer has been studied.Little systematic work has been carried out on the dynamics of both solvent and polymer in the same systan. 

The matrix elements in Equation 7 represent the mixing of vibrational states with the electronic states via the dynamic part of nuclear motion. The degree of this mixing is determined by the value of these matrix elements. At the same time, the sum of these matrix elements describes the coupling of various vibrational states through the nuclear motion operator. If there is no degeneracy or closeness of... 

The molecular potential energy is an energy calculated for static nuclei as a function of the positions of the nuclei. It is called potential energy because it is the potential energy in the dynamical equations of nuclear motion. 

For the purpose of any dynamical calculation it will generally be necessary to have the potential energy as an analytical function of the internal coordinates. This will certainly be true if the equations of nuclear motion are to be solved analytically, and even if they are solved numerically one needs a method for rapid evaluation of the potential at any point on the surface and this is only possible if an explicit analytical function is available. 

A quantum mechanical theory is in principle needed to describe molecular phenomena in both few-atom and many-atom systems. In some cases a single electronic state is involved, and it is possible to gain valuable insight using only classical molecular dynamics, which can be relatively easy to apply even for a system of many atoms. A quantum mechanical description of molecular phenomena is however clearly needed for electronic states, insofar these have pronounced wavemechanical properties. The need for a quantum description of nuclear motions in molecular dynamics is less apparent, but it is required in some important situations. If we consider a generic interaction between two species A(a) and B(j3) leading to formation of two others, C(7) and D(6), all of them in the specified quantum states, so that... 

As we are interested in the low energy states close to the bottom of the wells, the amplitude of nuclear motion is small compared to the overall average value of the nuclear displacement. Thus the criterion for smallness comes from the small deviation qp of the displacement from the bottom of the minimum point. Ultimately, we should include nuclear motion as a part of the dynamic problem so that the parameter qp will become a dynamic variable associated with the ground harmonic oscillator state 10) in well p. However, this will not be considered further here. 

J0H82] N.R. Johnson, in Proc. of the 1982 Inst, for Nuclear Study Intnl. Symposium on Dynamics of Nuclear Collective Motion, ed. by K. Ogawa and K. Tanahe (Inst, for Nuclear Study, Univ. of Tokyo, 1982), p. 144. 

The theoretical foundation for reaction dynamics is quantum mechanics and statistical mechanics. In addition, in the description of nuclear motion, concepts from classical mechanics play an important role. A few results of molecular quantum mechanics and statistical mechanics are summarized in the next two sections. In the second part of the book, we will return to concepts and results of particular relevance to condensed-phase dynamics. 

Thus the studies of the QA-involved electron tunneling reactions have shown that analysis of the reaction rate vs — AG° dependences provides information about the character of nuclear motions coupled to electron transfer in RC of photosynthetic bacteria. In particular, as kT becomes smaller than ha the dependence of electron transfer rate on — AG° becomes sensitive to the magnitude of ha. In other words, it is possible to find out from these experiments what aspects of protein and cofactor dynamics are important for the reaction. 

Molecular dynamics (MD) has been traditionally linked to purely classical modeling, e.g. based on empirical force fields, or with classical trajectory calculations, based on predetermined potential energy surfaces. Unfortunately, such potentials are extremely expensive to evaluate for chemically interesting systems. Nevertheless, attempts to conduct ab initio molecular dynamics, in which the classical description of nuclear motion is combined with quantum-mechanical determination of the forces, dates back several decades. 

One has to emphasize that Eqs. (82) and (83) do not involve the Born-Oppenheimer approximation although the nuclear motion is treated classically. This is an important advantage over the quantum molecular dynamics approach [47-54] where the nuclear Newton equations (82) are solved simultaneously with a set of ground-state KS equations at the instantaneous nuclear positions. In spite of the obvious numerical advantages one has to keep in mind that the classical treatment of nuclear motion is justified only if the probability densities (R, t) remain narrow distributions during the whole process considered. The splitting of the nuclear wave packet found, e.g., in pump-probe experiments [55-58] cannot be properly accounted for by treating the nuclear motion classically. In this case, one has to face the complete system (67-72) of coupled TDKS equations for electrons and nuclei. 

The time scale of ET observed in these systems falls into the time scale of solvent relaxation and of nuclear motions and the reaction will be severely influenced by these dynamics. It is also the aim of the present study to investigate the role of intermolecular interaction, namely, solvent effects to the reaction as well as the effects of inter- and intramolecular dynamics. 

In addition to the thermal bath of nuclear motions, important groups of solids— metals and semiconductors provide continua of electronic states that can dominate the dynamical behavior of adsorbed molecules. For example, the primary relaxation route of an electronically excited molecule positioned near a metal surface is electron and/or energy transfer involving the electronic degrees of freedom in the metal. In this section we briefly outline some concepts from the electronic structure of solids that are needed to understand the interactions of molecules with such environments. 

Consider now the electron transfer process. In contrast to the problem discussed in Sections 16.3 and 16.4, of electron transfer between molecular donor and acceptor states, where the role of nuclear motion was critical for converting a two-state dynamics into a rate process, in the present situation a rate exists even in the absence of nuclear relaxation because of the presence of a continuum of metal levels. We will start by considering this problem, disregarding nuclear motion. 

Equations (17,8) and (17.9) were obtained under the assumption that electron transfer takes place in the absence of nuclear motions. How do such motions, that were found to play a central role in molecular electron transfer, affect the dynamics in the present case In analogy to Eq. (16.52) we can now write for electron transfer to the metal... 

The exploration of ultrafast molecular and cluster dynamics addressed herein unveiled novel facets of the analysis and control of ultrafast processes in clusters, which prevail on the femtosecond time scale of nuclear motion. Have we reached the temporal boarders of fundamental processes in chemical physics Ultrafast molecular and cluster dynamics is not limited on the time scale of the motion of nuclei, but is currently extended to the realm of electron dynamics [321]. Characteristic time scales for electron dynamics roughly involve the period of electron motion in atomic or molecular systems, which is characterized by x 1 a.u. (of time) = 24 attoseconds. Accordingly, the time scales for molecular and cluster dynamics are reduced (again ) by about three orders of magnitude from femtosecond nuclear dynamics to attosecond electron dynamics. Novel developments in the realm of electron dynamics of molecules in molecular clusters pertain to the coupling of clusters to ultraintense laser fields (peak intensity I = lO -lO W cm [322], where intracluster fragmentation and response of a nanoplasma occurs on the time scale of 100 attoseconds to femtoseconds [323]. 

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