Initial conditions, dynamics

This section will look at formation and fluid data gathering before significant amounts of fluid have been produced hence describing how the static reservoir is sampled. Data gathered prior to production provides vital information, used to predict reservoir behaviour under dynamic conditions. Without this baseline data no meaningful reservoir simulation can be carried out. The other major benefit of data gathered at initial reservoir conditions is that pressure and fluid distribution are in equilibrium this is usuaily not the case once production commences. Data gathered at initial conditions is therefore not complicated... 

Vibrational motion is thus an important primary step in a general reaction mechanism and detailed investigation of this motion is of utmost relevance for our understanding of the dynamics of chemical reactions. In classical mechanics, vibrational motion is described by the time evolution and l t) of general internal position and momentum coordinates. These time dependent fiinctions are solutions of the classical equations of motion, e.g. Newton s equations for given initial conditions and I Iq) = Pq. 

As in classical mechanics, the outcome of time-dependent quantum dynamics and, in particular, the occurrence of IVR in polyatomic molecules, depends both on the Flamiltonian and the initial conditions, i.e. the initial quantum mechanical state I /(tQ)). We focus here on the time-dependent aspects of IVR, and in this case such initial conditions always correspond to the preparation, at a time of superposition states of molecular (spectroscopic) eigenstates involving at least two distinct vibrational energy levels. Strictly, IVR occurs if these levels involve at least two distinct... 

In view of the foregoing discussion, one might ask what is a typical time evolution of the wave packet for the isolated molecule, what are typical tune scales and, if initial conditions are such that an entire energy shell participates, does the wave packet resulting from the coherent dynamics look like a microcanonical... 

Figure B3.3.12. Sulphur atoms in liquid iron at the Earth s core conditions, simnlated by first-principle Car-Parrinello molecular dynamics, (a) Initial conditions, showing a mannally-prepared initial cluster of snlphur atoms, (b) A short tune later, indicating spontaneous dispersal of the snlphur atoms, which mingle with the surroundmg iron atoms. Thanks are dne to D Alfe and M J Gillan for this figure. For fiirtlier details see [210. 211].

It is convenient to analyse tliese rate equations from a dynamical systems point of view similar to tliat used in classical mechanics where one follows tire trajectories of particles in phase space. For tire chemical rate law (C3.6.2) tire phase space , conventionally denoted by F, is -dimensional and tire chemical concentrations, CpC2,- are taken as ortliogonal coordinates of F, ratlier tlian tire particle positions and velocities used as tire coordinates in mechanics. In analogy to classical mechanical systems, as tire concentrations evolve in time tliey will trace out a trajectory in F. Since tire velocity functions in tire system of ODEs (C3.6.2) do not depend explicitly on time, a given initial condition in F will always produce tire same trajectory. The vector R of velocity functions in (C3.6.2) defines a phase-space (or trajectory) flow and in it is often convenient to tliink of tliese ODEs as describing tire motion of a fluid in F with velocity field/ (c p). 

Election nuclear dynamics theory is a direct nonadiababc dynamics approach to molecular processes and uses an electi onic basis of atomic orbitals attached to dynamical centers, whose positions and momenta are dynamical variables. Although computationally intensive, this approach is general and has a systematic hierarchy of approximations when applied in an ab initio fashion. It can also be applied with semiempirical treatment of electronic degrees of freedom [4]. It is important to recognize that the reactants in this approach are not forced to follow a certain reaction path but for a given set of initial conditions the entire system evolves in time in a completely dynamical manner dictated by the inteiparbcle interactions. 

The chaotic nature of individual MD trajectories has been well appreciated. A small change in initial conditions (e.g., a fraction of an Angstrom difference in Cartesian coordinates) can lead to exponentially-diverging trajectories in a relatively short time. The larger the initial difference and/or the timestep, the more rapid this Lyapunov instability. Fig. 1 reports observed behavior for the dynamics of a butane molecule. The governing Newtonian model is the following set of two first-order differential equations ... 

It is appropriate to consider first the question of what kind of accuracy is expected from a simulation. In molecular dynamics (MD) very small perturbations to initial conditions grow exponentially in time until they completely overwhelm the trajectory itself. Hence, it is inappropriate to expect that accurate trajectories be computed for more than a short time interval. Rather it is expected only that the trajectories have the correct statistical properties, which is sensible if, for example, the initial velocities are randomly generated from a Maxwell distribution. 

Coordin ates of atom s can he set by n orm al translation orrotation of HyperCh cm molecules, fo set initial velocities, however, it is necessary to edit th e H l. file explicitly. The tin it o f velocity in the HIN file is. An gstrom s/picosecon d.. Areact.hin file and a script react.scr are in eluded with HyperChem to illustrate one simple reacting trajectory. In order to have these initial velocities used in a trajectory the Restart check box of the Molecular Dynamics Option s dialog box must he checked. If it is n ot, the in itial velocities in the HIN file will be ignored and a re-equilibration to the tern peratiire f of th e Molecular Dyn am ics Option s dialog box will occur. This destroys any imposed initial conditions on the molecular dynamics trajectory. 

A classical molecular dynamics trajectory is simply a set of atoms with initial conditions consisting of the 3N Cartesian coordinates of N atoms A(X, Y, Z ) and the 3N Cartesian velocities (v a VyA v a) evolving according to Newton s equation of motion ... 

In a typical dynamic trajectory, the initial position is well controlled but the endpoint of the trajectory is unknown. For chemical reaction dynamics, we are interested in trajectories that link known initial (reactant) and final (product) states so that both the initial conditions and the final conditions of the trajectory are fixed. 

In contrast to dissipative dynamical systems, conservative systems preserve phase-space volumes and hence cannot display any attracting regions in phase space there can be no fixed points, no limit cycles and no strange attractors. There can nonetheless be chaotic motion in the sense that points along particular trajectories may show sensitivity to initial conditions. A familiar example of a conservative system from classical mechanics is that of a Hamiltonian system. 

The first component on the right-hand side controls dynamic development of the response in the same way as in Eq. (3.8), and the other two control spectral exchange due to collisions. Solution of Eq. (3.26) should satisfy the initial condition... 

A formulation of electronic rearrangement in quantum molecular dynamics has been based on the Liouville-von Neumann equation for the density matrix. Introducing an eikonal representation, it naturally leads to a general treatment where Hamiltonian equations for nuclear motions are coupled to the electronic density matrix equations, in a formally exact theory. Expectation values of molecular operators can be obtained from integrations over initial conditions. 

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