Condition, initial

Trajectories can in principle be initiated anywhere on the PES. The initial geometry needs to be defined (the x-, y-, and z-coordinates of every atom) along with the initial velocity of every atom. If one uses normal modes, assigning the velocities is a matter of deciding how much energy (or how many quanta) to place into each mode, along with the phase of that vibration. 

Taking the most literal approach, one might start the trajectories with reactant(s) and follow them over TSs, through intermediates and finishing at product(s). Undoubtedly, some trajectories started at reactants will proceed to products, but most of them will bounce around the neighborhood of the reactant for quite some time. It s only when the trajectory heads toward the TS that it may eventually cross over, and most trajectories that begin at reactants will not be headed in that direction. 

In order to obtain trajectories that are reactive, most MD studies initiate the trajectories in the neighborhood of the TS. A trajectory is first run in one direction, say, toward product. Next, aU of the velocities are reversed, and a second trajectory is again started at the TS and followed toward reactant. The union of these two trajectories defines a full trajectory from reactant to product 

A single trajectory indicates one path, one way for a molecular system to evolve from reactant to product This one trajectory has limited value, but a sampling of trajectories, if done in a statistically meaningful way, can provide insight toward real molecular systems. Typically, an MD simulation will have hundreds to thousands of trajectories, though fewer might be run if limited interpretation is acceptable. 

There are a mrmber of different techniques for generating statistical initial conditions for the trajectories. We discuss here only one technique to give a sense of how this sampling is created. Interested readers are directed toward more extensive discussions in Refs. 5 and 6. 

The kinetics of the diffusion-controlled reaction A -h B 0 under study is defined by the initial conditions imposed on the kinetic equations. Let us discuss this point using the production of geminate particles (defects) as an example. Neglecting for the sake of simplicity diffusion and recombination (note that even the kinetics of immobile particle accumulation under steady-state source is not a simple problem - see Chapter 7), let us consider several equations from the infinite hierarchy of equations (2.3.43)  

These two- and three-particle densities could be used as the initial conditions for the diffusion-controlled particle recombination, when the particle source is switched off. In terms of the correlation functions these initial conditions are n(0) = no, 

Strictly speaking, particle diffusion and recombination occurring in a course of irradiation might affect equations (4.1.9) to (4.1.11), but for a short time (small doses) we can neglect it. Equations (4.1.9) and (4.1.10) describe the random distribution of similar particles entering different geminate pairs and 

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As the conditions of pressure and temperature vary, the phases in which hydrocarbons exist, and the composition of the phases may change. It is necessary to understand the initial condition of fluids to be able to calculate surface volumes represented by subsurface hydrocarbons. It is also necessary to be able to predict phase changes as the temperature and pressure vary both in the reservoir and as the fluids pass through the surface facilities, so that the appropriate subsurface and surface development plans can be made. 

The four vertical lines on the diagram show the isothermal depletion loci for the main types of hydrocarbon gas (incorporating dry gas and wet gas), gas condensate, volatile oil and black oil. The starting point, or initial conditions of temperature and pressure, relative to the two-phase envelope are different for each fluid type. 

The initial condition for the dry gas is outside the two-phase envelope, and is to the right of the critical point, confirming that the fluid initially exists as a single phase gas. As the reservoir is produced, the pressure drops under isothermal conditions, as indicated by the vertical line. Since the initial temperature is higher than the maximum temperature of the two-phase envelope (the cricondotherm - typically less than 0°C for a dry gas) the reservoir conditions of temperature and pressure never fall inside the two phase region, indicating that the composition and phase of the fluid in the reservoir remains constant. 

The initial temperature of a gas condensate lies between the critical temperature and the cricondotherm. The fluid therefore exists at initial conditions in the reservoir as a gas, but on pressure depletion the dew point line is reached, at which point liquids condense in the reservoir. As can be seen from Figure 5.22, the volume percentage of liquids is low, typically insufficient for the saturation of the liquid in the pore space to reach the critical saturation beyond which the liquid phase becomes mobile. These... 

An oil reservoir which exists at initial conditions with an overlying gas cap must by definition be at the bubble point pressure at the interface between the gas and the oil, the gas-oil-contact (GOC). Gas existing in an initial gas cap is called free gas, while the gas in solution in the oil is called dissolved or solution gas. 

In a reservoir at initial conditions, an equilibrium exists between buoyancy forces and capillary forces. These forces determine the initial distribution of fluids, and hence the volumes of fluid in place. An understanding of the relationship between these forces is useful in calculating volumetries, and in explaining the difference between free water level (FWL) and oil-water contact (OWC) introduced in the last section. 

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... 

It should be noted that the recovery factor for a reservoir is highly dependenf upon the development plan, and that initial conditions alone cannot be used to determine this parameter. 

The previous section showed that the fluids present in the reservoir, their compressibilities, and the reservoir pressure all determine the amount of energy stored in the system. Three sets of initial conditions can be distinguished, and reservoir and production behaviour may be characterised in each case ... 

Physically, why does a temi like the Darling-Dennison couplmg arise We have said that the spectroscopic Hamiltonian is an abstract representation of the more concrete, physical Hamiltonian fomied by letting the nuclei in the molecule move with specified initial conditions of displacement and momentum on the PES, with a given total kinetic plus potential energy. This is the sense in which the spectroscopic Hamiltonian is an effective Hamiltonian, in the nomenclature used above. The concrete Hamiltonian that it mimics is expressed in temis of particle momenta and displacements, in the representation given by the nomial coordinates. Then, in general, it may contain temis proportional to all the powers of the products of the... 

It was assumed that, apart from a vanishingly small number of exceptions, the initial conditions do not have an effect on these averages. However, since tire limitmg value of the time averages caimot be computed, an... 

The statistics for the initial conditions, aj(0), are detennined by tlie equilibrium distribution obtained from the... 

The solutions of such partial differential equations require infomiation on the spatial boundary conditions and initial conditions. Suppose we have an infinite system in which the concentration flucPiations vanish at the infinite boundary. If, at t = 0 we have a flucPiation at origin 5C(f,0) = AC (f), then the diflfiision equation... 

Equation (A3.3.57) must be supplied with appropriate initial conditions describing the system prior to the onset of phase separation. The initial post-quench state is characterized by the order parameter fluctuations characteristic of the pre-quench initial temperature T.. The role of these fluctuations has been described in detail m [23]. Flowever, again using the renomialization group arguments, any initial short-range correlations should be irrelevant, and one can take the initial conditions to represent a completely disordered state at J = xj. For example, one can choose the white noise fomi (i /(,t,0)v (,t, 0)) = q8(.t -. ), where ( ) represents an... 

The fiindamental problem of understanding phase separation kinetics is then posed as finding the nature of late-time solutions of detemiinistic equations such as (A3.3.57) subject to random initial conditions. 

Northrup S H and Hynes J T 1980 The stable states picture of chemical reactions. I. Formulation for rate constants and initial condition effects J. Chem. Phys. 73 2700-14... 

There are significant differences between tliese two types of reactions as far as how they are treated experimentally and theoretically. Photodissociation typically involves excitation to an excited electronic state, whereas bimolecular reactions often occur on the ground-state potential energy surface for a reaction. In addition, the initial conditions are very different. In bimolecular collisions one has no control over the reactant orbital angular momentum (impact parameter), whereas m photodissociation one can start with cold molecules with total angular momentum 0. Nonetheless, many theoretical constructs and experimental methods can be applied to both types of reactions, and from the point of view of this chapter their similarities are more important than their differences. 

One consequence of perfonning the stabilization procedure is that the initial conditions that correspond to the current g (R) are changed each time stabilization is perfomied. However this does not matter as long the mitial g (R) value corresponds to the limit 0 as then all one needs is for g (R) to be small (i.e., die actual value is not important). 

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. 

The definition of initial conditions is generally limited in precision to within experimental uncertainties A and A p, more fiindamentally related by the Fleisenberg principle A q A= li/4ji. Therefore, we need to... 

Altematively, in the case of incoherent (e.g. statistical) initial conditions, the density matrix operator P(t) I 1>(0) (v(01 at time t can be obtained as the solution of the Liouville-von Neumann equation ... 

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... 

Here, = q ([) ) are the wave fiinctions of the spectroscopic states and the coefficients are detennined from the initial conditions... 

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... 

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