Mapping phase space coordinates

Because the energy is fixed, the range of available phase-space coordinates is finite, and so eventually the two branches must intersect. The reconstructed separatrix consists of a closed curve of points formed by the first. intersection of W- with (see Figure 16A). The two lines formed by map-... 

A set of points M is said to be a -dimensional manifold if each point of M has an open neighborhood, which has a continuous 1 1 map onto an open set of of R , the set of all w-tuples of real numbers. Consider an w-dimensional Riemannian manifold with metric G. In an arbitrary coordinate system x, .. . , x", the volume -form is generally given by u> = dx a a dx . Here, g is the determinant of the metric in this basis, and a denotes the wedge or antisymmetric tensor product. For a flow field on the manifold prescribed by x = x) with density f x, t), a continuity equation for f x, t) can be obtained by considering the number of ensemble members >T t) within a volume Q of phase space given by... 

The advection problem is thus described by a periodically driven non-autonomous Hamiltonian dynamical system. In such case, besides the two spatial dimensions an additional variable is needed to complete the phase space description, which is conveniently taken to be the cyclic temporal coordinate, r = t mod T, representing the phase of the periodic time-dependence of the flow. In time-dependent flows ip is not conserved along the trajectories, hence trajectories are no longer restricted to the streamlines. The structure of the trajectories in the phase space can be visualized on a Poincare section that contains the intersection points of the trajectories with a plane corresponding to a specified fixed phase of the flow, tq. On this stroboscopic section the advection dynamics can be defined by the stroboscopic Lagrangian map... 

Asymptotically, hyperspherical coordinates become inadequate since the energetically allowed space contains fewer and fewer grid points. It is therefore necessary to map the wave function onto other coordinates (e.g., Jacobi coordinates). However, in the semiclassical treatment of the problem this is not possible since the wave function is known only in a restricted phase space, i.e., in either (6, ) or (p, 0, < >) space. It is therefore necessary either to carry out the projection in these coordinates by using variable grid methodology or to introduce a mixed Jacobi-hyperspherical coordinate treatment. This latter procedure is possible since we can express the Hamiltonian as... 

Let us give a brief explanation on what this intersection means for chemical reactions. In Figure 3.10(a), a schematic picture is shown for a potential curve of a ID reaction coordinate. This model potential is of the dissociation processes. Because of the interaction with other degrees of freedom, the potential curve is time dependent. Suppose that the interaction is periodic in time and take the Poincare map of its movement. Then the flow in the phase space of the reaction coordinate looks like that shown in Figure 3.10(b). 

Next considering the case of reactive motion at the same energy E, we realize that the situation at hand is not very different. The phase space of qi is still elliptical. The phase space of is not elliptical, but it is a simple closed curve, and it still therefore has the same topology as a one-dimensional sphere (every point on a closed curve can be uniquely mapped onto a sphere). Thus, the phase space of reactive motion consists of foliated tori that span both sides of the potential barrier. These reactive tori will be skinny when sliced along the ( 2 Pz) compared to the trapped tori, because they have less energy in the vibrational coordinate and more in the reaction coordinate. In Figure 8 these are labeled Qab j... 

First we consider a system with two degrees of freedom (N = 2). Suppose we have two closed curves yi and y2 phase space, both of which encircle a tube of trajectories generated by Hamilton s equations of motion. These curves can be at two sequential times (tj, or they can be at two sequential mappings on a Poincare map These curves are associated with domains labeled (Dj, D2), which are the projections of the closed curves upon the coordinate planes (pj, qj. Because both the mapping and the time propagation are canonical transformations, the integral invariants (J-j,. 2) are preserved (constant) in either case. There are two of them, of the form ... 

Thus, a dynamic system is objectively mapped in the phase space of m coordinates and proper OT momenta x, Pi. ... 

Single surface calculations with proper phase treatment in the adiabatic representation with shifted conical intersection has been performed in polai coordinates. For this calculation, the initial adiabatic wave function tad(9, 4 > o) is obtained by mapping t, to) ittlo polai space using the relations,... 

Single surface calculations with proper phase treatment in the adiabatic representation with shifted conical intersection has been performed in polar coordinates. For this calculation, the initial adiabatic wave function bad(< , to) is obtained by mapping 4 a and R Rq = qcas < x At this point, it is necessary to mention that in all the above cases the initial wave function is localized at the positive end of the R coordinate where the negative and positive ends of the R coordinate are considered as reactive and nonreactive channels. 

An example of this procedure is shown in Fig. 1. This example shows the build-up of the 2D potential of Ti2S projected along the short c axis, but the principle is the same for creating a 3D potential. The potential is a continuous function in real space and can be described in a map (Fig. 1). On the other hand, the structure factors are discrete points in reciprocal space and can be represented by a list of amplitudes and phases (Table 1). In this Fourier synthesis we have used the structure factors calculated from the refined coordinates of Ti2S °. 

Figure 1 Fourier synthesis of the projected potential map of Xi2S along the c-axis. Amplitudes and phases of the structure factors are calculated from the refined atomic coordinates of Ti2S and listed in Table 1. The space group of Xi2S is Pnnm and unit cell parameters a= 11.35, fc=14.05 and c=3.32 A.

A Patterson map, different for each space group, is a unique puzzle that must be solved to gain a foothold on the phase problem. It is by finding the absolute atomic coordinates of a heavy atom, for both small molecule and macromolecular crystals, that initial estimates (later to be improved upon) can be obtained for the phases of the structure factors needed to calculate an electron density map. 

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