Coordinates canonical

To achieve the desired separation of the reactive degree of freedom from the bath modes, we use time-dependent normal form theory [40,99]. As a first step, the phase space is extended through the addition of two auxiliary variables a canonical coordinate x, which takes the same value as time t, and its conjugate momentum PT. The dynamics on the extended phase space is described by the Hamiltonian... 

The most convenient way to plot a projection of an w-dimensional system in an m-dimensional linear projective subspace is to use multiple orthogonal projection, in which the directions of projection rays are parallel to the normal vectors (qi, q>,. .., qn-m) defining the projective subspace. Under this projection, a point is first projected in the direction of qi, then of q/, and so on. A convenient set of canonical coordinates describing the projective subspace is given by... 

Equation (9) describes a linear stoichiometric variety of dimension R. A reactive projection is defined as a multiple orthogonal projection from a C-dimensional space to a (C - R - 1)-dimensional subspace, where the directions of the R projection rays follow the directions of (qi, q>,. .., q ) defined in Eq. (10). Such projection causes the stoichiometric variety to disappear, leaving a reaction-invariant projection. The set of canonical coordinates defining the projective subspace can be found by substituting Eq. (10) into Eqs. (2)—(4) [7]. 

The origin of the canonical coordinate system is moved into the experimental domain by the transformation... 

We shall place ourselves in the Hamiltonian framework. We consider a 2n-dimensional phase space F endowed with canonical coordinates qi,..., qn,pi,..., pn. The flow in the phase space is determined by a smooth Hamiltonian function F[(q,p,t) via the Hamilton s equations... 

In the quantum mechanical case we start with a Hamilton operator H which we assume to be obtained from the Weyl quantization of a classical Hamilton function H(q, p). Like in the previous section, q = pi, p2, , pf) and p = pi, p2,. .., pd) denote the canonical coordinates and momenta, respectively, of a Hamiltonian system with d DoEs. Eor convenience, we again choose atomic units, so that q and p are dimensionless. We denote the... 

Such a system generally does not have analytically integrable equations of motion. However, we may apply Hamilton s equations of motion, solve them numerically, and thus generate a unique trajectory for each set of initial conditions we choose. The resulting dynamics generally exhibits a variety of interesting phenomena. First, the frequency of motion in each mode is no longer a constant [as would be the case if we had f(q, qi) = 0] but depends on the instantaneous values of the canonical coordinates ( p, ) ... 

For a multi-particles system, the canonical coordinates associated to tiie... 

Going a step further we may again obtain methods which exactly preserve the canonical measure based on canonical coordinate splitting, resulting in systems of the form... 

For applications, it is instructive to know the explicit expression for the Poisson bracket of two functions in canonical symplectic coordinates. Let pi,..., be canonical coordinates in a symplectic space Then... 

Kamalin, S. A., and Perelomov, A. M. "Construction of canonical coordinates on polarized coadjoint orbits of Lie groups. Comm. Math. Phys. 97 (1985), 553-568. 

We will first show how one can obtain the time-correlation function expression for the susceptibility in a classical statistical ensemble of particles which exhibits a linear response to an externally applied perturbation. This will be followed by an outline of the argument that leads to the generalized Langevin equation for the time-dependence of an arbitrary function of the molecular canonical coordinates. In both cases, derivations with minor modifications have been presented previously in numerous reviews, monographs, etc. However, the results are employed in a large fraction of current descriptions of dynamical processes in dense phases and thus, it seems worthwhile to again show the basic ideas underlying the formalism. 

The numerical evaluation of the parameters in equations (40)-(41) is discussed in Section 7. Equations (32) and (39)-(42) demonstrate that the topology of the potential energy surfaces in the vicinity of a conical intersection can be determined directly from the characteristic parameters, g, h and = (g/ -I-)/2, p, 6, z will be referred to as the canonical coordinates for R a point of conical intersection,... 

From Figure 2(a) the (R) in the Jacobi basis, r = r, R, y, are seen to be quite large which is not unexpected since Cl is near the conical intersection. However, in Figure 2(b) f /CR) in the canonical coordinate basis, z = 9, p, (and z), are small. This is in fact the correct result as can be seen from equations (46) and (47). In the canonical coordinate basis, the singularity is contained entirely in (1/p) In Figure 2(a)... 

Another class of methods that has been used to remove sampling difficulties is based on what is often called the multicanonical ensemble These methods have also been called entropy sampling methods,for reasons that are made clear below. It is easiest to understand the multicanonical methods by considering the full classical canonical coordinate-momentum distribution... 

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