Scaling/ scaled coordinates

Let us continue with the atom-diatom collinear collision model, this time allowing for the possibility of the reaction A -r BC —> AB -i- C. We first introduce mass-scaled coordinates, as these are especially convenient to describe rearrangements, using... 

One nice thing about H in mass-scaled coordinates is that it is identical to the Hamiltonian of a mass point movmg in two dimensions. This is convenient for visualizing trajectory motions or wavepackets, so the mass-scaled coordinates are commonly used for plotting data from scattering calculations. 

Another reason why mass-scaled coordinates are useful is that they simplify the transfomiation to the Jacobi coordinates that are associated with the products AB + C. If we define. S as the distance from C to the centre of mass of AB, and s as the AB distance, mass scaling is accomplished via... 

Here we have introduced scaled coordinates where L is the box length (assumed cubic). 

Finally, we shall look briefly at the form of the non-adiabatic operators. Taking the kinetic energy operator in Cartesian form, and using mass-scaled coordinates where Ma is the nuclear mass associated with the ath... 

The advantage of using scaled coordinates is that they are independent of the size of t] simulation box. Thus the energy of the configuration in a different-sized box (with si<... 

Step 5 Calculate the full-scale coordinates of the flame tip X,= djR... 

We scale coordinates and momenta to remove 7 dependence of the Hamiltonian (3)... 

Monte Carlo method within this ensemble [29,30,32,33] differs from the canonical ensemble (N,V,T) in that the configuration variables are c]successive steps of the Markov chain not only q,N> must be perturbed but also the volume. For computational reasons it is convenient to introduce scaled coordinates ... 

Let us introduce the scaled coordinates, defined this time as ... 

We use mass-scaled coordinates and set h = 1, kB = 1.) Let r0>r be the coordinate of an arbitrary point still lying in the region where the harmonic approximation for V(x) is correct with the (arbitrary) given accuracy. Then, if the instanton starts out at t = 0, the time it takes to reach the point r0 equals... 

In contrast to (2.39) and (2.40), only a single mass occurs in this Hamiltonian. The picture of a billiard ball with mass fj, rolling on a two-dimensional PES, which we will frequently adopt in the following chapters, is based on mass-scaled coordinates. 

The scaling transformation which converts Eq. (98) to Eq. (101) can be introduced in a simpler manner from the passive viewpoint by using two coordinate systems the physical coordinate system, defined by r, pr, and a scaled coordinate system (Cizek and Paldus, 1977). The details are given in Appendix B, Section XIII. 

While in the bulk the phases of the growing concentration waves are random, and also the directions of the wavevectors q are controlled by random fluctuations in the inital states, a surface creates a boundary condition, and working out adynamic extension [45,129,132,133,144,156] of the model in Sect. 2.1 Eqs. (7)-(10) one finds that under typical conditions wavevectors oriented perpendicular to the walls occur, with phases such that the maxima of the waves occur at the walls (Fig. 28). In terms of a normalized order parameter i /(Z, R, x) where x is a scaled time and Z, R, are scaled coordinates perpendicular and parallel to the walls, Z=z/2 b, R=q/2 b, V /=(( )-( )crit)/(( )coex-( )crit), this dynamic extension is the Cahn-Hilliard equation [291-294]... 

To perform the isobaric-isothermal MC simulation [122], we perform Metropolis sampling on the scaled coordinates r, = L 1qi (qi are the real coordinates) and the volume V (here, the particles are placed in a cubic box of size L = /V). The trial moves from state x with the scaled coordinates r with volume V to state x with the scaled coordinate r and volume V are generated by uniform random numbers. The enthalpy is accordingly changed from Ti(E(r, V), V) to 7i E r, V), V) by these trial moves. The trial moves will be accepted with the probability... 

For the remainder of this chapter we work with Z-scaled coordinates X xjZ and p pZ. This transformation is canonical and yields... 

Thus, during dynamics, the particles could leave the central box. At equilibrium, it is not necessary for the particles to be brought back into the central box. However, when this must be done, the PBC procedure, which is similar to minimum imaging, can be performed. In this procedure, the particle coordinates q,- are converted to scaled coordinates s,-. These are then brought into the ntral cubic box by means of the dnint operation, and then unsealed using h to give back q, in the central cell. Because unshifted particle coordinates along the trajectory are often required (to calculate, e.g., mean-squared displacements), it is not necessary to perform PBC under equilibrium conditions. 

If we introduce scaled coordinates as defined in Sect. 4.1 and expand the evolution operator to first order in the mass ratio /x, the resulting expression for the quantum-classical evolution operator in unsealed coordinates is ... 

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