Mass-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... 

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... 

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. 

Figure 5. RPO s of the collinear IHI system on the LEPS A surface of Ref. (8,10). Equipotential contours are at 0.1, 0.5 and 0.9 eV relative to the bottom of the asymptotic HI well. The mass scaled coordinates x,y are defined in Fig. 4. The RPO s are the heavy lines denoting trajectories that have their two turning points on the same equipotential lines.

In our most straightforward implementation of VTST for gas-phase reactions, rather than allow arbitrary orientations of the dividing surface, we consider a one-parameter sequence of dividing surfaces that are defined in terms of a reaction path [12,13]. This procedure is applicable to complex problems, and it immediately provides a practical improvement over the conventional choice of placing the dividing surface at the saddle point. A robust choice for the reaction path is the minimum energy path (MEP), that is, the path of steepest descent in the mass-scaled coordinates [14]. The coordinates on this path are denoted q (j ) as a function of a progress variable s, and the path is defined by... 

In terms of these mass-scaled coordinates, the hamiltonian for the system, with the motion of its center of mass omitted, is... 

In mass-scaled cartesians the three cartesian coordinates of atom A are scaled by (m /y) where m is the mass of A and y is an arbitrary convenient mass. Jn mass-weighted cartesians the coordinates of atom A are weighted by m. Mass-scaled coordinates have units of length mass-weighted coordinates have units of mass length, and are usually given as u S or aQ, where 1 u = 1 universal ( C) atomic mass unit == 1822.887 mg, and 1 aQ = 1 bohr = 0.5291771 8. 

Fig. 1. The potential energy as a function of the distance along the calculated MEP (through mass-scaled coordinates with p =...

We denote the entrance and exit reaction channels by a and 3, respectively. In space-fixed coordinates, denoted by (xyz), the inter-nuclear distance vector of the reactant molecule is r and the vector of the colliding atom with respect to the centre of mass of the diatomic molecule is Ry. The angle between y and Ry is y. We define the mass-scaled coordinates... 

Here is a sufficiently large value of R that the atom and diatom in arraAgement 1 do not interact and y is the system reduced mass common to all arrangements (2-3) i.e., J,i/ J i mass scaled coordinates. 

Now, an average nuclear mass M is introduced and coefficients ba = Ha/M are defined, relating the reduced masses of the above coordinates Ra to the average mass. Defining mass-scaled coordinates, Rq, = 6q Rq, leads to the following kinetic energy operator... 

An atom-diatom coUinear system can be described in terms of the diatom separations, r a and r y, and the distances of the atoms from the diatom center of mass, R a and R y. The use of Jacobi mass scaled coordinates results in a kinetic enei gy term with the single mass... 

The subscripted i terms are reduced masses. The use of mass scaled coordinates casts the 3 body coUinear problem into that of a single particle moving on a potential eneigy surface skewed by the angle p,... 

The angle p becomes smaUer as the central atom mass, Mb decreases relative to the masses of A and C. CoUinear hyperspherical coordinates are defined in terms of the mass scaled coordinates as ... 

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