Jacobi coordinates reactant

If reactant coordinates are to be used for the propagation, then this function is propagated forward in time and analyzed [151] after it has reached the product region at large values of r. If state-to-state reactive probabilities and cross sections are required, the initial wavepacket must be transformed to the product Jacobi coordinates and the propagation must be performed in these coordinates. The general form of the transformation from reactant to product Jacobi coordinates is... 

The Jacobi coordinates used for the propagation of the system are not convenient for such an analysis, since they were based on reactants A and BC and therefore include the BC distance and the A-BC distance rather than the AB and AB C distances. It is therefore convenient to replace the coordinates in Eq. (4.66),... 

The total reaction probability is typically obtained fiom the reactive flux calculated at the dividing surface placed at a point-of-no-retum.[70,71] This surface is often located in the product channel, but not necessarily at the asymptote where the S-matrix elements are completely converged. Consequently, such calculations can be conveniently carried out in reactant Jacobi coordinates and the computational costs are no more expensive than that for inelastic scattering. Implemented for the Chebyshev propagation, the reaction probability is given as below [72]... 

The Hamiltonian expressed in the reactant Jacobi coordinates for a given total angular momentum J can be written as... 

The time-dependent wavefunction satisfying the Schrodinger equation (d/dr) ) = HW(t) can be expanded in terms of BF (body-fixed) rovibrational eigenfunctions defined using the reactant Jacobi coordinates as... 

The theory presented here is for calculating the initial state-specific total reaction probabilities and cross sections for a diatom-diatom reaction AB + CD - A + BCD in full dimensions. The Hamiltonian expressed in the reactant Jacobi coordinates shown in Fig. 1 for a given total angular momentum J can be written as... 

Coupled-channel equations arise in scattering dynamics when all but one of the degrees of freedom of the system are expanded in a square integral basis (of "channels"). The coupled channel equations are then solved numerically and describe motion in the unbound, or scattering coordinate. The principal difficulty of any reactive scattering calculation is that the coordinate system which best describes the asymptotic motions of reactants differs from the coordinate system best suited for products. Consequently, computational methods commonly use different coordinate systems in different parts of configuration space. Boundary conditions are expressed in terms of Jacobi coordinates (sometimes referred to as "cartesian coordinates"), where in the A -BC arrangement... 

Reactive scattering, in which the products of a collision are chemically different from the reactants, is formally similar to inelastic scattering. However, there are complications that arise from the fact that a basis set that efficiently describes the reactants is usually inefficient to describe the products and vice versa. Even the coordinate system to be used requires some care for example, Jacobi coordinates... 

In what follows, we use the mass-scaled reactant channel body-fixed Jacobi coordinates J2, r, 7 in order to describe the A - - BC collisions for the total angular momentum J = 0 (planar collisions). The two Jacobi distances are denoted as R (distance of A from the center-of-mass of BC) and r (BC internuclear distance not to be confused with the electronic coordinates as stated in the introduction), respectively. The Jacobi angle (angle between R and r) is denoted by 7. The body-fixed 2 axis is defined to be parallel to R and BC lies in the (x, z) plane. 

Figure 8.1 Product and reactant Jacobi coordinates for a triatomic reaction system.

In this approach, the non-zero part of the body-fixed nuclear wave packet (ij/i t = 0) = qi t = 0) -h ipi t = 0)) is first constructed in grid basis using reactant Jacobi coordinates for the convenience of defining the initial rovi-... 

After this and for the convenience of extracting state-to-state dynamical quantities, the initial nuclear wave packet is immediately transferred from reactant Jacobi coordinates to product Jacobi coordinates by ... 

The starting point of this approach is solving the time-dependent Schrodinger equation formulated in terms of reactant Jacobi coordinates R, r, rj, 0i, 02, 9)... 

This new PES was used in a recent full-dimensional QM calculations [107], which solves the nuclear Schrodinger equation with the Chebyshev propagator [42] in the OH-CO Jacobi coordinates. Such calculations are extremely challenging due to the three heavy atoms in the system and the large number of quantum states supported by the HOCO well. As a result, only the 7 = 0 partial wave was considered. The initial state in the reactant asymptote was represented by a... 

It should be noted that the initial and final wave packets are usually expressed in the Jacobi coordinates of their own arrangements, which results in the difficult coordinate problem in state-to-state reactive scattering as we mentioned previously. One may either choose the product Jacobi coordinate [47,53,65,151], or reactant Jacobi coordinate to propagate the initial wave packet, and there also exist two other methodologies but may be both named as reactant coordinate-based (RGB) method the first one is to employ interpolation schemes for the coordinate transformation [41,89,126,127,156], and the second one is realized by projection of both reactant and product wave packets to an intermediate coordinate [43,127], Alternatively, in the reactant-product decoupling (RPD) method [6,7,96], both the reactant and product coordinates are used, and they are divided and combined by a complex absorbing potential. 

In the original RPD approach, the source term p(t) = (1 -is saved and transformed from reactant to product Jacobi coordinates at every propagation time step. By using the multiple-step reactant-product decoupling (MRPD) scheme, one can be saving and transforming the source term at every M time [68,70,71]. 

A projection plane is often defined as Rv = Rvo for the vth (v = or y) arrangement using the corresponding product Jacobi coordinate, and the time correlation function is always carried out on this projection plane but the projection action can be carried out in either reactant or product Jacobi coordinate. The final product wave packet f(Rv) is also defined in the SF frame due to the merits already mentioned above. 

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