Close-coupling equations

The time-independent Schrodinger equation for the bound as well as the continuum wavefunction is 

Differential equations for the Xn are obtained in the usual way inserting (3.4) into (3.3), multiplying with (pn (r) from the left, and integration over r yields the set of coupled equations,  

Equation (3.5) is a set of ordinary differential equations of second order for each radial wavefunction Xri(-R) the different expansion functions Xn (R Ef n) are coupled to all other functions by the real and symmetric potential matrix V. 

In principle, Equation (3.5) represents an infinite set of coupled equations. In practice, however, we must truncate the expansion (3.4) at a maximal channel n which turns (3.5) into a finite set that can be numerically solved by several, specially developed algorithms (Thomas et al. 1981). The required basis size depends solely on the particular system. The convergence of the close-coupling approach must be tested for each system and for each total energy by variation of n until the desired cross sections do not change when additional channels are included. Expansion (3.4) should, in principle, include all open channels (k 0) as well as some of the closed vibrational channels (k% 0). Note, however, that because of energy conservation the latter cannot be populated asymptotically. 

In order to get acquainted with the potential matrix let us assume an interaction potential of the simple form 

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Alexander M H and Manolopoulos D E 1987 A stable linear reference potential algorithm for solution of the quantum close-coupled equations in molecular scattering theory J. Chem. Phys. 86 2044-50... 

B2.2.9.1 CLOSE-COUPLING EQUATIONS FOR ELECTRON-ATOM (ION) COLLISIONS... 

Numerical solution of this set of close-coupled equations is feasible only for a limited number of close target states. For each N, several sets of independent solutions F.. of the resulting close-coupled equations are detennined subject to F.. = 0 at r = 0 and to the reactance A-matrix asymptotic boundary conditions,... 

The close-coupling equations are also applicable to electron-molecule collision but severe computational difficulties arise due to the large number of rotational and vibrational channels that must be retained in the expansion for the system wavefiinction. In the fixed nuclei approximation, the Bom-Oppenlieimer separation of electronic and nuclear motion pennits electronic motion and scattering amplitudes f, (R) to be detemiined at fixed intemuclear separations R. Then in the adiabatic nuclear approximation the scattering amplitude for ... 

Quantally, this problem is in the usual close-coupled equations category. For analysis of product angular distributions, it seem reasonable that a two-state analysis might suffice, provided that the different product channels are not strongly coupled themselves and that the absorption into channels other than the one considered is weak. 

The natural choice of the reaction coordinate R, mentioned just before Section 3.1, for describing the channels with the asymptotic arrangement A -(- B is the distance between the centers of mass of A and B. This defines the conceptually simplest set of close-coupling equations. However, the corresponding potential matrix elements Vn,/(R) are difficult to interpret since many channels are coupled to each other strongly in general. Thus, no single potential is expected to predict the physics of the processes under consideration. 

This coupling potential is smooth everywhere, which allows numerical calculations with high precision. There is no nonadiabatic coupling since the basis functions [0< )( 2C) are independent of p in each sector. The solution I Wf/o, 2C) is connected smoothly, in principle, from sector to sector by a unitary frame transformation from the /th set of channels to the (/ + l)st set [97-99]. The coordinate system is transformed from the hyperspherical to the Jacobi coordinates at some large p, beyond which the conventional close-coupling equations are employed for determining the asymptotic form of the wavefunction appropriate for the scattering boundary condition [100]. 

The results of HSCC calculations have proved much more rapid convergence with the number of coupled channels than the conventional close-coupling equations in terms of the independent-particle coordinates or the Jacobi coordinates based on them. This is considered to be because of the particle-particle correlations considerably taken into account already in the choice of the hyperspherical coordinate system. The results suggest an approximate adiabaticity with respect to the hyperradius p, even when the mass ratios might appear to violate the conditions for the adiabaticity, for example, for Ps- with three equal masses. Then, it makes sense to study an adiabatic approximation with p adopted as the adiabatic parameter. 

Each rotational state is coupled to all other states through the potential matrix V defined in (3.22). Initial conditions Xj(I 0) are obtained by expanding — in analogy to (3.26) — the ground-state wavefunction multiplied by the transition dipole function in terms of the Yjo- The total of all one-dimensional wavepackets Xj (R t) forms an R- and i-dependent vector x whose propagation in space and time follows as described before for the two-dimensional wavepacket, with the exception that multiplication by the potential is replaced by a matrix multiplication Vx-The close-coupling equations become computationally more convenient if one makes an additional transformation to the so-called discrete variable representation (Bacic and Light 1986). The autocorrelation function is simply calculated from... 

Consider the multichannel radial close-coupling equations, without Lagrange multipliers for orthogonality,... 

This is transfer covariant if all quadratically integrable functions are represented in the same orbital basis. Requiring fps to be orthogonal to all radial factor riPa(r)) enforces a unique representation, but introduces Lagrange multipliers in the close-coupling equations. An alternative is to require... 

The A-matrix can be matched at r to external channel orbitals, solutions in principle of external close-coupling equations, to determine scattering matrices. Radial channel orbital vectors, of standard asymptotic form for the A -matrix,... 

Close-coupling equations were solved using the hybrid modified log-derivative / Airy propagator of Alexander and Manolopulos [67]. The agreement... 

Figure 7. How close coupled equations are obtained in chemical dynamics...

The standard indirect approach for calculations in the continuum is to solve the close coupling equations [20,148]. The first investigation of... 

Solution of the Time-Independent Close-Coupled Equations... 

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