Dual transformation for wave packet dynamics: Application to Coulomb systems

A dual transformation technique that can deal with awkward Coulomb potentia ls is developed for electronic wave packet dynamics. The technique consists of the variable transformation of the Hamiltonian and the transformation o f the wave function with a normalization constraint. The time evolution is carried out by the alternating-direction implicit method. The operation of the transformed Hamiltonian on the wave function is implemented by using th ree- and five-point finite difference formulas. We apply it to the H atom a nd a realistic three-dimensional (3D) model of H-2(+). The cylindrical coor dinates rho and z are transformed as rho = f(xi) and z = g(zeta), where xi and zeta are scaled cylindrical coordinates. Efficient time evolution schem es are provided by imposing the variable transformations on the following r equirements: The transformed wave function is zero and analytic at the nucl ei; the equal spacings in the scaled coordinates correspond to grid spacing s in the cylindrical coordinates that are small near the nuclei (to cope wi th relatively high momentum components near the nuclei) and are large at la rger distances thereafter. No modifications of the Coulomb potentials are i ntroduced. We propose the form f(xi) = xi[xi(n)/(xi(n) + alpha(n))](nu). Th e parameter alpha designates the rho-range where the Coulomb potentials are steep. The n = 1 and nu = 1/2 transformation provides most accurate result s when the grid spacing Delta xi is sufficiently small or the number of gri d points, N-xi, is large enough. For small N-xi, the n = 1/2 and nu = 1 tra nsformation is superior to the n = 1 and nu = 1/2 one. The two transformati ons are also applied to the dissociation dynamics in the 3D model of H-2(+) . For the n = 1/2 and nu = 1 transformation, the main features of the dynam ics are well simulated even with moderate numbers of grid points. The valid ity of the two transformations is also enforced by the fact that the missin g volume in phase space decreases with decreasing Delta xi. (C) 1999 Americ an Institute of Physics. [S0021-9606(99)30145-8].