An example of three dimensional dynamics

As an example of the split-operator method, we here consider the time propagation of the vibrational wavefunction of a triatomic molecule, such as NO2 (Sec. 5.4). The two bond lengths of the molecule are ri and r2, and the bond angle is / . The total kinetic energy operator TV is represented in a convenient form by taking the Jacobi coordinates (r, R, 6) as depicted in Fig. 3.4, in which the length R is the distance between the upper O atom and the centroid of the N and the lower O atoms, and r is the bond length of one NO moiety. It is 

We apply the split-operator method Eq. (3.7) to separately handle the potential and kinetic energy terms of the Hamiltonian. We further apply the split-operator scheme to separate the kinetic energy term into the two exponentially noncommutative parts Tr + Tr and Tg to obtain a numerical short-time propagation method. 

We note that Tg, Eq. (3.11) cannot be simply written in the form of Eq. (3.4). In such a case the general prescription is to introduce a suitable orthogonal basis set that diagonalizes the operator under study, or is to enable efficient evaluation of its operation (using the so-called discrete variable representation [240]). In the former approach, even a wave function spatially confined (such as would be the case when the angular coordinate represents molecular vibration) would be a coherent sum of a potentially large number of spatially global basis functions. Here we outline a spatially local method devised by Dateo and Metiu [106] based on the Fourier transform. 

The time propagation step in the split-operator scheme using the operator Tg of Eq. (3.11) is approximated as 

For a bound bandwidth limited wavefunction, must approach zero for large n. Therefore a discretized version of Eq. (3.19) can be used iteratively to obtain ip t + At) for all n from values at larger k and y (t) of a previous 

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