Vibronic secular matrix

When the eigenfunctions of ho are taken as basis functions, the vibronic secular matrix is highly sparse, i.e. most of the matrix elements are zero. The degree of sparsity increases with the number of modes considered. For / vibrational modes, and with typically n basis functions per mode, the fraction R of non-zero entries per row or column is... 

Thus, the eigenvalue problems of the individual Hamiltonians cannot be solved separately. Rather, the multi-mode vibronic secular matrix has to be diagonalized as a whole (see Chapter 7 for a discussion of the corresponding numerical techniques). 

To calculate numerically the quantum dynamics of the various cations in time-dependent domain, we shall use the multiconfiguration time-dependent Hartree method (MCTDH) [79-82, 113, 114]. This method for propagating multidimensional wave packets is one of the most powerful techniques currently available. For an overview of the capabilities and applications of the MCTDH method we refer to a recent book [114]. Additional insight into the vibronic dynamics can be achieved by performing time-independent calculations. To this end Lanczos algorithm [115,116] is a very suitable algorithm for our purposes because of the structural sparsity of the Hamiltonian secular matrix and the matrix-vector multiplication routine is very efficient to implement [6]. 

The vibronic energy levels and vibrational amplitude functions X <+), and Xd > where a = 2,5e, or 5a) may be found by si ving the secular equation, Eq. (18). In this matrix equation,... 

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