MCTDH method

If the PES are known, the time-dependent Schrbdinger equation, Eq. (1), can in principle be solved directly using what are termed wavepacket dynamics [15-18]. Here, a time-independent basis set expansion is used to represent the wavepacket and the Hamiltonian. The evolution is then carried by the expansion coefficients. While providing a complete description of the system dynamics, these methods are restricted to the study of typically 3-6 degrees of freedom. Even the highly efficient multiconfiguration time-dependent Hartree (MCTDH) method [19,20], which uses a time-dependent basis set expansion, can handle no more than 30 degrees of freedom. 

Unfortunately, the resources required for these numerically exact methods grow exponentially with the number of degrees of freedom in the system of interest. Without the use of clever algorithms to optimize the basis set used [106,107], this limits the range of systems treatable to 4-6 degrees of freedom (3-4 atoms). For larger systems, the MCTDH method [19,20,108] provides a... 

In the past decade, vibronic coupling models have been used extensively and successfully to explain the short-time excited-state dynamics of small to medium-sized molecules [200-202]. In many cases, these models were used in conjunction with the MCTDH method [203-207] and the comparison to experimental data (typically electronic absorption spectra) validated both the MCTDH method and the model potentials, which were obtained by fitting high-level quantum chemistry calculations. In certain cases the ab initio-determined parameters were modified to agree with experimental results (e.g., excitation energies). The MCTDH method assumes the existence of factorizable parameterized PESs and is thus very different from AIMS. However, it does scale more favorably with system size than other numerically exact quantum... 

Multiconfiguration time-dependent Hartree (MCTDH) method, direct molecular dynamics ... 

Similarly to the parent MCTDH method [4,5], the G-MCTDH variant [1-3] uses a wavefunc-tion ansatz corresponding to a multiconfigurational form. For coupled electronic states, the wavefunction takes the form j l ) = s)> with component wavefunctions... 

Note that each of the modes, or particles k can be multidimensional. Indeed a key aspect of the G-MCTDH method is the combination of (potentially many) degrees of freedom in such multidimensional Gaussian particles. 

To illustrate the potential of the hybrid method in describing the role of an intramolecular bath in the decay dynamics induced by a conical intersection, we consider the model of Ref. [7,8] for the S2-S1 Cl in pyrazine. Fig. 1 shows the wavepacket autocorrelation function C(t) = ( k(O)l (t) for an increasing number of bath modes. G-MCTDH hybrid calculations for 4 core (primary) modes plus nb bath (secondary) modes are compared with reference calculations by the standard MCTDH method. 

Fig. 1. The autocorrelation function C(t) = (U (O)l I (i) is shown for a wavepacket initially prepared on the upper diabatic surface [7]. Panels (a) and (b) C(t) for the four core modes calculated by the standard MCTDH method for the model Hamiltonian Hy of Eq. (9), shown on different scales in the two panels. Panel (c) G-MCTDH calculation (bold line) as compared with standard MCTDH calculation (dotted line) for the composite system with four core modes (combined into two 2-dimensional particles

In the calculations reported below, the MCTDH method has been employed for all calculations involving more than a single 2/i electronic state, i.e., involving PJT interactions. As a drawback, vibronic line spectra are not directly obtained from this (as with any wave-packet propagation) method. The spectral envelope is, however, easily obtained as a Fourier transform according to Ref. [26] ... 

M.H. Beck, et al.. The multiconfiguration time-dependent Hartree (MCTDH) method A highly efficient algorithm for propagating wavepackets, Phys. Rep.-Rev. Seet. Phys. Lett. 324 (1) (2000)... 

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 Multiconfiguration Time-Dependent Hartree (MCTDH) Method... 

The MCTDH method [79-82,113,114] uses a time development of the wavefunc-tion expanded in a basis of sets of variationally optimized time-dependent functions called single-particle functions (SPFs). A set of SPFs is used for each particle, where each particle represents a coordinate or a set of coordinates called combined mode. Indeed, when some modes are strongly coupled, and when there are many degrees of freedom, it is more efficient to combine sets of coordinates together as a... 

Interestingly, the LVC and QVC Hamiltonians of Sect. 4 are already in this form, allowing a powerful use of the MCTDH method and enabling us to include as many as 10 to 13 modes in the dynamics. 

Beck, M.H., Jackie, A., Worth, G.A. and Meyer, H.D. (2000) The multiconfigurational time-dependent Hartree (MCTDH) method a highly efficient algorithm for propagating wavepackets, P/ryi. Rep. 324,1-105 and references therein. 

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