Time-dependent Hartree method

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 RPA formalism that we have just presented only applies at zero temperature. It is possible, however, to derive similar eigenvalue equations for the excitation frequencies x(q) by means of the time-dependent Hartree method (Fredkin and Werthamer, 1965 Hiiller, 1974 Jansen et al., 1984). The TDH equations are valid for finite temperature in the limit of T— 0 K, they become identical to the RPA equations. The TDH matrix that replaces the RPA matrix in the eigenvalue equations (126) can be written as... 

Phonons and librons in herringbone and pinwheel commensurate and incommensurate N2 monolayers on graphite were investigated based on quantum-mechanical mean-field and time-dependent Hartree methods for the ground state [19]. The latter method includes on a systematic basis rotation-translation coupling which is neglected within the mean-field approximation, and it is able to treat motions with larger (but still finite) amplitudes around... 

Meyer, H.D. and Worth, G.A., Quantum molecular dynamics propagating wave-packets and density operators using the multiconfiguration time-dependent Hartree method, Theor. Chem. Acc., 109, 251-267, 2003. 

When considering the femtosecond photoionization dynamics of complex systems, a completely exact evaluation of the time and energy resolved photoelectron spectrum is often not really necessary. Approximative schemes which require significantly lower computational effort are valuable in such cases. Within the nonperturbative formalism, Meier et al. have proposed an efficient computational scheme which incorporates the multi-configuration time-dependent Hartree method.An approximate method which is based on a classical-trajectory description of the nuclear dynamics has been elaborated by Hartmann, Heidenreich, Bonacic-Koutecky and coworkers and applied, among other systems,to the time-resolved photoionization spectroscopy of conical intersections in sodium fluoride clusters. 

The Mean-Field Method, the Random-Phase Approximation and the Time-Dependent Hartree Method... 

M. Sala, F. Gatti and S. Guerin, Coherent destruction of tunneling in a six-dimensional model of NHD2 a computational study using the multi-configuration time-dependent Hartree method , J. Chem. Phys. 141, 164326 (2014)... 

Joubert Doriol L, Gatti F, lung C, Meyer H-D (2008) Computation of vibrational energy levels and eigenstates of fluoroform using the multiconfiguration time-dependent Hartree method. J Chem Phys 129 224109... 

Full dimensional quantum dynamics calculations on polyatomic molecules will hopefully lead to a deeper understanding on molecule/surface interactions mechanisms, as the six-dimensional quantum dynamics calculations have already done for H2/metal surfaces. These kind of simulations are required to describe accurately processes involving the breaking of a X-R bond, beyond dissociation of CH4 on H and CH3. A description along the lines of the MCTDH (multi-configurational time-dependent Hartree) method would be worth exploring. For these polyatomic molecules, a better description of the molecule-lattice coupling is also desirable. 

Beck MH, Jackie A, Worth GA, Meyer HD (2000) The multiconfiguration time-dependent Hartree method a highly efScient algorithm for propagating wavepackets. Phys Rep 324 1... 

Meyer H-D (2012) Studying molecular quantum dyntimics with the multiconfiguration time-dependent Hartree method. Whey Intentiscip Rev Comput Mol Sci 2 351... 

Vendrell O, Meyer H-D (2011) Multilayer multiconfiguration time-dependent Hartree method implementation and applications to a Henon-Heiles Hamiltonian and to pyrazine. J Chem Phys 134 044135... 

The above is a powerful generalization of the time-dependent Hartree method used by Gerber and co-workers [24, 33-35] in which a single Hartree product F is used. 

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