Non-Adiabatic Quantum Molecular Dynamics

From a case study [5], the different excitation mechanisms (electronic and vibrational) as well as related relaxation phenomena (phase transitions and fragmentation) in atom-cluster collisions will be presented in Sect. 3. We show the gradual change of these mechanisms as a function of the impact energy in a wide range for the collision system Nag+ + Na. 

Charge transfer (ct) represents one of the most frequently studied phenomena in the field of ion-atom scattering [6] and has been intensively for ion-surface interactions [7] as well. In order to close this gap, there has been a great experimental effort on ct in cluster collisions [8-13]. Here in Sect. 4, we present two systematic investigations of recent experiments of ct measuring 1) integral ct cross sections for various alkali clusters [14,15], and 2) the laser-enhanced charge transfer for small sodium clusters [16]. 

Coupled equations of motions (bom) for a mixed system of classical and quantum dof can be derived by means of the Hamilton operator 

Within the na-qmd approach, electronic (quantum) and nuclear (classical) dof are treated by combining time-dependent (td) density functional theory (dft) with molecular dynamics (md). The basic theorem of tddft [24-26] states that for a system of interacting particles the many-particle state and, thus, any observable are uniquely determined by the time-dependent single-particle density p(r, t) which can be written identically as the density 

Note that here and later on r denotes the single-particle coordinate whereas R is still used as abbreviation for all nuclear positions as in Eq. (1). The potential (5) consists, on one hand, of an external potential V(r,R), which in our case is time-dependent owing to the atomic motion R( ). On the other hand, there are electron-electron interaction terms, namely the Hartree and the exchange-correlation term, which depend both via the density p on the functions tpj. The exchange-correlation potential VIC is defined within the so-called adiabatic local density approximation [25] which is the natural extension of the lda from stationary dpt. It is assumed to give reliable results for problems where the time scale of the external potential (in our case typical collision times) is larger than the electronic time scale. 

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Saalmann U, Schmidt R (1996) Non-adiabatic quantum molecular dynamics basic formalism and case study Z. fr. Phys D 38 153. doi 10.1007/s004600050077... 

In its most fiindamental fonn, quantum molecular dynamics is associated with solving the Sclirodinger equation for molecular motion, whether using a single electronic surface (as in the Bom-Oppenlieimer approximation— section B3.4.2 or with the inclusion of multiple electronic states, which is important when discussing non-adiabatic effects, in which tire electronic state is changed [15,16, YL, 18 and 19]. 

A comer-stone of a large portion of quantum molecular dynamics is the use of a single electronic surface. Since electrons are much lighter than nuclei, they typically adjust their wavefiinction to follow the nuclei [26]. Specifically, if a collision is started in which the electrons are in their ground state, they typically remain in the ground state. An exception is non-adiabatic processes, which are discussed later in this section. 

Information about critical points on the PES is useful in building up a picture of what is important in a particular reaction. In some cases, usually themially activated processes, it may even be enough to describe the mechanism behind a reaction. However, for many real systems dynamical effects will be important, and the MEP may be misleading. This is particularly true in non-adiabatic systems, where quantum mechanical effects play a large role. For example, the spread of energies in an excited wavepacket may mean that the system finds an intersection away from the minimum energy point, and crosses there. It is for this reason that molecular dynamics is also required for a full characterization of the system of interest. 

Discrete Fourier transform (DFT), non-adiabatic coupling, Longuet-Higgins phase-based treatment, two-dimensional two-surface system, scattering calculation, 153-155 Discrete variable representation (DVR) direct molecular dynamics, nuclear motion Schrodinger equation, 364-373 non-adiabatic coupling, quantum dressed classical mechanics, 177-183 formulation, 181-183... 

Muller and Stock [227] used the vibronic coupling model Hamiltonian, Section III.D, to compare surface hopping and Ehrenfest dynamics with exact calculations for a number of model cases. The results again show that the semiclassical methods are able to provide a qualitative, if not quantitative, description of the dynamics. A large-scale comparison of mixed method and quantum dynamics has been made in a study of the pyrazine absorption spectrum, including all 24 degrees of freedom [228]. Here a method related to Ehrenfest dynamics was used with reasonable success, showing that these methods are indeed able to reproduce the main features of the dynamics of non-adiabatic molecular systems. 

Since the dielectric continuum representation of the solvent has significant limitations, the molecular dynamics simulation of PCET with explicit solvent molecules is also an important direction. One approach is to utilize a multistate VB model with explicit solvent interactions [34-36] and to incorporate transitions among the adiabatic mixed electronic/proton vibrational states with the Molecular Dynamics with Quantum Transitions (MDQT) surface hopping method [39, 40]. The MDQT method has already been applied to a one-dimensional model PCET system [39]. The advantage of this approach for PCET reactions is that it is valid in the adiabatic and non-adiatic limits as well as in the intermediate regime. Furthermore, this approach is applicable to PCET in proteins as well as in solution. 

D. F. Coker and S. Bonella (2006) Linearized non-adiabatic dynamics in the adiabatic representation. In David Micha and Irene Burghardt, editors, Quantum dynamics of complex molecular systems. Springer-Verlag, Berlin, p. 307... 

Using the H-atom Rydberg tagging time-of-flight crossed molecular beam technique, Dr. Ren studied the reaction resonance and non-adiabatic effects at a full quantum resolved level in the F + H2 system. Through state-to-state resolved experiments, he provided the first conclusive evidence of reaction resonances in the F %,2) + H2 -> HF + H reaction. The dramatic difference between the dynamics for the F( P3/2) + H2(j = 0,1) reactions represents a textbook example of the role of reactant rotational level in resonance phenomena in this benchmark system. Dr. Ren has also carried out a very high-resolution experimental study on the dynamics of the isotope substituted reaction, F( P3/2) -I- HD -> HF -I- D, with spectroscopic accuracy (a few cm ). These findings provided a very clear physical picture for reaction resonances in this benchmark system, which has eluded us for more than 30 years. 

NON-ADIABATIC MOLECULAR DYNAMICS AND QUANTUM SOLVENT EFFECTS... 

Non-adiabatic molecular dynamics Quantum solvent eflfects... 

The methodology of molecular quantum dynamics applied to non-adiabatic systems is presented from a time-dependent perspective in Chap. 4. The representation of the molecular Hamiltonian is first discussed, with a focus on the choice of the coordinates to parametrize the nuclear motion and on the discrete variable representation. The multi-configuration time-dependent Hartree (MCTDH) method for the solution of the time-dependent Schrddinger equation is then presented. The chapter ends with a presentation of the vibronic coupling model of Kdppel, Domcke and Cederbaum and the methodology used in the calculation of absorption spectra. 

From the theoretical point of view, this relaxation process has been the subject of a large number of quantum dynamics investigations, based on reduced and full dimensional models. Farly works [13-17] reported three- and four-mode models and showed that a simple two-state four-dimensional model provides a qualitatively correct simulation of the UV absorption spectrum [17], These models were used to simulate various spectroscopic signals, including time-resolved transient absorption [18-20], and ionization [21] spectra, fluorescence [22] and resonance Raman spectra [23]. Worth et al. [24-27] performed accurate quantum dynamics simulations based on a model including the twenty-four vibrational modes of the molecule using the MCTDH method. These benchmark results have then been used to test various approximate methods for the simulation of non-adiabatic dynamics of molecular systems [28 0]. 

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