Full-dimensional

Manthe U, Seideman T and Miller WH 1993 Full-dimensional quantum-mechanical calculation of the rate-constant for the H2 + OH H2O + H reaction J. Chem. Phys. 99 10 078-81... 

Zhang D H and Zhang J Z H 1994 Full-dimensional time-dependent treatment for diatom-diatom reactions—the H2+OH reaction J. Chem. Phys. 101 1146... 

As stated earlier, the main motivation for using either PCA or PCA is to construct a low-dimensional representation of the original high-dimensional data. The notion behind this approach is that the effective (or essential, as some call it [33]) dimensionality of a molecular conformational space is significantly smaller than its full dimensionality (3N-6 degrees of freedom for an A-atom molecule). Following the PCA procedure, each new... 

Figure 25. Electron-transfer rate the electronic coupling strength at T = 500 K for the asymmetric reaction (AG = —3ffl2, oh = 749 cm ). Solid line-present full dimensional results with use of the ZN formulas. Dotted line-full dimensional results obtained from the Bixon-Jortner formula. Filled dotts-effective ID results of the quantum mechanical flux-flux correlation function. Dashed line-effective ID results with use of the ZN formulas. Taken from Ref. [28].

The point q = p = 0 (or P = Q = 0) is a fixed point of the dynamics in the reactive mode. In the full-dimensional dynamics, it corresponds to all trajectories in which only the motion in the bath modes is excited. These trajectories are characterized by the property that they remain confined to the neighborhood of the saddle point for all time. They correspond to a bound state in the continuum, and thus to the transition state in the sense of Ref. 20. Because it is described by the two independent conditions q = 0 and p = 0, the set of all initial conditions that give rise to trajectories in the transition state forms a manifold of dimension 2/V — 2 in the full 2/V-dimensional phase space. It is called the central manifold of the saddle point. The central manifold is subdivided into level sets of the Hamiltonian in Eq. (5), each of which has dimension 2N — 1. These energy shells are normally hyperbolic invariant manifolds (NHIM) of the dynamical system [88]. Following Ref. 34, we use the term NHIM to refer to these objects. In the special case of the two-dimensional system, every NHIM has dimension one. It reduces to a periodic orbit and reproduces the well-known PODS [20-22]. 

An overview of the time-dependent wavepacket propagation approach for four-atom reactions together with the construction of ab initio potential energy surfaces sufficiently accurate for quantum dynamics calculations has been presented. Today, we are able to perform the full-dimensional (six degrees-of-freedom) quantum dynamics calculations for four-atom reactions. With the most accurate YZCL2 surface for the benchmark four-atom reaction H2 + OH <-> H+H2O and its isotopic analogs, we were able to show the following ... 

A more general description of the effects of vibronic coupling can be made using the model Hamiltonian developed by Koppel, Domcke and Cederbaum [65], The basic idea is the same as that used in Section III.C, that is to assume a quasidiabatic representation, and to develop a Hamiltonian in this picture. It is a useful model, providing a simple yet accurate analytical expression for the coupled PES manifold, and identifying the modes essential for the non-adiabatic effects. As a result it can be used for comparing how well different dynamics methods perform for non-adiabatic systems. It has, for example, been used to perform benchmark full-dimensional (24-mode) quantum dynamics calculations... 

Mechanical Calculations of the Rate Constant for the H2 + OH — H + H20 Reaction Full-Dimensional Results and Comparison to Reduced Dimensionality Models. 

M. D. Coutinho Neto, A. Viel, and U. Manthe, The ground state tunneling splitting of malonaldehyde Accurate full dimensional quantum dynamics calculations. J. Chem. Phys. 121, 9207 9210 (2004). 

Exciting new developments, not discussed in the review are the extension of time-dependent wavepacket reactive scattering theory to full dimensional four-atom systems [137,199-201], the adaptation of the codes to use the power of parallel computers [202], and the development of new computational techniques for acting with the Hamiltonian operator on the wavepacket [138]. 

These equations involve two variables, a and / , and two parameters, p and ku. This is at least a more economical representation than the full dimensional forms where we had five parameters—the rate constants and p0. 

A very perceptive treatment of chemical reaction dynamics, called the reaction path Hamiltonian analysis, states that the reactive trajectory is determined as the minimum energy path, and small displacements from that path, on the potential-energy surface [64-71]. The usual analysis keeps the full dimensionality of the reacting system, albeit with a focus on motion along and orthogonal to the minimum energy path. It is also possible to define a reaction path in a reduced dimensionality representation. 

R. Schinke In the case of HNO and HO2, we calculated the number of states and simply extrapolated this number into the continuum. We believe that this is the best what can be done, provided a global potential eneigy surface and full dimensionality dynamics calculations for this potential are available. Because of the much smaller number of states, for HCO this procedure is less well defined. In our final analysis (Ref. 33 of our paper in this volume) we tested the extrapolation from the bound to the continuum region and an estimation of the density of states based on a Dunham expansion of the term energies and found that both recipes give essentially the same result. 

Fig. 8 Time-evolving XT state populations obtained from quantum dynamical (MCTDH) calculations for the 2-state model of Sec. 5.1, for different levels of the HEP hierarchy as compared with the full-dimensional (24-mode) result. Panel (a) shows the H approximation (3 modes) as compared with the II1 1 1 approximation (6 modes) and the II1 2 1 approximation (6 modes). Panel (b) presents a comparison with the approximation including a Markovian closure as described...

A final word concerns the n-body problem with n > 3. Here the main problem is the rate of convergence of the EHF and corr series expansions, which we have discussed in Section IV.C. Although we may consider the knowledge of the potential-energy surface in its full dimensionality to be of fundamental importance in the case of four or five atoms, we suspect that the same may not be true for systems with a larger number of atoms, since the main role in the system chemical reactivity may then be attributed to three, four, or five atoms which define the active molecular center. Currently under way are studies for the H03 and 04 systems, and we hope, by using the DMBE... 

Till recently, computations of vibronic spectra have been limited to small systems or approximated approaches, mainly as a consequence of the difficulties to obtain accurate descriptions of excited electronic states of polyatomic molecules and to computational cost of full dimensional vibronic treatment. Recent developments in electronic structure theory for excited states within the time-dependent density functional theory (TD-DFT) and resolution-of-the-identity approximation of coupled cluster theory (R1-CC2) and in effective approaches to simulate electronic spectra have paved the route toward the simulation of spectra for significantly larger systems. 

The results presented in this section all depend strongly on the assumption which allowed us to terminate the trajectories. For instance, certain reactive trajectories, if they were free to go on, could come back to the startii point of the reaction. Conversely, certain non-reactive trajectories, after the first process of ring opening and closure, could yield a cyclopropane molecule possessing a more suitable amount of CH2 vibration energy and the isomerization reaction could now be possible (Fig. 12a). Furthermore, the treatment of the dynamical problem in its full dimensionality might well make the unreactive region between the two reactive bands disappear. 

R.B. Gerber and M.A. Ratner, Self-consistent field methods for vibrational excitations in polyatomic molecules, Adv. Chem. Phys., 70 (1988) 97. P. Jungwirth and R.B. Gerber, Quantum dynamics of large polyatomic systems using a classically based separable potential method, J. Chem. Phys., 102 (1995) 6046 Quantum dynamics of many atom systems by classically based separable potential (CSP) method Calculations for T (Ar),2 in full dimensionality, J. Chem. Phys., 102 (1995) 8855. 

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