Bottlenecks Hamiltonian systems

We also found [41] that besides total energy, the velocity across the transition state plays a major role in many-dof systems to migrate the reaction bottleneck outward from the naive dividing surface S(qi = 0). A similar picture has been observed by Pechukas and co-workers [37] in 2D Hamiltonian systems, that is, as energy increases, pairs of the periodic orbit dividing surfaces (PODSs) appearing on each reactant and product side migrate outwards, toward reactant and product state, and the outermost... 

The evaluation of the action of the Hamiltonian matrix on a vector is the central computational bottleneck. (The action of the absorption matrix, A, is generally a simple diagonal damping operation near the relevant grid edges.) Section IIIA discusses a useful representation for four-atom systems. Section IIIB outlines one aspect of how the action of the kinetic energy operator is evaluated that may prove of general interest and also is of relevance for problems that require parallelization. Section IIIC discusses initial conditions and hnal state analysis and Section HID outlines some relevant equations for the construction of cross sections and rate constants for four-atom problems of the type AB + CD ABC + D. 

The quantum system. The quantum system in the laboratory knows its own Hamiltonian and solves its own Schrodinger equation with full precision and as rapidly as possible when exposed to a laser field e(t). The quantum system acts as an excellent analog computer, circumventing the field design difficulties of Hamiltonian uncertainty and computational bottlenecks. 

To discuss the separation of time scales, we begin with the argument that a system that reaches the continuum via a narrow bottleneck can exhibit more than one time scale [45a,b,f, 51]. Particular attention will be given to the question of when this will be the case. The argument begins by considering the time evolution in the bound subspace. As is well known [52,53, 54], one can confine attention to the bound levels by the introduction of an effective Hamiltonian H in which the coupling to the continuum is accounted foT by a rate operator T ... 

The relative cost of ab initio calculations depends on many variables, such as the Hamiltonian, basis set, accuracy requirement, size, and density of the system (see Appendix 2). The Fock or KS matrix diagonalization step during the solution of Eq. [25] can become the calculation bottleneck with a large basis set, when, for example, more than 1000 basis functions are used. Such a number of functions may correspond to about 100 atoms per cell, when a local basis set is used, but this is the usual size of plane wave calculations, even with a small unit cell. As many crystalline systems are highly symmetric, taking advantage of symmetry is, therefore, important for reducing computational time. 

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