Computations integration step

In order to solve the classical equations of motion numerically, and, thus, to t)btain the motion of all atoms the forces acting on every atom have to be computed at each integration step. The forces are derived from an energy function which defines the molecular model [1, 2, 3]. Besides other important contributions (which we shall not discuss here) this function contains the Coulomb sum... 

In Table 1 the CPU time required by the two methods (LFV and SISM) for 1000 MD integration steps computed on an HP 735 workstation are compared for the same model system, a box of 50 water molecules, respectively. The computation cost per integration step is approximately the same for both methods so that th< syieed up of the SISM over the LFV algorithm is deter-... 

The computer model consists of the numerical integration of a set of differential equations which conceptualizes the high-pressure polyethylene reactor. A Runge-Kutta technique is used for integration with the use of an automatically adjusted integration step size. The equations used for the computer model are shown in Appendix A. 

Under normal circumstances, the use of a characteristic velocity equation of the type shown above can cause difficulties in computation, owing to the existence of an implicit algebraic loop, which must be solved, at every integration step length. In this the appropriate value of L or G satisfying the value of h generated in the differential mass balance equation, must be found as shown in the information flow diagram of Fig. 3.54. 

Much effort has been devoted to producing fast and efficient numerical integration techniques, and there is a very wide variety of methods now available. The efficiency of an integration routine depends on the number of function evaluations, required to achieve a given degree of accuracy. The number of evaluations depends both on the complexity of the computation and on the number of integration step lengths. The number of steps depends on both the na-... 

It was only recently recognized (38) that such constraints, even when applied to a 1arge number of bonds simultaneously, need not appreciably increase the machine time required to do one integration step. Of course the mass-modified system does not have the same dynamics as the original system, and the rigid-bond system has neither the same dynamics nor the same statistical properties however, accurate dynamics is needed only in the bottlenecks— correct statistical properties are sufficient elsewhere. In view of the near-harmonicity of the bonded vibrations, it is probable that their effect on the statistical properties could be computed as a perturbation to the statistical properties of a rigid-bond system. 

The MCSCF optimization process is only the last step in the computational procedure that leads to the MCSCF wave function. Normally the calculation starts with the selection of an atomic orbital (AO) basis set, in which the molecular orbitals are expanded. The first computational step is then to calculate and save the one- and two-electron integrals. These integrals are commonly processed in different ways. Most MCSCF programs use a supermatrix (as defined in the closed shell HF operator) in order to simplify the evaluation of the energy and different matrix elements. The second step is then the construction of this super-matrix from the list of two-electron integrals. The MCSCF optimization procedure includes a step, where these AO integrals are transformed to MO basis. This transformation is most effectively performed with a symmetry blocked and ordered list of AO integrals. Step... 

Table 9.2 Quasi-steady versus constant versus dynamic solution domains. The darker middle boxes denote the timescale, r, for the given transient phenomena. Phenomena with timescales to left of the computational time step, At, can be considered quasi-steady. Phenomena with timescales to the right of the integration time, T, can be considered constant.

In the computer model the simultaneous equations (5), (6) and (8) are combined into a single third-order equation for Ce, which is solved iterately for each integration step. For relevant values of the various parameters, the equation has a single positive solution. 

The computational cost of this recalculation procedure is roughly the same as that of the direct double summation with respect to the trajectories. If the propagation duration is long, the re-expansion should be performed every 200 300 time integration step. This means that this new algorithm can reduce the computational cost of semiclassical calculations by more than two orders of magnitude in comparison with the direct implementation of semiclassical methods into the OCT. 

For this class of problems, the implicit approach is the basic advantage of the new method. Numerical stability is thus guaranteed, and the size of the integration step is only limited by accuracy considerations. The concomitant decrease in the number of integration steps is the principal gain in computing efficiency. 

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