Numerically exact solution

If V(R) is known and the mahix elements ffap ate evaluated, then solution of Eq. nO) for a given initial wavepacket is the numerically exact solution to the Schrddinger equation. 

Solving the gas dynamics expressions of Kuhl et al. (1973) requires numerical integration of ordinary differential equations. Hence, the Kuhl et al. paper was soon followed by various papers in which Kuhl s numerical exact solution was approximated by analytical expressions. 

VTST has also been applied to systems with two degrees of freedom coupled to a dissipative bath." Previous results of Berezhkovskii and Zitserman which predicted strong deviations from the Kramers-Grote-Hynes expression in the presence of anisotropic friction for the two degrees of freedom " were well accounted for. Subsequent numerically exact solution of the Fokker-Planck equation further verified these results. 

Obtaining the exact rate (which is independent of qds), necessitates a real time propagation. A numerically exact solution is feasible for systems with a few degrees of freedom,already discussed above, there is still a way to go before one can rigorously implement the time evolution in a liquid. 

Figure 4.26. Comparison of the numerically exact solution (solid lines) and the effective time approximation (circles) with respect to the quadratic SNR e = 0.1 (1), 0.3 (2), 0.5 (3). Low-frequency case flio = 1CT4.

In addition to analytical solutions the possibility exists to obtain numerous exact solutions using numerical methods with help from computers. The advantage of numerical methods lies primarily with their application for complicated cases, e.g. for non-constant diffusion coefficients, for which there are no analytical solutions. 

Pseudostate calculations have the advantage over Born and optical-potential methods that they constitute a numerically-exact solution of a problem. The problem is not identical to a scattering problem but can be made quite realistic for useful classes of scattering phenomena by an appropriate basis choice. The state vectors, or equivalently the set of half-off-shell T -matrix elements, for such a calculation contain quite realistic information about the ionisation space. 

In a second set of calculations we have studied two kinds of transitions in Habsorption from the ground ro-vibrational state of X to all discrete levels of the first excited electronic state B +, and the emission from the ground ro-vibrational state of B 1Ej to all discrete levels of the ground electronic state X 1E+. The potentials and the transition moments have been taken from the numerically exact solutions of the electronic Schrodinger equation for Hi obtained by Wolniewicz (21). 

Methods based on an ASC have a long history in quantum-mechanical (QM) calculations with continuum solvent [60, 61, 77], where they are generally known as polarizable continuum models (PCMs). However, PCMs have seen little use in the area of biomolecular electrostatics, for reasons that are unclear to us. In the QM context, such methods are inherently approximate, even with respect to the model problem defined by Poisson s equation, owing to the volume polarization that results from the tail of the QM electron density that penetrates beyond the cavity and into the continuum [13, 14, 89], The effects of volume polarization can be treated only approximately within the ASC formalism [14, 15, 89], For a classical solute, however, there is no such tail and certain methods in the PCM family do afford a numerically exact solution of Poisson s equation, up to discretization errors that are systematically eliminable. Moreover, ASC methods have been generalized to... 

Figure 6.4 Plots of the numerically exact solution (continuous line) and the first order composite solutions (Eqs. 6.118) for e = 0.5.

Figure 6.4 shows the comparison between the composite asymptotic solution and the numerically exact solution for e = 0.5. It is useful to note that the asymptotic solution agrees quite well with the exact solution even when e = 0.5. 

We do this because the sin(e) and cos(e) functions do not tend to infinity, as does the tanCe) function. Use the Newton-Raphson formula given in Appendix A to show that the iteration equation to obtain the numerically exact solution for the eigenvalue is... 

Write a program and perform the computations for the numerically exact solutions to give values of the first eigenvalue for Bi = 0.01, 0.1, and 1 equal to 0.099833639, 0.3110528, and 0.86033, respectively, using relative percentage error of less than l.d-06, that is, the Nth iterated solution will be accepted as the numerically exact solution when... 

Over the last 20 years powerful methods to solve the TDSE have been developed [46, 47]. These are based on using a grid-based representation of the wavefunction and Hamiltonian and have provided detailed descriptions of non-adiabatic events. Unfortunately, such numerically exact solutions of the TDSE require huge computer resources as they scale exponentially with the number of degrees of freedom and approximations must be introduced to treat systems with more than 20 atoms, which include the majority of photochemistry. 

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