Numerical integration, vibrational

Step size is critical in all sim tilation s. fh is is th c iricrcm en t for in tc-grating th c equation s of motion. It uitim atcly deterrn in cs the accuracy of the numerical integration. For rn olecu les with high frequency motion, such as bond vibrations that involve hydrogens, use a small step size. 

The first few bound vibrational states in eaeh potential were determined by the Nu-merov algorithm [30], in conjunction with a technique, similar to Hajj s [31], which allows to change both the starting and the ending point of the numerical integration... 

Despite the fact that numerical integration is involved, pseudoanalytical procedures have been developed for calculation of first and second energy derivatives. This means that density functional models, like Hartree-Fock models are routinely applicable to determination of equilibrium and transition-state geometries and of vibrational frequencies. 

In particular, Shapiro and others calculated state-to-state photodissociation cross sections from vibrationally excited states of HCN and DCN [58], N2O [59], and O3 [60]. Eor instance, the detailed product-vibrational state distributions and absorption spectra of HCN(DCN) were compared [58]. These results were obtained employing a half-collision approximation, where the photodissociation could be depicted as consisting of two steps, that is, absorption of the photon and the dissociation, as well as an exact numerical integration of the coupled equations. In particular, it was predicted that large isotope effects can be obtained in certain regions of the spectrum by photodissociation of vibrationally excited molecules. 

For H2, accurate theoretical calculations3 of the vibration-rotation energy levels have been done by solving the radial differential equation (4.11) using numerical integration. The potential-energy function used is that found from a 100-term variational electronic wave function. 

Hamilton s equations form a set of coupled first-order differential equar tions which under normal conditions can be numerically integrated without any problems. The forces —dVi/dR and —dVi/dr and the torque —dVj/d7, which reflect the coordinate dependence of the interaction potential, control the coupling between the translational (R,P), the vibrational (r,p), and the rotational (7,j) degrees of freedom. Due to this coupling energy can flow between the various modes. The translational mode becomes decoupled from the internal motion of the diatomic fragment (i.e., dP/dt = 0 and dR/dt =constant) when the interaction potential diminishes in the limit R — 00. As a consequence, the translational energy... 

Elements for Bound-Continuum Vibrational Transitions Calculated by Numerical Integration and Basis Set Expansion Techniques. 

We shall proceed as follows. We shall first diagonalize the Schrbdinger problem [Eq. (3.46)] with respect to the vibrational and rotational quantum numbers (Section 5.1). We arrive in this way at a Schrodinger equation in the variable p with an effective potential function for each vibration—rotation state. A least squares procedure that includes the numerical integration of the Schrodinger equation for this effective Hamiltonian will be used to determine the harmonic force field and the doubleminimum inversion potential function for ( NHa, NHs), ( ND3, NTa) and NH2D, ND2H (Section 5.2). 

Once a potential energy function is chosen or determined for a molecule, there are three major components to a trajectory study the selection of initial conditions for the excited molecule, the numerical integration of the classical equations of motion, and the analysis of the trajectories and their final conditions. The last item may include the time at which the trajectory decomposed to products, the nature of the trajectory s intramolecular motion, i.e., regular or irregular, and the vibrational, rotational and translational energies of the reaction products. 

The difference equation or numerical integration method for vibrational wavefunctions usually referred to as the Numerov-Cooley method [111] has been extended by Dykstra and Malik [116] to an open-ended method for the analytical differentiation of the vibrational Schrodinger equation of a diatomic. This is particularly important for high-order derivatives (i.e., hyperpolarizabilities) where numerical difficulties may limit the use of finite-field treatments. As in Numerov-Cooley, this is a procedure that invokes the Born-Oppenheimer approximation. The accuracy of the results are limited only by the quality of the electronic wavefunction s description of the stretching potential and of the electrical property functions and by the adequacy of the Born-Oppenheimer approximation. 

For diatomic molecules, if the potential curve is known, numerical vibrational wavefunctions can be determined by the standard Numerov-Cooley technique [49] and for which computer programmes are available [50], The vibrational averaging for any state is then simply the numerical integration of , where v> is the vibrational wavefunction for the state of interest (usually the ground state). This, obviously, requires knowledge of the property P for a number of intemuclear separations (R). Alternatively, simple perturbation... 

The system of vibrational master equations has been numerically integrated with the same initial condition described in Sect. 2.3.1. Figure 15 reports the N distribution obtained in JVE with allowance for recombination at different times. One can note that at a time of 10" 3 s the N distribution with recombination is very similar to that calculated without recombination. The only part affected by recombination and by the presence of atoms is the tail of N -distribution, which is strongly overestimated in JVE without recombination (see Fig. 8). This is due to the chemical deactivation, which increases the depopulation of higher vibrational levels. These considerations suggest that the kjj values (which depend on the tail of N distribution) are smaller than the corresponding values calculated without atoms. 

The principal feature which distinguishes the numerical integration of complex-valued trajectories from real-valued ones lies in the flexibility one has in choosing the complex time path along which time is incremented. Although the quantities from which the classical S-matrix is constructed are analytic functions and thus independent of the particular time path,9 there are practical considerations that restrict the choice. Thus although translational coordinates behave as low order polynomials in time, so that nothing drastic happens to them when t becomes complex, the vibrational coordinate is oscillatory—... 

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