Numerov-Cooley

We have used the Numerov-Cooley method of numerical integration to solve the Schrodinger equation... 

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. 

The relevant equations for the derivative Numerov-Cooley (DNC) method closely follow Cooley s [111] presentation. Let R be the radial coordinate or bond displacement coordinate, P R) a radial eigenfunction, and U R) the potential function. The one-dimensional Schrodinger equation is then... 

In Standard Numerov-Cooley, integration or step-by-step use of Eqn. (101) begins both at close-in and at far-out extremes where the values of P and K are near zero, assuming a bound state. These are guessed to be very small values. Then, the inward and outward functions that are obtained are matched in slope and value at some midway point by iterative adjustment of the energy. In Eqn. (101), the zero-order energy E is known already and so the process requires no iteration. This also means the integration needs to be done in only one direction. The F that is found will be a mixture of the true derivative wavefunction and the zero-order wavefunction, and so the last step is a projection step to ensure orthonormality. 

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 RKR potential may be tested against the input G(v) and B(v) values by exact solution of the nuclear Schrodinger equation [see Wicke and Harris, 1976, review and compare various procedures, e.g., Numerov-Cooley numerical integration (Cooley, 1961), finite difference boundary value matrix diagonaliza-tion (Shore, 1973), and the discrete variable representation (DVR) (Harris, et al., 1965)]. G(v) + y00 typically deviates from EVjj=o by < 1 cm-1 except near dissociation. Bv may be computed from Xv,J=o(R) by... 

Table 5.3 Electronic and vibrational contributions to the parallel components of the polarizability Uzz and first hyperpolarizability Pzzz obtained using the Numerov-Cooley intergration scheme at the CASSCF and CASPT2 levels of theory...

The ZPVA and pure vibrational (pv) corrections from the Numerov-Cooley intergration... 

Ingamells, Papadopoulos and Sadlej have introduced a new semi-numerical method based on Numerov-Cooley integration to calculate the vibrational... 

Numerov-Cooley and DNC are ideal for one-dimensional oscillators since anharmonicity of any sort presents no complication and because it can be applied to the full manifold of bound vibrational states. It would be extremely valuable to have this same capability for multidimensional vibrational problems. A possibility is to use a self-consistent or effeaive potential for the mode-mode interaction and then treat each mode numerically with Numerov-Cooley. This would be an approximation, hopefully a good approximation, and it would lend itself to direct differentiation. 

C. E. Dykstra and D. J. Malik,. Chem. Phys., 87, 2806 (1987). Derivative Numerov-Cooley Theory. A Method for Finding Vibrationsil State Properties of Diatomic Molecules. 

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