Derivative Numerov-Cooley

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... 

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. 

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. 

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. 

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