Morse potential matrix elements

Berrondo, M Palma, A., and L6pez-Bonilla, J. L. (1987), Matrix Elements for the Morse Potential Using Ladder Operators, Int lJ. Quant. Chem. 31,243. 

Matsumoto, A. (1988), Generalized Matrix Elements in Discrete and Continuum States for the Morse Potential, J. Phys. B 21, 2863. 

Matrix elements of the other primitive basis functions are readily obtained by differentiation. It is possible to demonstrate that the form of the matrix elements is very much the same as for the simpler matrix elements that arise with the Morse potential. Thus, one finds for... 

Finally, for the Morse potential, one has to consider the derivatives of the exponential operator in the matrix elements ... 

The simple treatment of the nearly free ion was based on the use of a fast Fourier transform of the effective potential measured along the helical axis. It is possible, however, to evaluate matrix elements of both the Coulomb interaction and the Morse potential in terms of basis functions that are bound in the x,y-plane and a plane wave along the helical axis. The purpose of this appendix is to outline these evaluations. 

An expansion of the Morse potential, for example, in a set of Gaussian functions is given by eq (C4) in Appendix C. Matrix elements of the Morse potential in terms of the Gaussian primitive basis functions are therefore simply three center overlap integrals [49], These matrix elements can be evaluated for each term in the sum and then converted to the final expression in a straightforward manner. 

Once a model potential is derived, it is possible to verify and refine this potential, making use of observed perturbation matrix elements and calculated overlap integrals between the vibrational levels of the two interacting electronic states. Although it is usually more convenient to input the analytic form of V(R) into a numerical integration program to calculate overlap integrals (Section 5.1.3), analytic expressions exist for harmonic and Morse (vi vj) factors. 

Finally, the normal coordinates need not be linear combinations of the internal extension coordinates. In the results reported in this chapter, we use Simons-Parr-Finlan (SPF) (68) or Morse coordinates (2) for describing the stretching degrees of freedom. The normal coordinates are then defined as the appropriate linear combination of these coordinates. When we expand the coordinate dependent terms of the Hamiltonian in a Taylor series to a given order, the normal coordinates based on the SPF or Morse coordinates lead to a more accurate representation of both the model potential and the G matrix elements than do expansions based on the usual internal extension coordinates. In the case of the SPF coordinates, these expansions are exact at fourth order. Likewise, an appropriate choice of bending coordinates can also provide a more rapid convergence of these terms (49). 

Here, the HC bond is taken to be a Morse oscillator (r is the stretch coordinate) while the HCC bending potential ( 6 is the HCC wag coordinate) is a harmonic oscillator with r-dependent force constant. The CC bond stretch coordinate R and the CCC bond angle are frozen at their equilibrium values. The stretch-bend G-matrix elements are ... 

A semi-classical treatment171-175 of the model depicted in Fig. 15, based on the Morse curve theory of thermal dissociative electron transfer described earlier, allows the prediction of the quantum yield as a function of the electronic matrix coupling element, H.54 The various states to be considered in the region where the zero-order potential energy curves cross each other are shown in the insert of Fig. 15. The treatment of the whole kinetics leads to the expression of the complete quenching fragmentation quantum yield, oc, given in equation (61)... 

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