Eigenvalues parabolic coordinates

The choice of parabolic coordinates in [35] and Equation (56) is motivated by our interest in exploiting the connection between the superintegrable harmonic-oscillator and atomic-hydrogen systems [33-35]. For instance, the well-known eigenfunctions and energy eigenvalues for the two-dimensional harmonic oscillators can be written immediately by borrowing them from [33] ... 

For the parabolic and prolate spheroidal coordinates, the changes in the eigenfunctions and eigenvalues are reduced to the replacement of m by the value of iJb. The respective energy parameters become... 

Fig. 5.8 (A) Classical parabolic free energy functions for diabatic reactant and product states of an electron-transfer reaction (DA D A ) as functions of a dimensionless reaction coordinate. The standard free energy change (AG ), reorganization energy (A), and activation free energy (AG ) are indicated vertical arrows). The two functions are assumed to have the same curvature, which will be the case if there is no change in entropy in the reaction. (B) Potential energies (solid curves) and the first few vibrational wavefunctions (dashed and dot-dashed curves) and eigenvalues (horizontal dotted lines) for two electronic states (DA and D A ), as functions of the dimensionless vibrational coordinate (u in Eq. 2.3.2) for a harmonic vibrational mode with the same frequency (v) in the two states. The quantum number (n) of each wavefunction is indicated.

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