Asymmetric One-Dimensional Hamiltonians

In fitting vibrational energy separations, Eq. (3.32) or an equivalent reduced equation is used. 

For purposes of comparison, it is possible to classify the various types of potential functions which may be represented by the functional form used in Eq. (3.32) with a few simple considerations. The restrictions we shall make are always to locate the origin in the minimum, or if more than one, in the deepest minimum second minima or inflection points are restricted to negative values of the coordinate Z and the positive values of Z always represent the most rapidly rising portion of the function. These restrictions do not eliminate any unique shape of potential function. Any other functions described by Eq. (3.32) are related to those already included by a simple translation of the origin or by rotation about the vertical axis. These operations, at most, change the eigenvalues by an additive constant. The different types of potential functions are summarized in Table 3.1. 

It is a simple matter to restrict the potential functions as mentioned above. 

the parameter A [Eq. (3.32)] is simply a scale factor and need not be considered further. If both B and C, the respective coefficients of Z2 and Z3, are restricted to positive values, the origin will be a minimum and second minima or inflection points, if present, will occur for negative values of Z. If we make the further restriction that 9C2 36B, the origin will be in the deepest minimum. The case 9C2 = 36B represents a symmetric double minimum potential function with the origin in the right well. 

With the above in mind, all asymmetric double-minimum potential functions represented by Eq. (3.32) may be described with parameters in the range 36B 9C2 32B. The maximum occurs at 

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