Barrier connection formula

The barrier connection formula presented in this section is valid uniformly for all energies, below and somewhat above the top of the barrier. We would also like to emphasize that while the connection formulas pertaining to a turning point are one-directional (N. Froman 1966a, Froman and Froman 2002), the barrier connection formula (4.30a,b) along with (4.31)-(4.35) is bi-directional. However, when the energy is close to a resonance energy, a careful discussion is required. 

We shall first briefly describe the phase-integral approximation referred to in item (i). Then we collect connection formulas pertaining to a single transition point [first-order zero or first-order pole of Q2(z) and to a real potential barrier, which can be derived by... 

Connection formula for a real, smooth, single-hump potential barrier... 

Al being the contour of integration shown in Figs. 5.1b,c when m / 0 and in Figs. 5.2b,c when m = 0. According to the connection formula (4.38a,b) for a real potential barrier the particular solution of the differential equation (2.32a,b), which to the left of the barrier is given... 

Molecules with the same formula but in which the distances between corresponding atoms are not all the same are called structural isomers (Figure 3-1). They are of two types. If their atomic connectivities are the same, they are diastereomers, and if their atomic connectivities are different, they are constitutional isomers. Some diastereomers can become superimposable by rotation about a bond, and they are called rotational isomers. Depending on the magnitude of the barrier to rotation, geometrical isomers (high barrier) and conformers (low barrier) are distinguished. 

By examining the structural formulas for cis- and rn7 r-l,2-dichloroethene, we see that they have the same molecular formula (C2H2CI2) and the same connectivity (both compounds have two central carbon atoms joined by a double bond, and both compounds have one chlorine and one hydrogen atom attached to each carbon atom). But, their atoms have a different arrangement in space that is not interconvertible from one to another (due to the large barrier to rotation of the carbon—carbon double bond), making them stereoisomers. Furthermore, they are stereoisomers that are not mirror images of each other therefore they are diastereomers and not enantiomers. 

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