Symmetric properties potential energy surfaces

This fact and the results of several experimental and theoretical studies suggest that the nuclei around A = 100 change their shapes rapidly but that they have complex potential energy surfaces. In particular, these nuclei are supposed to be soft with respect to y deformations. However, recent investigations on odd-mass nuclei revealed properties of classical symmetric rotors. A good example is Y6q, the isotone of 98Sr and 100Zr,... 

The Hamiltonian is also invariant under the permutation of the variable sets of all identical particles and it is natural to require, if possible, that the potential energy surface is invariant under permutations of the variable sets of the identical nuclei. But in any case it is essential that the permu-tational properties of the various parts of the decoupled wavefunction be well specified in order that they are properly symmetric or antisymmetric, according to particle type, when spin variables are included. 

In this chapter we have discussed symmetry aspects of molecular structure, with emphasis on the symmetry of small distortions from a symmetric reference structure and on the interconversions between isometric structures, which mostly involve large distortions. In the language of configuration space, such interconversions take the form of pathways between equivalent points. In Sections 2.4.3, 2.6.1 and 2.6.3-5 we have described specific examples of such pathways and related them to the molecular potential energy surface. A more detailed discussion of the use of symmetry arguments in deriving properties of the energy surface is included in Chapter 5. 

Figure 9. Transition probability amplitude, to- o( ) assisted by the 3-rd effective complex resonance potential energy surface, (a) phase, (b) absolute value, (c) trajectory. The phase changes by tt in resonances. The phase drops by tt between resonances, when t E) intersects the origin. The position of resonance states is plotted in squares (open squares -symmetric resonance states, full squares - antisymmetric resonance states). Note that due to the symmetry properties of the problem, transition is not assisted by antisymmetric resonances.

In fact, the barrier for proton transfer in the maleate anion appears to lie below the zero-point vibrational energy level (W. M. Westler, private communication). Thus, vibrationally averaged properties of the maleate anion will correspond to a symmetrically bridged Cjv transition-state structure rather than to either of the asymmetrically bridged equilibrium structures in Fig. 5.22. For present purposes this interesting feature of the potential surface can be ignored. 

Unfortunately there is as yet no known way to obtain the repulsion energy from properties of the separate molecules. An attempt has been made to characterise the repulsive surface of a molecule by performing IMPT calculations between the molecule and a suitable test particle, such as a helium atom. Because the helium atom has only one molecular orbital and is spherically symmetrical, such calculations can be done much more easily than calculations involving two ordinary molecules. From the data for the repulsion between molecule A and the test particle, and between B and the test particle, it may be possible to construct a repulsive potential between A and B. Some limited progress has been made with this idea. An alternative approach has been based on the suggestion that the repulsion energy is closely correlated with the overlap between the molecular wavefunctions, but this seems likely to be more useful as a guide to the form of analytic models than as a direct route to accurate potential functions. 

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