Energy levels patterns

Figure Al.2.8. Typical energy level pattern of a sequence of levels with quantum numbers nj for the number of quanta in the symmetric and antisymmetric stretch. The bend quantum number is neglected and may be taken as fixed for the sequence. The total number of quanta (n + n = 6) is the polyad number, which...

Figure Al.2.12. Energy level pattern of a polyad with resonant collective modes. The top and bottom energy levels conespond to overtone motion along the two modes shown in figure Al.2.11. which have a different frequency. The spacing between adjacent levels decreases until it reaches a minimum between the third and fourth levels from the top. This minimum is the hallmark of a separatrix [29, 45] in phase space.

In experimental data sueh evenly spaeed energy level patterns are seldom seen most eommonly, one finds spaeings En+i - En that deerease as the quantum number n inereases. In sueh eases, one says that the progression of vibrational levels displays anharmonieity. 

Molecular Orbitals Possess Specific Topology, Symmetry, and Energy-Level Patterns... 

These exhibit the expected degeneracies but not the expected energy splittings of the HMO energy-level pattern in Fig. 3.48. For example, if one takes... 

Figure 6.3 Energy level pattern and spectroscopic labeling of states from the schematic shell model. The angular momentum coupling is indicated at the left side and the numbers of nucleons needed to fill each orbital and each shell are shown on the right side. From M. G. Mayer and J. H. D. Jenson, Elementery Theory of Nuclear Shell Structure, Wiley, New York, 1955.

Figure 6.5 Energy level pattern and filling for the exotic nucleus nLi in the schematic shell model.

Hiickel noted that if electron pairs are filled into the energy-level pattern 32 or 33, a closed-shell structure (all electrons paired) will result only when the total number of pairs is odd (total number of electrons = 4re + 2, = 0, 1, 2,...) if the number of pairs is even (total number of electrons = 4re, re = 0, 1, 2,...),... 

Another intriguing ion, hexachlorobenzene dication (46), a four tt electron system, has been observed by Wasserman and his collaborators. As predicted by the simple molecular orbital energy-level pattern, the ion has two unpaired electrons.25... 

Fig. 5.2. Zeeman energy level pattern for the OD A2T,+, v = 0, IV = 1, J = 3/2 level as a function of magnetic field.

It is now instructive to make use of the weak field results to obtain an energy level pattern for the J = 1 rotational level. The complete zero-field matrix is as follows. 

Figure 8.39. Behaviour of the energy levels of HF in the J = 1 level in magnetic fields from 0 to 8 kG. The electric field was 2952 V cm-1. The vertical arrows indicate the five transitions measured by de Leeuw and Dymanus [89]. For the zero field energy level pattern, see figure 8.38.

In Figure 6.5(a), for monocyclic Cm, among the three columns of energy-level patterns the central column shows the energy levels of the a orbital system. Of the 40 valence electrons of Cm, 20 electrons occupy the lowest 10 valencc-cf orbitals. Note that these 10 levels show characteristic... 

C4H4 We shall by-pass the C4H4-case for a moment as mentioned earlier, difficulties are to be expected here because of the anticipated presence (confirmed by the energy-level pattern of Fig. 5-2) of non-bonding orbitals giving rise to an electronic configuration involving unpaired (parallel) spins—a triplet state. 

This type of energy-level pattern was evident in the MO energy-spectrum of the alternant hydrocarbon, butadiene, discussed in Chapter Two ( 2.7) and in the energy levels of the [n]-annulenes (n even) of Fig. 5-2. 

As an illustration of these ideas, and of the 4p + 2 rule, let us consider the energy-level patterns, given by equation (5-2), in the annulenes, C H , where n runs from 3 to 8. These are shown in Fig. 5-2 in which the energy-level... 

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