Energy level schemes

Energy level schemes are very instructive representations for describing the line splitting of simple spin systems such as AX or AMX spin systems and for visualizing the quantum mechanical allowed and forbidden transitions. The relative population of each state can be calculated and the signal enhancement by polarization transfer using a double-quantum, forbidden transition calculated. 

AX spin system with J x 0 energy level scheme and spectrum, a and p denotes the spin state (a = antiparallel to Bq). 

Transitions in a AX spin system coi = allowed transitions with m = 1 (total magnetic spin quantum number), coq, C02 = forbidden transitions with a total magnetic quantum change of 0 and 2 respectively. 

However the description of complex pulse sequences soon becomes cumbersome and less instructive and consequently the main application of the energy level scheme is for describing processes such as the population changes associated with a transition in a SPI experiment. NMR-SIM can be used to construct energy level schemes for a spin system in the state of thermal equilibrium. 

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Figure B2.5.12 shows the energy-level scheme of the fine structure and hyperfme structure levels of iodine. The corresponding absorption spectrum shows six sharp hyperfme structure transitions. The experimental resolution is sufficient to detennine the Doppler line shape associated with the velocity distribution of the I atoms produced in the reaction. In this way, one can detennine either the temperature in an oven—as shown in Figure B2.5.12 —or the primary translational energy distribution of I atoms produced in photolysis, equation B2.5.35.

Figure B2.5.12. Hyperfine structure energy level scheme and spectrum for the...

Before we look at the various methods of pumping we shall consider the types of energy level scheme encountered in lasing materials. 

Laser action fakes place befween excifed levels of fhe neon atoms, in a four-level scheme, fhe helium atoms serving only fo mop up energy from fhe pump source and fransfer if fo neon atoms on collision. The energy level scheme is shown in Figure 9.12. 

The spectroscopy of ion lasers is generally less well understood than that of neutral atom lasers because of the lack of detailed knowledge of ion energy-level schemes. Indeed, ion lasers were first produced accidentally and attempts to assign the transitions came later. 

Figure 9.1 8 Energy level scheme for a dye molecule showing nine processes important in laser action...

Figure 9.30 Vibrational energy level scheme for multiphoton dissociation...

Fig. 2.35. Typical EEL spectrum and corresponding energy-level scheme.

Recent observations of fluorescence in NpF6 and PuF6 (46) are consistent with the energy-level scheme proposed. However, comparison of the calculated level structure with high-resolution spectra of PuFg (44) confirms that much of the observed structure is vibronic in character, built on electronic transitions that are forbidden by the inversion symmetry at the Pu site. 

Fig. 15 Energy level scheme of the isotopic line splitting of V9 in terms of the free molecules ( Sg and 81 87 as the main contributions) and of the orthorhombic crystal with natural abundance of isotopomers, after [109], Numerical values are observed wavenumbers (in cm ), values in brackets came from MD simulations on free rings [131] and in the case of au from LD calculations [116, 117]...

Scheme 3). The qualitative energy levels (Scheme 4) show the number of valence electrons necessary to obtain closed-shell electronic structures. Each orbital in the. y-orbital set is assumed to be occupied by a pair of electrons since the 5-orbital energies are low and separate from those of the p-orbital ones, especially for heavy atoms. The total number of valence electrons for the closed-shell structures... 

The i-orbital array of three and four-membered rings is of the Hiickel conjugation. (Scheme 2). The splitting patterns of the orbital energy levels (Scheme 3) show that the total number of valence electrons for the closed-shell structures is 4Af + 2 for the three- N= 0) and four-membered rings (N= 0, 1). 

All molecules of a particular type have orbital energy level schemes that are qualitatively similar but differ in the number of valence electrons for example, BH2 and NHj belong to the same diagram. 

Figure 10. Electron excitations in radicals (a) Collective representation of one-electron transitions of the A, B, and C types if denotes MO (b) LCI energy-level scheme (Jablonski diagram) for doublet and quartet states indicating why with radicals fluorescence (- - -) but not phosphorescence is observed. Spin-forbidden transitions are represented by dashed lines.

Figure 13. Orbital energy-level scheme for the biphenyl anion radical (A) and biphenyl cation radical (C) based on the SCF calculations (59) by the Pople and Longuet-Higgins method. Pairing of MO s is indicated. Thick lines with arrows represent the five lowest transition energies. All entries are given in eV.

The colors of fireworks displays are produced by emission from atomic ions as described in Chapter 7. The explosions of fireworks promote electrons to excited states. The energy level scheme of every element is different, so fireworks manufacturers can change colors by incorporating different elements. Sodium ions emit... 

Finally, the rotational partition function of a diatomic molecule follows from the quantum mechanical energy level scheme ... 

Figure 3.5. Energy level scheme of two hypothetical molecules...

FIGURE 8.2 (a) Simplified energy-level scheme of a carotenoid molecule. The solid arrow represents the... 

If we assume coupling with single pure dyz, dxz, and dxy orbitals, we have AEyz = 16(, AEXZ = 19C, AExy = -1100(, which is qualitatively consistent with the expected MO energy level scheme. 

Fig. 20. Energy level schemes for the O- ion in (a) tetragonal symmetry and (b) orthorhombic symmetry.

Undoubtedly, the most controversial spectrum in surface studies has been that of the O- species. As indicated in Appendix C, the theoretical spectrum depends upon the energy level configuration for the 2p orbitals. For the energy level scheme shown in Fig. 20a the principal components of the g tensor are approximately given by... 

Figure 2.2 (a,b) Projection of terbium phthalocyaninato derivative showing the twist angle in the Tb(lll) ion square-antiprismatic coordination site and energy level scheme for the ground J = 6 multiples... 

Figure 2.3 (a-d) Energy level scheme derived from Ishikawa s model for the ground J = 6 multiplet for terbium and J = 15/2 for erbium in LnPc2 and LnW10 complexes. 

High-quality experimental data to be fitted, for example, magnetic data (/T vs T curves) and spectroscopic data, in order to obtain an accurate picture of the energy level scheme. 

Further evidence that this is the correct energy level scheme to be used for C2 comes from the fact that the molecule is diamagnetic. The molecular orbital configurations for these molecules can be written as... 

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