Electronic and vibrational states

Before we proceed to these details we must describe some aspects of the theory of the electronic and vibrational states of diatomic molecules. To this end we return to the master equation displayed at the end of chapter 3, and develop the consequences of some of the terms contained therein. This is a huge subject, described in many textbooks, and at any level of detail which one might require. In this chapter we present what we consider to be the minimum required for a satisfactory understanding of what follows in later chapters. What is satisfactory is a subjective matter for the reader, and in many cases there are aspects to be explored in much greater depth than is to be found here. Some of these aspects are presented in later chapters, but here we deal with the essential fundamentals. 

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Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

This lineshape analysis also implies tliat electron-transfer rates should be vibrational-state dependent, which has been observed experimentally [44]- Spin-orbit relaxation has also been identified as an important factor in controlling tire identity of botli electron and vibrational-state distributions in radiationless ET reactions. 

For most purposes only the Stokes-shifted Raman spectmm, which results from molecules in the ground electronic and vibrational states being excited, is measured and reported. Anti-Stokes spectra arise from molecules in vibrational excited states returning to the ground state. The relative intensities of the Stokes and anti-Stokes bands are proportional to the relative populations of the ground and excited vibrational states. These proportions are temperature-dependent and foUow a Boltzmann distribution. At room temperature, the anti-Stokes Stokes intensity ratio decreases by a factor of 10 with each 480 cm from the exciting frequency. Because of the weakness of the anti-Stokes spectmm (except at low frequency shift), the most important use of this spectmm is for optical temperature measurement (qv) using the Boltzmann distribution function. 

E s are the unperturbed energies of the electronic and vibrational states, respectively, and Bm is a constant energy factor which depends on the M excited state. It appears from Eq. (6.5) that ungerade symmetry of inverse energy mismatch between the relevant levels. 

In order to determine the physical mechanism of initial ET including other rapid kinetics in photosynthetic RCs, it is necessary to construct a vibronic model that comprises the electronic and vibrational states of the system. It is also important to take into account temperature effect in both experiments and theories. In particular, we should stress that most of MO calculations carried out for RCs are based on the crystallographic structures. However, the structure at room temperature may be different from that obtained from the X-ray analysis,... 

The energy gap between electronic states is much greater than that between vibrational states, which in turn is much greater than that between rotational states. As a result, we are able to adequately describe the effects of electronic transitions within molecules by considering quantised electronic and vibrational states. 

Absorption of ultraviolet and visible light by molecules results in electronic transitions in which changes in both electronic and vibrational states occur. Such transitions are called vibronic transitions. 

The fundamental principle of PES is the photo-electric effect. A molecule M in the gas phase is irradiated with monochromatic UV light which is usually generated by a helium discharge source (Hel 21.22 eV, 58.43 nm Hell 40.81 eV, 30.38 nm). Electrons can be ejected when their binding energy is lower than the photon energy leaving behind a radical cation M+ in a certain electronic and vibrational state. 

Example The electronic and vibrational states of the oxygen molecular ion could be perfectly resolved by PES (Fig. 2.20), thus allowing to directly read out the Franck-Condon factors and to identify the (0 <— 0) transitions corresponding to... 

In microwave spectroscopy, the energy of the radiation lies in the range of fractions of a cm-1 through several cm 1 such energies are adequate to excite rotational motions of molecules but are not high enough to excite any but the weakest vibrations (e.g., those of weakly bound Van der Waals complexes). In rotational transitions, the electronic and vibrational states are thus left unchanged by the excitation process hence /ej = /ef and %Yl... 

For purely rotational transitions, the initial and final electronic and vibrational states are the same. Moreover, the electronic and vibrational states are not summed over in the analog of the above development because one is interested in obtaining an expression for a particular Xiv Vie ==> %fv Vfe electronic-vibrational transition s lineshape. As a result, the... 

For a molecule in a given electronic and vibrational state, it is convenient to define the permanent dipole operator d = (i/r // i/r), where v/) is a product of the electronic and vibrational states. This vector operator depends on the angles that specify the orientation of the molecule with respect to the external field axis. For diatomic molecules, d is directed along the intermolecular axis. The Stark shifts of the molecule in a DC electric field can (almost always) be found by treating the molecule as a rigid rotor and diagonalizing the matrix of the operator... 

As a final point, let us consider tire transition not simply between electronic states, but between wave functions described as products of (decoupled) electronic and vibrational states. That is, we consider wave functions A of the form... 

The effect of vibrational excitation is examined in Fig. 4. Shown here is the difference diffraction pattern of the molecule in the Si electronic state with excitation to vibration 16a8, vs. the vibrationless level 0° of the S, electronic state. The gray-scale indicates the difference in the total diffraction signals of the two vibronically excited states, divided by the diffraction signal of the molecule in the ground electronic and vibrational state. Important to note is that this difference pattern has a strong feature at a = 0°, i.e. along the direction of the laser... 

Ordinarily M will be in its ground electronic and vibrational state. The lines in the photoelectron spectrum will thus be due to M+ being produced in different electronic and vibrational states. The photoelectron spectrum consists of a number of bands each band corresponds to removal of an electron from a different MO of M and production of a different electronic state of M +. For diatomics and for polyatomics that are not too large, vibrational structure is resolved. The strongest vibrational line in a band is given by the vertical transition from M to M + for example, the 0 = 0—>3... 

In this section we deal with the second aspect of PglES, and the first aspect is treated later, when the population of different final electronic and vibrational states is discussed. 

Vibronic spectra reflect changes in the electronic and vibrational state of a molecule at the same time. It is possible to calculate the geometry of the excited species and the potential hypersurface close to the equilibrium state. For this, a spectrum is required with sufficiently well resolved vibronic structure to carry... 

The validity conditions for the semiclassic adiabatic approach in the description of the systems with orbitally non-degenerate levels are elucidated in the basic works of Bom and Oppenheimer (comprehensive discussion can be found in Refs. [6,7]). In these systems, the slow nuclear motion can be separated from the fast electronic one. The situation is quite different in the JT systems where, in general, this separation is impossible due to hybridization of the electronic and vibrational states. Nevertheless, in many important cases the adiabatic approach can serve as a relatively simple and at the same time powerful tool for the theoretical study of the JT systems giving accurate quantitative results and clear insight on the physical nature of the physical phenomena. 

Resonance-enhanced MPI, when combined with PES to yield REMPI-PES, permits determination of the electronic and vibrational states in which the ions are created. Since the final ionic state distribution is largely determined by the resonant state, this distribution provides information about the electronic and vibrational character of the resonant state. The dynamic processes which occur in the excited states can also be elucidated. In... 

In Eq. (Ill), the average is taken over QD and SM electronic and vibrational states. Then, making use of the Heisenberg equation, we obtain the following expression for the photocurrent [45,52] ... 

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