Discrete levels spectra

This expression assumes a system with a discrete level structure for systems with both a discrete and a continuous portion to their spectrum the expression consists of a sum over the discrete states and an integral over the continuous states.) Flere, ifi (v) is a solution of the time-independent Sclirodinger equation,... 

The theory of electron-transfer reactions presented in Chapter 6 was mainly based on classical statistical mechanics. While this treatment is reasonable for the reorganization of the outer sphere, the inner-sphere modes must strictly be treated by quantum mechanics. It is well known from infrared spectroscopy that molecular vibrational modes possess a discrete energy spectrum, and that at room temperature the spacing of these levels is usually larger than the thermal energy kT. Therefore we will reconsider electron-transfer reactions from a quantum-mechanical viewpoint that was first advanced by Levich and Dogonadze [1]. In this course we will rederive several of, the results of Chapter 6, show under which conditions they are valid, and obtain generalizations that account for the quantum nature of the inner-sphere modes. By necessity this chapter contains more mathematics than the others, but the calculations axe not particularly difficult. Readers who are not interested in the mathematical details can turn to the summary presented in Section 6. 

The question arises how does one distinguish experimentally between these two types of photodissociation This question can be answered from consideration of the absorption spectrum. The predissociative state is bound, and, therefore, is characterized by a set of discrete levels. The indirect channel implies the appearance of resonant structure in the photodissociation cross section as a function of the frequency of the incident radiation. Hence, discrete structure in the absorption spectrum indicates the indirect nature of the photodissociation. For example, analysis of the absorption spectrum of C2N2 leads to the conclusion that the process C2N2 (C- -IIu)+ hv -+ CNCX rtj +CN(A II) at V = 164 nm is an indirect photodissociation process (8). 

Nano-scale and molecular-scale systems are naturally described by discrete-level models, for example eigenstates of quantum dots, molecular orbitals, or atomic orbitals. But the leads are very large (infinite) and have a continuous energy spectrum. To include the lead effects systematically, it is reasonable to start from the discrete-level representation for the whole system. It can be made by the tight-binding (TB) model, which was proposed to describe quantum systems in which the localized electronic states play an essential role, it is widely used as an alternative to the plane wave description of electrons in solids, and also as a method to calculate the electronic structure of molecules in quantum chemistry. 

Bound electronic states exhibit a discrete spectrum of rovibrational eigenstates below the dissociation energy. The interaction between discrete levels of two bound electronic states may lead to perturbations in their rovibrational spectra and to nonradiative transitions between the two potentials. In the case of an intersystem crossing, this process is often followed by a radiative depletion. Above the dissociation energy and for unbound states, the energy is not quantized, that is, the spectrum is continuous. The coupling of a bound state to the vibrational continuum of another electronic state leads to predissociation. 

In practical applications, the continuum is often approximated by a discrete spectrum. To this end, one conveniently introduces a potential wall at long internuclear separations and solves for the artifically bound states.171,172 Alternatively, basis set expansion techniques can be employed.195,196 In either case, the density of states depends on external conditions, that is, the size of the box or the number of basis functions. This dependence on external conditions has to be accounted for by the energy normalization. Instead of employing a single continuum wave function with proper energy E in Eq. [240], one samples over the discrete levels with energy E -... 

The recent experimental core level spectrum by Cavell and Allison129 in Fig. 50 is very similar to that of N2 (Fig. 45), with a weak Is 3(1 jAr) satellite with 2% relative intensity at 7.2 eV above the main line, a strong 1 s 3(1 n 2 ri) satellite with 10% relative intensity at 12.2 eV, followed by a prominent discrete and continuous satellite spectrum at higher energies. The total relaxation shift is 16 eV, the MO HF ljs level lies well above the low-lying nn satellites and the limited range of the shake-up and shake-off spectrum shown in Fig. 50 only accounts for about one third of the relaxation shift (through Eq. (22)). 

Let us briefly discuss the relationship between approaches which use basis sets and thus have a discrete single-particle spectrum and those which employ the Hartree-Fock hamiltonian, which has a continuous spectrum, directly. Consider an atom enclosed in a box of radius R, much greater than the atomic dimension. This replaces the continuous spectrum by a set of closely spaced discrete levels. The relationship between the matrix Hartree-Fock problem, which arises when basis sets of discrete functions are utilized, and the Hartree-Fock problem can be seen by letting the dimensions of the box increase to infinity. Calculations which use discrete basis sets are thus capable, in principle, of yielding exact expectation values of the hamiltonian and other operators. In using a discrete basis set, we replace integrals over the continuum which arise in the evaluation of expectation values by summations. The use of a discrete basis set may thus be regarded as a quadrature scheme. 

The discrete line spectrum of hydrogen shows that only certain energies are possible that is, the electron energy levels are quantized. In contrast, if any energy level were allowed, the emission spectrum would be continuous. 

The very different spectra of iodine obtained under continuum and discrete resonance-Raman conditions are illustrated in Fig. 11 for resonance with the B state, whose dissociation limit is 20,162 cm . In the case illustrated of discrete resonance-Raman scattering, Xl =514.5 nm, and specific re-emission results from an initial transition from the v" = 1 vibrational, J" = 99 rotational level of the X state to the v = 58, J = 100 level of the B state, i.e. the transition is 58 - l" R(99). Owing to the rotational selection rule for dipole radiation, AJ = 1, a pattern of doublets appears in the emission. Clearly, the continuum resonance-Raman spectrum of iodine (Xl = 488.0 nm) is very different from the discrete case spectrum. The structure, which arises from the 0,Q, and S branches of the multitude of vibration-rotation transitions occurring, can be analysed in terms of a Fortrat diagram, as done for gaseous bromine (67). 

The discrete level structure is crucial for the lowest energy excitations, whose wavelength is comparable to Rq, and which represent collective excitations of the entire cluster. The eigenmomenta k e are defined by boundary condition (i) for the LDM, with je kneRo) = 0, where j -) are the spherical Bessel functions. The compressional density fluctuations in a liquid drop give a phonon-like discrete spectrum for all clusters sizes [84, 85, 128]... 

While no inelastic rotational cross-sections have as yet been reported for hydrogen colliding with Kr or Xe, McKellar (29) has measured a predissociation width of 0.11 cm for the (n,A,v,j,J)= (0,2,1,2,0) level of H2-Kr. As is apparently the case (see above) for H2 Ar, for which such widths have not yet been resolved, this value is somewhat (ca. 50%) larger than the prediction implied by the potential of Table I (see Table IV). Thus, it seems reasonable to expect that an analogously refined potential for H2 Kr could be obtdined from a simultaneous analysis of this observed level width and the discrete infrared spectrum. 

Despite its great success in accounting for the spectral lines of the H atom, the Bohr model failed to predict the spectrum of any other atom, even that of helium, the next simplest element. In essence, the Bohr model predicts spectral lines for the H atom and other one-electron species, such as He" (Z = 2), Li (Z = 3), and Be (Z = 4). But, it fails for atoms with more than one electron because in these systems, electron-electron repulsions and additional nucleus-electron attractions are present as well. Nevertheless, we still use the terms ground state and excited state and retain one of Bohr s central ideas in our current model the energy of an atom occurs in discrete levels. 

Molecules possess discrete levels of vibrational energy. Vibrations in molecules can be excited by interaction with waves and with particles. In electron-energy loss spectroscopy (EELS, sometimes HREELS for high resolution EELS), a beam of monochromatic, low energy electrons falls on the surface, where it excites lattice vibrations of the substrate, molecular vibrations of adsorbed species and even electronic transitions. An energy spectrum of the scattered electrons reveals how much energy the electrons have lost to vibrations, according to the formula... 

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