Transition, vertical

Figure Al.6.27. Equipotential contour plots of (a) the excited- and (b), (c) ground-state potential energy surfaces. (Here a hamionic excited state is used because that is the way the first calculations were perfomied.) (a) The classical trajectory that originates from rest on the ground-state surface makes a vertical transition to the excited state, and subsequently undergoes Lissajous motion, which is shown superimposed, (b) Assuming a vertical transition down at time (position and momentum conserved) the trajectory continues to evolve on the ground-state surface and exits from chaimel 1. (c) If the transition down is at time 2 the classical trajectory exits from chaimel 2 (reprinted from [52]).

The electron alfinity and ionization potential can be either for vertical excitations or adiabatic excitations. For adiabatic potentials, the geometry of both ions is optimized. For vertical transitions, both energies are computed for the same geometry, optimized for the starting state. 

Another technique for obtaining an ionization potential is to use the negative of the HOMO energy from a Hartree-Fock calculation. This is called Koopman s theorem it estimates vertical transitions. This does not apply to methods other than HF but gives a good prediction of the ionization potential for many classes of compounds. 

All of the predicted excitation energies are in good agreement with the experimental values. It should also be noted that the experimental excitation energy for the third state measured the adiabatic transition rather than the vertical transition, so this value must be assumed to be somewhat lower than the true vertical excitation energy. A larger basis set is needed to produce better agreement with experiment. 

In a combined experimental/computational study, the vibrational spectra of the N9H and N7H tautomers of the parent purine have been investigated [99SA(A) 2329]. Solvent effects were estimated by SCRF calculations. Vertical transitions, transition dipole moments, and permanent dipole moments of several low-lying valence states of 2-aminopurine 146 were computed using the CIS and CASSCF methods [98JPC(A)526, 00JPC(A)1930]. While the first excited state of adenine is characterized by an n n transition, it is the transition for 146. The... 

In addition to the block, which consists of four vertical transitions, corresponding to the four-level model (j, l < 1 m = n = 0), there are also four pairs of transitions, which correspond to the two spectral doublets P-Q (j = 0,1 m = 0 / = 1 n = 1) and Q-R (j = 1, m = 1 l = 0,1 n = 0). So, each of these doublets is doubly degenerate and transforms with increase of x 1 independently of the other and of the spectral triplet of the four-level problem. Transitions between levels j = l = 1, m = n = 1 are forbidden optically and they are not connected by relaxation with the other ones. Therefore they do not appear in the spectrum even if interaction of the rotator with the orienting field is taken into account thus they may be excluded from further consideration. 

This is the expression of the uncoupled Hartree Fock polarizability per unit cell [33], Since in this case, there is no coupling between the different vertical transitions, the coupling, and thus the matrices Ai and B, are responsible for the field-induced electron reorganizational effects. 

Figure 14. Tunneling to the alternative state at energy can be accompanied by a distortion of the domain boundary and thus populating the ripplon states. All transitions exemplified by solid lines involve tunneling between the intrinsic states and are coupled linearly to the lattice distortion and contribute the strongest to the phonon scattering. The vertical transitions, denoted by the dashed lines, are coupled to the higher order strain (see Appendix A) and contribute only to Rayleigh-type scattering, which is much lower in strength than that due to the resonant transitions.

Figure 1.3. Real-time femtosecond spectroscopy of molecules can be described in terms of optical transitions excited by ultrafast laser pulses between potential energy curves which indicate how different energy states of a molecule vary with interatomic distances. The example shown here is for the dissociation of iodine bromide (IBr). An initial pump laser excites a vertical transition from the potential curve of the lowest (ground) electronic state Vg to an excited state Vj. The fragmentation of IBr to form I + Br is described by quantum theory in terms of a wavepacket which either oscillates between the extremes of or crosses over onto the steeply repulsive potential V[ leading to dissociation, as indicated by the two arrows. These motions are monitored in the time domain by simultaneous absorption of two probe-pulse photons which, in this case, ionise the dissociating molecule.

Figure 5.4. Potential energy surfaces for out-of-plane bending of cyclooctatetraene and its negative ion. The arrow indicates a vertical transition upon photodetachment to the transition state of the neutral.

FIGURE 4.1 Illustration of adiabatic and vertical ionization potentials. Adiabatic I.P. refers to the energy difference between the lowest quantum states of the molecule and its positive ion. Often, Franck-Condon (vertical) transitions lead to a higher value, the vertical ionization potential. 

Nuclei move much more slowly than the much-lighter electrons, so when a transition occurs from one electronic state to another, it takes place so rapidly that the nuclei of the vibrating molecule can be assumed to be fixed during the transition. This is called the Franck-Condon principle, and a consequence of it is that an electronic transition is represented by a vertical arrow such as that shown in Figure 2.5 that is, an electronic transition occurs within a stationary nuclear framework. Thus the electronic transition accompanying the absorption of a photon is often referred to as a vertical transition or Franck-Condon transition. 

Radiative transitions may be considered as vertical transitions and may therefore be explained in terms of the Franck-Condon principle. The intensity of any vibrational fine structure associated with such transitions will, therefore, be related to the overlap between the square of the wavefunctions of the vibronic levels of the excited state and ground state. This overlap is maximised for the most probable electronic transition (the most intense band in the fluorescence spectrum). Figure... 

Fig. 2.4. Top Potential energy diagrams with vertical transitions (Franck-Condon principle). Bottom shape of the absorption bands the vertical broken lines represent the absorption lines that are observed for a vapor, whereas broadening of the spectra is expected in solution (solid line).

Bulk silicon is a semiconductor with an indirect band structure, as schematically shown in Fig. 7.12 c. The top of the VB is located at the center of the Brillouin zone, while the CB has six minima at the equivalent (100) directions. The only allowed optical transition is a vertical transition of a photon with a subsequent electron-phonon scattering process which is needed to conserve the crystal momentum, as indicated by arrows in Fig. 7.12 c. The relevant phonon modes include transverse optical phonons (TO 56 meV), longitudinal optical phonons (LO 53.5 meV) and transverse acoustic phonons (TA 18.7 meV). At very low temperature a splitting (2.5 meV) of the main free exciton line in TO and LO replicas can be observed [Kol5]. 

If we write the change of environmental energy (i.e., inner-sphere plus medium contributions) accompanying a vertical transition from precursor complex to the upper potential surface with electronic wavefunction... 

Electron transfer reactions, treated by continuum theory, suggested that the Franck-Condon barrier (the barrier for the vertical transition of electrons), which is about four times the activation barrier for the isotopic electron transfer in solution, is due to Bom continuum solvation processes. Specific contributions for the activation of ions come from the solvent continuum far from the ion the important contribution from the solvent molecules oriented toward the central ion in the first and second solvation shells is neglected. ... 

The probability of a particular vertical transition from the neutral to a certain vibrational level of the ion is expressed by its Franck-Condon factor. The distribution of Franck-Condon factors, /pc, describes the distribution of vibrational states for an excited ion. [33] The larger ri compared to ro, the more probable will be the generation of ions excited even well above dissociation energy. Photoelectron spectroscopy allows for both the determination of adiabatic ionization energies and of Franck-Condon factors (Chap. 2.10.1). 

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