Virtual states

Molecules initially in the J = 0 state encounter intense, monochromatic radiation of wavenumber v. Provided the energy hcv does not correspond to the difference in energy between J = 0 and any other state (electronic, vibrational or rotational) of the molecule it is not absorbed but produces an induced dipole in the molecule, as expressed by Equation (5.43). The molecule is said to be in a virtual state which, in the case shown in Figure 5.16, is Vq. When scattering occurs the molecule may return, according to the selection mles, to J = 0 (Rayleigh) or J = 2 (Stokes). Similarly a molecule initially in the J = 2 state goes to... 

Figure 5.16 Raman and Rayleigh scattering processes involving virtual states Fq and Fj...

The mechanism for Stokes and anti-Stokes vibrational Raman transitions is analogous to that for rotational transitions, illustrated in Figure 5.16. As shown in Figure 6.3, intense monochromatic radiation may take the molecule from the u = 0 state to a virtual state Vq. Then it may return to u = 0 in a Rayleigh scattering process or to u = 1 in a Stokes Raman transition. Alternatively, it may go from the v = state to the virtual state Fj and return to V = (Rayleigh) or to u = 0 (Raman anti-Stokes). Flowever, in many molecules at normal... 

The reason why the spacings are equal, and not the 1-0, 2-1, 3-2,... anharmonic intervals, is explained in Figure 9.21. The laser radiation of wavenumber Vg takes benzene molecules into the virtual state Fj from which they may drop down to the v = level. The resulting Stokes scattering is, as mentioned above, extremely intense in the forward direction with about 50 per cent of the incident radiation scattered at a wavenumber of Vg — Vj. This radiation is sufficiently intense to take other molecules into the virtual state V2, resulting in intense scattering at Vg — 2vj, and so on. 

The similarity between a two-photon absorption and a Raman scattering process is even closer. Figure 9.27(a) shows that a Raman transition between states 1 and 2 is really a two-photon process. The first photon is absorbed at a wavenumber to take the molecule from state 1 to the virtual state V and the second photon is emitted at a wavenumber Vj,. 

In a two-photon absorption process the first photon takes the molecule from the initial state 1 to a virtual state V and the second takes it from V to 2. As in Raman spectroscopy, the state V is not an eigenstate of the molecule. The two photons absorbed may be of equal or unequal energies, as shown in Figures 9.27(b) and 9.27(c). It is possible that more than two photons may be absorbed in going from state 1 to 2. Figure 9.27(d) illustrates three-photon absorption. 

Figure 9.27 Multiphoton processes (a) Raman scattering, (b) absorption of two identical photons, (c) absorption of two different photons and (d) absorption of three identical photons. V and V are virtual states...

Figure 9.28 A two-photon (or more) absorption process may be monitored by (a) measuring total, undispersed fluorescence or (b) counting the ions produced by a further photon (or photons). T is a virtual state...

Iang87] Langton, C.G., Virtual state machines in cellular automata . Complex Systems 1 (1987) 257. 

The last state in Fig. 11.1 that has not yet been discussed is the state of the neat liquid compound X. For liquid compounds this is the relevant initial state for solubility, but almost aU drug-Uke compounds are solid at room temperature. In this case the neat liquid is a virtual state of a supercooled liquid which can hardly be accessed experimentally. However, it is an interesting intermediate state because it allows us to split the calculation of solubility into two separate steps, which are conceptually and for some methods computationally easier to handle than the complete step from the crystaUine state of the drug to the liquid state of the drug dissolved in water. In the first step we only have to transfer the compound from its neat crystalline state to its neat liquid state. The free energy of this fusion transfer is usually called AG s (or if considered in the opposite direction). 

Laser-based methods of identification are extremely powerful they are able to provide species and structural information, as well as accurate system temperature values. Spontaneous Raman scattering experiments are useful for detection of the major species present in the system. Raman scattering is the result of an inelastic collision process between the photons and the molecule, allowing light to excite the molecule into a virtual state. The scattered light is either weaker (Stokes shifted) or... 

The detection probability for a given trajectory depends on the fragment orientation (its Mj value) and the nature of the probe transition. All of these images were obtained via the two-photon Ilg XAS) transition. Five rotational branches are thus possible O, P, Q, R and S. The amplitudes for each of these two-photon transitions can be obtained from a sum of paired, Mj-dependent, one-photon amplitudes.37 The O branch, for example, consists of a contribution from a parallel P-type transition to a 7A virtual state, followed by a perpendicular P-type transition to the final 1ffs Rydberg (which is assumed to be ionized promptly). The product of those two transition amplitudes must be summed with another product in which the first transition is perpendicular and the second is parallel. The P and R branches consist of four contributions each and the Q branch has six such terms in its transition amplitude. The required one-photon amplitudes are taken from Ref. 37. 

The Raman effect arises when a photon is incident on a molecule and interacts with the electric dipole of the molecule. In classical terms, the interaction can be viewed as a perturbation of the molecule s electric field. In quantum mechanics the scattering is described as an excitation to a virtual state lower in energy than a real electronic transition with nearly coincident de-excitation and a change in vibrational energy. The scattering event occurs in 10 14 seconds or less. The virtual state description of scattering is shown in Figure 1. 

The quality of the TD-DFT results is determined by the quality of the KS molecular orbitals and the orbital energies for the occupied and virtual states. These in turn depend on the exchange-correlation potential. In particular, excitations to Rydberg and valence states are sensitive to the behavior of the exchange-correlation potential in the asymptotic region. If the exchange-correla-... 

D level lies well outside virtual states of B... 

Fig. 11.4. Schematic of two-photon excitation compared to one-photon excitation. The dashed line represents the virtual state that mediates the absorption.

In conventional fluorescence spectroscopy, a fluorophore is excited by absorption of one photon whose energy corresponds to the energy difference between the ground state and the excited state. Excitation is also possible by the simultaneous absorption of two photons of lower energy (i.e. of longer wavelength) via a short-lived virtual state (Figure 11.4)3). For instance, absorption of two photons in the red can... 

In open shell metals, these empty states can be d- or f-states somewhat hybridized with band states (see Chap. A). In a metal, these states may be pulled down into the conduction band (as a virtual state, see Chap. A) in a compound, presenting a ligand valence band (insulator or semiconductor), they may be pulled down to an energy position coinciding with or very near to this valence band (as a true impurity level). The two possible final states (Eqs. 22 a and 22 b) explain the occurrence of a split response one of the crystal band electrons occupies either the outer hole level P (Eq. 22 a) or the more bandlike hole B " (Eq. 22 b). 

When pulled down from beyond Ep, the quasi-localized state may be seen as being a virtual state in the conduction band (e.g., a 5f state in the (s, p, d) conduction band). In this situation, the probability of occupation of the state by a conduction electron, is clearly related to the WsJA+ ratio. 

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