Adiabatic excitation

The present paper is organized as follows In a first step, the derivation of QCMD and related models is reviewed in the framework of the semiclassical approach, 2. This approach, however, does not reveal the close connection between the QCMD and BO models. For establishing this connection, the BO model is shown to be the adiabatic limit of both, QD and QCMD, 3. Since the BO model is well-known to fail at energy level crossings, we have to discuss the influence of such crossings on QCMD-like models, too. This is done by the means of a relatively simple test system for a specific type of such a crossing where non-adiabatic excitations take place, 4. Here, all models so far discussed fail. Finally, we suggest a modification of the QCMD system to overcome this failure. 

Energy Level Crossings with Non-Adiabatic Excitations... 

Thus, neither BO nor QCMD can describe the non-adiabatic excitation at the crossing. However, as studied in [7], there is yet another feature of the QCMD model that could turn out to be useful here and might help to include the non-adiabatic process. After the crossing the adiabatic limit of QCMD is, in a sense, not uniquely determined ... 

Since QCMD reproduces the BO solution, we again have [g] = q o ignoring the non-adiabatic excitation process at the crossing. Consequently, we have to modify the very concept of QCMD bundles. 

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. 

These studies discuss vertical and adiabatic excitation energies but the photophysical behavior requires calculations along the PES and at highly distorted geometries, which are more difficult to carry out in the presence of solvent. Some theoretical work has been done in this area, but it is quite limited. 

In spectroscopy we may distinguish two types of process, adiabatic and vertical. Adiabatic excitation energies are by definition thermodynamic ones, and they are usually further defined to refer to at 0° K. In practice, at least for electronic spectroscopy, one is more likely to observe vertical processes, because of the Franck-Condon principle. The simplest principle for understandings solvation effects on vertical electronic transitions is the two-response-time model in which the solvent is assumed to have a fast response time associated with electronic polarization and a slow response time associated with translational, librational, and vibrational motions of the nuclei.92 One assumes that electronic excitation is slow compared with electronic response but fast compared with nuclear response. The latter assumption is quite reasonable, but the former is questionable since the time scale of electronic excitation is quite comparable to solvent electronic polarization (consider, e.g., the excitation of a 4.5 eV n — n carbonyl transition in a solvent whose frequency response is centered at 10 eV the corresponding time scales are 10 15 s and 2 x 10 15 s respectively). A theory that takes account of the similarity of these time scales would be very difficult, involving explicit electron correlation between the solute and the macroscopic solvent. One can, however, treat the limit where the solvent electronic response is fast compared to solute electronic transitions this is called the direct reaction field (DRF). 49,93 The accurate answer must lie somewhere between the SCRF and DRF limits 94 nevertheless one can obtain very useful results with a two-time-scale version of the more manageable SCRF limit, as illustrated by a very successful recent treatment... 

In the excited state, the redistribution of electrons can lead to localized states with distinct fluorescence spectra that are known as intramolecular charge transfer (ICT) states. This process is dynamic and coupled with dielectric relaxations in the environment [16]. This and other solvent-controlled adiabatic excited-state reactions are discussed in [17], As shown in Fig. 1, the locally excited (LE) state is populated initially upon excitation, and the ICT state appears with time in a process coupled with the reorientation of surrounding dipoles. 

Figure 13. Comparison of quantum (thick hues), QCL (thin lines), and SH (dashed lines) results as obtained for the one-mode two-state model IVa [205], Shown are (a) the adiabatic excited-state population P i), (b) the corresponding diabatic population probability and (c) the...

Figure 14. Statistical error of the adiabatic excited-state population at times r = 10 fs ( ), 30 fs -f), and 50 fs (X x x) for Model IVa [205], plotted as a function of the number of iterations N. The full lines represent fits to a 1 / /N dependence.

Table 1 Equilibrium geometries, harmonic vibrational frequencies (cm ), and adiabatic excitation energies (eV) for singlet methylene (i, and states) ...

TABLE 3. Total energies (E), adiabatic excitation energies (Te), reduced excitations level (REL) values, and dipole moments (/r) of the ground and low-lying excited states of the CH radical, as obtained with the aug-cc-pVTZ (E, Te, and REL) and aug-cc-pVDZ (/r) basis sets [110,111]. Experimental data and nuclear geometries used in the CC/EOMCC calculations are taken from Refs. [113-119]. 

N. V. Vitanov and B. Girard. Adiabatic excitation of rotational ladder by chirped laser pulses. Phys. Rev. A, 69(3) 033409 (2004). 

To provide a specific example of die method, near UV experiments have led to assignments of the vertical and adiabatic excitation energies for die I B PAg transition in A-diazene (HN=NH), where the Bg state is open-shell. Table 14.4 compares sum-method predictions at the UHF and BLYP levels of theory to diese experimental values, and also to published results at the MRCI level of theory. For diis system, die HF results are systematically too high, and the DFT too low (cf. the sum method prediction for A2 phenylnitrene in Table 14.1), but are competitive with the much more expensive MRCI results. Note that all three levels do quite well at predicting the difference in verdcal and adiabatic excitation energies. 

An electronic transition involves excitation of an electron from the ground state wave function to one of the excited state wave functions. An adiabatic excitation is one that involves adjustment of the nuclear geometry to minimize the energy of the excited molecular system. A vertical excitation is one that occurs so rapidly that the ground state geometry does not have time to change. This latter type of excitation is usually adequate for modeling UV-Vis spectra. 

Table 5 indicates that HPHF method yields slightly better results than single Cl with same basis functions. The adiabatic excitation energies calculated for the respective states by Mukheijee et al [56] and by Takeshita and Mukheijee [55] are displayed in Table 6. 

TABLE 6. Adiabatic excitation energies for the 2 A and 1 B states of hydrogen peroxide... 

Kohn and Hattig [40] have presented a quite extensive study on the performance of the CC2 method for adiabatic excitation energies, excited state structures, and excited state harmonic frequencies. The systems studied include 7 diatomic molecules, 8 triatomic molecules, and 5 larger molecules. The aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets were used. The results in general are quite encouraging, and studies of this sort with CCSD and, to the extent that they are possible, with higher level methods would be most welcome. 

Substitution of hydrogen H(1) by a methyl group has been found to have a significant impact on the excited electronic state of C, in contrast to the observations for G (see Sections 10.3.3.2.3 and 10.3.3.2.4). In the case of Me-keto C, the ROKS method does not describe the bright tttt state but a dark hit state [41, 42], Stabilization of a dark state by methylation has also been suggested by the REMPI spectrocopic measurements of He et al. [34], The optimized Sj structure closely resembles the tttt structure of the unmethylated species. However, the vertical and adiabatic excitation energies of Me-keto C are higher by 0.4 eV compared to H-keto C (see Table 10-1). 

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