Coupling potential energy

Figure 55. Two-dimensional coupled potential energy surfaces and the wavepacket motion, (a) Si — S2 surfaces and (b) Si — So surfaces. The black, gray, and white circles and dotted lines indicate the locations of the FC region. Si - S2 conical intersection minimum, 5MR Si — So conical intersection minimum, and seam lines, respectively. The solid arrows indicate the schematic wavepacket pathway in the case of natural photoisomerization starting from the vibrational ground state. Taken from Ref. [49].

Figure 56. Changes of the wavepacket populations on the Si - So coupled potential energy surfaces as a function of time duration from the FC region. Taken from Ref. [48].

As a last example of a molecular system exhibiting nonadiabatic dynamics caused by a conical intersection, we consider a model that recently has been proposed by Seidner and Domcke to describe ultrafast cis-trans isomerization processes in unsaturated hydrocarbons [172]. Photochemical reactions of this type are known to involve large-amplitode motion on coupled potential-energy surfaces [169], thus representing another stringent test for a mixed quantum-classical description that is complementary to Models 1 and II. A number of theoretical investigations, including quantum wave-packet studies [163, 164, 172], time-resolved pump-probe spectra [164, 181], and various mixed... 

Let us finally discuss to what extent the MFT method is able to (i) obey the principle of microreversibility, (ii) account for the electronic phase coherence, and (iii) correctly describe the vibrational motion on coupled potential-energy surfaces. It is a well-known flaw of the MFT method to violate quantum microreversibility. This basic problem is most easily rationalized in the case of a scattering reaction occurring in a two-state curve-crossing system, where the initial and final state of the scattered particle may be characterized by the momenta p, and pf, respectively. We wish to calculate the probability Pi 2 that... 

A further important property of a MQC description is the ability to correctly describe the time evolution of the electronic coefficients. A proper description of the electronic phase coherence is expected to be particularly important in the case of multiple curve-crossings that are frequently encountered in bound-state relaxation dynamics [163]. Within the limits of the classical-path approximation, the MPT method naturally accounts for the coherent time evolution of the electronic coefficients (see Fig. 5). This conclusion is also supported by the numerical results for the transient oscillations of the electronic population, which were reproduced quite well by the MFT method. Similarly, it has been shown that the MFT method in general does a good job in reproducing coherent nuclear motion on coupled potential-energy surfaces. 

The excellent performance of the mapping formulation for this model encouraged us to consider an extended model of the benzene cation, for which no quantum reference calculations are available [227]. The model comprises 16 vibrational DoF and five coupled potential-energy surfaces, thus accounting for... 

Describing complex wave-packet motion on the two coupled potential energy surfaces, this quantity is also of interest since it can be monitored in femtosecond pump-probe experiments [163]. In fact, it has been shown in Ref. 126 employing again the quasi-classical approximation (104) that the time-and frequency-resolved stimulated emission spectrum is nicely reproduced by the PO calculation. Hence vibronic POs may provide a clear and physically appealing interpretation of femtosecond experiments reflecting coherent electron transfer. We note that POs have also been used in semiclassical trace formulas to calculate spectral response functions [3]. 

The wavepacket calculation for the femtosecond pump-probe experiment presented in Fig. 16 (bottom) is the result of the first consistent ab initio treatment for three coupled potential-energy surfaces in the complete three-dimensional vibrational space of the Naa molecule. In order to simulate the experimental femtosecond ion signal, the experimental pulse parameters were used duration A/fWhm = 120 fs, intensity I - 520 MW/cm2, and central... 

Fig. 2 The model of coupled potential energy surfaces used to explain the vibronic spectral features in Re dioxo complexes. Solid and dashed lines represent adiabatic and diabatic surfaces, respectively. The lowest adiabatic surface corresponds to the electronic ground state used to calculate the luminescence spectra. The crystal field energies for all three Ai states are given along the vertical dashed line...

Coalson, R.D. (1989). Time-dependent wavepacket approach to optical spectroscopy involving nonadiabatically coupled potential energy surfaces, Adv. Chem. Phys. 73, 605-636. 

Wavepacket calculations in curvilinear coordinates on coupled potential energy surfaces and a Wigner transform of the Boltzmann operator Wagner A.F. 

In the continuation of our discussion we face two tasks. First, we need to replace the simple two-parabola model described above by a realistic model that uses input from the energetic and dynamic properties of the solvent. Second, we have to provide a reliable description of the process that takes place when the electronic states become nearly degenerate, that is, of the electronic transition itself, taking into account the quantum mechanical nature of motion on two coupled potential energy surfaces. 

The Born-Oppenheimer adiabatic approximation represents one of the cornerstones of molecular physics and chemistry. The concept of adiabatic potential-energy surfaces, defined by the Born-Oppenheimer approximation, is fundamental to our thinking about molecular spectroscopy and chemical reaction djmamics. Many chemical processes can be rationalized in terms of the dynamics of the atomic nuclei on a single Born Oppenheimer potential-energy smface. Nonadiabatic processes, that is, chemical processes which involve nuclear djmamics on at least two coupled potential-energy surfaces and thus cannot be rationalized within the Born-Oppenheimer approximation, are nevertheless ubiquitous in chemistry, most notably in photochemistry and photobiology. Typical phenomena associated with a violation of the Born-Oppenheimer approximation are the radiationless relaxation of excited electronic states, photoinduced uni-molecular decay and isomerization processes of polyatomic molecules. 

The studies so far made for this ESHAT employed the ab initio configuration interaction (Cl) and/or complete active space SCF (CAS-SCF) methods and examined only the two lowest coupled potential energy surfaces (and nuclear wavepacket dynamics simplified [228]). However, because of the fact that the relevant excited states are highly quasi-degenerate in reality, and in view of the complicated nature of electron dynamics associated with proton transfer as described above, it is worthwhile to re-examine the mechanism from the nonadiabatic electron wavepacket dynamics. 

Li ZH, Valero R, Truhlar DG (2007) Improved direct diabatization and coupled potentied energy surfaces for the photodissociation of ammonia. Theor Chem Acc 118 9... 

In this contribution our quantum-dynamical treatment of nuclear motion on coupled potential energy surfaces (PESs) is surveyed. The present general introduction deals with the early history of the field (i.e. from 1977 to 1991), followed by a more systematic classification of types of surface intersections and a phenomenology of the ensuing nuclear dynamics. 

In non-adiabatic dynamics it is necessary to treat the nuclei as moving over a set of coupled potential energy surfaces rather than the single surface of classical molecular dynamics. The surfaces can then approach to form avoided crossings or meet as conical intersections that provide pathways where the initially excited molecule can cross back to the ground electronic state in a non-radiative manner. This crossing is particularly efficient at a conical intersection, which is why these features play a central role in the mechanistic description of photochemistry, in a similar way to the role played by the transition state in thermal chemistry. 

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