Conical intersections numerical calculations

The numerical part is based on two circles, C3 and C4, related to two different centers (see Fig. 13). Circle C3, with a radius of 0.4 A, has its center at the position of the (2,3) conical intersection (like before). Circle C4, with a radius 0.25 A, has its center (also) on the C v line, but at a distance of 0.2 A from the (2,3) conical intersection and closer to the two (3,4) conical intersections. The computational effort concentrates on calculating the exponential in Eq, (38) for the given set of ab initio 3 x 3 x matrices computed along the above mentioned two circles. Thus, following Eq, (28) we are interested in calculating the following expression ... 

To summarize, it has been found that the SH method is able to at least qualitatively describe the complex photoinduced electronic and vibrational relaxation dynamics exhibited by the model problems under consideration. The overall quality of SH calculations is typically somewhat better than the quality of the mean-field trajectory results. In particular, this holds in the case of several curve crossings (see Fig. 2) as well as when the dynamics and the observables of interest are essentially of adiabatic nature— for example, for the calculation of the adiabatic population dynamics associated with a conical intersection (see Figs. 3 and 12). Furthermore, we have briefly discussed various consistency problems of a simple quasi-classical SH description. It has been shown that binned electronic population probabilities and no momentum adjustment for classically forbidden transitions help us to improve this matter. There have been numerous suggestions to further improve the hopping algorithm [70-74] however, the performance of all these variants seems to depend largely on the problem under consideration. 

Pump-probe experiment is an efficient approach to detect the ultrafast processes of molecules, clusters, and dense media. The dynamics of population and coherence of the system can be theoretically described using density matrix method. In this chapter, for ultrafast processes, we choose to investigate the effect of conical intersection (Cl) on internal conversion (IC) and the theory and numerical calculations of intramolecular vibrational relaxation (IVR). Since the 1970s, the theories of vibrational relaxation have been widely studied [1-7], Until recently, the quantum chemical calculations of anharmonic coefficients of potential-energy surfaces (PESs) have become available [8-10]. In this chapter, we shall use the water dimer (H20)2 and aniline as examples to demonstrate how to apply the adiabatic approximation to calculate the rates of vibrational relaxation. 

In an alternative approach, exact (numerical) time-dependent quantum wave-packet methods have been employed since the early eighties of the last century to explore the d3mamics of ob-initio-haseA models of conical intersections, see Refs. 6-8 for reviews. It has been found by these calculations that the fundamental dissipative processes of population and phase relaxation at femtosecond time scales are clearly expressed already in fewmode systems, if a directly accessible conical intersection of the PE surfaces is involved. The results strongly support the idea that conical intersections provide the microscopic mechanism for ultrafast relaxation processes in polyatomic molecules. " More recently, these calculations have been extended to describe photodissociation and photoisomerization processes associated with conical intersections. The latter are particularly relevant for our understanding of the elementary mechanisms of photochemistry. 

QCL calculations performed for conical intersections cannot yet compete in accuracy and efficiency with more established methods. Recalling, though, that all numerical implementations of the QCL equation have been suggested within the last few years, it is clear that the QCL approach still holds a large potential to be discovered. 

In the preceding sections we show that, by postulating simple VB structures on a photochemical reaction path, one can deduce not only that a conical intersection may be involved but also the nature of the branching space of the conical intersection. For problems such as 3 orbitals with 3 electrons or 4 orbitals with 4 electrons it is simple to manipulate the VB matrix elements to make these deductions. By the time one gets to 6 orbitals with 6 electrons there are very many possibihties. So one has to leam " by extracting the VB structures from the ab initio data. For the 6 orbitals with 6 electron case, we use the MMVB method to do this. Once the more important structures are identified this way, we can perform the manipulations analytically to confirm the result by comparison with numerical data. Finally, for 8 orbitals with 8 electrons we were able to show that one may also extract the VB data from the MMVB method and come to understand the nature of the conical intersection. However, it is rather tedious to do the calculations analytically and this work has never been carried out. 

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