Quasi-energy surface

The topology of the quasi-energy surfaces thus shows which appropriate delays and peak amplitudes induce desired atomic population and photon transfers. In the adiabatic regime, these loops can be classified into topologically inequivalent classes. If the evolution is adiabatic, all paths of a given class lead to the same end effect. This property underlies the robustness of the process. 

Finally, in brief, we demonstrate the influence of the upper adiabatic electronic state(s) on the ground state due to the presence of a Cl between two or more than two adiabatic potential energy surfaces. Considering the HLH phase, we present the extended BO equations for a quasi-JT model and for an A -1- B2 type reactive system, that is, the geometric phase (GP) effect has been inhoduced either by including a vector potential in the system Hamiltonian or... 

This reaction describes a quasi equilibrium in the sense that it counts only those transition states moving forward (i.e., toward products) along the reaction coordinate. One can picture18 reactants moving along a potential energy surface such as that in Fig. 7-5. The transition state resides in one sense at the maximum of this surface in another sense, at the minimum point. This is called a saddle point or a col (from the French word for mountain pass). 

After 28 years the perepoxide quasi-intermediate was supported by a two-step no intermediate mechanism [71, 72]. The minimum energy path on the potential energy surface of the reaction between singlet molecular oxygen ( A and dg-teramethylethylene reaches a vaUey-ridge inflection point and then bifurcates leading to the two final products [73]. 

Nevertheless, very-long-lived quasi-stationary-state solutions of Schrodinger s equation can be found for each of the chemical structures shown in (5.6a)-(5.6d). These are virtually stationary on the time scale of chemical experiments, and are therefore in better correspondence with laboratory samples than are the true stationary eigenstates of H.21 Each quasi-stationary solution corresponds (to an excellent approximation) to a distinct minimum on the Born-Oppenheimer potential-energy surface. In turn, each quasi-stationary solution can be used to construct an alternative model unperturbed Hamiltonian //(0) and perturbative interaction L("U),... 

The quasi-classical SH model employs the simple and physically appealing picture in which a molecular system always evolves on a single adiabatic potential-energy surface (PES). When the trajectory reaches an intersection of the electronic PESs, the transition probability pk t to the other PES is calculated... 

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 method of moments of coupled-cluster equations (MMCC) is extended to potential energy surfaces involving multiple bond breaking by developing the quasi-variational (QV) and quadratic (Q) variants of the MMCC theory. The QVMMCC and QMMCC methods are related to the extended CC (ECC) theory, in which products involving cluster operators and their deexcitation counterparts mimic the effects of higher-order clusters. The test calculations for N2 show that the QMMCC and ECC methods can provide spectacular improvements in the description of multiple bond breaking by the standard CC approaches. 

As we have seen, the appearance of strongly bound chemisorbed particles on the semiconductor surface gives rise to a surface charge, which, in turn, leads to a bending of the energy bands inside the semiconductor. Of course, a similar bending of the bands may arise as well in the particular case of a quasi-isolated surface. 

Figure 2. To the left, quasi-diabatic potential energy surfaces in the B3LYP/cc-pvtz Dunning s basis set. AA represents a cis state (solid line) BB a trans state (solid line) AB is the excited diradical state spin singlet (dashed line) Triplet is the diradical state S=1 (dotted line). To the right, extrapolated diabatic potential energy surfaces for the same states. The angle used to plot energy entries is a = 2 0. All calculations were done with Gaussian 98 [23].

Not shown to the quasi-diabatic potential energy surfaces in Fig. (3) there is a adiabatic potential energy surface. This is distinguished by the maximum at the crossing point nil. The system has a saddle-point structure. In the regions about the cis and trans attractors there is no difference between them. Between 2%1 i... 

The harmonic approximation is unrealistic in a dynamical description of the dissociation dynamics, because anharmonic potential energy terms will play an important role in the large amplitude motion associated with dissociation. An accurate potential energy surface must be used in order to obtain a realistic dynamical description of the dissociation process and, as in the quasi-classical approach for bimolecular collisions, a numerical solution of the classical equations of motion is required [2]. 

The decay of Nal can be described in an alternative way [K.B. Mpller, N.E. Henriksen, and A.H. Zewail, J. Chem. Phys. 113, 10477 (2000)]. In the bound region of the excited-state potential energy surface, one can define a discrete set of quasi-stationary states that are (weakly) coupled to the continuum states in the dissociation channel Na + I. These quasi-stationary states are also called resonance states and they have a finite lifetime due to the coupling to the continuum. Each quasi-stationary state has a time-dependent amplitude with a time evolution that can be expressed in terms of an effective (complex, non-Hermitian) Hamiltonian. 

In-situ growth monitoring, in particular by reflective high energy electron diffraction (RHEED), has provided some fundamental information on the surface and nucleation properties of nitrides. Early RHEED studies by Hughes et al [34] and Hacke et al [49] were completed by Smith et al [50], The observed surface reconstructions for (0001) and (0001) GaN surfaces have allowed the modelling of the quasi-equilibrium surface, which has been calculated to be preferentially Ga-terminated [51], Feuillet et al [52] have followed the evolution of surface lattice constants in RHEED for the nucleation of GaN on AIN or InN on GaN (and vice versa) and extracted a wide range of information on the character of nucleation and misfit relaxation. 

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