Spin-coherent state

Spin-Hamiltonians that are linear in the spin operators, HSpm propagate spin-coherent states exactly,... 

Second, the mapping approach to nonadiabatic quantum dynamics is reviewed in Sections VI-VII. Based on an exact quantum-mechanical formulation, this approach allows us in several aspects to go beyond the scope of standard mixed quantum-classical methods. In particular, we study the classical phase space of a nonadiabatic system (including the discussion of vibronic periodic orbits) and the semiclassical description of nonadiabatic quantum mechanics via initial-value representations of the semiclassical propagator. The semiclassical spin-coherent state method and its close relation to the mapping approach is discussed in Section IX. Section X summarizes our results and concludes with some general remarks. 

Finally, it is noted that there exist alternative mappings of spin to continuous DoF. For example. Ref. 219 discusses various mappings that (like the Holstein-Primakoff transformation) represent a spin system by a single-boson DoF. The possibility of utilizing spin coherent states for this purpose is discussed in Section IX. 

A further simplification of the semiclassical mapping approach can be obtained by introducing electronic action-angle variables and performing the integration over the initial conditions of the electronic DoF within the stationary-phase approximation [120]. Thereby the number of trajectories required to obtain convergence is reduced significantly [120]. A related approach is discussed below within the spin-coherent state representation. 

Let us start with a brief review of spin-coherent state theory. For simplicity we focus on a two-level (or spin 1/2) system. The coherent states for a two-level system with basis states /i), /2) can be written as [136, 139]... 

To discuss the semiclassical spin-coherent state propagator, we consider a general transition amplitude ( / e which can be expressed as an integral... 

The path integral representation of the spin-coherent state propagator is formally given by [139]... 

Employing the stationary-phase approximation to the path integral, the semiclassical spin-coherent state propagator is obtained [139, 140, 143, 281, 282] ... 

Within the theoretical framework of time-dependent Hartree-Fock theory, Suzuki has proposed an initial-value representation for a spin-coherent state propagator [286]. When we adopt a two-level system with quantum Hamiltonian H, this propagator reads... 

Another possibility to introduce a semiclasscial initial value representation for the spin-coherent state propagator is to exploit the close relation between Schwinger s representation of a spin system and the spin-coherent state theory [100, 133-135]. To illustrate this approach, we consider an electronic two-level system coupled to Vvib nuclear DoF. Within the mapping approach the semiclassical propagator for this system is given by... 

In order to express the propagator (134) in terms of spin-coherent states, we introduce the following parameterization of the complex electronic variables [133] ... 

Figure 44. Diabatic population (a) and modulus of the autocorrelation function (b) of the initially prepared state for Model IVa. The full line is the quantum result, the dashed-dotted line is the result of the semiclassical spin-coherent state propagator, and the dashed line depicts the result of Suzuki s propagator. The semiclassical data have been normalized. Panel (c) shows the norm of the semiclassical wave functions.

The results obtained for the three-mode Model IVb are depicted in Fig. 45. As was found for the semiclassical mapping approach, the spin-coherent state propagators can only reproduce the short-time dynamics for the electronic population. The autocorrelation function, on the other hand, is reproduced at least qualitatively correctly by the semiclassical spin-coherent state propagator. 

For a nonadiabatic system with spin-orbit interaction, the validity of the semiclassical approximation (based on the spin-coherent state representation) has been discussed in detail in Ref. 147. 

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