Wigner distribution dynamics

We stress that level spacings alone do not suffice to make statements about quantum chaos integrable systems are known which give Wigner distributions [561, 562]. Simple systems are also known whose classical dynamics is chaotic, but whose nearest level spacings differ markedly from the Wigner distribution [563]. For atoms, such systems can have the property that one series is dominant, i.e. has a much higher density of... 

Using a GVB-OA-CAS(2/2) S wave function and a double-f quahty basis set, we simulated ethylene photochemistiy following n- n excitation. The AIMS simulations treat the excitation as instantaneous and centered at the absorption maximum. Hence, the initial-state nuclear basis functions are sampled from the groimd-state Wigner distribution in the harmonic approximation. Ten basis functions are used to describe the initial state. Overall, approximately 100 basis functions are spawned diuing the dynamics, and we follow the dynamics up to 0.5 ps (picoseconds) (using a time-step of 0.25 fs [femtoseconds]). 

Irving and Zwanzig that the theorem expressed by Eq. 1.11 also holds in quantum-statistical mechanics if the function f is the Wigner distribution function and if the dynamical variable a is a polynomial of a degree not higher than the second in the momenta. 

The variational solution of the decoupled equation of motion (Eq. Ills) for the Wigner distribution function might serve as a starting point for further studies of exchange and correlation in the dielectric function. Its connection with several other approximations has been examined, showing that many of them are particular cases or additional approximations to this variational approach. The improvement upon the RPA from dynamical exchange effects, and the fact that all checked sum rules are satisfied, gives... 

Abstract We present a general theoretical approach for the simulation and control of ultrafast processes in complex molecular systems. It is based on the combination of quantum chemical nonadiabatic dynamics on the fly with the Wigner distribution approach for simulation and control of laser-induced ultrafast processes. Specifically, we have developed a procedure for the nonadiabatic dynamics in the framework of time-dependent density functional theory using localized basis sets, which is applicable to a large class of molecules and clusters. This has been combined with our general approach for the simulation of time-resolved photoelectron spectra that represents a powerful tool to identify the mechanism of nonadiabatic processes, which has been illustrated on the example of ultrafast photodynamics of furan. Furthermore, we present our field-induced surface hopping (FISH) method which allows to include laser fields directly into the nonadiabatic... 

It is clear that the strong form of the QCT is impossible to obtain from either the isolated or open evolution equations for the density matrix or Wigner function. For a generic dynamical system, a localized initial distribution tends to distribute itself over phase space and either continue to evolve in complicated ways (isolated system) or asymptote to an equilibrium state (open system) - whether classically or quantum mechanically. In the case of conditioned evolution, however, the distribution can be localized due to the information gained from the measurement. In order to quantify how this happ ens, let us first apply a cumulant expansion to the (fine-grained) conditioned classical evolution (5), resulting in the equations for the centroids (x = (t), P= (P ,... 

Classical and semiclassical moment expressions. The expressions for the spectral moments can be made classical by substituting the classical distribution function, g(R) = exp (— V(R)/kT), for the quantum expressions. Wigner-Kirkwood corrections are known which account to lowest order for the static quantum corrections, Eq. 5.44 [177, 292]. For the second and higher moments, dynamic quantum corrections must also be made [177]. As was mentioned in the previous Chapter, such semiclassical corrections are useful in supplementing quantum computations of the spectral moments at large separations where the quantum effects are small the computational effort of quantum calculations, which is substantial at large separations, may thus be avoided. 

In this section we advocate a far more advantageous route to studying conceptual features of the classical-quantum correspondence, and indeed for each mechanics independently, in which phase space distributions are used in both classical and quantum mechanics, that is, classical Liouville dynamics50 in the former and the Wigner-Weyl representation in the latter. This approach provides, as will be demonstrated, powerful conceptual insights into the relationship between classical and quantum mechanics. The essential point of this section is easily stated using similar mathematics in both quantum and classical mechanics results in a similar qualitative picture of the dynamics. 

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