Operational forces ensemble

The operators Fk(t) defined in Eq.(49) are taken as fluctuations based on the idea that at t=0 the initial values of the bath operators are uncertain. Ensemble averages over initial conditions allow for a definite specification of statistical properties. The statistical average of the stochastic forces Fk(t) is calculated over the solvent effective ensemble by taking the trace of the operator product pmFk (this is equivalent to sum over the diagonal matrix elements of this product), so that = Trace(pmFk) is identically zero (Fjk(t)=Fk(t) in this particular case). The non-zero correlation functions of the fluctuations are solvent statistical averages over products of operator forces,... 

Differentiation of (41), using the formalism defined for the microcanonical ensemble, in terms of a generalized force operator, yields... 

The pulse sequence starts with a preparation period P, which usually allows the ensemble of spins - still partially perturbed by the pulses applied in the preceding scan -to return back to the equilibrium state. The preparation period may also be used to force this ensemble of spins to a defined non-equilibrium state according to the operators needs. 

Chemical defense includes individual and collective protection and decontamination. The goal of individual and collective protection is to use clothing ensembles and respirators as well as collective filtration systems and shelters to insulate forces from chemical agents. Decontamination is essential to return personnel and equipment to normal operating conditions. 

The central point to simulate decoherence is to generate independent MD simulations on the PESs of the two electronic states of interest. To have a nonzero overlap at time 1 = 0 (a coherent superposition of states), these MD simulations will share the same initial conditions (positions and momentum). In later times the classical nuclei will feel different forces on the two PESs and will diverge over time. Once the two trajectories have been carried out, the function can be calculated using the values of xy, X2j, Py, xy, and IsK 2 evaluated along the diverging trajectories. This operation is repeated with different initial conditions to sample the desired thermodynamics ensemble. 

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