Density matrices from propagator

A similar expression applies to the density matrix, from its correspondence with the propagator. For example,... 

Let us investigate the change of the various components of the density matrix under propagation from time t to t + 5t. We first consider the action of jSfo- Due to this operator, the density matrix element p (t) attains a phase factor... 

Figure 11 Simulation of the fid of a spin set. (A) Individual density matrix is calculated at each exchange point. (B) Eigencoherence representation of the density matrix is propagated from the beginning of the time slice for each detection point.

An alternative relax-and-drive procedure can be based on a strictly unitary treatment where the advance from Iq to t is done with a norm-conserving propagation such as provided by the split-operator propagation technique.(49, 50) This however is more laborious, and although it conserves the norm of the density matrix, it is not necessarily more accurate because of possible inaccuracies in the individual (complex) density matrix elements. It can however be used to advantage when the dimension of the density matrix is small and exponentiation of matrices can be easily done.(51, 52)... 

Fig. 6. All paths leading from the initial to the final points in time t contribute an interfering amplitude to the path sum describing the resultant probability amplitude for the quantum propagation. In this double slit free particle case, two paths of constant speed are local functional stationary points of the action, and these two dominant paths provide the basis for a (semiclassical) classification of subsets of paths which contribute to the path integral. In the statistical thermodynamic path expression, the path sum is equal to the off-diagonal electronic thermal density matrix...

By knowing the trajectory of a spin set, its individual density matrix can be calculated at any time points. The key to the simulation is the determination of the propagating matrix (see Section 3.5). The FID and spectrum of a spin set upon the individual trajectory (one scan) can be determined from the actual values of the time-dependent density matrix. 

Further propagation should be calculated from this matrix. The detected signal can be calculated from the transformed density matrix as ... 

An important consequence of the inclusion of an intraatomic repulsion term is the possibility of modeling magnetic features in partially filled, narrow bands. Some elements of the two-particle density matrix, for instance , derive from the propagator matrix and give an indication of the local spin features. The use of the additional operators requires some further algebra. The metric measures... 

To be self-contained as much as possible we here make a very brief review of our practice in the application of the FSSH scheme A diabatic representation is used for the dynamics of electron wavefunction. The electron wavefunction is propagated coherently on a reference nuclear trajectory throughout the dynamics. The part of density matrix created by the wave-function is utilized to determine the probability for an otherwise classically continuous path to hop from one PES to another within every time interval [t, t + At]. To be more precise, the switching probability is given as... 

Equation 10.86 shows that the approximation leading toEq. 10.83 becomes worse at the increase of the propagation time of the doorway state (or better of the corresponding density matrix) on the final PES. This also explains why in many cases such a classical approximation reproduces low-resolution spectra quite well, since, for this latter a short time propagation is needed (due to the fact that the dipole correlation function is assumed to decay rapidly with time), while it fails in shaping the fine vibrational details of the spectrum which arise from partial revivals of the correlation function and then need a long-time dynamics. 

A full discussion of the reasons for the success of the pair action approach, some estimates of the order of accuracy, and the measures to enhance its performance can be found in Ceperley s review [28], However, the exact density matrix contains not only pair terms but also triplet terms, quadruplet terms, and so on. These terms are needed in the study of situations at very high densities. In relation to this, although some clever tricks can be used to deal with the pair case, the next extension to deal with triplet terms is far from straightforward, because of the increase in the number of variables needed (e.g., for a homogeneous and isotropic fluid from four variables in the pair case to 13 in the triplet case) [28]. Furthermore, it is difficult to anticipate what the order of accuracy ) of a triplet action will be. This situation contrasts sharply with that of the higher-order propagators, for which the order of accuracy is always known. Another problem in this context is that the usefulness of pair actions cannot be extended to deal with non-symmetri-cally spherical potentials (e.g., molecular fluids). 

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