Reduced density-matrix distribution densities

For simplicity, we shall commonly refer to the Q-electron distribution function as the 2-density and the 2-electron reduced density matrix as the 2-ntatrix. In position-space discussions, the diagonal elements of the 2-ntatrix are commonly referred to as the 2-density. In this chapter, we will also refer to the diagonal element of orbital-space representation of the Q-vaatnx as the 2-density. 

Fig. 4 Snapshots of the probability distribution evolution for two CO molecules on a copper(lOO) surface subject to non-adiabatic coupling. The dynamics is initiated with two quanta of vibrational excitation in one of the CO molecules, here the qi mode. The labels full and fact, refer to the choice of vibrational basis to represent the reduced density matrix in eqn (19). The left panels show the system in the absence of intermode coupling and the strong intermode coupling regime is depicted in the right panels. Reproduced with permission from ref. 97.

In this field, the density operator plays an important and uncontested role. It allows for more than just the above-given series expansion it can be used for consistent approximations by integrating out (read take the partial trace) of a series of states we are not particularly interested in, leading to the so-called reduced density matrix. It can also be used to find representations in spaces, for instance, the Wigner representation [35], that give more insight into the quantum distribution functions, and provide in some cases distribution functions that are more close to the classical. 

Processes request tasks (atom quartets) by calling the function get quartet, which has been implemented in both a dynamic and a static version. The dynamic work distribution uses a manager-worker model with a manager process dedicated to distributing tasks to the other processes, whereas the static version employs a round-robin distribution of tasks. When the number of processes is small, fhe sfafic scheme achieves the best parallel performance because the dynamic scheme, when run on p processes, uses only p - 1 processes for compulation. As the number of processes increases, however, the parallel performance for the dynamic task distribution surpasses that of the static scheme, whose efficiency is reduced by load imbalance. Wifh fhe entire Fock and density matrix available to every process, no communication is required during the computation of the Fock matrix other than the fetching of tasks in the dynamic scheme. After all ABCD tasks have been processed, a global summation is required to add the contributions to the Fock matrix from all processes and send the result to every process. 

Such a form of quasi-equilibrium distribution takes place due to the fact of the availability of two invariants of motion. In Equation 25 parameters a and p linked to the operators Hz and Hss are thermodynamically conjugative parameters for the Zeeman energy and the energy of spin-spin interactions respectively. We can expand the exponent in Equation 25 in jxjwers of xT-Lz and f Hss and keep only the linear terms. As we shall see later such a linearization corresponds to the high temperature approximation. In the linear approximation in x Hz and Hss, the density matrix is reduced to... 

In SIMCA the distribution of the object in the inner model space is not considered, so the probability density in the inner space is constant and the overall PD appears as shown in Figs. 29, 30 for the enlarged and reduced SIMCA models. In CLASSY, Kernel estimation is used to compute the PD in the inner model space, whereas the errors in the outer space are considered, as in SIMCA, uncorrelated and with normal multivariate distribution, so that the overall distribution, in the inner and outer space of a one-dimensional model, looks like that reported in Fig. 31. Figures 32, 33 show the PD of the bivariate normal distribution and Kernel distribution (ALLOC) for the same data matrix as used for Fig. 31. Although in the data set of French wines no really important differences have been detected between SIMCA (enlarged model), ALLOC and CLASSY, it seems that CLASSY should be chosen when the number of objects is large and the distribution on the components of the inner model space is very different from a rectangular distribution. 

In order to overcome this drawback, the concept of selective blending was exploited. Selective blending of PPE with low-viscous PS allowed one to control the microstructure, to refine the phase size, and to adjust the foaming characteristics of the individual phases of PPE/SAN blends. Appropriate blend compositions allowed simultaneous nucleation and cooperative expansion of both phases, generally leading to bimodal cell size distributions in the micron range. Due to cell nucleation and growth in both blend phases, the density could be further reduced when compared to PPE/SAN blends. Moreover, the presence of coalesced foam structure and particularly macroscopic defects could be avoided, and the matrix of the foamed structure was formed by the heat resistant PPE/PS phase. 

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