Correlation dynamic

It is possible to divide electron correlation as dynamic and nondynamic correlations. Dynamic correlation is associated with instant correlation between electrons occupying the same spatial orbitals and the nondynamic correlation is associated with the electrons avoiding each other by occupying different spatial orbitals. Thus, the ground state electronic wave function cannot be described with a single Slater determinant (Figure 3.3) and multiconfiguration self-consistent field (MCSCF) procedures are necessary to include dynamic electron correlation. 

Spatially Correlated Dynamics in a Simulated Glass-Forming Polymer Melt Analysis of Clustering Phenomena. 

The parallel-replica method also correctly accounts for correlated dynamical events (there is no requirement that the system obeys TST), unlike the other AMD methods. This is accomplished by allowing the trajectory that made the transition to continue for a further amount of time Afcorr > Tcorr, during which recrossings or follow-on events may occur. The simulation clock is then advanced by Afcorr, the new state is replicated on all processors, and the whole process is repeated. 

It is convenient to divide a set of fluctuation-controlled kinetic equations into two basic components equations for time development of the order parameter n (concentration dynamics) and the complementary set of the partial differential equations for the joint correlation functions x(r, t) (correlation dynamics). Many-particle effects under study arise due to interplay of these two kinds of dynamics. It is important to note that equations for the concentration dynamics coincide formally with those known in the standard kinetics... 

Fig. 4.1. The schematic relation between concentration dynamics and correlation dynamics in...

Substitution of equation (2.3.62) into a set of equations (4.1.13) to (4.1.16) for noncharged (neutral) particles (Uvil r) = 0) does not affect equations (4.1.18) and (4.1.19) whereas the linear equation (4.1.23) describing the correlation dynamics splits now into three integro-differential equations. Main stages of the passage from general equations (4.1.14)—(4.1.16) for the joint densities to those for the joint correlation functions have been demonstrated earlier, see (4.1.20) and (4.1.21). Therefore let us consider only those terms which are affected by the use of superposition approximation. Hereafter we use the relative coordinates f=f — f(, f = r 2 — r[ and... 

The shortened Kirkwood superposition approximation (2.3.64) differs from the complete one, equation (2.3.63), by the additional condition imposed on the correlation function of similar particles Xv(r,t) = 1 at any time t. Its substitution into equation (5.1.4) and taking into account equation (5.1.5) leads, as one could expect, to the linearized equation (4.1.23) for the correlation dynamics. Therefore, the applicability range of the linearized kinetic... 

All said above demonstrates how new degrees of freedom arises when more refined approximation (2.3.64) is used - the complete superposition approximation instead of the standard equation (2.3.64), but it does not prove existence of effects themselves. Solution of non-linear equations for the correlation dynamics is discussed in next Sections of this Chapter. 

A comparison with the correlation dynamics of the A + B —> 0 reaction, equations (5.1.33) to (5.1.35), shows their similarity, except that now several terms containing functionals J[Z have changed their signs and several singular correlation sources emerged. The accuracy of the superposition approximation in the diffusion-controlled and static reactions was recently confirmed by means of large-scale computer simulations [28]. It was shown to be quite correct up to large reaction depths r = 3 studied. 

The uncorrelated particle distribution (4.1.12) is used, as standard initial conditions for the correlation dynamics. After the transient period the solution (for the stable regime) becomes independent on the initial conditions. For both the joint correlation functions boundary conditions at large distances X (oo, t) = Y(oo, t) = 1 has to be fulfilled due to the correlation weakening. The black sphere model imposes the additional boundary condition (5.1.39) for the correlation function Y(r,t). 

This statement is not self-evident and needs some comments. A role of concentration degrees of freedom in terms of the formally-kinetic description was discussed in Section 2.1.1. Stochastic approach adds here a set of equations for the correlation dynamics where the correlation functions are field-type values. Due to very complicated form of the complete set of these equations, the analytical analysis of the stationary point stability is hardly possible. In its turn, a numerical study of stability was carried out independently for the correlation dynamics with the fixed particle concentrations. 

As it was said above, there is no stationary solution of the Lotka-Volterra model for d = 1 (i.e., the parameter k does not exist), whereas for d = 2 we can speak of the quasi-steady state. If the calculation time fmax is not too long, the marginal value of k = K.(a, ft, Na,N, max) could be also defined. Depending on k, at t < fmax both oscillatory and monotonous solutions of the correlation dynamics are observed. At long t the solutions of nonsteady-state equations for correlation dynamics for d = 1 and d = 2 are qualitatively similar the correlation functions reveal oscillations in time, with the oscillation amplitudes slowly increasing in time. 

The behaviour of the correlation functions shown in Fig. 8.5 corresponds to the regime of unstable focus whose phase portrait was earlier plotted in Fig. 8.1. For a given choice of the parameter k = 0.9 the correlation dynamics has a stationary solution. Since a complete set of equations for this model has no stationary solution, the concentration oscillations with increasing amplitude arise in its turn, they create the passive standing waves in the correlation dynamics. These latter are characterized by the monotonous behaviour of the correlations functions of similar and dissimilar particles. Since both the amplitude and oscillation period of concentrations increase in time, the standing waves do not reveal a periodical motion. There are two kinds of particle distributions distinctive for these standing waves. Figure 8.5 at t = 295 demonstrates the structure at the maximal concentration... 

More complicated case of standing waves emerges in the regime of chaotic oscillations. Here the equations for the correlation dynamics are able to describe auto-oscillations (for d — 3). However, a noise in concentrations changes stochastically the amplitude and period of the standing waves. It results finally in the correlation functions with non-monotonous behaviour. Despite the fact that the motion of both concentrations and of the correlation functions is aperiodic, the time evolution of the correlation functions reveals several distinctive distributions shown in Fig. 8.6. 

This statement could be proved in the manner similar to that used in Section 8.2. It is important to note that the correlation dynamics of the Lotka and Lotka-Volterra model do not differ qualitatively. A stationary solution exists for d = 3 only. Depending on the parameter k, different regimes are observed. For k kq the correlation functions are changing monotonously (a stable solution) but as k < o> the spatial oscillations of the correlation functions (unstable solution) are observed. In the latter case a solution of non-steady-state equations of the correlation dynamics has a form of the non-linear standing waves. In one- and two-dimensional cases there are no stationary solutions of the Lotka model. 

Figure 8.7 shows the dependence of the reaction rate K(t) for different k values. For the space dimension d = 3 the obtained results could be easily interpreted as follows there exists the marginal value of kq (Statement 2). For k = 0.05 the inequality k > o holds. The stable stationary solution exists for the correlation dynamics and due to decay of the concentration motion... 

First let us review static and dynamic electron correlation. Dynamic (dynamical) electron correlation is easy to grasp, if not so easy to treat exhaustively. It is simply the adjustment by each electron, at each moment, of its motion in accordance with its interaction with each other electron in the system. Dynamic correlation and its treatment with perturbation (Mpller-Plesset), configuration interaction, and coupled cluster methods was covered in Section 5.4. 

Is there no rigorous way to separate static and dynamic electron correlation Dynamic correlation is present in any system with two or more electrons, but static correlation requires degenerate or near-degenerate partially-filled orbitals, a feature absent in normal closed-shell molecules. So in this sense they are separate phenomena. In another sense they are intertwined methods that go beyond the Hartree-Fock in invoking more than one determinant, namely Cl and its coupled cluster variant, improve the handling of both phenomena. 

Zhang, Q., Stelzer, A. C., Fisher, C. K., and Al-Hashimi, H. M. (2007). Visualizing spatially correlated dynamics that directs RNA conformational transitions. Nature 450, 1263—1267. 

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The power of NMR methods for studying dynamics of complexes is beautifully illustrated by an earlier study of the complex of DHFR with methotrexate.147 In this case a correlated dynamic rotation of a carboxylate group on the ligand and Arg57 of the protein was detected as illustrated in Fig. 28. 

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