The correlation hole

Consider, for example, the case where each chain is labeled at one end only the first monomer of each chain is deuterated. We take one such 

Qualitatively, we may say that Sn(r) shows a correlation hole of width R and of average depth 1/7V a/Ro From an experimental 

The structure of 11(9) for a chain labeled at one end is shown in Fig. II.8 (from Ref. 7). Note that for 9 - 0 the scattering intensity vanishes. For long wavelength fluctuations, the label density pi is simply p/N, and the total density p does not fluctuate in our model there is no scattering. 

This corresponds to the following sum rule in direct space  

In more general terms, this sum rule tells us that the depletion in the correlation hole corresponds exactly to one monomer unit 

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This function also gives an accurate description of the behavior of a linear chain in a good solvent (the expansion of the chain size is scaled by the x variable) except for very high values of x, corresponding to short distances between units. These short distances are dominated by the correlation hole effect due to EV [16,26]. 

As mentioned above, LSD yields a reasonable description of the exchange-correlation hole, because it satisfies several exact conditions. However, since the correlation hole satisfies a zero sum rule, the scale of the hole must be set by its value at some value of . The local approximation is most accurate at points near the electron. In fact, while not exact at m = 0, LSD is highly accurate there. Thus the on-top hole provides the missing link between the uniform electron gas and real atoms and molecules [18]. 

K. S. Schweizer and J. G. Curro, Macromolecules, 21, 3070 (1988). Integral Equation Theory of Polymer Melts Intramolecular Structure, Local Order, and the Correlation Hole. 

In this work we are particularly concerned with the spatial extent of the effects of this self-pairing correlation on the motions of the electrons and with whether or not the net effect of this correlation for any one particle, the correlation hole, may be localized to one particular region of space. 

In an electronic system it is essential to distinguish between the correlation of electrons with identical spin, Fermi correlation, from that of electrons with opposite spins. Coulomb correlation. The correlation term/(ri,r2) as defined in eqn (E7.10) will measure both types of correlation (McWeeny 1960). However, the limiting value of the correlation hole as expressed in eqn (E7.11) arises only from the correlation between electrons of the same spin, the Fermi hole. That part of/(r, r2) that refers to the correlation between electrons of opposite spin contributes zero when integrated over all space. If the co-... 

Thus, only if the correlation hole for each of the JV( 2) particles in Ci is totally contained within Q for all values of rj in 2 will D2(ii, 2) correspond to the exact number of pairs which can be formed from a distinct set of JV( 2) particles. That is, in this limiting situation the N Q) electrons (where JV( 2) is necessarily an integer) generate a pure pair population with... 

Complete localization is, of course, possible only for an isolated system. What is remarkable, however, is the extent to which the electrons of atoms in an ionic molecule approach this limit of perfect localization, with /(fi) values in excess of 95 per cent not being uncommon. In systems, such as the fluorides and chlorides of lithium and sodium displayed in Fig. E7.2, the atomic surface of zero flux is found to minimize the fluctuation in the atomic populations and, thus, the magnitude of the correlation hole per particle is an extremum for such atoms. The properties of the number and pair densities for these... 

Since the correlation hole is everything that is not in the exchange hole, and since the whole exchange-correlation hole integrates to -1, we can immediately say that we have... 

In the correlation hole concept, the S(q) function does not exhibit a maximum. However, upon combination with a monotonically decreasing P(q)... 

The exact DGE for the correlation hole is not known except in the high-density limit, so away from this limit approximate gradient-corrected models have to be used instead. Perdew and coworkers [153,158] constructed a real-space model for the spherically-averaged GGA correlation hole starting from an accurate analytic representation A(,(rs, u) of the LSDA correlation hole, adding a gradient correction u), and... 

Around the same time, Becke [184] proposed a real-space normalized model for the correlation hole. He started from the observation that, for a given coupling strength A, the spin-polarized components of the spherically averaged pair densities have well-defined short-range behavior near the reference point [185] ... 

The correlation hole is multipole-like for each position of one electron any charge scooped out by the Coulomb repulsion near to that position must be compensated by a build-up of charge somewhere else to give no net charge. 

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