Density matrices exchange-correlation holes

This long-range correlation effect shows up in both the first-order density matrix and the exchange-correlation hole for finite systems [19]. We concentrate here on the exchange-correlation hole. The general asymptotic form of the pair density is then... 

In view of this interpretation of eqn ( 33.4) we might expect that this normalisation condition is essentially due to the exchange part of the exchange-correlation hole and that it might be useful to divide this density into an exchange hole and a correlation hole . Guided by the product form of the (real) two-particle density for the single-determinant model which involves the density function p and the density matrix pi ... 

The exchange-correlation energy can thus be obtained by integrating the electron-electron interaction over the A variable and subtracting the Coulomb part. The right-hand side of eq. (B.18) can be written in terms of the second-order reduced density matrix eq. (6.14), and the definition of the exchange-correlation hole in eq. (6.21) allows the Coulomb energy to be separated out. 

One of the most useful tools for the detectionof free volume holes or voids, free volume, and free volume distribution, at an atomic scale, is positron annihilation lifetime spectroscopy (PALS). The technique exploits the fact that the positively charged positron (e" "), the antiparticle to the electron, preferentially samples regions of low positive charge density. When injected in a polymer matrix, thermal-ized positrons can combine with an electron to form a bound state, known as positronium (Ps). This species can only exist in a void and it rapidly annihilates on contact with the electron cloud of a molecule. For polymer studies using PALS, it is ortho-positronium (oPs, a triplet state) that is of main interest. The oPs spin exchanges with electrons of opposite spin on the walls of the cavity and it is annihilated. Thus, the oPs lifetime, T3, gives a measure of the mean free volume cavity radius whereas the relative intensity of the oPs component, 1, can be related to the number of cavities. A semiempirical equation has been derived that correlates T3 with the cavity radius, r [81] ... 

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