Compressibility sum rule

In 1974 Finnis showed that the q - 0 limit could be evaluated directly by using the compressibility sum rule of the free-electron gas, which relates the long wavelength behaviour of the dielectric constant to its compressibility. He found that... 

Beyond the noninteracting limit only the vacuum part of the 2-loop contribution to the polarization function has been evaluated [2()4,205]. Moreover, the screening length n rf(0,0) is related to the energy density via the compressibility sum rule [206],... 

The first order terms in (323) vanish due to norm conservation, Eq.(309). The second order terms can be rewritten by use of the compressibility sum rule (284),... 

Typically lattice-induced strain results in the bonds around one cation being stretched and the bonds around another cation being compressed as found in BaRuOs (10253) by Santoro et al. (1999, 2000). When this happens, the valence sum rule will be violated around the cations in question but the valence still distributes itself as uniformly as possible among the bonds, so that the experimental bond valences determined from the bond lengths remain as close as possible to the theoretical bond valences. For this reason the BSI is typically smaller than the GII for lattice-induced strains, though the opposite is true for compounds with electronically induced strain where the valence sum rule remains well obeyed. 

Lattice-induced strains are characterized by large values of the GII because the environments around some atoms are stretched and around other atoms are compressed but, since the valence is still distributed as uniformly as possible among the bonds, the BSI remains small. This contrast with the electronically driven distortions discussed in Chapter 8 where the GII is small (the valence sum rule is obeyed) but the BSI is necessarily large. 

Eq. (Vn.26) is not entirely rigorous because it neglects certain anomalies in the viscosity of the solution which renormalize 17, in the denominator however, these corrections are minor. It is also possible to express the osmotic compressibility dc/dll in terms of the pair correlation (r) through the sum rule... 

Before passing on to specific results three characteristics of the TDLDA should be noted. First, at zero frequency it is exact within density functional theoryl and therefore provides an explicit test of the LDA itself. Second, the cross section satisfies the f-sum rule and the compressibility sum rulel ... 

There are a number of tools that are useful in validating either a measured or a proposed structure. The most widely used is the valence sum rule (2) using bond valences calculated from the observed bond lengths. Experimental uncertainties will mean that the atomic valence and bond valence sum are rarely exactly the same, but for a well-determined structure the difference is usually around 0.05 vu. Larger differences are often found, indicating that some bonds are compressed or stretched by the steric constraints imposed on the structure (Sect. 8). In some cases the differences can be quite large, but for a strained structure to be in equilibrium both stretched and compressed bonds must be present, which is key to verifying the presence of steric strains. 

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