Lieb-Oxford bounds

We see that PW86 includes both the correct uniform gas limit and the GEA by having the right s2 dependence. It still has curve-crossing problems, but not as severe as LM. It violates the Lieb-Oxford bound, Eq. (66), and the exact condition... 

Here k = 0.804 is a nonempirical parameter chosen to satisfy the Lieb-Oxford bound in its local form F s) 1.804 for any s. The value of p= Pmb(tt /3) = 0.21951 is determined from the condition that the second-order gradient term for exchange cancel that for correlation (i.e. This choice, rather than p = /tak =... 

Among others, three conditions are of particular importance. The first constraint is related to the behavior in the small x region, where the GGA exchange functional should reduce to in order to recover the correct uniform gas limit. The second condition was defined by Levy, who showed that some scaling properties can be satisfied if the asymptotic form of the functional for large x is x , where a > Vi [68,69]. The last condition is the so called "Lieb-Oxford bound" [70], which states that ... 

The behavior of some of the most common functionals with respect to these three constraints is reported in table II. The B functional does not obey neither the Levy condition nor the Lieb-Oxford bound, but its numerical performances are better than those provided by the PW functional, which respects all the above mentioned constraints. 

In addition, the Lieb-Oxford bound condition (Lieb and Oxford 1981),... 

The Lieb-Oxford bound that establishes the most negative value that the exchange energy may attain for a given electron density, namely. 

Finally, we want to satisfy the Lieb-Oxford bound of (1.120) and (1.122), which LSD respects. We can achieve this, and also recover the limit of (1.215), with the simple form... 

The Lieb-Oxford bound of (1.122) will be satisfied for all densities n(r) if and only if... 

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