Analysis of the Orbital-Dependent Correlation

Given the motivation for implicit correlation functionals the first question to be addressed is that of dispersion forces. As none of the semi-empirical functionals of Sect. 2.5 can deal with these long-range forces, the present discussion focuses on the second order correlation functional (2.82) as the simplest first-principles functional. 

Consider two neutral atoms A and B, separated by a sufficiently large distance R, so that no molecular orbitals are formed. For this system, the 

if one takes the atoms apart, only those Slater integrals in Ec survive which do not link orbitals from different atomic centers at the same point r. Consequently, neither nor the exchange component of E contribute 

If one expands (2.108) in powers of 1/f and re-introduces the frequency integration inherent in the ring diagram, one ends up with an expression [47] which is much more familiar. 

Here aA(m) denotes the atomic polarizability (for the case of closed subshells), evaluated at imaginary frequency. 

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