Frequency moments of the dielectric function for an inhomogeneous electron gas

A unified treatment of the arbitrary frequency moments Iqq(n)of the longitudinal dielectric matrix epsilonqq'( omega ) and of the moments Fqq(n)of epsilonqq'-1( omega ) is given for the inhomogeneous electron gas. The Iqq(n)are in the RPA, whereas the Fqq(n)are exact in the non-relativistic limit excluding phonons. Furthermore, the author derives formulae allowing the Fqq(n)to be calculated from Iqq(n)and vice versa for a given approximation, which obeys the Kramers-Kronig relations. Using inequalities between the moments the author derives an inequality relating the static dielectric function of a non-conductor and the electron density at the nuclear sites.