Sum rule constraints on Kubo-transformed correlation functions

We show how the Kubo transform can be inverted in the time domain and then use this result to investigate the sum rule constraints on a Kubo-transformed correlation function c ˜ AB ( t ) = 1 β ∫ 0 β d λ 〈 A ( - i λ ℏ ) B ( t ) 〉 that arise from the values of the static equilibrium properties c AB ( n ) ( 0 ) = [ d n 〈 A ( 0 ) B ( t ) 〉 / d t n ] t = 0 . We find, perhaps not surprisingly, that these sum rules only depend on the behavior of c ˜ AB ( t ) for times on the order of βℏ. The implications of this finding are discussed in light of the recent use of these sum rules to assess the quality of approximate Kubo-transformed correlation functions for liquid hydrogen at 14 K and liquid water at 298 K.