All three expectation values on the right hand side of Eq. 24 match the general E gen c of Eq. 15, and thus, we can use the gauge strategy from Sec. III [E gen c → e gen c (r)] to find an expression for each energy density term on the left hand side of Eq. 24. This involves writing every expectation value in Eq. 24 in terms of the simplest possible reduced quantity from Eqs. 10-13. Starting with e λ c,ee (r), we recall that any expectation value of Vee (Eq. 24a) explicitly depends on the pair density. Hence, e λ c,ee (r) can be expressed analogously to the DFT correlation energy density (Eq. 19),