Explicit construction of the energy density functional within an orbit

Explicit construction of the energy density functional within an orbit 

Let vis write down the explidt expression for the energy functional [W] in terms of the 1-matrix the 2-matrix Df xi,x2 xi,x2) and the one particle 

In order to obtain from [W] the fvmctional E[f] appearing in the variational prin-riple described by Eq. (24), we have to apply local-scaling transformations to the wavefunction M, or in view of Eq. (42), to the 1-matrix LA, to the 2-matrix Df and to the density pg. The pertinent expressions for these transformations are [21]  

In addition, let us separate the 1-matrix appearing in the right-hand side of Eq. (43) into its local and non-local components  

Substituting Eqs. (48) and (49) in Eq. (42), after some simple algebra we obtain [21] 

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