Projected DFT energy

Based on the above discussion, we postulate the projected DFT energy in the form:

$$E_{\mbox{\rm\scriptsize {DFT}}}^N= \frac{\oint_C dz z^{-N-... ...riptsize {$z=0$}}}$}\,z^{-N-1}\langle\Phi\vert\Phi(z)\rangle}.$$ (34)

At variance with the Hamiltonian-based HFB theory, the projected DFT energy may depend on the integration contour $C$. Moreover, the numerator in Eq. (34) is, in general, not equal to the residue at $z$=0 as in Eq. (15). Consequently, both the transition energy density and the contour $C$ define the projected energy in DFT. Since the projected DFT energy (34) must be real, in view of condition (32), we restrict further considerations only to contours which are symmetric with respect to the real $z$-axis. Accordingly, only the upper-half contour $C_+$ above the real axis can be considered and $\oint_C dz\cdots =2{\mbox{\rm\scriptsize {R}}e}[\oint_{C_+} dz\cdots ]$.