Transition energy density

In the DFT approach, the Hamiltonian of the system does not appear explicitly; hence, the projected energy cannot be calculated as its expectation value in the projected state. However, since the DFT energy density is most often postulated, not derived, we can apply the same philosophy to the projected energy, i.e., we can postulate the projected functional. In doing so, we have to guarantee that it reverts to the projected HFB energy (15) when the system is described by a Hamiltonian. In the present study, we do not discuss the construction of the projected DFT functional, but simply assume, as in most calculations up to now, that the DFT transition energy density $E_{\mbox{\rm\scriptsize {DFT}}}(\rho_z,\chi_z,\bar{\chi}_z)$ is the same as the DFT energy density $E_{\mbox{\rm\scriptsize {DFT}}}(\rho,\chi,\chi^*)$, but with densities $\rho$, $\chi$, and $\chi^*$ (30) replaced by the transition densities $\rho _z$, $\chi_z$, and $\bar{\chi}_z$ (27). This guarantees that in the limit of $z\rightarrow 1$, the projected functional gets back to the usual form.

Since the overlap (25) and HFB transition energy density (26) depend only on the shift parameter $z$ and not on its complex conjugation $z^*$, it is natural to restrict further considerations to the DFT transition energy density parametrized in the same way, i.e.,

$$E_{\mbox{\rm\scriptsize {DFT}}}^*(z)=E_{\mbox{\rm\scriptsize {DFT}}}(z^*) .$$ (32)

Moreover, by construction, the DFT transition energy density depends only on $z^2$, and therefore it must be a symmetric function of $z$,

$$E_{\mbox{\rm\scriptsize {DFT}}}(-z)=E_{\mbox{\rm\scriptsize {DFT}}}(z) .$$ (33)