The DFT sum rules

The HFB sum rules derived in Sec. 2.3 are based on the linearity of the Hamiltonian, by which a matrix element involving the HFB state is a sum of matrix elements calculated for all the PNP components (18). In order to derive the analogous sum rules for the projected DFT energies, one can only use properties of the underlying transition energy density. To this end, we recall that in the HFB theory, the mixing of particle numbers corresponds to the broken U(1) gauge symmetry, and that the PNP actually corresponds to expanding the HFB state in irreducible representations of this group. This observation can be extended to the DFT transition energy density, expanded in these same irreducible representations, with the projected DFT energies being the expansion coefficients. The resulting sum rules must follow from the closure relations on the group manifold.

These general remarks can be expressed in an explicit form in the following way. By using integration contours that are circles of radius $\vert z_0\vert$ around the origin, $z=z_0e^{i\phi}$, we have the following expression for the projected DFT energy (34)

$$\langle\Psi_N\vert\Psi_N\rangle E_{\mbox{\rm\scriptsize {DF... ...{2\pi} \int_0^{2\pi} {\rm d}\phi e^{-iN\phi}{\cal{}E}(\phi) ,$$ (48)

where by ${\cal{}E}(\phi)$ we denoted the part of the integrand that does not depend on $N$, i.e.,

$${\cal{}E}(\phi) = \langle\Phi\vert\Phi(z)\rangle E_{\mbox... ...o_z,\chi_z,\bar{\chi}_z) \quad\mbox{at}\quad z=z_0e^{i\phi} .$$ (49)

Hence, the DFT projected energy is given by a Fourier transform of ${\cal{}E}(\phi)$. Since the Fourier components constitute a complete set of functions on a circle, $\sum_{N=0}^{\infty}e^{-iN\phi}=2\pi\delta(0)$, we obtain the DFT sum rule,

$$\langle\Phi\vert\Phi(z_0)\rangle E_{\mbox{\rm\scriptsize {... ...le\Psi_N\vert\Psi_N\rangle E_{\mbox{\rm\scriptsize {DFT}}}^N ,$$ (50)

which is the analogue of the HFB sum rule for matrix elements (24). For $z_0$=1, we obtain the DFT counterpart of the HFB sum rule (20):

$$E_{\mbox{\rm\scriptsize {DFT}}}(\rho,\chi,\chi^*) = \sum_{... ...le\Psi_N\vert\Psi_N\rangle E_{\mbox{\rm\scriptsize {DFT}}}^N .$$ (51)

We note that in the above derivations, $z_0$ is an arbitrary complex number; its modulus fixes the radius of integration contour, while its phase gives the point on the circle that fixes the starting point of the integral in Eq. (48). This starting point has obviously no importance for the value of the integral. The sum rule (50) gives, therefore, a representation of the DFT transition energy density in terms of a series expansion in $z_0$, which converges only on the ring between the poles. For each such ring, the projected DFT energies $E_{\mbox{\rm\scriptsize {DFT}}}^N$ are different, and the DFT transition energy density is thus equal to a different series expansion. It is obvious that these different values of the projected DFT energies do not contradict the continuity of the DFT transition energy density. In this way, all projected DFT energies for arbitrarily chosen contours of integration correspond to this same common DFT energy functional.