Symmetry-covariant energy density

Treatment of space symmetries in the construction of EDFs is another kettle of fish. Here, we base our considerations on the derivation before separation of symmetries, which we introduced above, and on the fact that invariance of the energy density itself is not a prerequisite for the invariance of the EDF. In fact, the EDF and total energy (1) are invariant with respect to symmetry $S$ also when the energy density is covariant with $S$, i.e.,

$${\cal H}^S(\vec {r}) = {\cal H}(S^+\vec {r}S),$$ (83)

where $S^+\vec {r}S$ denotes the space point transformed by symmetry $S$ (see also discussion in Ref. [32]). Indeed, due to the fact that space integrals are invariant, we have

$$\int{\rm d}^3\vec {r}{\cal H}(S^+\vec {r}S) = \int{\rm d}^3r{\cal H}(\vec {r}),$$ (84)

which guarantees invariance of the EDF and total energy.

For the space-inversion symmetry, we have

$${\cal H}^P(\vec {r}) \equiv {\cal H}\left[\rho^P(\vec {r}\sigma,\vec {r}'\sigma')\right]$$

$$= {\cal H}\left[\rho(-\vec {r}\sigma,-\vec {r}'\sigma')\right]$$ (85)

$${\cal H}(P^+\vec {r}P) \equiv {\cal H}(-\vec {r}) ,$$ (86)

and the covariance condition (83) reads

$${\cal H}\left[\rho(-\vec {r}\sigma,-\vec {r}'\sigma') \right] = {\cal H}(-\vec {r}) .$$ (87)

It is now essential to realize that the arguments of the density matrix, $\rho(-\vec {r}\sigma,-\vec {r}'\sigma')$ on which the energy densities in Eq. (87) depends, are the same on both sides of Eq. (87). The covariance condition then tests only the parity of all other operators that may appear in the definition of local densities. Therefore, to each primary density $\rho_{nLvJ}(\vec {r})$ (23) we may attribute $P$-parity corresponding to the $P$-parity of the operator $K_{nL}$ only, which is equal to $(-1)^{n}$, see Eq. (26) and columns denoted by $P$ in Tables 3 and 4. This attribution is performed regardless of space-inversion properties of the nonlocal densities, i.e., regardless of whether the parity of the many-body state is conserved or broken. Similarly, $P$-parities of secondary densities $\rho_{mI,nLvJ,Q}(\vec {r})$ (24) are equal to $(-1)^{n+m}$. Construction of the $P$-covariant energy density (83) can now be realized by multiplying densities that have identical $P$-parities.

Construction of a rotationally covariant energy density can be performed in an entirely analogous way. We must only ensure, that all tensor operators used in constructing all terms of the energy density are always coupled to total angular momentum (rank) zero. This coupling proceeds regardless of any transformation properties of nonlocal densities with respect to rotation, because again, their rotated space arguments appear on both sides of the covariance condition (83).

It is obvious that this is the correct procedure to follow when the rotational symmetry is not broken, and nonlocal densities $\rho(\vec {r},\vec {r}')$ and $\vec {s}(\vec {r},\vec {r}')$ are scalar and vector functions of their arguments, respectively. In fact, this is how we refer to these densities throughout the entire paper, seemingly forgetting that the rotational symmetry can be broken, and that these functions can then have no good tensor properties with respect to rotation. Nevertheless, in view of the covariance condition (83), these rotational properties of broken-symmetry nonlocal densities are irrelevant for the construction of the energy density.