shell is assumed [1, 2]. The physical density ρ(r) which can be mapped onto such a
non-interacting density ρ s (r) is said to be non-interacting pure state v-representable,
or PS-VR for brevity [21]. Naturally, in the case of non-interacting particles, such a
pure state wavefunction is represented by a single KS determinant and one may
speak of a determinantal v-representability (D-VR) as well [21].
A general proof of the existence of such a KS potential v s (r) and of the PS-VR
property has never been achieved for an arbitrary physical density. By contrast,
rigorous theoretical arguments have been given in favor of an alternative representation of an arbitrary fermionic density ρ(r) by an ensemble (weighted sum) of a
finite number (M ) of the densities ρ K (r) originating from the same physical external
potential v ext (r) [19–21]:
ρ r
ð Þ ¼
X M
K¼1
λ K ρ K r
ð Þ, λ K ! 0,
X M
K¼1
λ K ¼ 1 :
ð2Þ
For the ensemble v-representable (E-VR) densities, the existence of a universal
density functional F[ρ] and its differentiability with respect to the density ρ(r) were
rigorously proved [20, 21], thus confirming the existence of the KS potential v s (r)
and the respective non-interacting KS system.
Initially, ensemble DFT was formulated for ground state ensembles [20, 21],
which implied that one could speak of averaging over degenerate electronic states.
It is natural to assume that the degeneracy is imposed by the symmetry of the
system. This seems a plausible assumption in the case of interacting particles,
although for the non-interacting fermions (such as the KS reference system) there
is a possibility of accidental degeneracy of several electronic configurations as was
demonstrated in first principles numeric experiments by Schipper et al. [26] and by
Morrison [28]. In these works it was shown that, when obtaining the KS potential
v s (r) from the known (nearly) exact density [62], the fractional occupation numbers
of several KS orbitals (i.e., the ensemble representation) have to be invoked. Thus,
certain physical (i.e., interacting) PS-VR densities (the target densities were
obtained from the accurate ab initio WFT calculations) can only be mapped onto
the non-interacting E-VR densities. Remarkably, these target densities were
obtained for molecular systems for which it was known that their electronic
structure is dominated by the non-dynamic electron correlation; [63] in particular,
the rectangular H 2 + H 2 system, the ground state of the C 2 molecule [26], and the
ground state of a series of Be-like atomic ions [28] were investigated. For these
atomic and molecular systems it is well established that, at the ab initio WFT level,
their ground state wavefunctions require a multi-reference description, which is
typically associated with the strong non-dynamic correlation [63].
The ensemble representation of the non-interacting KS reference system leads
naturally to the fractional occupation numbers of KS orbitals:
100
M. Filatov
non-interacting density ρ s (r) is said to be non-interacting pure state v-representable,
or PS-VR for brevity [21]. Naturally, in the case of non-interacting particles, such a
pure state wavefunction is represented by a single KS determinant and one may
speak of a determinantal v-representability (D-VR) as well [21].
A general proof of the existence of such a KS potential v s (r) and of the PS-VR
property has never been achieved for an arbitrary physical density. By contrast,
rigorous theoretical arguments have been given in favor of an alternative representation of an arbitrary fermionic density ρ(r) by an ensemble (weighted sum) of a
finite number (M ) of the densities ρ K (r) originating from the same physical external
potential v ext (r) [19–21]:
ρ r
ð Þ ¼
X M
K¼1
λ K ρ K r
ð Þ, λ K ! 0,
X M
K¼1
λ K ¼ 1 :
ð2Þ
For the ensemble v-representable (E-VR) densities, the existence of a universal
density functional F[ρ] and its differentiability with respect to the density ρ(r) were
rigorously proved [20, 21], thus confirming the existence of the KS potential v s (r)
and the respective non-interacting KS system.
Initially, ensemble DFT was formulated for ground state ensembles [20, 21],
which implied that one could speak of averaging over degenerate electronic states.
It is natural to assume that the degeneracy is imposed by the symmetry of the
system. This seems a plausible assumption in the case of interacting particles,
although for the non-interacting fermions (such as the KS reference system) there
is a possibility of accidental degeneracy of several electronic configurations as was
demonstrated in first principles numeric experiments by Schipper et al. [26] and by
Morrison [28]. In these works it was shown that, when obtaining the KS potential
v s (r) from the known (nearly) exact density [62], the fractional occupation numbers
of several KS orbitals (i.e., the ensemble representation) have to be invoked. Thus,
certain physical (i.e., interacting) PS-VR densities (the target densities were
obtained from the accurate ab initio WFT calculations) can only be mapped onto
the non-interacting E-VR densities. Remarkably, these target densities were
obtained for molecular systems for which it was known that their electronic
structure is dominated by the non-dynamic electron correlation; [63] in particular,
the rectangular H 2 + H 2 system, the ground state of the C 2 molecule [26], and the
ground state of a series of Be-like atomic ions [28] were investigated. For these
atomic and molecular systems it is well established that, at the ab initio WFT level,
their ground state wavefunctions require a multi-reference description, which is
typically associated with the strong non-dynamic correlation [63].
The ensemble representation of the non-interacting KS reference system leads
naturally to the fractional occupation numbers of KS orbitals:
100
M. Filatov