# The structure of the density-potential mapping
Part I: Standard density-functional theory

Markus Penz  Basic Research Community for Physics, Innsbruck, Austria  Erik I.
Tellgren  Hylleraas Centre for Quantum Molecular Sciences, Department of
Chemistry, University of Oslo, Norway  Mihály A. Csirik  Hylleraas Centre for
Quantum Molecular Sciences, Department of Chemistry, University of Oslo,
Norway  Department of Computer Science, Oslo Metropolitan University, Norway
Michael Ruggenthaler  Max Planck Institute for the Structure and Dynamics of
Matter, Hamburg, Germany  Andre Laestadius<EMAIL_ADDRESS>Department of Computer Science, Oslo Metropolitan University, Norway
Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry,
University of Oslo, Norway

###### Abstract

The Hohenberg–Kohn theorem of density-functional theory (DFT) is broadly
considered the conceptual basis for a full characterization of an electronic
system in its ground state by just the one-body particle density. Part I of
this review aims at clarifying the status of the Hohenberg–Kohn theorem within
DFT and Part II at different extensions of the theory that include magnetic
fields. We collect evidence that the Hohenberg–Kohn theorem does not so much
form the basis of DFT, but is rather the consequence of a more comprehensive
mathematical framework. Such results are especially useful when it comes to
the construction of generalized DFTs.

###### Contents

  1. I Introduction
  2. II Preliminaries
  3. III Representability of densities
  4. IV The Hohenberg–Kohn theorem
  5. V The unique-continuation property
  6. VI Hierarchy of density functionals
  7. VII Density-potential mappings from differentials
  8. VIII Linking to a reference system: the Kohn–Sham scheme
  9. IX Density-potential mixing and regularized DFT
  10. X Abstract density-potential mapping
  11. XI Summary
  12. XII Outlook

##  I Introduction

The theorem of Hohenberg and Kohn [1] (HK) is usually presented as the
theoretical justification of density-functional theory (DFT). It states that
the one-body particle density uniquely (up to an additive constant) determines
the scalar potential of a non-relativistic many-electron system in its ground
state. The Mathematical analysis of ground-state DFT was pioneered by Lieb
[2], using tools from convex analysis. In it, some important problems,
especially in relation with differentiability of the involved functionals that
map densities to energies, were left unanswered and remained as open
questions. Lammert [3] then demonstrated that the key functional of DFT is
indeed non-differentiable, but it remained unclear to what extent this
threatens the foundations of DFT and its algorithmic realization, the
Kohn–Sham scheme employed for practical calculations. Regularization as a
means to overcome non-differentiability has been applied to DFT [4] (Section
IX) and its extension, current DFT (CDFT) [5; 6]. The existence of functional
derivatives through regularization also avoids the problem of
$v$-representability that usually haunts DFT, i.e., that not every reasonable
density is the solution to a certain potential (Section III).

A central result in this work is a very convenient and novel formulation of
the HK theorem that restructures it into two sub-theorems, HK1 and HK2
(Section IV):

  * •

(HK1) If two potentials share a common ground-state density then they also
share a common ground-state wave function or density matrix.

  * •

(HK2) If two potentials share any common eigenstate and if that eigenstate is
non-zero almost everywhere (a property that is guaranteed if the the unique-
continuation property (UCP) holds; see Section V) then they are equal up to a
constant.

Combining HK1 and HK2, one obtains the classical HK theorem and with it a
well-defined density-potential mapping. The proof of HK1 will be shown to be
immediate from just the formulation of “ground-state energy”. Consequently, it
is also easily attainable in an abstract or extended formulation of DFT
(Section X). The situation for HK2, on the other hand, is more complicated
but, as will be demonstrated, it holds true with certain restrictions in the
standard DFT setting. It is known _not_ to hold in paramagnetic CDFT [7] and
has, to the best of our knowledge, an unknown status in total CDFT. In Part II
of this review, we will exemplify how different DFTs follow this structure
and, maybe more importantly, pinpoint why this route might fail.

After analyzing its basic structure, the status of the HK theorem within DFT
is scrutinized. If only the ground state of a system is the matter of
interest, a constrained-search approach seems to be sufficient for the
formulation of DFT, and the usual type of constrained-search functional [8; 9]
even _implicitly includes_ HK1 (Section VI). Besides being a mathematically
more transparent formulation than the HK theorem, the constrained-search
formalism is also a better starting point for deriving approximate density
functionals. Nonetheless, the full HK theorem remains important for going
beyond the bare minimum needed to set up a ground-state theory. For example,
the HK theorem implies that the ground-state density determines not only the
ground state but, by fixing the scalar potential, also all excited states.
This becomes relevant when thermostatistical properties are considered.
Furthermore, in order to be able to define the Kohn–Sham scheme (Section
VIII), one actually demands more than just the HK result, relying on
differentiability of the energy functional that in turn would imply the
_whole_ HK result (Section VII). Consequently, in a (Moreau–Yosida)
regularized setting, the Kohn–Sham scheme can be rigorously formulated and
even proven to converge in finitely many dimensions [10; 11; 12], and HK
becomes just a by-product.

Although we will do our best to orient the reader within the rich subject that
is DFT, the scope of this review is limited. We will mainly focus on, in our
opinion, matters closely related to the HK mapping and properties of the exact
functional(s). Many excellent reviews and textbooks are available on the
subject [13; 14; 15; 16; 17; 18]. For the interested reader, we also point out
the very recent article based on a round-table discussion [19].

##  II Preliminaries

Density-functional theory is an approach to describe particles that obey the
laws of quantum mechanics, but that avoids their full description by a wave
function and instead switches to reduced quantities like the one-particle
density. In its basic form discussed here, the focus is solely on the ground-
state properties of the quantum system. For the configuration space of a
single particle we always choose $\mathbb{R}^{3}$ with the additional spin
degree-of-freedom for spin-$\frac{1}{2}$ particles. The Hamiltonian comprises
three parts,

$H[v]=T+W+V[v],$

relating to the kinetic energy, the Coulomb repulsion, and the external scalar
potential, respectively. The internal parts will be collected as $H_{0}=T+W$.
The kinetic-energy operator is $T=-\frac{1}{2}\sum_{i=1}^{N}\nabla_{i}^{2}$ in
standard DFT, where atomic units are employed. Notation-wise, we use small
letters for one-body objects. The external potential contribution $V[v]$ is
always defined from a one-body potential $v(\mathbf{r})$ and is of an additive
form,

$V[v](\underline{\mathbf{r}})=\sum_{i=1}^{N}v(\mathbf{r}_{i}),$

where $\underline{\mathbf{r}}=(\mathbf{r}_{1},\dots,\mathbf{r}_{N})$. For
later reference we also define
$\underline{\sigma}=(\sigma_{1},\dots,\sigma_{N})$ for the spin degrees-of-
freedom. The full quantum-mechanical description of a system in its ground
state is achieved by determining the eigenstate $\psi_{0}$ of $H[v]$ that has
the correct symmetry and the lowest eigenvalue $E_{0}$ (ground-state energy),

$H[v]\psi_{0}=E_{0}\psi_{0}.$ (1)

If such a lowest eigenstate is not unique, we speak of _degeneracy_ , a case
that will often appear in the discussion below and that leads to several
complicacies. Then a valid ground state can also be given as a statistical
mixture of the pure ground states $\psi_{k}$ in the form of a density matrix
$\Gamma=\sum_{k}\lambda_{k}|\psi_{k}\rangle\langle\psi_{k}|$ with
$\lambda_{k}\in[0,1]$ and $\sum_{k}\lambda_{k}=1$. It is natural to require
states of finite kinetic energy,

$\langle\psi|T|\psi\rangle=\frac{1}{2}\sum_{i=1}^{N}\sum_{\underline{\sigma}}\int_{\mathbb{R}^{3N}}|\nabla_{i}\psi|^{2}\,\mathrm{d}\underline{\mathbf{r}}<+\infty,$

and we define the basic set for wave functions

$\mathcal{W}=\\{\psi\mid\,\psi\;\text{anti-
symmetric},\,\langle\psi|T|\psi\rangle<+\infty\\}.$

In cases where density matrices $\Gamma$ are considered, we require
$\psi_{k}\in\mathcal{W}$ for all their components.

The one-particle density of a given $\psi$ as the basic variable of standard
DFT is

$\rho_{\psi}(\mathbf{r}_{1})=N\sum_{\underline{\sigma}}\int_{\mathbb{R}^{3(N-1)}}|\psi|^{2}\,\mathrm{d}\mathbf{r}_{\perp},$
(2)

where we used the shorthand notation
$\mathbf{r}_{\perp}=(\mathbf{r}_{2},\dots,\mathbf{r}_{N})$, and it is
$\rho_{\Gamma}(\mathbf{r})=\sum_{k}\lambda_{k}\rho_{\psi_{k}}(\mathbf{r})$ for
a given mixed state $\Gamma$. Since $\Gamma$ already includes the squared wave
function from Eq. (2), the mapping $\Gamma\mapsto\rho_{\Gamma}$ is linear.
Note that whenever we talk about a “density”, this will be assumed to be a map
$\rho:\mathbb{R}^{3}\to\mathbb{R}_{\geq 0}$ that is normalized to the particle
number $N$, $\int\rho(\mathbf{r})\,\mathrm{d}\mathbf{r}=N$, like it is
automatically the case for $\rho_{\psi}$ and $\Gamma_{\psi}$ if $\psi,\Gamma$
are normalized to 1.

The density alone suffices to give an expression for the potential energy
contribution. The resulting integral over the single-particle configuration
space will be written like an inner product $\langle\cdot,\cdot\rangle$, to
wit,

$\displaystyle\langle\psi|V[v]|\psi\rangle$
$\displaystyle=\sum_{i=1}^{N}\sum_{\underline{\sigma}}\int_{\mathbb{R}^{3N}}v(\mathbf{r}_{i})|\psi|^{2}\,\mathrm{d}\underline{\mathbf{r}}$
(3)
$\displaystyle=N\sum_{\underline{\sigma}}\int_{\mathbb{R}^{3}}v(\mathbf{r}_{1})\int_{\mathbb{R}^{3(N-1)}}|\psi|^{2}\,\mathrm{d}\mathbf{r}_{\perp}\,\mathrm{d}\mathbf{r}_{1}$
$\displaystyle=\int_{\mathbb{R}^{3}}v(\mathbf{r})\rho_{\psi}(\mathbf{r})\,\mathrm{d}\mathbf{r}=\langle
v,\rho_{\psi}\rangle.$

The notation $\langle v,\rho\rangle$ thus expresses a dual pairing between two
$L^{p}$ spaces or a combination of such, one for densities and the other one
for potentials. These density and potential spaces are the topic of the next
section. Without going into technicalities, the space $L^{p}(\mathbb{R}^{n})$,
$1\leq p\leq\infty$, can be thought of as all functions $f(\mathbf{r})$ that
have a finite $L^{p}$ norm

$\|f\|_{L^{p}}=\left(\int_{\mathbb{R}^{n}}|f(\mathbf{r})|^{p}\,\mathrm{d}\mathbf{r}\right)^{1/p}<\infty,$

where in the case $p=\infty$ a supremum norm is employed instead.

##  III Representability of densities

The notion of “representability” is ubiquitous and conceptually important in
DFT. It generally refers to the situation that any density of a certain class
comes from a well-defined construction. Such a construction can simply be how
a density is calculated from an $N$-particle wave function of finite kinetic
energy following Eq. (2) and we then call the density “$N$-representable”. Or
one demands that the density should be that of an actual ground-state solution
of a Schrödinger equation with some given external potential $v$ and one calls
it “$v$-representable”. However, this definition of $v$-representability is a
bit naive since the set of permitted potentials to choose from was not even
specified [3]. One could argue that any potential that can be put into the
Schrödinger equation should be considered, but then the dual pairing $\langle
v,\rho\rangle$ appearing in Eq. (3) between the spaces of densities and
potentials might be “lost”, which has consequences for the density functionals
defined later in Section VI. So in order to talk about $v$-representability,
we will first have to choose a basic density space that includes the
$N$-representable densities.

The task of determining $N$-representable density classes was originally
tackled by Gilbert [20] and Harriman [21]. In the first work,
differentiability of the density was required, whereas in the second work no
further conditions on the density were assumed. Here, we rely on the version
by Lieb [2, Theorem 1.2] that gives the following class of $N$-representable
densities,

$N\text{-}\mathsf{rep}=\left\\{\rho\mid\rho(\mathbf{r})\geq
0,\smallint\rho\,\mathrm{d}\mathbf{r}=N,\nabla\sqrt{\rho}\in
L^{2}(\mathbb{R}^{3})\right\\}.$

The benefit of the additional constraint $\nabla\sqrt{\rho}\in L^{2}$ is that
one can always find a wave-function that not only gives the desired density
but also has finite kinetic energy and is thus in $\mathcal{W}$ (and in
addition is properly normalized). Lieb [2] further showed that
$N\text{-}\mathsf{rep}$ is convex and included in $X=L^{1}(\mathbb{R}^{3})\cap
L^{3}(\mathbb{R}^{3})$. This space $X$ is the basic density space in terms of
$L^{p}$ spaces, so by Eq. (3) this automatically yields a corresponding
potential space that is its dual,
$X^{*}=L^{3/2}(\mathbb{R}^{3})+L^{\infty}(\mathbb{R}^{3})$. Any element $v\in
X^{*}$ can thus be written as $v=v_{1}+v_{2}$ with $v_{1}\in
L^{3/2}(\mathbb{R}^{3})$ and $v_{2}\in L^{\infty}(\mathbb{R}^{3})$. Potentials
of Coulomb type, $v(\mathbf{r})=Cr^{-1}$, $r=|\mathbf{r}|$, are for example
elements of this $X^{*}$ (by virtue of
$\int_{0}^{R}|v(\mathbf{r})|^{3/2}r^{2}\,\mathrm{d}r<\infty$ for any finite
$R>0$ and $|v(\mathbf{r})|<\infty$ for $r>R$).

The issue of “$v$-representability” is much more profound. To date there is no
explicit description for the set of all $v$-representable densities
$v\text{-}\mathsf{rep}$. This issue is known as the “$v$-representability
problem”. We already noted that $v\text{-}\mathsf{rep}$ should contain all
densities that are a ground-state density for some potential $v\in X^{*}$. For
a glimpse of what densities have to be included in this set we refer to the
illustrative construction of Englisch and Englisch [22]. At this point one has
to differentiate between several levels of $v$-representability. We defined
$v\text{-}\mathsf{rep}$ as coming from a ground state of a Schrödinger
equation with some given external potential $v$. Within DFT we usually
consider two settings, the full system that contains a (Coulomb) interaction
$W$ and the Kohn–Sham system that does not. So whenever we talk about
$v$-representability, this can be amended by the attributes “interacting” or
“non-interacting” and it is not obvious at this point if the two classes are
equal, overlap, or are even disjoint. After all, the sets are not explicitly
known. Within each class we also have the possibility of ground-state
degeneracy. Then, instead of ground-state wave functions, the more general
concept of density matrices comes into play. The resulting notions are then
“pure-state $v$-representability” and “ensemble $v$-representability”. In the
second case such a density $\rho$ is then the convex combination of pure-state
$v$-representable densities $\rho_{k}$ that come from the degenerate ground-
states $\psi_{k}$ of $H[v]$, i.e., $\rho=\sum_{k}\lambda_{k}\rho_{k}$
($\lambda_{k}\in[0,1],\sum_{k}\lambda_{k}=1$). In the first case only
densities from pure states are allowed, but they might still individually come
from a set of degenerate ground-state wave functions. It was demonstrated by
Englisch and Englisch [22] by giving explicit examples that there are
$N$-representable densities that are not ensemble $v$-representable (an
obvious example is a density that vanishes on a set of positive measure,
however, for more elaborate examples we refer to Section 3.2 in Ref. 22). Levy
[23] and Lieb [2] gave arguments that an ensemble $v$-representable density
does not have to be pure-state $v$-representable. An explicit example for such
a density $\rho\in v\text{-}\mathsf{rep}_{\mathrm{ens}}\setminus
v\text{-}\mathsf{rep}_{\mathrm{pure}}$ was found within a finite-lattice
system of cuboctahedral symmetry [24]. So we can symbolically note that

$v\text{-}\mathsf{rep}_{\mathrm{pure}}\subsetneqq
v\text{-}\mathsf{rep}_{\mathrm{ens}}\subsetneqq
N\text{-}\mathsf{rep}\subsetneqq X.$ (4)

In Garrigue [25] it was demonstrated that the set
$v\text{-}\mathsf{rep}_{\mathrm{pure}}$ is path-connected. There are further
topological relations between the sets appearing in Eq. (4) that are worth
mentioning. Since every $\rho\in v\text{-}\mathsf{rep}_{\mathrm{ens}}$ is a
convex combination $\rho=\sum_{k}\lambda_{k}\rho_{k}$ with $\rho_{k}\in
v\text{-}\mathsf{rep}_{\mathrm{pure}}$, it holds

$v\text{-}\mathsf{rep}_{\mathrm{ens}}\subseteqq\operatorname{conv}v\text{-}\mathsf{rep}_{\mathrm{pure}}\subsetneqq
N\text{-}\mathsf{rep}\subsetneqq X,$

where $\operatorname{conv}$ is the convex hull of a set. So while
$v\text{-}\mathsf{rep}_{\mathrm{pure}}$ is definitely not convex because of
the mentioned counterexamples, $v\text{-}\mathsf{rep}_{\mathrm{ens}}$ might
still be (to our understanding this is not known). Lastly,
$N\text{-}\mathsf{rep}$ is the closure of
$v\text{-}\mathsf{rep}_{\mathrm{ens}}$ within $L^{1}\cap L^{3}$, which means
that any $\rho\in N\text{-}\mathsf{rep}$ can be approximated arbitrarily well
by densities in $v\text{-}\mathsf{rep}_{\mathrm{ens}}$ when distance is
measured in the $L^{1}\cap L^{3}$-norm [2, Theorem 3.14]. With the notion of
the “subdifferential” from Section VII, this result can be established as a
direct consequence of the Brøndsted–Rockafellar theorem [26, Corollary 2.44].
Still, potentials that lead to densities that are arbitrarily close could be
very far apart in the potential space $X^{*}$. On the other hand, it has been
suggested that $v\text{-}\mathsf{rep}_{\mathrm{pure}}$ is not dense in
$N\text{-}\mathsf{rep}$ (see Conjecture 3.8 in Ref. 27).

##  IV The Hohenberg–Kohn theorem

The classical HK theorem [1] states the existence of a well-defined density-
potential mapping for ground states. For a given potential $v$,

$\displaystyle E[v]$
$\displaystyle=\inf\left\\{\langle\psi|H_{0}+V[v]|\psi\rangle\mid\psi\in\mathcal{W},\|\psi\|=1\right\\}$
(5) $\displaystyle=\inf\left\\{\langle\psi|H_{0}|\psi\rangle+\langle
v,\rho_{\psi}\rangle\mid\psi\in\mathcal{W},\|\psi\|=1\right\\}$

is the _ground-state energy_ by the Rayleigh–Ritz variation principle. If a
minimizer exists then $\psi$ and $\rho_{\psi}$ are the corresponding _ground
state_ and _ground-state density_ that might not be unique in the case of
degeneracy. If a minimizer does not exist, there is still always a sequence
$\psi_{i}$ in $\mathcal{W}$ with $\|\psi_{i}\|=1$ such that
$\langle\psi_{i}|H_{0}+V[v]|\psi_{i}\rangle$ converges to $E[v]$. In Eq. (5),
$v$ should be selected from a class that makes $E[v]$ bounded below. See Reed
and Simon, Section X.2, for an extensive discussion on such potentials [28]. A
further demand on $v$ will later be that it guarantees a ground state that is
non-zero (almost everywhere), a property needed in the proof of the second
part of the HK theorem (HK2) below.

In Eq. (5) the problem of solving a partial-differential equation, the
stationary Schrödinger equation (1), has been transformed into a variational
problem: finding a minimizer for Eq. (5). The route backwards is also feasible
and any such minimizer is also a distributional solution to the Schrödinger
equation [29, Theorem 11.8].

We will now demonstrate that simply by virtue of the structure of $E[v]$,
where density and potential are combined in the term $\langle v,\rho\rangle$
that makes no explicit reference to the wave function while the remaining part
$\langle\psi|H_{0}|\psi\rangle$ (or $\operatorname{Tr}(H_{0}\Gamma)$, if
density matrices are used to describe the state) does not depend on $v$, we
can already define a mapping from ground-state densities to ground-state wave
functions or density matrices. This, then, is already half of a HK theorem,
that we will already give in a variant for ensemble $v$-representable
densities.

######  Theorem 1 (HK1).

Let $\Gamma_{1}$ be a ground state of $H[v_{1}]$ and $\Gamma_{2}$ a ground
state of $H[v_{2}]$. If $\Gamma_{1},\Gamma_{2}\mapsto\rho$, i.e., if these
states share the same density, then $\Gamma_{1}$ is also a ground state of
$H[v_{2}]$ and $\Gamma_{2}$ is also a ground state $H[v_{1}]$.

###### Proof 1.

Since we assumed the existence of ground states $\Gamma_{1},\Gamma_{2}$ for
the potentials $v_{1},v_{2}$, the infimum in Eq. (5), when varied over density
matrices, is actually a minimum. Further, the potential-energy contribution
$\langle v,\rho\rangle$ is fixed because $\rho$ is given and can be taken out
of the minimum,

$\displaystyle E[v_{1}]$
$\displaystyle=\min_{\Gamma^{\prime}_{1}\mapsto\rho}\operatorname{Tr}(H_{0}\Gamma^{\prime}_{1})+\langle
v_{1},\rho\rangle$ $\displaystyle=\operatorname{Tr}(H_{0}\Gamma_{1})+\langle
v_{1},\rho\rangle$ (6a) $\displaystyle E[v_{2}]$
$\displaystyle=\min_{\Gamma^{\prime}_{2}\mapsto\rho}\operatorname{Tr}(H_{0}\Gamma^{\prime}_{2})+\langle
v_{2},\rho\rangle$ $\displaystyle=\operatorname{Tr}(H_{0}\Gamma_{2})+\langle
v_{2},\rho\rangle$ (6b)

For completeness, we also give the same expression for a general $v$ in case
the state is pure.

$E[v]=\min_{\psi\mapsto\rho}\langle\psi|H_{0}|\psi\rangle+\langle
v,\rho\rangle\\\ $ (7)

Here, the notation “$\Gamma\mapsto\rho$” and “$\psi\mapsto\rho$” means
variation over all states in $\mathcal{W}$ with density $\rho$. But the
remaining minima in Eq. (6) are then completely determined by the fixed
ground-state density and we can always choose $\Gamma_{1}=\Gamma_{2}$ [primes
removed] as a valid ground state. Thus the density alone already defines the
ground state, irrespective of the potential $v_{1}$ or $v_{2}$. ∎

As highlighted before, the above proof relies purely on the specific structure
of the energy function $E[v]$ that allows the potential part to be taken as a
separate, additive contribution that depends solely on the density. This idea
is due to Paul E. Lammert (during discussion at the workshop “Do Electron
Current Densities Determine All There Is to Know?” in Oslo, 2018). In contrast
to this, the usual proofs of this part of the HK theorem additionally depend
on the _linear_ structure of the density-potential pairing. Moreover, such
proofs are almost always performed indirectly (reductio ad absurdum), with a
few notable exceptions [30; 31]. For completeness, we will give an additional,
more traditional proof, yet one that is direct and does not work by raising a
contradiction.

###### Proof 2.

By the variational principle, we have

$\displaystyle E[v_{1}]$
$\displaystyle=\operatorname{Tr}(H[v_{1}]\Gamma_{1})\leq\operatorname{Tr}(H[v_{1}]\Gamma_{2}),$
$\displaystyle E[v_{2}]$
$\displaystyle=\operatorname{Tr}(H[v_{2}]\Gamma_{2})\leq\operatorname{Tr}(H[v_{2}]\Gamma_{1}).$

Exploiting the shared density $\rho$, this may be written as

$\begin{split}E[v_{1}]&=\operatorname{Tr}(H_{0}\Gamma_{1})+\langle
v_{1},\rho\rangle\\\ &\leq\operatorname{Tr}(H_{0}\Gamma_{2})+\langle
v_{1}+v_{2}-v_{2},\rho\rangle\\\ &=E[v_{2}]+\langle
v_{1}-v_{2},\rho\rangle\end{split}$

and analogously as

$E[v_{2}]\leq E[v_{1}]+\langle v_{2}-v_{1},\rho\rangle.$

Combining the inequalities gives

$E[v_{1}]-E[v_{2}]=\langle v_{1}-v_{2},\rho\rangle$

and from

$\operatorname{Tr}(H[v_{2}]\Gamma_{1})=\operatorname{Tr}(H[v_{1}]\Gamma_{1})-\langle
v_{1}-v_{2},\rho\rangle$

that $\operatorname{Tr}(H[v_{2}]\Gamma_{1})=E[v_{1}]$. So $\Gamma_{1}$ is also
a ground state of $H[v_{2}]$. Likewise,
$\operatorname{Tr}(H[v_{1}]\Gamma_{2})=E[v_{2}]$, so $\Gamma_{2}$ is also a
ground state of $H[v_{1}]$, as required. ∎

HK1 holds generally for mixed or pure ground states. The same proofs remain
valid when the theorem is specialized to a statement about pure states
$\Gamma_{i}=|\psi_{i}\rangle\langle\psi_{i}|$. An immediate but maybe
surprising consequence that is often referred to as the basis of DFT is that a
ground-state density ${\rho_{\mathrm{gs}}}$ alone already determines a ground
state. This result has been coined a _weak HK-like result_ before [32] and it
will be used to define the HK1 functionals on
$v\text{-}\mathsf{rep}_{\mathrm{pure}}$ and
$v\text{-}\mathsf{rep}_{\mathrm{ens}}$ in Eq. (9) below. The ground state
(associated with ${\rho_{\mathrm{gs}}}$) is pure if ${\rho_{\mathrm{gs}}}\in
v\text{-}\mathsf{rep}_{\mathrm{pure}}$ but has to be an ensemble if
${\rho_{\mathrm{gs}}}\in v\text{-}\mathsf{rep}_{\mathrm{ens}}\setminus
v\text{-}\mathsf{rep}_{\mathrm{pure}}$. Any state that is a minimizer in Eq.
(6) is really a ground state for _all_ potentials that share the same ground-
state density. That all those potentials are in fact equal (up to a constant)
is then the statement of HK2, the second part of the HK theorem. It will be
formulated for eigenstates, in case of an ensemble we are free to just take
any of its components.

######  Theorem 2 (HK2).

If two potentials share any common eigenstate and if that eigenstate is non-
zero almost everywhere, then the potentials are equal up to a constant.

###### Proof.

If $v_{1},v_{2}$ share a common eigenstate $\psi$ it holds

$\displaystyle(H_{0}+V[v_{1}])\psi=E[v_{1}]\psi,$
$\displaystyle(H_{0}+V[v_{2}])\psi=E[v_{2}]\psi.$

Subtraction of the two equations and moving all potential parts that do not
depend on $\mathbf{r}_{1}$ to the right-hand side gives

$\displaystyle(v_{1}(\mathbf{r}_{1})-v_{2}(\mathbf{r}_{1}))\psi=$
$\displaystyle(E[v_{1}]-E[v_{2}])\psi$ (8)
$\displaystyle-\sum_{i=2}^{N}(v_{1}(\mathbf{r}_{i})-v_{2}(\mathbf{r}_{i}))\psi.$

Since we assumed $\psi$ non-zero almost everywhere, we can then divide by
$\psi$ and get $v_{1}(\mathbf{r}_{1})-v_{2}(\mathbf{r}_{1})=\mathrm{constant}$
(almost everywhere) because the right-hand side does not depend on
$\mathbf{r}_{1}$. ∎

Since HK2 states that sharing _any_ common eigenstate for two potentials means
that they are equal (up to a constant), this of course implies that the
potentials share _all_ eigenstates because they yield exactly the same
Hamiltonian (up to an additive constant that just shifts the spectrum). The
special requirement that the wave function is non-zero (almost everywhere) is
guaranteed for a large class of potentials by the _unique-continuation
property (UCP) from sets of positive measure_. This property will be further
discussed in Section V. That zeroes (nodes) in the wave function _are_ still
allowed on a set of measure zero is important here, since the fermionic many-
particle wave functions will exhibit nodal surfaces when particle positions
agree. Outside of the continuum setting, for example in finite-lattice
systems, such a UCP is _not_ at hand and there are actual counterexamples to
HK2, were two different potentials share a common eigenstate [24].

The complete HK result is then obtained by combining the two theorems above.
We will assume here that the potential is from the mentioned class that
guarantees a non-zero ground state. We should remember that such or similar
restrictions will always come into play if we want to show validity of a
density-potential mapping in other settings. The statement will be formulated
for densities in $v\text{-}\mathsf{rep}_{\mathrm{ens}}$, so it automatically
holds for $v\text{-}\mathsf{rep}_{\mathrm{pure}}$ as well.

######  Corollary 3 (HK).

If two potentials share a common ensemble $v$-representable ground-state
density, then they are equal up to a constant.

###### Proof.

By HK1 there is a density matrix
$\Gamma=\sum_{k}\lambda_{k}|\psi_{k}\rangle\langle\psi_{k}|$ that is a ground
state for both potentials. Since at least one $\lambda_{k}\neq 0$ and since
the corresponding $\psi_{k}$ is a ground-state wave function for both
Hamiltonians, the proof can be completed by HK2. ∎

This structuring into two separate theorems was already used in Kohn, Savin,
and Ullrich [33], just in the reverse order, for a brief argument about DFT
with magnetization. Historically, the HK theorem was first given only for the
non-degenerate case and was only later extended to include degeneracy [22;
34]. The proof presented here does not suffer from any limitation to non-
degenerate ground states.

A final note is directed towards more general DFTs that will be briefly
discussed in Section X and especially in the forthcoming Part II of this
review. For other types of potentials, like vector potentials, the statement
in the HK theorem would not necessarily be that the potentials are equal “up
to a constant”, but for example “up to a gauge transformation”. The set of
gauge transformations that are possible without affecting the physical
properties of the system then have to be specified within the respective
theory.

##  V The unique-continuation property

In this section, we summarize some important results on the unique-
continuation property (UCP) of solutions to the Schrödinger equation that is
heavily used in the context of (mathematical formulation of) HK-type theorems.
The current understanding is that the UCP cannot be avoided in a rigorous
proof of a HK-type theorem. The setting will be slightly more general than
before and allow for dimensionality $d$ of the spatial part of the single-
particle configuration space $\mathbb{R}^{d}$. The $N$-particle configuration
space is then $\mathbb{R}^{n}$ with $n=dN$.

Roughly speaking, the desired UCP result states that under certain conditions
on the potentials building up the operators $V$ and $W$ and if a solution
$\psi$ to the (distributional) equation $H[v]\psi=0$ vanishes on a set of
positive measure, then $\psi$ vanishes everywhere. That the right hand side is
zero comes as no restriction here, since the energy $E$ can always be absorbed
into the scalar potential $v$. The usual literature on the UCP shows _strong_
UCP, which means that $\psi$ is assumed to _vanish to infinite order_ at a
point $\underline{\mathbf{r}}_{0}\in\mathbb{R}^{n}$ and then the statement
follows. A function $f(\mathbf{r})$ is said to vanish to infinite order at
$\underline{\mathbf{r}}_{0}\in\mathbb{R}^{n}$ if for all $k\geq 1$ there is a
$c_{k}$ such that

$\int_{|\underline{\mathbf{r}}-\underline{\mathbf{r}}_{0}|<\epsilon}|f(\underline{\mathbf{r}})|^{2}\,\mathrm{d}\underline{\mathbf{r}}<c_{k}\epsilon^{k}$

for every $0<\epsilon<1$. Now a very convenient result by Regbaoui [35] shows
that the UCP on sets of positive measure actually follows from such a strong
UCP if the potentials are in $L^{n/2}_{\mathrm{loc}}$. This work apparently
built on de Figueiredo and Gossez [36] that again rests on an early estimate
for general Sobolev spaces by Ladyzenskaya and Ural’tzeva [37, Lemma 3.4]. The
result and its proof have been repeated in Lammert [38]. For us that means
that any strong UCP can also be used as a UCP on sets of positive measure
which is the one needed for the proof of HK2. Yet the traditional strong UCP
results, like most notably in Jerison and Kenig [39], also give dimension-
dependent constraints on the potentials like $L^{n/2}_{\mathrm{loc}}$, which
approaches $L^{\infty}$ for growing particle number and is thus too
restrictive for our use where singular potentials need to be considered. The
saving idea recently came from Garrigue [31] and was also extended to more
complex systems [40; 41]: To take the special $N$-body structure of the
potentials into account and thus avoid any dependence of the constraints on
the particle number $N$.

######  Theorem 4 (Garrigue’s UCP).

Suppose that the potentials are in $L_{\mathrm{loc}}^{p}(\mathbb{R}^{d})$ with
$p>2$ for $d=3$ and $p=\max(2d/3,2)$ else. If a solution $\psi$ to the
Schrödinger equation vanishes on a set of positive measure or if it vanishes
to infinite order at any point, then $\psi=0$.

The most relevant case here is obviously $d=3$ which means that the potentials
need to be in $L_{\mathrm{loc}}^{p}(\mathbb{R}^{3})$ with $p>2$ but exactly
$p=2$ is not enough yet. This clearly does not fit our potential space
$X^{*}=L^{3/2}(\mathbb{R}^{3})+L^{\infty}(\mathbb{R}^{3})$, so while this UCP
result is the best one available, it cannot be used for a HK2 theorem that
covers the whole potential space of DFT in the formulation discussed here.
Lieb [2] also remarked on the UCP in the context of the HK theorem, which “is
believed to hold” for potentials in $X^{*}$, however in a weaker form that is
not sufficient for the current purpose. So whenever we state that the HK holds
in standard DFT, we actually mean under the given restrictions on the
potentials.

##  VI Hierarchy of density functionals

The first part of the HK theorem, HK1, analogously holds in many different
varieties of DFT (that will be explored in Part II), simply because its
validity just depends on the form of the energy functional. HK1 then ensures
that we can map from pure-state $v$-representable ground-state densities
${\rho_{\mathrm{gs,pure}}}$ to ground-state wave functions
$\psi[{\rho_{\mathrm{gs,pure}}}]$ and from ensemble $v$-representable ground-
state densities ${\rho_{\mathrm{gs,ens}}}$ to ground-state density matrices
$\Gamma[{\rho_{\mathrm{gs,ens}}}]$. This makes it possible to define the HK1
functionals

$\displaystyle
F_{\mathrm{HK1,pure}}[{\rho_{\mathrm{gs}}}]=\langle\psi[{\rho_{\mathrm{gs}}}]|H_{0}|\psi[{\rho_{\mathrm{gs}}}]\rangle$
$\displaystyle\,\,\,\,\text{on}\;v\text{-}\mathsf{rep}_{\mathrm{pure}}$ (9a)
and $\displaystyle
F_{\mathrm{HK1,ens}}[{\rho_{\mathrm{gs}}}]=\operatorname{Tr}(H_{0}\Gamma[{\rho_{\mathrm{gs}}}])$
$\displaystyle\,\,\,\,\text{on}\;v\text{-}\mathsf{rep}_{\mathrm{ens}}$ (9b)

as the energy contribution only from the internal parts $H_{0}$ of the
Hamiltonian. The universal nature of such functionals, being independent of
any external $v$, justifies the usual attribution as universal functionals. It
is then possible to determine also the internal energy contributions for any
state with density ${\rho_{\mathrm{gs}}}$ just from ${\rho_{\mathrm{gs}}}$. To
get the total ground-state energy (5) with the help of the functional above,
it is enough to vary over $v$-representable densities alone, instead of the
much larger set of wave functions. We can write

$\displaystyle E[v]$
$\displaystyle=\inf\\{\langle\psi|H_{0}|\psi\rangle+\langle
v,\rho_{\psi}\rangle\mid\psi\in\mathcal{W},\|\psi\|=1\\}$ (10)
$\displaystyle=\inf_{{\rho^{\prime}_{\mathrm{gs}}}}\\{\langle\psi[{\rho^{\prime}_{\mathrm{gs}}}],H_{0}\psi[{\rho^{\prime}_{\mathrm{gs}}}]\rangle+\langle
v,{\rho^{\prime}_{\mathrm{gs}}}\rangle\\}$
$\displaystyle=\inf_{\rho^{\prime}_{\mathrm{gs}}}\\{F_{\mathrm{HK1,pure}}[{\rho^{\prime}_{\mathrm{gs}}}]+\langle
v,{\rho^{\prime}_{\mathrm{gs}}}\rangle\\}\quad\text{on}\;X^{*},$

or equivalently with $F_{\mathrm{HK1,ens}}$. We see already that there is a
certain ambiguity in which density functional to use in the definition of
$E[v]$. The other density functionals presented here will all have the
property that they give the correct ground-state energy when applied in Eq.
(10) which makes them all _admissible_ functionals [42]. Yet, they will differ
with respect to their mathematical properties and we thus aim for the one with
the best features.

The first problem here is that the densities to be considered in the
variational problem are limited to those that are actual ground-state
densities ($v\text{-}\mathsf{rep}$), because else
$F_{\mathrm{HK1}}[{\rho_{\mathrm{gs}}}]$ is left undefined, and we already
learned in Section III that $v\text{-}\mathsf{rep}$ is not an explicitly
characterized set. Apart from that, HK1 just states the existence of a map
${\rho_{\mathrm{gs}}}\mapsto\psi$ or $\Gamma$ without giving any hints towards
a constructive scheme. A first step to overcome these problems is to inspect
Eq. (7). This suggests the definition of another pair of density functionals
that goes under the name of “constrained search”,

$\displaystyle
F_{\mathrm{CS,pure}}[\rho]=\inf_{\psi\mapsto\rho}\langle\psi|H_{0}|\psi\rangle$
$\displaystyle\quad\text{on}\;N\text{-}\mathsf{rep}\;\text{and}$ (11a)
$\displaystyle
F_{\mathrm{CS,ens}}[\rho]=\inf_{\Gamma\mapsto\rho}\operatorname{Tr}(H_{0}\Gamma)$
$\displaystyle\quad\text{on}\;N\text{-}\mathsf{rep}.$ (11b)

The domain is now the larger, convex, and explicitly defined
$N\text{-}\mathsf{rep}$ in both cases. Note that the literature mostly denotes
those functionals as $F_{\mathrm{CS,pure}}=F_{\mathrm{LL}}$ (“Levy–Lieb” [9;
2]) and $F_{\mathrm{CS,ens}}=F_{\mathrm{DM}}$ (from “density matrix” [2]). A
recent, comprehensive study of these functionals can be found in Lewin, Lieb,
and Seiringer [43]. Since the density is limited to the set
$N\text{-}\mathsf{rep}$ that guarantees finite kinetic energy, the infima in
Eq. (11) are always attained, though not necessarily by a possible ground
state (if $\rho$ is not $v$-representable), and can thus be replaced by minima
in both cases [2, Theorem 3.3]. The convex combination of pure-state
projections into density matrices translates to the functionals, so that
$F_{\mathrm{CS,ens}}$ is the convex envelope of $F_{\mathrm{CS,pure}}$ [24,
Proposition 18, the article treats DFT on a lattice but the statement and the
proof of proposition remains exactly the same in the continuum case.]. This
automatically ensures that $F_{\mathrm{CS,ens}}$ is convex, a fact that can
also be concluded from observing that $\Gamma\mapsto\rho$ linear [2, Section
4.B].

Since these density functionals appear in the optimization problem that
determines the ground-state energy and density, like in Eq. (10), convexity is
of great importance because only for a convex functional can we be sure that
identifying any _local_ minimum also means that a _global_ minimum has been
found. So while we now know that $F_{\mathrm{CS,ens}}$ is convex, the previous
functional $F_{\mathrm{HK1,pure}}$ does not even have a convex domain and
therefore cannot be convex. Levy [23] and Lieb [2] also gave arguments for the
non-convexity of $F_{\mathrm{CS,pure}}$. Since
$F_{\mathrm{CS,ens}}=\operatorname{conv}F_{\mathrm{CS,pure}}$, any density
where $F_{\mathrm{CS,ens}}[\rho]\neq F_{\mathrm{CS,pure}}[\rho]$ already shows
non-convexity of $F_{\mathrm{CS,pure}}$. But this is equivalent to saying that
$\rho$ is ensemble $v$-representable while it is _not_ pure-state
$v$-representable, so $\rho\in v\text{-}\mathsf{rep}_{\mathrm{ens}}\setminus
v\text{-}\mathsf{rep}_{\mathrm{pure}}$ [24, Proposition 21].

Note especially, that HK1 was necessary to define $F_{\mathrm{HK1}}$, but is
not needed any more for the constrained-search functional $F_{\mathrm{CS}}$.
Being able to define a universal constrained-search functional, one that is
independent of the potential like in Eq. (11), already fully facilitates the
proof of HK1 and thus implies this result. A potential-independent
constrained-search functional _already implicitly includes HK1_. This
implication was proven by Levy [9] along the lines of the usual HK proof and
is mentioned in textbooks like Parr and Yang [18, after their Eq. (3.4.4)] and
Tsuneda [44, after Eq. (4.5)]. Speaking generally though, a constrained search
is just as feasible if the constrained-search functional also depends on the
external potential $v$ (although it would not be universal), so indeed this
approach is more general than relying on HK1. Such a case turns up in CDFT
when the current variable is the total current that itself depends on the
vector potential (see Part II of this review for more on this).

By employing the constrained-search functional, the ground-state energy from
Eq. (5) can now be rewritten again as

$\displaystyle E[v]$
$\displaystyle=\inf\\{\langle\psi|H_{0}|\psi\rangle+\langle
v,\rho_{\psi}\rangle\mid\psi\in\mathcal{W},\|\psi\|=1\\}$
$\displaystyle=\inf\\{F_{\mathrm{CS,pure}}[\rho_{\psi}]+\langle
v,\rho_{\psi}\rangle\mid\psi\in\mathcal{W},\|\psi\|=1\\}$
$\displaystyle=\inf_{\rho}\\{F_{\mathrm{CS,pure}}[\rho]+\langle
v,\rho\rangle\\}\quad\text{on}\;X^{*},$

or equivalently with $F_{\mathrm{CS,ens}}$, where minimization is now
performed over $N\text{-}\mathsf{rep}$.

When looking at non-interacting systems, the definitions of
$F_{\mathrm{HK1}}$, Eq. (9), and $F_{\mathrm{CS}}$, Eq. (11), involve only the
kinetic-energy operator $T$ instead of $H_{0}$. We will then denote these
functionals with a zero superscript,
$F_{\mathrm{HK1}}^{0},F_{\mathrm{CS}}^{0}$, etc., that indicates that non-
interacting systems are considered. A further functional then comes into play
that is defined like $F_{\mathrm{CS,pure}}$, but where only Slater
determinants are considered as wave functions. We define on
$N\text{-}\mathsf{rep}$,

$F^{0}_{\mathrm{SD}}[\rho]=\inf_{\phi\mapsto\rho}\left\\{\langle\phi|T|\phi\rangle\mid\text{{\hbox{\phi}}
is a Slater determinant}\right\\}.$

The usual name in the literature is $F^{0}_{\mathrm{SD}}=T_{S}$. This
functional is of importance because it is the one used in Kohn–Sham theory
which will be discussed in Section VIII. In their original article, Kohn and
Sham [45] implicitly set $F^{0}_{\mathrm{SD}}=F_{\mathrm{HK1,pure}}^{0}$ for
all non-interacting pure-state $v$-representable densities, which has been
noted to be wrong because of possible degeneracy [2, Section 4.C]. On the
other hand, for non-degenerate ground states $\phi$, which by necessity are
always determinants in non-interacting systems, it holds that
$F^{0}_{\mathrm{SD}}[\rho_{\phi}]=F_{\mathrm{CS,pure}}^{0}[\rho_{\phi}]=F_{\mathrm{HK1,pure}}^{0}[\rho_{\phi}]$,
and else $F^{0}_{\mathrm{SD}}\geq F_{\mathrm{CS,pure}}^{0}$. Nevertheless, for
practical purposes, $F^{0}_{\mathrm{SD}}$ usually takes up the role of the
density functional when defining the energy functional in a non-interacting
setting.

The transformation from any density functional $F_{\bullet}$ for an
interacting system from above to the energy functional,

$E[v]=\inf_{\rho}\\{F_{\bullet}[\rho]+\langle
v,\rho\rangle\\}\quad\text{on}\;X^{*},$ (12)

is called the _convex conjugate_ or Legendre–Fenchel transformation [26,
Section 2.1.4]. There is also a way to reverse the transformation and we
define

$F[\rho]=\sup_{v}\\{E[v]-\langle v,\rho\rangle\\}\quad\text{on}\;X.$ (13)

This $F$ is the famous _Lieb functional_ [2], yet another density functional,
but this time the last one to be defined in standard DFT. It is the
_biconjugate_ of any $F_{\bullet}$ considered before. Defined this way, both
$E$ and $F$ are lower-semicontinuous and $E$ is concave while $F$ is convex
and has the property $F\leq F_{\bullet}$ [26, Proposition 2.19]. Actually, as
a biconjugate, $F$ is the largest convex and lower semicontinuous functional
that fulfills $F\leq F_{\bullet}$ which makes it the _convex envelope_ of
$F_{\bullet}$. The domain is now the whole $X=L^{1}(\mathbb{R}^{3})\cap
L^{3}(\mathbb{R}^{3})$, but automatically $F[\rho]=\infty$ for all densities
that are not in $N\text{-}\mathsf{rep}$ [2, Theorem 3.8], while at the same
time $F[\rho]<\infty$ if $\rho\in N\text{-}\mathsf{rep}$ [2, Theorem 3.9 and
the following Remark]. Let the _effective domain_ ‘$\operatorname{dom}$’ of a
convex functional be the elements from its domain where it is finite, then
this means that $\operatorname{dom}F=N\text{-}\mathsf{rep}$. Having reached
$F$, it does not matter any more which (admissible) functional has been used
in Eq. (12), which means the convex envelopes of all the functionals above
agree. Conversely, the Legendre–Fenchel transformation can also be utilized to
go back from $F$ to $E$ [26, Theorem 2.22],

$E[v]=\inf_{\rho}\\{F[\rho]+\langle v,\rho\rangle\\}.$ (14)

We already noted that $F$ is convex and lower-semicontinuous, which are both
important properties if we want to use the variational problem
$E[v]=\inf_{\rho}\\{F[\rho]+\langle v,\rho\rangle\\}$ to find a minimizing
density. The same properties come into play when defining the minimizers by
differentiation in Section VII. From the definition of $F$ it follows directly
that

$E[v]\leq F[\rho]+\langle v,\rho\rangle,$

a version of the Young inequality. Equality in the above estimate holds if the
density _is_ the ground-state density ${\rho_{\mathrm{gs}}}$ for the potential
$v$,

$E[v]=F[{\rho_{\mathrm{gs}}}]+\langle v,{\rho_{\mathrm{gs}}}\rangle.$

For $F_{\mathrm{CS,pure}}$ the converse holds too: If
$E[v]=F_{\mathrm{CS,pure}}[\rho]+\langle v,\rho\rangle$ then $\rho$ is a
ground-state density within $v\text{-}\mathsf{rep}_{\mathrm{pure}}$ for the
potential $v$ and further
$F_{\mathrm{HK1,pure}}[\rho]=F_{\mathrm{HK1,ens}}[\rho]=F_{\mathrm{CS,pure}}[\rho]=F_{\mathrm{CS,ens}}[\rho]=F[\rho]$
[2, Theorem 3.10]. But what about using the more general functional $F$ for
the variational principle like in Eq. (14)? Can we find a real ground state
like this or will this variational principle yield additional artificial
solutions because it is too general? Because it is the convex envelope of the
other functionals, it cannot produce a functional value below the ground-state
energy, but it could produce a minimizing density where there are no
$v$-representable ground-state densities! The problem is solved if we allow
for ensembles of ground states: An “amusing fact” in Lieb [2, Eq. (4.5)] gives
$F=F_{\mathrm{CS,ens}}$ on $N\text{-}\mathsf{rep}$, which effectively means
$F=F_{\mathrm{CS,ens}}$ since we can just set
$F_{\mathrm{CS,ens}}[\rho]=\infty$ outside of its domain
$N\text{-}\mathsf{rep}$ to achieve equality globally on $X$. So any minimizer
of $F+\langle v,\cdot\rangle$ is also one of $F_{\mathrm{CS,ens}}+\langle
v,\cdot\rangle$ and it is further the convex combination of ground states for
the potential $v$. Consequently, when talking about ground states in the
context of the functional $F$, we will always actually mean _ensembles_ of
possibly degenerate ground states.

When comparing the functionals on $X$, we just set them to $\infty$ whenever
we are outside their domains. The following hierarchy can be set up and is
further laid out in Table 1.

$F=F_{\mathrm{CS,ens}}\leq\begin{array}[]{c}F_{\mathrm{CS,pure}}(\leq
F^{0}_{\mathrm{SD}})\\\ F_{\mathrm{HK1,ens}}\end{array}\leq
F_{\mathrm{HK1,pure}}.$

Here, $F^{0}_{\mathrm{SD}}$ appears in parentheses since it only comes into
play in the non-interacting setting where we can perform the same type of
transformations and have $F^{0}[\rho]$ and $E^{0}[v]$.

$F_{\bullet}$ | convex | domain | convex
---|---|---|---
$F_{\mathrm{HK1,pure}}$ | no |  | $v\text{-}\mathsf{rep}_{\mathrm{pure}}$ | no |
$F_{\mathrm{HK1,ens}}$ | ? |  | $v\text{-}\mathsf{rep}_{\mathrm{ens}}$ | ? | $\Downarrow$ cl
$F_{\mathrm{CS,pure}}$ | no | $\Downarrow$ conv | $N\text{-}\mathsf{rep}$ | yes
$F_{\mathrm{CS,ens}}$ | yes | $N\text{-}\mathsf{rep}$ | yes |
$F$ | yes |  | $L^{1}\cap L^{3}$ | yes |
Table 1:  The table shows the relations between the functionals discussed in
Section VI. From $F_{\mathrm{HK1,pure}}$ to $F_{\mathrm{HK1,ens}}$ the domain
gets extended to $v\text{-}\mathsf{rep}_{\mathrm{ens}}$ while they agree on
$v\text{-}\mathsf{rep}_{\mathrm{pure}}$. From $F_{\mathrm{HK1,ens}}$ to
$F_{\mathrm{CS,pure}}$ the domain gets closed (cl) within $L^{1}\cap L^{3}$
and from $F_{\mathrm{CS,pure}}$ to $F_{\mathrm{CS,ens}}$ the functional itself
gets convexified (conv) while the domain remains the same. Finally, $F$ is
just equal to $F_{\mathrm{CS,ens}}$ on $N\text{-}\mathsf{rep}$.

##  VII Density-potential mappings from differentials

In the previous section it was stated that in order to get the ground-state
density of any system we have to find a solution to the variational problem

$E[v]=\inf_{\rho}\\{F[\rho]+\langle v,\rho\rangle\\},$ (15)

now relying on the density functional $F$ from Eq. (13). To find the global
minimum of a _convex and lower-semicontinuous_ functional we can perform
differentiation, i.e., demand that the differential of $F[\rho]+\langle
v,\rho\rangle$ with respect to $\rho$ must equal zero at the position of a
ground-state density ${\rho_{\mathrm{gs}}}$.

$\rho$$\rho_{0}$$\infty$ Figure 1: Example of a convex and lower-
semicontinuous function with a discontinuity at $\rho_{0}$ and some elements
from the subdifferential displayed as linear continuous tangent functionals at
$\rho_{0}$, represented by dashed lines.

The suitable notion of differentiation here is the subdifferential
$\underline{\partial}F$ that gives the _set_ of all linear continuous tangent
functionals to a convex functional $F$ at a given density $\rho$,

$\underline{\partial}F[\rho]=\\{v\in X^{*}\mid\forall\rho^{\prime}\in
X:F(\rho)\leq F(\rho^{\prime})+\langle v,\rho-\rho^{\prime}\rangle\\}.$

It is always well-defined, since the set $\underline{\partial}F[\rho]$ can
contain many elements, in case the functional $F$ has a kink (like the example
shown in Fig. 1), or can even be empty. Finally, if it contains exactly one
element, we found a _unique_ potential yielding that ground-state density. In
any case, the variational problem (15) has a minimizer ${\rho_{\mathrm{gs}}}$
if and only if the following condition is fulfilled [26, Proposition 2.33],

$\left.\underline{\partial}(F[\rho]+\langle
v,\rho\rangle)\right|_{\rho={\rho_{\mathrm{gs}}}}\ni
0\Longleftrightarrow\langle
v,\cdot\rangle\in-\underline{\partial}F[{\rho_{\mathrm{gs}}}].$ (16)

In what follows, we identify $v$ with the functional $\langle v,\cdot\rangle$
whenever the context implies a functional on density space instead of a
potential on configuration space, so Eq. (16) can be written
$v\in-\underline{\partial}F[{\rho_{\mathrm{gs}}}]$. The potential as the
subdifferential of the density functional means that potentials $v$ are from
the dual of the space of densities like already noted in Section III. This
general principle is not always respected in more complex versions of DFT, as
we will see in Section X and discuss further in Part II of this review.

If the set $\underline{\partial}F[{\rho_{\mathrm{gs}}}]$ is non-empty then
there is at least one potential $v\in X^{*}$ that yields the given ground-
state density. The set of all densities where
$\underline{\partial}F[{\rho_{\mathrm{gs}}}]\neq\emptyset$ is called the
domain of the subdifferential, so it follows that
$\operatorname{dom}\underline{\partial}F=v\text{-}\mathsf{rep}_{\mathrm{ens}}$.
Note that by a theorem of convex analysis [26, Corollary 2.44],
$\operatorname{dom}\underline{\partial}F$ is dense in $\operatorname{dom}F$,
so $v\text{-}\mathsf{rep}_{\mathrm{ens}}$ is dense in $N\text{-}\mathsf{rep}$,
a fact already expressed with $N\text{-}\mathsf{rep}$ being the closure of
$v\text{-}\mathsf{rep}_{\mathrm{ens}}$ in Section III.

The meaning of a valid HK theorem for a class of densities is that they can
all be mapped as ground-state densities back to a unique potential (modulo a
constant) and consequently
$-\underline{\partial}F[{\rho_{\mathrm{gs}}}]=\\{v+c\mid c\in\mathbb{R}\\}$.
By eliminating the (physically unimportant) constant potentials from the
potential space, the subdifferential of a $v$-representable density is
precisely $-\underline{\partial}F[{\rho_{\mathrm{gs}}}]=\\{v\\}$ if the HK
theorem holds. If, on the other hand, $F$ is assumed differentiable, then the
directional derivative $-\frac{\delta}{\delta\rho}F[{\rho_{\mathrm{gs}}}]=v$
anyway always maps to a unique potential. One thus has a well-defined map from
densities in $v\text{-}\mathsf{rep}$ to the corresponding potentials, exactly
the content of the HK theorem! But where did it enter? The HK theorem is here
a _consequence_ from the assumption of differentiability of $F[\rho]$ at
$v$-representable densities. The situation will be summarized diagrammatically
in Section XI.

Because any potential $v$ that we determine by Eq. (16) will also be the
maximizer in the conjugate variational problem

$F[{\rho_{\mathrm{gs}}}]=\sup_{v}\\{E[v]-\langle
v,{\rho_{\mathrm{gs}}}\rangle\\},$

we can just as well say the same with the superdifferential of the concave
functional $E$,

$\left.\overline{\partial}(E[v^{\prime}]-\langle
v^{\prime},{\rho_{\mathrm{gs}}}\rangle)\right|_{v^{\prime}=v}=0\Longleftrightarrow{\rho_{\mathrm{gs}}}\in\overline{\partial}E[v].$
(17)

The right hand side, ${\rho_{\mathrm{gs}}}\in\overline{\partial}E[v]$, means
to find a density (or possibly many) that comes from a wave-function that
minimizes the total energy including $v$. It is thus a conceptual shortcut to
map from potentials to ground-state densities without any reference to an
underlying wave function or Schrödinger equation. The situation of a set
$\overline{\partial}E[v]$ with more than one element is known from
degeneracies of the Hamiltonian $H_{0}+V[v]$, where different linearly
independent ground states with eventually different densities all have the
same eigenvalue.

We showed in this section the important role of the generalized concepts of
sub/superdifferentials in the context of DFT, because indeed the functionals
from Section VI can _not_ be assumed differentiable as van Leeuwen [46] has
demonstrated for the $F_{\mathrm{HK1}}$ functionals and Lammert [3] for
$F_{\mathrm{CS}}$. The reason for non-differentiability even of
$F_{\mathrm{CS}}$ is that at any $\rho$ the functional $F[\rho+\delta\rho]$ is
infinite for various, arbitrarily small shifts $\delta\rho$ that lead out of
$N\text{-}\mathsf{rep}$, even if the normalization of the density is kept
constant. This happens by infinitely increasing the internal energy through
tiny oscillations of the density. A possible way to prevent that is to limit
the density space $X$ so that such shifts $\delta\rho$ are not possible any
more and Lammert [3] actually shows this for the Sobolev space
$H^{2}(\mathbb{R}^{3})$ when $\rho$ is also assumed $v$-representable. Another
way is to establish a coarse-grained model for DFT in which $F$ really becomes
differentiable and every density is ensemble $v$-representable [47]. In the
following section, in accordance with the vast majority of the literature, we
will assume functional differentiability of $F$ and consequently
$v$-representability. This strong assumption can be justified a posteriori, as
discussed later in Section IX, when a regularization procedure is applied.

##  VIII Linking to a reference system: the Kohn–Sham scheme

In Section VI it was noted that a functional might be introduced for an
interacting or a non-interacting system. This means the respective Hamiltonian
has the internal part $T+\lambda W$ with $\lambda\in\\{0,1\\}$. We will now
write $F^{1}$ and $F^{0}$ to differentiate clearly between those two
situations. We then introduce the difference functional
$F_{\mathrm{Hxc}}=F^{1}-F^{0}$, which just corresponds to the internal-energy
difference between the interacting and the non-interacting system and that
will later be linked to the Hartree-exchange-correlation potential
$v_{\mathrm{Hxc}}$. This potential effectively compensates for the Hartree-
mean-field interaction as well as ‘exchange’ and ‘correlation’ effects. The
idea behind introducing this auxiliary non-interacting system is that the
energy difference between the (numerically tractable) non-interacting system
and the (numerically unfeasible) interacting system is small and can be
efficiently approximated. Since the reference system is non-interacting,
$F^{0}_{\mathrm{SD}}$ can be employed for $F^{0}$ if degeneracy for the ground
state does not have to be taken into account, like it was mentioned in Section
VI, and this switchover is performed in most practical situations. Then the
energy functional for the full system is

$\displaystyle E^{1}[v]$ $\displaystyle=\inf_{\rho}\\{F^{1}[\rho]+\langle
v,\rho\rangle\\}$
$\displaystyle=\inf_{\rho}\\{F^{0}[\rho]+F_{\mathrm{Hxc}}[\rho]+\langle
v,\rho\rangle\\}$
$\displaystyle=\inf_{\phi}\\{\langle\phi|T|\phi\rangle+F_{\mathrm{Hxc}}[\rho_{\phi}]+\langle
v,\rho_{\phi}\rangle\\}.$

In the last step the variation is changed from $N\text{-}\mathsf{rep}$
densities to single Slater determinants $\phi$, the minimizer – if it exists –
is then the Kohn–Sham Slater determinant. In order to link this to a partial
differential equation for the orbitals $\varphi_{i}$ constituting $\phi$, the
Kohn–Sham equation, variation of the energy expression above with respect to
$\varphi_{i}$ is performed under the constraint that all the $\varphi_{i}$
stay normalized. This means
$\rho_{\phi}(\mathbf{r})=\sum_{i=1}^{N}\sum_{\sigma}|\varphi_{i}(\mathbf{r}\sigma)|^{2}$
always stays in $N\text{-}\mathsf{rep}$, but generally the issue of non-
differentiability from Section VII remains. The resulting equation is a one-
particle Schrödinger equation with effective potential $v_{s}$ and eigenstates
$\varphi_{i}$,

$\left(-\tfrac{1}{2}\nabla^{2}+v_{s}(\mathbf{r})\right)\varphi_{i}(\mathbf{r}\sigma)=\varepsilon_{i}\varphi_{i}(\mathbf{r}\sigma).$
(18)

On the other hand this approach does not lead to the effective potential
$v_{s}$ for the Kohn–Sham equation right away, but requires the additional,
computationally challenging step of extracting the effective potential from
the variation of $F_{\mathrm{Hxc}}$ with respect to the orbitals (OEP integral
equation[48]).

To have a well defined $F_{\mathrm{Hxc}}[\rho]=F^{1}[\rho]-F^{0}[\rho]$, the
$\rho$ must be both, interacting and non-interacting $v$-representable. Both
systems then share the same ground-state density $\rho$ when the different
external potentials

$v\in-\underline{\partial}F^{1}[\rho]\quad\text{and}\quad
v_{s}\in-\underline{\partial}F^{0}[\rho]$ (19)

are assigned to them. That the density $\rho$ is simultaneously interacting
_and_ non-interacting $v$-representable is tacitly assumed here, else one of
the subdifferentials above is empty. This means that actually the
$v$-representability problem from Section III shows up at this point. A
purported solution [49; 50; 51] rests on an ill-founded notion of
differentiability where the functionals are extended to distributions, but
with an incorrect application of the calculus of distributions (see, e.g., Eq.
(0.24) in Gonis [49]).

The usual rationale of DFT is to assume that the potentials from Eq. (19)
exist and are unique (modulo a constant; after all the latter is the content
of the HK theorem). The difference $v_{\mathrm{Hxc}}=v_{s}-v$ is then known as
the Hartree-exchange-correlation potential: what needs to be added to the
fixed external potential $v$ in order to simulate all interactions in an non-
interacting system. Note that such missing effects from interactions do not
stem exclusively from the $W$-term in $F^{1}$, but also from the different
kinetic energy contributions between the interacting and non-interacting
system. Nevertheless, the usual understanding is that most of the kinetic
energy contributions can already be captured by a non-interacting system (with
an uncorrelated wave function) and that they thus practically cancel between
$F^{1}$ and $F^{0}$ when we calculate

$v_{\mathrm{Hxc}}[\rho]=v_{s}-v\in\underline{\partial}F^{1}[\rho]-\underline{\partial}F^{0}[\rho].$
(20)

At this point a problematic discrepancy is introduced, since the
subdifferential is not linear and thus $v_{\mathrm{Hxc}}$ and
$\underline{\partial}F_{\mathrm{Hxc}}$ need not match. If $v_{\mathrm{Hxc}}$
cannot be determined as $\underline{\partial}F_{\mathrm{Hxc}}$ we are left
with the necessity of individually solving the inverse problems $\rho\mapsto
v$ and $\rho\mapsto v_{s}$ in Eq. (20) for both systems, interacting and non-
interacting. In practice this means one cannot benefit from finding good
approximations to $F_{\mathrm{Hxc}}$ which are the most important elements of
applied DFT.

A possible circumvention lies in a conceptual shift from describing a system
in terms of energies to forces. The ground state is then characterized by a
certain force-balance equation that can be equally found in non-equilibrium
settings, just with an additional dynamical term [52; 53]. At a density that
is simultaneously interacting and non-interacting $v$-representable and where
the wave function has a sufficient regularity, the force-balance equation can
be employed to derive $v_{\mathrm{Hxc}}$ as the solution of a Poisson equation
instead of a functional derivative [54]. An alternative derivation for this
was already given earlier using line integrals describing the work it takes to
move an electron from infinity against the force field of the exchange-
correlation hole charge [55; 56].

Yet, we will proceed here for the sake of argument by _assuming_
differentiability for now. Since the functional derivative
$\frac{\delta}{\delta\rho}$ is linear and it holds

$\displaystyle v_{\mathrm{Hxc}}[\rho]$
$\displaystyle=v_{s}-v=-\frac{\delta}{\delta\rho}F^{0}[\rho]+\frac{\delta}{\delta\rho}F^{1}[\rho]$
(21)
$\displaystyle=\frac{\delta}{\delta\rho}(F^{1}[\rho]-F^{0}[\rho])=\frac{\delta}{\delta\rho}F_{\mathrm{Hxc}}[\rho].$

Also, several important properties that the Hxc potential needs to have are
automatically fulfilled when they are functional derivatives [57], which is
especially relevant for functional approximations to $v_{\mathrm{Hxc}}$.

The Kohn–Sham scheme is now introduced in order to find an unknown ground-
state density ${\rho_{\mathrm{gs}}}$ of an interacting system by starting from
an initial guess $\rho_{0}$ and by using $v_{\mathrm{Hxc}}$ (in practice a
suitable approximation to it) as the connection between the interacting system
and a non-interacting reference system. To this end, rewrite Eq. (19) with
assumed differentiability as
$\frac{\delta}{\delta\rho}F^{1}[{\rho_{\mathrm{gs}}}]+v=0$ and
$\frac{\delta}{\delta\rho}F^{0}[{\rho_{\mathrm{gs}}}]+v_{s}=0$ and set the two
equations equal,

$\frac{\delta}{\delta\rho}F^{1}[{\rho_{\mathrm{gs}}}]+v=\frac{\delta}{\delta\rho}F^{0}[{\rho_{\mathrm{gs}}}]+v_{s}.$

Now, apart from the fixed external potential $v$ of the interacting system,
all variables in this equation still remain generally unknown: the effective
potential of the non-interacting system $v_{s}$ and, especially, the density
$\rho$ of both systems that we would like to determine. The trick lies in
introducing sequences $\rho_{i}\to{\rho_{\mathrm{gs}}}$, $v_{i}\to v_{s}$ and
define an update rule,

$v_{i+1}=v+\frac{\delta}{\delta\rho}F^{1}[\rho_{i}]-\frac{\delta}{\delta\rho}F^{0}[\rho_{i}]=v+v_{\mathrm{Hxc}}[\rho_{i}].$
(22)

We see immediately that if $\rho_{i}$ has converged to the correct ground-
state density ${\rho_{\mathrm{gs}}}$ of the interacting system, then
$v+\frac{\delta}{\delta\rho}F^{1}[\rho_{i}]=0$ and the remaining equation
tells us that indeed $v_{i+1}$ is the potential that gives the same density
${\rho_{\mathrm{gs}}}$ in the non-interacting system. The next step after Eq.
(22) in the Kohn–Sham iteration lies in determining the density $\rho_{i+1}$
that comes from $v_{i+1}$ in the non-interacting system (which is comparably
easy achieved by solving the corresponding Kohn–Sham equation (18)) and then
iterate. Convergence problems are a big issue within this iteration scheme and
have been dealt with by either damping the iteration step from
$\rho_{i}\to\rho_{i+1}$ to $\rho_{i}\to\rho_{i}+\mu(\rho_{i+1}-\rho_{i})$,
$\mu\in(0,1)$, or mixing several of the previous steps $\\{\rho_{i}\\}$ into
the result $\rho_{i+1}$ [58; 59; 60]. Guaranteed convergence has been studied
and proven for the finite-lattice case [10; 11; 12] by combining an optimal
damping step and a regularization technique [4; 12], the latter truly making
$F$ differentiable and $E$ a strictly concave functional. This solves the
problem of defining $v_{\mathrm{Hxc}}$ in Eq. (21) and yields a curvature
bound on $F$ that is needed for guaranteed convergence. The regularization
method is briefly explained in Section IX below. For the Kohn–Sham iteration
in continuum DFT the convergence is still an open problem, a direct
generalization of the finite-lattice case has been found to be insufficient
[61]. In practical applications that suffer from convergence issues,
imaginary-time propagation in time-dependent DFT has recently been found as a
viable alternative to find a Kohn–Sham ground state [62].

##  IX Density-potential mixing and regularized DFT

The full HK theorem guarantees a unique inversion from densities to
potentials, but the whole discussion, especially regarding the necessary
conditions for showing HK2, probably already made us a little bit sceptical
about its validity in different settings. We will thus introduce a method that
always guarantees a bijective mapping, not between densities and potentials,
but between _quasidensities_ (called _pseudo-densities_ in the original work
on regularization [4]) and potentials. The basic idea is simple: If for some
reason we cannot guarantee a unique (injective) mapping from potentials to
ground-state densities $v\mapsto\rho[v]$, meaning that different $v\neq
v^{\prime}$ map to the same $\rho[v]=\rho[v^{\prime}]$, then let us try it for
$v\mapsto\rho_{\varepsilon}[v]=\rho[v]-\varepsilon v$, where at least in the
previous example we would have
$\rho_{\varepsilon}[v]\neq\rho_{\varepsilon}[v^{\prime}]$ for sure. One could
argue that this could just as easily introduce new problems for injectivity,
like having $v\neq v^{\prime}$ such that
$\rho_{\varepsilon}[v]=\rho_{\varepsilon}[v^{\prime}]$, but we will show in
the following that this cannot be the case for the functionals considered
here. Remember that the mapping $v\mapsto\rho[v]$ can be defined by the
superdifferential of $E$, $\rho[v]=\overline{\partial}E[v]$, as explained in
Eq. (17). So what is the corresponding functional $E_{\varepsilon}$ such that
$\rho_{\varepsilon}[v]=\rho[v]-\varepsilon
v=\overline{\partial}E_{\varepsilon}[v]$? The superdifferential retains the
linear nature of a derivative if only concave functionals are added, so we can
look for a convex functional $\phi$ such that
$\overline{\partial}(-\phi)[v]=-\underline{\partial}\phi[v]=-v$. In a general
space, such a question proves hard [61], but it is easy to see that in the
usual space $L^{2}$ of square-integrable functions the norm square gives
exactly what we need, $\phi[v]=\tfrac{1}{2}\|v\|^{2}=\tfrac{1}{2}\langle
v,v\rangle$. In any case, we have established
$E_{\varepsilon}=E-\varepsilon\phi$ and
$\rho_{\varepsilon}[v]=\overline{\partial}E_{\varepsilon}[v]$ with such a
convex $\phi$. But in many cases, not only for the mentioned $L^{2}$ space,
the functional $\phi$ is not only convex, but _strictly convex_ , meaning that
any local minimizer is not only global but even unique. But this feature
transfers to $E_{\varepsilon}$ if $-\varepsilon\phi$, as a _strictly concave_
functional, is added to $E$. Consequently, $E_{\varepsilon}$ is also strictly
concave and any maximizing potential in

$F_{\varepsilon}[x]=\sup_{v}\\{E_{\varepsilon}[v]-\langle v,x\rangle\\}$ (23)

is necessarily unique (not just up to a constant). This means we can always
uniquely map $v\mapsto x=\rho[v]-\varepsilon v$ and back. We wrote $x$ now to
make clear that this is a quasidensity, a mixture between a density and its
associated potential. As such it is neither necessarily normalized nor
positive, just a general element of the density space, $x\in X$. By what we
learned in Section VII, the _quasidensity-potential mapping_ can also be
directly defined by $-\underline{\partial}F_{\varepsilon}[x]=\\{v\\}$ for all
$x$ without any “$v$-representability” restriction for $x$. Consequently, the
mapping is defined for all $x$ in the density space $X$ and thus bijective.

The whole maneuver of passing from $F$ to $F_{\varepsilon}$ corresponds to a
regularization strategy called Moreau–Yosida regularization [4; 12] by which
not only the concave $E$ transforms into a strictly concave $E_{\varepsilon}$,
but also the $F_{\varepsilon}$ defined by Eq. (23) is finally differentiable
if the spaces $X,X^{*}$ have some additional properties [5, Theorem 9]. The
only problem is that this requires the space $X$ to be reflexive, which it is
not in our current formulation as introduced in Section III, since it includes
the non-reflexive $L^{1}$ in its definition. So a different choice for the
basic spaces, like $X=L^{2}$ on a bounded domain [4] or $X=L^{3}$ as a larger
alternative to our space[5] has to be taken.

This section demonstrated how such a regularization that facilitates a unique
(quasi)density-potential mapping can be used to fully circumvent any reference
to the HK theorem. But to avoid confusion we will _not_ say that in a
regularized setting the HK theorem “holds” even though a unique and well-
defined (quasi)density-potential mapping exists. It is interesting to note
that the popular Zhao–Morrison–Parr method for density-potential inversion
already implicitly employs Moreau–Yosida regularization and a limit procedure
$\varepsilon\to 0$ [63].

##  X Abstract density-potential mapping

The presented form of HK1 allows for an abstraction and thereby for
generalizations. Therein, the density is generalized to any system-inherent
quantity that seems suitable to describe other system parameters that we are
interested in. This could be the density together with the spin density, a
current-quantity etc. On the other side, we select a generalized form of the
potential that enters the Hamiltonian and that is able to steer the “density-
quantity” by coupling to it. Such a framework was developed in Laestadius  _et
al._ [5], building on Banach spaces and their duals for density and potential
quantities. This enables us to employ the regularization technique from
Section IX to obtain a well-defined Kohn–Sham iteration scheme.

In order to be more concrete, let $\mathbf{x}$ be the density quantity
describing a state that will in general include many components, like
different densities, currents etc., and $\mathbf{v}$ the collection of
external potentials acting on them. At this point we do not even assume that
$\mathbf{x}$ and $\mathbf{v}$ have the same number or type of components like
a dual structure between densities and potentials would impose. Instead of a
linear pairing $\langle\mathbf{v},\mathbf{x}\rangle$ for the coupling to the
external potential we can introduce an arbitrary functional
$f[\mathbf{v},\mathbf{x}]$. Then the _only_ necessary condition left for an
abstract HK1 is that the ground-state energy expression has the form

$\displaystyle\tilde{F}[\mathbf{x}]=\inf_{\psi\mapsto\mathbf{x}}\left\\{\langle\psi|H_{0}|\psi\rangle\right\\},$
$\displaystyle
E[\mathbf{v}]=\inf_{\mathbf{x}}\\{\tilde{F}[\mathbf{x}]+f[\mathbf{v},\mathbf{x}]\\}.$
(24)

Since $\tilde{F}[\mathbf{x}]$ is independent of $\mathbf{v}$, the critical
argument in the first proof of HK1 still holds and thus two potentials that
share a common $\mathbf{x}$ in the ground state will also share a common
ground-state wave function or density matrix. Consequently, HK1 is secured in
any such formulation of DFT, while the situation for HK2 quite generally is
more problematic. Even if the coupling between $\mathbf{v}$ and $\mathbf{x}$
that enters the energy functional in Eq. (24) is linear like in
$f[\mathbf{v},\mathbf{x}]=\langle\mathbf{v},\mathbf{x}\rangle$, the critical
step (8) in the proof of HK2 will involve more degrees-of-freedom on the
potential side and the argument may fail.

In the literature, the presented situation with linear coupling corresponds to
what Schönhammer, Gunnarsson, and Noack [64] call $\\{a\\}$-functional theory.
Similarly, Higuchi and Higuchi [65; 66] allow for a more general choice of
basic variables in DFT next to the usual density. Xu  _et al._ [67] derived
conditions that need to be fulfilled to also have a HK2 in such a general
setting. One can then try and extend DFT and the Kohn–Sham scheme
systematically to predict further system parameters, if good approximative
functionals can be found.

A first example would be the spin-resolved functional that has the usual one-
particle density $\rho=\rho_{\uparrow}+\rho_{\downarrow}$ and the spin-density
$\rho_{\uparrow}-\rho_{\downarrow}$ as basic variables,
$\mathbf{x}=(\rho_{\uparrow}+\rho_{\downarrow},\rho_{\uparrow}-\rho_{\downarrow})$.
An alternative possible choice would clearly be
$\mathbf{x}=(\rho_{\uparrow},\rho_{\downarrow})$ [18, Section 8.1]. The energy
functional is $E[v]=\inf_{\mathbf{x}}\\{\tilde{F}[\mathbf{x}]+\langle
v,\rho\rangle\\}$, with $v$ just the usual scalar potential that couples to
the one-particle density $\rho$. The involved spaces for densities and
potentials are not dual in this example, since they involve a different number
of components. But by choosing an $F_{\mathrm{Hxc}}[\mathbf{x}]$ that depends
on the spin-resolved density, the Hxc-potential as its derivative (and with it
the effective potential of the Kohn–Sham system) must be from the dual space
of $\mathbf{x}$ and thus include components that act on the different spin-
components individually.

A second example is CDFT and its variants that will be thoroughly discussed in
Part II of this review. The paramagnetic current density of a given state
$\psi\in\mathcal{W}$ is defined as

$\mathbf{j}^{\mathrm{p}}_{\psi}(\mathbf{r}_{1})=N\sum_{\underline{\sigma}}\int_{\mathbb{R}^{3(N-1)}}\mathrm{Im}\left\\{\psi^{*}\nabla_{1}\psi\right\\}\,\mathrm{d}\mathbf{r}_{\perp}.$

Then the amended density quantity is
$\mathbf{x}=(\rho,\mathbf{j}^{\mathrm{p}})$ which couples linearly to
$\mathbf{v}=(v+\frac{1}{2}|\mathbf{A}|^{2},\mathbf{A})$ [68]. Since by this
the potential-energy contribution amounts exactly to the linear pairing
$f[\mathbf{v},\mathbf{x}]=\langle\mathbf{v},\mathbf{x}\rangle$ that allows to
define a potential-independent constrained-search functional, HK1 holds.

This means one can continue along the lines started in this work and try to
generalize many concepts and results from above to such extended DFTs. This
includes the definition of representable densities (Section III), different
functionals (Section VI), functional differentiability (Section VII), setting
up a Kohn–Sham scheme (Section VIII), as well as regularisation (Section IX),
since also there the existence of a full HK theorem was hardly ever assumed.

$\begin{array}[]{ccccccc}&\lx@intercol\hfil\text{$F$
differentiable}\hfil\lx@intercol&&&\text{$F_{\varepsilon}$ from}\\\
&\lx@intercol\hfil\text{at $\rho\in
v\text{-}\mathsf{rep}$}\hfil\lx@intercol&&&\text{regularization}\\\
&&\Downarrow&&&&\Downarrow\\\
-\underline{\partial}F[\rho]=\\{v+c\\}&\hbox{\multirowsetup$\Longleftrightarrow$}&\hbox{\multirowsetup
HK}&\hbox{\multirowsetup$\Longleftrightarrow$}&\text{unique
density-}&\hbox{\multirowsetup$\Longrightarrow$}&\text{unique
\emph{quasi}density-}\\\ \text{at $\rho\in
v\text{-}\mathsf{rep}$}&&&&\text{potential mapping}&&\text{potential
mapping}\\\ &&\Updownarrow\\\
&\lx@intercol\hfil$\downbracefill$\hfil\lx@intercol\\\
&\text{HK1}&&\text{HK2}\\\ &\Uparrow&&\Uparrow\\\
\lx@intercol\hfil$\text{potential-
independent}$\hfil\lx@intercol&\lx@intercol$\text{UCP
result}$\hfil\lx@intercol\\\ \lx@intercol\hfil$\text{constrained-search
fctl.}$\hfil\lx@intercol\end{array}$ Figure 2: Logical implications between
the different statements relating to a “unique density-potential mapping” and
the HK theorem in standard DFT.

##  XI Summary

We will give a brief summary of the structure of the density-potential mapping
and its relation to the HK theorem. Following the last section on abstract
DFTs, at least the HK1 result does not only hold for standard DFT (that maps
one-particle densities to scalar potentials), but it holds for all variants of
DFTs that offer the required structures. This will be especially useful with
foresight towards CDFT, the topic of the second part of this review.

In standard DFT, with a setting that yields the unique-continuation property
that in turn prevents the ground-state density from being zero on a set of
non-zero measure (Section V) and due to the simple relation (8) in the proof
of HK2, a full HK result can be established. In any higher DFT this proof
strategy potentially fails. The status of HK1, on the other hand, is much less
critical, since this result holds automatically whenever a _potential-
independent_ (“universal”) constrained-search functional can be set up. But
also in cases where the constrained-search functional depends on the external
potential, a valid statement like in the HK theorem, that two potentials that
share a common ground-state density are equal up to gauge changes, is still
possible in general. The more general way how to think and talk about a HK
result is by calling it a “unique density-potential mapping” and we explained
how such a mapping can be established as the subdifferential of the density
functional $F$ at $v$-representable densities. If the potentials in the
resulting subdifferential are equal up to a gauge transformation, then this is
just the HK result again. Assuming full differentiability of $F$ implies a
one-element subdifferential, so there would not even be any room for gauge
changes, and a unique density-potential mapping would be the result once more.
This property of differentiability of the density functional $F$ is desirable
also in the context of Kohn–Sham theory in order to be able to link the
functional $F_{\mathrm{Hxc}}$ to the Hxc potential like in Eq. (21).

But since differentiability is _not_ a property of the usual DFTs, a
regularization strategy was devised and briefly explained in Section IX. This
yields a unique _quasi_ density-potential mapping, where quasidensities are
actually mixtures between ground-state densities and their potentials. The
mixing parameter $\varepsilon$ could be set to zero to retrieve the
unregularized theory together with the problem of non-differentiability. The
whole structure is laid out diagrammatically in Figure 2.

##  XII Outlook

In this outlook, we first want to collect the problems that still remain open
within the foundations of standard DFT and that will surely be the topic in
upcoming works. Considering Lieb’s mathematical formulation of DFT, summarized
above in Section III and VI, there are two main issues. Firstly, HK2 is
guaranteed only for eigenstates that are non-zero almost everywhere, a
property that is secured by the UCP explained in Section V. But the potential
space required for this does not cover all potentials from the Lieb setting
and a sufficiently general UCP result is not available to date. Secondly, the
issue of $v$-representability, explained in Section III, still remains open.
While regularization as described in Section IX formally allows us to
circumvent this problem, it has not yet been put to practical use. Since the
overlap between interacting and non-interacting $v$-representability is poorly
understood, this has direct implications for Kohn–Sham theory. But even with
$v$-representability assumed, convergence of the Kohn–Sham self-consistent
field iterations in the standard setting is still an open problem. Both
issues, availability of UCP and $v$-representability, relate to the function
spaces for densities and potentials. Possibly, with a more refined choice of
these spaces, full $v$-representability or even differentiability of $F$ might
be achievable. However, it also cannot be ruled out that non-differentiability
is fundamental to DFT.

This non-differentiability of $F$, that has been repeatedly stressed in this
work, implies that the exchange-correlation potential cannot be found as a
functional derivative with respect to the density, as it is usually assumed in
standard DFT. Orbital-dependent functionals [69] can be formally viewed as
relying on the HK1 map $\rho\mapsto\phi$ to obtain the Kohn–Sham wave function
from a density. Non-differentiability of $F[\rho]$ might then be represented
in the noninteracting wave function $\phi[\rho]$, which may benefit the
functional approximations if they rely directly on the Kohn–Sham orbitals. The
lack of differentiability also favors approaches based on forces instead of
energies, as mentioned in Section VIII. However, practical functionals that
are derived from this approach remain unexplored and there is still a
dependence on $v$-representability.

It is interesting to note which useful structures of DFT carry over to
“higher” density-functional theories, and in Part II we will discuss density-
functional theory for systems involving magnetic fields. While one of its
flavours, paramagnetic CDFT, already briefly discussed in Section X, still
allows for a constrained-search functional (HK1), the realization of a full
density-potential mapping is highly problematic. For this reason, in the
classical formulation of paramagnetic CDFT [70] the HK2 result that different
potentials lead to different ground states was just tacitly assumed with the
words: “Let $\psi$ and $\psi^{\prime}$ be the two different ground states
corresponding to the two sets of fields [$(v,\mathbf{A})$ and
$(v^{\prime},\mathbf{A}^{\prime})$].” Later, Capelle and Vignale [7] even
found counterexamples to HK2 which shows that a density-potential mapping
cannot be constructed in paramagnetic CDFT. But this clearly does not mean
that in different versions of CDFT the density-potential mapping is impossible
to achieve in general. A formulation utilizing the total current will be
studied as well, but here the constrained-search functional would depend on
$\mathbf{A}$ and thus HK1 is not available in the fashion as it was presented
here. So while for paramagnetic CDFT the HK2 fails, for total (physical) CDFT
already HK1 does not hold. Overall, the existence of a well-defined density-
potential mapping in CDFT is still an open issue that will be considered in
the second part of this review.

## Acknowledgement

EIT, MAC and AL thank the Research Council of Norway (RCN) under CoE
(Hylleraas Centre) Grant No. 262695, for AL and MAC also CCerror Grant No.
287906 and for EIT also “Magnetic Chemistry” Grant No. 287950, and MR
acknowledges the Cluster of Excellence “CUI: Advanced Imaging of Matter” of
the Deutsche Forschungsgemeinschaft (DFG), EXC 2056, project ID 390715994. AL
and MAC was also supported by the ERC through StG REGAL under agreement No.
101041487. The authors thank Centre for Advanced Studies (CAS) in Oslo, since
this work includes insights gathered at the YoungCAS workshop “Do Electron
Current Densities Determine All There Is to Know?”, held July 9-13, 2018, in
Oslo, Norway.

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