# Entropic Density Functional Theory

Ahmad Yousefi
Department of Physics
SUNY at Albany
Albany, NY 12222
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
& Ariel Caticha
Department of Physics
SUNY at Albany
Albany, NY 12222
<EMAIL_ADDRESS>

###### Abstract

A formulation of the density functional theory is constructed on the
foundations of entropic inference. The theory is introduced as an application
of maximum entropy for inhomogeneous fluids in thermal equilibrium. It is
shown that entropic inference produces the variational principle of DFT when
information about the expected density of particles is imposed. This process
introduces a family of trial density-parametrized density matrices from which
the preferred density matrix is found using the method of quantum maximum
entropy. As illustrations some known approximation schemes of the theory are
discussed.

_Keywords_ Entropic inference $\cdot$ Maximum entropy $\cdot$ Density
functional theory $\cdot$ Inhomogeneous fluids

##  1 Introduction

The Density Functional Theory (DFT) is an approach to the many-body particle
systems. The theory was first introduced in 1964 in the context of quantum
mechanics by Hohenberg and Kohn [1]. The article not only showed that for an
electron gas at zero temperature, the electronic density distribution uniquely
characterizes the ground state, but also proved that there exists a density
functional variational equation which yields the equilibrium density as its
solution, equal to the density calculated using the time independent
Schrödinger equation.
This equivalence of the Hohenberg-Kohn (HK) equation and the time independent
Schrödinger equation, extended to a new equivalence only a year later, when
Mermin [2] proved that for an electron gas at finite temperature in the
presence of an external potential, there exists a density functional
variational equation which gives the same equilibrium density as gives the
grand canonical ensemble formalism of statistical mechanics.
Unlike the usual historical order of theories in physics, the classical DFT
emerged years after the quantum DFT. Evans formulated the finite temperature
DFT in the classical regime [3] with motivation from Mermin’s work [2]. We
showed that from an entropic inference point of view Evans’ variational
principle of classical DFT is a Legendre transform of the principle of maximum
entropy (MaxEnt) [4, 5]. Before we proceed to expand the work to quantum
regime, we shall look at the historical background : in 1929, Paul Dirac
announced that the non-relativistic quantum mechanics is complete and
approximation schemes are desired to simplify the sophisticated quantum-
mechanical calculations [6]. Physicists and chemists followed Dirac’s advice
and developed the mathematical frame work to solve the Schrödinger equation of
atoms, molecules, and solids over the years 1930’s, 40’s, 50’s, and 60’s.
Meanwhile the first digital computers were made during the 40’s and they had
rapidly grown to powerful machines capable of performing quantum calculations
of atoms and single molecules by 1965 [7]. At the time, computers were able to
give an approximate solution to the Schrödinger equation of many-particle
systems using the Hartree-Fock approximation method for molecules and atoms,
while solid state physicists preferred to borrow analytical techniques from
quantum field theory to study the electronic correlation effects in materials,
which evolved to the whole branch of statistical physics of fields [8].
Readers may refer to [7], [9], and [10] for a deeper dive into the history of
DFT, one may conclude that the increasing global demand for quantum
calculations, together with the exponential growth of computer technology left
the theoretical aspects of DFT neglected compared to the enormous developments
in its applications. Consequently the theory is mostly known as a
computational modeling method for investigation of electronic structure of
quantum mechanical systems.
This paper, however, is primarily concerned with the DFT itself as a
theoretical framework from which one of the most effective computational
techniques have emerged. We introduce the density functional formalism, as a
straightforward application of the maximum entropy principle. we will show
that the DFT variational principle is nothing but the principle of maximum
entropy subject to some appropriate constraints, rewritten under an
appropriate Legendre Transformation. such a study integrates the DFT with
statistical mechanics [11, 12, 13, 14, 15, 16] and quantum mechanics [17, 18]
as systematic applications of entropic inference and may offer a more general
framework for approximations from which other theories such as Mean Field
Theory, Dynamic DFT, Ginzburg-Landau Theory, etc. can be recovered. Another
advantage of such a study is that it clarifies the ambiguity of what is exact
and what is approximate in the DFT calculations, this clarification may help
computational physicists and chemists to make better decisions as to which
parts of the codes are modifiable and how they may want to compare their
results to experiments and to other computational methods.
In section 2 we review the quantum MaxEnt as a tool to update density matrices
in response to new information, and we show that any quantum MaxEnt is
recognized as a geometrical contact structure which is invariant under
Legendre transformations; and then we show how the MaxEnt is used to find
optimal approximation of a density matrix and we prove the Bogolyubov
inequality.
In section 3 we construct the DFT formalism and show that the DFT variational
equation is the Legendre transform of the MaxEnt where the inference method is
used to pick the best density matrix from a trial family of density
parametrized density matrices.
In section 4 we recover the almost constant density and famous Kohn-Sham model
as illustrations.
And finally in section 5 after a brief conclusion we lay down the pathway for
future theoretical research based on this paper.

##  2 Quantum MaxEnt

Although the method of entropic inference has its roots in the work of
Boltzmann, Maxwell, and Gibbs, it was Jaynes who first showed that
thermodynamics, as fundamental as it is should be interpreted as a scheme for
inference about nature [19, 20, 21, 22]. From Jaynes’ point of view -following
Shannons axioms [23]\- entropy was the amount of information that is missing
in a probability distribution. In 1980 Shore and Johnson [24] suggested a new
interpretation of the MaxEnt, the idea was to axiomatize the updating method
of probabilities and they proposed a variational principle for that matter. in
1998 Skilling [25] justified the proposed variational principle by a simple
idea: in order to find the preferred posterior, one must rank all the
probability distributions in the order of preference and then pick the
highest.
In the first two decades of the twenty first century, Caticha took the
information theory to the next level by deriving the Kullback-Leibler
divergence as the unique relative entropy $S_{r}[p|q]$ which allows an ideal
agent to update their degree of rational belief from the prior probability $q$
to the posterior probability $p$ [11, 12, 13, 14, 15, 16]. In this approach,
information is the constraint on the probabilities in the updating process
rather than anything stored in a system or in a probability distribution.
In quantum mechanics, the most informative state of knowledge about a system
is represented by a state vector $|\psi\rangle$. When we have incomplete
information about the system, we call it a mixed state. Mixed states are
treated with a density operator represented by the density matrix $\hat{\rho}$
instead of a single wave function. The density matrix assigns probabilities
$p_{i}$ to each member of an orthonormal basis set $|\psi_{i}\rangle$:

$\hat{\rho}=\sum_{i,j}p_{i}\delta_{i,j}|\psi_{i}\rangle\langle\psi_{j}|\,.$
(1)

One can show that the expected value of any Hermitian operator $\hat{A}$ is
given by

$\langle\hat{A}\rangle=Tr\hat{\rho}\hat{A}\,.$ (2)

From the entropic inference point of view, the task of MaxEnt is to update a
prior density matrix $\hat{\sigma}$ subject to new information in form of
expected values of Hermitian operators $\langle\hat{A}_{i}\rangle=A_{i}$.
Vanslette [26] showed that the Umegaki entropy [27] is the unique tool to
update density matrices in response to such new information; i.e. the
posterior density matrix $\hat{\rho}^{*}$ is found by maximizing the quantum
entropy of $\hat{\rho}$ relative to $\hat{\sigma}$

$S_{r}[\hat{\rho}|\hat{\sigma}]=-Tr\hat{\rho}[log\hat{\rho}-log\hat{\sigma}]$
(3)

subject to the information and the normalization of the posterior

$\delta\Big{[}S_{r}[\hat{\rho}|\hat{\sigma}]-\alpha_{0}[Tr\hat{\rho}-1]-\sum_{i}\alpha_{i}[Tr\hat{\rho}\hat{A}_{i}-A_{i}]\Big{]}_{\hat{\rho}={\hat{\rho}}^{*}}=0\,.$
(4)

Such a maximization yields to the posterior density matrix

$\hat{\rho}^{*}=\frac{1}{Z}e^{log\hat{\sigma}-\sum\alpha_{i}\hat{A}_{i}}$ (5)

where the partition function $Z$ is the normalization factor defined as

$Z\equiv e^{\alpha_{0}}=Tre^{log\hat{\sigma}-\sum\alpha_{i}A_{i}}\,.$ (6)

And therefore the maximized entropy is given by

$S\equiv S_{r}[\hat{\rho}^{*}|\hat{\sigma}]=\sum\alpha_{i}A_{i}+logZ\,.$ (7)

###  2.1 Contact Structure of MaxEnt

It had been shown that the macroscopic thermodynamic variables produce a
contact structure [28, 29]. Earlier we expanded the idea [4] and showed that
any classical MaxEnt with $m$ constraints can be geometrically interpreted as
a $2m+1$-dimensional manifold $\mathbb{T}$ equipped with a contact form
$\omega_{c}$ which vanishes on the physically relevant submanifold
$\mathbb{M}$, here we shall prove that the quantum MaxEnt has a similar
geometrical structure.
Take the complete differential of equation (7) to find that

$dS=\sum\alpha_{i}dA_{i}+d\alpha_{i}A_{i}+\frac{dZ}{Z}\,.$ (8)

Then use the matrix exponential differential identity [30]

$\frac{d}{dt}e^{X(t)}=e^{X}\frac{1-e^{ad_{X}}}{ad_{X}}\frac{dX}{dt}$ (9)

in which $ad_{X}Y=[X,Y]$ for $X$ and $Y$ belonging to the same Hilbert space,
to have

$dS=\sum_{i}\alpha_{i}dA_{i}-\sum_{j}\sum_{n=1}^{\infty}\frac{(-Z)^{n}}{(n+1)!}\langle
ad_{\hat{\rho}}^{n}\hat{A}_{j}\rangle d\alpha_{j}\,.$ (10)

Note that the the expected value of the first adjoint action $\langle
ad_{\hat{\rho}}\hat{A}_{j}\rangle=Tr\hat{\rho}[\hat{\rho},\hat{A}]$ vanishes
by the cyclic property of trace and similarly for higher orders $\langle
ad_{\hat{\rho}}^{n}\hat{A}_{j}\rangle=0$, and we end up with the vanishing
one-form

$\omega_{q}\equiv dS-\sum_{i}\alpha_{i}dA_{i}=0\,.$ (11)

Now, similar to the classical case [4], for any quantum MaxEnt with $m$
constraints we define the contact structure $\\{\mathbb{T},\omega_{q}\\}$ in
which $\mathbb{T}$ is a $2m+1$-dimensional manifold with coordinates
$\\{q_{0},q_{1},\dots,q_{m};p_{1},\dots,p_{m}\\}$. This means that the
physically relevant manifold $\mathbb{M}$ is a m-dimensional submanifold on
$\mathbb{T}$ locally represented by the vanishing one-form $\omega_{q}$; or
equivalently, $\mathbb{M}$ is determined by $m+1$ equations

$q_{0}\equiv S(\\{A_{i}\\})$ (12) $p_{i}\equiv\alpha_{i}=\frac{\partial
S}{\partial A_{i}}\,.$ (13)

Now we can state that the Legendre transformations are coordinate
transformations on $\mathbb{T}$ under which the contact form $\omega_{q}$ is
invariant:

$q_{0}\rightarrow q_{0}-\sum_{j=1}^{l}p_{j}q_{j}$ (14) $q_{i}\rightarrow
p_{i},\quad p_{i}\rightarrow-q_{i}\quad\textrm{for }1\leq i\leq l\,.$ (15)

The existence of the prescribed contact structure makes it legitimate to
utilize Legendre transformations in MaxEnt regardless of thermal equilibrium
and the physical significance of the so called "free energy" functions.

###  2.2 Optimal Approximation of Density Matrix: the Bogolyubov Inequality

Here we prescribe the last prerequisite for tackling the DFT formalism.
Caticha and Tseng [31] showed that the classical MaxEnt provides a method for
approximating an exact probability distribution by the best member of a family
of available trial distributions. Here we expand the method to the quantum
case where the MaxEnt is used to find the optimal approximation of an exact
density matrix from a family of parametrized density matrices: The relative
entropy $S_{r}[\hat{\rho}|\hat{\sigma}]$ provides a ranking of all
$\hat{\rho}$’s relative to a single $\hat{\sigma}$ [26, 32], on the other hand
the relative entropy $S_{r}[\hat{\rho}|\hat{\sigma}]$ is non-positive and
assumes zero only for $\hat{\rho}=\hat{\sigma}$ [33]

$S_{r}[\hat{\rho}|\hat{\sigma}]\leq 0\quad\textrm{with equality iff
}\hat{\rho}=\hat{\sigma}\,.$ (16)

Now consider $\hat{\rho}\equiv\hat{\rho}_{\theta}$ a family of density
matrices parametrized by variables $\theta=\\{\theta_{1},\theta_{2},\dots\\}$.
The member of the trial family $\hat{\rho}^{*}$ which best approximates
$\hat{\sigma}$ is the one which maximizes the entropy of $\hat{\rho}_{\theta}$
relative to $\hat{\sigma}$

$\frac{\partial}{\partial\theta}\Big{[}S_{r}[\hat{\rho}_{\theta}|\hat{\sigma}]\Big{]}_{\theta=\theta^{*}}=0\,.$
(17)

In a special case where $\hat{\sigma}$ and $\hat{\rho}_{\theta}$ are
exponential density matrices defined by

$\hat{\sigma}=\frac{1}{Z}e^{-\beta\hat{H}}\quad\textrm{and
}\hat{\rho}_{\theta}=\frac{1}{Z_{\theta}}e^{-\beta\hat{H}_{\theta}}\,,$ (18)

the inequality (16) reduces to

$F\leq F_{\theta}+\langle\hat{H}-\hat{H}_{\theta}\rangle_{\theta}$ (19)

in which $F=\frac{-logZ}{\beta}$, $F_{\theta}=\frac{-logZ_{\theta}}{\beta}$,
and $\langle.\rangle_{\theta}\equiv Tr(\hat{\rho}_{\theta}(.))$.
The equation (19) is known as the Bogolyubov inequality and has been
frequently used in statistical mechanics to find the best estimate of a
density matrix [34].

##  3 Density Functional Formalism

Any equilibrium ensemble formalism of statistical mechanics is a MaxEnt which
constitutes of a trinity of elements: (i) the Hilbert space associated with
the time independent Schrödinger equation of the system, (ii) uniform prior
density matrix, (iii) inferential constraints. The density functional
formalism is not an exception, the time independent Shrödinger equation for an
electron gas of $N$ particles in the absence of magnetic field is given by

$\hat{H}|\psi\rangle=E|\psi\rangle\,,$ (20)

where

$\hat{H}_{v}=\hat{H}^{(0)}+\hat{V}=\hat{T}+\hat{U}+\hat{V}\equiv\sum_{i=1}^{N}\hat{p}_{i}^{2}+\frac{1}{2}\sum_{j\neq
k}^{N}\frac{1}{|\hat{x}_{j}-\hat{x}_{k}|}+\sum_{l=1}^{N}v(\hat{x}_{l})\,,$
(21)

and $|\psi\rangle$ is an antisymmetrized product of N two-spinor orbitals.

###  3.1 Constructing the Trial States

We are interested in the thermal properties of an inhomogeneous fluid,
therefore we need trial states that describe both thermal equilibrium and
inhomogeneity, the former is imposed by one constraint on the expected value
of energy and the latter is incorporated by one constraint on the expected
value of density at each point of space. Therefore we maximize the entropy

$S_{r}[\hat{\rho}|\hat{\mathbb{I}}]=Tr\hat{\rho}log\hat{\rho}\,,$ (22)

subject to constraints

$Tr\hat{\rho}=1\,,$ (23) $Tr\hat{\rho}\hat{H}=E\,,$ (24)
$Tr\hat{\rho}\hat{n}_{x}=n(x)\quad\textrm{where}\int d^{3}xn(x)=N\,,$ (25)

where $\hat{n}_{x}\equiv\sum_{i=1}^{N}\delta(\hat{x}_{i}-x)$. Note that the
density function $n(x)$ plays the role of parameters $\\{\theta_{i}\\}$ in
equation (18), under the generalization that the parameters $n(x)$ are
continuous here.
Note that there is one constraint for each point in $\mathbb{R}^{3}$ in line
(25), but the expected density function $n(x)$ is not arbitrary, it must
conform to the fact that the number of particles $N$ is fixed by definition of
the wave function. Here we deal with a MaxEnt under some constraints, the
constraints themselves are limited together under a condition imposed by the
definition of the Hilbert space. Proceed with the MaxEnt analog to equation
(5) to find the trial density matrix

$\hat{\rho}_{n}=\frac{1}{Z_{v}}e^{-\beta\hat{H}_{v}-\int
d^{3}x\alpha(x)\hat{n}_{x}}$ (26)

where

$Z_{v}(\beta;\alpha]=Tre^{-\beta\hat{H}_{v}-\int
d^{3}x\alpha(x)\hat{n}_{x}}\,,$ (27)

and the Lagrange multipliers $\alpha(x)$ are determined implicitly by
$\frac{\delta logZ_{v}}{\delta\alpha(x)}=-n(x)$. Now we can calculate the
entropy of $\hat{\rho}_{n}$ relative to the identity density matrix to define
the trial entropy $S_{v}(E;n]$ as

$S_{v}(E;n]\equiv S_{r}[\hat{\rho}_{n}|\mathbb{I}]=\beta E+\int
d^{3}x\alpha(x)n(x)+logZ_{v}(\beta;\alpha]\;,$ (28)

in which the Lagrange multiplier function $\alpha(x)$ is bound to the
condition

$\int d^{3}xn(x)=-\int d^{3}x\frac{\delta
logZ_{v}(\beta;\alpha]}{\delta\alpha(x)}=N\;.$ (29)

Remark: Note that in $\hat{\rho}_{n}$, $Z_{v}$, and $S_{v}$, the Lagrange
multiplier $\alpha(x)$ is not an arbitrary function; the only permissible
$\alpha(x)$’s are those which keep the number of particles fixed in (29).
Remark: An important symmetry in the DFT formalism is that $\hat{\rho}_{n}$
and $Z_{v}(\beta;\alpha]$ depend on the external potential $v(x)$ and the
Lagrange multipliers $\alpha(x)$ only through the particular combination

$\bar{\alpha}(x)\equiv\beta v(x)+\alpha(x)\;.$ (30)

Substitute $\bar{\alpha}$ in $Z_{v}$ to find that

$Z_{v}(\beta,\alpha]=Tre^{-\beta\hat{H}^{(0)}-\int
d^{3}x\bar{\alpha}(x)\hat{n}_{x}}\equiv Z(\beta,\bar{\alpha}]\;,$ (31)

and

$-n(x)=\frac{\delta logZ(\beta;\bar{\alpha}]}{\delta\bar{\alpha}(x)}\;.$ (32)

The symmetry implies that changing $\alpha(x)$ and $v(x)$ will not affect
expected density $n(x)$, as long as $\bar{\alpha}(x)$ remains unchanged; as we
shall see later, the core argument behind the DFT theorem is based on this
symmetry.
Remark: A shift in $\alpha(x)$ can be compensated by shifting the external
potential to keep $\bar{\alpha}(x)$ unchanged, therefore if $\alpha(x)$ is
permissible in equation (29), so is $\alpha(x)+\textit{const.}$ .

###  3.2 The Equilibrium Density Matrix

Now that we have constructed our desired family of density matrices
$\hat{\rho}_{n}$, it is time to approximate the canonical density matrix of
the system by the best fitting member of $\hat{\rho}_{n}$’s. The canonical
density matrix is defined as

$\hat{c}=\frac{1}{Z_{v}(\beta)}e^{-\beta\hat{H}_{v}}\,,$ (33)

where $Z_{v}(\beta)=Tre^{-\beta\hat{H}_{v}}$ .
Calculate the entropy of $\hat{\rho}_{n}$ relative to $\hat{c}$ to find that

$S_{r}[\hat{\rho}_{n}|\hat{c}]=S_{v}(E;n]-\beta E-logZ(\beta)\,.$ (34)

Maximization of the relative entropy

$\delta S_{r}[\hat{\rho}_{n}|\hat{c}]=0\quad\textrm{for fixed }\beta$ (35)

gives the closest $\hat{\rho}$ to $\hat{c}$.
In analogy to equation (17) the preferred $n(x)$ is found by maximizing the
relative entropy $S_{r}[\hat{\rho}_{n}|\hat{c}]$. Use equation (34), and add a
new Lagrange multiplier term $\alpha_{eq}$ to write the maximization of
$S_{r}[\hat{\rho}|\hat{c}]$ as

$\frac{\delta}{\delta n(x^{\prime})}\bigg{[}S_{v}(E;n]-\beta
E+\alpha_{eq}[N-\int d^{3}xn(x)]\bigg{]}_{n(x)=n_{eq}(x)}=0\quad\textrm{for
fixed $\beta$}\,.$ (36)

Note that the term $\alpha_{eq}[N-\int d^{3}xn(x)]$ takes care of the fact
that $n(x)$ is not an arbitrary function and it must conform to the fact that
the number of particles $N$ is fixed.
Up to here, we have shown that there exists a trial density functional entropy
$S_{v}(E;n]$ which assumes its maximum value at equilibrium density function
$n_{eq}(x)$, but so far it is not guaranteed that this variational principle
is equivalent to the DFT principle which was derived in 1965 [2].

###  3.3 The DFT Theorem

Substitute equation (28) in (36) and use the identity

$\frac{\delta}{\delta n(x^{\prime})}logZ_{v}(\beta;\alpha]\Big{|}_{\beta}=\int
d^{3}x\frac{\delta
logZ_{v}(\beta;\alpha]}{\delta\alpha(x)}\frac{\delta\alpha(x)}{\delta
n(x^{\prime})}$ (37)

to find that

$\int d^{3}x\Bigg{[}\frac{\delta
logZ_{v}(\beta;\alpha]}{\delta\alpha(x)}+n(x)\Bigg{]}\frac{\delta\alpha(x)}{\delta
n(x^{\prime})}=\int d^{3}x\Bigg{[}\alpha_{eq}-\alpha(x)\Bigg{]}\frac{\delta
n(x)}{\delta n(x^{\prime})}.$ (38)

The RHS of equation (38) vanishes by the equation (29), and $\frac{\delta
n(x)}{\delta n(x^{\prime})}=\delta(x-x^{\prime})$, therefore the equation (38)
boils down to

$\alpha(x)=\alpha_{eq}$ (39)

and consequently

$\frac{\delta
Z_{v}(\beta;\alpha]}{\delta\alpha(x)}\Big{|}_{\alpha(x)=\alpha_{eq}}=-n_{eq}(x)\,.$
(40)

Next, substitute (39) in (26) to find the equilibrium density matrix equal to
that of the canonical formalism

$\hat{\rho}_{eq}=\frac{1}{Z_{v,eq}}e^{-\beta\hat{H}_{v}-\alpha_{eq}\int
d^{3}x\hat{n}_{x}}=\frac{1}{Z_{v,eq}}e^{-\beta\hat{H}_{v}-\alpha_{eq}N}=\hat{c}\,,$
(41)

where $Z_{v,eq}(\beta,\alpha_{eq})=Tre^{-\beta\hat{H}_{v}-\alpha_{eq}N}$ .
Define the density functional potential $\Omega_{v}(\beta;n]$ as a Legendre
transform of $S_{v}(E;n]$ (up to multiplication by $\frac{-1}{\beta}$)

$\Omega_{v}(\beta;n]\equiv-\frac{S_{v}(E;n]-\beta E}{\beta}=-\int
d^{3}x\frac{\alpha(x)}{\beta}n(x)-\frac{1}{\beta}logZ_{v}(\beta;\alpha]$ (42)

to rewrite equation (36) as

$\frac{\delta}{\delta
n(x^{\prime})}\Bigg{[}\Omega_{v}(\beta;n]+\frac{\alpha_{eq}}{\beta}\int
d^{3}xn(x)\Bigg{]}_{n(x)=n_{eq}(x)}=0\quad\textrm{at fixed }\beta\,,$ (43)

which means that the equilibrium density $n_{eq}(x)$ is found by minimization
of $\Omega(\beta;n]$ at fixed temperature and under the condition of fixed
number of particles.
Remark: note that the combination of the fact
$\alpha(x)=-\beta\frac{\delta\Omega_{v}}{\delta n(x)}$ with equation (39) give
the "core integro-differential equation of DFT" [3]:

$\nabla\frac{\delta\Omega(\beta;n]}{\delta n(x)}\Big{|}_{n(x)=n_{eq}(x)}=0\,.$
(44)

Which implies that

$\Omega_{v,eq}(\beta;n_{eq}]\leq\Omega_{v}(\beta;n]$ (45)
$\Omega_{v,eq}(\beta;n_{eq}]=-\frac{\alpha_{eq}}{\beta}\int
d^{3}xn_{eq}(x)-\frac{1}{\beta}logZ_{v}^{*}(\beta,\alpha_{eq})\,.$ (46)

To proceed further, combine equations (20), (21), and (42) to find that

$\Omega_{v}(\beta;n]=\int
d^{3}xv(x)n(x)+\langle\hat{K}+\hat{U}\rangle-\frac{S_{v}(E;n]}{\beta}\;.$ (47)

Next, define the intrinsic density functional potential $F_{v}$ as

$F_{v}(\beta;n]\equiv\langle\hat{K}+\hat{U}\rangle-\frac{S_{v}}{\beta}\;$ (48)

to have

$\Omega_{v}(\beta;n]=\int v(x)n(x)+F_{v}(\beta;n]\;.$ (49)

We are now ready to state the finite temperature DFT theorem.

Theorem (Density Functional)  The intrinsic functional potential $F_{v}$ is a
functional of density $n(x)$ and is independent of the external potential
$v(x)$:

$\frac{\delta F_{v}(\beta;n]}{\delta v(x^{\prime})}=0\quad\textrm{for fixed
$\beta$ and $n(x)$.}$ (50)

Proof  The crucial observation behind the DFT formalism is that
$\hat{\rho}_{n}$ and $Z_{v}(\beta;\alpha]$ depend on the external potential
$v(x)$ and the Lagrange multiplier function $\alpha(x)$ only through a
particular combination $\bar{\alpha}$ defined in (30): substitute (28) and
(30) in (48) to get

$\beta F_{v}(\beta;n]=logZ(\beta;\bar{\alpha}]+\int
d^{3}x\bar{\alpha}(x)n(x),$ (51)

where
$Z(\beta;\bar{\alpha}]=Z_{v}(\beta;\alpha]=Tr_{c}e^{-\beta(\hat{K}+\hat{U})-\int
d^{3}x\bar{\alpha}(x)\hat{n}_{x}}\;.$ The functional derivative of $\beta
F_{v}$ at fixed $n(x)$ and $\beta$ is

$\frac{\delta(\beta F_{v}(\beta;n])}{\delta
v(x^{\prime})}\Big{|}_{\beta,n(x)}=\int
d^{3}x^{\prime\prime}\frac{\delta}{\delta\bar{\alpha}(x^{\prime\prime})}\Big{[}logZ(\beta;\bar{\alpha}]+\int
d^{3}x\bar{\alpha}(x)n(x)\Big{]}\frac{\delta\bar{\alpha}(x^{\prime\prime})}{\delta
v(x^{\prime})}\Big{|}_{\beta,n(x)}\;.$ (52)

Since $n(x^{\prime})=-\frac{\delta
logZ(\beta;\bar{\alpha}]}{\delta\bar{\alpha}(x^{\prime})}$, keeping $n(x)$
fixed is achieved by keeping $\bar{\alpha}(x)$ fixed:

$\frac{\delta\bar{\alpha}(x^{\prime\prime})}{\delta
v(x^{\prime})}\Big{|}_{\beta,n(x)}=\frac{\delta\bar{\alpha}(x^{\prime\prime})}{\delta
v(x^{\prime})}\Big{|}_{\beta,\bar{\alpha}(x)}=0\;,$ (53)

so that

$\frac{\delta F_{v}(\beta;n]}{\delta v(x^{\prime})}\Big{|}_{\beta,n(x)}=0\;,$
(54)

which concludes the proof; Now we can drop the index $v$ , and write down the
intrinsic potential $F(\beta;n]$ as

$F(\beta;n]\equiv F_{v}(\beta;n]\;.$ (55)

Now define the chemical potential

$\mu\equiv\frac{-\alpha_{eq}}{\beta}\,,$ (56)

Remark: Note that a change of the external potential $v(x)$ can be compensated
by a suitable change of the multiplier $\alpha(x)$ in such a way as to keep
$\bar{\alpha}(x)$ fixed, such changes in $v(x)$ will have no effect on $n(x)$.
Therefore keeping $n(x)$ fixed on the left hand side of (51) means that
$\bar{\alpha}(x)$ on the right side is fixed too.
Now we can substitute equation (49) in (43), and define the chemical potential

$\mu\equiv\frac{-\alpha_{eq}}{\beta}\;,$ (57)

to have

$\frac{\delta}{\delta n(x^{\prime})}\Bigg{[}\int
d^{3}xv(x)n(x)+F(\beta;n]-\mu\int d^{3}xn(x)\Bigg{]}_{n(x)=n_{eq}(x)}=0\;.$
(58)

We can also substitute (49) in (44) to find

$v(x)+\frac{\delta F}{\delta n(x)}\Big{|}_{eq}=\mu,$ (59)

which allows us to interpret $\mu_{in}(x;n]\equiv\frac{\delta F}{\delta n(x)}$
as intrinsic chemical potential of the system.

##  4 Approximation Schemes

In previous section we proved that there exists a density functional
$\Omega_{v}(\beta,n]$ defined as

$\Omega_{v}(\beta;n]=\int d^{3}xv(x)n(x)+F(\beta;n]$ (60)

which assumes its minimum at equilibrium density. Exact calculation of
$F(\beta;n]$ requires calculation of the canonical partition function of the
system, this is what an approximation scheme of statistical mechanics must
avoid to begin with, therefore all different DFT models are different
approximations of $F(\beta;n]$. In this section, as an illustration of the
method, we reformulate some well-known approximations.

###  4.1 Free Energy Approximation for Almost Constant Density

Similar to [1] and [2], consider an electron gas with mean density $n_{0}$,
imposed to a small electric field produced by an external positive charge
distribution $n_{ext}(x)$, write down all the relevant variables in the same
form as a function of the perturbation parameter $\lambda$:

$\displaystyle n_{ext}(x)=$ $\displaystyle 0+\lambda n_{ext}^{(1)}(x)+\dots$
(61) $\displaystyle n(x)=$ $\displaystyle n_{0}+\lambda n^{(1)}(x)+\dots$ (62)
$\displaystyle\alpha(x)=$ $\displaystyle 0+\lambda\alpha^{(1)}(x)+\dots$ (63)
$\displaystyle\Omega_{v}=$
$\displaystyle\Omega_{v}^{(0)}+\lambda\Omega_{v}^{(1)}+\lambda^{2}\Omega_{v}^{(2)}+\dots$
(64)

Note that we discussed in a remark earlier that shifting $\alpha(x)$ by a
constant doesn’t change the physics, therefore we have chosen
$\alpha^{(0)}=0$. On the other hand the density of the unperturbed electron
gas is $n_{0}$ therefore comes the choice of $n^{(0)}=n_{0}$ in the equation
(62).
To calculate $\Omega_{v}(\beta;n]$, combine (62) and (63) with equation (42)
to find

$\Omega_{v}(\beta;n]\approx-\int
d^{3}x\frac{\lambda\alpha^{(1)}(x)}{\beta}\Bigg{[}n_{0}+\lambda
n^{(1)}(x)\Bigg{]}-\frac{1}{\beta}logZ_{v}(\beta;\alpha]\;.$ (65)

In order to expand $Z_{v}(\beta;\alpha]$ in terms of $\lambda$, write down the
Taylor expansion of the functional around $\alpha(x)=0$

$\displaystyle logZ_{v}(\beta;\alpha]=$ $\displaystyle
logZ_{v}(\beta,0]\Big{|}_{v(x)=0}+\int d^{3}x\frac{\delta
logZ_{v}}{\delta\alpha(x)}\Bigg{|}_{v(x)=0}\delta\alpha(x)$ (66)
$\displaystyle+\frac{1}{2}\int
d^{3}xd^{3}x^{\prime}\frac{\delta^{2}logZ_{v}}{\delta\alpha(x)\delta\alpha(x^{\prime})}\Bigg{|}_{v(x)=0}\delta\alpha(x)\delta\alpha(x^{\prime})+\dots\;.$

Since we have set $n(x)=n_{0}$ in the absence of external potential, taking
the functional derivative of $Z_{v}$ given in (27) yields

$\displaystyle\frac{\delta logZ_{v}}{\delta{\alpha(x)}}\Big{|}_{v(x)=0}$
$\displaystyle=-n_{0}$ (67)
$\displaystyle\frac{\delta^{2}logZ_{v}}{\delta\alpha(x)\delta\alpha(x^{\prime})}\Big{|}_{v(x)=0}$
$\displaystyle=n_{0}^{(2)}(x,x^{\prime})=n_{0}^{(2)}(|x-x^{\prime}|)\;,$ (68)

where $n_{0}^{(2)}(|x-x^{\prime}|)$ is the density correlation function of
uniform electron gas at temperature $\beta$ and mean density $n_{0}$. The
zeroth order term

$Z_{v}(\beta,0]\Big{|}_{v(x)=0}=Tre^{-\beta\hat{H}^{(}0)}=Z_{N}(\beta)$ (69)

is the canonical partition function of a uniform electron gas in absence of
external potential.
Equation (63) tells that

$\delta\alpha(x)\equiv\lambda\alpha^{(1)}(x)\,.$ (70)

Substitute (67), (68), and (69), in equation (66) to find that

$\displaystyle logZ_{v}(\beta;\alpha]\approx$ $\displaystyle
logZ_{N}(\beta)-\lambda\int d^{3}xn_{0}\alpha^{(1)}(x)$ (71)
$\displaystyle+\frac{\lambda^{2}}{2}\int
d^{3}xd^{3}x^{\prime}n_{0}^{(2)}(|x-x^{\prime}|)\alpha^{(1)}(x)\alpha^{(1)}(x^{\prime})+\dots\;.$

Substitute (71) in (65) to get

$\displaystyle\Omega_{v}(\beta;n]\approx$
$\displaystyle\frac{1}{\beta}logZ_{N}(\beta)$ (72)
$\displaystyle+\lambda^{2}\Bigg{[}-\frac{1}{\beta}\int
d^{3}x\alpha^{(1)}(x)n^{(1)}(x)\Bigg{]}+\lambda^{2}\Bigg{[}\frac{1}{2\beta}\int
d^{3}xd^{3}x^{\prime}\alpha^{(1)}(x)\alpha^{(1)}(x^{\prime})n_{0}^{(2)}(|x-x^{\prime}|)\Bigg{]}\;.$

Now that we are done with the perturbative calculations, we can take back
$\lambda\rightarrow 1$ and consequently use $\alpha^{(1)}(x)=\alpha(x)$ and
$n^{(1)}(x)\approx n(x)-n_{0}$, to have

$\displaystyle\Omega_{v}(\beta;n]\approx$
$\displaystyle\frac{1}{\beta}logZ_{N}(\beta)$ (73)
$\displaystyle+\Bigg{[}-\frac{1}{\beta}\int
d^{3}x\alpha(x)[n(x)-n_{0}]\Bigg{]}+\Bigg{[}\frac{1}{2\beta}\int
d^{3}xd^{3}x^{\prime}\alpha(x)\alpha(x^{\prime})n_{0}^{(2)}(|x-x^{\prime}|)\Bigg{]}\;.$

Now according to equation 43 we must minimize $\Omega_{v}(\beta;n]$ subject to
the condition $\int d^{3}xn(x)=N$. So we define the chemical potential
$\mu\equiv-\frac{\alpha_{eq}}{\beta}$ to find an approximation for the DFT
variational principle which is solely written in terms of the thermodynamic
properties of a uniform electron gas

$\displaystyle\frac{\delta}{\delta n(x^{\prime\prime})}\Bigg{[}$
$\displaystyle\int d^{3}x\mu(n(x))[n(x)-n_{0}]$ (74)
$\displaystyle+\frac{1}{2}\int
d^{3}xd^{3}x^{\prime}\mu(n(x))\mu(n(x^{\prime}))n_{0}^{(2)}(|x-x^{\prime}|)-\mu\int
d^{3}xn(x)\Bigg{]}_{n(x)=n_{eq}(x)}=0\;.$

Once again, recall equation (59) and approximate the intrinsic density
potential with chemical potential of a uniform electron gas
$\mu_{in}(x;n]\approx\mu(n(x))$ to find that

$\displaystyle\frac{\delta}{\delta n(x^{\prime\prime})}\Bigg{[}$
$\displaystyle\int d^{3}xv(x)n(x)+\int d^{3}x\mu(n(x))n_{0}$ (75)
$\displaystyle-\frac{1}{2}\int
d^{3}xd^{3}x^{\prime}\mu(n(x))\mu(n(x^{\prime}))n_{0}^{(2)}(|x-x^{\prime}|)\Bigg{]}_{n(x)=n_{eq}(x)}=0\;.$

This means that we reduced analysis of an inhomogeneous gas to that of a
uniform gas for the case of almost constant density.

###  4.2 Kohn-Sham Model

Another approach to approximate $\Omega_{v}$ is the Kohn-Sham Model [35] for
slowly varying potential, the condition of slowly varying potential is of
course one degree loser than the almost constant density, in a sense that
although the density does not change dramatically from one volume element to
the next, but the differences may accumulate to make a considerable difference
of density between far away coordinates.
The novelty of the Kohn-Sham model is that instead of calculating approximate
value of $\Omega_{v}$, they showed that for a slowly varying density, the DFT
variational principle gives an equilibrium density equal to that of a non-
interacting electron gas in the presence of an effective potential. This means
that the Kohn-Sham model simplifies an interacting many-body Schrödinger
equation, to that of a single particle in the presence of an effective
potential.
Let us see how the Kohn-Sham model fits in our framework. First split
$F(\beta;n]$ into the Coulomb interaction and the rest: regarding the long
range nature of the Coulomb interaction we can approximate it with the
classical one:

$F(\beta;n]=\int
d^{3}xd^{3}x^{\prime}\frac{n(x)n(x^{\prime})}{|x-x^{\prime}|}+G(\beta;n]\;,$
(76)

where $G(\beta;n]$ is independent of $v(x)$ just like $F(\beta;n]$. Compare
equation (48) with (76) to find

$G(\beta;n]=\langle\hat{K}\rangle-S_{v}(E;n]\;.$ (77)

Again, split $G(\beta;n]$ into that of a non-interacting electron gas and the
rest

$G(\beta;n]=G_{s}(\beta;n]+F_{xc}(\beta;n]\;,$ (78)

where

$G_{s}(\beta;n]=\langle\hat{K}\rangle_{0}-S_{0,v}(E;n]\;,$ (79)

and the non-interacting trial kinetic energy $\langle\hat{K}\rangle_{0}$, and
non-interacting trial entropy $S_{0,v}$ are found by

$\displaystyle\langle.\rangle_{0}=Tr\hat{\rho}_{0,n}(.)$ (80)
$\displaystyle\hat{\rho}_{0,n}=\frac{e^{-\beta\hat{H}_{0}-\int\alpha(x)\hat{n}(x)}}{Tre^{-\beta\hat{H}_{0}-\int\alpha(x)\hat{n}(x)}}$
(81) $\displaystyle\hat{H}_{0}=\hat{K}+\hat{V}$ (82) $\displaystyle
S_{0,v}=-Tr\hat{\rho}_{0,n}log\hat{\rho}_{0,n}$ (83)

where $\hat{\rho}_{0,n}$ is the non-interacting trial density matrix and
$\hat{H}_{0}$ the non-interacting Hamiltonian operator.
Substitute equations (78) and (76) in (60) to have an almost exact
approximation for the functional $\Omega_{v}$

$\Omega_{v}(\beta;n]=\int d^{3}xv(x)n(x)+\frac{1}{2}\int
d^{3}xd^{3}x^{\prime}\frac{n(x)n(x^{\prime})}{|x-x^{\prime}|}+F_{xc}(\beta;n]\;.$
(84)

The expression (84) is almost exact and not exact in a sense that the Coulomb
potential is approximated by its classical counterpart, otherwise the
expression $F_{xc}(\beta;n]$ is exactly the difference between the intrinsic
free energy $F(\beta;n]$ and the intrinsic free energy of a non-interacting
electron gas with the same density $F_{0}(\beta;n]$

$F_{xc}(\beta;n]=F(\beta;n]-F_{0}(\beta;n]\;.$ (85)

The non-interacting intrinsic free energy is

$F_{0}(\beta;n]=\int d^{3}xf_{0}(n(x))n(x)\;,$ (86)

where $f_{0}(n(x))$ is the intrinsic free energy per particle for a uniform
non-interacting electron gas with density $n=n(x)$. Although, it is not
guaranteed that one can write a $F(\beta;n]$ in such a form, indeed one can
not, because exchange correlation energy is non-local by definition, but it is
a reasonable approximation for a thermal system to assume that entanglement of
particles only appear in short distances, so that we can approximate the total
exchange correlation energy with sum of that of volume elements

$F_{xc}(\beta;n]=\int d^{3}xf_{xc}(n(x))n(x)\;,$ (87)

where $f_{xc}(n(x))$ is the exchange correlation free energy per particle for
a uniform electron gas with density $n=n(x)$.
Combine equations (87), (84), (78), and (58) to get

$0=\phi(x)+\frac{\delta G_{s}(\beta;n]}{\delta n(x)}+\mu_{xc}(n(x))-\mu\;,$
(88)

where

$\phi(x)\equiv v(x)+\int
d^{3}x^{\prime}\frac{n(x^{\prime})}{|x-x^{\prime}|}\;,$ (89)

and

$\mu_{xc}(n(x))=\frac{d[n(x)f_{xc}(n(x))]}{d[n(x)]}\;.$ (90)

Equation (88) is identical to the corresponding equation for a system of non-
interacting electrons in the presence of the effective potential
$\phi(x)+\mu_{xc}(x)$. Its solution is therefore determined by the following
system of equations:

$\big{[}-\frac{1}{2}\nabla^{2}+\phi(x)+\mu_{xc}(n(x))\big{]}\psi_{i}=\epsilon_{i}\psi_{i}\;,$
(91)
$n(x)=\sum_{i=1}^{N}\frac{|\psi_{i}(x)|^{2}}{e^{\beta(\epsilon_{i}-\mu)}+1}\;.$
(92)

##  5 Final Remarks

In this paper, we reformulated the quantum DFT from the MaxEnt point of view.
We showed that the variational principle of DFT is a special case of quantum
entropic inference, where the method is incorporated to treat an inhomogeneous
electron gas. we clarified that the practical advantage of DFT is that it
enables one to analyze an inhomogeneous electron gas by just knowing
characteristics of the homogeneous gas. As a matter of fact we did not even
mention the second quantization in the theoretical framework of DFT. Although
the use of the grand canonical formalism for approximation of $F_{xc}$ (or
equivalently of $\mu_{xc}$) is intuitively justifiable which means that we
have considered the rest of the system to behave like a thermal and particle
bath for each volume element.
There are three different approximations in the theory which finally lead to
the famous Kohn-Sham equations (91) and (92): i) the Coulomb interaction is
treated classically ii) the volume elements of the system are assumed to
contribute additively to the exchange correlation part of the intrinsic free
energy iii) the expected number of particles in a volume element is negligible
compared to the total number of particles.
The future research can be pursued in three different directions i) to
approach the dynamic DFT [36] by generalizing the idea to non-equilibrium
systems following the theory of maximum caliber [37] ii) to revisit the
objective of section 2.2 and assign an optimal probability distribution to the
family of trial density matrices instead of picking the best one in order to
find a formulation for the Weighted DFT [38, 39] iii) to choose parameters
other than the density of particles to construct other families of
parametrized density matrices to tackle upper hand equilibrium or stationary
theories such as Ginzburg-Landau Theory and Mean Field Theory.

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