In the original form of density functional theory (DFT),
suggested by Thomas and Fermi (TF)
and made formally exact by Hohenberg and Kohn,
the energy of a manybody quantum system is minimized directly
as a functional of the density.
Its modern incarnation uses the KohnSham (KS) scheme,
which employs the orbitals of a fictitious noninteracting system,
such that only a small fraction of the total energy need be approximated.
With the goal to simulate ever larger molecular and
solid state systems, interest is rapidly reviving
in finding an orbitalfree approach to DFT.
The major bottleneck in modern calculations is
the solution of the KS equations, which can be avoided with a
pure density functional for the kinetic energy
of noninteracting fermions, T_{s}.
Despite decades of effort, no generally applicable approximation for T_{s}
has been found.
We consider the potential as a more
natural variable to use in deriving approximations to quantum systems.
In exact potential functional theory (PFT)
we obtain the groundstate (gs) energy from
where
is the kinetic energy operator,
the interelectronic repulsion,
the onebody potential, and Ψ are
Nparticle wavefunctions.
Defining
with Ψ_{ν} denoting the gs wavefunction of potential ν(r),
the exact gs energy is given by
n denoting the gs density.
In practice, this requires two separate approximations,
F^{A}[ν] and
n^{A}[ν]
yielding a direct approximation
In a recent letter ^{[1]}
we go beyond those results
by considering explicit potential functional approximations
to interacting and noninteracting systems of electrons.
We show that (i) the universal functional, F[ν], is determined entirely
from knowledge of the density as a functional of the potential,
such that only one approximation is required, namely
n^{A}[ν],
(ii) the variational principle imposes a condition relating energy and density
approximations,
(iii) a simple condition guarantees satisfaction of the variational
principle,
(iv) with an orbitalfree approximation to the noninteracting density as a functional
of the potential, T_{s} is automatically determined,
i.e., there is no need for a separate approximation,
and (v) satisfaction of the variational principle improves accuracy of
approximations.
We deduce
an approximation to F from any n^{A}[ν](r)
by introducing a coupling constant in the onebody potential,
ν^{λ}(r) =
(1λ)ν_{0}(r) +λν(r),
where ν_{0}(r) is some reference potential.
Via the HellmannFeynman theorem (and for the choice ν_{0}=0)
we obtain
This establishes that the universal functional is determined
solely by the knowledge of the density as functional
of the potential.
Moreover, insertion of n^{A}(r) on the right
defines an associated
approximate F^{cc}[n^{A}[ν]], where cc denotes coupling constant.
Furthermore, consider the variational principle.
In PFT, this yields^{[2]}
with a possibly different value from
E_{ν}^{A,dir}
for a given pair of approximations {F
^{A}, n
^{A}}.
We also show that this minimization over trial potentials
can be avoided, such that
F^{cc}[n^{A}[ν]]
does yield
E_{ν}^{A,var}, if, and only if,
This condition is satisfied by the exact response function,
but not necessarily by an approximation n^{A}[ν];
furthermore, enforcing Eq. (6) on n^{A}[ν] improves upon the accuracy.
Of much more practical use is the
application of these results to the noninteracting electrons in a
KS potential, mimicking interelectronic interactions via
the Hartree (ν_{H})
and exchangecorrelation (ν_{XC}) contribution:
For a given
which determines
this equation can be easily solved by standard iteration techniques,
bypassing the need to solve the KS equations via
a given
where the subscript S denotes singleparticle quantities.
However, once selfconsistency is achieved,
we need
T_{s} to extract the total energy
of the interacting electronic system.
All our derivations apply equally to
the noninteracting problem, so we deduce:
which is the analog of Eq. (4) for noninteracting
electrons in the external potential ν(r)
(which is called ν_{s}(r)
when it is the KS potential of some interacting system).
This defines a kinetic energy approximation determined solely
by the density approximation,
T_{s}^{cc}[n_{s}^{A}[v]].
This result eliminates the need for constructing
approximations to the noninteracting kinetic energy,
T_{s}.
To illustrate the power of these results, we
consider a simple example of noninteracting,
spinless fermions in a onedimensional box with potential ν(x).
In Fig. (1) we plot exact and approximate kinetic energy densities (KED)
demonstrating that our couplingconstant approach from Eq.(8)
evaluated on n_{s}^{A}[ν] from Ref.
[3]
yields both locally and globally more accurate results,
and required no separate approximation for the KED.
 Figure 1: Top panel: Exact (ex) KED of the couplingconstant (cc) and the direct (dir) approach (black and blue) with the corresponding semiclassical (sc) approximations (red and green) for one particle in ν(x) = 5 sin^{2}(πx), 0 < x < 1 . Bottom panel: Absolute errors.
Potential functional approximations to the density, n_{s}^{A}[ν],
are presently being developed via a systematic asymptotic expansion
in terms of the potential, which has already
been found in simple cases
^{[3],
[4]}.
The leading terms are local approximations of the TF type,
and the leading corrections, which are relatively simple,
nonlocal functionals of the potential, greatly improve
over the accuracy of local approximations in a systematic
and understandable way. However, a practical realization
of the presented approach awaits generalpurpose approximations to
n_{s}^{A}[v](r)
for an arbitrary threedimensional
case. We have shown here that it no longer awaits the corresponding kinetic
energy approximations.
References
 [1]
 A. Cangi, D. Lee, P. Elliott, K. Burke and E. K. U. Gross,
Phys. Rev. Lett. 106, 236404 (2011).
 [2]
 E.K.U. Gross and C.R. Proetto,
J. Chem. Theory Comput. 5, 844 (2009).
 [3]
 P. Elliott, D. Lee, A. Cangi, and K. Burke,
Phys. Rev. Lett. 100, 256406 (2008).
 [4]
 A. Cangi, D. Lee, P. Elliott, and K. Burke,
Phys. Rev. B 81, 235128 (2010).
To top
Most formulations of spin density functional theory (SDFT) restrict the magnetization
vector field to have global collinearity. Nevertheless, there exists a wealth of strong
noncollinearity in nature, for example molecular magnets, spinspirals, spinglasses
and all magnets at finite temperatures.
The local spin density approximation (LSDA) can be extended to these noncollinear
cases^{
[1]
} but this extension has the undesirable property of having the exchangecorrelation
(XC) field parallel to the magnetization density at each point in space. When used in conjunction with the equation of motion for the spin magnetization in the absence of spin currents
and external fields^{
[2],
[3]
} , this local collinearity eliminates the torsional term, resulting in no
time evolution. This severe shortcoming of LSDA, where the physical prediction is qualitatively wrong, opens up an important new direction for the development of functionals where
this time evolution is correctly described.
Towards this goal we have taken two approaches. The first extends the KohnSham optimized effective potential (OEP) method to the noncollinear case^{
[3],
[4]
} , while the second is to develop a new XC functional^{[5]}
based on spinspiral waves which goes beyond the LSDA by using gradients of the magnetization density. In both cases, we require the KohnSham (KS) equation for twocomponent spinors Φ_{i}
, which has the form of a Pauli equation. For noninteracting electrons moving in an
effective scalar potential v_{s} and a magnetic vector field B_{s} it reads as (atomic units are used
throughout):
This equation can be derived by minimizing the total
energy which, in SDFT, is given as a functional of the density
and the magnetization density
 Fig. 1: Fully noncollinear magnetization density and B field obtained using the LSDA and exchangeonly EXX functionals for an unsupported Crmonolayer in Néel state. Arrows indicate the direction and information about the magnitude (in atomic units) is given in the colour bar.
For a given external scalar potential v_{ext} and magnetic field B_{ext} this total energy reads
where
is the Hartree energy. The XC potential and XC magnetic field are given by
respectively. The exact functional form of
E_{xc}[ρ,
m]
is unknown and has to be approximated in practice.
Assuming that the densities [ρ,m] are noninteracting
(v,B)representable
one may, equivalently, minimize the totalenergy functional (2) over
the effective scalar potential and magnetic field. Thus the conditions
must be satisfied.
If we were to use a functional that depends explicitly on spinor valued wavefunctions
we can stay within a single global reference frame, in contrast to the case where functionals
originally designed for collinear magnetism are used in a noncollinear context by introducing
a local reference frame at each point in space. The most commonly used orbital functional is
the EXX energy given by
where the label occ indicates that the summation runs only over occupied states.
From the conditions of Eq. (4)
and using the functional form given in Eq. (5)
(although generalization to other orbital functionals is straightforward), the
OEP equations for the noncollinear systems were derived^{[3, 4]}.
In order to explore the impact of treating noncollinear magnetism in the way outlined
above and at the same time to ensure that our numerical analysis be as accurate as
possible, we implemented the OEP equations for the fully noncollinear case within
the fullpotential linearized augmented plane wave (FPLAPW) method implemented
within the ELK code^{[6]
}
This method is then applied to study the noncollinear spin
magnetism in an unsupported Cr (111) monolayer. In Fig.1 are
shown the
magnetization density and B field calculated
using both the LSDA and the OEP method.
A striking feature of the OEP B field is that, unlike its LSDA counterpart,
it is not locally parallel to the magnetization density and this will produce manifestly
different spindynamics. This is because the equation of motion for the spin
magnetization reads
where J_{s} is the spin current and γ the
gyromagetic ratio. In the timeindependent LSDA and conventional GGA,
m ( r )
and
B_{xc}
( r) are locally
collinear, as is clear from Fig.
1, and therefore m(r) ×
B_{xc}(r) vanishes.
This also holds true in the adiabatic approximation of time dependent SDFT which,
by Eq. (6), implies that these functionals cannot properly
describe the dynamics of the
spin magnetization. In contrast, already at the static level, for the EXX functional
m(r) × B_{x}(r) does not vanish. In fact, in
the ground state of a noncollinear ferromagnet without external magnetic field,
m(r) ×
B_{xc}
(r) exactly cancels the divergence of the spin
current,∇·J_{s}, i.e. these terms
are equally important, and it is essential to have a proper description
of m(r) ×
B_{xc
}(r).
These results indicate that a
timedependent generalization of our method could open the way to an abinitio
description of spin dynamics. How well this functional really performs in describing
the spin dynamics remains a question for future investigations.
To top
 Fig. 2: Illustration of spin spiral waves along one spatial coordinate for two different choices of wavevector q=k _{1,2}. The angle Θ is determined by s while the angle φ=q · r. Image reproduced from the thesis of F. Eich.
Unfortunately OEP calculations are computationally demanding, leading us to
develop^{[5]}
a new semilocal XC functional for noncollinear magnetism.
This is analogous to the case in DFT, where going from the LDA to the generalized
gradient approximations (GGAs) led to additional accuracy. To derive a new functional,
our starting point was the spinspiral wave (SSW) phase of the electron gas, which
has magnetization:
where s is the spin projection on the zaxis and q is the SSW wavevector, it is illustrated in Fig. 2. Using quantum montecarlo or the
random phase approximation (RPA), the XC energy for such a state
can be calculated and parameterized. To then construct a
functional, we will use gradients of the magnetization to
create effective s and q
variables which can then be inserted into our parameterization.
One particular choice is to use quantities:
 Fig. 3: The magnitude (heat map) and direction (arrows) of the magnetization density (left) and XC magnetic field (right) of the Cr monolayer.
giving effective
s and
q:
The functional can then be written as:
where
is the noncollinear LSDA functional discussed earlier.
S_{xc}
is the enhancement factor based on the parameterization of the SSW energies,
and is designed so that the total functional reduces to LSDA in the appropriate limit.
It can also be shown that this functional respects the zerotorque theorem.
To test this functional, it was implemented in ELK^{
[6]}
and applied to the same case
studied by OEP EXX above. In Fig. 3, we plot the
selfconsistent magnetization density
m(r)
and the XC magnetic field
B_{XC}(r)
for a 2d slice in the plane of the Cr monolayer.
To demonstrate that this functional does lead to locally noncollinear terms,
in Fig.4 we plot the zcomponent of
m(r) ×
B_{XC}(r)
in the
region around an atom of Chromium. Unlike the LSDA case and similar to the
OEP case, this quantity is nonzero implying nonzero spin currents in the groundstate.
For TDDFT this functional shows much promise as it contains nonzero local torques
on the magnetization density (missing in LSDA) while still a semilocal functional and
hence computationally less demanding that OEP.
 Fig. 4: Heat map of the zcomponent of
References
 [1]
 J. Kuebler, K.H. Hoeck, J. Sticht and A. R. Williams,
J. Phys. F18, 469 (1993).
 [2]
 K. Capelle, G. Vignale and B. L. Gyoerffy,
Phys. Rev. Lett.87, 206403 (2001).
 [3]
 S. Sharma, J. K. Dewhurst, C. AmbroschDraxl, S. Kurth,
N. Helbig, S. Pittalis, S. Shallcross,
L. Nordstroem and E.K.U. Gross
Phys. Rev. Lett.98, 196405 (2007)
 [4]
 S. Sharma, S. Pittalis, S. Kurth, S. Shallcross,
J. K. Dewhurst and E.K.U. Gross
Phys. Rev. B76, 100401 (Rapid Comm.) (2007)
 [5]
 F. G. Eich and E. K. U. Gross, arXiv:1212.3658 [condmat.strel] (2013).
 [6]
 http://elk.sourceforge.net
To top
Because of the small mass ratio between electrons and nuclei,
standard electronic structure calculations treat the former
as being in their ground state, but routinely account for
the finite temperature of the latter, as in ab initio molecular
dynamics^{[1]}.
But as electronic structure methods are applied in ever more esoteric areas, the need
to account for the finite temperature of electrons increases.
Phenomena where such effects play a role include rapid heating of solids via
strong laser fields^{ [2]},
dynamo effects in giant planets
^{[3]
},
magnetic^{
[4]
}
and superconducting phase transitions
^{
[5],
[6]
}
, shock waves
^{
[7]
}
, warm dense matter
^{
[8]
}
, and hot plasmas
^{
[9]
}.
Within density functional theory, the natural framework for
treating such effects was created by Mermin
^{
[10]
}.
Application of that work to the KohnSham (KS) scheme
at finite temperature also yields a natural approximation:
treat KS electrons at finite temperature, but use groundstate
exchangecorrelation (XC) functionals. This works
well in recent calculations
^{
[7]
,
[8]
}, where inclusion
of such effects is crucial for accurate prediction.
This assumes that finitetemperature effects on
exchangecorrelation
are negligible relative to the KS contributions,
which may not always be true.
In the present work we establish the basic rules that will allow the building of
finitetemperature exchangecorrelation functionals neyond these standard approximations
^{
[11]
}.
Central to the thermodynamic description of manyelectron systems
is the grandcanonical potential, defined as the statistical
average of the grandcanonical operator
where Ĥ, Ŝ, ^N, τ and μ are the Hamiltonian, entropy, and particlenumber operators,
temperature and chemical potential, respectively. In detail,
where ^T and ^V_{ee} are the kinetic energy and the
Coulomb electronelectron interaction operators, and ^V represents
an external scalar potential v(r). The entropy operator is given by
where k is the Boltzmann constant and
is a statistical operator, with  Ψ_{
N,i}⟩ and w_{N,i} being orthonormal
Nparticle states and statistical weights, respectively, with the latter satisfying the
(normalization) condition Σ_{N,i}w_{N,i} = 1.
The statistical average of an operator ^A is obtained as
To create a DFT at finite temperature, Mermin
^{
[10]
}
rewrites this as (in modern parlance)
where the minimizing n(r) is the equilibrium
density n^{0}(r), and
is the finitetemperature analog of the universal HohenbergKohn functional,
defined through a constrained search
^{
[15]}.
This depends only on τ and not on μ.
We denote
as the minimizing statistical operator in Eq. (4), and define the density functionals:
i.e., each density functional is the trace of its operator
over the minimizing ^Γ for the given τ and density.
Because it arises so often in this work, we have defined the
"kentropy" K^{τ}[n].
Next consider a system of noninteracting
electrons at the same temperature τ
and denote its onebody potential as v_{s}(r).
All the previous considerations apply, and we choose
v_{S}(
r)
to make its density match that of the interacting problem. This defines the
KS system at finite temperature. The corresponding functionals will be labeled
with the subscritp
s.
We have found that the exchange (x) and
correlation (c) parts of the grandcanonical potential,
as well as the kentropy and the potential energy (U)
are bound to fulfil the exact inequalities
^{
[11]
}
:
and no approximation should violate these basic rules.
Some of the most important results in groundstate DFT
come from uniform scaling of the coordinates
^{
[17]
}.
In the following considerations, when we refer explicitly to wavefunctions, we shall restrict to
wavefunctions having finite norm on their entire domain of definition.
Under normpreserving homogeneous scaling of the coordinate
r→γr,
with γ > 0, to the scaled wave function
^{
[17]
}.
corresponds the scaled density n_{γ}(r) =
γ^{3}n(γr).
A simple 1D sketch of the scaling procedure is given in this figure:
Writing Ψ^{γ}(r_{1},....r_{N}) =
⟨r,...,r_{N}Ψ^{γ}⟩
in terms of the (representationfree) element Ψ^{γ}⟩
of Hilbert space, the scaled statistical operator is defined as^{
[11]
}
where the statistical weights are hold fixed,
i.e., the scaling only acts on the states.
With the above definition, the statistical
average of an operator whose purestate expectation value
scales homogeneously , scales homogeneously
^{
[17]
}
as well.
In particular, we have:
The scaling behavior of the density functionals is, however, more subtle.
First consider the noninteracting functionals in some detail.
We observe that
For noninteracting electrons, the statistical operator at
a given temperature that is the minimizer for a given scaled
density is simply the scaled statistical operator, but at a quadratically scaled
temperature, an effect that is obviously absent in the groundstate
theory.
Next, we consider the interacting case.
The analysis of the scaling relations allows to derive a set
of inequalities that provide tight constraints on the functionals,
and are used in nonempirical functional construction in the ground state.
We obtain the following:
which are reversed if
γ < 1.
Lastly, we consider the adiabatic coupling constant for finite temperature,
its relationship to scaling,
and derive the adiabatic connection formula.
Define
with
being the corresponding minimizing
.
Of course, noninteracting functionals are not affected by a coupling constant modification.
Eq. (10) implies that the exchange and Hartree density functionals have a linear dependence
on λ. Employing the minimization property of Eq. (13) and the HellmannFeynman
theorem, we find^{ [11]
}
Eq.(14) is the finitetemperature adiabatic connection formula, whose zerotemperature limit played a central role in groundstate DFT.
(Eq. 5), and the scaling inequalities can be
combined, analogously to Ref.
[17]
to show that
U_{xc}^{τ}[n](λ) is
monotonically decreasing in λ , as shown in the schematic drawing below.
 Fig. 1. Geometrical interpretation of the adiabatic connection formula. The shaded area between the curve and the λ axis represents Ω_{xc}^{τ}[n]. All magnitudes are negative, in agreement with the inequalities in Eq. (5).
So far, all results presented have been exact. To see them
in practice, consider the finitetemperature local density approximation (LDA) to
Ω_{xc}^{τ}[n]
where ω_{xc}^{unifτ}(n)
is the XC grand canonical potential density of
a uniform electron gas of density n.
Because a uniform electron gas is a quantum mechanical system, its energies
satisfy all our conditions, guaranteeing
by construction that LDA satisfies all the
exact conditions listed here. In the Jacob's ladder of functional construction^{
[13]
},
more sophisticated approximations should also
satisfy these conditions.
To give one simple example of the usefullness of our results, Eq. (10) implies
is a dimensionless measure of the local
temperature. Thus the largest fractional deviations from groundstate results
should occur (in LDA) in regions of lowest density, but these contribute less
in absolute terms.
In summary,
there is a present lack of approximate density functionals for finite temperature.
We have derived many
basic relations needed to construct such approximations, and expect future
approximations to either build these in, or be tested against them.
In principle, such approximations should already be implemented in
hightemperature DFT calculations, at least at the LDA level, as a check
that XC corrections due to finite temperature do not alter calculated results.
If they do, then more accurate approximations than LDA will be needed to
account for them.
To top
References
 [1]
 R. Car and M. Parrinello,
Phys. Rev. Lett. 55, 2471 (1985).
 [2]
 J. Gavnholt,
A. Rubio,
T. Olsen,
K. S. Thygesen,
and
J. Schiøtz,
Phys. Rev. B 79,
195405 (2009).
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 R. Redmer,
T. R. Mattsson,
N. Nettelmann,
and M. French,
Icarus 21, 798
(2011).
 [4]
 N. M. Rosengaard
and
B. Johansson,
Phys. Rev. B 55,
14975 (1997).
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 G. Profeta,
C. Franchini,
N. N. Lathiotakis,
A. Floris,
A. Sanna,
M. A. L. Marques,
M. Lüders,
S. Massidda,
E. K. U. Gross,
and
A. Continenza,
Phys. Rev. Lett. 96,
047003 (2006).
 [6]
 P. Cudazzo,
G. Profeta,
A. Sanna,
A. Floris,
A. Continenza,
S. Massidda, and
E. K. U. Gross,
Phys. Rev. Lett. 100,
257001 (2008).
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 S. Root,
R. J. Magyar,
J. H. Carpenter,
D. L. Hanson,
and T. R.
Mattsson, Phys. Rev. Lett.
105, 085501
(2010).
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 A. Kietzmann,
R. Redmer,
M. P. Desjarlais,
and T. R.
Mattsson, Phys. Rev. Lett.
101, 070401
(2008).
 [9]
 M. W. C. Dharmawardana
and M. S.
Murillo, Phys. Rev. E
77, 026401
(2008).
 [10]
 N. D. Mermin,
Phys. Rev. 137,
A1441 (1965).
 [11]
 S. Pittalis,
C. R. Proetto,
A. Floris,
A. Sanna,
C. Bersier,
K. Burke, and
E. K. U. Gross,
Phys. Rev. Lett. **** ,
****** (2011).
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 R. G. Dandrea,
N. W. Ashcroft,
and A. E.
Carlsson, Phys. Rev. B
34, 2097 (1986).
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 C. Fiolhais,
F. Nogueira, and
M. A. Marques, eds.,
A Primer in Density Functional Theory
(Springer, 2003).
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 D. C. Langreth and
J. P. Perdew,
Solid State Commun. 31,
567 (1979).
 [15]
 R. Parr and
X. Yang,
Densityfunctional Theory of Atoms and Molecules
(Oxford University Press, 1989).
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 M. Greiner,
P. Carrier, and
A. Görling,
Phys. Rev. B 81,
155119 (2010).
 [17]
 M. Levy and
J. P. Perdew,
Phys. Rev. A 32,
2010 (1985).
To top
The adiabatic connection fluctuation dissipation (ACFD) formula within timedependent
density functional theory (TDDFT) is an exact expression for the correlation energy within
DFT. The first approximation, obtained by setting the exchangecorrelation (XC) kernel to
zero, yields the socalled random phase approximation (RPA) which incorporates longrange
correlation effects such as the van der Waals forces. It also removes static correlation errors
or fractional spin errors when breaking chemical bonds.
For a unique solution the RPA should be solved selfconsistently, which involves
the construction of the corresponding RPA correlation potential. In this work we have
calculated selfconsistent RPA potentials for diatomic molecules and investigated the RPA
correlation potentials at different interatomic distances. By increasing the interatomic
distance correlation effects become more important and the exact correlation potential is
known to exhibit strong correlation features such as a sharp step and a peak at the
bondmidpoint.
Within the ACFD framework the exact correlation energy is written as
where
χ_{s} is the noninteracting
KS density response function and
χ_{λ}
is the scaled interacting density response function. The scaling refers to a system with a
linearly scaled Coulomb interaction λv(
r,
r') plus a fictitious potential
which keeps the density fixed as λ is changed. The parameter λ runs between
0 (noninteracting KS case) and 1 (fully interacting case). We have used the short hand notation
for any twopoint functions
f and
g.
Within TDDFT the function
χ^{λ} reads
The scaled XC kernel
f_{xc}^{λ}
is defined as the functional derivative of the scaled XC potential
v_{xc}^{λ} with respect to
the density n, evaluated at the ground state density.
In the RPA, f_{xc}^{λ} = 0
and thus corresponds to the simplest approximation within the ACFD formalism. Within
RPA the λintegral in Eq. (1) can be evaluated analytically with the result
Diagrammatically Eq. (3) is equal to an infinite summation of ringdiagrams.
The RPA correlation potential v_{c}
can be obtained as the functional derivative of Eq. 3
with respect to the density. If we let V_{s} signify the total KS potential,
G_{s} the noninteracting KS Green function and χ_{s}=
iG_{s}G_{s}, the functional derivative
can be obtained via the chain rule
The result is the wellknown linearized ShamSchlüter (LSS) equation
Here, we have used the notation (
r_{1},t
_{1}) = 1 etc. and introduced
Λ(3,2;1)=
iG
_{s}(3,1)G
_{s}(1,2).
The correlation part of the selfenergy Σ
_{c} in
the RPA is given by
where
In Fig. 1 we show the RPA correlation potential of 1D LiH
(Z_{Li}=3.6, Z_{H
}=1.2). The
solid, dotted and dashed curves represent the correlation potentials for 2, 3, and 8
Bohr interatomic distances, respectively. The build up of the peak at the bond midpoint
is apparent. However, a step due to the derivative discontinuity, as is present in the
exact KS correlation potential, is not observed.
 Fig. 1: The correlation potential along the bond with the bond midpoint at zero. The system is 1D LiH with a soft coulomb potential. The Li atom is located at 1, 1.5, and 4 Bohr, respectively. The H atom is located at 1, 1.5, and 4 Bohr, respectively.
Fig. 2 shows the correlation
potential of the three dimensional LiH for bond distances 3 (solid), 6 (dotted), and 8
Bohr (dashed). The same features as in the 1D case are also found in the results for
the 3D correlation potentials. In the region of the Li atom (10 to 0) the potential qualitatively
resembles that of the 1D system in Fig. 1. We see a well with a peak close to the
nucleus. Also in the region of the H atom (0 to 10) figures 1 and 2 show the same structure.
A well emerges with increasing bond distance.
The difference between 1D and 3D is found only at the bond midpoint.
A peak emerges only for the 1D system (Fig. 1). In contrast, the 3D system (Fig. 2)
exhibits a peak at zero only for small and intermediate bond distances (3 and 5 Bohr).
The orbital and potential basis functions are simply not diffuse enough to extend to the
bond midpoint.
 Fig. 2: The correlation potential of LiH along the bond axis. The bond midpoint is at zero. The Li atom is located at 1.5, 2.5, and 4 Bohr, respectively. The H atom is located at 1.5, 2.5, and 4 Bohr, respectively.
We have further analyzed the RPA functional in the context of fractional charge and spin by calculating the total energy at different particle numbers. Fig. 3 shows the total energy as a function of the number of electrons at the H atom. The number of electrons at the Li atom will then be four minus the x value. The exact functional (black) has a minimum at 1.0, because it dissociates LiH into a neutral H atom and a neutral Li atom.
 Fig. 3: The total energy of dissociated LiH as a function of number of electrons at the H atom for EXX (blue), RPA (red), and exact (black).
In Fig. 3 we see that EXX (blue) and RPA (red) do not dissociate LiH into the neutral atoms.
In both cases there is a surplus of electronic charge at the H atom. However, in the case
of RPA the surplus (0.16 electrons) is much smaller than in the case of EXX (0.4 electrons).
The large improvement may be related to the peak that is present in the RPA correlation potential.
The total energy in the dissociation limit is shown to be largely improved in the RPA showing
only a small fractional spin error.
References
 [1]

M. Hellgren, D.R. Rohr, E.K.U. Gross, Journal of Chemical Physics 136, 034106 (2012).
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Density functional theory (DFT) owes its great success to the fact that the highly intricate manybody wave function is replaced by the much simpler electronic density. In addition, the density is calculated
from a system of noninteracting electrons known as the KohnSham (KS) system. The effective KS potential is composed of the external, the Hartree and the socalled exchangecorrelation (XC) potential v_{xc}, where the latter is given by the functional derivative of the XC energy E_{xc} with respect to the density, and incorporates all the effects of the Coulomb interaction beyond the Hartree level. The difficulties in using DFT lies in finding good approximations to E_{xc}. Many works have therefore been devoted to study its exact properties. One of the most intriguing features is the existence of derivative discontinuities at integral particle numbers
^{[1]}, a property which turns out to have several important physical implications.
It is not hard to see that a derivative discontinuity in
E_{xc} gives rise to a singularity in v_{xc} in the form of a jump by a constant
Δ_{xc}, i.e., v_{xc}^{+}(r)=v_{xc}
^{}(r)+Δ_{xc}, where ± refers to N→ N_{0}^{±},
and N_{0} is an integer. A classic example where the jump in v_{xc} becomes important is in the dissociation of
closedshell molecules composed of openshell atoms. In order to get the correct
dissociation limit a step in v_{xc} has to develop
over the atom with the lower ionization energy (a similar situation but with closedshell
fragments is depicted in Fig. 1).
 Fig. 1: Left: f_{xc} and v_{xc} of a dissociating HeBe^{2+} system. Right: Discontinuity of the Be^{2+} atom.
With the advent of timedependent DFT (TDDFT) the variation of
v_{xc} with respect to the density, i.e., the XC
kernel f_{xc}(r,r',ω),
has become another central quantity of interest. The reason is the possibility of extracting
excitation energies from the linear density response function, which in TDDFT can be
expressed_{[2]}
in terms of the KS response function χ_{s}, the Coulomb
interaction v,
and f_{xc}
In this work we have evaluated the discontinuities of f_{xc
} in order to increase the understanding of its exact behavior
^{
[3]
}.
The discontinuities of f_{xc} turned out
to be of a more complex nature than those of v_{xc}
,
where the general structure is an r and ωdependent function
g_{xc}(r,ω), which may diverge.
In order to obtain an exact expression for the discontinuity we study the limiting
behavior of E_{xc} as a function of particle
number N around an integer N_{0}.
The constant jump in v_{xc} can then formally be
written as
where
f(r)=∂n(r)/
∂N is the Fukui function.The same idea can then be
used for the static XC kernel and we find an expression for
g_{xc}
After performing a common denominator approximation to the response
functions it is easy to see that the second term exhibits a diverging behavior
which depends on the difference between the ionization energy
I and the
KS affinity
A_{s}. This divergency can be seen in
f_{
xc} of the dissociating system in Fig. 1
where the calculations have been made in the exactexchange approximation,
which is known to have a discontinuity. The right panel of Fig. 1 displays
g_{
xc}, calculated in the same approximation but
defined on an ensemble which allows for fractional charges. A striking similarity can be
seen and we can conclude that the peaked structure in the XC kernel of a dissociating
system is just the discontinuity of the kernel.
A diverging behavior of the kernel is exactly what is needed in order to describe
chargetransfer (CT) excitations. The reason is that Eq. (1) involves only matrix
elements of
f_{xc}
between socalled excitation functions, i.e., products of occupied and unoccupied
KS orbitals. As the distance between the fragments increases these products vanish
exponentially and thus there is no correction to the KS eigenvalue differences unless
the kernel diverges.
The formulation of TDDFT for fractional charges has to allow the particle number to
change in time and we have therefore proposed an ensemble of the form
where
Ψ_{k}(t) is the
kparticle manybody state at
time
t and
α_{k}(t) are given TD probabilities. A RungeGross
theorem can be proved also in this case allowing for an analysis of the same kind as
in the static case. The function
g_{xc}
acquires an ωdependence and Fig. 2 illustrates the difference compared to the
static case showing mainly an increase of the exponential growth. This increase is,
however, crucial for the description of CT excitations, as seen in Fig. 3, where AEXX
uses
f_{x}(ω=0) and TDEXX
uses
f_{x}(ω
_{
KS}).
 Fig. 2: v_{xc} and f_{xc} at ω=0 and ω_{KS} of a dissociating HeBe system.
 Fig. 3: CT excitation energies of the system in Fig. 2 in different approximations.
Apart from playing an important role for the excitation energies also the orbitals are
affected by the discontinuity. Within chemical reactivity the Fukui function
f(r) as defined above is an important reactivity
indicator. The Fukui function can be expressed as ^{
[4]
}
and we see that a discontinuity of
f_{
xc} does not
cancel when determining
f^{+}.
This contribution can be very large for certain molecules as shown in Fig. 4.
 Fig. 4: The Fukui function f^{+} of a 1D Be^{2+} atom.
References
 [1]

J. P. Perdew et al., PPhys. Rev. Lett. 49, 1691 (1982)
 [2]

E. K. U. Gross and W. Kohn, Phys. Rev. Lett. 55, 2850 (1985)
 [3]

M. Hellgren, E.K.U. Gross, Phys. Rev. A 85, 022514 (2012).
 [4]
 M. Hellgren,
E.K.U. Gross, J. Chem. Phys. 136, 114102 (2012)
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