Levy-Lieb constrained-search formulation as a minimization of the
  correlation functional
Levy-Lieb constrained-search formulation as a minimization of
the correlation functional.
Luigi Delle Site∗
Max-Planck-Institute for Polymer Research
Ackermannweg 10, D 55021 Mainz Germany.
Abstract
The constrained-search formulation of Levy and Lieb, which formally defines the exact
Hohenberg-Kohn functional for any N -representable electron density, is here shown to be equivalent
to the minimization of the correlation functional with respect to the N − 1 conditional probability
density, where N is number of electrons of the system. The consequences and implications of such
a result are here analyzed and discussed via a practical example.
PACS numbers: 03.65. w, 71.10. w, 71.15.Mb
∗Electronic address: dellsite@mpip-mainz.mpg.de
http://arxiv.org/abs/0704.0372v1
mailto:dellsite@mpip-mainz.mpg.de
I. INTRODUCTION
The Hohenberg-Kohn (HK) theorem [1] has opened new perspectives to the calculations
of electronic-based properties of condensed matter [2], and, an aspect often disregarded, has
given profound new insights into the general understanding of quantum mechanics. In fact
the 3N -dimensional Schrödinger problem for the ground state of an electronic system:
HNψ(r1, ...r2) = E0ψ(r1, ...r2);HN =
i=1,N
∇2i ) +
i=1,N
v(ri) +
where v(ri) is the external potential,
is the electron-electron Coulomb term, E0
is the energy of the ground state and ψ(r1, ...rN) is the 3N -dimensional antisymmet-
ric ground state wavefunction [3], is transformed into a ”manageable” variational prob-
lem in three dimensions where the central role is played by the electron density: ρ(r) =
ψ∗(r, r2, .....rN)ψ(r, r2, .....rN)dr2....drN , where ΩN−1 is the N − 1 spatial domain.
In explicit terms the variational problems is written as:
E0 =MinρE[ρ] (2)
where
Ω ρ(r)dr = N (Ω being the spatial domain of definition) and E[ρ] = T [ρ] + Vee[ρ] +
Vext[ρ] is the energy functional composed respectively by the kinetic, electron-electron po-
tential and the external potential functional. However in its original formulation the HK
theorem and the related variational problem have got a restricted field of applicability; it
is valid only if the electron density ρ(r) is v-representable, that is if ρ(r) is the density cor-
responding to an antisymmetric wavefunction of the ground-state of an Hamiltonian of the
form of Eq.1. It follows that the correct formulation of the variational problem becomes:
E0 =MinρEv[ρ] (3)
where v refers to the v-representability of ρ(r). As discussed in Ref.[2], there are no general
conditions for a density to be v-representable and this makes the use of the HK theorem and
its associated variational principle not practical. A generalization of the HK theorem which
does not require ρ(r) to be v-representable was found, in parallel, by M.Levy [4] and E.Lieb
[5] and it is usually known as the Levy constrained-search formulation or Levy-Lieb
constrained-search formulation [6]; in this paper we adopt the latter terminology. We
also notice that recently P.Ayers [7] has further clarified this concept and developed an
axiomatic treatment of the Hohenberg-Kohn functional. In the following we briefly describe
the crucial aspects of the abovementioned approach which are relevant for the current work.
The starting point of the theory is the distinction between the ground state wavefunction, ψ,
and a wavefunction ψλ that also integrates to the ground state electron density ρ(r). Since
ψ is the ground state wavefunction, we have:
〈ψλ |HN |ψλ〉 ≥ 〈ψ |HN |ψ〉 = E0. (4)
Taking into account that Vext[ρ] is a functional of ρ only, Eq.4 can be written as:
〈ψλ |T + Vee|ψλ〉 ≥ 〈ψ |T + Vee|ψ〉 (5)
where T and Vee are respectively the kinetic and Coulomb electron-electron operator as
defined in Eq.1. The meaning of Eq.5 is that ψ is the wavefunction that minimize the
kinetic plus the electron-electron repulsion energy and integrates to ρ. It follows that the
initial variational problem of Eq.2 can be transformed in a double hierarchical minimization
procedure which formally allows for searching among all the ρ’s which are N -representable,
i.e. it can be obtained from some antisymmetric wavefunction; this is a condition which
is much weaker and more controllable than the v-representability. In explicit terms such a
formulation is written as:
E0 =Minρ
Minψλ→ρ 〈ψλ |T + Vee|ψλ〉+
v(r)ρ(r)dr
. (6)
The inner minimization is restricted to all wave functions ψλ leading to ρ(r), while the outer
minimization searches over all the ρ’s which integrate to N . The original HK formulation can
then be seen as a part of this new one once its universal functional, F [ρ] = 〈ψ |T + Vee|ψ〉
is written as:
F [ρ] =Minψ→ρ 〈ψ |T + Vee|ψ〉 . (7)
The purpose of this work is to show that F [ρ] can be determined solely by a minimiza-
tion with respect to the N − 1 conditional probability density of the electron correlation
functional. This latter will be shown to be composed by the non local Fisher information
functional [8] and the electron-electron two-particle Coulomb term. The advantage of this
representation is manifold; it further clarifies the connection of electronic properties to the
Fisher theory and shows that the knowledge of such a functional is the crucial ingredi-
ent in density functional based approaches; it also identifies the Weizsacker kinetic term,
∫ |∇ρ(r)|2
dr, as necessary component of the universal functional F [ρ] and, in practical terms,
offers an objective criterion of evaluation of ”approximate” exchange and correlation func-
tional, i.e. among two functionals, the physically better founded is the ”smaller” one. In
order to show the practical aspects of our idea we illustrate a potential application.
II. THE NEW REPRESENTATION
Before writing the functional in the conditional probability density formalism, we need
to define such a quantity. Let us consider a generic fermionic wavefunction ψ(r1, ....rN), for
simplicity we consider a real wavefunction, but the extension to a complex one can be also
done [9]; we do not consider the spin dependence explicitly, however this will not influence
the main conclusions. Then the N -particle probability density is [10, 11]:
Nψ∗(r1, ....rN )ψ(r1, ....rN) = Θ
2(r1, ...., rN) (8)
and this can formally decomposed as [10, 11]:
Θ2(r1, ...., rN) = ρ(r1)f(r2, ......., rN/r1) (9)
where ρ(r1) is the one particle probability density (normalized to N) and f(r2, ......., rN/r1)
is the N − 1 electron conditional (w.r.t. r1) probability density, i.e. the probability density
of finding an N −1 electron configuration, C(r2, ......., rN), for a given fixed value of r1. The
function f satisfies the following properties:
f(r2, ........, rN/r1)dr2.......drN = 1∀r1
(ii) f(r1, ..ri...rj−1, rj+1..., rN/rj) = 0; for i = j; ∀i, j = 1, N
(iii) f(r1, .....ri.., rj...rk−1, rk+1.., rN/rk) = 0; for i = j; ∀i, j 6= k (10)
The property (iii) of Eq.10 assures us that f reflects the fermionic character of an electronic
wavefunction. In fact it says that if any two particles are in the same ’state’ ”r” the
probability of that specific global configuration is zero. In principle, together with condition
(ii), this is a way to mimic the antisymmetric character of the fermionic wavefunction since
for fermions |ψ(r1, ...ri, ...rj, ...rN)|
2 = 0; for i = j, ∀i, j.It must be noticed that condition
(iii) is complementary to (ii). With this formalism the term 〈ψ |T + Vee|ψ〉 can be written
as (see Refs.[9, 10, 12]):
〈ψ |T + Vee|ψ〉 =
|∇ρ(r)|2
, ...., rN/r)|
, ....., rN/r)
....drN
(N − 1)
, ....., rN/r)
|r− r
....drN
dr(11)
where we have identified r1 with r and made use of the property of electron indistinguisha-
bility, thus r could be identified with any of the ri (and the same for r
identified here with
r2) without changing the results; a further consequence is that the Coulomb expression (last
term on the r.h.s.) is written as the sum of N − 1 identical terms for the generic r and r
particles. Using Eq.11 the Levy-Lieb constrained-search formulation can then be written as:
E0 =Minρ
Minf (Γ[f, ρ]) +
|∇ρ(r)|2
v(r)ρ(r)dr
where
Γ[f, ρ] =
, ...., rN/r)|
, ....., rN/r)
....drN
(N − 1)
, ....., rN/r)
|r− r
....drN
dr. (13)
In this way we have transferred the problem from from ψ to f which means that the focus
is now on Γ[f, ρ], i.e., as discussed in Ref.[12], the correlation functional.
III. A PRACTICAL EXAMPLE: THE PARAMETRIC EXPONENTIAL FORM
In our previous work [12], we have proposed an approximation for f based on a two-
particle factorization:
f = ΠNi=2hi(EH(r, ri)) = Π
(N−1)E(r)e−EH (r,ri) (14)
where
e−E(r) =
e−EH (r,ri)dri. (15)
here EH(r, ri) =
ρ(r)ρ(ri)
|r−ri|
, N is the number of particle, and ω the volume corresponding to
one particle. Such an approximation, due to its simplicity, allows us to write an analytic
expression of the Fisher functional which can be used in a straightforward way in numerical
calculations. However it does not satisfy the condition (iii) of Eq.10, and, for this reason,
in order to use it into the Levy-Lieb constrained-search scheme it must be extended. The
expression we propose here is the following:
f(r2, ...rN/r) = Πn=2,Ne
E(r)−γEH (r,rn) × Πi>j 6=1e
−βEH(ri,rj) (16)
with:
e−E(r) =
Πn=2,NΠi>j 6=1e
−γEH (r,rn)−βEH (ri,rj)dr2.....drN (17)
Here γ and β are two free parameters. As it can be easily verified this expression of f
satisfies all the requirements of Eq.10. The meaning of f as expressed in Eq.16 is that the
probability of finding a certain configuration for the N − 1 particles, having fixed particle
r1 = r, depends not only on the fixed particle and its interaction with the N − 1 other
particles as before, but also on the mutual arrangements of the N − 1 particles (it has
also to be kept in mind that using the particle indistinguishability the formalism can be
applied to any ri as a fixed particle). The parameters γ and β express how important
the N − 1 mutual interactions are with respect to the interactions with r. Being now f a
biparametric function, one can use the Levy-Lieb constrained search in our formulation and
find the optimal values for γ and β. This practical example shows two different aspects of
our formulation; basically we have shown that indeed it is possible to build a function f and
actually it can be chosen in a way that its optimal expression can be determined via the
constrained-search formulation. It must be noticed that this form of f is still rather simple
since the spins are not explicitly considered when constructing the function and thus one
cannot distinguish between the exchange and the correlation part of the electron-electron
interaction as it is done in standard Density Functional Theory; as a consequence one should
expect only an overall average description of these two terms which are here incorporated into
the global correlation. However the construction of a more complete expression of f , which
takes care of the effects of the spins, is the subject of current investigation. This emphasizes
once more the merit of the general procedure shown here, that is different expressions of f ,
with different degrees of complexity, can be proposed and their relative validity checked by
the constrained-search procedure.
IV. DISCUSSION AND CONCLUSIONS
As anticipated in the introduction, the consequences of Eqs.12,13 are rather interesting.
The Levy-Lieb variational principle can be reformulated as: The universal functional F [ρ]
is the one with the minimum correlation functional with respect to the electron conditional
probability density. This new interpretation of the HK universal functional tells us that
only an accurate description of the correlation effects, considering the Weizsacker term as
a necessary term, leads to an accurate description of the whole energy functional; such a
criterion is necessary and sufficient. It is obvious that it is necessary; without knowing
Γ[f, ρ], F [ρ] cannot be known; it is sufficient because once Γ[f, ρ] or better f(r2, ...rN/r1)
is (in principle) known than the whole energy functional is known explicitely. Clearly, the
”true” f(r2, ...rN/r1) is very difficult if not impossible to obtain [13], however it can be
sufficiently well described on the basis of mathematical requirements and physical intuition
as done for example in Ref.[12] and as shown in the previous section. From this point of
view, Eqs.12,13, can be seen as an objective criterion to design, on the basis of physical
intuition and fundamental mathematical requirements, valid energy functionals. In fact, as
done in Ref.[12] and in the previous section, one can construct well-founded expressions
for f keeping in mind the physical meaning of the electron correlation effects and the
necessary related mathematical prescriptions of Eq.10. Next one can make use of Eqs.12,13
and choose among different functional forms of f , the one giving the ”smaller” Γ. It
must be noticed that in this work we do not claim that finding a functional form of f is
easier or more rigorous than to find an exchange-correlation functional in standard Density
Functional Theory; it represents an alternative or complementary approach to the latter.
However, the approach based on f allows one to express in a more direct way, via the choice
of different forms of f , the physical principles related to the electron correlation effects and
to have an explicit form of the correlation term for the kinetic functional which is of great
advantage for kinetic functional based methods (see e.g. Refs.[14, 15]).An important aspect
linked to the statement above is that the term: 1
|∇rf(r
,....,rN/r)|
,.....,rN/r)
....drNdr, is
the well known non local Fisher information functional about which a vast literature
is available (see e.g. [10, 16, 17] and references therein); this term is very often linked to
the electron correlation functional and electronic properties(see Refs.[18, 19]), our work
further clarifies this connection, suggesting that the results known from the analysis of
the Fisher functional could be employed in this context. In conclusion we have shown an
alternative view of the Levy-Lieb constrained search approach and provided an example
which clarifies the practical advantage of our idea; in this sense the present work it is not
merely a marginal new formal contribution to a rather well-known method, but gives a new
powerful insight into the field of applicability for realistic systems.
Acknowledgments
I would like to thank Luca Ghiringhelli for a critical reading the manuscript.
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	Introduction
	The new representation
	A practical Example: The parametric exponential form of f
	Discussion and Conclusions
	References
