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Blair, Alexander and Kroukis, Aristeidis and Gidopoulos, N.I. (2015) 'A correction for the HartreeFock
density of states for jellium without screening.', Journal of chemical physics., 142 (8). 084116.
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A correction for the HartreeFock density of states for jellium without screening
Alexander I. Blair, Aristeidis Kroukis, and Nikitas I. Gidopoulos
Citation: The Journal of Chemical Physics 142, 084116 (2015); doi: 10.1063/1.4909519
View online: http://dx.doi.org/10.1063/1.4909519
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/8?ver=pdfcov
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THE JOURNAL OF CHEMICAL PHYSICS 142, 084116 (2015)
A correction for the HartreeFock density of states for jellium
without screening
Alexander I. Blair, Aristeidis Kroukis, and Nikitas I. Gidopoulos
Department of Physics, Durham University, South Road, Durham DH1 3LE, United Kingdom
(Received 14 December 2014; accepted 5 February 2015; published online 26 February 2015)
We revisit the HartreeFock (HF) calculation for the uniform electron gas, or jellium model, whose
predictions—divergent derivative of the energy dispersion relation and vanishing density of states
(DOS) at the Fermi level—are in qualitative disagreement with experimental evidence for simple
metals. Currently, this qualitative failure is attributed to the lack of screening in the HF equations.
Employing Slater’s hyperHartreeFock (HHF) equations, derived variationally, to study the ground
state and the excited states of jellium, we ﬁnd that the divergent derivative of the energy dispersion
relation and the zero in the DOS are still present, but shifted from the Fermi wavevector and energy
of jellium to the boundary between the set of variationally optimised and unoptimised HHF orbitals.
The location of this boundary is not ﬁxed, but it can be chosen to lie at arbitrarily high values of
wavevector and energy, well clear from the Fermi level of jellium. We conclude that, rather than
the lack of screening in the HF equations, the wellknown qualitative failure of the groundstate
HF approximation is an artifact of its nonlocal exchange operator. Other similar artifacts of the HF
nonlocal exchange operator, not associated with the lack of electronic correlation, are known in the
literature.
C
2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4909519]
INTRODUCTION
The uniform electron gas, or jellium model, is an arche
typal example in solidstate physics and manybody theory.
Its treatment, in the Hartree Fock (HF) approximation, can
be found in classic textbooks,
1–6
where, we learn that the HF
equations applied to the ground state of the jellium admit
plane wave solutions with energywavevector dispersion rela
tion given by
ε(k) =
k
2
2
−
k
F
π
*
,
1 +
k
2
F
− k
2
2k k
F
ln
k
F
+ k
k
F
− k
+

. (1)
k
F
is the Fermi wavevector, k
3
F
= 3π
2
(N/V). The single
particle energy ε(k) is the sum of the freeelectron energy,
k
2
/2, and the singleparticle exchange energy. The Fermi
wavevector k
F
is often expressed in terms of the mean radius
per particle r
s
=
3
9π/4k
3
F
;
5
for typical values of r
s
in metals,
the two terms in (1) are comparable in size.
It is well known in the literature that the dispersion relation
(1) has a logarithmically divergent derivative at the Fermi
energy, shown in Fig. 1. Another marked diﬀerence between
the free electron result and the HF solution for jellium, evident
in Fig. 1, is the considerably increased bandwidth of the HF
dispersion. Finally, it is well known that in the HF approxi
mation, the density of states (DOS) for jellium vanishes at the
Fermi level (Fig. 1), since the DOS is inversely proportional to
the derivative of the dispersion. The zero in the DOS at the
Fermi level suggests that jellium is a semimetal, in obvious
disagreement with experimental evidence for simple metals,
such as sodium or aluminium, which are described accurately
by the jellium model.
In the literature, the qualitatively wrong description of
jellium in the HF approximation is attributed to the long
range of the Coulomb repulsion.
1–6
It is well known that the
ﬂawed description can be corrected by the introduction of
electronic many body correlation eﬀects,
1–7
which screen the
bare Coulomb potential and thus eliminate the unphysical
divergent derivative of the dispersion relation, the zero in the
DOS at the Fermi level, and also reduce the bandwidth of the
HF dispersion relation of jellium.
In an eﬀort to understand whether HF’s lack of screening
actually plays a role, we revisit the HF study of jellium, at
tempting to correct the qualitative errors of the HF description,
but without including any form of electronic correlation. For
this purpose, we employ Slater’s hyperHF (HHF) theory for
the ground and the excited states of an Nelectron system.
8
Speciﬁcally, we use the singleparticle HHF equations by Gi
dopoulos and Theophilou,
9,10
who considered an N electron
system described by a Hamiltonian H and then variationally
optimised the average energy
n
⟨Φ
n
HΦ
n
⟩ of all conﬁgura
tions (Nelectron Slater determinants Φ
n
) constructed from a
basis set of R spinorbitals, R ≥ N.
THE HYPERHARTREEFOCK EQUATIONS
FOR JELLIUM
The aim in HHF theory is to obtain approximations, at the
HF level of description, for the ground and the excited states
of an Nelectron system. These states are represented by N
electron Slater determinants, constructed from a common set
of spinorbitals. Obviously, to have the ﬂexibility to describe
excited states, the number of spinorbitals (R) must exceed
the number of electrons. For example, say we are interested
to approximate the ground and excited states of the helium
00219606/2015/142(8)/084116/5/$30.00 142, 0841161 © 2015 AIP Publishing LLC
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0841162 Blair, Kroukis, and Gidopoulos J. Chem. Phys. 142, 084116 (2015)
FIG. 1. Solid lines show groundstate HF results for jellium, compared to
freeelectron results in dotted lines. (r
s
/a
0
= 4, ε
0
F
= k
2
F
/2.) Top: Energy
vs wave vector dispersion relation ε(k). The logarithmic divergence in the
derivative, dε/dk , is marked with a triangle (N). Bottom: DOS, showing the
unphysical zero at the Fermi level for jellium.
atom. In the HF ground state of He, the 1s orbital (ϕ
1s
) is
doubly occupied. To study a couple of excited states, we need
at least one more spinorbital and the next one is ϕ
↑
2s
. With
the three available spinorbitals, 1s
↑
,1s
↓
,2s
↑
, we can form
three conﬁgurations for the He atom (twoelectron Slater deter
minants): Φ
1
= [1s
↓
,1s
↑
], Φ
2
= [1s
↓
,2s
↑
], Φ
3
= [1s
↑
,2s
↑
]. In
HHF theory, we variationally optimise the three common spin
orbitals simultaneously, by minimising the sum of the expec
tation values
3
i=1
⟨Φ
i
HΦ
i
⟩. The minimisation leads to the
HHF singleparticle equations for the three spinorbitals. It
turns out that these equations resemble the groundstate HF
equations for the lithium atom (three electrons) but with a
weakened Coulomb repulsion between the three electrons, to
keep balance with the nuclear charge which has remained that
of the He nucleus.
In general, in HHF theory
8,9
for an N electron system,
one considers a set of R orthonormal spinorbitals, with R
≥ N. On this spinorbital basis set one may deﬁne, D = R!/
(N !(R − N)!), Nelectron Slater determinants.
The derivation of the singleparticle HHF equations in
Ref. 9 is based on Theophilou’s variational principle,
11
D
n=1
⟨Φ
n
HΦ
n
⟩ ≥
D
n=1
E
(0)
n
, (2)
where {E
0
n
} are the D lowest eigenvalues of the Nelectron
Hamiltonian H.
An extension of the variational principle, with unequal
weights in the sums in (2) was proposed by Theophilou,
12
and independently by Gross, Oliveira, and Kohn.
13
These vari
ational principles can be derived as special cases from the
Helmholtz variational principle in statistical mechanics.
14,15
In
particular, the inequality in (2) arises as the high temperature
limit of the Helmholtz variational principle.
Optimisation of the R spinorbitals {ϕ
i
} to minimise the
sum of the energies on the l.h.s. of (2) leads to the follow
ing singleparticle equations for the spatial part of the spin
orbitals
9
(in atomic units):
−
1
2
∇
2
+ V
ext
(r)
ϕ
i
(r)
+
1
Λ
R
j=1
J
j
(r) − δ
s
j
, s
i
K
j
(r)
ϕ
i
(r) = λ
i
ϕ
i
(r), (3)
where,
Λ =
R − 1
N − 1
, (4)
and
J
j
(r)ϕ
i
(r) ≡
d
3
r
′

r − r
′

ϕ
j
(r
′
)
2
ϕ
i
(r), (5)
K
j
(r)ϕ
i
(r) ≡
d
3
r
′

r − r
′

ϕ
∗
j
(r
′
)ϕ
i
(r
′
)ϕ
j
(r) (6)
are the Coulomb and exchange operators, respectively. V
ext
sig
niﬁes the attractive potential of the nuclear charge. For R = N,
Eq. (3) reduce to the familiar groundstate HF equations. In
Eq. (3), the orbitals ϕ
i
, with i = 1, . . . , R, are correctly repelled
electrostatically by a charge of N − 1 electrons. In contrast to
the groundstate HF case,
16
this holds true even for the orbitals
that are not occupied in the HHF groundstate Slater determi
nant as long as these orbitals are variationally optimised, i.e.,
for ϕ
i
, with N < i ≤ R. Furthermore, the orbitals that are left
variationally unoptimised, i.e., ϕ
i
, with i > R, are repelled by
a charge of N − 1 + (1/Λ) electrons. In the HHF equations, the
wellknown asymmetry in the treatment of the variationally
optimised and unoptimised orbitals by the nonlocal exchange
operator
16
is still present, but softened (for large Λ), compared
with groundstate HF. We note that for R > N, Koopmans’
theorem
19,20
ceases to hold for the HHF equations.
The HHF equations (3) have the form of groundstate
HF equations for a virtual system of Relectrons, where the
electronic Coulomb repulsion is multiplied by 1/Λ: 1/r − r
′

→ Λ
−1
/r − r
′
. Therefore, the calculation of the optimal spin
orbitals to represent the ground and excitedstates of an N
electron system in the HHF approximation is reduced to the
calculation of the groundstate HF orbitals of a ﬁctitious sys
tem with a greater number of electrons R ≥ N, and scaled
down electronic Coulomb repulsion. A related approach is the
“superhamiltonian method” by Katriel.
22,23
Finally, before applying the HHF equations to jellium,
we remark that correlated, approximate eigenstates of the
Hamiltonian H can be obtained by diagonalising the matrix
⟨Φ
n
HΦ
m
⟩,
9
where Φ
n
are the Nelectron HHF Slater deter
minants. This conﬁgurationinteraction method employs the
HHF spinorbitals, which are optimised to represent on equal
footing the ground and the excited states of H.
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0841163 Blair, Kroukis, and Gidopoulos J. Chem. Phys. 142, 084116 (2015)
Solution of the HHF equations for jellium
Similarly to the HF ground state, the HHF equations for
jellium admit plane wave solutions. It follows that the ground
state Nelectron Slater determinant and its total energy are the
same in the HF and HHF approximations.
Although the HF and HHF equations for jellium admit
the same solution for the orbitals, the dispersion relations for
the singleparticle energies ε(k) and λ(k) diﬀer. In particular,
the HHF dispersion, λ(k), results from an optimisation that
involves a broader range of wavevectors than the HF dispersion
ε(k).
Following the standard treatment in textbooks,
1–6
it is
straightforward to work out directly the solution of the HHF
Eq. (3). Here, we exploit the similarity of Eq. (3) with ground
state HF equations of an Relectron system, to obtain that the
HHF dispersion relation, λ(k), will be given by (1) with the
singleparticle exchange energy scaled down by the factor 1/Λ,
λ(k) =
k
2
2
−
k
R
Λπ
*
,
1 +
k
2
R
− k
2
2k k
R
ln
k
R
+ k
k
R
− k
+

. (7)
k
R
is the Fermi wave vector of the virtual Relectron system,
k
3
R
= 3π
2
R
V
. (8)
Dividing k
R
/k
F
and taking the thermodynamic limit, N,
V → ∞, with the ratio Λ ﬁxed, we obtain the following:
k
R
= Λ
1/3
k
F
. (9)
Substitution of the above into Eq. (7) yields the desired expres
sion for the single particle energy levels of jellium, in terms
of Λ and the Fermi wavevector k
F
of the actual Nelectron
system,
λ(k) =
k
2
2
− Λ
−2/3
k
F
π
×
(
1 +
Λ
2/3
k
2
F
− k
2
2Λ
1/3
k k
F
ln
Λ
1/3
k
F
+ k
Λ
1/3
k
F
− k
)
. (10)
The dispersion relation in Eq. (10) reduces to the groundstate
HartreeFock result for Λ = 1, and to the free electron disper
sion relation, λ(k) = k
2
/2, in the limit Λ → ∞ (see Fig. 2).
For increasing Λ, the bandwidth of the HHF dispersion, λ(k),
decreases compared to the groundstate HF dispersion, ε(k).
For Λ → ∞, the exchange term in HHF dispersion vanishes
and λ(k) reduces to the freeelectron result.
Importantly, the wave vector at which the logarithmically
divergent derivative occurs is shifted from k
F
to k
R
, such that
the divergence no longer occurs at the Fermi energy of the
physical Nelectron system, when the number of optimised
orbitals is R > N .
The DOS, g(λ)δλ, can be obtained directly from Eq. (10)
2
and is given by the parametric equation,
g(λ(k)) =
V k
2
π
2
(dλ/dk)
=
V k
2
π
2
k −
1
Λπ
*
,
k
R
k
−
k
2
R
+ k
2
2k
2
ln
k
R
+ k
k
R
− k
+

−1
.
(11)
FIG. 2. Excitedstate HF results for jellium, for various Λ = R/N , compared
to free electron results (dotted lines). (r
s
/a
0
= 4, ε
0
F
= k
2
F
/2.) Top: Energy vs
wave vector dispersion relations λ(k). When Λ = 1, λ(k) = ε(k). Triangles (N)
mark logarithmic divergence in dλ/dk at Fermi level of ﬁctitious Relectron
system. Bottom: DOS g (λ(k)), showing the zero at the Fermi level of the
ﬁctitious Relectron system.
g(λ(k)) is expressed in terms of k
R
(rather than Λ
1/3
k
F
) to keep
the notation simple. For any ﬁnite Λ, the DOS still vanishes.
However, as shown in Fig. 2, the zero in the DOS occurs at the
Fermi energy of the ﬁctitious Relectron system, λ(k
R
), rather
than the Fermi energy of the physical system λ(k
F
).
DISCUSSION
In metals, screening is an important eﬀect that reduces the
range of the eﬀective repulsion between electrons, shielding
any charge at distances greater than a characteristic screening
length. In the literature of manybody theory
1,3–6
and solid
state physics,
2–4
where jellium is a paradigm, the qualitatively
ﬂawed description of metals by the HF approximation is
attributed to the longrange nature of the Coulomb interaction,
which, combined with the neglect of correlation, deprives from
the HF equations the ﬂexibility to model the phenomenon
of screening. This understanding of HF’s failure is further
supported by the softening of the divergence in the slope of
ε(k), after replacing the bare Coulomb potential in the HF
nonlocal exchange term by a screened Coulomb potential.
2
On the other hand, in the theoretical chemistry literature, it
is well known that the HF nonlocal exchange term, in ﬁnite sys
tems, gives rise to several counterintuitive results, reminiscent
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