![Table 1: Total energy differences ∆E (in eV) from the HF energy and Kohn Sham bandgaps (in eV) for xOEP, LFX and CEDA potentials. Last column gives experimental bandgap values; from Refs. [18, 47].](/figures/table-1-total-energy-differences-e-in-ev-from-the-hf-energy-3elpsvzt.webp)
![Table 2: Magnetic moments (in µB) for transition metal monoxides for xOEP, LFX and CEDA. Experimental values from Ref. [47].](/figures/table-2-magnetic-moments-in-ub-for-transition-metal-23aiefwy.webp)
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Hollins, T.W. and Clark, S.J. and Refson, K. and Gidopoulos, N.I. (2017) 'A local Fock-exchange potential in
KohnSham equations.', Journal of physics : condensed matter., 29 (4). 04LT01.
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A local Fock-exchange potential in Kohn-Sham
equations
T.W. Hollins
1
, S.J. Clark
1
, K. Refson
2,3
and N.I. Gidopoulos
1
1
Department of Physics, Durham University, South Road,
Durham, DH1 3LE, United Kingdom.
2
Department of Physics, Royal Holloway, University of London,
Egham TW20 0EX, United Kingdom.
3
ISIS Facility, Science and Technology Facilities Council,
Rutherford Appleton Laboratory, Didcot OX11 0QX, United
Kingdom.
Abstract.
We derive and employ a local potential to represent the Fock exchange operator in
electronic single-particle equations. This local Fock-exchange (LFX) potential is very
similar to the exact exchange (EXX) potential in density functional theory (DFT). The
practical software implementation of the two potentials (LFX and EXX) yields robust
and accurate results for a variety of systems (semiconductors, transition metal oxides)
where Hartree Fock and popular approximations of DFT typically fail. This includes
examples traditionally considered qualitatively inaccessible to calculations that omit
correlation.
A local Fock-exchange potential in Kohn-Sham equations 2
1. Introduction
The exchange symmetry in quantum mechanics refers to the invariance of a quantum
system when two identical particles exchange positions. For electronic systems in
particular, this symmetry leads to Pauli’s exclusion principle that makes necessary the
use of antisymmetric, or fermionic, wave functions.
However, in the theory of electronic structure, the term “exchange” is often used in
the narrower context of an effective description that treats the interacting electrons as
independent particles, assigning a spin-orbital (φ
i
) to each one of them. The N-electron
system is represented by a Slater determinant (Φ) built on the set of spin-orbitals {φ
i
}
and the electrons do not interact directly with each other but each electron lies in the
effective field of the other N − 1 electrons. The two principal examples of such effective,
independent-particle descriptions are the Hartree-Fock (HF) approximation [1] and the
Kohn-Sham (KS) scheme [2] in density functional theory (DFT) [3]. Based on the
reference state Φ of the model, the definition of the exchange energy is given in the
context of such an independent-particle theory,
E
x
[Φ] = −
1
2
X
σ
Z Z
dr dr
0
|ρ
σ
Φ
(r, r
0
)|
2
|r − r
0
|
. (1)
where ρ
σ
Φ
(r, r
0
) with σ =↑, ↓ is the one-particle reduced density matrix of the reference
Slater determinant Φ.
Hence, the exchange energy is defined differently in HF theory, where Φ is the HF
Slater determinant Φ
HF
, from DFT where Φ is the KS Slater determinant Φ
KS
. The
two different definitions lead naturally to different representations of the exchange term
(local in KS, nonlocal in HF) in the corresponding single-particle equations.
In HF the nonlocal Fock exchange term acts in a different way on the occupied and
on the virtual orbitals and this asymmetry can lead to counterintuitive and unphysical
behaviour of the HF orbitals and their energies. For example, the virtual HF orbitals
appear to be repelled by a charge of N rather than N −1 electrons[4], a fact that has led
to the interpretation of the virtual orbital energies as (negative) electron affinities, in an
obvious extension of Koopmans’ theorem [5]. On the other hand, if we view that a virtual
HF orbital represents an excitation of a ground state orbital, then the repulsion of the
virtual orbital by a charge of N rather than N − 1 electrons becomes a qualitative error
arising from the self-repulsion of the occupied orbital – accommodating the electron
before excitation – with the virtual orbital hosting to the electron after excitation.
Ref.[6] discusses a similar case of self-repulsion (“ghost-interaction”) in ensemble DFT
for excited states.
The self-repulsion of the virtual orbitals and the asymmetry of the action of the
Fock operator on occupied and virtual orbitals leads to too high single-particle excitation
energies and to poor band-structures with too large band gaps. In metals, the density of
states of the uniform electron gas is found to vanish at the Fermi energy[7], predicting
wrongly the behaviour of an ideal metal to be almost insulating. (However, see Ref. [8].)
Another qualitative error of these spurious self-repulsions is that in unrestricted HF
A local Fock-exchange potential in Kohn-Sham equations 3
theory there are no unfilled shells, i.e., the highest occupied and the lowest unoccupied
orbitals are never degenerate, even in systems with odd number of electrons[9]. Finally,
it seems counterintuitive that all the occupied HF orbitals turn out to have the same
asymptotic decay[10].
The asymmetry in the HF treatment of occupied and virtual orbitals is well known
and there is significant work to correct it, e.g. by making rotations in the Hilbert space of
virtual orbitals[11]. An elegant solution to this problem is to employ a common, local,
multiplicative, single-particle exchange potential ˆv
x
= v
x
(r), that treats all orbitals,
occupied and virtual, in a symmetric way. Indeed, with the self-interaction-free “exact
exchange” (EXX) potential in KS theory (see Ref.[12] and references therein) single-
particle properties, such as ionisation potentials[12, 15], electron affinities[12], single-
particle excitation energies and band structures[12, 16, 17, 18] are obtained accurately.
We note that the EXX potential cannot be written explicitly in terms of the electron
density, but it must be obtained indirectly from the density, by solving a Fredholm
integral equation of the first kind, known as the equation for the optimized effective
potential (OEP) method [13, 14, 12]. In the terminology of KS theory, when correlation
is omitted the EXX potential is also referred to as the exchange optimized effective
potential (xOEP).
By comparison, in semi-local KS approximations (such as the local density
approximation, LDA, and other related approximations) the cancellation of self-
interactions is incomplete making the asymptotic decay of the exchange potential too
fast, thus leading to inferior single-particle properties. In fact, the enforcement of the
correct asymptotic behaviour is sufficient to improve considerably the accuracy of single-
particle properties in these approximations [19].
Given that the exchange energy and exchange potential are not measurable
quantities, the question arises about their dependence on the reference state of the
effective model. For example, are there physical limits where this dependence can be
expected to be either strong or weak? This question will be investigated in the following,
after obtaining the local exchange potential (“local Fock exchange”, LFX) corresponding
optimally to HF’s nonlocal Fock exchange term.
2. The local Fock exchange potential
The KS system is a virtual system of noninteracting electrons defined to have the same
ground state charge density as the interacting system of interest. The KS system is
determined by solving single-particle (KS) equations featuring the KS potential, which
forces the noninteracting electrons to have the required density.
In DFT, the KS potential (functional derivative w.r.t. density of the noninteracting
kinetic energy functional) is the sum of the electron-nuclear attraction, v
en
(r), the
Hartree potential
R
dr
0
ρ(r
0
)/|r − r
0
|, and the exchange and correlation potential, v
xc
(r);
the latter is the functional derivative w.r.t. the density of the exchange and correlation
energy. In particular the xOEP is the functional derivative of the exact exchange energy,
A local Fock-exchange potential in Kohn-Sham equations 4
given by the Fock expression (1) in terms of the KS orbitals.
Ref. [20] gives an alternative (to DFT) and direct way to obtain ab initio
approximations for the KS potential, by minimizing an appropriate energy difference
(T
Ψ
[v], see the variational principle (4) in [20]). We note that the formalism in [20]
assumes some knowledge of the interacting ground state Ψ. Also, we point out that
although the theoretical scheme developed in Ref. [20], is based on wave function theory
rather than DFT, it still results in the usual KS single-particle equations employing the
(exact or approximate) KS potential.
In the following, to derive the local potential that best simulates the nonlocal Fock
exchange term, we follow Ref. [20], and use the HF ground state Slater determinant Φ
HF
in place of the interacting state Ψ. Then, we search among all effective Hamiltonians
H
v
characterised by a local potential v(r),
H
v
=
N
X
i=1
(
−
∇
2
i
2
+ v(r
i
)
)
, (2)
for the one (H
v
MP0
) which adopts optimally Φ
HF
as its own approximate ground
state. The optimisation is achieved using the Rayleigh Ritz variational principle and
minimising, over all local potentials v(r), the non-negative energy difference,
T
HF
[v] ≥ 0, with T
HF
[v]
.
= hΦ
HF
|H
v
|Φ
HF
i − E
v
. (3)
E
v
is the ground state energy of H
v
. The functional derivative of T
HF
[v] is equal to the
difference of the HF density and the density of the effective system [20],
δT
HF
[v]
δv(r)
= ρ
HF
(r) − ρ
v
(r). (4)
At the minimum, the two densities are equal and the optimal potential v
MP0
has the
same density as HF. The difference of v
MP0
and the sum of the Hartree potential and
the electron-nuclear attraction defines the local Fock-exchange (LFX) potential:
v
LFX
(r)
.
= v
MP0
(r) − v
en
(r) −
Z
dr
0
ρ
HF
(r
0
)
|r − r
0
|
. (5)
Compared with the exchange optimised effective potential (xOEP) case, the functional
derivative in (4) is easier to obtain, as it does not involve the calculation of the orbital
shifts[18, 21].
The local potential with the HF density was studied in the past but also recently,
in Ref. [22, 23], where it was found that it provides an almost perfect approximation to
xOEP, avoiding mathematical issues in the implementation of xOEP with finite basis
sets.
However, from our variational derivation we argue that LFX is actually more than
a mere approximation to xOEP: both potentials are optimal in Rayleigh-Ritz energy
minimizations where we search for the effective Hamiltonian H
v
that, either adopts the
HF ground state Φ
HF
optimally as its own approximate ground state (LFX), or, whose
ground state minimizes the HF total energy (xOEP). In either case, if the restriction
of a local potential were relaxed, the HF Hamiltonian would be the minimizing one.