Abstract. The paper by Kohn and Sham (KS) is
important for at least two reasons. First, it is the basis
for practical methods for density functional calculations.
Second, it has endowed chemistry and physics with an
independent particle model with very appealing features.
As expressed in the title of the KS paper, correlation
e€ects are included at the level of one-electron equa-
tions, the practical advantages of which have often been
stressed. An implication that has been less widely
recognized is that the KS molecular orbital model is
physically well-founded and has certain advantages over
the Hartree–Fock model. It provides an excellent basis
for molecular orbital theoretical interpretation and
prediction in chemistry.
Key words: Density functional theory – Kohn–Sham –
Electron correlation – Molecular orbital theory
1 Introduction
The title paper by Kohn and Sham (KS) [1] has of course
to be considered in conjunction with its predecessor, the
paper by Hohenberg and Kohn (HK) [2]. The HK paper
established for a many-particle system with some two-
particle interaction, where all particles move in a given
local potential, m(r), and with a restriction to systems
that have nondegenerate ground states, that there is
a one-to-one mapping between the potential, m(r), the
particle density, q(r), and the ground-state wavefunc-
tion, Y
0
,
q…r† $ m…r† $ W
0
: …1†
If Y
0
is a functional of the density, then so are all
properties, since any property may be determined as the
expectation value of the corresponding operator,
^
O say,
O‰qŠ ˆ W
0
‰qŠh j
^
O W
0
‰qŠj i. In particular the kinetic energy
is also a functional of the density, T[q], the electron–
electron interaction energy, W[q], and the total energy,
E[q]. HK also established the existence of the total
energy functional, E
m
[q], for which the ground-state
energy, E
0,
of the system with external potential m, is a
lower bound,
E
m
‰qŠ ˆ W‰qŠh
^
T ‡
^
V ‡
^
W




W‰qŠi
ˆ T ‰qŠ ‡
Z
qm dr ‡ W ‰qŠ  E
0
: …2†
It is easy to generalize Eqs. (1) and (2) if the ground state
is degenerate [3]. A particularly elegant definition of the
functional F[q] = T[q] + W[q], which automatically
covers the case of ground-state degeneracy, has been
provided in Levy’s constrained search formulation [4]
F
L
‰qŠ ˆ T ‰qŠ ‡ W ‰qŠ ˆ min
W!q
W
^
T ‡
^
W




W


; …3†
where the minimum is to be searched over all possible
wavefunctions that yield the given q as the density.
There is no denying the great importance of these
theorems, but it may still be argued that the step taken in
the KS paper has been as important: it certainly has been
for chemistry. Equation (2) holds the promise of a very
ecient route to total energies of many-electron systems,
from a Euler–Lagrange equation for the density, if good
approximations for T[q] and W[q] could be found. This
would e€ectively reduce the very high dimensional
problem of the calculation of the many-particle wave-
function, Y
0
, to the determination of just the simple
function in 3D real space, q(r); yet, Eq. (2) has found
very little practical application. The reason is that it is
very dicult to develop suciently accurate density
functionals, in particular, for the kinetic energy. This is a
problem that already plagued the Thomas–Fermi ap-
proach, and although the HK theorems give a much
more theoretically sound basis to density functional
theory (DFT), its practical importance might not have
Perspective
Perspective on ‘‘Self-consistent equations including exchange
and correlation e€ects’’
Kohn W, Sham LJ (1965) Phys Rev A 140:133–1138
E.J. Baerends
Scheikundig Laboratorium der Vrije Universiteit, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands
Received: 16 February 1999 / Accepted: 22 June 1999 / Published online: 9 September 1999
Theor Chem Acc (2000) 103:265–269
DOI 10.1007/s002149900067

risen above that of the Thomas–Fermi model if it had
not been for the KS one-electron model.
So it is the second step, made in the KS paper, that
has been essential for DFT to become the widely applied
method for electronic structure calculations that it is
today. We will comment briefly on the historical context
of the KS paper. Then a few remarks on practical
(computational) aspects of the KS equations will be
made and finally the conceptual implications of the KS
one-electron or molecular orbital (MO) model and its
importance for chemistry now and in the future will be
commented on.
2 Historical context
In the many years that have passed since the publication
of the KS paper, the way that their results are usually
presented, and where the emphasis is put, has of course
shifted. It is now customary to stress from the outset that
the KS theory introduces a system of noninteracting
electrons, moving in a local potential, m
s
(r). The ground-
state wavefunction of the KS system – a single Slater
determinant of the lowest N orbitals – will yield precisely
the same electron density as the exact interacting
electron system with potential m(r). So the KS Hamilto-
nian,
^
H
s
, is just a sum of one-electron Hamiltonians,
^
h
s
, and the wavefunction of the KS system is a simple
one-determinantal wavefunction,
^
H
s
ˆ
X
i
^
h
s
…i† ˆ
X
i
ÿ
1
2
r
2
…i† ‡ m
s
…r
i
†
 
^
h
s
…1†/
i
…1† ˆ e
i
/
i
…1†
W
s
ˆ /
1
…1†;/
2
…2†; . . . ;/
N
…N†j j
q
s
…r† ˆ
X
N
iˆ1
X
s
/
i
…r; s†j j
2
ˆ q
exact
…r†
…4†
The KS paper does not even mention Y
s
, but it
concentrates on the orbital equations and right away
specializes on systems with slowly varying density. It
uses the local density approximation (LDA) from the
outset. Considerable attention is given to corrections to
be introduced, dependent on the gradient of the density,
to take variation of the density into account. It is
historically understandable, in view of the importance at
the time (and for some 10–15 years after the publication
date) of Slater’s exchange approximation [5], that
extensive comparison is made to this exchange approx-
imation, which led to a potential proportional to q(r)
1/3
.
It is pointed out that the variation procedure that is
inherent to the KS approach leads to an exchange
potential that is two-thirds of Slater’s, a point that had
been shown earlier by Ga´ spa´ r [6]. Slater next introduced
a constant a in his exchange potential; this could be
determined according to various criteria. There has been
considerable debate over this constant, which has now
long subsided, the KS treatment with its explicit
inclusion of a correlation functional and potential
having been generally adopted. In their paper KS also
discuss the possibility that one could use a nonlocal
exchange potential, and add only a local correlation
potential. Although they stress (in the title of the paper)
that they introduce self-consistent equations that include
correlation e€ects, it is, with hindsight, interesting to
observe that KS only mentioned in a note added in proof
that the paper has actually achieved the possible
replacement of the many-electron problem with an
exactly equivalent set of self-consistent one-electron
equations. In this note they introduce the local ex-
change–correlation potential m
xc
(r)=dE
xc
[q]/dq(r) which
features in the exact m
s
,
m
s
…r† ˆ m…r† ‡ m
Coul
…r† ‡ m
xc
…r† ; …5†
where m
Coul
=òq(r
2
)/r
12
dr
2
. The exchange–correlation
energy, E
xc
, which is a crucial quantity in DFT, is
defined only within the context of the KS one-electron
model. We will comment later on E
xc
more extensively,
but note at this point that it is exactly this feature, of
treating the complicated many-electron system in prin-
ciple exactly with only the computational expense of
a self-consistent-field calculation, which has been so
appealing in KS theory. However enticing the promise of
computational simplicity and eciency is, the possibility
to treat correlation at the one-electron level has been
hard to accept for a quantum chemistry community that
was steeped in the belief that correlation was by
definition everything that could not be covered at the
one-electron level.
3 Computational considerations
The KS equation shares with Slater’s Xa method the
replacement of the nonlocal exchange operator of
Hartree–Fock (HF) theory by a local potential. This
particular feature of Slater’s exchange approximation
had been extensively exploited in solid-state physics, and
initially the interest in quantum chemistry almost
exclusively focussed on the eciency of the scattered-
wave technique borrowed from physics for solving the
Xa and LDA one-electron equations [7]. The required
mun-tin approximation of the potential, however,
proved to be too severe in molecules, prohibiting reliable
bond energy and structure determinations. The obvious
alternative is to use basis sets [8, 9], as was common in
quantum chemistry, and evaluate the matrix elements of
m
xc
by numerical integration. The approximate Diop-
hantine method was introduced by Ellis for this purpose
[10]. Although this method was suciently precise to
calculate bond energies and structures, it was not
capable of high numerical precision. In the mid 1980s
the problem of 3D numerical integration of molecular
integrands, with their characteristic singularities at the
nuclear sites, was solved simultaneously, in somewhat
di€erent ways, by Becke [11] and by Boerrigter et al.
[12].
Precise 3D numerical integration a€ords solution of
the KS equations with a basis-set-expansion method. It
would, however, make a KS calculation more expensive
266