References are as follows: LYP,
67
B3PW91,
97
PBE,
59
B88,
63
KS,
4
HK.
3
Paper
Web of Science
Google Scholar
LYP
43 123
49 703
B3PW91
42 642
52 028
PBE
30 575
37 771
B88
24 766
28 529
KS
21 670
31 251
HK
15 222
27 317
densities from other sources, TFD energies have errors of
around 10%, too large for computational purposes. Further
more, Teller deduced
13
that ThomasFermi theory cannot bind
molecules. TFD is useful for rough estimates of atomic prop
erties only.
Shell structure is a consequence of the Pauli exclusion
principle and, as such, arises from electron pairs in orthonor
mal orbitals arranged in Slater determinants. HartreeFock
(HF) theory is the simplest realization. Given the Hamiltonian
operator,
H
, for
N
electrons in an external (nuclear) potential
v
ext
:
H
= −
1
2
i
∇
2
i
+
i
v
ext
(
r
i
)
+
1
2
j
=
i
i
1

r
j
−
r
i

,
(1)
we minimize the energy of a Slater determinant with respect
to variations in the occupied spin orbitals
ψ
i
σ
. The result is
the famous HF orbital equation
−
1
2
∇
2
ψ
iσ
(1)
+
v
ext
(1)
ψ
iσ
(1)
+
v
el
(1)
ψ
iσ
(1)
−
j
ψ
∗
jσ
(2)
ψ
iσ
(2)
r
12
d
2
ψ
jσ
(1)
=
ε
iσ
ψ
iσ
(1)
,
(2)
v
el
(1)
=
ρ
(2)
r
12
d
2
,
ρ
=
σ
ρ
σ
,
ρ
σ
=
i

ψ
iσ

2
,
with the summation over
j
in the exchange term over orbitals
of parallel spin only. The total HF energy is given by
E
HF
= −
1
2
σ
i
ψ
∗
iσ
∇
2
ψ
iσ
+
v
ext
ρ
+
1
2
ρ
(1)
ρ
(2)
r
12
d
1
d
2
+
E
X
,
(3)
E
X
= −
1
2
σ

∑
i
ψ
∗
iσ
(1)
ψ
iσ
(2)

2
r
12
d
1
d
2
,
where
E
X
is the HartreeFock
exchange
energy. The first
three terms are the total kinetic energy, the interaction
energy with the external potential, and the classical Coulomb
selfinteraction energy, respectively.
HF theory, while immensely more useful than TFD, is
still not accurate enough for energy predictions in chem
istry. Bond energies are significantly underestimated. Some
molecules, F
2
, for example, are not even bound at the Hartree
Fock level. Thus
post
HF methods, adding to HartreeFock
numerous other determinants involving excited or “virtual”
orbitals, are generally required for viable chemical com
putations. PostHF technology is well developed (see, e.g.,
Refs.
14
and
15
for overviews) and capable of very high accu
racy, but the development and computational costs are severe.
Simply put, it is
complicated
, and the computertime scaling
with system size
N
is several orders larger than for Hartree
Fock depending on the method (i.e., the formal scaling
16
of
HF is
N
4
, while postHF methods scale like
N
5
and higher as
their sophistication increases). I will refer to HF and postHF
methods as
wavefunction
methods, and the theory as
wave
function theory
(WFT). The more common terminology,
ab
initio
theory, is regrettable as it denigrates densityfunctional
theory. As we shall see in Secs.
III
and
IV
, DFT is as
“
ab initio
” as WFT.
In
the
mid
1900s,
computation
of
HF
orbitals
in
condensedmatter systems was intractable. The problem is
the orbitaldependent, nonlocal, exchange operator in Eq.
(2)
.
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 Fall '15
 Quantum Chemistry, Kinetic Energy, density functional theory, Axel D. Becke, EXC