Appendix D

Explicit Density Functionals for the Kinetic

Energy: Thomas-Fermi Models and Beyond

The theorem of Hohenberg and Kohn provides a justification of early density func-
tional models which relied on a representation of the complete ground state energy
E0 in terms of the density,

E0 = E[n0 ] . (D.1)

The first density functional of this type was the model of Thomas and Fermi (TF),
which was established in the years 1927/28 [13, 14]. These authors considered a
uniform gas of noninteracting electrons, the homogeneous electron gas (HEG) of
Sect. 4.3, in order to derive a representation of the kinetic energy in terms of the
density.
Their result can be derived by the Green’s function techniques utilized in Sect. 4.3
for the discussion of the xc-energy of the HEG. In order to provide some alterna-
tive to this approach, however, a more elementary route for the derivation of the
TF functional is taken in this Appendix. The Schrödinger equation for the single-
particle states of the noninteracting electron gas reads

h̄2 ∇ 2
− φi (rr σ ) = εi φi (rr σ ) . (D.2)
2m
The solutions of (D.2) are given by

φk s (rr σ ) = C eikk·rr χs (σ ) (quantum number i ≡ k s) , (D.3)

with the Pauli spinors χs (σ ) and the eigenvalues

h̄2 k 2
εk = . (D.4)
2m
Normalizable solutions can only be obtained if k is real. However, even in the case
of real k the norm of φk s is infinite, as soon as the complete space is considered.
Moreover, the differential equation (D.2) allows arbitrary real values of k , so that
438 D Explicit Density Functionals for the Kinetic Energy: Thomas-Fermi Models and Beyond

one finds more than countably many states. It is thus necessary to regularize the
problem by an additional boundary condition which ensures the normalizability of
the φk s and at the same time discretizes the spectrum. For this regularization one
chooses a cubic box with sides of length L. Requiring periodic boundary conditions
for all three Cartesian directions,

φk s (x + L, y, z, σ ) = φk s (x, y + L, z, σ ) = φk s (x, y, z + L, σ ) = φk s (x, y, z, σ ) , (D.5)

leads to a quantization (i.e. discretization) of all components of k ,

2π
ki = αi with αi = 0, ±1, ±2, . . . (i = 1, 2, 3) . (D.6)
L
√
Normalization to 1 inside the box is obtained for C = 1/ L3 ,

L
L
L
2π i (α  −α )·rr

0
dx
0
dy
0
dx e L
∑ χs (σ )χs (σ ) = L3 δα α  δss . (D.7)
σ =↑,↓

The single-particle states which are properly normalized within a cubic box are thus
given by

eikk ·rr 2π
φk s (rr σ ) = χs (σ ) k= α with αi = 0, ±1, ±2, . . . . (D.8)
L3/2 L
In the ground state of the noninteracting homogeneous electron gas each level
is filled with one spin-up and one spin-down electron. The number of levels which
are occupied is determined by the number of particles in the box. The eigenvalue
of the energetically highest occupied state is identified with the Fermi energy εF .
Consequently, the density of the system is
∞
n0 = ∑ Θ (εF − εi ) ∑ φi∗ (rr σ )φi (rr σ )
i=1 σ =↑,↓
∞
= ∑ Θ (εF − εk ) ∑ φk∗s (rr σ )φk s (rr σ )
α1 ,α2 ,α3 =0 σ =↑,↓
 
∞ 2 k2
h̄ k 2
= ∑ Θ εF −
2m L3
. (D.9)
α1 ,α2 ,α3 =0

Similarly one obtains for the kinetic energy per volume element
∞

Ts (V ) 1 −h̄2 ∇ 2
V
=
V ∑ Θ (εF − εi ) ∑ d 3 r φi∗ (rr σ )
2m
φi (rr σ )
i=1 σ =↑,↓ V
∞
L
L
L
1 h̄2 k 2
= 3 ∑
L α1 ,α2 ,α3 =0
Θ (εF − εk ) ∑
0
dx
0
dy
0
dz φk∗s (rr σ )
2m k s
φ (rr σ )
σ =↑,↓
D Explicit Density Functionals for the Kinetic Energy: Thomas-Fermi Models and Beyond 439
 
∞
h̄2 k 2 2 h̄2 k 2
= ∑ Θ ε F −
2m L3 2m
. (D.10)
α1 ,α2 ,α3 =0

At this point all expressions have been evaluated to a point at which the limit L → ∞
can be taken, which leads back to the electron gas of infinite extension. In this limit
the spacing between adjacent momenta k becomes infinitesimally small, so that the
summation over all discrete values of k goes over into an integration over k . The
volume element of this k -integration is obtained from the volume in k -space which
is associated with each of the discrete k -values. For each of the Cartesian directions
two neighboring ki -values differ by 2π /L, so that the k -space volume per discrete
k -value is (2π /L)3 ,
 3
2π L
Δ ki = Δ αi =⇒ Δ α1 Δ α2 Δ α3 = Δ 3k
L 2π
∞  3

L
∑
L→∞
−→ d3k . (D.11)
α1 ,α2 ,α3 =0 2π

Introducing the Fermi momentum

√
2mεF
kF := , (D.12)
h̄
the density and kinetic energy density are now easily evaluated using spherical co-
ordinates,
 3

L 2
n0 = d 3 k Θ (kF − |kk |) 3
2π L
kF3
= (D.13)
3π 2
 3

Ts (V ) L 2 h̄2 k 2
= d 3 k Θ (kF − |kk |) 3
V 2π L 2m
h̄2 kF5
= . (D.14)
10π 2 m
Finally, one can invert the relation between n0 and kF ,
# $1/3
kF = 3π 2 n0 , (D.15)

to end up with the desired relation between the kinetic energy density ts and the
density n0 ,

Ts (V ) h̄2 (3π 2 n0 )5/3

ts ≡ = . (D.16)
V 10π 2 m
440 D Explicit Density Functionals for the Kinetic Energy: Thomas-Fermi Models and Beyond

In order to apply this result to atoms, Thomas and Fermi (TF) relied on the lo-
cal density approximation discussed in Sect. 4.3. In this approximation the energy
density ts (rr ) of the actual inhomogeneous system is replaced by the energy density
of the electron gas, Eq. (D.16), evaluated with the local density n(rr ). The complete
kinetic energy is then given by

3(3π 2 )2/3 h̄2

TsTF = d 3 r n(rr )5/3 . (D.17)
10m
This expression is manifestly a density functional. As it is derived from the non-
interacting gas it represents an approximation for the Kohn-Sham kinetic energy
functional Ts [n], introduced in Sect. 3.1.
The total energy functional of Thomas and Fermi neglected all exchange and
correlation effects, so that only the direct Coulomb repulsion (Hartree energy) and
the coupling to the external potential remain,

e2 n(rr ) n(rr  )
E TF [n] = TsTF [n] + d3r d 3 r + d 3 r vext (rr )n(rr ) . (D.18)
2 |rr − r  |

Applications can be based directly on the variational equation (2.38), reflecting the
minimum principle (2.28). The welcome feature is the fact that the variational ap-
proach reduces the many-particle problem to a form which is independent of the
particle number.
Considerable effort was expended in order to improve this model. The first and
most important step was the inclusion of exchange by Dirac in 1930 [131]. Dirac
followed the path of Thomas and Fermi and considered the exchange energy of the
uniform electron gas. The exact exchange energy of the gas in the cubic box of
volume V = L3 has the form
e2 ∞
2 i,∑
Ex (V ) = − Θ (εF − εi )Θ (εF − ε j )
j=1


φi∗ (rr σ )φ j (rr σ )φ j∗ (rr  σ  )φi (rr  σ  )
× ∑ d3r d 3 r
|rr − r  |
. (D.19)
σ ,σ  =↑,↓ V

Insertion of the states (D.8) of the uniform gas yields for the exchange energy per
volume element (after an appropriate shift of r  by r )

$
3  ei(kkα −kkβ )·rr

Ex (V ) #
ex ≡ = −e ∑ Θ (kF − |k α |) Θ kF − |k β |
2
k k d r . (D.20)
V αβ
L6 |rr  |

One can now use the fact that for L → ∞ the summation over all integers α can be
replaced by an integration over k , Eq. (D.11), to obtain




d3k d3q ei(kk−qq)·rr
ex = −e2 Θ (kF − |kk |)Θ (kF − |qq|) d 3 r . (D.21)
(2π )3 (2π )3 |rr  |
D Explicit Density Functionals for the Kinetic Energy: Thomas-Fermi Models and Beyond 441

Next, the r  -integration can be carried out by introducing a suitable intermediate reg-

ularization factor e−μ |rr | in the integral and taking the limit μ → 0 after integration
(compare Eq. (4.144) and the subsequent discussion in Sect. 4.4.1),

d3k d3q 4π
ex = −e2 Θ (kF − |kk |)Θ (kF − |qq|) . (D.22)
(2π )3 (2π )3 (kk − q )2

One now first performs the q -integration. Choosing the z-axis of the coordinate sys-
tem for q so that it is parallel to k , the q -integration can be done in spherical coor-
dinates,


kF
+1
e2 d3k 1
ex = − Θ (kF − k) q2 dq d cos(θ )
π (2π )3 0 −1 k2 + q2 − 2kq cos(θ )

kF
kF
e2
= kdk qdq [ln |k − q| − ln(k + q)] .
2π 3 0 0

The remaining integrations are straightforward, after splitting the range of the inner
integration over q into the subregimes [0, k] and [k, kF ],

e2 4
ex = − k . (D.23)
4π 3 F
Insertion of the Fermi momentum (D.15) then leads to

e2 # 2 $4/3
ex = − 3π n0 . (D.24)
4π 3
Using the local density approximation, one finally arrives at the density functional

3(3π 2 )1/3 e2
ExD [n] = − d 3 r n(rr )4/3 . (D.25)
4π

ExD [n] is an approximation for the exact exchange energy functional Ex [n] of DFT.
As is clear from its construction, ExD [n] is nothing but the LDA for exchange,
Eqs. (4.99), (4.109), in modern terminology. Adding this term to the energy (D.18)
constitutes the Thomas-Fermi-Dirac model.
The next step towards extending the TF model was taken by von Weizsäcker in
1935 [174]. Von Weizsäcker observed that one can express the kinetic energy of a
single particle in terms of the density. In fact, if there is only one particle bound by
some potential, the corresponding ground state orbital

φi (rr σ ) = ϕ0 (rr ) χs (σ )

may be chosen real, so that its kinetic energy may be written as1

1 The surface term does not contribute in the partial integration since a normalizable orbital decays

sufficiently rapidly for |rr | → ∞.

442 D Explicit Density Functionals for the Kinetic Energy: Thomas-Fermi Models and Beyond

−h̄2 ∇ 2 ∇ϕ0 (rr )]2

[h̄∇
Ts = ∑ d 3 r φi∗ (rr σ )
2m
φi (rr σ ) = d3r
2m
. (D.26)
σ =↑,↓

The corresponding density is given by

n(rr ) = ∑ |φi (rr σ )|2 = ϕ0 (rr )2 .

σ =↑,↓

Insertion into (D.26) leads to the von Weizsäcker functional

h̄2 ∇n(rr )]2

[∇
TsvW [n] = d3r . (D.27)
m 8n(rr )

This density functional also represents the exact kinetic energy in the case of a
noninteracting 2-particle system in which both particles occupy the same orbital
ϕ0 , but have opposite spins. TsvW [n] thus agrees with the exact Ts [n] of Kohn-Sham
theory for a single particle and a spin-saturated pair of two particles.
The expression for TsvW [n] also indicates how the TF kinetic energy can be ex-
tended in order to better account for the inhomogeneity of real systems: obviously,
the gradient of the density is the simplest purely density-dependent measure of the
inhomogeneity in a many-particle system. The only parameter-free expression for ts
which (i) depends only locally on ∇ n and (ii) does not depend on the characteristics
of the external potential (as for instance on some preferred axis) is the functional
(D.27). It is thus no surprise that a systematic derivation of gradient corrections for
the kinetic energy, either using some form of the so-called commutator expansion
[173] or following the lines of Sect. 4.4, leads to an expression which differs from
TsvW [n] only by an overall prefactor λ = 1/9 (for all details, including higher order
gradient corrections [175, 733, 194, 195, 734], see Chap. 5 of [7]). Adding λ TsvW [n]
to E TF [n] + ExD [n] constitutes the Thomas-Fermi-Dirac-Weizsäcker model.
Without going into detail, we list some further extensions of the TF-model:
• First correlation contributions were introduced by Wigner as early as 1934 [138]
(see Sect. 4.3.4).
• Gradient corrections to the Dirac exchange energy were calculated subsequently,
but were found to lead to a divergent behavior for small and large separations
from the nucleus in atoms—compare Sect. 4.4.3.
Nonetheless, the endeavors to improve TF-type density functionals were essentially
abandoned until recently, since the explicitly density-dependent representation of Ts
used in these models does not allow to reproduce shell structure.
Renewed interest in functionals of the type (D.1) has been stimulated by the N 3 -
scaling of the Kohn-Sham approach with system size: if one wants to perform calcu-
lations for truly large quantum systems without any periodicity or other symmetry
(e.g. disordered solids or huge (bio)molecules), an N 3 -scaling is still prohibitive. In
this case use of a kinetic energy density functional (KEDF) is highly attractive. In
view of the limitations of the TF-type semi-local functionals a fully nonlocal ansatz
is chosen for modern KEDFs [735–749]. The general form of these approximations
D Explicit Density Functionals for the Kinetic Energy: Thomas-Fermi Models and Beyond 443

is2

Tsnl [n] = TsTF [n] + TsvW [n]

# $
h̄2 3(3π 2 )2/3
+ d 3 r d 3 r n(rr )α wαβ ξγ (rr , r  ), r − r  n(rr  )β , (D.28)
10m
with the 2-body Fermi wavevector
! "1/γ
 (3π 2 n(rr ))γ /3 + (3π 2 n(rr  ))γ /3
ξγ (rr , r ) = (D.29)
2

(the structure of (D.28) can be motivated by scaling arguments [750]). By construc-

tion the functional Tsnl [n] can be exact for the electron gas with ∇ n = 0 and for
a 2-particle system, if the density-dependent kernel w is chosen appropriately. So,
obviously one has the requirement
# $
wαβ ξγ , r − r  = 0

in the electron gas limit. Moreover, in order to recover the exact linear response re-
sult for the weakly inhomogeneous electron gas, Eq. (4.156), the kernel has to satisfy
a differential equation, which allows to determine its shape. In fact, this differential
equation can even be solved analytically [749], which, in spite of the nonlocality
of ξγ (rr , r  ), leads to an N ln(N) scaling of the computational effort with the system
size. KEDFs can therefore provide the basis for multiscale modelling.
Selfconsistent calculations with KEDFs are usually based on pseudopotentials.
The pseudopotentials have to be local, as projecting out part of the all-electron
Hilbert space is not possible, if no states are involved. However, an accurate de-
scription by local pseudopotentials can only be expected for simple metals. Ap-
plications of KEDFs to bulk aluminum, aluminum surfaces and aluminum clusters
[742, 743, 746, 749] demonstrated that the functional (D.28) accurately reproduces
the geometry, energetics (including vacancy formation) and density profiles of the
full Kohn-Sham solutions. In particular, one finds very accurate results for the rel-
ative energies of different crystal structures [746]. KEDFs perform even better for
sodium [742, 743].

2 Sometimes, even several nonlocal kernels of the form (D.28) are superposed [745],

∑ λαβ n(rr)α wαβ n(rr  )β  ,

αβ

in order to allow for more flexibility. In this case ∑αβ λαβ = 1 is required.