Chapter 16
Semilocal Approximations for the Kinetic Energy
Fabien Tran1 and Tomasz A. Wesolowski2
1
Institute of Materials Chemistry, Vienna University of Technology
Getreidemarkt 9/165-TC, A-1060 Vienna, Austria
[email protected]
2
Department of Physical Chemistry, University of Geneva
30, quai Ernest-Ansermet, CH-1211 Geneva 4, Switzerland
[email protected]
Approximations to the non-interacting kinetic energy Ts [ρ], which take the form
of semilocal analytic expressions are collected. They are grouped according to the
quantities on which they explicitly depend. Additionally, the approximations for
quantities related to Ts [ρ] (kinetic potential and non-additive kinetic energy), for
which the analytic expressions for the “parent” approximation for the functional
Ts [ρ] are unknown, are also given.
Contents
16.1 Notation and conventions . . . . . . . . . . .
16.2 Known exact functionals . . . . . . . . . . . .
16.3 Local density approximation — LDA . . . . .
16.4 Gradient expansion approximation — GEAn
16.5 Generalized gradient approximation — GGA
16.6 Other semilocal approximations . . . . . . . .
16.7 N - and r-dependent approximations . . . . .
16.8 Miscellaneous . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . .
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429
430
430
431
432
437
439
440
441
16.1. Notation and conventions
Atomic units [me = e = = 1/(4πε0 ) = 1] are used in all formulas. The notation
for functions and constants that will be used throughout the text is the following:
• ρ = ρ(r) – the electron density.
1/3 4/3 • s = |∇ρ| / 2 3π 2
ρ
– the dimensionless reduced density gradient.
• x = (5/27) s2 – a quantity proportional to the square of s.
2/3
2.8712 – the Thomas-Fermi constant.
• CF = (3/10) 3π 2
429
F. Tran & T. A. Wesolowski
430
1/3
• b = 2 6π 2
– a conversion factor.
• Capital T – integrated kinetic energy.
• Small t – kinetic-energy density per volume unit.
The analytic expressions of the non-interacting kinetic-energy functionals Ts [ρ]
will be given for the spin-compensated case (ρ↑ = ρ↓ ). For spin-polarized electron densities, the corresponding expression for Ts [ρ↑ , ρ↓ ] can be easily obtained by
applying the extension formula of Oliver and Perdew:1
1
(Ts [2ρ↑ ] + Ts [2ρ↓ ])
2
Ts [ρ↑ , ρ↓ ] =
(16.1.1)
The labels used for the approximations reflect the name given by the authors, the
most common convention used in the literature, or the names of the authors.
16.2. Known exact functionals
For two types of systems, the exact analytic form of Ts [ρ] is known:
• Thomas and Fermi:2,3
TsTF [ρ]
ρ5/3 (r)d3 r
= CF
(16.2.2)
The Thomas-Fermi functional2,3 is exact for the homogeneous electron gas. Applying it for inhomogeneous systems leads to an approximation known as the
Thomas-Fermi functional or the local density approximation (LDA) functional.
• von Weizsäcker:4
TsW [ρ]
1
=
8
2
|∇ρ(r)| 3
d r
ρ(r)
(16.2.3)
This functional is exact for one-electron and spin-compensated two-electron
systems.
Applying it for other systems leads to an approximation known as the von
Weizsäcker functional.
16.3. Local density approximation — LDA
The label LDA is sometimes used in a more general way for any approximation
which depends solely on the electron density like the TF functional [Eq. (16.2.2)].
In addition to TF, a few such functionals were proposed in the literature.
GaussianLDA
Lee and Parr:5
TsGaussianLDA [ρ]
3π
= 5/3
2
ρ5/3 (r)d3 r
(16.3.4)
Semilocal Approximations for the Kinetic Energy
431
Note that the coefficient 3π/25/3 2.9686 is about 3% larger than the coefficient
CF of the TF functional [Eq. (16.2.2)].
Fuentealba and Reyes:6
⎡
⎛
1/3
ρ(r)
⎢
⎜
TsZLP [ρ] = c1 ρ5/3 (r) ⎣1 − c2
ln ⎝1 +
2
ZLP
⎞⎤
c2
1
ρ(r)
2
⎟⎥ 3
1/3 ⎠⎦ d r
(16.3.5)
where c1 = 3.2372 and c2 = 0.00196. It was constructed following the “conjointness conjecture”7 applied to the ZLP8 approximation for the exchange-correlation
energy. Note that in Eq. (9) in Ref. 6 the symbol ρ should be replaced by ρσ for
the given coefficients. For the numerical verification see Ref. 9.
LP97
Liu and Parr:10
TsLP97 [ρ]
= 3.26422
ρ
5/3
(r)d r − 0.02631
3
+0.000498
2
ρ
4/3
3
(r)d r
3
ρ
11/9
3
(r)d r
(16.3.6)
Note that the coefficient in front of the third term was given incorrectly in Ref. 10.
16.4. Gradient expansion approximation — GEAn
The gradient expansion approximation (GEA) until the nth order:
n
n GEAn
Ts
ti (ρ(r), ∇ρ(r), . . .) d3 r
[ρ] =
Ts,i [ρ] =
i=0
(16.4.7)
i=0
where only the terms for i even are non-zero. The analytical form of the terms up
to i = 6 have been derived:
t0 = tTF = CF ρ5/3
(16.4.8)
2
1 W
1 |∇ρ|
t =
(16.4.9)
9
72 ρ
2
2
4
∇2 ρ
(3π 2 )−2/3 1/3
9 ∇2 ρ |∇ρ|
1 |∇ρ|
(16.4.10)
−
+
ρ
t4 =
540
ρ
8 ρ
ρ2
3 ρ4
2 −4/3
3
2
∇ ∇2 ρ 2
3π
2575 ∇2 ρ
249 |∇ρ| ∇4 ρ
−1/3
t6 =
13
+
+
ρ
45360
ρ2
144
ρ
16 ρ2
ρ
2 2 2
2
1499 |∇ρ|
∇ ρ
1307 |∇ρ| ∇ρ · ∇ ∇2 ρ
+
−
18 ρ2
ρ
36 ρ2
ρ2
2
4
6
343 ∇ρ · ∇∇ρ
8341 ∇2 ρ |∇ρ|
1600495 |∇ρ|
(16.4.11)
+
+
−
18
ρ2
72 ρ
ρ4
2592
ρ6
t2 =
F. Tran & T. A. Wesolowski
432
These terms were obtained by various authors: t0 by Thomas and Fermi,2,3 t2
by Kompaneets and Pavlovskii11 and by Kirzhnits,12 t4 by Hodges,13 and t6 by
Murphy.14 t2 and t4 were obtained after integration by part of 15,16
2
1 |∇ρ|
1
(16.4.12)
+ ∇2 ρ
72 ρ
6
⎛
2
2 2 2
2
|∇ρ|
2 −2/3
4
∇
∇ρ · ∇ ∇ ρ
(3π )
∇ ρ
∇ ρ
tJ4 =
− 14
−7
ρ1/3 ⎝12
− 30
2
4320
ρ
ρ
ρ
ρ2
⎞
2
2 2
4
∇ρ
·
∇
|∇ρ|
140 |∇ρ| ∇ ρ 92
|∇ρ|
+
+
− 48 4 ⎠
(16.4.13)
3
3
3
ρ
3
ρ
ρ
tJ2 =
respectively. Choosing n = 0, 2, and 4 in Eq. (16.4.7) leads to approximations with
the following labels: GEA0 which is just the TF functional [Eq. (16.2.2)], GEA2
which is also denoted with TF 19 W, and GEA4.
16.5. Generalized gradient approximation — GGA
The general form of GGA functionals reads
GGA
3
[ρ] = f (ρ(r), |∇ρ(r)|) d r = CF ρ5/3 (r)F (s(r))d3 r
Ts
(16.5.14)
where F (s) is the so-called enhancement factor. The enhancement factor of the von
Weizsäcker functional [Eq. (16.2.3)] is given by
5 2
(16.5.15)
s
3
while for the GEA truncated at the 0th and 2nd orders [Eqs. (16.4.7)-(16.4.9)], the
enhancement factors are
F W (s) =
F GEA0 (s) = 1
F GEA2 (s) = 1 +
5 2
s
27
(16.5.16)
(16.5.17)
respectively.
A large group of approximations in the GGA family were constructed following the “conjointness conjecture” of Lee, Lee, and Parr7 according to which the
enhancement factor of an exchange GGA functional can be used (with possible reoptimization of the free coefficients) for the kinetic energy. The following convention
is used for labeling the conjoint functionals: if not only the analytic form but also
the free coefficients are the same as in the exchange functional, then the functional
is called conjoint X, where X stands for the name of the “parent” exchange functional. In the case of reoptimization of the coefficients, the standard convention
applies (see Sec. 16.1).
Semilocal Approximations for the Kinetic Energy
TFλW
433
The functionals in this family differ only in the value of the constant λ:
TsTFλW [ρ] = TsTF [ρ] + λTsW [ρ]
(16.5.18)
GEA0 and GEA2 correspond to λ = 0 and 1/9, respectively. Other values of λ were
proposed in the literature, e.g., λ = 1.290/9 (modified 2nd order gradient expansion
proposed by Lee et al.17 ). See Ref. 18 for a compilation of the different values of λ
proposed in the literature. The enhancement factor of the TFλW functional is
P82
5
F TFλW (s) = 1 + λ s2
3
(16.5.19)
5 s2
27 1 + s6
(16.5.20)
Pearson:19
F P82 (s) = 1 +
DKPadé
DePristo and Kress:20
F DKPadé (x) =
LLP
1 + 0.95x + 14.28111x2 − 19.57962x3 + 26.64765x4
1 − 0.05x + 9.99802x2 + 2.96085x3
(16.5.21)
Lee, Lee, and Parr:7
F LLP (s) = 1 +
0.0044188b2s2
1 + 0.0253bs arcsinh(bs)
(16.5.22)
This is the enhancement factor of the exchange functional B88 of Becke21 refitted
for the kinetic energy.
OL1 and OL2
Ou-Yang and Levy:22
5 2
20 2 −1/3
s
s + 0.00677
3π
27
3
(16.5.23)
2 1/3
2
3π
s
5
0.0887
F OL2 (s) = 1 + s2 +
1/3
27
CF 1 + 8 (3π 2 ) s
(16.5.24)
F OL1 (s) = 1 +
The coefficients in front of the third terms in OL1 and OL2 were inverted in the
original paper of Ou-Yang and Levy (caption of Table I in Ref. 22). The correctness
of the coefficients given here was confirmed by numerical values of Ts [ρ] .23–25 The
correct scaling properties of Ts [ρ] require that the coefficient in front of the last
terms in OL1 and OL2 are not free but must be non-negative.
P92
Perdew:26
F P92 (s) =
1 + 88.396s2 + 16.3683s4
1 + 88.2108s2
(16.5.25)
F. Tran & T. A. Wesolowski
434
T92
Thakkar:23
F T92 (s) = 1 +
LC94
0.0055b2s2
0.072bs
−
1 + 0.0253bs arcsinh(bs) 1 + 25/3 bs
(16.5.26)
Lembarki and Chermette:27
F LC94 (s) =
2
1 + 0.093907s arcsinh(76.32s) + 0.26608 − 0.0809615e−100s s2
1 + 0.093907s arcsinh(76.32s) + 0.57767 · 10−4 s4
(16.5.27)
This is the enhancement factor of the exchange functional PW91 of Perdew and
Wang28 refitted for the kinetic energy.
FR95A Fuentealba and Reyes:6
F FR95A (s) = 1 +
0.004596b2s2
1 + 0.02774bs arcsinh(bs)
(16.5.28)
This is the enhancement factor of the exchange functional B88 of Becke21 refitted
for the kinetic energy.
FR95B Fuentealba and Reyes:6
1/15
F FR95B (s) = 1 + 2.208s2 + 9.27s4 + 0.2s6
(16.5.29)
This is the enhancement factor of the exchange functional PW86 of Perdew and
Wang29 refitted for the kinetic energy.
Vitos, Skriver, and Kollár:30
VSK98
F
VJKS00
1 + 0.95x + 3.564x3
(x) =
1 − 0.05x + 0.396x2
(16.5.30)
1 + 0.8944s2 − 0.0431s6
(s) =
1 + 0.6511s2 + 0.0431s4
(16.5.31)
VSK98
Vitos et al.:31
F
VJKS00
E00 Ernzerhof:32
F
TW02
E00
135 + 28s2 + 5s4
(s) =
135 + 3s2
(16.5.32)
Tran and Wesolowski:25
F TW02 (s) = 1 + κ −
κ
1 + μκ s2
(16.5.33)
Semilocal Approximations for the Kinetic Energy
435
where κ = 0.8438 and μ = 0.2319. This is the enhancement factor of the exchange
functional PBE of Perdew, Burke, and Ernzerhof 33,34 refitted for the kinetic energy.
PBEn Karasiev, Trickey, and Harris:35
F PBEn (s) = 1 +
n−1
i=1
(n)
Ci
s2
1 + a(n) s2
i
(16.5.34)
This is the enhancement factor of the exchange functional mPBE of Adamo and
Barone36 refitted for the kinetic energy. Three approximations (n = 2, 3, and 4) of
(n)
the above general form were considered in Ref. 35. The coefficients Ci and a(n)
are given in Table 16.1.
Table 16.1. Coefficients of the enhancement factor of
PBEn [Eq. (16.5.34)].
Functional
PBE2
PBE3
PBE4
exp4
(n)
C1
2.0309
−3.7425
−7.2333
(n)
C2
50.258
61.645
(n)
C3
−93.683
a(n)
0.2942
4.1355
1.7107
Karasiev, Trickey, and Harris:35
2
4
F exp4 (s) = C1 1 − e−a1 s + C2 1 − e−a2 s
(16.5.35)
where C1 = 0.8524, C2 = 1.2264, a1 = 199.81, and a2 = 4.3476.
CR
Constantin and Ruzsinszky:37
2
5
1 + a1 + 27
s + a2 s4 + a3 s6 − a4 s8
CR
F (s) =
3
1 + a1 s2 + a5 s4 + 40β−5
a4 s6
(16.5.36)
Three approximations (corresponding to β = 1/5, 1/6, and 0.185) of the above
general form were considered in Ref. 37. The corresponding sets of coefficients ai
are given in Table 16.2.
Table 16.2. Coefficients of the enhancement
factor of CR [Eq. (16.5.36)].
β
1/5
1/6
0.185
a1
a2
a3
a4
a5
1.122609
0.900085
−0.227373
0.014177
0.731298
1.301786
3.715282
0.343244
0.032663
2.393929
1.293576
2.161116
−0.144896
0.025505
1.444659
F. Tran & T. A. Wesolowski
436
MGE2 Constantin et al.:38
F MGE2 (s) = 1 + κ −
κ
1 + μκ s2
(16.5.37)
where κ = 0.804 and μ = 0.23889. This is the enhancement factor of the exchange
functional PBE of Perdew, Burke, and Ernzerhof.33,34
Conjoint B86A Lacks and Gordon:39
F B86A (s) = 1 + 0.00387
b 2 s2
1 + 0.004b2s2
(16.5.38)
This is the enhancement factor of the exchange functional B86A of Becke.40
Conjoint PW86 Lacks and Gordon:39
1/15
F PW86 (s) = 1 + 1.296s2 + 14s4 + 0.2s6
(16.5.39)
This is the enhancement factor of the PW86 exchange functional of Perdew and
Wang.29
Conjoint B86B Lacks and Gordon:39
F
B86B
(s) = 1 + 0.00403
b 2 s2
4/5
(1 + 0.007b2s2 )
(16.5.40)
This is the enhancement factor of the exchange functional B86B of Becke.41
Conjoint DK87 Lacks and Gordon:39
F
DK87
1 + 0.861504bs
7b2 s2
(s) = 1 +
.
4
1/3
324(36π ) 1 + 0.044286b2s2
(16.5.41)
This is the enhancement factor of the exchange functional DK87 of DePristo and
Kress.42
Conjoint B88 Tran and Wesolowski:25
F B88 (s) = 1 +
1/3
where Cx = (3/4) (3/π)
tional B88 of Becke.21
0.0042
b 2 s2
21/3 Cx 1 + 0.0252bs arcsinh(bs)
(16.5.42)
. This is the enhancement factor of the exchange func-
Conjoint PW91 Lacks and Gordon:39
F PW91 (s) =
−100s2
1 + 0.19645s arcsinh(7.7956s) + 0.2743 − 0.1508e
s2
1 + 0.19645s arcsinh(7.7956s) + 0.004s4
(16.5.43)
Semilocal Approximations for the Kinetic Energy
437
This is the enhancement factor of the exchange functional PW91 of Perdew and
Wang.28
Conjoint xFit Lacks and Gordon:39
1
10−8 + 0.1234 2
xFit
s + 29.790s4 + 22.417s6
F
(s) =
1
+
1 + 10−8 s2
0.024974
0.024974
+12.119s8 + 1570.1s10 + 55.944s12
(16.5.44)
This is the enhancement factor of the exchange functional xFit of Lacks and Gordon.43
Conjoint PBE Perdew et al.:44
F PBE (s) = 1 + κ −
κ
1 + μκ s2
(16.5.45)
where κ = 0.804 and μ = 0.21951. This is the enhancement factor of the exchange
functional PBE of Perdew, Burke, and Ernzerhof.33,34
16.6. Other semilocal approximations
TB78
Tal and Bader:45
TsTB78 [ρ]
=
TsTF [ρs ]
+
TsW [ρs ]
+
M
TsW [ρr,A ]
(16.6.46)
A=1
where
ρr,A (r) = ρ(RA )e−2ZA |r−RA |
(16.6.47)
and
ρs (r) = ρ(r) −
M
ρr,A (r)
(16.6.48)
A=1
M , RA , and ZA are the number, positions, and charges of the nuclei, respectively.
CN84 Cummins and Nordholm:46
TsCN84 [ρ]
=
tCN84 (r)d3 r
(16.6.49)
where
tCN84 (r) = max tTF (r), tW (r)
(16.6.50)
F. Tran & T. A. Wesolowski
438
PG85
Pearson and Gordon:47
TsPG85 [ρ]
where
t
PG85
n−1
=
tPG85 (r)d3 r
(16.6.51)
if t2 (r) t0 (r)
if t2 (r) > t0 (r)
(16.6.52)
1
1
TsmGEA4 [ρ] = TsTF [ρ] + TsW [ρ] + Ts,4 [ρ]
9
2
(16.6.53)
i=0
(r) =
t0 (r)
t2i (r) + 12 t2n (r)
mGEA4 Allan et al.:48
PP88
Plindov and Pogrebnya:49
TsPP88 [ρ]
=
TsTF [ρ]
1
+ TsW [ρ] +
9
t4 (r)
1+
1 t4 (r)
8 t2 (r)
d3 r
MGGA Perdew and Constantin:50
F MGGA = F W + F GE4−M − F W fab F GE4−M − F W
(16.6.54)
(16.6.55)
where
F
GE4−M
F GEA4
=
2
1 + 1+F54s2
(16.6.56)
3
and
fab (z) =
⎧
0
⎪
⎨
⎪
⎩
1
a/(a−z)
1+e
ea/z +ea/(a−z)
b
z0
0<z<a
(16.6.57)
za
Equation (16.6.56) is the enhancement factor of a modified 4th order GEA,
where
F GEA4 and F4 are the enhancement factors of the functionals TsGEA4 =
t0 + tJ2 + t4 d3 r and Ts,4 = t4 d3 r, respectively [see Eqs. (16.4.7)−(16.4.13)].
In Eq. (16.6.57), a = 0.5389 and b = 3 are two parameters which were optimized.
GDS08
Ghiringhelli and Delle Site:51
GDS08
W
Ts
[ρ] = Ts [ρ] + ρ(r) (A + B ln(ρ(r))) d3 r
(16.6.58)
where A = 0.860 and B = 0.224.
MGEA4 Lee et al.:17
TsMGEA4 [ρ]
=
(t0 (r) + 1.789t2 (r) − 3.841t4 (r)) d3 r
(16.6.59)
Semilocal Approximations for the Kinetic Energy
439
RDA Karasiev et al.:52
TsRDA [ρ]
TsW [ρ]
=
+
t0 (r)Fθm0 (r)d3 r
(16.6.60)
where
Fθm0
= A0 + A1
κ̃4a
1 + β1 κ̃4a
2
+ A2
κ̃4b
1 + β2 κ̃4b
4
+ A3
κ2c
1 + β3 κ2c
(16.6.61)
with
κ̃4a =
!
s4 + ap2
(16.6.62)
κ̃4b =
!
s4 + bp2
(16.6.63)
κ2c = s2 + cp
(16.6.64)
2/3 5/3 ρ
and p = ∇2 ρ/ 4 3π 2
. The constants in Eqs. (16.6.61)-(16.6.64) are
A0 = 0.50616, A1 = 3.04121, A2 = −0.34567, A3 = −1.89738, β1 = 1.29691,
β2 = 0.56184, β3 = 0.21944, a = 46.47662, b = 18.80658, and c = −0.90346.
16.7. N - and r-dependent approximations
In the approximations collected below, the analytic expressions for the kinetic energy depend explicitly not only on ρ (and its derivatives) but also on the number
of electrons N and/or on the position (i.e., distance r from the nucleus). The
r-dependent expressions are applicable only for mono-atomic systems.
HCD84 Haq, Chattaraj, and Deb:53
TsHCD84 [ρ]
GD94
1
−
40
r · ∇ρ(r) 3
d r
r2
(16.7.65)
Ghosh and Deb:54
TsGD94 [ρ]
F97
=
TsTF [ρ]
=
TsTF [ρ]
3
+
4
1/3 3
ρ4/3 (r)
d3 r
rρ1/3 (r)
π
r 1 + 0.043
(16.7.66)
Fuentealba:55
TsF97 [ρ]
= CF
ρ5/3 (r)
1
1−
2/3
0.213e−8.8(ρ(r)/2) r2
d3 r
(16.7.67)
F. Tran & T. A. Wesolowski
440
Nagy, Liu, and Parr:56
2
NLP99
−2
3
−1
3
Ts
[ρ] = 0.179883 r ρ(r)d r + 0.0622411
r ρ(r)d r
NLP99
ABSP80
−8.88819 · 10
−4
+7.29627 · 10
−6
r
r
Acharaya et al.:57
TsABSP80 [ρ]
=
TsW [ρ]
−2/3
−1/2
3
3
ρ(r)d r
4
3
ρ(r)d r
1.412
+ 1 − 1/3 TsTF [ρ]
N
GR82 Gázquez and Robles:58
1.303 0.029
2
GR82
W
1 − 1/3 + 2/3 TsTF [ρ]
Ts
[ρ] = Ts [ρ] + 1 −
N
N
N
(16.7.68)
(16.7.69)
(16.7.70)
GB85 Ghosh and Balbás:59
TsGB85 [ρ]
0.153 0.071
=
+
+ 0.1369 + 1/3 − 2/3
N
N
2
r · ∇ρ(r) ∇ ρ(r)
d3 r
×
−
r2
2
TsW [ρ]
TsTF [ρ]
(16.7.71)
16.8. Miscellaneous
Approximation for the kinetic potential The modified Thomas-Fermi (MTF)
model of Chai and Weeks:60
vtMTF ([ρ], r) =
5
α ∇2 ρ(r)
CF ρ2/3 (r) −
3
4 ρ(r)
(16.8.72)
where α = 1/2. Note that there exists no functional TsMTF such that vtMTF =
δTsMTF /δρ, because the second term in Eq. (16.8.72) can not be obtained as a
functional derivative.
Approximation for the non-additive kinetic energy Approximation to the
bi-functional Tsnad[ρA , ρB ] by Lastra, Kaminski, and Wesolowski:61
5/3
5/3
5/3
nad(NDSD)
(ρA (r) + ρB (r)) − ρA (r) − ρB (r) d3 r
Ts
[ρA , ρB ] = CF
+ f (ρB (r), ∇ρB (r))ρA (r) vtlimit ([ρB ], r)d3 r + C[ρB ]
(16.8.73)
Semilocal Approximations for the Kinetic Energy
441
where
vtlimit ([ρB ], r) =
1 ∇2 ρB (r)
1 |∇ρB (r)|2
−
8 ρ2B (r)
4 ρB (r)
(16.8.74)
−1 −1 max
λ(−sB +smin
λ(−s
+s
)
B
B ) + 1
B
f (ρB , ∇ρB ) = e
1− e
+1
−1
min
× eλ(−ρB +ρB ) + 1
(16.8.75)
= 0.3, smax
= 0.9 and ρmin
= 0.7), and C[ρB ] is
(with the constants λ = 500, smin
B
B
B
a ρA -independent functional. The acronym NDSD (Non-Decomposable approximation to the potential involving only Second Derivatives) reflects the fact that there
nad(NDSD)
exists no such functional approximation T̃s [ρ] from which Ts
[ρA , ρB ] can
nad(NDSD)
be obtained as Ts
[ρA , ρB ] = T̃s [ρA + ρB ] − T̃s [ρA ] − T̃s [ρB ].
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