Table Of ContentSemiclassical atom theory applied to solid-state physics
Lucian A. Constantin,1 Aleksandrs Terentjevs,2 Fabio Della Sala,2,1 Pietro Cortona,3 and Eduardo Fabiano2,1
1Center for Biomolecular Nanotechnologies @UNILE,
Istituto Italiano di Tecnologia, Via Barsanti, I-73010 Arnesano, Italy
2Istituto Nanoscienze-CNR, Euromediterranean Center for Nanomaterial
Modelling and Technology (ECMT), Via per Arnesano 73100 Lecce, Italy
3Laboratoire Structures, Propri´et´es et Mod´elisation des Solides, CNRS UMR 8580,
E´cole Centrale Paris, Grand Voie des Vignes, F-92295 Chˆatenay-Malabry, France
(Dated: January 26, 2016)
6
1 Using thesemiclassical neutralatom theory,we extendto fourth order themodified gradient ex-
0 pansionoftheexchangeenergyofdensityfunctionaltheory. Thisexpansioncanbeappliedbothto
2 largeatomsandsolid-stateproblems. Moreover,weshowthatitcanbeemployedtoconstructasim-
ple and non-empirical generalized gradient approximation (GGA) exchange-correlation functional
n
competitivewithstate-of-the-artGGAsforsolids, butalsoreasonablyaccurateforlargeatomsand
a
J ordinary chemistry.
5
PACSnumbers: 71.10.Ca,71.15.Mb,71.45.Gm
2
]
r I. INTRODUCTION The two gradient expansions discussed above have
e
been employed to develop GGA functionals accurate ei-
h
t Density functionaltheory (DFT)1–3 is one of the most ther for solid-state (e.g. the PBEsol functional of Ref.
o 7), or for chemistry (e.g. the APBE of Ref. 34). Nev-
popular computational approaches to material science
.
t and condensed-matter physics. However, the final accu- ertheless, the exchange enhancement factor Fx (defined
a by E [n] = dr ǫLDAF (s)) of both both PBEsol and
m racy of DFT calculations depends on the approximation x x x
APBE behavRes as
usedfortheexchange-correlation(XC)functional,which
-
d describesthequantumeffectsontheelectron-electronin- µ2
n teraction. Thus, the developmentandtesting ofnew XC Fx(s→0)→1+µs2+νs4+O s6 withν =− , (3)
κ
o functionals have been active research fields during the (cid:0) (cid:1)
c last decades4–6. where κ = 0.804 is fixed from the Lieb-Oxford
[
Model systems are fundamental tools for the devel- bound37,38, and µ is the pertinent second-order co-
1 opment of non-empirical DFT functionals. One popu- efficient. The importance of the fourth-order term
v larmodelis the electrongaswithslowly-varyingdensity. in this development has been already discussed in
4 Performing a second-order gradient expansion (GE2) of literature14,39–41. It is important for intermediate val-
9
the exchange energy density ǫ , this model gives ues of the reduced gradient (0.3 / s / 1) as those often
4 x
6 encounteredin bulk solids. Indeed, functionals whichre-
0 ǫx =ǫLxDA(1+µGE2s2) , (1) coverGE2,butwiththe s4 terminthe Taylorexpansion
. ofexchangesettozero,e.g. theonesinRefs10,14,showa
1
0 where ǫLxDA = −(3/4)(3/π)1/3n4/3 is the exchange en- quite different behavior with respect to PBEsol(see also
6 ergy density in the local density approximation(LDA)2, results in Ref. 15). On the other hand, the relevance
1 n is the electron density, s = |∇n|/[2(3π2)1/3n4/3] is ofhigher-orderterms inthe modifiedgradientexpansion
: the dimensionless reduced gradient, and µGE2 = 10/81 has not yet been investigated.
v
i is the GE2 coefficient. The slowly varying density InthisworkweconsiderthisissueandweusetheSCA
X regime is considered a paradigm for solid-state physics, theorytointroduceanextensionofthemodifiedgradient
r and the GE2 has been successfully used as a key tool expansion to fourth-order (MGE4). We show that this
a
to develop Generalized Gradient Approximation (GGA) expansionisappropriateforbothsemiclassicalatomsand
functionals7–15 as well as meta-GGA functionals16–22. solid-stateproblems. Moreover,asimpleGGA,basedon
Another important model system is the semiclassical MGE4, is constructed. These achievements will empha-
neutral atom (SCA), whose theory was established sev- sizethe importanceofthe SCAmodelforDFT,notonly
eral years ago23–29. This model has been recently used when one is concerned with finite systems34, but also in
to derive a modified second-order gradient expansion the popular field of solid-state physics.
(MGE2) for exchange30
ǫ =ǫLDA(1+µMGE2s2) , (2) II. THEORY
x x
where µMGE2 = 0.26. This expansion has been shown In Ref. 30, the MGE2 coefficient was derived by re-
tobe relevantfortheaccurateDFT descriptionofatoms quiring that the expansion given in Eq. (2) has to be
and molecules30–36. large-Z asymptotically exact to the 1st degree. Thus, in
2
practice,anenergyconstraintwasappliedandtheMGE2
µ
was forcedto recoverthe correctlowestcoefficientof the
semiclassical expansion of the exchange energy. 0.20
An alternative way to derive the MGE2, is to im-
pose that, in the slowly-varying density region of non- 0.10
relativistic large neutral atoms (i.e. with Z → ∞), the o
modified gradientexpansionrecoversthe exactexchange ati
r0.00
energy. Thus, for Z →∞, we have
ǫexactdr= ǫLDA(1+µs2+νs4+...)dr. (4) -0.10 ν
Z x Z x
V V
The integration is performed on the slowly-varying den- -0.20
0 0.1 0.2 0.3 0.4
sity region V, defined by the condition −1 ≤ q ≤ 1,
-1/3
where q = ∇2n/{4(3π2)2/3n5/3}. Note that this region Z
dominates for an atom with an infinite number of elec-
FIG. 1: Right-hand-side ratios of Eqs. (6) and (7) (i.e. µ
trons and it is also the only one where a gradient ex-
and ν values) computed for different noble-gas atoms up to
pansion makes sense. The use of the reduced Laplacian
Z = 290. The extrapolation to Z → ∞ was done using a
q in the definition of V is motivated by the fact that parabolic fit as in Refs. 30,33,36.
slowly-varying density regions of atoms cannot be de-
fined in terms of s only, because the reduced gradient is
small also near the nuclear cusp42, where the density is istic SCA enhancement factor in the slowly-varyingden-
rapidly varying. Instead, they are well identified by con- sity limit. With this choice we also obtain to reduce the
sideringthereducedLaplacianq andusingthe condition importance of high-density regions, which instead dom-
|q| ≈ s2 / 1 (conversely q → −∞ near the nucleus and inate the MGE2 behavior, since they are the ones that
q →∞ in the tail). mostly contribute to the energy density, Thus, we can
The semiclassical theory of atoms is based on the achievea more balanceddescriptionofthe whole slowly-
Thomas-Fermi scaling31, which implies the following varying regime (including low-density regions, which are
scaling rules for the density and the reduced gradients: rather important in real bulk solids). We recall that the
enhancement factor is not an observable and is defined
nλ(r)=λ2n(λ1/3r) , rsλ(r)=λ−2/3rs(λ1/3r) , only up to a gauge transformation43. Nevertheless, the
sλ(r)=λ−1/3s(λ1/3r) , qλ(r)=λ−2/3q(λ1/3r) , enhancement factor Fxexact = ǫexxact/ǫLxDA of the con-
(5) ventional exact exchange energy density is well defined
with λ → ∞, while the nuclear charge behaves as Z → and has a clear physical meaning (i.e. it measures the
λZ, in order to preserve the total charge neutrality. In interaction between an electron and the true exchange
Eq. (5), rs = (3/4πn)1/3 is the Wigner-Seitz radius. hole). Note also that the non-uniqueness problem is re-
Usingthesescalingrelations,weobtain(inthelimitλ→ ducedintheslowly-varyingdensitylimit,wheretheexact
∞) exchange hole has a semilocal expansion44–46 which be-
comes unique for the uniform electron gas. Thus, Fexact
µMGE2 = lim V(ǫexxact[Z]−ǫLxDA[Z])dr . (6) can be safely used as a reference for our scope. x
Z→∞R V ǫLxDA[Z]s[Z]2dr Following Eq. (6), we define an effective fourth-order
R coefficient (in the spirit of Eq. (3)) as
Here[Z]denotesthatallquantitiesarecomputedforthe
non-relativistic atom with Z electrons. dr(Fexact[Z]−FMGE2[Z])
νMGE4 = lim V x x . (7)
The values of the ratio on the right hand side of Eq. Z→∞R drs[Z]4
V
(6) for different noble-gas atoms, up to Z = 290, are R
reported in Fig. 1 together with the extrapolation to We remark that any fourth-order gradient expansion of
Z → ∞. We obtain µMGE2 = 0.262, that is practically the exchange energy diverges for atoms, because of the
the same result as in Ref. 30. exponential decay of the density. Using the integration
In principle, Eq. (4) could be also used to obtain technique proposed in Eq. (7), we remove this difficulty
higher-order results. Nevertheless, such an approach and we focus on the slowly-varying density regime, that
is prone to large oscillations because the integral of istheonlyonewhereagradientexpansioniswelldefined.
ǫexact−ǫLDA(1+µMGE2s2) is small by construction, as Finally,wehighlightthatthefourth-ordergradientex-
x x
it is that of ǫLDAs4, for Z → ∞. Thus, in order to pansionof exchange depends, in general,onboth the re-
x
extend the modified gradient expansion to fourth order, ducedgradientandthe Laplacian. However,aLaplacian
we need to impose an additional constraint beyond the dependence is beyond the scope of this paper. Instead,
energy one. We do this by requiring that the modified usingEq. (7),weaimtoextractaneffectivefourth-order
gradientexpansionnotonly reproducesthe SCAasymp- coefficient in the spirit of Eq. (3).
totic energy (which yields MGE2), but also gives a real- The values of the ratio on the right hand side of Eq.
3
(7) for different noble-gas atoms, up to Z = 290, are 1e+04
A
reported in Fig. 1 together with the extrapolation to D 1e+03
Lx
Z → ∞. The behavior with Z is regular, showing the ε 1e+02
2
r
physical meaningfulness of Eq. (7). Extrapolation to π 1e+01
4
Z →∞ gives - 1e+00
s 1.2
t
n 0.9
νMGE4 =−0.195. (8) die 0.6 s
a
r 0.3
g
Note that this limit value is independent on the exact d. 0.0 q
e-0.3
valuesusedtodefinetheboundariesofregionV. Indeed, R
0.09
the same νMGE4 is obtained using −0.8 ≤ q ≤ 0.8 or x
−1.2≤q ≤1.2 (plots not reported). pro 0.06 GE4
ThecoefficientofEq. (8),togetherwithµMGE2,define apF0.03 GE2
the modified fourth-order gradient expansion (MGE4), act-0.00 MGE4
with the following enhancement factor exF-0.03 MGE2
-0.06
0 1 2 3 4
FMGE4 =1+µMGE2s2+νMGE4s4 . (9) r (au)
x
FIG. 2: Radial LDA exchange energy density (top panel),
This reproduces, as close as possible, the conventional
reduced gradients s and q (middle panel), and (lower panel)
exactexchangeenhancementfactorintheslowly-varying the deviation, with respect to the exact conventional one, of
density regime of non-relativistic large neutral atoms. the exchange enhancement factors of several gradient expan-
Note that νMGE4 is rather different from the fourth- sions, as functions of thedistancer from thenucleus,for the
order coefficient that is implicitly employed in APBE non-relativistic noble atom with 290 electrons.
(−(µMGE2)2/κ=−0.084).
The main features of the MGE4 can be seen in Fig. 2
where we plot, for the non-relativistic noble atom with
A. Assessment of MGE4 for jellium cluster models
290 electrons and in the region V defined before: the
radial LDA exchange energy density, the reduced gradi-
entssandq,andthedeviation,withrespecttotheexact
The MGE4 is accurate by construction for the slowly-
conventional one, of the exchange enhancement factors
varying density regime of large non-relativistic neutral
of several gradient expansions. Here GE4 is the con-
atoms. To test it on a different model, we consider its
ventionalfourth-ordergradientexpansiondefined by the
performance for jellium clusters. These systems satisfy
enhancementfactorFxGE4 =1+10/81s2+146/2025q2− the uniform-electron-gas scaling47 (n (r) = n(λ−1/3r))
73/405qs2 + Ds4, where D = 0 is the best numerical λ
and, in the limit of a large number of electrons,they are
estimation for this parameter16. We observe that:
representative for solid-state systems.
i) In the inner atomic core (i.e. for r / 1), MGE2 is
In Fig. 3 we have plotted the relative error ∆E =
very accurate and, because both s and q are relatively x
(Eexact−Eapprox)/ELDA,computedoverthevolume(V
small, the forth-order terms in gradient expansions are x x x
defined as in Eq. (7)), for jellium clusters of bulk pa-
not much significant. Consequently, MGE4 is as accu-
rameter r = 4 and up to Z = 2048 electrons. The
rate as MGE2. On the other hand, the GE2 and GE4 s
restrictionoftheintergaldomaintothevolumeV allows
exchange enhancement factors are smaller, on average,
to remove the non-integrable region for the fourth-order
than the exact one and they do not describe with high
terms (i.e. the tail of the density) and to consider solely
accuracy this energetical region. In fact, despite both s
theslowly-varyingdensityregion. TheLDA,MGE2,and
and q are not large in this region, only q is very close to
GE4 results are also reported in the figure.
zero,while s≈0.3. Therefore,the conventionalgradient
expansions do not work very well in this regime. Note It can be seen that MGE4 behaves similarly to GE4,
that the inner-core high-density region gives the main
whichisderivedfromtheslowly-varyingdensitylimitbe-
contribution to the exchange energy (99.3% of it). havior. Actually, MGE4 gives even the best results for
ii) In the outer atomic core (i.e. for 1 / r / 4), largerclusters. Fortheselattersystems,MGE4alsoout-
the reduced gradientss and q start to increase,but they performs MGE2, which is instead more accurate for the
are still smaller than 1, thus the density is still slowly- smallest clusters. To our knowledge, MGE4 is the only
varying. This atomic region is not very important for expansionwhichisrealisticforatoms(thatarecharacter-
the total exchange energy of atoms, but it is a model for izedbytheThomas-Fermiscaling47 n (r)=λ2n(λ1/3r))
λ
solid-state problems (where the high-density limit is not andjellium models (whicharemodels forsolid-stateand
common). While MGE2 fails in this region, the MGE4 are based on the uniform-electron-gas scaling47 n (r) =
λ
and GE2 are very accurate. n(λ−1/3r)).
4
1e-01 1.5
or1.4 APBE
ct SG4
∆|Ex1e-02 LGMDEGA4E2 ment fa11..23 PMBGEEso4l
| MGE4 e
c
n1.1
a
h
1e-03 En1.0
0.1 0.2 0.3 0.4 0.5 0.9
-1/3 0.0 0.5 1.0 1.5 2.0
Z s
FIG. 3: Relative exchange errors |∆Ex| = |(Exexact − FIG.5: Enhancementfactorsofdifferentfunctionalsasfunc-
Exapprox)/ExLDA|forjelliumclustersofbulkparameterrs=4 tions of the reduced gradient s.
and up to Z =2048.
pletely constructed from the SCA model. Its enhance-
PBEsol
6e-03 mentfactoris reportedinFig. 5. Forsmallvalues ofthe
APBE reduced gradient s, it is close to the APBE one, since in
P|4e-03
I this case MGE2 and MGE4 are very similar (they coin-
∆
| cide in the limit of very small s values). However,unlike
2e-03
APBE,the SG4 functional recoversMGE4 until s≈0.6.
0e+00 For larger values of the reduced gradient the SG4 en-
0.02 0.04 0.06
µ hancement factor is between the APBE and the PBEsol
1
ones.
FIG. 4: Absolute error (Ha) for the exchange ionization po- To complete the SG4 functional we need to comple-
tential in the SCA limit as a function of the parameter µ1. ment it with a correlation functional. This must de-
The errors of the PBEsol and APBE functionals are also re- scribe accurately the SCA correlation expansion E ≈
c
ported for reference. 0.02072Zln(Z)+BZ+... (where B =0.038 is the best
estimateforthefirst-ordercoefficient48),andrecoverthe
APBE correlation in the core of a large atom (i.e. for
B. Construction of a generalized gradient r →0 ands→0; note that, for exchange,SG4→APBE
s
approximation functional based on MGE4
in this limit). Hence, we consider the simple correlation
energy per particle
To demonstrate the practicalutility of MGE4, we em-
ployittoconstructasimplegeneralizedgradientapproxi- ǫSG4 =ǫLDA+φαt3H(r ,ζ,t), (11)
c c s
mation (GGA) functional named the SemiclassicalGGA
where t = |∇n|/(2k φn) is the reduced gradient for
at fourth-order (SG4). Being based on MGE4, we ex- s
correlation49, with k = (4k /π)1/2 being the Thomas-
pectthattheSG4functionalperformswellforbothlarge s F
Fermi screening wave vector (k = (3π2n)1/3), φ =
atomsandsolid-statesystems;moreover,wewillseethat F
it is rather accurate also for ordinary chemistry. (1+ζ)2/3+(1−ζ)2/3 /2 is a spin-scaling factor, ζ =
h i
The SG4 exchange enhancement factor takes the form
(n −n )/n is the relative spin polarization, and H is a
↑ ↓
localized PBE-like gradient correction49,50 where we use
κ (1−µ s2/κ ) κ
FSG4 =1+κ +κ − 1 1 1 − 2 ,
x 1 2 1−(µ1s2/κ1)5 1+µ2s2/κ2 β =β0+σt(1−e−rs2) . (12)
(10)
where the condition κ +κ = 0.804 is fixed from the InordertorecovertheaccurateLDAlinearresponse34,51,
1 2
Lieb-Oxford bound37, while µ1 +µ2 = µMGE2 = 0.26 wefixβ0 =3µMGE2/π2. Moreover,wefixtheparameter
and κ = −µ2/νMGE4 are imposed to recover MGE2 σ = 0.07 fitting to jellium surface exchange-correlation
2 2
andMGE4,respectively. NotethatinaTaylorexpansion energies52 (in analogy to PBEsol7) and α = 0.8 min-
around s = 0, the fourth term on the right hand side of imizing the information entropy function described in
Eq. (10)contributesonlywithµ s2+O(s10),whereasthe Ref. 53. We recall that the spin-correction factor
1
MGE4behavioriscompletelydescribedbythelastterm. φαt3 is always equal to one for spin-unpolarized sys-
Such a simple splitting allows a better understanding of tems (e.g. nonmagnetic solids), being importantonly in
the physics behind the functional. It remains only one the rapidly-varying spin-dependent density regime (e.g.
free parameter not fixed by the previous slowly-varying small atoms)53.
density conditions. We fix it to µ = 0.042 by fitting to Equations (10) and (11) define the SG4 exchange-
1
the exchange ionization potential in the SCA limit (see correlation functional which satisfies, with no empirical
Fig. 4). parameters, many exact properties, including the con-
In such a way, the SG4 exchange functional is com- straints derived from the SCA theory.
5
1e-01
TABLE I: Mean absolute errors for equilibrium lattice con-
E|x1e-02 LAPBDPEBAsEol sfotarnstesvearnadl cbhuelkmimstorydutleisotsf a[insedtetoafil2,9atboumlkizamtiaotnerieanlseragineds
∆ SG4
| of main-group molecules (G2/97) and transition metal com-
1e-03
plexes (TM10AE), metal-organic interfaces (SI12), bond
lengths involving H atoms (MGHBL9) and not involving H
atoms (MGNHBL11), interaction energies of hydrogen bond
1e-02 anddipole-dipolecomplexes(HB6+DI6),interactionenergies
E|x of dihydrogen bond complexes (DHB23)]. The best (worst)
∆
| results of each line are in boldface (underlined).
1e-03
APBE PBE SG4 PBEsol WC
0.1 0.2 0.3 0.4 0.5
Z-1/3 Lattice constants (m˚A)
simple metals 34.1 31.4 41.7 55.6 58.4
FIG. 6: Relative exchange errors (∆Ex = |Exexact − transition metals 63.7 44.9 23.9 25.6 23.7
Eapprox|/ELDA) for noble-gas atoms up to Z = 290 elec-
x x semiconductors 102.3 85.3 21.0 32.7 31.5
trons(upperpanel)andforjelliumclusters(lowerpanel)with
ionic solids 95.6 76.0 20.8 20.0 18.8
rs=4, up toZ =2048.
insulators 34.0 27.8 8.0 8.5 8.0
total MAE 66.0 53.0 24.9 31.0 30.7
III. RESULTS LC20 70.3 55.9 23.1 34.0 31.8
bulk moduli (GPa)
In this section we present a general assessment of the simple metals 1.3 1.1 0.7 0.2 0.54
performance of the SG4 functional for solid-state prob- transition metals 26.3 21.2 20.0 20.2 18.9
lems, which is the main topic of this work. For com- semiconductors 18.7 16.9 5.8 8.2 8.0
pleteness, severalatomic andmolecular tests are alsore- ionic solids 9.6 8.5 6.2 3.9 4.8
ported. Finally, we consider some application examples,
insulators 18.6 15.3 4.9 6.2 6.1
to show the practical utility of MGE4 and the related
total MAE 14.8 12.4 7.9 8.2 8.0
SG4 functional in condensed-matter physics.
cohesive energies (eV)
simple metals 0.09 0.05 0.21 0.15 0.13
transition metals 0.32 0.21 0.39 0.62 0.51
A. General assessment
semiconductors 0.29 0.13 0.09 0.28 0.20
ionic solids 0.19 0.14 0.20 0.07 0.06
At first, we consider a general assessment of the ex-
insulators 0.12 0.16 0.36 0.57 0.47
change only SG4 functional. This will allow a more
total MAE 0.21 0.14 0.24 0.33 0.27
direct evaluation of the importance of the MGE4 re-
covery. In Fig. 6 we show the relative errors ∆E = Molecular tests (kcal/mol ; m˚A)
x
|Eexact−Eapprox|/ELDA for noble atoms (upper panel) G2/97 8.9 14.8 15.7 37.7 27.6
x x x
and jellium clusters (lower panel), for several exchange TM10AE 11.1 13.0 11.9 18.3 15.8
functionals. The PBEsol exchange, which is not based SI12 5.9 3.7 2.6 3.8 3.3
on the SCA theory, is not accurate for atoms33, while MGHBL9 10.0 11.5 10.3 14.5 13.9
APBE and SG4 are very accurate. This result is not
MGNHBL11 9.2 7.6 3.7 5.2 6.1
highly surprising, since both these functionals are con-
HB6+DI6 0.4 0.4 0.5 1.3 0.9
structed to recover the SCA theory. Nevertheless, it is
DHB23 0.8 1.0 1.1 1.8 1.6
interesting to note that a good performance is obtained
notonlyforverylargeatoms,butalsoforthemoderately
small ones. In the case of jellium clusters, APBE is the
best for Z < 100, while SG4 becomes more accurate for
larger values of Z. According to the liquid-drop model benchmark set for lattice constants includes, as a sub-
theory of jellium spheres47, this means that it describes set, the LC20 benchmark set of Refs. 21,58, which is
accuratelytheexchangequantumeffectspresentinthese also reported in Table I. The comparison is done with
systems. APBE34, that is the other non-empirical XC functional
Next, wediscussthe performanceofthe fullexchange- based on the SCA theory, as well as with the PBEsol7,
correlation SG4 functional for some basic solid-state PBE49, and Wu-Cohen (WC)40 functionals, which are
tests. For completeness, several molecular tests are also amongthemostpopularGGAsforsolids(anotherpopu-
reported. In Table I we show the results of SG4 calcula- larsolid-statefunctionalistheAM0554–56(notreported),
tions forthe lattice constants,bulk moduli, andcohesive which performs similarly to PBEsol and WC).
energies of a set of 29 bulk solids (see Section V). Our It can be seen that SG4 works remarkably well for
6
solids. It outperforms APBE (and PBE) and is of-
ten even better than the state-of-the-art GGA for solids TABLE II: (111) surface energies (J/m2) and monovacancy
formation energies (eV)inseveralsimpleandtransition met-
PBEsol and WC. The comparison of the SG4 results
als. Mean absolute errors (MAE) are reported in the last
for lattice constants and bulk moduli with the APBE line. Values in best agreement with experiments60–67 are in
ones shows the relevance of MGE4 for solid-state sys-
boldface.
tems. WehighlightthattheSG4resultfortheLC20test
(MAE=23.1m˚A)alsocompeteswiththeonesofthebest Metal PBEsol SG4 Exp.
meta-GGAs for solids. From literature, we found indeed Surface energies (J/m2)
the following MAEs for the LC20 test set: TPSS = 43 Al 0.96 1.06 1.14
m˚A58, revTPSS = 32 m˚A58, SCAN = 16 m˚A21. This is Ca 0.52 0.54 0.50
a remarkable performance of the SG4 functional for the
Sr 0.40 0.41 0.42
equilibrium lattice constants of bulk solids, suggesting
Cu 1.61 1.67 1.79
that the MGE4 gradient expansion can also be a useful
Pt 1.83 1.89 2.49
tool for further meta-GGA development.
Rh 2.45 2.51 2.70
The results for the cohesive energies display a quite
different trend. Actually, this property involves a differ- Au 0.98 1.01 1.50
ence between results from bulk and atomic calculations. Pd 1.69 1.72 2.00
Thus, the best results are found for the PBE functional, MAE 0.27 0.23
whichprovidesthebesterrorcancellation(notethatPBE Monovacancy energy (eV)
is the best neither for solid-state nor for atoms). The Cu 1.25 1.35 1.28
SG4 functional performs overall similarly as the APBE
Ni 1.73 1.83 1.79
one,beingslightlypenalizedbytheneedtoincludesmall
Pd 1.49 1.59 1.85
atoms’calculations. Nevertheless,SG4definitelyoutper-
Ir 1.88 2.04 1.97
formsPBEsol,whichyieldsaquitepoordescriptionofall
Au 0.65 0.78 0.89
atoms.
Cohesive energy results can be rationalized even bet- Pt 1.02 1.15 1.35
ter looking at the outcome of several molecular tests. MAE 0.19 0.13
These tests are also useful to obtain a more comprehen-
sive assessment of the performance of the functionals,
eventhoughwerecallthatthe focusofthepresentpaper
is onsolid-state properties. Moreover,the comparisonof These tests involve a comparison between bulk energies
SG4 with PBEsol provides a hint of the relevance of the inadelocalizedelectronicsystem(metal)andtheenergy
SCA theory underlying the SG4 construction. ofthequitelocalizedsurface/vacancy. Thus,theymaybe
Inspection of the lowest panel of Table I shows that the idealplaygroundforthe SG4functionalwhichshows
SG4 is quite accurate for molecular tests. It is compa- a goodperformance for bulk, being simultaneously quite
rable with PBE for atomization and non-covalent ener- accurate also for confined systems thanks to the under-
gies (within chemical accuracy), and very accurate for lying SCA theory.
geometryand interactionenergiesat interfaces. The lat- Indeed, SG4 performs remarkably well for both prob-
ter results are especially interesting, since these tests re- lems, yielding MAEs of 0.23 J/m2 and 0.13 eV, which
quire a delicate balance between the description of dif- compare favorablywith those of PBEsol(0.27 J/m2 and
ferent density regimes51, which is important for broad 0.19 eV). WC is close to PBEsol but slightly worse
applicability at the GGA level10,51,59. In particular, the (MAEsare0.29J/m2 and0.22eV);PBEandAPBEare
molecular bond lengths in the MGNHBL11 test (that systematically worse than PBEsol and are not reported.
do not imply bonds with hydrogen atoms) are best de- Notably, the improvement is also systematic, since SG4
scribed by semilocal functionals with low non-locality51 is alwayscloser to the experimental values than PBEsol,
(e.g. PBEsol), while the ones in the MGHBL9 test re- with the only exception of Ca and Cu for surface and
quire a large amount of non-locality (APBE works at monovacacy formation energies, respectively.
best). SG4 appears to be able to capture well both situ-
ations and yields a total mean absolute error (MAE) for
geometry of 6.7 m˚A, better than both APBE (9.5 m˚A) C. Structure of boehmite and diaspore crystals
and PBE (9.4 m˚A).
In Table III we list the structural parameters, as de-
fined in Fig. 7, computed for the boehmite and diaspore
B. Surface and monovacancy formation energies crystals. These systems consist of layers of aluminum
hydroxides bound together by hydrogen bonds. We re-
InTableII wereportthe surfaceenergiesofthreesim- call that layeredsolids are becoming increasingly impor-
plemetalsandfivetransitionmetals,aswellasthemono- tant in materials science applications, thanks to their
vacancy formation energies in several transition metals. anisotropic behavior. However, an accurate description
7
Boehmite Diaspore
TABLE IV: Calculated lattice constants (˚A) for ice Ih and
β-AgIaswellasthecorrespondinglatticemismatch. Experi-
mentaldataaretakenfrom Ref. 71. Thebest result for each
line is highlighted in bold.
PBE PBEsol SG4 Exp.
a (ice Ih) 4.42 4.29 4.33 4.50
b (β-AgI) 4.68 4.56 4.57 4.59
mismatch 5.7% 6.1% 5.4% 2.2%
FIG.7: Structuralparametersofboehmiteanddiasporecrys- bofβ-AgI,whichareusedtodefine the latticemismatch
tals.
2(b−a)
f = . (13)
b+a
TABLE III: Lattice parameters (a, b, c) and various atomic
distances for the boehmite and diaspore crystals. All data
Thelatticemismatchf isanimportantquantityinmany
arein˚A.Thereferencedataaretakenfrom Refs. 68,69. The
applications,sinceitdeterminesthegrowthrateoficeon
best result for each line is highlighted in bold.
a β-AgI surface(for example,β-AgI is usedas seedcrys-
PBEsol SG4 Exp. tal to produce artificial rainfall). However, the compu-
Boehmite tational determination of the lattice mismatch is a quite
hard task, since it involves the simultaneous calculation
a 2.868 2.863 2.868
ofthelatticeconstantsoftwomaterialswithquitediffer-
b 11.839 11.858 12.234
ent electronic properties. For this reason its calculation
c 3.714 3.713 3.692
is a challenge for semilocal density functionals71,72.
Al-O 1.911 1.909 1.907
In Table IV we report the computed lattice constants
OH 1.027 1.019 0.970 andthecorrespondinglatticemismatch,asobtainedfrom
H···O 1.545 1.565 1.738 several functionals. In this case we also report PBE
Diaspore results, since for this problem PBE is one of the best
a 4.360 4.362 4.401 GGAs, performing better than PBEsol. Indeed, we see
b 9.411 9.399 9.425 that PBE yields quite good results for the lattice con-
stant of both materials, showing errors below 0.1 ˚A in
c 2.848 2.842 2.845
both cases. However, because the errors for ice Ih and
Al-O 1.917 1.915 1.915
β-AgI lattice constantshaveoppositesigns,the finallat-
OH 1.039 1.031 0.989
tice mismatch is computed rather inaccurately. On the
H···O 1.524 1.535 1.676
other hand, PBEsol performs well for β-AgI, but yields
a muchlargererrorfor ice Ih. Thus,the resulting lattice
mismatch is definitely overestimated. A better balance
in the performance is seen instead in the SG4 case. This
oftheequilibriumstructureofthesesystemsrequiresthe functionalisinfactthe bestforβ-AgIandbetweenPBE
ability to describe both covalentandnon-covalentbonds and PBEsol for ice Ih. Hence, it finally yields a lattice
inthebulkwithsimilaraccuracy. Thisisaquitedifficult mismatch of 5.4%. This value is still too large with re-
task for GGAs70. spectto the experimentalone (2.2%),but improveswith
In general, the PBEsol functional is among the best respect to the other GGAs.
GGAs for boehmite and diaspore70. It describes with
good accuracy the covalent bonds, but it suffers of
some limitations in the description of the non-covalent IV. SUMMARY AND CONCLUDING
ones. The SG4 functional preserves the good features of REMARKS
PBEsol and provides, in addition, small but important
and systematic improvements for hydrogen bonds.
InthispaperwehaveusedtheSCAtheorytointroduce
a modified fourth-order gradient expansion (MGE4),
whichisaccurateforatomsandsolid-statesystems. The
D. Ice lattice mismatch problem MGE4includesandextendsthe well-knownMGE2. The
extension over MGE2 is obtained by using, in the gra-
Theicelatticemismatchproblemisapopularproblem dient expansion construction, an additional constraint,
in solid-state physics71,72. It involves the calculation of beyond the energy one, which provides an improved de-
thelatticeconstantaoficeIhandofthelatticeconstant scription of slowly-varying regions of different materials.
8
To implement the new constraint, an integration over atoms (MGNHBL1184) bonds.
slowly-varying density regions was introduced in order
to reduce computational noise and avoid the divergence • Non-covalent interactions: interaction energies
of fourth order terms in rapidly-varying density regions. of hydrogen-bond and dipole-dipole complexes
To exploit the goodfeatures of MGE4 we have used it (HB6+DI685) as well as of dihydrogen bond com-
as a base to construct a simple GGA functional, named plexes (DHB2386).
SG4. This functional is free of parameters fitted on
real systems and satisfies several exact properties, in-
• Solid-statetests: Equilibriumlatticeconstantsand
cluding those relevant for the SCA model, the Lieb-
bulk moduli of 29 solids, including Al, Ca, K, Li,
Oxford bound, the LDA linear response behavior, and
Na,Sr,Ba(simplemetals);Ag,Cu,Pd,Rh,V,Pt,
the rapidly-varying as well as the high-density limits of
Ni(transitionmetals);LiCl,LiF,MgO,NaCl,NaF
correlation. TheSG4functionalperformswellforabroad
(ionic solids); AlN, BN, BP, C (insulators); GaAs,
range of problems in solid-state physics, still preserving,
GaP, GaN, Si, SiC, Ge (semiconductors). Refer-
thanks to the SCA underlying theory, a reasonable per-
ence data to construct this set were taken from
formance also for molecular tests. Due to this eclectic
Refs. 58,87–90.
character, the SG4 functional is particularly promising
for problems involving multiple electronic structure fea-
All calculations for molecular systems have been per-
tures, such as surface energies, vacancies, non-covalent formed with the TURBOMOLE program package91,92,
interactions in bulk solids, and interfaces. usingadef2-TZVPPbasisset93,94. Calculationsconcern-
The results of the present work highlight, through the
ingsolid-statetests havebeen performedwiththe VASP
power of MGE4 for different problems, the importance program95, using PBE-PAW pseudopotentials. We re-
of the underlying SCA model as a reference system in
mark that the use of the same pseudopotential for all
DFT also for solid-state systems. This was the primary
the functionals may lead to inaccuracies in the final re-
goal of the present work, since the relevance of the SCA
sults. Nevertheless, the use of PAW core potentials en-
model system was often overlookedin the literature and sures good transferability for multiple functionals56,57,
the utility of this model has been often considered to be
since the core-valenceinteractionis recalculatedfor each
limited to the atomic and molecular framework.
functional. Indeed, test calculations employing differ-
Toconcludewerecallthatgradientexpansionsareba-
ent variants of the PAW potentials (GGA-PAW) have
sic tools for the contruction of non-empirical DFT func-
shown that the estimated convergence level of our cal-
tionals, even beyond the semilocal XC level16,44,45,73–76. culations is about 1 m˚A for lattice constants, 0.5 GPa
Thus,in the future, further studies mayfocus onthe use
for bulk moduli, and 0.01 eV for cohesive energies. All
ofMGE4toconstructandoptimizehighlyaccuratefunc-
BrillouinzoneintegrationswereperformedonΓ-centered
tionals of different ranks, beyond the simple SG4 that
symmetry-reduced Monkhorst-Pack k-point meshes, us-
we have presented here mainly to illustrate the practical
ing the tetrahedronmethod with Blo¨chlcorrections. For
utility of the SCA theory.
all the calculations a 24×24×24 k-mesh grid was ap-
plied and the plane-wave cutoff was chosen to be 30%
larger than maximum cutoff defined for the pseudopo-
V. COMPUTATIONAL DETAILS
tential of each considered atom. The bulk modulus was
obtained using the Murnaghan equation of state. The
All atomic calculations used to derive the MGE4 ex- cohesive energy, defined as the energy per atom needed
pansion have been performed using the Engel code77 to atomize the crystal, is calculated for each functional
with the exact exchange functional. Further tests of the fromtheenergiesofthecrystalatitsequilibriumvolume
MGE4 on jellium clusters have been carriedout employ- andthe spin-polarizedsymmetry-brokensolutionsof the
ing accurate LDA Kohn-Sham densities78. constituent atoms. To generate symmetry breaking so-
The SG4 functional has been tested on different lutions, atoms were placed in a large orthorhombic box
datasets, including with dimensions 13×14×15 ˚A3.
Calculations for the examples reported in subsections
• Atomization and interaction energies: atomiza-
tion energies of small molecules (G2/9779,80), at- IIIB,IIIC,andIIIDhavebeenperformedusingthesame
computational setups as in Refs. 62,64,70,71.
omization energies of small transition metal com-
plexes (TM10AE53,81,82), small gold-organic inter- Acknowledgments. We thank Prof. K. Burke for
faces (SI1283). usefuldiscussionsandTURBOMOLEGmbHfor provid-
ing the TURBOMOLE program. E. Fabiano acknowl-
• Structural properties: bond lengths involving H edges a partial funding of this work from a Centrale-
atoms(MGHBL984), bond lengths notinvolvingH Sup´elec visiting professorship.
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