Ground-statecorrela/onenergywithinthe
ACFDT:Theoryandimplementa/on
Dario Rocca!
!
!
Université de Lorraine and CNRS!
Nancy!
!
!
!
!
Goal:Implementa/onof“quantumchemical”methods
withinQuantumEspresso
q  Random phase approximation !
q  Random phase approximation with exchange:
SOSEX and approximate TDHF!
q  MP2 !
• 
• 
So far implemented as post-DFT!
Implementation as post-HF challenging!
Outline
q  Randomphaseapproxima/on
q  Theory
q  Algorithms
q  Implementa/on
q  Numericalexamples
q  BeyondRPA:AC-SOSEXandeh-TDHF
q  Numericalexamplesinalocalizedbasissetimplementa/on
q  Comparisonofplane-waveandlocalizedbasissetresults
Outline
q  Randomphaseapproxima/on
q  Theory
q  Algorithms
q  Implementa/on
q  Numericalexamples
q  BeyondRPA:AC-SOSEXandeh-TDHF
q  Numericalexamplesinalocalizedbasissetimplementa/on
q  Comparisonofplane-waveandlocalizedbasissetresults
Correla/onenergywithintheadiaba/cconnec/on
fluctua/onanddissipa/ontheorem(ACFDT)
Independent-electron polarizability!
Polarizability of a fictitious system whose
electrons interact with a scaled Coulomb
potential λvc!
Random-phaseapproxima/on(RPA)
TDDFT or Bethe-Salpeter Equation!
with a kernel that contains the Hartree contribution only!
After analytical integration over λ the correlation energy becomes!
Outline
q  Randomphaseapproxima/on
q  Theory
q  Algorithms
q  Implementa/on
q  Numericalexamples
q  BeyondRPA:AC-SOSEXandeh-TDHF
q  Numericalexamplesinalocalizedbasissetimplementa/on
q  Comparisonofplane-waveandlocalizedbasissetresults
Challengesinthenumericalcalcula/onofRPA
correla/onenergies
v valence states !
c conduction states !
q Toevaluateχ0itisnecessarytocomputeseveral
conduc/onstatesϕc. Convergenceisslowwith
respecttothisparameter.
q Itisnecessarytocomputeanintegralbetween0
and+∞.
q Itisnecessarytostoreinmemoryandcompute
thelogarithmofthelargematrixχ0.
Elimina/onoftheemptystateswithDFPT
techniques
€
Elimina/onoftheemptystateswithDFPT
techniques
€
Elimina/onoftheemptystateswithDFPT
techniques
€
Elimina/onoftheemptystateswithDFPT
techniques
€
Elimina/onoftheemptystateswithDFPT
techniques
€
Lanczosalgorithmforresponsefunc/ons
q Thealgorithmitera/velyreducestheverylargematrixHto
asmalltridiagonalmatrixT
q ThetridiagonalmatrixTdoesnotdependoniu:OnceTis
evaluatedwecaneasilycomputeχijforseveralvaluesofiu
q ThesamealgorithmcouldbeusedinGWcodesand,in
thepseudo-Hermi9anversion,alsointheTDDFPTcode
Rocca, R. Gebauer, Y. Saad and S. Baroni, J. Chem. Phys. (2008)
-P. Umari, G. Stenuit, and S. Baroni, Phys. Rev. B (2010)
-H. V. Nguyen, T. A. Pham, D. Rocca, and G. Galli, Phys.Rev, B-RC (2012)
-D.
Construc/onoftheop/malbasisset:Afew
observa/ons
Let us consider the static response function (iu=0)!
The auxiliary basis set elements need to accurately represent the products!
€
The products of valence (v) and
conduction (c) states have a strong
linear dependence !
keeping into account the weight!
D. Rocca, J. Chem. Phys. (2014), special issue Advances in DFT Methodology
Construc/onoftheop/malbasisset:Afew
observa/ons
Let us consider the static response function (iu=0)!
Small contribution (because of the weight)!
Important Contribution!
€
Just an example!
D. Rocca, J. Chem. Phys. (2014), special issue Advances in DFT Methodology
Construc/onoftheop/malbasisset:Afew
observa/ons
Let us consider the static response function (iu=0)!
Small contribution (because of the weight)!
Important Contribution!
€
Just an example!
Kinetic energy only!
D. Rocca, J. Chem. Phys. (2014), special issue Advances in DFT Methodology
Construc/onoftheop/malbasissetbyitera/ve
diagonaliza/onofaχ0containingonlythekine/cenergyterm
q Thisprocedureeliminatesthelineardependenceofvalence-conduc/onstate
productskeepingintoaccounttheenergyweight.
q Thekine/cenergyisdiagonalinreciprocalspace(propor/onaltoG2)andtheinverse
iseasilycomputed.
q Thismethodisanumericallyefficientapproxima/onoftheprojec/vedielectric
eigenpoten/almethod(Wilson,Gygi,andGalli,PRB2008;NguyenanddeGironcoliPRB2009).
D. Rocca, J. Chem. Phys. (2014), special issue Advances in DFT Methodology
Outline
q  Randomphaseapproxima/on
q  Theory
q  Algorithms
q  Implementa/on
q  Numericalexamples
q  BeyondRPA:AC-SOSEXandeh-TDHF
q  Numericalexamplesinalocalizedbasissetimplementa/on
q  Comparisonofplane-waveandlocalizedbasissetresults
Implementa/on
Generation of the auxiliary basis set!
Lanczos procedure!
Calculation of the correlation
energy in post processing!
Genera/onoftheauxiliarybasisset
Some similarities with solve_linter.f90!
but no linear system to be solved!
Iterative diagonalization of a Hermitian!
matrix!
Lanczosalgorithm
h_psi or h_psiq !
lr_dot (for RPA only) !
Necessary for the post-processing !
Addi/onaldetails
q Calcula/onsareperformedinanonself-consistentwaystar/ngfromLDAor
GGAorbitalsandenergies
q Normconservingpseudopoten/alsonly
q Finitegapsystems
Outline
q  Randomphaseapproxima/on
q  Theory
q  Algorithms
q  Implementa/on
q  Numericalexamples
q  BeyondRPA:AC-SOSEXandeh-TDHF
q  Numericalexamplesinalocalizedbasissetimplementa/on
q  Comparisonofplane-waveandlocalizedbasissetresults
RPAbindingcurveofthet-shapedconfigura/onof
thebenzenedimer
q 60Rycut-off
q 165000plane-waves
forthewavefunc/ons
q 1.3millionsofPWs
forthecharge/poten/al
q Curvesareshownfor
differentsizesofthe
op/malbasisset
q RPAgivesthecorrect
equilibriumdistanceand
80%ofthecoupled
clusterbindingenergy
ONLY 1000 BASIS VECTORS ARE SUFFICIENT TO CONVERGE THE
BINDING CURVE WITHIN 0.05 kcal/mol!
Convergenceofthecorrela/onenergydifference
asafunc/onofthebasissetsize(d=3.9Å)
q ΔEcisthedifferencebetweenthecorrela/onenergyofthedimerand
twicethecorrela/onenergyofthemonomer
q Thequan/tythatactuallyconvergesrapidlyasafunc/onofthebasis
setsizeistheENERGYDIFFERENCE
Applica/onstosolids:lanceparametersandbulk
moduli(inparenthesis)
Covalent bonds!
Method
LDA
RPA@LDA
GGA
RPA@GGA
Experiment
Si
10.20 (92.10)
10.11 (102.1)
10.33 (85.70)
10.21 (97.30)
10.26 (99.00)
!
Weak interactions!
Ar
Kr
Ne C
SiC
6.67 (449.7)
8.21 (216.1)
9.32 (6.71)
10.09 (6.15) 7.27 (8.25)
6.68 (441.1)
8.17 (222.4)
10.23 (1.58)
10.83 (2.24) 8.97 (0.26)
6.73 (416.2)
8.26 (204.4)
11.22 (0.70)
12.12 (0.61) 8.69 (1.17)
6.73 (417.0)
8.21 (215.0)
10.00 (2.79)
10.72 (2.98) 8.18 (1.47)
6.74 (443.0)
8.24 (225.0)
10.04 (2.66)
10.60 (3.34) 8.22 (0.99)
The RPA preserves the accuracy of traditional DFT methods for covalent
solids while is significantly more reliable for weakly interacting solids!
F. Kaoui and D. Rocca, J. Phys. Condensed Matter (2016)
Outline
q  Randomphaseapproxima/on
q  Theory
q  Algorithms
q  Implementa/on
q  Numericalexamples
q  BeyondRPA:AC-SOSEXandeh-TDHF
q  Numericalexamplesinalocalizedbasissetimplementa/on
q  Comparisonofplane-waveandlocalizedbasissetresults
Time-dependentHartreeFock(TDHF)
Bethe-Salpeter Equation!
with a kernel that contains the Hartree and exchange contributions!
q  Expression of the correlation energy hard to simplify!
q  Elimination of the empty states impossible and equations not
suitable for the optimal basis set representation!
Aprac/calapproxima/on:Theelectron-hole(eh)
exchangekernel
The exchange kernel can be split in two parts!
Aprac/calapproxima/on:Theelectron-hole(eh)
exchangekernel
The exchange kernel can be split in two parts!
Electron-hole approximation!
q  The approximate kernel contains valence-conduction state products only
and can be efficiently represented by the optimal basis set.!
q  Logarithmic formulas similar to RPA can be obtained.!
q  This approximation leads to expressions for the correlation energy that
correctly reduce to MP2 to the second order.!
q  The f-sum rule is satisfied even when using DFT states as starting point.!
q  Within the optimized effective potential framework this approximation is
exact for a system of two electrons (Hesselman and Gorling PRL 2011).!
q  In numerical applications, this kernel significantly improves over RPA
results.!
Equa/onsforthecorrela/onenergyobtainedfrom
theapproximateelectron-holeexchangekernel
Random Phase Approximation (RPA)!
Adiabatic connection second order screened !
exchange (AC-SOSEX)!
Electron-hole TDHF (eh-TDHF) !
Logarithmic
equations
similar to
RPA!
These expressions allow for the elimination of the empty states and are
suitable for the optimal basis set representation !
Outline
q  Randomphaseapproxima/on
q  Theory
q  Algorithms
q  Implementa/on
q  Numericalexamples
q  BeyondRPA:AC-SOSEXandeh-TDHF
q  Numericalexamplesinalocalizedbasissetimplementa/on
q  Comparisonofplane-waveandlocalizedbasissetresults
Mean Absolute Error [kcal/mol]
Reac/onenergies(localizedbasisset)
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
SD
F
SE
H
D
SO
-T
CC
eh
C-
A
A
RP
E
PB
X
q Testsetofreac/onenergiesfromA.HesselmannPRA85,012517(2012)
q Extrapola/ontocompletebasissetlimit
q Errorinkcal/molcomputedwithrespecttoCCSD(T)results
q Theeh-TDHFmethodpresentsanaccuracysimilartoCCSD
B. Mussard, D. Rocca, G. Jansen, and J. G. Angyan, J. Chem. Theory Comput., submitted
0.1
0.08
0.06
AC-SOSEX
eh-TDHF
CCSD
0.04
0.02
0
-0.02
-0.04
H2
CH4
NH3
H 2O
C 2H 2
C 2H 4
C 2H 6
CO
HCHO
CH3OH
H2O2
H2CCO
C 2H 4O
CH3OH
C2H5OH
HNCO
HCONH2
CO2
HCOOH
NH2CONH2
HCOOCH3
CCSD(T)
(ERPA
) / |ECCSD(T)
| [%]
tot - Etot
tot
Totalenergiesofmolecules(localized
basisset)
q Moleculesfromthepreviouslyconsideredtestsetofreac/onenergies
q Rela/veerrorcomputedwithrespecttoCCSD(T)
q Withtheexcep/onofH2theeh-TDHFapproxima9onismoreaccurate
thanCCSD
B. Mussard, D. Rocca, G. Jansen, and J. G. Angyan, J. Chem. Theory Comput., submitted
Outline
q  Randomphaseapproxima/on
q  Theory
q  Algorithms
q  Implementa/on
q  Numericalexamples
q  BeyondRPA:AC-SOSEXandeh-TDHF
q  Numericalexamplesinalocalizedbasissetimplementa/on
q  Comparisonofplane-waveandlocalizedbasissetresults
Mean Absolute Error [kcal/mol]
SD
CC
F
H
D
-T
eh
SE
SO
C-
A
X
Localized basis set !
SD
CC
F
H
D
-T
eh
SE
SO
C-
A
A
RP
E
PB
q TheuseofPAW,asimplementedin
PHonon,mightbethesolu9on
A
E
q Ratherdifferentresultsalreadyat
thePBElevel(evenupto2-3kcal/mol)
q Possibleexplana/ons:Basisset
convergence,extrapola/ontechniques,
andeffectsofthepseudopoten/als
RP
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
PW!
PB
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
Mean Absolute Error [kcal/mol]
Reac/onenergies:localizedbasissetvs.
plane-wavebasisset
X
Conclusions
q  Theimplementa/onIshowedhasaffini/eswiththePHonon,TDDFPT
andGWcodes.
q  ItispossibletouseaLanczoslibrarysharedwithGWandTDDFPT
q  UseofsomePhononcodesubrou/nesbasedonamodifiedsolve_linter
GWcorrec/onsforthebenzenemolecule:600
eigenpoten/als
Bandindex
GWPDEP(eV)
GWkine/c-PDEP(eV) Difference(eV)
1
-24.72373
-24.72201
-0.00172
2
-22.12861
-22.16208
0.03347
3
-22.13058
-22.16548
0.03490
4
-18.55981
-18.55866
-0.00115
5
-18.56192
-18.56144
-0.00048
6
-16.64635
-16.64672
-0.00037
7
-15.05978
-15.05905
-0.00073
8
-14.42094
-14.41936
-0.00158
9
-13.93097
-13.92884
-0.00213
10
-13.93267
-13.93126
-0.00141
11
-11.97800
-11.97851
0.00051
12
-11.86021
-11.85558
-0.00463
13
-11.85498
-11.85421
-0.00077
14
-9.08178
-9.08117
-0.00061
15
-9.08079
-9.07991
-0.00088
€
GWcorrec/onsforthebenzenemolecule:100
eigenpoten/als
Bandindex
GWPDEP(eV)
GWkine/c-PDEP(eV) Difference(eV)
1
-24.52845
-24.53984
0.01139
2
-21.90946
-21.91368
0.00422
3
-21.90845
-21.91313
0.00468
4
-18.37895
-18.38489
0.00594
5
-18.37894
-18.39093
0.01199
6
-16.45100
-16.46620
0.01520
7
-14.88200
-14.89410
0.01210
8
-14.15117
-14.14245
-0.00872
9
-13.70358
-13.69991
-0.00367
10
-13.70397
-13.70120
-0.00277
11
-11.78619
-11.79756
0.01137
12
-11.60987
-11.59932
-0.01055
13
-11.60808
-11.59782
-0.01026
14
-8.92014
-8.91023
-0.00991
15
-8.92049
-8.90963
-0.01086
€
GWcorrec/onsfortheC29H36nanopar/cle
100eigenpoten/als
Bandindex
GWPDEP(eV)
GWkine/c-PDEP(eV)
Difference(eV)
76
-7.76940
-7.68951
-0.07989
77
1.22383
1.25920
-0.03537
600eigenpoten/als
€
Bandindex
GWPDEP(eV)
GWkine/c-PDEP(eV)
Difference(eV)
76
-8.06783
-8.05619
-0.01164
77
0.84406
0.82456
0.01950
Eigenvaluesandeigenvectorsofχ0astheop/mal
basisset:Itera/vediagonaliza/on
ONLY FEW EIGENPAIRS ARE
NECESSARY!
H. F Wilson, F. Gygi, and G. Galli, PRB (2008)
Eigenvaluesandeigenvectorsofχ0astheop/mal
basisset:Itera/vediagonaliza/on
ONLY FEW EIGENPAIRS ARE
NECESSARY!
It is not necessary to know explicitly the
matrix in order to apply it to a vector!
Iterative diagonalization of χ0 !
Drawbacks:
q Theresponsefunc/onhastobediagonalizedforeach
valueoftheimaginaryenergyiu
q Every/metheresponsefunc/onmatrixisappliedtoa
vectoranon-Hermi/anlinearsystemhastobesolved
D. Lu, Y. Li, D. Rocca, G. Galli, PRL (2009)
Useofafrequencyindependentop/malbasisset
andtheLanczosalgorithm
First step: Diagonalization of χ0
in the static case (iu=0) !
Second step: The static eigenvectors of
χ0 are used as a basis set and the
dynamical effects are introduced by the
Lanczos algorithmm!
q Thesta/ceigenpoten/alsofχ0areagoodbasissetalsoatfinite
frequency.ThishasbeendemonstratedforGWcalcula/ons
q However,aconsiderableeffortiss/llnecessarytobuildthebasisset
H.-V. Nguyen, T. A. Pham, D. Rocca et G. Galli, PRB-Rap. Comm. (2012)
T. A. Pham, H.-V. Nguyen, D. Rocca et G. Galli, PRB (2013)
Convergenceofthecorrela/onenergydifference
asafunc/onofthebasissetsize(d=3.9Å)
q ΔEcisthedifferencebetweenthecorrela/onenergyofthedimerand
twicethecorrela/onenergyofthemonomer
q Thequan/tythatactuallyconvergesrapidlyasafunc/onofthebasis
setsizeistheENERGYDIFFERENCE
ConvergencewithrespecttothenumberofLanczos
stepsandthekine/cenergycutoff(d=3.9Å)
q ByincreasingtheLanczosstepsfrom
30to40theenergyshitsbyonly0.005
kcal/mol
q Thismeansthatthe165000x165000
HamiltonianHcanbeapproximatedby
a30x30tridiagonalmatrixT.
q Thekine/cenergycut-offused(60
Ry)forthewavefunc/onsmightbe
responsibleofanerrorof≈0.01kcal/
mol
RPAbindingcurveofthesandwichconfigura/onof
thebenzenedimer
q 60Rycut-off
ONLY 1000 BASIS VECTORS ARE SUFFICIENT TO
CONVERGE THE BINDING CURVE WITHIN 0.05 kcal/mol!
q 123000plane-waves
forthewavefunc/ons
q Almost1millionof
PWsforthecharge/
poten/al
q Curvesareshownfor
differentsizesofthe
op/malbasisset
q RPAgivesthecorrect
equilibriumdistanceand
64%ofthecoupled
clustersbindingenergy
RPAbindingcurveoftheparallel-displaced
configura/onofthebenzenedimer(R=3.4Å)
q 60Rycut-off
q 143000plane-waves
forthewavefunc/ons
q 1.1millionsofPWs
forthecharge/poten/al
q Curvesareshownfor
differentsizesofthe
op/malbasisset
q RPAgivesthecorrect
equilibriumdistance
and68%ofthecoupled
clustersbindingenergy
ONLY 1000 BASIS VECTORS ARE SUFFICIENT TO CONVERGE THE BINDING CURVE
WITHIN 0.05 kcal/mol!
Applica/ontobulksilicon:Convergencewith
respecttotheauxiliarybasissetsize
Total Energy (Ry)!
50 basis vectors
already give a good
estimate of the
structural properties !
Lattice Parameter (Bohr)!
Method
RPA@LDA
Calculation
Basis set size
50
100
150
200
250
a0 (Bohr)
B (GPa)
B’
10.10
10.10
10.10
10.11
10.11
102.9
102.3
102.2
102.2
102.1
4.50
4.48
4.49
4.49
4.49
Total Energy (Ry)
-16.15272
-16.18685
-16.19806
-16.20356
-16.20681
Applica/ontobulksilicon(covalentbonding):
ComparisonofLDA,GGA,RPA@LDA,andRPA@GGA
Etot(a)-Etot(a0) (Ry)!
!
The results of LDA,
GGA, RPA@LDA,
RP@GGA are similar !
Lattice Parameter (Bohr)!
Method
LDA
RPA@LDA
GGA
RPA@GGA
Experiment
a0(Bohr)
10.20
10.11
10.33
10.21
10.26
B (GPa)
92.1
102.1
85.7
97.3
99.2
B’
4.39
4.49
4.20
4.34
4.15
Applica/ontosolidkrypton(weakinterac/ons):
ComparisonofLDA,GGA,RPA@LDA,andRPA@GGA
Total
E
Energy
(Ry)!)!
tot(a)-E
tot(a0) (Ry
LDA and GGA
! poorly
describe the structural
properties with respect
to experiment while
RPA significantly
improves the results!
Lattice Parameter (Bohr)!
Method
LDA
LDA-RPA
GGA
GGA-RPA
Experiment
a0(Bohr)
10.09
10.83
12.12
10.72
10.60
B (GPa)
6.15
2.24
0.61
2.98
3.34
B’
6.61
6.42
7.42
7.58
7.20
Eigenvaluesandeigenvectorsofχ0astheop/mal
basisset:Itera/vediagonaliza/on
Startfromini/al
poten/alsUi
Applyχ0kintoUi
Orthogonalize
{χ0U}i
DavidsonorCG
λiconverged?
Output
Workinprogress:Applica/onstozeolites
q Adsorp/onof
methanein
mordenite
q Periodicmodelwith
about400electrons
perunitcell