Developments in Electronic Structure Theory.
Bastien Mussard
Laboratoire de Chimie TheoriqueInstitut du Calcul et de la Simulation
Sorbonne Universites, Universite Pierre et Marie Curie
[email protected]/pagesperso/mussard/
A short presentation (1)
What I do : Quantum MechanicsI Wavefunction theories (WFT)
I Density Functional theory (DFT)
I Range Separation methods (RS)
Main research interestI Weak interactions / London dispersion forces / VdW interactions
I Random Phase Approximation (RPA)- formalism (new approximations, spin-unrestricted generalization, . . .)- local orbitals (designed new localized orbitals, selection of excitation)- gradients of the energy (forces, dipoles, optimization of geometry, . . .)- . . .
Numerous collaborationsI High Harmonics Generation spectra (Electron dynamics)
I Quantum Monte Carlo methods (QMC)
A short presentation (2)
Notable Developments
I Developments in MOLPRO
Commercial codeVersatile (HF,DFT,CI,CC,MCSCF,
MRCI,MRPT,MRCC,FCIQMC,DMRG,Gradients and Properties)
∼ 4 000 000 lines of codeIn charge of the RPA codewww.molpro.net
I Developments in CHAMP
Free codeVMC, DMCOptimization of wavefunctions, . . .Fast calculation of determinantsCHAMP@Cornell
I Developped personal codes,from scratchCalculations on grid, parallel codes
Technical side of the work
I Management of codes (SVN/GIT/BugZilla)
I Cleaning-upOptimizing pre-existing codes (use of memory, . . .)
I HPC
I Interfaces between codes
Context of my main research interest
I Electronic structure calculations.
I post-DFT calculations in the context of Range Separation.
I In the WFT part, for the treatment of London dispersion forces,we use the Random Phase Approximation.
Range Separation : Motivation [Savin, Rec.dev. (1996)]
Get the accurate answer for the right reason at a reasonable cost.
WFT (the wavefunction Ψ(xN) is the key quantity ; determinants)
6 rather costly (size of Hilbert space)
4 improvement of the description is systematic (add more determinants)
DFT (the density n(r) is the key quantity)
4 cheap methods ; applicable to big systems ; impressive successes
6 need to approximate Exc [n] ; not systematical
Range Separation (a rigourous combination of both approaches)
WFT suffers at short range from the e-e coalescence.
DFAs are (semi)local approximation, best at short-range.
The idea : split the e-e interaction into long- and short-range,treat the long-range with WFT, the short-range with DFT.
1.0
1.1
1.2
1.3
1.4
−40 −20 20 40
θ (degree)
L = 1L = 2L = 3L = 4L = ∞
Range Separation : Realisation [Savin, Rec.dev. (1996)]
Split of the Coulomb interaction ; rigourous mix of WFT and DFT methods.
1
r= v lr
ee(r) + v sree(r)
V lree long-range e-e interaction
E srHxc[n] short-range density functional
1/r
vlree(r)
vsree(r)
I In a variational way :
Eexact = minΨ
{〈Ψ|T + Vne + V lr
ee|Ψ〉+ E srHxc[nΨ]
}e.g. MCSCF+DFT, CI+DFT,. . .-> for the static (strong) correlation.[Fromager,Toulouse,Jensen JCP 2007]
[Sharkas,Savin,Jensen,Toulouse JCP 2012]
I With a perturbative treatment, from a mono-determinental reference (RSH) :
Eexact = minΦ
{〈Φ|T + Vne + V lr
ee|Φ〉+ E srHxc[nΦ]
}+ E lr
c
e.g. with RPA -> for long range dynamical correlation (vdW dispersion).[Angyan,Gerber,Savin,Toulouse PRA 2005]
[Toulouse,Gerber,Jansen,Savin,Angyan PRL 2009]
Range Separation : Examples [Savin, Rec.dev. (1996)]
Ar2, aug-cc-pV5Z basis, µ = 0.5 bohr−1
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
6 7 8 9 10 11 12 13
Inte
rac
tio
n e
ne
rgy
(m
ha
rtre
e)
Internuclear distance (bohr)
AccuratePBE
(sr)PBE
I RSH reference contains no dispersion.
I In RPA, it is important to add exchange effects for dispersion interactions.
Range Separation : Examples [Savin, Rec.dev. (1996)]
Ar2, aug-cc-pV5Z basis, µ = 0.5 bohr−1
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
6 7 8 9 10 11 12 13
Inte
rac
tio
n e
ne
rgy
(m
ha
rtre
e)
Internuclear distance (bohr)
AccuratedRPA
(lr)dRPARPAx
(lr)RPAx
I RSH reference contains no dispersion.
I In RPA, it is important to add exchange effects for dispersion interactions.
Random Phase Approximation : Origins [Bohm,Pines PR (1951)]
The original intention is to decouple by a canonical transformation the particle andfield variables in the description of the Uniform Electron Gaz (UEG).
The physics : mix of long-range organized behavior (collective plasma oscillations)and of short-range screened explicit interactions.
This emerges naturally by the canonical transformation, and involves a cut-offdistance, over which long-range behavior correctly describes the system, and underwhich it no longer captures the main physics : the short-range interaction is thenpredominant.
Hence, in RPA, the energy is the sum of oscillator energies and of a short-rangecorrection. This seems ideal in a range separation context.
What is “random” in RPA ?
Only the particles in phase with the oscillating field contribute to the oscillationsthe other particles, having a random phase , are neglected.
Random Phase Approximation : formalisms and versions
[Angyan,Liu,Toulouse,Jansen JCTC (2011)]
[BM,Rocca,Jansen,Angyan JCTC (submitted)]AC-FDT equation
ERPAc =
1
2
∫ 1
0
dα
∫ ∞−∞
dω
2πTr(Vee
[�RPAα (iω)− �0(iω)
]) ∣∣∣∣ (�RPAα )−1 = �−1
0 − fHx,α
Versions (include/exclude exchange)
I direct or exchange RPA
I single-bar or double-bar
This yields different, non-equivalentversions of RPA
Formulations (analytical/numerical integrals)
I density-matrix formulation ERPAc = 1
2
∫dα tr (Pc,α V)
I dielectric-matrix formulation ERPAc = 1
2
∫dω
2πtr (log(ε(iω))− ε(iω) + 1)
I plasmon formula ERPAc = 1
2
∑ωRPA − ωTDA
I Ricatti equations and rCCD ERPAc = 1
2 tr(BT)
+Approximations (in each formulations, for each versions...) SOSEX, SO2
Random Phase Approximation : formalisms and versions
[Angyan,Liu,Toulouse,Jansen JCTC (2011)]
[BM,Rocca,Jansen,Angyan JCTC (submitted)]AC-FDT equation
ERPAc =
1
2
∫ 1
0
dα
∫ ∞−∞
dω
2πTr(Vee
[�RPAα (iω)− �0(iω)
]) ∣∣∣∣ (�RPAα )−1 = �−1
0 − fHx,α
Versions (include/exclude exchange)
I direct or exchange RPA
I single-bar or double-bar
This yields different, non-equivalentversions of RPA
Formulations (analytical/numerical integrals)
I density-matrix formulation ERPAc = 1
2
∫dα tr (Pc,α V)
I dielectric-matrix formulation ERPAc = 1
2
∫dω
2πtr (log(ε(iω))− ε(iω) + 1)
I plasmon formula ERPAc = 1
2
∑ωRPA − ωTDA
I Ricatti equations and rCCD ERPAc = 1
2 tr(BT)
+Approximations (in each formulations, for each versions...) SOSEX, SO2
Random Phase Approximation : formalisms and versions
[Angyan,Liu,Toulouse,Jansen JCTC (2011)]
[BM,Rocca,Jansen,Angyan JCTC (submitted)]AC-FDT equation
ERPAc =
1
2
∫ 1
0
dα
∫ ∞−∞
dω
2πTr(Vee
[�RPAα (iω)− �0(iω)
]) ∣∣∣∣ (�RPAα )−1 = �−1
0 − fHx,α
Versions (include/exclude exchange)
I direct or exchange RPA
I single-bar or double-bar
This yields different, non-equivalentversions of RPA
Formulations (analytical/numerical integrals)
I density-matrix formulation ERPAc = 1
2
∫dα tr (Pc,α V)
I dielectric-matrix formulation ERPAc = 1
2
∫dω
2πtr (log(ε(iω))− ε(iω) + 1)
I plasmon formula ERPAc = 1
2
∑ωRPA − ωTDA
I Ricatti equations and rCCD ERPAc = 1
2 tr(BT)
+Approximations (in each formulations, for each versions...) SOSEX, SO2
Random Phase Approximation : Feynman diagrams
dRPA-I
E dRPA-Ic =
1
2tr〈ab|ij〉lr(T lr
drCCD)ia,jb
+ +. . .
SOSEX
ESOSEXc =
1
2tr〈ab||ij〉lr(T lr
drCCD)ia,jb
[Gruneis,et. al. JCP 2009]
+ +
+ +. . .
Not that good for weak interactions.
RPAx-II
ERPAx-IIc =
1
4tr〈ab||ij〉lr(T lr
rCCDx)ia,jb
[McLachlan,Ball RMP 1964]
[a lot of diagrams]
RPAx-SO2
ERPAx-SO2c =
1
2tr〈ab|ij〉lr(T lr
rCCDx)ia,jb
[Szabo,Ostlund JCP 1977]
[Toulouse et. al. JCP 2011]
+ +
+ +
+ +. . .
It is our method of choice.
New developments
I Basis Set Convergence studyand Three-point extrapolation scheme.
[Franck,BM,Luppi,Toulouse JCP (2015)]
I Spin-Unrestricted generalizationand Calculations on DBH24/08 and AE49.
[BM,Reinhardt,Angyan,Toulouse JCP (2015)]
I Development and Implementation ofthe Gradients of RSH+RPA methods.
[BM,Szalay,Angyan JCTC (2014)]
I Development of Localized Virtual Orbitalsand Application for RPA calculations.
[BM,Angyan TCA (2015)]
Basis Set Convergence [Franck,BM,Luppi,Toulouse JCP (2015)]
Partial wave expansion of the WF [Gori-Giorgi,Savin PRA (2006)]Ψ(r12)
Ψ(0)= 1 +
r12
2+ . . .
=∑
c`P`(cosθ) with c` ∼ `−2
Ψµ(r12)
Ψµ(0)= 1 +
µr 212
3√π
+ . . .
=∑
c`P`(cosθ) with c` ∼ e−α`
1.0
1.1
1.2
1.3
1.4
−40 −20 20 40
θ (degree)
L = 1L = 2L = 3L = 4L = ∞
Her1
r2θ
1.00
1.02
1.04
1.06
−40 −20 20 40
θ (degree)
L = 1L = 2L = 3L = 4
Principal number expansion (wrt one-particule basis) Ψ(r1, r2) =∑
ci φi (r1) φi (r2)
0.21
0.23
0.25
0.27
0.29
−180 −120 −60 0 60 120 180
θ (degree)
HFVDZVTZVQZV5ZV6Z
Hylleraas
Her1
r2θ
0.21
0.23
0.25
0.27
0.29
−180 −120 −60 0 60 120 180
θ (degree)
RSHVDZVTZVQZV5ZV6Z
0.263
0.264
0.265
−120 −60 0 60 120
Spin-Unrestricted RPA [BM,Reinhardt,Angyan,Toulouse JCP (2015)]
Implemented most of the variants with anospinflip/spinflip block structure.
Normal distributions of errors(kcal/mol) of calculations on AE49.
⟩I Mean errors of post-RSH calculation
are better.
I Post-RSH calculations dist. of errorshave sharper distributions .
Similar results were obtained on theDBH24/08 dataset.
We argue that RPAx-SO2 is a goodmethod for a wide range of
applications.
0
0.04
0.08
0.12
-40 -20 0 20
HF+MP2
0
0.04
0.08
0.12
-40 -20 0 20
RSH+MP2
0
0.04
0.08
0.12
-40 -20 0 20
HF+dRPA-I
0
0.04
0.08
0.12
-40 -20 0 20
RSH+dRPA-I
0
0.04
0.08
0.12
-40 -20 0 20
HF+RPAx-SO2
0
0.04
0.08
0.12
-40 -20 0 20
RSH+RPAx-SO2
>cc-pVQZ (post-RSH : srPBE, µ = 0.5)
Analytical Gradients : Lagrangian formalism [BM,Szalay,Angyan JCTC (2014)]
Given an energy E [κ,V(κ),T(κ)]
you have rules for V : ∂E/∂V = 0
you have rules for T : R[T] = 0
energie
energie
(c)(b)(a)
parametreparametreparametre
energie
lagrangien
T TV
R[T] = 0E[V]
E[T] E[T]
L[T]
energie
energie
(c)(b)(a)
parametreparametreparametre
energie
lagrangien
T TV
R[T] = 0E[V]
E[T] E[T]
L[T]
dE
dκ=∂E
∂κ+∂E
∂V
∂V
∂κ×+∂E
∂T
∂T
∂κ
(in∂E
∂κthere is h(κ), (µν|σρ)(κ) and S(κ)
)For non-variational methods
The solution is to work with an alternative object that is variational :
L[κ,V,T,λ] = E [κ,V,T] + tr(λR[T])
∂L∂V
=∂E
∂V= 0 4
∂L∂λ
= R[T] = 0 4
∂L∂T
=∂E
∂T+ tr
(λ∂R
∂T
)= 0
energie
energie
(c)(b)(a)
parametreparametreparametre
energie
lagrangien
T TV
R[T] = 0E[V]
E[T] E[T]
L[T]
The RSH+RPA energy is : E = minΦ
{〈Φ|T + Vne + V lr
ee|Φ〉+ E srHxc[nΦ]
}+ E lr,RPA
c
The equations were derived (sr and lr terms) and implemented in MOLPRO.
Analytical Gradients of RSH+RPA [BM,Szalay,Angyan JCTC (2014)]
Bond Lengths of simple molecules> aug-cc-pVQZ
-0.08
-0.06
-0.04
-0.02
0.00
H2/H
−HHF/F−H
H2O/O
−HHOF/O
−HHNC/N
−HNH3/N
−HN2H2/N
−HHNO/N
−HC 2
H2/C−H
HCN/C−H
C 2H4/C−H
CH4/C−H
N2/N
−NCH2O/C−H
CH2/C−H
CO/C−O
HCN/C−N
CO2/C−O
HNC/C−N
C 2H2/C−C
CH2O/C−O
HNO/O
−NN2H2/N
−NC 2
H4/C−C
F 2/F−F
HOF/O
−F
∆r
(Å)
RHF+dRPA-ILDA+dRPA-IRHF+SOSEXLDA+SOSEX
Mean Absolute Errors
Bond Lengths of simple moleculesRHF+dRPA-I 0.016 RHF+SOSEX 0.021
LDA+dRPA-I 0.013 LDA+SOSEX 0.014
Interaction Energies Intermonomer distancesMP2 0.76 MP2 0.075
LDA+MP2 0.63 LDA+MP2 0.053
LDA+dRPA-I 0.35 LDA+dRPA-I 0.023
Interaction Energies> aug-cc-pVTZ
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
C2H4. .
. F2
NH3. .
. F2
C2H2. .
. ClF
HCN. .
. ClF
NH3. .
. Cl 2
H2O. .
. ClF
NH3. .
. ClF
∆(∆
E)
(kca
l.m
ol−1) MP2
LDA+MP2LDA+dRPA-I
Intermonomer distances> aug-cc-pVTZ
-0.20
-0.15
-0.10
-0.05
0.00
0.05
C2H4. .
. F2
NH3. .
. F2
C2H2. .
. ClF
HCN. .
. ClF
NH3. .
. Cl 2
H2O. .
. ClF
NH3. .
. ClF
∆R
(Å)
MP2LDA+MP2LDA+dRPA-I
Projected Oscillator Orbitals [BM,Angyan TCA (2015)]
The objective is to construct a set of localized virtual orbitals by multiplying theset of occupied Localized Molecular Orbitals (LMOs) with solid spherical harmonicfunctions. The orthogonality to the occupied space is ensured by projection.
Dipolar oscillator orbital [Foster, Boys RMP (1960)]
|iα〉= (1− P)︸ ︷︷ ︸projector
(rα − D iα)︸ ︷︷ ︸
harmonic
|i〉︸︷︷︸LMO
where D iα = 〈i |rα|i〉
C-H bond in CH2=O :∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
Projected Oscillator Orbitals : Key features [BM,Angyan TCA (2015)]
Dipolar oscillator orbital
|iα〉= (1− P)(rα − D iα) |i〉 where D i
α = 〈i |rα|i〉
This formulation is best employed with Boys’ LMOs, see :
|iα〉= rα |i〉−∑|m〉 〈m|rα|i〉
The POOs are non-orthogonal, with overlap :
S ijαβ = 〈i |rαrβ |j〉 −
∑〈i |rα|m〉〈m|rβ |j〉
The fock matrix element are found to be :
f iiαβ = 12δαβ + 1
2
∑(fim〈m|rαrβ |i〉+ 〈i |rαrβ |m〉fmi )−
∑〈i |rα|m〉fmn〈n|rβ |i〉
The two-elec integrals are written with a multipole expansion of the lr interaction :
〈iαj |klβ〉 =∑〈iα|rγ |k〉Lijγδ〈j |rδ|lβ〉+ . . .
A nice result greatly simplifies the equations :
〈iα|rβ |j〉 = 〈i |rαrβ |j〉 −∑〈i |rα|m〉〈m|rβ |j〉 = S ij
αβ
Projected Oscillator Orbitals : RPA and C6 [BM,Angyan TCA (2015)]
The working equations are the local RPA Ricatti equations (i.e. local rCCD) withthe local excitation approximation and spherical average approximation.
RPA correlation energy
ERPA,lrc =
occ∑ij
4
9s i s j tr
(LijTij
)where s i =
∑Siiαα
C6 coefficients
E (2),lrc =
occ∑ij
4
9
s i s j
∆i + ∆jtr(LijLij
)=
occ∑ij
8
3
s i s j
∆i + ∆j
Fµdamp(D ij)
D ij 6
where ∆i = fii − f i/s i
-50
0
50
100
150
H2
HF
H2O
N2
CO
NH3
CH4
CO2
H2CO
N2O
C2H2
CH3OH
C2H4
CH3NH2
C2H6
CH3CHO
CH3OCH3
C3H6
HCl
HBr
H2S
SO2
SiH
4Cl 2
COS
CS2
CCl 4
Err
orin
the
mol
ecula
rC
6co
effici
ents
(%) LDA
PBE
RHF
RSHLDA
> aug-cc-pVTZ (RSH : µ = 0.5)
MA%E
LDA 59.8PBE 56.7RHF 15.2RSHLDA 11.8
Other Works
Efficient calculations of determinants in QMC usingthe Sherman-Morrison-Woodbury formula.
Fractional occupation number calculationsand Instabilities in the RPA problem.
[BM,Toulouse (in prep)]
Electron dynamics for High Harmonics Generation spectra[Coccia,BM,Labeye,Caillat,Taieb,Toulouse,Luppi (submitted)]
In real space : relationship betweenthe response functions, exchange holes and localized orbitals.
[BM,Angyan CTC (2015)]
Collaborators : Janos Angyan, Roland Assaraf, Odile Franck, Georg Jan-sen, Eleonora Luppi, Peter Reinhardt, Dario Rocca, Julien Toulouse,Cyrus Umrigar.
www.lct.jussieu.fr/pagesperso/mussard/
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