Post Hartree-Fock local-correlation method for …Post Hartree-Fock local-correlation method for crystals S. Casassa C. Pisani, L. Maschio, M. Halo Theoretical Chemistry Group, Dipartimento

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Post Hartree-Fock local-correlation method forcrystals

S. Casassa

C. Pisani, L. Maschio, M. Halo

Theoretical Chemistry Group, Dipartimento di Chimica I.F.M., Università di Torino

Excellence Center: Nanostructured Interfaces and Surfaces (NIS)

Ab-initio Modelling in Solid State Chemistry MSSC2007, Torino, September 2007

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Electron Correlation (1)• Lowdin definition (1959):

? Ecorr = E0 − EHF

→ the difference between the exact energyof the fundamental state E0 and the Hartree-Fock energy EHF calculated by means of thevariational method

within the HF framework we can miss..

• STATIC correlation

? when the fundamental configuration is not a good approximation of theelectronic ground state of the system

• DYNAMIC correlation

? always: due to the lack of INSTANTANEOUS correlation in electrons motion


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Electron Correlation (2)

♣ DENSITY FUNCTIONAL METHODS

• semi-empirical receipts

• need of parametrization by means of ab-initio results

• accurate but often not accurate enough:→ it can’t account for dispersive interactions between remote parts of the system


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Formation energies of a set of 50 molecules (calc. vs exp.)By courtesy of Martin Schütz

DFTHybrid DFT


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Electron Correlation (3)

♣ Møller-Plesset Perturbation theory, at second order (MP2)

♠ it is inadequate in many respects: in particular, it is non-variational

♥ is the simplest post-HF correlation technique,

♥ MP2 energy E(2) is size consistent;

♥ it provides an adequate treatment of long-range interactions;

♥ it permits the assessment of techniques, basis sets, etc., before introducing amore adequate treatment of short-range interactions (MP4, CCSD,..);

♥ the MP2 correlation energy estimates can be corrected using the simple GrimmeSCS (Spin-Component-Scaled) MP2 formula [Grimme, J. Chem. Phys. 118 (2003)9095], which has proved very efficient in a molecular context [Hill, Platts, Werner,PCCP 8 (2006) 4072].


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Crystalline Argon: energy (log. scale) versus lattice parameterBy courtesy of Denis Usvyat

PBE, MP2, LDA


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The CRYSCOR Strategy

MOLPRO06

(Pulay) Werner, Schütz, Manby,Knowles..

http://www.molpro.net

CRYSTAL06

Dovesi, Saunders, Roetti, Orlan-do, Zicovich-Wilson, Pascale, Ci-valleri...

http://www.crystal.unito.it

CRYSCORTeam:? C. Pisani, R. Dovesi, L. Maschio, S. Casassa, Università di Torino, Torino, Italy;? C.M. Zicovich-Wilson, Universidad Autonoma, Cuernavaca, Mexico? M. Schütz and D. Usvyat, Universität Stuttgart, Stuttgart, Germany;

Functionality:

♥ MP2 single point energy♥ Fast evaluation of Integrals

→ Periodic Density Fitting→ Multipolar Expansion→ Lennard-Jones extrapolation

♥ MP2 correction to Density Matrix


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Main ideas

MOLPRO06



♥ local correlation

♥ non canonical MP2 equations

♥ density fitting technique

♥ various levels of sophisticati-on: MP2, CCSD(T), ..

Background

CRYSTAL06

Dovesi, Saunders, Roetti, Or-lando, Zicovich-Wilson, Pascale,Civalleri...


♥ geometrical and structuralanalysis of periodic systems

♥ accurate HF solution

♥ full use of symmetry

♥ local representation of occu-pied manifold (Wannier Func-tion)


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Main ideas

MOLPRO06







Background

CRYSTAL06








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Local-correlation approach for MOLECULESTheory: Meyer, Pulay, Saebø, (1976). MOLRPO: Werner, Knowles, Hetzer, Manby, Schütz...

exploitation of the short-range character of the dynamic correlation

1. BASIS SETboth occupied and virtual spaces must be described by a set of localizedfunction

→ ♠ orthonormality can be lost

2. TRUNCATION STRATEGY

• the number of excited configurations is limited by adopting a distance criterium• for each excited configuration the virtual space is reduced by adopting a

domain criterium

3. non-canonical MP2 equations:K i j

ab+∑

cd[ facTi jcdSdb+ SacT

i jcd fdc] −

∑cd[Sac

∑k( facT

k jcd + T ik

cd fk j)Scb] = 0

→ ♠ solution achieved by minimization of the Hylleraas functional

→ ♥ the problem scales with N, the basis set dimension!


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Formation energies of a set of 50 molecules (calc. vs exp.)By courtesy of Martin Schütz

Coupled ClusterLocal Coupled Cluster


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Local-correlation approach for CRYSTALS

1. BASIS SET

• Occupied HF manifold: local, ortho-normal, translationally invariant, sym-metry adapted Wannier Function

• Virtual space: non-orthonormal, red-undant but localized Projected AtomicOrbitals

WFs PAOsω (i j k ...N) χ (a b c ...M)⊗

|χ >= (1 − P)|µ >


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Main ideas

MOLPRO06







Background

CRYSTAL06






♥ local representation of oc-cupied manifold (WannierFunction)


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Wannier Functions1. unitary transformation of the crystalline orbitals, Ψs(~r ,~k):

ωs(r − g) = V2π3

∫BZ

eik g Ψs(r, k) dk

2. localization by minimization of the Marzari and Vanderbilt functional:

Ω =∑

s[〈 ωs|r2|ωs 〉 − 〈 ωs|r|ωs 〉2]

3. symmetry adaptation procedure


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Local-correlation approach for CRYSTALS2. TRUNCATION STRATEGY:

• design a proper domain for each WF• ignore very distant W-W pair excitations• exploit the translational symmetry

→ ♥ the problem scales with N, the size of the irreducible part of the crystal cell!


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Main ideas

MOLPRO06




♥ non canonical MP2 equati-ons



Background

CRYSTAL06








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Local-correlation approach for CRYSTALS

3. EQUATIONS

Ri j

ab= K i j

ab+∑cd

[ facTi jcdSdb+ SacT

i jcd fdc] −

∑cd

[Sac

∑k

( facTk jcd + T ik

cd fk j)Scb] = 0

E2 =∑

i j

∑ab∈(i j )

K i j

ab(2 T i j

ab− T i jba)

K i j

ab= (i a| j b) =

∑µρνσ

cWFiµ cPAO

aρ cWFjν cPAO

bσ (µρ|νσ)

∑i j →

∑i0 jJ

∑ab→

∑aA bB


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Periodic Local MP2 equations

Ri j

ab= K i j

ab+∑cd


i jcd fdc] −

∑cd

[Sac

∑k

( fikTk jcd + T ik

cd fk j)Scb] = 0

E2 =∑

i j

∑ab∈(i j )

K i j

ab(2 T i j

ab− T i jba)

K i j

ab= (i a| j b) =

∑µρνσ

cWFiµ cPAO

aρ cWFjν cPAO

bσ (µρ|νσ)

• WFs Fock matrix


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Ri j

ab= K i j

ab+∑cd


i jcd fdc] −

∑cd

[Sac

∑k

( fikTk jcd + T ik

cd fk j)Scb] = 0

E2 =∑

i j

∑ab∈(i j )

K i j

ab(2 T i j

ab− T i jba)

K i j

ab= (i a| j b) =

∑µρνσ

cWFiµ cPAO

aρ cWFjν cPAO

bσ (µρ|νσ)

• PAOs Overlap and Fock matrices


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Ri j

ab= K i j

ab+∑cd


i jcd fdc] −

∑cd

[Sac

∑k

( facTk jcd + T ik

cd fk j)Scb] = 0

E2 =∑

i j

∑ab∈(i j )

K i j

ab(2 T i j

ab− T i jba)

K i j

ab= (i a| j b) =

∑µρνσ

cWFiµ cPAO

aρ cWFjν cPAO

bσ (µρ|νσ)

• Amplitudes.. which are the unknowns


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Ri j

ab= K i j

ab+∑cd


i jcd fdc] −

∑cd

[Sac

∑k

( fikTk jcd + T ik

cd fk j)Scb] = 0

E2 =∑

i j

∑ab∈(i j )

K i j

ab(2 T i j

ab− T i jba)

K i j

ab= (i a| j b) =

∑µρνσ

cWFiµ cPAO

aρ cWFjν cPAO

bσ (µρ|νσ)

• Maximum WF-WF distance


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Ri j

ab= K i j

ab+∑cd


i jcd fdc] −

∑cd

[Sac

∑k

( fikTk jcd + T ik

cd fk j)Scb] = 0

E2 =∑

i j

∑ab∈(i j )

K i j

ab(2 T i j

ab− T i jba)

K i j

ab= (i a| j b) =

∑µρνσ

cWFiµ cPAO

aρ cWFjν cPAO

bσ (µρ|νσ)

• Maximum WF-WF distance (range of∑

i j )

• Size of WF domains


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Ri j

ab= K i j

ab+∑cd


i jcd fdc] −

∑cd

[Sac

∑k

( fikTk jcd + T ik

cd fk j)Scb] = 0

E2 =∑

i j

∑ab∈(i j )

K i j

ab(2 T i j

ab− T i jba)

K i j

ab= (i a| j b) =

∑µρνσ

cWFiµ cPAO

aρ cWFjν cPAO

bσ (µρ|νσ)



• Basis set µ (representation of WFs and PAOs in terms of AOs)


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Ri j

ab= K i j

ab+∑cd


i jcd fdc] −

∑cd

[Sac

∑k

( fikTk jcd + T ik

cd fk j)Scb] = 0

E2 =∑

i j

∑ab∈(i j )

K i j

ab(2 T i j

ab− T i jba)

K i j

ab= (i a| j b) =

∑µρνσ

cWFiµ cPAO

aρ cWFjν cPAO

bσ (µρ|νσ)




• Truncation of WF and PAO tails (|cWFiµ | > tow; |cPAO

aν | > toq)


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Ri j

ab= K i j

ab+∑cd


i jcd fdc] −

∑cd

[Sac

∑k

( fikTk jcd + T ik

cd fk j)Scb] = 0

E2 =∑

i j

∑ab∈(i j )

K i j

ab(2 T i j

ab− T i jba)

K i0 jJaA bB

= (i a| j b) =∑

µM ρR νNσScWF

i0 µM cPAOaA ρR cWF

jJ νN cPAObBσS(µMρR|νNσS)




• Truncation of WF and PAO tails (|cWFiµ | > tow; |cPAO

aν | > toq)

• Exact, density-fitting and multipolar treatment of K integrals andpre-screening of exact K integrals


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2-electron integrals calculation1. for distant WW-pairs: multipolar evaluation

2. for strong and weak WW-pairs: density-fitting technique

• Systematic exploitation of translational and point-group symmetry• Systematic use of Reciprocal Space techniques• Use of Dipole-Corrected Product Distributions


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Main ideas

MOLPRO06






Background

CRYSTAL06



→ implemented in CRYSCOR by L. Maschio and D. Usvyat


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Df-technique (1)

K i j

ab= (ωi χa|ω j χb)

ρia = (ωiχa)

ΦP(r) = fitting function

MOLECULAR CASE

ρia ≈ ρia =∑P dia

PΦP(r)

∆w = (ρia − ρia|w12|ρia − ρia)

w12 =1

r12

JPQ =∫

dr1∫ΦP(r1) 1

r12ΦQ(r2) dr2

JiaP=∫

dr1∫ΦP(r1) 1

r12ρia(r2) dr2

diaP=∑Q Jia

Q[J−1]QP

K i jab = (ρia|ρ jb) + (ρia|ρ jb) − (ρia|ρ jb)

K i jab =∑P dia

PJ jbP


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Df-technique (1)

K i j

ab= (ωi χa|ω j χb)

ρia = (ωiχa)

ΦP(r) = fitting function

MOLECULAR CASE


PΦP(r)

∆w = (ρia − ρia|w12|ρia − ρia)

w12 =1

r12

JPQ =∫

dr1∫ΦP(r1) 1

r12ΦQ(r2) dr2

JiaP=∫

dr1∫ΦP(r1) 1

r12ρia(r2) dr2

diaP=∑Q Jia

Q[J−1]QP

K i jab = (ρia|ρ jb) + (ρia|ρ jb) − (ρia|ρ jb)

K i jab =∑P dia

PJ jbP

FITTING DOMAIN for each different product distribution


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Df-technique (2)

MOLECULAR CASE PERIODIC CASE

indices: i, j, a, b, P ... indices: i I, j J, a A, b B, P P ...


PΦP(r) ρia A =

∑PP dia A

PP ΦPP(r)

diaP=∑Q Jia

Q[J−1]QP di aA

PP =∑QQ Ji aA

QQ [J−1]QQPP

K i jab =∑P dia

PJ jbP

˜K i0 jJ

aA bB =∑PP di aA

PP J jJ bBPP


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Df-technique (2)

MOLECULAR CASE PERIODIC CASE

indices: i, j, a, b, P ... indices: i I, j J, a A, b B, P P ...


PΦP(r) ρia A =

∑PP di aA

PP ΦPP(r)

diaP=∑Q Jia

Q[J−1]QP di aA

PP =∑QQ Ji aA

QQ [J−1]QQPP

infinite matrix ....?

K i jab =∑P dia

PJ jbP

˜K i0 jJ

aA bB =∑PP di aA

PP J jJ bBPP


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Df-technique (3): how to overcome the ”infinite” problem

1. calculation of the JiaAPP and J

P0QQ integrals

2. skip to reciprocal space

→ for each k point of the Monkhorst set:

• Fourier Transform: JPQ(k) =∑

Q JP0QQ exp[ikRQ]

JiaAP

(k) =∑

P JiaAPP exp[ikRP]

• matrix inversion: [J−1]QP(k)• coefficients calculation: diaA

P(k) =

∑Q Ji aA

Q(k) [J−1]QP(k)

• integrals evaluation:˜

K i jJaA bB(k) =

∑P diaA

P(k) J jJ bB

P(k)

3. ANTI FOURIER TRANSFORM:˜

K i jJaA bB =

1Nk

∑k˜

K i jJaA bB(k) exp[−ikRJ]


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P0QQ integrals


→ for each k point of the Monkhorst set :


Q JP0QQ exp[ikRQ]

JiaAP

(k) =∑

P JiaAPP exp[ikRP]


P(k) =

∑Q Ji aA

Q(k) [J−1]QP(k)


K i jJaA bB(k) =

∑P diaA

P(k) J jJ bB

P(k)


K i jJaA bB =

1Nk

∑k˜


good calibration of the Monkhorst net is needed!


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P0QQ integrals




Q JP0QQ exp[ikRQ]

JiaAP

(k) =∑

P JiaAPP exp[ikRP]


P(k) =

∑Q Ji aA

Q(k) [J−1]QP(k)


K i jJaA bB(k) =

∑P diaA

P(k) J j bL

P(k)


K i jJaA bB =

1Nk

∑k˜


L = B - J


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P0QQ integrals




Q JP0QQ exp[ikRQ]

JiaAP

(k) =∑

P JiaAPP exp[ikRP]

• matrix inversion: [J−1]QP(k)• coefficients calculation: di aA

P(k) =

∑Q Ji aA

Q(k) [J−1]QP(k)


K i jJaA bB(k) =

∑P diaA

P(k) J j bL

P(k)


K i jJaA bB =

1Nk

∑k˜


convergence PROBLEMS in the summation over FF integrals!

→ use of the original dipole-corrected method→ fitting functions: Poisson (and few Gaussian)


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Df-techique results


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Df-techique results

K i j

ab= (i a| j b) =

∑µρνσ

cWFiµ cPAO

aρ cWFjν cPAO

bσ (µρ|νσ)


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The CRYSCOR Structure


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The CRYSCOR Structure


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Functionality LiH: cohesive energy and lattice parameter

♥ MP2 and Grimme correction to HFenergy♥ Lennard-Jones contribution to energy

♥ MP2 correction to HF density matrix

♥ fixing of indices for geometry optmi-zation


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Functionality LiH: energy contribution vs WW-pair distance





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Functionality LiH: electron density profile





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Functionality and successfully tested systems




• Some study cases: LiH [S. Casassa, M. Halo, L. Maschio, C. Roetti and C. Pisani, Theor.Chem. Acc., (2006)] and MgO bulk [poster: M. Halo et al.a]

• CRYSCOR TUTORIAL: LiH, H2O polymer, Argon adsorbed on MgO


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The perfectTest


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Conclusion and Perspectives

♥ Refinement and standardization of the basic program

? Beta-version of the code in few months

♥ Tests

? different systems? assessing basis set quality

♥ Extension to other local correlation schemes (CCSD, MP4, ..)

♣ Acknowledgments

? all of you for your kind attention

?


Documents

Post Hartree-Fock local-correlation method for …Post Hartree-Fock local-correlation method for crystals S. Casassa C. Pisani, L. Maschio, M. Halo Theoretical Chemistry Group, Dipartimento