mssc2016 hbs civalleri fin - · pdf filebartolomeo civalleri department of chemistry ... (m...

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MSSC2016 (Torino) 4-9/09/2016 B. Civalleri

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B. Civalleri

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One-electron Hamiltonians:

HF & DFT Bartolomeo Civalleri

Department of Chemistry NIS Centre of Excellence

University of Torino [email protected]

MSSC2016 Ab initio Modelling in Solid State Chemistry Torino, 4–9/09/2016

“…and he dreamed that there was a ladder set up on the earth, the top of it reaching to heaven; and the angels of God were ascending and descending on it”

Genesis 28, 10-12


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Quantum Chemistry and Molecules/Crystals

Structure and composition

Physical and chemical properties

The problem is of evaluating quantum mechanically the ground state electronic structure and total energy of a system of interacting electrons for a given nuclear configuration

HΨ(r)=EΨ(r) Quantum mechanics

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Brief overview of HF and DFT methods: merits and limits DFT methods in CRYSTAL17 Validation of DFT methods for molecules and solids

Outline

Which model Hamiltonian for solids? Hartree-Fock, DFT, …? Which representation for Ψ? Plane waves, Gaussians, …?

(see next lecture)

Questions to answer


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Fundamental approximations

•  Time independent Schrödinger equation

•  Born-Oppenheimer approximation

•  Relativistic effects are usually neglected

•  Neglect of higher order effects (e.g. spin-orbit interaction)

•  No excited states ⇒ Ground state (E0, Ψ0, ρ0)

ttit

∂

Ψ=Ψ

),(),(H rr !

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HΨ(r;R) = EΨ(r;R)

2e

1T2

N

ii

= − ∇∑

Vext = −Za

Ra − ria

M

∑i

N

∑

Vee =1

ri − rjj>i

N

∑i

N

∑

nnVM M

a b

a b a a b

Z Z>

=−∑∑ R R

Te Vext Vee +++H = Vnn

For a system of N electrons at a given nuclear configuration (M nuclei of charge Za in Ra)

Kinetic energy:

Electrons-nuclei interaction:

Electron-electron interaction:

Nuclear repulsion:

Ψ(r1,…,rN ;R1,…,RM )

Ab-‐ini&o

Too complex!

Non relativistic Schrödinger equation


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Tools: brain, chalk & dashboard

Approximations

Computers

HF DFT

MP2 MP3

MP4

LDA GGA

B3LYP PBE

CI CISD CC CCSD(T)

Schrodinger EQUATION

QUANTUM

Theory: finding out the rules

HΨ=EΨ

What do we need?

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Energy & Variational Principle

For any legal wavefunction (N-electron, antisymmetric, normalized) the energy is:

ΨΨ≡ΨΨ= ∫ ∗ HdHE ˆˆ r

⇒ the energy is a functional of Ψ: [ ]Ψ≡EE

Search all Ψ to minimize E ⇒ the ground state (E0, Ψ0)

[ ] [ ]00 Ψ≥Ψ EE

[ ]Ψ=→ΨEE

Nmin0


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The Hartree-Fock (HF) approximation - I

Given a complete orthonormal set of {ψi(x)} of one-electron spin-orbitals (x = r,α ; r,β), the exact ground state wavefunction can be expressed as a linear combination of all N-electron Slater determinants (antisymmetric) from {ψi(x)} :

( )10 ,..., N

SDi ii i i

C≡

Ψ = Φ∑

Φ iSD =

1N !

ψi11( ) ! ψiN

1( )! " !

ψi1N( ) ! ψiN

N( )The HF approximation considers just a single-determinant: ΦSD

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The Hartree-Fock (HF) approximation - II

Ψ0 ≈ ΦSD =

1N !

ψ1

HF x1( ) ! ψN

HF x1( )! " !

ψ1

HF xN( ) ! ψN

HF xN( )

The HF method provides the N spin-orbitals ψHFi(x) (i=1,2,…,N) which

define the “best” single-determinant approximation of Ψ0:

EHF = ΦSD H ΦSD ≥ E0

EHF = minΦSD→N

E ΦSD$%

&'The HF equations (SCF)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2222 1 2 1 1

1 2 1 2

1ˆ2

j ii i ext i j i i

jf V d d

ψ ψρψ ψ ψ εψ

∗⎛ ⎞⎡ ⎤= − ∇ + + − =⎜ ⎟⎢ ⎥ ⎜ ⎟− −⎢ ⎥⎣ ⎦ ⎝ ⎠

∑∫ ∫r rr

r r r r r r rr r r r

Mean-field theory → no electron correlation effects (Ec=E0-EHF)


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In Hartree-Fock theory the real electron-electron interaction is replaced by an average interaction.

The HF wave function accounts for ≈99% of the total energy.

The ≈1% lost correlation energy is however crucial for: Bond energy Reaction intermediates Dispersive interactions

???

“Electrons moving through the density swerve to avoid one another, like shoppers in a mall” J.P. Perdew. Results of the swerving motion are: •  a reduction of the potential energy of mutual Coulomb repulsion (negative exchange-correlation energy) •  a small positive kinetic energy contribution to the correlation energy.

The electron correlation error

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Beyond Hartree-Fock

Many methods/approximations applicable, e.g.: •  MBPT (Møller-Plesset): MP2, MP3, MP4 •  Configuration interaction (CI): CIS, CISD •  Coupled Cluster (CC): CCSD(T), CC2

•  GW •  QMC

To learn more on Post-HF methods ⇒ L. Maschio’s lecture (Thursday)

Generally expensive (N3 ⇒ N5, N6, N7) but systematically improvable Some of them are now available for periodic systems (e.g. CRYSCOR)

( )10 ,..., N

SDi ii i i

C≡

Ψ = Φ∑


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( , , )x y zρ It is a function of three spatial variables irrespective of the system complexity

Does the electron density contain all the ingredients of a given physical system?

ΨH2O(r1,…,r10;R1,…,R3)

H2O: 10 e- and 3 nuclei

Do we really need to know Ψ(r1,…,rN; R1,…,RM) ?

Vext = −Za

Ra − ria

M

∑i

N

∑

It is an observable X-‐ray diffrac3on

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The Density Functional Theory (DFT) The electron density contains all the ingredients of a given physical system

[ ]0 ( )min ( )

NE E

ρρ

→=

rr

Given an “external potential” Vext(r), the exact ground state energy E0 and the corresponding exact electron density ρ0(r) are obtained by minimizing the functional:

where

ρ(r)dr =N∫under the constraint:

E ρ(r)!" #$= F ρ(r)!" #$+ Vext (r)ρ(r)dr∫•  the system is uniquely described by Vext(r) •  F[ρ(r)]=T[ρ(r)]+Vee[ρ(r)] is the same for all systems (universal) •  F[ρ(r)] is NOT KNOWN …unfortunately

Hohenberg-Kohn (1964) established the basics of the Density Functional Theory (DFT):

The energy is a functional of ρ(r)

E ≡ E ρ(r)[ ]


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DFT: the Kohn-Sham formalism Kohn and Sham (1965) proposed to write the density in terms of a set of orthonormal single-particle functions for non-interacting particles:

Allow us to recast the energy functional as:

ρ r( ) ≡ ρs r( ) = ψi r( )2

i=1

N

∑

( ) ( ) ( ) ( ) ( )KSS ext H xcE T E E Eρ ρ ρ ρ ρ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= + + +⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦r r r r r

The kinetic energy (non-interacting electrons) and the classical Coulomb energy (Hartree potential)

Ts = −12

ψi r( ) ∇i2 ψi r( )

i

N

∑ EH [ρ] =12

ρ(r1)ρ(r2)r1 − r2

∫ dr1dr2

Exc[ρ(r)] is the exchange-correlation energy

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Kohn-Sham equations

hKS = − 12∇2 +Vext (r )+

ρ(r ')r − r '

dr '∫ +Vxc (r )

By resorting to a Slater determinant (antisymmetric) from {ψi(x)}, the ground state density ρ0(r) can be obtained by solving self-consistently a coupled set of one-electron pseudo-Schrödinger equations, the Kohn-Sham equations:

( ) ( )ˆKSi i ih ψ εψ=r r

No approximations

If we knew Exc[ρ(r)] we could solve for the exact ground state energy and density!

KS equations are similar to HF equations (non-local potential) Similar cost (N3) but including correlation energy

VXC(r ) =δEXC [ρ(r )]δρ(r );


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Exc[ρ(r)] is the exchange-correlation energy •  it contains the rest of the total energy: (i.e. Fermi, Coulomb and “kinetic” correlation, Self-Interaction (SIE))

•  it is usually splitted into two components: •  it is not known exactly → approximations (DFA)

Kohn-‐Sham & the unknown XC func&onal

[ ] [ ] [ ]( ) ( ) ( )xc x cE r E r E rρ ρ ρ= +

)()( Heesxc EVTTE −+−=

XC func. acronym: author’s name

(usually) or place

B-LYP B3-LYP

PW91 PBE, PBE0

A.D. Becke R.G Parr J.P. Perdew

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Alphabeth Soup – Peter Elliott The quest of finding more accurate and reliable Exc[ρ(r)] functionals has originated a swarm of approximations:

•  Different ingredients •  Functional forms

•  Empirical (many fitted parameters) •  Non-empirical (many constraints)

from K. Burke JCP 136 (2012) 150901


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“Jacob’s Ladder” classification of DFT

[ ] ( )( )LDA unifxc xcE dρ ρ ε ρ= ∫ r r

[ ] ( )( ) ,GGA GGAxc xcE dρ ρ ε ρ ρ= ∇∫ r r

[ ] ( )2( ) , , ,mGGA mGGAxc xcE dρ ρ ε ρ ρ ρ τ= ∇ ∇∫ r r

The ascent of the ladder consists in embedding increasingly complex (costly) ingredients into Exc[ρ(r)].

J.P. Perdew and K. Schmidt in Density Functional Theory and Its Application to Materials, edited by V. Van Doren et al. (AIP, 2001)

Chemical Accuracy

Hartree world

LDA

GGA

Meta-GGA

Hyper-GGA

Heaven-GGA

RPA, DHyb, …

[ ] ( )( ) , , ,HGGA HGGA Exactxc xc XE dρ ρ ε ρ ρ τ ε= ∇∫ r r

[ ] ( )2( ) , , , ,DHyb DHyb Exact PTxc xc X CE dρ ρ ε ρ ρ τ ε ε= ∇∫ r r

Hybrid

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The Jacob’s Ladder in CRYSTAL

3rd rung: m-GGA: (CRYSTAL14 (M. Causà))

4th rung: Hybrids: (since CRYSTAL98). Extended to Range-Separated Hybrids (RSH) Extended to mGGA hybrids

5th rung: Double-Hybrids (CRYSTAL14) combining CRYSTAL & CRYSCOR

Since CRYSTAL14, DFA for all of the five rungs of the Jacob’s Ladder are available

1st rung: LDA

Chemical Accuracy

Hartree world

LDA

GGA

Meta-GGA

Hyper-GGA

Heaven-GGA

RPA, DHyb, …

2nd rung: GGA available since CRYSTAL98


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4th rung: Hyper-GGA (Hybrids)

Global hybrids: include a constant amount of HF exchange: B3LYP PBE0 B97

E.g.:

Local hybrids: the HF exchange contribution is included through a position-dependent function, a(r), the so-called local mixing function:

Range-separated hybrids: The amount of HF exchange included depends on the distance between electrons

( )= + − +1GH HF DFA DFAXC X X CE aE a E E

( ) ( )( )⎡ ⎤= + − +⎣ ⎦1LH HF DFA DFAXC X X CE a E a E Er r

Exexact = −

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d!r1∫ d

!r2ψi

∗(!r1)ψ j

∗(!r2)ψi (

!r2)ψ j (

!r1)!

r1 −!r2

∫i , j∑

Occupied orbitals are included through a HF-like exact exchange term

E.g.:

( ) ( )( )

τ

τ= Wa

rr

r

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Range-Separated Hybrids (I)

( ) ( ) ( ) ( )11 SR SR LR LRerf r erf r erf r erf rr r r r

ω ω ω ω− −= + +

SR MR LR

Splitting of the Coulomb operator 1/r into different ranges (A. Savin)

The amount of HF exchange included depends on the distance between electrons

ω is the length scale of separation


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Range-Separated Hybrids: erf(ωr)

( ) ( ) ( )= + − + − + −, , , , , ,RSH DFA SR HF SR DFA MR HF MR DFA LR HF LR DFAXC XC SR X X MR X X LR X XE E c E E c E E c E E

Accordingly, three families of RSH can be defined:

cSR≠0, cMR=0, cLR=0 ⇒ Screened Coulomb RSH (SC-RSH)

[HSE06, HSEsol]

cSR=0, cMR ≠ 0, cLR=0 ⇒ Middle-range corrected RSH (MC-RSH)

[HISS]

cSR≠0, cMR=0, cLR ≠ 0 ⇒ Long-range corrected RSH (LC-RSH)

[LC-ωPBE, LC-ωPBEsol, ωB97, ωB97-X ]

( ) ( ) ( ) ( )11 SR SR LR LRerf r erf r erf r erf rr r r r

ω ω ω ω− −= + +

SR MR LR

Most common formulations of RSH are based on the error function

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5th rung: Double-Hybrids (The best of both worlds)

See: Goerigk-Grimme Wires Comput Mol Sci

4 (2014) 576

•  Include long-range correlation effects (i.e. vdW-dispersion)

•  Less basis set dependent than MP2

•  Higher cost than other rungs (N5 vs N3)

Inclusion of virtual KS-orbitals usually through a MP2-like 2nd-order PT

•  Non-local HF-exchange contribution

•  Non-local correlation contribution

Different ways of including the PT2 correction:

Global Density Scaled

Range Separated

LDA GGA

mGGA

DFT WFT HF

PT2 0 1 (∞)


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5th rung: from Global Hybrids to Double Hybrids

Global hybrids: include a certain amount of HF-type exchange

Double Hybrid Func&onal Parameters Exchange Correla&on a b

B2-‐PLYP B88 LYP 0.53 0.27 B2GP-‐PLYP B88 LYP 0.65 0.36 mPW2-‐PLYP mPW LYP 0.55 0.25

( )= + − +1GH HF DFA DFAXC X X CE aE a E E

Double hybrids: not only the HF exchange contribution is included but also a MP2-like term for correlation (i.e. virtual orbitals) (Truhlar, Grimme 2006)

PT2: Gorling-Levy perturbation theory (GL2≈MP2) Cost as MP2, but less basis set dependent Parameters (a and b) are fitted to thermochemical data (Grimme)

SCF Post-SCF

( ) ( )= + − + − +DH HF DFA DFA PTXC X X C CE aE a E b E bE 21 1

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A rigorous Double Hybrid: Density-Scaled 1DH

Double Hybrid Func&onal Parameters Exchange Correla&on λ λ2

DS1DH-‐BLYP B88 LYP 0.80 0.64 1DH-‐BLYP B88 LYP 0.55 0.30 B2-‐PLYP B88 LYP 0.53 0.27 (0.28)

1DH: by neglecting the density-scaling, the one-parameter double-hybrid approximation is obtained thus making a link with standard DHs

Rigorously proved through the adiabatic connection that a=λ and b=λ2 (λ=coupling strength constant). Only one parameter is needed.

[Sharkas, Toulouse & Savin JCP 134 (2011) 064113]

DS1DH: Density-Scaled one-parameter Double-Hybrid

SCF Post-SCF

( ) [ ] [ ]λλλ λ ρ ρ λ ρ λ⎡ ⎤= + − + − +⎣ ⎦

1 , 2 2 211DS DH HF DFA DFA DFA PT

XC X X C C CE E E E E E

( ) [ ] ( ) [ ]λ λ λ ρ λ ρ λ= + − + − +1 , 2 2 21 1DH HF DFA DFA PTXC X X C CE E E E E


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Range-Separated Double Hybrids RS-DH include the PT2 correlation correction at long-range according to a range separation scheme of the Coulomb operator (Savin, Stoll, Gill):

I. C. Gerber and J. Ángyán, CPL, 415, 100 (2005) J. G. Ángyán, I. C. Gerber, A. Savin, J. Toulouse, PRA 72, 012510 (2005).

* Chai/Head-Gordon, JCP (2009) Cx=0.64 Css=0.53,Cos=0.45

SCF Post-SCF

EXCRS−DH = EX

DFA,sr ρ!" #$+EXHF ,lr +EC

DFA,sr ρ!" #$+ECPT 2,lr

Double Hybrid Func&onal Parameters Exchange Correla&on µ

RSHLDA+MP2 µLDA µPW92 0.5 RSHPBE+MP2 µPBE µPBE 0.5 ωB97X-‐2 * µB97X B97 0.3

E. Goll, H.-J. Werner, and H. Stoll, PCCP 7, 3917 (2005). E. Goll, H.-J. Werner, H. Stoll, et al. Chem. Phys. 329, 276 (2006).

Not only PT2 but also CCSD:

1r12

=1−erf µr12( )

r12+erf µr12( )r12

for both exchange and correlation (Savin, Toulouse, Werner, Ángyán, Fromager,…)

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* RSH methods based on the PBE functional and the Henderson-Janesko-Scuseria (HJS) X-hole model (as in VASP, NWChem). No-bipolar expansion for exchange

Availability: Total energy and analytic gradients, IR int, Piezoelectricity, CPHF

Self-Consistent-Hybrid scheme (G. Galli) (new in CRYSTAL17)

More recent Minnesota functionals will be available (in collaboration with D. Truhlar)

Summary of XC functionals available in CRYSTAL LDA GGA/mGGA Hybrids GGA/mGGA

(GH & RSH) Double-Hybrids

Slater PBE-xc PBE0 B2-PLYP VWN B88 B3LYP B2GP-PLYP PZ LYP PBEsol0, WC1LYP mPW2-PLYP

PWLSD PBEsol-xc B97 VBH-x SOGGA-x M05, M05-‐2X / M06 family DS1DH

WC-x PW92-xc HSE06 / HSEsol / HISS * RS-DH

WL-c LC-ωPBE / LC-ωPBEsol * M06-L ωB97 / ωB97-X *

LC-BLYP (Hirao) * B97 family RSHXLDA *


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HF and DFT: a few remarks

HF •  Catches an important part of physics (mean-field theory) •  Does not include correlation energy (Ec=E0-EHF) •  Self interaction free •  No dispersive interaction

DFT •  It is in principle exact •  In practice, it makes more or less justified approximations •  Results may be better than those obtained with HF •  Which Exc functional? (…must be validated -> benchmarks) •  Self interaction error (hybrid functionals) •  Problems with spin-polarized systems and strongly correlated systems •  No dispersive interaction (Double Hybrids or empirical and non-empirical corrections) (…wait until next lecture on Thursday…)

mssc2016 hbs civalleri fin - · pdf filebartolomeo civalleri department of chemistry ... (m...

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