mssc2016 HBS civalleri fin - · PDF fileBartolomeo Civalleri Department of Chemistry ... (M...
Transcript of mssc2016 HBS civalleri fin - · PDF fileBartolomeo Civalleri Department of Chemistry ... (M...
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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 = −
12
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…)