E LECTRONIC S TRUCTURE WITH DFT: GGA AND BEYOND Jorge Kohanoff Queen’s University Belfast United...
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Transcript of E LECTRONIC S TRUCTURE WITH DFT: GGA AND BEYOND Jorge Kohanoff Queen’s University Belfast United...
ELECTRONIC STRUCTURE WITH DFT:GGA AND BEYOND
Jorge KohanoffQueen’s University Belfast
United Kingdom
Kohn-Sham equations:
This PDE must be solved self-consistently, as the KS potential depends on the density, which is constructed with the solutions of the KS equations.
DENSITY FUNCTIONAL THEORY (DFT):KOHN-SHAM EQUATIONS
)()()]([2
22
rrr KSnn
KSnKSV
m
N
n
KSnKS
1
2)()()( rrr
]['|'|
)'()(][ XCextKS dVV
r
rr
rr
The inhomogeneous electron gas is considered as locally homogeneous:
LDA XC hole centred at r, interacts with the electron also at r. The exact XC hole is centred at r’
This is partially compensated by multiplying the pair correlation function with the density ratio (r)/(r’)
EXCHANGE AND CORRELATION IN DFT:
THE LOCAL DENSITY APPROXIMATION (LDA)
rrr dE LDAXCXC )]([)(][
'|'|
)',(~][ r
rr
rrdXCLDA
XC
XC energy density of the HEG
)'(
)())(|,'(|~)',(~
r
rrrrrr
HEG
XCLDAXC gg
Location of the XC hole (Jones and Gunnarsson, 1982)
EXCHANGE AND CORRELATION IN DFT:
THE LOCAL DENSITY APPROXIMATION (LDA)
Favors more homogeneous electron densities Overbinds molecules and solids (Hartree-Fock underbinds) Geometries, bond lengths and angles, vibrational frequencies
reproduced within 2-3% Dielectric constants overestimated by about 10% Bond lengths too short for weakly bound systems (H-bonds, VDW) Correct chemical trends, e.g. ionization energies
Atoms (core electrons) poorly described (HF is much better) XC potential decays exponentially into vacuum regions. It should
decay as –e2/r. Hence, it is poor for dissociation and ionization Poor for metallic surfaces and physisorption Very poor for negatively charged ions (self-interaction error) Poor for weakly bound systems: H-bonds (), VDW (non-local) Band gap in semiconductors too small (~40%) Poor for strong on-site correlations (d and f systems, oxides, UO2)
LDA-LSDA: TRENDS AND LIMITATIONS
Inhomogeneities in the density Self-interaction cancellation Non-locality in exchange and correlation Strong local correlations
Gradient expansions Weighted density approximation Exact exchange in DFT (OEP local vs HF non-local) DFT-HF hybrids Self-interaction correction Van der Waals and RPA functionals LSDA+U Multi-reference Kohn-Sham GW approximation (Many-body)
BEYOND THE LDA
EXC expanded in gradients of the density
where is the spin polarization
s=||/2kF is the density gradient
And FXC is the enhancement factor
First-order term is fine, but higher-order terms diverge. Only by some re-summation to ∞-order the expansion converges.
GGA: FXC is designed to fulfil a number of exactly known properties, e.g. Perdew-Burke-Ernzerhof (PBE)
1. Exchange: uniform scaling, LSDA limit, spin-scaling relationship, LSDA linear response, Lieb-Oxford bound
2. Correlation: second-order expansion, hole sum rule, vanishes for rapidly varying densities, cancels singularity at high densities
GRADIENT EXPANSIONS:GENERALIZED GRADIENT
APPROXIMATION
rrrr dsFE XCLSDAXCGGA )](,,[)](,[)(][
r+dr
(r+dr)
Improves atomization and surface energies Favors density inhomogeneities Increases lattice parameters of metals Favors non-spherical distortions Improves bond lengths Improves energies and geometries of H-bonded systems There is error cancellation between X and C at short range
XC potential still decays exponentially into vacuum regions Some improvement in band gaps in semiconductors What was correct in LDA is worsened in GGA Still incorrect dissociation limit. Fractionally charged
fragments Inter-configurational errors in IP and EA
Error cancellation between X and C is not complete at long-range. X hole is more long-ranged than XC hole
PROPERTIES OF THE GGA
Combine GGA local exchange with Hartree-Fock non-local exchange:
Parameter fitted to experimental data for molecules (~0.75), or determined from known properties.
PBE0, B3LYP, HSE06
Properties:
1. Quite accurate in many respects, e.g. energies and geometries
2. Improve on the self-interaction error, but not fully SI-free
3. Improve on band gaps
4. Improve on electron affinities
5. Better quality than MP2
6. Fitted hybrids unsatisfactory from the theoretical point of view
HYBRID FUNCTIONALS
][][)1(][][ GGAC
HFX
GGAXHYBRID EEEE
Self-interaction can be removed at the level of classical electrostatics:
Potential is state-dependent. Hence it is not an eigenvalue problem anymore, but a system of coupled PDEs
Orthogonality of SIC orbitals not guaranteed, but it can be imposed (Suraud)
Similar to HF, but the Slater determinant of SIC orbitals is not invariant against orbital transformations
The result depends on the choice of orbitals (localization)
SELF-INTERACTION CORRECTION (SIC)
'
|'|
)'()(
2
1rr
rr
rrddEH
N
n
nnHSIC ddEE
1
'|'|
)'()(
2
1rr
rr
rr
'|'|
)'()()()( r
rr
rrr dVV n
Hn
SIC
Perdew-Zunger 1982Mauri, Sprik, Suraud
Van der Waals (dispersion) interactions: are a dynamical non-local correlation effect
Dipole-induced dipole interaction due to quantum density fluctuations in spatially separated fragments
Functional (Dion et al 2004):
Expensive double integral Efficient implementations (Roman-Perez and Soler
2009) Good approximations based on dynamical response
theory Beyond VDW: Random Phase Approximation (Furche)
DYNAMICAL CORRELATION: VDW
1(r,t) 2(r,t)
1(t) E1(t).2(t)
')'()',()( rrrrrr ddEVDW = VDW kernel fully non-local. Depends on (r) and (r’)
Empirical approaches:
With f a function that removes smoothly the singularity at R=0, and interferes very little with GGA (local) correlation.
Grimme (2006): C6 parameters from atomic calculations. Extensive parameterization: DFT-D.
Tkatchenko and Scheffler (2009): C6 parameters dependent on the density.
Random Phase Approximation (RPA): captures VDW and beyond. Can be safely combined wit exact exchange (SIC). Infinite order perturbation (like Coupled clusters in QC).
Furche (2008); Paier et al (2010); Hesselmann and Görling (2011)
DYNAMICAL CORRELATION: VDW AND BEYOND
66 )(
)(IJ
IJIJP
IJIJIJVDW R
RCRfE
Strong onsite Coulomb correlations are ot captured by LDA/GGA
These are important for localized (d and f) electronic bands, where many electrons share the same spatial region: self-interaction problem
Semi-empirical solution: separate occupied and empty state by an additional energy U as in Hubbard’s model:
This induces a splitting in the KS eigenvalues:
STRONG STATIC CORRELATION: LSDA+U
jji
iULSDAULSDA ffUNNUEE
2
1)1(
2
1fi=orbital occupations
fUf
E LSDAi
i
ULSDAi 2
12/
2/
U
ULSDAi
emptyi
LSDAi
occi
SUMMARY OF DFT APPROXIMATIONS
ELECTRONIC STRUCTURE OF UO2
Using the quantum-espresso package (http://www.quantum-espresso.org/)
• Pseudopotentials• Plane wave basis set
PROPERTIES
fluorite structure fcc, 3 atoms un unit cellLattice constant = 10.26 BohrElectronic insulator. Eg=2.1 eVElectronic configuration of U: [Rn]7s26d15f3
U4+: f2
5f-band partially occupied (2/7)UO2: splitted by crystal field:
t1u(3)+t2u(3)+ag(1)Still partially occupied (2/3) Jahn-Teller distortion opens gap.
PSEUDOPOTENTIAL
CONVERGENCE WITH ENERGY CUTOFF
ENERGY-VOLUME CURVE
GGA(PBE) DENSITY OF STATES
GGA+U DENSITY OF STATES
GGA+U DENSITY OF STATES: DISTORTED
An important family of Room Temperature Ionic Liquids (Green solvents)
Competing electrostatic vs dispersion interactions
Large systems studied with DFT, within LDA or GGA [Del Popolo, Lynden-Bell and Kohanoff, JPCB 109, 5895 (2005)]
Force fields fitted to DFT-GGA calculations[Youngs, Del Popolo and Kohanoff, JPCB 110, 5697 (2006)]
Electrostatics well described in DFT (LDA or GGA)
Dispersion (van der Waals) interactions are absent in both, LDA and GGA
VAN DER WAALS FOR IMIDAZOLIUM SALTS
DFT IMIDAZOLIUM SALTS
RESULTS FOR SIMPLE DIMERS
Ar2 and Kr2(C6H6)2
Bond lengths: 5-10% too long
Binding energies: 50-100% too large
M. Dion et al, PRL 92, 246401 (2004)
RESULTS FOR SOLIDS
Polyethylene Silicon
• Reasonable results for molecular systems
• Keeps GGA accuracy for covalent systems
General purpose functional
SOLID FCC ARGON (E. ARTACHO)
Some overbinding, and lattice constant still 5% too large
… but much better than PBE (massive underbinding and lattice constant 14% too large)
THE DOUBLE INTEGRAL PROBLEM
(q1,q2,r12) decays as r12-6
Ecnl = (1/2) d3r1 d3r2 (r1) (r2) (q1,q2,r12) can be truncated for
r12 > rc ~ 15Å
In principle O(N) calculation for systems larger than 2rc ~ 30Å
But... with x ~ 0.15Å (Ec=120Ry) there are ~(2106)2 = 41012 integration points
Consequently, direct evaluation of vdW functional is much more expensive than LDA/GGA
FACTORING (Q1,Q2,R12)
)(),()()(
)()()(2
1
)()()(2
1
),,()()(2
1
)()()(),,(
,
,122121
12212121
,12211221
rrrr
kkk
rrrr
rrrr
qp
kd
rdd
rqqddE
rqpqprqq
nlc
G. Román-Pėrez and J. M. Soler, Phys. Rev. Lett. 2009
Expand in a basis set of functions p(q)
FT
INTERPOLATION AS AN EXPANSION
otherwise0
if)(
)(
)(
)()()(
nini
ni
ijnij ji
j
i
iii
ii
xxxxx
xx
xp
xffxpfxf
20 grid points are sufficient
Smoothening of required at small q
Basis functions p(q) interpolate between grid points
• Lagrange polynomials: grid given by zeros of orthogonal polynomials• Cubic splines: grid points defined on a logarithmic mesh
O(NLOGN) ALGORITHM
do, for each grid point i
find i and i
find qi=q(i ,i )
find i = i p(qi )
end do
Fourier-transform i k
do, for each reciprocal vector k
find uk = (k) k
end do
Inverse-Fourier-transform uk ui
do, for each grid point i
find i , i , and qi
find i , i /i , and i / i
find vi
end do
Implemented into SIESTA, but not SIESTA-specific:
Input: i on a regular grid
Output: Exc , vixc on the grid
No need for supercells in solids
No cutoff radius of interaction
ALGORITHM EFFICIENCY
MessageIf you can simulate a system with LDA/GGA,
you can also simulate it with vdW-DFT
System Atoms
CPU time in GGA-XC
CPU time in vdW-XC
vdW/GGA overhead
Ar2 2 0.75 s (44%) 7.5 s (89%) 400%
MMX polymer 124 1.9 s (2%) 10.6 s (16%) 17%
DWCN 168 11.9 s (0.6%) 109 s (5.2%) 4%
IMIDAZOLIUM CRYSTALS: VOLUMES
LDA PBE VDW EXP
[emim][PF6] 877.4 1088.3 1059.1 1023.9
[bmim][PF6] 513.4 636.6 620.0 605.0
[ddmim][PF6] 1710.9 2258.0 2095.5 2000.6
[mmim][Cl] 607.9 750.5 728.6 687.6
[bmim][Cl]-o 843.8 1039.9 1019.8 961.1
[bmim][Cl]-m 836.3 1051.9 1024.6 966.7
Error -13.5% +8.5% +5%
IMIDAZOLIUM CRYSTALS: HEXAFLUOROPHOSPATES
bmimpf6
emimpf6
ddmimpf6
[bmim][PF6]
[ddmim][PF6]C4-C4 C1”-
C1”C12-C12’
C4’-C4’ C2-C4’ C4’-C1” P-P C2-P C1”-P C4’-P C12’-P Error
LDA 5.18 4.28 4.83 4.52 4.23 6.00 6.66 3.55 3.95 4.27 4.35 6.0%
PBE 5.62 4.24 5.44 5.12 5.10 6.56 6.89 4.00 4.75 4.96 4.78 7.0%
VDW 5.47 4.29 5.23 4.90 4.79 6.44 6.80 3.82 4.22 4.68 4.65 3.3%
EXP 5.47 4.36 5.01 4.85 4.61 6.25 6.45 4.23 4.01 4.62 4.63
C5-C5 C1”-C1” C4’-C4’ C2-C4’ C4’-C1” P-P C2-P C1”-P C4’-P Error
LDA 4.32 4.24 3.73 4.61 3.64 5.05 3.70 3.88 4.75 7.0%
PBE 4.64 4.85 3.99 5.07 3.97 5.69 4.16 4.21 4.95 2.6%
VDWE 4.60 4.55 4.05 4.80 4.07 5.64 4.21 4.20 4.91 2.6%
VDW 4.54 4.60 4.04 4.80 3.95 5.62 3.90 4.11 4.87 2.0%
EXP 4.56 4.52 4.05 5.08 3.86 5.54 3.95 4.29 4.94
IMIDAZOLIUM CRYSTALS: HEXAFLUOROPHOSPATES
IMIDAZOLIUM CRYSTALS: CHLORIDES
[bmim][Cl]-m
C5-C5 C1’-C1’ C2-C1’ C2-Cl C1’-Cl
LDA 3.07 3.54 3.37 3.34 3.81
PBE 3.94 3.86 3.63 3.48 3.74
VDW 3.34 3.82 3.56 3.56 3.77
C5-C5 C1”-C1” C4’-C4’ C2-C4’ C4’-C1” C2-Cl C1”-Cl C4’-Cl
LDA 4.87 4.50 5.17 3.25 3.83 3.28 3.51 3.70
PBE 5.26 5.14 5.72 3.63 3.81 3.38 3.86 3.89
VDW 5.21 4.97 5.42 3.53 3.84 3.46 3.84 3.94
EXP 5.19 4.96 5.33 3.48 3.62 3.39 3.82 3.80
[mmim][Cl]
[BMIM][CL]: POLYMORPHISM
[bmim][Cl]-Monoclinic [bmim][Cl]-Orthorhombic
Energy difference per neutral ion pair
LDA PBE VDW FF(CLaP)
E(M-O) [eV] -0.05 -0.06 -0.01 0.08
IMIDAZOLIUM CLUSTERS: TRIFLATES
[bmim][Tf] [bmim][Tf]2
LDA PBE VDW FF(CLaP)
E [eV] -1.411 -1.053 -1.453 -1.457
Dimer association energy
LDA PBE VDW FF(CLaP)
Error 7.2% 2.3% 1% --
Geometry
CONCLUSIONS
1. The non-empirical van der Waals functional of Dion et al. (DRSLL) improves significantly the description of the geometry of imidazolium salts.
2. Volumes are improved respect to PBE, but still overestimated by 5%.
3. Energetics is also improved. It is similar to that of empirical force fields such as CLaP.
4. The cost of calculating the van der Waals correlation correction is 10 times that of PBE. However, in a self-consistent calculation for 100 atoms the overhead is only 20%.
THE THEORETICAL LANDSCAPE
Size (number of atoms)
1000
10000
1000000
100000100
High
Low
Treat relevant part of the system quantum-mechanically, and the rest classically.
The problem is how to match the
two regions. Easy for non-bonded
interactions, more difficult for
chemical bonds
One can also treat part of the
system as a polarizable
continuum, or reaction field (RF)
QM/MM
QUANTUM MECHANICS IN A LOCAL BASIS
TIGHT BINDING
TIGHT BINDING
TIGHT BINDING MODELS FOR WATER
Ground-up philosophy
Water molecule
1.Minimal basis. On-site energies to reproduce band structurea) 1s orbital for H: Hs
b) 2s & 2p orbitals for O: Os, Op
2.O-H hopping integrals: a) Values at equilibrium length to reproduce HOMO-LUMO gap: tss, tsp
b) GSP functional form. Cut-off between first and second neighbours
3.Charge transfer: Hubbard terms fitted to reproduce dipole moment: UO, UH
4.O-H pair potential: GSP form, fitted to reproduce bond length and symmetric stretching force constant
5.Crystal field parameter spp selected to reproduce polarizability
A. T. Paxton and J. Kohanoff, J. Chem. Phys. 134, 044130 (2011)
TIGHT BINDING MODELS FOR WATER
Ground-up philosophy
Water dimer
6.O-O hopping integrals: tss, tsp , tpp, tpp GSP form. Cut-off after first neighbours
7.O-O pair potential Various forms (GSP, quadratic) to reproduce binding energy curve
8.Fitting procedurea) By hand (intuitive)b) Genetic algorithm
This is the end of the fitting
All the rest are predictions
ICE-XI
• DFT ice sinks in water!• Polarizable model marginal• Point charge model is fine
TIGHT-BINDING LIQUID WATERA. T. PAXTON AND J. KOHANOFF, J. CHEM. PHYS. 134, 044130 (2011)
TIGHT BINDING MODEL FOR TIO2Band structure: A. Y. Lozovoi, A. T. Paxton and J. Kohanoff
Rutile
Anatase
DFT TB
O on-site energies Os, Op and O-O hopping integrals: tss, tsp , tpp, tpp
WATER/TIO2 INTERFACES (SASHA LOZOVOI)
Single water adsorption: dissociation
WATER/TIO2 INTERFACES (SASHA LOZOVOI)
A water layer on TiO2
WATER/TIO2 INTERFACES (SASHA LOZOVOI)
Bulk water on TiO2
• 4 TiO2 layers (480 atoms)
• 184 water molecules (552 atoms)
• 1032 atoms
• TBMD 1 ps (one week)
WATER/TIO2 INTERFACES (SASHA LOZOVOI)
z-density profile of bulk water between TiO2 surfaces
PROTON TRANSFER IN WATER
Chemical bonds broken and formed
ORGANIC MOLECULES (TERENCE SHEPPARD)
Benzylacetone in water
BENZYLACETONE AT WATER/TIO2 INTERFACE
Sasha Lozovoiand
Terence Sheppard
SUMMARY In developing a force field there are 3 main aspects to
consider. In order of relevance: Designing the model Choosing the properties to be reproduced Choosing a methodology to fit those properties
Fitting procedures cannot work out their magic if the model is not good enough
To become model-independent we need to introduce electrons explicitly: ab initio and DFT methods
These are expensive. Simplifications like semi-empirical, tight-binding and QM/MM methods bridge the gap
Further simplifications lead to extremely efficient bond-order potentials (BOP)