ELECTRONIC STRUCTURE WITH DFT:GGA AND BEYOND

Jorge KohanoffQueen’s University Belfast

United Kingdom

j.kohanoff@qub.ac.uk

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)