Electronic structure theory: Fundamentals to frontiers.

2. Density functional theory

MARTIN HEAD-GORDON, Department of Chemistry, University of California, and

Chemical Sciences Division, Lawrence Berkeley National Laboratory

Berkeley CA 94720, USA

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Outline

1. Basics

2. Limitations of standard functionals 3.  Range-separated functionals

Branches of the family tree

•  Wavefunction-based electronic structure theory: •  Minimize the energy by varying the wavefunction •  Tremendously complicated unknown function:

•  Modeling the wavefunction yields “model chemistries”

•  Density functional theory •  The unknown is very simple: •  Hohenberg-Kohn theorem guarantees that: •  True functional is unknown and probably unknowable •  Modeling the functional gives DFT model chemistries.

! = !

!r( )

! = !

!r1, !

r2 ,...,!rn( )

E = E !

!r( ){ }

A brief overview of density functional theory

•  First Hohenberg-Kohn theorem (1965): •  1:1 mapping between ground state electron densities and Hamiltonians.

•  Proof by contradiction: let H1 and H2 have the same ρ(r) •  Use Ψ2 as trial function in H1 problem •  Use Ψ1 as trial function in H2 problem

•  Contradiction:

E1 !2{ } = !2 T̂ + V̂ ee !2 + v1"# dr > !1 T̂ + V̂ ee !1 + v1"# dr

!1 T̂ + V̂ ee !1 > !2 T̂ + V̂ ee !2

!2 T̂ + V̂ ee !2 > !1 T̂ + V̂ ee !1

E2 !1{ } = !1 T̂ + V̂ ee !1 + v2"# dr > !2 T̂ + V̂ ee !2 + v2"# dr

E1 !2{ } > E1

E2 !1{ } > E2

A brief overview of density functional theory

•  First Hohenberg-Kohn theorem (1965): •  1:1 mapping between ground state electron densities and Hamiltonians. •  Ground state energy E is determined directly from the Hamiltonian •  Hence E is given in terms of the density, ρ(r).

•  A formal construction exists for the exact functional, •  Constrained search over all wavefunctions yielding ρ(r) (!!!)

•  So, in practice the functional must be modeled. •  Given a functional, and an external potential (nuclear field) ρ(r) is

found by minimizing over allowed densities.

E = E !

!r( ){ }

Construction of model density functionals

•  To model kinetic, exchange, correlation functionals…

•  1A) Find a model problem where the functional can be obtained…

•  H atom? Uniform electron gas?

•  1B) Assume a form for the functional and fix the parameters by…

•  Known exact conditions (e.g. get model problems right) •  Minimizing the errors on known data

•  2) Transfer the functional to problems of interest •  Test, test, test…. •  If validation is encouraging enough, predict…

•  In one of your problems, you will extract the kinetic energy functional that solves the uniform electron gas problem. It is not much more difficult to extract the corresponding exchange functional.

•  These are the main ingredients of the Thomas-Fermi model which is a Hohenberg-Kohn density functional.

Kohn-Sham density functional theory

•  Largest (unknown) energy contribution is the kinetic energy. •  No satisfactory kinetic energy functional yet exists.

•  Kohn-Sham framework (a beautiful sidestep): •  Use the kinetic energy of a non-interacting system with the

same electron density (a Hartree-Fock type wavefunction). •  Leaves exchange and electron correlation (XC) to specify. •  Kohn-Sham computational cost: similar to Hartree-Fock. •  Still cheap enough to apply to large systems.

Modern Kohn-Sham density functionals

•  Local density approximation (LDA): 1960’s, 1970’s •  Functional depends only on the density at each point, ρ(r) •  LDA overbinds as much as Hartree-Fock (mean field) underbinds!

•  Generalized gradient approximations (GGA’s): 1988 •  Functional depends on density ρ(r) and its gradients at each r •  Greatly improved results! 4-6 kcal/mol error for BLYP, PBE etc.

•  Exact exchange mixing (adiabatic connection): 1992 •  Mix some Hartree-Fock exchange with GGA’s (Becke) •  Best yet! 2-3 kcal/mol error for B3LYP

Classes of Kohn-Sham density functionals

•  Local spin density approx

•  Example: SVWN

•  Generalized gradient approx

•  Example: BLYP

•  Hybrid density functionals •  Wave function exchange •  Example: B3LYP

223 atomization energies Mean abs errors (kcal/mol)

G3/99 test set

EXC = dr ! XC" # r( ){ }

EXC = dr ! XC" # r( ),$# r( ){ }

1966

1985

1993

Multiple choice questions….

•  In Kohn-Sham DFT, which energy contribution is not strictly a functional of the electron density? –  (a) electron-nuclear attraction –  (b) exchange-correlation –  (c) kinetic energy

•  Which of the following properties is obeyed by B3LYP? –  (a) variationality –  (b) exact for the uniform electron gas –  (c) exact for 1-electron systems –  (d) size-consistency

Outline

1. Basics

2. Limitations of standard functionals 3.  Range-separated functionals

Challenges for density functionals

•  Accuracy: lack of systematic improvability confronts…

•  (1) Limitations of the exchange functional

•  Self-interaction

•  (2) Limitations of the correlation functional

• London forces

•  Strong correlations

B3LYP dissociation of H2+ (0.65Å to 3Å)

-80

-70

-60

-50

-40

-30

-20

-10

0

rela

tive

ener

gy (

kca

l/m

ol)

HF

B3LYP

3Å 0.65Å

B3LYP dissociation of H2+ (3Å to 13Å)

-60

-50

-40

-30

-20

-10

0

rela

tive

ener

gy (

kca

l/m

ol)

HF

B3LYP

3Å 13Å

Alkali halide dissociation curves

B3LYP

products have fractional charges -- due to electronegativity difference

Charge transfer states in time-dependent DFT

CT states are too low & lack Coulomb attraction!

!CT r( ) " IPZnBC + EABC #1 / R" 2.7eV

BLYP/6-31G*

Importance of long-range exact exchange

•  Ground state potential energy surfaces •  Diatomic cation dissociation problem (H2

+, Ar2+, etc)

•  Barrier height problems: generally too low •  Electrons tend to be too delocalized

•  Charge-transfer excited states •  D-A Coulomb attraction is missing! •  Magnitude of CT states is greatly underestimated •  Contaminates the TDDFT spectrum of large molecules

Reducing self-interaction: Range-separation long-range exchange via erf(ωr)

•  erf(ωr): long-range. Do exactly. •  erfc(ωr): short-range. Do GGA.

•  Key contributions: •  Savin (1996): concept •  Gill et al (1996): solved short-range LSDA exchange •  Hirao et al (2001): long-range corrected (LC) functional •  Handy, Gerber & Angyan, Scuseria, Perdew, Yang, …

•  One can view this as justified within a generalized Kohn-Sham framework, or via adiabatic connection.

1r12

=erfc(! r12 )

r12

+erf (! r12 )

r12

Dispersive effects: e.g. supramolecular interactions

•  fullerene-porphyrin dimer

•  binding is 31 kcal/mol

•  GGA’s give little or no binding energy

Y. Jung, MHG, Phys. Chem. Chem. Phys. 8, 2831 (2006)

Recovering Van der Waals interactions: Empirical dispersion (-D) corrections

•  Additional non-local correlation energy contribution:

•  C6i are atomic C6 factors; f damps at short-range

•  Greatly improves dispersion-dominated interactions: •  R. Ahlrichs, R. Penco, G. Scoles, Chem. Phys. 19, 119 (1977) •  Q. Wu and W.T. Yang, J. Chem. Phys. 116, 515 (2002) •  S. Grimme, J. Comput. Chem. 25, 1463 (2004); 27, 1787 (2006) • 

•  Not actually a density functional, but.... •  Computationally free •  Physically reasonable (but double counting problem)

Edisp = !C6

ij

Rij6

i< j

atoms

" fdamp Rij( ) C6ij = C6

iC6j fdamp = 1+ a(Rij / Rr )

!12"# $%!1

Recovering Van der Waals interactions: Double hybrid functionals (assigned paper)

•  Gorling-Levy perturbation theory motivates mixing 2nd order perturbation theory (for correlation) with semilocal correlation functionals....

•  Physically, PT2 includes non-local long-range correlation that is missing in semilocal functionals...

•  But, there is again a double counting problem...

Strongly correlated molecules

Cope rearrangement Oxygen-evolving complex: Mn4O4

No easy answers for strong correlations....

•  Either requires a tremendously powerful correlation functional, or, ...

•  lies beyond generalized Kohn-Sham theory. For instance using a multi-configuration reference wave-function....

•  While this is an important challenge, it is one that is not yet satisfactorily answered today...

Outline

1. Basics

2. Limitations of standard functionals 3.  Range-separated functionals

Functional ingredients.... and parameters...

•  B97 XC density functional: 12 linear parameters (M=4)

•  Long-range exact exchange: 1 non-linear parameter (ω)

•  Short-range exact exchange: 1 linear parameter (cX)

s! = "#! / #!4 /3EB97 = dr! "#

LSDA cj# f s#

2( )$% &'j

j=0

M

(

EXLR!HF = !

12

dr1" #i r1( )# j r1( ) dr2

erf $r12( )r12

" #i r2( )# j r2( )ij%

EXSR!HF = !

cX2

dr1" #i r1( )# j r1( ) dr2

erfc $r12( )r12

" #i r2( )# j r2( )ij%

2 types of non-local correlation corrections

•  Empirical atom-atom dispersion (-D): 1 parameter (a)

•  Similar to R. Ahlrichs, W.T. Yang, S. Grimme... •  Computational cost is zero, but not a density functional

•  Or: Double hybrid perturbation theory: 2 parameters

•  Includes effect of unoccupied orbitals •  Significantly more computational expense

Edisp = !C6

ij

Rij6

i< j

atoms

" fdamp Rij( ) C6ij = C6

iC6j fdamp = 1+ a(Rij / Rr )

!12"# $%!1

EPT 2 = cOSEOS(2) + cSSESS

(2)

4 long-range corrected B97 functionals (Jeng-Da Chai)

•  ωB97: 100% long-range exact exchange (13 parameters)

•  ωB97X: adds some short-range exact exchange (14)

•  ωB97X-D: adds empirical dispersion (15)

•  ωB97X-2: adds non-local second order correlation (16)

EXC!B97 = EX

LR"HF + EXSR"B97 + EC

B97

EXC!B97X = EX

LR"HF + cXEXSR"HF + EX

SR"B97 + ECB97

EXC!B97X"2 = EX

LR"HF + cXEXSR"HF + EX

SR"B97 + ECB97 + EC

PT 2

EXC!B97X"D = EX

LR"HF + cXEXSR"HF + EX

SR"B97 + ECB97 + EC

disp

Why must these functionals be “trained”?

•  All parameters should be determined self-consistently... subject to constraints that preserve the LDA limit –  hence cannot adopt existing B97 values

•  For ωB97 and ωB97X: –  GGA parameters: short-range exchange; semi-local correlation –  Range separator: compromise across problems of interest

•  For ωB97X-D: –  Additionally minimize the correlation double-counting error

Training set: 412 data points (Jeng-Da Chai)

•  Bond-breaking energies: G3/99 dataset (296) –  Curtiss, Raghavachari, Redfern, Pople, JCP 112, 7374 (2000)

•  Barrier heights for simple chemical reactions (76) –  Zhao, Truhlar et al, JPC A 108, 2715 (2005), 109, 2012 (2006)

•  Non-covalent interactions (22) –  Jurecka, Sponer, Cerny, Hobza, PCCP 8, 1985 (2006)

•  Absolute atomic energies (18) –  Chakravorty, Gwaltney, Davidson, Parpia, Fischer, PR A 47, 3649 (1993)

Comparison of optimizable functionals: All trained identically (Jeng-Da Chai)

HCTH*: 12 parameter GGA (like ωB97 with ω=0) B97*: 13 parameter hybrid (like ωB97X with ω=0)

ωB97: 13 parameter range-separated. ωopt=0.4

ωB97X: 14 parameter range-separated hybrid ωopt=0.3, cX=0.16 ωB97X-D: 15 parameter, with dispersion ωopt = 0.2, cX = 0.22

–  All are exact for the uniform electron gas (constraints)...

–  What is the value of range separation? And dispersion?

223 G3/99 atomization energies (Jeng-Da Chai)

Training set data

GGA hybrid range-separated family

38 non-hydrogen transfer barriers (Jeng-Da Chai)

Training set data

range-separated family hybrid GGA

22 intermolecular interactions (Jeng-Da Chai)

Training set data

range-separated family hybrid GGA

Test set data

Test performance for energies (Jeng-Da Chai)

Alanine tetrapeptide conformational energies

•  Compare against basis set limit MP2 •  27 conformations •  Calculations by Daniel Lambrecht

Conclusions and open issues

•  For molecular problems, particularly where self-interaction is significant, range-separated functionals are a significant improvement over hybrids –  ωB97, ωB97X, and ωB97X-D are widely useful –  though significant weaknesses remain... –  and further testing & comparison is desirable (e.g. vs M06)

•  Challenges include –  strong correlation (unresolved) –  can self-interaction can be further reduced? –  increased exact exchange degrades performance for metals*