Density Functional Theory (DFT)

CHEM 430

·,-

ti ¼ ... .. ..

Method Selection

10 ms, thousands of atoms: protein folding, drug binding

100 fs, 10 atoms: photochemistry

Corr lated M

Coarse-Gr

1 ms+, 1 million atoms: dynamics of large proteins, cell membranes, viruses

Density

Functional

Theory

10 ps, 100 atoms: chemical

Computational cost

Approximate Solutions to Schrodinger's Equation

Schrodinger's equation is nearly impossible to solve, so various

approximate methods are used

X1(1) x1(2)

l'-l')HF = %2

:(1) %2(2)

xN (I) xN (2)

Slater determinant: Use wavefunction of

noninteracting electrons for interacting system

Density functional theory: The ground-state

electron density contains all information in the

ground-state wavefunction

• Electron repulsion makes this

problem difficult

• Approach 1: Use approximate

wavefunction forms (QM)

• Approach 2: Use the electron density

as the main variable (OFT)

• Approach 3: Approximate the energy

using empirical functions (MM)

!��

, . . :

�· �t It 01 t ..

,.,..,.

Molecular mechanics: Use empirical

functions and parameters to describe the

energy (e.g. harmonic oscillator for chemical

bond)

Why Density Functional

Theory?• Wavefunction based methods in QC to get correlation

energy are quite time-consuming– MP2 scales N5, CCSD scales N6

• DFT techniques provide a means for recoveringcorrelation energy at a fraction of the computationalcost– Can be made to scale linearly with the size of the molecular

system.

– Can be applied to 100s to 1000s of atoms

– Can be formulated similarly to the standard SCF procedure

• By far, DFT is the most popular QC method usedtoday.

DFT: The Big Picture• DFT improves upon Hartree-Fock by including an

approximate treatment of the correlated motions ofelectrons (these are treated in Hartree-Fock in only anaveraged sense)

• Treatment of electron correlation is much cheaper than incorrelated wavefunction methods like MP2, CCSD, CCSD(T)

• Although there is a loose ordering of density functionalsfrom “less sophisticated” to “more sophisticated” (e.g., LDAto GGA to hybrid, etc.), in practice there is no reliable wayto improve your computation by going to the “next better”functional. By contrast, this is possible with wavefunctionmethods: one almost always has CCSD(T)>CCSD>MP2>HF.

Hohenberg & Kohn Theorems

• First Hohenberg-Kohn theorem: The groundstate properties of a many-electron systemdepend only on the electronic density n(x,y,z)

• Second Hohenberg-Kohn theorem: Thecorrect ground state density for a system isthe one that minimizes the total energythrough the functional E[n(x,y,z)]

• A functional is just a function that depends ona function

Walter Kohn was awarded with the Nobel Prize in Chemistry in 1998 for his develop-ment of the density functional theory.

Walter Kohn receiving his Nobel Prize from HisMajesty the King at the Stockholm Concert

Hall.

The Nobel Prize medal.

There is a Catch ....

• No one knows the exact form of the exact

kinetic energy or the exchange-correlation

functional

• No real systematic

means for finding

“truth”.

• Non-convergent QC “Truth”

B3LYP

LSDA

HTCH

BP86

BPW91

BLYP

Electron density• Basic premise of DFT: All the intricate motions and pair correlations in a

many-electron system are somehow contained in the total electrondensity alone

¦

N

ii rr

1

2)()( \U

• By locating cusps in density and measuring the slope at those cusps, youcan find the nuclei; therefore, construct the one-electron and nuclearrepulsion terms of the Hamiltonian and that means the density containsthe same information about the molecule as the Hamiltonian itself

Functional

• Function of a function

Density is a function of r

• Energy is a function of density

- Functional

8 □ OFTI PBEI B3LYP

E[p(r)]

FIG. I. Numbers of papers when DFT is searched as a topic in Web of Knowledge (grey), B3LYP citations (blue), and PBE citations (green, on top of blue).

What is DFT?

• In DFT, the energy of a molecular system isderived as a functional of the electron density

• If one knows the exact form of each of theseterms, the energy and properties of a systemby just plugging in the density– No basis sets, no determinants, no complicated

computational procedures…

Form of the Density Functional• So what’s the density functional actually look

like?• The Coulomb interaction for a given density

interacting the nuclei is very straightforward tocompute; so is the Coulomb interaction of thedensity with itself (J term)

• Coulomb (J) terms are great, but we also need toaccount for electron antisymmetry (exchangeeffects) and electron correlation effects

• Additionally, not clear how to compute kineticenergy as a function of the density

Form of the Density Functional

Kohn & Sham (KS)

• Compute the kinetic energy of a density byassuming that the density corresponds to awavefunction consisting of a single Slaterdeterminant (“non-interacting limit”): we knowhow to compute the kinetic energy of a Slaterdeterminant (orbitals) --- looks same as Hartree-Fock Theory

• This procedure is called Kohn-Sham DFT and isthe most common approach (although doesn’twork well for extremely large systems due tocomputational cost)

Kohn-Sham DFT

• Replace the exact kinetic energy with that of the kinetic energy

of a non-interacting many electron system.

• Many-electron interacting system -> Non-interacting reference system

• Mapped through an effective potential - adiabatic connection

• The wavefunction for the reference system is a Slater determinant!

– Same functional form as HF theory, can use the same iterative SCF

procedure to attack the problem.

Kohn-Sham Kinetic Energy

p(T)

N 1 - h \ c/>i - 2 v2 ¢i) ,

- I: ¢i ( f) 2 .i==l

EKS-DFT [p] = Ts [p] + EeN [p] + J[p] + Exe [p],

Exc[P] = (T[p] - Ts[P]) + (Eee[P] - J[p]).

Exchange-Correlation Functional

EKS-DFT[P]

Exc[P]

Ts [p] + EeN [p] + J[p] + Exe [p],

( T [p] - Ts [p] ) + ( E ee [p] - J [p]) •

• We can compute every piece of a Kohn-Sham DFTenergy exactly except for the "exchange­correlation" piece, Exc [p].• Unfortunately the exact exchange-correlation energy functional is not known and is probably so complicated that even if it were known it would not be computationally useful

• Hence, use various approximate exchange­correlation functionals (S-VWN, B3LYP, etc.)

Interpreting KS-DFT

• As a consequence of obtaining a solution through SCF, we get

KS orbitals

– Not the same as HF orbitals

• Qualitatively, the shape of the orbitals are useful for analysis

• Application of this wavefunction for computing properties

provides good results.

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Spin-polarized KS-DFT

• One of the nice features of DFT is the fact that there

is a natural partitioning of alpha and beta spin

electrons

– The electron density already has this feature

• So… to handle open shell molecules, the unrestricted

formalism provides the proper description.

– No spin contamination

• There is no valid restricted open shell formalism.

Observations on KS DFT

• Cost is similar to HF (similar equations) butquality can be better because correlation isbuilt in through the correlation functional

• Cost can actually be cheaper than HF if wereplace the expensive, long-range exchangeintegrals (K terms) from HF with a shorter-range exchange potential (which howevermight not be as accurate…)

Anatomy of a Functional

• Since there is no known exact forms for the

exchange correlation functional, approximate forms

have been determined

• Most functionals have a separate components for the

exchange and correlation energy

• No systematic means for finding the exact functional

• Many have parameterized factors incorporated.

Hierarchy of Exchange-Correlation Functionals

• Local density approximation (LDA): Functional depends only on the (local) density at a given point. Example: S-VWN

• Gradient-corrected approximation (GGA): Functional depends on local density and its gradient. Examples: PW91 and LYP correlation functionals, B88 exchange functional

• Meta-GGA: Functional depends on density, its gradient, and its second derivative. Example: M06-L

• Hybrid DFT: Mixes in Hartree-Fock exchange. Most popular example: B3LYP (hybrid GGA); Also M05-2X and M06-2X (hybrid meta-GGA’s)

• Double-Hybrid DFT: Hybrid DFT that also mixes in some MP2 correlation. Example: B2-PLYP

NI. ��, 1------t Gk,,uie

� 14t,...

"C 1'0ll1U

...___"A

i------i L.U

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u

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OFT-Jacob's Ladder

Exact Solution

Hybrid GGA (HGGA) and HMGGA

Meta GGA (MGGA)

Generalized Gradient Approximation (GGA) _._ ___

Local Density Approximation (LDA)

•

J.P. Perdew, K. Schmidt AIP Conf Proc. 577, 1 (2001}

)> n

n

C

� Q)

n

Hartree

Hartree world

• "null-null"

• "HF-null"=100% exact exchange (ala HF) and no correlation

"There's really no need for confusion. Part 95 of section 33 of article Q

in the formula uite clearl states ... "

LDA

Local Density Approximation

• Local functional (depends only on p)

EtDA= -C x f p3 (r)dr

3 3 3

Cx =

-4 1r

4

2

o��--�-�

0 0.5

r

1.5 2

f'igurc 10.1: R.udial density of the :-lo atom, both exactly and from an LL)A calcull\tion

• Correlation can not be done analytically ... so it takes on various crazy forms ...

• LSDA-For open shell systems

Local Density Approximation

• For the correlation energy, again use the uniformelectron gas as a model. Its correlation energywas determined numerically by Monte Carlosimulations and fit to an analytic form by Vosko,Wilk, and Nusair (VWN), to give εc

VWN. L(S)DAusually implies VWN correlation

• More technical name for L(S)DA is S-VWN (Slater exchange plus Vosko, Wilk, Nusair correlation)

• Electron correlation can be overestimated by a factor of 2 when using VWN. Bond strengths too large. Need better approximations

GGA

Generalized Gradient Approximation

• Eis a function of density and its gradient

EJp] = J e�A (P)fx(s)

• Semi-local functional (depends on p and derivatives)

• "s" is the dimensionless or reduced density gradient• f)s) is the exchange enhancement factor• UEG limit requires f

x(O)=l

All GGA and meta-GGA are corrections to LOA (all revert to the UEG at zero density gradient)

Generalized Gradient Approximations (GGA’s)

• The uniform electron gas isn’t such a great modelbecause in molecules, the electron density can varyrapidly over a small region of space

• One way to improve over LSDA is to make thefunctional depend on both the density and thegradient of the density

• This leads to “gradient corrected” functionals or“generalized gradient approximation” (GGA)functionals

• Also sometimes called “nonlocal” functionals but this isa bad name that is fortunately falling out of favor now:the density and its gradient still supply only localinformation

Examples of GGA's

• Exchange: PW86, B88 ("B"), PW91

• Correlation: LYP

x2

EB88 [p] == ELDA [p] _ (3 pl /3 . ' x x 1 + 6/3xsinh- 1x

IVPI X == 4/3p

• GGA's improve over LDA but are still not necessarily very accurate; one reason is the exchange potential doesn't necessarily have the correct qualitative behavior

• B88 has the correct -1/r asymptotic behavior of the energy density but not the overall exchange potential

LDA versus GGA

• Self-interaction error

• Static correlation error

30

0

-30

-60

200

100

0

-100

A

C

H; binding curve

3 5 7

R (Angstrom)

H2

binding curve

9

HF­B3LYP­

LOA-

3 5 7

R (Angstrom)

9

B H atom with fractional charge

0.0 0.25 0.5 0.75 1.0

Fractional charge (e-)

D H atom with fractional spins

[1,0J [3/4, 1/4] [1/2, 1/2) [1/4,3/4] [O, 1 J

Fractional spins [f3,a)

Fig. 1. DFT approximations fail The dissociation of H2 + molecule (A) and H2 molecule (C) are shown for calculations with approximate functionals: Hartree-Fock (HF), local density approximation (LDA), and B3LYP. Although OFT gives good bonding structures, errors increase with the bond length. The huge errors at dissociation of H2 + exactly match the error of atoms with fractional charges (B), and for H2 they exactly match the error of atoms with fractional spins (D). The understanding of these failures leads to the characterization of the delocalization error and static correlation error that are pervasive throughout chemistry and physics, explaining a host of problems with currently used exchange-correlation functionals.

Meta-GGA

• There exists a self-correlation error unless kinetic energy density is included

i

• Functional -r(p) not known➔functional derivative of -r-dependent functionals is

problematic

Differentiate with respect to orbital expansion coefficients instead of

functional differentiation with respect to the density

• Example: TPSS• Satisfies U EG and other theoretical constraints• Correct exchange energy of the H atom• MAEs of G2/97 atomization energies of ""S kcal/mol

Hybrid GGA

• LDA and GGA overbind, HF underbinds ...

• Most famous HGGA=B3LYP

• a=0.20, b=0. 72, c=0.81

• Utlizes LYP correlation functional

• Semi-empirical (fit to atomization energies ... )

• PBEO

• Non-empirical

What is the right amount of exact (HF) exchange ?

Adiabatic Connection Formula

fl A

Exe =

Jo (WA Vxc(A) WA)d,,\

• If Vxc

is linear in A, can approximate as

1 A 1 A

Exe� -(\J!o Vxc(O) \J!o) + -(\Jf 1 Vxc(l) \Jf 1) 2 2

A=O: only exchange, no correlation; '¥ 0

is a Slater determinant, can use HF equation for exchange energy

A=l: Real system, unknown solution. Can approximate with LSDA and add to A=O part to give "half and half"

EH+H == �EHF + � (ELSDA + ELSDA

)XC 2 X 2 X C

Range-Separated Hybrids

• Long-range corrected methods• Divide inter-electronic interaction 1/r12 into a short-range and a

long-range part• Short range = modified DFT functional• Long range = Hartree-Fock (exact exchange)• Range parameter ω varies from system to system (0.4 is typical)

12

12

12

12

12

)()(11rrerf

rrerf

rZZ

��

EXC = aEXSR-DFT +bEX

LR-HF +cDECDFT

Dispersion

• Local functionals cannot give an asymptotic dispersion interaction of Londonform

• Asymptotic interaction energy of local functionals falls off exponentiallyinstead

• Empirical dispersion correction

66

RC

�

66)(ij

ij

ijijdampdisp RC

RfE ¦!

�

• Interatomic C6ij coefficients obtained by fitting to accurate referencemolecular C6 coefficients

• Damping function fdamp required in order to tame the R-6 divergences atsmall internuclear separations

• Example: B97-D (B97=Becke’s 1997 flexible 10 parameters HGGA)

Double Hybrid GGA

• Include exchange energy from MP2 to improve dispersion and improved

recovery of exact exchange

- HGGA underestimate dispersion ... MP2 overestimates dispersion ...

- Includes virtual orbitals so improves description of dispersion

• Formal computer scaling is higher than HF and KS-OFT

• Examples: B2GP-PLYP, mPW2PLYP

E =aEDFT +(l-a)EHF +cMDFT +(1-c)Mn

XC X X C C

Obituary: Density Functional Theory (1927-1993)

Peter M W. Gill

School of Chemistry, University of Nottingham, Nottingham NG7 2RD, United Kingdom. Manuscript received: 7 March 2002 (e-mail: Peter.Gill@nottingham.ac.uk).

Final version: 14 March 2002.

Aust. J. Chem. 2001, 54, 661-662

What Do You Need to Specifyto Run a DFT Computation

• Molecule• Molecular charge• Spin multiplicity• Basis set• Exchange functional: S, B, B3, etc.• Correlation functional: LYP, PW91, etc.

Hybrid functionals and

efficiency• Computing exact exchange is

very inefficient

– Has very non-local character

• When only the Coulomb energy

needs to computed, advanced

techniques like Coulomb fitting, J-

Engine, and CFMM can be used.

• This is why mGGA’s are

important.

Numerical Grids in DFT

• The performance and accuracy of DFT

methods are intimately tied to the numerical

integration grid.

– Number of points, cutoffs, type of grid

• Pruned Grids

– Vary the angular grid at different radial positions

– Greatly reduces the number of points needed with

little loss in accuracy

– Gaussian’s (75,302) pruned, SG-1

Accuracy of Various Functionals

Reproduced from Boese, Martin and Handy, J. Chem. Phys., 119, 3005 (2003)

SCC-DFTB

• Modification of the original non-iterative DFTB approach to a

second-order expansion of the charge density relative to a

reference density.

• Slater-orbital basis used to expand the wavefunction

• Two-centered integrals are parameterized and

pseudopotentials are used for core contributions.

• Mulliken charges are used to estimate the charge fluctuations

ace N

E[B = L(wilHol'1Ti) + 1/2 L 'Yaf3�qa�q/3 + Erepi a.B

L Cvi(Hµv - ciSµv) = 0 Yµ, iN

Hµv == {¢µ 1Hl¢v) + l/2Sµv L(,a( + 1{3()1:l.q(, YµEa, VE(3 '

M. Elstner, D. Porezag, G. Jungnickel, J. Elsner, M. Haugk, T. Frauenheim, S. Suhai, and G. Seifert, Phys. Rev. B 58, 7260 (1998).

Mean absolute errors ( N comparisons) for a standard validation set of mostly

organic compounds (C, H, N, O).

120

0.27

0.45

1.8

2.0

2.9

OM3

90155189151172241112ν (cm-1)

0.370.280.250.270.260.3553µ (D)

3.820.260.320.420.350.4652IP (eV)

1.32.21.82.11.92.6 101θ (degree)

1.51.61.21.11.71.4 242R (pm)

7.73.13.54.25.56.3 140ΔHf

(kcal/mol)

N AM1MNDO DFTB OM2 OM1 PM3Propertya

a) Heats of formation ΔHf, bond lengths R, bond angles θ, vertical ionization

potentials IP, dipole moments µ, vibrational wavenumbers ν.

Reproduced from ACS presentation of Walter Thiel (Sept 2006)

Choice of Methodology

• Ab initio

No experimental parameters

Derived from first principles and physical constants (e.g. speed of light, charge of electron, Planck's constant)

Based on electrons and their coordinates

• Density Functional Theory

Hohenberg and Kohn ➔ground state electronic energy is determined by the electron density

The relationship between the electron density and energy is unique ➔ Relationship UNKNOWN!

Ha rtree-Fock MP2 CCSD CCSD(T) DFT

Observations DFT• DFT good for geometries, often not as good for energies• B3LYP works really well and is (usually) hard to beat• Minnesota functionals (M05-2X, M06-2X, M06-L, etc) seem to work

well also (but can be sensitive to the numerical integration grid)• Barrier heights often underestimated• Totally fails for non-covalent interactions (just like Hartree-Fock); fix

with DFT-D or XDM or vdW-DFT• Can have large errors for excitation energies to Rydberg excited

states (fix with asymptotically-corrected functionals like CAM-B3LYP) (DFT for excited states is called time-dependent DFT, or TDDFT)

• Can totally fail for charge-transfer states (fix with “range-separated hybrids” that include HF exchange at long range only)

• Can get wrong energetic ordering of spin states of metal systems (just like Hartree-Fock)