Density Functional Theory (DFT)
Spring 2016
CHEM 430
1
Method selection
100 fs, 10
atoms:
photochemistry
Correlated Method
(MP2, CCSD)
Molecular detail
Atomistic MM
10 ms, thousands
of atoms:
protein folding,
drug binding
Density
Functional
Theory
10 ps, 100
atoms: chemical
reactions
Coarse-grain
1 ms+, 1 million
atoms: dynamics of
large proteins, cell
membranes, viruses
Computational cost
2
Approximate solutions to Schrödinger equation
Schrödinger’s equation is nearly impossible to solve, so approximate
methods are used.
2
æ
Ñ
ç -å i - å Z I + å 1
ç
è i 2 i,I R I - r̂i i¹ j r̂i - r̂j

HF

1 1
 2 1

ö
÷ Y ( ri ) = EY ( ri )
÷
ø
1 2  1 N 
 2 2   2 N 

 N 1  N 2   N N 
Slater determinant: Use wavefunction of
noninteracting electrons for interacting system
 0 r   0 r1 , r2 , r3 ,, rN   E
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 (DFT)
• Approach 3: Approximate the energy
using empirical functions (MM)
E ( R I ) ® kIJ ( RIJ - RIJ0 )
2
Molecular mechanics: Use empirical
functions and parameters to describe the
energy (e.g. harmonic oscillator for chemical 3
bond)
Scaling
Method and basis set
Number of basis functions
CBS
“Exact”
solution to
HΨ=E Ψ
6Z
5Z
QZ
TZ
DZ
Hartree-Fock
MP2
DFT
CCSD
CCSD(T)
Full CI
Increasing Cost
• Scaling
• Memory, computer processing time
4
Electron density
• Basic premise of DFT: All the intricate motions and pair correlations in a
many-electron system are somehow contained in the total electron
N
density alone
 (r )    i (r )
2
i 1
• By locating cusps in density and measuring the slope at those cusps, you
can find the nuclei; therefore, construct the one-electron and nuclear
repulsion terms of the Hamiltonian and that means the density contains
the same information about the molecule as the Hamiltonian itself
5
Functional
• Function of a function
– Density is a function of r
• Energy is a function of density
– Functional
E[  (r )]
6
Thomas-Fermi
• Kinetic and exchange energies of a system of many electrons locally
modeled by uniform electron gas energy densities
• Assume electrons distributed uniformly in small volume element and
electron density can vary from one volume element to the next
• Failscannot self-consistently
reproduce atomic shell structure,
cannot bind molecules
• T-Fexpression for kinetic energy (and potential energy of nuclearelectron and electron-electron interaction)
• Diracexpression for exchange energy
– TFD theory
7
Uniform electron gas
• Fictitious system of N noninteracting fermions (think as having same spin
and mass as electrons but zero charge) but subject them to a very special
external potential V such that the density (ρ) is the same as for the real
system ρ0
• Large number of N electrons in a cube of volume V where there is a
uniform spread of positive charge (makes the system neutral)
– UEG defined as the limit N∞, V∞, with the density ρ=N/V
remaining finite
• Can obtain KE ala Thomas Fermi and exchange ala Dirac
8
THE theorems of DFT
1) The external potential v(r) is determined, to within an additive constant,
by the electron density ρ(r)
2) For a trial density ρ1(r), such that ρ1(r)≥0 and integral of ρ1(r)dr=N,
E0≤E[ρ1], where E[ρ1] is the energy functional
• Hohenberg and Kohn, Phys. Rev. 136, B864 (1964)
9
Theorem 1
• The external potential v(r) is determined, to within an additive constant,
by the electron density ρ(r)
• i.e., The external potential, and hence the total energy, are unique
functionals of the electron density ρ(r)
• Proof by contradiction…
10
Theorem 2
• For a trial density ρ1(r), such that ρ1(r)≥0 and integral of ρ1(r)dr=N,
E0≤E[ρ1], where E[ρ1] is the energy functional
• Variational principle introduced into DFT
• If N interacting electrons move in an external potential Vext(r), the ground
state electron density minimizes the functional
• Proof…
Define F such that…
Then…
From variational principle…
But if we minimize F we get…
Add in
and we get…
E[ r0 ] £ E0
Combine with what we had above and… E[ r0 (r)] = E0
Since Ψ0 yields the ground state
density ρ0
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Now what?
• From Hohenberg-Kohn, there exists a functional of the density which
would give the exact ground state energy for the exact density
• So the Holy Grail exists…
– But how do we find it
r ® uext ® Y 0 ® everything!
12
Kohn-Sham equations
• Imagine a fictitious system of N noninteracting fermions but subject them
to a special external potential V such that ρ is the same as for the real
system
2


2

 i
• Wavefunction for fictitious particles
i
1 2
(   VS ) i   i i
2
E[  (r )]  TS [  (r )]   VS (r )  (r )dr
• For real system
E[  (r )]  T [  (r )]   Vdr  Vee [  ]
• Substitute TS and regroup
E[  (r )]  TS [  ]   Vdr  J [  ]  Vee [  ]  (T [  ]  TS [  ])  J [  ]
13
Kohn-Sham equation cont’d
E[  (r )]  TS [  ]   Vdr  J [  ]  Vee [  ]  (T [  ]  TS [  ])  J [  ]
• Gather final three terms…
TS  
J ( ) 
1
2  i* 2 i

2 i
1  (1)  (2)
d1d 2
2   r12
14
KS DFT SCF
• Kohn-Sham effective potential
ueff µ u xc
• Exchange-correlation potential
d Exc [ r ]
u xc (r) =
dr (r)
• Since υeff is dependent on ρ, the KS equations must be solved iteratively
• If you set υxc to HF exchange potential, then you have the HF equations
• KS orbitals represent density while HF orbitals represent an approximation
of the wavefunction
• HF variational, DFT only variational if exact energy functional is used
16
DFT
• Unique relationship between electron density and energy
• Need to find the best functional for exchange and correlation energy
E  ET  EV  E XC
E XC  E X  EC
• Analytic form of Exc unknown, so different DFT functionals correspond to
different educated guesses of Exc
• Parameterized based on: laws of physics, test sets
• Variety of test sets: Gn sets, TAE109/03, NBTH, kinetics…
• Different functionals for different properties, molecules, etc.
Full CI
Hartree-Fock
MP2
CCSD
CCSD(T)
DFT
18
Side note: adiabatic connection
F l [n] = min Y®n Y T̂ + lV̂ee Y
• λ=1 Physical system
(Interacting system with kinetic
energy T and Coulomb
interaction energy Vee)
• λ=0 Kohn-Sham system
(Noninteracting KS reference
system with kinetic energy T0
and no Coulomb interaction)
• λ=1 is just Ex+Uc
• λ=0 is just Ex
• Area between curve and x-axis is Exc
19
DFT-Jacob’s Ladder
Exact Solution
Hybrid GGA (HGGA) and HMGGA
Meta GGA (MGGA)
Accuracy
Simplicity
Double Hybrid (DH)
Generalized Gradient Approximation (GGA)
Local Density Approximation (LDA)
Hartree
J.P. Perdew, K. Schmidt AIP Conf. Proc. 577, 1 (2001).
20
Hartree world
0
• “null-null”
• “HF-null”=100% exact exchange (ala HF) and no correlation
21
LDA
1
Local Density Approximation
• Local functional (depends only on ρ)
4
3
E XLDA  C X   (r )dr
3 3
CX   
4  
1
3
• Correlation can not be done analytically…so it takes on various crazy forms…
• LSDA—For open shell systems
E
LSDA
X
1
3
4
4

 ,    2 CX    3   3 (r ) dr


22
HF versus LDA
• HF underbinds and LDA overbinds
•
(atomization energies)
Atomization energy=energy required to dissociate a molecule entirely to free atoms, i.e., break all bonds
23
GGA
2
Generalized Gradient Approximation
• E is a function of density and its gradient
• Semi-local functional (depends on ρ and derivatives)
• “s” is the dimensionless or reduced density gradient
• fx(s) is the exchange enhancement factor
• UEG limit requires fx(0)=1
– All GGA and meta-GGA are corrections to LDA (all revert to the UEG at
zero density gradient)
24
GGA
• Examples: B86
Β = 0.0036 and γ=0.004 (empirical parameters)
c1=2(6π2)1/3 and c2 is proportional to Cx
• Examples: PBE
κ = 0.804 (guarantees Lieb-Oxford)
γ=0.21951
Exc £ 2.28 ExLDA
2
b
c
(c
s)
FxB86 (s) =1+ 2 1 2
1+ g (c1s)
2
g
s
FxPBE (s) =1+ k
1+ g s 2
25
LDA versus GGA
• Self-interaction error
• Static correlation error
26
Meta-GGA
3
• There exists a self-correlation error unless kinetic energy density is included
t s = å Ñyis
2
i
• Functional τ(ρ) not knownfunctional derivative of τ-dependent functionals is
problematic
– Differentiate with respect to orbital expansion coefficients instead of
functional differentiation with respect to the density
• Example: TPSS
• Satisfies UEG and other theoretical constraints
• Correct exchange energy of the H atom
• MAEs of G2/97 atomization energies of ~5 kcal/mol
27
Hybrid GGA
4
B3LYP
LSDA
EXC
= EXC
+ 0.20(EXexact - EXLSDA )+ 0.72DEXB88 + 0.81DECLYP
• 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…)
• PBE0
• Non-empirical
PBE 0
EXC
= 0.25EXexact + 0.75DEXPBE + ECPBE
What is the right amount of exact (HF) exchange???
28
29
GGA and HGGA failures
• Overstabilization of molecular radicals, poor treatment of charge transfer
processes, inability to account for dispersion interactions (nonlocal effects,
extending beyond intra-atomic to inter-atomic distances)
– Cannot be treated by local functionals depending on ρ , Ñρ, Ñ 2ρ and/or τ (local
use information only at each integration point r1)
– Nonlocal effects must be sensed by explicitly sampling at each integration point all
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other points r2 not equal r1
Range-separated
• Long-range corrected methods
• Divide inter-electronic interaction 1/r12 into a short-range and a longrange part
• Short range=modified functional
• Long range=HF
• Range parameter ω varies from system to system
1 1  erf (r12 ) erf (r12 )


r12
r12
r12
EXC = aEXSR-DFT + bEXLR-HF + cDECDFT
31
Dispersion
• Local functionals cannot give an asymptotic dispersion interaction of London
form  CR
• Asymptotic interaction energy of local functionals falls off exponentially
instead
• Empirical dispersion correction
6
6
Edisp   f damp ( Rij )
j i
C6ij
Rij
6
• Interatomic C6ij coefficients obtained by fitting to accurate reference
molecular C6 coefficients
• Damping function fdamp required in order to tame the R-6 divergences at
small internuclear separations
• Example: B97-D (B97=Becke’s 1997 flexible 10 parameters HGGA)
32
5
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-DFT
• Examples: B2GP-PLYP, mPW2PLYP
EXC = aEXDFT + (1- a)EXHF + cDECDFT + (1- c)DECPT
33
DFT Families
DFT Family
Sample Functionals
Gradient of
the Electron
Density
GGA/GGE
BLYP, BP86, BPW91, HCTH,
mPWLYP, mPWPW91,
PBEPBE, TPSSKCIS, M06-L





MGGA
BB95, PBEKCIS, VSXC





HGGA
B3LYP, B3P86, B971, B97-2,
B98, X3LYP, PBE1PBE,
mPW1LYP, B1LYP





HMGGA
PBE1KCIS, TPSS1KCIS, M06,
M06-2X, mPW1B95, BMK,
MPW1KCIS





RSGGA
LC-ωPBE, ωB97





RSHGGA
ωB97X, wB97XD, CAM-B3LYP





DHGGA
B2-PLYP, B2GPPLYP,
mPW2PLYP





GGA – Generalized Gradient Approximation
GGE – Generalized Gradient Exchange
HGGA – Hybrid GGA
MGGA – Meta GGA
HartreeFock
Exchange
Kinetic
Energy
Density
Range
Separation
of Exchange
PT2
Correlation
HMGGA – Hybrid Meta GGA
RSGGA – Range Separated GGA
RSHGGA-Range Separated Hybrid GGA
DHGGA – Double Hybrid GGA
34
Numerical grids
• Numerical integration of the functionalsaccuracy of DFT calcs depends
on the number of points used in the numerical integration
Gaussian09 (If comparing energies, must use same grid for all)
– All grids are “pruned”
– Optimized to use the minimal number of points to give desired
accuracy
– (x,y)…x is radial shells and y is angular points per a shell
– Total number of integration points is x times y
• “fine” is default
– Enhances calc accuracy at minimal additional cost
– (75,302)…~7000 points per an atom
– “ultrafine”
(99,590)
– “superfine” (150,974) first row atoms, (225,974) beyond
– “coarse”
(35,110)
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Other DFTs
• Plane wave methods
(solid state calcs)
• TD-DFT
(excited states and excitations)
• Carr-Parinello Dynamics
Software
• Gaussian, Molpro, NWChem, Qchem, GAMESS, Turbomol, Spartan, Psi,
ACES,…
36
Accuracy and reliability
• Ab initio
– Within ± 1 kcal mol-1 of experimental value (main group)
– Within ± 3 kcal mol-1 of experimental value (transition metal)
– Gold standard: CCSD(T)/aug-cc-pwCV5Z-DK (or similar large basis set)
• DFT
• Cross your fingers
• Being aware of the functional development
37
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!
Full CI
Hartree-Fock
MP2
CCSD
CCSD(T)
DFT
38