<date/time>
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Limitations of DFTStefano Borini
Quick recap of DFT - 0
Wavefunction: a description of our n-particle system that can be probed with an operator to find physical information about the system
̂
H
Ψ
(
x
1,
x
2,
...
,
x
n
)
=
E
Ψ
(
x
1,
x
2,
...
,
x
n
)
The wavefunction is a function of all electrons coordinates
The electronic density is a spatial (3D) information which- depends on the wavefunction- tells us a lot about the ground state
Quick recap of DFT - 1
E
[
ρ
]
=
F
[
ρ
]
+
∫
ρ
(
r
)
V
ext
(
r
)
dr
Hohenberg-Kohn
Universal functional(not dependent from system)
Non-universal functionalDepends on the systemthrough external potential
There is a 3D density ρ0(r) which gives the lowest energy E = E0
How do we find it ?
E
[
ρ
]
=
F
[
ρ
]
+
∫
ρ
(
r
)
V
ext
(
r
)
dr
E
[
ρ
]
=
T
s
[
ρ
]
+
1
2
∬
ρ
(
r
)
ρ
(
r
'
)
r
−
r
'
dr
dr
'
+
E
XC
[
ρ
]
+
∫
ρ
(
r
)
V
ext
(
r
)
dr
kinetic energy of non-interacting electron gas with density
T
s
[
ρ
(
r
)
]
=
−
1
2
∑
1
N
〈
ψ
i
∣
∇
2
∣
ψ
i
〉
AKA J[p] Coulomb interaction of many-charges cloud with itself
“Under the carpet”
density/potential interaction
ρ
(
r
)
=
∑
i
=
1
N
∣
ψ
i
∣
2
That is: we don't know thedensity, but we express it ona basis of N non-interacting orbitals
Quick recap of DFT - 2
Let's ignore the problem with Exc for a moment.
That's easy: derivative with respect to ρ
∫
ρ
(
r
)
dr
=
N
Finding the optimal density ρ0 that minimizes the energy ?
with constraint
δ
E
[
ρ
]
δ
ρ
Lagrange multipliers
δ
δ
ρ
[
E
[
ρ
]
−
μ
∫
ρ
dr
]
=
0
δ
E
[
ρ
]
δ
ρ
[
−
μ
∫
ρ
dr
]
=
0
Quick recap of DFT - 3
Let's do the derivative
E
[
ρ
]
=
T
s
[
ρ
]
+
1
2
∬
ρ
(
r
)
ρ
(
r
'
)
r
−
r
'
dr
dr
'
+
E
XC
[
ρ
]
+
∫
ρ
(
r
)
V
ext
(
r
)
dr
Quick recap of DFT - 4
δ
E
[
ρ
]
δ
ρ
=
μ
δ
E
[
ρ
]
δ
ρ
=
δ
T
s
[
ρ
]
δ
ρ
+
∫
ρ
(
r
'
)
r
−
r
'
dr
'
+
δ
E
XC
[
ρ
]
δ
ρ
+
V
ext
(
r
)
=
μ
δ
T
s
[
ρ
]
δ
ρ
+
V
eff
(
r
)
=
μ
V
XC
(
r
)
Behold ! non-interacting electrons in a black magic potential !
So, what does that mean?
An effective, mean field potential exists
V
eff
=
V
ext
+
V
xc
so that when applied to a system ofnon-interacting electrons
it gives the exact same density and energy as the real system with interacting particles.
ρ
(
r
)
=
∑
i
=
1
N
∣
ψ
i
(
r
)
∣
2
We just moved our ignorance around,but in a more manageable entity
Quick recap of DFT - 6
We can define a one-electron Schroedinger equation using this potential
(
−
1
2
∇
i
2
+
V
eff
(
r
)
−
ϵ
i
)
ψ
i
(
r
)
=
0
Which gives us the orbitals. From the orbitals we get the density
ρ
(
r
)
=
∑
i
=
1
N
∣
ψ
i
(
r
)
∣
2
V
eff
[
ρ
]
=
∫
ρ
(
r
'
)
r
−
r
'
dr
'
+
V
XC
[
ρ
]
+
V
ext
(
r
)
From the density the new potential
From the new potentiala new Schroedinger equation
Quick recap of DFT - 5
Final energy
E
[
ρ
]
=
T
s
[
ρ
]
+
1
2
∬
ρ
(
r
)
ρ
(
r
'
)
r
−
r
'
dr
dr
'
+
E
XC
[
ρ
]
+
∫
ρ
(
r
)
V
ext
(
r
)
dr
∑
i
=
1
N
ϵ
i
=
∑
〈
ψ
i
∣
−
1
2
∇
i
2
+
V
eff
(
r
)
∣
ψ
i
〉
=
T
s
[
ρ
]
+
∫
ρ
(
r
)
V
eff
(
r
)
dr
E
[
ρ
]
=
∑
i
=
1
N
ϵ
i
−
∫
ρ
(
r
)
V
eff
(
r
)
dr
+
1
2
∬
ρ
(
r
)
ρ
(
r
'
)
r
−
r
'
dr
dr
'
+
E
XC
[
ρ
]
+
∫
ρ
(
r
)
V
ext
(
r
)
dr
E
[
ρ
]
=
∑
i
=
1
N
ϵ
i
+
1
2
∬
ρ
(
r
)
ρ
(
r
'
)
r
−
r
'
dr
dr
'
+
E
XC
[
ρ
]
−
∫
ρ
(
r
)
V
xc
(
r
)
dr
DFT is an exact and efficient approachbut it sweeps complexity under the rug (XC)
DFT is successful if the XC is successful
Said in another way...
System made of interacting electrons in a regular potential
System made of non-interacting electrons in a black magic potential(which includes the XC term)
That isHow well the XC functional reproduces all the things we ignore in a simplified system of non-interacting electrons
Limitation n. 1
Functionals
Functionals
E
[
ρ
]
=
T
s
[
ρ
(
r
)
]
+
1
2
∬
ρ
(
r
)
ρ
(
r
'
)
r
−
r
'
dr
dr
'
+
E
XC
+
∫
ρ
(
r
)
V
ext
(
r
)
dr
Exchange arises from Pauli exclusion principle (requirement for antisymmetry of the WF)
Correlation includes complex many-body interactions
In ab-initio approachesexchange is addressed through antisymmetric functions (Slater determinants)correlation (static+dynamic) through multi-determinant wavefunctions
In DFT, simple XC functionals perform amazingly well in describing those effects.Errors may occur not from an inexact theory, but a limited XC functional.
E
XC
=
E
X
+
E
C
Digression: wavefunction QM
In “traditional” wavefunction QM we proceed like this
1. we define one-electron orbitals out of a linear combination of an atomic basis set
ψ
i
(
r
)
=
∑
j
c
ij
χ
j
2. we use these orbitals to build a Slater determinant: polyelectronic, antisymmetric wavefunction (includes exchange energy due to antisymmetry)
Φ
HF
=
∥
ψ
1
ψ
1
∥
optimizing the one-electron orbitals.
3. We define a linear combination of electronic distributions in the orbitals, building other determinants, and either optimizing the coefficients, or both the coefficient and the orbitals
Ψ
MR
=
∑
i
C
i
Φ
i
We can improve the wavefunction to perfection: infinite atomic basis set withall possible electronic distributions represented in the MR wavefunction.
Computationally impossible, but conceptually simple
Correlation
Hartree Fock includes exchange, but completely misses correlation
Correlation energy can be divided into
Static: emerging from close degeneracies of configurations
Dynamic: emerging from instantaneous electronic interactions
We can describe both terms with a multireference expansion. However, while static correlation requires few terms, dynamic requiresa larger amount of them, and it is one of the holy grail of accurate calculations
Classic example: Hydrogen molecule
Correlation: how an electron being in x1 affects the probability of another electron in x2
Exchange is also a form of correlation, but here we talk about what is not accounted by exchange
Non-correlated electrons
P
(
x
1
;
x
2
)
=
P
(
x
1
)
P
(
x
2
)
Correlated electrons
P
(
x
1
;
x
2
)
=
N
(
N
−
1
)
∫
∣
Ψ
(
x
1
;
x
2
;
x
3,
...
,
x
n
)
∣
2
dx
3
...
dx
n
CorrelationEnergy
Correlation
Why the difference ?
Hartree-Fock determinant =
σ
g
=
N
g
(
1s
A
+
1s
B
)
σ
u
=
N
u
(
1s
A
−
1s
B
)
∥
1
σ
g
1
σ
g
∥
=
N
⋅
1
σ
g
(
1
)
α
(
1
)
1
σ
g
(
2
)
β
(
2
)
−
1
σ
g
(
1
)
β
(
1
)
1
σ
g
(
2
)
α
(
2
)
N
⋅
1
σ
g
(
1
)
1
σ
g
(
2
)
(
α
(
1
)
β
(
2
)
−
β
(
1
)
α
(
2
)
)
Unpacking the sigmas and getting rid of the spin
N
⋅
(
N
g
(
1s
A
(
1
)
+
1s
B
(
1
)
)
)
⋅
(
N
g
(
1s
A
(
2
)
+
1s
B
(
2
)
)
)
N
⋅
(
N
g
2
(
1s
A
(
1
)
1s
A
(
2
)
+
1s
B
(
1
)
1s
A
(
2
)
+
1s
A
(
1
)
1s
B
(
2
)
+
1s
B
(
1
)
1s
B
(
2
)
)
)
N
⋅
(
N
g
2
(
A
(
1
)
A
(
2
)
+
B
(
1
)
B
(
2
)
+
B
(
1
)
A
(
2
)
+
A
(
1
)
B
(
2
)
)
)
Ionic form
covalent form
N
⋅
(
N
g
2
(
A
(
1
)
A
(
2
)
+
B
(
1
)
B
(
2
)
+
B
(
1
)
A
(
2
)
+
A
(
1
)
B
(
2
)
)
)
Ionic form
covalent form
Covalent form and ionic form have the same weightThis weight cannot change with respect to distance.A dissociated H2 is covalent (two atoms, one electron each).Ionic forms should have zero weight at large distances.We miss this flexibility.At long distances, the interaction decreases, and so thesplitting between sigmag and sigmau. Smaller splitting means these two orbitals are getting more and more degenerate: their energy becomes the same.
If our wavefunction is made of two configurations
Ψ
=
c
1
∥
1
σ
g
1
σ
g
∥
+
c
2
∥
1
σ
u
1
σ
u
∥
Then we gain this flexibility in the coefficients c1 and c2
Static correlation
This was just static correlation. Arises from degeneracy or close degeneracy of orbitals.Dynamic correlation arises from description of instantaneous interaction vs. an average field. We need a large Configuration Interaction to address this.Final limit: Full CI. Extremely expensive.
However, the direction is clear: a larger basis set and a larger CI always systematically improve the energy towards the true energy of the system.
Dynamic correlation
What about DFT?How does it address correlation (static and dynamic)Do we have a direction for systematic improvement ?
In wavefunction based ab-initio methods, a systematic method to improve the quality of results exists.Not so in DFT
There is no systematic way of improving adefective XC functional
Limitation n. 2
Improving is hard
Improvement can be sought in two ways
Rule obedience: build it so that good rules are honored “a priori”Fitting: parametrize the XC, then “fit” to a set of experimental results. Fitting values are “magic numbers” in the resulting XC functional
However, the XC is technically universalFind the right one, and it works everywhere
Some rules
Lieb-Oxford bound
E
x
≥
E
xc
≥
−
1.68
e
2
∫
d
3
r
n
(
r
)
4
/
3
satisfied by LDA and some GGA
Size consistency
E
[
ρ
A
+
ρ
B
]
=
E
[
ρ
A
]
+
E
[
ρ
B
]
For two non-overlapping, non-interacting densities
Most functionals are size consistent
Exchange decay
The exchange potential should decay as 1/rThe correlation as 1/r4
In LDA they decay exponentially.
Typical Errors in DFT
Underestimatebarriers of chemical reactionsband gapsion dissociation energiescharge transfer
Overestimatebinding energies of charge transfer complexesresponse to electric fields
Fail to describedegenerate or near-degenerate statesbond breakingstrongly correlated material
Root cause: Delocalization error.Coulomb term dominates, pushing electrons apart
Root cause: Static correlation error.Similar error found in Hartree Fock
Static Correlation Error
Overestimates energy on dissociation
Root cause: near degeneracies at the wavefunction level
Example: stretching of H2
Note that the problem is similar to the HF level case.In a purely ab-initio approach this is a well-known problemof a single determinant wavefunction.
Static Correlation Error
Dissociation limit for H2: singlet state with separated atoms
Limitations of the functionalsand their origin
Local Density Approximation
So we need a value to put into Exc
We define a box of “jellium” with uniform electron density ρ
For each value kp of the density, there's an analytical expression that givesthe corresponding exchange-correlation value.
We can divide this value by N (total number of electrons) toget the exchange-correlation “per electron”.
ϵ
xc
[
ρ
]
=
E
xc
[
ρ
]
/
N
ρ
(
r
)
=
k
ρ
Jellium: uniform electronic charge and uniform positive charge,with volume of the box growing to infinity
Credits: Rubén Pérez Departamento de Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid, Spain
Local Density Approximation
Now, we can use these values in our DFT treatment.
E
xc
LDA
=
∫
ϵ
xc
(
ρ
(
r
)
)
ρ
(
r
)
dr
Procedure:take our real systemdivide it into small, infinitesimal cubesfor each cube, put some charge inside divide charge by the volume of the cube to get density in that cubefor the density in that cube, get the value εxc for a Jellium having that densityscale it properly with the actual amount of charge in the cubesum up all the contributes
That is
LDA Performance
Exchange (5-10%)Bond lengths (1%)Elastic constants (5%)Energy barriers (100%)
Underestimate
Overestimate
Correlation (100-200%)Atomization energies (15%)
Favors close-packed structuresPoor description of magnetic systemsPoor description of van der Waals and Hydrogen bondsNegative ions unstable
Other
Densities are usually good.
Overestimation of correlation offsets underestimation of exchange, leading toan overall error of 7% for EXC
So why does LDA work ?
Exact properties of the xc-hole maintained.The electron-electron interaction depends only on the spherical average of the xc-hole. This is reasonably well reproduced The errors in the exchange and correlation energy densities tend to cancel
Some things that do not work in this approach •Van der Waals interactions: due to mutual dynamical charge polarisation of the atoms not properly included in any existing approximations to Exc • Excited states: DFT is a ground state theory (ways forward: time-dependent DFT, GW, ...) • Non Born-Oppenheimer processes (i.e, non-radiative transitions between electronic states) • Self-interaction problem: each electron lives in the field created by all electrons including itself (ways forward: SIC, hybrid DFT)
Self interaction problem
In Hartree Fock, self-interaction is canceled by the equal (and opposite) exchange termNot necessarily so in DFT
E
[
ρ
]
=
T
s
[
ρ
]
+
1
2
∬
ρ
(
r
)
ρ
(
r
'
)
r
−
r
'
dr
dr
'
+
E
XC
[
ρ
]
+
∫
ρ
(
r
)
V
ext
(
r
)
dr
A Coulomb contribute would arise from an electron interacting with itself
That contribute does not vanish for a one-electron case
J
[
ρ
]
Exact
LDA
PBE
Exchange
0.3125
0.2680
0.3059
Correlation
0.0000
0.0222
0.0060
In the multi-electron case for each one-electron density:
E
xc
[
ρ
i
]
+
J
[
ρ
i
]
=
0
E
xc
[
ρ
]
+
J
[
ρ
]
=
0
E
x
[
ρ
i
]
+
J
[
ρ
i
]
=
0
E
c
[
ρ
i
]
=
0
Condition for self-interaction free functional in a one-electron case
Can be split:
Self interaction problem
J is fully non-localIts contribute cannot be recovered by semi-local functionals
Delocalization Error
H2+
(H
H)
+
HF
B3LYP
LDA
Delocalization Error
H2+
(H
H)
+
HF
B3LYP
LDA
Delocalization Error
H2+
(H
H)
+
A linear combination of two different Slater determinants,with different occupation and equal (at dissociation limit) coefficient.
Looking only at density, at dissociation limit we have two H atoms with half electron each.
Ψ
=
c
1
∥
1s
H
1
1
∥
+
c
2
∥
1s
H
2
1
∥
What is the supposed behavior for fractional charge ?
Delocalization Error
The better a XC functional approximates the required linear interpolation between integers charges, the better will reduce the delocalization error.Error can be convex (above) or concave.Cancellation may arise if Exchange and Correlation compensate out.
For an exact functional, the curve E/N must be a series of linear segments, with discontinuities in the derivative at integer charges
Same error!
Additional trouble
Functionals that are well behaved for one-electron self-interaction error are not necessarily well-behaved for the N-electron cases.
Delocalization error and self-interaction error are part of the same problem:Inability of the functional to behave linearly for fractional charge.
For any external potential and number of electrons, two conditions must be satisfied:
1. linear variation for non-integer charges2. realistic energies at integer charges
Semilocal approximations satisfy 2 but not 1.
Hartree-Fock satisfies 1 (but not 2) only for one electron cases.
Delocalization Error
Energy is too low if the electron is delocalized over two centers.
Related to self-interaction error.
Convex errors results in:
Reaction barriers too lowPolarizabilities too highOverestimate conductanceUnderestimate band gapsFavors delocalization
Wrong derivative with respect to charge has strong implications
Delocalization Error
Meaning of Kohn-Sham energies and orbitals
In Hartree-Fock formalism, orbitals and energies have physical meaning
Koopmans theorem
For closed-shell systems, the orbital energy corresponds to the ionization energyof an electron from that orbital.
IP
i
=
E
−
E
i
+
=
−
ϵ
i
Under a few assumption, it is exact in the context of restricted Hartree-Fock theory.In fact, orbital energies are qualitatively good approximations of the ionization energies.
Meaning of Kohn-Sham energies and orbitals
What about Hartree Fock orbitals ?
Brillouin theorem
The Hartree Fock orbitals are those orbitals which give an antisymmetrizedwavefunction (Slater determinant) which does not interact with any single excitations.
Orbitals where any given electron interacts with all others electrons seen as an average field
also,
Kohn-Sham orbitalsSame symmetry of Hartree-FockIdentical of HF on all practical purposes for MO considerations
Kohn Sham energies are not orbital energies.They have no such physical meaning.They are not a good approximation to ionization energies.They are simply Lagrange parameters to fulfill the constraint.
Meaning of Kohn-Sham energies and orbitals
Janak's Theorem
ϵ
i
=
∂
E
∂
n
i
Although technically
ϵ
N
(
N
)
=
−
I
if the XC functional is exact
Local Density
LDA
GGA
meta-GGA
hyper/hybrid-GGA
fully non local
XC Approximation Ladder
XC func dependsonly on the density
Gradient dependent(density first derivative)
XC func dependson the density andits first derivative
E
xc
[
ρ
]
E
xc
[
ρ
,
∇
ρ
]
“Second derivative”methods
XC func depends on the first and secondderivatives of the density
Completely non-local.Uses exact exchange.
E
xc
[
ρ
,
∇
ρ
,
∇
2
ρ
]
Non-local methods
Semi-Local
depend on the density at a given pointand its ”immediate proximity”
Bestiary: PW86, BP, LYP, PW91, PBE, RPBE
Bestiary: TPSS
Exchange Correlation Hole
Conditional probability of finding an electron in r2 given that there's an electron at r1
If we had the exact Exchange Correlation hole, it should normalize so that
∫
P
xc
(
r
1,
r
2
)
dr
2
=
−
1
∫
P
x
(
r
1,
r
2
)
dr
2
=
−
1
∫
P
c
(
r
1,
r
2
)
dr
2
=
0
We can be even more specific
Generalized Gradient Approximation (GGA)
XC Functional depends on the density and the gradient of the density
E
xc
=
∫
ϵ
xc
(
ρ
,
∇
ρ
)
ρ
but actually is Generalized Expansion Approximation with tricks
cutoff at integrationto one
Exchange:
Generalized Gradient Approximation (GGA)
Correlation
Generalized Gradient Approximation (GGA)
Improved exchange
Improved correlation,but potential artifactsdue to cutoff
GGA Performance
Exchange (0.5%) (LDA 5%)Elastic constants (5%)Energy barriers (30%) (LDA 100%)
Underestimate
Overestimate
Correlation (5%) (LDA 100%)Bond lengths (1%) (LDA 1% shorter)Atomization energies (4%) (LDA 15%)
Improves van der Waals and Hydrogen bondsImprove negative ions
Other
Better than LDA, but still not chemical accuracy (error = 1kcal/mol)
Fails
Weak/long range interactionsvan der Waals/Hydrogen bonds better, but not good enough.
END