1
Carlo Adamo
Density Functional Theory
Concepts and models
Équipe de Chimie Théorique et ModélisationInstitut de Recherche de Chimie Paris
Ecole Nationale Supérieure de Chimie de Paris Chimie ParisTech
1
K
i ig c
x r
2
Hartree-Fock Model
The basic idea of Hartree-Fock calculations:
• solve Ĥ E for a system of M nuclei and N electrons using:
1. the full Hamiltonian within the Born-Oppenheimer approximation:
1 12
1 1 1 1 1
1 1ˆ2
N M N N N M MI JI
ii I i i j i I J IIi ij IJ
Z ZZH
r r r
2. a trial wavefunction consisting of one Slater determinant:
1 2... ...SD
i j N
3. molecular orbitals expressed as linear combinations of basis functions:
basis function with a fixed form
coefficient in linear expansion (called molecular orbital coefficients)
artificial spin function
Hartree-Fock models is the basis for virtually all quantum chemical methods
3
The basic idea of Hartree-Fock calculations:
• solve Ĥ E for a system of M nuclei and N electrons using:
1. the full Hamiltonian within the Born-Oppenheimer approximation:
1 12
1 1 1 1 1
1 1ˆ2
N M N N N M MI JI
ii I i i j i I J IIi ij IJ
Z ZZH
r r r
2. a trial wavefunction consisting of one Slater determinant:
1 2... ...SD
i j N
3. molecular orbitals expressed as linear combinations of basis functions:
1
K
i ig c
x r
4. variational optimization of using the molecular orbital coefficients as variational parameters
Hartree-Fock Model
4
Hartree-Fock Energy
to use the variational method, we need a relationship between the energy and the MO coefficients
* ˆE H d for any normalized wavefunction:
in Hartree-Fock theory:
1 2... ...SD
i j N
and if you work it out, the energy becomes:
2* 1
1 1 11 1
* * * *1 1 2 2 1 1 2 2
1 2 1 21 1 12 12
2
1
2
N MI
a aa I Ii
N Na a b b a b b a
a b
ZE d
r
d d d dr r
x x x
x x x x x x x xx x x x
5
Hartree-Fock Energy
2* 1
1 1 11 1
* * * *1 1 2 2 1 1 2 2
1 2 1 21 1 12 12
2
1
2
N MI
a aa I Ii
N Na a b b a b b a
a b
ZE d
r
d d d dr r
x x x
x x x x x x x xx x x x
to use the variational method, we need a relationship between the energy and the MO coefficients
recall that molecular orbitals are linear combinations of basis functions
1
K
i ig c
x r
so, now we have a direct relationship between the Hartree-Fock energy and molecular orbital coefficients
6
1 1 1
1
2
N N N
aa ab aba a b
E h J K
Hartree-Fock Energy
or in standard short-hand notation:
one electron energy
Coulomb integral
exchange integral
two electron energy
2* 1
1 1 11 1
* * * *1 1 2 2 1 1 2 2
1 2 1 21 1 12 12
2
1
2
N MI
a aa I Ii
N Na a b b a b b a
a b
ZE d
r
d d d dr r
x x x
x x x x x x x xx x x x
in terms of the MOs, the Hartree-Fock energy is:
7
Electron Correlation: Summary
• electron correlation is the electron-electron energy ‘missing’ in an Hartree-Fock calculation
0corr HFE E E there are two types of electron correlation
1. Dynamic correlation: • accounts for fact that electrons move such that they avoid other electrons
2. Static correlation: • in systems with multiple resonance states, electrons can avoid each other by occupying different resonance states
• captured by using a multi-determinant wavefunction
• captured by using multiple Slater determinants in the wavefunction
• each Slater determinant is made of a unique set of molecular orbitals and represents a different electronic
configuration
• each Slater determinant is built by ‘exciting’ electrons within the Hartree-Fock determinant
• the molecular orbitals in each Slater determinant are not re-optimized
density functional theory treats the electron density quantum mechanically
8
Density vs. Wavefunction
Wavefunction:
• contains all of the information about the system
• not an observable quantity
• function of 4N variables (3 spatial, 1 spin per electron)
Density:
• function of only 3 spatial variables
• physically measurable quantity
• connected to the wavefunction
2 d r r r
does the electron density contain the same information that is contained in the wavefunction?
• for a one electron wavefunction:
The ingredients (1):
Density : It provides us information about how something is distributed/spread on a given space (volume)Electron Density :It tells us where the electrons are likely to exist
Electron density is an « observable »
2* )()()()( rrrr
bis(diiminosuccinonitrilo)nickelexp DFT
9
The ingredients (2):
1234
14916Y=f(X)
A function (f) maps a set of numbers to another set of
numbers
A functional (F) is a function of a function
Y
A function which maps a set of functions to a set of
numbersEx. F[A(X),B(X),C(X),….]
A(X)B(X)C(X)D(X)
2013F1
234
X
YX
Y=X2
Example : Fxc[(r),(r)]10
Why DFT?
From a pragmatic point of view: E. Bright Wilson, 1965
• If one knows the exact electron density, (r), then the cusps ofthis density would occur at the positions of the nuclei.
• Furthermore, a knowledge of |(r) | at the nuclei would give theirnuclear charges.
• Thus the full Schrödinger Hamiltonian was known because it iscompletely defined once the position and charge of the nucleiare given.
• In principle, the wavefunction and E are known, and thuseverything is known.
The knowledge of the density is all that is necessary for a complete determination of all molecular properties.
11
Can the total energy be expressed as a function of the density?
HE ˆ
21211
rr11 rrrrrrrrrr
'11
ddr
dVdH ext ),(1
)()(),(2
1ˆ2
121
21
N2N21N2'111 xxxxr...xxrrr dddN ...)...,(*),(...),( 1111
NN11 xxxxxrr ddddNN
...)...,(...2
)1(),( 321
2
222
Critical ingredients: kinetic energy and electron-electron interaction
12
][)()(
2
1
12
Jddr
2121 rrrr
][)()(][ˆ eeext EdVTHE rrr
Early attempts: Thomas-Fermi (1927)
1) Exact kin. en. (T[]) is substituted by the kin. en. of a homogenous electrons gas
cF=constante de Fermi
e-/e- (Coulomb)Nuclei-electrons interaction(Coulomb)
][)()(][ eeTFTF EdvTE rrr
2) The external potential (Vext[]) is that generated by the nuclei3) The electron-electron interaction is the Coulomb repulsion
rr
r dZ
A
A AR
rr dc f3/5
13
Thomas-Fermi (1927): how does it works?
212121 )()(
2
1)(][ rr
rr
rrr
rr ddd
ZTE
A
ATFTF
AR
classical Coulomb interactions
Not very well :in TF theory no molecular system is stable relative to dissociation
into consitutents fragments
non bonding theorem
(Teller (1962) / Balazs (1967) / Lieb (1973) / Simon (1977))
What is missing?
Exchange
Correlation
and furthermore … TTF is local
14
Add exchange : Thomas-Fermi-Dirac model (TFD)
rrrrrrr
rrr
dcJdvdcE
KJdvTE
xfTF
TFTF
3/43/5 )(][)()()(
][][)()(][
cf = 3/10(3p2)2/ 3cx = 3/4(3/p)1/ 3
Add gradient corrections to kinetic energy :Thomas-Fermi-Dirac-Weizsacker model (TFDW or TFD-lW)
rrrrrrr
rrr dcJdvddcE xfTF
3/4
2
3/5 )(][)()()(
)(
8
1)(
l
Everything is expressed as a function of the density (or gradient of the density)
15
HF
A different approach:
Slater and the X model (or HFS model)
Cx = constant 0.75 < < 1
Exchange only; no correlation
Works EXTREMELY well (and it is still used especially for solid state)
)1()1()1()1(22
1
2/,1
21 ii
Naaa
A A
A KJRr
Z
)1()1()1()1(2
1 21 iiiXHFcoulomb
A A
A VVRr
Z
SLATER
3/1rcV xXHF
16
Why HFS is better than TFD?
17
X performances
18
The Hohenberg and Kohn theorems (1964)
• All properties of the many-body system are determined by the ground state density 0(r)
• Each property is a functional of the ground state density 0(r) which is written as f [0]
)()(
)(
0i
0
rr
rr
extv
19
0(r) define the external potential Vext(r) (except for constant) and thus all properties
r
Hohenberg and Kohn: theorem I
20
21
First Hohenberg-Kohn Theorem
the ground state energy of a system is uniquely determined by the ground state electron density
12
1 1 1 1
1 1ˆ2
N N M N NI
a ai i I i j iiI ij
ZH
r r
consider two different Hamiltonians:
12
1 1 1 1
1 1ˆ2
N N M N NI
b bi i I i j iiI ij
ZH
r r
0, 0, 0,ˆa a a aH E
0, 0, 0,ˆb b b bH E
the two Hamiltonians differ in the electron-nuclear attraction term:
• electron-nuclear attraction is called external potential in DFT lingo
• each Hamiltonian has a unique ground state wavefunction and ground state energy
22
First Hohenberg-Kohn Theorem
can two different wavefunctions give us the same electron density??
*0, 0, 0,
ˆa b a bE H d r
*0, 0, 0,ˆ ˆ ˆa b a b b bE H H H d r
* *0, 0, 0, 0, 0,ˆ ˆ ˆa b a b b b b bE H H d H d r r
we know:
1 1 1 1
ˆ ˆN M N M
I Ia b a ba b
i I i IiI iI
Z ZH H
r r
*0, 0, 0,
ˆb b b bH d E r
0, 0 0,a bd d r r r r r
based on the variational principle:
23
First Hohenberg-Kohn Theorem
can two different wavefunctions give us the same electron density??
* *0, 0, 0, 0, 0,ˆ ˆ ˆa b a b b b b bE H H d H d r r *0, 0, 0, 0,a a b b b bE d E r
0, 0 0,a a b bE d E r r
if we do the same for E0,b:
0, 0 0,b b a aE d E r r
* *0, 0, 0, 0, 0a a b bd d r r r r r r r
in both cases we assumed that:
24
First Hohenberg-Kohn Theorem
can two different wavefunctions give us the same electron density??
0, 0 0,a a b bE d E r r
0, 0 0,b b a aE d E r r+
0, 0, 0 0, 0,a b a b b a b aE E d E E r r0
0, 0, 0, 0,a b b aE E E E
clearly, the sum of two energies cannot be less than the sum of the same two energies
therefore, our assumption that two different wavefunctions with different energies can give the same density is wrong
the ground state energy of a system is uniquely determined by the ground state electron density
25
First Hohenberg-Kohn Theorem
the ground state energy of a system is uniquely determined by the ground state electron density
the first Hohenberg-Kohn theorem is an existence theorem
• it tells us that a unique relationship exists between the ground stateenergy and the ground state electron density
• it does not tell us what that relationship is
we do know that the energy is a functional of the density
E E r• a functional is a function whose argument is another function
• it can be shown that all other ground state properties of the system can be expressed as functionals of the electron density, too
In principle, one can find all other properties and they are functionals of 0(r).
r
r
E
E
rrr
rrr
dV
dVVTE
extHK
extHK
F
int
26
27
Second Hohenberg-Kohn Theorem
the ground state density is a variational quantity
This means:
• if we pick any arbitrary density it will give us an upper bound on the true ground state density
• if we compare two densities, the one with the lower energy is the better one
• we can employ linear variation techniques to optimize the density, like we did with wavefunction methods
Proof:
• we know there is unique correspondence between the density and the wavefunction
• so, if we pick an arbitrary trial density, we will get an arbitrary trial wavefunction
• the variational theory tells us that for any trial wavefunction:
*0
ˆtrial trial trialE H d E
trial trial
28
Energy Functionals
First HK Theorem: • there exists an exact functional relationship between the ground state density and the ground state energy
Second HK Theorem: • the electron density can be optimized variationally
to use variational optimization procedures, we must know how E and are related!!!
)()(
)(
0i
0
rr
rr
extv
There is an infinite number of w.f. yielding 0
DFT- HK
QC
• How do I know, given an arbitrary function (r), that it is a density coming from an antisymmetric N-body wavefunction Ψ(r1,...,rN)?
All observables
N-representability
• How do I know, given an arbitrary function (r), that it is the ground statedensity of a local potential v(r)?
How to get the wf from the density?
Solved: any square-integrable nonnegative function satisfies it
V-representability Constrained-search formalism (Levy-Lieb)Definition of the universal functional
How to get the ground state density (r)?
29
Constrained search formalism (Levy and Lieb (1977))
eeextVVTE ˆˆˆminmin0
drVVTE extee )(ˆˆminmin0 r
Double minimization procedure:
12
eeVTF ˆˆmin
Definition of the universal functional :
eeVTF ˆˆmin
Wavefunctions giving density ( r )
30
Double minimisation or how to find the tallest child in a school?
From Parr&Yang
min
min
1. Find the tallest in each classroom
2. Find the tallest of the tallest
31
Various extensions have been proposed:
Extensions to spin dependent systems (Barth, Hedin, 1972)
[ , ]E n n
Extension to relativistic systems (Vignale, Kohn, 1988)
[ ( )]E j rExtension to finite temperatures
[ ] [ ] [ ]F n E n TS n
Time-Dependent DFT (Runge, Gross, 1984)
32
iiieffiKS rvH
2
2
1ˆ
Kohn and Sham ansatz (1965)
Replace the interacting-particles hamiltonian with one that it can be solved more easily
KS hamiltonian: an hamiltonian describing N non-interacting particlesassumed to have the same density as the true interacting system.
NN ...det! 3212/1
If you don’t like the answer, change the question
33
What have we gained so far?
Apparently Nothing: The only result is that the density determines the potential
We are still left with the original many-body problem
34
Kohn-Sham Energy Functional
Kohn and Sham suggested decomposing the total energy into kinetic and potential energy contributions
classical non classicalne ee eeE T V V V
?T • accounts for the kinetic energy of the electrons
1
MI I
neI I
ZV d
R rr
r R
• accounts for nuclear-electron Coulombic attraction
1 2 1 21 2
1
2classicaleeV d d
r r
r rr r
• accounts for average electron-electron Coulombic repulsion
?non classicaleeV • accounts for all electron-electron interactions not in classicaleeV• includes exchange and instantaneous electron-electron
correlation
35
Kohn-Sham Energy Functional
Kohn and Sham recognized that it’s easier to calculate the kinetic energy if we have a wavefunction
2
*
2iT d
r
but we don’t know the ground state wavefunction (if we did, we wouldn’t bother with DFT)
So, they suggested:
• instead of using the real wavefunction and density, let’s use a Slater determinant wavefunction built up from a set of one-electron orbitals (like molecular orbitals) to build up an artificial electron system that represents the ground state density
• ‘artificial system’ is often called the non-interacting reference system
• the ‘one-electron orbitals’ are called Kohn-Sham orbitals
• the Kohn-Sham orbitals are orthonormal:
* 1 if
0 otherwisei j iji j
d
36
Kohn-Sham Energy Functional
the Kohn-Sham orbitals give us an easy way to construct the density and get an approximate value of the kinetic energy
kinetic energy of N one-electron orbitals:
2*
1 2
Ni
rs i ii
T d
r
density from N one-electron orbitals:
*1
N
i ii
r r r
N d r r
the density integrates to give the total number of electrons:
37
Kohn-Sham Energy Functional
a Slater determinant wavefunction built up from Kohn-Sham orbitals will never give the exact ground state kinetic energy
2*
2i
exactT d
r2
*
1 2
Ni
rs i ii
T d
r
(the closest approximation to the real wavefunction you can get with a single Slater determinant is the Hartree-Fock wavefunction)
so, Kohn and Sham suggested splitting up the kinetic energy
exact rsT T T
exact ground state kinetic energy kinetic energy of the
reference system
correction accounting for kinetic energy not contained in Trs
38
Kohn-Sham Energy Functional
classical non classicalne ee eeE T V V V
incorporating the orbital-based expressions into the total energy functional:
gives:
classical non classicalrs ne ee eeE T V V V T
terms we have explicit expressions for with everything that
Kohn-Sham orbitals we don't know how to solve
classicalrs ne ee xcE T V V E
this is the Kohn-Sham total energy functional
39
Kohn-Sham Energy Functional
classicalrs ne ee xcE T V V E
2*
1 2
Ni
rs i ii
T d
r
1
MI I
neI I
ZV d
R rr
r R
*1
N
i ii
r r r
with Kohn-Sham orbitals, we can calculate:
we don’t have an expression for Exc:
• Exc is called the exchange correlation functional
• it acts as a repository for all contributions to the energy that we do not know how to calculate exactly
• if we had an exact expression for Exc we could calculate the energy and density exactly
1 2 1 21 2
1
2classicaleeV d d
r r
r rr r
40
The Exchange-Correlation Functional
the 1st Hohengberg-Kohn theorem tells us that an exact functional relationship between energy and the density
classicalrs ne ee xcE T V V E
this means an exact form of Exc must exist, and if we had it, the total energy functional would be exact
unfortunately, we do not know the exact form of Exc
• the accuracy of DFT calculations hinges on the accuracy various approximations to Exc
• the development of exchange-correlation functionals is a major area of research in modern theoretical chemistry
• we’ll talk about about exchange-correlation functionals in greater detail later
• for now, let’s just treat the exchange-correlation functional in a generic sense
41
The Kohn-Sham Orbitals
how do we solve for the Kohn-Sham orbitals?
the second Hohenberg-Kohn theorem tells us that the ground state density minimizes the energy
this means:
we’ll use variational theory
• we need connections between the energy, orbitals, and density
• we need a way to ‘optimize’ the orbitals
• expanding the Kohn-Sham orbitals as linear combinations of basis functions:
a variational treatment of the orbitals will involve
• using linear variation methods to get the set of coefficients that give us a density that minimizes the energy
basis function with a fixed form
coefficient in linear expansion
1
K
i ig c
x r
artificial spin function
42
The Kohn-Sham Orbitals
how do we solve for the Kohn-Sham orbitals?
in terms of Kohn-Sham orbitals:
classicalrs ne ee xcE T V V E
2* *
1 1
2*1 2 1 1
1 1
2
1
2
N Mi I
i i i ii I I
N
i i xci
ZE d d
d d E
r r r r r rr R
rr r r r
r r2
2
1 2*1 2
1 1 1 1
1
2 2
N M NIi
i i xci I iI
ZE d d d d E
r r r
r r r r r rr R r r2
43
The Kohn-Sham Orbitals
how do we solve for the Kohn-Sham orbitals?
we have connections between the orbitals, density and energy:
2* *
1 1
2*1 2 1 1
1 1
2
1
2
N Mi I
i i i ii I I
N
i i xci
ZE d d
d d E
2
r r r r r rr R
rr r r r
r r
*1
N
i ii
r r r
we also have a connection to the orbital coefficients:
1
K
i ig c
x r
44
The Kohn-Sham Orbitals
we have to apply the apply the linear variational approach to the Kohn-Sham DFT energy
with the constraint that the Kohn-Sham orbitals remain orthonormal
classicalrs ne ee xcE T V V E
*i j ijd
if you do this (we won’t), you find out that the ‘best’ set of Kohn-Sham orbitals are given by:
221 2 1 11 1 1 2
1
2
MI
xc i i iI I
Zd V
r
r r rr R r r
Kohn-Sham operator Kohn-Sham orbital i
energy of Kohn-Sham orbital i
45
The Kohn-Sham Operator
the Kohn-Sham orbitals are eigenfunctions of the Kohn-Sham operator
221 2 1 11 1 1 2
1
2
MI
xc i i iI I
Zd V
r
r r rr R r r
Kohn-Sham operator Kohn-Sham orbital i
energy of Kohn-Sham orbital i
the first three terms account for: • the kinetic energy of the electron in i• the Coulombic attraction between the
nuclei and the electron in i• the static Coulombic repulsion between
the electron in i and the total electron density
the last term accounts for: • exchange repulsion, electron correlation, self-interaction energy and kinetic energy
correction associated with the electron in i
46
Comparison with Hartree-Fock
the Kohn-Sham orbitals are eigenfunctions of the Kohn-Sham operator
221 2 1 11 1 1 2
1
2
MI
xc i i iI I
Zd V
r
r r rr R r r
2 *2
2 2 2 2 211 1 1 1
1 1 12 12
( )
2
Mb b aI
a a b a aI b a b aI
d dZ
r r r
x x x x x
x x x x
the Hartree-Fock molecular orbitals are eigenfunctions of the Fock operator
these two expressions are very similar:
• the kinetic energy, nuclear-electron, and static Coulomb electron-electron repulsion terms in each operator are identical (but they may be written slightly differently)
• the last terms differ:
• in the Fock operator, only exchange interactions are considered
• in the Kohn-Sham operator, all remaining interactions are considered
can never give exact results
can give exact results, if Vxc is exact
47
Solving for the Kohn-Sham Orbitals
we want to solve
1 1 1ˆ ( )KS i i if r r r
1 1 11 1
ˆ ( )K K
KSi i if c c
r r r
multiply on the left by * and integrate:
* *1 1 1 1 1 1 11 1
ˆ
KS
K KKS
i i i
SF
c d f c d
r r r r r r r
FKS is an element of the Kohn-Sham matrix
S is an element of the overlap matrix
Note: there will be K of these equations because there are K basis functions
using a linear combination of basis functions
this is completely analogous to what we do in the Hartree-Fock method
1
K
i ig c
x r
48
Solving for the Kohn-Sham Orbitals
by converting to a matrix form, we can solve for the orbitals with a self-consistent procedure like we did with Hartree-Fock
* *1 1 1 1 1 1 11 1
ˆ
KS
K KKS
i i i
SF
c d f c d
r r r r r r r
FKSC = SC
C is a K x K matrix whose columns define the coefficients, ci:
is a diagonal matrix of the orbital energies, i:
1
2
K
0ε
0
11 12 1
21 22 2
1 2
K
K
K K KK
c c c
c c c
c c c
C
1 2 K
49
221 2 1 11 1 1 2
1
2
MI
xc i i iI I
Zd V
r
r r rr R r r
Solving for the Kohn-Sham Orbitals
once again, the Kohn-Sham matrix elements depend on the Kohn-Sham orbitals coefficients
*1
N
i ii
r r r
* *1 1 1
K K N
i i i ii
P
c c
r r r
P is the density matrix, it quantifies the amount of electron density ‘shared’ between basis functions and
50
once again, the Kohn-Sham matrix elements depend on the Kohn-Sham orbitals coefficients
Solving for the Kohn-Sham Orbitals
Solution: • solve with iterative techniques
• make an initial guess of C1• build FKS1 and solve for C2
• use C2 to build FKS2 and solve for C3...
• use CN to build FKSN and solve for CN+1
• stop when CN and CN+1 are the same (or very similar)
• this self-consistent field (SCF) approach was also used in Hartree-Fock
• final matrix C defines the Kohn-Sham orbitals that give the density that minimizes the energy
51
Kohn-Sham Density Functional Theory
isn’t this just Hartree-Fock?
No, in Hartree-Fock we try to solve the Schrödinger equation using a trial wavefunction consisting of a single Slater determinant
1 1 1ˆ a a af x x x
1 2... ...HF
i j N
* ˆHF HF HFE H d since the Hartree-Fock wavefunction is necessarily approximate, this can
never give the exact ground state energy
52
in Kohn-Sham DFT we only use the Kohn-Sham orbitals
to give us the density:
*1
N
i ii
r r r
that minimizes the Kohn-Sham energy functional:
2
1 2*1 2
1 1 1 1 2
1
2 2
N M NIi
i i xci I iI
ZE d d d d E
r r r
r r r r r rr R r r
Kohn-Sham Density Functional Theory
isn’t this just Hartree-Fock?
1 1 1ˆ ( )KS i i if r r r
if we have an exact form of Exc, the total energy functional is exact and we can get the exact ground state energy and density with DFT
53
Meaning of KS Eigenvalues
The ionization potential is
i
iif2
)()( rr f= occupation number (could be fractional)
kkkkk
vNk
Nkk fdffdf
EEEI
1
0
1
0
1
Using the mean-value approximation 5.0kkI
This is the so-called Janak-(Slater) transition state theorem
It could be considered as the equivalent of the Koopman’s theorem
54
Exchange Correlation Functionals
classicalrs ne ee xcE T V V E
the accuracy of Kohn-Sham DFT rests on the availability of accurate exchange-correlation functionals
although the 1st Hohenberg-Kohn theorem tells that an exact form of Excmust exist, we don’t know what it is
The exchange-correlation functional should describe:
1. exchange (Pauli) repulsion between electrons of the same spin
2. electron correlations from instantaneous electron-electron interactions
3. kinetic energy corrections
in modern Kohn-Sham DFT, we use approximate exchange-correlation functionals that account for a portion of these interactions
The problem: Exc exact is unknown
The theory is exact, the functionals are approximated
55
Self-Interaction Energy
1 2 1 21 2
1
2classicaleeV d d
r r
r rr r
electron density of the hydrogen atom
For the Hydrogen atom:
• there is only one electron
• we represent the electron with a probability distribution and average it over all space
• so the electron makes a contribution to the density at r1 and r2
• leads to a spurious electron repulsion
• the situation extends to multi-electron systems
• the self-interaction error must be cancelled out
• in Hartree-Fock, exchange cancels out the self-interaction
• in DFT, we use the exchange-correlation functional to cancel out the self-interaction
rrr dVEJTE extXCSHK )()(
JVTT
JTEE
eeS
SHKXC
EHF Hi 12i1,N
J ij Kij i, j1,N
Consequences of the use of approximate vxc : self interaction error (SIE)
In exact KS as in HF: there is no Coulomb interaction of one electron with itself
SIE(N) J i Exc i ,0
i
But in approximate DFT this is not the case:
56
Effects of self interaction error
He2+
He2+
CCSD(T)
BLYP
LDA
PW91
(He++He)
2(He+0.5)
57
How to get an approximate xc functional?
Contains information on the many-body system of interacting electrons
The easiest way: Local Density Approximation – LDA
• Assume the functional is the same as a model problem –the homogeneous electron gas
• Separate the exchange and correlation contributions: Exc = Ex + Ec
• Exc can be calculated as a function of the density only
58
LDA
rrrrrrr ddE CXXCLDAXC ))(())(()())(()(][ homogeneous electron gas exchange-
correlation energy per particle(at the point r)
probability of finding the particle at r
model problem : the homogeneous electron gas
The value of the xc energy depends only on the local density.
The e- density () may vary as a function of r, but is single-valued, and the fluctuations in away from r do not
affect the value of Excat r.
59
• around each electron other electrons tend to be excluded
Definition of “x-c hole”:
• Excis the interaction of the electron with the “hole” : it involves only a spherical average
Exchange hole in Ne atomGunnarsson, et. al.
Very non-spherical!
Spherical average very closeto the hole in a homogeneouselectron gas!
nucleus electron
Spherical averagearound electron
Is the Local Density Approximation physically sounding?
)( r'r,xc
r'r
r'r,rr dd][
r'-
)( xcxcE
60
Exchange-correlation (x-c) hole in silicon
• Calculated by Monte Carlo methods
Hood et al
Exchange Correlation
Hole is reasonably well localized near the electronSupports a local approximation
61
LDA
rrrrrrr ddE CXXCLDAXC ))(())(()())(()(][ model problem : the homogeneous electron gas
Get an expression for them
62
3/1)(r xLDAX C
LDA : Exchange partDerived by Bloch et Dirac (1929/1930) for
homogeneous electron gas
rrr dE XLDAX ))(()(][
(per particle)
Usually called Slater exchange functional
Functional form identical to that of Slater (HFS)
LDA : Correlation part• No explicit formulation• Approximate analytical expression to reproduce accurate quantum Monte-Carlo (Ceperly & Alder, 1980)results for a homogeneous electron gas• Most used LDA approximation for correlation Volsko, Wilk et Nusair (1980): VWN.
bx
Q
Q
xbxx
xX
bx
bx
Q
Q
b
xX
xALDAC 2
arctan)2(2
)ln()(2
arctan2
)(ln
202
00
02
63
Local Spin Density Correlation Functional
• Not for the faint of heart:
709921.1
5198421.0
/2)1()1('
),49671.0,88026.0,6231.3,357.10,11125.0,0168869.0(
),62517.0,3662.3,1977.6,1189.14,20548.0,01554535.0(
),4294.0,6382.1,5876.3,5957.7,21370.0,0310907.0(
)1(')'1()'1(],[
3/43/4
2/1
2/1
2/1
444
zz
scorrm
scorrp
scorru
zz
mpus
LSD
C
f
rG
rGe
rGe
feer
)(((2
11log)1(2),,,,,,(
4321
2
143211 rbbrbrbarraarbbbbaaGcorr
LDA: how it works
From A. V. Morozov
65
Discontinuty of thepotential for the fillingof a electronic shell
rr
xcxc
xcxc
Ev
• A: High density, large kinetic energy, LDA approximation unimportant
• B: Small density gradient, LDA is good
• C:large gradient, LDA fails
66
LDA: how it works
Exchange-Correlation Hole
• Due to phenomena of exchange there is a depletion of density (of the same spin) around each electron.
• Mathematically described as
• The exchange correlation energy written as
rr ,xc
rr
rr
rrrddxcExc
,
Properties of the hole
• Subject of much research.
0'',
1'',
0',
',',',
rrr
rrr
rr
rrrrrr
d
d
c
x
x
cxxc
The LDA must obey these.
Negative quantity determining thedecrease in the probability of finding anelectron of the same spin at position r’when one electron is known to be atreference position r
Why is this important?
• Huge error made to the integral would occur if the hole is not normalised correctly.
• The LDA has this correct – it is the correct expression for a proper physical system.
• In fact, only need the spherical average of the hole is needed.
• Different densities for different spins : split the total density
Open shell systems and Local Spin Density Approximation(LSDA)
rrrr dE XCLSDAXC ))(),(()(],[
)()()( rrr
Measure of spin polarization :
)(
)()(
r
rr
Remark : in principle, since the external potential is spin independent there is no need to split the different spin densities
70
How to ameliorate the LDA?
Atoms, molecules or solids are not a homogeneous electron gas :
Include non local effects
GEA (Gradient Expansion Approximation): F(r), (r)
',
3/23/2, ...),(),()(],[
rrr dCdE XCXC
GEAXC
Taylor expansion
Note: mathematically speaking GEAs are still local
How do they work? Not a great improvementReason : Exchange correlation hole properties not satisfied
71
GGA idea
• A brute force fix.
• If x(r,r’)>0, set it to zero.• If sum rule violated, truncate the hole.
• Resulting expressions look like:
rr
r
3/4
3/4
n
ns
dnsFLSDAEGGAE xx
Impose the fulfillment of the properties of the exchange-correlation hole
rdfEGGAXC ),,,(],[ GGAC
GGAX
GGAXC EEE
GGA (Generalized Gradient Approximation)
rr dsFEELDAX
GGAX )()(
3/4
)(
)()(
3/4 r
rr
swhere s is the reduced density
S high for high gradient or small density regions (far from nuclei)S small for small gradients (bonding region)S intermediate for high gradient and small density (near the nuclei)
Measure local inhomogenity
73
ss
sF B
1
2
sinh61
Becke, 1988 (B ou B88)
Example of commonly used exchange functionals
74
• Chemistry stable– Empirical
– =0.0042, fitted to exchange energies of He ... Rn.– Gives correct asymptotic form in exponential tails.
Example of commonly used exchange functionals
Problem: How to get a GGA?
Functional form can get extremely complex (especially for correlation functionals)
75
• The physics stable:– Principled, parameter free
– Numerous analytic properties
– Slow varying limit should give LDA response. This requires Fx →s2 , =0.21951
– Density scaling, n(r)→l3n(lr), Ex→l Ex
804.0,/1 2
s
sF
• Parametrized functionals –semi-empirical-use adjusted parameters to reproduced exact or experimental data (ex. atomicenergies)
• Non parametrized functionalsimpose physical constrains (uniform electron gas limit, asymptotic behavior)
Fxauthors Fc
authors
B
PW91
PBE
mPW
HCTH
B97
Becke (1988)
Perdew et Wang (1991)
Perdew, Burke, Ernzerhof (1996)
Adamo, Barone (1997)
Handy et al. (1999)
Becke
P86
LYP
PW91
PBE
Perdew (1986)
Parr et al. (1988)
Perdew et Wang (1991)
Perdew, Burke, Ernzerhof (1996)
Brief (and extremely non exhaustive) list of commonly used GGA functionals
Over parametrisation Semi-empirical Good properties
« universal » (worse) chemical properties
76
Exchange, Ex
LSDA GGA Meta-GGA
X1951
Dirac1930
G96
B86 B88
PW91PBE1996
RPBE1999
revPBE1998
xPBE2004
PW86
mPW
TPSS2003
BR89
PKZB1999
CS1975
Correlation, Ec
LSDA GGA Meta-GGA
W38
xPBE2004
PW86
PBE1996PW91
LYP1988
B95
TPSS2003
PKZB1999
B88VWN1980
PZ81
PW92CA Data1980
79
Some theoretical contraints
Size consistency : E(AB)=E(A)+E(B)Virial TheoremSelf-interaction errorJanak Theorem
Lieb-Oxfod boundCoordinate scaling
Hirao JCP 1999
80
Lieb-Oxford Limit
Bonding region
Adamo JCP 2002
• Exchange-correlation functionals must be numerically integrated– not as robust as analytic methods
• Energies and gradients are 1-3 times the cost of Hartree-Fock
• Frequencies are 2-4 times the cost of HF• Some of this computational cost can be
recuperated for pure density functionals by employing the density fitting approximation for the Coulomb interaction.
Calculating Exc Terms
HF LDA GGA Exp.
H2 3.64 4.90 4.55 4.73
H2O 6.72 11.58 10.15 10.06
HF 4.21 7.03 6.16 6.11
O2 1.43 7.59 6.24 5.25
F2 -1.60 3.34 2.30 1.69
CH4 14.22 20.03 18.21 18.17
Atomisation energies (in eV) of several molecules: theory vs experiment.
Property HF LDA GGA Exp.
a0, Å 3.58 3.53 3.57 3.567
Ea, eV -5.2 -8.87 -7.72 -7.55
K0, GPa 471 455 438 442
Solids : diamond
82
fully nonlocal+ explicit dependence on unoccupied orbitals
rung 5
hybrid functionals
+ explicit dependence on occupied orbitals
rung 4
meta-GGAs + explicit dependence on kinetic energy density
rung 3
GGAs +explicit dependence on gradients of the density
rung 2
LDAlocal density onlyrung 1
John Perdew Jacob Ladder*
*DFT conference Menton 2000
Philippe Ratner
EARTH (Hartree theory)
HEAVEN (chemical accuracy)
How to improve GGA results?
83Update 2007:
Rung 4bis: Hyper GGAs
meta-GGA or dependent
Fxc, , occ,1i2
i kinetic energy density
Introduce quasi-local information
84
How to improve the exchange energy : Hybrid Functionals
How does it work? Very bad!Mean average error 32 kcal/mole for the G2 ensemble (50 molecules) while aMAE of 5-7 kcal/mol is obtained with standard GGA
KSC
exactXXC EEE
Idea: HF exchange is exact. Therefore we could think to combine exact (HF) exchange with a (GGA ) correlation functional (Lie & Clementi 1974)
Actually combining a percentage of HF exchange with a GGA exchange works much better!!!
Theoretical justification : adiabatic connection (Becke 1993 – Half and Half)
In practise : HF% between 20 and 30%
85
The Adiabatic Connection
What do we know about ?
86
What does Wl look like?From Yang
87
Use of Adiabatic Connection
None of these functionals is self-interaction free,
even with the use
88
9110
3 )1( PWccLSDc
Bxx
HFxxo
LSDxx
Bxc EaEEaEaEaE
PBE0
PBEc
PBEx
HFx
PBExc EEEE 4
3
4
10
B3LYP
(Becke, JCP 1993)3 parameters fitted on G2 (ionization and atomization energies)
(Adamo, Scuseria JCP 1999)No fitted parameters
Normally hybrid functionals outperform all GGA and meta-GGA functionals and they are the reference for chemical applications
89
Method Distance(Å)
D0(kcal/mol)
Dipole moment (D)
Harmonic freq. (cm-1)
HF & post-HF
HF 0.022 82.0 0.29 144
MP2 0.014 23.7 0.28 99
CCSD[T] 0.005 11.5 0.10 31
LDA & GGA
LSDA 0.017 43.5 0.25 75
BPW 0.014 6.0 0.11 69
BLYP 0.014 9.6 0.10 59
LGLYP 0.013 7.2 0.10 60
PWPW 0.012 8.6 0.12 66
mPWPW 6.7 0.11 65
Hybrid 3 parameters
B3LYP 0.004 2.4 0.08 31
B3PW 0.008 4.8 0.08 45
mPW3PW 0.008 2.6 0.08 37
Hybrid ab-initio
B1LYP 0.005 3.1 0.08 33
B1PW 0.010 5.4 0.10 48
LG1LYP 0.005 4.0 0.10 45
Performance of selected functionals
diatomic molecules
90
MAE for harmonic frequencies(cm-1)
(G2 set,>50 organic molecules)
Level of thoery Errorr
HF and post-HF
HF/6-311G(3df,2p) 144
MP2/6-31G(d,p) 99
CCSD/6-311G(3df,2p) 31
LSDA
SVWN/6-31G(d,p) 75
GGA
BLYP/6-311G(d,p) 59
BPW91/6-311G(d,p) 69
PWPW91/6-311G(d,p) 66
mPWPW91/6-311G(d,p) 66
Hybrid functionals
B1LYP/6-311G(d,p) 33
B1PW91/6-311G(d,p) 48
mPW1PW91/6-311G(d,p) 39
B3LYP/6-311G(d,p) 31
B3PW91/6-311G(d,p) 45
mPW3PW91/6-311G(d,p) 37
Performance of selected functionals
91
92
Rutile - GTO
93
TS 13 P14
BHTTS 15
G. Talarico, P. H. M. Budzelaar, V. Barone and C. Adamo Chem. Phys. Lett. 329, 99, (2000).G. Talarico, V. Barone, P. H. M. Budzelaar and C. Adamo, J. Phys. Chem. A, 105 (2001) 9014G. Talarico, V. Barone, L. Joubert and C. Adamo Int. J. Quantum Chem. 91 (2003) 474
The XC functional can be crucial for chemical understanding
R1 = H, R2 = iso-propyl ;R1 = tert-butyl, R2 = iso-propR=growing chain
N
Al
N
R1
R2
R
R2
The catalyst
Homogeneous catalysis of ethylenegrowing chain
termination
94
Different functionals different CHEMICAL
answers.
insertion channel
BLYP
BP86
B3LYP
B3PW91
B1LYP
mPW0
B98
PBE0
B1Bc95
VSXC
MP2
best est.
15 20 25 30 35
E#ins
E(kcal/mol)
fun
ctio
na
l
BLYP
BP86
B3LYP
B3PW91
B1LYP
mPW0
B98
PBE0
B1Bc95
VSXC
MP2
best est.
5 10 15 20 25
E#
BHT
E(kcal/mol)
func
tion
al
termination channel
BP86 termination most probable(low m.w.)VSXC insertion most probable(higher m.w.)
95
Known breakdowns of DFT
LocalizationSelf Interaction Error (SIE)
Non-dynamic correlation effects
reaction barriers, CT complexes, Rydberg excitations, IP from Koopmans, band gap, bond length alternation, magnetic properties, vdW interactions,…
(Un)Known causes?
Many efforts to assess the reliability of DFT by a trial-and-error approach
All problems coming from the approximate
nature of the XC contribution
among others
The theory is exact, the functionals are approximate
96
Challenges in DFT
Better functionals (e.g. CR, MR-05)Error corrections (e.g. SIC, ad-hoc parametrization)Static correlation (e.g. MC-DFT, RSH)Localization vs delocalization
97
In chemical language
98
Many functionals on the market
adapted from K. Burke
Known breakdowns of DFA
LocalizationSelf Interaction Error (SIE)
Non-dynamic correlation effectsPoor performing functionals
reaction barriers, CT complexes, Rydberg excitations, IP from Koopmans, band gap, bond length alternation, magnetic properties,
vdW interactions,…
(Un)Known causes?
Many efforts to assess the reliability of DFA by a trial-and-error approach
All problems coming from the approximate nature of the XC
contribution
among others
From DFT to DFA
fully nonlocal+ explicit dependence on unoccupied orbitals
rung 5
hybrid functionals + explicit dependence on occupied orbitals
rung 4
meta-GGAs + explicit dependence on kinetic energy density
rung 3
GGAs +explicit dependence on gradients of the density
rung 2
LDAlocal density onlyrung 1
EARTH (Hartree theory)
HEAVEN (chemical accuracy)
The Jacob’s Ladder of John Perdew
Update 2007:Rung 4bis: Hyper GGAs
No simple rules for a priori reliabilityNo systematic route to improvementToo many functionals to choose fromStatistical approaches
It’s tail must decay like -1/r
It must have sharp steps for stretched
bonds
It keeps H2 in singlet state as
R→∞
Adapted from a poem by John Godfrey Saxe (1816–1887)
What we know about the « true » functional is only a part
Exploring DFT as blind men?
102
Some theoretical contraints
Lieb-Oxfod boundCoordinate scaling
Hirao JCP 1999Using (some) of the known theoretical constraints todefine the exchange-correlation functional
The starting point: the PBE functional
Focusing on the exchange
Analytical formula of Becke86
1) Recovering the uniform gas limit
2/22, xxx EEE2) The exact exchange energy obeys
the spin-scaling relationship
3) Recovering the LSD linear response(small density variations)
21)( ssFx
1)0( xF
Perdew-Burke-Ernzerhof PRL 1996
2
2
11
s
sF PBEx
rdrFE unifxGGAxGGAx 33/4)()(],[
=0.21951,k=0.804 and s=||/2r(3p2)1/3
3 more theoretical conditions for correlation
4) Lieb-Oxford limit rdEx 33/4679.1,
HOF IP EA PA BDE NHTBH HTBH NCIE All0
5
10
15
100
120
MA
D (
kca
l/mol
BENCHMARK SET
LDA BLYP PBE
First-principles vs semiempirical
PBE vs BLYP
BLYP 4 parameters
HOF = heat of formation;IP = ionization potential,
EA = electron affinityPA = proton affinity,
BDE = bond dissociation energy, NHTBH, HTBH = barrier heights for reactions,
NCIE = the binding in molecular clusters data from Goddard III
Alternative to PBE ex
PBE Perdew PRL 1996 k= 0.804 =0.21951revPBE Yang PRL 1996 k=1.245, =0.21951RPBE Hammer PRB 1999 k= 0.804 =0.21951mPBE Adamo JCP 2002 C1=0.804, C2=-0.015, a=0.157 PBEsol Perdew PRL 2008 k=0.804 =0.1235SOGGA Truhlar JCP 2008 k=0.552, =0.1235
2exp11 sF RPBEx
2
2
2
22
2
1 111
as
sC
as
sCFmPBEx
2
2,
11
s
sF PBEsolrevPBEx
Bonding region
Lieb-Oxford Limit
RPBEx
PBEx
SOGGAx FFF 2
111
106
JCP 2002
Alternative to PBE exch
How to improve the exchange : Hybrid Functionals
How does it work? Very bad!Mean average error 32 kcal/mole for the G2 ensemble (50 molecules) while aMAE of 5-7 kcal/mol is obtained with standard GGA
KSC
exactXXC EEE
Idea: HF exchange is exact. Therefore we could think to combine exact (HF) exchange with a (GGA ) correlation functional (Lie & Clementi 1974)
Actually combining a percentage of HF exchange with a GGA exchange works much better!!!
Theoretical justification : adiabatic connection (Becke 1993 – Half and Half)
In practise : HF% between 20 and 30%
107
108
From Yang
The Adiabatic Connection
109
Adiabatic Connection
9110
3 )1( PWccLSDc
Bxx
HFxxo
LSDxx
Bxc EaEEaEaEaE
3 parameters fitted on G2 (atomization energies) with the PW91 correlation
Becke, JCP 1993
First-principles hybrids: the PBE0 model
Perdew’s Hypothesis(JCP 1996)
Our working-horse functional
1,, 1 nDFAxxDFAxchybxc EEEE lll
)10(.... 11102, llll nnxc ccceE
DFAxxDFAxchybxchybxc EEnEdEE 11
0 ,ll
1n
The optimum integer n is derived from a pertubation theory on l-dependence of Exc,l
The lowest order n=4
DFAxxDFAxchybxc EEEE 41
DFA=BLYP then B1LYP (CPL 1997)DFA=PBE then PBE0 (JCP 1999)
PBE0: a fair play
PBE0 = about 3000 citations
Called PBE1PBE, later PBEh
Called PBE0
LYPcc
LSDc
Bxx
HFxxo
LSDxx
LYPBxc EaEEaEaEaE 10
3 )1(
PBE0
PBEc
PBEx
HFx
PBExc EEEE 4
3
4
10
B3LYP
3 additional parameters fitted on G2 (ionization and atomization energies)
No fitted parameters
Normally hybrid functionals outperform all GGA and meta-GGA functionals and they are the reference for chemical applications
112
PBE0 vs B3LYP
HOF IP EA PA BDE NHTBH HTBH NCIE All0
4
8
12
20
24
BENCHMARK SET
MA
D (
kca
l/mo
l
BLYP PBE B3LYP PBE0
First-principles vs semiempirical
PBE0 vs B3LYP
B3LYP 7 parameters
HOF = heat of formation;IP = ionization potential,
EA = electron affinityPA = proton affinity,
BDE = bond dissociation energy, NHTBH, HTBH = barrier heights for reactions,
NCIE = the binding in molecular clusters
B3LYP & PBE0 closer than BLYP & PBE
data from Goddard III
Illas et al J. Chem. Phys. 129, 184110 (2008)
JPBE0-6971-762-77
2339-1313
-49170
3-35-16
-346-492
Illas et al. J. Chem. Phys. 129, 114103 (2008)
J=2(E(BS)-E(T))
PBE0 vs B3LYP
EPR parameters: g-tensor (Dipeptide analogue of glycyl radical, HCO–GlyR–NH2 )
JCP 2004
Critical GS Properties: EPR parameters
Excited state properties: vertical absorptions
A large series of organic dyes
JCTC 2008, ACR 2009, JCTC 2010
HF PBE PBE0 LC-PBE LC-wPBECAM-B3LYP0
20
40
60
80
100
120
140
160
180
MA
E (
nm
)
All Aq AB IG
Large benchmark 483 molecules, 614 valence exc. 28 functionals: JCTC 2009
Valence excitations TD-DFT/6-31+G(d,p)/PCM
Excited state properties: vertical absorptions
5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5
5
6
7
8
9
10
11
HOMOC
alc
. (e
V)
Exp. (eV)
6-31G 6-31G(d) 6-31G(d,p) 6-31+G(d,p) 6-31+G(2d,p) 6-31+G(2d,2p) 6-31++G(2df,2pd) 6-31++G(3df,3pd) Valence Rydberg
Our suggestion PBE0 up to e(HOMO) + 1eVOur suggestion PBE0 up to e(HOMO) + 1eV
BENZENE
PBE0
JPC A 2007
Excited state properties: vertical absorptions
300 350 400 450 500300
350
400
450
500
l the
or(n
m)
lexp
(nm)
PBE0 SLR-PBE0
Coumarins
JCP 2006
TD-PBE0/6-31+G(d,p)/PCM// TD-PBE0/6-31+G(d,p)/PCM
Analytical TD-DFT first derivatives
Excited state properties: vertical emission
Solids
Kresse JCP 2007Kresse JPCM 2008
Bulk moduli & atomization energy
B3LYP>PBE>PBE0≈HSE
GAP (semiconductor, insulators)
HSE
Perovskite KNiF3
KNiF3 : [(Ni2F11K12Ni10)25+ + 430 point charges] 0
K2NiF4 : [(Ni2F11K16Ni6)21+ + 1154 point charges] 0
La2CuO4: [(Cu2O11La12Cu6)30+ + 1005 point charges] 0
All electron
ECP
Critical GS Properties: magnetic coupling
Proportional to the difference in energy between different spin states
JCP 2004
1.62 1.64 1.66 1.68 1.70 1.72 1.74 1.76 1.78 1.80 1.82 1.840
5
10
15
20
25
30
35
40
45
J (m
eV
)
Pd
Hybrid functionals:GGA+ HF exchange
exchange functionals
LDA, GGA and correlation functionals
Exp.
KNiF3
JCP 2004
PBE0
Critical GS Properties: magnetic coupling
123
Why does PBE0 work?
In short: PBE0 as good as B3LYP
One more rung: double hybrids
Grimme’s ideas: add some PT2 contribution in a B3LYP style(JCP 2006)
222 )1( PTcPLYPB
cHFxx
BLYPxx
PLYPBxc cEbEEaEaE
MP2 contribution computed with B2LYP orbitalsSame number of parameters as B3LYP
Savin’s contributions: rigourous derivation + from 3 to 1 parameter(JCP 2011)
22/1
2,1 ][][][)1( MPcccxHFx
DHDSxc EEEEEE llll l
l
l cc EE /1If the correlation does not scale with l
2221 ][)1(][)1( MPccxHFx
DHDSxc EEEEE llll l
A further step PBE0-DH
22/1
2,1 ][][][)1( MPcccxHFx
DHDSxc EEEEEE llll l
l
Imposing the Levy-Görling limit
2331 ][)1(][)1( MPccxHFx
DHLSxc EEEEE llll l
2/1
0][lim GLcc EE ll
i i
xi
ij ji
eejiHFc
MPc
GLc
fvvEEE
ˆˆˆ
4
122
22 MPc
GLc EE 0HFcE
Linear scaled one-parameter hybrid ][)1(][ 2/1 ll l cMPcc EEE
JCP 2011a, JCP2011b
0.0 0.2 0.4 0.6 0.8 1.04.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
MA
E (
kca
l/mol
)
l
PBE
MP2
PBE0-DH
JCP 2011a
20
4
1
4
7
2
1
2
1 MPccx
HFx
DHPBExc EEEEE
Using the simplest approx to the adiabatic integral(as in the Becke Half&Half)
Theoretical l almost the optimal value for atomization energies
A further step PBE0-DH
CRISMAT 27/10/2011
G2 (148 molecules)
MAE RMS MAX
B3LYP 3.0 4.4 -20.0 (SiF4)
B2PLYP 2.6 4.5 -12.3 (SiF4)
PBE0 5.3 6.7 -21.6 (SiF4)
PBE0-DH 5.0 6.7 -24.8 (O3)
NCB31 MAE RMS MAX
B3LYP 0.93 1.44 4.48 (par C6H6)
B2PLYP 0.53 0.77 1.89 (sand. C6H6)
PBE0 0.70 1.08 2.69 (par C6H6)
PBE0-DH 0.49 0.74 1.94 (NH3F2)
HT HAT NS UA DBH24/total
B3LYP 5.1 (5.1) 7.6 (7.6) 3.8 (3.8 2.6 (1.2) 4.8 (4.4)
B2PLYP 2.1 (1.3) 2.5 (2.1) 2.4 (2.4) 1.3 (-0.6) 2.1 (1.3)
PBE0 4.7 (4.7) 6.4 (6.4) 2.3 (2.3) 2.7 (0.4) 4.0 (3.5)
PBE0-DH 1.5 (1.5) 1.6 (1.2) 1.2 (0.3) 2.3 (-1.2) 1.6 (0.4)
Some results
JCP 2011data in kcal/mol
G2=Atomization EnergiesNCB=Non-Covalent BondingHT= Hydrogen TransferHAT=Heavy Atom TransferNS=Nucleofilic SubstitutionsUA=Unimolecular Reactions
CRISMAT 27/10/2011
set PBE0 PBE0-DH B2PLYPBH76 4.12 1.68 2.24S22 2.37 1.65 1.79RG6 0.42 0.37 0.47HEAVY28 0.65 0.52 0.69ACONF 0.64 0.49 0.50SCONF 0.53 0.47 0.59CYCONF 0.58 0.41 0.22
Some preliminary results
data in kcal/mol
Testing on the large GMTKN30 set (2059 energy data)
S22 noncovalently bound dimersBH76 barrier heightsRG6 rare gas dimersHEAVY28 heavy element hydridesACONF alkane conformersSCONF sugar conformersCYCONF cysteine conformes
References data: exp, CCSD(T)/CBS or W1-W4