DFT adamo reseau - French National Centre for Scientific Research · 2017. 1. 25. · 7kh edvlf lghd ri +duwuhh )rfn fdofxodwlrqv vroyh + (Ö< < iru d v\vwhp ri 0qxfohl dqg 1hohfwurqv

1

Carlo Adamo

Density Functional Theory

Concepts and models

[email protected]

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


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


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


can two different wavefunctions give us the same electron density??

*0, 0, 0,

â b a bE H d r

*0, 0, 0,ˆ ˆ â b a b b bE H H H d r

* *0, 0, 0, 0, 0,ˆ ˆ â 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



* *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



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


25



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 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


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


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


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



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


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



in terms of Kohn-Sham orbitals:


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



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


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


*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


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


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


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


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

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


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


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

Documents

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