Transcript A new implementation of an accurate self-interaction-corrected

A new implementation of an accurate self-interaction-corrected
correlation energy functional based on an electron gas with a gap
JULIEN TOULOUSE1, ANDREAS SAVIN2 and CARLO ADAMO1
Complexes
1 Laboratoire d’Electrochimie et de Chimie Analytique (UMR 7575) – Ecole Nationale Supérieure de Chimie de Paris, 11 rue
Pierre et Marie Curie, 75231 Paris Cedex 05, France.
2 Laboratoire de Chimie Théorique (CNRS), Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France.
Modélisation des Systèmes
ABSTRACT - Density functional theory (DFT) is a very effective method for the computation of the electronic structure of atoms, molecules or solids. In practical applications of this theory, only the exchange-correlation contribution to the total energy
needs to be approximated. Whereas a large number of approximations have been proposed for the exchange part, there are less correlation functionals, more difficult to model. However, Krieger, Chen, Iafrate and Savin [1,2] have recently designed a metaGGA correlation functional called KCIS that satisfied a large number of rigorous physical conditions. Furthermore, this functional, based on the idea of a uniform electron gas with a gap in the excitation spectrum [3], contains no empirical parameters. In
order to test this functional and to build new accurate DFT models, we have in this work implemented KCIS in a self-consistent way in the quantum computation package GAUSSIAN. The search for the best exchange functionals which can been used with
KCIS has leaded to two new hybrid models with the Becke 88 (B) and mPBE exchange functionals: B0KCIS and mPBE0KCIS. These models, both including 25 % of exact exchange, contain only one empirical parameter in the exchange part. These two
functionals have been tested over a varied set of physico-chemical properties and have turned out to have performances better or at least equivalent to those provided by semi-empirical exchange-correlation functionals like B3LYP. A detailed analysis of
the results suggests that the best improvements brought by our models concern properties where the correlation contribution plays an important role like for atomization energies, energetic reaction barriers and magnetic properties.
Introduction
Density Functional Theory
In order to obtain more accurate molecular properties, new approximations
to the DFT exchange-correlation functional Exc are expected, especially for
the correlation contribution which is the most difficult part to model.
The KCIS correlation functional (1)
 Within the Kohn-Sham approach to DFT, the electronic energy is the sum
of several contributions:
E[ra,rb ] = Ts[ra,rb ] + J[r] + v(r)r(r) dr + Exc[ra,rb ]
This approach is exact but the exchange-correlation functional Exc is unknown.
 implemented a new, promising correlation functional named KCIS in a
self-consistent way in the quantum chemistry software GAUSSIAN, as well
as its second derivatives with respect to the required variables,
more variables
  
a can be optimized on experimental data (ACM1) or fixed theoretically to 1/4
(ACM0).
2
Test of the KCIS functional

tW

r  eGGAGAP
(r  ,0, r  )
c
  a,b t 

occupied

3
8,2
4,6
5,1
B0LYP
PBE0
0,009
3,1
mPBE0KCIS
3,3
0,011
KCIS
real system
PBE
0,015
PKZB
HOMO
occupied
uniform gas with a gap
uniform gas
mPBE
 The slowly-varying limit: lim Ec   c1 (r)  c2 (r) r 2 dr
Ionization
corrections
Which exchange functionals with KCIS?
 KCIS satisfies several theoretical conditions, in particular:
 r 0
EF
gradient
Mean absolute errors (MAE) on atomization energies of 55 covalent molecules
belonging to the G2 set (6-311+G(3df,2p) basis set):
0,018
PBE
LYP

unoccupied
G[r]
Mean absolute errors (MAE) on correlation energies of atoms from H to Ar
(6-311+G(3df,3pd) basis set and Hartree-Fock densities):
EcKCIS ra , rb  F ra , rb , ra , rb , ta , tb dr
real system
unoccupied

hyb
Exc
 a Exexact  Ex  Exc
The KCIS correlation functional (2)
 KCIS is a parameter-free meta-GGA correlation functional:
HOMO
 In KCIS, the correlation energy is calculated on the basis of a uniform
electron gas with a gap [3]:
meta-GGA : exc(ra,rb ,ra,rb ,2ra,2rb ,ta,tb )

Ionization
occupied
uniform gas
GGA : exc(ra,rb ,ra,rb )
 evaluated the accuracy of these new DFT models over a wide set of
physico-chemical properties.
1
EF
occupied
 A further improvement: Hybrid functionals or Adiabatic Connection Models

unoccupied
gradient corrections
and different levels of approximations have been proposed for exc:
LSD : exc(ra,rb )
 identified the best exchange functionals that can be used with the KCIS
functional in order to construct new accurate DFT models,
F  reGGAGAP
(r a , rb , r) 
c
unoccupied
 Exc can be expressed by the general formula: Exc[ra,rb ] = r(r)exc(r)dr
In this context, we have in this work :
where
(Krieger, Chen, Iafrate and Savin [1,2] , 1999)
 Usually, correlation functionals Ec are simply constructed from the
uniform electron gas:
B0KCIS
 The rapidly-varying limit: lim Ec  0
3,0
r 
0
 Saturation under uniform scaling to the high-density limit: lim Ec r   cst
0,005
0,01
 
0,015
0,02
0
MAE (Hartree)
KCIS is one of the most accurate correlation functionals.
4
5
Test of B0KCIS: Molecular geometries
Test of B0KCIS: Weak interactions
0,020
B0KCIS
0,015
E (eV)
0,007
0,006
0,008
0,010
PBE0
0,005
0,000
exact
PBE
-0,005
0,002
0,004
0,006
0,008
0,01
0,012
2,5
2,75
3
3,25
3,5
3,75
7
6
5
4
3
2
1
ClCH3… Cl-
8
g eb e g nb nr s (rn )
3h
direct contribution
spin polarization
vinyl radical ()
methyl radical ()
Reaction coordinate
H
PBE
0,0
H
C
0,9
PBE0
H
3,6
B0KCIS
4
5
6
C
Exp.
PBE
PBE0
B0LYP
B0KCIS
107,6 Gauss
102,2
110,2
112,7
107,6
H
C
H
7
8
9
10
Like the other hybrid functionals, B0KCIS gives accurate excitation
energies, even for high-lying excited states (Rydberg states).
9
H
Exp.
PBE
PBE0
B0LYP
B0KCIS
 The KCIS correlation functional is generally more accurate than the other
semi-empirical or theoretical functionals, like LYP or PBE.
 B0KCIS is globally more accurate than B0LYP for properties which
strongly depend on the correlation contribution of the functional like
atomization energies, energetic reaction barriers and magnetic properties.
 B0KCIS gives similar results for properties more depending on the
exchange contribution of the functional like molecular geometries, weak
interactions and excitation energies.
2,5
MP2
3
Conclusion
The isotropic hyperfine coupling constant of a nucleus n is connected to the spin
density at the nucleus:
an 
B0LYP
2
8
ClCH3 + Cl-
-3,8
1
Experimental excitation energies (eV)
Test of B0KCIS: Magnetic properties
ClCH3 +Cl- (6-311G(d,p) basis
PBE
B0LYP
PBE0
B0KCIS
8
0
Like B0LYP, B0KCIS can’t describe the dispersive interactions which
can only be properly treated with an accurate exchange contribution.
Test of B0KCIS: Chemical reactivity
Cl- … CH3Cl
9
4
7
[Cl … CH3…Cl]E
10
d(He-He) (Å)
Compared to other hybrid functionals, B0KCIS gives similar
results for molecular geometries.
Cl- + CH3Cl
6
0
2,25
MAE (Å)
Symmetric SN2 reaction: Cl- + CH3Cl
Energy
set)
10
Vertical excitation energies for the singlet states of H2CO (6-311G(d,p) basis
set):
B0LYP
0
8
Test of B0KCIS: Excitation energies (TDDFT)
He …He van der Waals dimer (uncontracted aug-cc-pV5Z basis set):
0,011
PBE
B0KCIS
6
Some accurate hybrid DFT models can be obtained by combining
KCIS with the Becke 88 (B) or mPBE exchange functionals.
Computed excitation energies (eV)
Mean absolute errors (MAE) on bond lengths of 32 molecules belonging to the
G2 set (6-311G(d,p) basis set):
B0LYP
4
MAE (kcal/mol)
 The Self-Interaction Correction (SIC): E c r ia ,0  0
PBE0
2
28,4 Gauss
24,6
29,2
31,7
24,9
 Prospects:
Construct more accurate DFT models by combining KCIS with some
recently developed meta-GGA exchange functionals involving the
laplacian of the electron density.
References:
-5
-4
-3
-2
-1
0
1
2
3
4
E (kcal/mol)
B0KCIS gives a realistic energetic reaction barrier.
10
B0KCIS gives very accurate hyperfine coupling constants for
simple radicals without spin polarization.
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[1] J. B. Krieger, J. Chen, G. J. Iafrate and A. Savin in Electron Correlations and Materials
Properties, A. Gonis and N. Kioussis (Eds) Plenum, New York (1999).
[2] S. Kurth, J. P. Perdew and P. Blaha, Int. J. Quant. Chem. 75, 889 (1999).
[3] J. Rey and A. Savin, Int. J. Quant. Chem. 69, 581 (1998).
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ENSCP/LECA/MSC, July 2002.