JULIEN TOULOUSE 1, ANDREAS SAVIN 2 and CARLO ADAMO 1 1 Laboratoire d’Electrochimie et de Chimie...
1
0 0,002 0,004 0,006 0,008 0,01 0,012 PBE PBE0 B0LYP B0K C IS M A E (Å ) 0,008 0,011 0,007 0,006 JULIEN TOULOUSE 1 , ANDREAS SAVIN 2 and CARLO ADAMO 1 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. A new implementation of an accurate self-interaction- corrected correlation energy functional based on an electron gas with a gap Introduction 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 meta-GGA 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. In order to obtain more accurate molecular properties, new approximations to the DFT exchange- correlation functional E xc are expected, especially for the correlation contribution which is the most difficult part to model. In this context, we have in this work : 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, identified the best exchange functionals that can be used with the KCIS functional in order to construct new accurate DFT models, evaluated the accuracy of these new DFT models over a wide set of physico-chemical properties. Density Functional Theory Within the Kohn-Sham approach to DFT, the electronic energy is the sum of several contributions: This approach is exact but the exchange-correlation functional E xc is unknown. E xc can be expressed by the general formula: E xc [ ] = (r) xc (r)dr and different levels of approximations have been proposed for xc : LSD : xc ( ) GGA : xc ( ) meta-GGA : xc ( ) E[ ] = T s [ ] + J[] + v(r)(r) dr + E xc [ ] more variables A further improvement: Hybrid functionals or Adiabatic Connection Models xc x exact x hyb xc E E E a E a can be optimized on experimental data (ACM1) or fixed theoretically to 1/4 (ACM0). The KCIS correlation functional (1) (Krieger, Chen, Iafrate and Savin [1,2] , 1999) Usually, correlation functionals E c are simply constructed from the uniform electron gas: unoccupied occupied EF gradient corrections HOMO Ionization unoccupied occupied In KCIS, the correlation energy is calculated on the basis of a uniform electron gas with a gap [3]: uniform gas real system unoccupied occupied EF gradient HOMO Ionizatio n unoccupied occupied uniform gas real system uniform gas with a gap corrections G[] The KCIS correlation functional (2) KCIS is a parameter-free meta-GGA correlation functional: r d F E KCIS c , , , , , , ) , 0 , ( ) , , ( , GGAGAP c W GGAGAP c F where KCIS satisfies several theoretical conditions, in particular: 0 lim c E r d c c Ec 2 2 1 0 ) ( ) ( lim The slowly-varying limit: The rapidly-varying limit: Saturation under uniform scaling to the high-density limit: The Self-Interaction Correction (SIC): 0 0 , i c E cst Ec lim Test of the KCIS functional Mean absolute errors (MAE) on correlation energies of atoms from H to Ar (6-311+G(3df,3pd) basis set and Hartree-Fock densities): 0 0,005 0,01 0,015 0,02 K CIS PK ZB LYP PBE M A E (H artree) 0,011 0,018 0,015 0,009 KCIS is one of the most accurate correlation functionals. Which exchange functionals with KCIS? Mean absolute errors (MAE) on atomization energies of 55 covalent molecules belonging to the G2 set (6- 311+G(3df,2p) basis set): 0 2 4 6 8 10 PBE m PBE B0LYP PBE0 m PBE0K C IS B0K C IS M A E (kcal/m ol) 0 2 4 6 8 10 PBE m PBE B0LYP PBE0 m PBE0K C IS B0K C IS M A E (kcal/m ol) 8,2 4,6 5,1 3,1 3,3 3,0 Some accurate hybrid DFT models can be obtained by combining KCIS with the Becke 88 (B) or mPBE exchange functionals. Test of B0KCIS: Molecular geometries Mean absolute errors (MAE) on bond lengths of 32 molecules belonging to the G2 set (6-311G(d,p) basis set): Compared to other hybrid functionals, B0KCIS gives similar results for molecular geometries. Test of B0KCIS: Weak interactions He … He van der Waals dimer (uncontracted aug-cc-pV5Z basis set): B0K C IS B0K C IS B0K C IS -0,005 0,000 0,005 0,010 0,015 0,020 2,25 2,5 2,75 3 3,25 3,5 3,75 4 d(H e-H e)(Å) E (eV) PBE PBE0 B0LYP exact -0,005 0,000 0,005 0,010 0,015 0,020 2,25 2,5 2,75 3 3,25 3,5 3,75 4 d(H e-H e)(Å) E (eV) -0,005 0,000 0,005 0,010 0,015 0,020 2,25 2,5 2,75 3 3,25 3,5 3,75 4 d(H e-H e)(Å) E (eV) PBE PBE0 B0LYP exact Like B0LYP, B0KCIS can’t describe the dispersive interactions which can only be properly treated with an accurate exchange contribution. Test of B0KCIS: Excitation energies (TDDFT) Vertical excitation energies for the singlet states of H 2 CO (6-311G(d,p) basis set): 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Experim entalexcitation energies(eV) C om puted excitation energies(eV ) PBE B0LYP PBE0 B0K CIS Like the other hybrid functionals, B0KCIS gives accurate excitation energies, even for high-lying excited states (Rydberg states). Test of B0KCIS: Chemical reactivity Symmetric S N 2 reaction: Cl + CH 3 Cl ClCH 3 +Cl - (6-311G(d,p) basis set) Energy Reaction coordinate Cl - + C H3Cl ClCH3 … Cl - [Cl … CH3 … Cl] - Cl -… CH3Cl Energy E ClCH3 + Cl - -5 -4 -3 -2 -1 0 1 2 3 4 M P2 PBE B 0L Y P PBE0 B0K C IS E (kcal/m ol) 2,5 -3,8 0,0 0,9 -5 -4 -3 -2 -1 0 1 2 3 4 M P2 PBE B 0L Y P PBE0 B0K C IS E (kcal/m ol) -5 -4 -3 -2 -1 0 1 2 3 4 M P2 PBE B 0L Y P PBE0 B0K C IS E (kcal/m ol) 2,5 -3,8 0,0 0,9 3,6 B0KCIS gives a realistic energetic reaction barrier. Test of B0KCIS: Magnetic properties The isotropic hyperfine coupling constant of a nucleus n is connected to the spin density at the nucleus: ) ( 3 8 n s n n e e n g g h a r direct contribution vinyl radical () C C H H H C C H H H spin polarization methyl radical () C H H H C H H H Exp. PBE PBE0 B0LYP B0K C IS 107,6 G auss 102,2 110,2 112,7 107,6 Exp. PBE PBE0 B0LYP B0K C IS 107,6 G auss 102,2 110,2 112,7 107,6 107,6 G auss 102,2 110,2 112,7 107,6 E xp. PBE PBE0 B0LYP B0K C IS 28,4 G auss 24,6 29,2 31,7 24,9 E xp. PBE PBE0 B0LYP B0K C IS 28,4 G auss 24,6 29,2 31,7 24,9 28,4 G auss 24,6 29,2 31,7 24,9 B0KCIS gives very accurate hyperfine coupling constants for simple radicals without spin polarization. Conclusion 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. Prospects: Construct more accurate DFT models by combining KCIS with some recently developed meta-GGA exchange functionals involving the laplacian of the electron density. ENSCP/LECA/MSC, July 2002. M odélisation des S ystèmes C o m p l e x e s References: [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). 1 2 3 4 5 6 7 8 9 1 0 11 1 2
-
Upload
shanna-preston -
Category
Documents
-
view
212 -
download
0
Transcript of JULIEN TOULOUSE 1, ANDREAS SAVIN 2 and CARLO ADAMO 1 1 Laboratoire d’Electrochimie et de Chimie...
0 0,002 0,004 0,006 0,008 0,01 0,012
PBE
PBE0
B0LYP
B0KCIS
MAE (Å)
0 0,002 0,004 0,006 0,008 0,01 0,012
PBE
PBE0
B0LYP
B0KCIS
MAE (Å)
0,008
0,011
0,007
0,006
JULIEN TOULOUSE1, ANDREAS SAVIN2 and CARLO ADAMO1
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.
A new implementation of an accurate self-interaction-corrected correlation energy functional based on an
electron gas with a gap
Introduction
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 meta-GGA 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.
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.
In this context, we have in this work :
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,
identified the best exchange functionals that can be used with the KCIS functional in order to construct new accurate DFT models,
evaluated the accuracy of these new DFT models over a wide set of physico-chemical properties.
Density Functional Theory Within the Kohn-Sham approach to DFT, the electronic energy is the
sum of several contributions:
This approach is exact but the exchange-correlation functional Exc is unknown.
Exc can be expressed by the general formula: Exc[] = (r)xc(r)dr
and different levels of approximations have been proposed for xc:
LSD : xc()
GGA : xc()
meta-GGA : xc()
E[] = Ts[] + J[] + v(r)(r) dr + Exc[]
more variables
A further improvement: Hybrid functionals or Adiabatic Connection Models
xcxexactx
hybxc EEEaE
a can be optimized on experimental data (ACM1) or fixed theoretically to 1/4 (ACM0).
The KCIS correlation functional (1)(Krieger, Chen, Iafrate and Savin [1,2] , 1999)
Usually, correlation functionals Ec are simply constructed from the uniform electron gas:
unoccupied
occupied EF
gradient corrections
HOMO
Ionization unoccupied
occupied
In KCIS, the correlation energy is calculated on the basis of a uniform electron gas with a gap [3]:
uniform gas real system
unoccupied
occupied EF
gradient
HOMO
Ionization unoccupied
occupied
uniform gas real system uniform gas with a gap
corrections
G[]
The KCIS correlation functional (2)
KCIS is a parameter-free meta-GGA correlation functional:
rdFE KCISc ,,,,,,
),0,(),,(,
GGAGAPc
WGGAGAPcFwhere
KCIS satisfies several theoretical conditions, in particular:
0lim
cE
rdccEc2
210
)()(lim The slowly-varying limit:
The rapidly-varying limit:
Saturation under uniform scaling to the high-density limit:
The Self-Interaction Correction (SIC): 00, icE
cstEc
lim
Test of the KCIS functional
Mean absolute errors (MAE) on correlation energies of atoms from H to Ar (6-311+G(3df,3pd) basis set and Hartree-Fock densities):
0 0,005 0,01 0,015 0,02
KCIS
PKZB
LYP
PBE
MAE (Hartree)
0 0,005 0,01 0,015 0,02
KCIS
PKZB
LYP
PBE
MAE (Hartree)
0,011
0,018
0,015
0,009
KCIS is one of the most accurate correlation functionals.
Which exchange functionals with KCIS?
Mean absolute errors (MAE) on atomization energies of 55 covalent molecules belonging to the G2 set (6-311+G(3df,2p) basis set):
0 2 4 6 8 10
PBE
mPBE
B0LYP
PBE0
mPBE0KCIS
B0KCIS
MAE (kcal/mol)
8,2
4,6
5,1
3,1
0 2 4 6 8 10
PBE
mPBE
B0LYP
PBE0
mPBE0KCIS
B0KCIS
MAE (kcal/mol)
8,2
4,6
5,1
3,1
3,3
3,0
Some accurate hybrid DFT models can be obtained by combining KCIS with the Becke 88 (B) or mPBE exchange functionals.
Test of B0KCIS: Molecular geometries
Mean absolute errors (MAE) on bond lengths of 32 molecules belonging to the G2 set (6-311G(d,p) basis set):
Compared to other hybrid functionals, B0KCIS gives similar results for molecular geometries.
Test of B0KCIS: Weak interactions
He …He van der Waals dimer (uncontracted aug-cc-pV5Z basis set):
B0KCISB0KCISB0KCIS
-0,005
0,000
0,005
0,010
0,015
0,020
2,25 2,5 2,75 3 3,25 3,5 3,75 4
d(He-He) (Å)
E (
eV)
PBE
PBE0
B0LYP
exact
-0,005
0,000
0,005
0,010
0,015
0,020
2,25 2,5 2,75 3 3,25 3,5 3,75 4
d(He-He) (Å)
E (
eV)
-0,005
0,000
0,005
0,010
0,015
0,020
2,25 2,5 2,75 3 3,25 3,5 3,75 4
d(He-He) (Å)
E (
eV)
PBE
PBE0
B0LYP
exact
Like B0LYP, B0KCIS can’t describe the dispersive interactions which can only be properly treated with an accurate exchange contribution.
Test of B0KCIS: Excitation energies (TDDFT)
Vertical excitation energies for the singlet states of H2CO (6-311G(d,p) basis set):
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
Experimental excitation energies (eV)
Com
pu
ted
exci
tati
on e
ner
gies
(eV
)
PBE
B0LYP
PBE0
B0KCIS
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
Experimental excitation energies (eV)
Com
pu
ted
exci
tati
on e
ner
gies
(eV
)
PBE
B0LYP
PBE0
B0KCIS
Like the other hybrid functionals, B0KCIS gives accurate excitation energies, even for high-lying excited states (Rydberg states).
Test of B0KCIS: Chemical reactivity
Symmetric SN2 reaction: Cl- + CH3Cl ClCH3 +Cl- (6-311G(d,p) basis set)
Reaction coordinate
Cl- + CH3Cl
ClCH3… Cl-
[Cl … CH3…Cl]-
Cl- … CH3Cl
Energy
E ClCH3 + Cl-
Reaction coordinate
Cl- + CH3Cl
ClCH3… Cl-
[Cl … CH3…Cl]-
Cl- … CH3Cl
Energy
E ClCH3 + Cl-
-5 -4 -3 -2 -1 0 1 2 3 4
MP2
PBE
B0LYP
PBE0
B0KCIS
E (kcal/mol)
2,5
-3,8
0,0
0,9
-5 -4 -3 -2 -1 0 1 2 3 4
MP2
PBE
B0LYP
PBE0
B0KCIS
E (kcal/mol)
-5 -4 -3 -2 -1 0 1 2 3 4
MP2
PBE
B0LYP
PBE0
B0KCIS
E (kcal/mol)
2,5
-3,8
0,0
0,9
3,6
B0KCIS gives a realistic energetic reaction barrier.
Test of B0KCIS: Magnetic properties
The isotropic hyperfine coupling constant of a nucleus n is connected to the spin density at the nucleus:
)(3
8n
snneen gg
ha r
direct contribution
vinyl radical ()
C C
H
H
H
C C
H
H
H
spin polarization
methyl radical ()
CHH
HCH
H
H
Exp.PBE
PBE0B0LYPB0KCIS
107,6 Gauss102,2110,2112,7107,6
Exp.PBE
PBE0B0LYPB0KCIS
107,6 Gauss102,2110,2112,7107,6
107,6 Gauss102,2110,2112,7107,6
Exp.PBE
PBE0B0LYPB0KCIS
28,4 Gauss24,629,231,724,9
Exp.PBE
PBE0B0LYPB0KCIS
28,4 Gauss24,629,231,724,9
28,4 Gauss24,629,231,724,9
B0KCIS gives very accurate hyperfine coupling constants for simple radicals without spin polarization.
Conclusion 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.
Prospects:
Construct more accurate DFT models by combining KCIS with some recently developed meta-GGA exchange functionals involving the laplacian of the electron density.
ENSCP/LECA/MSC, July 2002.
Modélisation des Systèmes Com
plex
es
References:[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).
1 2 3
4 5 6
7 8 9
10 11 12
