Chemical Physics Letters 380 (2003) 12–20

www.elsevier.com/locate/cplett

A mean-field self-interaction correctionin density functional theory: implementation

and validation for molecules

Ilaria Ciofini a, Henry Chermette b, Carlo Adamo a,*

a Ecole Nationale Sup�eerieure de Chimie de Paris, Laboratoire d�Electrochimie et Chimie Analytique,

UMR CNRS-ENSCP n� 7575, 11 rue P. et M. Curie, F-75231 Paris Cedex 05, Franceb Laboratoire de Chimie Physique Th�eeorique Bat. 210, Universit�ee Claude Bernard Lyon I,

CNRS UMR 5182, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France

Received 8 April 2003; in final form 27 August 2003

Published online: 23 September 2003

Abstract

The implementation and the validation of a simple, self-consistent method to correct the self-interaction error in

density functional theory approaches is presented. This model is based on an average density self-interaction correction.

The main advantage of the method rests on its simplicity and favorable scaling with the size of the system. We have

tested this method on one of the classical pathological problem for density functional theory: the direct estimation of

the ionization potential from orbital eigenvalues. The proposed method provides good values for delocalized molecular

systems, while large deviations, are obtained for atomic or strongly localized molecular systems.

� 2003 Elsevier B.V. All rights reserved.

1. Introduction

Density functional theory (DFT), combining

good performances and low computational costs,

is nowadays one of the most powerful and reliable

tools of quantum chemistry for the computation ofthe electronic structure of atoms, molecules or

solids [1]. In the application of this theory within

the Kohn–Sham formalism, the total electronic

energy can be exactly expressed as a functional of

* Corresponding author. Fax: +33-1-4427-6750.

E-mail address: adamo@ext.jussieu.fr (C. Adamo).

0009-2614/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/j.cplett.2003.08.084

the electron density, the only contribution that

needs to be approximated being the exchange-

correlation term [2]. The quest for improved

approximations to this contribution has then be-

comes one of the main streams in theoretical DFT

development in order to obtain accurate molecularproperties (see for instance [3,4]). While modern

exchange-correlation functionals provide numeri-

cal performances close to the so-called �chemical

accuracy� for a wide class of molecular properties,

they still fail in a number of pathological cases.

Activation energies for some reactions, like SN2

and proton/hydrogen transfer, dissociation ener-

gies of two center–three electron systems, ioniza-

ed.

I. Ciofini et al. / Chemical Physics Letters 380 (2003) 12–20 13

tion potentials, charge transfer systems, are just

few examples [5–8]. Most of these problems arise

from the approximate nature of commonly used

functionals, which include the spurious interaction

of an electron with itself, the so-called self-inter-

action (SI) error. The reason for this effect is thatthe SI contained in the classical Coulomb energy

of the Kohn–Sham (KS) Hamiltonian is only

partially cancelled by the exchange-correlation

contribution [2,9]. Approximate functionals have

therefore the wrong asymptotic behavior for finite

systems: they decrease exponentially rather than

�1=r for neutral systems. One of the consequences

is that the eigenvalue of the highest occupied KSorbital does not correspond to the ionization po-

tential (IP), as it should. While some efforts has

been done in determining SI-free correlations

functionals [10,11], yet there is no exchange func-

tional that is completely SI free, if one excepts the

self-interaction correction (SIC) formalism of

Perdew and Zunger [12]. This approach uses an

orbital dependent exchange-correlation potential,which is, in practical implementations, computa-

tionally too demanding, so that its use as so far

restricted to small model systems or simplified

Hamiltonians [13,14]. For this reason, various at-

tempts have been made to develop simplified SIC

scheme, based mainly on a mean field approxi-

mation. The most successful approaches are those

based on the optimized effective potential (OEP)[15,16] and in particular that developed by Krie-

ger, Li and Iafrate, which involves an integral

equation for the averaged SIC field [17]. Even

if such approach has been successfully applied in

a number of different cases (see for instances

[18,19]), still it require a large amount of computer

resources in order to evaluate the orbital-depen-

dent Coulomb part.In this context, approximated mean field ap-

proaches could be a promising way out, coupling a

reduced computational effort with an almost

complete removing of the SI error. We have de-

veloped a simplified, yet effective, a posteriori

correction based on the Slater�s transition state

technique, which provides reliable results in a

number of selected cases [7,20]. More recently, anaverage-density SIC (ADSIC) approach has been

proposed by Legrand, Suraud and Reinhard,

based on the screening of both exchange-correla-

tion and Coulomb contribution through a sub-

traction from the total density of a fraction

proportional to 1=N , N being the total number of

electron [21]. This simple, self-consistent approach

is technically not expensive and, at the same time,still retains a number of theoretical features, like

the correct behavior for the asymptotic potential

and a variational formulation [21]. Despite its

promising features, the ADSIC approach has been

applied only to the study the electronic features of

some sodium clusters, leaving all its potentialities

in molecular applications unexplored [21].

The aim of this Letter is to investigate the limitsof the applicability of such an approach in the field

of molecular applications. To this end the evalu-

ation of the vertical ionization potentials (IPs) for

some test cases, ranging from atoms to relatively

large conjugate systems, has been chosen as a

difficult playground. While in principle, the first

IPs could be directly calculated using the DFT

extension of the Koopmans theorem [22], the KSHOMO energies provided by modern approximate

functionals exhibit a shift of several electron-volts

(eV) [23–25], due to the incorrect asymptotic be-

havior of the exchange-correlation potential (for a

recent discussion on this point see [26]). This effect

is directly related to the SI error [26]. In this

respect, IPs belong to key properties to validate

any new SI corrections.

2. Computational details

In the Kohn–Sham (KS) approach to DFT the

total exact energy can be written as [2]

EKS½qa; qb� ¼ Ts½fuirg� þ J ½q� þZ

qðrÞvðrÞdr

þ Exc½qa; qb�; ð1Þ

where fuirg (r ¼ a or b) are the spin orbitals. The

first term in Eq. (1) is the kinetic energy of a sys-

tem of non-interacting particle, the second is

Coulomb interaction and the third is the interac-

tion energy between the electron density qðrÞ andthe external potential mðrÞ. These terms are allknown exactly, which is not the case for the

last contribution, the Exc term, containing all the

14 I. Ciofini et al. / Chemical Physics Letters 380 (2003) 12–20

remaining contribution to both the kinetic energy

and the electron–electron interaction, and usually

is approximated by some functional form, Eapproxxc .

The starting point for any SIC is the Perdew–

Zunger approach [12]:

ESICxc ½qa;qb� ¼Eapprox

xc ½qa; qb�

�Xr¼a;b

XNr

i¼1

Eapproxxc ½qr

i ; 0��

þ J ½qri ��;

ð2Þ

where the second sum runs on all the occupied spin

orbitals. The corresponding local spin–orbital

potential is then

vixc½qr; qb� ¼ oEapproxxc

oqri

�Z

qri ðr0Þ

jr � r0j dr0: ð3Þ

Since the exchange-correlation energy and

potential are dependent on each spin–orbital, thiscorrection procedure is expensive. A more conve-

nient way is to average the resulting potential over

the different spin–orbitals in order to have an or-

bital-independent potential. Different approaches

have been proposed, as the KLI [17], the Slater [27]

or the globally averaged method (GAM) [28], the

average being done on the local densities in the first

two cases or on the orbital occupation in the lastone. An even more crude approach can be ob-

tained by subtracting a fraction 1=N from the total

density, so that qðN � 1Þ=N can be considered

as the density of the ðN � 1Þ electrons seen by

the spectator electron. This approach rests on

the original idea of Fermi and Amaldi [29], but the

inclusion of the exchange-correlation term makes

it more suitable for dealing with current approxi-mate functionals. The energy correction in such a

case, takes a really simple form [21]:

EADSICxc ½qa; qb� ¼ Eapprox

xc ½qa; qb� � NJqN

h i

�Xr¼a;b

NrEapproxxc

qr

Nr; 0

� �; ð4Þ

whereas the potential is

vADSICxc ½qr� ¼ oEapproxxc

oqr

�Z

qrðr0Þ=Njr � r0j dr0: ð5Þ

This model, even in its crudeness, still retains a

number of properties, and, in particular the poten-

tial (5) has the correct 1=r asymptotic behavior, can

be variationally related to an energy potential and,

of course, give zero for one-electron potential [21].

We have implemented the ADSIC approach in

one of the development versions of the GAUSSIANAUSSIAN

03 code [30], and for three different exchangefunctionals: the local approach (LDA) i.e., Slater

exchange coupled with the Vosko, Wilk and

Nusair correlation and two generalized gradient

approximations (GGA), namely Becke88 and

Perdew, Burke and Ernzerhof. These exchange

functionals were coupled with the GGA correla-

tion of Lee, Yang and Parr, and the one of Per-

dew, Burke and Ernzerhof respectively. The LYPcorrelation functional is by construction SI-free,

whereas, in principle, both the VWN and the PBE

correlations are not. Nevertheless SI error plays a

minor role in correlation [7]. The resulting func-

tional we use will be referred to as LDA, BLYP

and PBE. Work is in progress to extend this

approach to hybrid functionals.

All the molecules have been optimized using the6-311G(d,p) basis set and energy evaluation have

been done with the same basis, eventually aug-

mented with a diffuse function on heavier atoms

(6-311+G(d,p) basis). Some tests have been also

carried out with the extended 6-311++G(3df,3pd)

basis set. See [31] for all the references on func-

tionals and basis sets.

3. Results and discussion

As first test, we have considered the IPs of some

atoms through the periodic table, even if we do not

expect that the ADSIC approach performs par-

ticularly well on such systems, due to its average

ansatz. For this reason, only 10 spherical atoms,almost all belonging to the first and second group

of the periodic table (configuration [core] ns1 or

[core] ns2) have been chosen, and the values have

been computed only with the BLYP functional.

The results are collected in Table 1, together with

the experimental values [32]. In the same table are

reported the IPs calculated with the DSCF ap-

proach, that is as difference between the neutraland the cationic species. As already known, the IPs

calculated as the �eHOMO without the SI correction

Table 1

Ionization potentials (eV) for some selected atoms, computed using the BLYP functional and the 6-311+G(d,p) basis set

Atoms No. electrons DSCF �eHOMO Experimentala

NoSIC ADSIC

Li 3 5.52 3.03 8.13 5.39

Be 4 8.98 5.47 11.75 9.32

Na 11 5.35 2.90 8.73 5.14

Mg 12 7.63 4.57 11.12 7.65

K 19 4.43 2.41 7.23 4.34

Ca 20 6.08 3.60 8.93 6.11

Rbb 37 4.41 2.41 6.28 4.18

Srb 38 5.94 3.63 8.02 5.70

aRef. [32].b LANL2DZ basis set.

I. Ciofini et al. / Chemical Physics Letters 380 (2003) 12–20 15

exhibit a large error with respect to the experi-mental data, ranging from )1.8 (Rb) to )3.9 (Be)

eV. These values well underline the wrong as-

ymptotic behavior, leading to underestimate IPs.

In contrast the ADSIC correction shifts the orbital

energies to lower values, so that the corresponding

IPs are overestimated, between +3.6 (Na) and +2.3

(Sr). It is also interesting to note that while the

eigenvalues for the highest occupied atomic orbitalare quite sensitive to the basis set for standard

functional, this is not the case for the ADSIC

approach. In fact the difference found for the Na

atom in going from the 6-311+G(d,p) to the large

6-311+G(2df,2p) basis is less than 0.0002 eV.

As next test, we have chosen an ensemble of

seven molecules, containing 14 and 15 main group

elements. These molecules contains a double bond,Y@X (X¼ Si, C; Y¼N, P), bonded to hydrogen

or methyl groups, so that the electronic properties

are expected in changing the X and Y atoms or the

substituents. Furthermore, some of such molecules

(trimethylsilanimine, phosphaethene, 1-phosphap-

ropene) are generally characterized by two IPs

corresponding to the ejection of an electron from

the non-bonding orbital (nY) or from the bondingpX@Y orbital. The two corresponding IPs have the

peculiarity of being close so that they are difficulty

assigned without theoretical help [25].

At this point, it is worthwhile to add some

comments on the use of KS orbital eigenvalues to

calculate IPs in the framework of DFT, since their

physical meaning was subject of extensive debate

[26,33]. In fact, while the direct relationship be-

tween the HOMO energy and the first IP has beendemonstrated several years ago (see for instance

[34]), only recently Perdew and co-corkers showed

that the KS orbital energies approximate very well

the relaxed IPs, establishing exact relations be-

tween the two quantities [26]. Following this phi-

losophy, we have computed not only the first, but

also the higher IPs using the orbital eigenvalues

corrected by the ADSIC method.All the calculations have been carried out at the

respectively (i.e., LDA, BLYP or PBE) optimized

geometries, obtained without the ADSIC correc-

tion. The results obtained with different function-

als (LDA, BLYP and PBE) and different

approaches (DSCF and ADSIC) are reported in

Table 2. All the values can be grouped following

the calculation methods for the IPs, that is DSCF,no SIC and ADSIC. The values obtained using the

uncorrected �eHOMO (and �eHOMO-1) energies (no

SIC approach) have a mean average error (MAE)

around 3.0 eV, ranging between the LDA

(MAE¼ 3.0 eV) and the BLYP (MAE¼ 3.6 eV)

deviations. As expected these IPs are significantly

underestimated. In contrast a much smaller error

is found for the DSCF method, with all the MAEsaround 0.7/0.6 eV, as expected on the basis of

previous calculations [25]. Close deviations have

been found for the ADSIC approach, the MAE

being between 0.9 eV (LDA and BLYP) and 0.7 eV

(PBE).

In one case (phosphapropene) we have verified

the basis set dependency of the ADSIC orbital

energies by using the large 6-311++G(3df,3pd) set,

ble

2

nizationpotential(eV)forsomeunsaturatedmolecules,computedeither

usingtheDSCFapproach

ortheHOMO

eigenvalues.Allthecalculationshavebeencarried

twiththe6-311G(d,p)basisset

Molecule

DSCF

�e H

OMO

Experim

entala

NoSIC

ADSIC

LDA

aBLYP

PBE

LDA

BLYP

PBE

LDA

BLYP

PBE

Ethylene

11.5

10.5

10.6

6.9

6.4

6.6

12.2

12.1

11.6

10.5

Isobutene

9.8

8.9

8.7

5.8

5.1

5.7

9.6

9.5

8.9

9.5

Silaethene

9.7

8.7

8.9

5.8

6.2

5.6

10.6

10.5

10.0

9.0

Dim

ethylsilaethene

9.0

7.8

7.9

5.0

4.7

4.8

8.7

8.6

8.1

8.3

Trimethylsilanim

ine(2A

0 )8.6

6.9

7.0

4.9

4.2

4.4

7.6

7.0

7.0

7.9

Trimethylsilanim

ine(2A

00 )9.1

9.7

9.9

7.9

7.0

7.2

10.3

10.0

9.6

8.3

Phosphaethene(2A

0 )11.1

10.3

10.4

7.0

6.7

6.8

8.0

11.8

11.2

10.7

Phosphaethene(2A

00 )11.0

10.1

10.3

6.9

6.6

6.8

8.4

12.0

11.5

10.3

1-Phosphapropene(2A

00 )10.1

9.3

9.5

6.4

6.1

6.3

10.7

10.6

10.1

9.8

1-Phosphapropene(2A

0 )10.5

9.9

10.0

6.7

6.5

6.6

10.8

10.8

10.3

10.4

aAllexperim

entalvalues

are

from

[25].

16 I. Ciofini et al. / Chemical Physics Letters 380 (2003) 12–20

Ta

Io ou

and we did not find any significant difference

(DeHOMO < 0:02 eV).

Two main features appear from our results. The

first is related to the IPs of methyl substituted

systems, which are better reproduced than those

non-methylated. For instance the PBE error on theethylene is 1.1 eV, whereas it is 0.6 eV for isobu-

tene, and the same behavior is found for the other

couple, silaethene and dimethylsilaethene (1.0 and

0.2 eV, respectively). This substitution effect is

clearly not present or significantly damped in the

considered GGA functionals and also in some

hybrid HF/KS approaches [25].

The second point concerns the order of the IPsfor three molecules: trimethylsilanimine, 1-phos-

phapropene, phosphaethene. For the first two

molecules the two IPs, corresponding to the 2A0

and 2A00 electronic states, are computed in the

correct order at ADSIC level, for all the consid-

ered functionals. In particular the 2A0 is more

stable than the 2A00 for trimethylsilanimine, while

the reverse is true for 1-phosphapropene. In thislast case the functional could play a not negligible

role, the two IPs being almost degenerate at LDA

level (10.7 vs 10.8 eV). Nevertheless, these results

suggest that the ADSIC approach provides not

only reliable �eHOMO (corresponding to the first

IP) but also �eHOMO-1 (second IP).

The situation is more complex for the third

molecule, namely the phosphaethene. The first twobands measured at 10.3 and 10.7 eV have been

assigned to the ionization of the pC@P and nP or-

bitals, respectively (states 2A00 and 2A0) [35]. Using

the optimized BLYP geometries these transition

are calculated at 6.6 and 6.7 eV from the BLYP,

corresponding to the 2A00 and 2A0 state, respec-

tively. When the ADSIC correction is applied, the

two transitions are calculated at 12.0 (2A00) and11.8 (2A0) eV, so that the reverse order is found.

Actually our calculations have been carried out

using BLYP geometries with the assumption that

SI does not affect in a significant way the structural

parameters. This could be not the case for the

phosphaethene, where the small gap between the

two IPs (0.4 eV) and their sensitivity to the geo-

metrical parameters [25] could explain the appar-ently wrong behavior obtained with the ADSIC

approach. To verify this point, we have calculated

I. Ciofini et al. / Chemical Physics Letters 380 (2003) 12–20 17

the HOMO and HOMO-1 orbital energies as a

function of the P–C distance, keeping frozen all

the others parameters. The results reported in

Fig. 1 show that for distances greater than 1.73 �AAthe correct order of the HOMO and HOMO-1

energies is restored.The results obtained for the atoms and the

chosen molecular systems are consistent with the

average approximations: more electrons are in-

volved and more delocalized the system is, more

the results are accurate. So adding methyl groups

to unsaturated carbons or changing these latter

with heavier atoms (Si or P) always leads to im-

proved results. At the same time, the ADSICseems to provide correct eigenvalues also for or-

bitals lower in energy than the HOMO.

These considerations induced us to investigate

some aromatic systems for which, at least in

principle, the ADSIC model should work at its

best. To this end, we have chosen 7 aromatic sys-

tems (5 and 6 member rings) containing not only

carbon atoms (benzene and fulvene), but also ni-trogen (pyridine, pyrimidine and pyrazine) and

sulfur (furane and thiophene). For these systems

different IPs have experimentally observed and

theoretically computed, thus providing a further

1.5 1.6 16

7

8

9

10

11

12

13

-ε(e

V)

Fig. 1. Plot of the calculated HOMO energies of pho

verification for our approach [24,26]. The results

obtained with different functionals and the DSCFand ADSIC approaches are collected in Table 3.

Without entering in a tedious analysis of all the

ionization potentials, we note that the ADSIC

approaches provides MAEs which are of the samequality of the DSCF method and one tenth of that

obtained with the uncorrected BLYP functional

from orbital eigenvalues (0.6 vs 4.0 eV, in average).

Furthermore all the considered functionals, LDA,

BLYP and PBE, give close deviations (MAE¼ 0.5,

0.6 and 0.7 eV, respectively). The data of Table 3

show that the ADSIC approach treats on the same

foot both valence and core orbitals, thus allowingaccurate calculations also for high energy transi-

tions. It also interesting to compare our results for

pyridine, furane and thiophene, with those recently

obtained by Chong and co-workers, using a sta-

tistical average orbital potential (SAOP), starting

from post-HF wavefunctions [26]. For these three

molecules the SAOP deviation is 0.5 eV, which is

also our best results at the BLYP/ADSIC level.The overall trends are summarized in Fig. 2

where the BLYP/ADSIC results are reported ver-

sus the experimental values for four molecules

(pyrimidine, furane, benzene and fulvene). From

.7 1.8 1.9

BLY

P b

ond

leng

th

A" BLYPA' BLYPA" BsicLYP A' BsicLYP

sphaethene as function of the PC bond length.

Table 3

Computed and experimental vertical ionization potentials (eV) for conjugated planar molecules

Molecule Electronic

state

PW86–PW91/

DSCFa

BLYP/

NoSIC

BLYP/

ADSIC

LDA/

ADSIC

PBE/

ADSIC

Experimentala

Benzene 1e1g 9.25 5.99 9.96 10.19 9.40 9.24

3e2g 11.36 8.10 11.48 11.45 10.86 11.49

1a2u 12.04 8.66 12.73 13.00 12.19 12.3

3e1u 13.29 10.03 13.52 13.54 12.94 13.8

1b1u 14.12 10.62 14.15 14.00 13.60 14.7

2b2u 14.27 11.03 14.39 14.57 13.89 15.4

3a1g 15.86 12.59 16.17 16.29 15.65 18.85

2e2g 17.91 14.53 18.02 18.02 17.58 19.2

2e1u 21.59 18.08 21.67 21.63 21.31 22.8

2a1g 24.50 20.77 24.59 24.63 24.27 25.9

Fulvene 1a2 8.31 4.65 8.91 9.12 8.33 8.55

2b1 9.37 6.14 10.05 10.30 9.49 9.54

7b2 11.39 8.46 11.55 11.55 10.96 12.1

11a1 12.08 8.78 12.10 12.11 11.51 12.8

10a1 12.66 8.86 12.78 12.84 12.24 13.6

6b2 12.88 9.05 12.85 12.98 12.25 14

Pyridine 1a2 9.69 5.66 9.39 9.33 8.71 9.6

11a1 9.42 6.37 10.46 10.70 9.88 9.75

2b1 10.35 6.95 11.06 11.30 10.48 10.81

7b2 12.04 8.68 12.14 12.11 11.52 12.61

1b1 13.08 9.52 13.21 13.21 12.62 13.1

10a1 13.17 9.64 13.72 13.91 13.16 13.8

6b2 13.71 10.31 13.89 14.01 13.29 14.5

Pyrimidine 7b2 9.33 5.70 9.52 9.55 8.87 9.73

2b1 10.54 6.95 10.69 10.67 10.09 10.5

11a1 10.49 7.07 11.29 11.54 10.70 11.2

1a2 11.26 7.75 12.01 12.27 11.41 11.5

1b1 13.97 9.87 13.50 13.48 12.90 13.9

10a1 13.43 10.12 13.84 13.87 13.22 14.5

6b2 13.84 10.32 14.67 14.88 14.09 14.5

Pyrazine 6ag 9.14 5.63 9.36 9.36 8.71 9.4

1b1g 10.18 6.80 10.92 10.95 10.33 10.2

5b1u 10.80 7.07 11.02 11.26 10.42 11.4

1b2g 11.64 8.05 12.31 12.58 11.72 11.7

3b3g 12.89 9.38 12.92 12.88 12.29 13.3

1b3u 13.94 10.31 14.31 14.32 13.69 14

4b2u 14.10 10.62 14.65 14.95 14.07 15

4b1u 15.40 11.89 15.31 14.98 14.72 16.2

Furane 1a2 8.87 5.36 9.85 10.01 9.25 8.89

2b1 10.21 6.52 11.05 11.22 10.45 10.25

9a1 12.55 8.66 12.91 12.95 12.31 13

8a1 13.20 9.44 13.53 13.56 13.56 13.8

6b2 13.38 9.81 13.71 13.65 13.65 14.4

5b2 14.46 10.50 14.53 14.47 14.47 15.25

1b1 14.79 10.67 15.20 15.42 15.42 15.6

Thiophene 1a2 8.87 5.56 9.75 9.96 9.20 8.87

3b1 9.25 5.96 10.25 10.43 9.72 9.52

11a1 11.70 8.21 12.33 12.42 11.82 12.1

2b1 12.48 8.95 13.10 13.11 12.57 12.7

18 I. Ciofini et al. / Chemical Physics Letters 380 (2003) 12–20

8 9 10 11 12 13 14 15 168

9

10

11

12

13

14

15

16

fulvene pyrimidine benzene furane

IP c

alc

(eV

)

Fig. 2. Correlation between experimental and computed BLYP/ADSIC ionization potentials for four different aromatic molecules.

Table 3 (continued )

Molecule Electronic

state

PW86–PW91/

DSCFa

BLYP/

NoSIC

BLYP/

ADSIC

LDA/

ADSIC

PBE/

ADSIC

Experimentala

7b2 12.77 9.27 13.23 13.30 12.70 13.3

6b2 13.42 9.42 13.32 13.57 12.83 14.3

MAEb 0.5 4.0 0.5 0.6 0.7

All the calculations have been carried out using the 6-311G(d,p) basis set.aAll the PW86-PW91 and experimental values are from [24].bMean average error on all the IPs.

I. Ciofini et al. / Chemical Physics Letters 380 (2003) 12–20 19

these plots, a linear correlation between the twodata sets is quite apparent, the correlation factor

(R) being always greater that 0.99.

In summary the ADSIC approach provides re-

sults as accurate as the standardDSCFapproach for

large, conjugate systems, using the corrected KS

orbital eigenvalues. This result is even more inter-

esting, since the latter approach require as many

energy calculations as cationic species are searched,besides the neutral ground state calculation,

whereas the ADSIC calculation provides the whole

ionization energy spectrum within one-shot calcu-

lation. In this context, the ADSIC approach is not

only much more accurate, but already twice faster

for a single ionization estimation. Furthermore, the

accuracy of the ADSIC model is close to that pro-

vided by more complex and resource-demandingDFT methods [26].

4. Conclusions

In this Letter, we have presented the validation

of a simple approach to correct the self-interaction

error present in the common approximate ex-

change-correlation functionals used in density

functional theory. This model rests on an average

density self-interaction correction (ADSIC), so

that the main advantages of the method with re-spect to other corrections are its simplicity and its

favorable scaling with the size of the system. At the

same time, it retains a number of theoretical fea-

tures, such as the correct asymptotic behavior. We

have tested the ADSIC approach on the direct es-

timation of the ionization potential from orbital

eigenvalues. On this difficult playground, the AD-

SIC approach provides good numerical values forall the molecular systems, while large deviations,

20 I. Ciofini et al. / Chemical Physics Letters 380 (2003) 12–20

due to its average ansatz, are obtained for atomic

or strongly localized molecular systems.

Acknowledgements

Authors thank E. Suraud for a preprint of [21].

I.C. and C.A. also thank CNRS for a financial

support from the ACI �Jeune Equipe 2002� project.This work has also been carried out within the

framework of the Cost Action D26 �IntegrativeComputational Chemistry� (action no. D26/0013/

02) and of the GdR DFT.

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