self-interaction error in density functional theory: a mean-field correction for molecules and large...

Chemical Physics 309 (2005) 67–76

www.elsevier.com/locate/chemphys

Self-interaction error in density functional theory: amean-field correction for molecules and large systems

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

a Ecole Nationale Sup�erieure de Chimie de Paris, Laboratoire d’Electrochimie et Chimie Analytique, UMR CNRS-ENSCP no. 7575,

11 rue P. et M. Curie, F-75231 Paris Cedex 05, Franceb Laboratoire de Chimie Physique Th�eorique, Universit�e Claude Bernard, Bat. 210, Lyon I and CNRS UMR 5182, 43 Boulevard du 11 Novembre 1918,

F-69622 Villeurbanne Cedex, France

Received 26 April 2004; accepted 5 May 2004

Available online 28 July 2004

Abstract

Corrections to the self-interaction error which is rooted in all standard exchange-correlation functionals in the density functional

theory (DFT) have become the object of an increasing interest. After an introduction reminding the origin of the self-interaction

error in the DFT formalism, and a brief review of the self-interaction free approximations, we present a simple, yet effective, self-

consistent method to correct this error. The model is based on an average density self-interaction correction (ADSIC), where both

exchange and Coulomb contributions are screened by a fraction of the electron density. The ansatz on which the method is built

makes it particularly appealing, due to its simplicity and its favorable scaling with the size of the system. We have tested the ADSIC

approach on one of the classical pathological problem for density functional theory: the direct estimation of the ionization potential

from orbital eigenvalues. A large set of different chemical systems, ranging from simple atoms to large fullerenes, has been con-

sidered as test cases. Our results show that the ADSIC approach provides good numerical values for all the molecular systems, the

agreement with the experimental values increasing, due to its average ansatz, with the size (conjugation) of the systems.

� 2004 Elsevier B.V. All rights reserved.

1. Introduction

Density functional theory (DFT), combining good

performances and low computational costs, has become

an invaluable tool for chemists and physicists in un-

derstanding the electronic structure of atoms, molecules

or solids and related properties [1]. In the framework ofthe Kohn–Sham (KS) approach to DFT, the quality of

the results is strictly related to the functional used to

evaluate the exchange and correlation energy, the only

contribution that needs to be approximated in the ex-

pression of the total KS energy [2]. The research for

improved approximations to this contribution has

therefore become one of the main streams in theoretical

DFT development (see for instance [3] and [4]). In such aquest for higher accuracy, some failures of the different

* Corresponding author. +33-4-7244-8427; fax: +33-4-7244-8004.

E-mail address: [email protected] (H. Chermette).

0301-0104/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.chemphys.2004.05.034

models have been considered as ‘‘pathological’’, that is

intrinsic to the DFT approach itself. Among others,

activation energies for SN2 and proton transfer reac-

tions, dissociation energies of two center-three electron

systems, ionization potentials and charge transfer sys-

tems, can be considered as representative examples [5–

8]. In many cases, these faults only depend on theapproximate nature of the used functionals, which leads

to the so-called self interaction (SI) error. This spurious

effect arises from the interaction of an electron with it-

self, and it is related to Coulomb energy of the Kohn–

Sham (KS) Hamiltonian which is not, in contrast to the

Hartree–Fock approach, totally cancelled by the ex-

change contribution [2,9]. One of the consequences is

that the eigenvalue of the highest occupied KS orbitaldoes not correspond to the ionization potential (IP), as

it should. The SI problem has been clearly identified in

the early DFT approaches and some solutions were

proposed by Fermi and Amaldi [10] or by Slater [11].

mail to: [email protected]

68 I. Ciofini et al. / Chemical Physics 309 (2005) 67–76

More complex is, instead, the handling of SI in the

framework of the KS approach. Some years ago Perdew

and Zunger [12] have introduced a self-interaction cor-

rection (SIC), which is yet the only approach giving a

correction for both exchange and correlation contribu-tions. In fact, while some of the most recent correlation

functionals are SI-free by construction, the handling of

the exchange part is still troublesome [13–15].

The more effective recipe for the elimination of the SI

error is the evaluation of an orbital dependent exchange-

correlation potential, which is, in practical implemen-

tations, computationally excessively demanding, so that

its use is so far restricted to small model systems orsimplified Hamiltonians [16,17]. For this reason, various

attempts have been made to develop simplified SIC

scheme, based mainly on mean field approximations,

since they couple a reduced computational effort with an

almost complete removing of the SI error. When they do

not involve SIC pseudopotentials [18–22], the most

successful approaches are those based on the optimized

effective potential (OEP) [23,24] and in particular thatdeveloped by Krieger, Li and Iafrate (KLI), which in-

volves an integral equation for the averaged SIC field

[25]. Even if such approach has been successfully applied

in a number of different cases, including molecular or

solid state problems [26–28], still it requires a significant

amount of computer resources in order to evaluate the

orbital-dependent Coulomb part [29].

We have recently implemented in a molecular code asimplified, yet effective, approach based on an average-

density SIC (ADSIC) approximation, which takes into

account the screening of both exchange-correlation and

Coulomb contribution through a subtraction from the

total density of a fraction proportional to 1=N , N being

the total number of electron [30]. This simple, self con-

sistent approach, first proposed by Legrand et al. [31]

and applied to a jellium model is technically not ex-pensive and, at the same time, still retains a number of

theoretical features, like the correct behavior for the

asymptotic potential and a variational formulation [31].

The first applications to molecular systems are quite

promising, but some potentialities of ADSIC are still

unexplored [30].

The aimof this paper is to investigate some of the limits

of such an approach in the field ofmolecular applications.To this end, the evaluation of the vertical ionization po-

tentials (IPs) for selected test cases, ranging from atoms to

large conjugate systems, has been chosen as a difficult

playground. Although the first IPs could be directly cal-

culated using the DFT extension of the Koopmans the-

orem [25,32], or more exactly of the Janak theorem [33],

the negative of the KS HOMO energy is too small with

respect to the experimental values [34], even if the shift,usually of several electronvolts, is rather constant [35–39].

It has been demonstrated that with the self interaction

correction included, the HOMO energies are much closer

to the first IPs, and that this effect is directly related to the

SI error [40]. In this respect, IPs belong to key properties

to validate any new SI corrections.

2. Theoretical background

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

exact energy can be written as [2]:

EKS qa; qb� �

¼ Ts uri

� �� þ J q½ � þ

ZqðrÞvðrÞdr

þ Exc qa; qb� �

; ð1Þ

where fuirgðr ¼ a or bÞ are the spin orbitals and qr are

the total density of spin r. The first term in Eq. (1) is the

kinetic energy of a system of non-interacting particle,

the second is the Coulomb interaction and the third isthe interaction energy between the electron density qðrÞand the external potential mðrÞ. These terms are all

known exactly, which is not the case for the last con-

tribution, the Exc term, containing all the remaining

contribution to both the kinetic energy and the electron-

electron interaction, and usually is approximated by

some functional form of the density, Eapproxxc . Even if the

choice of the functional form is completely arbitrary,some physical constraints which have to be respected by

the exact exchange-correlation functionals (potential)

are well known (see [41] for an interesting summary on

this point). In particular, the exact potential should

manifest a derivative discontinuity in the bulk region, as

a constant shift [42]:

vþxcðrÞ � v�xcðrÞ ¼oExc

oN

� �þ

v

�� oExc

oN

� ��

v

�

� eLUMOð � eHOMOÞ; ð2Þ

where vþxcðrÞ and v�xcðrÞ are the right and left derivatives

of the vxcðrÞ potential with respect to the number of

electrons, evaluated at N � d, d being a positive infini-

tesimal. At the same time vxcðrÞ should fall off as 1=r inthe asymptotic region [43]:

vxc ! � 1

r; r ! 1 ð3Þ

Because LDA and GGA XC functionals are not self-

interaction free, and they do not cancel the SI contained

in the Coulomb potential, they do not exhibit the correct

asymptotic decay [44]. Moreover, because the correla-

tion contribution of the XC energy and potential are

short ranged, describing only the so-called dynamic

correlation, only the exchange potential controls theasymptotic decay of the potential. Several attempts to

improve exchange functionals have borne on the one

hand on the exchange potential itself, in order to obtain

model potential with the correct asymptotic behavior,

[25,44–48], and on the other hand on self-interaction

corrections to the XC energy [12,16,17,31,45,49–55].

I. Ciofini et al. / Chemical Physics 309 (2005) 67–76 69

In a recent work, Della Sala and G€orling [56] have

demonstrated that in molecular systems, the asymptotic

region of the Kohn–Sham exchange potential is even

more complicated (than so far assumed), because the 1=rdecay has to be replaced by a C � 1=r, C being a con-stant, in directions belonging to nodal surfaces of the

HOMO of the molecule (this is also shown in [29]). The

C constant is directly related to the energy difference

between the HOMO and the highest occupied MO be-

low the HOMO possessing no nodal surface in the same

region. They have introduced a localized Hartree–Fock

method (LHF) allowing to approximately determine a

KS exchange potential by localizing the (non-local)Hartree–Fock potential [44]. Furthermore, they have

shown that the KLI procedure can be considered as an

approximation of the LHF method. These methods lead

to qualitatively correct SI free KS orbitals, whose ei-

genvalues can be compared to ionization energies or

electronic affinities, and be used for property calcula-

tions, e.g., TDDFT calculations of optical spectra.

The self interaction correction method of Perdew andZunger (PZ) effectively yields a discontinuity for the

highest occupied state when N is increased though in-

tegral occupancy [12]. This approach is de facto the

starting point for any SIC:

ESICxc qa; qb

� �¼ Eapprox

xc qa; qb� �

�Xr¼a;b

�XNr

i¼1

Eapproxxc qr

i ; 0� �

þ J qri

� �; ð4Þ

where the second sum runs on all the occupied spin

orbitals of density qri . The self-interaction error is thus

explicitly subtracted for each orbital from the standard

KS exchange-correlation energy. The correspondinglocal spin–orbital potential is then:

vSICxc;i qa; qb� �

¼ dEapproxxc

dqri

�Z

qri ðr0Þr� r0j j dr

0 � dExc½qri ; 0�

dqri

: ð5Þ

Since the exchange-correlation energy and potential are

dependent on each spin–orbital, this correction proce-

dure is computationally expensive. Moreover, SIC are

not invariant with respect to unitary transformations of

the orbitals. This leads to the necessity to employ orbital

localization techniques [27,57]. Other SIC techniques

may be found in [58–61].

A more convenient way is to average the resultingpotential over the different spin–orbitals in order to have

an orbital-independent potential. This is the philosophy

of the Krieger–Li–Iafrate approach [25]. Here the po-

tential, mKLIxc , is an explicit functional of the orbitals:

vKLIxc;i qa; qb

� �¼

Xr

Xi

qri

qrvSIC;rxc;i

��Z

qri vKLI;r

xc;i ðrÞ�

� vSIC;rxc;i ðrÞ�dr�; ð6Þ

where the vSIC;rxc;i is expressed by Eq. (5). The correction

thus obtained is, in practical applications, equivalent to

the PZ approximation, but has the advantage of coming

from a single local potential. In particular, atomic ex-

change correlation potentials, with the correct asymp-totic behavior have been obtained, giving very accurate

ionization potentials [62]. However, the SIC potential

needs to be determined through an iterative scheme, as it

appears in both sides of Eq. (6). The KLI scheme can be

further simplified by neglecting the orbital-dependent

part. The result is the so-called Slater potential [11,25]:

vSlaterxc qa; qb� �

¼Xr

Xi

qri

qrvSIC;rxc;i ðrÞ: ð7Þ

Of course, the Coulomb part, J ½qri �, still requires

some computational efforts.

Further simplification of this model is its adaptation

to the free electron gas [11], leading to the Dirac–Slater

exchange or the Xa exchange-correlation potentials [63],

which are not SI free, as it is known.An even more crude simplification of Eqs. (4) and (5)

can be obtained by subtracting a fraction 1=N from the

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

the density of the (N � 1) electrons seen by the spectator

electron. This approach rests on the original idea of

Fermi and Amaldi [10], but the inclusion of the ex-

change-correlation term makes it more suitable for

dealing with current approximate functionals. The en-ergy correction in such a case, takes a really simple form

[31]:

EADSICxc qa; qb

� �¼ Eapprox

xc qa; qb� �

� NJqN

h i

�Xr¼a;b

NrEapproxxc

qr

Nr; 0

� �; ð8Þ

whereas the potential is

vADSICxc qr½ � ¼ dEapprox

xc

dqr�Z

qrðr0Þ=Nr� r0j j dr0 � dExc½qr=N ; 0�

dqr:

ð9ÞThis simple model retains a number of properties,

and, in particular the potential (9) has the correct 1=rasymptotic behavior, can be variationally related to an

energy potential and, of course, give zero for one-elec-

tron potential [31]. In the same vein, Lundin and Eri-

ksson [58] corrected only the density of the localized

electrons of their system (f electrons).

3. Computational details

We have implemented the ADSIC approach in one of

the development versions of the Gaussian 03 code [64] for

the generalized gradient approximation (GGA) for the

exchange based on the Becke 88 functional [65]. Thiscontribution has been next coupled with the GGA


correlation of Lee, Yang and Parr [66], a functional by

construction SI-free, so to have the BLYP approach.

All the molecules have been optimized using the 6-

311G(d,p) basis set and energy evaluation have been done

with the same basis, eventually augmented with a diffusefunction on heavier atoms (6-311+G(d,p) basis) [67,68].

Some tests have been also carried out with the extended 6-

311++G(3df,3pd) basis set, while the smaller 6-31G basis

has been used for fullerenes. In this case the geometries of

the most stable isomers obtained with a 6-31G basis were

used. All fullerenes structures were optimized at the PM3

level. ForC78, all isomers respecting the isolated pentagon

rule have been computed to check the variation of theproperties with the isomer symmetry. The reliability of

such structures was verified by carrying out geometry

optimization on themost stable isomer at the BLYP level.

Since no significant difference was found, we expected no

difference in the DFT properties.

4. Results and discussion

As first test, we have computed the IPs, from the

highest orbital eigenvalue, for some atomic systems,

even if we do not expect that the ADSIC approach

performs particularly well on such systems, due to its

average hypothesis. For this reason, only 10 spherical

atoms, all belonging to the first and second group of the

periodic table (configuration [core] n1 or [core] n2) havebeen chosen. The results are reported in Fig. 1 as

function of the number of electrons, together with the

experimental values [69] and the corresponding uncor-

rected energies (labeled noSIC on the plot). The parallel

behavior of the computed values for the two sets, with

and without SIC, is evident from the plots of Fig. 1.

From one hand, the IPs calculated as the �eHOMO

without the SI correction exhibit a large error with re-

3 4 11 12 19 20 37 382

4

6

8

10

12 noSIC ADSIC exp

IP(e

V)

n. electrons

Fig. 1. Experimental (from [68]) and theoretical ionization potential

for some selected atoms, ranging between Li (no. electrons¼ 3) and Sr

(no. electrons¼ 38). The values have been computed as �eHOMO, using

the 6-311+G(d,p) basis set.

spect to the experimental data, ranging from )1.8 (Rb)

and )3.9 (Be) eV. On the other hand, the ADSIC cor-

rection shifts the orbital energies to lower values, so that

the corresponding IPs are overestimated, between +3.6

(Na) and +2.3 (Sr). Nevertheless, we have to notice thatwhile the eigenvalues for the highest occupied atomic

orbital are quite sensitive to the basis set for standard

functional, this is not the case for the ADSIC approach.

For instance, the differences found for the considered

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

311+G(2df,2p) basis is always less than 0.0003 eV.

As next test, we have chosen an ensemble of five

molecules, containing 14 and 15 main group elements.These molecules have been chosen since they contain a

double bond, N@X (X¼ Si, C), bonded to hydrogen or

methyl groups, and so the electronic properties are ex-

pected changing with the X atoms or the substitutes in a

regular way. Furthermore, one of such molecules,

namely trimethylsilanimine, is characterized by two IPs

corresponding to the ejection of an electron from the

non-bonding orbital (nY ) or from the bonding pX¼Y

orbital. The two corresponding IPs have the peculiarity

of being close so that they are difficulty assigned without

theoretical help [39].

All the calculations have been carried out at the, re-

spectively, BLYP optimized geometries, obtained with-

out the ADSIC correction, and the results are reported

in Table 1. In order to have a better picture of the

performances of the ADSIC approach, the IPs havebeen calculated either using the orbital energies and the

so-called DSCF method, where the total energy differ-

ence between neutral and charged molecules.

The values obtained using the uncorrected �eHOMO

(and �eHOMO�1) energies (no SIC approach) have a

mean average error (MAE) of 3.6 eV. As already found

for the atoms, these IPs are significantly underestimated.

In contrast a much smaller error is found for the DSCFmethod, with the MAE around 0.7 eV, as expected on

the basis of previous calculations [39]. Close deviations

have been found for the ADSIC approach, the MAE

being 0.9 eV.

In one case (silaethene) we have checked the basis set

dependency of the ADSIC orbital energies by using the

large 6-311++G(3df,3pd) set, and we did not find any

significant difference (DeHOMO < 0:02 eV).More in general, it appears from our results that the

ADSIC approach provides, for such systems, IPs with

the same accuracy than the DSCF method: when the

latter fails, the ADSIC method seems to be neither

better, not worse than DSCF with respect to the exper-

iments. In the case of the two states of the trimethylsi-

lanimine (2A0 and 2A00), they are found significantly too

separated than experiment suggests with both methods.This is probably due to a multideterminant nature of (or

at least of one of) these states, which is poorly described

by standard GGAs and, consequently, by the ADSIC

Table 1

Ionization potential (eV) for some unsaturated molecules, computed either using the DSCF approach (first column) or the HOMO eigenvalues

(second and third columns)

Molecule DSCF �eHOMO Exp.a

NoSIC ADSIC

Ethylene 10.5 6.4 12.1 10.5

Isobutene 8.9 5.1 9.5 9.5

Silaethene 8.7 6.2 10.5 9.0

Dimethylsilaethene 7.8 4.7 8.6 8.3

Trimethylsilanimine (2A0) 6.9 4.2 7.0 7.9

Trimethylsilanimine (2A00) 9.7 7.0 10.0 8.3

All the calculations have been carried out with the 6-311G(d,p) basis set.aAll experimental values are from [32].


approach (for multideterminantal DFT approaches see

[70]).

The results obtained for the atoms and the chosen

molecular systems are consistent with the average ap-

proximations: more electrons are involved 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) alwaysleads to improved results. At the same time, the ADSIC

seems to provide correct eigenvalues also for orbitals

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 two different sets. The first one is composed

by 7 aromatic systems, relatively large (5 and 6 memberrings) and containing not only carbon atoms (benzene

and fulvene), but also nitrogen (pyridine, pyrimidine

and pyrazine) and sulphur (furane and thiophene). For

these systems several IPs have experimentally observed

and theoretically computed, thus providing a further

verification for our approach [38,40]. The second set is

composed by three different fullerenes, namely C60, C70

and C78 (see Fig. 2), whose IPs have been recently ex-perimentally determined [71]. While the first two fulle-

renes are symmetric, the last one, C78, is not, so that a

different behavior of the ADSIC approach is expected.

The results obtained for the first set of aromatic

molecules are collected in Table 2. Without entering in a

tedious analysis of all the ionization potentials, we note

that the ADSIC approaches provides Mean Average

Errors (MAE) which are about one tenth of that ob-tained with the uncorrected BLYP functional (0.6 vs 4.0

eV, in average) and close to the results provides by the

DSCF approach (MAE¼ 0.5 eV, see [35]). Furthermore,

the data of Table 2 show that the ADSIC approach

treats on the same foot both r and p valence orbitals,

thus allowing accurate calculations also for high energy

transitions. It also interesting to compare our results for

pyridine, furane and thiophene, with those recently ob-tained by Chong and co-workers, using a statistical

average orbital potential (SAOP), starting from post-

HF wavefunctions [41]. For these three molecules the

SAOP deviation is 0.5 eV, which is also our results at the

BLYP/ADSIC level. The overall trends are summarized

in Fig. 3 where the BLYP/ADSIC results are reported

versus the experimental values for four molecules (py-

rimidine, furane, benzene and fulvene). From these

plots, a linear correlation between the two data sets is

quite apparent, the correlation factor (R) being alwaysgreater that 0.99.

Finally we did compute both the ionization potentials

and the electron affinities of three fullerenes in what we

suppose to be their most stable isomers. The computed

values are reported in Table 3 in comparison with the

available experimental data [62,72,73]. It is worthwhile

to note that, in some cases (e.g., C78), the experimental

values refer to a mixture of isomeric forms for a givenCn. Nevertheless the values for a given isomer are nor-

mally estimated within 0.1eV from the average mixture

value [74].

The computed SIC IP of C60 and C70 compare well to

the experimental ones (deviation of 0.5 eV), whereas the

less symmetric C78 deviates more strongly from the ex-

periment (0.7–1.0 eV). This cannot be related to the fact

that the experimental data refer to a Boltzman distri-bution of different isomers, since our calculations con-

sidering all stable isomers show a limited spread (Table

3). For what concerns the EA, the values computed

from the LUMO eigenvalue of the neutral systems

(�eLUMO in Table 3) strongly overestimate the EA both

when neglecting and including the SIC. Nevertheless

when electronic relaxation is allowed, that is when

computing the EAs as SOMO eigenvalue of the C�n

anionic systems (�eSOMO in Table 3), the ADSIC results

become in a 0.5 eV agreement with the experimental

data whereas the non-corrected values do strongly de-

viates (1.5–2.0 eV). Furthermore, our results are closed

to those obtained with parameterized semi-empirical

approaches [71] or with DFT/DSCF method [75]. In this

last case, the IP for the C60 computed by Martin and co-

workers using the B3LYP/6-31G(d) approach and theDSCF method is 7.14 eV, in excellent agreement with

our ADSIC value (7.15 eV) [75].

Fig. 2. Sketches of the considered molecules: (a) small organic compounds; (b) fullerenes: C60 (left), C70 (center), C78 (right).


In summary the ADSIC approach provides results as

accurate as the standard DSCF approach for large,

conjugate systems, using the corrected KS orbital ei-

genvalues. This result is even more interesting, since the

latter approach requires as many energy calculations as

cationic species are searched, besides the neutral ground

state calculation, whereas the ADSIC calculation pro-

vides the whole ionization energy spectrum within one-shot calculation. In this context, the ADSIC approach is

not only accurate but twice faster for single ionization

estimation. Furthermore, the accuracy of the ADSIC

model is close to that provided by more complex and

resource-demanding DFT methods [40].

SIC methods lead to orbital patterns where on the

one hand the occupied orbitals energies are shifted

downwards with respect to LDA or GGA orbitals, be-coming close to corresponding HF eigenvalues. On the

other hand the unoccupied orbitals energies are signifi-

cantly lower than corresponding HF eigenvalues, be-

cause the latter obey to a exchange potential which acts

differently on occupied and unoccupied orbitals, leading

to artificially overestimated HOMO–LUMO gaps.

Therefore, SIC methods deliver more realistic gaps, ly-

ing between (underestimated) LDA/GGA gaps and

(overestimated) HF ones. Consequently SIC methods

should lead to significantly better properties than those

calculated at the LDA, GGA, metaGGA KS level (on

the one hand) or HF level (on the other hand). This is

visible in properties like IPs, even with approximate SIC

models such as the ADSIC used in this work, and

should stem within TDDFT calculations (work in pro-

gress). At the KLI level, Patchkovskii et al. [27] haveproven that magnetic properties are significantly im-

proved.

Whereas SIC orbitals are significantly better than

standard KS orbitals, this is not necessarily the case of

the SIC energy. Indeed, the SIC energy functionals have

lost the non-dynamic correlation part which is present in

LDA/GGA exchange functionals. Therefore energetic

properties (reaction energies, barriers, etc.) may be(barriers) or may be not (reactions) improved, according

to the difference in electronic density delocalization be-

tween reactants, products and transition states: if this

difference is large, SIC are welcome and lead to im-

proved energy differences (see for instance Patchkovskii

and Ziegler in [76]). In most cases, SIC geometry opti-

mizations are rather poor, and deviate like, but even

more strongly than HF geometries [58]. For that reason,

Table 2

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

Molecule Elec. state NoSICa ADSIC expa

Benzene 1e1g 5.99 9.96 9.24

3e2g 8.10 11.48 11.49

1a2u 8.66 12.73 12.3

3e1u 10.03 13.52 13.8

1b1u 10.62 14.15 14.7

2b2u 11.03 14.39 15.4

3a1g 12.59 16.17 18.85

2e2g 14.53 18.02 19.2

2e1u 18.08 21.67 22.8

2a1g 20.77 24.59 25.9

Fulvene 1a2 4.65 8.91 8.55

2b1 6.14 10.05 9.54

7b2 8.46 11.55 12.1

11a1 8.78 12.10 12.8

10a1 8.86 12.78 13.6

6b2 9.05 12.85 14

Pyridine 1a2 5.66 9.39 9.6

11a1 6.37 10.46 9.75

2b1 6.95 11.06 10.81

7b2 8.68 12.14 12.61

1b1 9.52 13.21 13.1

10a1 9.64 13.72 13.8

6b2 10.31 13.89 14.5

Pyrimidine 7b2 5.70 9.52 9.73

2b1 6.95 10.69 10.5

11a1 7.07 11.29 11.2

1a2 7.75 12.01 11.5

1b1 9.87 13.50 13.9

10a1 10.12 13.84 14.5

6b2 10.32 14.67 14.5

Pyrazine 6ag 5.63 9.36 9.4

1b1g 6.80 10.92 10.2

5b1u 7.07 11.02 11.4

1b2g 8.05 12.31 11.7

3b3g 9.38 12.92 13.3

1b3u 10.31 14.31 14

4b2u 10.62 14.65 15

4b1u 11.89 15.31 16.2

Furane 1a2 5.36 9.85 8.89

2b1 6.52 11.05 10.25

9a1 8.66 12.91 13

8a1 9.44 13.53 13.8

6b2 9.81 13.71 14.4

5b2 10.50 14.53 15.25

1b1 10.67 15.20 15.6

Thiophene 1a2 5.56 9.75 8.87

3b1 5.96 10.25 9.52

11a1 8.21 12.33 12.1

2b1 8.95 13.10 12.7

7b2 9.27 13.23 13.3

6b2 9.42 13.32 14.3

MAEb 4.0 0.5

All the calculations have been carried out using the 6-311G(d,p) basis set.aAll the BLYP and experimental values are from [31].bMean average error on all the IPs.


most papers describing SIC properties use molecular

geometries optimized with standard GGA, metaGGA

or hybrid XC functionals. This has been the case in the

present work. The poorness of geometries obtained with

model exchange potentials has early been also under-

lined in many papers [77,78].

Table 3

Computed and experimental ionization potentials (IP, eV) and electron affinities (EA, eV) for the three considered fullerenes

IP

E (kcal/mol) �eHOMO Exp.a

NoSIC ADSIC

C60Ih 0 5.37 7.18 7.57

C70D5h 0 5.3 6.97 7.36

C78C2v 0 4.77 6.33 7.26

D3h 3.2 (3.8) 4.79 6.34

C2v 7.6 (5.6) 5.01 6.57

D3 9.7 (6.3) 4.74 6.30

D3h 26.8 (17.5) 5.08 6.63

EA

�eLUMO �eSOMO Exp.b

NoSIC ADSIC NoSIC ADSIC

C60Ih 3.48 5.28 0.29 2.09 2.65

C70D5h 3.46 5.11 0.45 2.10 2.73

C78C2v 3.89 5.43 0.93 2.49 3.1

D3h 4.05 5.61 1.23 2.66

C2v 3.79 5.34 0.84 2.39

D3 3.84 5.41 0.96 2.51

D3h 3.41 4.95 0.53 2.07

All the calculations have been carried out using the 6-31G basis set.aRef. [65].bRef. [66].

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

)

IP exp (eV)

Fig. 3. Correlation between experimental and computed BLYP/AD-

SIC ionisation potentials for four different aromatic molecules.


The reason of that disappointing feature is that the

removal of the SI energy removes the non-dynamical (or

left-right, or quasi-degeneracy) correlation energy in-

cluded in the exchange energy functional, whereas nosignificant change is obtained for the (short ranged)

dynamical correlation energy, described by the correla-

tion energy functional. Whereas it is now widely ad-

mitted than the long ranged non-dynamical correlation

energy is described by the exchange part of the ex-

change-correlation energy functional, as a consequence

of the structure of the (Fermi) exchange hole [79–84], it

has been argued more recently that it is the SI error

which mimics long range (left-right) non-dynamic cor-

relation [85,86].

5. Conclusions

In this paper, we have presented the validation of a

simple approach to correct the self interaction error

present in the common approximate exchange-correla-

tion functionals used in density functional theory. This

model rests on an average density self-interaction cor-

rection (ADSIC), so that the main advantages of the

method with respect to other corrections are its sim-plicity and its favorable scaling with the size of the

system. At the same time, it retains a number of theo-

retical features, such as the correct asymptotic behavior.

We have tested the ADSIC approach on the direct es-

timation of the ionization potential from orbital eigen-

values. On this difficult playground, the ADSIC

approach provides good numerical values for all the

molecular systems, while large deviations, due to itsaverage ansatz, are obtained for atomic or strongly lo-

calized molecular systems.

6. Note added in proofs

Eq. (5) and (9) of this paper were wrongly reported in

Ref. [30] (one missing term), but correctly programmed.The correct form is given here.


Acknowledgements

Authors thank Thomas Heine (University of Dres-

den) help concerning fullerene structures. 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 ‘‘Integrative Computational Chemistry’’ (action n.

D26/0013/02) and a support from the CNRS GdR

‘‘DFT’’. The CINES is acknowledged for a grant of time

(project cpt2130).

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