Electron Pair Localization Function (EPLF) for Density Functional Theory and ab Initio Wave Function-Based Methods: A New Tool for Chemical Interpretation

Abstract

We present a modified definition of the Electron Pair Localization Function (EPLF), initially defined within the framework of quantum Monte Carlo approaches [ Scemama, A. ; Caffarel, M. ; Chaquin, P. J. Chem. Phys. 2004, 121, 1725 ] to be used in Density Functional Theories (DFT) and ab initio wave-function-based methods. This modified version of the EPLF—while keeping the same physical and chemical contents—is built to be analytically computable with standard wave functions or Kohn−Sham representations. It is illustrated that the EPLF defines a simple and powerful tool for chemical interpretation via selected applications including atomic and molecular closed-shell systems, σ and π bonds, radical and singlet open-shell systems, and molecules having a strong multiconfigurational character. Some applications of the EPLF are presented at various levels of theory and compared to Becke and Edgecombe’s Electron Localization Function (ELF). Our open-source parallel software implementation of the EPLF opens the possibility of its use by a large community of chemists interested in the chemical interpretation of complex electronic structures.

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Published:February 02, 2011
r2011 American Chemical Society
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ARTICLE
pubs.acs.org/JCTC
Electron Pair Localization Function (EPLF) for Density Functional
Theory and ab Initio Wave Function-Based Methods: A New Tool for
Chemical Interpretation
Anthony Scemama,*,†Michel Caffarel,†Robin Chaudret,‡and Jean-Philip Piquemal‡
†Laboratoire de Chimie et Physique Quantiques, CNRS-IRSAMC, Universit? e de Toulouse, France
‡UPMC Univ Paris 06, UMR 7616, Laboratoire de Chimie Th? eorique, case courrier 137, 4 place Jussieu, F-75005, Paris, France, and
CNRS, UMR 7616, Laboratoire de Chimie Th? eorique, case courrier 137, 4 place Jussieu, F-75005, Paris, France
b
S Supporting Information
ABSTRACT: We present a modified definition of the Electron Pair Localization Function (EPLF), initially defined within the
frameworkofquantumMonteCarloapproaches[Scemama,A.;Caffarel,M.;Chaquin,P.J.Chem.Phys.2004,121,1725]tobeusedin
DensityFunctionalTheories(DFT) andabinitiowave-function-basedmethods.ThismodifiedversionoftheEPLF—whilekeeping
the same physical and chemical contents—is built to be analytically computable with standard wave functions or Kohn-Sham
representations.ItisillustratedthattheEPLFdefinesa simpleandpowerfultoolforchemicalinterpretationviaselectedapplications
including atomic and molecular closed-shell systems, σ and π bonds, radical and singlet open-shell systems, and molecules having a
strongmulticonfigurationalcharacter.SomeapplicationsoftheEPLFarepresentedatvariouslevelsoftheoryandcomparedtoBecke
and Edgecombe’s Electron Localization Function (ELF). Our open-source parallel software implementation of the EPLF opens the
possibility of its use by a large community of chemists interested in the chemical interpretation of complex electronic structures.
1. INTRODUCTION
Nowadays, when dealing with theoretical chemical interpreta-
tion, quantum chemists rely on two main strategies. The first
consists of the traditional direct interpretation of the wave
function through its projection onto molecular orbitals (MO)
or valence bond (VB) structures (the so-called Hilbert space
partitioning). The second uses a geometrical direct-space de-
scriptioninordertopartitiontheelectronicdensityintodomains
within the ordinary 3D space. The design of such interpretative
techniques, initiated by Daudel et al.,1was popularized by Bader,
who introduced the Quantum Theory of Atoms in Molecules
(QTAIM).2Along with QTAIM, Bader introduced the concept
of topological analysis, offering an atom-based partition of the
molecular space grounded on the gradient dynamical system
theory and using a local function, here the Laplacian of the
electron density. Through the years, much effort has been
devoted to the designof alternative local functions. For example,
Becke and Edgebombe introduced the Electron Localization
Function(ELF),3offering accesstochemicallyintuitive domains
beyondatomiccentersencompassingbonds,lonepairs,etc.Ever
since, its usefulness has been demonstrated by Silvi and Savin,4
who extensively developed its topological analysis, although no
partition of space is unique.5
The problem of getting an accurate description of chemical
bonding gets more and more difficult as the complexity of
the wave function goes beyond the single determinant
approximation.6Therefore, an additional natural orbital approx-
imation was added to the ELF formalism7to extend it to the
correlated level, but its general applicability to any quantum
chemical method is still subject to intense development. In that
context, other methods were introduced such as the electron
localizabilityindicator(ELI,seeref8andreferencestherein),the
analysis of electronic probability distributions,9,10and the Elec-
tron Pair Localization Function (EPLF).11
In this work, we shall focus on this latter function, EPLF,
whose main feature is giving direct access to the local (spatial)
electroniccorrelationsbetweenspin-likeandspin-unlikeelectro-
nic pairs. EPLF was first introduced within the framework of
quantum Monte Carlo (QMC) approaches where introducing
simple and direct estimators of such local electronic correlations
is particularly easy. In practice, it has been proposed to build an
indicator—the electron pair localization function—based on a
suitable combination of the average distances between an
electron of a given spin located at point r and the closest spin-
like and spin-unlike electrons. EPLF has been shown to be
particularly interesting to get new insights into the nature of
the pairing and localization of electrons and, particularly, to
understand more deeply the role of the dynamical and non-
dynamical near-degeneracy correlation effects.12,13From a fun-
damental point of view, such a result is not surprising, since the
EPLFisactuallyrelatedtotheconditionalprobabilitiesoffinding
an electron at point r2with spin σ or σh, knowing that an electron
of a given spin σ is located at some point r1. Indeed, having such
quantities at our disposal is known to be sufficient to define
an exact electronic structure theory (e.g., the exact exchange-
correlation energy of DFT can be in principle derived from such
conditional probabilities, ;see, e.g., ref 14). The advantage of
havingdefinedtheEPLFwithinaQMCcomputationalschemeis
that such a function can be easily calculated at various levels of
approximation. Indeed, by generating QMC probability densities
Received:October 15, 2010

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associated with various trial wave functions, the average distances
betweenelectronsand, thus, the EPLF functioncanbe evaluatedat
the Hartree-Fock, DFT, CASSCF, CI, VB, etc. levels of approx-
imation. It is also possible to evaluate the EPLF at the fixed-node
diffusion Monte Carlo level, a particularly accurate QMC approx-
imationrecoveringthemajorpartofstaticanddynamicalcorrelation
effects,evenifinsomecasesthequalityofthefixed-nodeerrorisnot
soeasytoassess,see,e.g.,ref15.Besidestheseadvantages,themain
drawback of calculating the EPLF with QMC is that simulations
need to be rather intensive to decrease sufficiently the statistical
errors of the EPLF values at each point r of the grid employed.
Indeed, a minimal resolution is needed to distinguish the subtle
changes in local properties.
Inthis work, we propose a modified form for the EPLFallowing
its exact computation (no statistical error) for the standard wave
functions of computational chemistry written as determinantal
expansions built from molecular orbitals expressed in some
Gaussian basis set. The approach can also be naturally applied to
DFT calculations based on a Kohn-Sham density expressed in a
determinantal form. As we shall see, the proposed modification of
the EPLF does not alter its chemical content. Using this modified
expression, the EPLF is much more rapid to compute since its
calculation requires only the evaluation of monoelectronic integrals
(see below). In particular, it avoids the use of Monte Carlo sampling,
which can be rather CPU-intensive for large systems, opening the pos-
sibility to perform full topological analyses in the near future. Accord-
ingly,onceintroducedintostandardcomputationalchemistrypackages,
webelievethattheEPLFwillbecomeaveryusefulandpowerfultoolfor
chemical interpretation accessible to a wide community of chemists.
2. EPLF: THE ORIGINAL DEFINITION
In the original definition of the EPLF, ref 11, the motivation
was to define a function of R3measuring locally the electron
pairinginamolecularsystem.Todothat,thefollowingdefinition
ofelectronpairingwasfirstintroduced:Anelectronilocatedatri
is said to be paired to an electron j located at rjif electron j is the
closest electron to i. Having defined such a pairing, it has been
proposed to define the amount of electron pairing at point r in
terms of a quantity inversely proportional to
?????
where d(r) can be interpreted as the average of the shortest
electron-electrondistanceatr,Ψ(r1,...,rN)beingtheN-electron
wave function, and rij= |ri- rj|.
Two different types of electron pairs are to be defined: pairs of
electrons having the same spin (σ) and pairs of electrons with
oppositespins(σ,σh).Hence,twoquantitiesneedtobeintroduced:
?????
dσσðrÞ¼
i¼1,N
The electron pair localization function is bound in the [-1,1]
interval and is defined as
EPLFðrÞ¼dσσðrÞ-dσσðrÞ
dσσðrÞþdσσðrÞ
dðrÞ¼
Ψ
*
X
i¼1,N
δðr-riÞ min
j6¼i
rij
?????Ψ
+
ð1Þ
dσσðrÞ¼
Ψ
*
X
?????
i¼1,N
δðr-riÞ
min
j6¼i; σi¼σjrij
?????Ψ
+
ð2Þ
Ψ
*
X
δðr-riÞ min
j; σi6¼σjrij
?????Ψ
+
ð3Þ
ð4Þ
Whenthepairingofspin-unlikeelectronsispredominant,dσσ(r)>
dσσ h(r) and EPLF(r) > 0. When the pairing of spin-like electrons
is predominant, dσσ(r) < dσσ h(r) and EPLF(r) < 0. When the
electron pairing of spin-like and spin-unlike electrons is equiva-
lent, EPLF(r) ∼ 0.
Thislocalizationfunctiondoesnotdependonthetypeofwave
function and can therefore measure electron pairing using any
kind of representation: Hartree-Fock (HF), Kohn-Sham
(KS),ConfigurationInteraction(CI),andMulti-Configurational
Self-Consistent-Field (MCSCF) as well as Slater-Jastrow, Diffu-
sionMonte Carlo (DMC), Hylleraas wave functions, etc. Due to
the presence of the min function in the definitions of dσσ(r) and
dσσ h(r), these quantities cannot be evaluated in an analytical way,
and quantum Monte Carlo (QMC) approaches appear to be the
most efficient way of computing the three-dimensional EPLF
grids via a statistical sampling of ∼Ψ2(r1,...,rN) in the case of
VariationalMonteCarlo(VMC)-typecalculationsor ∼Ψ(r1,...,rN)
Φ0(r1,...,rN) (Φ0fixed-node ground-state wave function) in the
case of the more accurate Fixed-Node Diffusion Monte Carlo
(FN-DMC)-type calculations11-13,16(for a detailed presentation
of these various versions of QMC approaches, see, e.g., ref 17).
3. EPLF: A MODIFIED DEFINITION SUITABLE FOR DFT
AND WAVE FUNCTION-BASED METHODS
Following preliminary developments,18we propose here to
introduce a modified definition of the EPLF which—in contrast
withtheoriginaldefinition—cannowbeanalyticallycomputable
forstandardwavefunctionsofquantumchemistry,thusavoiding
the need for statistical sampling. To do that, we propose to
express the min function appearing in the average distances in
terms of Gaussian functions. More precisely, we introduce the
following exact representation:
min
j6¼i
rij¼ lim
γfþ¥
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
-1
γln fðγ;rijÞ
r
ð5Þ
with
fðγ;rijÞ¼
X
j6¼i
e-γr2
ij
ð6Þ
Now, our basic approximation consists in replacing, for γ large,
the integrals
?????
appearing in eq 1 with
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
i¼1
Ψ
*
X
N
i¼1
δðr-riÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
-1
γln fðγ;rijÞ
r
!?????Ψ
?????Ψ
+
ð7Þ
-1
γln
Ψ
?????
*
X
N
δðr-riÞfðγ;rijÞ
+
v
u
u
t
ð8Þ
The expectation values of the minimum distances are now given
by
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dσσðrÞ ∼
γlarge
-1
γln fσσðγ;rÞ
r
ð9Þ
dσσðrÞ ∼
γlarge
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
-1
γln fσσðγ;rÞ
r
ð10Þ

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with the two-electron integrals:
*
fσσðγ; rÞ¼
Ψ
?????
X
?????
N
i¼1
δðr-riÞ
X
N
j6¼i;σi¼σj
e-γjri-rjj2
?????Ψ
?????Ψ
+
ð11Þ
fσσðγ; rÞ¼
Ψ
*
X
N
i¼1
δðr-riÞ
X
N
j;σi6¼σj
e-γjri-rjj2
+
ð12Þ
When the wave function Ψ has a standard form (sum of
determinants built from molecular integrals φ’s), such integrals
can be easily obtained in terms of the following elementary
contributions:
Z
which in turn can be evaluated as generalized overlap integrals.
Letusnowdiscussourbasicapproximationconsistingingoing
from eq 7 to eq 8. This approximation can be written in a more
compact way as
ffiffiffiffiffiffiffiffiffiffi
where the symbol ÆQæ denotes the integration of QΨ2over all
particle coordinates except the ith one. For a given electronic
configuration (r1,...,rN) and γ large enough, f is dominated by a
single exponential, namely, e-γ|ri-rjmin|2, where |ri- rjmin| is the
distance between the reference electron i located at riand the
closest electron labeledjmin. Thevalidityof our basic approxima-
tion is directly related to the amount of fluctuations of the
quantity fwhen various electronic configurations are considered.
Notethatforagivenelectroni,thedistance|ri-rj|canvaryalot,
but it is much less the case for |ri- rjmin|, where the electron
number jmincan be different from one configuration to another.
When these fluctuations are small, the ratio in eq 14 is close to 1
and the approximation is of good quality. To see what happens
for larger fluctuations, let us write
φiðrÞ φkðrÞ
dr0φjðr0Þ φlðr0Þ e-γjr-r0j2
ð13Þ
Æ
p
-ln f
ffiffiffiffiffiffiffiffiffiffiffiffiffi
p
æ
-lnÆ fæ
∼
γlarge1
ð14Þ
f ¼ fminþδf
ð15Þ
A simple calculation leads to
Æ
p
ffiffiffiffiffiffiffiffiffiffiffi
-ln f
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
æ
-lnÆ fæ
¼1þO½ðδfÞ2?ð16Þ
showingthatatfirstorderinthefluctuationstheratioisstillequal
to 1, illustrating the validity of our approximation.
Alastpointtodiscussisthevalueofγtobechoseninpractice.
Because of our approximation, the limit γfþ¥ cannot be taken
since the ratio in eq 14 goes to zero.19Therefore, the value of γ
has to be large enough to discriminate between the closest
electron located at rjminfrom the other ones located at larger
distancesofelectroni,whilestayingintheregimewheretheratio
in eq 14 stays close to one. We have found that a value of γ
depending on r and chosen on physical grounds allows systema-
tic recovery of the essential features of the EPLF images
calculated with QMC, that is to say, with the exact expression
of the min function. To be effective, the discrimination of the
closest electron with the other ones must be properly imple-
mented. To do that, the value of γ is adapted to keep the leading
exponential e-γ|ri-rjmin|2significantly larger than the subleading
exponentiale-γ|ri-rjnext-min|2associatedwiththesecondclosestelectron
jnext-min.First,wedefineasphereΩ(ri)centeredonriwitharadius
dΩ(ri).Then,locally,werepresentoursystemmadeoftheelectron
locatedatrianditstwoclosestneighborsbyamodelsystemofthree
independent particles. If one calculates the probability of finding all
three particles inside the sphere, one finds
Z
If the density F(r) issupposed as constant and equalto F(ri),the
radius dΩ(ri) of the sphere can be set such that PΩis equal to a
fixed value:
?
Then, γ(ri) is chosen in order to set a constant ratio κ between
the width of e-γrij2and the radius of the sphere
k¼
We obtain an expression of γ(ri) which depends on the electron
density:
?
Inoursimulations,wehavefoundthattheEPLFimagesobtained
with QMC are properly recovered using PΩ= 0.001 and κ = 50.
PΩðriÞ¼
1
3
ΩðriÞdr FðrÞ
!3
ð17Þ
dΩðriÞ¼
4π
9PΩ-1=3FðriÞ
?-1=3
ð18Þ
ffiffiffiffiffiffiffiffiffiffiffiffi
2γðriÞ
p
dΩðriÞð19Þ
γðriÞ¼k2
2
4π
9PΩ-1=3FðriÞ
?2=3
ð20Þ
Figure 1. ELF and EPLF radial values for the argon atom as a function
of the distance to the nucleus.
Figure2. ELFandEPLFvaluesintheCH3S-anionalongtheC-Saxis
computed using a Hartree-Fock and a BLYP determinant.

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4. SOME APPLICATIONS
As discussed previously, the modified form of the EPLF pre-
sentedhereis aimedatproviding thesamechemicalinformationas
theoriginalQMC-basedEPLFscheme,butwithoutthestatistical
noise inherited from the QMC approach. Therefore, the inter-
ested reader can refer to existing recent publications that deal
with the QMC-EPLF analysis of covalent, ionic, and multicenter
bonds.11-13,16,20We focus in this section on some illustrative
applications highlighting the specific capabilities of the EPLF as
compared to Becke and Edgecombe’s ELF.
4.1. Closed-Shell Single-Determinant Systems. A first nat-
uralexampletolookatisthecaseofaclosed-shellatomdescribed
at the Hartree-Fock (HF) level. Using Dunning’s cc-pVDZ
atomic basis set,21the radial values of the EPLF and ELF for the
argon atom are displayed in Figure 1. It is noted that both
functions display three maximum values corresponding to the
n = 1, n = 2, and n = 3 values of the principal quantum number.
Furthermore, these maxima are essentially located at the same
place. The gross features of the atomic shell structure are thus
described in a similar way by both approaches. However, there is
also a striking difference: The magnitudes of the two secondary
maxima corresponding to the two most external shells are
essentially identical in the ELF case but very different for the
EPLF, where the outermost one is much smaller. Note that
having such a difference is not surprising since EPLF is, in
contrast with ELF, directly connected to electron pairing. The
pairing of antiparallel electrons is likely to be the strongest in the
firstshell,weakerinthesecondshell,andtheweakestinthemost
diffuse third shell.
EPLF and ELF were computed for the CH3S-methanethio-
late anion, using a Hartree-Fock determinant and a Kohn-
Shamdeterminant.The6-31þ2G**atomicbasisset22,23wasused
forbothdeterminants,andtheBLYPfunctional24,25wasusedfor
the DFT calculation. Figure 2 compares the one-dimensional
plots of the EPLF and ELF along the C-S axis of the tetrahedral
CH3S-.Asfortheargonatom,thetopologiesoftheEPLFandELF
functions are comparable for both the Hartree-Fock and the
Kohn-Sham determinants. Going from the Hartree-Fock to the
BLYPlevel,thevaluesoftheELFareessentiallythesameinthecore
domains, become slightly smaller in the C-S bonding region, and
become slightly larger on the rest of the C-S axis. As the EPLF
exhibits the same trend, we conclude that for closed-shell single
determinants the EPLF and ELF give qualitatively similar results.
The ELF and EPLF were computed for the ethylene molecule
using a HF/cc-pVDZ wave function. The isosurfaces ELF = 0.75
and EPLF = 0.12 are represented in Figure 3. These images are
qualitatively similar, even if the core domains seem to be larger
using the EPLF. This is due to the fact that the EPLF values are
higher in the first atomic shells (as in the argon example), while
the ELF has more comparable values among the shells.
To have a more quantitative visualization of the similarities
anddifferencesbetweentheELFandtheEPLF,acorrelationplot
relating the values of both functions is presented in Figure 4.
Three different regimes can be observed. First, a regime corre-
sponding to the core domain where the EPLF takes its larger
values. In this region, an almost perfect one-to-one correspon-
dence is observed, thus illustrating the similarity between both
localization functions. In contrast, in the valence region where
the (EPLF,ELF) points are scattered, it seems to be no longer
true. In fact, this is not really the case since the majorityof points
are found to be almost aligned along the left side of the envelope
of points. To illustrate this, the median line (same number of
points on each side) is represented. Finally, a last regime
corresponding to the region where the ELF and EPLF values
are small (say, ELF smaller than 0.05) can be defined. In such a
regime, the two localization functions turn out to be fully
decorrelated. However, the underlying configurations corre-
spond to regions in space where the electronic densities are
(very) small, and this case is not of great chemical interest. As a
conclusion, in all chemically interesting regimes, the correlation
between ELF and EPLF is high. We have found that such a
conclusionisvalidnotonlyforthiscasebutalsoforallmolecules
described by a closed-shell single determinant wave function. In
thiscase,thequalitativeinformationthatcanbeobtainedfroman
ELF and an EPLF calculation is essentially the same. This can be
understoodbynotingthatforaclosed-shellmonoconfigurational
wave function the R electrons are independent from the β
electrons, so localizing electrons is essentially equivalent to
localizing antiparallel electron pairs.
4.2. Open-Shell Hartree-Fock. A wave function for the
HC2•radical was obtained at the restricted open-shell Hartree-
Fock level (ROHF), using the cc-pVDZ atomic basis set. Both
ELF and EPLF were computed, and the results are displayed in
Figure 5. This example points out the main difference between
ELF and EPLF: the localization region of the unpaired electron
exhibits a maximum for ELF (high electron localization) and a
minimum for EPLF (low electron pairing). EPLF can identify
Figure 3. ELF = 0.75 (top) and EPLF = 0.12 (bottom) isosurfaces of
the ethylene molecule.
Figure 4. Correlation between the ELF and the EPLF in the ethylene
molecule.

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ARTICLE
clearlydomainsofelectronpairing(lonepairs,coredomains,and
bonds), and it can additionally characterize localized unpaired
electrons similarly to spin density.
4.3. Multiconfigurational Wave Function. A wave function
for the singlet state of the ozone molecule was first calculated at
the HF/cc-pVDZ level. ELF and the EPLF were both calculated
and give similar qualitative results (Figure 6).
Then,acompleteactivespacewavefunctionwitheightelectrons
ineightorbitals(CAS(8,8))wasprepared,andEPLFwascalculated
(Figure 7). The EPLF obtained from the CAS wave function is
Figure 5. ELF (top) and EPLF (bottom) contour plots of the HC2•radical in the molecular plane. Red values are the lowest and blue values are the
highest.

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significantly different from the EPLF obtained from the HF wave
function. In the HF framework, the O-O bonding domains are
more connected to each other than to the lone pair domains of the
centraloxygenatom.WiththeCASwavefunction,eachO-Obond
domain is more connected to the lone pair domains than to each
other. This example shows that EPLF is an alternative to ELF in
closed shell systems where a multiconfigurational method is re-
quired, as EPLF is well-defined for such cases.
4.4. Open-Shell Singlet. When the ethylene molecule is
twisted with an angle of 90? along the C-C axis, the π bond
breaks. Each one of the π electrons localizes on a carbon atom,
giving rise to an open-shell singlet (see Figure 8), degenerate
with the triplet state. In order to preserve the spin symmetry, a
CAS(2,2) wave function was computed to describe the singlet
state. With such a wave function, the spin density is not able to
localizetheunpairedelectronssincetheRone-electrondensityis
equal to the β one-electron density in every point of space. The
EPLF reveals the presence of these unpaired electrons by local
minimumvaluesofthefunctionclosetothecarbonatoms,inthe
plane perpendicular to the C-H bonds.
5. SOFTWARE
To realize the EPLF and ELF calculations presented in this
paper, a code was written using the IRPF90 Fortran generator.26
This code is interfaced with the Gaussian 03,27GAMESS,28and
Molpro29programs. As the calculation of EPLF is more expen-
sivethan the calculation of ELF, theprogram has been efficiently
parallelized (for both EPLF and ELF calculations) using the
message passing interface (MPI) library30and exhibits a linear
speedup property with the number of cores. The EPLF code is
licensedundertheGNUGeneralPublicLicense,andthesourcefiles
can be downloaded from the Web at http://eplf.sourceforge.net.
6. CONCLUSION
In this work, we have introduced a modified version of the
EPLF analytically computable for standard wave functions and
DFT representations. When compared to the original EPLF
defined in a QMC framework, essentially the same images are
recovered. A systematic comparison of our analytical EPLF with
the Electronic Localization Function (ELF) of Becke and Edge-
combehasbeenmade.Forclosed-shellsystems,theEPLFresults
are shown to closely match the ELF ones. However, for other
situations, the two localization functions may differ significantly
(radicals, systems with strong static correlations, etc). The major
advantage of the reformulated EPLF is that it can be easily
computed for any kind of electronic structure method defined
from single or multideterminantal wave functions. Further
development will focus on the topological analysis of the EPLF,
which will provide the possibility of computing various proper-
tiesintegratedfromapartitionofthethree-dimensionalspace.As
oursoftwareisavailableforfree,itshouldopenthepossibilityfor
any chemist to use EPLF for the understanding of complex
electronic structures.
’ASSOCIATED CONTENT
b
S
Supporting Information.
atom. This material is available free of charge via the Internet at
http://pubs.acs.org.
EPLF vs ELF for the xenon
’AUTHOR INFORMATION
Corresponding Author
*E-mail: scemama@irsamc.ups-tlse.fr.
’ACKNOWLEDGMENT
Support from the French “Centre National de la Recherche
Scientifique(CNRS)”,Universit? edeToulouse,andUniversit? eParis
6isgratefullyacknowledged.WewouldliketothankIDRIS(CNRS,
Orsay), CCRT (CEA/DAM, Ile-de-France), and CALMIP
(Universit? e de Toulouse) for providing us with computational
resources.
’REFERENCES
(1) Daudel, R.; Odiot, S.; Brion, H. J. Physique Rad. 1954, 15, 804–
809.
(2) Bader, R.F.W.Atomsinmolecules: Aquantumtheory;Clarendon
Press: Oxford, U. K., 1990; p 438.
Figure 6. ELF = 0.61 (top) and EPLF = 0.123 (bottom) isosurfaces of
the singlet state of the ozone molecule (Hartree-Fock).
Figure 7. EPLF = 0.123 isosurface the singlet state of the ozone
molecule (CAS-SCF).
Figure 8. EPLF contour plot and isosurface of the singlet state of the
twisted H2C•-•CH2biradical. Red values of the EPLF are the lowest,
and green values are the highest.

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ARTICLE
(3) Becke, A. D.; Edgecombe, K. E. J. Chem. Phys. 1990, 92, 5397–
5403.
(4) Silvi, B.; Savin, A. Nature 1994, 371, 683–686.
(5) Savin, A. J. Chem. Sci. 2005, 117, 473–475.
(6) Silvi, B. J. Phys. Chem. A 2003, 107, 3081–3085.
(7) Feixas, F.; Matito, E.; Duran, M.; Sol? a, M.; Silvi, B. J. Chem.
Theory Comput. 2010, 6, 2736–2742.
(8) Bezugly, V.; Wielgus, P.; Kohout, M.; Wagner, F. R. J. Comput.
Chem. 2010, 31, 1504–1519.
(9) Canc? es, E.; Keriven, R.; Lodier, F.; Savin, A. Theor. Chim. Acta
2004, 111, 373–380.
(10) Scemama,A.;Caffarel,M.;Savin,A.J.Comput.Chem.2007,28(1),
442–454.
(11) Scemama, A.; Chaquin, P.; Caffarel, M. J. Chem. Phys. 2004,
121, 1725–1735.
(12) Scemama, A.; Caffarel, M.; Ramírez-Solís, A. J. Phys. Chem. A
2009, 113, 9014–9021.
(13) Caffarel, M.; Scemama, A.; Ramírez-Solís, A. Theor. Chim. Acta
2010, 126, 275–287.
(14) Parr, R.; Yang, W. Density-Functional Theory of Atoms and
Molecules; Oxford University Press: Oxford, U. K., 1989; p 344.
(15) Caffarel,M.;Hern? andez-Lamoneda,R.;Scemama,A.;Ramírez-
Solís, A. Phys. Rev. Lett. 2007, 99, 153001.
(16) Amador-Bedolla, C.; Salom? on-Ferrer, R.; Lester, W. A., Jr.;
V? azquez-Martínez, J. A.; Aspuru-Guzik, A. J. Chem. Phys. 2007, 126,
204308.
(17) Hammond,B.L.;Lester,W.A.,Jr.;Reynolds,P.J.MonteCarlo
Methods in Ab Initio Quantum Chemistry. Lecture and Course Notes in
Chemistry; World Scientific: Singapore, 1994; Vol. 1.
(18) Alary, F.; Heully, J.-L.; Scemama, A.; Garreau-de Bonneval, B.;
Chane-Ching, K.; Caffarel, M. Theor. Chim. Acta 2010, 126, 243–255.
(19) For a given electronic configuration (r1,...,rN), f goes as exp-
(-γ|r - rmin|2) when γ becomes large. Accordingly, in this limit, Æ(-ln
f)1/2æ goes to a finite positive constant given by Æ|r - rmin|2æ. On the
other hand, Æfæ can be written as the integral of exp(-γ|r - rmin|2) over
all possible configurations, which can be rewritten as 1/(γ)1/2(γ)1/2
exp(-γ|r - rmin|2), that is, at large γ, the product of 1/(γ)1/2times a
Dirac distribution. Finally, in the large-γ limit, the ratio (Æ(-ln f)1/2æ)/
((-lnÆf)1/2æ) goes to zero as ∼1/(γ)1/2.
(20) De La Lande, A.; Salahub, D. R.; Maddaluno, J.; Scemama, A.;
Pilme, J.; Parisel, O.; Gerard, H.; Caffarel, M.; Piquemal, J.-P. J. Comput.
Chem. 2010, [Online] DOI: 10.1002/jcc.21698.
(21) Dunning, T. H., Jr. J. Chem. Phys. 1989, 90, 1007–1023.
(22) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 1973, 28, 213–
222.
(23) Francl,M.M.;Pietro,W.J.;Hehre,W.J.;Binkley,J.S.;Gordon,
M. S.; DeFrees, D. J.; Pople, J. A. J. Chem. Phys. 1982, 77, 3654–3665.
(24) Becke, A. D. Phys. Rev. A 1988, 38, 3098–3100.
(25) Lee, C.; Yang, W.; Parr, R. G. Phys. Rev. B 1988, 37, 785–789.
(26) Scemama, A. ArXiv e-prints 2009, [cs.SE], 0909.5012.
(27) Frisch, M. J. et al. Gaussian 03; Gaussian, Inc.: Wallingford,
CT, 2004.
(28) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.;
Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.;
Su, S.; Windus, T. L.; Dupuis, M.; Montgomery, J. A., Jr. J. Comput.
Chem. 1993, 14 11, 1347–1363.
(29) Werner, H. J. et al. MOLPRO, version 2009.1. http://www.
molpro.net (accessed Dec 7, 2009).
(30) Gropp, W.; Lusk, E.; Doss, N.; Skjellum, A. Parallel Comput.
1996, 22, 789–828.