Nuclear Fukui functions from nonintegral electron number calculations

Nuclear Fukui Functions FromNonintegral Electron NumberCalculationsCARLOS CARDENAS,1 EDUARDO CHAMORRO,1

MARCELO GALVAN,2 PATRICIO FUENTEALBA3

1Departamento de Ciencias Quımicas, Facultad de Ecologıa y Recursos Naturales,Universidad Andres Bello (UNAB), Avenida Republica 275, 8370146 Santiago, Chile2Departamento de Quımica, Universidad Autonoma Metropolitana Iztapalapa,DF 09340 Mexico DF, Mexico3Departamento de Fısica, Facultad de Ciencias, Universidad de Chile, Las Palmeras 3425,Nunoa, 7800024 Santiago, Chile

Received 21 May 2006; accepted 5 July 2006Published online 2 October 2006 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.21202

ABSTRACT: Numerical results for the nuclear Fukui function (NFF) based on anonintegral number of electrons methodology (NIEM) are reported for a series ofsimple diatomic molecules. A comparison with those obtained from othermethodologies is focused on the estimation of the error associated with a finitedifference approximation for the evaluation of the NFF. The dependence of NFFs on thetype and size of the basis set is also discussed. The NIEM values are in close agreementwith those obtained from a finite difference approximation using �N � �1 with largebasis sets. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem 107: 807–815, 2007

Key words: chemical reactivity; DFT; noninteger electrons; Janak DFT; diatomicmolecules

Introduction

D ensity functional theory (DFT) has become auseful mathematical framework for the de-

velopment of a theory of chemical reactivity [1].

Based on the Hohenberg–Kohn (HK) theorems [2],it uses the electron density �(r) as the basic variableof molecular systems instead of the wave function.Henceforth, several response functions of the den-sity or related quantities (i.e., reactivity indexes)have been associated with susceptibilities of thesystem properties to change against external per-turbations [3]. Electronic and nuclear reactivity de-scriptors have been defined on the basis of Taylorseries expansions of the energy functional withinthe four ensembles of DFT, providing a formal basisfor a deeper understanding of chemical reactivityand chemical reaction processes within a perturba-tive approximation [4–6]. A complete hierarchy of

Correspondence to: C. Cardenas; e-mail: [email protected]

Contract grant sponsor: Fondecyt (Chile).Contract grant numbers: 1030173; 1050294; 7030009.Contract grant sponsor: Millennium Nucleus for Applied

Quantum Mechanics and Computational Chemistry (Mideplan-Conicyt, Chile).

Contract grant number: P02-004-F.Contract grant sponsor: Universidad Andres Bello (UNAB).Contract grant numbers: DI16-04; 10-05/I.

International Journal of Quantum Chemistry, Vol 107, 807–815 (2007)© 2006 Wiley Periodicals, Inc.

global, local, and nonlocal electronic descriptorsconstitutes the basic ingredients with which to dis-cuss chemical reactivity [7–9]. These electronic re-sponses basically represent responses given interms of the density-related properties without anexplicit consideration of the nuclei framework.Global electronic response quantities, such as theelectronic chemical potential [10]

� � � �E�N�

��r�

,

chemical hardness [11]

� � � ��

�N��r�

,

and chemical softness [11]

S � ��N��

��r�

,

are global responses of the system to global pertur-bations [12]. Local electronic descriptors, such asthe electron density

��r� � � �E��r��

N

,

local softness [8, 13]

s�r� � ��r��

��r�

,

and Fukui function [14]

f�r� � ��r��N �

��r�

� � ��

��r��N

,

account for local (global) responses of the systemagainst global (local) perturbations [15, 16]. Nonlo-cal response functions, such as the linear responsefunction,

��r, r�� r��r��

N

,

and the softness

s�r, r�� r��r��

and hardness

��r, r�� 2F��

��r��r��kernels characterizes local responses against localperturbations [8, 9, 17]. In the absence of otherfields, the external potential �(r) arises only fromthe spatial configuration of the nuclei in the system,

��r� � �

Z/�r R�.

F[�] stands for the kinetic energy and the electron–electron repulsion functionals. The responses tochanges in the external potential, �(r), are associ-ated with changes in nuclear configuration, whichis the most obvious variable in a chemical reaction[18–20]. Although the electron density determinesall ground-state properties of a molecular system,the response of the nuclei to a perturbation in Nremains unknown, and a complicated response ker-nel translates electron density changes into externalpotential changes. Cohen and Ganduglia-Pirovano[19] introduced an alternative to this problem de-fining a nuclear Fukui function (NFF) as the changeof the Hellman–Feynman force on the nucleus , F,due to a perturbation in the number of electrons ata constant external potential,

� � ��F

�N��r�

,

where

F ��E�R

� Z

�

�R� ��r�

�r Ra�dr

� Z

�

�R��

Z�

�R R�� (1)

�, a vectorial quantity, does not directly measurethe change in the external potential ��(r); instead, itdescribes the onset of that change [21]. In the limitof very low temperature, the density matrix in thegrand canonical ensemble is a piecewise function ofthe number of electron [18]; consequently, any de-rivative with respect to N will be a discontinuous

CARDENAS ET AL.

808 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 107, NO. 4

function of N. Therefore, the derivative respect to Nmust be taken just above or below the correct inte-gral number of electrons of the system, which per-mits us to define left-hand () and right-hand (�)nuclear Fukui functions

��/ � ��F

�/

�N ��r�

.

In the case of diatomic molecules and using internalcoordinates as the reference system, the sign of theNFFs has a simple meaning. In the case of �

�, apositive sign indicates a bond lengthening withionization, whereas the same sign in �

predicts ashortening of the bond length. This tendency tochange the bond length can be rationalized in thelight of the Berlin division of the molecular space[22]. The density rearrangement due to changes inthe number of electrons implies an intramolecularcharge transfer. The bond length changes dependon whether charge is arriving or leaving the bond-ing regions. Using Maxwell relations, Baekelandt etal. [23] showed that the nuclear Fukui function canbe also identified as the configurational contribu-tion to the change in the chemical potential

� � ��F

�N��r�

� � ��

�R�

N

, (2)

where R designates the position of the nucleus .Further nuclear-related reactivity indexes, based ona nuclear kernel softness quantity, have been de-fined and its relations with electronic descriptorsexplored [24–31]. An extension of this formalism tothe spin polarized DFT has been recently presented[32, 33]. It is clear that these nuclear reactivity hier-archies complement and extend the usefulness ofDFT as applied to gain insight into the reactivitywithin molecular chemical systems, despite its re-maining a perturbative approximation.

In the present work, and with an ongoing inter-est in the computational exploration of nuclear re-activities [26–28, 31], we focus on the evaluation ofthe NFF using numerical calculations of the deriv-ative with respect to the number of electrons on thebasis of Janak’s theorem [34]. Three approximatemethods have already been reported to calculatethe NFF: (i) a finite difference approximation to Eq.(2) [28]; (ii) the derivative of the chemical potentialwith respect to nuclear displacements [31]; and (iii)approximate analytical methods developed in anal-ogy to the coupled Hartree–Fock approach to the

electronic Fukui function [26]. The finite differenceapproximation has been the most widely usedmethod for the calculation of NFF and, althoughthere are no a priori reasons, this method usuallyassumes that the forces on atoms change linearlywithin the range ��N� � 0–1.

Methodologies based on a noninteger occupa-tion scheme are well known for the evaluation ofhardness tensors [35] and electronic Fukui func-tions [36]. In the present work, we focus on theevaluation of the NFF comparing for the first timenumerical results based on noninteger approxima-tion with those obtained from other methodologies.Our goal is to elucidate the error associated with afinite difference approximation in the estimation ofnuclear reactivity descriptors. An analysis of thedependence of the NFF on the type and size of thebasis set are also discussed.

DFT Formalism for NonintegerNumber of Electrons

The HK theorem establishes that the ground-state energy is a functional of the electron density ifand only if it is v-representable. Additionally, theconstrained search of Levy [37] and Lieb [39] ex-tends the range of densities to those N-repre-sentables. The N-representability is a weaker con-dition of the v-representability; nevertheless, bothformulations are strictly applicable to systems withan integer number of electrons. This clearly repre-sents a “problem” for the chemical reactivity theoryin the context of the DFT, because there are manyderivatives with respect to the electron number�/�N. Perdew [40] proposed an elegant approachfor noninteger electron numbers. These investiga-tors showed that the ground-state energy and den-sity of a system with noninteger electron number atzero temperature could be represented by an en-semble of two pure ground states ��Z and ��Z�1 with N and N � 1 electrons, respectively. In con-trast, it is possible to deal with noninteger electronsystems by taking an extension of the KS equationsto fractional orbital occupation (ni), i.e., the Janakenergy functional [34]:

E�� TJ�� J�� EXC�� r��r�dr

NUCLEAR FUKUI FUNCTIONS FROM NONINTEGRAL ELECTRON NUMBER CALCULATIONS

VOL. 107, NO. 4 DOI 10.1002/qua INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 809

� �i

ni � *i�x��12 �2� i�x�dx � J�� EXC��

� � ��r��r�dr

��r� � �i

ni �s

� i�x��2. (3)

Note that in this context, the energy depends onboth the set of occupation numbers ni and the set ofKS orbitals i. Thus, by minimizing the energy withrespect to these two sets of variables, one obtainsthe ground state. In Eq. (3), TJ[�] is the functional ofkinetic energy for the noninteracting nonintegerelectron number system, which differs from Ts[�] inthe partially occupied orbitals. Janak’s equationsprovide us with a scheme that interpolates betweeninteger numbers of electrons. This provides astraightforward method to diminish the error inevaluating the derivatives with respect to N.

However, the interpretation of results based onsuch a method still deserves attention, because itsimplementation finally rests on a mathematical ar-tifice [41]. The KS equations were deduced for in-teger electron number and the correct procedure todeal with open systems that can have a time-aver-aged fractional number of electrons is to use thezero temperature Kohn–Sham–Mermin equations.Indeed, a variational evaluation of Eq. (3) with re-spect to both orbitals and occupation numbers, con-serving the orthogonality among orbitals, yieldsagain the correct KS equations and ground-stateoccupation scheme [34].

Computational Details

To calculate the NFF, the forces on atoms wereevaluated for 11 different electron numbers(�N � 0.0, �0.1, �0.2, �0.3, �0.4, and �0.5)and interpolated to a fifth-order polynomial func-tion. The NFF of Eq. (2) was obtained from thecoefficient of the linear term. The change in elec-tron number was determined by the change inoccupation numbers,

dN � �i

dni,

keeping constant the population of inner orbitalsand varying only the occupation of the frontier

orbitals. It is well known that high-degree polyno-mials tend to oscillate between the points they areinterpolating. Since the derivative of a polynomialis itself a polynomial, it oscillates between thepoints of constraint. However, in most cases theforce is practically linear with the number of elec-trons, and the coefficient of the linear term can betaken as a good approximation to the derivative.We have compared our scheme with a finite differ-ence of high order, and the values are almost thesame.

All electron DFT calculations were performedusing the Becke exchange functional [42] and thePerdew 86 correlation functional [40] with the so-called ET-pVQZ basis set as implemented in theADF package [43]. This high-quality basis set in-cludes core and valence polarization additional toquintuplet zeta Slater-type orbitals (STOs). This ba-sis set allows us to approximate to the basis setlimit. In each case, the geometry is the equilibriumgeometry for the system with the correct number ofelectrons. To minimize errors in the energy gradi-ents, the evaluation of the integrals was performedusing seven significant figures. Additionally, a ba-sis set convergence study was carried out.

Results and Discussion

Table I reports the left- and right-hand side de-rivatives of the NFFs for a series of simple diatomicmolecules. The series exhibits a broad range inbonding nature from ionic to covalent interactions.We include the NFFs evaluated with finite differ-ences using both the fractional occupation schemeexplored here and the one with a step correspond-ing to �N � �1 at the same level of calculation. Forpurposes of comparison, we include previously re-ported results [26] obtained from a coupled per-turbed Hartree–Fock (CP-HF) analytical methodand from the negative gradients of the chemicalpotential.

As was explained above, lengthening (shorten-ing) the bond distance with the ionization process isassociated with negative (positive) left-hand NFFsand positive (negative) right-hand NFFs. From Ta-ble I, the NFFs obtained from noninteger electronmethodology (NIEM) agree very well with theother methods, except in the cases of SiS and PNmolecules for left-hand NFF. The discrepancy in theSiS molecule is caused by a numerical instability inthe analytical algorithm to evaluate the forces,which yields small positive and negative values of

CARDENAS ET AL.


the force in the range 1 � �N � 0. This oscillationis an unphysical result because the response of thesystem when an electron leaves is unique. We re-port the left-hand NFF for SiS only for complete-ness. Balawender and Geerlings [26] found thesame incongruity with respect to the sign of left-hand NFF for SiS. For all diatomic molecules inTable I, the right-hand NFF is positive; i.e., the bondlength is increased when the system takes an elec-tron.

In Table I the series is organized in increasingorder according to NIEM values. For the left-handNFFs, the predicted order is not the same for dif-ferent methods of calculation. The best correspon-dence occurs between NIEM and �N � �1 for bothtypes of NFFs. This is due in part to the fact thatCP-HF and �� values were obtained with a differ-ent level of calculation (HF/6-31��G**), so thesevalues are not directly comparable. The order cor-respondence is better for left-hand NFF than forright-hand NFF, which reflects the technical diffi-culty associated with the calculation of the anionproperties. Nevertheless, the observed differences

in NFF along the series are small, especially in theleft-hand one.

We have analyzed the correlation between twonumerical methods: the NFFs calculated as finitedifferences with �N � �1 and NFFs obtained fromNIEM. In Figure 1, the correlation coefficients forthese linear correlations are 0.985 and 0.990 for theleft-hand and right-hand NFF, respectively. In thefirst case, the slope is �0.85, indicating that a finitedifference approximation with a big �N tends tounderestimate the left-hand NFF. We are assumingthat an evaluation of the NFF through an interpo-lation among noninteger electron number systemsis better than a finite differences approximationbased on forces on cation and anion.

With the aim of gaining more insights with re-spect to the dependence of the NFF on the methodof calculation, we have also studied the effect of thesize of basis set for the same series of molecules ofTable I. Tables II and III present the NFF calculatedusing SZ, DZ, DZP, TZP, and p-QZP STO basis setswith �N � �1 for left-hand and right-hand deriv-atives, respectively. For each basis set, the NFFs

TABLE I ______________________________________________________________________________________________NFF for diatomic molecules.

� ��

NIEMa �N � 1b CP-HFc ��d NIEM �N � 1 CP-HF ��

LiF 0.05119 0.06325 0.05649 0.05545 AlH 0.0077 0.00554 0.00512 0.00609HF 0.04407 0.07522 0.0553 0.0554 LiCl 0.01675 0.01362 0.01278 0.01245BeO 0.03225 0.04486 0.09816 0.08233 LiH 0.01679 0.00834 0.00633 0.00629LiCl 0.02977 0.03567 0.0368 0.03219 LiF 0.01839 0.01378 0.01279 0.01215N2 0.01617 0.03209 0.04472 0.05363 AlCl 0.02906 0.02324 0.02201 0.02615LiH 0.00968 0.02797 0.02747 0.02904 AlF 0.03072 0.0192 0.01761 0.01961AlH 0.00922 0.00718 0.01437 0.01559 BeO 0.03344 0.02926 0.03817 0.04368PN 0.02518 0.01139 0.13527 0.11459 BCl 0.04497 0.03934 0.03656 0.04075AlF 0.03301 0.03416 0.04315 0.04469 SiS 0.04567 0.04444 0.05047 0.05447AlCl 0.03467 0.03201 0.03625 0.04251 SiO 0.05925 0.04186 0.03875 0.04473Cl2 0.03759 0.03415 0.05106 0.05548 BF 0.06759 0.04339 0.02074 0.03009ClF 0.05610 0.04957 0.05145 0.07275 PN 0.07187 0.05891 0.04475 0.05731BF 0.05748 0.05885 0.04559 0.07005 Cl2 0.14734 0.13857 0.15249 0.15522F2 0.06310 0.06310 0.09626 0.01189 ClF 0.18381 0.16645 0.13164 0.14587BCl 0.06708 0.06608 0.05628 0.06623 N2 0.23379 0.19321 0.08806 0.08600SiO 0.00309 0.00077 0.00110 0.01599 F2 0.32915 0.31505 0.34621 0.35711BH 0.02314 0.01232 0.01171 0.01550 BH 0.00807 0.00546 0.00421 0.00596SiS 0.02225 0.01047 0.07833 — HF 0.08658 0.02398 0.02110 0.02137

a NIEM values from polynomial interpolation for systems with noninteger number of electrons.b �N � � 1 values from finite difference approximation with �N � � 1 (gradients in ionic forms).c CP-HF values from the analytical coupled perturbed Hartree–Fock methodology (Ref. [26]).d �� values from the approximation of the NFF by the negative gradient of the frontier molecular orbital energy (HOMO–LUMO)(Ref. [26]).



were evaluated at the corresponding equilibriumgeometries. The series is in increasing order accord-ing to p-QZP basis set results. The norm and sign ofboth left-hand and right-hand NFFs show conver-gent behavior at the biggest TZP and p-QZP basissets, except in the case of BH and SiO for the left-hand NFF and the BH and HF systems for theright-hand one. The case of BH is remarkable be-cause the high-quality basis p-QZP yields a NFF

value smaller by one order of magnitude than thoseobserved for the smaller basis sets. In the case ofleft-hand NFF, the order of magnitude and the signare conserved for the TZP and p-QZP basis set withexception of a minor change for the AlF molecule,whereas the predicted order changes for the SZ,DZ, and DZP basis sets, and the sign is incorrectlypredicted in few cases for the SZ basis set. For theright-hand NFF, all basis sets predict the correct

FIGURE 1. Correlation between the NFF calculatedthrough finite differences with step size of �N � � 1and the interpolation scheme for noninteger number ofelectrons. (a) ��. (b) � excluding SiS. NFF was evalu-ated at BP86/p-vqz level of calculation using Slater-type orbitals (STO).

FIGURE 2. Dependence of the NFF on basis set type.(a) Correlation between left side NFF evaluated atBP86/p-vqz level of calculation using Slater-type orbit-als (STO) and Slater-type orbitals expanded as Gauss-ian function (STO-G). (b) Same correlation for right sideNFF.

CARDENAS ET AL.


sign, but the order along the series (of systems)becomes more sensitive, as can be noted from theTZP and p-QZP results. In summary, the NFF isbasis set dependent for small basis, but conver-gence is reached (with few exceptions) for TZP andhigher basis sets.

Note that the electronic component of the forcecalculated using the Hellman–Feynman theorem atthe position of nuclei tends to infinity. Hence, itcould be expected that the correct behavior of thedensity near the nuclei play an important role in thecalculated values of the NFFs. The correct electrondensity must fulfill the cusp condition in the nuclei,which is not the case for a density constructed withmolecular orbitals expanded in Gaussian functions.Therefore, it is interesting to compare our resultswith similar calculations using an equivalentGaussian basis set. Table IV presents NFF valuesfrom a finite differences approximation with �N ��1 calculated using the p-QZP basis set expandedin both Gaussian and Slater atomic orbitals. As canbe immediately noted, excluding the above-men-tioned problematic case of SiO, the values for theNFF do not depend on the fulfillment of the cuspcondition. Nevertheless, this result is partially arti-ficial because most of the electronic structure codes,

included Gaussian 98 and ADF, evaluate the forcesby means of analytical expressions for energy gra-dients [44]. Figure 2 presents the observed correla-tions between Slater and Gaussian basis set for thecalculation of the NFFs. The slope is near to one,and the intercept is small with coefficients of 0.98and 0.99 for the left-hand and right-hand NFFs,respectively. In this case, the gradient is calculatedby analytical methods; the NFFs are essentially in-dependent on whether the basis set appropriatelydescribes the cusp condition.

Concluding Remarks

Numerical results for the NFF based on NIEMhave been calculated for a series of simple diatomicmolecules. A comparison to those obtained fromother methodologies [26, 31] has been focused onthe estimation of the error associated to a finitedifference approximation for the evaluation of theNFF. The dependence of the NFF on the type andsize of the basis set has been also discussed. Ingeneral, the NIEM NFFs values are in good agree-ment with those obtained from finite differenceswith �N � �1 evaluated at higher basis sets.

TABLE II ______________________________________________________________________________________________Dependence of the left side NFF on basis size.

SZa DZ DZP TZP p-QZP

HF 0.06268 0.08516 0.08272 0.07513 0.07521BeO 0.10877 0.33276 0.07723 0.07236 0.07397LiF 0.05938 0.06161 0.06083 0.06485 0.06324LiCl 0.05463 0.03647 0.03667 0.03586 0.03606N2 0.08284 0.02173 0.01725 0.03313 0.03209LiH 0.03877 0.02780 0.02698 0.02815 0.02798AlH 0.01006 0.00908 0.00801 0.00746 0.00718SiS 0.00869 0.00699 0.01437 0.01216 0.01045PN 0.00946 0.00643 0.01597 0.01296 0.01147AlCl 0.02877 0.01984 0.03267 0.03349 0.03200Cl2 0.01080 0.01613 0.03560 0.03481 0.03408AlF 0.01765 0.01946 0.02971 0.03360 0.03416ClF 0.02429 0.02738 0.04589 0.04684 0.04957BF 0.02952 0.05330 0.05790 0.05906 0.05885F2 0.03213 0.02777 0.05448 0.06093 0.06310BCl 0.06189 0.05800 0.06889 0.06803 0.06609BH 0.00263 0.01268 0.00320 0.00025 0.01232SiO 0.01759 0.00934 0.00231 0.00084 0.00077

a SZ, DZ, DZP, TZP, p-QZP stand for single zeta, double zeta, double zeta plus polarization, triple zeta plus polarization, andeven-tempered quadruple zeta plus polarization. NFF was evaluated at BP86/p-vqz level of calculation using Slater-type orbitals(STO) and finite approximation with �N � 1.



TABLE III _____________________________________________________________________________________________Dependence of the right-side NFF on basis size.

SZa DZ DZP TZP p-QZP

AlH 0.02744 0.00871 0.00819 0.00607 0.00554LiH 0.00568 0.00986 0.01028 0.00960 0.00834LiCl 0.01802 0.01190 0.01445 0.01388 0.01361LiF 0.01003 0.01235 0.01450 0.01518 0.01377AlF 0.07106 0.01921 0.02176 0.01855 0.01921AlCl 0.05451 0.02314 0.02733 0.02427 0.02325BeO 0.00447 0.22465 0.01233 0.03102 0.03064BCl 0.07420 0.05600 0.05378 0.05017 0.03934SiO 0.09005 0.04827 0.04949 0.04525 0.04185BF 0.11524 0.06778 0.07799 0.06461 0.04339SiS 0.06380 0.04561 0.05063 0.04675 0.04446PN 0.09757 0.08377 0.07586 0.06521 0.05883Cl2 0.15401 0.10586 0.15262 0.14448 0.13867ClF 0.22936 0.15871 0.18265 0.17751 0.16645N2 0.14643 0.24789 0.25228 0.22697 0.19322F2 0.40116 0.23477 0.31579 0.28695 0.31504BH 0.02347 0.02277 0.02544 0.02014 0.00546HF 0.43342 0.09017 0.09176 0.04558 0.02398

a SZ, DZ, DZP, TZP, p-QZP stand for single zeta, double zeta, double zeta plus polarization, triple zeta plus polarization, andeven-tempered quadruple zeta plus polarization. NFF was evaluated at BP86/p-vqz level of calculation using Slater-type orbitals(STO) and finite approximation with �N � 1.

TABLE IV _____________________________________________________________________________________________Dependence of the NFF on kind of basis set.

� a ��

Slater Gaussian Slater Gaussian

HF 0.07522 0.07577 AlH 0.00554 0.00603LiF 0.06325 0.05414 LiH 0.00834 0.00951BeO 0.04485 0.00822 LiCl 0.01362 0.01376LiCl 0.03567 0.03438 LiF 0.01378 0.01443N2 0.03209 0.03204 AlF 0.01920 0.02148LiH 0.02797 0.02826 AlCl 0.02324 0.02544AlH 0.00718 0.00743 BeO 0.02925 0.02952SiS 0.01047 0.01002 BCl 0.03934 0.04543PN 0.01139 0.01122 SiO 0.04186 0.04794AlCl 0.03201 0.03196 BF 0.04339 0.06890Cl2 0.03415 0.03453 SiS 0.04444 0.04654AlF 0.03416 0.03344 PN 0.05891 0.06664ClF 0.04957 0.04930 Cl2 0.13857 0.14450BF 0.05885 0.05822 ClF 0.16645 0.17860F2 0.06310 0.06388 N2 0.19321 0.21683BCl 0.06608 0.06596 F2 0.31505 0.33253BH 0.01232 0.01257 BH 0.00546 0.00847SiO 0.00077 0.00013 HF 0.02398 0.07563

a NFF was evaluated at BP86/p-vqz level of calculation using Slater-type and Gaussian-type orbitals and finite approximation with�N � 1.

CARDENAS ET AL.


ACKNOWLEDGMENTS

C. C. is indeed grateful to UNAB for a Ph.D.fellowship.

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Nuclear Fukui functions from nonintegral electron number calculations