research article nucleophilicity index based on atomic...

Hindawi Publishing CorporationJournal of ChemistryVolume 2013 Article ID 684134 6 pageshttpdxdoiorg1011552013684134

Research ArticleNucleophilicity Index Based on Atomic Natural Orbitals

Dariusz W Szczepanik and Janusz Mrozek

Department of Computational Methods in Chemistry Faculty of Chemistry Jagiellonian UniversitySt Ingardena 3 30-060 Cracow Poland

Correspondence should be addressed to Dariusz W Szczepanik szczepadchemiaujedupl

Received 30 May 2013 Accepted 21 August 2013

Academic Editor K R S Chandrakumar

Copyright copy 2013 D W Szczepanik and J Mrozek This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A simple method of evaluating a semilocal (regional) nucleophilicity is introducedThe concept involves use of the natural orbitalsfor atomic populations to identify the most ldquoreactive populationrdquo of electrons on particular atom in molecule The results of testcalculations considering the regioselectivity problem in electrophilic aromatic substitution to the benzene derivatives are presentedand briefly discussed

1 Introduction

There is a wide variety of reactivity predictors for electron-transfer-controlled reactions among which the Fukui func-tion originating from the density functional theory (DFT) isone of the most fundamental ones [1 2] Within the orbital-based calculations one of the simplest methods to interpretand implement is the condensed Fukui index [3ndash6] (a coarse-grained atom-by-atom representation of the Fukui function)which can be determined from electron population analysisThere are many arguments [7 8] in favor of the condensedFukui index based on the Hirshfeldrsquos populational scheme[9] however other population analysis methods includingthe Mullikenrsquos [10ndash13] and Lowdinrsquos [14] as well as the NPAschemes [15] are also in common use (eg [16ndash18])

In this paper we briefly introduce and test a new simplemethod of evaluating the relative (with respect to particularatom in two ldquohomologousrdquo molecules) reactivity of nucle-ophiles based on the natural orbitals for atomic populationof electrons [19] and involving their energies as well asoccupation numbers The method is related to the Fukuifunction concept and its approximation within the frame-work of FrontierMolecular Orbital (FMO) theory [20ndash22] Inthis workwe take into consideration the well-known problem(somewhat trivial but illustrative) of regioselectivity predic-tion in the electrophilic aromatic substitution to the fol-lowing benzene derivatives fluorobenzene (C

6H5F) aniline

(C6H5NH2) phenol (C

6H5OH) nitrobenzene (C

6H5NO2)

benzoic acid (C6H5COOH) and benzaldehyde (C

6H5CHO)

In general functional groups ndashOH and ndashNH2are clas-

sified as electron donating and strongly activating in theelectrophilic substitution reactions while functional groupsndashNO2 ndashCOOH and CHO remove electron density from

the benzene ring and thus strongly deactivate the moleculeFunctional groups from the former class tend to be orthoparadirecting while those from the latter one direct electrophilesto attack the benzene molecule at the meta position In flu-orobenzene (likewise in other benzene halides) the benzenering is weakly deactivated due to inductive withdrawal ofelectrons by electronegative atom F However the resonancedonation of nonbonding electrons of fluorine atom to thebenzene ring causes that the most preferable positions ofelectrophilic attack are ortho and para

2 Method Details

Let us assume the closed-shell molecular system with119873 elec-trons doubly occupying 119899 lowest molecular orbitals |120593119900⟩ =

|1205931⟩ |120593

119899⟩ equiv |120593

lowast⟩ generated as linear combinations of

orthogonalized atomic orbitals (OAO) |120594⟩

1003816100381610038161003816120593119894 ⟩ = sum

120572

1003816100381610038161003816120594120572⟩ ⟨120594120572 | 120593119894⟩ = sum

120572

1003816100381610038161003816120594120572⟩119862120572119894 (1)

2 Journal of Chemistry

or in matrix form1003816100381610038161003816120593119900⟩ =

1003816100381610038161003816120594⟩ ⟨120594 | 120593119900⟩ =

1003816100381610038161003816120594⟩C119900 (2)

where the rectangular matrix C119900 groups the relevant LCAOMOexpansion coefficients It follows directly from the super-position principle of quantummechanics that the conditionalprobability of ldquofinding electronrdquo from 119894th occupied molecularorbital on 120572th atomic orbital reads

119875119894 (120572) = ⟨120594120572 | 120593119894⟩ ⟨120593119894 | 120594120572⟩ = ⟨120594120572

10038161003816100381610038161003816119894

10038161003816100381610038161003816120594120572⟩ =

10038161003816100381610038161198621205721198941003816100381610038161003816

2 (3)

Hence the LCAO MO expansion coefficient 119862120572119894

can beregarded as the conditional-probability amplitude Alterna-tively one can relate 119862

120572119894to the amplitude of the probability

of ldquofinding electronrdquo from 120572th atomic orbital on 119894th canonicalmolecular orbital Then

119875120572 (119894) = ⟨120593119894 | 120594120572⟩ ⟨120594120572 | 120593119894⟩ = ⟨120593119894

10038161003816100381610038161003816120572

10038161003816100381610038161003816120593119894⟩ =

10038161003816100381610038161198621205721198941003816100381610038161003816

2 (4)

Similarly by replacing the operator 120572with the operator

119883

projecting onto the subspace of all OAOs centered on atom119883 we get the conditional probability of the event that theelectron from electron population on atom119883 can be ascribedto 119894th molecular orbital

119875119883 (119894) =N

119883⟨120593119894

10038161003816100381610038161003816119883

10038161003816100381610038161003816120593119894⟩ =N

119883

119883

sum

120572

⟨120593119894

10038161003816100381610038161003816120572

10038161003816100381610038161003816120593119894⟩

=N119883

119883

sum

120572

10038161003816100381610038161198621205721198941003816100381610038161003816

2

(5)

where normalization constant reads

N119883= [

119899

sum

119894

119875119883 (119894)]

minus1

= [

119899

sum

119894

119883

sum

120572

10038161003816100381610038161198621205721198941003816100381610038161003816

2]

minus1

(6)

The corresponding 119899 times 119899 matrix P119883(120593

o120593

o) of elements rep-

resenting projections of 119895th occupied MO onto 119894th occupiedMO through the subspace of AOs assigned to atom119883 [23]

P119883(120593119900120593119900)

= 119875119883(119894 119895) =N

119883

119883

sum

120572

⟨120593119894

10038161003816100381610038161003816120572

10038161003816100381610038161003816120593119895⟩ =N

119883

119883

sum

120572

119862120572119894119862120572119895

(7)

is obviously not diagonal It follows directly from orthonor-mality of molecular orbitals that regarding the whole (molec-ular) electron population119873

P (120593119900120593119900) = sum119883

P119883(120593119900120593119900)

=N⟨120593119900

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119883

119883

1003816100381610038161003816100381610038161003816100381610038161003816

120593119900⟩ =N ⟨120593

119900| 120593119900⟩ =

1

1198991119899

(8)

The representation of orbitals |120579119883⟩

1003816100381610038161003816120579119883⟩ =1003816100381610038161003816120593119900⟩ ⟨120593119900| 120579119883⟩ =

1003816100381610038161003816120593119900⟩C120579119883 (9)

in which off-diagonal elements of matrix (7) are zeroes119875119883(119894 119895) = 120575

119894119895 one can straightforwardly obtain fromdiago-

nalization of P119883(120593119900120593119900)

P119883(120593119900120593119900) = C120579119883p120579119883C120579dagger119883 (10)

The diagonal matrix of electron probabilities p120579119883can be used

to calculate the electron populations n120579119883of the corresponding

natural orbitals for electron population of atom119883 |120579119883⟩

n120579119883= 119873119883p120579119883 where 119873

119883=2

119899

sum

119894

119883

sum

120572

10038161003816100381610038161198621205721198941003816100381610038161003816

2 (11)

stands for the electron population on atom 119883 The corre-sponding matrix of the 1-electron orbital ldquoenergiesrdquo 120576120579

119883(ie

expectation values of 119865120593 in |120579119883⟩) can be determined as

follows120576120579

119883= ⟨120579119883

1003816100381610038161003816100381611986512059310038161003816100381610038161003816120579119883⟩ = C120579dagger

119883⟨120593119900 1003816100381610038161003816100381611986512059310038161003816100381610038161003816120593119900⟩C120579119883= C120579dagger119883120576119900C120579119883

(12)

Here 119865120593 is the Fock operator and diagonal matrix 120576119900collects orbital energies of occupied canonical MOs Sincenatural orbitals |120579

119883⟩ are not the eigenvectors of 119865120593 the

matrix of orbital energies (12) is not diagonal Orbital ener-gies of both frontier canonical molecular orbitals HOMO(120576lowastequiv 120576

HOMOequiv 120576119899119899 ) and LUMO (120576

LUMOequiv 120576119899+1119899+1

) areknown to be useful qualitative indicators of chemical reac-tivity Thus it is of our special interest to investigate how theexpectation values of operator 119865120593 within representation of|120579119883⟩managewith evaluation of the reactivity of the particular

atom119883 However due to nonorthogonality of natural-orbitalsets for different atoms only the relative changes of 120576120579

119883for the

same119883 in two homologous species should be comparedOne of the standard frontier-orbital treatments of chem-

ical reactivity is the regioselectivity problem for examplein the electrophilic aromatic substitution to the benzenederivatives The standard FMO theory analysis of the effectof substituent groups involves the electron populations ofthe highest occupied molecular orbital (HOMO) whereaswithin the newly proposed ldquoreverse scenariordquo we focus on theelectron population of particular atom first and then analyzethe highest ldquoenergiesrdquo of occupied atomic NOs

3 Numerical Results

To examine the presented methodology we have used state-functions calculated at RHF(ROHF for cations)STO-3GRHF(ROHF)6-31Glowast as well as DFTB3LYPaug-cc-pVDZtheory levels using the standard ab initio quantum chemistrypackage GAMESS [24] for the following benzene derivativesfluorobenzene aniline phenol nitrobenzene benzoic acidand benzaldehyde The highest energies of occupied naturalorbitals and the corresponding electron populations werecomparedwith two standardMO-based atomic descriptors ofreactivity the atomic index of nucleophilicity [25] involvingonly the highest occupied MO |120593lowast⟩

119877119883=sum119883

120572

10038161003816100381610038161198621205721198991003816100381610038161003816

2

(1 minus 120576119899119899)=sum119883

120572

1003816100381610038161003816119862lowast

120572

1003816100381610038161003816

2

(1 minus 120576lowast) (13)

Journal of Chemistry 3

Table 1 Atomic indices of nucleophilicity (13) and condensed atomic Fukui indices (14) for selected benzene derivatives For asymmetricspecies the position refers to carbon atom with larger electron population Methods RHFROHF basis sets STO-3G 6-31Glowast

Molecule Position STO-3G 6-31Glowast

119877M119883

119877L119883

119865M119883

119865L119883

119877M119883

119877L119883

119865M119883

119865L119883

Fluorobenzeneortho 0083 0083 0063 0074 0076 0076 0039 0065meta 0046 0046 0037 0040 0057 0058 0013 0056para 0236 0236 0136 0185 0250 0247 0142 0227

Anilineortho 0093 0093 0075 0094 0099 0099 0077 0103meta 0032 0031 0013 0008 0033 0035 minus0025 0012para 0205 0205 0104 0140 0218 0214 0137 0207

Phenolortho 0103 0103 0084 0104 0104 0103 0065 0096meta 0026 0026 0017 0012 0030 0033 minus0017 0021para 0221 0221 0124 0168 0238 0235 0142 0220

Nitrobenzeneortho 0000 0000 0036 0039 0180 0179 0034 0038meta 0000 0000 0020 0020 0182 0180 minus0004 0023para 0000 0000 0052 0068 0003 0006 0050 0083

Benzoic Acidortho 0050 0050 0042 0047 0143 0142 0038 0047meta 0065 0065 0017 0015 0220 0217 minus0002 0022para 0222 0222 0051 0067 0010 0012 0042 0074

Benzaldehydeortho 0051 0051 0050 0057 0017 0019 0046 0058meta 0069 0069 0016 0014 0226 0223 minus0008 0016para 0227 0227 0061 0080 0117 0117 0053 0089

and the Fukui nucleophilic reactivity index [26] involvingelectron populations on atom 119883 in cation 119873+

119883 and neutral

molecule119873119883

119865119883= 119873119883minus 119873+

119883 (14)

Both indices were calculated only within basis sets STO-3Gand 6-31Glowast since larger basis sets (especially those includingdiffuse functions eg aug-cc-pVDZ) are well known to dra-matically lose their ldquoatomic attributenessrdquo of AOs and con-sequently many population-type descriptors usually assumecompletely unreasonable values Two alternative proceduresof population analysis were used in calculation of indices (13)and (14) the Mullikenrsquos scheme (superscript M) [10ndash13] andthe Lowdinrsquos one (superscriptL) [14] involving the standardldquogeometricalrdquo orthogonalization of atomic orbitals

The results presented in Table 1 clearly indicate that onlyindex 119877

119883calculated within extended basis set allows one

to correctly predict the position of electrophilic attack inall molecules under consideration The Fukui nucleophilicreactivity index 119865

119883properly copes with orthopara directing

but it completely fails with respect to meta directing Itis worth notice that indicator 119877

119883 involving only electron

populations and energies of HOMO is far less sensitive topopulation analysis method than index 119865

119883 Furthermore if

one excludes the results formeta directing groups there is nosignificant difference between 119877

119883calculated within STO-3G

and 6-31Glowast basis setsWe performed a similar analysis using the newly pro-

posed method based on natural orbitals |120579119883⟩ Table 2 collects

orbital occupation numbers 119899lowast119883and the corresponding rela-

tive ldquoenergetic effectsrdquoΔ120576lowast119883 calculated as a difference between

the highest energy of occupied natural orbitals of particularcarbon atoms in benzene and its derivative

Δ120576lowast

119883= 120576lowast

119883(derivative) minus 120576lowast

119883(benzene) (15)

Even a cursory analysis of occupation numbers in Table 2allows one to correctly predict the preferential orthoparaor meta directing of substituent groups regardless of basissets used in calculations However evaluation of energeticeffects provides more detailed information about activat-ingdeactivating influence of substituent groups on particularcarbon atoms For example ndashNH

2is properly recognized

as the best activator and the most activated carbon atomis in position para (activated for about 11 kcalsdotmolminus1) whilesubstituent group ndashNO

2strongly deactivates themolecule for

about 20ndash25 kcalsdotmolminus1 and the most active population ofelectrons is then located on carbon atom in position metaFor the majority of cases orbital energies properly predict theposition of electrophilic substitution

In calculations of natural orbitals from Table 2 we usedthe standard Lowdinrsquos orthogonalization procedure to obtainorthogonal AOs However it was of our interest to find how998779120576lowast

119883and 119899

lowast

119883depend on the orthogonalization procedure

Table 3 contains the same data as Table 2 but is calcu-lated within representation of ldquophysicallyrdquo orthogonalizedatomic orbitals (superscript P) [27ndash29] At first glance onecan observe a small improvement of electron populationsfrom DFTB3LYPaug-cc-pVDZ Also in contrast to cal-culations involving ldquogeometricalrdquo orthogonalization Δ120576lowastP

119883


Table 2 Occupation numbers (11) and differences between the highest NO ldquoenergiesrdquo (15) in [kcalsdotmoLminus1] for benzene and its derivativescalculated within representation of ldquogeometricallyrdquo orthogonalized atomic orbitals For asymmetric species the position refers to carbonatom with larger electron population Methods RHFSTO-3G 6-31Glowast and DFTB3LYPaug-cc-pVDZ

Molecule Position STO-3G 6-31Glowast Aug-cc-pVDZ119899lowastL119883

Δ120576lowastL119883

119899lowastL119883


119899lowastL119883


Fluorobenzeneortho 1039 minus25 1033 minus82 0786 minus73meta 0985 minus73 0958 minus100 0760 minus75para 1020 minus13 1013 minus42 0778 minus37

Anilineortho 1057 38 1073 82 0803 90meta 0983 minus14 0949 minus16 0760 36para 1035 67 1047 104 0794 110

Phenolortho 1046 50 1048 35 0793 33meta 0979 minus37 0949 minus44 0758 minus09para 1035 43 1037 46 0788 48

Nitrobenzeneortho 0971 minus237 0928 minus272 0738 minus230meta 0999 minus196 0992 minus208 0763 minus191para 0968 minus205 0930 minus242 0744 minus206

Benzoic acidortho 0974 minus70 0936 minus122 0737 minus112meta 1005 minus63 0999 minus85 0767 minus90para 0978 minus75 0940 minus120 0748 minus109

Benzaldehydeortho 0989 minus67 0954 minus148 0739 minus145meta 1004 minus66 0996 minus109 0768 minus119para 0984 minus63 0943 minus131 0746 minus133

Table 3 Occupation numbers (11) and differences between the highest NO ldquoenergiesrdquo (15) in [kcalsdotmolminus1] for benzene and its derivativescalculated within representation of ldquophysicallyrdquo orthogonalized atomic orbitals For asymmetric species the position refers to carbon atomwith larger electron population Methods RHFSTO-3G 6-31Glowast and DFTB3LYPaug-cc-pVDZ

Molecule Position STO-3G 6-31Glowast Aug-cc-pVDZ119899lowastP119883

Δ120576lowastP119883

119899lowastP119883


119899lowastP119883








for benzaldehyde in STO-3G basis set correctly predictmeta directing of substituent group In general howevercalculated energies are similar to those from Table 2 (averagedistances between the respective energies calculated within

ldquogeometricallyrdquo and ldquophysicallyrdquo orthogonalized AO repre-sentations are 04 (STO-3G) 08 (6-31Glowast) and 09 (aug-cc-pVDZ) [kcalsdotmolminus1]) and thus allow one to draw almost thesame conclusions about reactivity of particular carbon atom


The energetic effects from Tables 2 and 3 are relativelysmall (but comparable with differences between stabilizationenergies of the corresponding Wheland intermediates [30])For comparison below we present several simple speciesordered with respect to relative nucleophilicities (from weakto very good nucleophiles) and the corresponding energiesΔ120576lowastL119883

[kcalsdotmolminus1] (of orbitals identified as lone pairs inall cases) calculated at DFTB3LYPTZP with model corepotential [31 32]

CH3COOH ltH

2O lt NH

3lt CH

3Ominus lt CH

3Sminus

00(reference) 118 611 2179 2324

4 Summary

In this work we have briefly introduced a simple method ofevaluating the relative nucleophilicity in energy scale Theconcept involves the use of natural orbitals for atomic popula-tion of electrons and their energies (ie expectation values ofFock operator) as well as occupation numbers to identify ldquothemost reactive population of electronsrdquo on particular atom (ormolecular fragment) Such scenario is directly related to thestandard FMO theory treatment involving atomic popula-tions of electrons of the highest occupied molecular orbital(HOMO) in the newly proposed approach we first focus onthe electron population of particular atom and then analyzeenergies of occupied natural orbital This strategy has beenexamined on the regioselectivity problem in the electrophilicaromatic substitution to the benzene derivatives Analysis ofthe results allows one to draw the conclusion that evaluationand comparison of relative chemical nucleophilicities ofatoms in an energy domain are more reliable and advan-tageous than analyses involving other popular MO-basedthis seems to be somehow obvious since in contradistinc-tion to the majority of condensed atomic indices energeticdescriptors converge systematically to the complete-basis-setlimit It has to be noticed however that electron populationof the highest occupied natural orbital of a particular atomin molecule is somewhat insensitive to basis set variationsand the corresponding energy seems to exhibit the basis setdependence quite consistent with the variational principle

The presented methodology is still in need of thoroughanalysis and examination Also it is of our special interestto take advantage of the approach based on natural atomicorbitals involving virtual molecular orbitals to evaluate theenergetic descriptor of electrophilicity of atoms and molecu-lar fragments

References

[1] R G Parr and W T Yang J ldquoDensity functional approach tothe frontier-electron theory of chemical reactivityrdquo Journal ofthe American Chemical Society vol 106 no 14 pp 4049ndash40501984

[2] P W Ayers and M Levy ldquoPerspective on lsquoDensity functionalapproach to the frontier-electron theory of chemical reactivityrsquordquoTheoretical Chemistry Accounts vol 103 no 3-4 pp 353ndash3602000

[3] W T Yang and W J Mortier ldquoThe use of global and localmolecular parameters for the analysis of the gas-phase basicity

of aminesrdquo Journal of the American Chemical Society vol 108no 19 pp 5708ndash5711 1986

[4] P Fuentealba P Perez and R Contreras ldquoOn the condensedFukui functionrdquoThe Journal of Chemical Physics vol 113 no 7Article ID 2544 8 pages 2000

[5] N Otero M Mandado and R A Mosquera ldquoRevisiting thecalculation of condensed Fukui functions using the quantumtheory of atoms in moleculesrdquoThe Journal of Chemical Physicsvol 126 no 23 Article ID 234108 6 pages 2007

[6] P Bultinck S Fias C van Alsenoy P W Ayers and R Carbo-Dorca ldquoCritical thoughts on computing atom condensed Fukuifunctionsrdquo The Journal of Chemical Physics vol 127 no 3Article ID 034102 11 pages 2007

[7] R F Nalewajski and R G Parr ldquoInformation theory atoms inmolecules andmolecular similarityrdquoProceedings of theNationalAcademy of Sciences of the United States of America vol 97 no16 pp 8879ndash8882 2000

[8] P W Ayers ldquoAtoms in molecules an axiomatic approach IMaximum transferabilityrdquoThe Journal of Chemical Physics vol113 no 24 pp 10886ndash10898 2000

[9] F L Hirshfeld ldquoBonded-atom fragments for describing molec-ular charge densitiesrdquo Theoretica Chimica Acta vol 44 no 2pp 129ndash138 1977

[10] R S Mulliken ldquoElectronic population analysis on LCAO-MOmolecular wave functions Irdquo The Journal of Chemical Physicsvol 23 no 10 Article ID 1833 8 pages 1955

[11] R S Mulliken ldquoElectronic population analysis on LCAO-MO molecular wave functions II Overlap populations bondorders and covalent bond energiesrdquo The Journal of ChemicalPhysics vol 23 no 10 Article ID 1841 6 pages 1955

[12] R S Mulliken ldquoElectronic population analysis on LCAO-MOmolecular wave functions IV Bonding and antibonding inLCAO and valence-bond theoriesrdquo The Journal of ChemicalPhysics vol 23 no 12 pp 2343ndash2346 1955

[13] R S Mulliken ldquoElectronic population analysis on LCAO-MO molecular wave functions III Effects of hybridization onoverlap and gross AO populationsrdquo The Journal of ChemicalPhysics vol 23 no 12 Article ID 2338 5 pages 1955

[14] P O Lowdin ldquoOn the non-orthogonality problem connectedwith the use of atomicwave functions in the theory ofmoleculesand crystalsrdquo The Journal of Chemical Physics vol 18 no 3Article ID 365 11 pages 1950

[15] A E Reed R B Weinstock and F Weinhold ldquoNatural popu-lation analysisrdquo The Journal of Chemical Physics vol 83 no 2Article ID 735 12 pages 1985

[16] S Arulmozhiraja and P Kolandaivel ldquoCondensed Fukui func-tion dependency on atomic chargesrdquoMolecular Physics vol 90no 1 pp 55ndash62 1997

[17] R K Roy K Hirao S Krishnamurty and S Pal ldquoMullikenpopulation analysis based evaluation of condensed Fukui func-tion indices using fractional molecular chargerdquo The Journal ofChemical Physics vol 115 no 7 pp 2901ndash2907 2001

[18] F de Proft C van Alsenoy A Peeters W Langenaeker andP Geerlings ldquoAtomic charges dipole moments and Fukuifunctions using the Hirshfeld partitioning of the electrondensityrdquo Journal of Computational Chemistry vol 23 no 12 pp1198ndash1209 2002

[19] D Szczepanik and J Mrozek ldquoOn quadratic bond-orderdecomposition within molecular orbital spacerdquo Journal ofMathematical Chemistry vol 51 no 6 pp 1619ndash1633 2013

[20] K Fukui T Yonezawa and H Shingu ldquoA molecular orbitaltheory of reactivity in aromatic hydrocarbonsrdquo The Journal ofChemical Physics vol 20 no 4 pp 722ndash725 1952


[21] K Fukui T Yonezawa and C Nagata ldquoA free-electron modelfor discussing reactivity in unsaturated hydrocarbonsrdquo TheJournal of Chemical Physics vol 21 no 1 pp 174ndash176 1953

[22] M Berkowitz ldquoDensity functional approach to frontier con-trolled reactionsrdquo Journal of the American Chemical Society vol109 no 16 pp 4823ndash4825 1987

[23] R F Nalewajski D Szczepanik and J Mrozek ldquoBond differ-entiation and orbital decoupling in the orbital-communicationtheory of the chemical bondrdquoAdvances in Quantum Chemistryvol 61 pp 1ndash48 2011

[24] M W Schmidt K K Baldridge J A Boatz et al ldquoGeneralatomic and molecular electronic structure systemrdquo Journal ofComputational Chemistry vol 14 no 11 pp 1347ndash1363 1993

[25] R Franke Theorecital Drug Design Methods Elsevier Amster-dam The Netherlands 1984

[26] J Melin P W Ayers and J V Ortiz ldquoRemoving electrons canincrease the electron density a computational study of negativefukui functionsrdquo The Journal of Physical Chemistry A vol 111no 40 pp 10017ndash10019 2007

[27] D Szczepanik and J Mrozek ldquoSymmetrical orthogonalizationwithin linear space of molecular orbitalsrdquo Chemical PhysicsLetters vol 512 pp 157ndash160 2012

[28] D Szczepanik and J Mrozek ldquoElectron population analysisusing a referenceminimal set of atomic orbitalsrdquoComputationaland Theoretical Chemistry vol 996 pp 103ndash109 2012

[29] D Szczepanik and J Mrozek ldquoOn several alternatives forLowdin orthogonalizationrdquo Computational and TheoreticalChemistry vol 1008 pp 15ndash19 2013

[30] G W Wheland ldquoA quantum mechanical investigation of theorientation of substituants in aromaticmoleculesrdquo Journal of theAmerican Chemical Society vol 64 no 4 pp 900ndash908 1942

[31] Y Sakai E Miyoshi M Klobukowski and S Huzinaga ldquoModelpotentials for main group elements Li through RnrdquoThe Journalof Chemical Physics vol 106 no 19 pp 8084ndash8092 1997

[32] T NoroM Sekiya and T Koga ldquoContracted polarization func-tions for the atoms helium through neonrdquoTheoretical ChemistryAccounts vol 98 no 1 pp 25ndash32 1997

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CatalystsJournal of


or in matrix form1003816100381610038161003816120593119900⟩ =

1003816100381610038161003816120594⟩ ⟨120594 | 120593119900⟩ =

1003816100381610038161003816120594⟩C119900 (2)

where the rectangular matrix C119900 groups the relevant LCAOMOexpansion coefficients It follows directly from the super-position principle of quantummechanics that the conditionalprobability of ldquofinding electronrdquo from 119894th occupied molecularorbital on 120572th atomic orbital reads

119875119894 (120572) = ⟨120594120572 | 120593119894⟩ ⟨120593119894 | 120594120572⟩ = ⟨120594120572

10038161003816100381610038161003816119894

10038161003816100381610038161003816120594120572⟩ =

10038161003816100381610038161198621205721198941003816100381610038161003816

2 (3)

Hence the LCAO MO expansion coefficient 119862120572119894

can beregarded as the conditional-probability amplitude Alterna-tively one can relate 119862

120572119894to the amplitude of the probability

of ldquofinding electronrdquo from 120572th atomic orbital on 119894th canonicalmolecular orbital Then

119875120572 (119894) = ⟨120593119894 | 120594120572⟩ ⟨120594120572 | 120593119894⟩ = ⟨120593119894

10038161003816100381610038161003816120572

10038161003816100381610038161003816120593119894⟩ =

10038161003816100381610038161198621205721198941003816100381610038161003816

2 (4)

Similarly by replacing the operator 120572with the operator

119883

projecting onto the subspace of all OAOs centered on atom119883 we get the conditional probability of the event that theelectron from electron population on atom119883 can be ascribedto 119894th molecular orbital

119875119883 (119894) =N

119883⟨120593119894

10038161003816100381610038161003816119883

10038161003816100381610038161003816120593119894⟩ =N

119883

119883

sum

120572

⟨120593119894

10038161003816100381610038161003816120572

10038161003816100381610038161003816120593119894⟩

=N119883

119883

sum

120572

10038161003816100381610038161198621205721198941003816100381610038161003816

2

(5)

where normalization constant reads

N119883= [

119899

sum

119894

119875119883 (119894)]

minus1

= [

119899

sum

119894

119883

sum

120572

10038161003816100381610038161198621205721198941003816100381610038161003816

2]

minus1

(6)

The corresponding 119899 times 119899 matrix P119883(120593

o120593

o) of elements rep-

resenting projections of 119895th occupied MO onto 119894th occupiedMO through the subspace of AOs assigned to atom119883 [23]

P119883(120593119900120593119900)

= 119875119883(119894 119895) =N

119883

119883

sum

120572

⟨120593119894

10038161003816100381610038161003816120572

10038161003816100381610038161003816120593119895⟩ =N

119883

119883

sum

120572

119862120572119894119862120572119895

(7)

is obviously not diagonal It follows directly from orthonor-mality of molecular orbitals that regarding the whole (molec-ular) electron population119873

P (120593119900120593119900) = sum119883

P119883(120593119900120593119900)

=N⟨120593119900

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119883

119883

1003816100381610038161003816100381610038161003816100381610038161003816

120593119900⟩ =N ⟨120593

119900| 120593119900⟩ =

1

1198991119899

(8)

The representation of orbitals |120579119883⟩

1003816100381610038161003816120579119883⟩ =1003816100381610038161003816120593119900⟩ ⟨120593119900| 120579119883⟩ =

1003816100381610038161003816120593119900⟩C120579119883 (9)

in which off-diagonal elements of matrix (7) are zeroes119875119883(119894 119895) = 120575

119894119895 one can straightforwardly obtain fromdiago-

nalization of P119883(120593119900120593119900)

P119883(120593119900120593119900) = C120579119883p120579119883C120579dagger119883 (10)

The diagonal matrix of electron probabilities p120579119883can be used

to calculate the electron populations n120579119883of the corresponding

natural orbitals for electron population of atom119883 |120579119883⟩

n120579119883= 119873119883p120579119883 where 119873

119883=2

119899

sum

119894

119883

sum

120572

10038161003816100381610038161198621205721198941003816100381610038161003816

2 (11)

stands for the electron population on atom 119883 The corre-sponding matrix of the 1-electron orbital ldquoenergiesrdquo 120576120579

119883(ie

expectation values of 119865120593 in |120579119883⟩) can be determined as

follows120576120579

119883= ⟨120579119883

1003816100381610038161003816100381611986512059310038161003816100381610038161003816120579119883⟩ = C120579dagger

119883⟨120593119900 1003816100381610038161003816100381611986512059310038161003816100381610038161003816120593119900⟩C120579119883= C120579dagger119883120576119900C120579119883

(12)

Here 119865120593 is the Fock operator and diagonal matrix 120576119900collects orbital energies of occupied canonical MOs Sincenatural orbitals |120579

119883⟩ are not the eigenvectors of 119865120593 the

matrix of orbital energies (12) is not diagonal Orbital ener-gies of both frontier canonical molecular orbitals HOMO(120576lowastequiv 120576

HOMOequiv 120576119899119899 ) and LUMO (120576

LUMOequiv 120576119899+1119899+1

) areknown to be useful qualitative indicators of chemical reac-tivity Thus it is of our special interest to investigate how theexpectation values of operator 119865120593 within representation of|120579119883⟩managewith evaluation of the reactivity of the particular

atom119883 However due to nonorthogonality of natural-orbitalsets for different atoms only the relative changes of 120576120579

119883for the

same119883 in two homologous species should be comparedOne of the standard frontier-orbital treatments of chem-

ical reactivity is the regioselectivity problem for examplein the electrophilic aromatic substitution to the benzenederivatives The standard FMO theory analysis of the effectof substituent groups involves the electron populations ofthe highest occupied molecular orbital (HOMO) whereaswithin the newly proposed ldquoreverse scenariordquo we focus on theelectron population of particular atom first and then analyzethe highest ldquoenergiesrdquo of occupied atomic NOs

3 Numerical Results

To examine the presented methodology we have used state-functions calculated at RHF(ROHF for cations)STO-3GRHF(ROHF)6-31Glowast as well as DFTB3LYPaug-cc-pVDZtheory levels using the standard ab initio quantum chemistrypackage GAMESS [24] for the following benzene derivativesfluorobenzene aniline phenol nitrobenzene benzoic acidand benzaldehyde The highest energies of occupied naturalorbitals and the corresponding electron populations werecomparedwith two standardMO-based atomic descriptors ofreactivity the atomic index of nucleophilicity [25] involvingonly the highest occupied MO |120593lowast⟩

119877119883=sum119883

120572

10038161003816100381610038161198621205721198991003816100381610038161003816

2

(1 minus 120576119899119899)=sum119883

120572

1003816100381610038161003816119862lowast

120572

1003816100381610038161003816

2

(1 minus 120576lowast) (13)




119877M119883

119877L119883

119865M119883

119865L119883

119877M119883

119877L119883

119865M119883

119865L119883








119883 and neutral


119865119883= 119873119883minus 119873+

119883 (14)

















Δ120576lowast

119883= 120576lowast


119883(benzene) (15)







119883and 119899

lowast



119883





119899lowastL119883


119899lowastL119883











119899lowastP119883


119899lowastP119883













CH3COOH ltH

2O lt NH

3lt CH

3Ominus lt CH

3Sminus

00(reference) 118 611 2179 2324

4 Summary



References












































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




119877M119883

119877L119883

119865M119883

119865L119883

119877M119883

119877L119883

119865M119883

119865L119883








119883 and neutral


119865119883= 119873119883minus 119873+

119883 (14)

















Δ120576lowast

119883= 120576lowast


119883(benzene) (15)







119883and 119899

lowast



119883





119899lowastL119883


119899lowastL119883











119899lowastP119883


119899lowastP119883













CH3COOH ltH

2O lt NH

3lt CH

3Ominus lt CH

3Sminus

00(reference) 118 611 2179 2324

4 Summary



References












































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of





119899lowastL119883


119899lowastL119883











119899lowastP119883


119899lowastP119883













CH3COOH ltH

2O lt NH

3lt CH

3Ominus lt CH

3Sminus

00(reference) 118 611 2179 2324

4 Summary



References












































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




CH3COOH ltH

2O lt NH

3lt CH

3Ominus lt CH

3Sminus

00(reference) 118 611 2179 2324

4 Summary



References












































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

research article nucleophilicity index based on atomic...

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