research article koopmans analysis of chemical...

Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 348415 14 pageshttpdxdoiorg1011552013348415

Research ArticleKoopmansrsquo Analysis of Chemical Hardness withSpectral-Like Resolution

Mihai V Putz

Laboratory of Structural and Computational Chemistry Biology-Chemistry Department West University of TimisoaraPestalozzi Street No 16 300115 Timisoara Romania

Correspondence should be addressed to Mihai V Putz mv putzyahoocom

Received 29 April 2013 Accepted 17 June 2013

Academic Editors A Avramopoulos and D Catone

Copyright copy 2013 Mihai V PutzThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Three approximation levels of Koopmansrsquo theorem are explored and applied the first referring to the inner quantum behavior of theorbitalic energies that depart from the genuine ones in Fock space when the wave-functionsrsquo Hilbert-Banach basis set is specified tosolve the many-electronic spectra of spin-orbitalsrsquo eigenstates it is the most subtle issue regarding Koopmansrsquo theorem as it bringsmany critics and refutation in the last decades yet it is shown here as an irrefutable ldquoobservationalrdquo effect through computationspecific to any in silico spectra of an eigenproblem the second level assumes the ldquofrozen spin-orbitalsrdquo approximation during theextracting or adding of electrons to the frontier of the chemical system through the ionization and affinity processes respectivelythis approximation is nevertheless workable for great deal of chemical compounds especially organic systems and is justified forchemical reactivity and aromaticity hierarchies in an homologue series the third and the most severe approximation regards theextension of the second one to superior orders of ionization and affinities here studied at the level of chemical hardness compact-finite expressions up to spectral-like resolution for a paradigmatic set of aromatic carbohydrates

1 Introduction

In modern structural chemistry [1] the Aufbau principlefor atomic periodicity [2] Huckel molecular orbital theory[3] reinforced by Hartree-Fock theory [4 5] and then com-pleted by the density functional theory (DFT) [6ndash12] are allcornerstones in modeling and predict physical behavior ofmany electronic systems from atoms to molecules and solidstates However the chemical regime of this wide physicalrange of many-electronic manifestation was often reduced tothe frontier electrons [13 14] targeting the chemical reactivityas a special and specific way of interaction thus defininga proper chemical orthogonal space while generalizing thecustom Cartesian-physical one [15] In this context thedensity functional theory (DFT) developed in the celebratedconceptual chemical reactivity theory where for instance theclassical concept of electronegativity [16 17] acquired newand fruitful shape and formulation [18 19] along promotingother informational indices as such chemical hardness [20ndash24] chemical action [25 26] and electrophilicity [27ndash29]

with the allied principles [30ndash36] which ultimately deliveredthe present rationalization of chemical bond by min-maxvariational principles in a coupled interrelation within thebonding chemical scenario [37 38] However apart fromthese somehow global frontier indices there remains theinquiring behavior of frontier orbitals themselves duringthe electronic charge transfer encountering in chemicalreactivity which at the limit obey the Koopmans theorem(KT) [39] since approximately not depending on the numberof electrons they host in the course of chemical reaction[14 40] Of course historically the Koopmans theorem wasdeveloped within Hartree-Fock theory giving the route topredict ionization potentials [41] while giving simple phys-ical interpretation to the eigen value of the Fock matrix andjustifying the existence of the ldquoorbitalsrdquo by their observableenergies [42 43]

Although criticized [44 45] because its inner definitionexcludes electronic relaxation effects at the orbitalsrsquo levelsthat is neglecting the electronic correlation Koopmanstheorem resisted through time since proving versatile ways

2 The Scientific World Journal

for avoiding or including the ldquomissing correlation infor-mationrdquo by remarkable methods Among most preeminentapproaches in this direction are

(i) the electron propagation theory (through consideringthe self-energy operator) [46]

(ii) disproving the existence of ionization potential as thelowest eigenvalues of KT but generalizing it to thearbitrarily close value to IP [43]

(iii) interpreting the self-consistent Hartree-Fock field ascoupled harmonic oscillators evolving in a nonlinearpotential [47]

(iv) variationally extending KT to restricted open-sellscanonical orbitals which nevertheless overestimatethe Aufbau principle [48]

(v) differentiating between vertical and adiabatic ioniza-tion potentials for the strongest line of the band andfor the 0 rarr 0 band transition respectively [49]

(vi) establishing the connectionwithDFT through Janakrsquostheorem and proving its reliability for largemolecularsystems (including fullerenes or boron nitride nan-otubes B

48N48) [50]

(vii) including the negative electron affinity extensionswithin DFT for halogenated small organic molecules[51]

(viii) establishing the direct connection of the frontierorbitals with the pi-electrons and of the electronictransfer of conjugated aromatic systems [52]

(ix) driving the electronic transfer in alfa-substitutedorganic polymers [53] providing with optical spec-tra analysis for intervalence complexes formed byorganic bridges between radical ions [54]

(x) till the modeling of anti-inflammatory activities ofclinical drugs acting through ionization processes inspecial [55] and by chemical reactivity indices andDFT in general [56ndash62]

It is the last context in which the present work is placedtoo it reviews the Koopmans method with the Hartree-Focktheory while emphasizing on the generality of the methodthere where a limitation was previously identified it willbe connected then with chemical hardness by means ofLUMO-HOMO gap that eventually cancels the correlationsand relaxation opposite effects appareling on the successivehighest occupied and lowest unoccupied molecular orbitalsit will end with an illustration on how chemical hardnessanalysis based onKoopmans superior orders orbitals is fittingwith compact-finite difference expression of it in the highestanalytical accuracy of spectral-like resolution (SLR) and howthese two faces of the chemical hardness generally asses thearomaticity hierarchy along a homological series of organicmolecules

2 Reviewing the Koopmans Method

21 Nature of Koopmansrsquo Theorem within Hartree-Fock For-malism Consider the many-electronic wave-function ket-vector in N-dimensional Hilbert spin-orbitalsrsquo space

10038161003816100381610038161003816Ψ(119873)

0⟩ =

10038161003816100381610038161205951 sdot sdot sdot 120595119886 sdot sdot sdot 120595119873⟩(1)

constructed from one-electronic Fock eigenstates

11989110038161003816100381610038161003816120595119886⟩ = 120576

119886

1003816100381610038161003816 120595119886⟩ 119886 = 1119873 (2)

with the Fock operator

119891 = ℎ +

119873

sum

119887=1

[119869119887minus 119887] (3)

written (in atomic units) in terms of the kinetic + potentialoperator

ℎ = minus1

2nabla2

1minussum

119860

119885119860

1199031119860

(4)

accounting for the one-electronic motion in the potential ofthe nucleinucleus in a molecularatomic system to it oneadds the resulting energy remaining from subtraction of theexchange influence

119870119887(1) = int1198892120595

lowast

119887(2) 119903minus1

12120595119887(1) = int1198892120595

lowast

119887(2) 119903minus1

12weierp12120595119887(2)

(5)

from the interelectronic Coulombic interaction

119869119887(1) = int1198892120595

lowast

119887(2) 119903minus1

12120595119887(2) (6)

Note in the exchange energy (5) the appearance of the inter-changing particle operator weierp

12which acts by interchanging

1 harr 2 particles on all wave functions on its right-had sideit also helps in identifying the Hartree-Fock potential in theFock operator rewritten as

119891 (1) = ℎ (1) + V119867119865(1)

= ℎ (1) +

119873

sum

119887=1

int1198892120595lowast

119887(2) 119903minus1

12(1 minus weierp

12) 120595119887(2)

(7)

The Scientific World Journal 3

Altogether this formalism allows for expressing the totaleigenenergy of the N-electronic system to be successivelywritten as follows

119864119873= ⟨Ψ(119873)

0

1003816100381610038161003816100381610038161003816100381610038161003816Ψ(119873)

0⟩

=

119873

sum

119886=1

⟨120595119886

10038161003816100381610038161003816ℎ10038161003816100381610038161003816120595119886⟩

+1

2

119873

sum

119886=1119887=1

[⟨120595119886

10038161003816100381610038161003816int 1198892120595

lowast

119887(2) 119903minus1

12120595119887(2)10038161003816100381610038161003816120595119886⟩

minus⟨120595119886

10038161003816100381610038161003816int 1198892120595

lowast

119887(2) 119903minus1

12weierp12120595119887(2)10038161003816100381610038161003816120595119886⟩]

=

119873

sum

119886=1

⟨120595119886

10038161003816100381610038161003816ℎ10038161003816100381610038161003816120595119886⟩

+1

2

119873

sum

119886=1119887=1

[int11988911198892120595

lowast

119886(1) 120595lowast

119887(2) 119903minus1

12120595119887(2) 120595119886(1)

minus int 11988911198892120595lowast

119886(1) 120595lowast

119887(2) 119903minus1

12weierp12120595119887(2) 120595119886(1)]

=

119873

sum

119886=1

⟨120595119886

10038161003816100381610038161003816ℎ10038161003816100381610038161003816120595119886⟩

+1

2

119873

sum

119886=1119887=1

[int11988911198892120595

lowast

119886(1) 120595lowast

119887(2) 119903minus1

12120595119887(2) 120595119886(1)

minus int 11988911198892120595lowast

119886(1) 120595lowast

119887(2) 119903minus1

12120595119887(1) 120595119886(2)]

=

119873

sum

119886=1

⟨120595119886

10038161003816100381610038161003816ℎ10038161003816100381610038161003816120595119886⟩

+1

2

119873

sum

119886=1119887=1

[int11988911198892120595

lowast

119886(1) 120595119886(1) 119903minus1

12120595lowast

119887(2) 120595119887(2)

minus int 11988911198892120595lowast

119886(1) 120595119887(1) 119903minus1

12120595lowast

119887(2) 120595119886(2)]

equiv

119873

sum

119886=1

⟨120595119886

10038161003816100381610038161003816ℎ10038161003816100381610038161003816120595119886⟩ +

1

2

119873

sum

119886=1119887=1

[⟨119886119886 | 119887119887⟩ minus ⟨119886119887 | 119887119886⟩]

equiv

119873

sum

119886=1

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

⟨119886119887 | 119886119887⟩

(8)

The issue appears while noticing the Fock operator functionaldependency on the occupied spin orbitals once the functions120595119887(2) are known (say as a basis set)119891 becomes a well-defined

Hermitic operator with infinite eigenstates and functions itallows therefore distinction between

(i) the first lowestN spin-orbitals occupied in the overallwave-function |Ψ(119873)

0⟩ = |120595

1sdot sdot sdot 120595119886sdot sdot sdot 120595119873⟩

(ii) the rest (from N up to infinity) virtual of unoccupiedorbitals formally denoted as 120595

119903 120595119904

In this computational context the orbitals extend their spec-trum with the general eigenenergies as follows

120576119894=1infin

= ⟨120595119894

1003816100381610038161003816100381611989110038161003816100381610038161003816120595119894⟩

= ⟨120595119894

1003816100381610038161003816100381610038161003816100381610038161003816

(ℎ +

119873

sum

119887=1

[119869119887minus 119887])

1003816100381610038161003816100381610038161003816100381610038161003816

120595119894⟩

= ⟨120595119894

10038161003816100381610038161003816ℎ10038161003816100381610038161003816120595119894⟩ +

119873

sum

119887=1

[⟨120595119894

10038161003816100381610038161003816119869119887

10038161003816100381610038161003816120595119894⟩ minus ⟨120595

119894

10038161003816100381610038161003816119887

10038161003816100381610038161003816120595119894⟩]

equiv ⟨11989410038161003816100381610038161003816ℎ10038161003816100381610038161003816119894⟩ +

119873

sum

119887=1

[⟨119894119894 | 119887119887⟩ minus ⟨119894119887 | 119887119894⟩]

equiv ⟨11989410038161003816100381610038161003816ℎ10038161003816100381610038161003816119894⟩ +

119873

sum

119887=1

⟨119894119887 | 119894119887⟩

(9)

The important point here is that when turning the lastequation into the orbital eigen-energies of the occupiedorbitals

120576119886=1119873

= ⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

119873

sum

119887=1

119887 = 119886

⟨119886119887 | 119886119887⟩ (10)

and of those left unoccupied

120576119903=119873+1infin

= ⟨11990310038161003816100381610038161003816ℎ10038161003816100381610038161003816119903⟩ +

119873

sum

119887=1

⟨119903119887 | 119903119887⟩ (11)

the summation upon the energies of the occupied spinorbitals yields the interesting result

119873

sum

119886=1

120576119886=

119873

sum

119886=1

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

119873

sum

119886=1119887=1

⟨119886119887 | 119886119887⟩ = 119864119873 (12)

Equation (12) does not exactly recovering the previous totalenergy of the N-occupied spin-orbitals Equation (8) whenthey were considered ldquofree (not depending)rdquo of computation(basis set) however this may be considered as in silicomanifestation of quantum ldquoobservabilityrdquo (once a basis setrepresentation applies) which destroys the quantum systemin itself rsquos (or eigen) manifestation Here the mathematicalproperties of eigenfunction computed upon a given basis onHilbert-Banach spaces determine the ldquoshiftrdquo or the ldquounre-alisticrdquo energies of orbitals since spanning those occupiedand unoccupied alike from the present dichotomy basicallyfollows all critics on theHartree-Fock formalism and of alliedmolecular orbital theory Koopmansrsquo ldquotheoremrdquo includedinstead there seems that such departure of the computedfrom the exact energy orbitals is inherent to quantumformalism and not necessary a weakness of the Hartree-formalism itself since it will appear to any quantum many-particle problem involving eigenproblems

Now returning to the previous occupied and unoccupiedorbital energy one may assume (Koopmansrsquo ansatz) that onthe frontier levels of a many-electronic system extractingor adding of an electron (or even few of them but lesserthan the total number of valence electrons) will not affect theremaining (or119873plusmn1119873plusmn1plusmn11015840 electronic orbitals) stateson successive levels and not successive electrons on levels (seeFigure 1)

This approach allows simplifying of the common termsand emphasizing only on the involving frontier orbitals


120576p|p⟩

120576s|s⟩

120576r|r⟩

120576c|c⟩

120576d|d⟩

120576LUMO(3)

120576LUMO(2)

120576LUMO(1)

120576HOMO(1)

120576HOMO(2)

120576HOMO(3)120576e|e⟩

120576

Figure 1The paradigmatic in silico spectra of the first three highestoccupied and lowest unoccupied molecular orbitals illustrating therespective successive ionization and affinities energies as providedby Koopmansrsquo theorem Note that KT implies ionization and affinityof one electron on successive levels and not of successive electronson levels see the marked occupied and virtual spin orbitals

participating in chemical reactivity Accordingly for the firstionization potential one successively obtains (see Figure 1)

1198681198751= 119864119873minus1minus 119864119873

= ⟨Ψ(119873minus1)

119888

1003816100381610038161003816100381610038161003816100381610038161003816Ψ(119873minus1)

119888⟩ minus ⟨Ψ

(119873)

0

1003816100381610038161003816100381610038161003816100381610038161003816Ψ(119873)

0⟩

=

119873

sum

119886=1

119886 = 119888

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

⟨119886119887 | 119886119887⟩

minus

[[

[

119873

sum

119886=1

119886 = 119888

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ + ⟨119888

10038161003816100381610038161003816ℎ10038161003816100381610038161003816119888⟩]]

]

+1

2

[[[

[

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

⟨119886119887 | 119886119887⟩ +

119873

sum

119886=1119887=1

119886 = 119888119887=119888

⟨119886119888 | 119886119888⟩

+

119873

sum

119886=1119887=1

119886=119888119887 = 119888

⟨119888119887 | 119888119887⟩

]]]

]

= minus⟨11988810038161003816100381610038161003816ℎ10038161003816100381610038161003816119888⟩ minus

119873

sum

119887=1

⟨119888119887 | 119888119887⟩

= minus120576119888= minus120576HOMO(1)

(13)

Remarkable in this analytics one starts with in se quantumexpression of total energies of the N and (119873 minus 1) systemsand ends up with a result characteristic to the computational(shifted) realm since recovering the orbital energy of the insilico state fromwhich the electronwas removed Yet onemayask how such in se to in silico quantum chemical passage ispossible the answer is naturally positive since the previousderivation associates with the ionization process which isbasically an observer intervention to the genuine quantumsystem from where the final result will reflect the energeticdeviation from in se to in silico as an irrefutable quantummanifestation of electronic system

Similarly for electronic affinity one will act on thein se quantum system to add an electron at the frontierlevel and under the ldquofrozen spin-orbitalsrdquo physical-chemicalassumption one gets the energetic turn from the genuine HFexpression to the in silico orbital energy on which the ldquoactionrdquowas undertaken (see Figure 1)

EA1= 119864119873minus 119864119873+1

= minus ⟨11990310038161003816100381610038161003816ℎ10038161003816100381610038161003816119903⟩ minus

119873

sum

119887=1

⟨119903119887 | 119903119887⟩

= minus 120576119903

= minus 120576LUMO(1)

(14)

These results are usually considered as defining the popularKoopmans theorem used for estimating the observablequantities as ionization potential and electronic affinity interms of ldquoartefactualrdquo computed orbital energies (first appro-ximation) and in the spin-orbitalic frozen framework duringthe electronic extraction or addition (the second approxima-tion)

22 Compact-Finite Chemical Hardness Koopmansrsquo Approa-ches In modern quantum chemical reactivity approacheschemical hardness plays a preeminent role due to fruitful


connections it establishes with principles of hard and softacids and bases (HSAB) andmaximumhardness prescriptionfor stabilizing a reactive compound [30ndash32 36 37] Mutatismutandis also gives a reliable measure for the degree of aro-maticity a chemical structure displays in isolate or interactionstates [63ndash66] While being defined within the conceptualdensity functional theory as the second-order derivative orthe total N-electronic energy with respect to the number ofchanging charges it recently acquired a significant extendedworking form by considering its compact-finite differenceunfolding up to the third ionization and affinity order ofelectronic removing and attaching processes [15 24 36 63]

2120578 =1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873⟩

cong 21198862

119864119873+1minus 2119864119873+ 119864119873minus1

2+ 1198872

119864119873+2minus 2119864119873+ 119864119873minus2

4

+ 1198882

119864119873+3minus 2119864119873+ 119864119873minus3

9

minus 1205722(1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873minus1⟩

+1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873+1⟩

)

minus 1205732(1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873minus2⟩

+1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873+2⟩

)

= 21198862

119864119873+1minus 2119864119873+ 119864119873minus1

2+ 1198872

119864119873+2minus 2119864119873+ 119864119873minus2

4

+ 1198882

119864119873+3minus 2119864119873+ 119864119873minus3

9

minus 1205722(21198862

119864119873minus 2119864119873minus1+ 119864119873minus2

2

+21198862

119864119873+2minus 2119864119873+1+ 119864119873

2)

minus 1205732(21198862

119864119873minus1minus 2119864119873minus2+ 119864119873minus3

2

+21198862

119864119873+3minus 2119864119873+2+ 119864119873+1

2)

= 21198862(1 + 2120572

2minus 1205732)119864119873+1+ 119864119873minus1

2

+ (811988621205732+ 1198872minus 411988621205722)119864119873+2+ 119864119873minus2

4

+ (1198882minus 911988621205732)119864119873+3+ 119864119873minus3

9

minus (21198862+1

21198872+2

91198882+ 211988621205722)119864119873

(15)

Note that this expansion may in principle correspondingto the parabolic expansion of the energy containing dou-ble charged cationsanions which considerably expand thechemical reactivity analysis towards considering ceding

accepting electronic pairing (dicationsdianions) or trans-ferring entire chemical bonds in molecular interactionswhile the accuracy depends on how one refines the finitedifferentiation schemes (see below) the limits are restrictedto the frontier or valence electrons in bonding

The last equation may be rewritten in terms of theobservational quantities as the ionization potential (IP) andelectronic affinity (EA) of the involved eigen-energies of 119894th(119894 = 1 2 3) order

IP119894= 119864119873minus119894minus 119864119873minus119894+1

EA119894= 119864119873+119894minus1

minus 119864119873+119894

(16)

through the energetic equivalents for the respective sums

119864119873+1+ 119864119873minus1

= (IP1minus EA1) + 2119864

119873

119864119873+2+ 119864119873minus2

= (IP1minus EA1) + (IP

2minus EA2) + 2119864

119873

119864119873+3+ 119864119873minus3


2minus EA2) + (IP

3minus EA3) + 2119864

119873

(17)

to provide the working expression [15 24 36 63]

120578IPminusEACFD = [119886

2(1 minus 120572

2+ 21205732) +1

41198872+1

91198882]IP1minus EA1

2

+ [1

21198872+2

91198882+ 21198862(1205732minus 1205722)]

IP2minus EA2

4

+ [1

31198882minus 311988621205732]IP3minus EA3

6

(18)

whose coefficients are given in Table 1 being obtained bymatching the previous expansion (15) with Taylor seriesexpansions in various orders (from second to tenth order)the results are not system dependent being susceptible to avariety of the boundary conditions [67]

It is worth remarking that when particularizing thisformula for the fashioned two-point central finite differencethat is when having 119886

2= 1 119887

2= 1198882= 1205722= 1205732= 0 of Table 1

one recovers the basic chemical hardness as prescribed by thecelebrated Pearson nucleophilic-electrophilic reactivity gap[20ndash22]

1205782C =

IP1minus EA1

2(19)

already used as measuring the aromaticity through themolecular stability against the reaction propensity [64 65]

At this point the third level of Koopmansrsquo approximationmay be considered namely through extending the secondpart of Koopmansrsquo theorem as given by the identification ofthe IP andEAwith the (minus) energies of the in silico highestoccupied (molecular) orbital (HOMO

1) and with the lowest

unoccupied (molecular) orbital (LUMO1) to superior levels

of HOMO119894=123

and LUMO119894=123

respectively

IP119894= minus120576HOMO(119894)

EA119894= minus120576LUMO(119894)

(20)


Table 1 Numerical parameters for the compact finite second (2C)-fourth (4C)- and sixth (6C)-order central differences standardPade(SP) schemes sixth (6T)- and eight (8T)-order tridiagonal schemeseighth (8P)- and tenth (10P)-order pentadiagonal schemes up tospectral-like resolution (SLR) schemes for chemical hardness of (18)[15 24 36 63]

Scheme 1198862

1198872

1198882

1205722

1205732

2C 1 0 0 0 04C 1333 minus0333 0 0 06C 1091 0273 0 0182 0SP 12 0 0 01 06T 15 minus06 02 0 08T 0967 0537 minus003 0237 08P 0814 0789 0 0292 00110P 0592 1155 0044 0372 0024SLR 0216 1723 0177 0502 0056

With this assumption one yields the in silico-superior order-freezing spin-orbitals compact-finite difference (CFD) form ofchemical hardness [15 24 36 63] as follows

120578LUMO-HOMOCFD

= [1198862(1 minus 120572

2+ 21205732) +

1

41198872+1

91198882]

times120576LUMO(1) minus 120576HOMO(1)

2

+ [1

21198872+2

91198882+ 21198862(1205732minus 1205722)]


4

+ [1

31198882minus 311988621205732]120576LUMO(3) minus 120576HOMO(3)

6

(21)

However one may ask whether this approximation is validand in which conditions This can be achieved by reconsid-ering the previous Koopmans first-order IP and EA to thesuperior differenceswithinHartree-Fock framework as suchfor the second order of ionization potential one gets (seeFigure 1)

IP2

= 119864119873minus2minus 119864119873minus1

=

119873

sum

119886=1

119886 = 119888

119886 = 119889

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887 = 119889

⟨119886119887 | 119886119887⟩

minus

119873

sum

119886=1

119886 = 119888

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

⟨119886119887 | 119886119887⟩

=

119873

sum

119886=1

119886 = 119888

119886 = 119889

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887 = 119889

⟨119886119887 | 119886119887⟩

minus

119873

sum

119886=1119886 = 119888

119886 = 119889

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ + ⟨119889

10038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887 = 119889

⟨119886119887 | 119886119887⟩

+1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886=119889119887 = 119889

⟨119889119887 | 119889119887⟩ +1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887=119889

⟨119886119889 | 119886119889⟩

= minus⟨11988910038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩ minus

119873

sum

119886=1119887=1

119887 = 119889

⟨119889119887 | 119889119887⟩

= minus120576119889= minus120576HOMO(2)

(22)

Note that this derivation eventually employs the equivalencyfor the Coulombic and exchange terms for orbitals of thesame nature (with missing the same number of spin orbitalssee Figure 1) However in the case this will be furtherrefined to isolate the first two orders of highest occupiedmolecular orbitals the last expression will be corrected withHOMO

1HOMO

2(Coulombic and exchange) interaction to

successively become

IP2= 119864119873minus2minus 119864119873minus1

= minus⟨11988910038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩

minus

1

2

119873

sum

119886=1119887=1

119886 = 119888

119886=119889119887 = 119889

⟨119889119887 | 119889119887⟩ +1

2⟨119889119888 | 119889119888⟩

+1

2

119873

sum

119886=1119887=1

119887 = 119888

119886 = 119889119887=119889

⟨119886119889 | 119886119889⟩ +1

2⟨119888119889 | 119888119889⟩

= minus⟨11988910038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩ minus

119873

sum

119886=1119887=1

119887 = 119889

⟨119889119887 | 119889119887⟩ + ⟨119888119889 | 119888119889⟩

= minus120576119889+ ⟨119888119889 | 119888119889⟩

= minus120576HOMO(2)+ ⟨HOMO1HOMO

2| HOMO

1HOMO

2⟩

(23)


However reloading this procedure for electronic affinityprocess too one gets

EA2= 119864119873+1minus 119864119873+2

= minus120576LUMO(2) + ⟨LUMO1LUMO

2| LUMO

1LUMO

2⟩

(24)

When combining (24) with its IP counterpart (23) withinthe chemical hardness extended CFD analysis of (18)there appears that the simple Koopmansrsquo orbitals energydifference is corrected by the HOMO

1HOMO

2versus

LUMO1LUMO

2as follows

IP2minus EA2= 120576LUMO(2) minus 120576HOMO(2)

+ (⟨HOMO

1HOMO

2| HOMO

1HOMO

2⟩

minus ⟨LUMO1LUMO

2| LUMO

1LUMO

2⟩)

(25)

This expression is usually reduced to the superior orderLUMO-HOMO difference

IP2minus EA2cong 120576LUMO(2) minus 120576HOMO(2) (26)

due to the energetic spectra symmetry of Figure 1 relayingon the bonding versus antibonding displacements of orbitalsspecific to molecular orbital theory Therefore with thepremise that molecular orbital theory itself is correct or atleast a reliable quantum undulatory modeling of multielec-tronic systems moving in a nuclei potential the above IP-EAdifferences in terms of Koopmansrsquo in silico LUMO-HOMOenergetic gaps hold also for superior orders

An illustrative analysis for homologues organic aromatichydrocarbons regarding how much the second and the thirdorders respectively of the IP-EA or LUMO-HOMO gapsaffect the chemical hardness hierarchies and therefore theirordering aromaticity will be exposed and discussed in thenext section

3 Application on Aromatic Basic Systems

It is true that Koopmans theorem seems having somelimitations for small molecules and for some inorganiccomplexes [44 45] however one is interested here in testingKoopmansrsquo superior ordersrsquo HOMO-LUMO behavior onthe systems that work such as the aromatic hydrocarbonsAccordingly in Table 2 a short series of paradigmaticorganics is considered with one and two rings and variousbasic ring substitutions or additions respectively [66]For them the HOMO and LUMO are computed withinsemiempirical AM1 framework [68] till the third order ofKoopmans frozen spin-orbitalsrsquo approximation they are thencombinedwith the various finite difference forms (from 2C toSLR) of chemical hardness as mentioned above (see Table 1)and grouped also in sequential order respecting chemicalhardness gap contributions (ie separately for LUMO1-HOMO1 LUMO1-HOMO1 LUMO2-HOMO2 andLUMO1-HOMO1 LUMO2-HOMO2 LUMO3-HOMO3)the results are systematically presented in Tables 3ndash5 The

results of Tables 3ndash5 reveal very interesting features inthe light of considering the aromaticity as being reliablymeasured by chemical hardness alone since both associatewith chemical resistance to reactivity or the terminus ofa chemical reaction according to the maximum chemicalhardness principle [30 31]

Moreover the benchmark-ordering hierarchywas chosenas produced by Huckel theory and approximation sinceclosely related with pi-electrons delocalized at the ringlevel as the main source of the experimentally recordedaromaticity of organic compounds under study [69] Notethat although computational method used here is of lowlevel it nevertheless responds to present desiderate hav-ing a non (orbitalic) basis-dependent computational outputand discussion whereas further (Hartree-Fock) ab initio(Moslashller-Plesset) perturbation methods and basis set depen-dency considerations as HF MP2 and DFT respectively forinstance can be further considered for comparative analysisIn these conditions themainKoopmansrsquo analysis of chemicalhardness or aromaticity behavior for the envisagedmoleculesleaves with relevant observations

(i) In absolutely all cases analytical or computationalthe first two molecules Benzene (I) and Pyrim-idine (II) are inversed for their chemical hard-nessaromaticity hierarchies respecting the bench-marking Huckel one meaning that even in the mostsimple case (say 2CLUMO1-HOMO1) double sub-stitution of carbon with nitrogen increases the ringstability most probably due to the additional pairingof electrons entering the pi-system as coming fromthe free valence of nitrogen atoms in molecular ringThis additional pair of electrons eventually affects byshielding also the core of the hydrocarbon rings thatis the sigma system of Pyrimidine (II) in a specificquantum way not clearly accounted by the Huckeltheory

(ii) The same behavior is recorded also for the couple ofmolecules I and III (Pyridine) however only for theSLR of chemical hardness computed with second andthe third orders of Koopmans frozen spin-orbitalsthis suggests the necessary insight the spectral likeresolution analysis may provide respecting the otherforms of finite compact differences in chemical hard-ness computation yet only when it is combined withhigher Koopmans HOMO and LUMO orbitals

(iii) In the same line of discussion only for the secondand the third Koopmans orders and only for theSLR chemical hardness development that is thelast columns of Tables 4 and 5 one records similarreversed order of the molecules 2-Napthol (VII) and1-Naphtol (VIII) with the more aromatic characterfor the last case when having the OH group moreclosely to the middy of the naphthalene structureit is explained as previous due to the electronicpair of chemical-bonding contribution more closeto the ldquocorerdquo of the system with direct influenceto increase the shielding electrons of the sigma


Table 2 Molecular structures of paradigmatic aromatic hydrocarbons [66] ordered downwards according with their Huckel first-orderHOMO-LUMO gap [69] along their first three highest occupied (HOMOs) and lowest unoccupied (LUMOs) (in electron-volts (eV))computationally recorded levels within semiempirical AM1 method [68]

FormulaNameCASIndex (mw [gmol])

MolecularStructure HOMO(1) HOMO(2) HOMO(3) LUMO(1) LUMO(2) LUMO(3)

C6H6Benzene71-43-2I (7811)

minus9652904 minus9653568 minus11887457 0554835 0555246 2978299

C4H4N2Pyrimidine289-95-2II (80088)

minus10578436 minus10614932 minus11602985 minus0234993 minus0081421 2543489

C5H5NPyridine110-86-1III (7910)


C6H6OPhenol108-95-2IV (9411)


C6H7NAniline62-53-3V (9313)


C10H8Naphthalene91-20-3VI (12817)

minus8710653 minus9340973 minus10658237 minus0265649 0180618 1210350

C10H8O2-Naphthol135-19-3VII (14417)


C10H8O1-Naphthol90-15-3VIII (14417)



Table 2 Continued



C10H9N2-Naphthalenamine91-59-8IX (14319)


C10H9N1-Naphthalenamine134-32-7X (14319)


Table 3 Chemical hardness values (in eV) as computed for molecules of Table 2 with first-order LUMO(1)-HOMO(1) gap order of (21) withparameters of Table 1

Molecule 1205782C 1205784C 1205786C 120578SP 1205786T 1205788T 1205788P 12057810P 120578SLR

I 510387 6379837 4903511 5512179 7003643 4434762 4030827 3542746 2971354II 5171722 6464652 4968699 5585459 7096751 4493719 4084414 3589844 3010856III 5035515 6294393 4837839 5438356 6909845 4375368 3976843 3495299 2931559IV 4756227 5945284 4569516 5136725 65266 4132695 3756273 3301437 2768964V 4486057 5607571 4309951 4844941 6155866 3897943 3542904 3113904 2611677VI 4222502 5278128 4056743 4560302 5794211 366894 3334759 2930963 2458242VII 4146325 5182906 3983556 447803 5689679 360275 3274597 2878086 2413893VIII 4104214 5130268 3943098 4432551 5631894 356616 324134 2848856 2389378IX 4026496 503312 3868431 4348616 5525247 349863 3179962 2794909 2344132X 3966748 4958435 3811029 4284088 544326 3446715 3132775 2753437 2309348

Table 4 Chemical hardness values (in eV) as computed for molecules of Table 2 with first-order LUMO(1)-HOMO(1) and second-orderLUMO(2)-HOMO(2) gaps of (21) with parameters of Table 1




Table 5 Chemical hardness values (in eV) as computed for molecules of Table 2 with first-order LUMO(1)-HOMO(1) second-orderLUMO(2)-HOMO(2) and third-order LUMO(3)-HOMO(3) gaps of (21) with parameters of Table 1



systems while leading with smoothly increased sta-bilization contribution (enlarging also the sigma-pichemical gap) yet this is manifested when all thespectral-like resolution complexity is considered inchemical hardness expression and only in superiorKoopmans orders (second and third) otherwise notbeing recorded However this result advocates themeaningful of considering the SLR coupled withsuperiorKoopmans analysis in revealing subtle effectsin sigma-pi aromatic systems

(iv) In the rest of cases the Huckel downward hierarchy ofTable 2 is recovered in Tables 3ndash5 in a systematic way

(v) When going from 2C to SLR chemical hardnessanalytical forms of any of Koopmans orders on thehorizontal axis through the Tables 3ndash5 one system-atically records an increasing of the average chemicalhardnessaromaticity values from 2C to 6T schemesof computations while going again down towards SLRscheme of Table 1

All in all one may compare the extreme 2C and SLRoutputs of Tables 3ndash5 for a global view for the Koopmansrsquobehavior respecting various orders and chemical hardnessschemes of (compact-finite forms) computations the resultis graphically presented in Figure 2 The analysis of Figure 2yields a fundamental result for the present study that is thepractical identity among

(i) all Koopmans superior orbitals-based chemical hard-ness computations

(ii) the simplest 2C and the complex SLR analytical formsfor compact-finite difference schemes of chemicalhardness for the superior HOMO-LUMO gap exten-sions

By contrary to someone expecting the first order of Koop-mans theorem being more systematic only in this order the2C values are practically doubled respecting SLR counterpartsuch double behavior becomes convergent when superiorKoopmans orders of valence orbitals are considered either insimpler or complex forms of 2C and SLR respectively

This may lead to the fruitful result according which theKoopmans theorem works better when superior HOMO-LUMO frozen spin orbitals are considered probably due tocompensating correlating effects that such extension impliessee analytical discussion in the last section In any case thepresent molecular illustration of Koopmansrsquo approximationsto chemical harness computation clearly shows that at leastfor organic aromatic molecules it works better for superiororders of ldquofreezingrdquo spin orbitals and is not limitative tothe first valence orbitals as would be the common beliefMoreover it was also clear that the Koopmans theorem finelyaccords also with more complex ponder of its superior orderorbitals in chemical hardness expansions equation (21) whensubtle effects in lone pairing electrons (since remained orbitalis frozen upon successive electronic attachmentremovalsonfrom it) or chemical bonding pair of electrons influencethe aromatic ring core towards increasing its shielding andthe overall molecular reactivity resistance All these concep-tual and computational results should be further extendedand tested on increased number of molecules enlarging theirvariety too as well as by considering more refined quantumcomputational frameworks as the density functional theoryand (Hartree-Fock) ab initio schemes are currently comparedand discussed for various exchange-correlation and parame-terization limits and refutations

4 Conclusions

Koopmansrsquo theorem entered on the quantum chemistry asa versatile tool for estimating the ionization potentials forclosed-shells systems and it waswidely confirmed for organicmolecular systems due to the inner usually separationbetween sigma (core) and pi (valence) subelectronic systemsallowing to treat the ldquofrozen spin orbitalsrdquo as orbitals notessentially depending on the number of electrons in thevalence shells when some of them are extracted (via ioniza-tion) or added (via negative attachments) this approximationultimately works for Hartree-Fock systems when electroniccorrelationmay be negligible or cancels with the orbital relax-ations during ionization or affinity processes respectivelynaturally it works less when correlation is explicitly counted


1

2

3

4

5120578

(eV

)

I II III IV V VI VII VIII IX XMolecule

2C1SLR12C1 2

2C1 2 3SLR1 2 3

SLR1 2

NN

NOH

OH OH NH2NH2

NH2

Figure 2 Representation of the 2C and SLR chemical hardness hierarchies for the set of molecules of Table 1 upon the first second and thirdorders of Koopmansrsquo theorem applications as presented din Tables 3 4 and 5 respectively

as in density functional theory where instead the exchangeenergies are approximated or merely parameterized so thatldquoloosingrdquo somehow on the genuine spin-orbital nature of themono-determinantal approach of the Hartree-Fock with anatural energetic hierarchy included Despite the debatingcontext in which Koopmans theorem is valid or associateswith a physical-chemical sense the present work gives someinsight in this matter by clarifying upon some key features ofKoopmans analysis namely

(i) the Hartree-Fock spin orbitals involved in Koop-mansrsquo theorem are of computational nature emergedthrough solving an eigen-problem in a given basisset so that being characterized by a sort of ldquoquantumshiftrdquo related with quantum uncertainty when the freesystem is affected by observationmdashhere by computa-tion so this behavior is at its turn computationallynaturally and not viewed as a conceptual error instructurally assessing a many-electronic structure

(ii) the Koopmans theorem not restrictedly refers to thefirst ionization potential and may be extended to suc-cessive ionization potentials (and electronic affinities)as far the valence shell is not exhausted by the pi-collective electrons such that the sigma-pi separationmay be kept reliable and the ldquofrozen spin-orbitalsrdquomay be considered as such through cancellation ofthe relaxation effects with the electronic correlations

both explicitly escaping to Hartree-Fock formalismthis was however here emphasized by the appearanceof the quantum terms of type ⟨HOMO

1HOMO

2|

HOMO1HOMO

2⟩ in (23) and ⟨LUMO

1LUMO

2|

LUMO1LUMO

2⟩ in (24) which were considered as

reciprocal annihilating in chemical hardnessrsquo IP-EAdifferences in (25) due to symmetrical bonding ver-sus antibonding spectra displacements in molecularorbital theorymdashas a simplified version of Hartree-Fock theory

(iii) the Koopmans theorem goes at best with chemi-cal harness or aromaticity evaluation by means ofLUMO-HOMO gaps when they manifested surpris-ingly the same for superior orders of IPs-EAs this wayconfirming the previous point

Application on a paradigmatic set of mono and doublebenzoic rings molecules supported these conclusions yetleaving enough space for furthermolecular set extensions andcomputational various frameworks comparison

Abbreviations

120578 Chemical hardnessCFD Compact-finite differenceEA Electronic affinityHOMO Highest occupied molecular orbital


HSAB Hard-and-soft acids and basesIP Ionization potentialKT Koopmans theoremLUMO Lowest unoccupied molecular orbitalSLR Spectral like resolutionDFT Density functional theory

Acknowledgment

This work was supported by the Romanian National Councilof Scientific Research (CNCS-UEFISCDI) through projectTE162010ndash2013 within the PN II-RU-TE-2010-1 framework

References

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[2] E ScerriThePeriodic Table Its Story and Its Significance OxfordUniversity Press New York NY USA 1st edition 2006

[3] C A Coulson B OrsquoLeary and R B MallionHuckel Theory forOrganic Chemists Academic Press New York NY USA 1978

[4] I N Levine Quantum Chemistry Prentice Hall EnglewoodCliffs NJ USA 1991

[5] C J Cramer Essentials of Computational Chemistry JohnWileyamp Sons Chichester UK 2002

[6] P Hohenberg and W Kohn ldquoInhomogeneous electron gasrdquoPhysical Review B vol 136 no 3 pp B864ndashB871 1964

[7] W Kohn and L J Sham ldquoSelf-consistent equations includingexchange and correlation effectsrdquo Physical Review A vol 140no 4 pp A1133ndashA1138 1965

[8] M Levy ldquoUniversal variational functionals of electron densi-ties first-order density matrices and natural spin-orbitals andsolution of the v-representability problemrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 76 no 12 pp 6062ndash6065 1979

[9] R G Parr andW YangDensity-FunctionalTheory of Atoms andMolecules Oxford University Press New York NY USA 1989

[10] N H March Electron Density Theory of Atoms and MoleculesAcademic Press New York NY USA 1992

[11] P W Ayers ldquoDensity per particle as a descriptor of coulombicsystemsrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 97 no 5 pp 1959ndash1964 2000

[12] M V Putz Quantum Theory Density Condensation andBonding Apple Academics Waretown NJ USA 2012

[13] PW Ayers and R G Parr ldquoVariational principles for describingchemical reactions the Fukui function and chemical hardnessrevisitedrdquo Journal of the American Chemical Society vol 122 no9 pp 2010ndash2018 2000

[14] T Stein J Autschbach N Govind L Kronik and R BaerldquoCurvature and frontier orbital energies in density functionaltheoryrdquo Journal of Physical Chemistry Letters vol 3 no 24 pp3740ndash3744 2012

[15] M V PutzChemical Orthogonal Spaces vol 14 ofMathematicalChemistry Monographs Faculty of Science University of Kragu-jevac Kragujevac Serbia 2012

[16] R S Mulliken ldquoA new electroaffinity scale together with dataon valence states and on valence ionization potentials andelectron affinitiesrdquo The Journal of Chemical Physics vol 2 no11 pp 782ndash793 1934

[17] R S Mulliken ldquoElectronic structures of molecules XI Elec-troaffinity molecular orbitals and dipolemomentsrdquoThe Journalof Chemical Physics vol 3 no 9 pp 573ndash585 1935

[18] R G Parr R A Donnelly M Levy and W E Palke ldquoElec-tronegativity the density functional viewpointrdquo The Journal ofChemical Physics vol 68 no 8 pp 3801ndash3807 1977

[19] M V Putz ldquoSystematic formulations for electronegativity andhardness and their atomic scales within density functionalsoftness theoryrdquo International Journal of Quantum Chemistryvol 106 no 2 pp 369ndash389 2006

[20] R G Parr and R G Pearson ldquoAbsolute hardness companionparameter to absolute electronegativityrdquo Journal of theAmericanChemical Society vol 105 no 26 pp 7512ndash7516 1983

[21] R G Pearson ldquoAbsolute electronegativity and absolute hard-ness of Lewis acids and basesrdquo Journal of the American ChemicalSociety vol 107 no 24 pp 6801ndash6806 1985

[22] R G Pearson Chemical Hardness Wiley-VCH WeinheimGermany 1997

[23] M V Putz Absolute and Chemical Electronegativity and Hard-ness Nova Science New York NY USA 2008

[24] M V Putz ldquoElectronegativity and chemical hardness differentpatterns in quantum chemistryrdquo Current Physical Chemistryvol 1 no 2 pp 111ndash139 2011

[25] M V Putz ldquoChemical action and chemical bondingrdquo Journal ofMolecular Structure (THEOCHEM) vol 900 no 1ndash3 pp 64ndash702009

[26] M V Putz ldquoChemical action concept and principlerdquo MATCHCommunications in Mathematical and in Computer Chemistryvol 66 no 1 pp 35ndash63 2011

[27] R G Parr L V Szentpaly and S Liu ldquoElectrophilicity indexrdquoJournal of the American Chemical Society vol 121 no 9 pp1922ndash1924 1999

[28] P K Chattaraj U Sarkar andD R Roy ldquoElectrophilicity indexrdquoChemical Reviews vol 106 no 6 pp 2065ndash2091 2006

[29] F de Vleeschouwer V van Speybroeck M Waroquier PGeerlings and F de Proft ldquoElectrophilicity and nucleophilicityindex for radicalsrdquoOrganic Letters vol 9 no 14 pp 2720ndash27242007

[30] RG Parr andPKChattaraj ldquoPrinciple ofmaximumhardnessrdquoJournal of the American Chemical Society vol 113 no 5 pp1854ndash1855 1991

[31] P K Chattaraj G H Liu and R G Parr ldquoThe maximumhardness principle in the Gyftopoulos-Hatsopoulos three-levelmodel for an atomic or molecular species and its positive andnegative ionsrdquoChemical Physics Letters vol 237 no 1-2 pp 171ndash176 1995

[32] R G Pearson Hard and Soft Acids and Bases DowdenHutchinson amp Ross Stroudsberg Pa USA 1973

[33] R G Pearson ldquoHard and soft acids and basesmdashthe evolution ofa chemical conceptrdquo Coordination Chemistry Reviews vol 100pp 403ndash425 1990

[34] R S Drago and R A Kabler ldquoA quantitative evaluation of theHSAB conceptrdquo Inorganic Chemistry vol 11 no 12 pp 3144ndash3145 1972

[35] R G Pearson ldquoReply to the paper ldquoa quantitative evaluation ofthe HSAB conceptrdquo by Drago and Kablerrdquo Inorganic Chemistryvol 11 no 12 p 3146 1972

[36] M V Putz N Russo and E Sicilia ldquoOn the applicability oftheHSAB principle through the use of improved computationalschemes for chemical hardness evaluationrdquo Journal of Compu-tational Chemistry vol 25 no 7 pp 994ndash1003 2004


[37] M V Putz ldquoMaximum hardness index of quantum acid-basebondingrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 60 no 3 pp 845ndash868 2008

[38] M V Putz ldquoQuantum parabolic effects of electronegativity andchemical hardness on carbon 120587-systemsrdquo in Carbon Bondingand Structures Advances in Physics and Chemistry M V PutzEd Springer London UK 2011

[39] T Koopmans ldquoUber die Zuordnung von Wellenfunktionenund Eigenwerten zu den Einzelnen Elektronen Eines AtomsrdquoPhysica vol 1 no 1ndash6 pp 104ndash113 1934

[40] J P Perdew R G Parr M Levy and J L Balduz Jr ldquoDensity-functional theory for fractional particle number derivativediscontinuities of the energyrdquo Physical Review Letters vol 49no 23 pp 1691ndash1694 1982

[41] P Politzer and F Abu-Awwad ldquoA comparative analysis ofHartree-Fock and Kohn-Sham orbital energiesrdquo TheoreticalChemistry Accounts vol 99 no 2 pp 83ndash87 1998

[42] J Katriel and E R Davidson ldquoAsymptotic behavior of atomicand molecular wave functionsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 77 no8 pp 4403ndash4406 1980

[43] M Ernzerhof ldquoValidity of the extended Koopmansrsquo theoremrdquoJournal of Chemical Theory and Computation vol 5 no 4 pp793ndash797 2009

[44] B J Duke and B OrsquoLeary ldquoNon-Koopmansrsquo moleculesrdquo Journalof Chemical Education vol 73 no 6 pp 501ndash504 1995

[45] C Angeli ldquoPhysical interpretation of Koopmansrsquo theorema criticism of the current didactic presentationrdquo Journal ofChemical Education vol 75 no 11 pp 1494ndash1497 1998

[46] G Seabra V G Zakrzewski and J V Orti ldquoElectron propa-gator theory of ionization energies and dyson orbitals for 120583-hydrido bridge-bonded molecules diborane digallane andgallaboranerdquo in Structures and Mechanisms G R Eaton W NLipscomb D CWiley and O Jardetzky Eds ACS SymposiumSeries chapter 7 American Chemcial SocietyWashington DCUSA 2002

[47] MMessina ldquoTheHartree-Fock self-consistent field an allegori-cal connection using two coupled harmonic oscillatorsrdquo Journalof Chemical Education vol 76 no 10 pp 1439ndash1443 1999

[48] B N Plakhutin and E R Davidson ldquoKoopmansrsquo theorem inthe restricted open-shell hartree-fock method 1 A variationalapproachrdquo Journal of Physical Chemistry A vol 113 no 45 pp12386ndash12395 2009

[49] Z B Maksic and R Vianello ldquoHow good is Koopmansrsquoapproximation G2(MP2) study of the vertical and adiabaticionization potentials of some small moleculesrdquo Journal ofPhysical Chemistry A vol 106 no 27 pp 6515ndash6520 2002

[50] J Luo Z Q Xue W M Liu J L Wu and Z Q YangldquoKoopmansrsquo theorem for large molecular systems within den-sity functional theoryrdquo Journal of Physical Chemistry A vol 110no 43 pp 12005ndash12009 2006

[51] M J G Peach F de Proft and D J Tozer ldquoNegative electronaffinities fromDFT fluorination of ethylenerdquo Journal of PhysicalChemistry Letters vol 1 no 19 pp 2826ndash2831 2010

[52] Y H Wei and H Y Cheng ldquoStabilized Koopmansrsquotheorem calculations on the 120587 anion states of 1458-tetrahydronaphthalenerdquo Journal of Physical Chemistry Avol 102 no 20 pp 3560ndash3564 1998

[53] M A de Oliveira H A Duarte J M Pernaut and W Bde Almeida ldquoEnergy gaps of 1205721205721015840-substituted oligothiophenesfrom semiempirical Ab initio and density functionalmethodsrdquo

Journal of Physical Chemistry A vol 104 no 35 pp 8256ndash82622000

[54] S F Nelsen M N Weaver J P Telo B L Lucht andS Barlow ldquoKoopmans-based analysis of the optical spectraof p-phenylene-bridged intervalence radical ionsrdquo Journal ofOrganic Chemistry vol 70 no 23 pp 9326ndash9333 2005

[55] K B Shakman and D A Mazziotti ldquoAssessing the efficacyof nonsteroidal anti-inflammatory drugs through the quantumcomputation of molecular ionization energiesrdquo Journal of Phys-ical Chemistry A vol 111 no 30 pp 7223ndash7226 2007

[56] C S Chen T H Feng and J S P Y Chao ldquoStabilizedKoopmansrsquo theorem calculations on the 120587 temporary anionstates of benzene and substituted benzenesrdquo Journal of PhysicalChemistry vol 99 no 21 pp 8629ndash8632 1995

[57] J Cioslowski P Piskorz and G Liu ldquoLonization potentialsand electron affinities from the extended Koopmansrsquo theoremapplied to energy-derivative densitymatrices the EKTMPn andEKTQCISD methodsrdquo Journal of Chemical Physics vol 107 no17 pp 6804ndash6811 1997

[58] S Z Lu X Y Li and J F Liu ldquoMolecular orbital analysis inevaluation of electron-transfer matrix element by Koopmansrsquotheoryrdquo Journal of Physical Chemistry A vol 108 no 18 pp4125ndash4131 2004

[59] R Vargas J Garza and A Cedillo ldquoKoopmans-like approx-imation in the Kohn-Sham method and the impact of thefrozen core approximation on the computation of the reactivityparameters of the density functional theoryrdquo Journal of PhysicalChemistry A vol 109 no 39 pp 8880ndash8892 2005

[60] L Kronik T Stein S Refaely-Abramson and R Baer ldquoExcita-tion gaps of finite-sized systems from optimally tuned range-separated hybrid functionalsrdquo Journal of Chemical Theory andComputation vol 8 no 5 pp 1515ndash1531 2012

[61] L S Rodrıguez Pirani M Gerones C O Della Vedova etal ldquoElectronic properties and dissociative photoionization ofthiocyanates Part II Valence and shallow-core (sulfur andchlorine 2p) regions of chloromethyl thiocyanate CH

2ClSCNrdquo

Journal of Physical Chemistry A vol 116 no 1 pp 231ndash241 2012[62] H Y Cheng and C W Chen ldquoEnergies and lifetimes of tem-

porary anion states of chloromethanes by stabilized Koopmansrsquotheorem in long-range corrected density functional theoryrdquoJournal of Physical ChemistryA vol 116 no 50 pp 12364ndash123722012

[63] M V Putz ldquoOn absolute aromaticity within electronegativityand chemical hardness reactivity picturesrdquoMATCH Communi-cations inMathematical and in Computer Chemistry vol 64 no2 pp 391ndash418 2010

[64] A Ciesielski T M Krygowski M K Cyranski M ADobrowolski and A T Balaban ldquoAre thermodynamic andkinetic stabilities correlated A topological index of reactiv-ity toward electrophiles used as a criterion of aromaticityof polycyclic benzenoid hydrocarbonsrdquo Journal of ChemicalInformation and Modeling vol 49 no 2 pp 369ndash376 2009

[65] P K Chattaraj U Sarkar and D R Roy ldquoElectronic structureprinciples and aromaticityrdquo Journal of Chemical Education vol84 no 2 pp 354ndash358 2007

[66] M V Putz ldquoCompactness aromaticity of atoms in moleculesrdquoInternational Journal of Molecular Sciences vol 11 no 4 pp1269ndash1310 2010

[67] S K Lele ldquoCompact finite difference schemes with spectral-likeresolutionrdquo Journal of Computational Physics vol 103 no 1 pp16ndash42 1992


[68] ProgramPackageHyperChem701 Hypercube Gainesville FlaUSA 2002

[69] M V Putz A M Putz L Pitulice and V Chiriac ldquoOnchemical hardness assessment of aromaticity for some organiccompoundsrdquo International Journal of Chemical Modeling vol 2no 4 pp 343ndash354 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy


Carbohydrate Chemistry

International Journal of


Journal of

Chemistry


Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of


Chromatography Research International


Applied ChemistryJournal of



Theoretical ChemistryJournal of


Journal of

Spectroscopy

Analytical ChemistryInternational Journal of


Journal of


Quantum Chemistry


Organic Chemistry International

ElectrochemistryInternational Journal of



CatalystsJournal of


for avoiding or including the ldquomissing correlation infor-mationrdquo by remarkable methods Among most preeminentapproaches in this direction are

(i) the electron propagation theory (through consideringthe self-energy operator) [46]

(ii) disproving the existence of ionization potential as thelowest eigenvalues of KT but generalizing it to thearbitrarily close value to IP [43]

(iii) interpreting the self-consistent Hartree-Fock field ascoupled harmonic oscillators evolving in a nonlinearpotential [47]

(iv) variationally extending KT to restricted open-sellscanonical orbitals which nevertheless overestimatethe Aufbau principle [48]

(v) differentiating between vertical and adiabatic ioniza-tion potentials for the strongest line of the band andfor the 0 rarr 0 band transition respectively [49]

(vi) establishing the connectionwithDFT through Janakrsquostheorem and proving its reliability for largemolecularsystems (including fullerenes or boron nitride nan-otubes B

48N48) [50]

(vii) including the negative electron affinity extensionswithin DFT for halogenated small organic molecules[51]

(viii) establishing the direct connection of the frontierorbitals with the pi-electrons and of the electronictransfer of conjugated aromatic systems [52]

(ix) driving the electronic transfer in alfa-substitutedorganic polymers [53] providing with optical spec-tra analysis for intervalence complexes formed byorganic bridges between radical ions [54]

(x) till the modeling of anti-inflammatory activities ofclinical drugs acting through ionization processes inspecial [55] and by chemical reactivity indices andDFT in general [56ndash62]

It is the last context in which the present work is placedtoo it reviews the Koopmans method with the Hartree-Focktheory while emphasizing on the generality of the methodthere where a limitation was previously identified it willbe connected then with chemical hardness by means ofLUMO-HOMO gap that eventually cancels the correlationsand relaxation opposite effects appareling on the successivehighest occupied and lowest unoccupied molecular orbitalsit will end with an illustration on how chemical hardnessanalysis based onKoopmans superior orders orbitals is fittingwith compact-finite difference expression of it in the highestanalytical accuracy of spectral-like resolution (SLR) and howthese two faces of the chemical hardness generally asses thearomaticity hierarchy along a homological series of organicmolecules

2 Reviewing the Koopmans Method

21 Nature of Koopmansrsquo Theorem within Hartree-Fock For-malism Consider the many-electronic wave-function ket-vector in N-dimensional Hilbert spin-orbitalsrsquo space

10038161003816100381610038161003816Ψ(119873)

0⟩ =

10038161003816100381610038161205951 sdot sdot sdot 120595119886 sdot sdot sdot 120595119873⟩(1)

constructed from one-electronic Fock eigenstates

11989110038161003816100381610038161003816120595119886⟩ = 120576

119886

1003816100381610038161003816 120595119886⟩ 119886 = 1119873 (2)

with the Fock operator

119891 = ℎ +

119873

sum

119887=1

[119869119887minus 119887] (3)

written (in atomic units) in terms of the kinetic + potentialoperator

ℎ = minus1

2nabla2

1minussum

119860

119885119860

1199031119860

(4)

accounting for the one-electronic motion in the potential ofthe nucleinucleus in a molecularatomic system to it oneadds the resulting energy remaining from subtraction of theexchange influence

119870119887(1) = int1198892120595

lowast

119887(2) 119903minus1

12120595119887(1) = int1198892120595

lowast

119887(2) 119903minus1

12weierp12120595119887(2)

(5)

from the interelectronic Coulombic interaction

119869119887(1) = int1198892120595

lowast

119887(2) 119903minus1

12120595119887(2) (6)

Note in the exchange energy (5) the appearance of the inter-changing particle operator weierp

12which acts by interchanging

1 harr 2 particles on all wave functions on its right-had sideit also helps in identifying the Hartree-Fock potential in theFock operator rewritten as

119891 (1) = ℎ (1) + V119867119865(1)

= ℎ (1) +

119873

sum

119887=1

int1198892120595lowast

119887(2) 119903minus1

12(1 minus weierp

12) 120595119887(2)

(7)



119864119873= ⟨Ψ(119873)

0

1003816100381610038161003816100381610038161003816100381610038161003816Ψ(119873)

0⟩

=

119873

sum

119886=1

⟨120595119886

10038161003816100381610038161003816ℎ10038161003816100381610038161003816120595119886⟩

+1

2

119873

sum

119886=1119887=1

[⟨120595119886

10038161003816100381610038161003816int 1198892120595

lowast

119887(2) 119903minus1

12120595119887(2)10038161003816100381610038161003816120595119886⟩

minus⟨120595119886

10038161003816100381610038161003816int 1198892120595

lowast

119887(2) 119903minus1

12weierp12120595119887(2)10038161003816100381610038161003816120595119886⟩]

=

119873

sum

119886=1

⟨120595119886

10038161003816100381610038161003816ℎ10038161003816100381610038161003816120595119886⟩

+1

2

119873

sum

119886=1119887=1

[int11988911198892120595

lowast

119886(1) 120595lowast

119887(2) 119903minus1

12120595119887(2) 120595119886(1)

minus int 11988911198892120595lowast

119886(1) 120595lowast

119887(2) 119903minus1

12weierp12120595119887(2) 120595119886(1)]

=

119873

sum

119886=1

⟨120595119886

10038161003816100381610038161003816ℎ10038161003816100381610038161003816120595119886⟩

+1

2

119873

sum

119886=1119887=1

[int11988911198892120595

lowast

119886(1) 120595lowast

119887(2) 119903minus1

12120595119887(2) 120595119886(1)

minus int 11988911198892120595lowast

119886(1) 120595lowast

119887(2) 119903minus1

12120595119887(1) 120595119886(2)]

=

119873

sum

119886=1

⟨120595119886

10038161003816100381610038161003816ℎ10038161003816100381610038161003816120595119886⟩

+1

2

119873

sum

119886=1119887=1

[int11988911198892120595

lowast

119886(1) 120595119886(1) 119903minus1

12120595lowast

119887(2) 120595119887(2)

minus int 11988911198892120595lowast

119886(1) 120595119887(1) 119903minus1

12120595lowast

119887(2) 120595119886(2)]

equiv

119873

sum

119886=1

⟨120595119886

10038161003816100381610038161003816ℎ10038161003816100381610038161003816120595119886⟩ +

1

2

119873

sum

119886=1119887=1

[⟨119886119886 | 119887119887⟩ minus ⟨119886119887 | 119887119886⟩]

equiv

119873

sum

119886=1

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

⟨119886119887 | 119886119887⟩

(8)




0⟩ = |120595



119903 120595119904


120576119894=1infin

= ⟨120595119894

1003816100381610038161003816100381611989110038161003816100381610038161003816120595119894⟩

= ⟨120595119894

1003816100381610038161003816100381610038161003816100381610038161003816

(ℎ +

119873

sum

119887=1

[119869119887minus 119887])

1003816100381610038161003816100381610038161003816100381610038161003816

120595119894⟩

= ⟨120595119894

10038161003816100381610038161003816ℎ10038161003816100381610038161003816120595119894⟩ +

119873

sum

119887=1

[⟨120595119894

10038161003816100381610038161003816119869119887

10038161003816100381610038161003816120595119894⟩ minus ⟨120595

119894

10038161003816100381610038161003816119887

10038161003816100381610038161003816120595119894⟩]

equiv ⟨11989410038161003816100381610038161003816ℎ10038161003816100381610038161003816119894⟩ +

119873

sum

119887=1

[⟨119894119894 | 119887119887⟩ minus ⟨119894119887 | 119887119894⟩]

equiv ⟨11989410038161003816100381610038161003816ℎ10038161003816100381610038161003816119894⟩ +

119873

sum

119887=1

⟨119894119887 | 119894119887⟩

(9)


120576119886=1119873

= ⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

119873

sum

119887=1

119887 = 119886

⟨119886119887 | 119886119887⟩ (10)


120576119903=119873+1infin

= ⟨11990310038161003816100381610038161003816ℎ10038161003816100381610038161003816119903⟩ +

119873

sum

119887=1

⟨119903119887 | 119903119887⟩ (11)


119873

sum

119886=1

120576119886=

119873

sum

119886=1

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

119873

sum

119886=1119887=1

⟨119886119887 | 119886119887⟩ = 119864119873 (12)





120576p|p⟩

120576s|s⟩

120576r|r⟩

120576c|c⟩

120576d|d⟩

120576LUMO(3)

120576LUMO(2)

120576LUMO(1)

120576HOMO(1)

120576HOMO(2)

120576HOMO(3)120576e|e⟩

120576



1198681198751= 119864119873minus1minus 119864119873

= ⟨Ψ(119873minus1)

119888

1003816100381610038161003816100381610038161003816100381610038161003816Ψ(119873minus1)

119888⟩ minus ⟨Ψ

(119873)

0

1003816100381610038161003816100381610038161003816100381610038161003816Ψ(119873)

0⟩

=

119873

sum

119886=1

119886 = 119888

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

⟨119886119887 | 119886119887⟩

minus

[[

[

119873

sum

119886=1

119886 = 119888

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ + ⟨119888

10038161003816100381610038161003816ℎ10038161003816100381610038161003816119888⟩]]

]

+1

2

[[[

[

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

⟨119886119887 | 119886119887⟩ +

119873

sum

119886=1119887=1

119886 = 119888119887=119888

⟨119886119888 | 119886119888⟩

+

119873

sum

119886=1119887=1

119886=119888119887 = 119888

⟨119888119887 | 119888119887⟩

]]]

]

= minus⟨11988810038161003816100381610038161003816ℎ10038161003816100381610038161003816119888⟩ minus

119873

sum

119887=1

⟨119888119887 | 119888119887⟩

= minus120576119888= minus120576HOMO(1)

(13)



EA1= 119864119873minus 119864119873+1

= minus ⟨11990310038161003816100381610038161003816ℎ10038161003816100381610038161003816119903⟩ minus

119873

sum

119887=1

⟨119903119887 | 119903119887⟩

= minus 120576119903


(14)





2120578 =1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873⟩

cong 21198862

119864119873+1minus 2119864119873+ 119864119873minus1

2+ 1198872

119864119873+2minus 2119864119873+ 119864119873minus2

4

+ 1198882

119864119873+3minus 2119864119873+ 119864119873minus3

9

minus 1205722(1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873minus1⟩

+1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873+1⟩

)

minus 1205732(1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873minus2⟩

+1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873+2⟩

)

= 21198862

119864119873+1minus 2119864119873+ 119864119873minus1

2+ 1198872

119864119873+2minus 2119864119873+ 119864119873minus2

4

+ 1198882

119864119873+3minus 2119864119873+ 119864119873minus3

9

minus 1205722(21198862


2

+21198862

119864119873+2minus 2119864119873+1+ 119864119873

2)

minus 1205732(21198862


2

+21198862

119864119873+3minus 2119864119873+2+ 119864119873+1

2)

= 21198862(1 + 2120572

2minus 1205732)119864119873+1+ 119864119873minus1

2

+ (811988621205732+ 1198872minus 411988621205722)119864119873+2+ 119864119873minus2

4

+ (1198882minus 911988621205732)119864119873+3+ 119864119873minus3

9

minus (21198862+1

21198872+2

91198882+ 211988621205722)119864119873

(15)





EA119894= 119864119873+119894minus1

minus 119864119873+119894

(16)


119864119873+1+ 119864119873minus1

= (IP1minus EA1) + 2119864

119873

119864119873+2+ 119864119873minus2


2minus EA2) + 2119864

119873

119864119873+3+ 119864119873minus3


2minus EA2) + (IP

3minus EA3) + 2119864

119873

(17)



2(1 minus 120572

2+ 21205732) +1

41198872+1


2

+ [1

21198872+2

91198882+ 21198862(1205732minus 1205722)]

IP2minus EA2

4

+ [1

31198882minus 311988621205732]IP3minus EA3

6

(18)



2= 1 119887

2= 1198882= 1205722= 1205732= 0 of Table 1


1205782C =

IP1minus EA1

2(19)





of HOMO119894=123

and LUMO119894=123

respectively

IP119894= minus120576HOMO(119894)

EA119894= minus120576LUMO(119894)

(20)



Scheme 1198862

1198872

1198882

1205722

1205732



120578LUMO-HOMOCFD

= [1198862(1 minus 120572

2+ 21205732) +

1

41198872+1

91198882]


2

+ [1

21198872+2

91198882+ 21198862(1205732minus 1205722)]


4

+ [1


6

(21)


IP2


=

119873

sum

119886=1

119886 = 119888

119886 = 119889

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887 = 119889

⟨119886119887 | 119886119887⟩

minus

119873

sum

119886=1

119886 = 119888

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

⟨119886119887 | 119886119887⟩

=

119873

sum

119886=1

119886 = 119888

119886 = 119889

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887 = 119889

⟨119886119887 | 119886119887⟩

minus

119873

sum

119886=1119886 = 119888

119886 = 119889

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ + ⟨119889

10038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887 = 119889

⟨119886119887 | 119886119887⟩

+1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886=119889119887 = 119889

⟨119889119887 | 119889119887⟩ +1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887=119889

⟨119886119889 | 119886119889⟩

= minus⟨11988910038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩ minus

119873

sum

119886=1119887=1

119887 = 119889

⟨119889119887 | 119889119887⟩

= minus120576119889= minus120576HOMO(2)

(22)


1HOMO


successively become


= minus⟨11988910038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩

minus

1

2

119873

sum

119886=1119887=1

119886 = 119888

119886=119889119887 = 119889

⟨119889119887 | 119889119887⟩ +1

2⟨119889119888 | 119889119888⟩

+1

2

119873

sum

119886=1119887=1

119887 = 119888

119886 = 119889119887=119889

⟨119886119889 | 119886119889⟩ +1

2⟨119888119889 | 119888119889⟩

= minus⟨11988910038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩ minus

119873

sum

119886=1119887=1

119887 = 119889

⟨119889119887 | 119889119887⟩ + ⟨119888119889 | 119888119889⟩

= minus120576119889+ ⟨119888119889 | 119888119889⟩


2| HOMO

1HOMO

2⟩

(23)



EA2= 119864119873+1minus 119864119873+2


2| LUMO

1LUMO

2⟩

(24)


1HOMO

2versus

LUMO1LUMO

2as follows


+ (⟨HOMO

1HOMO

2| HOMO

1HOMO

2⟩

minus ⟨LUMO1LUMO

2| LUMO

1LUMO

2⟩)

(25)
















C6H6Benzene71-43-2I (7811)






C6H6OPhenol108-95-2IV (9411)











Table 2 Continued

























4 Conclusions



1

2

3

4

5120578

(eV

)


2C1SLR12C1 2

2C1 2 3SLR1 2 3

SLR1 2

NN

NOH

OH OH NH2NH2

NH2






1HOMO

2|

HOMO1HOMO


1LUMO

2|

LUMO1LUMO





Abbreviations




Acknowledgment


References
































































2ClSCNrdquo




















Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



119864119873= ⟨Ψ(119873)

0

1003816100381610038161003816100381610038161003816100381610038161003816Ψ(119873)

0⟩

=

119873

sum

119886=1

⟨120595119886

10038161003816100381610038161003816ℎ10038161003816100381610038161003816120595119886⟩

+1

2

119873

sum

119886=1119887=1

[⟨120595119886

10038161003816100381610038161003816int 1198892120595

lowast

119887(2) 119903minus1

12120595119887(2)10038161003816100381610038161003816120595119886⟩

minus⟨120595119886

10038161003816100381610038161003816int 1198892120595

lowast

119887(2) 119903minus1

12weierp12120595119887(2)10038161003816100381610038161003816120595119886⟩]

=

119873

sum

119886=1

⟨120595119886

10038161003816100381610038161003816ℎ10038161003816100381610038161003816120595119886⟩

+1

2

119873

sum

119886=1119887=1

[int11988911198892120595

lowast

119886(1) 120595lowast

119887(2) 119903minus1

12120595119887(2) 120595119886(1)

minus int 11988911198892120595lowast

119886(1) 120595lowast

119887(2) 119903minus1

12weierp12120595119887(2) 120595119886(1)]

=

119873

sum

119886=1

⟨120595119886

10038161003816100381610038161003816ℎ10038161003816100381610038161003816120595119886⟩

+1

2

119873

sum

119886=1119887=1

[int11988911198892120595

lowast

119886(1) 120595lowast

119887(2) 119903minus1

12120595119887(2) 120595119886(1)

minus int 11988911198892120595lowast

119886(1) 120595lowast

119887(2) 119903minus1

12120595119887(1) 120595119886(2)]

=

119873

sum

119886=1

⟨120595119886

10038161003816100381610038161003816ℎ10038161003816100381610038161003816120595119886⟩

+1

2

119873

sum

119886=1119887=1

[int11988911198892120595

lowast

119886(1) 120595119886(1) 119903minus1

12120595lowast

119887(2) 120595119887(2)

minus int 11988911198892120595lowast

119886(1) 120595119887(1) 119903minus1

12120595lowast

119887(2) 120595119886(2)]

equiv

119873

sum

119886=1

⟨120595119886

10038161003816100381610038161003816ℎ10038161003816100381610038161003816120595119886⟩ +

1

2

119873

sum

119886=1119887=1

[⟨119886119886 | 119887119887⟩ minus ⟨119886119887 | 119887119886⟩]

equiv

119873

sum

119886=1

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

⟨119886119887 | 119886119887⟩

(8)




0⟩ = |120595



119903 120595119904


120576119894=1infin

= ⟨120595119894

1003816100381610038161003816100381611989110038161003816100381610038161003816120595119894⟩

= ⟨120595119894

1003816100381610038161003816100381610038161003816100381610038161003816

(ℎ +

119873

sum

119887=1

[119869119887minus 119887])

1003816100381610038161003816100381610038161003816100381610038161003816

120595119894⟩

= ⟨120595119894

10038161003816100381610038161003816ℎ10038161003816100381610038161003816120595119894⟩ +

119873

sum

119887=1

[⟨120595119894

10038161003816100381610038161003816119869119887

10038161003816100381610038161003816120595119894⟩ minus ⟨120595

119894

10038161003816100381610038161003816119887

10038161003816100381610038161003816120595119894⟩]

equiv ⟨11989410038161003816100381610038161003816ℎ10038161003816100381610038161003816119894⟩ +

119873

sum

119887=1

[⟨119894119894 | 119887119887⟩ minus ⟨119894119887 | 119887119894⟩]

equiv ⟨11989410038161003816100381610038161003816ℎ10038161003816100381610038161003816119894⟩ +

119873

sum

119887=1

⟨119894119887 | 119894119887⟩

(9)


120576119886=1119873

= ⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

119873

sum

119887=1

119887 = 119886

⟨119886119887 | 119886119887⟩ (10)


120576119903=119873+1infin

= ⟨11990310038161003816100381610038161003816ℎ10038161003816100381610038161003816119903⟩ +

119873

sum

119887=1

⟨119903119887 | 119903119887⟩ (11)


119873

sum

119886=1

120576119886=

119873

sum

119886=1

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

119873

sum

119886=1119887=1

⟨119886119887 | 119886119887⟩ = 119864119873 (12)





120576p|p⟩

120576s|s⟩

120576r|r⟩

120576c|c⟩

120576d|d⟩

120576LUMO(3)

120576LUMO(2)

120576LUMO(1)

120576HOMO(1)

120576HOMO(2)

120576HOMO(3)120576e|e⟩

120576



1198681198751= 119864119873minus1minus 119864119873

= ⟨Ψ(119873minus1)

119888

1003816100381610038161003816100381610038161003816100381610038161003816Ψ(119873minus1)

119888⟩ minus ⟨Ψ

(119873)

0

1003816100381610038161003816100381610038161003816100381610038161003816Ψ(119873)

0⟩

=

119873

sum

119886=1

119886 = 119888

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

⟨119886119887 | 119886119887⟩

minus

[[

[

119873

sum

119886=1

119886 = 119888

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ + ⟨119888

10038161003816100381610038161003816ℎ10038161003816100381610038161003816119888⟩]]

]

+1

2

[[[

[

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

⟨119886119887 | 119886119887⟩ +

119873

sum

119886=1119887=1

119886 = 119888119887=119888

⟨119886119888 | 119886119888⟩

+

119873

sum

119886=1119887=1

119886=119888119887 = 119888

⟨119888119887 | 119888119887⟩

]]]

]

= minus⟨11988810038161003816100381610038161003816ℎ10038161003816100381610038161003816119888⟩ minus

119873

sum

119887=1

⟨119888119887 | 119888119887⟩

= minus120576119888= minus120576HOMO(1)

(13)



EA1= 119864119873minus 119864119873+1

= minus ⟨11990310038161003816100381610038161003816ℎ10038161003816100381610038161003816119903⟩ minus

119873

sum

119887=1

⟨119903119887 | 119903119887⟩

= minus 120576119903


(14)





2120578 =1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873⟩

cong 21198862

119864119873+1minus 2119864119873+ 119864119873minus1

2+ 1198872

119864119873+2minus 2119864119873+ 119864119873minus2

4

+ 1198882

119864119873+3minus 2119864119873+ 119864119873minus3

9

minus 1205722(1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873minus1⟩

+1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873+1⟩

)

minus 1205732(1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873minus2⟩

+1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873+2⟩

)

= 21198862

119864119873+1minus 2119864119873+ 119864119873minus1

2+ 1198872

119864119873+2minus 2119864119873+ 119864119873minus2

4

+ 1198882

119864119873+3minus 2119864119873+ 119864119873minus3

9

minus 1205722(21198862


2

+21198862

119864119873+2minus 2119864119873+1+ 119864119873

2)

minus 1205732(21198862


2

+21198862

119864119873+3minus 2119864119873+2+ 119864119873+1

2)

= 21198862(1 + 2120572

2minus 1205732)119864119873+1+ 119864119873minus1

2

+ (811988621205732+ 1198872minus 411988621205722)119864119873+2+ 119864119873minus2

4

+ (1198882minus 911988621205732)119864119873+3+ 119864119873minus3

9

minus (21198862+1

21198872+2

91198882+ 211988621205722)119864119873

(15)





EA119894= 119864119873+119894minus1

minus 119864119873+119894

(16)


119864119873+1+ 119864119873minus1

= (IP1minus EA1) + 2119864

119873

119864119873+2+ 119864119873minus2


2minus EA2) + 2119864

119873

119864119873+3+ 119864119873minus3


2minus EA2) + (IP

3minus EA3) + 2119864

119873

(17)



2(1 minus 120572

2+ 21205732) +1

41198872+1


2

+ [1

21198872+2

91198882+ 21198862(1205732minus 1205722)]

IP2minus EA2

4

+ [1

31198882minus 311988621205732]IP3minus EA3

6

(18)



2= 1 119887

2= 1198882= 1205722= 1205732= 0 of Table 1


1205782C =

IP1minus EA1

2(19)





of HOMO119894=123

and LUMO119894=123

respectively

IP119894= minus120576HOMO(119894)

EA119894= minus120576LUMO(119894)

(20)



Scheme 1198862

1198872

1198882

1205722

1205732



120578LUMO-HOMOCFD

= [1198862(1 minus 120572

2+ 21205732) +

1

41198872+1

91198882]


2

+ [1

21198872+2

91198882+ 21198862(1205732minus 1205722)]


4

+ [1


6

(21)


IP2


=

119873

sum

119886=1

119886 = 119888

119886 = 119889

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887 = 119889

⟨119886119887 | 119886119887⟩

minus

119873

sum

119886=1

119886 = 119888

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

⟨119886119887 | 119886119887⟩

=

119873

sum

119886=1

119886 = 119888

119886 = 119889

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887 = 119889

⟨119886119887 | 119886119887⟩

minus

119873

sum

119886=1119886 = 119888

119886 = 119889

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ + ⟨119889

10038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887 = 119889

⟨119886119887 | 119886119887⟩

+1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886=119889119887 = 119889

⟨119889119887 | 119889119887⟩ +1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887=119889

⟨119886119889 | 119886119889⟩

= minus⟨11988910038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩ minus

119873

sum

119886=1119887=1

119887 = 119889

⟨119889119887 | 119889119887⟩

= minus120576119889= minus120576HOMO(2)

(22)


1HOMO


successively become


= minus⟨11988910038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩

minus

1

2

119873

sum

119886=1119887=1

119886 = 119888

119886=119889119887 = 119889

⟨119889119887 | 119889119887⟩ +1

2⟨119889119888 | 119889119888⟩

+1

2

119873

sum

119886=1119887=1

119887 = 119888

119886 = 119889119887=119889

⟨119886119889 | 119886119889⟩ +1

2⟨119888119889 | 119888119889⟩

= minus⟨11988910038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩ minus

119873

sum

119886=1119887=1

119887 = 119889

⟨119889119887 | 119889119887⟩ + ⟨119888119889 | 119888119889⟩

= minus120576119889+ ⟨119888119889 | 119888119889⟩


2| HOMO

1HOMO

2⟩

(23)



EA2= 119864119873+1minus 119864119873+2


2| LUMO

1LUMO

2⟩

(24)


1HOMO

2versus

LUMO1LUMO

2as follows


+ (⟨HOMO

1HOMO

2| HOMO

1HOMO

2⟩

minus ⟨LUMO1LUMO

2| LUMO

1LUMO

2⟩)

(25)
















C6H6Benzene71-43-2I (7811)






C6H6OPhenol108-95-2IV (9411)











Table 2 Continued

























4 Conclusions



1

2

3

4

5120578

(eV

)


2C1SLR12C1 2

2C1 2 3SLR1 2 3

SLR1 2

NN

NOH

OH OH NH2NH2

NH2






1HOMO

2|

HOMO1HOMO


1LUMO

2|

LUMO1LUMO





Abbreviations




Acknowledgment


References
































































2ClSCNrdquo




















Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


120576p|p⟩

120576s|s⟩

120576r|r⟩

120576c|c⟩

120576d|d⟩

120576LUMO(3)

120576LUMO(2)

120576LUMO(1)

120576HOMO(1)

120576HOMO(2)

120576HOMO(3)120576e|e⟩

120576



1198681198751= 119864119873minus1minus 119864119873

= ⟨Ψ(119873minus1)

119888

1003816100381610038161003816100381610038161003816100381610038161003816Ψ(119873minus1)

119888⟩ minus ⟨Ψ

(119873)

0

1003816100381610038161003816100381610038161003816100381610038161003816Ψ(119873)

0⟩

=

119873

sum

119886=1

119886 = 119888

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

⟨119886119887 | 119886119887⟩

minus

[[

[

119873

sum

119886=1

119886 = 119888

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ + ⟨119888

10038161003816100381610038161003816ℎ10038161003816100381610038161003816119888⟩]]

]

+1

2

[[[

[

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

⟨119886119887 | 119886119887⟩ +

119873

sum

119886=1119887=1

119886 = 119888119887=119888

⟨119886119888 | 119886119888⟩

+

119873

sum

119886=1119887=1

119886=119888119887 = 119888

⟨119888119887 | 119888119887⟩

]]]

]

= minus⟨11988810038161003816100381610038161003816ℎ10038161003816100381610038161003816119888⟩ minus

119873

sum

119887=1

⟨119888119887 | 119888119887⟩

= minus120576119888= minus120576HOMO(1)

(13)



EA1= 119864119873minus 119864119873+1

= minus ⟨11990310038161003816100381610038161003816ℎ10038161003816100381610038161003816119903⟩ minus

119873

sum

119887=1

⟨119903119887 | 119903119887⟩

= minus 120576119903


(14)





2120578 =1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873⟩

cong 21198862

119864119873+1minus 2119864119873+ 119864119873minus1

2+ 1198872

119864119873+2minus 2119864119873+ 119864119873minus2

4

+ 1198882

119864119873+3minus 2119864119873+ 119864119873minus3

9

minus 1205722(1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873minus1⟩

+1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873+1⟩

)

minus 1205732(1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873minus2⟩

+1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873+2⟩

)

= 21198862

119864119873+1minus 2119864119873+ 119864119873minus1

2+ 1198872

119864119873+2minus 2119864119873+ 119864119873minus2

4

+ 1198882

119864119873+3minus 2119864119873+ 119864119873minus3

9

minus 1205722(21198862


2

+21198862

119864119873+2minus 2119864119873+1+ 119864119873

2)

minus 1205732(21198862


2

+21198862

119864119873+3minus 2119864119873+2+ 119864119873+1

2)

= 21198862(1 + 2120572

2minus 1205732)119864119873+1+ 119864119873minus1

2

+ (811988621205732+ 1198872minus 411988621205722)119864119873+2+ 119864119873minus2

4

+ (1198882minus 911988621205732)119864119873+3+ 119864119873minus3

9

minus (21198862+1

21198872+2

91198882+ 211988621205722)119864119873

(15)





EA119894= 119864119873+119894minus1

minus 119864119873+119894

(16)


119864119873+1+ 119864119873minus1

= (IP1minus EA1) + 2119864

119873

119864119873+2+ 119864119873minus2


2minus EA2) + 2119864

119873

119864119873+3+ 119864119873minus3


2minus EA2) + (IP

3minus EA3) + 2119864

119873

(17)



2(1 minus 120572

2+ 21205732) +1

41198872+1


2

+ [1

21198872+2

91198882+ 21198862(1205732minus 1205722)]

IP2minus EA2

4

+ [1

31198882minus 311988621205732]IP3minus EA3

6

(18)



2= 1 119887

2= 1198882= 1205722= 1205732= 0 of Table 1


1205782C =

IP1minus EA1

2(19)





of HOMO119894=123

and LUMO119894=123

respectively

IP119894= minus120576HOMO(119894)

EA119894= minus120576LUMO(119894)

(20)



Scheme 1198862

1198872

1198882

1205722

1205732



120578LUMO-HOMOCFD

= [1198862(1 minus 120572

2+ 21205732) +

1

41198872+1

91198882]


2

+ [1

21198872+2

91198882+ 21198862(1205732minus 1205722)]


4

+ [1


6

(21)


IP2


=

119873

sum

119886=1

119886 = 119888

119886 = 119889

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887 = 119889

⟨119886119887 | 119886119887⟩

minus

119873

sum

119886=1

119886 = 119888

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

⟨119886119887 | 119886119887⟩

=

119873

sum

119886=1

119886 = 119888

119886 = 119889

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887 = 119889

⟨119886119887 | 119886119887⟩

minus

119873

sum

119886=1119886 = 119888

119886 = 119889

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ + ⟨119889

10038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887 = 119889

⟨119886119887 | 119886119887⟩

+1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886=119889119887 = 119889

⟨119889119887 | 119889119887⟩ +1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887=119889

⟨119886119889 | 119886119889⟩

= minus⟨11988910038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩ minus

119873

sum

119886=1119887=1

119887 = 119889

⟨119889119887 | 119889119887⟩

= minus120576119889= minus120576HOMO(2)

(22)


1HOMO


successively become


= minus⟨11988910038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩

minus

1

2

119873

sum

119886=1119887=1

119886 = 119888

119886=119889119887 = 119889

⟨119889119887 | 119889119887⟩ +1

2⟨119889119888 | 119889119888⟩

+1

2

119873

sum

119886=1119887=1

119887 = 119888

119886 = 119889119887=119889

⟨119886119889 | 119886119889⟩ +1

2⟨119888119889 | 119888119889⟩

= minus⟨11988910038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩ minus

119873

sum

119886=1119887=1

119887 = 119889

⟨119889119887 | 119889119887⟩ + ⟨119888119889 | 119888119889⟩

= minus120576119889+ ⟨119888119889 | 119888119889⟩


2| HOMO

1HOMO

2⟩

(23)



EA2= 119864119873+1minus 119864119873+2


2| LUMO

1LUMO

2⟩

(24)


1HOMO

2versus

LUMO1LUMO

2as follows


+ (⟨HOMO

1HOMO

2| HOMO

1HOMO

2⟩

minus ⟨LUMO1LUMO

2| LUMO

1LUMO

2⟩)

(25)
















C6H6Benzene71-43-2I (7811)






C6H6OPhenol108-95-2IV (9411)











Table 2 Continued

























4 Conclusions



1

2

3

4

5120578

(eV

)


2C1SLR12C1 2

2C1 2 3SLR1 2 3

SLR1 2

NN

NOH

OH OH NH2NH2

NH2






1HOMO

2|

HOMO1HOMO


1LUMO

2|

LUMO1LUMO





Abbreviations




Acknowledgment


References
































































2ClSCNrdquo




















Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



2120578 =1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873⟩

cong 21198862

119864119873+1minus 2119864119873+ 119864119873minus1

2+ 1198872

119864119873+2minus 2119864119873+ 119864119873minus2

4

+ 1198882

119864119873+3minus 2119864119873+ 119864119873minus3

9

minus 1205722(1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873minus1⟩

+1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873+1⟩

)

minus 1205732(1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873minus2⟩

+1205972119864

1205971198732

100381610038161003816100381610038161003816100381610038161003816|119873+2⟩

)

= 21198862

119864119873+1minus 2119864119873+ 119864119873minus1

2+ 1198872

119864119873+2minus 2119864119873+ 119864119873minus2

4

+ 1198882

119864119873+3minus 2119864119873+ 119864119873minus3

9

minus 1205722(21198862


2

+21198862

119864119873+2minus 2119864119873+1+ 119864119873

2)

minus 1205732(21198862


2

+21198862

119864119873+3minus 2119864119873+2+ 119864119873+1

2)

= 21198862(1 + 2120572

2minus 1205732)119864119873+1+ 119864119873minus1

2

+ (811988621205732+ 1198872minus 411988621205722)119864119873+2+ 119864119873minus2

4

+ (1198882minus 911988621205732)119864119873+3+ 119864119873minus3

9

minus (21198862+1

21198872+2

91198882+ 211988621205722)119864119873

(15)





EA119894= 119864119873+119894minus1

minus 119864119873+119894

(16)


119864119873+1+ 119864119873minus1

= (IP1minus EA1) + 2119864

119873

119864119873+2+ 119864119873minus2


2minus EA2) + 2119864

119873

119864119873+3+ 119864119873minus3


2minus EA2) + (IP

3minus EA3) + 2119864

119873

(17)



2(1 minus 120572

2+ 21205732) +1

41198872+1


2

+ [1

21198872+2

91198882+ 21198862(1205732minus 1205722)]

IP2minus EA2

4

+ [1

31198882minus 311988621205732]IP3minus EA3

6

(18)



2= 1 119887

2= 1198882= 1205722= 1205732= 0 of Table 1


1205782C =

IP1minus EA1

2(19)





of HOMO119894=123

and LUMO119894=123

respectively

IP119894= minus120576HOMO(119894)

EA119894= minus120576LUMO(119894)

(20)



Scheme 1198862

1198872

1198882

1205722

1205732



120578LUMO-HOMOCFD

= [1198862(1 minus 120572

2+ 21205732) +

1

41198872+1

91198882]


2

+ [1

21198872+2

91198882+ 21198862(1205732minus 1205722)]


4

+ [1


6

(21)


IP2


=

119873

sum

119886=1

119886 = 119888

119886 = 119889

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887 = 119889

⟨119886119887 | 119886119887⟩

minus

119873

sum

119886=1

119886 = 119888

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

⟨119886119887 | 119886119887⟩

=

119873

sum

119886=1

119886 = 119888

119886 = 119889

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887 = 119889

⟨119886119887 | 119886119887⟩

minus

119873

sum

119886=1119886 = 119888

119886 = 119889

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ + ⟨119889

10038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887 = 119889

⟨119886119887 | 119886119887⟩

+1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886=119889119887 = 119889

⟨119889119887 | 119889119887⟩ +1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887=119889

⟨119886119889 | 119886119889⟩

= minus⟨11988910038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩ minus

119873

sum

119886=1119887=1

119887 = 119889

⟨119889119887 | 119889119887⟩

= minus120576119889= minus120576HOMO(2)

(22)


1HOMO


successively become


= minus⟨11988910038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩

minus

1

2

119873

sum

119886=1119887=1

119886 = 119888

119886=119889119887 = 119889

⟨119889119887 | 119889119887⟩ +1

2⟨119889119888 | 119889119888⟩

+1

2

119873

sum

119886=1119887=1

119887 = 119888

119886 = 119889119887=119889

⟨119886119889 | 119886119889⟩ +1

2⟨119888119889 | 119888119889⟩

= minus⟨11988910038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩ minus

119873

sum

119886=1119887=1

119887 = 119889

⟨119889119887 | 119889119887⟩ + ⟨119888119889 | 119888119889⟩

= minus120576119889+ ⟨119888119889 | 119888119889⟩


2| HOMO

1HOMO

2⟩

(23)



EA2= 119864119873+1minus 119864119873+2


2| LUMO

1LUMO

2⟩

(24)


1HOMO

2versus

LUMO1LUMO

2as follows


+ (⟨HOMO

1HOMO

2| HOMO

1HOMO

2⟩

minus ⟨LUMO1LUMO

2| LUMO

1LUMO

2⟩)

(25)
















C6H6Benzene71-43-2I (7811)






C6H6OPhenol108-95-2IV (9411)











Table 2 Continued

























4 Conclusions



1

2

3

4

5120578

(eV

)


2C1SLR12C1 2

2C1 2 3SLR1 2 3

SLR1 2

NN

NOH

OH OH NH2NH2

NH2






1HOMO

2|

HOMO1HOMO


1LUMO

2|

LUMO1LUMO





Abbreviations




Acknowledgment


References
































































2ClSCNrdquo




















Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



Scheme 1198862

1198872

1198882

1205722

1205732



120578LUMO-HOMOCFD

= [1198862(1 minus 120572

2+ 21205732) +

1

41198872+1

91198882]


2

+ [1

21198872+2

91198882+ 21198862(1205732minus 1205722)]


4

+ [1


6

(21)


IP2


=

119873

sum

119886=1

119886 = 119888

119886 = 119889

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887 = 119889

⟨119886119887 | 119886119887⟩

minus

119873

sum

119886=1

119886 = 119888

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

⟨119886119887 | 119886119887⟩

=

119873

sum

119886=1

119886 = 119888

119886 = 119889

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887 = 119889

⟨119886119887 | 119886119887⟩

minus

119873

sum

119886=1119886 = 119888

119886 = 119889

⟨11988610038161003816100381610038161003816ℎ10038161003816100381610038161003816119886⟩ + ⟨119889

10038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩ +

1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887 = 119889

⟨119886119887 | 119886119887⟩

+1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886=119889119887 = 119889

⟨119889119887 | 119889119887⟩ +1

2

119873

sum

119886=1119887=1

119886 = 119888119887 = 119888

119886 = 119889119887=119889

⟨119886119889 | 119886119889⟩

= minus⟨11988910038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩ minus

119873

sum

119886=1119887=1

119887 = 119889

⟨119889119887 | 119889119887⟩

= minus120576119889= minus120576HOMO(2)

(22)


1HOMO


successively become


= minus⟨11988910038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩

minus

1

2

119873

sum

119886=1119887=1

119886 = 119888

119886=119889119887 = 119889

⟨119889119887 | 119889119887⟩ +1

2⟨119889119888 | 119889119888⟩

+1

2

119873

sum

119886=1119887=1

119887 = 119888

119886 = 119889119887=119889

⟨119886119889 | 119886119889⟩ +1

2⟨119888119889 | 119888119889⟩

= minus⟨11988910038161003816100381610038161003816ℎ10038161003816100381610038161003816119889⟩ minus

119873

sum

119886=1119887=1

119887 = 119889

⟨119889119887 | 119889119887⟩ + ⟨119888119889 | 119888119889⟩

= minus120576119889+ ⟨119888119889 | 119888119889⟩


2| HOMO

1HOMO

2⟩

(23)



EA2= 119864119873+1minus 119864119873+2


2| LUMO

1LUMO

2⟩

(24)


1HOMO

2versus

LUMO1LUMO

2as follows


+ (⟨HOMO

1HOMO

2| HOMO

1HOMO

2⟩

minus ⟨LUMO1LUMO

2| LUMO

1LUMO

2⟩)

(25)
















C6H6Benzene71-43-2I (7811)






C6H6OPhenol108-95-2IV (9411)











Table 2 Continued

























4 Conclusions



1

2

3

4

5120578

(eV

)


2C1SLR12C1 2

2C1 2 3SLR1 2 3

SLR1 2

NN

NOH

OH OH NH2NH2

NH2






1HOMO

2|

HOMO1HOMO


1LUMO

2|

LUMO1LUMO





Abbreviations




Acknowledgment


References
































































2ClSCNrdquo




















Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



EA2= 119864119873+1minus 119864119873+2


2| LUMO

1LUMO

2⟩

(24)


1HOMO

2versus

LUMO1LUMO

2as follows


+ (⟨HOMO

1HOMO

2| HOMO

1HOMO

2⟩

minus ⟨LUMO1LUMO

2| LUMO

1LUMO

2⟩)

(25)
















C6H6Benzene71-43-2I (7811)






C6H6OPhenol108-95-2IV (9411)











Table 2 Continued

























4 Conclusions



1

2

3

4

5120578

(eV

)


2C1SLR12C1 2

2C1 2 3SLR1 2 3

SLR1 2

NN

NOH

OH OH NH2NH2

NH2






1HOMO

2|

HOMO1HOMO


1LUMO

2|

LUMO1LUMO





Abbreviations




Acknowledgment


References
































































2ClSCNrdquo




















Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of





C6H6Benzene71-43-2I (7811)






C6H6OPhenol108-95-2IV (9411)











Table 2 Continued

























4 Conclusions



1

2

3

4

5120578

(eV

)


2C1SLR12C1 2

2C1 2 3SLR1 2 3

SLR1 2

NN

NOH

OH OH NH2NH2

NH2






1HOMO

2|

HOMO1HOMO


1LUMO

2|

LUMO1LUMO





Abbreviations




Acknowledgment


References
































































2ClSCNrdquo




















Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


Table 2 Continued

























4 Conclusions



1

2

3

4

5120578

(eV

)


2C1SLR12C1 2

2C1 2 3SLR1 2 3

SLR1 2

NN

NOH

OH OH NH2NH2

NH2






1HOMO

2|

HOMO1HOMO


1LUMO

2|

LUMO1LUMO





Abbreviations




Acknowledgment


References
































































2ClSCNrdquo




















Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of













4 Conclusions



1

2

3

4

5120578

(eV

)


2C1SLR12C1 2

2C1 2 3SLR1 2 3

SLR1 2

NN

NOH

OH OH NH2NH2

NH2






1HOMO

2|

HOMO1HOMO


1LUMO

2|

LUMO1LUMO





Abbreviations




Acknowledgment


References
































































2ClSCNrdquo




















Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


1

2

3

4

5120578

(eV

)


2C1SLR12C1 2

2C1 2 3SLR1 2 3

SLR1 2

NN

NOH

OH OH NH2NH2

NH2






1HOMO

2|

HOMO1HOMO


1LUMO

2|

LUMO1LUMO





Abbreviations




Acknowledgment


References
































































2ClSCNrdquo




















Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



Acknowledgment


References
































































2ClSCNrdquo




















Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




























2ClSCNrdquo




















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 koopmans analysis of chemical...

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