Carbon bonding and structures advances in physics and chemistry

Remarkably, expression 1.4 is so close in form with a frontier formulation ofthe “chemical” energy of a system, as being the energy engaged or responsible forthe chemical reaction taking

A comprehensive book series which encompasses the complete coverage of carbonmaterials and carbon-rich molecules from elemental carbon dust in the interstellar medium,

to the most specialized industrial applications of the elemental carbon and derivatives

A great emphasis is placed on the most advanced and promising applications ranging fromelectronics to medicinal chemistry The aim is to offer the reader a book series which notonly consists of self-sufficient reference works, but one which stimulates further researchand enthusiasm

Series Editors

Dr Prof Franco Cataldo Professor Paolo Milani

Via Casilina 1626/A, Department of Physics

00133 Rome, Italy University of Milan

Via Celoria, 26

20133, Milan, Italy

VOLUME5:

CARBONBONDING AND STRUCTURES

ADVANCES INPHYSICS ANDCHEMISTRY

Volume Editor

Dr Mihai V PutzChemistry DepartmentWest University of Timis¸oaraStr Pestalozzi, No 16RO-300115, Timis¸oaraRomania

For further volumes:

http://www.springer.com/series/7825

Mihai V Putz

Laboratory of Computational and Structural Physical Chemistry

Chemistry Department

West University of Timis¸oara

Pestalozzi 16, Timis¸oara, RO300115

Springer Dordrecht Heidelberg London New York

Library of Congress Control Number: 2011934966

# Springer Science+Business Media B.V 2011

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At the beginning it was Carbon; at the beginning of complex nature, complex life,and even conscience While Hydrogen belongs to the moving Universe, Helium andCarbon are the protagonists of the universal nucleogenesis, assure the Universe’scombustion, and ultimately support its evolution As such, the Carbon was limitedlyinterpreted as belonging exclusively to the organic life base or to the life itself as werecognize it Otherwise, Carbon may be part of the very-short list of the PeriodicTable, i.e {H, He, C, O, N}, that may assure for appreciable extent the innermachinery of the observed word On the other side, Carbon has at least one specialfeature in each natural science (Physics, Chemistry, Biology) that makes it worthyfor being in depth explored either theoretically as well as in current laboratorystructural design, respectively:

l In Physics, Carbon is the preeminent resistant structure to the phenomenon of(Bose-Einstein) condensation, while being at the base of polymeric structures;

l In Chemistry, Carbon marks the unique four allotropic forms as the simplesubstance, diamond, graphite, and fullerene, each of these opening entire scien-tific chapters, plethora of nano-structures and every-day life applications;

l In Biology, Carbon assures through its tetravalent flexible bonds the backbone ofpolypeptides, the skeleton of amino-acids and bio-molecules themselves untilthe most advanced bio-responsive nano-materials

l In Technology, Carbon, besides providing the actual challenging nano-materialsand benchmark, it also opens the gates towards its relative Silicon element basedcomposite, and hybrids

As a consequence, the Carbon versatility seems to assure the messengerinformation within and in between the Natures’ levels of manifestation or on itsartifacts The present volume, while approaching many parts of abovementionedfundamental research directions, brings in the International Year of Chemistry 2011homage to the miracle of Carbon as a key element in the vast actual fields ofmodeling structure and bonded nanosystems with implication in all natural sciencesand challenging technologies It was possible through the exquisite contributions

v

of eminent scientists and professors from major continents as Europe, North andSouth Americas, and Asia that give their best understanding of the Carbon phenom-enology and advanced implication nowadays I do thank them all for the consistenteffort they encompassed in writing high-class scientific reports in providing theaudience with a broad perspectives and gates to be next open in making the Carbonstructure and bonding our home and reliable future!

Special thanks are due to Professor Franco Cataldo, the main coordinator ofthe Springer Carbon Materials Series, for kind invitation for pursuit the presenteditorial project, as well to the Springer Chemistry Team and to its Senior EditorSonia Ojo for supporting all stages towards the publication of the presentvolume .on the Carbon copies!

Mihai V Putz

1 Quantum Parabolic Effects of Electronegativity

and Chemical Hardness on Carbonp-Systems 1Mihai V Putz

2 Stiff Polymers at Ultralow Temperatures 33Hagen Kleinert

3 On Topological Modeling of 5|7 Structural Defects Drifting

in Graphene 43Ottorino Ori, Franco Cataldo, and Ante Graovac

4 The Chemical Reactivity of Fullerenes and Endohedral

Fullerenes: A Theoretical Perspective 57Sı´lvia Osuna, Marcel Swart, and Miquel Sola`

5 High Pressure Synthesis of the Carbon Allotrope Hexagonite

with Carbon Nanotubes in a Diamond Anvil Cell 79Michael J Bucknum and Eduardo A Castro

6 Graph Drawing with Eigenvectors 95Istva´n La´szlo´, Ante Graovac, Tomazˇ Pisanski, and Dejan Plavsˇic´

7 Applications of Chemical Graph Theory to Organic Molecules 117Lionello Pogliani

8 Structural Approach to Aromaticity and Local Aromaticity

in Conjugated Polycyclic Systems 159Alexandru T Balaban and Milan Randic´

vii

9 Coding and Ordering Benzenoids and Their Kekule´ Structures 205Bono Lucˇic´, Ante Milicˇevic´, Sonja Nikolic´, and Nenad Trinajstic´

10 Prochirality and Pro-RS-Stereogenicity Stereoisogram

Approach Free from the Conventional “Prochirality”

and “Prostereogenicity” 227Shinsaku Fujita

11 Diamond D5, a Novel Class of Carbon Allotropes 273Mircea V Diudea, Csaba L Nagy, and Aleksandar Ilic´

12 Empirical Study of Diameters of Fullerene Graphs 291Tomislav Dosˇlic´

13 Hardness Equalization in the Formation Poly

Atomic Carbon Compounds 301Nazmul Islam and Dulal C Ghosh

14 Modeling of the Chemico-Physical Process of Protonation

of Carbon Compounds 321Sandip K Rajak, Nazmul Islam, and Dulal C Ghosh

15 Molecular Shape Descriptors: Applications

to Structure-Activity Studies 337Dan Ciubotariu, Vicentiu Vlaia, Ciprian Ciubotariu, Tudor Olariu,

and Mihai Medeleanu

16 Recent Advances in Bioresponsive Nanomaterials 379Cecilia Savii and Ana-Maria Putz

Index 437

Alexandru T Balaban Texas A&M University at Galveston,

MARS, 5007 Avenue U, Galveston, TX 77551, USA

balabana@tamug.edu

Michael J Bucknum INIFTA, Theoretical Chemistry Division,

Suc 4, C.C 16, Universidad de La Plata, 1900 La Plata,

Buenos Aires, Argentina

mjbucknum@gmail.com

Eduardo A Castro INIFTA, Theoretical Chemistry Division,

Suc 4, C.C 16, Universidad de La Plata, 1900 La Plata,

Buenos Aires, Argentina

eacast@gmail.com

Franco Cataldo Actinium Chemical Research,

Via Casilina 1626/A, 00133 Rome, Italy

franco.cataldo@fastwebnet.it

Ciprian Ciubotariu Department of Computer Sciences,

University “Politehnica”, P-ta Victoriei No 2, 300006,

Timis¸oara, Romania

Dan Ciubotariu Department of Organic Chemistry, Faculty of Pharmacy,

“Victor Babes” University of Medicine and Pharmacy, P-ta Eftimie

Murgu No 2, 300041, Timis¸oara, Romania

dciubotariu@mail.dnttm.ro

Mircea V Diudea Faculty of Chemistry and Chemical Engineering,

“Babes-Bolyai” University, Arany Janos Str 11, 400028 Cluj, Romania

diudea@gmail.com

Tomislav Dosˇlic´ Faculty of Civil Engineering, University of Zagreb,

Kacˇic´eva 26, 10000 Zagreb, Croatia

doslic@master.grad.hr

ix

Shinsaku Fujita Shonan Institute of Chemoinformatics and MathematicalChemistry, Kaneko 479–7, Ooimachi, Ashigara-Kami-Gun,

Ante Graovac Department of Chemistry, Faculty of Science,

University of Split, Nikole Tesle 12, HR-21000 Split, Croatia

NMR Center, The “Ruđer Bošković” Institute, HR-10002 Zagreb, Croatia

IMC, University of Dubrovnik, Branitelja Dubrovnika 29, HR-20000

Istva´n La´szlo´ Department of Theoretical Physics, Institute of Physics,

Budapest University of Technology and Economics, H-1521 Budapest, Hungarylaszlo@eik.bme.hu

Bono Lucˇic´ The Rugjer Bosˇkovic´ Institute, Bijenicˇka 54,

P.O.B 180, HR-10 002 Zagreb, Croatia

lucic@irb.hr

Mihai Medeleanu Department of Organic Chemistry, University

“Politehnica”, P-ta Victoriei, No 2, 300006, Timis¸oara, Romania

mihai.medeleanu@chim.upt.ro

Ante Milicˇevic´ The Institute for Medical Research and Occupational Health,Ksaverskac 2, P.O.B 291, HR-10 002 Zagreb, Croatia

antem@imi.hr

Csaba L Nagy Faculty of Chemistry and Chemical Engineering,

“Babes-Bolyai” University, Arany Janos Str 11, 400028 Cluj, Romania

nc35@chem.ubbcluj.ro

Sonja Nikolic´ The Rugjer Bosˇkovic´ Institute, Bijenicˇka 54,

P.O.B 180, HR-10 002 Zagreb, Croatia

sonja@irb.hr

Tudor Olariu Department of Organic Chemistry,

Faculty of Pharmacy, “Victor Babes” University of Medicine

and Pharmacy, P-ta Eftimie Murgu No 2, 300041,

Timis¸oara, Romania

rolariu@umft.ro

Ottorino Ori Actinium Chemical Research,

Via Casilina 1626/A, 00133 Rome, Italy

ottorino.ori@alice.it

Sı´lvia Osuna Institut de Quı´mica Computacional and Departament de Quı´mica,Universitat de Girona, Campus Montilivi, 17071 Girona, Catalonia, Spainsilvia.osuma@udg.edu

Tomazˇ Pisanski Department of Theoretical Computer Science, Institute

of Mathematics, Physics and Mechanics, University of Ljubljana,

Jadranska 19, SI-1000 Ljubljana, Slovenia

tomaz.pisanski@fmf.uni-lj.si

Dejan Plavsˇic´ NMR Center, The “Ruđer Bošković” Institute,

HR-10002 Zagreb, Croatia

dplavsic@irb.hr

Lionello Pogliani Dipartimento di Chimica, Universita` della Calabria,

via P Bucci, 87036 Rende (CS), Italy

lionp@unical.it

Ana-Maria Putz Laboratory of Inorganic Chemistry, Institute

of Chemistry of Timis¸oara Romanian Academy, Ave Mihai Viteazul,

No 24, Timis¸oara, RO 300223, Romania

putzanamaria@yahoo.com

Mihai V Putz Laboratory of Computational and Structural Physical Chemistry,Chemistry Department, West University of Timis¸oara,

Pestalozzi 16, Timis¸oara, RO 300115, Romania

mv_putz@yahoo.com; mvputz@cbg.uvt.ro; www.mvputz.iqstorm.ro

Sandip K Rajak Department of Chemistry, University of Kalyani,

Kalyani 741235, India

sandip1ku@gmail.com

Milan Randic´ National Institute of Chemistry, P.O Box 3430,

1001 Ljubljana, Slovenia

mrandic@msn.com

Cecilia Savii Laboratory of Inorganic Chemistry, Institute of Chemistry

Timis¸oara of Romanian Academy, Ave Mihai Viteazul,

No 24, Timis¸oara, RO 300223, Romania

ceciliasavii@yahoo.com

Miquel Sola` Institut de Quı´mica Computacional and Departament de Quı´mica,Universitat de Girona, Campus Montilivi, 17071 Girona, Catalonia, Spainmiquel.sola@udg.edu

Marcel Swart Institut de Quı´mica Computacional and Departament de Quı´mica,Universitat de Girona, Campus Montilivi, 17071 Girona, Catalonia, SpainInstitucio´ Catalana de Recerca i Estudis Avanc¸ats (ICREA),

Pg Lluı´s Companys 23, 08010 Barcelona, Spain

marcel.swart@udg.edu

Nenad Trinajstic´ The Rugjer Bosˇkovic´ Institute, Bijenicˇka 54,

P.O.B 180, HR-10 002 Zagreb, Croatia

trina@irb.hr

Vicentiu Vlaia Department of Organic Chemistry, Faculty of Pharmacy,

“Victor Babes”, University of Medicine and Pharmacy,

P-ta Eftimie Murgu No 2, 300041, Timis¸oara, Romania

vlaiav@gmail.com

Quantum Parabolic Effects of Electronegativity

Mihai V Putz1

Abstract The fundamental issue of conceptually assessment of the totalpi-electronic energy is here addressed towards the possibility in assuming theelectronegativity and chemical hardness within a quantum parabolic energeticeffect that closely resembles other pi-equivalent energy expressions within thesemi-empirical computation framework as better as the carbon-based systemincreases its complexity On the other side, the present analysis affirms electro-negativity as the quantum observable for the states that represent full particleexistence, while chemical hardness posses the second quantization degree ofuncertainty in observation, although through the present study an alternativedefinite H€uckel based resonance integral expression is advanced

Chemistry in general and quantum chemistry in special is nowadays affirmed as themost intriguing application of the physical and quantum mechanics principles,respectively This because, beside offering among the first application of thequantum theory through offering the consistent picture of the chemical bonding

by means of molecular orbital theory, it arrives to be developed in the powerfulcomputational chemistry that allows the so called molecular design beingperformed with so many application in bio-, eco-, toxico-, and pharmaco-logywhile drastically reducing the experimental costs, risks and time

However, conceptually, modeling the chemical bonding seems to combine atbest the main feature of the quantum characterization of Nature as illustrated in theflowing Figure1.1 Basically, starting with a collection ofN- electrons that evolve

1 Laboratory of Computational and Structural Physical Chemistry, Chemistry Department, West University of Timis¸oara, Pestalozzi 16, Timis¸oara, RO 300115, Romania

e-mail: mv_putz@yahoo.com ; mvputz@cbg.uvt.ro ; www.mvputz.iqstorm.ro

M.V Putz (ed.), Carbon Bonding and Structures: Advances in Physics and Chemistry,

Carbon Materials: Chemistry and Physics 5, DOI 10.1007/978-94-007-1733-6_1,

# Springer Science+Business Media B.V 2011

1

in a given (nuclei) potentialVðrÞ they are qualitatively represented within the firstquantization scheme by the celebrated one-electronic wave functions’ði ¼ 1; NÞ(Slater1929) that eventually combine (viz Hartree-Fock factorization combinedwith superposition principles) to produce the so called molecular orbitals

C
’ði ¼ 1; NÞ(Hartree 1957; Slater 1963); The quantitative realm is finallygained since the second quantization allows converting the molecular orbital many-electronic nature into the allied electronic density as prescribed by the basicprinciple of the Density Functional Theory (Parr1983; Kohn et al.1996; Parr andYang1989; Dreizler and Gross1990; March1991)

Remarkably, expression (1.4) is so close in form with a frontier formulation ofthe “chemical” energy of a system, as being the energy engaged or responsible forthe chemical reaction taking place or, in other terms, the energy endorsed in thesystems’ chemical bond that can be consumed for further reactivity, affinity, orligation This can be immediately become more apparent once the electronegativityand chemical hardness definitions (1.5) and (1.6) are further explicated in theirdifferential counterparts (Parr et al.1978; Parr and Pearson1983):

Note that, within the frontier view, the electronegativity and chemical hardnessmay be considered as two “orthogonal” (thus independent) chemical descriptors,see the HOMO-LUMO midlevel vs gap of Eqs.1.7and1.8, and can be thereforefurther used as 2D realization of the reaction coordinates to build up the chemicalorthogonal space (COS) within which the chemical bond and reactivity isdescribed

Moreover, the present frontier picture involves, in fact, the frozen core tion according with the Koopmans’ (1934) theorem Consequently, the presentendeavor, like to explore to which extent this theorem is applicable to the chemicalsystems having delocalized or p- electrons available to engage in chemical reactiv-ity; even more, we like to quest whether the energy (1.4) may be correlated and inwhich degree with the common semi-empirical energetic contribution to the fron-tier or semi-classical or chemical domain of increasingly complex molecules; fromsimple groups to rings, fused rings and nanostructures In the case of relevantresults, apart of offering a sort of practical energetic consequence of the Koopmanstheorem, i.e affirming the viable parabolic quantum effect of electronegativity andchemical hardness on the total energy of the system, the present study will assessthe orthogonal basis set wf ;  w?j g as a viable quantum set of indicators for thechemical reactivity space (Putz2011a) Whether and in which degree it is sufficient

assump-or universal fassump-or chemical reactivity will be responded by this and subsequentcommunication

1.2 Parabolic Principles of Electronegativity

and Chemical Hardness

The parabolic energetic relationship (1.4) is here tested against the physical tional principle to see with which extend it is capable to unfold the popularchemical reactivity principles, while providing a consistent chemical bondingscenario To this end one starts with setting the total energy variation

as a working tool in modeling the dynamical equilibrium for natural systems Next,one expands the left-hand side of (1.9) within the total energy functional depen-dencyE ¼ E½N; VðrÞ, in the spirit of parabolic form (1.4), yet adding the explicitexternal potential influence

dE ¼ wdN þ ðdNÞ2þ

ð

rðrÞdVðrÞdr (1.10)through the author’s identified chemical action (see Putz2003,2009aand the nextdiscussion)

Now, when Eqs.1.9and1.10are combined, one realizes that:

• either there is no action on the system (dN ¼ dV ¼ 0) so that no chemicalphenomena is recorded since the physical variational principle (1.9) is fulfilledfor whatever electronegativity and chemical hardness values in (1.10);

• or there is no electronic system at all (w¼  ¼ rðrÞ ¼ 0)

Therefore, it seems that the variational physical principle of Eq 1.9 do notsuffice to encompass the limiting cases of equilibrium, when is about the chemicalfrontier or valence domain; in passing, such apparently odd behavior of variationalprinciple is nothing else than another illustration the chemical principles are notreducible to physical ones but their complement (Putz 2011b) This is also thepresent case when the double variational procedure on the total energy is needed,i.e through applying the additional differentiation on physical energy expansion(1.10), within the so called “chemical variational mode” (and denoted as d½) wherethe total differentiation will be taken only over the scalar-global (extensive w; ; dN)and local (intensive rðrÞ; VðrÞ) but not over the vectorial (physical – as the coordi-nate itself r) quantities, and yields (Putz2011c)

d dE½  ¼ d½wdN þ d½ðdNÞ2 þ

ð

d rðrÞdVðrÞ½ dr (1.12)

Now, thechemical variational principle applied to Eq.1.12takes the form

when certain amount of charge transfer and the system’s potential fluctuations(departing from equilibrium) are involved in producing chemical reactivity and/orbinding

dN ¼ dNj j ¼ ct: 6¼ 0; dVðrÞ 6¼ 0 (1.14)Now, Eq 1.12 with condition (1.13) releases the quantitative basis of theindividual reactivity principles:

• for electronegativity contribution we have the general inequality:

 d½wdN  0 ,  dNj jd½w  0 ) d½w  0 (1.15)containing both the equality and minimum electronegativity fluctuations aroundchemical equilibrium (Mortier et al.1985; Tachibana1987; Tachibana and Parr

• for chemical action contribution there remains the sufficient exact equality

 d Vðr0Þ

ðrðrÞdr

dw¼ 0 ! dCA¼ 0 ! Dw<0 ! d ¼ 0 ! D>0 (1.19)

as corresponding to the encountering (or the electronegativity equality) stage,followed by chemical action minimum variation (i.e the global minimum ofbonding interaction), then by the charge fluctuation stage (due to minimum orresidual electronegativity), ending up withthe polarizability stage (or HSAB) andwith the final steric (due to maximum or residual hardness) stage Nevertheless,from Eq.1.19one observes the close laying chemical action with electronegativityinfluence in chemical reactivity and bonding principles.

Having conceptually advocating on electronegativity and chemical hardnessdifferent influences on various levels of quantum reactivity of atoms and molecules,the global scenario of reactivity may be advanced implying five stages of chemicalbonding hierarchy by referring to the principles resumed in Table1.1:

(i) The encountering stage, associated with the charge flow from the moreelectronegativity regions to the lower electronegativity regions in a molecularformation, is thus dominated by the difference in electronegativity betweenreactants and consumed when the electronegativity equalization principle isfulfilled among all constituents of the products: it is thecovalent binding step(Mortier et al.1985; Sanderson1988);

(ii) Theglobal optimization stage, associates with the variational principle ofthe total energy of ground/valence state in bonding that can be resumed

by the corresponding chemical action principle (Putz 2003,2009a, 2011a;Putz and Chiriac 2008) that adjust the applied potential and the responseelectronic density to be convoluted/coupled in optimum/unique way,i.e establishing the global minima on the potential surface of the system

Table 1.1 Synopsis of the basic principles of reactivity towards chemical

equilibrium with environment in terms of electronegativity, chemical action,

and chemical hardness (Putz 2008b , 2011a )

Chemical

Principle Principle of Bonding

dw ¼ 0 Electronegativity equality:

“Electronegativity of all constituent atoms in a bond

or molecule have the same value” (Sanderson 1988 )

dC A ¼ 0 Chemical action minimum variation:

Global minimum of bonding is attained by optimizing the convolution of the applied potential with the response density (Putz 2003 , 2008a , 2009a , 2011a )

Dw<0 Minimum (residual) electronegativity:

“the constancy of the chemical potential is perturbed by the electrons of bonds bringing about a finite difference in regional chemical potential even after chemical equilibrium is attained globally” (Tachibana et al 1999 )

d  ¼ 0 Hard-and-soft acids and bases:

“hard likes hard and soft likes soft” (Pearson 1973 ,

1990 , 1997 ) D>0 Maximum (residual) hardness:

“molecules arranges themselves as to be as hard as possible”

(Pearson 1985 , 1997 )

(iii) Thecharge fluctuation stage, relies on the fact that partial fractional instead

of integer charges are associated with atoms-in-molecules; therefore, even ifthe chemical equilibrium is attained globally the electrons involved in bondsacts as foreign objects between pairs of regions, at whatever level ofmolecular partitioning procedure, grounded by the quantum fluctuations inspecial and by quantum nature of the electron in general; it produces thedegree ofionicity occurred in bonds (Tachibana and Parr1992; Tachibana

et al.1999);

(iv) Thepolarizability stage, in which the induced ionicity character of bonds ispartially compensated by the chemical forces through the hardness equaliza-tion between the pair regions in molecule; at this point theHSAB principle(Pearson1973,1990,1997; Chattaraj and Schleyer1994; Chattaraj and Maiti

2003; Putz et al.2004) is involved as a second order effect in charge transfer –see the step (i) above;

(v) Thesteric stage, where the second order of quantum fluctuations provides afurther amount of finite difference, this time in attained global hardness, that istransposed in relaxation effects among the nuclear and electronic distributions

so that the remaining unsaturated chemical forces to be dispersed by tion of the molecular structure; this is covered by the maximum hardnessprinciple and the fully stabilization of the molecular system in a given envi-ronment (Pearson1985,1997; Chattaraj et al.1991,1995; Putz2008a).Having this way proved the efficiency of the parabolic energy expression in terms

stabiliza-of electronegativity and chemical hardness indices, with the regulatory effects inchemical reactivity principles, the next step consists in discussing their observabil-ity character in order to can be employed as viable quanto-computational toolslinking the density with many-electronic information and with the energetic para-bolic behavior

and Chemical Hardness

As Fig.1.1suggests the second quantization stays as the key step in recovering theobservable quantities in chemical domain, since assuring the passage from orbital todensity description of open systems As such, one would next proceed withexpressing the electronegativity and chemical hardness within the framework ofsecond quantization as well, through relaying on the generalparabolic Hamiltonian(Surja´n1989),

built within the fermionic Fock space with the help of creation and annihilationparticle operators

^1 ¼ 0j i 0h j þ 1j i 1h j ¼^a^aþþ^aþ^a ¼ ^a; ^a
þ

(1.25)while fulfilling the dot product rules

Now, the passage from the orbital to density picture may be immediatelyillustrated with the aid of the second quantization presented rules by employingthe inner-normalization of the Eq.1.1under the form

1¼ Ch 0 j C0i ¼ Ch 0j^1 Cj 0i

¼ Ch 0jð^a^aþþ^aþ^aÞ Cj 0i ¼ Ch 0j^a^aþjC0i þ Ch 0j^aþ^a Cj 0i

¼ 0 j Cjh 0ij2þ 1 j Cjh 0ij2¼ ð1  r0Þ þ r0; r02 ½0; 1; ð1:28Þfor unperturbed frontier molecular state Cj 0i with associated eigen-energy E0for agiven valence system, reciprocally related by the conventional eigen-equation

^

H Cj 0i ¼ E0jC0i (1.29)Observe that here the molecular orbital state is placed on the frontier domainsuch that further reactivity will be accounted by means of expressing the ionizationand affinity actions, namely as (Putz2009b,2011d)

obtaining therefore the perturbed frontier states through the perturbation factor l

In these conditions, the frontier indices of electronegativity and chemical hardnessare formed from the perturbed energy

EI$Al2<

:¼ CIlH CAl

CIl CAl (1.32)and electronic density

rI$A l2< :¼ CIl aþ^a CA

Explicit dependency on the perturbation factor is thus necessary onboth perturbed density and energy, in order the involved derivatives @l=@rl

and @ Eh i=@l be appropriately formulated and combined in working Eqs.l 1.34

C0

h j ^H^aþ^a Cj 0i ¼ Ch 0j ^H 1j i 0 j 0h i 1 j Ch 0i ¼ E0hC0j 1i 1 j Ch 0i ¼ E0r0;

(1.42)

(1.48)

while for the chemical hardness the combination of Eqs.1.40,1.41,1.46, and1.47

in 1.35 produces the frontier orbital chemical hardness with its density limits(Putz2010a,2011a,d)

• chemical hardness does not manifest as a quantum index (or it has the zerovalue) for densities that are not integer representation of fermionic existence; inother words it has no quantum observable character for electronic states unlessthey are fully equivalent with (integer) particle manifestations; on the other side,for such integer density states, i.e the second and the third branches of the limit(1.49), the chemical hardness has not definite (universally observable) quantity;

if the result in such cases is infinite it act like a field (like the electronegativity,i.e like a super-potential since the electronegativity is seen as the minus of thechemical potential); if it is zero then in all cases the chemical hardness is notobservable and the parabolic form itself of the frontier energy is superfluous; forthe non-zero results chemical hardness preserves the parabolic form of thefrontier energy (1.4) with the meaning registering its curvature, i.e how fast itchanges from donor to acceptor character, in accordance with the high of theHOMO-LUMO gap of Eq.1.8

Overall, beside the fact the quantum observable character of the chemical hardnessremain an open issue, being neither informed nor definitely confirmed by thepresent analysis, there was this way nevertheless argue on the parabolic sufficiency

on the quantum expansion in the frontier energy; in other terms, the secondquantization firmly prescribes the manifestation of the electronegativity as anenergy for the quantum states that characterizes full existence of the particles,being this virtually accompanied by the generally not definite second order contri-bution coming from the chemical hardness observable indeterminacy Even shorter,for precisely defined particle-quantum states their energy may be represented as asuperposition of an observable and a not observable (hidden variable) chemicalcontribution; such unique manifestation may be regarded as a special or comple-mentary uncertainty principle for chemical behavior, in the spirit of aboveenounced physics-to-chemistry non-reductionism

However, there is clear that if the second order or parabolic manifestation ofchemical phenomena inscribes quantity with non-observable character, the cubic oreven higher order of quantum energy manifestation may be conceptually discarded

There remains however to establish the specific behavior of the assumed parabolicquantum effect on chemical reactivity in an quantitative manner – an issue that will

be in next addressed, after surveying the most common semi-empirical orbitalpictures, as another level of treating the frontier effects of orbitals, this time on

an inter-orbitalic basis

1.4.1 General Mono-Electronic Molecular Orbitals’ Equations

For better understanding how the semi-empirical methods were introduced, worthreconsidering the molecular eigen-equation (1.29) under the so called independent-electron problem

^

Hieffci¼ Eici (1.50)with the aid of effective electron Hamiltonian partitioning

As such, viewed as the linear combination over theatomic orbitals, the resultedMO-LCAO wave-function

ci¼X

replaced in Eq.1.50and followed by integration over the electronic space allowsfor matrix version of Eq.1.50

by the matrix of coefficients (C); the resulted “Hamiltonian” is called Fock operator,while the associated eigen-problem is consecrated as the Hartree-Fock equation

Pmn¼Xocc i

J ¼ (mm|vv) as the Coulomb integral describing repulsion between two electronswith probabilities f2mand f2n

Moreover, the Hartree-Fock Eq.1.63with implementations given by Eqs.1.64

and1.65 are known as Roothaan equations (Roothaan 1951) and constitute thebasics for closed-shell (or restricted Hartree-Fock, RHF) molecular orbitalscalculations Their extension to the spin effects provides the equations for theopen-shell (or unrestricted Hartree-Fock, UHF) known also as the Pople-NesbetUnrestricted equations (Pople and Nesbet1954)

1.4.2 Semiempirical Approximations

The second level of approximation in molecular orbital computations regards thevarious ways the Fock matrix elements of Eq 1.65 are considered, namely theapproximations of the integrals (1.66) and of the effective one-electron Hamiltonianmatrix elementsHmn

The main route for such endeavor is undertaken through neglecting at differentdegrees certaindifferential overlapping terms (integrals) – as an offset ansatz –although with limited physical justification – while the adjustment with experiment

is done (post-factum) by fitting parameters – from where the semiempirical name

of such approximation Practically, by emphasizing the (nuclear) centers in theelectronic overlapping integral (1.57)

Smn¼

ð

fAmðr1ÞfB

nðr1Þdr1 (1.67)the differential overlap approximation may be considered by two situations

• Byneglecting the differential overlap (NDO) through the mono-atomic orbitalicconstraint

fmfn¼ fmfmdmn (1.68)leaving with the simplified integrals

• Byneglecting the diatomic differential overlap (NDDO) of the bi-atomic orbitals

fAmfBn ¼ fA

mfAndAB (1.71)that implies the actual simplifications

1.4.2.1 NDO Methods

The basic NDO approximation was developed by Pople and is known as the CompleteNeglect of Differential Overlap CNDO semiempirical method (Pople et al 1965;

Pople and Segal1965,1966; Pople and Beveridge1970) It employs the mation (1.68)–(1.70) such that the molecular rotational invariance is respectedthrough the requirement the integral (1.70) depends only on the atoms A or Bwhere the involved orbitals reside – and not by the orbitals themselves That is theintegral gAB in (1.69) is seen as the average electrostatic repulsion between anelectron in any orbital of A and an electron in any orbital of B

although the spin effects are not at all considered since no exchange integralinvolved This is in fact the weak point of the CNDO scheme and it is to be slightlyimproved by the next semiempirical methods.

The exchange effect due to the electronic spin accounted within the IntermediateNeglect of Differential Overlap (INDO) method (Slater1960) through considering

in Eqs 1.75 and 1.77 the exchange one-center integrals gAA K ¼ mn mnð j Þ isevaluated as

The INDO method may be further modified in parameterization of the spin effects

as developed by Dewar’s group and lead with the Modified Intermediate Neglect ofDifferential Overlap (MINDO) method (Pople and Beveridge1970; Baird and Dewar

1969; Dewar and Hasselbach1970; Dewar and Lo1972; Bingham et al.1975a,b,c,

d; Dewar et al.1975; Murrell and Harget1971) whose basic equations look like

XB A

Apart of specific counting of spin effects, other particularity of MINDOrespecting the CNDO/INDO is that all the non-zero two-center Coulombintegrals are set equal and parameterized by the appropriate one center twoelectrons integralsAAandABwithin the Ohno-Klopman expression (Ohno1964;Klopman1964)

gAB

MINDO¼ sðAsAjsBsBÞ ¼ sð AsAjpBpBÞ ¼ pð ApAjpBpBÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

r2

ABþ14

HMINDOmn ¼ Imþ In

with the parameter bMINDOAB being now dependent on the atoms-in-pair ratherthan the average of atomic pair involved As in INDO, the exchange terms, i.e.,the one-center-two-electron integrals, are computed employing the atomic spectraand theGk,Fk, Slater-Condon parameters, see Eq.1.81(Pople et al.1967) Finally,worth mentioning that the MINDO (also with its MINDO/3 version) improves uponthe CNDO and INDO the molecular geometries, heats of formation, being particu-larly suited for dealing with molecules containing heteroatoms.

in above Fock (1.86) and (1.87) expressions

Now, regarding the (Coulombic) two-center-two-electron integrals of type(1.73) appearing in Eqs.1.86and1.87there were indentified 22 unique forms foreach pair of non-hydrogen atoms, i.e., the rotational invariant 21 integralsðss ssj Þ,

ss pj sps

ð Þ, ss pð j pppÞ, , pð spsjpspsÞ, pð pppjppppÞ, , spð sjspsÞ, spð pjsppÞ, .,

pppsjspp

ð Þ, pð ppsjpppsÞ, and the 22nd one that is written as a combination of two

of previously ones, namelyðpppp 0jpppp 0Þ ¼ 0:5 p½ð pppjppppÞ  pð pppjpp 0pp 0Þ, with

the typical integral approximation relaying on the Eq (1.84) structure, howeverslightly modified as

HMNDOmn ¼bMNDOm þ bMNDO

n

containing the atomic adjustable parameters bMNDOm and bMNDOn for the orbitals fmand fnof the atoms A and B, respectively The exchange (one-center-two-electron)integrals are mostly obtained from data on isolated atoms (Oleari et al 1966).Basically, MNDO improves MINDO through the additional integrals consideredthe molecular properties such as the heats of formations, geometries, dipolemoments, HOMO and LUMO energies, etc., while problems still remaining withfour-member rings (too stable), hypervalent compounds (too unstable) in general,and predicting out-of-plane nitro group in nitrobenzene and too short bond length(~0.17 A˚ ) in peroxide – for specific molecules

The MNDO approximation is further improved by aid of the Austin Model 1(AM1) method (Dewar et al 1985; Dewar and Dieter1986; Stewart 1990) thatrefines the inter-electronic repulsion integrals

sAsAjsBsB

ð ÞAM1¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

r2

ABþ14

of molecules (e.g., for phosphorous compounds), still provides inaccurate modeling

of phosphorous-oxygen bonds, too positive energy of nitro compounds, while the

peroxide bond is still too short In many case the reparameterization of AM1 underthe Stewart’s PM3 model (Stewart1989a,b) is helpful since it is based on a widerplethora of experimental data fitting with molecular properties The best use of PM3method lays in the organic chemistry applications.

Going to systematically implement the transition metal orbitals in semiempiricalmethods the INDO method is augmented by Zerner’s group either with non-spectroscopic and spectroscopic (viz fitting with UV spectra) parameterization(Del Bene and Jaffe´ 1968a, b, c), known as ZINDO/1 and ZINDO/S methods,respectively (Ridley and Zerner1976; Bacon and Zerner1979; Stavrev et al.1995;Stavrev and Zerner 1995; Cory et al 1997; Anderson et al 1986, 1991) Theworking equations are formally the same as those for INDO except for the energy

of an atomic electron of Eq.1.77that now uses only the ionization potential instead

of electronegativity of the concerned electron Moreover, for ZINDO/S the coreHamiltonian elementsHmmis corrected

Beyond of either NDO or NDDO methods the self-consistent computation ofmolecular orbitals can be made by the so called ab initio approach directly relaying

on the HF equation or on its density functional extension, as will be in next sketched

The H€uckel method is simple and has been in use for decades (H€uckel1931a,b)

It is based on the s-p separation approximation while accounting for the pi-electronsonly, i.e the atomic orbitals involved refer to those 2p for Carbon atoms as well to

the 2pzand 3pzorbitals for the second and third period elements as (N,O, F) and(S, Cl) respectively; further discussion on the d-orbitals involvement may be alsoundertaken, yet the method essence reside in non explicitly counting on theelectronic repulsion with an effective, not-defined, mono-electronic Hamiltonian,

as the most simple semi-empirical approximation In these conditions, for themono-electronic Hamiltonian matrix elements two basic assumptions are advanced,namely:

• In the case of hydrocarbures (C containing only p-systems) one has:

• In the case heteroatoms (X) are present in the system one has to consider theCoulombic parameter hX correlating with the electronegativity differencebetween the heteroatom X and carbon, along the resonance parameterkCX thatmay include correlation with the binding energy; the form of matrix elements ofmonoelectronic effective Hamiltonian looks therefore as

EparaðbolicÞffi w Npþ  N2

when identified the reactive frontier electrons with the pi-electrons in the system,

DN ¼ N Even more, the present discussion permits the identification of the

Coulombic and resonance integrals in terms or electronegativity and chemicalhardness

para-EpiðmoleculeÞ ffi ETotalðmoleculeÞ  EBindðmoleculeÞ  EHeatðmoleculeÞ

• The total energy is relative to a sum of atomic energies for semi-empiricalcomputations

ETotalðmoleculeÞ ¼moleculeX

atoms

• Binding energy is the energy of the molecular atoms separated by infinity minusthe energy of the stable molecule at its equilibrium bond length

EBindðmoleculeÞ ¼ E1ðatomsÞ  EequilibriumðmoleculeÞ (1.103)

• The heat of formation is calculated by subtracting atomic heats of formationfrom the binding energy:

EHeatðmoleculeÞ ¼ EBindðmoleculeÞ moleculeX

Table 1.2 The Butadiene p-system, with DN¼N p ¼4, frontier energetic quantities, ionization potential ( IP), electron affinity (EA), electronegativity (w), and chemical hardness () of Eqs 1.7 and 1.8 – in electron volts (eV), and the resulted parabolic energy of Eq 1.98 , alongside with the p-related energy based on the H €uckel simplified (with Coulomb integrals set to zero, a ¼ 0) expression of ( 1.97 ) for the experimental/H €uckel method and on the related energy form of

Eq 1.101 and the other semi-empirical methods ( CNDO, INDO, MINDO, MNDO, AM1, PM3, ZINDO) as described in the previous section – expressed in kilocalories per mol (kcal/mol); their ratio in the last column reflects the value of the actual departure of the electronegativity and chemical hardness parabolic effect from the pi-bonding energy, while for the first ( Exp/H€ uckel) line it expresses the resonance contribution (and a sort of b factor integral) in ( 1.97 ) for the p-bond

in this system; the eV to kcal/mol conversion follows the rule 1 eV ffi 23.069 kcal/mol

a Calculated as the negative of the HOMO and LUMO energies (University Illinois 2011 )

Table 1.3 The same quantities as those reported in Table 1.2 , here for the Benzene p-system, with DN¼N p ¼6

INDO 13.48267 4.58566 4.448502 9.034166 3135.63 22231.9 7.09009 MINDO3 9.179751 1.24984 3.964955 5.214796 1616.597 18289.3 11.3135

a From National Institute of Standard and Technology (NIST 2011a )

b From interpolation data presented in Fig 1.3

c From the H €uckel total p-energy: 2 (2+2)¼8b[a.u.] 627.71 .~5021.68 kcal/mol, see Cotton ( 1971a )

the bivariate correlation between the obtained parabolic- and pi- energies are inFigs 1.2, 1.4, 1.5, and 1.6 represented From these results one may note thesystematic increasing of theEparaðbolicÞvs.Epicorrelation up to its almost parallel

Table 1.4 The same quantities as those reported in Table 1.2 , here for the Naphthalene p-system, with DN¼N p ¼10

ZINDO-1 7.512728 6.39221 0.560258 6.952471 7890.081 33558.4 4.25324 ZINDO-S 7.868645 0.04134 3.913653 3.954993 3659.046 24987.1 6.82887

a From National Institute of Standard and Technology (NIST 2011b )

b From the H €uckel total p-energy: 2 (2.303 + 1.618 + 1.303 + 1.000 + 0.618) ¼ 13.684b[a.u.] 627.71 .~8589.58 kcal/mol, see Cotton ( 1971b )

Table 1.5 The same quantities as those reported in Table 1.2 , here for the Fullerene p-system, with DN¼Np¼60

a From De Vries et al ( 1992 )

b From Yang et al ( 1987 )

c From H €uckel total p-energy: 93.161602 b[a.u.] 627.71 .~ 58478.5 kcal/mol, see Haddon

et al ( 1986 ); Haymet ( 1986 ); Fowler and Woolrich ( 1986 ); Byers-Brown ( 1987 )

Fig 1.2 The bivariate correlation of the parabolic- with p-energies as reported in Table 1.2 for Butadiene system

in Lide ( 2004 )

Fig 1.5 The bivariate correlation of the parabolic- with p- energies as reported in Table 1.4 for Naphthalene system

Fig 1.4 The bivariate correlation of the parabolic- with p- energies as reported in Table 1.3 for Benzene system