Density Functionals of Chemical Bonding

Transcription

1 Int. J. Mol. Sci. 008, 9, ; DOI: /ijms Special Issue The Chemical Bond and Bonding OPEN ACCESS Intenational Jounal of Molecula Sciences ISSN Review Density Functionals of Chemical Bonding Mihai V. Putz Laboatoy of Computational and Stuctual Physical Chemisty, Chemisty Depatment, West Univesity of Timişoaa, Pestalozzi Steet No.6, Timişoaa, RO-3005, Romania s: o Web: Received: 30 Apil 008; in evised fom: 9 June 008 / Accepted: 0 June 008 / Published: 6 June 008 Abstact: The behavio of electons in geneal many-electonic systems thoughout the density functionals of enegy is eviewed. The basic physico-chemical concepts of density functional theoy ae employed to highlight the enegy ole in chemical stuctue while its extended influence in electonic localization function helps in chemical bonding undestanding. In this context the enegy functionals accompanied by electonic localization functions may povide a compehensive desciption of the global-local levels electonic stuctues in geneal and of chemical bonds in special. Becke-Edgecombe and autho s Makovian electonic localization functions ae discussed at atomic, molecula and solid state levels. Then, the analytical suvey of the main wokable kinetic, exchange, and coelation density functionals within local and gadient density appoximations is undetaken. The hieachy of vaious enegy functionals is fomulated by employing both the paabolic and statistical coelation degee of them with the electonegativity and chemical hadness indices by means of quantitative stuctue-popety elationship QSPR analysis fo basic atomic and molecula systems. Keywods: density functional theoy, electonic localization function, kinetic enegy, exchange enegy, coelation enegy, exchange-coelation enegy, electonegativity, chemical hadness, QSPR.

2 Int. J. Mol. Sci. 008, Intoduction In Walte Kohn s lectue, with the occasion of eceiving his Nobel Pize in Chemisty [], back in 998, fo density functional theoy DFT theoy [-4], thee was fomulated a quite povoking assetion affiming that, heuistically, the geneal eigen-wave-function Ψ,..., N associated to a system of N electons fails to be a legitimate scientific concept when N N Nevetheless, this affimation may be at any time tuned in a theoem, eventually as Kohn s zeo DFT theoem, with a poof following the van Vleck pesciption of the so called exponential wall, leaving with the applicability limits of the conceptually eigen-wave function of multi-electonics systems. Howeve, befoe poceeding to demonstation, thee must be noted that such eality limitation chaacteized by eigen-wave-function of the multi electonic-systems is not tansfeable at the quantum mechanics postulates, but poviding an altenative quantum scheme, thus paalleling Schödinge equation, howeve in a moe geneally integated level. The demonstation of the non-epesentability of the eigen-function fo systems containing moe than N electons involves two aspects: the accuacy of epesentation by using the eigen-wave function and the possibility of measuing it. Regading the accuacy of the Ψ,..., N epesentation thee is widely known that it associates with the density of pobability smoothly appoaching unity, witten in the libeal fom accoding to Kohn: Ψ *,... N Ψ,..., N ε, ε 0 Now, consideing a collection of N molecules the total density of pobability of this multimolecula system and implicit a multi-electonic one will consequently be: * * N ' Ψ Ψ... Ψ Ψ N ' ε exp εn ' exp[ 0 N' ]. Fo N 0 3 molecules in whateve aggegates, e.g. solids, clustes, supe-molecules, o biological maco-molecules, the total density of pobability will esult fom as exp[-0] 5 0-5, meaning that it is detemined with much less accuacy compaed with the degee of individual eigen-function localization pecision in. Since each molecula system has at least one electon thee follows the theshold limit of N electons fom whee the lost in associated wavefunction natue is ecoded. This esult, elaying on the exponential fom, justifies the title of exponential wall fo the wavefunction limitation. Then, going to the measuable issue of such eigenfunctions, let s ask how many bits ae necessay fo ecoding its quantum dimension? Assume, again, the woking wavefunction Ψ,..., N fo all the N electons in a concened system. The N electons in system have a total of 3N space vaiables in the configuation space; let s now assume an aveage of q bits necessay in measuing a single vaiable fom the total of 3N; thee esults a total infomation of 3N B q 3 bits fo ecoding stoing the total eigen-wave-function of the system. Howeve, a simple evaluation of the dimension 3 shows that fo a minimum of q3 bits/vaiable and fo the above consecated minimum limit of NN electons in the system the total yield of necessay bits fo ecoding is about of ode a tuly non-ealistic dimension. This can be immediately visualized if one ecalls that the total numbe of bayons i.e. all femions and elementay paticles of potonic and

3 Int. J. Mol. Sci. 008, 9 05 neutonic type, but not limited only to these estimated in entiely Univese summing up all existing atoms and fee nuclei in the plasma state, but not only limited to these gives a esult of about 0 80 ode. Definitely, the concept of eigen-wave function must be enlaged o modified in such a manne that the quantum desciption does not be blocked by the exponential wall: fom whee we can stat? Fistly, as was exposed, the eigen-wave function in the configuation space multiplies in an exponential manne the vaiables accounting fo the numbe and the position of the electons; thus, the configuation space must be avoided. Then, the density of pobability must be efomulated as such the exponential wall fo a poly-electonic system be avoided while peseving the dependency of the total numbe of electons N. Fotunately, the above descibed conceptual poject was unfolded in 963 when Walte Kohn met in Pais at École Nomale Supéieue, duing his sabbatical semeste, the mate Piee Hohenbeg who was woking at the desciption of the metallic alloys specially the Cu x Zn -x systems by using quantum taditionally methods of aveaging cystalline peiodic field. Studies of this type of poblems often stat fom the level of the unifom electonic density efeential upon which specific petubation teatments ae applied. Fom this point Kohn and Hohenbeg made two cucial futhe steps in efomulation of the quantum pictue of the matte stuctue: one efeed at the electonic density, and anothe at the elation between electonic density with the extenally applied potential on the electonic system; they wee consecated in the so called Hohenbeg-Kohn KH theoems [, 5-7]. They wee the fundaments of new emeging quantum density functional theoy that mostly impacted the efomation of the quantum chemisty itself and its foegound pinciples of stuctue and tansfomation. The pesent wok likes to eview some fundamental aspects of density functional theoy highlighting on the pime conceptual and computational consequences in electonic localization and chemical eactivity.. Pimay Density Functional Theoy Concepts.. Hohenbeg-Kohn theoems The fist Hohenbeg-Kohn HK theoem gives space to the concept of electonic density of the system in tems of the extensive elation with the N electons fom the system that it chaacteizes [8]: d N. 4 The elation 4 as much simple it could appeas stands as the decisive passage fom the eigen-wave function level to the level of total electonic density [9-]: * N Ψ,,..., N Ψ,,..., N d... dn. 5 Fistly, Eq. 5 satisfies Eq. 4; this can be used also as simple immediate poof of the elation 4 itself. Then, the dependency fom the 3N-dimensions of configuation space was educed at 3 coodinates in the eal space, physically measuable. Howeve, still emains the question: what epesents the electonic density of Eq. 5? Definitely, it neithe epesents the electonic density in the configuation space no the density of a single electon,

4 Int. J. Mol. Sci. 008, since the N-electonic dependency as multiplication facto of the multiple integal in 5. What emains is that is simple the electonic density of the whole concened system in space point. Such simplified intepetation, appaently classics, peseves its quantum oots though the aveaging integal ove the many-electonic eigenfunction Ψ,..., N in 5. Altenatively, the explicit nondependency of density on the wave function is also possible within the quantum statistical appoach whee the elation with patition function of the system the global measue of the distibution of enegetic states of a system is mainly consideed. The majo consequence of this theoem consists in defining of the total enegy of a system as a function of the electonic density function in what is known as the density functional [8, 9, ]: E [ ] F HK [ ] + C A[ ], 6 fom whee the name of the theoy. The tems of enegy decomposition in 6 ae identified as: the Hohenbeg-Kohn density functional F HK [ ] T[ ] + Vee[ ] 7 viewed as the summed electonic kinetic T [] and electonic epulsion V [ ee ], and the so called chemical action tem [-4]: C A [ ] V d, 8 being the only explicit functional of total enegy. Although not entiely known the HK functional has a emakably popety: it is univesally, in a sense that both the kinetic and inte-electonic epulsion ae independent of the concened system. The consequence of such univesal natue offes the possibility that once it is exactly o appoximately knew the HK functional fo a given extenal potential V emain valuable fo any othe type of potential V applied on the concened many-electonic system. Let s note the fact that V should be not educed only to the Coulombic type of potentials but is caying the ole of the geneic potential applied, that could beg of eithe an electic, magnetic, nuclea, o even electonic natue as fa it is extenal to the system fixed by the N electons in the investigated system. Once in game the extenal applied potential povides the second Hohenbeg-Kohn HK theoem. In shot, HK theoem says that the extenal applied potential is detemined up to an additive constant by the electonic density of the N-electonic system gound state. In mathematical tems, the theoem assues the validity of the vaiational pinciple applied to the density functional 6 elation, i.e. [6] E [ ] E[ ] δe[ ] 0 9 fo evey electonic test density aound the eal density of the gound state. The poof of vaiational pinciple in 9, o, in othe wods, the one-to-one coespondence between the applied potential and the gound state electonic density, employs the eduction ad absudum pocedue. That is to assume that the gound state electonic density coesponds to two extenal potentials V, V fixing two associate Hamiltonians H, H to which two eigen-total enegy E, E and two eigen-wave functions Ψ, Ψ ae allowed. Now, if eigen-function Ψ is consideed as the tue one fo the gound state the vaiational pinciple 9 will cast as the inequality: * * * E [ ] Ψ H Ψdτ < Ψ H Ψdτ Ψ H + H H Ψdτ 0 which is futhe educed, on univesality easons of the HK functional in 6, to the fom:

5 Int. J. Mol. Sci. 008, [ V V ] d E[ ] < E[ ] +. On anothe way, if the eigen-function Ψ is assumed as being the one tue gound state wavefunction, the analogue inequality spings out as: E[ ] < E[ ] + [ V V ] d. Taken togethe elations and geneate, by diect summation, the evidence of the contadiction []: E ] + E [ ] < E [ ] + E [ ]. 3 [ The emoval of such contadiction could be done in a single way, namely, by abolishing, in a evese phenomenologically ode, the fact that two eigen-functions, two Hamiltonians and espectively, two extenal potential exist fo chaacteizing the same gound state of a given electonic system. With this statement the HK theoem is fomally poofed. Yet, thee appeas the so called V-epesentability poblem signaling the impossibility of an a pioi selection of the extenal potentials types that ae in bi-univocal elation with gound state of an electonic system [5-8]. The poblem was evealed as vey difficult at mathematical level due to the equivocal potential intinsic behavio that is neithe of univesal no of efeential independent value. Fotunately, such pincipial limitation does not affect the geneal validity of the vaiational pinciple 9 egading the selection of the enegy of gound state level fom a collection of states with diffeent associated extenal potentials. That because, the poblem of V-epesentability can be cicumvented by the so called N- contingency featues of gound state electonic density assuing that, aside of the N integability condition 4, the candidate gound state densities should fulfill the positivity condition an electonic density could not be negative [7, 8]: 0, R, 4 as well as the non-divegent integability condition on the eal domain in elation with the fact that the kinetic enegy of an electonic system could not be infinite since the light velocity estiction: R / d <. 5 Both 4 and 5 conditions ae easy accomplished by evey easonable density, allowing the employment of the vaiational pinciple 9 in two steps, accoding to the so called Levy-Lieb double minimization algoithm [9]: one egading the intinsic minimization pocedue of the enegetic tems especting all possible eigen-functions folding a tial electonic density followed by the extenal minimization ove all possible tial electonic densities yielding the coect gound state GS enegy density functional + + Ψ Ψ * EGS min min T Vee V dτ Ψ * Ψ T + V Ψd + V d ee τ min min Ψ min F [ ] + [ ] min E[ ]. HK C A One of the most impotant consequences of the HK conveys the ewiting of the vaiational pinciple 9 in the light of above N-contingency conditions of the tial densities as the woking Eule type equation: 6

6 Int. J. Mol. Sci. 008, δ{ E [ ] μn[ ]} 0 7 fom whee, thee follows the Lagange multiplication facto with the functional definition: δe[ ] μ δ 8 V this way intoducing the chemical potential as the fundamental quantity of the theoy. At this point, the whole chemisty can sping out since identifying the electonic systems electonegativity with the negative of the density functional chemical potential [9]: χ μ. 9 Up to now, the Hohenbeg-Kohn theoems give new conceptual quantum tools fo physicochemical chaacteization of an electonic sample by means of electonic density and its functionals, the total enegy and chemical potential electonegativity. These positive density functional pemises ae in next analyzed towads elucidating of the quantum natue of the chemical bond and eactivity [0]... Optimized enegy-electonegativity connection Back fom Pais, in the winte of 964, Kohn met at the San Diego Univesity of Califonia his new post-doc Lu J. Sham with who popose to extact fom HK & theoems the equation of total enegy of the gound state. In fact, they popose themselves to find the coespondent of the stationay eigenequation of Schödinge type, employing the elationship between the electonic density and the wave function. Thei basic idea consists in assuming a so called obital basic set fo the N-electonic system by eplacing the integation in the elation 5 with summation ove the vitual uni-electonic obitals ϕ i, i, N, in accodance with Pauli pinciple, assuing theefoe the HK fame with maximal spin/obital occupancy []: N ni ϕi, 0 ni, ni N. 0 i Then, the tial total eigen-enegy may be ewitten as density functional of eq. 6 natue expanded in the oiginal fom [, 3]: E [ ] F HK [ ] + C A[ ] T [ ] + V ee [ ] + C [ ] whee, the contibution of the efeential unifom kinetic enegy contibution N * Ts [ ] niϕi ϕi d i, with the infeio index s efeing to the spheical o homogeneous attibute togethe with the classical enegy of Coulombic inte-electonic epulsion J[ ] dd 3 A { T[ ] T [ ] + V [ ] J[ ] } C [ ] T [ ] + J[ ] + + s s N * niϕ i ϕ i d + dd + Exc[ ] + V d i ee i A

7 Int. J. Mol. Sci. 008, wee used as the analytical vehicles to elegantly poduce the exchange-coelation enegy E xc containing exchange V ee [ ] J[ ] and coelation T[ ] Ts[ ] heuistically intoduced tems as the quantum effects of spin anti-symmety ove the classical inteelectonic potential and of coected homogeneous electonic movement, espectively. Next, the tial density functional enegy will be optimized in the light of vaiational pinciple 7 as pescibed by the HK theoem. The combined esult of the HK theoems will eventually funish the new quantum enegy expession of multi-electonic systems beyond the exponential wall of the wave function. An instuctive method fo deiving such equation assume the same types of obitals fo the density expansion 0, * Nϕ ϕ 4 that, without diminishing the geneal validity of the esults, since peseving the N-electonic chaacte of the system, highly simplifies the analytical discouse. Actually, with the tial density 4 eplaced thoughout the enegy expession in has to undego the minimization pocedue 7 with the pactical equivalent integal vaiant: δ E[ ] μn[ ] * δϕ d 0. * 5 δϕ Note that, in fact, we chose the vaiation in the conjugated uni-obital ϕ * in 5 poviding fom 4 the useful diffeential link: δ Nϕ δϕ *. 6 Now, unfolding the equation 5 with the help of elations and 4, togethe with fundamental density functional pesciption 4, one fistly gets [4]: δ N * * * [ ] [ ] 0 * ϕ ϕ d J Exc NV ϕ ϕ d μnϕ ϕ d 7 δϕ By pefoming the equied patial functional deivations especting the uni-obital ϕ * and by taking account of the equivalence 6 in deivatives elating J [] and Exc[] tems, equation 7 takes the futhe fom: N δj[ ] δexc ϕ + Nϕ + Nϕ + NV ϕ μnϕ 0. 8 δ δ Afte immediate suppessing of the N facto in all the tems and by consideing the exchangecoelation potential with the fomal definition [4, 9]: δexc[ ] Vxc, δ 9 V equation 8 simplifies as [5, 6]: + ϕ μϕ V + d + Vxc. 30 Moeove, once intoducing the so called effective potential: V eff V + d + Vxc 3 the esulted equation ecoves the taditional Schödinge shape:

8 Int. J. Mol. Sci. 008, V eff μϕ ϕ. 3 The esult 3 is fundamental and equally subtle. Fistly, it was poved that the joined Hohenbeg- Kohn theoems ae compatible with consecated quantum mechanical postulates, howeve, still offeing a genealized view of the quantum natue of electonic stuctues, albeit the electonic density was assumed as the foegound eality. In these conditions, the meaning of functions ϕ is now unambiguously poducing the analytical passage fom configuation 3N-D to eal 3D space fo the whole system unde consideation. Nevetheless, the debate may still emain because once equation 3 is solved the basic functions ϕ geneating the electonic density 4 and not necessaily the eigen-functions of the oiginal system due to the pactical appoximations of the exchange and coelation tems appeaing in the effective potential 3. This is why the functions ϕ ae used to be called as Kohn-Sham KS obitals; they povide the obital set solutions of the associate KS equations [3]: + Veff i i i, i, N ϕ μ ϕ 33 once one econsides electonic density 4 back with geneal case 0. Yet, equations 33, apat of deliveing the KS wavefunctions ϕ i, associate with anothe famous physico-chemical figue, the obital chemical potential μ i, which in any moment can be seen as the negative of the obital electonegativities on the base of the elation 9. Going now to a summative chaacteization of the above optimization pocedue woth obseving that the N-electonic in an abitay extenal V-potential poblem is conceptual-computationally solved by means of the following self-consistent algoithm: i. It stats with a tial electonic density 0 satisfying the N-contingency conditions 4 and 5; ii. With tial density the effective potential 3 containing exchange and coelation is calculated; iii. With computed V eff the equations 33 ae solved foϕ i, i, N ; ϕ the new density 0 is ecalculated; iv. With the set of functions { i } i, N v. The pocedue is epeated until the diffeence between two consecutive densities appoaches zeo; ϕ, μ χ ; vi. Once the last condition is achieved one etains the last set { i i i } i, N vii. The electonegativity obital obseved contibutions ae summed up fom 33 with the expession: N N * χi niϕi + Veff ϕi d Ts[ ] + Veff d ; 34 i i viii. Replacing in 34 the unifom kinetic enegy, T s[] fom the geneal elation the density functional of the total enegy fo the N-electonic system will take the final figue [9, 4]: N E[ ] χi dd + { Exc[ ] Vxc d}. 35 i showing that the optimized many-electonic gound state enegy is diectly elated with global o summed ove obseved o aveaged o expected obital electonegativities. One can obseve fom 35 that even in the most optimistic case when the last two tems ae hopefully canceling each othe thee still emains a classical coection to be added on global electonegativity in total enegy. O, in othe tems, electonegativity alone is not enough to bette descibe the total enegy of a many-electonic system, while its coection can be modeled in a global almost classical way. Such consideations

9 Int. J. Mol. Sci. 008, stessed upon the accepted semiclassical behavio of the chemical systems, at the edge between the full quantum and classical teatments. Howeve, analytical expessing the total enegy equies the use of suitable appoximations, wheeas fo chemical intepetation of bonding the electonic localization infomation extacted fom enegy is compulsoy. This subject is in next focused followed by a eview of the popula enegetic density functionals and appoximations. 3. Electonic Localization Poblem 3.. Fom global functional to localization function. Localization in solids The application of Hohenbeg-Kohn theoems consecate the cucial contibutions of the so called spheical o homogeneous kinetic and of the exchange-coelation enegy tems in a multi-electonic system s gound state. Howeve, the spheical electonic case coesponds to the non petubed electonic system fo which the Thomas-Femi TF model was aleady advanced thoughout totally ignoing exchange- coelation tems fom the total enegy shape: E TF [ ] TTF [ ] + J[ ] + C A[ ]. 36 Such a efeential pictue is most useful in establishing the unifom electonic distibution by indicating the occupation of the all-possible electonic levels in a semiclassical quantum fame without explicit exchange-coelation involvement. Actually, the Femi sphee in a momentum space finely defines the total homogeneous kinetic enegy as: p F τ s, 37 m0 while the quantum natue of the kinetic enegy 37 is coveed by involving the quantum Heisenbeg uncetainty 3 Δ pxδp yδpzδxδyδz h 38 in unifom density computation. This suggests that the density of states in the Femi volume of the impulse p F has to be nomalized to the invese of the cube powe of Planck constant h, while the density of electons is eached by multiplying the density of states with the electon multiplicity /+ fo evey occupied state. The obtained density-femi impulse elationship: 3 F 3 4 p 3 / 3 d π d pf h 39 3 h 8π allows the Thomas-Femi kinetic enegy unfolding as the density functional [9, 7-3]: 3 5 / 3 TTF [ ] τ s d CTF d 5 3h 3, C TF.87[ a. u.] 40 0m0 8π with the help of which the total Thomas-Femi enegy functional takes the fom: 5 / 3 ETF [ ] CTF d + dd + V d 4 that can be seen as the fist appoximation fo the density functional total enegy. Fom physical point of view woth noted that the kinetic TF enegy exactly coesponds to the total enegy of the fee electons in a cystal, V0 in 4, equivalently with the fact that the electons ae not feeling the nuclei, i.e. electostatic attactions ae excluded, being as close each othe to avoid / 3 / 3

10 Int. J. Mol. Sci. 008, ecipocal epelling. Such pictue suggests that fee electons ae completely non-localized leaving with the condition of complete cancellation of the electonic inte-epulsion; this featue may be putted fomally as [3]: e λ e, λ 0. 4 Howeve, the model in which the valence electons ae completely fee and ae neithe feeling the attaction no the epulsion is cetain not popely descibing the natue of the chemical bond. In fact, this limitation was also the main objection bought to Thomas-Femi model and to the atomic o molecula appoximation of the homogeneous electonic gas o jellium model in solids. Nevetheless, the lesson is well seved because Thomas-Femi desciption may be egaded as the infeio exteme in quantum known stuctues while futhe exchange-coelation effects may be added in a petubative manne. The idea of intoducing exchange and coelation effects as a petubation of the homogeneous electonic system could be consideed fom the intepolation of the enegetic tems fo 0 λ in 4. Paamete λ is defined as a paamete of the electonic coupling, with a slightly adiabatically scaling of the petubation fom the homogeneous electonic systems, λ0, until the maximal inte-electonic inteaction, λ in accodance with Pauli pinciple. Theefoe, the oveall intepolation [ ] d λ will be spead ove the tems which contain the intemediate degee of exchange and coelation inteactions; since it accounts fo the electonic inte-epulsion while indexing the electonic pesence/absence in a given spatial egion the degee of electonic localization is in this way funishes. The coupling paamete λ will seve as a switche between the efeential Thomas-Femi unifom case and the full inteaction though the density limit: lim λ. λ 43 Actually, the density 43 has a majo ole in defining exchange-coelation functionals. To see that let s fistly conside the conditional electonic density g, ; λ indicating that the electonic density in is conditioned by the pesence localization of anothe electon any fom the total N in the system in. Mathematically, this is expessed by using the conditional pobabilities: g, ; λ 44 fulfilling the Pauli pinciple by means of the integation ule: g, ; λ d d 0 45 saying that the spatial aveage of the electonic ecipocal constaint vanishes. This behavio opens the possibility in intoducing the conditional pobability of electonic holes, h, ; λ g, ; λ, 46 poviding the associate integation ule [33]: h, ; λ d 47 consecating a sot of negative nomalization of the exchange and coelation density of holes: xc, ; λ h, ; λ. 48 0

11 Int. J. Mol. Sci. 008, Now, once this exchange-coelation hole density is mediated ove the coupling facto λ the aveaged exchange-coelation density of holes is geneated: xc, xc, ; λ dλ 49 0 allowing the fomal witing of exchange-coelation density functional fom 9 as a genealized vesion of the inte-electonic inteaction tem 3: xc, Exc[ ] d d d 50 Rxc, with the help of the intoduced adius of the λ-mediated exchange-coelation density of holes: xc,, R, : xc d 5 The adius R xc could be consideed as a functional of density 43 with the leading tem being defined in the shot limit of the distance, i.e. being of the inte-paticle aveage adius ode, / 3 4π 3 3 / 3 0 0, 5 3 4π known as the Wigne adius fo indexing the volume of a sphee containing localizing a single electon fom the total of N belonging to the density family 43, although, also othe quantities accounting fo electon localization such as the domain aveaged Femi hole of Ponec [34a] o the electon shaing index also known as delocalization index [34b] have been ecently poposed see also the fothcoming discussion. In these conditions, the invese adius 5 could be expessed aound the invese of the Wigne adius in a gadient density expansion [3]: R xc +..., F0 [ ] + F[ ] + F[ ] i i 53 while, by consideing it back in exchange-coelation enegy 50 poduces, afte the integation by pats, the genealized gadient density functional: E [ ] [ ] 0 4[ ]... xc G d + G d + G + d The estiction to the fist tem of the seies 54 coesponds to the cases whee the spatial distance of vaiation in electonic density highly exceeds the coesponding Wigne adius 5 this way poducing the famous local density appoximation LDA [34c]: LDA Exc [ ] exc[ ] d 55 with e xc being the exchange and coelation density pe paticle, that can be futhe appoximated see bellow as [35-40]: e xc In fact, the LDA stands as the immediate step afte TF appoximation; it can be extended also fo systems with un-pai spins by the so called local spin density appoximation LSDA [4-50]: E xc [ + ] E [ ] + [ ] xc E xc, 57 N i [ ] ex[ ] ec[ ] [ a. u.]

12 Int. J. Mol. Sci. 008, 9 06 while futhe inclusion of the gadient tems in 54 establishes the geneal gadient appoximation GGA. Woth noting that when undetaken GGA, beside the gadient tems aising in exchange-coelation enegy, the gadient coection of the kinetic enegy functional has to be as well consideed poviding tems of which the standad one takes the von Weizsäcke fom [5, 5]: τ W While moe analytical discussions about vaious appoximations and density functionals ae bellow pesented in a sepaate chapte, hee we would like only to pesent the pactical diffeence between the local and gadient density appoximations fo a solid state case. Fo instance, Figue pesents the band stuctue and the density of states DOS fo the oxide of the Cobalt tansitional metal CoO calculated with eithe LSDA and GGA appoximations [53]. Regading the enegy bands thee can be noted that, aound the Femi level E F, LSDA appoach is less elevant in indicating the enegetically gap especting the GGA computation. The diffeence is even moe dastic in DOS epesentations fo employed appoximations in d obital sepaation t g a g +e g due to the cental ion of cobalt tigonal symmety coodination. In fact, with LSDA a stong mixing of the obitals a g and e g is ecoded while in the case of GGA-DOS the bands with the symmety a g ae up and down shifted fo the espective down and up spin pojections esulting a clea sepaation fom states with e g symmety. Nonetheless, at the level of bands stuctue of the solids and cystals an inevitable localization paadox emeges namely, to use the eal 3D electonic densities in funish a localization desciption in the ecipocal enegy space. Figue. Left: the anti-feomagnetic stuctue CoO; ight: the band stuctue and the density of state DOS in LSDA and GGA appoximations, espectively [53]. The uppe and down aows ae associated with the spin obital pojections. R3 m Recently, it was found a way to avoid the electonic localization paadox though intoducing specific electonic localization functions ELF in eal space. Nevetheless, an ELF should elay on

13 Int. J. Mol. Sci. 008, 9 06 combination of the gadient and homogeneous enegetic density functionals, in accodance with Pauli pinciple, shaping fo instance as [54]: τ s τ W ELF + [ ] 59 TTF by emphasizing the excess of the kinetic enegy diffeence τ s - τ W nomalized to the efeential kinetic TF homogeneous behavio. Figue. The localization domains fo Li left and Sc ight cystals based on the electonic localization function ELF 59 [55]. Woth emaking that the localization function 59 acts like a sot of density, with values between 0 and coesponding with maximum delocalization and localization, espectively. This heuistically poposal has the meit to give an analytical eflection of the qualitative valence shell electon pai epulsion VSEPR geometic model [56], with the immediate consequence in topological chaacteization of the chemical bond [57]. In solid state case, the eliability of above ELF in descibing the chemical bond in eal space is illustated in Figue fo the Li and Sc cystals. Atomic and molecula levels ae in next section illustated with which occasion futhe ELF chaacteization and developments ae pesented. 3.. Localization in atoms and molecules The definitions that ae cuently used in the classification of chemical bonds ae often impecise, as they ae deived fom appoximate theoies. Based on the topological analysis local, quantummechanical functions elated to the Pauli Exclusion Pinciple may be fomulated as localization attactos of bonding, non-bonding, and coe types. Bonding attactos lie between nuclei coe attactos and chaacteize shaed electon inteactions. The spatial aangement of bond attactos allows fo an absolute classification of ionic vesus covalent bond to be deived fom electonic density combined functions [58]. Most moden classifications of the chemical bond ae based on Lewis theoy and ely on molecula-obital and valence-bond theoies with schemes involving linea combination of atomic obitals LCAO. Howeve, electon density alone does not easily eveal the consequences of the Pauli

14 Int. J. Mol. Sci. 008, Exclusion Pinciple on bonding natue. While VSEPR theoy indicates that the Pauli pinciple is impotant fo undestanding chemical stuctues, it has been efomulated in tems of maxima of electonic density s Laplacian [56]. Next, the exchange-coelation density functional concept was employed to achieve the coodinate-space dynamical coelation in an inhomogeneous electon gas. This way the exchange-coelation enegy 50 futhe e-expesses like [3, 54] αα Exc hxc dd, 60 αα by accounting fo the fou α -spin types of inteactions though the hole functions [40, 59] αα αα P, h, α 6 α whee P, stands fo the two-body o pai pobability density o coelation pobability of the abitay electons and, defined in tems of the N-body wave function Ψ as follows [9]: P, N N Ψ,, 3,, N Ψ,, 3,, N d3 dn. 6 Within the density functional theoy the electons of a pai of electons o a bond can be consideed as belonging to an inhomogeneous continuum gas. In analytical tems this was tanslated as the ELF 59 index as combining the homogeneous and inhomogeneous behavios of a many-electonic-nuclei system. Nevetheless, ecently, the Makovian analytical shape of an ELF was shown to have the geneal qualitative fom [60]: g ELF f h 63 with the limiting constains 0, >> lim ELF 64, << assuing the fulfillment of the Heisenbeg and Pauli pinciples. To claify this [60], we make ecouse to the Heisenbeg pinciple, compised in ELF. When density gadient dominates, >> then g>>h in 63, and f should accounted fo the infinite eo in assigning of momentum, theefoe indicating a pecisely spatial localization of electons; thus f and ELF 0. In such, the meaning of ELF is associated with the eo in spatial localization of electons, being zeo when the electons ae pecisely located. On the contay, when >> then h>>g in 63, and the esulting f0 indicates the minimum eo in defining of momentum and should povide the maximum uncetain of spatial distibution; in such f0 and ELF, whee stands hee fo 00% of coodinate localization eo. In this context, when the invese of diffeence in local kinetic tems is involved, the ELF is intepeted as the eo in localization of electons within taps athe than whee they have peaks of spatial density, as is fequently misintepeted in liteatue [6], albeit ecent extensions of ELF have used the coelated Hatee-Fock wavefunctions, though the conditional pai pobability, howeve not using the kinetic enegy appoach [6-64]. Among vaious classes of Makovian ELFs the most epesentative and efficient one was poposed as having the fom [60]:

15 Int. J. Mol. Sci. 008, with the components: 3 g ELF exp 65 h [ ] [ ] 3 / 3 g ϕ, 3π [ ] 5/ 3 i i 8 h 66 0 being esponsible fo the gadient g and the homogenous h density distibutions, espectively. In this fame, the ELF infomation pescibe that as it has values close to zeo as the bette electonic localization is poviding, accoding with the limits 64. Going to a paticula application of this scheme the atomic level is fistly pesented fo the special case of Li atom. The main stages consist in: Choosing the basis of the atomic functions [65]: f Li exp. 698, 5/ 0. f Li exp such that to fulfill the natual adial nomalization conditions 0 [ f ] Li d, n, n 68 Geneating the othonomal obital eigen-waves, hee accoding with the Gam-Schmidt algoithm among shells and sub-shells: Li Li Li Li ϕ f, ϕ f, s Li Li p Li Li [ ϕ αϕ ] ϕs Cs p s 5/.6 exp exp ensuing the additionally constaints: 0 Li Li 0 Li ϕ s ϕs d 0, [ ϕ s ] d. 70 Geneating the woking oveall electonic density Li Li [ ϕ ] [ ϕ ] Li s + s 7 that satisfies the spatial adial global N-integal condition 4: 0 Li d 3 7 The electonic density 7 is then used fo computation of the Makovian ELF 65, while thei compaison is in Figue 3 illustated. Fom the Figue 3, thee appeas that the smooth delocalization of electons of Li epesented by density stuctue is emoved by the electonic localization function by clealy indicating whee ae the egions whee the electonic ealm is with less uncetainty detected. This way the ELF indicates meely whee the electonic tansitions behave like a step-function. In this espect, ELF can be egaded as the complement of electonic density being a bette indicato of the egions whee the bonding may aise. Fo instance, in the case of Li atomic stuctue, the fact that the ELF does not displays localization ove the second shell due to its values appoaching unity in this ange indicates a natual tendency fo eleasing the outemost electon to the vitual neighbohood atoms with

16 Int. J. Mol. Sci. 008, uncompleted last shells while peseving its delocalization featue acoss the bond. As such the lithium hydide LiH bond is expected to be fomed with a cetain degee of ionicity in esonance + - with its covalence: LiH Li H. Figue 3. Compaison of the adial density given by 7 with the electon localization function 65 with components 66 fo the simplified self-consistent appoximation fo Li atomic stuctue [66]. The eliability of ELF to quantify the local tendency of atoms to fom bonds and aggegates can be futhe exemplified to diatomic molecules, while the paticula cases of HF, HCl, HB and HI stuctues ae consideed in Figue 4. In the bonding egion, i.e. in the space between the hydogen and halogen atomic centes in H-X molecules thee ae epesented both the electon densities, computed upon above ecipe [65], and the associate Makovian ELFs fo the concened atoms-in-molecules AIM. Figue 4 clealy shows that while the cossing of hydogen and halogen adial densities does not povide the ight bonding egion in HCl, HB, and HI cases, the coesponding ELFs coss-lines of AIM finely indicate the fontie of atomic basins in hydacids thus confiming the ELF eliability in identifying chemical bonds and bonding. One can equally say that in the cossing vicinity of AIM-ELFs the electons ae at the same time completely localized fo bonding with ELF X ELF H 0 and completely delocalized fo atomic systems with ELF X,H, accoding to above the ELF definition and pesent signification. In othe wods it can be alleged that ELF application on chemical bond helps in identifying the molecula egion in which the electons undego the tansition fom the complete delocalization in atoms to complete localization in molecula bonding behavio.

17 Int. J. Mol. Sci. 008, Figue 4. Compaative analysis of the chage density contous, electonic localization functions ELFs, and adial densities fo the H dashed lines, F, Cl, B, and I full lines atoms in molecula combinations HF, HCl, HB, and HI, espectively [4]. Actually, it also poves that localization issue of ionic and covalent classification of bonds may be solved by a continuous quantum eality. Such a featue gives, nevetheless, an in-depth undestanding of the quantum natue of the chemical bond by associating the mysteious paiing of electons issue to an analytical function able to distinguish the naow egions of molecula space

18 Int. J. Mol. Sci. 008, whee the Heisenbeg and Pauli pinciples ae jointly satisfied though the ELF s exteme values. Even moe such shap diffeentiation between 0 and in atomic and molecula ELF values offes the futue possibility in quantifying the chemical bond and bonding in the fame of quantum infomation theoy [67]. 4. Popula Enegetic Density Functionals Since the tems of total enegy ae involved in bonding and eactivity states of many-electonic systems, i.e. the kinetic enegetic tems in ELF topological analysis o the exchange and coelation density functionals in chemical eactivity in elation with eithe localization and chemical potential o electonegativity, woth pesenting vaious schemes of quantification and appoximation of these functionals fo bette undestanding thei ole in chemical stuctue and dynamics. 4.. Density functionals of kinetic enegy When the electonic density is seen as the diagonal element, the kinetic enegy may be geneally expessed fom the Hatee-Fock model, though employing the single deteminant, ', as the quantity [68]: T[ ] [ ', ' ] d ' ; 73 it may eventually be futhe witten by means of the themodynamical o statistical density functional [69]: 3 3 Tβ k BT d d β 74 that suppots vaious specializations depending on the statistical facto paticulaization β. Fo instance, in LDA appoximation, the tempeatue at a point is assumed as a function of the density in that point, β β ; this may be easily eached out by employing the scaling tansfomation to be [70] 3 λ λ λ T [ λ ] λ T[ ], λ ct., 75 poviding that 3 / 3 β C, 76 a esult that helps in ecoveing the taditional Thomas-Femi enegetic kinetic density functional fom 5 / 3 T [ ] C d, 77 while the indeteminacy emained is smeaed out in diffeent appoximation fames in which also the exchange enegy is evaluated. Note that the kinetic enegy is geneally foeseen as having an intimate elation with the exchange enegy since both ae expessed in Hatee-Fock model as deteminantal values of, ', see below. Actually, the diffeent LDA paticula cases ae deived by equating the total numbe of paticle N with vaious ealization of the integal N, ' dd ' 78

19 Int. J. Mol. Sci. 008, by ewitting it within the inte-paticle coodinates fame: ', s ' 79 as: N + s /, s / dds 80 followed by spheical aveaged expession: N π Γ, s ds ds 8 with s Γ, s +... β 8 The option in choosing the Γ, s seies 8 so that to convege in the sense of chage paticle integal 8 fixes the possible cases to be consideed [68]:. the Gaussian esummation uses: s Γ, s Γ, exp G s β 83. the tigonometic unifom gas appoximation looks like: sin t t cost 5 Γ, s ΓT, s 9, t s 6 t β 84 In each of 83 and 84 cases the LDA-β function 76 is fistly eplaced; then, the paticle integal 8 is solved to give the constant C and then the espective kinetic enegy density functional of 77 type is deliveed; the esults ae [68]:. in Gaussian esummation: T LDA 3π 5/ 3 G d, 5/ wheeas in tigonometic appoximation / 3 3 T LDA 3 5/ 3 TF π d 86 0 one ecoves the Thomas-Femi fomula type that closely esembles the oiginal TF 40 fomulation. In next one will conside the non-local functionals; this can be achieved though the gadient expansion in the case of slowly vaying densities that is assuming the expansion [5]: T d τ τ d m 0 [ ] + [ τ τ ] m + m d m τ 0 m d τ 87 The fist two tems of the seies espectively coves: the Thomas Femi typical functional fo the homogeneous gas / 3 τ / 0 6π 88 0 and the Weizsäcke elated fist gadient coection:

20 Int. J. Mol. Sci. 008, τ τ W They both coectly behave in asymptotic limits: τ 0 τ... << fa fom nucleus τ 9τ τ W... >> close to nucleus 8 90 Howeve, an inteesting esummation of the kinetic density functional gadient expansion seies 87 may be fomulated in tems of the Padé-appoximant model [7]: τ τ 0 P4, 3 x 9 with x + a x + a3x + 9b3 x P4,3 x x + b x + b3 x and whee the x-vaiable is given by 0 τ 5 x, 93 τ / 3 6π / while the paametes a, a 3, b, and b 3 ae detemined by fitting them to epoduce Hatee-Fock kinetic enegies of He, Ne, A, and K atoms, espectively [7]. Note that Padé function 9 may be egaded as a sot of genealized ELF susceptible to be futhe used in bonding chaacteizations. 4.. Density functionals of exchange enegy Stating fom the Hatee-Fock famewok of exchange enegy definition in tems of density matix [73],, ' K[ ] dd ' 4, 94 ' within the same consideation as befoe, we get that the spheical aveaged exchange density functional K π Γ, s dsds 95 takes the paticula foms [68]:. in Gaussian esummation: K LDA 4 / 3 G d ; / and in tigonometic appoximation ecoveing the Diac fomula: / 3 K LDA / 3 D d π Altenatively, by paalleling the kinetic density functional pevious developments the gadient expansion fo the exchange enegy may be egaded as the density dependent seies [74]: K K n 0 n

21 Int. J. Mol. Sci. 008, d n 0 k n d k 98 while the fist tem epoduces the Diac LDA tem [75, 76]: / / 3 k π and the second tem contains the density gadient coection, with the Becke poposed appoximation [77]: 4 / 3 k b a 00 + d 8/ 3 whee the paametes b and d ae detemined by fitting the k 0 +k exchange enegy to epoduce Hatee-Fock countepat enegy of He, Ne, A, and K atoms, and whee fo the a exponent eithe.0 o 4/5 value funishes excellent esults. Howeve, woth noting that when analyzing the asymptotic exchange enegy behavio, we get in small gadient limit [77]: 7 k << k / 3 6π / 43π wheeas the adequate lage-gadient limit is obtained by consideing an abitay damping function as multiplying the shot-ange behavio of the exchange-hole density, with the esult: 4 / 5 / 5 k >> c 0 whee the constant c depends of the damping function choice. Next, the Padé-esummation model of the exchange enegy pescibes the compact fom [74]: 0 k0 k 9 P4,3 x 03 with the same Padé-function 9 as peviously involved when dealing with the kinetic functional esummation. Note that when x0, one diectly obtains the Ghosh-Pa functional [78]: 0 k k Moeove, the asymptotic behavio of Padé exchange functional 03 leaves with the convegent limits: k x 0 SMALL GRADIENTS 9 7 / 3 4 / 3 43π 6π k 05 π... x LARGE GRADIENTS Once again, note that when paticulaizing small o lage gadients and fixing asymptotic long o shot ange behavio, we ae ecoveing the vaious cases of bonding modeled by the electonic localization ecipe as povided by ELF s limits 64.

22 Int. J. Mol. Sci. 008, 9 07 Anothe inteesting appoach of exchange enegy in the gadient expansion famewok was given by Batolotti though the two-component density functional [79]: 4 / 3 K[ ] C N d + D N d, / 3 06 whee the N-dependency is assumed to behave like: C D C N C +, D N / 3 / 3 N N 07 while the intoduced paametes C, C, and D wee fond with the exact values [80-8]: 3 / 3 C / 3 π 4 / 3 3 / 3 3 π C π, D 4 π Woth obseving that the exchange Batolotti functional 06 has some impotant phenomenological featues: it scales like potential enegy, fulfills the non-locality behavio though the powes of the electon and powes of the gadient of the density, while the atomic cusp condition is peseved [83]. Howeve, density functional exchange-enegy appoximation with coect asymptotic long ange behavio, i.e. satisfying the limits fo the density lim exp a σ σ 09 and fo the Coulomb potential of the exchange chage, o Femi hole density at the efeence point σ limu X, σ α o, β o... spin states 0 in the total exchange enegy σ K[ ] σ U X d, σ was given by Becke via employing the so called semiempiical SE modified gadient-coected functional [77]: xσ K SE 4 / 3 K 0 β σ d, 0 0 ] σ + γxσ σ K dk, x σ 4 / 3 σ to the woking single-paamete dependent one [84]: xσ K B 88 4 / 3 K 0 β σ d σ + 6βxσ sinh xσ 3 whee the value β 0.004[ a. u.] was found as the best fit among the noble gases He to Rn atoms exchange enegies; the constant a σ is elated to the ionization potential of the system. Still, having diffeent exchange appoximation enegetic functionals as possible woth explaining fom whee such ambiguity eventually comes. To claify this, it helps in ewiting the stating exchange enegy 94 unde the fomally exact fom [85]: K [ ] σ k[ σ ] g[ xσ ] d 4 Whee the typical components ae identified as: σ / k[ ] AX, A X, 5 4π while the gadient containing coection gx is to be detemined. / 3

23 Int. J. Mol. Sci. 008, 9 07 Fistly, one can notice that a sufficiency condition fo the two exchange integals and 4 to be equal is that thei integands, o the exchange potentials, to be equal; this povides the leading gadient coection: U X x g0 x 6 k[ x] with x following fom x by not unique invesion. Unfotunately, the above integity condition fo exchange integals to be equal is not also necessay, since any additional gadient coection g x g0 x + Δg x 7 fulfills the same constaint if it is chosen so that 4 / 3 Δg x d 0 8 o, with the geneal fom: 4 / 3 f x d Δg x f x 4 / 3 9 d being fx an abitay function. Nonetheless, if, fo instance, the function fx is specialized so that f x g0 x 0 the gadient coecting function 7 becomes: U X d g x α 4 / 3 X AX d ecoveing the Slate s famous X α method fo exchange enegy evaluations [86, 87]: 4 / 3 K[ ] α X AX d. Nevetheless, the diffeent values of the multiplication facto α X in can explain the vaious foms of exchange enegy coefficients and foms above. Moeove, following this conceptual line the above Becke 88 functional 3 can be futhe eaanged in a so called Xα-Becke88 fom [88]: XB88 4 / 3 / 3 xσ K α XB σ + d 3 σ + 6β XB xσ sinh xσ whee the paametes α XB and β XB ae to be detemined, as usually, thoughout atomic fitting; it may lead with a new wokable valuable density functional in exchange family Density functionals of coelation enegy The fist and immediate definition of enegy coelation may be given by the diffeence between the exact and Hatee-Fock HF total enegy of a poly-electonic system [89]: Ec[ ] E[ ] EHF [ ] 4 Instead, in density functional theoy the coelation enegy can be seen as the gain of the kinetic and electon epulsion enegy between the full inteacting λ and non-inteacting λ 0 states of the electonic systems [90]: λ 0 0 [ ] λ + λ λ + λ E c ψ T λv ee ψ ψ T λv ee ψ. 5

24 Int. J. Mol. Sci. 008, In this context, taking the vaiation of the coelation enegy 5 especting the coupling paamete λ [9, 9], λ λ Ec [ ] λ δec [ ] λ Ec [ ] + d, 6 λ δ by employing it though the functional diffeentiation with especting the electonic density, λ λ Vc [ ] λ λ δ Ec [ ] λ Vc [ ] Vc + d, 7 λ δ δ one obtains the equation to be solved fo coelation potential V λ δ λ c E [ ] / c δ ; then the coelation enegy is yielded by back integation: λ λ Ec [ ] Vc,[ ] d 8 fom whee the full coelation enegy is eached out by finally setting λ. When esticting to atomic systems, i.e. assuming spheical symmety, and neglecting the last tem of the coelation potential equation above, believed to be small [90], the equation to be solved simply becomes: λ Vc [ ] λ λ λ Vc [ ] Vc 9 λ that can eally be solved out with the solution: λ p+ p Vc Apλ 30 with the integation constants A p and p. Howeve, since the equation 9 is a homogeneous diffeential one, the linea combination of solutions gives a solution as well. This way, the geneal fom of coelation potential looks like: λ p+ p Vc Apλ. 3 p This pocedue can be then iteated by taking futhe deivative of 7 with espect to the density, solving the obtained equation until the second ode coection ove above fist ode solution 3, λ p+ p p+ p p V λ λ c Ap + Ap. 3 p p By mathematical induction, when going to highe odes the K-tuncated solution is iteatively founded as: V λ c K p k A pk λ pk+ p p k 33 poducing the λ-elated coelation functional: K λ pk+ p k Ec [ ] Apkλ p k k 34 and the associate full coelation enegy functional λ expession: K p k Ec[ ] Apk. p k k 35 As an obsevation, the coelation enegy 35 suppots also the immediate not spheically molecula genealization [90]: K l m n k Ec[ ] Almnk x x x. 36 k lmn k