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The behavior of electrons in general manyelectronic systems throughout the density functionals of energy is reviewed. The basic physicochemical concepts of density functional theory are employed to highlight the energy role in chemical structure while its extended influence in electronic localization function helps in chemical bonding understanding. In this context the energy functionals accompanied by electronic localization functions may provide a comprehensive description of the globallocal levels electronic structures in general and of chemical bonds in special. BeckeEdgecombe and author’s Markovian electronic localization functions are discussed at atomic, molecular and solid state levels. Then, the analytical survey of the main workable kinetic, exchange, and correlation density functionals within local and gradient density approximations is undertaken. The hierarchy of various energy functionals is formulated by employing both the parabolic and statistical correlation degree of them with the electronegativity and chemical hardness indices by means of quantitative structureproperty relationship (QSPR) analysis for basic atomic and molecular systems.
In Walter Kohn’s lecture, with the occasion of receiving his Nobel Prize in Chemistry [
Nevertheless, this affirmation may be at any time turned in a theorem, eventually as
The demonstration of the nonrepresentability of the eigenfunction for systems containing more than
Regarding the accuracy of the Ψ(
Now, considering a collection of
For
This result, relaying on the exponential form (2), justifies the title of „exponential wall” for the wavefunction limitation.
Then, going to the measurable issue of such eigenfunctions, let’s ask how many bits are necessary for recording its quantum dimension? Assume, again, the working wavefunction Ψ(
Definitely, the concept of eigenwave function must be enlarged or modified in such a manner that the quantum description does not be blocked by the exponential wall: from where we can start? Firstly, as was exposed, the eigenwave function in the configuration space multiplies in an exponential manner the variables accounting for the number and the position of the electrons; thus, the configuration space must be avoided. Then, the density of probability must be reformulated as such the exponential wall for a polyelectronic system be avoided while preserving the dependency of the total number of electrons
Fortunately, the above described conceptual project was unfolded in 1963 when Walter Kohn met in Paris (at École Normale Supérieure), during his sabbatical semester, the mate Pierre Hohenberg who was working at the description of the metallic alloys (specially the Cu_{x}Zn_{1x} systems) by using quantum traditionally methods of averaging crystalline periodic field. Studies of this type of problems often start from the level of the uniform electronic density referential upon which specific perturbation treatments are applied. From this point Kohn and Hohenberg made two crucial further steps in reformulation of the quantum picture of the matter structure: one referred at the electronic density, and another at the relation between electronic density with the externally applied potential on the electronic system; they were consecrated in the so called HohenbergKohn (KH) theorems [
The present work likes to review some fundamental aspects of density functional theory highlighting on the primer conceptual and computational consequences in electronic localization and chemical reactivity.
The first HohenbergKohn (HK1) theorem gives space to the concept of
The relation (4) as much simple it could appears stands as the decisive passage from the eigenwave function level to the level of total electronic density [
Firstly,
However, still remains the question: what represents the electronic density of
The major consequence of this theorem consists in defining of the total energy of a system as a function of the electronic density function in what is known as the density functional [
from where the name of the theory. The terms of energy decomposition in (6) are identified as: the HohenbergKohn density functional
Although not entirely known the HK functional has a remarkably property: it is universally, in a sense that both the kinetic and interelectronic repulsion are independent of the concerned system. The consequence of such universal nature offers the possibility that once it is exactly or approximately knew the HK functional
Once “in game” the external applied potential provides the second HohenbergKohn (HK2) theorem. In short, HK2 theorem says that “the external applied potential is determined up to an additive constant by the electronic density of the
In mathematical terms, the theorem assures the validity of the variational principle applied to the density functional (6) relation, i.e. [
The proof of variational principle in (9), or, in other words, the onetoone correspondence between the applied potential and the ground state electronic density, employs the
On another way, if the eigenfunction Ψ_{2} is assumed as being the one true ground state wavefunction, the analogue inequality springs out as:
Taken together relations (11) and (12) generate, by direct summation, the evidence of the contradiction [
The removal of such contradiction could be done in a single way, namely, by abolishing, in a reverse phenomenologically order, the fact that two eigenfunctions, two Hamiltonians and respectively, two external potential exist for characterizing the same ground state of a given electronic system. With this statement the HK2 theorem is formally proofed.
Yet, there appears the so called
That because, the problem of
Both (14) and (15) conditions are easy accomplished by every reasonable density, allowing the employment of the variational principle (9) in two steps, according to the so called
One of the most important consequences of the HK2 conveys the rewriting of the variational principle (9) in the light of above
Up to now, the HohenbergKohn theorems give new conceptual quantum tools for physicochemical characterization of an electronic sample by means of electronic density and its functionals, the total energy and chemical potential (electronegativity). These positive density functional premises are in next analyzed towards elucidating of the quantum nature of the chemical bond and reactivity [
Back from Paris, in the winter of 1964, Kohn met at the San Diego University of California his new postdoc Lu J. Sham with who propose to extract from HK1 & 2 theorems the equation of total energy of the ground state. In fact, they propose themselves to find the correspondent of the stationary eigenequation of Schrödinger type, employing the relationship between the electronic density and the wave function.
Their basic idea consists in assuming a so called orbital basic set for the
Then, the
Next, the trial density functional energy (21) will be optimized in the light of variational principle (17) as prescribed by the HK2 theorem. The combined result of the HK theorems will eventually furnish the new quantum energy expression of multielectronic systems beyond the exponential wall of the wave function.
An instructive method for deriving such equation assume the same types of orbitals for the density expansion (20),
Actually, with the trial density (24) replaced throughout the energy expression in (21) has to undergo the minimization procedure (17) with the practical equivalent integral variant:
Note that, in fact, we chose the variation in the conjugated uniorbital φ^{*}(
Now, unfolding the
By performing the required partial functional derivations respecting the uniorbital φ^{*}(
After immediate suppressing of the
Moreover, once introducing the so called
The result (32) is fundamental and equally subtle. Firstly, it was proved that the joined HohenbergKohn theorems are compatible with consecrated quantum mechanical postulates, however, still offering a generalized view of the quantum nature of electronic structures, albeit the electronic density was assumed as the foreground reality. In these conditions, the meaning of functions φ(
Yet,
Going now to a summative characterization of the above optimization procedure worth observing that the
It starts with a trial electronic density (20) satisfying the
With trial density the effective potential (31) containing exchange and correlation is calculated;
With computed
With the set of functions
The procedure is repeated until the difference between two consecutive densities approaches zero;
Once the last condition is achieved one retains the last set
The electronegativity orbital observed contributions are summed up from (33) with the expression:
Replacing in (34) the uniform kinetic energy,
showing that the optimized manyelectronic ground state energy is directly related with global or summed over observed or averaged or expected orbital electronegativities. One can observe from (35) that even in the most optimistic case when the last two terms are hopefully canceling each other there still remains a (classical) correction to be added on global electronegativity in total energy. Or, in other terms, electronegativity alone is not enough to better describe the total energy of a manyelectronic system, while its correction can be modeled in a global (almost classical) way. Such considerations stressed upon the accepted semiclassical behavior of the chemical systems, at the edge between the full quantum and classical treatments.
However, analytical expressing the total energy requires the use of suitable approximations, whereas for chemical interpretation of bonding the electronic localization information extracted from energy is compulsory. This subject is in next focused followed by a review of the popular energetic density functionals and approximations.
The application of HohenbergKohn theorems consecrate the crucial contributions of the so called “spherical” or homogeneous kinetic and of the exchangecorrelation energy terms in a multielectronic system’s ground state. However, the spherical electronic case corresponds to the non perturbed electronic system for which the ThomasFermi (TF) model was already advanced throughout totally ignoring exchange correlation terms from the total energy shape:
Such a referential picture is most useful in establishing the uniform electronic distribution by indicating the occupation of the allpossible electronic levels in a semiclassical quantum frame (without explicit exchangecorrelation involvement). Actually, the Fermi sphere in a momentum space finely defines the total homogeneous kinetic energy as:
From physical point of view worth noted that the kinetic TF energy exactly corresponds to the total energy of the
However, the model in which the (valence) electrons are completely free and are neither “feeling” the attraction nor the repulsion is certain not properly describing the nature of the chemical bond. In fact, this limitation was also the main objection brought to ThomasFermi model and to the atomic or molecular approximation of the homogeneous electronic gas or jellium model in solids. Nevertheless, the lesson is well served because ThomasFermi description may be regarded as the “inferior” extreme in quantum known structures while further exchangecorrelation effects may be added in a perturbative manner.
The idea of introducing exchange and correlation effects as a perturbation of the homogeneous electronic system could be considered from the interpolation of the energetic terms for 0≤λ≤1 in (41). Parameter λ is defined as a
The coupling parameter λ will serve as a switcher between the referential ThomasFermi uniform case and the full interaction through the density limit:
Actually, the density (43) has a major role in defining exchangecorrelation functionals. To see that let’s firstly consider the conditional electronic density
Now, once this exchangecorrelation hole density is mediated over the coupling factor λ the averaged exchangecorrelation density of holes is generated:
The radius
In these conditions, the inverse radius (51) could be expressed around the inverse of the Wigner radius in a gradient density expansion [
The restriction to the first term of the series (54) corresponds to the cases where the spatial distance of variation in electronic density highly exceeds the corresponding Wigner radius (52) this way producing the famous
In fact, the LDA stands as the immediate step after TF approximation; it can be extended also for systems with unpair spins by the so called
Worth noting that when undertaken GGA, beside the gradient terms arising in exchangecorrelation energy, the gradient correction of the kinetic energy functional has to be as well considered providing terms of which the standard one takes the von Weizsäcker form [
While more analytical discussions about various approximations and density functionals are bellow presented in a separate chapter, here we would like only to present the practical difference between the local and gradient density approximations for a solid state case.
For instance,
Regarding the energy bands there can be noted that, around the Fermi level
Nonetheless, at the level of bands structure of the solids and crystals an inevitable
Recently, it was found a way to avoid the electronic localization paradox through introducing specific electronic localization functions (ELF) in real space. Nevertheless, an ELF should relay on combination of the gradient and homogeneous energetic density functionals, in accordance with Pauli principle, shaping for instance as [
Worth remarking that the localization function (59) acts like a sort of density, with values between 0 and 1 corresponding with maximum delocalization and localization, respectively. This heuristically proposal has the merit to give an analytical reflection of the qualitative
The definitions that are currently used in the classification of chemical bonds are often imprecise, as they are derived from approximate theories. Based on the topological analysis local, quantummechanical functions related to the Pauli Exclusion Principle may be formulated as “localization attractors” of bonding, nonbonding, and core types. Bonding attractors lie between nuclei core attractors and characterize shared electron interactions. The spatial arrangement of bond attractors allows for an absolute classification of ionic
Most modern classifications of the chemical bond are based on Lewis’ theory and rely on molecularorbital and valencebond theories with schemes involving linear combination of atomic orbitals (LCAO). However, electron density alone does not easily reveal the consequences of the Pauli Exclusion Principle on bonding nature. While VSEPR theory indicates that the Pauli principle is important for understanding chemical structures, it has been reformulated in terms of maxima of electronic density’s Laplacian −∇^{2}ρ(
Within the density functional theory the electrons of a pair of electrons or a bond can be considered as belonging to an inhomogeneous continuum gas. In analytical terms this was translated as the ELF (59) index as combining the homogeneous and inhomogeneous behaviors of a manyelectronicnuclei system.
Nevertheless, recently, the Markovian analytical shape of an ELF was shown to have the general qualitative form [
In this context, when the inverse of difference in local kinetic terms is involved, the ELF is interpreted as the
Among various classes of Markovian ELFs the most representative and efficient one was proposed as having the form [
Going to a particular application of this scheme the atomic level is firstly presented for the special case of Li atom. The main stages consist in:
Choosing the basis of the atomic functions [
such that to fulfill the natural (radial) normalization conditions
Generating the orthonormal orbital eigenwaves, here according with the GramSchmidt algorithm among shells and subshells:
Generating the working overall electronic density
The electronic density (71) is then used for computation of the Markovian ELF (65), while their comparison is in
From the
The reliability of ELF to quantify the local tendency of atoms to form bonds and aggregates can be further exemplified to diatomic molecules, while the particular cases of HF, HCl, HBr and HI structures are considered in
One can equally say that in the crossing vicinity of
In other words it can be alleged that
Actually, it also proves that localization issue of ionic and covalent classification of bonds may be solved by a “continuous” quantum reality. Such a feature gives, nevertheless, an indepth understanding of the quantum nature of the chemical bond by associating the mysterious pairing of electrons issue to an analytical function able to distinguish the narrow regions of molecular space where the Heisenberg and Pauli principles are jointly satisfied through the ELF’s extreme values. Even more such sharp differentiation between 0 and 1 in atomic and molecular ELF values offers the future possibility in quantifying the chemical bond and bonding in the frame of quantum information theory [
Since the terms of total energy are involved in bonding and reactivity states of manyelectronic systems, i.e. the kinetic energetic terms in ELF topological analysis or the exchange and correlation density functionals in chemical reactivity in relation with either localization and chemical potential or electronegativity, worth presenting various schemes of quantification and approximation of these functionals for better understanding their role in chemical structure and dynamics.
When the electronic density is seen as the diagonal element ρ(
For instance, in LDA approximation, the temperature at a point is assumed as a function of the density in that point, β(
Actually, the different LDA particular cases are derived by equating the total number of particle
The option in choosing the Γ(
the Gaussian resummation uses:
the trigonometric (uniform gas) approximation looks like:
In each of (83) and (84) cases the LDAβ function (76) is firstly replaced; then, the particle integral (81) is solved to give the constant
in Gaussian resummation:
whereas in trigonometric approximation
one recovers the ThomasFermi formula type that closely resembles the original TF (40) formulation.
In next one will consider the nonlocal functionals; this can be achieved through the gradient expansion in the case of slowly varying densities – that is assuming the expansion [
The first two terms of the series respectively covers: the Thomas Fermi typical functional for the homogeneous gas
They both correctly behave in asymptotic limits:
However, an interesting resummation of the kinetic density functional gradient expansion series (87) may be formulated in terms of the Padéapproximant model [
Starting from the HartreeFock framework of exchange energy definition in terms of density matrix [
in Gaussian resummation:
and in trigonometric approximation (recovering the Dirac formula):
Alternatively, by paralleling the kinetic density functional previous developments the gradient expansion for the exchange energy may be regarded as the density dependent series [
Next, the Padéresummation model of the exchange energy prescribes the compact form [
Moreover, the asymptotic behavior of Padé exchange functional (103) leaves with the convergent limits:
Once again, note that when particularizing small or large gradients and fixing asymptotic long or short range behavior, we are recovering the various cases of bonding modeled by the electronic localization recipe as provided by ELF’s limits (64).
Another interesting approach of exchange energy in the gradient expansion framework was given by Bartolotti through the twocomponent density functional [
Worth observing that the exchange Bartolotti functional (106) has some important phenomenological features: it scales like potential energy, fulfills the nonlocality behavior through the powers of the electron and powers of the gradient of the density, while the atomic cusp condition is preserved [
However, density functional exchangeenergy approximation with correct asymptotic (long range) behavior, i.e. satisfying the limits for the density
Still, having different exchange approximation energetic functionals as possible worth explaining from where such ambiguity eventually comes. To clarify this, it helps in rewriting the starting exchange energy (94) under the formally exact form [
Where the typical components are identified as:
Firstly, one can notice that a sufficiency condition for the two exchange integrals (111) and (114) to be equal is that their integrands, or the exchange potentials, to be equal; this provides the leading gradient correction:
Unfortunately, the above “integrity” condition for exchange integrals to be equal is not also necessary, since any additional gradient correction
Nonetheless, if, for instance, the function
Nevertheless, the different values of the multiplication factor α
The first and immediate definition of energy correlation may be given by the difference between the exact and HartreeFock (HF) total energy of a polyelectronic system [
Instead, in density functional theory the correlation energy can be seen as the gain of the kinetic and electron repulsion energy between the full interacting (λ = 1) and noninteracting (λ = 0) states of the electronic systems [
In this context, taking the variation of the correlation energy (125) respecting the coupling parameter λ [
When restricting to atomic systems, i.e. assuming spherical symmetry, and neglecting the last term of the correlation potential equation above, believed to be small [
However, since the
This procedure can be then iterated by taking further derivative of (127) with respect to the density, solving the obtained equation until the second order correction over above first order solution (131),
By mathematical induction, when going to higher orders the
As an observation, the correlation energy (135) supports also the immediate not spherically (molecular) generalization [
Nevertheless, for atomic systems, the simplest specialization of the relation (135) involves the simplest density moments 〈ρ〉 =
Unfortunately, universal atomic values for the correlation constants
the fitting of (137) with the HF related correlation energy (124) reveals the atomicworking correlation energy with the form [
The last formula is circumvented to the highdensity total correlation density approaches rooting at their turn on the ThomasFermi atomic theory. Very interesting, the relation (139) may be seen as an atomic reflection of the (solid state) high density regime (
Instead, within the low density regime (
However, extended parameterization of the local correlation energy may be unfolded since considering the fit with an LSDA (ρ_{↑} and ρ_{↓}) analytical expression by
More specifically, we list bellow some nonlocal correlation density functionals in the low density (gradient corrections over LDA) regime:
the Rasolt and Geldar paramagnetic case (ρ_{↑} = ρ_{↓} = ρ/
with
The gradient corrected correlation functional reads as [
The
The openshell (OS) case provides the functional [
with the spindependency regulated by the factor ζ = (ρ_{↑} − ρ_{↓})/(ρ_{↑} + ρ_{↓}) approaching zero for closedshell case, while the specific coefficients are determined through a scaledminimization procedure yielding the values:
Finally, Perdew and Zunger (PZ) recommend the working functional [
with the numerical values for the fitting parameters founded as: α
Another approach in questing exchange and correlation density functionals consists in finding them both at once in what was defined as exchangecorrelation density functional (29). In this regard, following the Lee and Parr approach [
Now, the second order density matrix in (158) can be expressed as
On the other side, the average
Inserting relations (158)(165) in (
Making use of the two possible multiplication of the series in (166), i.e. either by retaining the α(
Now, laying aside other variants and choosing the simple (however meaningfully) density dependency
These functionals are formally exact for any κ albeit the resumed functions
Going now to the specific models, let’s explore the type I of exchangecorrelation functionals (168). Firstly, they can further undergo simplification since the reasonable (atomic) assumption according which
Within this frame the best provided model is of
When the condition (169) for κ is abolished the Wignerlike model results, again, having the best approximant exchangecorrelation model as the Padé form [
Turning to the IItype of exchangecorrelation functionals, the small density condition (169) delivers the gradient corrected
Still, a Padé approximant for the gradientcorrected Wignertype exchangecorrelation functional exists and it was firstly formulated by Rasolt and Geldar [
Finally, worth noting the Tozer and Handy general form for exchangecorrelation functionals viewed as a sum of products of powers of density and gradients [
However, since electronegativity and chemical hardness closely relate with chemical bonding, their relation with the total energy and component functionals is in next at both conceptual and applied levels explored.
Employing the KohnSham
For proper characterization of the chemical systems the quantum (Ehrenfest) version of the fundamental Newtonian law [
Now, combining the last two equations projected on the reaction path we successively obtain:
At this point, taking for the derivative in (183) the finite correspondence in what regarding the chemical potential formal (absolute) definition (181),
The remaining issue is to clarify the chemical potential related force meaning in above equation. In this respect, by considering the electronegativitychemical potential relationship (19), the associate electronegativity energy equation becomes:
With these considerations, the chemical reactivity energy
What there was actually proofed is that the quadratic chemical reactivity equations for total energy in both finite and differential fashions may be derived employing the KohnSham (as Schrödinger reminiscence) equation for chemical potential eigenvalue combined with the chemical quantum version of the Ehrenfest theorem involving the force concept and its activereactive peculiar property for the chemical potential and electronegativity, respectively. No particular assumptions were considered being all above arguments only on first principles grounded. Thus, the exposed demonstration is of general value cutting much discussion in the last decades on the viability of the second order truncation in the total energy expansion in terms of chemical reactivity indices, viz. electronegativity and chemical hardness concepts.
However, a computational test for this behavior is in next addressed.
The general bivariate equation linking the energy (various) functionals with electronegativity and chemical hardness, either for atomic and molecular systems, works out with form:
the degree of correlation itself between the employed energy functional and the couple of electronegativitychemical hardness structural indices; this is measured by
the degree of parabolic dependency by checking whether the chemical hardness coefficient (
At this point worth nothing that as σ_{π} → −1 as better the energy fulfils the correlation shape with the established parabolic
Finally, the sigmapi index (193) can be used in defining another reactivity index, namely
With these, the
At a glance,
However, for atomic analysis,
The situation is somehow changed for molecular systems, see
Finally, in
the correlation energy appears to provide acceptable parabolic shapes in both atomic and molecular cases, with better bilinear regression for molecular analysis, while strongly depending on the electronegativity and chemical hardness atomic models and scales;
the kinetic energy, while displaying poor parabolic shape at atomic level behaves with negative chemical hardness in molecular systems, probably due the positive contribution in bonding that compete with stabilization (localization) of the electrons within internuclear basin;
exchange and exchangecorrelation functionals reveal similar reactive (parabolic) efficiency as well as close bivariate regression correlation factors for both atomic and molecular cases, leaving with the impression that the exchange contribution is dominant in exchangecorrelation functionals since cancelling somehow the behavior of the correlation part of the functional.
overall, the total energy, although with correlation factors in the range of its components’ regressions does not fit with parabolic reactive theoretical prescription (189), at least for present employed set of atoms and molecules.
However, all revealed parabolic features have appeared on the statistical correlation basis and not upon an individual atomic or molecular analysis. At this point we can refer to studies [
The present dichotomy, while confronting the conceptual and computational behavior of the energetic functionals respecting the parabolic bilinear expansion in terms of electronegativity and chemical hardness, gives the indication that the derivation of the energetic functionals (and of the total energy in special) starting from electronegativity and chemical hardness functionals [
Why another quantum theory of the matter? Why the
The present paper aimed to unitarily presents the quantum fundaments of DFT as well to prospect directions in electronic bonding and reactivity characterization by using the density functionals and localization functions. Such theoretical tools allow the rationalization of the electronic structures while offering an analytical understanding and prediction of the experimental data along of the modeled reality.
In DFT, the density and their combinations in the density functionals of the total energy plays a primordial role. It fulfills the
Going to link the inner structural information with the manifested chemical reactivity the various energetic density functionals of atoms and molecules were presented, emphasizing on different levels of approximation and of quantum containing information. Nevertheless, their values were assumed merely as properties of particular systems when combined with associate reactive peculiarities quantified by electronegativity and chemical hardness. A theoretical link, based on first quantum mechanical principles, was also provided at a level of chemical potentialchemical force couple for the electronegativity and chemical hardness actions, respectively.
Actually, the parabolic shape of the energetic dependence on electronegativity and chemical hardness (
The numerical results of such correlation on selected, however limited, set of atomic and molecular systems, were intrigued since there was found out that only the correlation functionals provide appropriate parabolic and statistical behavior among considered atoms and molecules, while all other total energy components and the total energy itself posse both surprisingly low parabolic and correlation factors with the electronegativity and chemical hardness structural parameters. Nevertheless, at atomic level the finitedifference approximation for electronegativity and chemical hardness scales was found as most suitable in providing a correct (−, +) sign combination in parabolic (
The kind support of the Molecular Diversity Preservation International Foundation – MDPI for publishing this work and for recognizing the seminal role of “Chemical Bond and Bonding” theoretical and computational forum in actual Chemistry, as well the involvement of the Romanian National Council of Scientific Research in Universities – CNCSIS in promoting excellence exploratory research programs are hearty saluted.
Left: the antiferromagnetic structure CoO; right: the band structure and the density of state (DOS) in LSDA and GGA approximations, respectively [
The localization domains for Li (left) and Sc (right) crystals based on the electronic localization function ELF (59) [
Comparison of the radial density given by (71) with the electron localization function (65) with components (66) for the simplified selfconsistent approximation (67)–(70) for Li atomic structure [
Comparative analysis of the charge density contours, electronic localization functions (ELFs), and radial densities for the H (dashed lines), F, Cl, Br, and I (full lines) atoms in molecular combinations HF, HCl, HBr, and HI, respectively [
Charts of energetic functionals showing their reactive efficiency (196) as the height of bars for the parabolic expansion and of their statistical correlation (192) as the width of bars, in terms of electronegativity and chemical hardness, resuming atomic analysis of
Atomic kinetic, exchange, and correlation, energies (in hartrees) from various schemes of computations. The exact values are computed with HartreeFock densities.
Atoms  

2.86168  2.56054  2.87850  2.87639  −1.0260  −0.884  −1.025  −0.0425  −0.0215  −0.0681  
7.43273  6.70062  7.50504  7.44941  −1.7812  −1.538  −1.775  −0.0454  −0.0486  −0.0815  
14.5730  13.1286  14.6466  14.4223  −2.6669  −2.312  −2.658  −0.0945  −0.0820  −0.1192  
24.5291  22.0720  24.5228  24.2089  −3.7438  −3.272  −3.728  −0.1247  −0.1197  −0.1625  
37.6886  34.0144  37.5988  37.2533  −5.0444  −4.459  −5.032  −0.1566  −0.1609  −0.2091  
54.4009  49.4771  54.3852  54.0643  −6.5971  −5.893  −6.589  −0.1850  −0.2050  −0.2567  
74.8094  67.8965  74.3573  74.1625  −8.1752  −7.342  −8.169  −0.2579  −0.2512  −0.3035  
99.4093  90.4598  98.6429  98.6959  −10.003  −9.052  −10.02  −0.332  −0.2996  −0.3510  
128.547  117.761  127.829  128.221  −12.108  −11.03  −12.14  −0.390  −0.3498  −0.3987  
161.859  148.809  161.093  161.718  −14.017  −12.79  −14.03  −0.398  −0.3892  −0.4137  
199.614  184.017  198.749  199.578  −15.994  −14.61  −16.00  −0.443  −0.4351  −0.4491  
241.877  223.443  240.868  242.008  −18.069  −16.53  −18.06  −0.480  −0.4809  −0.4863  
288.854  267.315  287.659  289.139  −20.280  −18.59  −20.27  −0.521  −0.5308  −0.5308  
459.482  426.865  457.321  460.117  −27.512  −25.35  −27.49  −0.714  −0.6901  −0.6710  
526.817  490.017  524.289  527.617  −30.185  −27.86  −30.15  −0.787  −0.7459  −0.7190 
: from Ref. [
: from Ref. [
: from Ref. [
: from
: from fitting equation
Atomic exchangecorrelation and total energies (in hartrees) from various schemes of computations. The exact values are computed with HartreeFock densities.
Atoms  

−1.0685  −1.0604  −1.0566  −1.0633  −1.0654  −2.9042  −3.0317  −2.8601  −2.9071  −2.9000  
−1.8266  −1.8048  −1.8134  −1.8093  −1.8108  −7.4781  −7.6473  −7.3704  −7.4827  −7.4742  
−2.7614  −2.7260  −2.7522  −2.7325  −2.7342  −14.6675  −14.8911  −14.4966  −14.6615  −14.6479  
−3.8685  −3.8126  −3.8415  −3.8215  −3.8177  −24.6538  −24.9158  −24.4097  −24.6458  −24.6299  
−5.2010  −5.1127  −5.1338  −5.1248  −5.1121  −37.8163  −38.1305  −37.5095  −37.8430  −37.8265  
−6.7821  −6.6400  −6.6440  −6.6558  −6.6321  −54.4812  −54.8681  −54.1287  −54.5932  −54.5787  
−8.4331  −8.3599  −8.3405  −8.3796  −8.3450  −75.0271  −75.4597  −74.5979  −75.0786  −75.0543  
−10.325  −10.327  −10.277  −10.350  −10.305  −99.741  −100.235  −99.247  −99.7581  −99.7316  
−12.498  −12.551  −12.466  −12.579  −12.524  −128.937  −129.522  −128.403  −128.9730  −128.9466  
−14.415  −14.462  −14.382  −14.488  −14.445  −162.257  −162.862  −161.624  −162.293  −162.265  
−16.437  −16.482  −16.424  −16.504  −16.484  −200.058  −200.705  −199.340  −200.093  −200.060  
−18.549  −18.566  −18.542  −18.583  −18.593  −242.357  −243.028  −241.533  −242.380  −242.350  
−20.801  −20.774  −20.791  −20.784  −20.830  −289.356  −290.063  −288.435  −289.388  −289.363  
−28.226  −28.115  −28.272  −28.092  −28.281  −460.196  −461.005  −458.963  −460.165  −460.147  
−30.972  −30.827  −31.037  −30.789  −31.035  −527.605  −528.452  −526.267  −527.551  −527.539 
: from Ref. [
: from
: from Ref. [
: from Ref. [
Values (in hartrees) of the structural indices electronegativity (χ), chemical hardness (η), in finitedifference [
Level  

Atoms  
0.45094  0.45866  1.21132  1.66189  0.57038  0.2172  
0.11099  0.16134  0.15105  0.08784  0.00412  0.00334  
0.12606  0.21794  0.44248  0.44579  0.00893  0.0047  
0.15656  0.14921  1.15362  1.34105  0.01526  0.00588  
0.22933  0.18339  2.76332  2.9695  0.02279  0.00684  
0.25616  0.27894  5.79566  4.91363  0.03139  0.0076  
0.27894  0.22566  10.6505  5.91694  0.04072  0.00816  
0.38221  0.25983  16.9129  4.37707  0.05061  0.00849  
0.39361  0.40132  23.7119  −0.08747  0.06079  0.00864  
0.10290  0.10621  0.23153  0.18743  0.00011  0.00005  
0.09555  0.18339  0.49871  0.53142  0.00018  0.00007  
0.11834  0.10327  1.04631  1.19882  0.00026  0.00008  
0.17200  0.12606  2.10805  2.30724  0.00036  0.00009  
0.30577  0.17120  11.5766  7.7692  0.00074  0.00012  
0.28299  0.29806  17.8831  9.08857  0.00088  0.00013 
Molecular kinetic, exchange, correlation, exchangecorrelation, and total energies (in hartrees) from various schemes of computations. The exact values are computed with HF densities.
Molecules  

1.140  1.125  −0.657  −0.648  −95·10^{−3}  −47·10^{−3}  −0.698  −0.691  −1.169  −1.178  
7.978  8.003  −2.125  −2.105  −219·10^{−3}  −93·10^{−3}  −2.212  −2.188  −8.068  −8.070  
40.050  40.141  −6.576  −6.536  −593·10^{−3}  −328·10^{−3}  −6.883  −6.836  −40.502  −40.515  
76.150  75.477  −8.910  −8.917  −664·10^{−3}  −365·10^{−3}  −9.292  −9.241  −76.448  −76.433  
100.137  99.242  −10.378  −10.385  −704·10^{−3}  −380·10^{−3}  −10.779  −10.720  −100.48  −100.455  
109.115  108.242  −13.094  −13.128  −945·10^{−3}  −506·10^{−3}  −13.665  −13.580  −109.559  −109.54  
149.843  148.369  −16.290  −16.358  −1110·10^{−3}  −599·10^{−3}  −16.958  −16.887  −150.384  −150.337  
198.892  196.729  −19.872  −19.951  −1302·10^{−3}  −697·10^{−3}  −20.661  −20.564  −199.599  −199.533 
: from Ref. [
: from Ref. [
: from Ref. [
Values (in hartrees) of the structural indices electronegativity (χ), chemical hardness (η), compute by means of the group method [
Level Molecule s  

0.26387  0.2370  0.26384  0.23704  0.26387  0.23705  
0.15626  0.192  0.19212  0.12818  0.00811  0.006596  
0.25616  0.2239  0.32216  0.29051  0.08468  0.03064  
0.26871  0.2331  0.39097  0.34859  0.09335  0.0229  
0.3077  0.2479  0.51964  0.44974  0.08493  0.01639  
0.25616  0.2789  5.79566  4.91363  0.03139  0.00761  
0.27894  0.2257  10.6505  5.91694  0.04072  0.00816  
0.36898  0.2598  16.9129  4.37707  0.05061  0.00849 
Coefficients in bilinear correlation of the energies of
Method of

QSPR results

Method of

QSPR results
 

Energ y  (χ, η)  a  b  c  σ_{π}  r  Energy  (χ, η)  a  b  c  σ_{π}  r 
194.59  741.55  −951.38  −0.0017  0.33  −15.30  −44.80  60.89  −0.0303  0.37  
51.73  2.09  31.51  7.2  0.62  −6.56  −0.19  −1.53  40.84  0.58  
186.38

−2315.7

5147.4

−0.001

0.35

−13.76

73.07

−128.0

−0.024

0.38
 
179.67  687.26  −881.14  −0.0019  0.33  −15.28  −44.56  60.69  −0.0306  0.37  
46.99  1.91  29.41  8.07  0.62  −6.56  −0.2  −1.51  39.11  0.57  
172.42

−2185.3

4874.1

−0.001

0.35

−13.72

72.04

−125.5

−0.024

0.38
 
193.83  736.94  −946.29  −0.0017  0.33  −15.28  −44.81  60.92  −0.0303  0.37  
51.59  2.07  31.36  7.33  0.62  −6.52  −0.19  −1.53  40.81  0.57  
185.59

−2312.1

5142

−0.001

0.35

−13.75

74.72

−132.4

−0.024

0.38
 
194.61  741.17  −951.43  −0.0017  0.33  −15.29  −44.53  60.67  −0.0306  0.37  
51.54  2.08  31.58  7.3  0.62  −6.58  −0.2  −1.50  38.78  0.57  
186.42

−2334.4

5197.4

−0.001

0.35

−13.72

71.4

−123.8

−0.024

0.38
 
−14.91  −43.62  59.38  −0.0312  0.37  −15.30  −44.80  60.95  −0.0304  0.37  
−6.37  −0.19  −1.49  43.17  0.57  −6.54  −0.2  −1.52  39.98  0.57  
−13.4

72.74

−128.78

−0.0243

0.38

−13.76

74.1

−130.7

−0.024

0.38
 
−13.59  −40.29  54.69  −0.0337  0.37  −194.98  −742.7  952.91  −0.0017  0.33  
−5.72  −0.17  −1.39  46.96  0.58  −51.91  −2.10  −31.54  7.15  0.62  
−12.24

68.77

−123.42

−0.026

0.38

−186.73

2316

−5148

−0.001

0.35
 
−14.89  −43.61  59.33  −0.0312  0.37  −195.55  −743.9  954.5  −0.0017  0.33  
−6.37  −0.19  −1.49  42.35  0.57  −52.27  −2.11  −31.56  7.1  0.62  
−13.39

71.88

−126.54

−0.0245

0.38

−187.24

2315.

−5143.

−0.001

0.35
 
−0.39  −1.20  1.52  −1.055  0.38  −194.26  −740.7  950.13  −0.0017  0.33  
−0.19  −0.0075  −0.03  595.98  0.59  −51.59  −2.09  −31.47  7.18  0.62  
−0.362

0.232

0.997

18.528

0.38

−186.1

2313.

−5142.

−0.001

0.35
 
−0.40  −1.13  1.55  −1.207  0.39  −194.99  −742.7  952.86  −0.0017  0.33  
−0.19  −0.0056  −0.03  1081.3  0.57  −51.94  −2.1  −31.53  7.15  0.62  
−0.356

0.851

−0.563

−0.778

0.41

−186.74

2315.

−5145

−0.001

0.35
 
−0.41  −1.10  1.44  −1.1808  0.40  −194.97  −742.7  952.81  −0.0017  0.33  
−0.22  −0.0063  −0.03  735.5  0.59  −51.92  −2.1  −31.54  7.15  0.62  
−0.374  −0.224  2.113  −42.24  0.42  −186.7  2315.  −5145.  −0.001  0.35 
,
: from associate energetic atomic values of
Coefficients in bilinear correlation of the energies of
Method of

QSPR results

Method of

QSPR results
 

Energy y  χ & η  a  b  c  σ_{π}  r  Energy y  χ & η  a  b  c  σ_{π}  r 
−192.71  821.61  238.59  0.00035  0.77  −190.58  811.48  237.96  0.0004  0.77  
40.446  8.125  4.501  0.068  0.89  40.199  8.012  4.497  0.07  0.89  
82.14

542.16

−977.9

−0.0033

0.56

81.4

537.74

−969.47

−0.003

0.56
 
18.18  −70.1  −38.02  0.0077  0.74  18.324  −70.48  −38.25  0.0077  0.74  
−5.234  −0.646  −0.805  1.93  0.89  −5.219  −0.65  −0.811  1.92  0.89  
−9.78

−48.27

94.96

−0.04

0.61

−9.81

−48.15

95.

−0.04

0.61
 
1.039  −4.06  −2.74  0.167  0.71  0.59  −2.27  −1.496  0.289  0.72  
−0.423  −0.035  −0.06  47.83  0.87  −0.225  −0.019  −0.033  94.08  0.86  
−0.71

−3.13

6.22

−0.63

0.64


18.95  −72.63  −40.1  0.0076  0.74  18.85  −72.45  −39.62  0.0075  0.74  
−5.457  −0.668  −0.846  1.897  0.89  −5.421  −0.666  −0.84  1.898  0.89  
−10.19  −50.16  98.71  −0.039  0.61  −10.13  −49.97  98.28  −0.04  0.61  
193.12

−824.17

−238.96

0.00035

0.77

193.06

−823.8

−239.04

0.0004

0.77
 
−40.67  −8.149  −4.514  0.068  0.89  −40.67  −8.145  −4.514  0.068  0.89  
−82.48  −544.35  981.7  −0.003  0.56  −82.46  −544.3  981.5  −0.003  0.56 