The first Hohenberg-Kohn (HK1) theorem gives space to the concept of electronic density of the system ρ(r) in terms of the extensive relation with the N electrons from the system that it characterizes [8]: (4) ∫ ρ ( r ) d r = N    .

The relation (4) as much simple it could appears stands as the decisive passage from the eigen-wave function level to the level of total electronic density [9–11]: (5) ρ ( r ) = N ∫ Ψ * ( r , r 2 , ... , r N ) Ψ ( r , r 2 , ... , r N ) d r 2 ... d r N .

Firstly, Eq. (5) satisfies Eq. (4); this can be used also as simple immediate proof of the relation (4) itself. Then, the dependency from the 3N-dimensions of configuration space was reduced at 3 coordinates in the real space, physically measurable.

However, still remains the question: what represents the electronic density of Eq. (5)? Definitely, it neither represents the electronic density in the configuration space nor the density of a single electron, since the N-electronic dependency as multiplication factor of the multiple integral in (5). What remains is that ρ(r) is simple the electronic density (of the whole concerned system) in „r” space point. Such simplified interpretation, apparently classics, preserves its quantum roots through the averaging (integral) over the many-electronic eigenfunction Ψ(r₁,...,r_N) in (5). Alternatively, the explicit non-dependency of density on the wave function is also possible within the quantum statistical approach where the relation with partition function of the system (the global measure of the distribution of energetic states of a system) is mainly considered.

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 [8, 9, 12]: (6) E [ ρ ] = F H K [ ρ ] + C A [ ρ ] ,

from where the name of the theory. The terms of energy decomposition in (6) are identified as: the Hohenberg-Kohn density functional (7) F H K [ ρ ] = T [ ρ ] + V e e [ ρ ]viewed as the summed electronic kinetic T[ρ] and electronic repulsion V_ee [ρ], and the so called chemical action term [12–14]: (8) C A [ ρ ] = ∫ ρ ( r ) V ( r ) d r ,being the only explicit functional of total energy.

Although not entirely known the HK functional has a remarkably property: it is universally, in a sense that both the kinetic and inter-electronic 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 for a given external potential V(r) remain valuable for any other type of potential V’(r) applied on the concerned many-electronic system. Let’s note the fact that V(r) should be not reduced only to the Coulombic type of potentials but is carrying the role of the generic potential applied, that could beg of either an electric, magnetic, nuclear, or even electronic nature as far it is external to the system fixed by the N electrons in the investigated system.

Once “in game” the external applied potential provides the second Hohenberg-Kohn (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 N-electronic system ground state”.

In mathematical terms, the theorem assures the validity of the variational principle applied to the density functional (6) relation, i.e. [6] (9) E [ ρ ¯ ] ≥ E [ ρ ] ⇔ δ E [ ρ ] = 0for every electronic test density ρ̄ around the real density ρ of the ground state.

The proof of variational principle in (9), or, in other words, the one-to-one correspondence between the applied potential and the ground state electronic density, employs the reduction ad absurdum procedure. That is to assume that the ground state electronic density ρ(r) corresponds to two external potentials (V₁, V₂) fixing two associate Hamiltonians (H₁, H₂) to which two eigen-total energy (E₁, E₂) and two eigen-wave functions (Ψ₁, Ψ₂) are allowed. Now, if eigen-function Ψ₁ is considered as the true one for the ground state the variational principle (9) will cast as the inequality: (10) E 1 [ ρ ] = ∫ Ψ 1 * H ^ 1 Ψ 1 d τ < ∫ Ψ 2 * H ^ 1 Ψ 2 d τ = ∫ Ψ 2 * [ H ^ 2 + ( H ^ 1 − H ^ 2 ) ] Ψ 2 d τwhich is further reduced, on universality reasons of the HK functional in (6), to the form: (11) E 1 [ ρ ] < E 2 [ ρ ] + ∫ ρ ( r ) [ V 1 ( r ) − V 2 ( r ) ] d r .

On another way, if the eigen-function Ψ₂ is assumed as being the one true ground state wavefunction, the analogue inequality springs out as: (12) E 2 [ ρ ] < E 1 [ ρ ] + ∫ ρ ( r ) [ V 2 ( r ) − V 1 ( r ) ] d r.

Taken together relations (11) and (12) generate, by direct summation, the evidence of the contradiction [2]: (13) E 1 [ ρ ] + E 2 [ ρ ] < E 1 [ ρ ] + E 2 [ ρ ] .

The removal of such contradiction could be done in a single way, namely, by abolishing, in a reverse phenomenologically order, the fact that two eigen-functions, 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 V-representability problem signaling the impossibility of an a priori selection of the external potentials types that are in bi-univocal relation with ground state of an electronic system [15–18]. The problem was revealed as very difficult at mathematical level due to the equivocal potential intrinsic behavior that is neither of universal nor of referential independent value. Fortunately, such principial limitation does not affect the general validity of the variational principle (9) regarding the selection of the energy of ground state level from a collection of states with different associated external potentials.

That because, the problem of V-representability can be circumvented by the so called N-contingency features of ground state electronic density assuring that, aside of the N – integrability condition (4), the candidate ground state densities should fulfill the positivity condition (an electronic density could not be negative) [17, 18]: (14) ρ ( r ) ≥ 0 ,    ∀ | r | ∈ ℜ ,as well as the non-divergent integrability condition on the real domain (in relation with the fact that the kinetic energy of an electronic system could not be infinite – since the light velocity restriction): (15) ∫ ℜ | ∇ ρ ( r ) 1 / 2 | 2 d r < ∞ .

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 Levy-Lieb double minimization algorithm [19]: one regarding the intrinsic minimization procedure of the energetic terms respecting all possible eigen-functions folding a trial electronic density followed by the external minimization over all possible trial electronic densities yielding the correct ground state (GS) energy density functional (16) E G S = min ρ [ min Ψ → ρ ( ∫ Ψ * ( T + V e e + V ) Ψ d τ ) ] = min ρ [ min Ψ → ρ ( ∫ Ψ * ( T + V e e ) Ψ d τ ) + ∫ ρ ( r ) V ( r ) d r ]     min ρ ( F H K [ ρ ] + C A [ ρ ] ) = min ρ ( E [ ρ ] ) .

One of the most important consequences of the HK2 conveys the rewriting of the variational principle (9) in the light of above N-contingency conditions of the trial densities as the working Euler type equation: (17) δ { E [ ρ ] − μ N [ ρ ] } = 0from where, there follows the Lagrange multiplication factor with the functional definition: (18) μ = ( δ E [ ρ ] δ ρ ) ρ = ρ ( V )this way introducing the chemical potential as the fundamental quantity of the theory. At this point, the whole chemistry can spring out since identifying the electronic systems electronegativity with the negative of the density functional chemical potential [9]: (19) χ = − μ .

Up to now, the Hohenberg-Kohn theorems give new conceptual quantum tools for physico-chemical 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 [20].

Back from Paris, in the winter of 1964, Kohn met at the San Diego University of California his new post-doc 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 eigen-equation 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 N-electronic system by replacing the integration in the relation (5) with summation over the virtual uni-electronic orbitals ϕ i , i = 1 , N ¯, in accordance with Pauli principle, assuring therefore the HK1 frame with maximal spin/orbital occupancy [21]: (20) ρ ( r ) = Σ i N n i | φ i ( r ) | 2 ,   0 ≤ ni ≤ 1 , Σ i ni = N .

Then, the trial total eigen-energy may be rewritten as density functional of eq. (6) nature expanded in the original form [22, 23]: (21) E [ ρ ] = F H K [ ρ ] + C A [ ρ ] = T [ ρ ] + V e e [ ρ ] + C A [ ρ ] = T s [ ρ ] + j [ ρ ] + { ( T [ ρ ] − T s [ ρ ] ) + ( V e e [ ρ ] − J [ ρ ] ) } + C A [ ρ ] = Σ i N ∫ n i φ i * ( r ) [ − 1 2 ∇ 2 ] φ i ( r ) d r + 1 2 ∫ ∫ ρ ( r 1 ) ρ ( r 2 ) r 12 d r 1 d r 2 + E x c [ ρ ] + ∫ V ( r ) ρ ( r ) d rwhere, the contribution of the referential uniform kinetic energy contribution (22) T s [ ρ ] = Σ i N ∫ n i φ i * ( r ) [ − 1 2 ∇ 2 ] φ i ( r ) d r ,with the inferior index „s” referring to the „spherical” or homogeneous attribute together with the classical energy of Coulombic inter-electronic repulsion (23) J [ ρ ] = 1 2 ∫ ∫ ρ ( r 1 ) ρ ( r 2 ) r 12 d r 1 d r 2were used as the analytical vehicles to elegantly produce the exchange-correlation energy E_xc containing exchange (V_ee[ρ] – J[ρ]) and correlation (T[ρ] – T_s [ρ]) heuristically introduced terms as the quantum effects of spin anti-symmetry over the classical interelectronic potential and of corrected homogeneous electronic movement, respectively.

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 multi-electronic 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), (24) ρ ( r ) = N φ * ( r ) φ ( r )that, without diminishing the general validity of the results, since preserving the N-electronic character of the system, highly simplifies the analytical discourse.

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: (25) ∫ δ ( E [ ρ ] − μ N [ ρ ] ) δ φ * δ φ * d r = 0 .

Note that, in fact, we chose the variation in the conjugated uni-orbital φ^*(r) in (25) providing from (24) the useful differential link: (26) δ ρ ( r ) = N φ ( r ) δ φ * ( r ) .

Now, unfolding the equation (25) with the help of relations (21) and (24), together with fundamental density functional prescription (4), one firstly gets [24]: (27) δ δ φ * ( r ) { − N 2 ∫ φ * ( r ) ∇ 2 φ ( r ) d r + J [ ρ ] + E x c [ ρ ] + N ∫ V ( r ) φ * ( r ) φ ( r ) d r − μ N ∫ φ * ( r ) φ ( r ) d r } = 0

By performing the required partial functional derivations respecting the uni-orbital φ^*(r) and by taking account of the equivalence (26) in derivatives relating J[ρ] and E_xc [ρ] terms, equation (27) takes the further form: (28) − N 2 ∇ 2 φ ( r ) + N φ ( r ) δ J [ ρ ] δ ρ + N φ ( r ) δ E x c δ ρ + N V ( r ) φ ( r ) − μ N φ ( r ) = 0.

After immediate suppressing of the N factor in all the terms and by considering the exchange-correlation potential with the formal definition [4, 9]: (29) V x c ( r ) = ( δ E x c [ ρ ] δ ρ ( r ) ) V ( r ) ,equation (28) simplifies as [25, 26]: (30) [ − 1 2 ∇ 2 + ( V ( r ) + ∫ ρ ( r 2 ) | r − r 2 | d r 2 + V x c ( r ) ) ] φ ( r ) = μ φ ( r ) .

Moreover, once introducing the so called effective potential: (31) V e f f ( r ) = V ( r ) + ∫ ρ ( r 2 ) | r − r 2 | d r 2 + V x c ( r )the resulted equation recovers the traditional Schrödinger shape: (32) [ − 1 2 ∇ 2 + V e f f ] φ ( r ) = μ φ ( r ) .

The result (32) is fundamental and equally subtle. Firstly, it was proved that the joined Hohenberg-Kohn 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(r) is now unambiguously producing the analytical passage from configuration (3N-D) to real (3D) space for the whole system under consideration. Nevertheless, the debate may still remain because once equation (32) is solved the basic functions φ(r) generating the electronic density (24) and not necessarily the eigen-functions of the original system due to the practical approximations of the exchange and correlation terms appearing in the effective potential (31). This is why the functions φ(r) are used to be called as Kohn-Sham (KS) orbitals; they provide the orbital set solutions of the associate KS equations [3]: (33) [ − 1 2 ∇ 2 + V e f f ] φ i ( r ) = μ i φ i ( r ) ,   i = 1 , N ¯once one reconsiders electronic density (24) back with general case (20).

Yet, equations (33), apart of delivering the KS wavefunctions φ_i(r), associate with another famous physico-chemical figure, the orbital chemical potential μ_i, which in any moment can be seen as the negative of the orbital electronegativities on the base of the relation (19).

Going now to a summative characterization of the above optimization procedure worth observing that the N-electronic in an arbitrary external V-potential problem is conceptual-computationally solved by means of the following self-consistent algorithm:

showing that the optimized many-electronic 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 many-electronic 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 Hohenberg-Kohn theorems consecrate the crucial contributions of the so called “spherical” or homogeneous kinetic and of the exchange-correlation energy terms in a multi-electronic system’s ground state. However, the spherical electronic case corresponds to the non perturbed electronic system for which the Thomas-Fermi (TF) model was already advanced throughout totally ignoring exchange- correlation terms from the total energy shape: (36) E T F [ ρ ] = T T F [ ρ ] + J [ ρ ] + C A [ ρ ] .

Such a referential picture is most useful in establishing the uniform electronic distribution by indicating the occupation of the all-possible electronic levels in a semiclassical quantum frame (without explicit exchange-correlation involvement). Actually, the Fermi sphere in a momentum space finely defines the total homogeneous kinetic energy as: (37) τ s ( r ) = p F 2 2 m 0 ,while the quantum nature of the kinetic energy (37) is covered by involving the quantum (Heisenberg) uncertainty (38) Δ p x Δ p y Δ p z Δ x Δ y Δ z ≅ h 3in uniform density computation. This suggests that the density of states in the Fermi volume of the impulse p_F has to be normalized to the inverse of the cube power of Planck constant h, while the density of electrons is reached by multiplying the density of states with the electron multiplicity 2(1/2)+1=2 for every occupied state. The obtained density-Fermi impulse relationship: (39) ρ ( r ) d r = 2 4 3 π p F 3 h 3 d r ⇒ p F ( r ) = ( 3 8 π ) 1 / 3 h ρ ( r ) 1 / 3allows the Thomas-Fermi kinetic energy unfolding as the density functional [9, 27–31]: (40) T T F [ ρ ] = 3 5 ∫ ρ ( r ) τ s ( r ) d r = C T F ∫ ρ ( r ) 5 / 3 d r ,   C T F = 3 h 2 10 m 0 ( 3 8 π ) 2 / 3 ≅ 2.871 [ a . u . ]with the help of which the total Thomas-Fermi energy functional takes the form: (41) E T F [ ρ ] = C T F ∫ ρ ( r ) 5 / 3 d r + 1 2 ∬ ρ ( r 1 ) ρ ( r 2 ) | r 12 | d r 1 d r 2 + ∫ V ( r ) ρ ( r ) d rthat can be seen as the first approximation for the density functional total energy (21).

From physical point of view worth noted that the kinetic TF energy exactly corresponds to the total energy of the free electrons in a crystal, V(r)=0 in (41), equivalently with the fact that the electrons are not “feeling” the nuclei, i.e. electrostatic attractions are excluded, being as close each other to avoid reciprocal repelling. Such picture suggests that free electrons are completely non-localized leaving with the condition of complete cancellation of the electronic inter-repulsion; this feature may be putted formally as [32]: (42) e 2 | r 1 − r 2 | → λ e 2 | r 1 − r 2 | , λ = 0.

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 Thomas-Fermi 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 Thomas-Fermi description may be regarded as the “inferior” extreme in quantum known structures while further exchange-correlation 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 parameter of the electronic coupling, with a slightly (adiabatically) scaling of the perturbation from the homogeneous electronic systems, λ=0, until the maximal inter-electronic interaction, λ=1 (in accordance with Pauli principle). Therefore, the overall interpolation ∫ 0 1 [ • ] d λ will be spread over the terms which contain the intermediate degree of exchange and correlation interactions; since it accounts for the electronic inter-repulsion while indexing the electronic presence/absence in a given spatial region the degree of electronic localization is in this way furnishes.

The coupling parameter λ will serve as a switcher between the referential Thomas-Fermi uniform case and the full interaction through the density limit: (43) lim λ → 1 ρ λ ( r ) = ρ ( r = r 1 ∩ r 2 ) .

Actually, the density (43) has a major role in defining exchange-correlation functionals. To see that let’s firstly consider the conditional electronic density g(r₁, r₂;λ) indicating that the electronic density in r₁ is conditioned by the presence (localization) of another electron (any from the total N in the system) in r₂. Mathematically, this is expressed by using the conditional probabilities: (44) g ( r 1 , r 2 ; λ ) = ρ ( r = r 1 ∩ r 2 ) ρ ( r 2 )fulfilling the Pauli principle by means of the integration rule: (45) ∫ ρ ( r 2 ) g ( r 1 , r 2 ; λ ) d r 2 = ∫ ρ ( r 1 ∩ r 2 ) d r 2 = 0saying that the spatial average of the electronic reciprocal constraint vanishes. This behavior opens the possibility in introducing the conditional probability of electronic holes, (46) h ( r 1 , r 2 ; λ ) = g ( r 1 , r 2 ; λ ) − 1 ,providing the associate integration rule [33]: (47) ∫ ρ ( r 2 ) h ( r 1 , r 2 ; λ ) d r 2 = − 1consecrating a sort of negative normalization of the exchange and correlation density of holes: (48) ρ x c ( r 1 , r 2 ; λ ) = ρ ( r 2 ) h ( r 1 , r 2 ; λ ) .

Now, once this exchange-correlation hole density is mediated over the coupling factor λ the averaged exchange-correlation density of holes is generated: (49) ρ x c ¯ ( r 1 , r 2 ) = ∫ 0 1 ρ x c ( r 1 , r 2 ; λ ) d λallowing the formal writing of exchange-correlation density functional from (29) as a generalized version of the inter-electronic interaction term (23): (50) E x c [ ρ ] = 1 2 ∬ ρ ( r 1 ) ρ x c ¯ ( r 1 , r 2 ) r 12 d r 1 d r 2 = − 1 2 ∫ ρ ( r 1 ) R x c ¯ ( r 1 , ρ ( r ) ) d r 1with the help of the introduced radius of the λ-mediated exchange-correlation density of holes: (51) R x c ¯ − 1 ( r 1 , ρ ( r ) ) : = − ∫ ρ x c ¯ ( r 1 , r 2 , ρ ( r ) ) r 12 d r 2

The radius R x c ¯ ( r ) could be considered as a functional of density (43) with the leading term being defined in the short limit of the distance, i.e. being of the inter-particle average radius order, (52) 4 π 3 r 0 3 = 1 ρ ( r ) ⇒ r 0 = ( 3 4 π ) 1 / 3 ρ ( r ) − 1 / 3 ,known as the Wigner radius for indexing the volume of a sphere containing (localizing) a single electron (from the total of N) belonging to the density family (43), although, also other quantities accounting for electron localization such as the domain averaged Fermi hole of Ponec [34a] or the electron sharing index (also known as delocalization index) [34b] have been recently proposed (see also the forthcoming discussion).

In these conditions, the inverse radius (51) could be expressed around the inverse of the Wigner radius in a gradient density expansion [32]: (53) R x c ¯ − 1 ( r , ρ ( r ) ) = F 0 [ ρ ( r ) ] + F 21 [ ρ ( r ) ] ∇ 2 ρ ( r ) + F 22 [ ρ ( r ) ] ∑ i = 1 N ( ∇ i ρ ( r ) ) ( ∇ i ρ ( r ) ) + ...while, by considering it back in exchange-correlation energy (50) produces, after the integration by parts, the generalized gradient density functional: (54) E x c = ∫ G 0 [ ρ ( r ) ] ρ ( r ) d r + ∫ G 2 [ ρ ( r ) ] ( ∇ ρ ( r ) ) 2 d r + ∫ [ G 4 [ ρ ( r ) ] ( ∇ 2 ρ ( r ) ) 2 + ... ] d r + ...

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 local density approximation (LDA) [34c]: (55) E x c L D A [ ρ ] = ∫ e x c [ ρ ( r ) ] ρ ( r ) d rwith e_xc being the exchange and correlation density per particle, that can be further approximated (see bellow) as [35–40]: (56) e x c [ ρ ] = e x [ ρ ] + e c [ ρ ] = ( − 0.458 r 0 ) + ( − 0 . 44 r 0 + 7.8 ) [ a . u . ] .

In fact, the LDA stands as the immediate step after TF approximation; it can be extended also for systems with un-pair spins by the so called local spin density approximation (LSDA) [41–50]: (57) E x c [ ρ = ρ ↑ + ρ ↓ ] = E x c [ ρ ↑ ] + E x c [ ρ ↓ ] ,while further inclusion of the gradient terms in (54) establishes the general gradient approximation (GGA).

Worth noting that when undertaken GGA, beside the gradient terms arising in exchange-correlation 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 [51, 52]: (58) τ W ( r ) = 1 8 | ∇ ρ ( r ) | 2 ρ ( r ) .

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, Figure 1 presents the band structure and the density of states (DOS) for the R 3 ¯m oxide of the Cobalt transitional metal (CoO) calculated with either LSDA and GGA approximations [53].

Regarding the energy bands there can be noted that, around the Fermi level E_F, LSDA approach is less relevant in indicating the energetically gap respecting the GGA computation. The difference is even more drastic in DOS representations for employed approximations in d orbital separation (t_2g=a_1g+e_g’) due to the central ion of cobalt trigonal symmetry coordination. In fact, with LSDA a strong mixing of the orbitals a_1g and e_g’ is recorded while in the case of GGA-DOS the bands with the symmetry a1g are up and down shifted for the respective down and up spin projections resulting a clear separation from states with e_g’ symmetry.

Nonetheless, at the level of bands structure of the solids and crystals an inevitable localization paradox emerges namely, to use the real 3D electronic densities in furnish a localization description in the reciprocal (energy) space.

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 [54]: (59) E L F = ( 1 + [ τ s ( r ) − τ W ( r ) T T F [ ρ ( r ) ] ] 2 ) − 1by emphasizing the excess of the kinetic energy difference τ_s(r)- τ_W(r) „normalized” to the referential kinetic TF homogeneous behavior.

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 valence shell electron pair repulsion (VSEPR) geometric model [56], with the immediate consequence in topological characterization of the chemical bond [57]. In solid state case, the reliability of above ELF in describing the chemical bond in real space is illustrated in Figure 2 for the Li and Sc crystals. Atomic and molecular levels are in next section illustrated with which occasion further ELF characterization and developments are presented.

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, quantum-mechanical functions related to the Pauli Exclusion Principle may be formulated as “localization attractors” of bonding, non-bonding, 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 versus covalent bond to be derived from electronic density combined functions [58].

Most modern classifications of the chemical bond are based on Lewis’ theory and rely on molecular-orbital and valence-bond 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 −∇²ρ(r) [56]. Next, the exchange-correlation density functional concept was employed to achieve the coordinate-space dynamical correlation in an inhomogeneous electron gas. This way the exchange-correlation energy (50) further re-expresses like [32, 54] (60) E x c = 1 2 ∑ α α ′ ∬ ρ ( r 1 ) ρ ( r 2 ) r 12 h X C α α ′ d r 1 d r 2 ,by accounting for the four α-spin types of interactions through the “hole” functions [40, 59] (61) h α α ′ ( r 1 , r 2 ) = P 2 α α ′ ( r 1 , r 2 ) ρ α ( r 1 ) − − ρ α ( r 2 )where P₂(r₁,r₂) stands for the two-body or pair probability density or correlation probability of the arbitrary electrons „1 and 2”, defined in terms of the N-body wave function Ψ as follows [9]: (62) P 2 ( r 1 , r 2 ) = N ( N − 1 ) ∬ ... ∫ Ψ * ( r 1 , r 2 , r 3 , ... , r N ) Ψ ( r 1 , r 2 , r 3 , ... , r N ) d r 3 ... d r N   .

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 many-electronic-nuclei system.

Nevertheless, recently, the Markovian analytical shape of an ELF was shown to have the general qualitative form [60]: (63) E L F = f − 1 ( g ( r ) h ( r ) )with the limiting constrains (64) lim E L F = { 0 , ∇ ρ ( r )   ≫   ρ ( r ) 1 , ∇ ρ ( r )   ≪   ρ ( r )assuring the fulfillment of the Heisenberg and Pauli principles. To clarify this [60], we make recourse to the Heisenberg principle, comprised in ELF. When density gradient dominates,∇ρ≫ρ then g≫h in (63), and f(∞) should accounted for the infinite error in assigning of momentum, therefore indicating a precisely spatial localization of electrons; thus f(∞) = ∞ and ELF→0. In such, the meaning of ELF is associated with the error in spatial localization of electrons, being zero when the electrons are precisely located. On the contrary, when ρ≫∇ρ then h≫g in (63), and the resulting f(0) indicates the minimum error in defining of momentum and should provide the maximum uncertain of spatial distribution; in such f(0)=1 and ELF→1, where 1 stands here for 100% of coordinate localization error.

In this context, when the inverse of difference in local kinetic terms is involved, the ELF is interpreted as the error in localization of electrons within traps rather than where they have peaks of spatial density, as is frequently misinterpreted in literature [61], albeit recent extensions of ELF have used the correlated (Hartree-Fock) wavefunctions, through the conditional pair probability, however not using the “kinetic energy approach” [62–64].

Among various classes of Markovian ELFs the most representative and efficient one was proposed as having the form [60]: (65) E L F = exp { − 3 2 [ g ( r ) h ( r ) ] 2 }with the components: (66) g ( r ) = 1 2 ∑ i [ ∇ φ i ( r ) ] 2 − 1 8 [ ∇ ρ ( r ) ] 2 ρ ( r ) , h ( r ) = 3 10 ( 3 π 2 ) 2 / 3 [ ρ ( r ) ] 5 / 3being responsible for the gradient (g) and the homogenous (h) density distributions, respectively. In this frame, the ELF information prescribe that as it has values closer to zero as the better electronic localization is providing, according with the limits (64).

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:

such that to fulfill the natural (radial) normalization conditions (68) ∫ 0 ∞ [ f n L i ( r ) ] 2 d r = 1 , n = 1 , 2

Generating the orthonormal orbital eigen-waves, here according with the Gram-Schmidt algorithm among shells and sub-shells: (69) φ 1 s L i ( r ) = f 1 L i ( r ) ,   φ 2 p L i ( r ) = f 2 L i ( r ) , φ 2 s L i ( r ) = C 2 s L i [ φ 2 p L i ( r ) − α φ 1 s L i ( r ) ] = − 1 . 2226 exp ( − 2 . 698 r ) + 0.373222 r 5 / 2 exp ( − 0.797 r )

The electronic density (71) is then used for computation of the Markovian ELF (65), while their comparison is in Figure 3 illustrated.

From the Figure 3, there appears that the smooth delocalization of electrons of Li represented by density structure is removed by the electronic localization function by clearly indicating where are the regions where the electronic realm is with less uncertainty detected. This way the ELF indicates merely where the electronic transitions behave like a step-function. In this respect, ELF can be regarded as the complement of electronic density being a better indicator of the regions where the bonding may arise. For instance, in the case of Li atomic structure, the fact that the ELF does not displays localization over the second shell (due to its values approaching unity in this range) indicates a natural tendency for releasing the outermost electron to the (virtual) neighborhood atoms with uncompleted last shell(s) while preserving its delocalization feature across the bond. As such the lithium hydride (LiH) bond is expected to be formed with a certain degree of ionicity in resonance with its covalence: LiH ↔ Li⁺H⁻.

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 Figure 4. In the bonding region, i.e. in the space between the hydrogen and halogen atomic centers in H-X molecules there are represented both the electron densities, computed upon above recipe [65], and the associate Markovian ELFs for the concerned atoms-in-molecules (AIM).

Figure 4 clearly shows that while the crossing of hydrogen and halogen radial densities does not provide the right bonding region in HCl, HBr, and HI cases, the corresponding ELFs cross-lines of AIM finely indicate the frontier of atomic basins in hydracids thus confirming the ELF reliability in identifying chemical bonds and bonding.

One can equally say that in the crossing vicinity of AIM-ELFs the electrons are at the same time completely localized (for bonding with ELF_X – ELF_H →0) and completely delocalized for atomic systems (with ELF_X,H →1), according to above the ELF definition and present signification.

In other words it can be alleged that ELF application on chemical bond helps in identifying the molecular region in which the electrons undergo the transition from the complete delocalization in atoms to complete localization in molecular bonding behavior.

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 in-depth 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 [67].

When the electronic density is seen as the diagonal element ρ(r₁) = ρ(r₁,r₁) the kinetic energy may be generally expressed from the Hartree-Fock model, through employing the single determinant ρ(r₁,r′₁), as the quantity [68]: (73) T [ ρ ] = − 1 2 ∫ [ ∇ r ' 1 2 ρ ( r 1 , r ' 1 ) ] r 1 = r ' 1 d r 1 ;it may eventually be further written by means of the thermodynamical (or statistical) density functional [69]: (74) T β = 3 2 ∫ ρ ( r ) k B T ( r ) d r = 3 2 ∫ ρ ( r ) 1 β ( r ) d rthat supports various specializations depending on the statistical factor particularization β.

For instance, in LDA approximation, the temperature at a point is assumed as a function of the density in that point, β(r) = β(ρ(r)); this may be easily reached out by employing the scaling transformation to be [70] (75) ρ λ ( r ) = λ 3 ρ ( λ r ) ⇒ T [ ρ λ ] = λ 2 T [ ρ ] ,   λ = c t . ,providing that (76) β ( r ) = 3 2 C ρ − 2 / 3 ( r ) ,a result that helps in recovering the traditional (Thomas-Fermi) energetic kinetic density functional form (77) T [ ρ ] = C ∫ ρ 5 / 3 ( r ) d r ,while the indeterminacy remained is smeared out in different approximation frames in which also the exchange energy is evaluated. Note that the kinetic energy is generally foreseen as having an intimate relation with the exchange energy since both are expressed in Hartree-Fock model as determinantal values of ρ(r₁,r′₁), see below.

Actually, the different LDA particular cases are derived by equating the total number of particle N with various realization of the integral (78) N = 1 2 ∫ ∫ | ρ ( r 1 , r ' 1 ) | 2 d r 1 d r 1 'by rewritting it within the inter-particle coordinates frame: (79) r = 0.5 ( r 1 + r 1 ' ) , s = r 1 − r 1 'as: (80) N = 1 2 ∬ | ρ ( r + s / 2 , r − s / 2 ) | 2 d r d sfollowed by spherical averaged expression: (81) N = 2 π   ∬ ρ 2 ( r ) Γ ( r , s ) d r s 2 d swith (82) Γ ( r , s ) = 1 − s β ( r ) + …

The option in choosing the Γ(r, s) series (82) so that to converge in the sense of charge particle integral (81) fixes the possible cases to be considered [68]:

the Gaussian resummation uses: (83) Γ ( r , s ) ≅ Γ G ( r , s ) = exp ( − s 2 β ( r ) )

In each of (83) and (84) cases the LDA-β function (76) is firstly replaced; then, the particle integral (81) is solved to give the constant C and then the respective kinetic energy density functional of (77) type is delivered; the results are [68]:

one recovers the Thomas-Fermi formula type that closely resembles the original TF (40) formulation.

In next one will consider the non-local functionals; this can be achieved through the gradient expansion in the case of slowly varying densities – that is assuming the expansion [52]: (87) T = ∫ d r [ τ ( ρ ↑ ) + τ ( ρ ↓ ) ]       = ∫ d r Σ m = 0 ∞ [ τ 2 m ( ρ ↑ ) + τ 2 m ( ρ ↓ ) ]   = ∫ d r Σ m = 0 ∞ τ 2 m ( ρ )   = ∫ d r τ ( ρ )      

The first two terms of the series respectively covers: the Thomas Fermi typical functional for the homogeneous gas (88) τ 0 ( ρ ) = 3 10 ( 6 π 2 ) 2 / 3 ρ 5 / 3and the Weizsäcker related first gradient correction: (89) τ 2 ( ρ ) = 1 9 τ W ( ρ ) = 1 72 | ∇ ρ | 2 ρ .

They both correctly behave in asymptotic limits: (90) τ ( ρ ) = { τ 0 ( ρ ) = τ 2 ( ρ ) … ∇ ρ ≪ ( far    from    nucleus ) 9 τ 2 ( ρ ) = τ W ( ρ ) = 1 8 | ∇ ρ | 2 ρ … ∇ ρ ≫ ( close    to    nucleus )

However, an interesting resummation of the kinetic density functional gradient expansion series (87) may be formulated in terms of the Padé-approximant model [71]: (91) τ ( ρ ) = τ o ( ρ ) P 4 , 3 ( x )with (92) P 4 , 3 ( x ) = 1 + 0.95 x + a 2 x 2 + a 3 x 3 + 9 b 3 x 4 1 − 0.05 x + b 2 x 2 + b 3 x 3and where the x-variable is given by (93) x = τ 2 ( ρ ) τ 0 ( ρ ) = 5 108 1 ( 6 π 2 ) 2 / 3 | ∇ ρ | 2 ρ 8 / 3 ,while the parameters a₂, a₃, b₂, and b₃ are determined by fitting them to reproduce Hartree-Fock kinetic energies of He, Ne, Ar, and Kr atoms, respectively [72]. Note that Padé function (92) may be regarded as a sort of generalized ELF susceptible to be further used in bonding characterizations.

Starting from the Hartree-Fock framework of exchange energy definition in terms of density matrix [73], (94) K [ ρ ] = - 1 4 ∬ | ρ ( r 1 , r ' 1 ) | 2 | r 1 − r ' 1 | d r 1 d r 1 ' ,within the same consideration as before, we get that the spherical averaged exchange density functional (95) K = π ∬ ρ 2 ( r ) Γ ( r , s ) d r s d stakes the particular forms [68]:

Alternatively, by paralleling the kinetic density functional previous developments the gradient expansion for the exchange energy may be regarded as the density dependent series [74]: (98) K = Σ n = 0 ∞ K 2 n ( ρ ) = ∫ d r Σ n = 0 ∞ k 2 n ( ρ )   = ∫ d r k ( ρ )while the first term reproduces the Dirac LDA term [75, 76]: (99) k 0 ( ρ ) = − 3 2 ( 3 4 π ) 1 / 3 ρ 4 / 3and the second term contains the density gradient correction, with the Becke proposed approximation [77]: (100) k 2 ( ρ ) = − b | ∇ ρ | 2 ρ 4 / 3 ( 1 + d | ∇ ρ | 2 ρ 8 / 3 ) awhere the parameters b and d are determined by fitting the k₀+k₂ exchange energy to reproduce Hartree-Fock counterpart energy of He, Ne, Ar, and Kr atoms, and where for the a exponent either 1.0 or 4/5 value furnishes excellent results. However, worth noting that when analyzing the asymptotic exchange energy behavior, we get in small gradient limit [77]: (101) k ( ρ ) → ∇ ρ ≪ k 0 ( ρ ) − 7 432 π ( 6 π 2 ) 1 / 3 | ∇ ρ | 2 ρ 4 / 3whereas the adequate large-gradient limit is obtained by considering an arbitrary damping function as multiplying the short-range behavior of the exchange-hole density, with the result: (102) k ( ρ ) → ∇ ρ ≫ c ρ 4 / 5 | ∇ ρ | 2 / 5where the constant c depends of the damping function choice.

Next, the Padé-resummation model of the exchange energy prescribes the compact form [74]: (103) k ( ρ ) = 10 9 k 0 ( ρ ) P 4 , 3 ( x )with the same Padé-function (92) as previously involved when dealing with the kinetic functional resummation. Note that when x=0, one directly obtains the Ghosh-Parr functional [78]: (104) k ( ρ ) = 10 9 k 0 ( ρ )     .

Moreover, the asymptotic behavior of Padé exchange functional (103) leaves with the convergent limits: (105) k ( ρ ) = { 10 9 ( k 0 + 15 17 7 432 π ( 6 π 2 ) 1 / 3 | ∇ ρ | 2 ρ 4 / 3 ) …       x → 0   ( SMALL    GRADIENTS )   − 12 π ρ 2 | ∇ ρ | 2 …     x → ∞     ( LARGE    GRADIENTS )  

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 two-component density functional [79]: (106) K [ ρ ] = C ( N ) ∫ ρ ( r ) 4 / 3 d r + D ( N ) ∫ r 2 | ∇ ρ | 2 ρ 2 / 3 d r ,where the N-dependency is assumed to behave like: (107) C ( N ) = C 1 + C 2 N 2 / 3 ,     D ( N ) = D 2 N 2 / 3while the introduced parameters C₁, C₂, and D₂ were fond with the exact values [80–82]: (108) C 1 = − 3 4 π 1 / 3 ,     C 2 = − 3 4 π 1 / 3 [ 1 − ( 3 π 2 ) 1 / 3 ] ,     D 2 = π 1 / 3 729     .

Worth observing that the exchange Bartolotti functional (106) has some important phenomenological features: it scales like potential energy, fulfills the non-locality behavior through the powers of the electron and powers of the gradient of the density, while the atomic cusp condition is preserved [83].

However, density functional exchange-energy approximation with correct asymptotic (long range) behavior, i.e. satisfying the limits for the density (109) lim r → ∞   ρ σ = exp ( − a σ r )and for the Coulomb potential of the exchange charge, or Fermi hole density at the reference point r (110) lim r → ∞ U X σ = − 1 r ,     σ = α ( o r ↑ ) ,   β ( o r ↓ ) … s p i n     s t a t e sin the total exchange energy (111) K [ ρ ] = 1 2 ∑ σ ∫ ρ σ U X σ d r   ,was given by Becke via employing the so called semiempirical (SE) modified gradient-corrected functional [77]: (112) K S E = K 0 − β ∑ σ ∫ ρ σ 4 / 3 x σ 2 ( r ) 1 + γ x σ 2 ( r ) d r   ,     K 0 = ∫ d r k 0 [ ρ ( r ) ] ,       x σ ( r ) = | ∇ ρ σ ( r ) | ρ σ 4 / 3 ( r )to the working single-parameter dependent one [84]: (113) K B 88 = K 0 − β ∑ σ ∫ ρ σ 4 / 3 ( r ) x σ 2 ( r ) 1 + 6 β x σ ( r ) sinh − 1 x σ ( r ) d rwhere the value β = 00042[a.u.] was found as the best fit among the noble gases (He to Rn atoms) exchange energies; the constant a_σ is related to the ionization potential of the system.

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 [85]: (114) K [ ρ ] = ∑ σ ∫ ρ σ ( r ) k [ ρ σ ( r ) ] g [ x σ ( r ) ] d r

Where the typical components are identified as: (115) k [ ρ ] = − A X ρ 1 / 3 ,     A X = 3 2 ( 3 4 π ) 1 / 3 ,while the gradient containing correction g(x) is to be determined.

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: (116) g 0 ( x ) = 1 2 U X ( r ( x ) )   k [ ρ ( r ( x ) ) ]  with r(x) following from x(r) by (not unique) inversion.

Unfortunately, the above “integrity” condition for exchange integrals to be equal is not also necessary, since any additional gradient correction (117) g ( x ) = g 0 ( x ) + Δ g ( x )fulfills the same constraint if it is chosen so that (118) ∫ ρ 4 / 3 ( r ) Δ g ( x ( r ) ) d r = 0or, with the general form: (119) Δ g ( x ) = f ( x ) − ∫ ρ 4 / 3 ( r ) f ( x ( r ) ) d r ∫ ρ 4 / 3 ( r ) d rbeing f(x) an arbitrary function.

Nonetheless, if, for instance, the function f(x) is specialized so that (120) f ( x ) = − g 0 ( x )the gradient correcting function (117) becomes: (121) g ( x ) = − 1 2 A X ∫ ρ ( r ) U X ( r ) d r ∫ ρ 4 / 3 ( r ) d r ≡ α Xrecovering the Slater’s famous X_α method for exchange energy evaluations [86, 87]: (122) K [ ρ ] = − α X A x ∫ ρ 4 / 3 ( r ) d r .

Nevertheless, the different values of the multiplication factor α_X in (122) can explain the various forms of exchange energy coefficients and forms above. Moreover, following this conceptual line the above Becke’88 functional (113) can be further rearranged in a so called Xα-Becke88 form [88]: (123) K X B 88 = α X B ∑ σ ∫ ρ σ 4 / 3 ( r ) [ 2 1 / 3 + x σ 2 ( r ) 1 + 6 β X B x σ ( r ) sinh − 1 x σ ( r ) ] d rwhere the parameters α_XB and β_XB are to be determined, as usually, throughout atomic fitting; it may lead with a new workable valuable density functional in exchange family.

The first and immediate definition of energy correlation may be given by the difference between the exact and Hartree-Fock (HF) total energy of a poly-electronic system [89]: (124) E c [ ρ ] = E [ ρ ] − E H F [ ρ ]

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 non-interacting (λ = 0) states of the electronic systems [90]: (125) E c λ [ ρ ] = 〈 ψ λ | ( T ^ + λ V ^ e e ) | ψ λ 〉 − 〈 ψ λ = 0 | ( T ^ + λ V ^ e e ) | ψ λ = 0 〉 .

In this context, taking the variation of the correlation energy (125) respecting the coupling parameter λ [91, 92], (126) λ ∂ E c λ [ ρ ] ∂ λ = E c λ [ ρ ] + ∫ ρ ( r ) r ⋅ ∇ δ E c λ [ ρ ] δ ρ ( r ) d r ,by employing it through the functional differentiation with respecting the electronic density, (127) λ ∂ V c λ [ ρ ] ∂ λ − V c λ [ ρ ] = r ⋅ ∇ V c λ + ∫ ρ ( r 1 ) r 1 ⋅ ∇ 1 δ 2 E c λ [ ρ ] δ ρ ( r ) δ ρ ( r 1 ) d r 1 ,one obtains the equation to be solved for correlation potential V c λ = δ E c λ [ ρ ] / δ ρ; then the correlation energy is yielded by back integration: (128) E c λ [ ρ ] = ∫ V c λ ( r , [ ρ ] ) ρ ( r ) d rfrom where the full correlation energy is reached out by finally setting λ = 1.

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 [90], the equation to be solved simply becomes: (129) λ ∂ V c λ [ ρ ] ∂ λ − V c λ [ ρ ] = r ∇ V c λthat can really be solved out with the solution: (130) V c λ = A p λ p + 1 r pwith the integration constants A_p and p.

However, since the equation (129) is a homogeneous differential one, the linear combination of solutions gives a solution as well. This way, the general form of correlation potential looks like: (131) V c λ = ∑ p A p λ p + 1 r p .

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), (132) V c λ = ∑ p 1 A p 1 λ p 1 + 1 r p 1 + ∑ p 2 A p 2 λ 2 p 2 + 1 r p 2 〈 r p 2 ρ 〉 .

By mathematical induction, when going to higher orders the K-truncated solution is iteratively founded as: (133) V c λ = ∑ p ∑ k = 1 K A p k λ p k + 1 r p 〈 r p ρ 〉 k − 1producing the λ-related correlation functional: (134) E c λ [ ρ ] = ∑ p ∑ k = 1 K 1 k A p k λ p k + 1 〈 r p ρ 〉 kand the associate full correlation energy functional (λ=1) expression: (135) E c [ ρ ] = ∑ p ∑ k = 1 K 1 k A p k 〈 r p ρ 〉 k .

As an observation, the correlation energy (135) supports also the immediate not spherically (molecular) generalization [90]: (136) E c [ ρ ] = ∑ l m n ∑ k = 1 K 1 k A l m n k 〈 x 1 x m x n ρ 〉 k .

Nevertheless, for atomic systems, the simplest specialization of the relation (135) involves the simplest density moments 〈ρ〉 = N and 〈rρ〉 that gives:

Unfortunately, universal atomic values for the correlation constants A_c0 and A_c1 in (137) are not possible; they have to be related with the atomic number Z that on its turn can be seen as functional of density as well. Therefore, with the settings

the fitting of (137) with the HF related correlation energy (124) reveals the atomic-working correlation energy with the form [90]: (139) E c = − 0.16569 N ln Z + 0.0000401 Z 〈 r ρ 〉

The last formula is circumvented to the high-density total correlation density approaches rooting at their turn on the Thomas-Fermi atomic theory. Very interesting, the relation (139) may be seen as an atomic reflection of the (solid state) high density regime ( r_s <1) given by Perdew et al. [75, 80, 93–96]: (140) E c P Z ∞ [ ρ ] = ∫ d r ρ ( r ) ( − 0.048 − 0.0116 r s + 0.0311 ln r s + 0.0020 r s ln r s )in terms of the dimensionless ratio (141) r s = r 0 a 0between the Wigner-Seitz radius r₀ = (3/4πρ)^1/3 , see relation (52), and the first Bohr radius a₀ = ħ² / me².

Instead, within the low density regime ( r_s ≥ 1) the first approximation for correlation energy goes back to the Wigner jellium model of electronic fluid in solids thus providing the LDA form [97, 98]: (142) E c W − L D A [ ρ ] = ∫ ε c [ ρ ( r ) ] ρ ( r ) d r ,where (143) ε c [ ρ ( r ) ] = − 0.44 7.8 + r sis the correlation energy per particle of the homogeneous electron gas with density ρ, see relations (55) and (56).