An approach to local electron energy in atoms through Rényi’s entropy

In this work, it is presented an approach to the local electron energy of the first thirty-six atoms in their ground state using Rényi’s entropy is defined in terms of the electron density in position and momentum spaces. To carry out our study, we calculated and compared the trends obtained from the local functionals of kinetic, exchange, and Coulomb energy, with Rényi’s entropy. Our results suggest that Rényi’s entropy could be a good option to perform studies on the local behavior of the energy in atoms and also in studies on the electron correlation phenomenon.


Introduction and theoretical background
At the end of the 1940s, Shannon published a paper entitled 'A mathematical theory of communication' [1], in this work Shannon tried to aboard the complex phenomenon of the transmission of information among several devices, in his works one can read the fundamentals of the process of information transfer, in these manuscripts, an important aspect that caps our attention is the fact that Shanon proposed that the amount of information that a system can receive or transmit is linked to the entropy, which has the following explicit form, where p(x), is a continuous probabilistic distribution, that is subject to p(x)dx = 1, 0 < p(x) ≤ 1 and being p(x) a probabilistic distribution, clearly it is dimensionless, Eq. (1) also fulfills that S ≥ 0. An interesting aspect of Eq. (1) is that this equation B N. Flores-Gallegos nelson.flores@academicos.udg.mx 1 Centro Universitario de los Valles, Universidad de Guadalajara, Carretera Guadalajara -Ameca Km. 45.5, C.P. 46600 Ameca, Jalisco, Mexico has the same form as the definition of entropy given by Gibbs and Boltzmann, therefore we can establish a direct link between the informational entropy, Eq. (1), and thermodynamic entropy as, on the other hand, and, information theory since its publication, this model has been applied in several areas of science and technology that at the beginning they have not a clear relationship with Shannon's model; nevertheless, currently, information theory is applied in the computer sciences, finances, physics, chemistry, genetics, astrophysics, among others. In this regard, perhaps the concept most applied in the areas mentioned before is the informational entropy, Eq. (1), and since the 1960s one of the main proposes of several mathematicians was to try to find a generalization of it, in this regard, perhaps was Rényi who presented the first proposal, [2], in this equation, p i is a probabilistic distribution subject to n i=1 p i = 1, 0 < p i ≤ 1, α = 1, and R α ≥ 0. Eq. (3) also can be written in terms of a continuous probabilistic distribution, as follows, where, p(x)dx = 1, 0 < p(x) ≤ 1, α = 1 and R α ≥ 0. Equations (3) and (4) can be reduced to Shannon's entropy when, and Rényi's entropy has been amply used in the last two decades in the fields of the atomic and molecular physics [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. In this regard, to implement Rényi's entropy into the field of theoretical chemistry, it is necessary to re-express Eqs. (3) and (4) in terms of the occupation numbers or in terms of the electron densities respectively, for example, Eq. (4) can be rewritten using the electron density in position space, ρ(r), as, where ρ(r) is subject to ρ(r)dr = N and N is the electron number of the system; nevertheless, if one examines Eq. (7) with care, we have the following problems, 1. Being the natural logarithm a transcendental function, it cannot act over a function with units, in this case, the electron density has units of the number of particles by an element of volume, N/V . 2. Electron density can take values in the range of 0 < ρ(r) ≤ ρ max , where ρ max is the the greatest value of the electron density, for example, if we consider an atom, the maximum value of the electron density will be located at the nucleus. 3. Based on the last point, then R α 0, which violates the entropy's general properties. 4. If one analyze the local profile of R α , we will find zones in where R α ≤ 0.
Considering the previous points, a solution to these problems is to rewrite the electron density, ρ(r) as, where ρ max is the maximum value of the electron density and has units of N/V therefore the term ρ(r)/ρ max is dimensionless, in this case we have that the constant k ρ max has units of volume because of the term dr, thus we can define, this equation is dimensionless and ρ(r) ρ max ∈ (0, 1], with this consideration, then we can rewrite Eq. (7) as, this equation can be approximated to any local density functional, for example, if we do α = 5/3, Eq. (7) takes the following form, in where the argument of the natural logarithm can be related to Thomas-Fermi's kinetic energy functional [21,22], defined as, in the same way, we can do α = 4/3 to get, in this case, we can associate the integral to Fermi-Dirac's exchange functional [23] defined as, or to Parr's Coulomb energy functional [24][25][26] that is defined in the following form, in this context, we can write the following expression for the local electron energy, which is the sum of Eqs. (12), (14) and (15). Consequently, we can write an equation for the local electron energy of an atom or a molecule as, where c i is a proportionality constant that is specific for each kind of energy. In general, we note that each local functional has the following form, while Rényi's entropy has the general form, this last equation can be related to Eq. (18) as follows, based on this last equation, then we can write an approximation to the local electron energy as follows, using α = 4/3 and 5/3, we can write Eq. (21) as, however, according to Białynicki-Birula and Mycielski [27] the total entropy, must be the sum of the entropies in position and momentum spaces, for this reason, we define Rényi's entropy in momentum space as, where π max is the maximum value of the electron density, as in the case of Rényi's entropy defined in position space, Eq. (19); Eq. (23) is strictly positive overall space, and fulfills the dimensionless criterion. Thus considering Eqs. (19) and (23) we write the total Rényi's entropy as, Another interesting aspect that is worth addressing, is the process of choosing the value of α in Eqs. (19) and (23), in general, the selection is done based on our heuristic criterion; notwithstanding, we consider that worthwhile to analyze each component of these equations. In such expressions, we note that the terms that will dictate the general behavior of Rényi's entropy in position and momentum spaces are the kernel of the integrals of Eqs. (19) and (23), ρ(r) ρ max α and π(r) π max α , meanwhile the term 1 1−α acts as a proportionality constant, in this regard, in Fig. 1 we present the radial profile distribution of the Ne atom in its ground state, in where we showed the effect of α using values of 1/3, 2/3, 4/3, 5/3, 7/3, 8/3.
In Fig. 1a  has a maximum at R ≈ 2.8 a.u. then α = 1/3 could not be option, in Fig. 1b one can observe that with α = 2/3 the trend of ρ(r) ρ max 2/3 presents a change in its slope at R ≈ 0.5 a.u., and how in the previous figure, this trend has not a good structure.
In Fig. 1c and d we present the trends using α = 4/3 and α = 5/3 respectively, in these figures, we noted that trends of ρ(r)  obtained we can say that the α = 4/3, 5/3 are the exponents that provide the most valuable chemical information, actually the maximums observed are related to the shell structure of the Ne atom.

Results of Rényi's entropy applied to atoms
In this section, we present the general trends of Eqs. (19), (22), (23), and (24) when they are applied to the first 36 atoms in their ground state. The wave functions used to calculate the electron densities in position and momentum spaces were obtained with Gaussian 09 [28] using CISD(full) and the cc-pvTZ basis set, all electron densities were calculated with a precision of 10 −9 a.u. [29,30].
In Fig. 3a we present the results of ρ R α , Eq. (19), using α = 4/3 and 5/3, exponents that are related to the exchange and kinetic energy respectively, in both trends one can observe that they change in their curvature at Z = 10, 18 which corresponds to the change in the period of the periodic table, and as is expected both trends scale up according to the electron number of the atoms, this behavior is associated to the increasing of electron interactions. In Fig. 3b we present the trends of Eq. (24), in this case, the changes of both trends are notably different in comparison with the previous ones, and one can appreciate clearly that such changes correspond to the different periods of the periodic table, in addition, we can say that Rényi's entropy is considerably more sensible in momentum space than in position space, another characteristic of Eqs. (19) and (23) that we can note in these trends, is that ρ R α (r) < π R α (q), consequently will be Rényi's entropy in momentum space that govern the general behavior of total Rényi's entropy, Eq. (24), this can be observed in Fig. 3c, in this figure, we compared R T and electron energy, in general, we note that meanwhile, the entropy increases the electron energy decreases because of the maximum entropy principle; notwithstanding, comparing the trend of R T with the correlation energy, see Fig. 3d, we noted that each trend presents changes in, practically, a specular way, this suggests that R T could be used as a measure of electron correlation.
Electron correlation, is a phenomenon that is related to the interaction of several particles, in this case, electrons, and according to physics, the size of the system has to be related to how the particles will interact, for this reason in Fig. 4a we compare the trends of correlation energy and atomic radii, in this figure, we noted that the effects of the electron correlation increase according to the atomic radii.
In Fig. 4b, we compare the trend of R T and with the trend of the atomic radii, and as in the case of Fig. 3d, we observe that R T varies according to the atomic radii. This is an interesting result because, in general, not all descriptors used commonly in the field of theoretical chemistry can be related to several physical aspects of the systems, in this regard, we consider that Rényi's entropy, Eqs. (19), (23) and (24), defined in position and momentum spaces, can be useful tools to complement studies on the electron correlation phenomenon.
Finally, in Fig. 4c,d, we present the trends of the exponential Rényi's entropy, Eq. (20). In the first case, we depicted exponential Rényi's entropy in position space and compare it with local electron energy, Eq. (16), in such figure one can observe that both trends have the same structure, while in Fig. 4d, we compare the trends of the total exponential Rényi's entropy, Eq. (20), in position and momentum spaces, with the local electron energy, and we noted that there is no significant change in the general trend of exp(R T ) in comparison with exp( ρ R α ); therefore, we can conclude that it is

Conclusions
In this work, we rewrote Rényi's entropy in position and momentum spaces. The resultant expressions fulfill all properties of the entropy; we applied our equations to the first 36 atoms in their ground state. Rényi's entropy has a dependence on a parameter α, which can take, several values except 1, this can represent a problem because in general there is no way to choose it; nevertheless, in this work, we found that most proper values of α are 4/3 and 5/3, with this values we found that Rényi's entropy in momentum space can be related to the correlation energy, and also is linked to the variation of the atomic radii. Finally, we also found that exponential Rényi's entropy can be related to the local electron energy.