Fermi–Dirac entropy as a measure of electron interactions

In this work, we analyzed two simple chemical reactions using Fermi–Dirac’s entropy defined in terms of the Löwdin’s occupation numbers, this definition of entropy has not been applied or explored enough in the field of quantum chemistry, in this vein, Fermi–Dirac’s entropy maybe a good option to perform works in which the main purpose will be the study or analysis of the effect of the electron interactions. The results presented in this work were complemented by the analysis of the kinetic energy, potential energy, and the variation of the electric potential, along the reaction path, the results obtained, suggest that Fermi-Driac’s entropy presented in this work can be a useful tool to perform and complement studies about electron–electron interactions.


Introduction and theoretical background
In general, Fermi-Dirac's distribution is used to carry out studies on electrical conductivity, or in studies more focused on the field of physical statistics, solid state physics, or solar cells [1][2][3][4][5]; unfortunately, this distribution and entropy definition has not been applied enough in the field of quantum chemistry to describe atoms, molecules, or chemical reactions; however, recently some authors have explored this concept [6,7].
Fermi-Dirac's distribution is defined as, 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 where k B is the Boltzmann's constant, T is the absolute temperature, ε i is the energy of a single particle in the state i and μ is the chemical potential, this distribution is normalized and fulfills that, form distribution (1), it is possible to write the entropy associated with this distribution as,n it is interesting to note that the first term of Eq. (3) is similar to the well-known Shannon's entropy [8], which is a measure related to the amount of information that a system has, while the second term is related to the concept of information channel, a discussion on these concepts and their relationship to the chemical bond has been recently done [9]. Equation (3) also can be rewritten in terms of a probabilistic distribution, p(x) = {x i }, as where the set {x i } fulfills with n i=1 x i = 1 and x i ∈ (0, 1], in addition we have that Eq. (4) also fulfills with the condition S F D > 0, and it is strictly positive overall space, which is one of the main conditions of entropy.
On the other hand, in quantum chemistry, one can describe the systems, such as atoms or molecules, using descriptors such as the electron density or through the density matrix, in this last case, it is common to recur to the occupation numbers; currently, there are several kinds of occupation numbers, for example, Löwdin occupation numbers [10], λ i , which are molecular occupation numbers that are restricted to λ i ∈ (0, 2] because of Pauli's double occupation principle, and they fulfill with n i=1 λ i = N , where N is the electron number of the system; in the same line, we can choose the Natural Population Analysis (NPA) or the Natural Bond Order (NBO) [11], the NPAs are atomic occupation numbers, while the NBOs are molecular occupation numbers. Thus Eq. (4), can be rewritten in terms of occupation numbers as, where λ i = λ i /N and fulfills with the condition n i=1 λ i = 1, which are necessary conditions in order to preserve the general properties of entropy, otherwise it is possible to find cases such that λ S F D 0, which is a violation of one of the general properties of entropy.
In the following section, we present an analysis of the reactions H 2 O + H and NH 3 + OH, using Eq. (5) with the Löwdin occupation numbers.

Results of Fermi-Dirac's entropy in two chemical reactions
In this section, we present the results of Eq. (5) applied to two chemical reactions, (i) H 2 O +H and (ii) NH 3 +OH. The results of the first one, are presented in Sect. 2.1, and the results of the second reaction are presented in Sect. 2.2. The IRC calculations were performed with Gaussian 09 [12], and for each point of both IRC trajectories, we calculated the Löwdin occupation numbers which were used to carry out the calculations of Fermi-Dirac entropy.

Reaction: H 2 O + H
In this section, we present an analysis of the following reaction, to do our analysis, we perform an IRC calculation with 346 points of this reaction using the functional M11 and MG3S basis set In Figs. 1and 2, we present the general trends obtained in this first reaction, in all graphs we used the variable R X, which is the reaction coordinate defined by Fukui in the 1970s in the following way We tentatively define the reaction coordinate as a curve passing through the initial and the transition points and orthogonal to energy equipotential contour surface. A reaction coordinate, defined as such, may be called 'intrinsic reaction coordinate.' [13,14].
In Fig. 1a, is depicted the general trend of the electron energy, as is expected, this reaction is symmetric and has its maximum at R X = 0, in general, we can infer that there is not too much to say or interpret about this reaction considering only an energetic point of view; nevertheless, when this reaction is analyzed through Eq. (5), see Fig. 1b, one can find an interesting trend, in this figure, we note that this trend has two maximums at R X ≈ −0.25, 0.25 that may be considered an interesting zone, in this regard, in Fig. 1c we compare λ S F D with the Kinetic Energy, in this figure we can observe that both trends present changes at the same coordinates, that is, at R X ≈ −1, 1 both trends present a change in their slope, and at R X ≈ −0.75, 0.75, the trend of kinetic energy has two maximums while the trend of λ S F D exhibits a change in its slope, in addition, note that the maximums of the kinetic energy permit to define a zone in − 0.75 < R X < 0.75 in such zone, we observe that while the trend λ S F D has two maximums, the trend of kinetic energy has two minimums, this is because of the maximum entropy principle; therefore, we can say that Fermi-Dirac's entropy is capable to reveals a zone in which the process of bond-breaking and bond-forming is carried out, in the same line; in Fig. fig:3a, we compare the trends of λ S F D and the Potential energy, as in the previous case, the Potential Energy trend has two minimums at R X ≈ −0.75, 0.75 which define the same zone observed in the Kinetic Energy, and also one can observe, that meanwhile at R X = 0 the entropy is minimum, while Potential Energy is maximum, with this observations we can say that λ S F D is more related to the kinetic effects of this process.  Fig. 2, we compare the trend of λ S F D with the variations of the electric potential of the different atoms involved in this chemical reaction. In Fig. 2, we present several comparisons of λ S F D with the variation of the electric potential of the atoms of this reaction. In Fig. 2a, we compare the trend of λ S F D with the variation of the oxygen electric potential, in this figure, one can observe that the electric potential presents a change in its slope at the same R X values observed in the Kinetic Energy and Potential Energy, in Fig. 2b we compare the trend of Eq. (5) with the trend of the electric potential of the Hydrogen 1, which is the atom that intervenes in the process of bond-breaking and bond-forming of the chemical bond, in this trend, one can observe the same zone, −0.75 < R X < 0.75 observed in the previous trends, and also a reduced zone in −0.5 < R X < 0.5 bounded by two minimums, such zone defined by the electric Fig. 2 Comparison of λ S F D among the atomic electric potential of the different atoms involved in this reaction potential, ties with the change of the trend of λ S F D which increases quickly to reach its maximum at R X = 0 while in the case of the Hydrogen atom No. 2 its electric potential trend decreases in the same way and presents a clear minimum at R X = 0, see Fig. 2c, in the same way, the general trend of the Hydrogen No. 3, see Fig. 2d, has changes in its slope located in the same zone of R X observed in previous figures.

Reaction: NH 3 + OH
In this part of this work, we present an analysis of the reaction NH 3 + OH; in this case, we performed an IRC calculation with 201 points using the functional M11 and the MG3S basis set, and for each point of the trajectory, we calculated the Löwdin's occupation numbers using CISD/6-311++G**.
One of the main reasons to analyze this reaction is because is a non-symmetric reaction from an energetic point of view, in comparison with the previous reaction, which is symmetric, see Figs. 1a and 3a.  Figs. 3 and 4 , we present the general trends obtained for this reaction. In Fig.  3a, we present the general trends of the electron energy and correlation energy of this reaction, a remarkable point of view of these trends, is that the electron energy has its maximum at R X = 0, how is expected, while the correlation energy trend has a minimum at R X ∼ 0.18, which does not tie with the electron energy how one could expect, and at R X ≈ 0.6, the slope of this trend changes significantly; however, this last change ties reasonably with the electron energy trend. In this context, worthwhile to investigate the possible physical meaning of the minimum of the electron correlation energy, whit this in mind, in Fig. 3b, we compare the trend of λ S F D and the electron correlation trend, in this figure, we observe that the maximum and minimum observed in the trends of energy tie the maximum of λ S F D and R X = 0 and the minimum of the correlation energy ties with the maximum of λ S F D which is a noteworthy aspect of λ S F D .
In other to inquire about the physical meaning of the maximum and minimum observed in the trends of the electron energy and correlation energy, we compared the trend of λ S F D with the kinetic energy and potential energy, see Fig. 3c, d, in the first case, we can observe that trend of the kinetic energy has, essentially, the same trend of the correlation energy, both trends have a minimum at the same position, and such minimum match with the minimum of λ S F D located at R X = 0.18, this, physically implies that at this point is where the electron cloud is more diffuse and not as one could infer based on the trend of the electron energy, which has its maximum at R X = 0.
In Fig. 3d, we compared the potential energy trend with λ S F D , in this figure we can appreciate that at R X ≈ 0.18 the potential energy has a maximum, this can be interpreted as a zone in which the nuclear effects are predominant, it is interesting to note that this maximum encompasses a zone at 0 < R X < 0.18, that is a zone in which the Fermi-Dirac's entropy has a maximum and a minimum, based on this observation we can say that in this zone is where the process of bond-breaking and bond-forming starts.
This interpretation can be supported by the analysis of the trends of the electric potential of the Nitrogen and Oxygen atoms, in Fig. 4a and b, in these figures, one can observe that the electric potential, in the first figure, we observe that the electric potential of the nitrogen atom has a local minimum while λ S DF exhibits a global maximum; on the other hand, in Fig. 4b we can note that the trend of the electric potential of the oxygen atom has a minimum at R X ≈ 0.18 with matches with the minimum λ S F D at the same place, thus, based on the trends presented, we can ensure that the zones found in the λ S F D trend are related to the bond-breaking and bondforming of this chemical reaction, in addition, we can say that through λ S F D it is possible to identify the places in where the most important effects of the electronelectron interactions are carried out.

Conclusions
In this work, we used Fermi-Dirac's entropy defined in terms of the Löwdin occupation numbers, this entropy is used to describe particles as Fermions, such as electrons. In our work we used and applied this entropy to describe two chemical reactions, (i) H 2 O + H, and (ii) NH 3 + OH. In both cases, we showed that the entropy expression used in this work can describe zones where the most important physical and chemical changes linked to electron transfer are carried out.
Finally, based on our results, we infer that our proposal may be a useful tool to perform or complement studies on electron-electron interactions related to the bondbreaking and bond-forming processes.