Examination of Minimizer of Fermi Energy in Notions of Sobolev Spaces

This study examines the well-known Thomas-Fermi equation as a Euler-Lagrange equation associated with the Fermi energy. The first integral of Thomas-Fermi equation and the behaviour of the solution near the saddle point of the equation has been determined. Then, drawing upon advanced ingredients of Sobolev spaces and weak solutions, an exact methodology is presented for the quantum correction near the origin of Thomas-Fermi equation. By this approach, the existence and uniqueness of the minimizer for the energy functional of the Thomas-Fermi equation have been proved. It has been demonstrated that by the definition of such a functional and the relevant Sobolev spaces, the Thomas-Fermi equation, particularly of a neutral atom, extends to the nonlinear Poisson equation. Accordingly, weak solutions for the more general Euler-Lagrange equation with more singularities are proposed.


INTRODUCTION
Thomas-Fermi equation as a special case of nonlinear Poisson equations arises from a statistical model of many-electron atoms.Physical notions of Thomas-Fermi equation result to local Fermi momentum, Fermi sphere, Thomas-Fermi energy density and finally, Thomas-Fermi model applied to ions (Lieb and Simon, 1977;Lieb, 2000;Schwabl, 2007).This physical approach gives again the self-consistent Thomas-Fermi equation.In this manner, an energy functional extremization yield Thomas-Fermi equation can be derived.
Near the origin, however, (Columb) potential is singular and the Thomas-Fermi energy is no longer reliable for the large nuclear charges and it is the main problem of this study.On the other hand, one special Euler-Lagrange equation as a minimizer of energy functional gives a nonlinear Poisson equation which is an extended version of Thomas-Fermi equation.This fact and in the sequel of applying the differential technique for the analytic solution to the Thomas-Fermi equation (Pearson and Richardson, 1983;Liao, 2003;Fatoorehchi and Abolghasemi, 2014;Hasan-Zadeh and Fatootehchi, 2017) encourage to refer the analytic approach.
Many of solutions of this equation consist of the approximate or numerical solutions of it (Parker, 1988;Zaitsev et al., 2004;Turkyilmazoglu, 2012) for example refer to Baaquie (1997), Bahuguna et al. (2002) and Fatoorehchi and Abolghasemi (2014).This problem for natural atoms has been examined by integral curves of an infinitesimal version of the corresponding differential equation (Baaquie, 1997;Lieb, 2000;Hagen, 2009) which we collect all of them in Theorem 1.In effect, some results about the procedure have been defined as the integration of the problem.
For quantum correction near the origin of Thomas-Fermi equation, in Theorem 2, the minimizer of the energy functional will be found by some notions of Sobolev spaces.Also, this minimizer is a solution of a boundary-value problem for the Euler-Lagrange equation associated with the Fermi energy functional which satisfies in the condition for the existence of the solution in the weak sense.
This approach is a motivation for the definition of weak solution which can be applied to the general Euler-Lagrange equation with more singularities, sometimes awkward.Also, the structure of the proof of the Theorem 2 can be extended to the Euclidean space ℝ " and then Euler-Lagrange partial differential equation for no smooth functions with the singularities on the sets with the nonzero measure.

APPROXIMATE EXAMINATION OF THOMAS-FERMI EQUATION
This study was conducted in Fouman Faculty of Engineering, College of Engineering, University of Tehran, Iran.

Preliminaries of thomas-fermi equation:
As mentioned in the works of Pearson and Richardson (1983) and Schwabl (2007) the Thomas-Fermi equation is a spherically symmetric version of the Poisson equation for the electric potential # $ % outside the nucleus of a many-electron atom: here  ( =  − , − # % is potential,  is the energy of the most energetic electrons,  is electronic charge and − is charge density.By statistical considerations, the Eq. ( 1) becomes a nonlinear ordinary differential equation: (2) The solution of ( 2) is sought for  > 0, but as  → 0, the potential of the concentrated source (nucleus) at the origin is  → − C% 4 2 , where  is the atomic number.For neutral free atoms, a boundary condition at infinitely is also defined.The surface of the atom corresponds to  → ∞ where  → 0 ( = 0).No net charge demands;  ( → 0 as  → ∞.Thus, for neutral atoms the problem is: Of Eq. (3) the differential equation for the infinitesimal difference  =  is: where, (∞) = 0. We can show that if  (0) > 0, then  () ≥ 0 for all .
Theorem 1: The first integral of Thomas-Fermi Eq. (3) with boundary conditions (4) and (5), also the path which satisfies the boundary condition (5) and the behaviour of the solution near the saddle point of the equation can be determined.In effect, it can be given an algorithm which defines the integration of the problem (3) with boundary conditions ( 4) and ( 5). .The first order equation to be studied is: And the mapping to  along an integral curve of ( 7) is: The general solution of the first order differential Eq. ( 6) is a one-parameter family of curves (, ) = .Only one curve of this family corresponds to the family  () of positive solutions of the Thomas-Fermi Eq. ( 6) for which  (∞) = 0; for any one of these curves  () is uniquely determined by its value  (0) at  = 0.For the desired curve in the (, ) -plane, it can be concentrated on the fourth quadrant, shown in Fig. 1.
The isoclines of zero slope 4 +  < 4 = 0 are drawn as well as some representative paths.There is one singular point  of interest where 4 `+  < 4 = 0, 4 `+ 3 `= 0, or  `= 144 and  `= −432.The locus of zero slope  a is the curve 4 +  < 4 = 0 and the locus of infinite slope  b is the line 3 +  = 0 (the transverse line in Fig. 1).
The intersections of these loci are the singular points of the differential equation.They are a node at the origin O and a saddle point at  (144, −432).In the lenticular region between these two loci, the slope  The behaviour near the origin can be obtained from the local form of (7).Many paths run into the origin between the isoclines and (it turns out) that on these paths ≫  ≫  < 4 , so that ( 7) is approximated by where,  a to be determined.Integrating of the mapping formula (8), 9W result in  =  a  9 + ⋯ =  9 .Thus, the origin of (, ) corresponds to  = 0 and the boundary condition (4) determines the constant of integration in the mapping back to , namely  a = 1.The only path which has a chance to satisfy the boundary condition at infinity is the exceptional path running from the origin to the saddle point at .For the study of the behaviour as the solution approaches the saddle point along this path, let  =  `+  * ,  =  `+  * .So of (9), The final algorithm for a definition of integration of the problem (3) with boundary conditions ( 4) and ( 5) has been depicted in Fig. 2.

PHYSICAL NOTIONS OF THOMAS-FERMI EQUATION
As the notations of the works of Liao or Hagen (Liao, 2003;Hagen, 2009), if an atom has a large nuclear charge , most of the electrons move in orbits with large quantum numbers.
Filling up all negative energy states with electrons of both spin directions produces some local particle density  ( ⃗) calculated from the classical local density For neutral atoms, the Fermi energy is zero and the density (11) will be recovered.By occupying each state of negative energy twice, the classical electron density is: The potential energy density associated with the levels of negative energy is: Integrating the paths along the exceptional path toward the origin.Determining the trajectory.
To find the kinetic energy density should be integrated: To make sure that the nuclear is not changed by the electrons, where has been obtained in Eq. ( 12) with boundary conditions ( 13) and ( 14).
All length scales of the electrons will now be specified in units of  "Ž , i.e.,  =  "Ž .In these units, the electron density ( 13 So that will be obtained the self-consistent Thomas-Fermi equation: The condition  > 0 excludes the nuclear charge from the equation, whose correct size is incorporated by the initial condition ( 16).Near the origin, the Eq. ( 17) starts out like () = 1 −  + ⋯, with a slope ~1.58807.For large, it goes to zero like ()~( 88 < .This power falls off is a weakness of the model since the true screened potential should fall off exponentially fast.The right-hand side by itself happens to be an exact solution of (17) but does not satisfy the desired boundary condition ( 16).

IMPROVEMENTS IN THE QUANTUM EFFECTS NEAR THE SINGULARITY
Statement of the problem: The Thomas-Fermi energy with exchanges corrections which obtained in above Section would be reliable for large Z only if the potential was smooth so that the semiclassical approximation is applicable.
Near the origin, however, the Coulomb potential is singular and this condition is no longer satisfied.Some more calculational efforts are necessary to account for the quantum effects near the singularity, based on the other observation (Baaquie, 1997).This problem can be solved in Theorem 1 by analytic ingredients, especially weak solutions.

METHODOLOGY
Weak solution: For approximate approaches, the Fermi energy and the Poisson Eq. ( 15) which result in the Thomas-Fermi Eq. ( 17) will be reviewed in the notions of functional analysis.For the quantum correction near the origin, in general, suppose  ⊆ (0, ∞) is a bounded, open interval and : ℝ × ℝ ×  ¬ → ℝ;  = (, , ) is a smooth Lagrangian.Also, it should be assumed that the function [.] have the explicit form: for smooth functions :  ¬ → ℝ satisfying the boundary condition  =  on .Also, suppose some particular smooth function , satisfying the requisite boundary condition  =  on , happens to be a minimizer of  [. ].
In this way, the nonlinear ODE, i.e., the Euler-Lagrange equation associated with the energy functional [.] defined by ( 18) can be solved by : So, on the contrary, it can be tried to find a solution of (19) by searching for minimizers of (18).
Let the Sobolev space  l,`( ) consists of all weak derivations of the locally summable functions :  → ℝ such that for each multi-index  with || ≤  and for all  (Bahuguna et al., 2002;Maz'ya, 2008).
Theorem 2: Singularity near the origin of Thomas-Fermi equation (and then singularity of the Coulomb potential) as a Euler-Lagrange equation associated with the Fermi energy can be improved in the sense of nonsmooth potential.
Proof: In fact, it will be shown that the weak solution of the Lagrangian which is a minimizer of the Euler-Lagrange equation is the key to this enigma.
But then since (. ) has a minimum for  = 0,  X (0) = 0; and  ∈  = { ∈  (,À ()| =     ℎ  } is a solution of the boundary-value problem: In view of ( 23) the second term on the right is zero and therefore [] ≤  [𝜔] for each  ∈ .Then this solution is unique, too.

CONCLUSION
In this study, the first integral of Thomas-Fermi equation and the behaviour of the solution near the saddle point of the equation was determined which result to an algorithm for the numerical procedure.Also, a novel method for the quantum correction near the origin presented.This consists of finding the (weak) solution of the Lagrangian associated with Thomas-Fermi equation which has the singularities, especially in the origin.Also, since the Lagrangian mapping corresponded to the Thomas-Fermi equation is convex then each weak solution is, in fact, a minimizer.
The main advantage of the proposed method is to offer a general solution to the problem that can be applied to the nonlinear Poisson equation.The proposed approach for Euler-Lagrange equations corresponding to multivariate functions can be used with a large set of singularities of that functions.

CONFLICT OF INTEREST
It should be noted that there is no financial support and there is no competitive interest in this area.

Fig. 1 :
Fig. 1: First integrals of thomas-fermi equation Proof: The Eq. (3) scales under stretching transformations as E F ~E< 4 to either side of this region 3[ 3W > 0. As usual, the singular point represents one exceptional solution  d of (3),  d = (88 F < .
`'""Ž(P) ( ⃗) = ( ⃗)(P) () paths in the , plane which also represent solutions of the Thomas-Fermi equation.Converting the original two-point boundary value problem to an initial value problem.Determining the trajectory   .Choosing point 144 − , −432 −  as the starting point.Integrating numerically in the direction curves entering the origin.Starting at the saddle point  `,  `.