Application of the Green Function Method for Calculating the Spatial Distribution of Electrons in the Finite System

The paper presents a numerical implementation of the Green’s function method for calculating the spatial distribution of the macroscopic number of free electrons in a finite system. The obtained results are compared with the results obtained by direct summation of the wave functions. The quantitative coincidence is demonstrated.


Introduction
In [1], [2], using several methodological approaches (an analytical method -the method of quasiclassical Green's functions, direct numerical summation of exact solutions for electronic wave functions), the spatial distribution of free electrons in the ground state in an infinite spherical well was obtained. It was shown that the spatial distribution of electrons has a large spatial scale, which is of the order of the size of the system and is much larger than another spatial scale -the Fermi length of an electron. It is interesting to check the accuracy of the numerical technique used in [1], [2] For this purpose, we present another numerical method for calculating the electron concentration based on the Green's function method. The obtained results are in a good quantitative agreement with the results obtained in [1], [2].

Construction of a Green function in the terms of two linearly independent solutions
A Green function of the Schrodinger equation is defined as a solution to the non-uniform equation ( This solution should satisfy boundary conditions specified by the quantum-mechanical problem considered. In the case of the spherically-symmetric potential ( ) Vr it is convenient to represent the Green function in the form of the partial-wave expansion [3] ( The common multiplier in Eq. (2) is chosen for convenience. Substituting this expansion into Eq. (1), we arrive to the following equation for the partial Green functions  quantum-mechanical problems. Thus, it terns out that a finite set of terms is actually involved in summation in expansion (2). The model potential (1) in the form of infinitely deep spherical well is adopted in the present study. The requirement of boundedness of a solution to Eq. (1) leads to the zero Dirichlet boundary condition at the origin for a solution to radial equation (3). The next boundary condition has the similar form and is imposed at the wall of the well.
The desired radial Green function The first solution ( ) 1 r  satisfies obviously condition (4). As concerning the second one The partial green function ( ) ,, l g r r k  can be represented [3] in the following form Eqs. (6)-(10) allow us to obtain the explicit expression for ( ) kr j kr kr j kr kr j kr kr y kr k j kR This expression is applied here for numerical calculations.

Spatial density distribution: expression involving Green function
Let us consider the following model. Fermions occupy all the states below the Fermi energy F E in the single-particle scheme. The density distribution can be rather simply related with the Green function. The most simple case is the problem with pure discrete spectrum. As it is clear from the Green's function spectral decomposition the spatial density is connected with an integral of function ( ) ,, GE rr  in the complex E -plane over a closed contour, that encloses all the poles related to the occupied states. In accordance with Cauchy's integral formula this integral is equal to ( ) Here the index j includes the whole set of quantum numbers indexing the states (11). The numerical calculations can be based on the above mentioned contour integration. It is worth to mention the following methodical aspect. Representation (2), (9) also allows to establish rather simply the similar conclusion. The eigenfunctions satisfy both the boundary conditions (4), (5). It means the coincidence of the solutions ( ) The consideration of the concrete model problem presented above allows one to trace clearly the interrelation of the two types of Green function representations --spectral decomposition and by means of two linearly independent solutions. The aim of the presented consideration includes not only the methodical aspect but also comprise practical recipe for calculation of the spatial density. We shall discuss some aspects, connected with numerical procedures realized here. Let us pay attention to the fact that it is sufficient to account for the pole term (14) of the Green function only. Contour integration in the complex E -plane can be transformed to contour integration in the complex k -plane, which occurs more convenient for numerical calculations.   Using the described technique, the electron concentration was calculated for two systems. These results are shown in Fig. 2 (blue curves). We also show similar calculations obtained by direct summation of wave functions (red curves). The quantitative coincidence is demonstrated. It indicates a good accuracy of the developed numerical methods for calculating the electron concentration.

Conclusion
The paper presents a computational method for calculating the spatial distribution of free electrons in the ground state in a spherical indefinite well. The method is based on the numerical calculation of the exact Green's function of the system, which is used to determine the electron concentration. The results obtained by the developed numerical technique are in a good numerical agreement with the results obtained earlier by direct summation of the wave functions. This result indicates a good accuracy of the developed numerical methods for calculating the electron concentration.