Variation Principle and Chemical Potential of TF Theory

It is of considerable importance to note that the density-potential relationship (3) of the TF theory follows from a variational principle for the total energy. To see this, we note first that the classical electrostatic potential energy U consists of the sum of two terms in an atomic ion, the electron-nuclear potential energy Ken and the electron-electron potential energy Kee. We can write 

To obtain the total energy E(Z, N) of a positive atomic ion we must simply add the kinetic energy T to U in equation (15). 

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