A compact numerical approximate solution for Friedel-Anderson and Kondo problem

A Friedel Artificially Inserted Resonance (FAIR) state is introduced to approximate the ground state of the Friedel-Anderson and Kondo problem. The ground state of the Friedel-Anderson problem can be expressed by eight Slater states and that of the Kondo problem can be expressed by four Slater states. Each Slater state is composed of one d-electron or a FAIR state, and n s-electrons that are orthogonal to the FAIR state. From the rotation of the s-electrons basis in Hilbert space and the optimization of those Slater states coefficients, the ground state (singlet state) and the first excite state (triplet state) can be obtained simultaneously. The excitation energy between the singlet and the triplet state is treated as Kondo energy. The ground energy from FAIR method of the Friedel-Anderson problem is far lower than that from the Mean Field Theory solution. And the Kondo energy in the Kondo problem is related to the coupling constant J by the exponential factor e-12Jr0 .;The FAIR method is then extended from 1-channel to multi-channel problem. A compact approximate ground state of the multi-channel Friedel-Anderson or Kondo problem is constructed from the product of 1-channel ground state of all the individual channels. The Parmenter's Anderson model, which considered the Coulomb interaction, Exchange interaction and rotationally invariant in both the spin space and the real space, is applied in the calculation. The FAIR solution shows much better approximation to the ground state than the Mean Field approach does. The FAIR method is also applied on the multi-channel Kondo impurity problem. The multi-channel Kondo impurity ground state, which satisfies the Hund's Rule requirement, is constructed. From the discussion of the magnetic anisotropy of the Kondo impurity, the emergence of the Kondo resonance for large-spin atoms (S>1/2) impurity is clearly related to the spin flip transition process between DeltaSZ = 1 degenerate Slater states. The sign and magnitude of the magnetic anisotropy play essential roles in the formation of the Kondo resonance.