Self-consistent field approach is needed. DFT with local
approximation does not work well for strongly correlated systems.
Approximation here: neglect the quasi-particle interactions, and use
one-electron Hamiltonian to describe the host metal conduction electrons
assume these conduction states to be characterized by DOS (of
one spin component) In the ground state these levels will be filled to . The partition function is
where the trace over all the states yields , and the additional exponential 2
comes from the spin degeneracy. Consider the Sommerfeld
expansion up to the third order, for the particle number we have , and The particle number doesn't change with temperature, which
yields Expand near
, For the internal energy we have , and the expression up
to the third order is consider the specific heat and insert the expression of and above, we have where and the coefficients of and might deviate from 1 and -1, depending on how the DOS
is treated numerically. On coupling to a magnetic field , the spin-dependent density of states
become hence the total magnetization is keep the weak field limit , we have therefore The total magnetization becomes By definition the magnetic susceptibility is Apply the Sommerfeld expansion up to second order, we have
where