On the Non-Orthogonality Difficulty in Ferromagnetism

In a many-electron system, one is faced to some difficulties when one-electron wave functions are not orthogonal to each other. For example, the inner product (\(\), \(\)) (\(\) is a Slater determinant made from these non-orthogonal wave functions) loses its meaning as the number of electrons becomes large. We avoid this difficulty by using the expansion coefficients instead of the inner products, in order to define the matrix elements of operators. Applying this method to a magnetic problem, in which there are some doubts about the validity of the Heisenberg model \( H = -2 ∑i>j Ji,jSᵢ · Sⱼ \), we can derive the energy of the ground state and the energy spectrum of spin waves. The results are written as