Basic density functional theory

Equation 1.34 is an eigenvalue problem. What is generally done is to expand [tex]\psi[/tex] in some finite basis, then you have a matrix eigenvalue problem for the coefficients of the basis function and the energy eigenvalue. Once you have this matrix you can pass it along to any diagonalization routine that can handle Hermitian matrices with complex entries, and that will produce both the eigenvalue and the eigenvectors. There are a variety of such methods which produce both the eigenvalues and eigenvectors at once, without prior knowledge of either.

(As a practical matter, if you have an extremely large basis set, such as with plane waves, then you need to use tricks based on the fact that the Hamiltonian matrix will be sparse and you are only interested in some number of the lowest eigenstates.)