There are two schools of thought on how to achieve the minimization of the total energy. The more prevalent approach in the physics community is to view the calculation directly as a problem in numerical minimization and to apply modern techniques for constrained numerical minimization. We shall return to this approach in the second half of the this course. The second school of thought, more prevalent in the chemistry community, is to derive the Lagrange-multiplier equations for constrained minimization and to then use numerical methods to solve the resulting equations. As we shall see, each approach has its advantages and disadvantages. In the end, though, both must lead to the same result.
We now derive Lagrange-multiplier equations for density functional theory, known as the Kohn-Sham equations.