## Variational principle

Putting everything together, we now have our expression for the total energy,

 (12)

where is some known function, is the potential energy field created by the nuclei, is the simple electron static interaction among the nuclei,

and (usually equal to two) is the number of electrons in orbital . Note that the expression (12) maps each possible choice of the set of electronic orbitals to a unique value for the energy of the system and thereby gives the total energy as a function of the orbital functions . Such an expression which returns a number as a function of other functions is called a functional and denoted with square brackets as we do in Eq. (12).

We now have a functional for the energy in terms of the orbitals, but which orbitals are the right ones to use? The answer is quite sensible: the correct orbitals are those which minimize the total energy in (12) while obeying the orthonormality constraints (6). Combined with this variational principle, Eq. (12) now gives a complete prescription for computing total energies, and thereby all of the properties mentioned in Sec. 2.

Tomas Arias 2004-01-26