Variational principle

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

$$E[\{\psi(\vec x)\}] = \sum_i f_i\, \int \psi_i^*(\vec x) \left( -\frac{\hbar^2}{2m}\nabla^2\psi_i(\vec x) \right) \, dV + \int V_{nuc}(\vec x) n(\vec x)\,dV$$ (12)

$$+ \frac{1}{2} \int \phi(\vec x) n(\vec x)\,dV + \int f_{xc}(n(\vec x)) \, dV + U_{nuc-nuc},$$

where $f_{xc}(\ldots)$ is some known function, $V_{nuc}(\vec x)$ is the potential energy field created by the nuclei, $U_{nuc-nuc}$ is the simple electron static interaction among the nuclei,

$$n(\vec x) = \sum_i f_i\, \vert\psi_i(\vec x)\vert^2,$$

and $f_i$ (usually equal to two) is the number of electrons in orbital $i$. Note that the expression (12) maps each possible choice of the set of electronic orbitals $\{\psi_i(\vec x)\}$ to a unique value for the energy of the system and thereby gives the total energy $E$ as a function of the orbital functions $\phi_i(\vec x)$. 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 $E$ 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.