Quantum mechanics

Eqs. (1,3,5) describe the potential energy of the system, but we yet have to determine the electron density $n(\vec x)$ and have yet to consider the kinetic energy of the electrons. Density functional theory determines both of these quantities.

Within density functional theory a set of quantum mechanical Kohn-Sham orbitals $\psi_i(\vec x)$ describes the electrons. These are the electronic orbitals that you learn about in introductory chemistry class, each of which usually contains two electrons (one spin-up and one spin-down). In general, these orbitals may be complex, so that we must also consider the complex conjugates of the orbitals, $\psi_i^*(\vec x)$. For the problems of interest in their course, the orbitals always turn out to be real so that $\psi_i^*(\vec x)=\psi_i^*(\vec x)$. Thus, if you are unfamiliar or rusty with complex numbers you can simply ignore the *'s. We include them for those in the course who are familiar with quantum mechanics and who may have interest in problems where the orbitals can be complex.

Within quantum mechanics, the square magnitude of each orbital gives the probability of finding an electron, when in that orbital, at any point in space, ${\mathcal P}(\vec x)=\psi_i^*(\vec x) \psi_i(\vec x) \equiv \vert\psi_i(\vec x)\vert^2$. The orbitals are not free to be any functions whatsoever, but must obey certain constraints. First, because the electron must be somewhere in space, the probability must add up to unity,

$$1 = \int {\mathcal P}(\vec x)\,dV = \int \psi_i^*(\vec x) \psi_i(\vec x)\,dV.$$ (6)

In addition to this normality constraint for each orbital $i$, the orbitals must be orthogonal to each other,

$$0 = \int \psi_i^*(\vec x) \psi_j(\vec x)\,dV \mbox{\ \ \ for $i \ne j$},$$ (7)

the condition by which density functional theory encodes the Pauli exclusion principle from elementary chemistry courses. Apart from these constraints, the orbitals are completely free. Thus, we may combine all relevant constraints into the orthonormality constraint,

$$\int \psi_i^*(\vec x) \psi_j(\vec x)\,dV = \left\{\begin{array}{cc}1&i=j\\ 0&i\ne j\end{array}\right. .$$ (8)

The electron density and total kinetic energy come directly from the orbitals. Because the square of each orbital gives the distribution of the electrons in that orbital, the total electron density will be the sum of squares of the orbitals time the number of electrons $f_i$ in or ``filling'' each orbital,

$$n(\vec x) = \sum_i f_i\, \vert\psi_i(\vec x)\vert^2.$$ (9)

(As mentioned above, usually there are $f_i=2$ electrons in each orbital.) The total kinetic energy of the electrons $T_{el}$ is similarly just the sum over orbitals of the number of electrons in each orbital times the elementary quantum mechanical expression for the kinetic energy of each orbital,

$$T_{el} = \sum_i f_i\, \int \psi_i^*(\vec x) \left( -\frac{\hbar^2}{2m}\nabla^2\psi_i(\vec x) \right) \, dV.$$ (10)

There arises from advanced quantum mechanics one final subtle point. The electron density defined in (9) is only an average. The actual density fluctuates, resulting in relatively small but important errors in Eqs. (5,10) due to correlations in these fluctuations. In theory, we may correct for these errors exactly, but this turns out to be quite difficult in practice. A very good approximation to this exchange-correlation correction, sufficient in practice to compute most properties to within a few percent, is the local density approximation

$$E_{xc}=\int f_{xc}(n(\vec x)) \, dV,$$ (11)

where $f_{xc}(\ldots)$ is a relatively simply function which we will provide later in the course².

That's it - this is all the quantum mechanics we need to predict accurately the behavior of matter!