Density Functional theory

In this chapter we make some brief remarks about the method that will be used to compute energies and forces, which is a formulation of quantum mechanics know as density functional theory (DFT). Consider the time-independent many-body Schrödinger equation for a system of $N$ electrons:

$$\hat{H} \Psi({\bf r}_1,\dots,{\bf r}_N) = E \Psi({\bf r}_1,\dots,{\bf r}_N),$$ (8.1)

where ${\bf r}_1,\dots,{\bf r}_N$ are the positions of the electrons, and for simplicity we are not including the spin variables. The hamiltonian is given by the sum of the kinetic, external potential and electron-electron operators:

$$\hat{H} = \hat{T} + \hat{V}_{ext} + \hat{V}_{ee},$$ (8.2)

defined by ^8.1

$$\hat{T} \Psi({\bf r}_1,\dots,{\bf r}_N) = -\frac{1}{2} \sum_{i=1}^N \nabla_i^2 \Psi({\bf r}_1,\dots,{\bf r}_N),$$ (8.3)

$$\hat{V}_{ext} \Psi({\bf r}_1,\dots,{\bf r}_N) = \sum_{i=1}^N v({\bf r}_i) \Psi({\bf r}_1,\dots,{\bf r}_N),$$ (8.4)

with $v({\bf r}_i)$ the value of the external potential felt by electron $i$ at position ${\bf r}_i$, and

$$\hat{V}_{ee} \Psi({\bf r}_1,\dots,{\bf r}_N) = \sum_{i,j=1; i<j}^N \frac{1}{\vert{\bf r}_i - {\bf r}_j\vert} \Psi({\bf r}_1,\dots,{\bf r}_N).$$ (8.5)

If the wavefunction $\Psi$ is normalised ^8.2, then the energy $E$ is obtained from:

$$E = \int_V d^3{\bf r}_1 \dots d^3{\bf r}_N \Psi^*({\bf r}_1,\dots,{\bf r}_N) \hat{H} \Psi({\bf r}_1,\dots,{\bf r}_N),$$ (8.6)

where $V$ is the volume of the system and $\Psi^*$ is the complex conjugate of $\Psi$. The probability that electron $1$ is in volume $d^3{\bf r}_1$, electron $2$ is in volume $d^3{\bf r}_2$, and so on is given by $\vert\Psi({\bf r}_1,\dots,{\bf r}_N)\vert^2 d^3{\bf r}_1 \dots d^3{\bf r}_N$. Now we ask the question: what is the total number of electrons in an element of volume $d^3{\bf r}$? This number, divided by the volume $d^3{\bf r}$, defines the electron density $\rho({\bf r})$. To compute this number, note that any of the $N$ electrons can be found in $d^3{\bf r}$, and therefore the total number of electrons in this volume is given by the probability that electron 1 is in this volume, regardless of where all the other electrons are, plus the probability that electron 2 is in this volume, and again regardless of the positions of all other electrons, and so on. Since the electrons are identical these probabilities $p({\bf r})d^3{\bf r}$ are all identical, and so the number of electrons in $d^3{\bf r}$ is given by $N p({\bf r})d^3{\bf r}$, which gives $\rho({\bf r}) = N p({\bf r})$. To obtain $p({\bf r})$ we simply need to integrate $\vert\Psi({\bf r}_1,\dots,{\bf r}_N)\vert^2$ over the positions of the remaining $N-1$ electrons, as we don't care where they are:

$$p({\bf r}) = \int_V d^3{\bf r}_2 \dots d^3{\bf r}_N \vert\Psi({\bf r}, {\bf r}_2,\dots,{\bf r}_N)\vert^2,$$ (8.7)

and therefore the electron density is:

$$\rho({\bf r}) = N \int_V d^3{\bf r}_2 \dots d^3{\bf r}_N \vert\Psi({\bf r}, {\bf r}_2,\dots,{\bf r}_N)\vert^2.$$ (8.8)

The total external potential energy of the system of electrons is given by:

$$V_{ext} = \int_V d^3{\bf r}_1 \dots d^3{\bf r}_N \Psi^*({\bf r}_1,\dots,{\bf r}_N)\sum_{i=1}^N v({\bf r}_i) \Psi({\bf r}_1,\dots,{\bf r}_N)$$

$$= \sum_{i=1}^N \int_V d^3{\bf r}_1 \dots d^3{\bf r}_N \Psi^*({\bf r}_1,\dots,{\bf r}_N)v({\bf r}_i) \Psi({\bf r}_1,\dots,{\bf r}_N).$$ (8.9)

Since the electrons are all identical, these integrals are also all identical, each one equal to:

$$\int_V d^3{\bf r} d^3{\bf r}_2 \dots d^3{\bf r}_N \Psi^*({\bf r},{\bf r}_2,\dots,{\bf r}_N)v({\bf r}) \Psi({\bf r},{\bf r}_2,\dots,{\bf r}_N).$$ (8.10)

Since we have $N$ of them, we obtain:

$$V_{ext} = N \int_V d^3{\bf r} d^3{\bf r}_2 \dots d^3{\bf r}_N \Ps... ...r},{\bf r}_2,\dots,{\bf r}_N)v({\bf r}) \Psi({\bf r},{\bf r}_2,\dots,{\bf r}_N)$$

$$= \int_V d^3{\bf r} \rho({\bf r}) v({\bf r}). \quad \quad \quad \... ...d \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad$$ (8.11)

Eq.  shows that the potential energy due to the external potential is a functional of the electron density, which we indicate with the notation $V_{ext} [\rho]$.