Introduction to density-functional theory

Density-functional theory is a simplification of the full electronic Schrödinger equation leading to an effective single-particle model. Mathematically this can be cast as an energy minimisation problem in the electronic density $\rho$. In the Kohn-Sham variant of density-functional theory the corresponding first-order stationarity conditions can be written as the non-linear eigenproblem

\[\begin{aligned} &\left( -\frac12 \Delta + V\left(\rho\right) \right) \psi_i = \varepsilon_i \psi_i, \\ V(\rho) = &\, V_\text{nuc} + V_\text{H}^\rho + V_\text{XC}(\rho), \\ \rho = &\sum_{i=1}^N f(\varepsilon_i) \abs{\psi_i}^2, \\ \end{aligned}\]

where $\{\psi_1,\ldots, \psi_N\}$ are $N$ orthonormal orbitals, the one-particle eigenfunctions and $f$ is the occupation function (e.g. DFTK.Smearing.FermiDirac, DFTK.Smearing.Gaussian or DFTK.Smearing.MarzariVanderbilt) chosen such that the integral of the density is equal to the number of electrons in the system. Further the potential terms that make up $V(\rho)$ are

• the nuclear attraction potential $V_\text{nuc}$ (interaction of electrons and nuclei)
• the exchange-correlation potential $V_\text{xc}$, depending on $\rho$ and potentially its derivatives.
• The Hartree potential $V_\text{H}^\rho$, which is obtained as the unique zero-mean solution to the periodic Poisson equation

  \[-\Delta V_\text{H}^\rho(r) = 4\pi \left(\rho(r) - \frac{1}{|\Omega|} \int_\Omega \rho \right).\]

The non-linearity is such due to the fact that the DFT Hamiltonian

\[H = -\frac12 \Delta + V\left(\rho\right)\]

depends on the density $\rho$, which is built from its own eigenfunctions. Often $H$ is also called the Kohn-Sham matrix or Fock matrix.

Introducing the potential-to-density map

\[D(V) = \sum_{i=1}^N f(\varepsilon_i) \abs{\psi_i}^2 \qquad \text{where} \qquad \left(-\frac12 \Delta + V\right) \psi_i = \varepsilon_i \psi_i\]

allows to write the DFT problem in the short-hand form

\[\rho = D(V(\rho)),\]

which is a fixed-point problem in $\rho$, also known as the self-consistent field problem (SCF problem). Notice that computing $D(V)$ requires the diagonalisation of the operator $-\frac12 \Delta + V$, which is usually the most expensive step when solving the SCF problem.

To solve the above SCF problem $ \rho = D(V(\rho)) $ one usually follows an iterative procedure. That is to say that starting from an initial guess $\rho_0$ one then computes $D(V(\rho_n))$ for a sequence of iterates $\rho_n$ until input and output are close enough. That is until the residual

\[R(\rho_n) = D(V(\rho_n)) - \rho_n\]

is small. Details on such SCF algorithms will be discussed in Self-consistent field methods.