# Introduction to density-functional theory

This chapter only gives a very rough overview for now. Further details can be found in the summary of DFT theory or in the Introductory Resources.

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.