Density Functional Theory

In the following we want to look a bit into the theory behind density functional theory (DFT) and especially its variant Kohn-Sham DFT, which has some links to Hartee-Fock theory. It will be a very brief introduction convering the main aspects. For a detailed but still easily digestible introduction I recommend Ref. [1].

Introduction

The (electronic) wave function is a very complex quantity, which depends for an \(N\) electron system on \(3N\) spatial and \(N\) spin coordinates. However in general one is not interested in the wave function itself, but uses it as an auxiliary quantity to obtain expectation values of observables. The idea of density functional theory (DFT) is to use the electron density instead of the wave function for this purpose. The density depends only on three spatial coordinates and so the hope was to derive less computational expensive expressions. But in order to do so, it had to be clarified if expectation values of ground state observables are unique functionals of the ground state electronic density. This is the content of the 1st Hohenberg-Kohn theorem [2], which can be proven as follows:

The potential \(V\) uniquely defines the Hamilton operator and it is known that the mapping of a potential \(V\) to an electronic density \(n(\vec{r})\) is surjective. That means that only one unique density belongs to a potential \(V\)
For the Hohenberg-Kohn theorem it has to be shown that also the mapping from a density to a potential is surjective.
1. First we have to recall that no two potentials \(V\) and \(V'\) exist, which give the same ground state wave function and differ in more than an additive constant. One can write down the Schrödinger equation for both potentials as
  \begin{equation} \left( \sum_i -\frac{1}{2}\nabla_i^2 + \sum_{i>j} \frac{1}{r_{ij}} + V(\vec{r}) \right)\psi_0 = E_0 \psi_0 \end{equation}
  
  \begin{equation} \left( \sum_i -\frac{1}{2}\nabla_i^2 + \sum_{i>j} \frac{1}{r_{ij}} + V'(\vec{r}) \right)\psi_0 = E_0 \psi_0 \end{equation}
  Subtraction of both equation gives
  \begin{equation} \left( V(\vec{r}) - V'(\vec{r}) \right)\psi_0 = \left( E_0 - E_0' \right) \psi_0 \quad , \end{equation}
  which is only fulfilled if \(V\) and \(V'\) are the same or differ in an additive constant.
2. In the second step one shows using the variational principle that no two different ground state wave functions \(\psi_0\) and \(\psi_0'\) exist, which give the same density. The energy expectation value for \(\psi_0\) can be written as
  \begin{equation} E_0 = \langle{\psi_0}|{\hat{H}}|{\psi_0}\rangle < \langle{\psi_0'}|{\hat{H}}|{\psi_0'}\rangle \end{equation}
  and is due to the variational principle a lower bound to the matrix element with the same Hamiltonian, but an other wavefunction.
  \begin{equation} \begin{split} \langle{\psi_0'}|{\hat{H}}|{\psi_0'}\rangle & = \langle{\psi_0'}|{\hat{H}'+\left(V-V'\right)}|{\psi_0'}\rangle \\ & = E_0' + \int n(\vec{r})'\left(V(\vec{r}-V'(\vec{r})\right) d\tau \end{split} \end{equation}
  
  \begin{equation} E_0 < E_0' + \int n'(\vec{r})\left(V(\vec{r}-V'(\vec{r})\right) d\tau \end{equation}
  Starting from the expectation value \(\langle{\psi_0'}|{\hat{H}'}|{\psi_0'}\rangle\) one obtains in an analogue way
  \begin{equation} E_0' < E_0 + \int n(\vec{r})\left(V'(\vec{r}-V(\vec{r})\right) d\tau \end{equation}
  The addition of both inequalities yields
  \begin{equation} E_0 + E_0' < E_0' + E_0 + \int \left( n'(\vec{r}) - n(\vec{r}) \right)\left(V'(\vec{r}-V(\vec{r})\right) d\tau \end{equation}
  If the densities \(n(\vec{r})\), \( n'(\vec{r})\) are the same, the inequality becomes the contradiction \(E_0 + E_0' < E_0' + E_0\). Therefore every ground state density can be mapped onto an unique ground state wavefunction.

From the variational principle also follows the 2nd Hohenberg-Kohn theorem, which states that the energy functional \(E[n(\vec{r})]\) is minimal for the exact electronic ground state density.

In the early days of DFT several models came out, which were only of limited use for chemical problems and wont be discussed here. One major problem of these early models was to find an accurate functional for the kinetic energy.

Kohn-Sham DFT

Walter Kohn and Lu Jeu Sham solved as described in Ref. [3] the problem to find an accurate description of the kinetic energy by reintroduction of orbitals.\ In their ansatz the density \(n(\vec{r})\) is computed from \(N\) one-particle wave functions \(\phi_j\)

\begin{equation} n(\vec{r}) = \sum_{j=1}^{N} \left| \phi_j(\vec{r}) \right|^2 \quad , \end{equation}

which are the \(N\) solutions of a Schrödinger equation of a non interacting system in an effective potential \(v_{eff}(\vec{r})\)

\begin{equation} \left( -\frac{1}{2}\nabla^2+v_{eff}(\vec{r}) \right)\phi_j(\vec{r}) = \varepsilon_j \phi_j(\vec{r}) \quad . \end{equation}

For this non interacting system an exact expression for the kinetic energy can be derived. It is analogue to the kinetic energy in Hartree-Fock and the effective potential \(v_{eff}(\vec{r})\) is adjusted in a way that the density of the non interacting system resembles the density of the true interacting system. This potential is written as

\begin{equation} v_{eff}(\vec{r}) = v(\vec{r}) + \int \frac{n(\vec{r}')}{\vec{r}-\vec{r}'}d^3r'+v_{xc}[n(\vec{r})] \quad , \end{equation}

where \(v(\vec{r})\) is an external potential for instance the potential arising from the nuclei. The second term

\begin{equation} J[n(\vec{r})]=\int \frac{n(\vec{r}')}{\vec{r}-\vec{r}'}d^3r' \end{equation}

describes the coulomb interaction of the electrons, but the most interesting part is the term

\begin{equation} v_{xc}[n(\vec{r})] \quad, \end{equation}

which is the exchange correlation functional. It describes exchange and electron correlation effects and additionally accounts for the part of the kinetic energy, which is missing, because we solve a non interacting model system instead of the true system. In principle the whole many body problem is casted into this functional. Because \(v_{eff}(\vec{r})\) depends on the density and is on the same time part of the Schrödinger equation to determine the density, we have a non linear problem. Like in HF it can be solved iteratively until the density is self consistent. On a formal level there are in general large similarities between KS-DFT and HF, so that it is quite easy to rewrite a HF program to do KS-DFT. The missing ingredient is the numerical integration of \(v_{xc}[n(\vec{r})]\). This is probably also a reason for the great success of KS-DFT. The well known machinery of HF could be reused.

KS-DFT is formal exact, but the exact form of \(v_{xc}[n(\vec{r})]\) is unknown and there is no systematic way to obtain it. Nevertheless there exist a great variety of approximated exchange correlation functionals, which can be classified into four major categories.

Local density approximation LDA: In this approximation \(v_{xc}[n(\vec{r})]\) depends solely on the value of the electron density \(n(\vec{r})\) in each point of space. For LDA analytical expressions can be easily derived from the model of the homogeneous electron gas. In solid state physics LDA functionals deliver satisfactory results, but for molecular systems they often fail and should not be used.
Generalized gradient approximation GGA: In this approximation \(v_{xc}[n(\vec{r})]\) depends on the value of the electron density \(n(\vec{r})\) and its first derivative \(\nabla n(\vec{r})\), so that one accounts for the inhomogeneous of the density, which is important for chemical systems. The results for chemical problems are much better in comparison to LDA and are often accurate enough.
Hybrid functionals: Hybrid functionals are linear combination of a HF like exchange part with an exchange part of a GGA functional:
\begin{equation} E_{xc} = a_0 E_x^{\rm{GGA}} + (1-a_0)E_x^{\rm{HF}} + E_c^{\rm{GGA}} \end{equation}
Hybrid functionals deliver often very satisfactory results for chemical problems. Examples for hybrid functionals are the B3-LYP and the PBE0 functional. Due to the the HF like exchange part they are approximately as costly as HF and therefore more expensive than pure GGA functionals.
meta-GGAs: In this approximation \(v_{xc}[n(\vec{r})]\) depends on the value of the electron density \(n(\vec{r})\) and its first and second derivative \(\nabla n(\vec{r})\), \(\nabla^2 n(\vec{r})\).

In all these classes the common used functionals are local or semi-local functionals of the density, which is the reason for their failure in describing dispersion/van-der-Waals interactions, which are based on non-local correlation effects. Also the description of (non local) charge transfer excitations is problematic. An other point is that is not ensured that self interaction is 100% excluded in KS-DFT. In HF the exchange term cancels out the coulomb interaction of an electron with itself, but in DFT this is not guaranteed.

Nevertheless KS-DFT is nowadays the method of choice, because of its good cost to accuracy ratio. It is as fast or even faster than HF and additionally recovers correlation effects, but one has to know its weaknesses. If van-der-Waals interactions play a major role one should better use wave function based methods like MP2 or use at least an empirical dispersion correction and if one wants to study charge transfer excitation it is wisely to double check with methods like CC2.

Remarks

As MP2 DFT can be accelerated using the Resolution-of-the-Identity/density fitting approximation.
Also in KS-DFT only a model system is solved, the orbitals are commonly used for interpretation of chemical problems, but in principle they have no physical meaning. However one can justify it by argue that the model system is not far from the real system and that in contrast to HF the orbitals are somehow adjusted to correlation effects.
It is common practice to use DFT for geometry optimization even if one does use wavefunction based methods for subsidiary calculations. In contrast to the energy the geometry is not so sensible to the used method and converges faster with the level of theory.

References

A Chemist’s Guide to Density Functional Theory. Second Edition, Wolfram Koch, Max C. Holthausen
P. Hohenberg and W. Kohn: "Inhomogeneous Electron Gas". Phys. Rev. 136 (1964) B864-B871
W. Kohn, L. J. Sham: "Self-Consistent Equations Including Exchange and Correlation Effects". Physical Review. 140 (1965) A1133–A1138

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