2.5 Density functional theory

Density functional theory (DFT) is a powerful, formally exact theory (see Refs. [8,9,3] and references within). It is distinct from quantum chemical methods in that it is a non-interacting theory and does not yield a correlated $N$-body wavefunction. In the Kohn-Sham[10] DFT, the theory is a one-electron theory and shares many similarities with Hartree-Fock. DFT has come to prominence over the last decade as a method potentially capable of very accurate results at low cost. In practice, approximations are required to implement the theory, and a significantly variable accuracy results. Calibration studies are therefore required to establish the likely accuracy in a given class of systems.

2.5.1 The Hohenberg-Kohn theorem

The Hohenberg-Kohn theorem[11] states that if $N$ interacting electrons move in an external potential $V_{ext}({\bf r})$, the ground-state electron density $n_{0}({\bf r})$ minimises the functional

$$E[n]=F[n]+\int n({\bf r})V_{ext}({\bf r})d{\bf r}$$ (2.17)

where $F$ is a universal functional of $n$ and the minimum value of the functional $E$ is $E_0$ the exact ground-state electronic energy.

The proof of equation 2.17 is straightforward. It is a proof only of existence; additional theory is required before a method can be implemented.

Levy[12] gave a particularly simple proof of the Hohenberg-Kohn theorem which is as follows:

A functional ${O}$ is defined as

$$O\left[n({\bf r}) \right] = \min_{\vert\Psi>\rightarrow n({\bf r})} \left<\Psi\vert\hat{O}\vert\Psi\right> \;\;\; ,$$ (2.18)

where the expectation value is found by searching over all wavefunctions, $\Psi$, giving the density $n({\bf r})$ and selecting the wavefunction which minimises the expectation value of $\hat{O}$.

$F[n({\bf r})]$ is defined by

$$F\left[n({\bf r}) \right] = \min_{\vert\Psi>\rightarrow n({\bf r})} \left<\Psi\vert\hat{F}\vert\Psi\right> \;\;\; ,$$ (2.19)

so that

$$\hat{F} = \sum_i-\frac{1}{2}\nabla_i^2 + \frac{1}{2}\sum_{i\neq j}\frac{1}{\vert{\bf r}_i-{\bf r}_j\vert} \;\;\; .$$ (2.20)

Considering an $N$-electron ground state wavefunction $\Psi_0$ which yields a density $n({\bf r})$ and minimises $<\Psi\vert\hat{F}\vert\Psi>$, then from the definition of the functional $E$

$$E[n({\bf r})] = F[n({\bf r})] + \int n({\bf r})V_{\rm ext}({... ...r})d{\bf r} = <\Psi\vert\hat{F}+V_{\rm ext}\vert\Psi> \;\;\; .$$ (2.21)

The Hamiltonian is given by $\hat{F}+V_{\rm ext}$, and so $E[n({\bf r})]$ must obey the variational principle,

$$E[n({\bf r})] \geq E_0 \;\;\; .$$ (2.22)

This completes the first part of the proof, which places a lower bound on $E[n({\bf r})]$.

From the definition of $F[n({\bf r})]$ equation 2.19 we obtain

$$F[n_0({\bf r})] \leq <\Psi_0\vert\hat{F}\vert\Psi_0> ,$$ (2.23)

since $\Psi_0$ is a trial wavefunction yielding $n_0({\bf r})$. Combining $\int n({\bf r})V_{\rm ext}({\bf r})d{\bf r}$ with the above equation gives

$$E[n_0({\bf r})] \leq E_0 \;\;\; ,$$ (2.24)

which in combination with equation 2.22 yields the key result

$$E[n_0({\bf r})] = E_0 \;\;\; ,$$ (2.25)

completing the proof.

2.5.2 The Kohn-Sham equations

Kohn and Sham[10] derived a coupled set of differential equations enabling the ground state density $n_0({\bf r})$ to be found.

Kohn and Sham separated $F[n({\bf r})]$ into three distinct parts, so that the functional $E$ becomes

$$E\left[n({\bf r}) \right] = T_s\left[n({\bf r}) \right] +\fr... ...bf r})] + \int n({\bf r})V_{\rm ext}({\bf r})d{\bf r} \;\;\; ,$$ (2.26)

where $T_s\left[n({\bf r})\right]$ is defined as the kinetic energy of a non-interacting electron gas with density $n({\bf r})$,

$$T_s\left[n({\bf r})\right] = -\frac{1}{2}\sum_{i=1}^N\int \psi_i^*({\bf r})\nabla^2\psi_i({\bf r})d{\bf r} \;\;\; ,$$ (2.27)

and not the kinetic energy of the real system. Equation 2.26 also defines the exchange-correlation energy functional $E_{XC}[n]$. Introducing a normalisation constraint on the electron density, $\int n({\bf r})d{\bf r}=N$, we obtain

$$\frac{\delta}{\delta n({\bf r})}\left[ E[n({\bf r})]-\mu\int n({\bf r})d{\bf r}\right] = 0$$

$$\Rightarrow \frac{\delta E[n({\bf r})]}{\delta n({\bf r})} = \mu \;\;\; .$$ (2.28)

Equation 2.28 may now be rewritten in terms of an effective potential, $V_{\rm eff}({\bf r})$,

$$\frac{\delta T_s\left[n({\bf r})\right]}{\delta n({\bf r})} + V_{\rm eff}({\bf r}) = \mu \;\;\; ,$$ (2.29)

where

$$V_{\rm eff}({\bf r}) = V_{\rm ext}({\bf r}) + \int\frac{n({\... ...\vert{\bf r}-{\bf r}'\vert}d{\bf r}' + V_{XC}({\bf r}) \;\;\;$$ (2.30)

and

$$V_{XC}({\bf r}) = \frac{\delta E_{XC}[n({\bf r})]}{\delta n({\bf r})} \;\;\; .$$ (2.31)

Crucially, non-interacting electrons moving in an external potential $V_{eff}({\bf r})$ would result in the same equation 2.29. To find the ground state energy, $E_{0}$, and the ground state density, $n_0$, the one electron Schrödinger equation

$$\left( -\frac{1}{2}\nabla_i^2+V_{\rm eff}({\bf r})-\epsilon_i\right)\psi_i({\bf r})=0 \;\;\;$$ (2.32)

should be solved self-consistently with

$$n({\bf r}) = \sum_{i=1}^N\vert\psi_i({\bf r})\vert^2 \;\;\; ,$$ (2.33)

and equations 2.29 and 2.30. A self-consistent solution is required due to the dependence of $V_{\mathrm{eff}}({\bf r})$ on $n({\bf r})$.

The above equations provide a theoretically exact method for finding the ground state energy of an interacting system provided the form of $E_{XC}$ is known. Unfortunately, the form of $E_{XC}$ is in general unknown and its exact value has been calculated for only a few very simple systems. In electronic structure calculations $E_{XC}$ is most commonly approximated within the local density approximation or generalised-gradient approximation.

2.5.3 The local density approximation

In the local density approximation (LDA), the value of $E_{XC}[n({\bf r})]$ is approximated by the exchange-correlation energy of an electron in an homogeneous electron gas of the same density $n({\bf r})$, i.e.

$$E_{XC}^{LDA}[n({\bf r})] = \int \epsilon_{XC}(n({\bf r}))n({\bf r})d{\bf r} \;\;\; .$$ (2.34)

The most accurate data for $\epsilon_{XC}(n({\bf r}))$ is from Quantum Monte Carlo calculations.[13]^2.3 The LDA is often surprisingly accurate and for systems with slowly varying charge densities generally gives very good results. The failings of the LDA are now well established: it has a tendency to favour more homogeneous systems and over-binds molecules and solids. In weakly bonded systems these errors are exaggerated and bond lengths are too short. In good systems where the LDA works well, often those mostly consisting of $sp$ bonds, geometries are good and bond lengths and angles are accurate to within a few percent. Quantities such as the dielectric and piezoelectric constant are approximately 10% too large.

The principle advantage of LDA-DFT over methods such as Hartree-Fock is that where the LDA works well (correlation effects are well accounted for) many experimentally relevant physical properties can be determined to a useful level of accuracy. Difficulties arise where it is not clear whether the LDA is applicable. For example, although the LDA performs well in bulk group-IV semiconductors it is not immediately clear how well it performs at surfaces of these materials.

2.5.4 Limitations

Despite the remarkable success of the LDA, its limitations mean that care must be taken in its application. For systems where the density varies slowly, the LDA tends to perform well, and chemical trends are well reproduced. In strongly correlated systems where an independent particle picture breaks down, the LDA is very inaccurate. The transition metal oxides XO (X=Fe,Mn,Ni) are all Mott insulators, but the LDA predicts that they are either semiconductors or metals. The LDA has been applied to high $T_{c}$ superconductors, but finds several to be metallic, when in reality they are insulating at 0K. [3]

The LDA finds the wrong ground state for in many simpler cases. For example, the LDA finds the wrong ground state for the titanium atom. The LDA does not account for van der Waals bonding, and gives a very poor description of hydrogen bonding. These phenomena are essential for most of biochemistry: the structure of DNA of depends critically on hydrogen bonding, as do the changes in the structure of most molecules on solvation.

The success of the LDA has been shown by QMC calculations to result from a real-space cancellation of errors in the LDA exchange and correlation energies. [14,15] This is illustrated in figure 2.1, where the exchange and correlation energy densities of silicon are compared with an accurate QMC calculation. The cancellation represents a difficulty when improvements to the LDA are attempted, as an improvement in only the exchange or correlation contributions may give worse results.

Figure 2.1: Contour plots of the LDA exchange and correlation energy densities compared with VMC calculations for Si in the diamond structure in the (110) plane: (a) $e_{x}^{\mathrm{VMC}}({\bf r}) - e_{x}^{\mathrm{LDA}}({\bf r})$; (b) $e_{c}^{\mathrm{VMC}}({\bf r}) - e_{c}^{\mathrm{LDA}}({\bf r})$. The chain of atoms and bonds are represented schematically. The contours are in atomic units. Note that in the bonding regions between silicon atoms, the error in the exchange energy density tends to cancel with the error in the correlation energy density. Figure courtesy R. Q. Hood. [15]

An obvious approach to improving the LDA is to include gradient corrections, by making $E_{XC}$ a functional of the density and its gradient:

$$E_{XC}^{GGA}[n({\bf r})] = \int \epsilon_{XC}(n({\bf r}))n({... ..._{XC}[n({\bf r}),\vert\nabla n({\bf r})\vert]d{\bf r} \;\;\; ,$$ (2.35)

where $F_{XC}$ is a correction chosen to satisfy one or several known limits for $E_{XC}$.

Clearly, there is no unique recipe for $F_{XC}$, and several dozen functionals have been proposed in the literature. They do not always represent a systematic improvement over the LDA and results must be carefully compared against experiment.^2.4The development of improved functionals is currently a very active area of research and although incremental improvements are likely, it is far from clear whether the research will be successful in providing the substantial increase in accuracy desired.