Hartree-Fock vs. density functional theory 
*

*What is the relationship/difference between the Hartree-Fock method and the density functional theory? It seems the basic formulations of them are very similar.


*Which one is more accurate?
 A: TLDR: HF approximates the wavefunction as a single Slater determinant, DFT reformulates the many body problem in terms of the electron density and utilises the Hohenberg-Kohn theorems to find the ground state properties of the system.
Both start from the same N-electron Hamiltonian:
\begin{equation}
\begin{split}
\hat{H} &= \sum_{i=1}^{N} -\frac{\hbar^{2}}{2m}\nabla_{i}^{2} + U_{\text{ion}}(\textbf{r}_{i}) + \frac{1}{2}\sum_{i\neq j}^{N} \frac{e^{2}}{4\pi\epsilon_{0}|\textbf{r}_{i} - \textbf{r}_{j}|} \\
&= \sum_{i=1}^{N} \hat{T} + \sum_{i=1}^{N}V_{Ne}(\textbf{r}_{i}) + \sum_{i\neq j} V_{ee}(\textbf{r}_{i}, \textbf{r}_{j})
\end{split}
\end{equation}
where
\begin{equation}
U_{\text{ion}}(\textbf{r}_{i}) = -\sum_{j=1}^{M}\frac{Ze^{2}}{4\pi\epsilon_{0}|\textbf{r}_{i} - \textbf{R}_{j}|}
\end{equation}
is the contribution interaction of electrons with the background charge due to $M$ atomic nuclei. If we wanted, we could also consider terms due to the interaction between nuclei or due to external fields and include this in the potential term, $V_{Ne}$. Given this Hamiltonian the goal is to solve the Schrodinger equation, usually for the ground state. By the Pauli principle the wavefunction must change sign under particle exchange, i.e.
\begin{equation}
\Psi(x_{1}, x_{2}, ..., x_{i}, x_{j}, ... x_{N}) = - \Psi(x_{1}, x_{2}, ..., x_{j}, x_{i}, ..., x_{N})
\end{equation}
where $x_{i} = (\textbf{r}_{i}, \sigma_{i})$ is a combined index for position and spin. This means that we can't just write the wavefunction as a separable product of single particle wavefunctions, e.g.
\begin{equation}
\Psi(x_{1}, x_{2}, ..., x_{N}) \neq \psi_{1}(x_{1})\psi_{2}(x_{2})...\psi_{N}(x_{N}),
\end{equation}
where the single-particle wavefunctions are a product of their spatial and spin parts, $\psi_{i}(x_{i}) = \phi_{i}(\textbf{r}_{i})\chi_{i}(\sigma_{i})$. Instead, we must write the many particle wavefunction as a linear combination of Slater determinants
\begin{equation}
\Psi(x_{1}, x_{2}, ..., x_{N}) = \sum_{\alpha_{1}, \alpha_{2}, ..., \alpha_{N}=1}^{\infty} c_{\alpha_{1}, \alpha_{2}, ..., \alpha_{N}}\Phi_{\alpha_{1}, \alpha_{2}, ..., \alpha_{N}}(x_{1}, x_{2}, ..., x_{N})
\end{equation}
where $\Phi_{\alpha_{1}, \alpha_{2}, ..., \alpha_{N}}(x_{1}, x_{2}, ..., x_{N})$ is known as a Slater determinant, given by
\begin{equation}
\Phi_{\alpha_{1}, \alpha_{2}, ..., \alpha_{N}}(x_{1}, x_{2}, ..., x_{N}) = \frac{1}{\sqrt{N!}}
\begin{vmatrix}
\psi_{\alpha_{1}}(x_{1}) & \psi_{\alpha_{1}}(x_{2}) & \cdots & \psi_{\alpha_{1}}(x_{N}) \\
\psi_{\alpha_{2}}(x_{1}) & \psi_{\alpha_{2}}(x_{2}) & \cdots & \psi_{\alpha_{2}}(x_{N}) \\
\vdots & \ddots & & \vdots \\
\psi_{\alpha_{N}}(x_{1}) & \psi_{\alpha_{N}}(x_{2}) & \cdots & \psi_{\alpha_{N}}(x_{N}) \\
\end{vmatrix}.
\end{equation}
The single-particle wavefunction $\psi_{\alpha_{j}}(x_{i})$ represents the $\alpha_{j}$ orbital for the $\text{i}^{\text{th}}$ electron.
Because of the antisymmetry properties of a determinant, each Slater determinant will be antisymmetric under particle exchange and, hence, the sum of Slater determinants will obey the Pauli exclusion principle. A question that people often struggle with is why do we have a (generally infinite) sum of Slater determinants? Well, a single Slater determinant is a single electron configuration. Since we don't know the occupation of orbitals a priori, we have to include all possible configurations in our solution to Schrodinger's equation.
Obviously, it is not feasible to solve Schrodinger's equation for infinitely many electron configurations and so we have developed various approximations to do so instead.
In the Hartree-Fock method, we approximate the wavefunction as a single Slater determinant
\begin{equation}
\Psi(x_{1}, x_{2}, ..., x_{N}) = \frac{1}{\sqrt{N!}}
\begin{vmatrix}
\psi_{1}(x_{1}) & \psi_{1}(x_{2}) & \cdots & \psi_{1}(x_{N}) \\
\psi_{2}(x_{1}) & \psi_{2}(x_{2}) & \cdots & \psi_{2}(x_{N}) \\
\vdots & \ddots & & \vdots \\
\psi_{N}(x_{1}) & \psi_{N}(x_{2}) & \cdots & \psi_{N}(x_{N}) \\
\end{vmatrix}
\end{equation}
where the single-particle wavefunctions $\psi_{i}(x_{j})$ are chosen to minimise the ground state energy, $E_{0} = \left<\Psi_{0}\right|\hat{H}\left|\Psi_{0}\right>$. We then use the calculus of variations to find the set of single-particle wavefunctions $\{\psi_{i}\}$ that will minimise $E_{0}$. Because we have assumed the many electron wavefunction is a single Slater determinant, we aren't able to fully account for electron correlation effects (that is, the position of one electron is correlated with the position of another). As such, the Hartree Fock energy, $E_{0}$ is often higher than the true energy of the system.
On the other hand, we can use density functional theory (DFT) to try and solve the N-electron Hamiltonian. Rather than thinking about a many electron wavefunction, the central quantity in DFT is the electron density
\begin{equation}
n_{0}(\textbf{r}) = \sum_{i=1}^{N} \int \text{d}\textbf{r}_{1}^{3}\int\text{d}\textbf{r}_{2}^{3}, ...\int\text{d}\textbf{r}_{N}^{3}\Psi_{0}^{\dagger}(x_{1}, x_{2}, ...x_{N})\delta(\textbf{r}_{i} - \textbf{r})\Psi_{0}(x_{1}, x_{2}, ...x_{N})
\end{equation}
where $n_{0}(\textbf{r})$ is the ground states electron density. DFT then relies on the first Hohenberg-Kohn theorem to show that the ground state energy of the system is uniquely determined by the ground state electron density. The problem of finding this ground state energy is then reformulated in terms of functionals (functions that map functions to real numbers) of the electron density
\begin{equation}
\begin{split}
E_{0}[n(\textbf{r})] &= T[n(\textbf{r})] + E_{ee}[n(\textbf{r})] + E_{Ne}[n(\textbf{r})] \\
&= F_{HK}[n(\textbf(r)] + + E_{Ne}[n(\textbf{r})]
\end{split}
\end{equation}
where $F_{HK}[n(\textbf(r)]$ is known as the Hohenberg-Kohn density
functional. We then try and minimise $E_{0}$ with a variational method utilising Lagrange multipliers, resulting in the Kohn-Sham equations. The benefit of using DFT over Hartree-Fock is that we can still include the effect of long-range, non-classical electron correlations.
$F_{HK}[n(\textbf(r)]$ is a universal function, independent of the exact system. If we knew $F_{HK}[n(\textbf(r)]$ then, in theory, we could solve the Schrodinger equation for an arbitrary electron configuration - unfortunately we don't, so we try and approximate the functional instead. The simplest approximation is the local density approximation (LDA) where we assume the energy depends on the charge density in an identical manner to a uniform electron gas. The next most complicated set of approximations are generalised gradient approximation (GGA) methods, where we assume the energy also has some dependence on (higher-order) derivatives of the charge density.
