TLDR: HF approximates the wavefunction as a single <PERSON> 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 <PERSON> equation, usually for the ground state. 526
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 <PERSON> 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 <PERSON> determinant, given by
\begin{equation} \Phi_{\alpha_{1}, \alpha_{2}, ..., \alpha_{N}}(x_{1}, x_{2}, ..., x_{N}) = \frac{1}{\sqrt{N! 616