====== Force-Balance Approach To Functional Approximations ======
The purpose of this wiki is to create a more or less consistent nomenclature within this project for all participants. Additionally, this is a way to create a collection of relevant sources:
  * Framework of the project
  * [[forcebalance:definitions|Definitions]]
  * [[forcebalance:calculation|Analytical calculations]]
  * [[forcebalance:theorems|Theorems]]
  * Citations
  * Additional information
  * Numerical calculations with code or prompt
  * Open Questions
  * etc.

===== Overview =====
Starting with the Hamiltonian
\begin{equation}
  \hat{H} (t)
  =
  \hat{H}_0 (t) + H_{\text{int}} (t)
\end{equation}
describing electorns in an external Potential $V(r,t)$ and an external vectorpotential \vector{A}(\vector{r},t).
Here the singele particle part of the Hamiltonian is
\begin{equation}
  H_0 (t)
  =
  \sum_{\sigma^\prime, \sigma^{\prime\prime}}
  \int \!\mathrm{d}r^\prime
  \hat{\Psi}^{\dagger}(\vec{r}^\prime,\sigma^\prime)
  h_{\sigma^\prime, \sigma^{\prime\prime}} (\vec{r},-i\nabla,\vec{s},t)
  \hat{\Psi}(\vec{r}^\prime,\sigma^{\prime\prime})
  \quad ,
\end{equation}
with
\begin{equation}
  h_{\sigma^\prime, \sigma^{\prime\prime}} (\vec{r},-i\nabla,\vec{s},t)
  =
  \dots
\end{equation}
and the interaction part of the Hamiltonian is
\begin{equation}
  H_{\text{int}} (t)
  =
  \frac{1}{2}
  \int \!\mathrm{d}x^{\prime}
  \int \!\mathrm{d}x^{\prime\prime}
  \nu (\vec{r}^\prime,\vec{r}^{\prime\prime})
  \hat{\Psi}^{\dagger}(x^{\prime})
  \hat{\Psi}^{\dagger}(x^{\prime\prime})
  \hat{\Psi} (x^{\prime\prime})
  \hat{\Psi} (x^\prime)
  \quad .
\end{equation}
From this the [[equation of motion|equation of motion(EoM)]] of the electron current $\frac{\partial}{\partial t}J(\vec{r},t)$ can be derived.

((in there describe: starting from the Heisenberg EoM and the rest is transforming the equation))

TODO: Main ideas, formulas (-> with additional "calculations nodes"), variables (-> "definition node"), Hamiltonian [(:cite:forcebalance:tchenkoue_force_2019)] and linkts to "assumption nodes"

[[Force based formulation]]

Overview on the used [[Node structure|nodes]].

===== Goal =====
The aim of the project is to describe the [[ks-dft|KS-DFT]] as well as the [[TDDFT|TDDFT]] by force fields.
The functionals to solve the system are described analytically by the force fields. This analytical description cannot be solved numerically so far. Therefore, these force fields are subdivided, some parts of which are solvable and others not. ...
"theorem nodes" for important pages

===== References, abbreviations and important variables =====
Introductory literature can be found [[:refnotes:forcebalance|here]]. (Be more precice where to find wich idea)
TODO: Short overview on the main formulas variables and abbreviations. Further abbreviations can be found here and important variables can be found here.
"complementary nodes"

=====Abandoned Links=====
  *equation of moiton