What is the current state of the art for solving higher dimensional (3-10)  parabolic PDEs in the complex domain with simple poles (of the form $ \frac{1}{|\vec{r}_1 - \vec{r}_2|}$) and absorbing boundary conditions?Specifically, I'm interested in solving the multi-electron Schrdinger equation:$ \left( \sum_i \sum_{j\neq i}\left[ -\frac{\nabla_i^2}{2 m} - \frac{Z_i Z_j}{|\vec{r}_i - \vec{r}_j|}  + V(\vec{r}_i, t) \right]\right)\psi = -i\partial_t \psi $For a diatomic molecule with more than 1 electron.
What is the current state of the art in solving higher dimensional parabolic PDEs (multi-electron Schrdinger equation)
parabolic pde;quantum mechanics;high dimensional
The solutions for the equation are in$$\psi \in \mathbb{C}^{3M}\times\mathbb{R}^+ \enspace .$$If the number of electrons is small enough you can just use any traditional method. Like a domain discretization method (Finite Difference, Finite Element, Boundary Element), or a pseudospectral method. Since solving this equation is not more difficult than solving a multidimensional wave equation.In the case of bigger systems some trick is necessary to get the solution. We replace the electron-electron interaction for the interaction of an electron with a cloud of electrons (a mean field approximation of the rest of them), and then solve in a self-consistent fashion (due to the nonlinearity that come from the mean field term). This is done in Hartree-Fock and Density Functional Theory (DFT). Where the original differential equation is transformed into a variational formulation.DFT is the most common method nowadays, and the advantage is that all the equations are formulated in terms of the electron density and not in terms of the wave equations. So, the equations lie in a 3 dimensional space. One book that describes both of these methods isThijssen, Jos. Computational physics. Cambridge University Press, 2007. Amazon link.