What is the current state of the art in solving higher dimensional parabolic PDEs (multi-electron Schrödinger equation)

Question

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 Schrödinger 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.

Answer 1

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 is

• Thijssen, Jos. Computational physics. Cambridge University Press, 2007. Amazon link.

Answer 2

You want to solve for 3 to 10 particle systems (3D per particle)? As far as I am aware, mean field theories do not work especially well for so few particles, but it seems there has been DFT work on diatomic molecules.

Is this a system where Born-Oppenheimer is valid? If so, I might be inclined to expand the electronic wavefunction using a linear combination of Slater determinants possibly using sparse grid or spectral sparse grids This paper perhaps could help.

Another option is to try using a tight-binding approach, although the fact that you mentioned absorbing boundary conditions suggests you may be thinking of problems involving ionization/dissociation. TB would mostly be useful if you were trying to approximate low level states.

Possibly something like the multi-configurational time-dependent Hartree-Fock method could work here MCTDHF.

Finally, you could look at quantum Monte Carlo methods. These are the methods by which exchange and correlation functional models for single atoms are obtained to do DFT calculations. It looks like there are poly-atomic extensions. (I'm out of link privileges).

Answer 3

If you have $M$ atoms, your wave function depends on $3M$ variables. If you wanted to discretize this function on a uniform mesh with $N$ nodes in each of these directions (or with $N$ one-dimensional shape functions), you'd need a total of $N^{3M}$ unknowns -- far too many for any interesting number of electrons $M$. To give just one example, if you'd just use $N=10$ nodes in each direction, and you had just 3 electrons, you'd already have a system of size $10^9$, very much at the limit of what one can do today.

From this consideration follows that it is not possible to consider the problem with all electrons at the same time -- you need to restrict yourself to one or two electrons at a time. This naturally leads to you to methods such as the Hartree Fock method that iterates over electrons while keeping the rest of the system fixed.

I don't know the field well enough but imagine that there are a number of highly cited and well written review papers on the topic.