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It seems that people usually use the Single Active Electron (SAE) approximation to deal with a multi-electron system, transforming the problem into a single electron problem. For example, in numerically solving the problem of a helium atom interact with laser fields, people usually approximate include the electron-electron effect by a pseudo-potential and essentially solve the one electron problem. So why it is difficult to even numerically solve the time-dependent multi-electron Schrödinger's equation? Is it much difficult than the classical n-body problem? I've seen there are a lot a huge classical $n$-body problem solved numerically in astronomy even in real time, for example here simulate in real time a collision of two galaxies involving 280000 particles interaction.

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    $\begingroup$ Besides the difficulty, there's also utility that drives innovation. Astrophysical $n$-body problems need time evolution. On the other hand, there's a lot you can do with a multi-electron atom that has little to no time-dependence, like finding energy levels. In other words, there are more applications involving steady states for atoms than for colliding galaxies. $\endgroup$ – user3224 May 20 '13 at 19:10
  • $\begingroup$ Maybe, but I think that's besides the point. Even stationary quantum computations are vastly more expensive. But even then, time dependent quantum computations are highly relevant -- they're just too expensive to do in almost all practical cases, and this explains why it hasn't been done in the past. $\endgroup$ – Wolfgang Bangerth May 20 '13 at 19:15
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Yes, it is much more difficult to do so. For the $N$ body problem, all you need to compute are the trajectories $\mathbf x_i(t), i=1\ldots N$ which are just $N$ functions of a single variable.

On the other hand, even for a single electron, the solution of the Schroedinger equation is a function $\Psi(x,y,z,t)$, i.e., a function of four variables. For two electrons, you are looking for a function $\Psi(x_1,y_1,z_1,x_2,y_2,z_3,t)$ describing the wave function as a function of the locations of the two electrons plus time. That's seven variables.

Now, if you remember how to solve ordinary differential equations such Newton's equations for the $N$ body problem, then you need to move every equation forward by time stepping from time $t$ to $t+\Delta t$ and compute the solution there. So, if you divide your time interval $[0,T]$ into $M$ intervals of length $\Delta t=T/M$ then the effort for every time step will be $N^2M$ using a naive implementation of the interactions of the $N$ bodies (you can use methods to achieve $N (\log N) M$ effort, but that's besides the point).

On the other hand, to find a function of 7 variables, assume that you subdivide the time interval into $M$ subintervals as above, but that you also do the same for the 6 spatial coordinates. Then there are a total of $M^7$ grid points to consider. And in general, for an $N$ body quantum system, you have $M^{3N+1}$.

It is now easy to verify that even for rather small numbers $N,M$, the effort $M^{3N+1}$ is vastly larger than $N^2M$, which explains why multibody quantum computations are so extremely much more expensive to do than $N$ body classical mechanics.

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    $\begingroup$ Great answer. I would only mention that just like there are faster methods than the naive $N^2 M$ for Newton equations, there are also faster methods than the naive $M^{3N+1}$ for the Schrödinger equation. $\endgroup$ – Ondřej Čertík May 20 '13 at 20:28
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    $\begingroup$ Yes, indeed. But by and large, you can't get rid of the combinatorial complexity. $\endgroup$ – Wolfgang Bangerth May 21 '13 at 3:21

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