deletions | additions       diff --git a/Schroedinger equation.tex b/Schroedinger equation.tex  index c6cc5ce..d621d87 100644  --- a/Schroedinger equation.tex  +++ b/Schroedinger equation.tex 
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\label{eq:SENd}  \hat{H}\Psi_j(x_1...x_N)=E_j\Psi_j(x_1...x_N)  \end{equation}  which provides a spatial wave function for a single particle in $N$ dimensions. This equivalence is not restricted to one-dimensional problems. One can generally map a problem of $N$ electrons in $d$ dimensions onto the problem of a single particle in $Nd$ dimensions, or indeed a problem with multiple types of particles (e.g. electrons and protons) in $d$ dimensions, in the same way. What we exploit in octopus Octopus  is the basic machinery for solving the Sch\"odinger equation iteratively, in an arbitrary dimension,  the spatial/grid bookkeeping, the ability to represent an arbitrary external potential,  and the intrinsic parallelization. In order to keep our notation relatively simple, we will continue to discuss the case of an originally one-dimensional problem with $N$ electrons. Grid-based resolutions of the full Schr\"odinger equation are not new, and have been performed for many problems with either few electrons (in particular H$_2$, D$_2$ and H$_2^+$ e.g. in \cite{Ranitovic21012014} or \cite{lein2002}) or model interactions \cite{luo2013}, including time dependent cases \cite{ramsden2012}. The time dependent propagation of the Schr\"odinger equation can be carried out in the same spirit, since the Hamiltonian is given explicitly and each ``single particle orbital'' represents a full state of the system. A laser or electric field perturbation can also be applied, depending on the charge of each particle (given in the input), and taking care to apply the same effective field to each particle along the polarization direction of the field (in 1D the diagonal of the hyper-cube).   Solving Eq.\ (\ref{eq:SENd}) leaves the problem of constructing a wave function which satisfies the antisymmetry properties of $N$ electrons in one dimension. In particular, for fermions, one needs to ensure that those spatial wave functions $\Psi_j$ which are not the spatial part of a properly antisymmetric wave function are removed as allowed solutions for the $N$-electron problem. A graphical representation of which wave functions are allowed is given by the Young diagrams (or tableaux) for permutation symmetries, symmetries~\cite{Young diagrams},  where each electron is asigned a box, and the boxes are then stacked in columns and rows. Each box is labeled with a number from 1 to $N$ such that the numbers increase from top to bottom and left to right. %% NB: this is basic quantum mechanics, possibly not the place to add it here. I recommend we chop it out and add a reference to some QM textbook, even if we feel that it is not simple or well explained anywhere. A summary of the operation and the output of octopus would be sufficient.  All possible decorated Young diagrams for three and four electrons are shown in Fig.\ \ref{fig:young}. Since there are two different spin states for electrons, our Young diagrams for the allowed spatial wave functions contain at most two columns. The diagram d) is not allowed for the wave function of three particles with spin $1/2$, and diagrams k) to n) are not allowed for four particles. To connect a given wave function $\Psi_j$ with a diagram one has to symmetrize the wave function according to the diagram. For example, for diagram b) one would perform the following operations  \begin{equation}