deletions | additions       diff --git a/Schroedinger equation.tex b/Schroedinger equation.tex  index f23a412..c949022 100644  --- a/Schroedinger equation.tex  +++ b/Schroedinger equation.tex 
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\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 is the basic machinery for solving the Sch\"odinger equation iteratively, the spatial/grid bookkeeping, 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$_2$ (or D$_2$) and H$_2^+$ e.g. in \cite{Ranitovic21012014} or \cite{lein2002}) or model interactions\cite{luo2013}, including time dependent cases\cite{ramsden2012}.  %% @article{Ranitovic21012014, author = {Ranitovic, Predrag and Hogle, Craig W. and Rivière, Paula and Palacios, Alicia and Tong, Xiao-Ming and Toshima, Nobuyuki and González-Castrillo, Alberto and Martin, Leigh and Martín, Fernando and Murnane, Margaret M. and Kapteyn, Henry}, title = {Attosecond vacuum UV coherent control of molecular dynamics}, volume = {111}, number = {3}, pages = {912-917}, year = {2014}, optdoi = {10.1073/pnas.1321999111}, journal = {Proceedings of the National Academy of Sciences} }  %% @article{lein2002, title = {Strong-field ionization dynamics of a model ${\mathrm{H}}_{2}$ molecule}, author = {Lein, Manfred and Kreibich, Thomas and Gross, E. K. U. and Engel, Volker}, journal = {Phys. Rev. A}, volume = {65}, issue = {3}, pages = {033403}, year = {2002}, optdoi = {10.1103/PhysRevA.65.033403}}  %% @article{luo2013, title = {Absence of dynamical steps in the exact correlation potential in the linear response regime}, author = {Luo, Kai and Elliott, Peter and Maitra, Neepa T.}, journal = {Phys. Rev. A}, volume = {88}, issue = {4}, pages = {042508}, year = {2013}, optdoi = {10.1103/PhysRevA.88.042508} }  %%@article{ramsden2012, title = {Exact Density-Functional Potentials for Time-Dependent Quasiparticles}, author = {Ramsden, J. D. and Godby, R. W.}, journal = {Phys. Rev. Lett.}, volume = {109}, issue = {3}, pages = {036402}, year = {2012}, optdoi = {10.1103/PhysRevLett.109.036402}}  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).