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I will add another recommendation: CP2K. It is not written C/C++, it is written in Fortran; this might require a bit of time to adapt if you want to actually modify the code and have not encountered Fortran before. However, CP2K can do both quantum and classical mechanics simulations, and it can scale well. For example, a presentation by Jürg Hutter from ...


The matrix elements $D_{PQ}$ can either be zero if the orbital $P$ or $Q$ are not occupied -- then $a_Q\vert x\rangle=0$ or $\langle x \vert a_P^\dagger = 0$ -- and otherwise, they are overlaps of $N-1$ electron states $a_P\vert x\rangle$. If you define a matrix $X$ whose columns are $a_P\vert x\rangle$, then you can write $D=X^\dagger X$, and a matrix of ...


A few years ago the XUV physics group at the AMOLF Institute in Amsterdam were (to my knowledge the first to be) able to directly image the orbitals of excited hydrogen atoms using photoionization microscopy. For more details see the paper, Hydrogen Atoms under Magnification: Direct Observation of the Nodal Structure of Stark States. A.S. Stolodna et al. ...


The first images of hydrogen s orbitals were obtained in 2013 by physicists in the Netherlands.

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