# Concepts Of Theoretical Solid State Physics by Altland & Simons

By Altland & Simons

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**Example text**

5. Si9 H12 cluster used to model a Si(100) substrate. The silicon atoms are plotted as the darker large balls while the hydrogen atoms are presented by the lighter small balls surface. At an ideal surface there are two broken Si-Si bonds per surface atom. In order to reduce the number of these dangling bonds that correspond to unsaturated orbitals, the Si(100) surface reconstructs by creating Si dimers at the surface. The Si9 H12 cluster just contains one of these surface dimers. The hydrogen atoms are added to the Si cluster in order to saturate the silicon dangling bonds that would be bonded to other silicon atoms.

R N ) = ψ1 (r 1 ) · . . 5) in which every one-particle state is only occupied once, as a ﬁrst crude guess for the true many-particle wave function. 5). Thus we obtain N d3 rψi∗ (r) − ΨH |H|ΨH = i=1 N + 1 2 i,j=1 d3 rd3 r ¯2 2 h ∇ + vext (r) ψi (r) 2m e2 |ψi (r)|2 |ψj (r )|2 + Vnucl−nucl . 6) with respect to more suitable single-particle functions ψi (r) under the constraint that the wave functions are normalized. This is a typical variational problem with the constraint taken into account via Lagrange multipliers.

In addition, it is energydependent. This constitutes the diﬀerence to a “true” potential and is the reason why it is called a “pseudo-potential”. 72) acts as a short-range repulsive potential. Note that the wave function ψps (r) has the same one-particle energy εv as the true valence wave function ψv (r). Furthermore, ψps (r) does not need to be orthogonal to the core states. Consequently, it does not have to have a nodal structure in the core region. Therefore one has the freedom to choose a smooth pseudo-wave function which is rather advantageous when the wave function is expanded in some set of basis functions.