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   -> Volume 3, Issue 18

Question: Wavelets for solide state calculations
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Alexander Schmid (

PostPosted: Mon Dec 02, 2002 1:28 pm    
Subject: Question: Wavelets for solide state calculations
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Question: Wavelets for solide state calculations

Dear Waveletters,

I am considering to compute the wavelet transform of the Schrödinger
equation in a solid. The procedure is the following:

Commonly the pseudopotential method [Ref 1] and the local density
approximation in the density functional formalism [Ref 2] are
carried out for solide state systems. The Schrödinger equation
for an electron with band index $n$ and wave vector ${f k}$ is
written as: $( -{hbar^2over 2m}
abla^2 + V_{eff} )|Psi_{n{f k}}> =
varepsilon_{n{f k}}|Psi_{n{f k}}>$ (1).
For a crystalline solid with a set of reciprocal lattice vectors
${f G}$ and a set of Bloch functions $|{f k}j>$ chosen as the
basis functions to expand the wave function $Psi$:
$|Psi_{n{f k}}> = sum_j c_j^{n{f k}}|{f k}j>$ (2),
where j is a general composite index, Eq. (1) becomes a generalized
matrix equation:
$sum_j <bf k}i|( -{hbar^2over 2m}
abla^2 +
V_{eff}) |{f k}j> c_j^{n{f k}} =
sum_jvarepsilon_{n{f k}}<{f k}i|{f k}j> c_j^{n{f k}}$.
In practice the expansion in Eq. (2) must be truncated at some point,
so that a small number of basis functions gives an accurate
description of the crystal wave function $|Psi_{n{f k}}>$.
Traditional density functional calculations use plane wave basis
sets [Ref 3], where even simple crystal structures (e.g. NaCl, zincblende)
requires a matrix size of $approx 2500$.

Therefore I am looking for a ( wavelet ) basis for $L^2(R^3)$, which

- is independent on the location of the ions in the solid
- is explicitly dependent on the wave vector ${f k}$
- has fast convergence.


[1] G. B. Bachelet, D. R. Hamann, M. Schl"uter,
Phys. Rev. B {f 26}, 4199 (1982)
N. Toullier, J. L. Martins,
Phys. Rev. B {f 43}, 1993 (1991)

[2] P. Hohenberg, W. Kohn,
Phys. Rev. {f 136}, B864 (1964)
W. Kohn, L. J. Sham
Phys. Rev. {f 140}, A1133 (1965)

[3] J. Ihm, A. Zunger, M. l. Cohen
J. Phys. C {f 12}, 4409 (1979)
W. E. Pickett
Comp. Phys. Rep. {f 9}, 115 (1989)

Thank you in advance,

Alexander Schmid
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