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AB-INITIO SIMULATIONS IN MATERIALS SCIENCE - VASP

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Born-Oppenheimer approximation I<br />

Hamiltonian of the coupled electron-ion system:<br />

N ions, coordinates<br />

R 1<br />

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R N<br />

R, momenta<br />

P 1<br />

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P N<br />

P, charges Z 1<br />

ZN , ¢<br />

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masses M I<br />

Mn ¢<br />

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N e electrons, coordinates<br />

r 1<br />

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r N<br />

r, momenta<br />

p 1<br />

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p N<br />

p, mass m<br />

H<br />

N<br />

∑<br />

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P 2 I<br />

2M I<br />

N e<br />

∑<br />

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p 2 i<br />

2m©∑<br />

ij<br />

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r i<br />

e 2<br />

r j<br />

IJ<br />

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r ©VNN<br />

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Z I Z J e 2<br />

R i<br />

R J<br />

R ©VNe<br />

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r<br />

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R I<br />

Z I e 2<br />

r i<br />

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Schrödinger equation<br />

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R ©VNe<br />

r<br />

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R<br />

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R<br />

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x<br />

r<br />

s <br />

full set of electronic positions and spin variables<br />

J. HAFNER, <strong>AB</strong>-<strong><strong>IN</strong>ITIO</strong> <strong>MATERIALS</strong> <strong>SIMULATIONS</strong> Page 5

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