### Abstract

It is shown that the Bloch function, as an element of the Hilbert space spanned by Bloch-periodic planes waves, can be also represented as an on-shell superposition of Bloch-periodic orbitals which, however, exist in a distribution sense. By using this result and the multipole expansion of the Bloch function at the origin, it is further shown that the integral eigenvalue equation of the Bloch function for a general periodic potential is equivalent (both necessary and sufficient) to an algebraic system of homogeneous linear equations for the coefficients of the multipole expansion of the Bloch function at the origin, akin to the much simpler case of a finite-range potential. In contrast to the Korringa-Kohn-Rostoker (KKR) equation, the contribution of the cell potential (whether of the muffin-tin or general form) introduces a supplementary structure dependence. However, the separation between structure and potential, typical for the KKR equation, can be restored by introducing various approximations.

Original language | English |
---|---|

Pages (from-to) | 1067-1084 |

Number of pages | 18 |

Journal | Physical Review B |

Volume | 37 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1988 |

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### ASJC Scopus subject areas

- Condensed Matter Physics

### Cite this

*Physical Review B*,

*37*(3), 1067-1084. https://doi.org/10.1103/PhysRevB.37.1067

**Eigenvalue equation for a general periodic potential and its multipole expansion solution.** / Badralexe, E.; Freeman, Arthur J.

Research output: Contribution to journal › Article

*Physical Review B*, vol. 37, no. 3, pp. 1067-1084. https://doi.org/10.1103/PhysRevB.37.1067

}

TY - JOUR

T1 - Eigenvalue equation for a general periodic potential and its multipole expansion solution

AU - Badralexe, E.

AU - Freeman, Arthur J

PY - 1988

Y1 - 1988

N2 - It is shown that the Bloch function, as an element of the Hilbert space spanned by Bloch-periodic planes waves, can be also represented as an on-shell superposition of Bloch-periodic orbitals which, however, exist in a distribution sense. By using this result and the multipole expansion of the Bloch function at the origin, it is further shown that the integral eigenvalue equation of the Bloch function for a general periodic potential is equivalent (both necessary and sufficient) to an algebraic system of homogeneous linear equations for the coefficients of the multipole expansion of the Bloch function at the origin, akin to the much simpler case of a finite-range potential. In contrast to the Korringa-Kohn-Rostoker (KKR) equation, the contribution of the cell potential (whether of the muffin-tin or general form) introduces a supplementary structure dependence. However, the separation between structure and potential, typical for the KKR equation, can be restored by introducing various approximations.

AB - It is shown that the Bloch function, as an element of the Hilbert space spanned by Bloch-periodic planes waves, can be also represented as an on-shell superposition of Bloch-periodic orbitals which, however, exist in a distribution sense. By using this result and the multipole expansion of the Bloch function at the origin, it is further shown that the integral eigenvalue equation of the Bloch function for a general periodic potential is equivalent (both necessary and sufficient) to an algebraic system of homogeneous linear equations for the coefficients of the multipole expansion of the Bloch function at the origin, akin to the much simpler case of a finite-range potential. In contrast to the Korringa-Kohn-Rostoker (KKR) equation, the contribution of the cell potential (whether of the muffin-tin or general form) introduces a supplementary structure dependence. However, the separation between structure and potential, typical for the KKR equation, can be restored by introducing various approximations.

UR - http://www.scopus.com/inward/record.url?scp=0012580182&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0012580182&partnerID=8YFLogxK

U2 - 10.1103/PhysRevB.37.1067

DO - 10.1103/PhysRevB.37.1067

M3 - Article

AN - SCOPUS:0012580182

VL - 37

SP - 1067

EP - 1084

JO - Physical Review B-Condensed Matter

JF - Physical Review B-Condensed Matter

SN - 1098-0121

IS - 3

ER -