### Abstract

A new approachone not based on multiple scattering or the variational principleto the problem of the band structure for a general (i.e., non-muffin-tin) periodic potential is presented. From the eigenvalue equation, the Bloch function is shown to be expandable as a multipole series around the origin and the coefficients are found to be given as functionals of the Bloch function and the cell potential. By introducing into this functional various representations of the Bloch function (as a superposition in which each individual term is not Bloch periodic), we obtain previously derived band-structure equations (some claimed to be exact). By deriving, however, a new representation for the Bloch function (as an on-shell superposition in which each term is Bloch periodic) and using this representation in the above functional, we obtain a new energy-band equation in which the potential can no longer be separated from the structure. When approximations are introduced, then Korringa-Kohn-Rostoker-type equations (in which the potential is separated from the structure) may be obtained. All the steps of this new approach are illustrated in the soluble case of a constant periodic potential. It is shown, in this case, that only the newly introduced band-structure equation is able to generate the correct eigenvalues.

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

Pages (from-to) | 1378-1388 |

Number of pages | 11 |

Journal | Physical Review B |

Volume | 36 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1987 |

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### ASJC Scopus subject areas

- Condensed Matter Physics

### Cite this

*Physical Review B*,

*36*(3), 1378-1388. https://doi.org/10.1103/PhysRevB.36.1378

**Energy-band equation for a general periodic potential.** / Badralexe, E.; Freeman, Arthur J.

Research output: Contribution to journal › Article

*Physical Review B*, vol. 36, no. 3, pp. 1378-1388. https://doi.org/10.1103/PhysRevB.36.1378

}

TY - JOUR

T1 - Energy-band equation for a general periodic potential

AU - Badralexe, E.

AU - Freeman, Arthur J

PY - 1987

Y1 - 1987

N2 - A new approachone not based on multiple scattering or the variational principleto the problem of the band structure for a general (i.e., non-muffin-tin) periodic potential is presented. From the eigenvalue equation, the Bloch function is shown to be expandable as a multipole series around the origin and the coefficients are found to be given as functionals of the Bloch function and the cell potential. By introducing into this functional various representations of the Bloch function (as a superposition in which each individual term is not Bloch periodic), we obtain previously derived band-structure equations (some claimed to be exact). By deriving, however, a new representation for the Bloch function (as an on-shell superposition in which each term is Bloch periodic) and using this representation in the above functional, we obtain a new energy-band equation in which the potential can no longer be separated from the structure. When approximations are introduced, then Korringa-Kohn-Rostoker-type equations (in which the potential is separated from the structure) may be obtained. All the steps of this new approach are illustrated in the soluble case of a constant periodic potential. It is shown, in this case, that only the newly introduced band-structure equation is able to generate the correct eigenvalues.

AB - A new approachone not based on multiple scattering or the variational principleto the problem of the band structure for a general (i.e., non-muffin-tin) periodic potential is presented. From the eigenvalue equation, the Bloch function is shown to be expandable as a multipole series around the origin and the coefficients are found to be given as functionals of the Bloch function and the cell potential. By introducing into this functional various representations of the Bloch function (as a superposition in which each individual term is not Bloch periodic), we obtain previously derived band-structure equations (some claimed to be exact). By deriving, however, a new representation for the Bloch function (as an on-shell superposition in which each term is Bloch periodic) and using this representation in the above functional, we obtain a new energy-band equation in which the potential can no longer be separated from the structure. When approximations are introduced, then Korringa-Kohn-Rostoker-type equations (in which the potential is separated from the structure) may be obtained. All the steps of this new approach are illustrated in the soluble case of a constant periodic potential. It is shown, in this case, that only the newly introduced band-structure equation is able to generate the correct eigenvalues.

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U2 - 10.1103/PhysRevB.36.1378

DO - 10.1103/PhysRevB.36.1378

M3 - Article

VL - 36

SP - 1378

EP - 1388

JO - Physical Review B-Condensed Matter

JF - Physical Review B-Condensed Matter

SN - 1098-0121

IS - 3

ER -