### Abstract

This article aims to dispel confusions about the definition of size consistency as well as some incompatibility that exists between different criteria for judging whether an electronic structure theory is size consistent and thus yields energies and other quantities having correct asymptotic size dependence. It introduces extensive and intensive diagram theorems, which provide unambiguous sufficient conditions for size consistency for extensive and intensive quantities, respectively, stipulated in terms of diagrammatic topology and vertex makeup. The underlying algebraic size-consistency criterion is described, which relies on the polynomial dependence of terms in the formalism on the number of wave vector sampling points in Brillouin-zone integrations. The physical meanings of two types of normalization of excitation amplitudes in electron-correlation theories, namely, the intermediate and standard normalization, are revealed. The amplitudes of the operator that describes an extensive quantity (the extensive operator) are subject to the intermediate normalization, while those of the operator that corresponds to an intensive quantity (the intensive operator) must be normalized. The article also introduces the extensive-intensive consistency theorem which specifies the relationship between the spaces of determinants reached by the extensive and intensive operators in a size-consistent method for intensive quantities. Furthermore, a more fundamental question is addressed as to what makes energies extensive and thus an application of thermodynamics to chemistry valid. It is shown that the energy of an electrically neutral, periodic, non-metallic solid is extensive. On this basis, a strictly size-consistent redefinition of the Hartree-Fock theory is proposed.

Original language | English (US) |
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Pages (from-to) | 727-746 |

Number of pages | 20 |

Journal | Theoretical Chemistry Accounts |

Volume | 129 |

Issue number | 6 |

DOIs | |

State | Published - Aug 1 2011 |

### Keywords

- Diagrams
- Electronic structure theory
- Extensive diagram theorem
- Intensive diagram theorem
- Size consistency
- Thermodynamic limit

### ASJC Scopus subject areas

- Physical and Theoretical Chemistry