### Abstract

The paper gives an overview, generalization and systematization of the different energy decomposition schemes we have devised in the last few years by using both the 3-D analysis (the atoms are represented by different parts of the physical space) and the Hilbert space analysis in terms of the basis orbitals assigned to the individual atoms. The so called "atomic decomposition of identity" provides us the most general formalism for analyzing different physical quantities in terms of individual atoms or pairs of atoms. (The "atomic decomposition of identity" means that we present the identity operator as a sum of operators assigned to the individual atoms.) By proper definitions of the atomic operators, both Hilbert-space and the different 3-D decomposition schemes can be treated on an equal footing. Several different but closely related energy decomposition schemes have been proposed for the Hilbert space analysis. They differ by exact or approximate treatment of the three- and four-center integrals and by considering the kinetic energy as a part of the atomic Hamiltonian or as having genuine two-center components, too. (Also, some finite basis correction terms may be treated in different manners.) The exact schemes are obtained by using the "atomic decomposition of identity". In the approximate schemes a projective integral approximation is also introduced, thus the energy components contain only one- and two-center integrals. The diatomic energy contributions have also been decomposed into terms of different physical nature (electrostatic, exchange etc.) The 3-D analysis may be performed either in terms of disjunct atomic domains, as in the case of the AIM formalism, or by using the so called "fuzzy atoms" which do not have sharp boundaries but exhibit a continuous transition from one to another. The different schemes give different numbers, but each is capable of reflecting the most important intramolecular interactions as well as the secondary ones - e.g. intramolecular interactions of type C-H⋯O.

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

Pages (from-to) | 4630-4646 |

Number of pages | 17 |

Journal | Physical Chemistry Chemical Physics |

Volume | 8 |

Issue number | 40 |

DOIs | |

Publication status | Published - 2006 |

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### ASJC Scopus subject areas

- Physical and Theoretical Chemistry
- Atomic and Molecular Physics, and Optics

### Cite this

*Physical Chemistry Chemical Physics*,

*8*(40), 4630-4646. https://doi.org/10.1039/b608822h

**Energy partitioning schemes.** / Mayer, I.

Research output: Contribution to journal › Article

*Physical Chemistry Chemical Physics*, vol. 8, no. 40, pp. 4630-4646. https://doi.org/10.1039/b608822h

}

TY - JOUR

T1 - Energy partitioning schemes

AU - Mayer, I.

PY - 2006

Y1 - 2006

N2 - The paper gives an overview, generalization and systematization of the different energy decomposition schemes we have devised in the last few years by using both the 3-D analysis (the atoms are represented by different parts of the physical space) and the Hilbert space analysis in terms of the basis orbitals assigned to the individual atoms. The so called "atomic decomposition of identity" provides us the most general formalism for analyzing different physical quantities in terms of individual atoms or pairs of atoms. (The "atomic decomposition of identity" means that we present the identity operator as a sum of operators assigned to the individual atoms.) By proper definitions of the atomic operators, both Hilbert-space and the different 3-D decomposition schemes can be treated on an equal footing. Several different but closely related energy decomposition schemes have been proposed for the Hilbert space analysis. They differ by exact or approximate treatment of the three- and four-center integrals and by considering the kinetic energy as a part of the atomic Hamiltonian or as having genuine two-center components, too. (Also, some finite basis correction terms may be treated in different manners.) The exact schemes are obtained by using the "atomic decomposition of identity". In the approximate schemes a projective integral approximation is also introduced, thus the energy components contain only one- and two-center integrals. The diatomic energy contributions have also been decomposed into terms of different physical nature (electrostatic, exchange etc.) The 3-D analysis may be performed either in terms of disjunct atomic domains, as in the case of the AIM formalism, or by using the so called "fuzzy atoms" which do not have sharp boundaries but exhibit a continuous transition from one to another. The different schemes give different numbers, but each is capable of reflecting the most important intramolecular interactions as well as the secondary ones - e.g. intramolecular interactions of type C-H⋯O.

AB - The paper gives an overview, generalization and systematization of the different energy decomposition schemes we have devised in the last few years by using both the 3-D analysis (the atoms are represented by different parts of the physical space) and the Hilbert space analysis in terms of the basis orbitals assigned to the individual atoms. The so called "atomic decomposition of identity" provides us the most general formalism for analyzing different physical quantities in terms of individual atoms or pairs of atoms. (The "atomic decomposition of identity" means that we present the identity operator as a sum of operators assigned to the individual atoms.) By proper definitions of the atomic operators, both Hilbert-space and the different 3-D decomposition schemes can be treated on an equal footing. Several different but closely related energy decomposition schemes have been proposed for the Hilbert space analysis. They differ by exact or approximate treatment of the three- and four-center integrals and by considering the kinetic energy as a part of the atomic Hamiltonian or as having genuine two-center components, too. (Also, some finite basis correction terms may be treated in different manners.) The exact schemes are obtained by using the "atomic decomposition of identity". In the approximate schemes a projective integral approximation is also introduced, thus the energy components contain only one- and two-center integrals. The diatomic energy contributions have also been decomposed into terms of different physical nature (electrostatic, exchange etc.) The 3-D analysis may be performed either in terms of disjunct atomic domains, as in the case of the AIM formalism, or by using the so called "fuzzy atoms" which do not have sharp boundaries but exhibit a continuous transition from one to another. The different schemes give different numbers, but each is capable of reflecting the most important intramolecular interactions as well as the secondary ones - e.g. intramolecular interactions of type C-H⋯O.

UR - http://www.scopus.com/inward/record.url?scp=33750581324&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750581324&partnerID=8YFLogxK

U2 - 10.1039/b608822h

DO - 10.1039/b608822h

M3 - Article

C2 - 17047759

AN - SCOPUS:33750581324

VL - 8

SP - 4630

EP - 4646

JO - Physical Chemistry Chemical Physics

JF - Physical Chemistry Chemical Physics

SN - 1463-9076

IS - 40

ER -