### Abstract

A new, efficient algorithm for the evaluation of the matrix elements of the CI Hamiltonian in the basis of spin-coupled v-fold excitations (over orthonormal orbitals) is developed for even electron systems. For this purpose we construct an orthonormal, spin-adapted CI basis in the framework of second quantization. As a prerequisite, spin and space parts of the fermion operators have to be separated; this makes it possible to introduce the representation theory of the permutation group. The v-fold excitation operators are Serber spin-coupled products of particle-hole excitations. This construction is also designed for CI calculations from multireference (open-shell) states. The 2N-electron Hamiltonian is expanded in terms of spin-coupled particle-hole operators which map any v-fold excitation on v-, v±1-, and v±2-fold excitations. For the calculation of the CI matrix this leaves one with only the evaluation of overlap matrix elements between spin-coupled excitations. This leads to a set of ten general matrix element formulas which contain Serber representation matrices of the permutation group S^{v} × S^{v} as parameters. Because of the Serber structure of the CI basis these group-theoretical parameters are kept to a minimum such that they can be stored readily in the central memory of a computer for v ≤ 4 and even for higher excitations. As the computational effort required to obtain the CI matrix elements from the general formulas is very small, the algorithm presented appears to constitute for even electron systems a promising alternative to existing CI methods for multiply excited configurations, e.g., the unitary group approach. Our method makes possible the adaptation of spatial symmetries and the selection of any subset of configurations. The algorithm has been implemented in a computer program and tested extensively for v ≤ 4 and singlet ground and excited states.

Original language | English (US) |
---|---|

Pages (from-to) | 3547-3576 |

Number of pages | 30 |

Journal | The Journal of Chemical Physics |

Volume | 72 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 1980 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Physical and Theoretical Chemistry

### Cite this

*The Journal of Chemical Physics*,

*72*(6), 3547-3576. https://doi.org/10.1063/1.439617

**An efficient approach to CI : General matrix element formulas for spin-coupled particle-hole excitations.** / Tavan, Paul; Schulten, Klaus.

Research output: Contribution to journal › Article

*The Journal of Chemical Physics*, vol. 72, no. 6, pp. 3547-3576. https://doi.org/10.1063/1.439617

}

TY - JOUR

T1 - An efficient approach to CI

T2 - General matrix element formulas for spin-coupled particle-hole excitations

AU - Tavan, Paul

AU - Schulten, Klaus

PY - 1980/1/1

Y1 - 1980/1/1

N2 - A new, efficient algorithm for the evaluation of the matrix elements of the CI Hamiltonian in the basis of spin-coupled v-fold excitations (over orthonormal orbitals) is developed for even electron systems. For this purpose we construct an orthonormal, spin-adapted CI basis in the framework of second quantization. As a prerequisite, spin and space parts of the fermion operators have to be separated; this makes it possible to introduce the representation theory of the permutation group. The v-fold excitation operators are Serber spin-coupled products of particle-hole excitations. This construction is also designed for CI calculations from multireference (open-shell) states. The 2N-electron Hamiltonian is expanded in terms of spin-coupled particle-hole operators which map any v-fold excitation on v-, v±1-, and v±2-fold excitations. For the calculation of the CI matrix this leaves one with only the evaluation of overlap matrix elements between spin-coupled excitations. This leads to a set of ten general matrix element formulas which contain Serber representation matrices of the permutation group Sv × Sv as parameters. Because of the Serber structure of the CI basis these group-theoretical parameters are kept to a minimum such that they can be stored readily in the central memory of a computer for v ≤ 4 and even for higher excitations. As the computational effort required to obtain the CI matrix elements from the general formulas is very small, the algorithm presented appears to constitute for even electron systems a promising alternative to existing CI methods for multiply excited configurations, e.g., the unitary group approach. Our method makes possible the adaptation of spatial symmetries and the selection of any subset of configurations. The algorithm has been implemented in a computer program and tested extensively for v ≤ 4 and singlet ground and excited states.

AB - A new, efficient algorithm for the evaluation of the matrix elements of the CI Hamiltonian in the basis of spin-coupled v-fold excitations (over orthonormal orbitals) is developed for even electron systems. For this purpose we construct an orthonormal, spin-adapted CI basis in the framework of second quantization. As a prerequisite, spin and space parts of the fermion operators have to be separated; this makes it possible to introduce the representation theory of the permutation group. The v-fold excitation operators are Serber spin-coupled products of particle-hole excitations. This construction is also designed for CI calculations from multireference (open-shell) states. The 2N-electron Hamiltonian is expanded in terms of spin-coupled particle-hole operators which map any v-fold excitation on v-, v±1-, and v±2-fold excitations. For the calculation of the CI matrix this leaves one with only the evaluation of overlap matrix elements between spin-coupled excitations. This leads to a set of ten general matrix element formulas which contain Serber representation matrices of the permutation group Sv × Sv as parameters. Because of the Serber structure of the CI basis these group-theoretical parameters are kept to a minimum such that they can be stored readily in the central memory of a computer for v ≤ 4 and even for higher excitations. As the computational effort required to obtain the CI matrix elements from the general formulas is very small, the algorithm presented appears to constitute for even electron systems a promising alternative to existing CI methods for multiply excited configurations, e.g., the unitary group approach. Our method makes possible the adaptation of spatial symmetries and the selection of any subset of configurations. The algorithm has been implemented in a computer program and tested extensively for v ≤ 4 and singlet ground and excited states.

UR - http://www.scopus.com/inward/record.url?scp=0001396032&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001396032&partnerID=8YFLogxK

U2 - 10.1063/1.439617

DO - 10.1063/1.439617

M3 - Article

AN - SCOPUS:0001396032

VL - 72

SP - 3547

EP - 3576

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 6

ER -