### Abstract

The most recent methods in quantum chemical geometry optimization use the computed energy and its first derivatives with an approximate second derivative matrix. The performance of the optimization process depends highly on the choice of the coordinate system. In most cases the optimization is carried out in a complete internal coordinate system using the derivatives computed with respect to Cartesian coordinates. The computational bottlenecks for this process are the transformation of the derivatives into the internal coordinate system, the transformation of the resulting step back to Cartesian coordinates, and the evaluation of the Newton-Raphson or rational function optimization (RFO) step. The corresponding systems of linear equations occur as sequences of the form y_{i}=M_{i}x_{i}, where M_{i} can be regarded as a perturbation of the previous symmetric matrix M_{i-1}. They are normally solved via diagonalization of symmetric real matrices requiring O(N^{3}) operations. The current study is focused on a special approach to solving these sequential systems of linear equations using a method based on the update of the inverse of the symmetric matrix M_{i}. For convergence, this algorithm requires a number of O(N^{2}) operations with an O(N^{3}) factor for only the first calculation. The method is generalized to include redundant (singular) systems. The application of the algorithm to coordinate transformations in large molecular geometry optimization is discussed.

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

Pages (from-to) | 7100-7104 |

Number of pages | 5 |

Journal | The Journal of Chemical Physics |

Volume | 109 |

Issue number | 17 |

DOIs | |

Publication status | Published - 1998 |

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### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

**Methods for geometry optimization of large molecules. I. An O(N ^{2}) algorithm for solving systems of linear equations for the transformation of coordinates and forces.** / Farkas, O.; Schlegel, H. Bernhard.

Research output: Contribution to journal › Article

^{2}) algorithm for solving systems of linear equations for the transformation of coordinates and forces',

*The Journal of Chemical Physics*, vol. 109, no. 17, pp. 7100-7104. https://doi.org/10.1063/1.477393

}

TY - JOUR

T1 - Methods for geometry optimization of large molecules. I. An O(N2) algorithm for solving systems of linear equations for the transformation of coordinates and forces

AU - Farkas, O.

AU - Schlegel, H. Bernhard

PY - 1998

Y1 - 1998

N2 - The most recent methods in quantum chemical geometry optimization use the computed energy and its first derivatives with an approximate second derivative matrix. The performance of the optimization process depends highly on the choice of the coordinate system. In most cases the optimization is carried out in a complete internal coordinate system using the derivatives computed with respect to Cartesian coordinates. The computational bottlenecks for this process are the transformation of the derivatives into the internal coordinate system, the transformation of the resulting step back to Cartesian coordinates, and the evaluation of the Newton-Raphson or rational function optimization (RFO) step. The corresponding systems of linear equations occur as sequences of the form yi=Mixi, where Mi can be regarded as a perturbation of the previous symmetric matrix Mi-1. They are normally solved via diagonalization of symmetric real matrices requiring O(N3) operations. The current study is focused on a special approach to solving these sequential systems of linear equations using a method based on the update of the inverse of the symmetric matrix Mi. For convergence, this algorithm requires a number of O(N2) operations with an O(N3) factor for only the first calculation. The method is generalized to include redundant (singular) systems. The application of the algorithm to coordinate transformations in large molecular geometry optimization is discussed.

AB - The most recent methods in quantum chemical geometry optimization use the computed energy and its first derivatives with an approximate second derivative matrix. The performance of the optimization process depends highly on the choice of the coordinate system. In most cases the optimization is carried out in a complete internal coordinate system using the derivatives computed with respect to Cartesian coordinates. The computational bottlenecks for this process are the transformation of the derivatives into the internal coordinate system, the transformation of the resulting step back to Cartesian coordinates, and the evaluation of the Newton-Raphson or rational function optimization (RFO) step. The corresponding systems of linear equations occur as sequences of the form yi=Mixi, where Mi can be regarded as a perturbation of the previous symmetric matrix Mi-1. They are normally solved via diagonalization of symmetric real matrices requiring O(N3) operations. The current study is focused on a special approach to solving these sequential systems of linear equations using a method based on the update of the inverse of the symmetric matrix Mi. For convergence, this algorithm requires a number of O(N2) operations with an O(N3) factor for only the first calculation. The method is generalized to include redundant (singular) systems. The application of the algorithm to coordinate transformations in large molecular geometry optimization is discussed.

UR - http://www.scopus.com/inward/record.url?scp=0032209189&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032209189&partnerID=8YFLogxK

U2 - 10.1063/1.477393

DO - 10.1063/1.477393

M3 - Article

VL - 109

SP - 7100

EP - 7104

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 17

ER -