The Calculation of ab Initio Molecular Geometries: Efficient Optimization by Natural Internal Coordinates and Empirical Correction by Offset Forces

Geza Fogarasi, Xuefeng Zhou, Patterson W. Taylor, Peter Pulay

Research output: Contribution to journalArticle

852 Citations (Scopus)


Two suggestions are made to increase the efficiency and accuracy of ab initio optimization of molecular geometries. To improve the convergence of the optimization, a set of internal coordinates, the natural valence coordinates, is suggested. These coordinates originate from vibrational spectroscopy and reduce both harmonic and anharmonic coupling terms in the potential function as much as possible in a purely geometrical definition. The natural valence coordinates are local, eliminate most redundancies, and conform to local pseudosymmetry. Special attention has been paid to ring systems. A computer program has been included in our program system TX90 to generate the natural internal coordinates automatically. The usefulness of these coordinates is demonstrated by numerous examples of ab initio geometry optimization. Starting with a geometry preoptimized by molecular mechanics and using a simple diagonal estimate of the Hessian in conjunction with the GDIIS optimization technique, we usually achieved convergence in 8-15 steps, even for large molecules. It is demonstrated that, due to the reduction in anharmonic couplings, natural coordinates are superior to Cartesian or other simple internal coordinates, even when an accurate initial Hessian is available. Constrained optimization and the location of transition states are also discussed. The gradient optimization method has been generalized to handle redundancies; this is necessary in some complex polycyclic molecules and is illustrated on, among others, the porphine molecule. To increase the accuracy of relatively low-level calculations, empirical corrections to ab initio SCF geometries are suggested in the form of “offset forces” acting along bonds. We recommend offset forces for the most important bonds, to be used with the 4-21G(*) and the 6-31G* basis sets. Based on 130 comparisons, the mean absolute error between theoretical and experimental bond lengths is reduced this way from 0.014 to 0.005 Å.

Original languageEnglish
Pages (from-to)8191-8201
Number of pages11
JournalJournal of the American Chemical Society
Issue number21
Publication statusPublished - Oct 1 1992


ASJC Scopus subject areas

  • Catalysis
  • Chemistry(all)
  • Biochemistry
  • Colloid and Surface Chemistry

Cite this