Long-Range Effects in Optimizing the Geometry of Stereoregular Polymers—IV: Explicit Determination of the Helical Angle DENIS JACQUEMIN, BENOÎT CHAMPAGNE Laboratoire de Chimie Théorique Appliquée, Facultés Universitaires Notre-Dame de la Paix, rue de Bruxelles, 61, B-5000 Namur, Belgium Received 24 February 2001; revised 3 April 2001; accepted 4 April 2001 ABSTRACT: We study the converging behavior of the analytical gradient of the energy per unit cell with respect to the helical angle in stereoregular polymers. Similarly to the other forces (with respect to nuclei coordinates or unit cell length), this gradient presents unique particularities. A technique based on multiple Taylor expansions is developed in order to evaluate accurately and directly the long-range contributions to this gradient. These LR contributions have analytical characteristics similar to the one of the stress-tensor, although the latter presents diverging sums. The method is tested on model system and is shown to be extremely efficient, allowing great improvement in accuracy at a negligible cost. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem 85: 539–545, 2001 Key words: Hartree–Fock; crystal orbital; helical polymer; geometry optimization Introduction I n order to tackle the properties of macromole- cules, two strategies may be chosen. In the oligomeric approach, one computes the desired properties on increasingly large oligomers and ex- trapolates to the polymeric (infinite chain) limit. Correspondence to: D. Jacquemin; e-mail: djacquem@scf. fundp.ac.be. Contract grant sponsor: FNRS-FRFC, “Loterie Nationale.” Contract grant number: 2.4519.97. In general, this oligomeric approach does not re- quire any new theoretical developments and is thus very convenient. However, in conjugated sys- tems, the saturation of the properties with respect to chain length may be extremely slow, especially for vibrational frequencies or (hyper)polarizabilities which are connected to high-order derivatives of the total energy. As a consequence, very long oligomers may be necessary in order to obtain ac- curate extrapolations. In the polymeric approach (sometimes referred to as crystal orbital (CO) tech- niques), one considers explicitly the translational symmetry of the infinite chain; this allows an eval- International Journal of Quantum Chemistry, Vol. 85, 539–545 (2001) © 2001 John Wiley & Sons, Inc.