Finding all critical points (CP's) of the electron density is a difficult problem, aggravated in crystalline systems because the number of CP's in each unit cell can be very large. We have designed a new and efficient algorithm that exhaustively exploits the traslational and point group symmetries of the crystal to authomatically locate all independent CP's.
The ionic wavefunctions obtained from a final aiPI calculation at the optimal geometry have been used in the critic program to determine the crystal wavefunction for each crystal. Ions up to 3 cell side lengths contribute to the wavefunction at any point. The program implements a new algorithm to locate automatically all CP's of the electronic density. The algorithm works by first determining the irreducible zone of the Wigner-Seitz cell (IWSZ). The IWSZ is the smallest polyhedron that reproduces the complete crystal by the exhaustive application of the space group symmetry operations. Vertices of the IWSZ are guaranteed to be CP's. Similarly, every edge and face of the IWSZ contains at least one CP, and further CP's could be found within the IWSZ. The search algorithm explores the corresponding one-, two- and three-dimensional regions. Every time a new CP is found in a region, the region is splitted into two or more new ones and a further search is performed on each. As for the searching criterion, we have found that minimizing the square of the density gradient is far more efficient and less error prone than other navigation techniques based on the gradient and hessian of the function.
The automatic routine has worked remarkably well in most crystals. Early runs that included an insuficient number of ions in the evaluation of the wavefunction were unable to locate some CP's in some crystals. Once this problem was solved, the only failures of the automatic procedure were: (a) false positives of points that showed rather large gradients and that were easily discarded by inspection; and (b) remarkably, Wyckoff's 3c position, which is a CP fixed by symmetry (see below), was considered doubtful in some crystals by the automatic discard criteria. Morse relationships and similarities between compounds have been analyzed to make sure that all CP's were found in each case.