Modifications are made to a previously published algorithm for constrained optimization in Cartesian coordinates (J. Comp. Chem., 13, 240, 1992) to incorporate both fixed and dummy atoms. Standard distance and angle constraints can now be specified with respect to dummy atoms, greatly extending the range of constraints that can be handled. Fixed atoms can be eliminated from the optimization space and so there is no need to calculate their gradients resulting in potentially significant savings of CPU time in ab initio computations. Several examples illustrate the range and versatility of the modified algorithm. (C) 1993 by John Wiley & Sons, Inc.