A first example chosen here for the sake of comparison is that of neutral alanine. A complete input file for optimization of this system at the HF/6-31G(d) level of theory can be found here. The total electronic energy of the system at its starting point is -321.8330969au. The following table summarizes the results obtained in geometry optimizations using the the three different coordinate systems:

coordinates opt. cycles final energy (Hartree) CPU-time (sec.) CPU-time (per step) internal 18 -321.8647897 840 47 cartesian 57 -321.8647910 2627 46 redundant 18 -321.8647911 833 46

One can see that the geometry optimization proceeds equally well in Z-Matrix and in redundant internal coordinates in this case. Optimization in cartesian coordinates is, in contrast, much less efficient. This is reflected in both the number of optimization cycles as well as the total CPU time needed for the geometry optimization (performed here on Xeon/2.6GHz Linux PCs). The CPU time needed for each of the optimization cycles is, however, almost identical for all three options. The HF/6-31G(d) optimized structure can be viewed by clicking on the alanine molecule.

A second, slightly larger, model system has been chosen with the end-capped alanyl alanine (see structure below). The N-terminal end is here capped with the acetyl group, the C-terminal end with an N-methyl amide. For quantum mechanical methods this dipeptide, containing 32 atoms and 90 independent structural degrees of freedom, might already be considered a medium sized to large molecule. A complete input file for optimization of this system at the HF/6-31G(d) level of theory can be found here. Optimization of this systems has been performed again using the HF/6-31G(d) method from the same starting structure with total energy of -738.6310951 Hartree in all three coordinate systems available in Gaussian. The following results have been obtained:

coordinates opt. cycles final energy (Hartree) CPU-time (sec.) CPU-time (per step) internal 43 -738.7149776 19313 449 cartesian >318 (-738.7149762) >144353 454 redundant 40 -738.7149777 17141 429

For this very flexible system the optimization in redundant internals slightly outruns the optimization in Z-Matrix coordinates, while the optimization in cartesian coordinates is incomplete even after 318 optimization cycles. Optimization in redundant internals is particularly effective in the endgame and uses less SCF cycles per geometry optimization step in this last part of the optimization . On average this leads to slightly lower CPU times per cycle than for optimizations in Z-Matrix coordinates. This result is typical for large, unsymmetric structures. A different relative performance can be expected for smaller or highly symmetric systems.

last changes: 28.10.2004, HZ
questions & comments to: zipse@cup.uni-muenchen.de