29.9.7 Gradients and frequency calculations

Geometry optimizations [15-17] and numerical frequency calculations [18-20] can be performed using analytical energy gradients [15-17] for local MP2. LMP2 geometry optimizations are particularly attractive for weakly bound systems, since virtually BSSE free structures are obtained (see section 29.9.8 and Refs. [21-23]). For reasons of efficiency it is strongly advisable to use the DF-LMP2 Gradient [17] for all geometry optimizations. Setting SCSGRD=1 on the DF-LMP2 command or DFIT directive activates the gradient with respect to Grimmes SCS scaled MP2 energy functional (see also section DFIT). Analytical energy gradients are not yet available for the multipole approximation of distant pairs, and therefore MULTP cannot be used in geometry optimizations or frequency calculations.

In geometry optimizations, the domains are allowed to vary in the initial optimization steps. When the stepsize drops below a certain threshold (default 0.01) the domains are automatically frozen. In numerical Hessian or frequency calculations the domains are also frozen. It is therefore not necessary to include SAVE and START options.

Particular care must be taken in optimizations of highly symmetric aromatic systems, like, e.g., benzene. In $D_{6h}$ symmetry, the localization of the $\pi$-orbitals is not unique, i.e., the localized orbitals can be rotated around the $C_6$ axis without changing the localization criterion. This redundancy is lost if the symmetry is slightly distorted, which can lead to sudden changes of the localized orbitals. If now the domains are kept fixed using the SAVE and START options, a large error in the energy might result. On the other hand, if the domains are not kept fixed, their size and quality might change during the optimization, again leading to spurious energy changes and divergence of the optimization.

The best way to avoid this problem is to use the MERGEDOM=1 option (see section 29.6). If this option is given, the domains for the $\pi$ orbitals will comprise the basis functions of all six carbon atoms, and the energy will be invariant with respect to unitary transformations among the three $\pi$ orbitals. Note that this problem does not occur if the symmetry of the aromatic system is lowered by a substituent.

Redundant orbital rotations can also lead to convergence difficulties of the Pipek-Mezey localization. This can be overcome by using

PIPEK,METHOD=2

or

PIPEK,METHOD=3

With METHOD=2, the second derivatives of the localization criterion with respect to the orbital rotations is computed and diagonalized, and rotations corresponding to zero eigenvalues are eliminated. This method converges quadratically. With METHOD=3 first a few iterations with the standard Pipek-Mezey method are performed, then the second-order method is invoked. This appears to be the most robust and accurate localization method.

Finally, we note that the LMP2 gradients are quite sensitive to the accuracy of the SCF convergence (as is also the case for MP2). If very accurate structures are required, or if numerical frequencies are computed from the gradients, the default SCF accuracy might be insufficient. We recommend in such cases to add an ACCU,14 directive (possibly even ACCU,16) after the HF command. Indicative of insufficient SCF accuracy are small positive energy changes near the end of the geometry optimization.