26.3.4 Gradients and frequency calculations

Geometry optimizations [4,5] and numerical frequency calculations [5] can be performed using analytical energy gradients [4] for local MP2. LMP2 geometry optimizations are particularly attractive for weakly bound systems, since virtually BSSE free structures are obtained (see section 22.3.3 and Refs. [6,7]). It should be noted, however, that the current implementation is not particularly efficient, and nothing has been done so far to achieve low-order scaling for large systems. 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.

A particular problem in LMP2 gradient calculations is the elimination of redundant basis functions in the domains (see Refs. [1,4]). If redundancies are present, gradient calculations require the elimination of individual basis functions (option DELBAS=1), which is less unique than the elimination of eigenvectors corresponding to small eigenvalues of the overlap matrix (option DELBAS=0). If the redundancies are exact, i.e., if the overlap matrix for a domain has zero eigenvalues, it is in principle irrelevant which function is deleted. In practice, however, the selection sometimes influences the numerical stability. On the other hand, if the overlap matrix has very small but nonzero eigenvalues, the computed energy slightly depends on which basis functions are eliminated. We tried very hard to make the selection algorithm as robust as possible, but pitfalls in certain cases cannot be fully excluded. Problems with the redundancy elimination normally occur only for very small molecules (2 or 3 atoms) with very small basis sets. If difficulties are encountered, it is recommended to use PRINT,DOMAINS to obtain more detailed information about domains and redundant functions. The default behaviour can be changed using the DELBAS, DELEIG, DELSHL, TYPECHECK, and THRLOC options.

Whenever possible, the domains should be made rotationally invariant, which can be achieved by eliminating shells of basis functions (see DELSHL option). Sometimes it may be necessary to modify the threshold THRLOC to obtain the desired result. In order to avoid problems when rotational invariance is not strictly fulfilled, we recommend always to use of the NOORIENT option in the geometry input for optimizations or frequency calculations.

The SAVE and START options should always be used to keep the domains fixed during geometry optimizations, frequency calculations, or whenever smooth potential energy functions are required. In optimizations with very large geometry changes, it may be useful to determine new domains at the optimized geometry and repeat the geometry optimization with these domains.

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 large 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 22.3.1). 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.

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 card. Indicative of insufficient SCF accuracy are small positive energy changes near the end of the geometry optimization.