LBFGS

Optimize the geometry using the L-BFGS function minimizer. This is based on the BFGS optimizer, but it does not store the inverse Hessian, instead it is calculated as needed. Because of this, the L-BFGS method uses very little storage, and is therefore suitable for optimizing very large systems.

The L-BFGS optimizer is the default if 100 or more variables are to be optimized.  For such systems, L-BFGS is more efficient than Baker's Eigenfollowing, however there is no guarantee that the L-BFGS method will produce a minimum energy.  In systems where two or more species are present, and only low-energy interactions bind the species together, the L-BFGS optimizer sometimes fails to produce a minimum energy system .  If this happens, or if for any other reason the L-BFGS is not wanted, add keyword EF or BFGS.  Both these methods are more reliable for difficult systems.

Notes on the L-BFGS method

The L-BFGS optimization procedure is a limited memory variation of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton algorithm.  Rather than storing the Hessian, the L-BFGS method stores only the gradient vectors for the the last few geometries calculated.  In MOPAC, a maximum of 12 gradient vectors are stored.  It then uses this set of vectors to approximate individual elements of the Hessian as they are needed.  It is good for large systems, such as proteins and other macromolecules and solids.  In practice, it also seems to be the best geometry optimizer for systems of thirty or more atoms.  Go on-line for more detail on the L-BFGS method

It uses the gradient only, by moving the atoms in the system in the downhill direction. Although the L-BFGS method does not use the ΔHf, for most systems it does minimize the ΔHf as a side effect of minimizing the gradient.  Although it is not explicit in the algorithm, the L-BFGS method, like the other geometry optimizers in MOPAC, does optimize the geometry to a ground state.  There has not yet been a worked example of it optimizing an unconstrained geometry to a transition state.

Using only the gradient and not the ΔHf has advantages, in that when the L-BFGS method makes a bad step, it uses the results of that step to work out a better step.  This leads to a more rapid descent to a stationary point on the Potential Energy Surface.

Problem with some large solid-state systems

An unsolved fault in the L-BFGS method as implemented in MOPAC is that, for solids with a very large number of atoms in the cluster, after the geometry has been almost completely optimized, the algorithm starts to move the system away from the minimum, and continues to drift it in a nonsensical manner.  This behavior is obviously undesirable. And at present, the cause of this behavior is unknown, however a workaround has been developed to handle this situation.

Workaround for this problem

During a geometry optimization that uses the L-BFGS optimizer, the system with the lowest ΔHf is stored.  At the beginning of a job, the ΔHf is calculated and it, and the starting geometry, are stored.  After each subsequent cycle, the ΔHf is compared with the value in the store.  If it is lower (more negative), then the current geometry and ΔHf are stored.  So that after every cycle the ΔHf and geometry in the store are always the best values calculated.  At the end of the run the working geometry will be replaced by whatever geometry was in storage if the ΔHf of the working geometry is more positive than the value for the stored geometry.  If that is done, then the ΔHf of the lowest-energy geometry will be re-evaluated. The final output will then refer to this geometry.

This workaround is used for all jobs that use the L-BFGS optimizer that end normally, or if the job runs out of time, or the SHUT command is issued.  If a large job is restarted using RESTART and 1SCF is specified, then the workaround will not be used.  For large jobs, if a 1SCF is wanted, use the final geometry from the previous run. 

For most jobs the final geometry will be the best, and the workaround just described would not be used, but if the lowest-energy geometry is not the last geometry calculated by L-BFGS, then the message: "CURRENT BEST VALUE OF HEAT OF FORMATION" will be printed.