# EigenFollowing

The current version of the EF optimization routine is a combination of the original EF algorithm of Simons et al. (J. Phys. Chem. 89, 52) as implemented by Baker (J. Comp. Chem. 7, 385) and the QA algorithm of Culot et al. (Theo. Chim. Acta 82, 189), with some added features for improving stability.

The geometry optimization is based on a second order Taylor expansion of the energy around the current point. At this point the energy, the gradient and some estimate of the Hessian are available. There are three fundamental steps in determining the next geometry based on this information:

• finding the "best" step within or on the hypersphere with the current trust radius.
• possibly reject this step based on various criteria.
• update the trust radius.

1.
For a minimum search the correct Hessian has only positive eigenvalues. For a Transition State (TS) search the correct Hessian should have exactly one negative eigenvalue, and the corresponding eigenvector should be in the direction of the desired reaction coordinate. The geometry step is parameterized as g/(s-H), where s is a shift factor which ensure that the step-length is within or on the hypersphere. If the Hessian has the correct structure, a pure Newton-Raphson step is attempted. This corresponds to setting the shift factor to zero. If this step is longer than the trust radius, a P-RFO step is attempted. If this is also too long, then the best step on the hypersphere is made via the QA formula. This three step procedure is the default. The pure NR step can be skipped by giving the keyword NONR. An alternative to the QA step is to simply scale the P-RFO step down to the trust radius by a multiplicative constant, this can be accomplished by specifying RSCAL.

2.
Using the step determined from 1), the new energy and gradient are evaluated. If it is a TS search, two criteria are used in determining whether the step is "appropriate". The ratio between the actual and predicted energy change should ideally be 1. If it deviates substantially from this value, the second order Taylor expansion is no longer accurate. RMIN and RMAX (default values 0 and 4) determine the limits on how far from 1 the ratio can be before the step is rejected. If the ratio is outside the RMIN and RMAX limits, the step is rejected, the trust radius reduced by a factor of two and a new step is determined. The second criteria is that the eigenvector along which the energy is being maximized should not change substantially between iterations. The minimum overlap of the TS eigenvector with that of the previous iteration should be larger than OMIN, otherwise the step is rejected. Such a step rejection can be recognized in the output by the presence of (possibly more) lines with the same CYCLE number. The default OMIN value is 0.8, which allows fairly large changes to occur, and should be suitable for most uncomplicated systems. See below for a discussion of how to use RMIN, RMAX and OMIN for difficult cases. The selection of which eigenvector to follow towards the TS is given by MODE=n, where n is the number of the Hessian eigenvector to follow. The default is MODE=1. These features can be turned off by giving suitable values as keywords, e.g. RMIN=-100 RMAX=100 effectively inhibits step rejection. Similarly setting OMIN=0 disables step rejection based on large changes in the structure of the TS mode. The default is to use mode following even if the TS mode is the lowest eigenvector. This means that the TS mode may change to some higher mode during the optimization. To turn off mode following, and thus always follow the mode with lowest eigenvalue, set MODE=0. If it is a minimum search the new energy should be lower than the previous.

The acceptance criteria used is that the actual/predicted ratio should be larger than RMIN, which for the default value of RMIN=0 is equivalent to a lower energy. If the ratio is below RMIN, the step is rejected, the trust radius reduced by a factor of two and a new step is predicted. The RMIN, RMAX and OMIN features has been introduced in the current version of EF to improve the stability of TS optimizations. Setting RMIN and RMAX close to one will give a very stable, but also very slow, optimization. Wide limits on RMIN and RMAX may in some cases give a faster convergence, but there is always the risk that very poor steps are accepted, causing the optimization to diverge. The default values of 0 and 4 rarely rejects steps which would lead to faster convergence, but may occasionally accept poor steps. If TS searches are found to cause problems, the first try should be to lower the limits to 0.5 and 2. Tighter limits like 0.8 and 1.2, or even 0.9 and 1.1, will almost always slow the optimization down significantly but may be necessary in some cases.

In minimum searches it is usually desirable that the energy decreases in each iteration. In certain very rigid systems, however, the initial diagonal Hessian may be so poor that the algorithm cannot find an acceptable step larger than DDMIN, and the optimization terminates after only a few cycles with the "TRUST RADIUS BELOW DDMIN" warning long before the stationary point is reached. In such cases the user can add DDMIN=0.0 and RMIN set to some negative value, say -10, thereby allowing steps which allow the energy to increase. An alternative is to use LET DDMIN=0.0.

The algorithm has the capability of following Hessian eigenvectors other than the one with the lowest eigenvalue toward a TS. Such higher mode following are always much more difficult to make converge. Ideally, as the optimization progresses, the TS mode should at some point become the lowest eigenvector. Care must be taken during the optimization, however, that the nature of the mode does not change all of a sudden, leading to optimization to a different TS than the one desired. OMIN has been designed for ensuring that the nature of the TS mode only changes gradually, specifically the overlap between to successive TS modes should be higher than OMIN. While this concept at first appears very promising, it is not without problems when the Hessian is updated.

As the updated Hessian in each step is only approximately correct, there is a upper limit on how large the TS mode overlap between steps can be. To understand this, consider a series of steps made from the same geometry (e.g. at some point in the optimization), but with steadily smaller step-sizes. The update adds corrections to the Hessian to make it a better approximation to the exact Hessian. As the step-size become small, the updated Hessian converges toward the exact Hessian, at least in the direction of the step. The old Hessian is constant, thus the overlap between TS modes thus does not converge toward 1, but rather to a constant value which indicate how well the old approximate Hessian resembles the exact Hessian. Test calculations suggest a typical upper limit around 0.9, although cases have been seen where the limit is more like 0.7. It appears that an updated Hessian in general is not of sufficient accuracy for reliably rejecting steps with TS overlaps much greater than 0.80. The default OMIN of 0.80 reflects the typical use of an updated Hessian. If the Hessian is recalculated in each step, however, the TS mode overlap does converge toward 1 as the step-size goes toward zero, and in this cases there is no problems following high lying modes.

Unfortunately setting RECALC=1 is very expensive in terms of computer time, but used in conjecture with OMIN=0.90 (or possibly higher), and maybe also tighter limits on RMIN and RMAX, it represents an option of locating transitions structures that otherwise might not be possible. If problems are encountered with many step rejections due to small TS mode overlaps, try reducing OMIN, maybe all the way down to 0. This most likely will work if the TS mode is the lowest Hessian eigenvector, but it is doubtful that it will produce any useful results if a high lying mode is followed. Finally, following modes other than the lowest toward a TS indicates that the starting geometry is not "close" to the desired TS. In most cases it is thus much better to further refined the starting geometry, than to try following high lying modes. There are cases, however, where it is very difficult to locate a starting geometry which has the correct Hessian, and mode following may be of some use here.