This section of the manual, as well as the source code to trace gradient extremals was written by Frank Jensen of Odense University.
A Gradient Extremal (GE) curve consists of points where the gradient norm on a constant energy surface is stationary. This is equivalent to the condition that the gradient is an eigenvector of the Hessian. Such GE curves radiate along all normal modes from a stationary point, and the GE leaving along the lowest normal mode from a minimum is the gentlest ascent curve. This is not the same as the IRC curve connecting a minimum and a TS, but may in some cases be close.
GEs may be divided into three groups: those leading
to dissociation, those leading to atoms colliding, and
those which connect stationary points. The latter class
allows a determination of many (all?) stationary points on
a PES by tracing out all the GEs. Following GEs is thus a
semi-systematic way of mapping out stationary points. The
disadvantages are:
Normally one is only interested in minima and TSs, but many higher order saddle points will also be found. Furthermore, it appears that it is necessary to follow GEs radiating also from TSs and second (and possibly also higher) order saddle point to find all the TSs.
A rather complete map of the extremals for the H2CO
potential surface is available in a paper which explains
the points just raised in greater detail:
There are two GE algorithms in GAMESS, one due to Sun
and Ruedenberg (METHOD=SR), which has been extended to
include the capability of locating bifurcation points and
turning points, and another due to Jorgensen, Jensen, and
Helgaker (METHOD=JJH):
The Sun and Ruedenberg method consist of a predictor step taken along the tangent to the GE curve, followed by one or more corrector steps to bring the geometry back to the GE. Construction of the GE tangent and the corrector step requires elements of the third derivative of the energy, which is obtained by a numerical differentiation of two Hessians. This puts some limitations on which systems the GE algorithm can be used for. First, the numerical differentiation of the Hessian to produce third derivatives means that the Hessian should be calculated by analytical methods, thus only those types of wavefunctions where this is possible can be used. Second, each predictor/corrector step requires at least two Hessians, but often more. Maybe 20-50 such steps are necessary for tracing a GE from one stationary point to the next. A systematic study of all the GE radiating from a stationary point increases the work by a factor of ~2*(3N-6). One should thus be prepared to invest at least hundreds, and more likely thousands, of Hessian calculations. In other words, small systems, small basis sets, and simple wave- functions.
The Jorgensen, Jensen, and Helgaker method consists of taking a step in the direction of the chosen Hessian eigenvector, and then a pure NR step in the perpendicular modes. This requires (only) one Hessian calculation for each step. It is not suitable for following GEs where the GE tangent forms a large angle with the gradient, and it is incapable of locating GE bifurcations.
Although experience is limited at present, the JJH method does not appear to be suitable for following GEs in general (at least not in the current implementation). Experiment with it at your own risk!
The flow of the SR algorithm is as follows: A predictor geometry is produced, either by jumping away from a stationary point, or from a step in the tangent direction from the previous point on the GE. At the predictor geometry, we need the gradient, the Hessian, and the third derivative in the gradient direction. Depending on HSDFDB, this can be done in two ways. If .TRUE. the gradient is calculated, and two Hessians are calculated at SNUMH distance to each side in the gradient direction. The Hessian at the geometry is formed as the average of the two displaced Hessians. This corresponds to a double- sided differentiation, and is the numerical most stable method for getting the partial third derivative matrix. If HSDFDB = .FALSE., the gradient and Hessian are calculated at the current geometry, and one additional Hessian is calculated at SNUMH distance in the gradient direction. This corresponds to a single-sided differen- tiation. In both cases, two full Hessian calculations are necessary, but HSDFDB = .TRUE. require one additional wavefunction and gradient calculation. This is usually a fairly small price compared to two Hessians, and the numerically better double-sided differentiation has therefore been made the default.
Once the gradient, Hessian, and third derivative is available, the corrector step and the new GE tangent are constructed. If the corrector step is below a threshold, a new predictor step is taken along the tangent vector. If the corrector step is larger than the threshold, the correction step is taken, and a new micro iteration is performed. DELCOR thus determines how closely the GE will be followed, and DPRED determine how closely the GE path will be sampled.
The construction of the GE tangent and corrector step involve solution of a set of linear equations, which in matrix notation can be written as Ax=B. The A-matrix is also the second derivative of the gradient norm on the constant energy surface.
After each corrector step, various things are printed to monitor the behavior: The projection of the gradient along the Hessian eigenvalues (the gradient is parallel to an eigenvector on the GE), the projection of the GE tangent along the Hessian eigenvectors, and the overlap of the Hessian eigenvectors with the mode being followed from the previous (optimzed) geometry. The sign of these overlaps are not significant, they just refer to an arbitrary phase of the Hessian eigenvectors.
After the micro iterations has converged, the Hessian eigenvector curvatures are also displayed, this is an indication of the coupling between the normal modes. The number of negative eigenvalues in the A-matrix is denoted the GE index. If it changes, one of the eigenvalues must have passed through zero. Such points may either be GE bifurcations (where two GEs cross) or may just be "turning points", normally when the GE switches from going uphill in energy to downhill, or vice versa. The distinction is made based on the B-element corresponding to the A-matrix eigenvalue = 0. If the B-element = 0, it is a bifurcation, otherwise it is a turning point.
If the GE index changes, a linear interpolation is performed between the last two points to locate the point where the A-matrix is singular, and the corresponding B-element is determined. The linear interpolation points will in general be off the GE, and thus the evaluation of whether the B-element is 0 is not always easy. The program additionally evaluates the two limiting vectors which are solutions to the linear sets of equations, these are also used for testing whether the singular point is a bifurcation point or turning point.
Very close to a GE bifurcation, the corrector step become numerically unstable, but this is rarely a problem in practice. It is a priori expected that GE bifurcation will occur only in symmetric systems, and the crossing GE will break the symmetry. Equivalently, a crossing GE may be encountered when a symmetry element is formed, however such crossings are much harder to detect since the GE index does not change, as one of the A-matrix eigenvalues merely touches zero. The program prints an message if the absolute value of an A-matrix eigenvalue reaches a minimum near zero, as such points may indicate the passage of a bifurcation where a higher symmetry GE crosses. Run a movie of the geometries to see if a more symmetric structure is passed during the run.
An estimate of the possible crossing GE direction is made at all points where the A-matrix is singular, and two perturbed geometries in the + and - direction are written out. These may be used as predictor geometries for following a crossing GE. If the singular geometry is a turning point, the + and - geometries are just predictor geometries on the GE being followed.
In any case, a new predictor step can be taken to trace a different GE from the newly discovered singular point, using the direction determined by interpolation from the two end point tangents (the GE tangent cannot be uniquely determined at a bifurcation point). It is not possible to determine what the sign of IFOLOW should be when starting off along a crossing GE at a bifurcation, one will have to try a step to see if it returns to the bifurcation point or not.
In order to determine whether the GE index change it is necessary to keep track of the order of the A-matrix eigenvalues. The overlap between successive eigenvectors are shown as "Alpha mode overlaps".
Things to watch out for: