Higher Order Saddle Points

 I am currently studying the transition state structures that arise during
 the inversion of 6-membered ring systems. These systems exhibit various
 degrees of the anomeric effect. Although the majority of the higher energy
 saddle points (half-chairs) are first order, a few contain a second (25-75
 cm^-1) imaginary vibrational frequency. I realize that flexible rotors,
 methyl groups, and floppy rings sometimes demonstrate this behavior.
 (Hehre, W.H. Practical Strategies for Electronic Structure Calculations;
 Wavefunction: Irvine, 1995, 96)
 However, I've noticed that the rms gradient for structures having 2
 imaginary frequencies is slightly higher than for those containing 1. Many
 times it is necessary to loosen the convergence criteria in order to get an
 optimization to a saddle point to finish. The GAMESS User's Guide states
 that an OPTTOL value of 0.0005 is the largest that a person would want to
 use.
 (GAMESS User's Guide, 4 May 98 edition, p 4-38)
 This means that the maximum component of the gradient has to be less than
 0.0005 and the rms gradient has to be less than (1/3)OPTTOL or 0.000167. My
 higher order saddle points would require an OPTTOL value in the range
 (0.0006, 0.0007). This would correspond to a rms gradient of (0.0002,
 0.00023).
 While I've read many good journal articles that report finding transition
 states for other ring systems, they rarely list the convergence criterion
 used in the calculation. The reason for this is obvious. The purpose of
 performing these calculations is not to produce and report gradients but to
 obtain and report useful properties of structures.
 My questions are listed below:
 1.) In general, are rms gradients for higher order saddle points of ring
 structures larger than those with only 1 imaginary frequency?
 2.) What is considered to be a maximum acceptable rms gradient for
 transition states of larger systems? (>20 atoms)
 My next question has to do with higher order saddle points, such as monkey
 saddle points and so on.
 3.) I've read papers by Hehre, Pople, Pulay, Murrell, and Laidler that
 discuss the properties of saddle points. However, I would like to know what
 others think about higher order saddle points. In other words, do higher
 order saddle points actually exist for real potential energy surfaces, or
 are they artifacts introduced by the limitations of various level of theory?
 I'll produce a summary of responses after an accepted time period.
 Sincerely,
 David Baker dbaker - at - siue.edu