Geometry Searches and Internal Coordinates Stationary points are places on the potential energy surface with a zero gradient vector (first derivative of the energy with respect to nuclear coordinates). These include minima (whether relative or global), better known to chemists as reactants, products, and intermediates; as well as transition states (also known as saddle points). The two types of stationary points have a precise mathematical definition, depending on the curvature of the potential energy surface at these points. If all of the eigenvalues of the hessian matrix (second derivative of the energy with respect to nuclear coordinates) are positive, the stationary point is a minimum. If there is one, and only one, negative curvature, the stationary point is a transition state. Points with more than one negative curvature do exist, but are not important in chemistry. Because vibrational frequencies are basically the square roots of the curvatures, a minimum has all real frequencies, and a saddle point has one imaginary vibrational "frequency". GAMESS locates minima by geometry optimization, as RUNTYP=OPTIMIZE, and transition states by saddle point searches, as RUNTYP=SADPOINT. In many ways these are similar, and in fact nearly identical FORTRAN code is used for both. The term "geometry search" is used here to describe features which are common to both procedures. The input to control both RUNTYPs is found in the $STATPT group. As will be noted in the symmetry section below, an OPTIMIZE run does not always find a minimum, and a SADPOINT run may not find a transtion state, even though the gradient is brought to zero. You can prove you have located a minimum or saddle point only by examining the local curvatures of the potential energy surface. This can be done by following the geometry search with a RUNTYP=HESSIAN job, which should be a matter of routine. quasi-Newton Searches Geometry searches are most effectively done by what is called a quasi-Newton-Raphson procedure. These methods assume a quadratic potential surface, and require the exact gradient vector and an approximation to the hessian. It is the approximate nature of the hessian that makes the method "quasi". The rate of convergence of the geometry search depends on how quadratic the real surface is, and the quality of the hessian. The latter is something you have control over, and is discussed in the next section. GAMESS contains different implementations of quasi- Newton procedures for finding stationary points, namely METHOD=NR, RFO, QA, and the seldom used SCHLEGEL. They differ primarily in how the step size and direction are controlled, and how the Hessian is updated. The CONOPT method is a way of forcing a geometry away from a minimum towards a TS. It is not a quasi-Newton method, and is described at the very end of this section. The NR method employs a straight Newton-Raphson step. There is no step size control, the algorithm will simply try to locate the nearest stationary point, which may be a minimum, a TS, or any higher order saddle point. NR is not intended for general use, but is used by GAMESS in connection with some of the other methods after they have homed in on a stationary point, and by Gradient Extremal runs where location of higher order saddle points is common. NR requires a very good estimate of the geometry in order to converge on the desired stationary point. The RFO and QA methods are two different versions of the so-called augmented Hessian techniques. They both employ Hessian shift parameter(s) in order to control the step length and direction. In the RFO method, the shift parameter is determined by approximating the PES with a Rational Function, instead of a second order Taylor expansion. For a RUNTYP=SADPOINT, the TS direction is treated separately, giving two shift parameters. This is known as a Partitioned Rational Function Optimization (P-RFO). The shift parameter(s) ensure that the augmented Hessian has the correct eigen- value structure, all positive for a minimum search, and one negative eigenvalue for a TS search. The (P)-RFO step can have any length, but if it exceeds DXMAX, the step is simply scaled down. In the QA (Quadratic Approximation) method, the shift parameter is determined by the requirement that the step size should equal DXMAX. There is only one shift parameter for both minima and TS searches. Again the augmented Hessian will have the correct structure. There is another way of describing the same algorithm, namely as a minimization on the "image" potential. The latter is known as TRIM (Trust Radius Image Minimization). The working equation is identical in these two methods. When the RFO steplength is close to DXMAX, there is little difference between the RFO and QA methods. However, the RFO step may in some cases exceed DXMAX significantly, and a simple scaling of the step will usually not produce the best direction. The QA step is the best step on the hypersphere with radius DXMAX. For this reason QA is the default algorithm. Near a stationary point the straight NR algorithm is the most efficient. The RFO and QA may be viewed as methods for guiding the search in the "correct" direction when starting far from the stationary point. Once the stationary point is approached, the RFO and QA methods switch to NR, automatically, when the NR steplength drops below 0.10 or DXMAX, whichever is the smallest. The QA method works so well that we use it exclusively, and so the SCHLEGEL method will probably be omitted from some future version of GAMESS. You should read the papers mentioned below in order to understand how these methods are designed to work. The first 3 papers describe the RFO and TRIM/QA algorithms. A good but slightly dated summary of search procedures is given by Bell and Crighton, and see also the review by Schlegel. Most of the FORTRAN code for geometry searches, and some of the discussion in this section was written by Frank Jensen of the University of Aarhus, whose paper compares many of the algorithms implemented in GAMESS: 1. J.Baker J.Comput.Chem. 7, 385-395(1986) 2. T.Helgaker Chem.Phys.Lett. 182, 503-510(1991) 3. P.Culot, G.Dive, V.H.Nguyen, J.M.Ghuysen Theoret.Chim.Acta 82, 189-205(1992) 4. H.B.Schlegel J.Comput.Chem. 3, 214-218(1982) 5. S.Bell, J.S.Crighton J.Chem.Phys. 80, 2464-2475(1984). 6. H.B.Schlegel Advances in Chemical Physics (Ab Initio Methods in Quantum Chemistry, Part I), volume 67, K.P.Lawley, Ed. Wiley, New York, 1987, pp 249-286. 7. F.Jensen J.Chem.Phys. 102, 6706-6718(1995). the nuclear Hessian Although quasi-Newton methods require only an approximation to the true hessian, the quality of this matrix has a great affect on convergence of the geometry search. There is a procedure contained within GAMESS for guessing a positive definite hessian matrix, HESS=GUESS. If you are using Cartesian coordinates, the guess hessian is based on pairwise atom stretches. The guess is more sophisticated when internal coordinates are defined, as empirical rules will be used to estimate stretching and bending force constants. Other angular force constants are set to 1/4. The guess often works well for minima, but cannot possibly find transition states (because it is positive definite). Therefore, GUESS may not be selected for SADPOINT runs. Two options for providing a more accurate hessian are HESS=READ and CALC. For the latter, the true hessian is obtained by direct calculation at the initial geometry, and then the geometry search begins, all in one run. The READ option allows you to feed in the hessian in a $HESS group, as obtained by a RUNTYP=HESSIAN job. The second procedure is actually preferable, as you get a chance to see the frequencies. Then, if the local curvatures look good, you can commit to the geometry search. Be sure to include a $GRAD group (if the exact gradient is available) in the HESS=READ job so that GAMESS can take its first step immediately. Note also that you can compute the hessian at a lower basis set and/or wavefunction level, and read it into a higher level geometry search. In fact, the $HESS group could be obtained at the semiempirical level. This trick works because the hessian is 3Nx3N for N atoms, no matter what atomic basis is used. The gradient from the lower level is of course worthless, as the geometry search must work with the exact gradient of the wavefunction and basis set in current use. Discard the $GRAD group from the lower level calculation! You often get what you pay for. HESS=GUESS is free, but may lead to significantly more steps in the geometry search. The other two options are more expensive at the beginning, but may pay back by rapid convergence to the stationary point. The hessian update frequently improves the hessian for a few steps (especially for HESS=GUESS), but then breaks down. The symptoms are a nice lowering of the energy or the RMS gradient for maybe 10 steps, followed by crazy steps. You can help by putting the best coordinates into $DATA, and resubmitting, to make a fresh determination of the hessian. The default hessian update for OPTIMIZE runs is BFGS, which is likely to remain positive definite. The POWELL update is the default for SADPOINT runs, since the hessian can develop a negative curvature as the search progresses. The POWELL update is also used by the METHOD=NR and CONOPT since the Hessian may have any number of negative eigenvalues in these cases. The MSP update is a mixture of Murtagh-Sargent and Powell, suggested by Josep Bofill, (J.Comput.Chem., 15, 1-11, 1994). It sometimes works slightly better than Powell, so you may want to try it. coordinate choices Optimization in cartesian coordinates has a reputation of converging slowly. This is largely due to the fact that translations and rotations are usually left in the problem. Numerical problems caused by the small eigen- values associated with these degrees of freedom are the source of this poor convergence. The methods in GAMESS project the hessian matrix to eliminate these degrees of freedom, which should not cause a problem. Nonetheless, Cartesian coordinates are in general the most slowly convergent coordinate system. The use of internal coordinates (see NZVAR in $CONTRL as well as $ZMAT) also eliminates the six rotational and translational degrees of freedom. Also, when internal coordinates are used, the GUESS hessian is able to use empirical information about bond stretches and bends. On the other hand, there are many possible choices for the internal coordinates, and some of these may lead to much poorer convergence of the geometry search than others. Particularly poorly chosen coordinates may not even correspond to a quadratic surface, thereby ending all hope that a quasi-Newton method will converge. Internal coordinates are frequently strongly coupled. Because of this, Jerry Boatz has called them "infernal coordinates"! A very common example to illustrate this might be a bond length in a ring, and the angle on the opposite side of the ring. Clearly, changing one changes the other simultaneously. A more mathematical definition of "coupled" is to say that there is a large off-diagonal element in the hessian. In this case convergence may be unsatisfactory, especially with a diagonal GUESS hessian, where a "good" set of internals is one with a diagonally dominant hessian. Of course, if you provide an accurately computed hessian, it will have large off-diagonal values where those are truly present. Even so, convergence may be poor if the coordinates are coupled through large 3rd or higher derivatives. The best coordinates are therefore those which are the most "quadratic". One very popular set of internal coordinates is the usual "model builder" Z-matrix input, where for N atoms, one uses N-1 bond lengths, N-2 bond angles, and N-3 bond torsions. The popularity of this choice is based on its ease of use in specifying the initial molecular geometry. Typically, however, it is the worst possible choice of internal coordinates, and in the case of rings, is not even as good as Cartesian coordinates. However, GAMESS does not require this particular mix of the common types. GAMESS' only requirement is that you use a total of 3N-6 coordinates, chosen from these 3 basic types, or several more exotic possibilities. (Of course, we mean 3N-5 throughout for linear molecules). These additional types of internal coordinates include linear bends for 3 collinear atoms, out of plane bends, and so on. There is no reason at all why you should place yourself in a straightjacket of N-1 bonds, N-2 angles, and N-3 torsions. If the molecule has symmetry, be sure to use internals which are symmetrically related. For example, the most effective choice of coordinates for the atoms in a four membered ring is to define all four sides, any one of the internal angles, and a dihedral defining the ring pucker. For a six membered ring, the best coordinates seem to be 6 sides, 3 angles, and 3 torsions. The angles should be every other internal angle, so that the molecule can "breathe" freely. The torsions should be arranged so that the central bond of each is placed on alternating bonds of the ring, as if they were pi bonds in Kekule benzene. For a five membered ring, we suggest all 5 sides, 2 internal angles, again alternating every other one, and 2 dihedrals to fill in. The internal angles of necessity skip two atoms where the ring closes. Larger rings should generalize on the idea of using all sides but only alternating angles. If there are fused rings, start with angles on the fused bond, and alternate angles as you go around from this position. Rings and more especially fused rings can be tricky. For these systems, especially, we suggest the Cadillac of internal coordinates, the "natural internal coordinates" of Peter Pulay. For a description of these, see P.Pulay, G.Fogarosi, F.Pang, J.E.Boggs, J.Am.Chem.Soc. 101, 2550-2560 (1979). G.Fogarasi, X.Zhou, P.W.Taylor, P.Pulay J.Am.Chem.Soc. 114, 8191-8201 (1992). These are linear combinations of local coordinates, except in the case of rings. The examples given in these two papers are very thorough. An illustration of these types of coordinates is given in the example job EXAM25.INP, distributed with GAMESS. This is a nonsense molecule, designed to show many kinds of functional groups. It is defined using standard bond distances with a classical Z-matrix input, and the angles in the ring are adjusted so that the starting value of the unclosed OO bond is also a standard value. Using Cartesian coordinates is easiest, but takes a very large number of steps to converge. This however, is better than using the classical Z-matrix internals given in $DATA, which is accomplished by setting NZVAR to the correct 3N-6 value. The geometry search changes the OO bond length to a very short value on the 1st step, and the SCF fails to converge. (Note that if you have used dummy atoms in the $DATA input, you cannot simply enter NZVAR to optimize in internal coordinates, instead you must give a $ZMAT which involves only real atoms). The third choice of internal coordinates is the best set which can be made from the simple coordinates. It follows the advice given above for five membered rings, and because it includes the OO bond, has no trouble with crashing this bond. It takes 20 steps to converge, so the trouble of generating this $ZMAT is certainly worth it compared to the use of Cartesians. Natural internal coordinates are defined in the final group of input. The coordinates are set up first for the ring, including two linear combinations of all angles and all torsions withing the ring. After this the methyl is hooked to the ring as if it were a NH group, using the usual terminal methyl hydrogen definitions. The H is hooked to this same ring carbon as if it were a methine. The NH and the CH2 within the ring follow Pulay's rules exactly. The amount of input is much greater than a normal Z-matrix. For example, 46 internal coordinates are given, which are then placed in 3N-6=33 linear combinations. Note that natural internals tend to be rich in bends, and short on torsions. The energy results for the three coordinate systems which converge are as follows: NSERCH Cart good Z-mat nat. int. 0 -48.6594935049 -48.6594935049 -48.6594935049 1 -48.6800538676 -48.6806631261 -48.6838361406 2 -48.6822702585 -48.6510215698 -48.6874045449 3 -48.6858299354 -48.6882945647 -48.6932811528 4 -48.6881499412 -48.6849667085 -48.6946836332 5 -48.6890226067 -48.6911899936 -48.6959800274 6 -48.6898261650 -48.6878047907 -48.6973821465 7 -48.6901936624 -48.6930608185 -48.6987652146 8 -48.6905304889 -48.6940607117 -48.6996366016 9 -48.6908626791 -48.6949137185 -48.7006656309 10 -48.6914279465 -48.6963767038 -48.7017273728 11 -48.6921521142 -48.6986608776 -48.7021504975 12 -48.6931136707 -48.7007305310 -48.7022405019 13 -48.6940437619 -48.7016095285 -48.7022548935 14 -48.6949546487 -48.7021531692 -48.7022569328 15 -48.6961698826 -48.7022080183 -48.7022570260 16 -48.6973813002 -48.7022454522 -48.7022570662 17 -48.6984850655 -48.7022492840 18 -48.6991553826 -48.7022503853 19 -48.6996239136 -48.7022507037 20 -48.7002269303 -48.7022508393 21 -48.7005379631 22 -48.7008387759 ... 50 -48.7022519950 from which you can see that the natural internals are actually the best set. The $ZMAT exhibits upward burps in the energy at step 2, 4, and 6, so that for the same number of steps, these coordinates are always at a higher energy than the natural internals. The initial hessian generated for these three columns contains 0, 33, and 46 force constants. This assists the natural internals, but is not the major reason for its superior performance. The computed hessian at the final geometry of this molecule, when transformed into the natural internal coordinates is almost diagonal. This almost complete uncoupling of coordinates is what makes the natural internals perform so well. The conclusion is of course that not all coordinate systems are equal, and natural internals are the best. As another example, we have run the ATCHCP molecule, which is a popular geometry optimization test, due to its two fused rings: H.B.Schlegel, Int.J.Quantum Chem., Symp. 26, 253-264(1992) T.H.Fischer and J.Almlof, J.Phys.Chem. 96, 9768-9774(1992) J.Baker, J.Comput.Chem. 14, 1085-1100(1993) Here we have compared the same coordinate types, using a guess hessian, or a computed hessian. The latter set of runs is a test of the coordinates only, as the initial hessian information is identical. The results show clearly the superiority of the natural internals, which like the previous example, give an energy decrease on every step: HESS=GUESS HESS=READ Cartesians 65 41 steps good Z-matrix 32 23 natural internals 24 13 A final example is phosphinoazasilatrane, with three rings fused on a common SiN bond, in which 112 steps in Cartesian space became 32 steps in natural internals. The moral is: "A little brain time can save a lot of CPU time." In late 1998, a new kind of internal coordinate method was included into GAMESS. This is the delocalized internal coordinate (DLC) of J.Baker, A. Kessi, B.Delley J.Chem.Phys. 105, 192-212(1996) although as is the usual case, the implementation is not exactly the same. Bonds are kept as independent coordinates, while angles are placed in linear combination by the DLC process. There are some interesting options for applying constraints, and other options to assist the automatic DLC generation code by either adding or deleting coordinates. It is simple to use DLCs in their most basic form: $contrl nzvar=xx $end $zmat dlc=.true. auto=.true. $end Our initial experience is that the quality of DLCs is not as good as explicitly constructed natural internals, which benefit from human chemical knowledge, but are almost always better than carefully crafted $ZMATs using only the primitive internal coordinates (although we have seen a few exceptions). Once we have more numerical experience with the use of DLC's, we will come back and revise the above discussion of coordinate choices. In the meantime, they are quite simple to choose, so give them a go. the role of symmetry At the end of a succesful geometry search, you will have a set of coordinates where the gradient of the energy is zero. However your newly discovered stationary point is not necessarily a minimum or saddle point! This apparent mystery is due to the fact that the gradient vector transforms under the totally symmetric representation of the molecular point group. As a direct consequence, a geometry search is point group conserving. (For a proof of these statements, see J.W.McIver and A.Komornicki, Chem.Phys.Lett., 10,303-306(1971)). In simpler terms, the molecule will remain in whatever point group you select in $DATA, even if the true minimum is in some lower point group. Since a geometry search only explores totally symmetric degrees of freedom, the only way to learn about the curvatures for all degrees of freedom is RUNTYP=HESSIAN. As an example, consider disilene, the silicon analog of ethene. It is natural to assume that this molecule is planar like ethene, and an OPTIMIZE run in D2h symmetry will readily locate a stationary point. However, as a calculation of the hessian will readily show, this structure is a transition state (one imaginary frequency), and the molecule is really trans-bent (C2h). A careful worker will always characterize a stationary point as either a minimum, a transition state, or some higher order stationary point (which is not of great interest!) by performing a RUNTYP=HESSIAN. The point group conserving properties of a geometry search can be annoying, as in the preceeding example, or advantageous. For example, assume you wish to locate the transition state for rotation about the double bond in ethene. A little thought will soon reveal that ethene is D2h, the 90 degrees twisted structure is D2d, and structures in between are D2. Since the saddle point is actually higher symmetry than the rest of the rotational surface, you can locate it by RUNTYP=OPTIMIZE within D2d symmetry. You can readily find this stationary point with the diagonal guess hessian! In fact, if you attempt to do a RUNTYP=SADPOINT within D2d symmetry, there will be no totally symmetric modes with negative curvatures, and it is unlikely that the geometry search will be very well behaved. Although a geometry search cannot lower the symmetry, the gain of symmetry is quite possible. For example, if you initiate a water molecule optimization with a trial structure which has unequal bond lengths, the geometry search will come to a structure that is indeed C2v (to within OPTTOL, anyway). However, GAMESS leaves it up to you to realize that a gain of symmetry has occurred. In general, Mother Nature usually chooses more symmetrical structures over less symmetrical structures. Therefore you are probably better served to assume the higher symmetry, perform the geometry search, and then check the stationary point's curvatures. The alternative is to start with artificially lower symmetry and see if your system regains higher symmetry. The problem with this approach is that you don't necessarily know which subgroup is appropriate, and you lose the great speedups GAMESS can obtain from proper use of symmetry. It is good to note here that "lower symmetry" does not mean simply changing the name of the point group and entering more atoms in $DATA, instead the nuclear coordinates themselves must actually be of lower symmetry. practical matters Geometry searches do not bring the gradient exactly to zero. Instead they stop when the largest component of the gradient is below the value of OPTTOL, which defaults to a reasonable 0.0001. Analytic hessians usually have residual frequencies below 10 cm**-1 with this degree of optimization. The sloppiest value you probably ever want to try is 0.0005. If a geometry search runs out of time, or exceeds NSTEP, it can be restarted. For RUNTYP=OPTIMIZE, restart with the coordinates having the lowest total energy (do a string search on "FINAL"). For RUNTYP=SADPOINT, restart with the coordinates having the smallest gradient (do a string search on "RMS", which means root mean square). These are not necessarily at the last geometry! The "restart" should actually be a normal run, that is you should not try to use the restart options in $CONTRL (which may not work anyway). A geometry search can be restarted by extracting the desired coordinates for $DATA from the printout, and by extracting the corresponding $GRAD group from the PUNCH file. If the $GRAD group is supplied, the program is able to save the time it would ordinarily take to compute the wavefunction and gradient at the initial point, which can be a substantial savings. There is no input to trigger reading of a $GRAD group: if found, it is read and used. Be careful that your $GRAD group actually corresponds to the coordinates in $DATA, as GAMESS has no check for this. Sometimes when you are fairly close to the minimum, an OPTIMIZE run will take a first step which raises the energy, with subsequent steps improving the energy and perhaps finding the minimum. The erratic first step is caused by the GUESS hessian. It may help to limit the size of this wrong first step, by reducing its radius, DXMAX. Conversely, if you are far from the minimum, sometimes you can decrease the number of steps by increasing DXMAX. When using internals, the program uses an iterative process to convert the internal coordinate change into Cartesian space. In some cases, a small change in the internals will produce a large change in Cartesians, and thus produce a warning message on the output. If these warnings appear only in the beginning, there is probably no problem, but if they persist you can probably devise a better set of coordinates. You may in fact have one of the two problems described in the next paragraph. In some cases (hopefully very few) the iterations to find the Cartesian displacement may not converge, producing a second kind of warning message. The fix for this may very well be a new set of internal coordinates as well, or adjustment of ITBMAT in $STATPT. There are two examples of poorly behaved internal coordinates which can give serious problems. The first of these is three angles around a central atom, when this atom becomes planar (sum of the angles nears 360). The other is a dihedral where three of the atoms are nearly linear, causing the dihedral to flip between 0 and 180. Avoid these two situations if you want your geometry search to be convergent. Sometimes it is handy to constrain the geometry search by freezing one or more coordinates, via the IFREEZ array. For example, constrained optimizations may be useful while trying to determine what area of a potential energy surface contains a saddle point. If you try to freeze coordinates with an automatically generated $ZMAT, you need to know that the order of the coordinates defined in $DATA is y y x r1 y x r2 x a3 y x r4 x a5 x w6 y x r7 x a8 x w9 and so on, where y and x are whatever atoms and molecular connectivity you happen to be using. saddle points Finding minima is relatively easy. There are large tables of bond lengths and angles, so guessing starting geometries is pretty straightforward. Very nasty cases may require computation of an exact hessian, but the location of most minima is straightforward. In contrast, finding saddle points is a black art. The diagonal guess hessian will never work, so you must provide a computed one. The hessian should be computed at your best guess as to what the transition state (T.S.) should be. It is safer to do this in two steps as outlined above, rather than HESS=CALC. This lets you verify you have guessed a structure with one and only one negative curvature. Guessing a good trial structure is the hardest part of a RUNTYP=SADPOINT! This point is worth iterating. Even with sophisticated step size control such as is offered by the QA/TRIM or RFO methods, it is in general very difficult to move correctly from a region with incorrect curvatures towards a saddle point. Even procedures such as CONOPT or RUNTYP=GRADEXTR will not replace your own chemical intuition about where saddle points may be located. The RUNTYP=HESSIAN's normal coordinate analysis is rigorously valid only at stationary points on the surface. This means the frequencies from the hessian at your trial geometry are untrustworthy, in particular the six "zero" frequencies corresponding to translational and rotational (T&R) degrees of freedom will usually be 300-500 cm**-1, and possibly imaginary. The Sayvetz conditions on the printout will help you distinguish the T&R "contaminants" from the real vibrational modes. If you have defined a $ZMAT, the PURIFY option within $STATPT will help zap out these T&R contaminants). If the hessian at your assumed geometry does not have one and only one imaginary frequency (taking into account that the "zero" frequencies can sometimes be 300i!), then it will probably be difficult to find the saddle point. Instead you need to compute a hessian at a better guess for the initial geometry, or read about mode following below. If you need to restart your run, do so with the coordinates which have the smallest RMS gradient. Note that the energy does not necessarily have to decrease in a SADPOINT run, in contrast to an OPTIMIZE run. It is often necessary to do several restarts, involving recomputation of the hessian, before actually locating the saddle point. Assuming you do find the T.S., it is always a good idea to recompute the hessian at this structure. As described in the discussion of symmetry, only totally symmetric vibrational modes are probed in a geometry search. Thus it is fairly common to find that at your "T.S." there is a second imaginary frequency, which corresponds to a non-totally symmetric vibration. This means you haven't found the correct T.S., and are back to the drawing board. The proper procedure is to lower the point group symmetry by distorting along the symmetry breaking "extra" imaginary mode, by a reasonable amount. Don't be overly timid in the amount of distortion, or the next run will come back to the invalid structure. The real trick here is to find a good guess for the transition structure. The closer you are, the better. It is often difficult to guess these structures. One way around this is to compute a linear least motion (LLM) path. This connects the reactant structure to the product structure by linearly varying each coordinate. If you generate about ten structures intermediate to reactants and products, and compute the energy at each point, you will in general find that the energy first goes up, and then down. The maximum energy structure is a "good" guess for the true T.S. structure. Actually, the success of this method depends on how curved the reaction path is. A particularly good paper on the symmetry which a saddle point (and reaction path) can possess is by P.Pechukas, J.Chem.Phys. 64, 1516-1521(1976) mode following In certain circumstances, METHOD=RFO and QA can walk from a region of all positive curvatures (i.e. near a minimum) to a transition state. The criteria for whether this will work is that the mode being followed should be only weakly coupled to other close-lying Hessian modes. Especially, the coupling to lower modes should be almost zero. In practise this means that the mode being followed should be the lowest of a given symmetry, or spatially far away from lower modes (for example, rotation of methyl groups at different ends of the molecule). It is certainly possible to follow also modes which do not obey these criteria, but the resulting walk (and possibly TS location) will be extremely sensitive to small details such as the stepsize. This sensitivity also explain why TS searches often fail, even when starting in a region where the Hessian has the required one negative eigenvalue. If the TS mode is strongly coupled to other modes, the direction of the mode is incorrect, and the maximization of the energy along that direction is not really what you want (but what you get). Mode following is really not a substitute for the ability to intuit regions of the PES with a single local negative curvature. When you start near a minimum, it matters a great deal which side of the minima you start from, as the direction of the search depends on the sign of the gradient. We strongly urge that you read before trying to use IFOLOW, namely the papers by Frank Jensen and Jon Baker mentioned above, and see also Figure 3 of C.J.Tsai, K.D.Jordan, J.Phys.Chem. 97, 11227-11237 (1993) which is quite illuminating on the sensitivity of mode following to the initial geometry point. Note that GAMESS retains all degrees of freedom in its hessian, and thus there is no reason to suppose the lowest mode is totally symmetric. Remember to lower the symmetry in the input deck if you want to follow non-symmetric modes. You can get a printout of the modes in internal coordinate space by a EXETYP=CHECK run, which will help you decide on the value of IFOLOW. * * * CONOPT is a different sort of saddle point search procedure. Here a certain "CONstrained OPTimization" may be considered as another mode following method. The idea is to start from a minimum, and then perform a series of optimizations on hyperspheres of increasingly larger radii. The initial step is taken along one of the Hessian modes, chosen by IFOLOW, and the geometry is optimized subject to the constraint that the distance to the minimum is constant. The convergence criteria for the gradient norm perpendicular to the constraint is taken as 10*OPTTOL, and the corresponding steplength as 100*OPTTOL. After such a hypersphere optimization has converged, a step is taken along the line connecting the two previous optimized points to get an estimate of the next hyper- sphere geometry. The stepsize is DXMAX, and the radius of hyperspheres is thus increased by an amount close (but not equal) to DXMAX. Once the pure NR step size falls below DXMAX/2 or 0.10 (whichever is the largest) the algorithm switches to a straight NR iterate to (hopefully) converge on the stationary point. The current implementation always conducts the search in cartesian coordinates, but internal coordinates may be printed by the usual specification of NZVAR and ZMAT. At present there is no restart option programmed. CONOPT is based on the following papers, but the actual implementation is the modified equations presented in Frank Jensen's paper mentioned above. Y. Abashkin, N. Russo, J.Chem.Phys. 100, 4477-4483(1994). Y. Abashkin, N. Russo, M. Toscano, Int.J.Quant.Chem. 52, 695-704(1994). There is little experience on how this method works in practice, experiment with it at your own risk!