SUMMARY: Question pertaining to Vibrational Frequencies

Hello,
 Thank you so much for all of your responses to my question on vibrational
 frequencies.  The original question is given below:
  I performed a minimization on a structure using Gaussian 98W
 (B3LYP/6-311+G**) and followed it up with a frequency calculation at the
 same level (of course).  I obtained a single negative frequency at -27cm-1.
  I then re-optimized the geometry at the same level but this time ignored
 symmetry.  The new E was lower than the previous by 0.03 kcal/mol.  The
 newfrequency calculation afforded a single negative frequency at -11 cm-1.
 David Young's "Computational Chemistry" book suggests (page 94) that
 frequency values with a range of about -20 to 20 cm-1 are essentially zero
 within the numerical accuracy of most software packages.  Is this true for
 G98 as well?
 >
 To sum it all up: Can I take a -11 cm-1 frequency to be inconsequential?
 or do I need to continue to play with the system until absolutely no
 negative frequencies are present?
 I had many requests for a summary of the answers.  I provide these below
 without the names of those responding (which I guess is good etiquette,
 could be wrong).
 My final solution? still working on it.  I think that my final take home
 message is the following: small negative frequencies suggest that you are
 close to a minimum.  Probably the energy at this point is NOT very different
 than the energy at the minimum that you are close to.  However, one should
 do what one can to eliminate this minimum.  Suggestions for doing this are
 given below.
 Thanks again and best regards to the group members!
 ******************************************************
 A frequncy of -11 cm-1 is a transition state. However, I suspect that your
 case
 is "undecided". The number of imaginary freqs gives the curvature of
 the PES
 at
 the stationary point in question. G98 thinks that at your stationary point,
 with
 the method (B3LYP) and basis set (6-311+G**)  you used the PES is mildly
 saddle-shaped, but not very different from a minimum. Another method/basis
 might
 give a distinct saddle or minimum. Borderline cases like this could require
 high
 computational levels to settle.
 ***************************
 This imaginary frequency should disappear! To my knowledge, this is not a
 problem of minimisation. May be you have a methyl group in your molecule
 that in badly oriented. Just changing its orientation is OK!
 ****************************
 I wouldn't worry about a single imaginary frequency as low as 11i cm**-1. It
 is so low (there's about 350 cm**-1 to a kcal/mole!) that it may well be
 caused by numerical noise from the integration. You could try
 integrals(grid=ultrafine) to see if it goes away, but as I said, it doesn't
 really matter.
 ****************************
 One of the most important things to look at, is the six lowest eigenvalues
 of the Hessian (or frequencies) in absolute sense. In an ideal world, where
 the QuantumChemical packages would be able to provide geometries with
 exactly
 zero gradients, these lowest six should all be zero.
 The fact that they are not in practice, is (among other factors
 maybe) due to the fact
 that the gradient is not exactly zero, but below a certain threshold.
 Therefore,
 if your -11 cm-1 is one of these six lowest frequencies, then you can
 ignore it.
 NOTE:
 Gaussian used to be one of the packages that would already filter the
 output,
 i.e. the six lowest frequencies were not displayed in the list of
 frequencies,
 as I seem to remember. However, at another place in the output, they
 were displayed.
 As I haven't done frequency calculations with Gaussian for a few years now,
 I'm not sure if this is (still) true. Just be careful.
 *****************************
   In my opinion, the small imaginary frequency can be looked as an error
 > from
 computation and it is difficult to avoid for some systems. But maybe it is a
 rotational excited state as well. So please try to change your structure
 according to the imaginary frequency to find if you can get rid of it. Of
 course you may try opt=tight, too.
 *******************************
 You are going to get a lot of responses saying the same thing:
 YES, a freq of -11 may be taken as zero and does not mean you
 have a saddle.  I have done dozens of freq calcs via Gaussian98
 at geoms optimized at the B3LYP/6-311++g** level and I usually
 get anywhere from one to 5 just barely negative frequencies.
 Perhaps more to the point, when it has been a t-state, the
 negative freq has always been < -100.
 *******************************
    While a small value like -11cm-1 can arise from a couple of sources
 I would not suggest you pass it off, especially if it is a mode which
 would have bearing on any of the chemistry you wish to describe.  On the
 other hand I would agree that making any prediction about the observed
 frequency of a mode less than 20 wavenumbers based on this calculation
 is not warranted.   So I will make a couple of general comments and help
 you see the next step.
   First, there is are two sets of "Frequencies listed" the "low
 frequencies"
 and then the normal mode listings including intensities, reduced masses etc.
 The latter can be given twice if you used the HPmodes option.  The first
 set are the lowest 10 modes from diagonalizing the full 3Natoms by
 3Natoms matrix and so there will be 6 low frequency modes, say under 10
 wavenumbers and they correspond to overall rotation and translation.
 Negative
 values at this point for these modes is not particularly worth note.  Then
 this matrix has the analytic rotation and translation modes projected out
 and
 the remaining 3NAtoms-6 modes are determined.  At this point only vibrations
 are left and negative frequencies of any size are harder to dismiss.
   Second, even with NoSymm you can end up with a symmetric structure because
 the gradient carries the symmetry of the structure and will not follow
 anti-symmetric modes downhill.  So if the negative frequency is not totally
 symmetric, i.e. is anything except A1 symmetry, you may only have gotten
 numerical noise in the re-optimization.  Re-optimizing to break the symmetry
 also requires moving the structure along to lower symmetry, best by
 displacing along that mode.  You can add the displacement vector from the
 FREQ analysis element by element to the structure.
   Third, DFT does have more of a problem with numerical precision because of
 numerical integration.  With a low frequency mode like this I have seen use
 of Int=Grid=UltraFine or one of the higher order angular grids, see the Int
 keyword, clean up the integration.  You would need to use OPT=ReadFC with
 this Grid followed by FREQ with this same grid to be sure you have it.
    If your negative frequency is on the Low Freq list and not a normal mode,
 you are likely fine and it is just numerical.  If the structure really is
 reported as C1 then this may well be real and you should go on to use a
 better
 integration grid to sort it out.  Or if the structure is still rated as
 symmetric
 and this mode breaks symmetry try breaking the symmetry and re-doing the
 optimization at the lower symmetry.
    Agreed that modes this low are not going to make large energy differences
 but if it is a dihedral it can result in opening up a structure by a few
 degrees, enough to change conclusions.
 *****************************
 i am by no means an expert, but during my calculations on the same level of
 theory, i had the same problem. what this imaginary freqency means is
 essentially that you have a very flat poteential energy surface and there
 are probably more structures to explore. so what you should do is alter the
 structure and perform a new minimization and see where it takes you.
 *******************************
 You can probably ignore this frequency. Some energy surfaces are fairly flat
 (especially when a molecule can undergo internal rotations), and frequency
 calculations "reveal" imaginary frequencies even when you have
 effectively
 optimized the geometry.
 But just to be sure, you might try the following - if the calculation is
 convenient to do, you could tighten the optimization criteria and repeat the
 geometry optimization + frequency calculation. (Sorry, but I don't know how
 you specify these criteria in Gaussian).
 ********************************
 >To sum it all up: Can I take a -11 cm-1 frequency to be inconsequential?
 Yup.
   or
 >do I need to continue to play with the system until absolutely no negative
 >frequencies are present?
 Nope.
 >
 >
 as you can see in your example the amount of additional
 stabilization left in these soft modes is pretty minimal and thus is
 not worth chasing.
 ***********************************
 I prefer to tighten the SCF and the opt convergence criteria, use
 calcfc, and so on to see if I can remove such -ve frequencies.
 This can waste a lot of time though, and sometimes you never get rid
 of them, so you accept you're as close to the minimum as you're
 going to get. I've had more numerical trouble with DFT than HF/MP*.
 The -11 is a bit of a problem if you want to use and compare
 the ZPE and thermochemistry.
 **********************************
 I, too, am uncomfortable with small imaginary frequencies. With  the
 standard HF methods, one could usually rewrite the z-matrix to better
 represent the internal coordinates to get rid of them or "nudge" the
 internal coordinate(s) for the imaginary freqs and re-optimize. However,
 for DFT methods, these tactics are not highly reliable. Nowadays, people
 (including myself) tend write the input in terms of xyz coordinates and
 let G98 decide on the redundant internal coordinates so modifying the
 internal coordinates are a bit more tricky. One possible solution is just
 to re-order the xyz coordinates (especially the first 3) so that G98 will
 re-orient the molecule differently in space. Since DFT is a grid-based
 methods, this "coordinate juggling" procedure will very slightly alter
 the
 gradients that are being computed so that in the long run your freqs will
 all turn out real (hopefully, 8^) ). Of course, this leads to the second
 solution which is to simply add the option "int=ultrafine" which will
 give you a finer grid for the DFT calculation. This solution will result
 in greater CPU time (dependent on size of molecule), but it keeps you
 > from having to "play games" to get rid of the imaginary freqs.
 ************************************
 Try opt=tight.
 ***********************************
 I have heard (second hand from Gaussian) that up to about
 -50 cm-1 may be considered effectively zero.  The problem is more acute
 with DFT methods compared to Hartree-Fock or post HF methods since the DFT
 calculations have greater accumulation of rounding errors due to the
 numerical integration that is required.
 **************************************
 Are you using the ultrafine grid? Try if you are no using!! And maybe this
 frequency will desapear!