I tested the following example from the manual, both with and without the frozen technique. What I don’t understand is that, when this technique is NOT included, the result contains 0 imaginary frequencies. However, when this technique is included, it gives 4 imaginary frequencies, although when looking at the output file there is not a negative value for the first four wavenumbers. I was wondering if you could explain as to how including the technique helped and/or overcome any issue while the plain calculation resulted in zero negative frequencies in the first place? Below is the example from the manual.

Thank you.

$molecule
0 1
C -0.0014570824 1.4001468208 0.0000000000
H -0.0030029737 2.4867609686 0.0000000000
C -1.2093554750 0.6986986619 0.0000000000
H -2.1525287416 1.2392252926 0.0000000000
C -1.2094237515 -0.6985901650 0.0000000000
H -2.1508487680 -1.2427180000 0.0000000000
C -0.0003285453 -1.3965954363 0.0000000000
H -0.0006460438 -2.4839940415 0.0000000000
C 1.2098250013 -0.6978846084 0.0000000000
H 2.1493830451 -1.2446769146 0.0000000000
C 1.2133329176 0.6987868208 0.0000000000
H 2.1550504173 1.2408406009 0.0000000000
$end
$rem
JOBTYPE freq
METHOD b3lyp
BASIS 6-31G*
!FRZN_OPT 1 ! Turns on the freq calculation with frozen Hessians
!FRZ_ATOMS 2 ! No. of frozen atoms
SYM_IGNORE true
SYMMETRY false
$end
!$frozen_opt
!1 11
!$end

The zeroed-out Hessian technique was developed for the frequency calculation of the constrained-optimized structures, where the Cartesian coordinates are kept fixed for some of the atoms. This can be done by the block,

As the coordinates of these atoms are locked in space, they can be imagined as particles with infinite mass. So, the normal modes associated with these atoms will have meaningless (imaginary) frequencies. Zeroing out the Hessian elements associated with those atoms will remove the contribution of the frozen atoms to the normal modes and reduces the number of frequencies computed. For the above example, I will have 3(N−n) − 6 = 24 frequencies which are useful. Although it removes the notorious vibrational modes for constrained optimizations of larger systems it will introduce some imaginary frequencies.

Note: This method was introduced in Q-Chem from v5.4 onwards, so any prior version will not have this functionality.

As John mentioned in the previous comment, a much better approach would be to use the harmonic-confining potentials, which gives you more realistic vibrational frequencies along with the correct thermodynamic behavior.