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I am trying to calculate normal modes frequency matrix using HESSIAN calculation in GAMESS. Since we know that number of normal modes for a non-linear molecule is 3N-5 where N is the number of atoms. So the expected size of the normal modes matrix is 3N x (3N-5). However, the normal mode matrix obtained from GAMESS output is of the size 3N x 3N. My suspicion is that the extra 5 modes are irrelevant. Could you help in eliminating the irrelevant modes present in the normal modes matrix?

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    $\begingroup$ +1 welcome to the site! I have never used GAMESS, but in general most codes output all possible normal mode frequencies and eigenvectors, and you will simply find that the "trivial" ones have zero frequency and correspond to translations/rotations. $\endgroup$
    – ProfM
    Feb 18 at 15:55
  • $\begingroup$ Thanks @ProfM! Is there a trivial way to identify those belonging to rotation / translation? For zero frequency, I think I can simply take magnitude of the vector and check for a threshold. $\endgroup$ Feb 18 at 16:31
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Firstly, a minor correction: a non-linear molecule has 3N-6 normal modes, not 3N-5 modes. Linear molecules have 3N-5 normal modes.

GAMESS automatically prints out some values in normal coordinate analysis section with the labels TRANS. SAYVETZ and ROT. SAYVETZ. If the total value of ROT SAYVETZ is high, that mode can be identified as a rotation. You will usually find three modes where the value is much higher than the other modes. Similarly, you will also find three modes where the TRANS. SAYVETZ values are higher than the other modes—these are the translational modes.

Take for example, the following hessian calculation for a methane molecule with RHF/STO-3G:

                            1           2           3           4           5
       FREQUENCY:        17.63       13.88       13.41        0.01        0.02  
        SYMMETRY:         A           A           A           A           A   
    REDUCED MASS:      1.00783     1.00783     1.00783     3.20626     3.20625
    IR INTENSITY:      0.00000     0.00000     0.00000     0.00000     0.00000

  1   C            X -0.00000062  0.00000165 -0.00000383  0.24972074 -0.00350567
                   Y  0.00010786  0.00006334  0.00038252  0.00369972  0.12078890
                   Z  0.00010115 -0.00026452 -0.00014254  0.00196065  0.21857645
  2   H            X -0.00000834  0.00000588  0.00000001  0.24972074 -0.00350564
                   Y  0.10072063 -0.21370832  0.56275746  0.00370176  0.12042435
                   Z  0.50092706 -0.28645159 -0.19853368  0.00195808  0.21811549
  3   H            X  0.01624197  0.13411438 -0.55903118  0.24971826 -0.00327108
                   Y -0.10569085 -0.03585727 -0.21488136  0.00369986  0.12087293
                   Z -0.47274098 -0.35855637 -0.05174247  0.00196502  0.21857135
  4   H            X -0.42497218  0.20048610  0.33156599  0.24972356 -0.00318826
                   Y  0.26760422  0.51780259 -0.07120695  0.00369558  0.12106658
                   Z -0.07647050  0.22917175  0.10076290  0.00196046  0.21877700
  5   H            X  0.40873302 -0.33459136  0.22742554  0.24972024 -0.00405763
                   Y -0.26220197 -0.26797649 -0.27511402  0.00370167  0.12079171
                   Z  0.04869166  0.41482209  0.14894743  0.00195904  0.21884190

 TRANS. SAYVETZ    X -0.00001299  0.00003496 -0.00008587  4.00334788 -0.05620032
                   Y  0.00172967  0.00102261  0.00615760  0.05931125  1.93640304
                   Z  0.00162428 -0.00419627 -0.00228068  0.03143177  3.50406459
               TOTAL  0.00237281  0.00431922  0.00656695  4.00391059  4.00390857

   ROT. SAYVETZ    X -1.83361199 -2.71960126 -0.70573303  0.00002107 -0.00095138
                   Y -2.75454586  1.57410354  1.09118835  0.00001413  0.00253515
                   Z  0.55339513 -1.17579963  3.09322769  0.00001122 -0.00200507
               TOTAL  3.35498167  3.35507641  3.35511679  0.00002774  0.00336933

                            6           7           8           9          10
       FREQUENCY:         0.02     1675.67     1675.71     1675.76     1903.68  
        SYMMETRY:         A           A           A           A           A   
    REDUCED MASS:      3.20624     1.17145     1.17144     1.17144     1.00783
    IR INTENSITY:      0.00000     0.15729     0.15726     0.15732     0.00000

  1   C            X -0.00228962  0.02051160 -0.10602996 -0.03230322  0.00000583
                   Y  0.21857303 -0.01433194  0.03003937 -0.10769565 -0.00000247
                   Z -0.12082370 -0.10991319 -0.02370384  0.00801459  0.00000863
  2   H            X -0.00228962  0.01934522 -0.10005879 -0.03047736  0.00001134
                   Y  0.21779885  0.07075199 -0.14825981  0.53174540 -0.12291500
                   Z -0.12038398  0.54258678  0.11697000 -0.03957536 -0.48268961
  3   H            X -0.00148401 -0.10877382  0.39011099  0.33553048 -0.00818422
                   Y  0.21896760  0.09146521  0.21886006  0.04723247  0.11165411
                   Z -0.12039232  0.53775742  0.03032550  0.07487384  0.48525943
  4   H            X -0.00289646  0.09903687  0.51205228  0.06694073 -0.40256538
                   Y  0.21826225  0.23367102 -0.15755299  0.45641880  0.28658560
                   Z -0.12114100 -0.00009411  0.14777796  0.21105006 -0.06256500
  5   H            X -0.00248834 -0.25383642  0.46037640  0.01263521  0.41066886
                   Y  0.21926322 -0.22524028 -0.27072112  0.24691727 -0.27529527
                   Z -0.12137747  0.22846769 -0.01283586 -0.34177702  0.05989237

 TRANS. SAYVETZ    X -0.03670555 -0.00000006  0.00000030  0.00000009 -0.00000000
                   Y  3.50400963  0.00000001 -0.00000018  0.00000029  0.00000000
                   Z -1.93696102  0.00000022  0.00000004 -0.00000018 -0.00000000
               TOTAL  4.00390419  0.00000022  0.00000036  0.00000035  0.00000000

   ROT. SAYVETZ    X  0.00346439 -0.00000688  0.00001430 -0.00005114  0.00000983
                   Y -0.00241844 -0.00006552  0.00007507  0.00000286  0.00000699
                   Z -0.00425789 -0.00000799  0.00015751 -0.00000316  0.00000177
               TOTAL  0.00599838  0.00006636  0.00017507  0.00005131  0.00001219

                           11          12          13          14          15
       FREQUENCY:      1903.72     3526.08     3786.75     3786.81     3786.86  
        SYMMETRY:         A           A           A           A           A   
    REDUCED MASS:      1.00783     1.00783     1.10831     1.10832     1.10832
    IR INTENSITY:      0.00000     0.00000     0.01106     0.01105     0.01103

  1   C            X  0.00001099 -0.00000009  0.02677698 -0.08574373  0.01340844
                   Y  0.00000823  0.00000918 -0.00066764 -0.01423712 -0.08970002
                   Z  0.00000259 -0.00000864 -0.08678066 -0.02634660  0.00482855
  2   H            X  0.00001730  0.49805561 -0.24145237  0.77303200 -0.12085446
                   Y  0.48260611 -0.00000580 -0.00002679 -0.00061172 -0.00383651
                   Z -0.12293992  0.00000780 -0.00371796 -0.00111509  0.00020558
  3   H            X  0.46943007 -0.16599274  0.02915648  0.06043341 -0.26459559
                   Y  0.16350986  0.45692960 -0.07713260 -0.17706917  0.72611878
                   Z -0.02974730 -0.10794079  0.01450202  0.04055924 -0.17223007
  4   H            X -0.24187554 -0.16604034 -0.26187415  0.02244575  0.07363741
                   Y -0.38436659 -0.13502778 -0.21392745  0.02062641  0.05558323
                   Z -0.20467169  0.44980727  0.70880064 -0.07187413 -0.19772872
  5   H            X -0.22770271 -0.16602146  0.15534115  0.16502476  0.15216060
                   Y -0.26184739 -0.32200543  0.29903632  0.32657340  0.29017729
                   Z  0.35732809 -0.34177157  0.31369786  0.34613446  0.31226050

 TRANS. SAYVETZ    X -0.00000000 -0.00000005  0.00000000 -0.00000000 -0.00000000
                   Y  0.00000000 -0.00000015  0.00000000  0.00000001 -0.00000001
                   Z -0.00000000 -0.00000011  0.00000002  0.00000001  0.00000003
               TOTAL  0.00000000  0.00000019  0.00000002  0.00000001  0.00000004

   ROT. SAYVETZ    X -0.00000431  0.00000009 -0.00003050 -0.00001171 -0.00001387
                   Y  0.00000157  0.00000004  0.00000835  0.00001245  0.00000426
                   Z -0.00000777 -0.00000002 -0.00000454 -0.00000416  0.00001240
               TOTAL  0.00000902  0.00000010  0.00003195  0.00001759  0.00001908

Look at the TOTAL value for TRANS. SAYVETZ and ROT. SAYVETZ for the first 6 modes. You can immediately see that modes 1,2 and 3 are rotational modes whereas 4,5 and 6 are translational modes. This leaves us with $\mathrm{15-6=9}$ normal vibrational modes.


There are better ways of doing this. One is to use the PROJCT=.TRUE. option in the $FORCE section. If the molecule is at a stationary point i.e. optimized, then the program will try to automatically detect the rotational and vibrational modes, and then it will set those modes to zero.

This is the top half of the same calculation on methane, with PROJCT=.TRUE. added:

                            1           2           3           4           5
       FREQUENCY:         0.00        0.00        0.00        0.00        0.00  
        SYMMETRY:         A           A           A           A           A   
    REDUCED MASS:      1.69869     1.80186     1.73447     1.50339     1.20902
    IR INTENSITY:      0.00000     0.00000     0.00000     0.00000     0.00000

  1   C            X  0.18544520 -0.06453281 -0.05424004  0.10323087  0.00731683
                   Y  0.04925272  0.18950956  0.04481326 -0.07303356  0.00934898
                   Z -0.01355662  0.00338180  0.18210652  0.11830950  0.12246754
  2   H            X  0.18543884 -0.06453163 -0.05424640  0.10323472  0.00732201
                   Y  0.17102362  0.54645964 -0.12047768  0.19706661 -0.01669391
                   Z  0.30594469 -0.06047719  0.50480017 -0.07937287 -0.13871332
  3   H            X  0.14296863 -0.40589743  0.16736579 -0.18745463 -0.02538987
                   Y -0.03158358  0.07880458  0.12489341 -0.10130832 -0.08197398
                   Z -0.29043848  0.05967641  0.18032955  0.44561985 -0.21383067
  4   H            X -0.07005260  0.08990000 -0.39044048  0.35496230  0.23609947
                   Y  0.17634788  0.03606998 -0.00419087 -0.42034225  0.43466668
                   Z -0.06971641  0.01432464  0.04328714  0.10696653  0.33461067
  5   H            X  0.48341172  0.12243297  0.06032215  0.14221914 -0.18874644
                   Y -0.11876257  0.09670174  0.17901903  0.03242832 -0.29856752
                   Z  0.00000000  0.00000000  0.00000000  0.00000000  0.50784215

 TRANS. SAYVETZ    X  2.97291332 -1.03450939 -0.86957749  1.65496336  0.11731634
                   Y  0.78959978  3.03808233  0.71840554 -1.17084448  0.14991190
                   Z -0.21731384  0.05421128  2.91939491  1.89663038  1.96335287
               TOTAL  3.08365152  3.20984311  3.12971855  2.77614613  1.97255955

   ROT. SAYVETZ    X -1.02133456  0.20985148  0.63412701  1.56702735 -2.53776525
                   Y -1.75722268  0.35123915 -1.77483533  1.08722959  1.43654908
                   Z  0.66974877  1.96336847 -0.90916933  1.48569414 -0.14327855
               TOTAL  2.13998114  2.00554789  2.09254544  2.41762900  2.91966684

                            6           7           8           9          10
       FREQUENCY:         0.00     1675.67     1675.71     1675.76     1903.68  
        SYMMETRY:         A           A           A           A           A   
    REDUCED MASS:      1.42881     1.17145     1.17144     1.17144     1.00783
    IR INTENSITY:      0.00000     0.15729     0.15726     0.15732     0.00000

  1   C            X  0.10085354  0.02051281 -0.10602969 -0.03230341  0.00000583
                   Y  0.12887488 -0.01433217  0.03003943 -0.10769562 -0.00000247
                   Z -0.00495326 -0.10991295 -0.02370508  0.00801445  0.00000863
  2   H            X  0.10085825  0.01934637 -0.10005854 -0.03047754  0.00001134
                   Y -0.23012421  0.07075461 -0.14828865  0.53174574 -0.12291536
                   Z -0.23844365  0.54257359  0.11698974 -0.03957411 -0.48268833
  3   H            X  0.37964396 -0.10878236  0.39013836  0.33553069 -0.00818369
                   Y  0.28531794  0.09146253  0.21886997  0.04723477  0.11165381
                   Z  0.22859715  0.53776212  0.03032439  0.07488276  0.48525737
  4   H            X  0.21437181  0.09904228  0.51203315  0.06694147 -0.40256659
                   Y  0.09528755  0.23367074 -0.15753935  0.45641011  0.28658735
                   Z  0.02686809 -0.00009233  0.14777372  0.21104766 -0.06256492
  5   H            X -0.29147235 -0.25384877  0.46036440  0.01263656  0.41066955
                   Y  0.36500460 -0.22523712 -0.27071632  0.24692273 -0.27529636
                   Z -0.03684942  0.22847129 -0.01283556 -0.34178301  0.05989307

 TRANS. SAYVETZ    X  1.61680082  0.00000000  0.00000000 -0.00000000 -0.00000000
                   Y  2.06601817 -0.00000000 -0.00000000  0.00000000  0.00000000
                   Z -0.07942214 -0.00000000  0.00000000 -0.00000000  0.00000000
               TOTAL  2.62464928  0.00000000  0.00000000  0.00000000  0.00000000

   ROT. SAYVETZ    X  0.93345047 -0.00000000 -0.00000000 -0.00000000  0.00000000
                   Y  1.28415215 -0.00000000  0.00000000 -0.00000000 -0.00000000
                   Z -1.97460041  0.00000000  0.00000000 -0.00000000 -0.00000000
               TOTAL  2.53365809  0.00000000  0.00000000  0.00000000  0.00000000

As you can see, frequencies of modes 1-6 have been set to zero, as GAMESS identified them as rotational and translational modes. However, this projection method is not particularly reliable, and might fail. Also, it cannot be used if the molecule is not at a stationary point (it will set one vibrational mode to the direction of the gradient).

So, an alternative is to use internal coordinates and use PURIFY=.TRUE. in the $FORCE section. Using internal coordinates in GAMESS is a real pain. If your molecule does not have weak interactions i.e. all bonds are traditional covalent bonds then adding NZVAR=1 in $CONTRL and using $ZMAT DLC=.TRUE. AUTO=.TRUE. $END will work. Otherwise, you have to define the weak bonds with NONVDW in $ZMAT. You should look into the GAMESS manual for more detailed explanations.

Setting PURIFY=.TRUE. with internal coordinates will also have the same effect, it will set the rotational and translational modes to zero.

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