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# Projection Of Harmonic Frequencies And Normal Coordinates

By default, the rotational and translational degrees of freedom are projected out of the harmonic force constant matrix. For calculations that are done at a stationary point, this projection accomplishes nothing, but for calculations at non-stationary points, the harmonic frequencies obtained from the projected and unprojected Hessian matrices will differ. For example, if one uses the ZMAT file for methane:

methane
H
C 1 R
H 2 R 1 TDA
H 2 R 1 TDA 3 T
H 2 R 1 TDA 4 T

R = 1.099509562213279
TDA = 109.471220634490706
T = 120.000000000000071

*ACES2(BASIS=PVDZ,FROZEN_CORE=ON,VIB=EXACT,CALC=MP2)

(which is the correct optimized geometry for the method), the first instance of harmonic frequencies printed gives

                     Normal Coordinate Analysis
----------------------------------------------------------------
Irreducible         Harmonic       Infrared        Type
Representation       Frequency      Intensity
----------------------------------------------------------------
(cm-1)         (km/mol)
----------------------------------------------------------------
----                 0.0139i        0.0000      TRANSLATION
----                 0.0000i        0.0000      TRANSLATION
----                 0.0000i        0.0000      TRANSLATION
----                 0.0106         0.0000       ROTATION
----                 0.0237         0.0000       ROTATION
----                 0.0668         0.0000       ROTATION
T2              1338.2805        11.4854       VIBRATION
T2              1338.2805        11.4854       VIBRATION
T2              1338.2805        11.4854       VIBRATION
E              1565.2549         0.0000       VIBRATION
E              1565.2549         0.0000       VIBRATION
A1              3082.4492         0.0000       VIBRATION
T2              3233.7968        17.0183       VIBRATION
T2              3233.7968        17.0183       VIBRATION
T2              3233.7968        17.0183       VIBRATION
----------------------------------------------------------------


where the (near) zero values correspond to translation and rotation, with residues due essentially only to roundoff error in the force constants. The projected frequencies are the "same"

   Vibrational frequencies after rotational projection  of
Cartesian force constants:
1               0.0000i
2               0.0000i
3               0.0000
4               0.0000
5               0.0000
6               0.0000
7            1338.2805
8            1338.2805
9            1338.2805
10            1565.2549
11            1565.2549
12            3082.4492
13            3233.7968
14            3233.7968
15            3233.7968


However, if one changes the bond length to 1.05 Angstroms, then:

                     Normal Coordinate Analysis
----------------------------------------------------------------
Irreducible         Harmonic       Infrared        Type
Representation       Frequency      Intensity
----------------------------------------------------------------
(cm-1)         (km/mol)
----------------------------------------------------------------
----               717.4116i        0.0000       ROTATION
----               717.4116i        0.0000       ROTATION
----               717.4116i        0.0000       ROTATION
----                 0.0000         0.0000      TRANSLATION
----                 0.0000         0.0000      TRANSLATION
----                 0.0128         0.0000      TRANSLATION
T2              1188.8434        17.0198       VIBRATION
T2              1188.8434        17.0198       VIBRATION
T2              1188.8434        17.0198       VIBRATION
E              1509.8098         0.0000       VIBRATION
E              1509.8098         0.0000       VIBRATION
A1              3527.9665         0.0000       VIBRATION
T2              3713.4315        11.9365       VIBRATION
T2              3713.4315        11.9365       VIBRATION
T2              3713.4315        11.9365       VIBRATION
----------------------------------------------------------------


and

   Vibrational frequencies after rotational projection  of
Cartesian force constants:
1               0.0000i
2               0.0000i
3               0.0000
4               0.0000
5               0.0000
6               0.0001
7            1188.8434
8            1188.8434
9            1188.8434
10            1509.8098
11            1509.8098
12            3527.9665
13            3713.4315
14            3713.4315
15            3713.4315


where the gradient along the bond distance gives rise to a spuriously nonzero frequency of T1 symmetry, which is removed by the projection. However, there are no "usual" normal modes of this symmetry, so these values are unchanged; this would not be the case for molecules in which the spurious harmonic frequencies occur in symmetries which match those of the "usual" normal modes.

Reaction Path Calculations

When a reaction path is studied, as for example in using Variational Transition State Theory (VTST), one might want to project out coordinates other than rotations and translations. This can also be done, currently by one of two mechanisms. If the coordinate representing the distance between two atoms in the ZMAT file is chosen for projection (as, for example, the first CH distance in the methane example above), one can use the following ZMAT file

methane
H
C 1 RX
H 2 R 1 TDA
H 2 R 1 TDA 3 T
H 2 R 1 TDA 4 T

RX = 1.200000000000000
R = 1.099454699336068
TDA = 109.471220634490692
T = 120.000000000000171

*ACES2(BASIS=PVDZ,FROZEN_CORE=ON,VIB=EXACT,CALC=MP2)

%projectdistance
1 2

(where the distance and angle have been optimized subject to the constraint that the fixed distance (RX) is 1.2 Angstroms). The string

%projectdistance
1 2

tells the program to project out the coordinate that corresponds to the distance between atoms 1 and 2 in the ZMAT file. The resulting frequencies are:

unprojected

                     Normal Coordinate Analysis
----------------------------------------------------------------
Irreducible         Harmonic       Infrared        Type
Representation       Frequency      Intensity
----------------------------------------------------------------
(cm-1)         (km/mol)
----------------------------------------------------------------
----                 0.0432i        0.0000       ROTATION
----                 0.0079         0.0000      TRANSLATION
----                 0.0148         0.0000      TRANSLATION
----                 0.0208         0.0000      TRANSLATION
----               445.4537         0.4434       ROTATION
----               445.4537         0.4434       ROTATION
A1              1306.8165         5.1440       VIBRATION
E              1395.6238        10.3143       VIBRATION
E              1395.6238        10.3143       VIBRATION
A1              1570.5188         0.4116       VIBRATION
A2              1570.5188         0.4116       VIBRATION
A1              2370.7944        24.5352       VIBRATION
A1              3125.6077         6.6311       VIBRATION
E              3243.7138        14.7983       VIBRATION
E              3243.7138        14.7983       VIBRATION
----------------------------------------------------------------


projected

   Vibrational frequencies after rotational projection  of
Cartesian force constants:
1               0.0000i
2               0.0000i
3               0.0000i
4               0.0000i
5               0.0000
6               0.0000
7               0.0000
8            1313.2381
9            1385.0526
10            1385.0526
11            1563.7786
12            1563.7786
13            3120.3112
14            3243.6987
15            3243.6987


where it is to be noted that there are now seven, rather than six, zero frequencies; the seventh constraint on the motion being that a change in the distance of the first two atoms is no longer permitted in the normal coordinates.

If more general constraints are applied, the program currently requires that the projected coordinate should be entered as follows:

%projectfromfcm
12.0000000000 0.0000000000 0.0000000000
-1.007825 0.0000000000 0.0000000000
0.0000000000 0.0000000000 0.0000000000
0.0000000000 0.0000000000 0.0000000000
0.0000000000 0.0000000000 0.0000000000

which is the Cartesian displacement corresponds to changing the distance between the two atoms but which does not move the center of mass. Other choices for the coordinate are possible, but must be chosen so as not to perturb the center of mass or introduce any rotations. Running both calculations indicated above should be done for practice, and one should verify that the projected frequencies are the same in both cases. CFOUR is partially supported by the U.S. National Science Foundation.