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Projection Of Harmonic Frequencies And Normal CoordinatesBy 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 Normal Coordinate Analysis
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Irreducible Harmonic Infrared Type
Representation Frequency Intensity
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(cm-1) (km/mol)
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---- 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
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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
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Irreducible Harmonic Infrared Type
Representation Frequency Intensity
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(cm-1) (km/mol)
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---- 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
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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 %projectdistance unprojected Normal Coordinate Analysis
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Irreducible Harmonic Infrared Type
Representation Frequency Intensity
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(cm-1) (km/mol)
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---- 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
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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 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.