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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 ---------------------------------------------------------------- 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 T
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
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 ----------------------------------------------------------------
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 |

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