Normal mode analysis (FREQ)

In GRRM17 program, you can perform a normal mode analysis (harmonic vibrational frequency calculation) by specifying the job type "FREQ". A standard normal mode analysis is done by diagonalization of the mass-weighted Hessian matrix. Components of translation and rotation of the entire system are projected out before the diagonalization.

An input file for formaldehyde is:

# FREQ/MP2/6-31G
 
0 1
C         -0.000000000000         -0.000000000000         -0.549482561269
O          0.000000000000          0.000000000000          0.708343639882
H          0.000000000000          0.934113144104         -1.131025039307
H         -0.000000000000         -0.934113144104         -1.131025039307
Options
Temperature = 298.15
Pressure = 1.0
Symmetry number = 2
 

Harmonic frequencies and normal modes will be printed at the bottom of xxx.log file. The Gibbs free energy assuming Harmonic oscillator, rigid rotor, ideal gas approximations will also be computed. Although GRRM17 will assign the symmetry number automatically, users can specify it manually. An example of the output is shown below.

Normal Mode Analysis at Stationary Point
=========================================================================
 
FREQFREQFREQFREQFREQFREQFREQFREQFREQFREQFREQFREQFREQFREQFREQFREQFREQFREQF
Geometry (Origin = Center of Mass, Axes = Principal Axis), SYMMETRY = C2v
C          0.000000000000         -0.000000000000         -0.631332224508
O          0.000000000000          0.000000000000          0.626493976643
H          0.000000000000         -0.934113144104         -1.212874702546
H          0.000000000000          0.934113144104         -1.212874702546
 
                       1               2               3
Freq.  :          1197.42908695   1283.52009214   1515.48659989
C  - x :          -0.1709840977   -0.0000000000   -0.0000000000
C  - y :           0.0000000000   -0.1444298634    0.0000000000
C  - z :          -0.0000000000   -0.0000000000   -0.2456368276
O  - x :           0.0405571750    0.0000000000    0.0000000000
O  - y :          -0.0000000000    0.0757143551   -0.0000000000
O  - z :          -0.0000000000    0.0000000000    0.2629348534
H  - x :           0.6961032804    0.0000000000    0.0000000000
H  - y :          -0.0000000000    0.2590299404    0.2139066140
H  - z :           0.0000000000   -0.6477709205   -0.6241056532
H  - x :           0.6961032804    0.0000000000   -0.0000000000
H  - y :          -0.0000000000    0.2590299404   -0.2139066140
H  - z :          -0.0000000000    0.6477709205   -0.6241056532
 
                       4               5               6
Freq.  :          1680.11803551   3019.78912654   3105.61827592
C  - x :           0.0000000000    0.0000000000    0.0000000000
C  - y :           0.0000000000    0.0000000000    0.1002946008
C  - z :          -0.1910788293    0.0581483034   -0.0000000000
O  - x :          -0.0000000000   -0.0000000000   -0.0000000000
O  - y :          -0.0000000000   -0.0000000000    0.0000125967
O  - z :           0.0738795606    0.0000720620    0.0000000000
H  - x :          -0.0000000000   -0.0000000000   -0.0000000000
H  - y :          -0.4184163879   -0.6148755105   -0.5971952741
H  - z :           0.5513103229   -0.3467527813   -0.3719251282
H  - x :          -0.0000000000   -0.0000000000   -0.0000000000
H  - y :           0.4184163879    0.6148755105   -0.5971952741
H  - z :           0.5513103229   -0.3467527813    0.3719251282
 
Thermochemistry at  298.150 K,    1.000 Atm
E(el)         =  -114.028691241420
ZPVE          =     0.026886845139
Enthalpie(0K) =  -114.001804396281
E(tr)         =     0.001416284860
E(rot)        =     0.001416284860
E(vib)        =     0.026922793795
H-E(el)       =     0.030699553420
Enthalpie     =  -113.997991688000
S(el)         =     0.000000000000
S(tr)         =     0.000057577109
S(rot)        =     0.000025730302 (Symmetry number = 2)
S(vib)        =     0.000000139497
G-E(el)       =     0.005819857897
Free Energy   =  -114.022871383523
 
FREQFREQFREQFREQFREQFREQFREQFREQFREQFREQFREQFREQFREQFREQFREQFREQFREQFREQF

Frequently used options

• Project
• Temperature
• Pressure
• Symmetry number
• ATOM MASS
• MO GUESS

Note on the Treatment of Low-Frequency Modes:

At a stationary point, diagonalization of the mass-weighted Hessian matrix yields 3N−6 (or 3N−5 for linear molecules) vibrational modes as eigenvectors with non-zero eigenvalues, together with 6 (or 5 for linear molecules) rotational and translational modes as eigenvectors with zero eigenvalues. This is true only when the geometry optimization and Hessian calculation are free from numerical errors and ambiguities. In practice, however, such numerically exact calculations are not feasible.

Consequently, the six (or five for linear molecules) lowest eigenvalues typically have small but finite values, corresponding to frequencies of a few cm^-1. These numerical errors can become more significant when loose geometry optimization criteria, loose SCF convergence criteria, and/or sparse DFT integration grids are used. For example, when the #p keyword is specified, Gaussian 16 reports these low-frequency values in its output.

Because these low-frequency modes are no longer purely rotational or translational, they contain small vibrational components. As a result, they also influence the eigenvectors and frequencies of the vibrational modes. Consequently, the computed vibrational frequencies—particularly the low-frequency ones—may vary depending on the algorithm used in the normal mode analysis, and therefore may differ among software packages.

In GRRM, the rotational and translational components are explicitly projected out before diagonalization of the mass-weighted Hessian matrix as is done in the generalized normal mode analysis developed for the reaction path Hamiltonian and variational transition state (VTST) theory [https://wires.onlinelibrary.wiley.com/doi/10.1002/wcms.65]. As a result, the vibrational modes obtained are purely vibrational, even at non-stationary points.