3. Frequency calculations
Frequency calculations can serve a number of different purposes:



To predict the IR and Raman spectra of molecules (frequencies and intensities).
To compute force constants for a geometry optimization. To identify the nature of
stationary points on the potential energy surface.
To compute zero-point vibration and thermal energy corrections to total energies as well
as other thermodynamic quantities of interest such and the enthalpy and entropy of the
system.
Energy calculations and geometry optimizations ignore the vibrations in molecular systems. In
this way, these computations use an idealized view of nuclear position. In reality, the nuclei in
molecules are constantly in motion. Because vibrational motions tend to be highly localized
within molecules, and the energy spacings associated with individual linkages tend to be
reasonably similar irrespective of remote molecular functionality, IR and Raman spectroscopies
have a long history of use in structure determination. Vibrational frequencies also have other
important uses, for example in kinetics and computational geometry optimization so their
accurate prediction has been a long-standing computational goal.
3.1. Vibrational analysis
First, consider a simple diatomic molecule. The PES is one-dimensional (diatomic has only one
degree of freedom - the bond length q) and typically looking somewhat like the solid curve
below:
V(q)
q
q0
q
As we know from physical chemistry, the vibrational problem for a diatomic is equivalent the
problem of motion of a particle with the reduced mass  on the PES. Schrodinger equation is:
 2 d 2

 V ( q )  ( q )  E ( q )

2
 2  dq

(3.1)
For small deviations from equilibrium (minimum) the potential V(q) can be approximated by a
quadratic function (parabola - the dashed curve above). Sounds familiar? It was already done
above in equation 2.4, except here the reference point q0 is the minimum:
 dV 
1  d 2V 
 (q  q0 )   2  (q  q0 ) 2  ...
V (q)  V (q0 )  
2  dq  q
 dq  x0
0
(3.2)
that means that the first derivative at q0 is zero, leaving only the constant and the quadratic term
in 3.2. To simplify, we will “rescale” the axes, so that:
V(q0) = 0
q0 = 0
and define the force constant k:
 d 2V 
k   2 
 dq  q0
Substituting to (3.1) we have a Schrodinger equation for the harmonic oscillator:
 2 d 2

 kq 2  ( q )  E ( q )

2
 2  dq

(3.3)
which has well known solutions. In particular, the energy levels of the linear harmonic oscillator
are quantized as:
1

En   n  h
2

, n = 0, 1, 2, …
(3.4)
where h is Planck constant and  is the vibrational frequency that depends on the force constant
and (reduced) mass:
1 k
(3.5)

2 
This is the frequency with which the molecule vibrates and is usually given as a wavenumber
~ 
in cm-1 (called “wavenumbers”).
1 
1 k
 
 c 2c 
(3.6)
For a polyatomic molecule with N atoms the PES, as we know, has 3N  6 (3N  5 for a linear
molecule) dimensions. Typically, the vibrational calculations are done in Cartesian coordinates
where the surface is 3N dimensional.
The potential energy
V(X)=V(X1, X2, …, X3N)
where X1 etc. are deviations from the equilibrium position (energy minimum) is again expanded
in the Taylor series up to the second order around the minimum, giving the Schrodinger
equation:
 2

 2

1 2
1 3N

H

X
X


ij
i
j  ( X )  E ( X )
2
2 i , j 1
i 1 m i X i

3N
(3.7)
in (3.7) mi are actual atomic masses and Hij is our old friend Hessian, the 3N by 3N matrix of
force constants:
H ij 
  2V

 X 1X 1
  2V
H   X X
2
1

...

  2V
 X X
 3N 1
 2V
X i X j
 2V
X 1X 2
 2V
X 2 X 2
...
 2V
q3 N q2
 2V
X 1X 3 N
 2V
...
X 2 X 3 N
...
...
 2V
...
X 3 N X 3 N
...










(3.8)
Equation of (3.7) is a system of coupled differential equations, which are solved, as coupled
equations usually are, by uncoupling them. That means transforming the Hessian to a new
coordinate system, called normal coordinates and denoted Q, where it is diagonal. The
transformation is written as:
3N
Qk   S kj1 X j ,
k = 1, 2, …, 3N
(3.9)
j 1
the coupled system of equations then becomes 3N independent equations:
 2 2

 2 2 2 Q k2  (Q k )  E (Q k ) ,

2
 2 Q k

k = 1, 2, …, 3N
(3.10)
for which we know the solutions, because each equation is nothing else than the Schrodinger
equation for one linear harmonic oscillator, the same as (3.3).
Several notes about the normal coordinates and vibrational frequencies:

The whole derivation of this section makes sense only when the molecule is in its energy
minimum or, alternatively, a saddle point. Both ensure that the first derivative in the
Taylor expansion of energy (3.2) is zero. If it is not zero, i.e. if the geometry is not
optimized, the whole thing goes right out the window !

Even though we have 3N normal coordinates for 3N equations, only 3N  6 (for a general
nonlinear polyatomic) are meaningful, the remaining 6 have zero vibrational frequencies,
because they correspond to the translations and rotations of the molecule as a whole.
In practice they may not be exactly zero: residual couplings between the external and
internal degrees of freedom are present for imperfectly optimized structures (and in
practice nothing is perfect). However, the external degrees of freedom can be projected
out of the Cartesian force fields to eliminate this problem.

Vibrational frequencies calculated for equilibrium structures are positive real numbers,
because the force constants are positive (second derivative is positive at a minimum of a
function.) However, for saddle point (transition states) one force constant is negative,
yielding, by equation (3.5), an imaginary frequency. Imaginary frequencies (usually
reported as negative values) are telltale signs of saddle points !

Normal mode coordinates contain the atomic masses: they are mass-weighted. Each
normal mode has its characteristic “reduced mass” just like the diatomic oscillator (eqn.
3.3).

Normal modes are complicated “mixtures” (combinations) of Cartesian coordinates and
motions of the individual atoms in the molecule. They can be delocalized, i.e. the
particular normal mode can involve movement of most or all atoms in the molecule.
The above procedure is fairly straightforward, once the Hessian matrix is available. Calculation
of the Hessian, i.e. the force constants, i.e. the second derivatives of the energy, is the important
and the hard part. Remember that the Hessian matrix contains 3N x 3N = 9N2 values and, even
though it is symmetric, requiring calculation of “only” 1/2(9N2 +3N) values, it is still a lot of
calculation.
Above we have already discussed analytical vs. numerical gradients and Hessians. Frequency
calculations are fairly straightforward with methods that have implemented analytical second
derivatives, more time consuming for methods with analytical gradients (the second derivatives
can be obtained by finite differentiation of the gradients), but limited to small molecules for
methods where not even analytical first derivatives are implemented.
Vibrational frequencies are traditionally of primary interest to chemists. However, vibrational
spectra, such as IR and Raman, are not made of frequencies only. Spectrum by definition is the
plot of intensity versus frequency. That means intensities are also important, in fact, for physical
chemists perhaps even more important, because more interesting physics is in the intensities than
in the frequencies.
First, as you know from P-chem, the selection rules for the harmonic oscillator are
n  1
(3.11)
therefore only a single quantum with the energy corresponding to the vibrational frequency, h,
can be absorbed or emitted.
IR absorption intensity for a particular normal mode Q is proportional to the change of the
molecular electric dipole moment e (don’t confuse with the reduced mass ) as the molecule
vibrated along the normal coordinate squared:
I IR
μ e

Q
2
(3.12)
Raman intensity is proportional to the change in polarizability e with respect to the particular
normal mode vibration:
I Raman
α e

Q
2
(3.13)
This means that for calculations of intensities, derivatives of dipole moments (IR) and
polarizability (Raman) must also be calculated. Since polarizability is a higher order property
calculating its derivatives is more demanding than calculating dipole derivatives: we can expect
Raman calculations to take longer than IR calculations (and also to be less accurate).
3.2. Vibrational calculations in Gaussian
Including the Freq keyword in the route section requests a frequency job. The other sections of
the input file are the same as those we've considered previously. Because of the nature of the
computations involved, frequency calculations are valid only at stationary points on the potential
energy surface. Thus, frequency calculations must be performed on optimized structures. For this
reason, it is necessary to run a geometry optimization at the same level of theory (!) prior to
doing a frequency calculation.
It can be done by including both Opt and Freq in the route section of the job, which requests a
geometry optimization followed immediately by a frequency calculation. However, the
optimization performed is always a full optimization: constraints cannot be applied.1
1
It may seem wrong to do constrained minimization before frequency calculations, since constrains would
violate the requirement of the fully optimized structure. In some special cases it is allowed and in fact
necessary, but only with extreme care !
Alternatively, you can give an optimized geometry as the molecule specification section for a
stand-alone frequency job or read the optimized geometry from the checkpoint file saved from
the previous optimization job.
Most conveniently, the optimization and frequency jobs can be linker together throught
--Link1-command (see the section 2.7 on multi-step jobs above)
It is worth repeating that a frequency job must use the same theoretical model and basis set as
produced the optimized geometry. Frequencies computed with a different basis set or procedure
have no validity. We'll be using the 6-31G(d) basis set for all of the examples and exercises in this
chapter. This is the smallest basis set that gives satisfactory results for frequency calculations.
For our first example, we'll look at the Hartree-Fock frequencies for formaldehyde. Here is the
route section from the input file:
%chk=formaldehyde.chk
# RHF/6-31G(d) Freq Geom=AllCheck Guess=Read Test
Here the geometry is taken from a checkpoint file that was saved from a previous optimization
job on formaldehyde (at the same level of theory !).
Frequencies and intensities
A frequency job predicts the frequencies, intensities (IR and Raman), and Raman depolarization
ratios and scattering activities for each spectral line. Note the units:
Harmonic frequencies (cm**-1), IR intensities (KM/Mole), Raman scattering
activities (A**4/AMU), depolarization ratios for plane and unpolarized
incident light, reduced masses (AMU), force constants (mDyne/A),
and normal coordinates:
1
2
3
B1
B2
A1
Frequencies -- 1335.2844
1383.7422
1679.1743
Red. masses -1.3685
1.3439
1.1034
Frc consts -1.4376
1.5161
1.8330
IR Inten
-0.3752
23.1314
8.5554
Raman Activ -0.7606
4.5229
12.8152
Depolar (P) -0.7500
0.7500
0.5916
Depolar (U) -0.8571
0.8571
0.7434
This display gives predicted values for the first 3 spectral lines for formaldehyde. The other ones
follow:
Frequencies
Red. masses
Frc consts
IR Inten
Raman Activ
Depolar (P)
Depolar (U)
--------
4
A1
2028.1696
7.2760
17.6340
150.0921
8.1478
0.3285
0.4945
5
A1
3161.2337
1.0490
6.1763
49.5883
137.6788
0.1830
0.3094
6
B2
3233.8392
1.1206
6.9046
135.3880
58.1390
0.7500
0.8571
The strongest IR line is the 4th, at 2028 cm-1 with the intensity of 150 KM/Mole.
Normal Modes
In addition to the frequencies and intensities, the output also displays the displacements of the
nuclei corresponding to the normal mode associated with that spectral line. The displacements
are presented as XYZ coordinates, in the standard orientation:
Standard orientation:
------------------------------------------------------------Center
Atomic
Atomic
Coordinates (Angstroms)
Number
Number
Type
X
Y
Z
------------------------------------------------------------1
6
0
0.000000
0.000000 -0.519645
2
8
0
0.000000
0.000000
0.664624
3
1
0
0.000000
0.924643 -1.099562
4
1
0
0.000000 -0.924643 -1.099562
-------------------------------------------------------------
The carbon and oxygen atoms are situated on the Z-axis, and the plane of the molecule coincides
with the YZ-plane.
Here is the first normal mode for formaldehyde (it is displayed right under the frequency and
intensity values for the mode “1”):
Atom
1
2
3
4
AN
6
8
1
1
X
0.17
-0.04
-0.70
-0.70
Y
0.00
0.00
0.00
0.00
Z
0.00
0.00
0.00
0.00
In the standard orientation, the X coordinates for all four atoms are 0. When interpreting normal
mode output, the relative signs and relative values of the displacements for different atoms are
more important than their exact magnitudes. For this normal mode, the two hydrogen atoms
undergo the vast majority of the vibration, in the negative X direction. Although the values here
suggest movement below the plane of the molecule, they are to be interpreted as motion in the
opposite direction as well. Remember that the signs are arbitrary (i.e. all pluses could be changed
to minuses and minuses to pluses) - the atoms they are oscillating back and forth around their
equilibrium positions. What is important, however, once again, are the relative signs of the
modes with respect to each other (i.e. if one is plus and the other minus - they have opposite
signs. Changing the signs will make plus into minus and vice versa, but they will still remain
opposite !)
In our diagram, the motion is illustrated by showing the paths of the nuclei in both directions.
Thus, the hydrogens are oscillating above and below the plane of the molecule in this mode.
3.3. Visualizing spectra and normal modes in Gabedit
Making sense of the normal mode output gets overwhelming pretty quickly (unless you can
visualize the structure and the vibrations in your head from a long set of x, y, z coordinates and
displacements).
The best way to figure out what the vibrations look like is to animate them.
Again, start with clicking on the Display Geometry/Orbitals/Density/Vibration icon
to open
the window, then right click and in the menu select Animation (near the bottom) and Vibration
In the new window (called Vibration), select File/Read/Read a Gaussian output file.
You will see the table of the normal modes on the right and their graphical representation in the
molecule window in the left (you may want to rotate the molecule and zoom in):
You can step through the individual modes to visualize them (arrows are the displacements) and
animate them using the Play button. The amplitude and speed can be changed using the Scale
factor, Time step etc.
To draw the spectra, click on Tools button at the top of the right window and select Draw IR
spectrum or Draw Raman spectrum.
You can manipulate the plot in various ways: right click on the plot to display the Menu. For
example, if you want to reverse the axes, right click, select Render/Directions and
uncheck Xreflect and Yreflect.