Chemistry Reference
In-Depth Information
TABLE 5.2 DFT-calculated Normal Modes for Gas-Phase CO
a
Eigenvector (
˚
)
Normal
Mode
Frequency
(cm
1
)
Atom
x
Y
Z
Mode Type
0.76
1
2132
C
0.00
0.00
CO stretch
O
0.00
0.00
0.65
2
59
C
0.65
0.00
0.00
Rotation
O
0.73
0.00
0.00
3
56
C
0.00
0.69
0.00
Rotation
O
0.00
0.70
0.00
4
0.10
i
C
0.00
0.00
0.65
Translation
O
0.00
0.00
0.76
5
4.5
i
C
0.00
0.71
0.00
Translation
O
0.00
0.70
0.00
6
7.9
i
C
0.75
0.00
0.00
Translation
O
0.66
0.00
0.00
a
The components of the eigenvectors are listed in
˚
rounded to two significant figures, with nonzero values
in bold for emphasis. The frequencies of modes 4, 5, and 6 are imaginary.
accuracy. Because this outcome is a generic result for plane-wave DFT calcu-
lations (or any other flavor of quantum chemistry that uses finite differences to
calculate normal modes), it is always important to understand how many zero-
frequency modes must exist for any particular collection of atoms. Once this
number is known, it is typically easy in a calculated set of normal modes to
identify these modes.
5.3 MOLECULES ON SURFACES
Now that we have reviewed the calculation of vibrational frequencies for
collections of atoms, let us move on to something more interesting: namely,
calculating the vibrational frequency of a CO molecule adsorbed on a surface.
As a simple example, we will look at CO adsorbed on the Cu(001) surface.
Experimentally, it is known that CO forms a well-ordered layer on this surface
with a c(2
2) structure (see Fig. 4.17 for a picture of this structure).
1
In this
structure, the ratio of CO molecules to metal atoms on the surface is 1 : 2. DFT
calculations very similar to those we discussed in Chapter 4 show that the top
site is the preferred adsorption site for this ordered structure, with the C atom
bonding to a Cu atom and the CO molecule oriented along the surface normal.
The calculated CO bond length for the adsorbed CO molecule is 1.158
˚
,
which is 0.015
˚
longer than the (calculated) gas-phase bond length. This
lengthening of the equilibrium bond length is reflected in the vibrational
frequency of the molecule.
