Chemistry Reference
In-Depth Information
molecule contribute, then the adsorption energy is
X
6
hn
CO
2
hn
i
2
:
E
ads
¼
(
E
CO
þ E
Cu
)
E
CO
=
Cu
þ
(5
:
14)
i
1
The zero-point energy terms on the right-hand side of this expression come
from the one (six) vibrational mode that exists for the gas-phase (adsorbed)
molecule. Inclusion of the zero-point energies results in
E
ads
0.65 eV, or a
difference of about 0.06 eV.
†
Compared to the net adsorption energy, this con-
tribution from the zero-point energies is not large. Most of this net zero-point
energy comes from the five modes that exist on the surface with nonzero fre-
quencies in Eq. (5.14), not the CO stretch. The zero-point energy correction to
the adsorption energy associated just with the change in the CO stretching fre-
quency between the gas phase and the adsorbed molecule is only
0.004 eV.
As a qualitative rule, zero-point energy effects become more important for
problems involving light atoms. Specifically, this means that zero-point
energies may be especially important in problems that involve H atoms. As
an example, we can compare the energy of an H atom on the surface of
Cu(100) with the energy of an H atom inside bulk Cu. There are two kinds
of interstitial sites that can be considered inside an fcc metal, as illustrated
in Fig. 5.2, the sixfold octahedral site and the fourfold tetrahedral site. We
computed the energy of atomic H in one of these interstitial sites relative to
the energy of atomic H on the Cu(100) surface using
DE ¼
(
E
H
=
Cu(100)
þ E
bulk
)
(
E
Cu(100)
þ E
H
=
bulk
),
(5
:
15)
where
E
bulk
and
E
Cu(100)
are the DFT-calculated energies of a supercell contain-
ing bulk Cu and a Cu(100) slab, respectively, and
E
H
=
bulk
and
E
H
=
Cu(100)
are the
energies of the same supercells once an H atom is introduced into the supercell.
Using this definition, a negative value of
DE
implies that the interstitial site
being considered is more energetically favorable than the surface site.
The results from these calculations are summarized in Table 5.4. The results
without including zero-point energies indicate that the bulk octahedral site is
slightly more energetically preferred than the surface site, but that the bulk
tetrahedral site is less favorable. The calculated zero-point energies for the sur-
face site and the bulk octahedral site are similar in magnitude, so the relative
†
Although the change in this energy from the zero point energies is arguably not large, there are
situations where it could be unwise to neglect it. This would be the case, for example, if you were
attempting to decide which of several possible adsorption sites was most energetically favorable
and the sites differed in (classical) energy by an amount similar to the zero point energy
corrections.
