COMMENTS
Comment on ‘‘Does lattice vibration drive diffusion in zeolites?’’
†J. Chem. Phys. 114, 3776 „2001…‡
Giuseppe B. Suffritti
a)
and Pierfranco Demontis
Dipartimento di Chimica, Universita ` di Sassari and Consorzio Interuniversitario, Nazionale per la Scienza
e Tecnologia dei Materiali (INSTM), Unita ` di Ricerca di Sassari, Via Vienna, 2, 07100 Sassari, Italy
Giovanni Ciccotti
INFM and Dipartimento di Fisica, Universita ` di Roma ‘‘La Sapienza,’’P. le A. Moro, 2,
I-00185 Roma, Italy
Received 18 September 2002; accepted 20 November 2002
DOI: 10.1063/1.1538182
In a recent paper,
1
Kopelevich and Chang propose a
method to estimate the effect of lattice vibrations as a driving
force for sorbate diffusion in zeolites—a class of mi-
croporous crystalline aluminosilicates. To achieve that, they
use lattice dynamics LDRef. 2 to define a generalized
Langevin equation for sorbate motion. The effects of lattice
vibration are then estimated by two parameters, involving
quantities present in the generalized Langevin equation and
easily computed from LD results. Strangely enough, the LD
formulation of the crystal, involving only the harmonic con-
tribution from the lattice potential energy, is used by Ko-
pelevich and Chang to discuss the relevance of the molecular
dynamics
3
MD simulations in the study of the diffusive
process, forgetting that MD includes also anharmonic effects.
Notwithstanding this extrapolation, although the general ap-
proach remains useful and interesting. However, the peculiar
way in which they treat the low-frequency vibrational modes
of the host crystal and in particular the ‘‘zero-frequency
phonons,’’ which are always present in the dynamics of a
crystal, is not convincing. The LD technique yields three
zero frequency modes, which correspond to three uniform
time independent solutions of the equation of motion and,
actually, they are not phonons. It is important to stress that,
as the zero-frequency modes leave the crystal unchanged,
they cannot influence the diffusion of sorbates. Therefore,
they do not contribute to the Langevin equation for the dif-
fusive motion of the sorbate, since they do not represent
vibrational modes. On the contrary, one has to include all the
properly weighted other phonons present in the crystal.
Among them, the numerically unstable low but not zero
frequency phonons could require some care. However, their
statistical weight, close to the one given by the Debye
distribution,
2
is small, so that they contribute very little to the
motion of the sorbate. Kopelevich and Chang, instead, con-
sider the coupling of zero frequency modes with the sorbate
molecule Eqs. 75 – 79 of Ref. 1 and derive equations of
motion, Eqs. 77 and 78, which lead to inconsistent results
such as the fictitious ‘‘resonant blow up’’ of the crystal
caused by the sorbed molecule. We will show that Eqs. 76
and 77 contain errors and that the correct equations to be
used do not contain any ‘‘blow up.’’Let us rewrite the model
Hamiltonian of Eq. 75 in Ref. 1,
H
T
=
1
2
MX
˙
2
+
1
2
j =1
3
Q
˙
j
2
+
0
X +
j =1
3
ˆ
j
X Q
j
, 1
where X is the vector representing the position of a point
particle sorbate, Q
j
( j =1,2,3) are the three zero frequency
modes of the crystal, and the potential X,Q is approxi-
mated as the Taylor’s development to the first order with
0
( X) = ( X, Q=0) and
ˆ
j
( X) = / Q
j
|
Q
j
=0
, j =1,2,3.
Moreover, for the sake of simplicity, we assume the total
mass of atoms in a zeolite cell m associated with the Q
j
’s)
as mass unit. Since we have reduced the 3 N normal modes
to 3, we have
ˆ
j
X =
Q
j
Q
j
=0
=-
X
j
X
j
=0
, j =1,2,3, 2
where the last equality results from the translational invari-
ance of the potential energy ,
X
j
+a
j
, Q
j
+a
j
= X
j
, Q
j
, j =1,2,3 3
for any a
j
. Taking a
j
infinitesimal, translational invariance
implies
X
j
+
Q
j
=0, j =1,2,3 4
from which the right-hand side of Eq. 2 follows. Transla-
tional invariance can be also imposed on the partial expan-
sion of the potential energy assumed in Eq. 1. We can write
X
j
, Q
j
= X
j
, Q
j
0
+
j =1
3
X
j
, Q
j
0
Q
j
Q
j
,
j =1,2,3, 5
where Q
j
0
=0 are the initial values of Q
j
’s ( j =1,2,3)] and
impose the invariance to the left-hand side of Eq. 5,
JOURNAL OF CHEMICAL PHYSICS VOLUME 118, NUMBER 7 15 FEBRUARY 2003
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