Transition state theory description of surface self-diffusion: Comparison
with classical trajectory results
Arthur F. Voter and Jimmie D. Doll
Chemistry Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
(Received 6 January 1984; accepted 7 February 1984)
We have computed the surface self-diffusion constants on four different crystal faces [fcc(lll),
fcc( 1(0), bcc( 110), and bcc(211)] using classical transition state theory methods. These results can
be compared directly with previous classical-trajectory results which used the same LennardJones 6-12 potential and template model; the agreement is good, though dynamical effects are
evident for the fcc( 111) and bcc( 110) surfaces. Implications are discussed for low-temperature
diffusion studies, which are inaccessible to direct molecular dynamics, and the use of ab initio
potentials rather than approximate pairwise potentials.
I. INTRODUCTION
The migration of atoms and clusters on a surface is an
important process in a variety of phenomena, including crystal growth, defect formation, epitaxial layer growth, and heterogeneous catalysis. Understanding the factors which influence the rate and mechanism of this migration is thus of
central importance, and has received considerable attention
in recent years.
Use of the field ion microscope 1 (FIM), which is capable
of observing a single adatom on a clean crystal face, has
yielded high-quality surface diffusion constants for a variety
of metal systems. I - 7 These diffusion constants generally exhibit Arrhenius behavior, with activation energies and
preexponentials which vary widely with the choice of metal
and the crystal face. These results thus allow a test of our
theoretical understanding of the microscopic features of the
adatom dynamics. Consequently, a number of theoretical
studies have appeared 3•6 •8 •9 which applied approximate dynamical theories (transition state theory) to the single-adatom
diffusion problem, using Morse or Lennard-Jones pairwise
potentials to describe the adatom-substrate and substratesubstrate interactions. These studies showed varied success
in comparing with the experimental FIM results, and in the
cases of disagreement, there was no way to discern whether
the approximate dynamical theory or the approximate potential energy function was responsible for the discrepancy.
Recently, McDowell and DolpO-13 performed classical
trajectory studies for Rb on Rh( 111), Rb on Rb( 1(0), W on
W( 11 0), and W on W(211) using Lennard-Jones 6-12 potentials obtained from bulk thermodynamic data,13 and found
reasonable agreement with experiment. Independently,
Mruzik and Pound 15 applied similar methods to a variety of
fcc faces and obtained comparable results. Within the assumption of classical mechanics, the diffusion parameters
obtained from these two studies represent the exact dynamical values for the Lennard-Jones system. It is the purpose of
the present paper to test the accuracy of an approximate
dynamical approach, classical transition state theory (TST),
by applying it to the same Lennard-Jones systems. The motivation for this test is as follows: If TST yields the same results as molecular dynamics, then future studies of this type
can benefit from the considerable computational savings afforded by TST. Further, since TST in its simplest form requires knowledge of the system's potential energy at only a
5832
J. Chem. Phys. 80 (11). 1 June 1984
few geometries, these energies can, in principle, be obtained
from ab initio electronic structure calculations, eliminating
the pairwise potential approximation.
This paper is organized as follows: Section II contains
the description of the Lennard-Jones 6-12 potential and template model used in both the dynamical studies l O- 13 and the
present study. Section III briefly describes the classical trajectory method, and how diffusion parameters are obtained
from it. Section IV describes the simple TST approach, while
Sec. V describes the Monte Carlo TST (MCTST) method,
which we use to obtain "exact" TST results. Section VI contains results and discussion and Sec. VII gives our conclusions.
II. THE LENNARD-JONES 6-12 POTENTIAL AND
TEMPLATE MODEL
All calculations described here employ a LennardJones 6-12 potential, in which the total potential energy at a
given geometry is given by
(1)
where rij is the distance between particles i andj, while Eij
and 0'ij are the Lennard-Jones parameters for that pair. In
the present work, the same E and 0' apply to all atom pairs,
since the systems are homonuclear. Lennard-Jones parameters for a number of metals have been obtained by Halichioglu and Pound 14 from an analysis of bulk thermodynamic data. We use their values for tungsten (E/kB = 12391
K, 0' = 2.562 A, while the values for rhodium (E/kB = 7830
K 0' = 2.47 A) were obtained by similar considerations, as
described previously. 12
While the energies and diffusion constants we quote
here will be appropriate for the Rb (fcc) and W (bec) systems,
it is important to note that the corresponding-states principle allows a transformation to enable comparison with any
Lennard-Jones system, independent of E and 0'. These transformations to dimensionless quantities are given as follows
for energy, distance, temperature, and time, respectively:
E* =EIE,
r* = riO',
T*
t
= kBT IE,
*=
(2)
(E/m~)1/2t.
0021-9606/84/115832-07$02.10
® 1984 American Institute of Physics
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5833
A. F. Voter and J. D. Doll: Theory of surface self-diffusion
Here kB is the Boltzmann constant and m is the particle
mass. Thus, e.g., the reduced diffusion constant is given by
D*= ( -
m
€UZ
)112D.
(3)
The infinite surface is represented by the template model described by 0011 and McDowell. 12 One or more moving
layers of atoms are affixed to a rigid template which has the
geometry appropriate for the desired crystal face. The number of atoms per layer is between 15 and 40, depending on the
system, and the total number of layers is 7 for bcc(211), and
4 for the other three surfaces. To match the conditions used
by 0011 and McDowell, the same spherical cutoff (2.20') and
minimum-image periodic boundary conditions l6 were employed.
III. MOLECULAR DYNAMICS
Given the Lennard-Jones potential and the template
model, the diffusion constants obtained from the molecular
dynamics (MD) simulations represent the "correct" value
against which to compare the TST results. Though this paper presents no new MD results, we briefly describe the
method and how the diffusion constant is extracted from
such a simulation. The reader is referred elsewhere 10- 13 for
the details.
The system, consisting of the adatom and approximately 100 atoms representing the surface, is first "warmed up"
to the desired temperature using a smart Monte Carlo 17
(SMC) procedure. This consists of integrating Hamilton's
equations of motion while periodically randomizing the momenta of all the moving particles according to a Maxwellian
velocity distribution. After the system is warm, a Boltzmann-distributed set of starting conditions for the actual trajectories are chosen as snapshots from the subsequent SMC
moves. From each of these - 500 initial conditions, classical
trajectories are integrated long enough to observe diffusive
hops between binding sites. The diffusion constant is then
obtained from the asymptotic behavior of the mean-squared
displacement of the adatom
D = lim (..::1r(t) IUt,
(4)
by plotting (..::1r(t) vs t, and fitting a line to the limiting slope
(averaged over all the trajectories). Here d is the dimensionality of the system, taken as 2 for most surfaces, and 1 for
channeled surfaces such as the bcc(211). As mentioned
above, the temperature dependence of the diffusion constant
is generally found to be Arrhenius,
D = Doe - EAlkBT,
(5)
so that the preexponential Do and the activation energy EA
may be obtained from a plot of In(D) vs 1/T. The MD simulations typkally required 150 h of CPU time per temperature
on a VAX 11/780 with a floating-point accelerator.
IV. SIMPLE TST
In the TST method, we appeal to the simplest possible
description of the surface diffusion process. We assume that
the motion of the adatom consists of independent, randomly
oriented hops between adjacent binding sites. This motion
obeys random-walk statistics, so the diffusion constant is
given by
(6)
where I is the distance between binding sites, and k hop is the
rate at which the adatom executes hops. In simple TST, we
take k hop from the TST result for a one-dimensional harmonic potential
k hop
= np voexp [ -
(Esaddle -
E min )lkB T],
(7)
where Vo is the harmonic frequency, Esaddle is the energy at
the transition state between two binding sites, and E min is the
energy at the binding site minimum. Because there is more
than one exit path, the variable np is set equal to the number
of binding sites accessible by a single hop. The Arrhenius
parameters are thus obtained by
Do = NpVoL
2
(8)
2n
and
EA
= Esaddle
-
E min •
(9)
In calculating E min , E sadd1e , and vo, we specify how
many layers of the template are free to move (typically
between zero and two), and apply the method of molecular
statics. The value for E min is found by placing the adatom in
a binding site, and then minimizing the total energy of the
template-adatom system as a function of the position of the
adatom and all moving-layer atoms. A similar procedure is
used to find E sadd1e , except that the adatom is constrained to
lie at the maximum in the minimum-energy pathway
between two binding sites, while all other degrees of freedom
are energy optimized. The position of this saddle point is
shown in Fig. 1 for each of the four surfaces. While symmetry considerations dictate the saddle point location for the
fcc(lOO), fcc(lll), and bcc(11O) surfaces, the asymmetric
placement of the second layer in the channeled bcc(211) surface necessitates a search to find the maximum along the x
direction. The harmonic frequency Vo is found from a finitedifference method, using E min , and the energy at a small
distance (approximately 0.1 bohr) along the direction leading to the saddle point. In finding this energy, the positions
of all movable atoms are again allowed to relax, with the
constraint that the adatom x and Y coordinates are fixed at
0.1 bohr along the reaction coordinate. Thus, in the simpleTST approach, the adatom is considered to be moving in a
pseudo-one-dimensional potential, in which the positions of
all other atoms in the system are relaxed adiabatically.
The computer time required to obtain EA and Do at the
simple-TST level was less than 5 CPU minutes on the VAX,
and the results are shown in Table I under the "STST" heading. No Do value is given for the W( 110) surface because of
the ambiguity involved in choosing a normal mode leading
from the minimum to the saddle point.
J. Chern. Phys., Vol. 80, No. 11, 1 June 1984
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5834
A. F. Voter and J. D. Doll: Theory of surface self-diffusion
V. MONTE CARLO TST
As described by Doll, 18,19 the assumptions of a harmonic potential may be eliminated by using a statistical procedure to
evaluate the TST rate constant, which we will term Monte Carlo transition state theory (MCTST). The classical TST rate
constant for escape from state A to state B may be expressed in statistical-mechanical form (in one dimension) as
kA
_ B
= ~(~(x-q)/xl>A>
2
where x is the reaction coordinate, x
(1O)
= q gives the location of the TST dividing surface, and
.
dx
X=-.
(11 )
dt
The pointed brackets indicate the usual canonical-ensemble average
J
Ye-tJHdx dp
(12)
where H is the Hamiltonian, /J = lIkB T, and p is the momentum. The subscript A in Eq. (10) indicates that the ensemble
average should be taken only over the phase space belonging to state A, so that Eq. (10) becomes
J:
kA _
!
dp [~E dx[t5(x - q)Ii:lexp! - /J [V(x) + p2/2m]}]
=
----~----~------------------------------B
dp [~E dxexp! - /J [V(x) + p2/2m 1l
00
J:
00
(13)
where we have assumed that the particle is in state A when x<q, and the infinitesimal € has been added to the integration limit
so that the whole Dirac-delta function is contained in the integration. Because the Hamiltonian is separable, we can integrate
out the momentum to yield
= _I (2k
_ T)I/2 (t5(X-q)A'
(14)
B_
2
1Tm
Equations (10) or (14) correspond to the average flux through the TST dividing surface, divided by two since half the particles
contributing to this fiux are going in the wrong direction. The TST assumption, of course, is that all particles passing q in the
+ x direction originated in state A and will thermalize in state B without recrossing q.
Extending this result to 3 dimensions is straightforward. Defining a general TST dividing surface by the equation
fiR)
= 0,
(15)
the average fiux through this surface will be given by (15 (f)lVfllsl), wheres is the velocity normal to the surface/,
8=
~.v.
(16)
IVfl
In performing the ensemble average, we can consider integrating the numerator in a piecewise fashion, dividing configuration
space into many infinitesimal boxes. Only those boxes which contain a piece of the TST surface will give rise to a nonzero
integral. Within each of these boxes, we make an orthogonal coordinate transformation so that the surface normal direction s
is one of the coordinates, Le., such that
R=s+s,
p= p. +p,
(17)
where
s· S = 0,
p•• it = o.
Our ensemble average
J. Chem. Phys., Vol. 80, No. 11, 1 June 1984
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A. F. Voter and J. D. Doll: Theory of surface self-diffusion
5835
exp{ -/3 [V(R) +p2/2m])c5[f(R)]IV/llsldRdp
f-=---=.. 00foo
=----=:=-----00----------OO
(c5[f(R)] IV/lisl > =
f- 00 f- 00 -/3
f: 00 -,8p~/2mlsldPs f: 00
Lox
f: 00
f: 00
exp{
[V(R)
(18)
+ p2/2m]) dR dp
thus becomes
e
L
BOXES
e -pp'-/2m tip
e - ,8p'/2m dp
c5[/(s,8)] IV/le -,8V(s,i)ds ciS
e - ,8V(R)d R
Since
oo e-,8p~/2m Isldps = (2kBT)1I2 foo e-,8p~/2m dp"
ffm-oo
f -00
we have
L (~U,
(c5[f(R)]IV/llsl> =
BOXES
=
ff
exp[
-P(P~/2m + p'I2m))dp, dji6[f(RJlIV/le- PV'RJdR
ff
e - ,8p'/2m dp e -
,8V(R)
dR
(2kBj'n f e-"""6[f(RJlIV/I dR
ffm
(19)
f e - ,8V(R)d R
I
Thus,
(20)
where the ensemble average is taken over the configuration
space belonging to state A, including the boundary surface.
The ensemble average in Eq. (20) may be readily computed for arbitrary potential using Monte Carlo.20 This is
accomplished by taking a Metropolis walk in configuration
space and accumulating the average value of the property
c5 if (R)) IVI I. That is, we take a step from R to R', and accept
or reject that step according to
P
= e-,8[V(R')- VIR))
(21)
employed. A number of parallel counting boxes were positioned perpendicular to the channel in the vicinity of the
saddle point. For a given temperature, the box accumulating
the least counts was used in Eq. (22) to compute the rate. The
position of this box showed a slight temperature dependence
in the range of temperatures investigated.
To restrict the adatom to a single binding site [as indicated in the ensemble average in Eq. (20)), we simply reject
Rh(100)
W(211)
Rh(111)
W(110)
accept
In practice, the delta function is simulated by a narrow
"box" of width w (typically 0.2 bohr), and the rate is computed as (assuming IV/I = 1)
_
1 (2kBT)1I2
(IB) ,
--
kA~B - -2
(22)
ffm
w
where IB is the fraction of the Monte Carlo steps that lie
inside the box. For the fcc(lll), fcc(IOO), and bcc(llO) surfaces, the counting box is easily defined by connecting the
boundary atoms of a binding site with planes which extend
infinitely in the z direction, as shown in Fig. 1. These planes
move along with the top layer of atoms (m is thus replaced by
the effective mass of this coordinate, 2m/3), and the system
is considered to be in the counting box whenever the distance
between the adatom and the nearest plane is less than w/2.
Box placement on the bcc(211) surface is not so simple, because the in-channel saddle point does not coincide with the
plane connecting the top-layer atoms on opposite sides of the
channel. For this surface, a variational TST approach21 was
f------1
10 bohr
FIG. 1. The template models used for each of the four surfaces. The top
layer is shown [two layers for W(211)], along with the adatom (shaded) in a
binding-site minimum. For one another binding site on each surface, the
position of the TST saddle points are indicated by dots, and straight lines
show the portion of the MCTST counting plane bordering that site (the
T = 0 positions are shown for W(211)).
J. Chern. Phys., Vol. 80, No. 11, 1 June 1984
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5836
A. F. Voter and J. D. Doll: Theory of surface self-diffusion
Metropolis steps which go more than w/2 outside the region
bounded by the counting plane. Alternatively, the adatom
may be allowed to move anywhere on the surface, in which
case the rate obtained by Eq. (22) should be doubled, since
each counting box now serves as a border to two states. (This
procedure would be inappropriate for a case in which binding sites are not symmetry equivalent). Each Metropolis
move consisted of a random step by the adatom and one
other moving atom. This procedure was chosen to increase
the rate of sampling in the important regions of configuration space, since the adatom position most directly affects
the property in the ensemble average. The step size was chosen so that roughly half the moves were rejected. If during
the Metropolis walk, the adatom "desorbed" (if its distance
from the surface exceeded the cutoff range), the run was discarded. Typically 15 runs, with 10 000-30 000 MC steps
each, were performed to obtain D at a given temperature,
requiring 10-20 CPU hours on the VAX. EA and Do were
then obtained from an Arrhenius plot, as shown in Figs. 2-5.
VI. RESULTS AND DISCUSSION
Table I shows the Arrhenius parameters from MCTST
and STST, along with the MD results of McDowell and
Doll. The MCTST calculations were performed in the same
temperature range as the MD simulations, to allow a direct
comparison, as shown in Figs. 2-5.
For a two-state system (A~B), it is well known 21 that
the TST rate constant is an upper bound to the dynamically
exact rate constant. This is because each reactive conversion
from state A to state B (or B to A) will give rise to one or more
crossings of the TST surface. For example, a system passing
from state A to state B may jiggle back and forth in the
transition-state region (e.g., due to thermal fluctuations) before coming to rest in B, causing many TST-surface crossings for a single reactive event. Similarly, a system may pass
from A to B, but then be quickly bounced back to A without
ever "residing" in state B; this gives rise to two crossings
with no reaction. Since TST makes the assumption that each
crossing corresponds to one reactive state change, these dynamical recrossing events act to lower the rate from that predicted by TST, and only if these events are nonexistent will
TST be exact.
-6
~~
"
'
. " ..
'
2
Rh on Rh(100)
,'.
-8
.......
.-~~..
-10
-12
'Ii
,
= DYNAMICS
-14
0
5
10
10000/T
FIG. 3. Arrhenius plot (as in Fig. 2) for Rh on Rh(IOO) with one layer free.
The dynamics results are from Ref. 13.
In computing a diffusion constant from TST, however,
there is an additional consideration arising from the multistate nature of the system. The TST diffusion constant
(D TST) contains the assumption that successive hops are uncorrelated. If, after an adatom makes a hop, there is a probability of quickly making another hop in a specific direction,
the diffusion constant will be affected. To be directionally
oriented, this secondary hop must occur within a few vibrational periods after the primary hop, or the memory of the
preceeding hop direction will be lost. The diffusion constant
will be increased if the average angle between successive
hops is greater than 90·, and decreased for an average angle
less than 90·. Thus, the TST diffusion constant is not an
upper bound on the exact diffusion constant.
We can think of the classically exact diffusion constant
(D exact) as resulting from a competition of effects which
modify D TST: bounce-back recrossings and transition-state
oscillations, which act to lower D, and directionally extended multiple hops, which act to raiseD. It is worth noting
that the existence of quick successive hops is not alone sufficient to affect D TST; there must be an anisotropy to the exit
direction of the second hop. It is easy to imagine, e.g., a
situation in which the adatom, once thermally energized,
makes many hops before becoming deenergized, but with
little directional correlation between these hops, leading to
only a small effect on D.
-6
Rh on Rh(lll)
-8
2
-6
-10
-8
2
1:= MCTST
-12
-10
][ - DYNAMICS
-14
o
5
10
-12
15
10000/T
FIG. 2. Surface self-diffusion constants for Rh on Rh(111), plotted in Arrheniusfonn. The units areD (bohrla.u.) and T(K), and all error bars are one
standard deviation. The dashed line represents the simple TST results, as
described in Sec. IV. The dynamics results are from Ref. 12. All values were
obtained with one moving layer, and the Arrhenius parameters may be
found in Table I.
-14
o
5
10
10000/T
FIG. 4. Arrhenius plot (as in Fig. 2) for W on W(110) with one layer free.
The dynamics results are from Ref. 10.
J. Chern. Phys., Vol. 80, No. 11, 1 June 1984
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5837
A. F. Voter and J. D. Doll: Theory of surface self-diffusion
low the TST value, depending on which of these effects is
stronger. Of the four surfaces we examined, the Rb( 111) and
W(llO) systems show an increase in EA. due to dynamical
effects, of2.4 and -6.5 kcal, respectively. For the Rb(l00)
and W(2l1) surfaces, any change in E A. is much smaller than
the error estimate assigned to EA.' For the channeled W(21I)
surface, there are either very few multiple hops below 3000
K, or they are exactly balanced by recross~gs, since Fig. 5
shows very close agreement between MCTST and MD.
The STST results are also shown in Figs. 2-5, and are
seen to give a quite good approximation to the MCTST results. [For W(IlO), Do for STST was arbitrarily set equal to
Do from MCTST.] At lower temperatures, where the adatom spends most of its time very near the binding-site minimum, STST should be in even better agreement with
MCTST. Because STST requires knowledge of the energy at
only a few geometries, ab initio calculations on finite clusters
should thus be feasible for predicting D. Moreover, because
dynamical effects are expected to be small at lower temperatures, these ab initio diffusion constants may be highly accurate in the temperature range employed in FIM experiments
(- 50-300 K), allowing a meaningful and direct comparison
between experiment and theory.
As in MD, the MCTST method presented here becomes
prohibitively expensive at very low temperatures (k B T <EA.)'
This is because the Metropolis walk will only rarely reach
the high-energy region of the TST counting box. However,
importance sampling techniques 24 can be used to overcome
this problem. In a forthcoming paper we describe such an
approach, and present MCTST results in the temperature
range employed in FIM experiments.
The MCTST and STST methods can be generalized to
treat surfaces with inequivalent binding sites. This would be
useful, e.g., in examining a surface on which a certain fraction of sites are poisoned with an immobile adsorbate. The
escape rate from a binding site adjacent to the poisoned site
would be different than from an unperturbed site, so that
-6
§
.&
-8
f-
-10
r-
-12
Won W(211)
,
~
!
-MCTST
'Ii
= DYNAMICS
-
~"
"
-140
10
5
10000/T
FIG. S. Arrhenius plot (as in Fig. 2) for W on W(2ll) with two moving
layers. The dynamics results are from Ref. 11.
The diffusional activation energy E A will only differ
from the TST prediction if the dynamical effects are temperature dependent. For two-state systems, recrossing effects tend to increase with temperature and are usually negligible at low temperatures. 22 This is because at low
temperatures, the particle typically has just enough energy
to pass over the barrier and is easily trapped in the other
state, while at higher temperatures, there is an increased
probability that the particle will have energy in excess of that
required to cross the barrier, and the particle may recross
before this energy is dissipated. For the multistate case, similar arguments should apply to the probability of successive,
correlated hops, predicting an increase in correlated hops
with increasing temperature. This trend has been observed
for the fcc( 111) surface in an MD study by Mruzik and
Pound, 15 and for the fcc( 100) and fcc( 111) surfaces in a study
by Tully, Gilmer, and Shugard23 using ghost-atom dynamics. The temperature-dependent recrossing effects will tend
to lower E A. , because D is decreased more at higher temperatures (causing a less-steep Arrhenius slope), while the temperature-dependent multiple hops will tend to raise EA' The
activation energy from MD may thus be either above or be-
TABLE I. Comparison of Arrhenius parameters from various methods. Multiplicative error estimates for the Do values are shown in parentheses. Error
estimates are one standard deviation.
Do(cm2 /s)
EA(kcal)
Surface
Rh on Rh(lll)
Rh on Rh(I00)
WonW(llO)
Won W(211)
Layers
free
STST
McrsT
0
I
2
4
6.2
5.2
4.9
4.8
5.6 ± 0.4
4.6±0.3
0
I
2
24.0
23.8
23.7
23.7 ± 0.7
21.3 ± 2.8
20.8 ± 3.7b
0
1
15.7
16.9
17.1 ± 0.3
16.2 ± 1.7
22.7 ± 3.D"
0
I
2
23.3
16.2
19.0
22.0±4.2
MD
6.8"
6.2 ±0.6·
20.5 ± 5.2d
STST
MCTST
5.6xlO- 4
4.0xI0- 4
3.1 X 1O- 4 (1.l2± I)
2.9x 1O- 4 (1.16± I)
3.3X 10-'
2.5x 10-'
3.7x 1O-'(1.l2± I)
3.7 X 10-'(1.78 ± I)
4.1 X 1O-'(2.34± I)b
!.lOX 10-'(1.05 ± I)
1.1 X 10- 3 (1.31 ± I)
3.6X 10- 3 (1.77 ± I t
2.8X 1O- 3 (1.99± I)
2.18X 10- 3 (2.2 ± I)d
1.5 X 10- 3
1.3X 10- 3
1.6X 10- 3
MD
7.4xlO-··
7.1 X 1O-4 (1.28± I)a
"Results from Ref. 10.
d Results from Ref. 11.
• Results from Ref. 12.
bResults from Ref. 13.
J. Chem. Phys., Vol. 80, No. 11, 1 June 1984
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5838
A. F. Voter and J. D. Doll: Theory of surface self-diffusion
MCTST (or STST) would be applied to find the escape rate
for each direction from each unique site. The diffusion constant would then be determined from the results of a pseudotrajectory, in which a site-to-site random walk is performed
where the probability of each possible hop (and the time
between hops) is determined from the precalculated escape
rates.
VII. CONCLUSIONS
We have computed surface self-diffusion constants for
four crystal faces [fcc( 111), fcc( 1(0), bcc( 110), and bcc(211)]
using classical transition state theory, and compared with
previous classical trajectory studies on the same LennardJones model systems. Exact transition state theory, implemented via a Monte Carlo procedure, produces fairly accurate results, though dynamical effects raise the activation
energy (from the TST value) for the fcc( 111) and bcc( 110)
systems. Simple TST, which requires computation of the energy at only three geometries, is found to be a reasonable
approximation to exact TST, and the agreement is expected
to be even closer at lower temperatures. For this reason, and
because dynamical effects are expected to be small at low
temperatures, it should be feasible to obtain accurate diffusion constants from ab initio calculations on model clusters.
ACKNOWLEDGMENT
AFV wishes to thank Steven M. Valone for helpful discussions regarding this work.
'For a recent review, see G. Ehrlich and K. Stolt, Annu. Rev. Phys. Chern.
31,603 (1980); also see E. W. Muller and T. T. Tsong, Field Ion Microscopy, Principles and Applications (Elsevier, New York, 1969).
2G. Ehrlich and F. G. Hudda, J. Chern. Phys. 44,1039 (1966).
3G. Ehrlich and C. F. Kirk, J. Chern. Phys. 48,1465 (1968).
4D. W. Bassett and M. J. Parsley, J. Phys. D 3,707 (1970).
'G. Ayrault and G. Ehrlich, J. Chern. Phys. 60, 281 (1974).
6D. W. Bassett and P. R. Webber, Surf. Sci. 70, 520 (1978).
7p. P. Cowan and T. T. Tsong, 22nd International Field Emission Symposium, Atlanta, 1975; W. R. Graham and G. Ehrlich, Thin Solid Films 25,
85 (1975); Phys. Rev. Lett. 31,1407 (1973); P. G. Flahive and W. R. Graham, Surf. Sci. 91, 463 (1980).
"P. Wynblatt and N. A. Gjostein, Surf. Sci. 22,125 (1970).
9J. R. Banavar, M. H. Cohen, and R. Gomer, Surf. Sci. 107, 113 (1981).
IOH. K. McDowell and J. D. Doll, Surf. Sci. 121, L537 (1982).
"J. D. Doll and H. K. McDowell, Surf. Sci. 123,99 (1982).
12J. D. Doll and H. K. McDowell, J. Chern. Phys. 77,479 (1982).
I3H. K. McDowell and J. D. Doll, J. Chern. Phys. 78, 3219 (1982).
I4T. Halichioglu and G. M. Pound, Phys. Status Solidi A 30,619 (1975).
15M. R. Mruzik and G. M. Pound, J. Phys. F 11,1403 (1981).
16J. Kushick and B. J. Berne, in Modern Theoretical Chemistry, edited by B.
J. Berne (Plenum, New York, 1977), Vol. 6.
17p. J. Rossky, J. D. Doll, and H. L. Freidman, J. Chern. Phys. 69, 4628
(1978); M. Rao, C. Pangali, and B. J. Berne, Mol. Phys. 37,1773 (1979).
'"J. D. Doll, J. Chern. Phys. 73, 2760 (1980).
19J. D. Doll, J. Chern. Phys. 74, 1074 (1981).
2°N. Metropolis, A. Rosenbluth, M. N. Rosenbluth, A. Teller, and E. Teller, J. Chern. Phys. 21, 1087 (1953).
2ID. G. Truhlar and B. C. Garrett, Acc. Chern. Res. 13,440 (1980).
22P. Pechukas, Annu. Rev. Phys. Chern. 32, 159 (1981); also see Ref. 21.
23J. C. Tully, G. H. Gilmer, and M. Shugard, J. Chern. Phys. 71, 1630
(1979).
24J. P. Valleau and S. G. Whittington, in Modern Theoretical Chemistry,
edited by B. J. Berne (Plenum, New York, 1977), Vol. 5.
J. Chern. Phys., Vol. 80, No. 11,1 June 1984
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