MSU

LIBRARIES

 

 

“

RETURNING MATERIALS:
Place in book drop to
remove this checkout from
your record. FINES will
be charged if book is
returned after the date
stamped below.

 

 

 

 

 

 

 

 

 

 

ORIENTATIONS AND ORDERING OF CH4, CFH3, CHF3,

AND CF. MOLECULES ON A GRAPHITE SUBSTRATE

William Ross Hammond

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Physics and Astronomy

1989

5360 87010

ABSTRACT

ORIENTATIONS AND ORDERING OF CH4, CFH3, CHF3,
AND CF, MOLECULES ON A GRAPHITE SUBSTRATE

By

William Ross Hammond

We study the energetics of single CH4, CFH3, CHF3, and CF; molecules phy-
sisorbed on a graphite substrate, and of molecular pairs in free space. The model
used in our calculations is based on the Lennard-Jones 12-6 and Coulomb pair
potentials. The ground state of single molecule systems is determined, and some
conjectures, although no conclusive determination, are made as to the ground state
structure of monolayer systems. We also discuss orientational ordering, commen-
surate vs incommensurate structures, substrate mediated effects, and the range at

which the interactions (Coulomb and Lennard-Jones) are significant.

ACKNOWLEDGMENTS

I would like to thank my fiancee, Mindi Macklin, and my parents for their
support and encouragement of my work on this thesis. I am also indebted to
Professor R. Tobin for a critical reading of this thesis and many valuable suggestions.
The chairman of my thesis committee, Professor S.D. Mahanti, also deserves special
thanks for showing great patience with me in the research and writing of this work

as I learned in the time-honored way of making, finding, and correcting mistakes.

TABLE OF CONTENTS

List of Tables
List of Figures
I. Introduction
II. Review of Experimental Data
III. Review of Theoretical Data
A. Methane on Graphite
B. Intermolecular Potentials for CH4, CFH3, CHF3, CF4
IV. Determination of Parameters
V. Methods of Numerical Calculations
A. Setup of Model

B. Description of Molecule-Substrate Interaction
C. Description of Intermolecular Interaction

VI. Results
A. Single Molecule
B. Pair of Molecules
Appendix A.
Appendix B.

Bibliography

iv

14
14
15

18
26
26
27
31
32
32
37
44
51

60

Table 1 .

Table 2.

Table 3.

Table 4.

Table 5.

LIST OF TABLES

SCF data

Parameters tested

Minimum intermolecular energy for pairs of CH4 and CF4
Pair energies for CHF;

Pair energies for CFH;

17

19

39

45

48

LIST OF FIGURES

Figure 1. Scale drawing of adsorbate molecules.
Figure 2. Methane on graphite ground state commensurate structure.

Figure 3a. The unit cell of graphite.
Figure 3b. Two layers of graphite.

Figure 4. Methane in the atop site on graphite.

Figure 5. CF.. on graphite commensurate high temperature solid phase.

Figure 6. CHF3 monolayer lattice proposed structures.
Figure 7. Euler angles.

Figure 8. Methane showing positions of atoms 1,2,3 and 4 with Euler
angles (0,0,0).

Figure 9. The Euler transformation matrix.
Figure 10. Definition of sites used in Table 3.
Figure 11. Pair CHF; Pair energies for 451 = d): =125°, r=4.5A.

Figure 12. Pair CFH; Pair energies for d); = d); =100°, r=4.0A.

of

11

28

29

29

39

52

56

SECTION I

INTRODUCTION

The study of interactions on surfaces has opened up experimental and greatly
broadened theoretical exploration of several areas of physics including the phe-
nomena of molecular physisorption“) and phase transitions in two dimensional
molecular overlayers.(2) Specific phenomena of interest include orientational order-
disorder transitions,(2'3) commensurate—incommensurate phase transitions,(2") sub-
strate mediated orientational ordering,(5) and two dimensional melting!” There
have been a large number of recent studies, both experimental and theoretical,
on physisorption and ordering of rare gas atoms and simple multiatom systems
such as nitrogen, oxygen, methane and CF4 on a graphite substrate. However, to
date we know of only a few experimental”) and no theoretical studies of relatively
more complicated systems such as CHF; or CFH3. These asymmetric molecules
posses larger multipole moments as compared to their symmetric counterparts CH4
and CF4; thus one may see dominance of the electrostatic contributions in both
the molecule-substrate and intermolecular interactions. One therefore expects the
orientations of a single molecule on a graphite substrate and the orientational struc-

tures of molecular overlayers of CHF3 and CFH3 to be completely different from

CH4 and CF4 .

In this thesis we make a careful investigation of the equilibrium position and
orientation of a single physisorbed molecule of CH,.F4_fl (where n=0,1,3,4) on a
graphite substrate to find out how these change with n. In addition we explore

the orientation and position of a pair of interacting molecules in order to get

some insight into the orientational and center of mass structure of monolayer or
submonolayer systems. We use data from models of the intermolecular interac-
tion of CHnF4_n(°’°'1°) and attempt to determine from the characteristics of these
molecules precisely how they will position and orient on a graphite substrate in the
ground state and possibly other low lying locally stable excited configurations. To
do this we calculate the interaction energy between a molecule and the substrate,
and between a pair of molecules using the Lennard-Jones 12-6 and the Coulomb
pair potential. In the single molecule case the total energy of an adsorbed molecule
is obtained by summing over all the contributions to the interaction between each
atom of the molecule and the carbon atoms of the graphite lattice. The interac-
tion between molecular pairs was obtained by summing over all contributions to the

interaction between each atom of one molecule and each atom of the other molecule.

In the second and third sections of this thesis we review of what is known
experimentally and theoretically about the systems under investigation. Section
IV details the determination of the parameters appearing in the intermolecular
interactions. In section V we describe the methods used for calculating equilibrium
position, orientation, ground state energy, and translational energy barriers for a
single molecule. In addition we discuss the method used to obtain the intermolecular
interaction energy for a pair of molecules for a large portion of the available phase
space. Sections VI gives the results of our study and then discusses and summarizes

them.

SECTION II

REVIEW OF EXPERIMENTAL DATA

The four molecules that we have chosen are structurally very similar.11 The
C-F bonds are about 30% longer than the C-H bonds, and the angles between
bonds of the asymmetric molecules (CHF3 and CFH3) differ from the symmetric
bond angle of 109.47° by less than one degree as shown in Figure 1. The electrical
characters of the asymmetric molecules, however, are very different from that of the
symmetric molecules in that CHF3 and CFH; have large dipole moments, while the

octapole moment is the first non-zero moment for CH4 and CF4.

The most well studied of these four systems (molecules adsorbed on graphite)
is methane.("°) This is also the simplest system since the ground state of sub-
monolayer to monolayer methane (shown in Figure 2) is commensurate with the
graphite substrate, and the only other solid phase of methane,* which is stable at
a higher temperature, is isotropically expanded by approximately 1%.“) Both CF4
and CHF3 (to the best of our knowledge there is no data on CFHa) form incom-
mensurate ground states,(7’13) and have other stable high temperature solid phases
that are not as simply related to the ground state as methane’s high temperature

phase is related to its ground state.

The ground state of methane on graphite is a triangular \f3— x J3 R30° com-

mensurate lattice with nearest neighbor distance 4.263A. Here we use the standard

 

* Although we have not noted it elsewhere, Smalley et al.” have mentioned a

compressed phase of methane at coverages above 0.9 monolayer.

 

 

Methane all bond angles 109.47°, CFH3 a = 10850

all bond lengths 1.091 A C-F bond length is 1,332 K
C-H bond length is 1.095

 

 

CHFs 0‘ =110~1° CF4 all bond angles 109.47°,

C-F bond length is 1.098 A an bond lengths 1.32 A
C-H bond length is 1.332 K

Figure 1. Scale drawing of adsorbate molecules.

Figure 2. Methane on graphite ground state commensurate structure.

nomenclatureu‘) where the base unit cell is the unit cell of graphite shown in Figure
3a, and the R30° implies that the methane lattice is rotated by 30° with respect
to the graphite lattice. This unit cell describes only one layer of the graphite;
natural graphite has an A-B-A scheme with the A and B layers related to each
other as shown in Figure 3b. It has been suggested that in the minimum energy
configuration, shown in Figure 4, the center of a methane molecule sits 3.3A above
the graphite plane with one C-H bond pointing straight up normal to the graphite
plane, and the other three C-H bonds pointing out and slightly down into the
substrate.“” The incommensurate phase of methane on graphite mentioned above
has only been noted in the last ten years“” due to the very small (1%) increase in
nearest neighbor separations. This phase is orientationally disordered and slightly
expanded as compared to the commensurate phase. Melting to the two dimensional

liquid phase occurs from this incommensurate phase which is stable between 50K

and 70K.

In contrast to CH4, the CE, on graphite system shows a large variety of
structures.(13) There are five “two-dimensional” solid phases of CF4 only one of
which is commensurate. The commensurate phase which is stable at higher tem-
peratures is shown in Figure 5. (The preferred orientation of a single molecule on
the substrate has three fiuorines pointing down into the centers of the surrounding
graphite hexagons, however, our calculations show that the intermolecular interac-
tions will rotate the preferred orientations of the molecules to those shown in the
figure.) The intermolecular separation is 4.923A, and the phase shown has a 2 x 2
triangular structure unlike the J3 x \f3— R30° triangular structure of methane. In

Figures 2 and 5 we have placed the CH4 and CF4 molecules at the atop sites.

 

 

 

 

Figure 3a. The unit cell of graphite. Seven cells are drawn (dashed
rhombuses) superimposed on the familiar graphite hexagons.

1.421 L

E

l

f

l/\
\

 

 

 

 

 

 

 

 

\
/
\/(
\

I L

 

 

 

 

 

Figure 3!). Two layers of graphite.

Carbon (methane)
to carbon (graphite)
distance 3.29

(2
\ /

Figure 4. Methane in the atop site on graphite.

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5. CF4 on graphite commensurate high temperature solid phase.

Although there is no direct experimental evidence of this positioning there are
some symmetry based theoretical arguments,(") and our own calculations strongly

support this placement.

The exact structure of the ground state of CF4 has not been identified, and is
referred to in the literature as a three peaked phase;(2'l3) the three peaks refer to
peaks seen in X—ray diffraction experiments. There have also been several LEEDH”
and neutron scatteringu” experiments on CF4, however the experimental resolu-
tion of the data has been too poor to make any further conclusions on the precise
structures, in particular on the orientations of the molecules in the different phases.
One of our aims is to use theoretical calculations to find our about the molecular

orientation of CF4 on the substrate.

As far as we know there is only one experimental study of CHF3 on graphite,
done by Knorr and Wieckert.(7) For coverages less than or equal to 0.92 (in units of
one molecule per 3 graphite hexagons i.e. the commensurate structure of methane
discussed earlier) and temperatures less than 85K, their results indicate a triangu-
lar orientationally disordered lattice incommensurate with the substrate, and with
slightly smaller intermolecular distances (CHF; has an area per molecule of 16.5A2

in the ground state) than the CF4 on graphite system (CF4 has an area per molecule
of 21A2 in the 2 X 2 phase). Below approximately 45K the molecules become ori—
entationally ordered and the unit cell is uniaxially compressed; the single Bragg
peak also splits into two peaks, indicating that one molecule in the triangular lat-
tice moves closer to one of the other two molecules and farther from the other.
This is similar to the ground state structure of 02 on graphite for coverages of

approximately 0.9 monolayer.(’°)

Figure 6a shows the triangular orientationally disordered phase of CHF3 along
with the rectangular unit cell containing two molecules proposed by Knorr et 0.1.”)
The rectangular unit cell is preferred by Knorr et al. in describing the distorted
triangular lattices of Figures 6b and 6c. The molecules in Figures 6 are represented
by circles approximately outlining the area beyond which there is no significant
probability of finding electrons from that molecule. Figures 6b and 6c show the two
orientationally ordered and uniaxially compressed structures proposed as possible
configurations of the ground state. The arrows indicate the direction of the dipole
moment which is parallel to the C—H bond and conjectured to be on the average
approximately parallel to the graphite plane. These two structures were found by
simply drawing molecules on a surface and minimizing the overlap while maintaining
their calculated area per molecule of 16.5A2 (the area of three graphite hexagons is
15.7A’). This method of proposing structures is not meant to provide a conclusive
determination of the structure of the ground state, but to give an initial guess that

can be checked by further experiment or theoretical calculations.

Both the structures given in Figures 6b and 6c have an in plane (parallel to the
substrate) antiferroelectric moment, however, the herringbone structure (Figure 6b)
is antiferroquadrupolar while the structure of Figure 6c is ferroquadrupolar. The
long—range antiferroelectric order appears below 45K. In a Cartesian coordinate
system with origin at the lower left (as drawn) vertex of the unit cell, the coordinates
of the molecule in the center of the cell are (0.75s, 0.5b) for the herringbone structure
and (0.3a, 0.5b) for the ferroquadrupolar structure. The electric dipole moments of
the molecules in the ferroquadrupolar structure are, as shown, either all parallel or

all antiparallel to the short axis of the unit cell, while the dipole moments in the

10

 

Figure 6a. Orientationally disordered structure.

b=o.97l'§a \ >8) b=o.97{§aC4-o— («—

Figure 6b. Uniaxially compressed Figure 6c. Uniaxially compressed
herringbone structure. ferroquadrupolar structure.

 

 

 

 

Figure 6. CHF3 monolayer lattice proposed Structures.

11

herringbone structure are, either all at 60° or all at 120° to the short axis of the unit
cell. Both structures are consistent with the splitting of their diffraction peak and

appear equally probable as candidates for the ground state of CHF3 on graphite.

Knorr and Wieckert”) have claimed that there are some physical similarities
between their CHF; on graphite system and a molecular dynamics study of diatomic
oxygen on graphite by S. Tang, S.D. Mahanti, and R. Kaliaao) showing the ener-
gies of a ferroquadrupolar structure and a herringbone structure to be quite close,
and that a high density of defects such as vortex-antivortex pairs may change the
ground state locally from the ferroquadrupolar to the herringbone structure. Knorr
et al. note, however that the experimental structure factor of CHF; on graphite
system is very consistently 2 :l: 0.4, whereas the calculated structure factor for the
ferroquadrupolar structure (Figure 6c) is 4.0 and for herringbone structure is 0.5.
However, the precise value of the structure factor is very sensitive to the exact posi-
tion of the molecules in the lattice, so it is possible for either configuration to shift

slightly and have a structure factor of approximately 2.0.

The three physisorbed systems we have discussed in this section are all very
different despite the internal structural similarity of the adsorbed molecules. Only
methane on graphite has had its ground state structure precisely identified, and
one of the few experimentally known facts about the ground state of the other two
systems is that they are, unlike methane, incommensurate with the substrate. Even
the ground state of methane on graphite has not been completely identified since
we do not know for certain what the preferred adsorption site is. Despite careful
experiments on these systems only conjectures about the ground state structure and

orientations of CR, and CHF; on graphite exist, and to the best of our knowledge

12

there is no experimental basis to conjecture about the ground state orientation or
structure of the CFH; on graphite system. In this thesis we investigate the energet-
ics of these four physisorbed systems to understand what the precise ground state
structure of a single physisorbed molecule is, why the ground state structures are
different for different molecules, what interactions drive the different structures, and
attempt to gain some insight into the ground state structures in the submonolayer

to monolayer region for each of the four systems.

13

SECTION III

REVIEW OF THEORETICAL MODELS
A. Methane on graphite

Methane is the most studied system that we have considered in this thesis. It
is also the only system among CH.,,F.,_fl (where n=0,1,3,4) where the characteristics
of the ground state have been carefully investigated.(°'“) In these studies the ground
state of methane on graphite was investigated using an atom-atom Lennard-Jones

potential

V = Z4€ij[(0£j/rej)” - (ail/Tor], (1)

where the i’s are atoms in an adsorbed methane molecule, and the j’s are the atoms

in the graphite substrate.

This potential was used to obtain both adsorbate-substrate and adsorbate-
adsorbate interaction energies. Phillips at 0.1.“) also investigated the energetics
with an exponential-sixth potential, where the repulsive part of the Lennard-J ones
potential is replaced by an exponential function. The exponential-sixth model, how-
ever, did not give results which agreed with experiment in general“) In particular
the preferred adsorption site for methane using the exponential-sixth potential was
found to be the bridge site. instead of the atop site found by the Lennard-Jones 12—6
potential, the latter being consistent with the prediction using symmetry arguments
for CF41”) The parameters for the Lennard-J ones potential used by Phillips et a1.
were taken from several experimental intermolecular studies and are discussed fur-

ther in section IV.

14

The results of Phillips and Hammerbacher agree well with experimentw‘) in
finding the equilibrium height of a methane molecule at 3.28A above the graphite
plane measured from carbon to carbon. They also calculate the minimum energy of
a single molecule due to the substrate as -1663 K / molecule. This agrees reasonably
well with the isosteric heat reported by Thomy and Duvalu'l) which is given as
3.3Kcal/mole, 13.8. 1660 :l: 25K at coverage 0.1 in units of one molecule per three
graphite hexagons. The above agreement gives us some confidence in both the
potential model and the parameters which we used in our own calculations, as will
be discussed further in sections IV (Determination of Parameters) and V (Methods

of Numerical Calculations).

B. Intermolecular Potentials for CH4, CFH3, CHF3, CF4

In considering the intermolecular interactions of CHnF4_,, we have chosen to
use an atom-atom potential as the best method of describing the potential inter-
actions since other simple methods do not adequately explain some experimental
results.(8) Unlike the case for a molecule adsorbed on graphite, there is substantial
amount of information available on intermolecular interactions using atom-atom
potentials.(°'°'23) In our calculations we use data from intermolecular studies of
CHnF4_,. to determine some of the parameters (the 6’3 and 0’s of equation 1, and
q the partial charge on an atom in a molecule), and we also compare the results of

our intermolecular pair calculations with results from the literature.

The self consistent field (SCF) model has been used recently to study CHnF4_fl

intermolecular systems.(°'23) The intermolecular interaction is expressed as a

15

pairwise sum of interatomic potentials between atoms of different molecules. The

interaction energy between two atoms 1' and j is given by:

 

.. —(R.-,- - 35 - 3,) '1in Co
Ah-zexp +———F.-~—, (2)
J ( Pi + P1“ ) "1' J Tij

where T',‘,' is the distance between the interacting atoms, q; and q, are the partial
charges on each atom, and the other constants in the equation are parameters
obtained from ab initio methods.(22) One problem, however, with the parameter
E, in the above equation is that it is defined differently in different papers. F2, is

a damping function which is defined as:

1 ~28Rvdw

E, = exp(-( — 1)2), rgj < 1.2813va (3a)
1‘5]-
in one paper,“” and in another(23) as:
1.28Rv w
Fij = ewp(-(—‘,:+ - 1)“), m S Rvdw (3b)
:1

In both definitions F is defined as 1 outside of the appropriate above noted ranges,
and Rm“, is related to the Van der Waals radius that will be discussed in section IV.
Further discussion of equation 2 will also be presented in section IV. It should be
noted that the above equation describes the physical interaction between an atom
- of one molecule and another atom of a different molecule; and it does not describe
any changes that might occur in the two molecules as they are brought near to each
other. In the above version of the equation all parameters are in atomic units, i.e.
1a.u. of length =0.53A, 1a.u. of charge =1e, and 1a.u. of energy =13.6eV. The
parameters from several molecules and the energy at the Van der Waals minimum

i.e. rij =R,,.,¢,,,21/6 are given in Table 1.

16

 

 

Table 1. SCF data

Atom q R0 p; O’.‘ C“ from V501? (K)
C “0.2469 7.2118 0. 18078 1 .42232 27.09 CF33 29.6
C 40.7649 7.2118 0.32383 0.60460 26.98 CHI"; 15.4
C '0.3374 7.2118 0.20794 1.50191 30.69 QH3CN 43.8
C +0.488 7.2118 0.28362 1.08998 47.36 CH3_C_N 31.1
H +0. 1473 5 .090 0.27846 0. 15619 3.352 CFI‘I3 0.75
H +0.0214 5.090 0.24426 0. 18180 3.338 CHF3 20.4
H +0 . 1564 5 . 090 0 . 27093 0 . 26402 4 . 644 CH301 7 . 4 _
H +0 . 201 5 . 090 0 . 26653 0 . 14604 5 . 858 CH; CH 33 . 8
F '0.1950 6.2362 0.22596 1.03083 11 .30 CPR; 15.2
F '0. 2621 6.2362 0.22242 1 .05472 11 . 26 CHF3 15.7

 

 

 

 

 

 

 

 

 

 

The V501.- of Table 1 shows the energy (in K) of a pair of atoms (what kind of
atoms is specified by the first column of the table) at a separation of R0. ( All entries
in Table 1 except Vsop are in the atomic units defined earlier.) R0 is defined as
R0 = Rudw21/5, thus Vscp should equal —e (the e of equation 1) as will be discussed
in the next section. Note that we obtain the tabulated energy only when both atoms
are taken from the indicated molecule. Also the 11’s, the partial charge on the atoms,
were ignored in the calculation of VSCF, which is given by equation 2, since in the
next section we wish to compare the energy of the non-Coulomb part of the SCF

interaction to the energy of the Lennard-J ones interaction.

17

SECTION IV

DETERMINATION OF PARAMETERS

Several experimental studies have been done on methane to determine the
Lennard-J ones parameters, 6 and a', (see equation 1) of the constituent carbon and
hydrogen. The results of these studies are summarized by Severin and Tildesleyu‘)
who studied the ground state properties of a 400A2 monolayer of methane on
graphite with periodic boundary conditions. Based on their calculations Severin
and Tildesley adjusted the parameters (6 and 0‘) to those noted in Table 2. Al-
though the intention of this thesis is not to further refine the values of these pa-
rameters, we: found it useful to run some tests with various values of e and 0 taken
from the literature. The values of the various 6’5 and 0’s we chose to test are also
given in Table 2. All possible combinations of these parameters were tested by
assigning them to a methane molecule and determining the potential energy in the
preferred orientation of the molecule—substrate interaction 3.3A above the atop and
center sites of the graphite substrate. The results of this investigation are given
in Table 2, and based on a comparison of these results with the isosteric heat of
Thomy and Duvalull (-1663K) we adjusted our choice of parameters used in this
(methane-graphite) interaction to those (parameters i.e. A, C, G, and M) noted in

the table.

The parameters we use (6’5 and 0’s) are all parameters for an atom-atom
potential, thus when we choose a value for a parameter (e.g. ec_,,aph,-¢e) we use

that value of that parameter in all of our systems (Le. for all n in CHnF4_.,, where

18

 

 

Table 2. Parameters tested
Parameters Combination Energy at Energy at
used tested atop site‘ center site“
1 ec-g=47.68"'f 1.0.11.1 153311 154011
a ec_9=51.1QBb 1.0.11.1 157111 157211
1.0.11.11 159211 158811
0 50-9: 3.30” A.C.F,L 159811 159311
0 ac_,= 3.355 1.0.5.11 180511 1598K
11 50-..: 3.40c 1.0.1.11 184911 18201
1.0.0.1 158711 159211
F eH_,=15.794 1.0.0.1 182811 1627K
0 cg_,=17.00°vf 1.0.0.11 185111 184411
11 eH_,=23.798" 1.0.0.1. 185811 1650K
A, C', G, M 188511 185411
I aH_,= 2.906 1.0.0.11 171211 187811
1 ag_,= 2.95 1.0.11.1 189011 189811
11 aH_,= 2.98a 1.0.11.1 194811 19341
L aH_,= 2.995 1.0.11.11 198011 195811
11 aH_,= 3.00f 1,0.11.L 199011 198811
11 aH_,= 3.10e 1.0.11.11 199911 19721
1.0.8.0 208511 200811
1.0.11.1 155811 157211
1.0.11.1 159811 18041
1.0.5.1 181711 182111
1.0.11.1. 182311 182511
1.0.1' 11 183011 183011
1.0.5.11 187411 185211
1.0 0.1 181211 1624K
1.0.0.1 185311 18591
1.0.0.1 187811 18751
1.0.0.1. 188211 18821
1.0.0.1 188911 188811
1.0.0.11 173711 171111
1.0.11.1 191511 191811
1.0.11.1 197311 196611
1.0.11.11 200511 199011
1.0.11.1. 201511 199811
1.0.11.11 202411 200411
1.0.11.11 209011 203811
1.11.5.1 157411 15981
1.0.1.1 181311 183011
1,11,11,11 183411 184811
1.1:.1'.L 184011 185111

 

 

 

 

 

 

H
CO

 

Table 2 (cont’d.).

 

 

 

 

 

 

 

Parameters Combination Energy at Energy at
used tested atop site‘ center site’

1.56-9-47.886J' 1.5.5.5 18481 18551
5 ec_,=51.1985 1.5.5.5 18901 18771
, 1.5.0.1 18281 18501

0 00092 3.306J’ 1.5.0.1 18701 18841
0 ac_,= 3.356 1.5.0.1 18921 17021
5 ac_,= 3.406 1.5.0.L 18991 17071
1.5.0.5 17081 17121

5 eflcv=15.794 1.5.0.5 17531 17381
0 egcg=17.00°J’ 1.5.5.1 19321 19431
5 cgcg=23.7985 1.5.5.1 19901 19911
1.5.5.1 20211 20181

1 aH_,= 2.906 1.5.5.L 203111 20231
1 ayf_g= 2.95 1.5.5.5 20411 20301
1 07{_9= 2.986 1.5.5.5 21071 20831
L 071.9: 2.995 5.0.5.1 15931 18031
5 aflcva 3.00f 5.0.5.1 18321 18351
5 UH;1= 3.106 5.0.5.1 18531 18521
5.0.5.0 18591 18581

5.0.5.5 18851 18811

5.0.5.5 17101 18831

5.0.0.1 18471 18581

5.0.0.1 18891 18901

5.0.0.1 17111 17071

5.0.0.1 17181 17131

5.0.0.5 17251 17181

5.0.0.5 17721 17421

5.0.5.1 19411 19491

5.0.5.1 20091 19971

5.0.5.1 20401 20221

5.0.5.1 20501 20291

5.0.5.5 20801 20381

5.0.5.5 21281 20891

5.0.5.1 18201 18381

5.0.5.1 18591 18701

5.0.5.1 18801 18881

5.0.5.L 18881 18911

5.0.5.5 18921 18951

5.0.5.5 17381 17181

5.0.0.1 18741 18901

5.0.0.1 17181 17241

 

20

 

Table 2 (cont’d.).

 

 

 

Parameters Combination Energy at Energy at
used tested atop site‘ center site‘
1 802,847.68“, 5.0.0.1 17381 17321
5 Cc_g=51.19Bb 5.0.0.1. 17451 17471
5.0.0.5 17521 17521
0 00-9: 3.30” 5.0.0.5 17991 17781
0 ac_,= 3.35“ 5.0.5.1 19781 19831
5 «0-9: 3.40c 5.0.5.1 20381 20311
5.0.5.1 20871 20581
5 cg-,=15.79" 5.0.5.1. 20771 20831
0 cH_,=17.00°'f 5.0.5.1 20871 20701
5 eH_g=23.798" 5.0.5.5 21531 21041
5.5.5.1 18381 18851
1 03-9 2.90c 5.5.5.1 18781 18971
1 83-9- 2.95 5.5.5.1 18971 17131
1 011-9: 2.98“ 5.5.5.1. 17041 17181
L 01.1-, 2.995 5.5.5.1 17101 17221
5 aH_g= 3.00f 5.5.5.5 17541 17451
5 aH_,= 3.10e 5.5.0.1 18921 17171
5.5.0.1 17341 17521
5.5.0.1 17581 17591
5.5.0.1 17831 17741
5.5.0.5 17701 17791
5.5.0.5 18171 18031
5.5.5.1 19981 20101
5.5.5.1 20541 20581
5.5.5.1 20851 20831
5.5.5.L 20951 20901
5.5.5.5 21041 20971
5.5.5. 21711 21311

 

 

 

 

 

I

0"

O

9..

H.

All energies are negative.
From Severin and Tildesleyu‘) for molecule-substrate

From

From
From
From

interaction.

Severin and Tildesley for intermolecular
interaction.

Bondi.“°)
mixing rules.(”
Severin and Tildesley (quoted but not used in
their work).

Value used in our calculations.

21

 

n : 0,1,2,3,4). This assumes that the character of each atom in CHnF...fl does
not change significantly with n. This assumption need not be true, as can be
inferred from Table 1, and following the example of others in the literature“""
we did not assume that interactions with a carbon atom from the substrate were
described by the same parameters as similar interactions with a carbon atom from a
CHnF4_,. molecule. However, our assumption provided at least an initial estimate
of several needed parameters. We have not yet specified the values of the parameters
for the fluorine-graphite and intermolecular interactions. The values of the 8’s
for the (intermolecular) carbon-carbon and hydrogen-hydrogen interactions were
taken from the work of Severin and Tildesley. However, the values of the 0’s for
the (intermolecular) carbon-carbon and hydrogen-hydrogen as well as the fluorine-
fluorine interactions were taken from the work of Bondi‘lo) (who tabulated values
of various 0’s using experimental data), to which the SCF calculations are fit. Test
runs on a pair of methane molecules at a separation of 4.26A (the intermolecular
distance of methane in the commensurate structure on graphite) showed that the

0’s of Bondi and the 0’s of Severin and Tildesley gave the same energies to within

5K per molecule (10K per pair of molecules).

A calculation of the value of e for the fluorine-fluorine interaction was made
by equating the F-F potential energy well depth of the Lennard-Jones potential to
the F-F potential energy well depth of the SCF model. By differentiating equation
1 and setting dV/drzO it is immediately obvious that the Lennard-Jones potential
minimum occurs at r = R0 = a x 21/ °. Substituting this value for r back into
equation 1 shows that the potential well depth is just equal to 8. Therefore 8 =

— m1,..(1' = R0) = —V50p(r = R0). In calculating the SCF energy (Vscp) we chose

22

to use equation 3b for the definition of E3 of equation 2 since the paper containing
this particular definition of E3 gave a more complete table of parameters. Definition
3a would make E3 = 0.98, and thus would not make much of a difference. Note
that Rvdw, the Van der Waals radius of equations 3, is exactly the same as or of

equation 1.0033)

As can be seen from Table 1 the SCF energy of the carbon-carbon and
hydrogen-hydrogen interactions vary across a very large range, while that of the
fluorine-fluorine interaction is very consistent (c =15.2K and 15.7K for CFH; and
CHF; respectively). Despite the apparent consistency of the potential well for flu-
orine, the large variations in V(SCF) for carbon and hydrogen indicate that the
potential well for the fluorine-fluorine interaction may not be very accurate, or may
not be unique depending on the system under consideration. We have used an
ad hoc, but simple method of obtaining another estimate of the value of c for the
fluorine-fluorine interaction from a knowledge of the 8’s for carbon and hydrogen,
and from the Cscp’s, the constant in the attractive part of the SCF potential given

in Table 1. The method involved multiplying the value of C(scp for p) by the

average ratios 0f Ecarbon/C(SCF for carbon) and Ell/C(SCF for H): i-e-

 

 

1 . ecarbon CH
85 = C SCF r F ‘l + i (4)
( f0 )2 C(SCF for carbon) C(SCF for H)

where the 8’s are the 8’s of equation 1 and the C ’s are obtained from the SCF data

in Table 1. The resulting value of 8 for fluorine is 18.0K.

Test runs with 5;» =15.5K and CF =18.0K showed that the two different pa-
rameters gave the same minimum energy value and minimum energy configuration

for a single molecule (of any kind, i.e. for any n =0,1,3,4 in CHnF4_,.) on the

23

substrate to within 0.1K and 0.01.31. Test runs of molecular pairs showed that the
total energy of a pair of molecules increased by less than 9K for CF4, less than
5K for CHFa, and less than 2K for CFH3. There was no significant change in
the minimum energy configuration of any molecular pairs with the change in 8p.
Based on a comparison of our potential well depth for CHF; pairs (-911K) with the
average of results from the literature”) (~1155K and -760K are quoted for an aver-
age of approximately -955K) we chose to use the value of 18.0K for 5p for further

calculations.

The minng rules for calculating the (1’s and 5’s between different kinds of
atoms (e.g. fluorine and hydrogen) are quite well known,(l) and have consistently

given results that agree well with experiment. The mixing rules are:
Uij = (031+ (ml/2, (50)

60' = (65611)”2- (5b)

The values obtained from these mixing rules are noted in Table 2. The atoms in the
molecules were also considered to have a partial charge since the electron affinities of
carbon, hydrogen, and fluorine are very different. The partial charges on the atoms
were taken from the intermolecular SCF models(°'23) mentioned earlier except for
the case of CE. which was not considered in the referenced works. However, based
on the assumption that the difference in the partial charge on a hydrogen atom
between CFH; and CH4 molecules (which is close to 0 as can be seen in Table 1) is
nearly equal to the difference in the partial charge on a fluorine atom in CHF3 and

CF4, the partial charge on an F in OR, was extrapolated to ~0.262e.

24

The only parameters needed in the model that have not yet been mentioned
are the bond lengths and angles which were obtained from references 8, 11, and
24, and the polarizability of graphite. The atomic polarizability of carbon which
we used in our calculations is given by E.M. Purcell in his undergraduate text

Electricity and Magnetism as 1.5 x 10"24 cm3.

25

SECTION V

METHODS OF NUMERICAL CALCULATIONS
A. Set-up of model

In this section we detail the precise physical setup of the model we used in
our calculations, and explain exactly what we calculated and how we calculated
it. For the physical setup of our model we first generated a 30 x 30 x 3 graphite
substrate which should be large enough to contribute all the significant energy of an
infinite lattice (this is discussed and tested later). The graphite lattice was formed
by first stepping down the Y-axis in units of one bondlength, assigning one carbon
atom to the first two of every three steps. With the Y-axis defined this way the X

coordinates of an atom must be:

1/2
X(n,m) = n X bondlength X (3)2 , (6)

 

where n is the number of atoms out from the Y-axis the atom in question is, and m
is the number of atoms out from the X-axis. The Y coordinates are then staggered.
If an atom is an even number of atoms out from the Y-axis then its Y coordinate
is the same as the Y coordinate of the atom on the Y-axis, Y(0). If an atom is an

odd number of atoms out from the Y-axis in the X direction, then:

Y(n,m) = Y(0,m) for n even (7a)
Y(n,m) = Y(0,m) — bandlength/Z for n and m odd (7b)
Y(n,m) = Y(O,m) + bandlength/Z for n odd and m even (70)

26

The Z coordinate of the atoms in the top graphite plane are 0, for the second plane Z
is -3.35A, and for the third plane Z is -6.7A. The bandlength in the above equations
is 1.421A. In natural graphite the planes are shifted in relation to each other in
an A-B-A scheme. To simulate this in the model the X coordinate of every atom
in the second plane is increased by bondlength x 9121: and the Y coordinate is
increased by one half bandlength; the resultant lattice is shown in Figure 3. Once
the lattice has been set up, the adsorbate can be added to the system. The adsorbed
molecule has six degrees of freedom, three translational and three rotational. The
three rotational coordinates used are the Euler angled”) 9, <0, and 0:, where 0 is
a rotation about the molecules Z-axis, d) is a rotationabout the Y'-axis, and w is
a rotation about the Z"-axis as defined in Figure 7. If the four atoms about the
carbon are labeled form one to four, then for (9,05,02) = (0,0,0) the positions of

atoms 1, 2, 3 and 4 are defined in figure 8.
B. Description of the molecule-substrate interaction

Once the position of the molecule is known in phase space (i.e., X, Y, Z,
9, <0 and w are all known, where the X, Y and Z always refer to the coordinates
of the carbon atom) the X, Y, and Z coordinates of all the constituent atoms
can be found using the Euler transformation matrix which is given in Figure 9.
The Lennard—Jones interaction is then summed pairwise between each atom of the

adsorbed molecule and the substrate.

The Coulomb contribution to the potential energy for ion—atom pairs is well
known,(1) and is given by:

1
VC = “259—, (8)

27

 

 

Ys’Yn

 

9’ ’09
Z ,Z

 

Figure 7. Euler angles

28

 

 

 

 

 

I,
V /

atom 3 atom 2

 

 

 

X

Figure 8. Methane showing positions of atom l, 2, 3 and 4 with

Euler angles (0,0,0).

 

0099 cost!) 0090) - sine sinO) ~0090 cos¢ sinto - sine 0090)
sine 009(1) 0090) + 0090 9in0) 0090 0090) - sinO 0094) sin0)

-si09 0090) sine sina)

 

Figure 9. The Euler transformation matrix.

29

 

0099 sintb

sine sinq)

009(1)

 

where a is the polarizability of the atom, q is the charge on the ion, and r is the
distance between the ion and the atom. In order to calculate the net interaction
between the different ions of an adsorbed molecule and an atom in the substrate it
is necessary to first calculate the net polarization of the atom due to the molecule.
To accomplish this, equation 8 is separated into cartesian coordinates (X, Y, Z)
and the cartesian components of the electric field at the atom, the E X’s, By ’9, and
E z’s, are summed separately. Equation 8 now becomes:

Va = £00.51)? + (2.131)” + (2.515?” x lag-8|. (9)
where the sum is over the ions in the molecule. The first factor Ga times the sum)
in equation 9 is the response (polarization) of the atom, and the second factor is
the magnitude of the net field at the atom due to the ions of the molecule (which
is parallel to the polarization of the atom). Since the two sums are exactly equal

to each other, equation 9 reduces directly to:
1
V0 = —-2-a[(2.-EX.)’ + (53.1595)2 + (2,-Ez, )21- (10)

Thus it is only necessary to calculate the cartesian component of the electric field
at each atom due to every ion and sum up the energies (multiplied by $41 and a
conversion factor since we want to use Kelvin units to compare with the energies

due to the Lennard-J ones interaction).

At any distance more than a few angstroms from a given molecule in the X
or Y directions, interactions between a second adsorbed molecule and the substrate
will probably interfere with the X and Y components of the polarization of the

graphite due to the first molecule. In order to get some idea of possible effects of

30

this interference effect the model has two cutoff distances for the coulomb potential,
one cutoff distance in any direction, and the other cutoff depending only on the
sum of the squares of the X and Y components of the distance. The effects of these

various cutoffs are discussed further in the next section.

C. Description of the intermolecular interaction

For speed of calculation of the intermolecular interaction the Cartesian coordi-
nates of the individual ions of the adsorbate calculated for the adsorbate-substrate
interaction are stored and sent to the subroutine calculating the intermolecular po-
tential. Once these parameters are determined the sums are done in the same way
as for the graphite lattice. The only difference is that every atom has its own partial

charge, and thus the Coulomb interaction is described by an ion-ion potential:

 

VC = 5.,- 71.8,, (11)
7“]

where the i’s and j’s are ions of different molecules.

31

SECTION VI

RESULTS
A. Single molecule

In this section we give the results of our energy calculations for a single
molecule on a substrate and the conclusions based on these results. As will be
discussed throughout this section our results agree quite well with all available the-
oretical and experimental data including adsorption site, adsorption potential, and
most favored orientation. We should also note that the energy values we quote are
generally given to 1K, which may be up to four digits of accuracy. As the results
given in Table 2 show, we can expect our results to be accurate to no more than
two or three digits. However, some of the energy differences we wish to quote here
require greater accuracy, and we therefore quote the additional digits in our results.
We note that the energy differences, except for the depth of the corrugation of the
substrate potential (see Table 2), are accurate to the quoted number of digits for

reasonable choices of Lennard-Jones and Coulomb parameters.

In order to test whether or not our physical set up (Le. the computer model
of the graphite lattice described in section V) was adequate to describe the system,
we ran several tests on an expanded lattice of dimensions (90 atoms, 90 atoms, 9
layers) or approximately (108A, 187A, 30A). (As a comparison our (30 atoms, 30
atoms, 3 layers) lattice measures approximately (34A, 60A, 10A) with a surface
area a little over 2000A2.) To the extent of this expanded lattice we found only the

Coulomb interaction from CH4 and CR, to go to zero with six digits of accuracy.

32

However, even though the interaction energies did not go to zero within our tested
accuracies, the interaction of a molecule with the smaller lattice provided 99% of
the molecule-substrate interaction energy of the larger lattice for all four molecules.
Furthermore all of the energy differences (such as between different sites, or for
different orientations) were the same to within four digits of accuracy for test runs
on both lattices. Since calculations with both lattices produced the same minimum
energy configurations for all molecules and the same minimum energies to within
approximately 1%, and because of the great computational time saved, we chose to
use the smaller lattice for our investigations. Thus the energy values of adsorbed
molecules (not the values of the energy differences) we quote throughout this section

may be one or two percent low, however, all physical conclusions based on these

values should be valid.

We found that if the interactions were cutoff outside of a distance of approx-
imately 12A the results of our calculations were quite consistent with the results
calculated using very large cutoffs on the enlarged lattice. At a cutoff distance of
14A (15A for CF4), which implies interacting with only about 500 graphite atoms,
all four single molecule systems have 98% of the total energy calculated for the
enlarged lattice (where we used a cutoff of 54A). The physical characteristic most
sensitive to short distance cutoffs is the vertical height of a molecule above the
graphite plane. At a cutoff of approximately 10A (with approximately 94% to 95%
of the energy at large cutoff) the calculated preferred height of a molecule begins
to increase, but the preferred (minimum energy) orientations remain the same for

both the symmetric and asymmetric molecules.

33

We found the minimum energy configuration of a single methane molecule
on graphite to be 3.29A above an atop site with the three tripod legs pointing
towards the centers of the surrounding graphite hexagons (see Figure 4). In this
configuration the potential energy, Le. the isosteric heat, is -1665K. This value
agrees very well with the value of ~1663K obtained by Phillips and Hammerbacherl”
The small difference is due to a combination of the inclusion of a larger volume of
the graphite substrate by Phillips et al. in their calculation of the Lennard-Jones
energy, and our own inclusion of the Coulomb interaction. The potential energies
for methane on a bridge site (which we found to be the least favored adsorption
site) and on a center site (which we found to be a saddle point) are —1654K and
-1642K respectively. Thus the corrugation potential in which a methane sits is not
very deep. Therefore it seems that there should be little favoring of a commensurate
structure over an incommensurate structure. However, a final analysis of the relative
stabilities of a commensurate and incommensurate structure will depend on the pair

energies and the pair potential minimum. These will be discussed later.

The rotational barrier for methane is 53K if the height above an atop site
is fixed at 3.29A, and 47K if it is allowed to relax up to Z=3.33A. The barriers
to rotation are found to be larger than the barriers to translation, which seems to
argue that descriptions of methane as a spherical molecule in the orientationally

disordered solid phase are not sufficient to describe the system.

CE; is somewhat more interesting than methane on graphite since CF 4 has
such a large number of solid phases, and because the ground state structure has
not been precisely identified. There are, however, a large number of similarities to

methane. We find that the preferred position of CF4 is also an atop site 3.47A above

34

the graphite plane, and in this configuration a molecule will have a potential energy
of -2448K. Unlike CH4, however, the center site is the least preferred adsorption site
for CF; and has a potential energy of -2350K when Z=3.53A. Over a bridge site the
potential energy is -2380K if Z is allowed to relax to 3.51A above the graphite plane.
The corrugation of the potential is approximately 100K, which suggests that there is
little favoring of a commensurate phase over an incommensurate phase. The barrier
to rotation about the Z-axis is 142K if held, and 117K if the molecule is allowed
to relax in the Z direction to 3.54.4. Despite a considerable amount of work on the
CR, on graphite system in the literature,(2’l3) we have found no data to compare
to our calculations of the adsorption potential (isosteric heat) or corrugation of the

potential.

We find CHF3 to be the most interesting in the CH,,F4_fl on graphite system,
because of the existence of two very different locally stable configurations which are
close in energy. The actual ground state has the CHF; in the atop site with the
Euler angles (see Figures 7 and 8) 0 = 30°, 4) 2 123°, 02 2 0°, and has Z=3.33A.
This orientation has one F pointing up, the H and two F’s approximately parallel to
the graphite plane and pointing at the centers of the surrounding graphite hexagons
and a potential energy of -2482K. (This configuration is similar to that found in
experiment.(7)) The local minimum has Euler angles (0,0,90), i.e. the H pointing up
and the three F’s pointing at the centers of the three surrounding graphite hexagons.
The energy of this configuration is -2434K at Z=3.49A above the graphite plane.
The rigid and relaxed rotational barriers are respectively 126K and 104K for the
global, and 39K and 6K for the local minimum. The preferred orientation at the

bridge site is 0 = 30°, 05 2 124°, 02 = 0° with Z=3.36A and an energy of -2436K.

35

In the center site CHF3’s preferred orientation is 0 = 0°, (0 = 124°, 02 = 0°
with Z=3.37A and an energy of -2414K. The corrugation of the potential is thus

intermediate to the corrugation potentials of CH, and CF4.

In CFH; there is a global and local minimum energy similar to that for CHF3,
and similarly, the “tipped” phase of CFH3 has a lower energy than does the ori-
entation with the single F pointing straight up. When the F points up there is
a local minimum with the three tripod legs in the usual position, a Z of 3.25A
and an energy of -1911.5K / molecule. In the “tipped” orientation CFH3 has angles
6 = 30°, 45 2 103°, to 2 0°, with Z=3.31A and an energy of ~2117K. The rigid and
relaxed barriers are respectively 60K and 52K for the local minimum, and 100K and
73K for the global minimum. The orientation of the minimum energy configuration
at the bridge site is 0 = 0°, (1) 2 105°, 02 = 0°, with Z=3.34A and an energy of
-2085K. For the center site the angles are 0 2 0°, 43 2 101°, 0: 2 0°, with Z=3.33A

and an energy of -2072K.

For all adsorbates we found the Lennard-Jones interaction to dominate over
the Coulomb. This was most pronounced for the symmetric molecules (which have
a net octapole moment) where the Coulomb interaction died off completely (within
an accuracy of five digits) inside of 14A and provided only about ~3K to the total
energy of methane and about -20K for CF4. CHF3 was able to get approximately
-100K from the Coulomb interaction when d) 2 0°, and ~420K when 03 = 180°. (The
minimum energy orientation for only the Coulomb interaction of CHF; and CFH3,
which both have net dipole moments, is with (I) = 180°.) These calculations were

done with the carbon atom of the molecule held fixed at Z=3.3A for CFH; and

36

Z=3.32A for CHF3. In both these molecules the carbon atom is close to, but not

at, the center of charge.

The Coulomb interaction for CFH3 provided approximately twice the total
energy as it did for CHF3. At <0 = 0° the Coulomb contribution to the total energy
was approximately -160K. Unlike the case for CHF3, the Coulomb interaction for
CFH; has a shallow local maximum near, but not at, d) 2 0°. The minimum energy

of the Coulomb interaction for CFH; is approximately -820K at 03 = 180°.

The dominating force of the single molecule-substrate interaction for the sym-
metric molecules is the Lennard-J ones interaction with almost no contribution from
the Coulomb interactions. However, for the asymmetric molecules the ground state
structure is determined by the competition between the Coulomb interaction which
wants to have as large a 05 as possible, and the steric repulsion between the substrate
and atom “one” (see Figure 8, atom one is the H in CHF3 and the F in CFH3),

thus the repulsion tries to push atom one of the molecule up, i.e. towards smaller

45’s.
B. Pair of molecules

In the previous section we investigated the energetics of a single molecule
adsorbed on a graphite substrate. We now investigate the energetics of a pair of
interacting molecules in free space to gain some insight as to how monolayer or
submonolayer systems of CHnF4_,, will act on a graphite substrate. We calculate
the most favored orientation and separation of a pair of molecules, and consider,

but do not explicitly calculate the substrate mediated interactions.

37

We have used the molecule-substrate cutoff ’s discussed earlier to attempt to
get some idea of the possible substrate mediated interactions. For CF4, CHF3,
and CFH; the Coulomb interactions with the substrate (which cause the substrate
mediated effects) act significantly over a range far larger than the intermolecular
nearest neighbor distances (which will be discussed later). However, the Coulomb
interaction for CH4 is quite small and dies off very rapidly. The Coulomb interaction
energy for methane goes to zero in approximately 10A, and when at the atop site,
approximately three quarters of the Coulomb energy, or 2.9K, (methane generally
gains between 3K and 4K from the Coulomb interaction) comes from the interaction
with the carbon atom above which the molecule sits 3.29A away (in the minimum
energy configuration). The extremely short range of this significant part of the
Coulomb interaction (the carbon atom beneath one methane molecule should be
beyond the range where a second methane molecule can significantly influence it)
implies that there should be little or no substrate mediated interference effects.
Having not made an explicit calculation of the substrate mediated effects, however,

we can not conclusively state that they are insignificant.

For methane the minimum energy occurs at a separation of 4.00A with one
H from each molecule pointing straight up and a tripod leg from one of the two
molecules pointing into the V of the other molecule. The interaction energy is -167K.
To see how the minimum energy separation corresponds to various commensurate
lattice separations on a graphite lattice (see Figure 10, where we place a molecule
at site 0 and another at site 11 where n=1,2,3,. . . and all lattice points equidistant
from site 0 are given the same 11), Table 3 shows the pair energy at most favored

orientation at the sites shown in Figure 10. At site four with a separation of 3.76A

38

 

Figure 10. Definition of sites used in Table 3. Sites equidistant from site 0

are assigned the same site number.

Table 3. Minimum intermolecular energy for pairs of CH4 and CF4

 

 

 

 

 

 

 

Site Separation Energy in K
number from site 0 in 1 for 054 for (:54
1 1.421 >0 >0
2 2.461 >0 >0
3 2.842 >0 >0
4 3.760 - 165 >0
5 4.263 -142 - 127
6 4.923 -68 - 198
7 5.124 -55 - 167
8 5.684 -29 -93
9 6.194 -18 -52
10 6.512 - 12 -38
1 1 7.105 -7 -22
12 7.384 —6 - 17
13 7.519 -5 - 15
14 7.912 -4 -l 1
15 8.526 -2 -7
16 8.644 -6

 

39

 

two molecules would have an interaction energy of ~165K. There is no way, however,
to form a cluster (let alone a monolayer) with three or more nearest neighbors all
having this separation (3.76A) without losing the commensuration energy. The first
site at which it is energetically reasonable for methane to form a commensurate
lattice is site five, which corresponds to the triangular J3 x \f3— R30° lattice shown
in Figure 2. Site five has an interaction energy of -142K which is approximately 30K
above the minimum of the pair potential. Since this is larger than the corrugation
of the molecule-substrate potential the incommensurate phase appears to be the
ground state of methane on graphite. This does not agree with experiment, however,
some calculations by Phillips et all” indicate that a combination of the substrate
mediated interaction and the zero point oscillations of methane (which we did not
calculate) will shift the equilibrium energy and position so that the commensurate

phase is the ground state.

We have also considered the orientational order-disorder transition of methane
on graphite. One odd thing about the system is that for CH4 at a separation of
4.00A the barrier against rotation due to an adjacent molecule is 23K. The barrier
against rotation at 4.263A is 29K. This occurs because the Lennard-Jones interac-
tion and the Coulomb interaction are opposing each other. When one interaction
is at a maximum, the other is at a minimum. The net effect is that the variations
in the potential tend to cancel each other out except that the Coulomb interaction
always varies more with orientation than does the Lennard-Jones potential. At a
separation of 4.263A the Lennard-Jones potential is much flatter than at 4.00A,

and thus the variation in the Coulomb potential is more evident.

40

Table 3, mentioned earlier in this section, gives some data on the pair inter-
action of both methane and CF4. For CF4 the minimum of the pair potential is
-224K, and the pair potential at site six is -198K. Since the difference (26K) is less
than the difference between the atop site and other sites it looks like the ground
state of CR, should be the commensurate phase. However, in the 2 X 2 structure
the CF4’9 are forced to orient themselves away from the minimum energy configu-
ration of the substrate-CE, interaction. This reduces the commensuration energy
to below 26 K / molecule, and thus it is probable that the ground state of CF4 on
graphite is incommensurate. We note here that the zero point oscillations which
apparently affect the minimum energy configuration of methane should not have as
large an effect on the minimum energy configurations of CF4 since a fluorine atom
has almost twenty times the mass of a hydrogen atom and the C—F bond lengths
are about 30% longer than the C—H bond lengths. Thus the moment of inertia of

CE, is over 100 times larger than the moment of inertia of CH4.

The CFH; and CHF; systems on graphite are quite complex and a complete
analysis of these systems is very difficult. The tables in appendix A show the en-
ergy of a pair of interacting molecules throughout a large portion of the available
phase space. We have isolated at least a local minimum for the intermolecular
pair potential at (carbon to carbon) separations of 4.0.4 for CFH3 and 4.5A for
CHF3. These minima were found by rotating both molecules through all Euler
angles at initial separations of 4.4.4 for CFH3 and 4.5A for CHF3, and then vary-
ing the separation until a minimum was found. This minimum energy separation
agrees well with the experimental results of Knorr and Weickert”) for the sepa-

ration of molecules of CHF; on a graphite. The potential well depths mentioned

41

earlier were found by simply calculating the pair energy at the minimum energy
pair configurations (i.e. <01 = 16°, 03; 2 196°, X1 =0, X2 =3.33A for CFH3 and
4); 2 2°, d1; = 208°, w; = 60°, X1 =0, X2 =3.76A for CHF; with all coordinates
not mentioned equal to zero) of an SCF calculation“) The tables in appendix A
do not show any configuration that can completely dominate the structure of a

molecular monolayer for either CFH; or CHF3.

Since it is possible that the orientations of molecules in clusters or monolay-
ers of CFH3 and CHF; on graphite may be determined by the molecule-substrate
interaction we have further investigated the intermolecular interaction of pairs of
molecules in the orientations preferred by the molecule-substrate interaction. Both
systems have two molecular orientations preferred by the graphite-adsorbate in-
teractions, namely having all 05’s equal to 0°, and having d) = 103° or (b = 123°
for CFH3 and CHF; respectively. The configuration with all ¢’s equal to 0° is
highly unfavorable, the intermolecular interactions being repulsive for both CHF;
and CFH3. The “tipped” configurations (where :0 = 123K for CHF; and d) = 103K
for CFH3), however, have some orientations that are very favorable (have very low
energy), as well as some orientations that are very unfavorable (have extremely
high interaction energy). The figures in appendix B show the interaction energies
of various orientations (0’9) with the 42’s held to 125° or 100° for CHF3 and CFH;
respectively which are close to the “tipped” configurations’ values. Maxims. and
minima as well as points of zero energy are shown to display regions of attractive
and repulsive potentials which cover nearly equal areas of the phase space. In a
triangular lattice each molecule will have six nearest neighbors spaced 60° apart.

Thus, even assuming that all molecules will have only one rotational degree of

42

freedom (for instance 0 with <0 and 01 set to the values preferred by the substrate
interaction, as shown in appendix B) finding the exact ground state orientations is

not a trivial problem, and we do not attempt to do this in this thesis.

Our present study does not represent a complete analysis of the systems we
have investigated, and in view of the uncertainty in our parameters the molecule-
substrate potentials are uncertain to several (we estimate 2 to 4) tens of degrees
(K). However, energy differences due to changes in orientations or intermolecular
configurations should be accurate to the digits we have quoted. We believe we have
found several new and interesting results including the preferred orientation of CF4
in the 2 X 2 commensurate phase, the preferred orientations and adsorption sites of
single CFH; and CHF3 molecules, and approximate values for the isosteric heats
and vertical heights above the graphite plane for CF4, CHF3, and CFH3. Our
results agree well with available experimental data such as the vertical height of
methane above the carbon plane, the preferred orientation of CH;, the (theoretical)
preferred adsorption site for CH4 and CF 4, and the isosteric heat of physisorbed
methane. Although we did not perform a complete calculation of the ground state
of molecular monolayers, our calculations suggest an incommensurate phase for the
ground state of CF4 on graphite, and-“tipped” phases for the ground states of CFH;
and CHF3. We can not predict any preferred orientational structure for the ground
states of large systems of the asymmetric molecules on graphite. We can, however,
conclude that there should be little favoring of an orientationally ordered phase over
a disordered phase, this conclusion is supported by the low (45K) temperature at

which orientational ordering of CHF3 on graphite is observed”)

43

APPENDIX A

The tables in this appendix give the pair energy minima of the asymmetric
molecules, CHF; and CFH3, for various configurations of the two molecules. The
configurations of the molecules are described by the Euler angles 0, d), and 0: defined
in section V of the thesis, where 91:0 is the direction from atom 1 to atom 2, 02:0
points away from atom 1, and A is the carbon to carbon separation of the molecules.
The three energies quoted in Tables 4 and 5 are the total energy, Emgn, at the
pair potential minimum, and the contributions to this energy from the Coulomb

interaction, EC, and the Lennard-Jones interaction, ELJ.

Although our pair calculations were done in free space for simplicity, the main
objective of this thesis is to consider systems physisorbed on graphite. Since all pair
configurations with both 05’s small are energetically unfavorable, and the molecule-
substrate interaction prevents any configurations with large ¢’s, we give here those
431 ’9 with “tipped” orientations in the region where the molecule—substrate interac-
tion is energetically favorable. For each molecule we also give one table with 01:0
and vary d); through 180°. In calculating the pair energies, we rotated molecule two
through all possible 0 and 45 in steps of 10° or 15°. For CFH; we constrained w; =0,
since the pair energies are not very sensative to changes in w and the molecule-
substrate interaction for “tipped” configurations has a very strong preference for
02:0. The A’s were chosen by making pair energy tables at initial guesses of A=4.5A
for CHFa, and A=4.4A for CFH3. The configuration corresponding to the largest
(most negative) entry from each set of tables was then tested at various A’s until
minima were found at 45.4 and 4.0A for CHF; and CFH; respectively. We note
that the experimental separation of CHF3 on graphite”) is close to our chosen A

and the experimental separation of CH4 is close to our chosen A for CFH3.

44

 

 

 

 

 

 

 

 

 

 

 

 

Table 4. Pair energies for CHF3.
Table 4a.
CHF3 (#1 =50° w] =00 A34.5A
01 02 452 012 Emir: EC ELJ
-15 60 150 O -701 -598 -103
O O 160 60 -693 -596 -97
15 300 150 O -701 -598 -103
30 270 140 90 -714 -570 -144
45 270 140 90 -719 -580 -139
60 240 140 60 -703 -517 -186
75 240 130 60 -707 -524 -183
90 240 130 60 -675 -497 -177
105 210 140 30 -639 -440 -198
120 210 130 30 -583 -381 -202
135 210 120 60 -563 -361 -202
150 210 100 60 -578 -391 -187
165 210 100 60 -587 -414 -174
180 180 110 60 -597 -439 -158
Table 4b.
CHF3 ¢1 =70° w; ‘00 A34.5fi
91 92 ¢2 W2 Emir: EC 301
-15 30 140 90 -624 -562 -62
O O 140 60 -614 -532 -83
15 330 140 30 -624 -582 -62
30 300 130 O -686 -587 -80
45 270 120 90 -717 -576 -142
60 270 120 90 ~701 -565 -136
75 240 130 60 -670 -479 -191
90 240 120 60 -685 -496 -189
105 210 130 60 -643 -486 -184
120 210 120 60 -613 -422 -192
135 210 110 60 -559 -361 -199
150 180 100 30 -521 -332 -189
165 180 90 60 -509 -325 -184
180 180 90 60 -505 -324 -181

 

 

 

 

 

 

 

 

45

 

 

Table 40.

 

 

 

 

 

 

 

 

 

 

 

 

CHF3 ¢1=90° w] '00 A'4.51
91 92 4’2 “’2 3min EC EL.)
-15 30 60 60 -584 -448 -137
0 330 70 60 -560 -439 -121
15 330 60 60 -584 -447 -137
30 300 100 90 -646 -571 -75
45 270 100 90 -685 -552 -133
60 270 100 90 -718 -560 -159
75 270 90 60 -654 -461 -192
90 240 110 90 -681 -496 -185
105 240 100 60 -658 -470 -188
120 210 110 60 -643 -483 -160
135 210 100 60 -591 -405 -186
150 210 100 80 -533 -361 -172
165 180 90 60 -481 -295 -186
180 180 90 60 -456 -270 -186
Table 4d.
CHF3 ¢1=110° w1=0° A34.5A
91 92 4’: W2 Emir: EC 301
-15 3O 4O 60 -623 -487 -137
O 330 50 60 -607 -494 ~113
15 330 40 60 -623 -487 -137
30 300 50 60 -669 -584 -85
45 300 40 60 -691 -555 -135
60 270 80 90 -712 -543 -169
75 270 60 60 -643 -446 -198
90 240 90 90 -686 -498 -188
105 240 90 60 -671 -489 '181
120 210 80 90 -657 -538 -119
135 210 90 60 -629 -479 -150
150 180 70 90 -570 -402 -168
165 180 90 60 -520 -342 -178
180 180 90 60 -489 -302 -187

 

 

 

 

 

 

 

 

46

 

 

Table 4e.

 

 

 

 

 

 

 

 

 

 

 

 

CHF3 ¢1=130° w] ‘-'00 A34.5A
91 92 452 “’2 3min EC ELJ
-15 30 10 90 -698 -600 -99

0 0 10 O -690 '611 -79

15 330 10 30 -699 -600 -99
30 300 20 60 '715 -584 '131
45 270 40 90 -703 -535 “168
60 270 50 90 -698 -523 -175
75 270 40 60 '628 -428 -200
90 240 70 90 -685 -495 -190
105 240 70 60 -678 -498 -180
120 210 60 90 -692 -566 -126
135 210 70 60 -662 -533 '130
150 180 60 90 -625 -469 -156
165 180 70 60 -570 -389 -181
180 180 70 60 -538 '346 -193

Table 41.

CHF3 91=0° w] 80” A84.5A
¢1 02 ¢2 “’2 Emir: EC ELJ

O 180 ‘170 0 -630 -437 -193

10 210 170 30 ~662 -469 -192
20 180 180 O -691 -512 '179
30 60 170 0 -708 -543 -166
40 O 170 60 -716 '578 '139
50 0 160 60 '693 -596 '97
60 O 150' 60 -652 -579 -72
70 O 140 60 -614 -532 -83
80 0 130 60 -584 -465 -119
90 330 70 60 -560 -439 -121
100 330 60 60 -577 -472 -199
110 30 50 60 -607 '495 -113
120 30 40 60 -641 -503 -138
130 0 10 0 -690 -611 -79
140 O 0 O ~725 -581 -144
150 210 10 30 -726 -556 -170
160 180 20 60 -720 -539 '181
170 180 30 60 -725 '551 -174
180 180 40 60 -725 -581 -144

 

 

 

 

 

 

 

 

47

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 5. Pair energies for CF53.
Table 5a.
CF33 ¢1=7O° wl aw; -O° A-4.0l
0] 02 952 Emir: EC ELJ
-15 15 100 ~490 -633 +143
0 O 110 -383 -681 +297
15 345 100 -490 ~633 +143
30 345 130 -636 -579 -57
45 255 120 -781 -620 -161
60 255 120 -828 -613 -215
75 240 110 -809 -606 -203
90 240 110 -765 -572 -193
105 225 110 -744 -633 -111
120 225 110 -724 -565 -159
135 220 110 -700 -546 -153
150 210 120 -663 -520 -142
165 210 110 -628 -550 -79
180 160 120 -611 -528 -83
Table 5b.
CF11; ¢1 =80° w, aw; I=0° A-4.01
0] 92 ¢2 Emir: EC EL]
-15 15 90 —319 -684 +365
0 O 100 -108 -722 +614
15 345 90 -319 -684 +365
30 330 90 -599 -596 -3
45 255 110 -732 -615 -117
60 255 110 -827 -616 -211
75 240 110 -814 -610 -204
90 240 110 -776 -587 -189
105 225 110 -720 -652 -68
120 225 100 -716 -590 -127
135 225 100 -675 -511 -164
150 210 120 -645 —510 -135
165 210 110 -618 -531 -87
180 210 110 -597 -520 -77

 

 

 

 

 

 

 

48

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 50.
CF33 ¢1=90° 0:] =07; =0o A34.OA
91 92 452 Emir: EC ELJ
~15 15 90 ~239 ~704 +465
0 ~~~ ~~~ >0 ~~- ~~~
15 345 90 ~239 ~704 +465
30 330 80 ~591 ~621 +30
45 285 80 ~717 ~635 ~82
60 255 100 ~823 ~616 ~207
75 240 100 ~819 ~614 -204
90 240 100 ~787 ~602 ~185
105 225 80 ~713 ~628 ~86
120 225 90 ~701 ~602 ~99
135 225 100 ~666 ~517 ~149
150 210 120 -626 ~508 ~118
165 210 110 ~608 ~526 ~82
180 180 50 ~590 ~471 ~119
Table 5d.
CF33 ¢1=100° w] 3(1); =00 A34.OA
91 92 452 3min EC ELJ
~15 15 80 ~327 ~691 +364
0 0 8O ~106 ~722 +616
15 345 80 ~327 ~691 +364
30 315 70 ~627 ~667 +40
45 285 70 ~732 ~628 ~104
60 255 90 ~819 ~614 ~204
75 240 90 ~823 ~616 ~207
90 240 90 ~793 ~609 ~184
105 225 70 ~719 ~606 ~112
120 225 80 ~687 ~598 ~90
135 225 90 ~657 ~528 ~129
150 210 70 ~623 ~554 ~69
165 180 50 ~60? ~477 ~130
180 180 50 ~605 ~471 ~134

 

 

 

 

 

 

 

49

 

 

Table 5e.

 

 

 

 

 

 

 

 

 

 

 

CFH; ¢1=110° to] 3002300 A3401
01 92 ¢2 Emin EC EL]
~15 15 70 -496 ~653 +157

0 0 80 ~381 ~660 +280
15 345 70 ~496 ~653 +157
30 315 60 ~692 ~641 ~50
45 285 60 ~749 ~608 ~141
60 255 80 ~814 ~611 ~204
75 240 80 ~827 ~616 ~211
90 240 80 ~797 ~609 ~188

105 225 70 ~723 ~652 ~71
120 225 70 ~679 ~58O ~99
135 225 80 ~647 ~527 ~119
150 210 60 ~631 ~502 ~130
165 180 50 ~626 ~499 ~127
180 180 50 ~624 ~48? ~136
Table Sf.
CFH3 01 =0° 0;] =01; 80° A=4.01
451 92 4’2 Emir: EC ELJ

0 180 140 ~801 ~518 ~213
10 180 150 ~806 ~586 ~220
20 180 160 ~816 ~603 ~213
30 180 160 ~813 ~630 ~183
4O 20 180 ~763 ~599 ~163
50 0 160 ~741 ~631 ~110
60 0 140 ~611 ~684 +74
70 0 110 ~384 ~681 +297
80 0 100 ~108 ~722 +614
90. ~~~ ~~~ >0 ~~~ ~~~

100 0 8O ~106 ~722 +616
110 0 80 ~381 ~660 +280
120 340 60 ~581 ~627 +46
130 60 10 ~747 ~628 ~119
140 60 0 ~807 ~640 ~168
150 140 20 ~846 ~648 ~199
160 150 30 ~828 ~622 ~206
170 200 30 ~807 ~638 ~169
180 180 30 ~771 ~643 ~128

 

 

 

 

 

 

 

50

 

 

 

APPENDIX B

The figures in this appendix show the pair energies of the asymmetric mole-
cules, CHF3 and CFH3, for various 0’9 with the 05’s and 02’s (the 0’s, 43’s, and
02’s used here are the Euler angles defined in section V of this thesis) set to the
orientations noted in the captions of Figures 11 and 12. These noted orientations are
close (within 2 degrees) to the minimum energy orientations of the single molecule-

substrate interactions.

Each drawing in Figures 11 and 12 shows two molecules with molecule one
represented by a single dark line pointing in the direction of the dipole moment
of the molecule. Molecule two is represented by a circle centered on the carbon
atom, and showing the approximate radius of the molecule. For each orientation
(9) of molecule one, molecule two is rotated through 360 degrees (in steps of 10°)
and all local energy maxima and minima are shown by drawing a radial line in the
direction of the dipole moment at the maximum or minimum and noting the total
pair energy of the configuration and the angle (02) at which this energy was found.
We also show the approximate angles at which the pair energy is zero, and, as an
aid to the eye, we have shaded the regions of positive energy (Le. if the dipole
moment of molecule two lies in a shaded region, as drawn, then the pair interaction
energy is positive). As can be seen from the figures, regions of positive and negative
energy occupy nearly equal areas of the available (as we have constrained it) phase
space. The intermolecular separations, r, of the molecules represented in Figures
11 and 12 are 4.5.4 and 4.0A for CHF3 and CFH3 respectively; these separations
were chosen to match the intermolecular separations which we have discussed in

appendix A.

51

 

 

 

4.51

255' 5=0
radius 1.381

   
    
 
  
   
  

. 180° E=+554

...........
..........

E=-433 0' _________________
512:2: 180' E=+546 3'

=-519 30'

”100' 5=0
5=0

A. 1%.—E:+?29- —————— 7 —————

...........
........

100.

  

E=0 Figure 110
,:;.....245. E=O

Figure lle

52

!— 4.5A ——1

 
 
 
 
      
  

I
l
radius 1.38A :

’- 50°
E=+398

 

70‘

E=324
' E=-550

Figure llh

E=+859 70°

5=0 285° 260° 5=-550
5=+2072 320°

.. 01 _____________

5:-173 20°°'/ 90
13:0 30 140' 5=0 I

E=+2367 70° Figure llj

53

I——-— 45A —-—|

   
  

I
. I
_ - 240 E=-468
E'+5139 31° radius 1.38A :

    

=0 ' ———————————— 130, --—-
5-- 9 10'

=0 15' .

5=+5292 70" 125° E=° Figure 111

265° 5=0

E=+10089 310° 230° E=~587
5=+100 10' _________ 110'
E=+10215 ° 120° E=0 Figure 111

260' E=0

5=+13490 310 v 220. 5=-590
E=+186 0' ______________ ____

E=+13963 60' 110' 5=0

    
  
 
   
  

Figure llm
5=+13490 300' 250' 5=0
210' E=—578
E=+206 o' ————————————— ————
170' E=—489 130'
180' 5=-490
5=+13319 50' -
05 5=0 Figure lln

200' E=-557

————————— 1 21V"

E=O Figure 110

5=+9327 so"

54

4.51

 

.. 250° E=0
‘reizii‘?s~ radius 1.38A

 
  
    
   
   
 

5=+52 350' A190' =-529

...........
...........
.............

E=+4589 50' ““*‘=52?252§s?:s§s§

E=+1882 290i 250' 5:0

...........
‘e \9 e - ~

. . .

....

5=+13 350' .

1 180° =-491

 

180' E=-453

5=0 75"

E=+206 310. ..'.-:‘:1:=ité:i:

180' E=-437

90' E=0

 

______1

Figure 11. CHF, pair energies for ¢,=¢,=125°, r=4.5A

55

[—— 4.0A ————|
I

- 5=+1597
300, .250

5=+10255 ..
E=0 5',
5=-55 0'
5=0 5'.

     
 
 
  

:1 180' E=+575545

E=+8265 290° 250° E=+1281

5=0 340' "
=-161 350" ________

170° E=+1034579

V
5=0 10' "
E=+8364 60"

100' 5=+1453
=+3735 290' 240' E=+5°9
5=0 325'

5=-412 350'

 
  
   

100‘ E=+949

250' 5=0
240° E=-48

170' E=+160167

   
 
  

  

E=+1077 6()" 90' E=+447
E=-368 280' 250' E=-512
225' E=O
E=-630 330" 4

' 160' E=+42734
5=0 50'

56

radius 1.3A

—--——_l —————

00

Figure 125

_——? —————

10'

Figure 12b

Figure 120

    
 

I 160' E=+11640 / 50°

E=+3650 60'

E=-546 2

50' E=-808
.4210. 13:0
7’ 170' E=+l648 ’70
5=0 60'
240' E=-807
) 19g'_5:+;12_2 _____________
v 1 80°
5=0 50'

240° E=-784
215' E=0

   

90.

130' E=+278 F, 12.
90' =+372 'gm 1

57

  
  
 

I radius 1.3A
5=0 305' 240‘E=-740
E=+3O 320' ’ 220 13:0
E=+0.4 350° _______
. 180' -+7065
E=+396 80° 130. E=+112
E=O 290° - ____
E=+125 310. 230 E 701

180' E=+10107

5' 5-
. 130' 5:27
E=+426 70 125. 5=0
5=0 280' _ -
5=+173 310' 23g .1323)"
5=+111 350' 10
180' 5=+10255
140' 5=0
5=+417 60° .130' 5=-140
1 0 5=0
5=0 280'

1
100

E=+176 310' 230° .E='646

E=+138 340' 210 5=0
1230'—5:+'7'25_9'—_

140' 5=0

E=+369 60° 13g 5=-239

.0
E=O 280°

E=+160 310' 220' E=-641
E=+150 340° 200' 5:0
1757515233"
. 140' E=O

95E=0

58

Figure 120

4.0.4
I radius 1.3A

 

  
   
 
 
   
  
  
  

E=~630
...,-;--.-:;;;}3;l';25 2m. E=O

170' E=+1844

' 145' 5=0

E=+246 50' "iiiiiiiiiiii E=~420

185' E=0
140' =-541

E=0 280f _,

   
 

220' E=-579

  

E=+177 0’

4:: 140. =-579
5=0 80'

 

Figure 12q

Figure 12r

Figure 129

Figure 12. CF11, pair energies for ¢l=¢2=IOO°, r=4.0A

59

30' 5:25 ———————— 1 7OT~<—-

————————————— —— —
V 180' 5=-55

LIST OF REFERENCES

10

11

12

13

14

15

16

17

18

19

LIST OF REFERENCES

Steele, W.A., The Interaction of gases with solid surfaces (Pergamon Press, New York,
1974)

Sinha, S.K., Ordering in Two Dimensions (North Holland, New York, 1980).

Coulomb, J.P., Phys. Rev. Lett. 43, 1878 (1979).

Sephens, Phys. Rev. Lett. 41, 961 (1978).

Phillips, J.M. and Hammerbacker, M.D., Phys. Rev. B 29, 5865 (1984).

Halperin, BI. and Nelson, D.R., Phys. Rev. B 19, 2457 (1979).

Knorr, K. and Weickert, H., Phys. Rev. B 37, 3524 (1988).

Phillips, J.M. and Hammerbacker, M.D., Phys. Rev. B 37, 5859 (1984).

Bohm, H.J., Scharf, P., Ahlrichs, R., and Schifl'er, H., J. Chem. Phys. 81. 1389 (1984).
Bondi, A., J. Phys. Chem. 68, 441 (1964).

CRC Handbook of Physics and Chemistry 64th edition, (CRC Press, Boca Raton, F L.,
1983.

Smalley, M.V., Huller, A., Thomas, R.K., and White, J.W., Mol. Phys. 44, 533 (1981)..
Zhang, Q.M., Kim, H.K., and Chan M.H.W., Phys. Rev. B 34, 8050 (1986).
Severin, ES. and Tildseley, D.J., Mol. Phys. 41, 1401 (1980).

Marlowe, L, Thomas, R.K., Trewern, T.D., and White, J.W., J. Phys. (Paris) Colloq. 33,
4(1977)

Vora, P., Phys. Rev. Lett. 43, 704 (1979).
Bak, P., Phys. Rev. B 27, 591 (1983).
Calisti, P.S., PhD. thesis, University of Marseille, (1981).

Lauter, J. Phys. (Paris) 43, 1659 (1982).

60

20

21

22

23

24

25

Tang, 5., Mahanti, S.D., and Kalia, R., Phys. Rev. B 32, 3148 (1985).
Tang, 5., Mahanti, S.D., and Kalia, R., Phys. Rev. Lett. 56, 484 (1986).
Tang, 5., Jin, W., Mahanti,S.D., and Kalia, R., Phys. Rev. B 39, 677 (1989).
Thomy, A., and Duval, X., J. Chim. Phys. 67, 1101 (1970).

Bohm, H.J., and Ahlrichs, R., J. Chem. Phys. 77, 2028 (1982).

Bohm, H.J., Meissner, C., and Ahlrichs, R., Mol. Phys. 53, 651 (1984).
Bol’shutkin, Acta Cryst. 828, 3542 (1972).

Goldstein, H., Classical Mechanics (Addison-Wesley, Reading, Mass, 1980).

61

"I71111111111117111