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UNIVERSITY OF MINNESOTA
This is to certify that I have examined this bound copy of a masters thesis by
Anthony Carlson
and have found that it is complete and satisfactory in all respects,
and that any and all revisions required by the final
examining committee have been made.
Traian Dumitrica
Name of Faculty Adviser
Signature of Faculty Adviser
Date
GRADUATE SCHOOL
An Extended Tight-Binding Approach for Modeling
Supramolecular Interactions of Carbon Nanotubes
A THESIS
SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL
OF THE UNIVERSITY OF MINNESOTA
BY
Anthony Carlson
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
Master of Science
Traian Dumitrica, Adviser
October 2006
c© Anthony Carlson October 2006
Contents
Chapter 1 Introduction 1
1.1 Fullerenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Research interest in carbon nanotubes . . . . . . . . . . . . . . . . 6
1.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 CNT-based applications . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Role of van der Waal forces . . . . . . . . . . . . . . . . . . . . . . 10
1.7 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Chapter 2 Nature of bonding and mathematical models 15
2.1 Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Atomistic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Electromagnetic Cohesion . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.1 Repulsive force . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.2 Attraction in physical bonding . . . . . . . . . . . . . . . . . 19
i
2.3.3 Van der Waals forces and non-ideal gasses . . . . . . . . . . 19
2.3.4 Classification of van der Waals forces . . . . . . . . . . . . . 20
2.4 Mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.1 Empirical . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.2 Semi-empirical . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4.3 First principles . . . . . . . . . . . . . . . . . . . . . . . . . 25
Chapter 3 Graphite structure and interlayer properties 26
3.1 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Graphite stacking patterns . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1 Corrugation of graphene planes . . . . . . . . . . . . . . . . 30
3.2.2 Graphite unit cell and Brillouin zone . . . . . . . . . . . . . 31
3.3 Graphene as extended molecules . . . . . . . . . . . . . . . . . . . . 31
3.3.1 Experimental exfoliation energy . . . . . . . . . . . . . . . . 33
3.3.1.1 Girifalco method . . . . . . . . . . . . . . . . . . . 33
3.3.1.2 Benedict method . . . . . . . . . . . . . . . . . . . 33
3.3.1.3 Zacharia method . . . . . . . . . . . . . . . . . . . 33
3.3.1.4 Exfoliation energy summary . . . . . . . . . . . . . 34
3.4 Z-axis compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5 Vibrational properties . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.6 Ab-initio studies of graphite . . . . . . . . . . . . . . . . . . . . . . 36
3.6.1 Hartree-Fock treatment of graphite . . . . . . . . . . . . . . 36
ii
3.6.2 Density Functional Theory treatment of graphite . . . . . . 37
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Chapter 4 Existing models for dispersion interactions in graphite 41
4.1 Empirical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Semi-empirical and first principles models . . . . . . . . . . . . . . 44
4.3 Motivation for what is to be done . . . . . . . . . . . . . . . . . . . 46
Chapter 5 Modeling electronic and repulsive interaction with a
tight binding formalism 48
5.1 Linear combination of atomic orbitals . . . . . . . . . . . . . . . . . 49
5.2 Schrodinger equation in LCAO approximation . . . . . . . . . . . . 49
5.3 The basis and atomic centered orbitals . . . . . . . . . . . . . . . . 50
5.4 Periodic solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.5 The Hamiltonian and the two-center approximation . . . . . . . . . 53
5.6 Slater-Koster parametrization . . . . . . . . . . . . . . . . . . . . . 55
5.6.1 Orbital decomposition . . . . . . . . . . . . . . . . . . . . . 55
5.6.2 s-p decomposition . . . . . . . . . . . . . . . . . . . . . . . . 56
5.6.3 Construction of tight binding matrices . . . . . . . . . . . . 59
5.7 Tight Binding Total Energy . . . . . . . . . . . . . . . . . . . . . . 60
5.8 Fitting TB parameters . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.8.1 Density functional based tight binding . . . . . . . . . . . . 63
5.9 Use of Porezag parametrization and current code . . . . . . . . . . 64
iii
Chapter 6 Quantum mechanical origin of dispersion forces 65
6.1 Main assumptions in modeling dispersion interactions . . . . . . . . 66
6.2 Microscopic dispersion theory (London) . . . . . . . . . . . . . . . . 67
6.2.1 System description . . . . . . . . . . . . . . . . . . . . . . . 67
6.2.2 The perturbed system . . . . . . . . . . . . . . . . . . . . . 68
6.2.3 Energy corrections in perturbation theory . . . . . . . . . . 69
6.2.4 Application to Hydrogen . . . . . . . . . . . . . . . . . . . . 70
6.2.5 Higher order corrections . . . . . . . . . . . . . . . . . . . . 73
6.2.6 Applying London’s dispersion theory in Calculating C6 for
graphite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.2.6.1 London’s form and polarizability . . . . . . . . . . 73
6.2.6.2 Slater-Kirkwood approximation . . . . . . . . . . . 75
6.2.6.3 Kirkwood approximation . . . . . . . . . . . . . . . 76
6.3 Macroscopic dispersion theory (Lifshitz + Hamaker) . . . . . . . . . 76
6.3.1 Lifshitz dispersion theory . . . . . . . . . . . . . . . . . . . . 77
6.3.1.1 Dielectric response and ǫ(iξ) . . . . . . . . . . . . . 77
6.3.1.2 Lifshitz dispersion theory . . . . . . . . . . . . . . 78
6.3.2 Lifshitz theory applied to the evaluation of Hamaker con-
stants for graphite . . . . . . . . . . . . . . . . . . . . . . . 79
6.3.3 Hamaker constant derivation . . . . . . . . . . . . . . . . . . 80
6.3.3.1 Point-Surface Interaction . . . . . . . . . . . . . . 81
iv
6.3.3.2 Surface-Surface Interaction . . . . . . . . . . . . . 82
6.4 Summary of calculated C6 coefficients . . . . . . . . . . . . . . . . . 84
Chapter 7 Damping functions 85
7.1 Hydrogen damping function . . . . . . . . . . . . . . . . . . . . . . 87
7.2 Damping functions in literature . . . . . . . . . . . . . . . . . . . . 87
Chapter 8 Tight-binding plus dispersion parametrization 90
8.1 Total energy definition . . . . . . . . . . . . . . . . . . . . . . . . . 91
8.2 Evaluation of tight-binding energy . . . . . . . . . . . . . . . . . . . 93
8.2.1 Orbital expansion . . . . . . . . . . . . . . . . . . . . . . . . 94
8.3 Parameter fitting procedure . . . . . . . . . . . . . . . . . . . . . . 95
8.3.1 Equilibrium interlayer spacing . . . . . . . . . . . . . . . . . 97
8.3.2 Phonon frequency calculation . . . . . . . . . . . . . . . . . 97
8.3.3 Compressibility calculation . . . . . . . . . . . . . . . . . . . 99
8.4 Optimization results . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Chapter 9 TBD description of carbon nanotube interactions 107
9.1 Tube-graphene interactions . . . . . . . . . . . . . . . . . . . . . . . 108
9.2 Tube-tube interactions . . . . . . . . . . . . . . . . . . . . . . . . . 109
9.3 Universal binding curve . . . . . . . . . . . . . . . . . . . . . . . . . 114
9.4 Nanotube loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
v
9.5 Nanotube bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
9.6 Multiwalled tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9.7 C60 and carbon nanotube interactions . . . . . . . . . . . . . . . . . 120
9.8 Tube-tube MD simulation . . . . . . . . . . . . . . . . . . . . . . . 122
Chapter 10 Conclusion and future work 125
Appendix A Tight-binding + dispersion molecular dynamics overview127
A.1 Calculation of Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 129
A.1.1 Ionic and van der Waal’s forces . . . . . . . . . . . . . . . . 130
A.1.2 Tight-binding band structure forces . . . . . . . . . . . . . . 131
A.2 Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . . 132
Appendix B Quantum Mechanics Overview 135
B.1 Observables and expectation values . . . . . . . . . . . . . . . . . . 136
B.2 Dirac Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
B.3 Time-dependant Schrodinger equation . . . . . . . . . . . . . . . . 137
B.4 Time-independent Schrodinger equation . . . . . . . . . . . . . . . 138
B.5 General Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 139
B.6 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
B.7 Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . . 140
B.8 One-Electron Approximation . . . . . . . . . . . . . . . . . . . . . . 141
B.9 Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
vi
B.10 Extended example . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Appendix C Non-orthogonal Hellmann-Feynman forces 146
C.1 Model Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
C.2 Force derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Appendix D Multipole Expansion 152
Appendix E Perturbation Theory 156
E.1 First order correction to energy . . . . . . . . . . . . . . . . . . . . 158
E.2 First order correction to the wavefunction . . . . . . . . . . . . . . 159
E.3 Second order correction to the energy . . . . . . . . . . . . . . . . . 160
Appendix F Optimization 163
vii
List of Figures
1.1 Graphite to CNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Description of chirality . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Armchair, zigzag, and chiral tubes . . . . . . . . . . . . . . . . . . . 4
1.4 Exponential growth in nanotube papers . . . . . . . . . . . . . . . . 6
1.5 Nanotube twine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Hexagonal lattice of graphene and AFM scan . . . . . . . . . . . . . 11
1.7 CNT bundle cross section . . . . . . . . . . . . . . . . . . . . . . . 12
2.1 Rare gas dimer energy curve . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Schematic representations of van der Waals energy contributions . . 21
3.1 Hexagonal lattice of graphene and AFM scan . . . . . . . . . . . . . 28
3.2 Graphite stacking patterns . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 AFM scan of graphite showing Pz orbital corrugation . . . . . . . . 30
3.4 Hexagonal lattice of graphene and AFM scan . . . . . . . . . . . . . 32
viii
3.5 Raman active interlayer modes . . . . . . . . . . . . . . . . . . . . 35
4.1 Interlayer energy plots from existing empirical models . . . . . . . . 42
5.1 Schematic of relevant s and p orbital orientations . . . . . . . . . . 56
5.2 S and P orbital decomposition . . . . . . . . . . . . . . . . . . . . . 57
6.1 Hydrogen dimer schematic . . . . . . . . . . . . . . . . . . . . . . . 70
6.2 Semi-infinite slabs schematic . . . . . . . . . . . . . . . . . . . . . . 78
6.3 Point-surface Hamaker integration . . . . . . . . . . . . . . . . . . . 81
6.4 Surface-Surface Hamaker integration . . . . . . . . . . . . . . . . . 83
7.1 Tang’s damping function . . . . . . . . . . . . . . . . . . . . . . . . 89
8.1 Cutoff function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
8.2 Tight-binding interlayer energy . . . . . . . . . . . . . . . . . . . . 94
8.3 Modified tight-binding interlayer energy . . . . . . . . . . . . . . . 96
8.4 Fitted tight-binding plus dispersion - I . . . . . . . . . . . . . . . . 101
8.5 Fitted tight-binding plus dispersion - II . . . . . . . . . . . . . . . . 102
8.6 Graphite energy landscape . . . . . . . . . . . . . . . . . . . . . . . 102
8.7 Hydrostatic pressure effects on the E2g(1) mode . . . . . . . . . . . 105
8.8 Molecular dynamics results of graphene . . . . . . . . . . . . . . . . 106
9.1 Lock-in orientations on graphite . . . . . . . . . . . . . . . . . . . . 109
9.2 Interaction energy scans of tubes on graphite . . . . . . . . . . . . . 110
ix
9.3 Tube-tube orientation . . . . . . . . . . . . . . . . . . . . . . . . . 111
9.4 Tube-tube θ, dz landscape . . . . . . . . . . . . . . . . . . . . . . . 111
9.5 Tube-tube interaction energy scans . . . . . . . . . . . . . . . . . . 113
9.6 Tube-tube cohesive energy . . . . . . . . . . . . . . . . . . . . . . . 114
9.7 Carbon nanotube/tube/graphene universal curve . . . . . . . . . . 115
9.8 Hexagonal lattice of graphene and AFM scan . . . . . . . . . . . . . 118
9.9 Energy scan on CNT bundle . . . . . . . . . . . . . . . . . . . . . . 118
9.10 Nested (5,5)‖(10,10) CNT pair . . . . . . . . . . . . . . . . . . . . . 119
9.11 Nested (5,5)‖(10,10) energy landscape . . . . . . . . . . . . . . . . 121
9.12 Nested (5,5)‖(10,10) energy translation and rotation . . . . . . . . . 121
9.13 C60 external to a (10,10) tube energy scan . . . . . . . . . . . . . . 122
9.14 C60 external to a (10,10) tube energy scan . . . . . . . . . . . . . . 123
9.15 Peapod micrograph . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
9.16 MD results of two (5,5) tubes . . . . . . . . . . . . . . . . . . . . . 124
A.1 Molecular dynamics flowchart . . . . . . . . . . . . . . . . . . . . . 129
A.2 Schematic of periodic boundary conditions . . . . . . . . . . . . . . 133
D.1 Two separated charge clouds A and B . . . . . . . . . . . . . . . . . 153
x
List of Tables
3.1 Summary of experimental graphite exfoliation energies . . . . . . . 34
3.2 Summary of DFT calculations of graphite . . . . . . . . . . . . . . 38
4.1 Results of LJ empirical models with experimental data comparison . 43
6.1 Summary of C6 dispersion coefficients for graphite . . . . . . . . . . 84
8.1 Energy convergence: K-point selection . . . . . . . . . . . . . . . . 93
8.2 Optimized parameters for the tight-binding plus dispersion model . 100
9.1 Tube-graphene and tube-tube interactions summary . . . . . . . . . 112
9.2 Comparison of CNT bundle results . . . . . . . . . . . . . . . . . . 117
B.1 Operators in quantum mechanics . . . . . . . . . . . . . . . . . . . 136
D.1 Tabulation of selected terms in the multipole expansion . . . . . . . 155
xi
1Introduction
1.1 Fullerenes
The element carbon is the first element in group IV of the Periodic Table. It has
the ability to chemically bond with itself and other elements readily via orbital
hybridization. The whole of organic chemistry is dedicated to the study of the
millions of carbon-based molecules. There is a diverse variety of carbon solids
including a few crystalline allotropes and many amorphus and semi-crystalline
solids (such as lonsdaleite, and chaoite), but the most recognized forms are the
crystalline forms of diamond and graphite.
In the diamond crystal, each carbon atom is tetrahedrally bonded to its four
1
nearest neighbors via sp3 hybridization. The most stable allotrope of carbon is
graphite, which is nearly iso-energetic with diamond (∆E ∼ 0.014 eV/atom [172]).
Graphite has an interesting layered structure composed of planar sheets, called
graphene. In graphene, carbon atoms are trigonally bonded to their three nearest
neighbors by strong sp2 bonds, forming a hexagonal network (see left side of Figure
1.1). To build graphite, graphene sheets stack on top of one another. There is no
chemical bonding between layers; rather these layers are weakly bound by van der
Waals (vdW) forces [133]. This distinction between chemical and physical (vdW)
bonding will be addressed later in § 2.3.
While these two allotropes of carbon have been known for quite some time, new
forms, termed fullerenes, have been identified. Fullerenes are hollow, all carbon
structures in the form of balls, tubes and other closed structures with walls that
are one-atomic layer thick. A notably stable soccer ball-like fullerene is C60, com-
monly referred to as a “bucky ball”. It is named after Richard Buckminster Fuller,
an American author and architect whose geodesic dome had a very similar con-
struction. Tubular fullerenes are referred to as carbon nanotubes (CNTs). These
elongated molecules with nanometer-size diameters have hemispherical capped
ends, and length-to-diameter ratios typically of about 1000, which makes them
truly one-dimensional structures. The wall of these tubes share the same hexago-
nal structure as that of individual graphene sheets, except that they are figuratively
rolled up as is shown in Figure 1.1. “Figuratively” is emphasized because it de-
scribes their structure, not their manufacture. This figurative connection is useful
also because many of the physical properties of nanotubes can be derived from
graphite’s properties. In addition, adjacent tubes interact via vdW forces rather
than chemical bond just as the adjacent sheets of graphene in the graphite crystal
Ignoring now the issue of the capped ends, tubes are classified based on their
curvature and chirality. In an unwrapped representation of the nanotube, the
chirality describes how the hexagonal structure of the graphene lattice is orientated
2
Figure 1.1: Graphite to CNT from [21]
with respect to the axis of the tube. Figure 1.2 again shows a graphene lattice 1.
The point in the top left is the origin. Picking this point up and curling the sheet
onto one of the labeled atoms that fall between the two vectors creates a unique
CNT. Due to the symmetry of the graphene lattice only those atoms between the
two vectors need to be considered for a unique tube. The vector pointing from
the origin to the atom picked to describe the CNT is known as the chiral vector
( ~Ch). This vector is conveniently defined in terms of the graphene lattice vectors
labeled ~a and ~b, shown in Figure 1.2, as
Figure 1.2: The chirality is described by the nanotube wrapping angle of thegraphene sheet
1The lattice vectors and unit cell shown Figure 1.2 are differen’t but equivalent to thosedescribed later in § 3.1
3
~Ch = n~a+m~b ≡ (n,m), (1.1)
where n and m are integers. The angle this vector forms with respect to vector
~a is the chiral angle (χ). Tubes with χ = 0◦ are known as zigzag tubes and
are described by the chiral vector (n, 0). The moniker of zigzag comes from the
sawtooth like structure that is found around the circumference of the tube as can
be seen in Figure 1.3. The other extreme is χ = 30◦, or armchair tubes. These
tubes are defined by the case when n = m, otherwise noted by (n, n). All other
tubes that fall between these limits are referred to as chiral tubes (n,m).
Figure 1.3: Armchair, zigzag, and chiral tubes
Nanotubes can exist as single layered structures known as single-walled carbon
nanotubes (SWCNTs) or be composed of several coaxial SWCNTs nested inside
one another and referred to as multi-walled carbon nanotubes (MWCNTs). The
smallest observed SWCNT diameter is ∼4 A (1 A = 1 × 10−10 m), but most
often form in diameters around 10 A [134, 160]. For a spatial reference, human
hair has a median diameter of ∼ 1 × 106A. MWCNTs diameters can get quite
4
large. Starting with the smallest observed SWCNT (4 A) at the core successive
SWCNT encapsulate one another with an average wall-to-wall spacing of 3.4-3.9
A, increasing with decreasing tube diameter [84]. MWCNTs with diameters larger
than 300 A in diameter, comprising up to 100 SWCNTs, are not uncommon [160].
As mentioned, the discovery of fullerenes is quite recent. Smalley et. al, in 1985
were the first to experimentally verify the existence of the C60 fullerene [99]. The
discovery of a MWCNT, via high-resolution transmission electron microscopy (HR-
TEM), goes to to Iijima in 1991 [74]. (This distinction is being debated as dis-
cussed in a recent editorial by Monthioux et al. [121] who points out that in 1952
a Russian journal published a TEM image of a MWCNT [141], and later in 1976
Endo et al. also published a TEM image of a MWCNT [129]. Neither of these ar-
ticles made a clear distinction of the nature of their photographic results as Iijima
did.) The first evidence of a SWCNT was published two years later in two papers
[12, 76].
1.2 Production
While trace amounts of fullerenes are found in nature, i.e. in the soot of a candle
flame, hight quality CNTs are produced artificially. Production methods include
electric arc discharge between two graphite electrodes [37, 75], laser vaporization
of graphite [57], chemical vapor deposition (CVD) of hydrocarbon over metallic
catalysts seeds [28, 155], and other novel methods. All these methods have differ-
ent yields, some preferentially produce SWCNTs (regarded as higher quality, as
opposed to MWCNTs). Differing lengths and diameters are present and the prod-
uct usually needs a post-production purification step. Early production of CNTs
resulted in tubes that where quite short in length, < 10 µm. Advanced methods
have been developed to produce longer CNTs, with reports of several centimeters
for some SWCNTs [179].
5
1.3 Research interest in carbon nanotubes
After Iijima’s 1991 paper in Nature there was an explosion of interest in carbon
nanotube research that continues today. Figure 1.4 shows the nearly exponential
growth of published papers on nanotubes, with the year 2005 averaging 7 peer-
reviewed articles published a day.
19931994199519961997199819992000200120022003200420050
500
1000
1500
2000
2500
Year of publication
Num
ber
of p
aper
s
Figure 1.4: Graph showing the rapid increase in the number of published pa-pers on nanotubes. Data points are the number of papers returned from asearch of “nanotube”, from the extensive ISI Web of Knowledge database athttp://portal.isiknowledge.com
Why is there such an intense interest in the field of carbon nanotubes? Compared
to other nano-scale objects CNTS are well defined elongated structures that are
basically one-dimensional systems. There were early theoretical predictions of
unique properties which were proposed before they were measured. These predic-
tions included exotic mechanical and electronic properties that have since spurred
much new theoretical and experimental work on these unique objects.
6
1.4 Properties
An early and fascinating prediction of CNT electronic properties was that depend-
ing on tube chirality and diameter, CNTs would be metallic, semiconducting, or
insulating [60, 149]. This theoretical prediction was experimentally verified a few
years later [158]. Metallic MWCNTs have been shown to have amazing elec-
trical transmission properties, experimentally carrying 109 A/cm2 compared to
∼ 105 A/cm2 for some metals [171]. An excellent in-depth review of other unique
CNT electronic phenomena including field emission, optical, and contact proper-
ties is summarized in [3].
The mechanical properties of CNTs have attracted as much or more interest than
their electronic properties. The carbon-carbon sp2 bond found in graphite, and
hence in CNTs, is one of the strongest in nature. The in-plane elastic constant
for graphite is very stiff C11 = C22 ≈ 1.06 TPa [2, 169]. Axially, CNTs share this
same stiffness as evident by their high Young’s modulus. Computational results
include a tight binding study that reports a Young’s modulus (E) of 1.22 TPa,
and a DFT study that reports E = 1.1 TPa (tight binding and DFT methods will
be discussed further in § 2.4) [69].
It is possible to perform tensile loading experiments on CNTs between two atomic
force microscopy (AFM) tips, monitored in real time with electron microscopy
[15, 173, 174]. Measurements on SWCNTs have yielded Young’s moduli around
∼ 1 TPa, ultimate tensile strength (UTS) of ∼ 66 GPa and strain at failure (ǫf )
of 2 − 19%2. The average experimental Young’s moduli match quite well with
theoretical studies mentioned above. The CNT specimens that were tested were
quite short and were possibly defect-free. A detailed theoretical study of failure
of ideal CNTs shows that brittle or ductile failure (depending mostly on chirality)
occurs at strain levels above 17%. [34]. The introduction of defects into CNTs
can greatly reduce their their ultimate tensile strength [124].
Only when one compares these quoted CNT material properties to other materials
7
does one realize their unique mechanical qualities. One can compare them to steel’s
material properties (e.g. high strength tempered 4340 steel alloy, E = 12 TPa,
UTS = 1.7 GPa, ǫf = 12% [22]). It is wiser to compare CNTs material proper-
ties to other fiber-like materials like carbon fiber rather than three-dimensional
solids such as steel. Carbon fibers are polycrystalline graphitic fibers used heavily
in polymer-matrix composites for aerospace and other high-performance appli-
cations. Typical values for the Young’s modulus, tensile strength and strain at
failure are E = 0.3-0.7 TPa, UTS = 1.5-4.8 GPa, ǫf = 0.1-2.0 % [19, 22]. For
the high end of these quoted values carbon nanotubes are ∼ 1.4, 13.6, 9.5 times
greater respectively, and considering that carbon fibers are ∼ 1.5 times denser
than carbon nanotubes (ρcnt = 1.3 g/cm3 [19]), the specific strength and stiffness
properties of CNTs are even more impressive.
Currently the mechanical properties of individual carbon nanotubes do not fully
transfer to macroscopic bulk applications. It is possible to take the short length
sections of CNTs and spin them into a nanotube yarn as shown in Figure 1.5
A, and this yarn can be twisted or braided into rope like structures as shown in
Figure 1.5 B and C [176]. The mechanical properties of these braided fibers have
an ultimate tensile strength an order of magnitude less than the individual CNTs.
While stiff and strong axially, CNTs are quite compliant radially. They are flex-
ible enough to be curled into tight circles (as seen in bucky paper discussed be-
low), kinking if bent too much, but reversibly and elastically popping back when
straightened, seemingly without any defects [43, 169]. Compared to carbon fibers,
which are extremely brittle in bending, CNTs show remarkable bending resilience.
2The discussion of material properties such as the ultimate tensile strength or the Young’smodulus, necessitates a definition of cross sectional area. For carbon nanotubes the cross sec-tional area is not well defined. Since SWCNTs are an atomic layer thick and basically circularin cross section the area can be defined as A = 2πRδR where R is the radius of the tube and δRis the wall thickness. Should this value be the diameter of the carbon nuclei, or some functionof the electronic cloud spatial extent? Traditionally the value of δR = 3.4 A has been employed.
8
Figure 1.5: Carbon nanotubes twisted and braided into ropes, from [176]
1.5 CNT-based applications
The outstanding mechanical properties of CNTs in nanoscale devices or composite
materials remains a powerful motivation for the research. Carbon nanotubes have
been successfully used as atomic force microscopy (AFM) tips [26, 58]. In a sim-
ilar apparatus to a AFM setup, CNT tweezers composed of two nanotubes have
been used to selectively pinch and manipulate a target nanoparticle by applying a
voltage bias across the two tubes [85]. These tweezers when closed stick together
until a common polarity voltage is applied to them. Other electromagnetic based
CNT devices have been fabricated, including relays and switches that usually take
the form of a CNT cantilevered over a well or a fixed-fixed CNT over a well. Ac-
tuation or closing happens upon the application of a voltage bias and subsequent
deformation of the tube to a closed state [86, 150]. A form of non-volatile random
access memory that has a similar structure to these switches has been described
in literature [144] and is being produced by Nantero for sale in 2007 [78].
A very interesting mechanical aspect encountered in MWCNTs is the possibil-
ity of very low friction between the walls, which can provide for a very smooth
relative linear or rotational motion between a nested pair. This idea motivated
9
the possibility of CNT bearings [29, 94, 176]. Both rotational and linear bearings
have been modeled and tested. Relative linear translation of an inner tube being
partially pulled out of its shell have been performed with an AFM tip attached to
the inner tube while the outer tube is held rigid. Repeated in and out motion of
the inner tube did not deform or induce defects in the structure as viewed under
an SEM. During this same experiment it was noted that the inner tube quickly
retracted inside its parent SWCNT when the tube broke free of the AFM tip.
This rapid retraction induced by vdW forces led to the idea of a MWCNT system
that could sustain mechanical oscillation of an inner tube in the GHz range. This
proposed system consists of a short capped carbon nanotube nested in a lengthier
counterpart [104, 177, 178].
1.6 Role of van der Waal forces
The unique mechanical properties of carbon nanotubes indirectly lead to an un-
usual role of the forces between them. Like adjacent sheets of graphite, nanotubes
only occasionally form chemical bonds between one another and instead interact
via vdW forces. These “supramolecular” interactions are normally weak and, for
most molecular species, are easily overcome by thermal agitation (see the case of
proteins). However, they turn out to be significant in case of nanotubes, building
up very strong attractive forces over the extensive aligned contacts.
Because of their hollow structure and large aspect ratio, forces or interaction be-
tween them can cause deformations of bending, torsion, flattening polygonization.
A purified form of raw CNTs known as bucky paper is shown in Figure 1.6 (a).
The spaghetti-like agglomeration with highly curved structures is particularly in-
triguing in view of the strong covalent carbon-carbon bonding, which confers to
CNTs high mechanical stiffness. This characteristic feature is attributed not to
thermodynamic fluctuations, but rather to vdW forces between tubes, which pre-
vent these loops from unfolding. Another outstanding example of this balance
10
of the strain energy from bending and the van der Waals attraction is shown in
Figure 1.5 (b) [113]. Here we see closed loops of SWCNT bundles turned onto
themselves and stuck in a stable loop. It was reported that there is a critical
radius of 0.03 µm for SWCNT loops at room temperature. We will address this
phenomenon later with some sample calculations.
(a) (b)
Figure 1.6: (a) Bucky paper composed of numerous CNT bundles, from [77] (b)Stable curled CNT bundles
The supramolecular vdW forces in CNT systems also cause SWCNTs to self-
organize in ropes or bundles [80, 160]. The tubes in a bundle organize into a
triangular lattice (see Figure 1.7) with an approximate wall to wall distance of ∼ 3
A [160]. Interestingly these bundles seem to grow together with all tubes being
essentially the same length, and it has been noted that all the tubes constituting
a bundle have essentially the same diameter and only one or two chiralities [28].
Another interesting manifestation of vdW interactions is the experimental finding
that large diameter SWCNTs can be found collapsed, losing their circular cross
section and having opposite inner faces sticking together [125]. For two SWCNTs
larger than 20 A in diameter there is a flattening of the once circular cross section
along the edge of contact between one another as predicted by an empirical model
[145], or on a flat surface such as graphite [70]. For nanotube bundles with tube
diameters larger than 20 A the circular cross section can even become hexagonal
11
Figure 1.7: Typical CNT bundle showing cross sectional packing pattern [80]
in character [109, 159].
1.7 Motivation
Nanometer scale devices, while currently not outside the realm of manipulation or
measurement, pose unique hurdles to the fuller understanding or their function-
ality due simply to their diminutive size. Computational models of nano-devices
can be utilized to perform digital experiments that offer further insight or moti-
vation for experimental work. The atomistic simulation of carbon nanotubes is
quite common at many levels of theory, but almost all of these models, even some
of the most advanced, lack the vdW attraction. We have seen the importance of
vdW forces in many of the examples discussed above. In order to model and gain
insight for example into the possibility of efficiently gluing nanotubes by weak
supramolecular forces in a self-assembly process or the performance of MWCNT
bearings and oscillators it is important to have a detailed understanding of these
forces and an accurate model to describe them.
As nanomechanical systems typically involve a substantial number of atoms, it is
computationally convenient to adopt a classical description of the covalent bond-
ing. However, there are several reasons to favor a quantum mechanical treatment:
12
Firstly the interactions of supramolecular bodies are intrinsically coupled with
their own mechanical properties. For example, the degree of deformation can en-
hance (as in the case of partial polygonization of nanotubes in the array bundles)
supramolecular attraction, or decrease it under other circumstances. Thus, the
main motivation is the need for an accurate description of the interatomic interac-
tions. It is known that classical potentials (typically fitted to the bulk form) often
do not yield reasonable results. For instance, the second generation bond–order
potential of Brenner [17] yields elastic moduli that deviate significantly from the
tight-binding and density functional theory data [69, 81].
Secondly, since the electronic subsystem is treated explicitly, the characteristic
corrugation between nanotubes can be realistically accounted for. The widely
used classical treatment of Girifalco et al. [54, 53] does not properly capture this
effect although it offers an overall good energetic description of binding beyond
the covalent range.
Finally, an additional motivation is the renewed interest in TB modeling at larger
scale using the recently proposed objective molecular dynamics scheme [35, 73].
This method heavily reduces the computational effort through a drastic reduction
in the number of atoms to be accounted for and thus makes tractable tight binding
plus vdW nanomechanical modeling.
The purpose of this work is to extend an accurate tight binding modeling of the
covalent bonding with the long-range vdW attraction. This pursuit starts with a
discussion of the nature and variety of bonding in Chapter 2, including a discussion
of the levels of theoretical and mathematical models available. This background
leads into Chapter 3 where the structure of graphite and CNTs is pursued be-
yond their brief introduction above. Chapter 3 also details those properties that
are related to the interlayer vdW interactions in graphite from experimental and
computational techniques. This data set is used to evaluate the performance of
existing graphite models that are reviewed in Chapter 4. The proposal for a tight-
13
binding plus dispersion model is presented in the last chapters along with a study
of intertube interactions in MWCNTs and CNT bundles, and is finished with an
implementation of tight-binding plus dispersion molecular dynamics.
14
“All things are made of
atoms, little particles that
move around in perpetual mo-
tion, attracting each other
when they are a little distance
apart, but repelling upon be-
ing squeezed into one an-
other”
Richard Feynman
2Nature of bonding and mathematical
models
In the study and simulation of atomistic scale systems it is an obvious prerequisite
to understand the physical models that describe the interaction between the atomic
constituents. This chapter serves as a general non-mathematical introduction to
the concept and variety of atomistic bonds relevant for graphite and CNTs, with
a particular focus on the variety of physical bonding. These concepts will be
expanded in later chapters.
15
2.1 Bonding
A bond in the everyday macroscopic sense is something that fasten things to-
gether. It is quite the same on the atomistic scale and can be described as an
energetically favorable state. This state is described by the stable and finite sep-
aration of at least two atoms under the mutual influence of one another. In order
to understand some fundamental aspects of a bond, consider the simple example
of two interacting rare gas atoms. The solid line plotted in Figure 2.1 shows the
generic form of the interaction energy of this two-atom system as a function of
internuclear separation. The interaction energy is defined as the energy in excess
or recess of this two-atom system with respect to the energy of the two atoms iso-
lated from each other ( i.e. the energy of isolated atoms is defined as zero). There
0.4 0.5 0.6 0.7 0.8 0.9 1−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Internuclear Separation
Ene
rgy
Repulsion
Attraction
r
Ecoh
Figure 2.1: Characteristic energy vs. internuclear separation for a simple rare gasdimer
is a distinct minima in this plot. The energy at this point is lower than that
of the atoms at infinite separation and corresponds to a energetically favorable
state, which is a bond by our previous definition. The magnitude of the energy
at this minima (Ecoh) is the cohesive energy, and can be thought of as the energy
required to break this bond. The distance r at which this minima occurs is simply
the equilibrium bond length.
The nature of a bond intrinsically contains two complementary components: an
16
attractive, and a repulsive, component. This statement can be supported by a few
obvious points. First, it is evident that solids exist so there must be an attractive
component to bring atoms together. At the same time we are aware of the finite
density of solids, implying a repulsive component to keep matter from collapsing
upon itself. The two labeled dashed lines in Figure 2.1 show these two components
of the total interaction energy. When added together they simply sum to the solid
total interaction energy curve.
This view of bonding, while not rigorous, paves the way for thinking about more
difficult concepts. It leads logically to the question of the nature and variety of
these energetic interactions. Are the bonds between two argon atoms the same
as the bond between two carbon atoms? Over what length and energy scales do
they form?
2.2 Atomistic Forces
The starting point in a discussion of relevant forces at the atomic scale is a list
of the fundamental forces as given by the standard model: strong nuclear, weak
nuclear, gravitation, and electromagnetic forces. To what extent do these forces
play a role in bonding between atoms? The strong nuclear force is responsible for
the cohesion of the nucleus and has an extremely short range, as does the weak
nuclear interaction. Both of these forces act on the scale of 10−3 A, which is a few
orders of magnitude less than even one of the smallest interatomic bond length of
∼ 0.8 A for H2 [115]. These forces play no role in the bonding between atoms.
Gravitation on the other hand acts over longer ranges than the strong and weak
nuclear force. Understood in Newton’s classical sense, the gravitational interaction
energy between bodies separated by a distance r is
Ug(r) =Gm1m2
r. (2.1)
17
The gravitational energy is quite small on the atomistic scale. Take for example
two argon atoms (6.63× 10−26 Kg) separated by 5 A, which is approximately the
nearest neighbor distance in solid argon [4]. At this distance the gravitational
potential energy between the two atoms is −5.8×10−52 J, whereas experimentally
the potential energy at this same distance is −5.9 × 10−22 J [89]. The gravita-
tional potential energy is thirty orders of magnitude less than the experimental
measured energy. By inductive reasoning we are left with the electromagnetic
force as the dominant force in atomic bonding [89]. The manner in which the
numerous charged particles that constitute atoms interact electromagnetically is
diverse and will be addressed in the following section.
2.3 Electromagnetic Cohesion
Atomic cohesion due to the interplay of electronic charge can generally, but not
exactly, be classified into chemical and physical bonding. The general length scales
for these bonds are 1.5-3.0 A, and 3.0-5.0 A respectively.
The chemical, or covalent bond, is quantum mechanical in nature and is character-
ized by a directional (i.e. anisotropic) short range bond that is immensely strong.
In this chemical bond there is an overlap of the electronic clouds and sharing of
electrons. Details on the quantum mechanics and computational modeling of co-
valent bonds are described in later in Chapter 5. The rest of this present chapter
is dedicated to further explanation of the physical bond.
The physical bond, while still electromagnetic in nature, is a weaker, generally
isotropic bond that equilibrates at a greater distance than covalent bonds and
generally does not contain any charge overlap. Before discussing more about
the physical bond, the repulsive contribution to bonding for both chemical and
physical bonding is addressed.
18
2.3.1 Repulsive force
For neutral species the origin of the repulsive force that occurs at small distances
is based on the Pauli exclusion principle which states that no two fermions can
occupy precisely the same quantum state [4]. Phenomenologically explained there
is a huge energy penalty when two particles are very close to one another. As two
atoms are brought together their electron clouds tend to diminish along the inter-
nuclear axis and the electronic screening of the nucleus that was supported before
degrades and a repulsive Coulomb force between the exposed nuclei pushes away.
This is a very short-range interaction and energetically decays approximately as
1/r12. Of course another form of repulsion is simple Coulomb repulsion between
like charged particles that interact over long distances according the inverse square
law Fc ∝ q1q2/r2.
2.3.2 Attraction in physical bonding
The attractive component of a physical bond can be classified into the attraction
due to Coulomb interaction of ions and that between neutral species. The physi-
cal attraction between neutral species is more nuanced than the simple Coulomb
repulsion/attraction. Current understanding of this physical attraction has its
historical roots in the study of gasses.
2.3.3 Van der Waals forces and non-ideal gasses
The ideal gas law, used to describe the properties of gasses, fails when the gas is
not sufficiently dilute. Gas particles in a dense gas are more concentrated, and
hence closer together. In denser gasses the gaseous constituents spend more time
in the range of physical bonds, and deviations from the ideal gas law occur due to
this interaction. In this regime the effects of physical bonding must be considered
to properly model the gas. In 1873 Johannes van der Waals proposed a new
equation of state that captures the ideal gas deviations attributed to the physical
19
attraction, and finite size of molecules. Its form is
(P +
a
V
)(V − b) = kBT. (2.2)
In this equation P is the pressure, V is the molar volume, kB and T are Boltz-
mann’s constant and temperature respectively. These four quantities are found
in the ideal gas law. The van der Waals equation has two parameters in addition
to these: a and b. Where a is a measure of the physical attraction between the
molecular constituents, and b is a measure of the molecular size. The attractive
physical force described by a has been given the general moniker “van der Waals
force”. The physical vdW force between neutral atoms/molecules is classifiable
into three distinct groups with equally distinct mechanistic origins.
2.3.4 Classification of van der Waals forces
The three effects that are collectively referred to as van der Waals forces can
be classified as orientational (Vo), inductive (Vi), and dispersive (Vd) interactions
[116]. Their sum is referred to as the total van der Waals energy as shown in equa-
tion (2.3). In their simplest incarnation are expressed in terms of the internuclear
separation r of two atoms/molecules
VvdW (r) = Vo(r) + Vi(r) + Vd(r), (2.3)
and the total vdW energy for a system of N particles is expressed as a pairwise
summation over all constituents
UvdW =1
2
N∑
i,j
VvdW (rij), (2.4)
where rij is the distance between atoms i and j and the factor 1/2 accounts for
20
double counting.
The orientational term (Vo) arises if both molecules have a permanent dipole
µ. Dipoles can favorably rotate into a head to tail state, and come together to
lower their interaction energy. Figure 2.2 (A) schematically shows two dipoles
favorably orientated. In the gas phase these molecules are rotating, and at high
temperature they are spinning so rapidly that the dipole effectively disappears
and the interaction energy will tend toward zero. In 1912 W.H. Keesom showed
that at finite temperature, since the probability of a given orientation of dipoles
can be determined by a Boltzmann factor, one can integrate over all orientations
and come to [116]
Vo(r) = − 1
r6
2µ21µ
22
(4πεo)2(3kBT ). (2.5)
µ1 and µ2 are the magnitudes of the dipoles on atoms 1 and 2 that are separated
by a distance r, εo is the permittivity of free space, and kB and T are Boltzmann’s
constant and temperature respectively. We see that the dependance is 1/r6 and in
the limit of high temperature the orientational dipole-dipole energy tends toward
zero.
+− +− +− +−+−+−
A B C
Figure 2.2: Schematic representations of van der Waals energy contributions.Elongated molecules are permanent dipoles, spherical molecules are non-polaratoms/molecules with an induced dipole. (A) - dipole/dipole orientationalinteraction, (B) - dipole/induced-dipole inductive interaction, (C) - induced-dipole/induced-dipole dispersive interaction
Induction is an effect between a dipolar molecule and a non-polar molecule, as
shown in Figure 2.2 (B). The electric field of the dipolar molecule induces a dipole
21
in the non-polar molecule resulting in an effective dipole-dipole effect, as discussed
above. The ability of a molecule or atom to become induced into a dipole is know
as its polarizability. The static polarizability is simply the proportionality between
the induced dipole and the electric field strength E . The interaction is given by
[27, 116]
Vi(r) =1
r6
1
(4πεo)2(−µ2
1α2 − µ22α1). (2.6)
Finally, the motivation for a third vdW term in addition to the orientation and
inductive effects was to explain the presence of non-ideal gas behavior in non-polar
gasses. The dispersion effect, the last of the three to be elucidated, turned out to
be a completely quantum mechanical phenomenon that is more or less correctly
explained by being a collective interaction of instantaneous dipoles that are mutu-
ally induced. Further details on the quantum mechanics of this phenomenon will
be presented in Chapter 6. We will see there that the general relationship and
dominant term for this dispersive interaction energy takes the form
Vd(r) ≈C
r6, (2.7)
where C is a dispersion constant which will be discussed much more later.
Among these three terms in the van der Waals energy, the dispersion term is
the dominate contribution in most cases [116]. For example HCL(g), which has a
permanent dipole moment, has a intermolecular dispersion contribution which is 6
times greater than the orientational term and 20 times greater than the induction
term [116]. It is for this reason that while there are three collective vdW terms,
they are often referred to as vdW or dispersion forces in a synonymous sense.
22
2.4 Mathematical models
We have introduced chemical and physical bonds and now address the methods
of modeling them as they pertain to a set of atoms. For the general purpose
of atomistic models Schrodinger’s wave formulation of non-relativistic quantum
mechanics is a sufficient level of theory to capture almost all physical properties.
In 1928 Paul Dirac said of this equation
“The underlying physical laws necessary for the mathematical the-
ory of a large part of physics and the whole of chemistry are thus
completely known, and the difficulty is only that the exact application
of these laws leads to equations much too complicated to be soluble.”
The high fidelity of Schrodinger’s formulation comes with a computational price,
and deviations from this theory are either necessary for efficiency, or the neces-
sary information can be attained with a reduced (albeit less exact) formulation
depending on the property being calculated. Computational atomistic models can
be broken into three main methods in increasing accuracy and decreasing compu-
tational speed: empirical, semi-empirical and first principles.
2.4.1 Empirical
Empirical models consist of assumed or educated guesses at the functional form
of the energy dependance expressed in terms of relative ionic coordination. These
functions contain free parameters that are fitted to experimental or first principles
data on the representative material. The worst case scenario for these type of
energy functions is that they reproduce the fitted data points only. The quality of
an empirical potential can be judged on its predictive powers and transferability.
For example if a potential is fitted with the equilibrium spacing and cohesive
energy of solid argon, and then goes on to predict other measurable quantities of
argon such as bulk modulus or phonon frequencies, it is said to be predictive. On
23
the other hand, the potential is said to be transferable if it is good at describing
properties in different crystal structures or phases. For example, if a potential
is fitted to the face centered cubic (FCC) structure and the potential accurately
describes the bulk modulus of the diamond structure, the potential has a degree
of transferability.
Empirical potentials can be classified by their arguments. A two-body potential
depends only on the relative orientation of two atomic constituents. Higher order
many-body potentials depend on the relative orientation of more than two atoms at
a time. A well known empirical two-body potential is the Lennard-Jones potential
otherwise known as the 6-12 potential. Two equivalent forms of this potential are
φ(r) = 4ǫ
[(σr
)12
−(σr
)6]
=C12
r12− C6
r6, (2.8)
where ǫ and σ are constants, as are C6 and C12 in the second form. These constants
have different values for different atoms.
Empirical potentials are computationally tractable. The functions can be readily
calculated and, for the same cpu time, a much larger system can be considered than
with higher order formulations of energy calculation. The number one drawback
is the lack of predictive power.
2.4.2 Semi-empirical
Semi-empirical methods are based on the first principles formulation of quantum
mechanics. In some models terms are deemed negligible and ignored, in others
certain aspects of the calculations are parameterized. The tight-binding (TB)
method is a popular and much used semi-empirical method. The details of this
method are covered in Chapter 5. This and other semi-empirical methods have the
advantage of describing quantum mechanical phenomena that cannot traditionally
be attained from empirical potentials, and they tend to be computationally nimble
24
compared to first principles methods.
2.4.3 First principles
First principles, or ab initio calculations represent the pinnacle of electronic struc-
ture calculations. Starting with the fundamental constants and Schrodinger’s
equation as a postulate these methods proceed to describe the nature of atomistic
systems to a degree that is almost irrefutable. Computational resources and meth-
ods have come a long way since the time of Paul Dirac’s quote, and some of these
complicated equations referred to have in fact become readily soluble. But the
shear complexity of the many electron problem governed by Schrodinger’s equa-
tion remains burdensome for large atoms or several 10s of main group elements.
While these methods are rigorous in describing physical effects they are computa-
tionally expensive. The methods applied in solving Schrodinger’s equation break
into two main types: Hartree-Fock (HF) based methods and density functional
theory (DFT) methods. While both make approximations to make calculations
possible, they represent the best available methods for atomistic modeling. As
we will see latter the most crucial approximations in these methods entail the
electron-electron interactions. These interactions are the basis of van der Waals
attraction and the more common of these first principles methods fail to capture
this effect as further discussed in § 3.6.
25
“Let no man ignorant of ge-
ometry enter here”
Above the door to Plato’s
Academy
3Graphite structure and interlayer
properties
Our goal as stated is the development of an accurate model of graphite with
an emphasis on the interlayer properties related to dispersion forces. With this
goal in mind this chapter reviews details of the graphite structure, followed by
experimental measurements focusing on properties that pertain to the dispersion
forces between graphene layers, and insights provided by ab-inito calculations.
26
3.1 Graphene
As mentioned earlier the graphene plane consists of carbon atoms trigonally bonded
to neighbors in a sp2 hybrid network of covalent bonds. The in-plane hybridiza-
tion results in σ bonds. Normal to the sp2 plane are half filled un-hybridized pz
orbitals, that form weak π bonds between neighboring sites in-plane. The con-
tribution of these π bonds to the in-plane binding energy is very small and are
usually referred to as non-bonding [83]. The σ bonds in graphene is one of the
strongest bonds known [148]. Figure 3.1 (a) shows a more detailed structure of
the graphene hexagonal lattice shown in the introduction. The solid light lines
between atoms in this figure represent the chemical bonds. For both graphite and
carbon nanotubes the carbon-carbon bond length is ac−c = 1.42 A [7, 51]. The
dark arrows denote the primitive lattice vectors and together with the other two
heavy lines form the unit cell. The primitive lattice vectors, ~a1 and ~a2, defined in
terms of the cartesian coordinate system are given by
~a1 =a
2
1√
3
0
and ~a2 =a
2
1
−√
3
0
, (3.1)
where a is the magnitude of the primitive lattice vector, a = |~a1| = |~a2| =√
3 ac−c = 2.46 A. Each primitive unit cell contains two atoms labeled 1 and
2 shown in Figure 3.1 (a). The position of any of the atoms of type 1 or 2 can
be referenced with respect to the tip of the primitive lattice vectors by their basis
vectors. The basis vectors for atoms 1 and 2 are, ~B1 = (1/3)~a1 + (2/3)~a2 and
~B2 = (2/3)~a1 + (1/3)~a2, and the positions of an arbitrary atom of type 1 or 2 is
~R1 = n1~a1 + n2~a2 + ~B1, and ~R2 = n1~a1 + n2~a2 + ~B2, where n1, n2 ∈ Z.
The structure of graphene sheets is known through direct measurement via nu-
merous experimental methods. One of the more illustrative measurements is an
atomic force microscope (AFM) scan of a graphite surface, as shown in figure 3.1
27
2
1
ac−c
~a2
~a1
y
x
(a) (b)
Figure 3.1: (a) 2 dimensional graphene layer. Heavy dark line indicates primitiveunit cell composed of two atoms 1 and 2. (b) AFM scan of graphite surface withoverlay of a orientation of a single hexagonal ring, from [67]
(b). The AFM measures force between the AFM tip and the sample. High force
feedback indicates a localization of electrons, and hence the force contours can
approximately be interpreted as electron densities. With this interpretation the
figure shows a strong localization of electrons along bonds and nearly no electrons
in the open areas of the hexagons. This is in direct accordance with the nature of
covalent bonding and the sharing of electrons along the bond direction.
3.2 Graphite stacking patterns
The graphite crystal is characterized by many sheets of graphene stacked upon
one another. There are of course many ways in which these planes can be aligned.
The simplest form of stacking is known as AAA or simple hexagonal, and is shown
schematically in the left hand side of Figure 3.2. In this stacking pattern all carbon
atoms have neighbors directly above and below themselves.
A second stacking possibility is shown in the right hand side of Figure 3.2 and is
known as the ABA stacking pattern. In this structure half of the carbon atoms
in a graphene sheet have neighbors directly above and below themselves, and
28
A
A
A
B
A
A
Zeq
Simple Hexagonal Bernal
Figure 3.2: Graphite stacking patterns. AAA on the left and ABA on the right.Dashed lines denote atoms that are aligned with the z-axis.
the other half are situated such that directly above and below them is the open
center of a hexagon. If one starts with an AAA stacking pattern and translates
every other layer by twice the bond length along one of the carbon-carbon bond
directions the result is an ABA stacking pattern. The ABA stacking pattern is
sometimes referred to as the Bernal form, the namesake of its founder [11]. A
third stacking pattern, which is not illustrated here, is the ABC or rhombohedral
stacking. In ABC stacking half of the carbon atoms are directly below a open area
in the above hexagon and directly above a carbon, and vice versa for the other
half of the carbon atoms. Of course there are graphitic structures in which there
is no apparent stacking pattern, where the various layers are randomly rotated or
offset. These disorder stacking patterns are referred to in literature as turbostatic
[8, 47].
Graphite found in nature is usually graphitic carbon. Graphitic carbon is charac-
terized by small domains of graphite that are lumped together in random orien-
tations. Some samples of large natural graphite crystals have been collected and
tested to find that approximately 90 percent of the graphene layers are stacked
in the ABA pattern, and mixed in with it is the less common rhombohedral or
29
ABC stacking [23]. A very pure form of graphite termed highly orientated py-
rolytic graphite (HOPG) is synthetically manufactured. These graphite crystals
are composed of nearly 100 % ABA stacked planes [83].
For HOPG the interlayer spacing is well set at Zeq = 3.354 A for room temperature
[7, 8, 51], and shrinks slightly at low temperature (3.336A at 4 K) [7]. For other
less refined natural graphite and disordered turbostatic graphite the interlayer
spacing increases slightly to Zeq = 3.43 A [24, 47].
3.2.1 Corrugation of graphene planes
The stacking patterns discussed in the previous section highlight that the ABA
stacking seems to be the most energetically favorable. The AAA stacking on the
other hand has not been experimentally identified and is the least energetically
favorable stacking (as shown by DFT-LDA calculations [24]). The reason for
this can be attributed mainly to the nature of the pz orbitals sticking out of
the graphene planes. In the AAA stacking all of these pz orbitals would be at
a maximum amount of overlap possible. Any other stacking pattern than AAA
would have less pz overlap and would be more energetically favorable. Another
AFM scan of a graphite surface, this time shown from an oblique angle, is shown
in Figure 3.3. The discrete nature of the pz orbitals is evident and will be referred
to as a corrugated surface.
Figure 3.3: AFM scan of graphite showing Pz orbital corrugation, from [67]
30
3.2.2 Graphite unit cell and Brillouin zone
The graphite unit cell is similar to that of graphene but with a third primitive
lattice vector given by
~a3 = 2Zeq
0
0
1
. (3.2)
The ABA graphite unit cell has four atoms. The first two share the same defini-
tion as those of graphene and the other two have basis vectors ~B3 = (1/3)~a1 +
(2/3)~a2 + (1/2)~a3 and ~B4 = ~a1 + (1/2)~a3. The reciprocal lattice vectors of the
direct hexagonal lattice are
~b1 =2π
a
1
1√3
0
~b2 =2π
a
1
−1√3
0
~b3 =π
Zeq
0
0
1
, (3.3)
where the primitive and reciprocal lattice vectors must satisfy ~bi ·~aj = δij [4]. The
first Brillouin zone is given by the Wigner-Seitz unit cell of the reciprocal lattice
and is shown in Figure 3.4 (b) with some high symmetry points labeled.
A graphene sheet by itself is a zero gap semiconductor as can be seen by the
touching of the π bands at K [165]. When the graphene sheets are brought together
the interaction between layers is such that these bands split and graphite becomes
a semi-metal [165].
3.3 Graphene as extended molecules
Due to the dichotomy in the magnitude of intra and interlayer bonding (i.e. very
strong and very weak) it makes sense to talk about the two separately. Linus
Pauling said of graphite, “each of the layers is a giant molecule, and the superim-
31
(a) (b)
Figure 3.4: (a) 2 dimensional graphene layer. Heavy dark line indicates primitiveunit cell composed of two atoms 1 and 2. (b) AFM scan of graphite surface withoverlay of a orientation of a single hexagonal ring, from [67]
posed layer molecules are held together only by weak van der Waals forces” [133].
If we restrict ourselves to looking only at the interlayer properties we can gain
some insight into the dispersion energy between layers. By treating the layers as
extended molecules the interaction energy between them can be plotted versus the
interlayer spacing. This energy curves looks very similar in form to the diatomic
energy curve shown early in Figure 2.1. In these type of plots, which we will see
more of, the energy is referred to as the interlayer energy per atom and is de-
fined as E(Z) = [E(Z)−E(∞)]/Natom, where the tilde denotes interlayer energy.
The location of the minima of this curve corresponds to the equilibrium interlayer
spacing, Zeq = 3.35 A. The well depth in this interlayer energy picture is the exfo-
liation energy (Exf = E(Zeq)) and is the energy to dissociate the graphene sheets.
We refer to this as the exfoliation energy because we reserve the term cohesive
energy to refer to the energy to disassociate all carbon atoms in graphite rather
than just the layers.
32
3.3.1 Experimental exfoliation energy
The exfoliation energy is a difficult quantity to experimentally measure and there
seem to be but three experimental methods to measure the exfoliation energy of
graphite. A short description of the methods is presented in chronological order
in the following sections.
3.3.1.1 Girifalco method
The early and often quoted value by Girifalco was done by a heat of wetting
experiment that was apart of a doctoral thesis. The general idea of this experiment
comes from a relation that the cohesive energy is a function of the surface area
and the heat evolved when wetting (adding water to graphite is exothermic). The
average value over 8 experiments is reported as 260 ergs/cm2. Using the number
density of carbon atoms in a graphene plane (ρ = 4/√
3a2 = 2.46 atoms/A2) the
exfoliation energy per atom via the heat of wetting method is 42.5 meV/atom.
3.3.1.2 Benedict method
The second measurement was an experiment done on collapsed multi-walled nan-
otubes MWCNT. Benedict et al. noticed collapsed MWCNT of large diameter
during TEM measurements. This collapse was noted to occur along the length of
the tube and the caps at the end of the tubes were “bulbed” out. They proposed
that the cohesion seen in the middle was balanced by the strain energy in the
bulbous ends. Using the mean curvature modulus of graphite with an elastic con-
tinuum theory for the strain energy description balanced with a LJ potential they
calculated a value for the exfoliation energy of graphite of 35 ± 15 meV/atom.
3.3.1.3 Zacharia method
The third experimental result was a recent set of experiments of thermal desorption
(TD) rates of polyaromatic hydrocarbons (PAH) physisorbed on HOPG surfaces
33
under ultra high vacuum [175]. The chemical structure of PAH’s is very similar to
that graphite, sharing similar orbital hybridization and bond lengths. Large PAH’s
are simply small flecks of graphite with hydrogen bonds on the edges. This group
utilized an Arrhenius rate equation to study the activation energies were found
for successively larger PAH’s based on their experimental data. In their analysis
the effects of the hydrogen on the PAH’s was subtracted and a generalization of
52 meV/atom was found to be the binding exfoliation energy.
3.3.1.4 Exfoliation energy summary
Table 3.1 summarizes these three experimental values. Within the bounds of their
reported errors the exfoliation energy ranges between 20 and 57 meV/atom. The
lower bound defined by the Benedict model is suspect based on their use of a
continuum model for the strain energy. It seems that the other two methods
might hold a little more weight, placing the exfoliation energy around the mid to
high 40’s of meV/atom.
Exf Method Reference (et al.)[meV/atom]
43 ± 5 Heat of wetting Girifalco [54]35 ± 15 Tube collapse Benedict [9]52 ± 5 Thermal desorption Zacharia [175]
Table 3.1: Summary of experimental graphite exfoliation energies in meV/atomlisted in chronological order
3.4 Z-axis compressibility
The z-axis compressibility (kz) is related to the c33 elastic constant via [161]
c33 = k−1z . (3.4)
This property is described by the curvature of interlayer energy curve around the
34
equilibrium separation. The formal definition of this property and its calculation
are shown later in § 8.3.3. Experimental results on HOPG via ultrasonic testing
have found the compressibility to be 2.74(±0.0075)×10−12cm2dyne−1 at standard
temperature and pressure [16, 128]. The compressibility is a strong function of
pressure and temperature [50]. For example at a pressure of 20 kbar and standard
temperature it decreases to 1.85 × 10−12 cm2 dyne−1. At standard pressure and
low temperature (4 K) it decreases to 2.46 × 10−12 cm2 dyne−1.
3.5 Vibrational properties
Two vibrational modes in graphite highlight the dichotomy of intra and interlayer
binding and the latter gives us another more insight into the nature of interlayer
properties. A representation of the atomic motion for two active Raman modes
in question are shown in Figure 3.5. In plane the strong covalent bonds lead to
a high frequency E2g(2) mode that has a frequency of 46.3 THz, shown in Figure
3.5 (A) [162].
(B)(A)
Figure 3.5: Raman active modes: E2g(1), and E2g(2). The relative motion shownhere is adopted from [127]
The more interesting Raman active mode to us, is the rigid interlayer shearing
mode E2g(1). This mode is dominated by the weaker dispersion interactions and
35
overlap of Pz orbitals. Figure 3.5 (B) illustrates this motion. This mode is obvi-
ously of a much lower frequency than the latter due to the weaker physical bonding
involved and has been repeatedly measured at 42 cm−1 = 1.26 THz 1[2, 62, 119].
The E2g(1) shear mode is governed by the curvature of the energy landscape de-
scribed by the relative translation of every other graphene sheet relative to the
others.
3.6 Ab-initio studies of graphite
The calculation of graphite properties via a first principles treatment is something
to consider in addition to the experimental results given above. In § 2.4.3 two first
principles methods were mentioned Hartree-Fock self consistent field (HF-SCF)
methods and density functional theory. Both of these methods have been applied
to the graphite structure. The details of these methods will not be completely
explained here, rather a quick overview of main ideas and important points relating
to dispersion energy will be made
3.6.1 Hartree-Fock treatment of graphite
The defining idea of Hartree-Fock methods lies treating the electron-electron in-
teraction in a mean field manner. A particular electron sees the average field of
the other electrons but is unaware of their instantaneous positions. In real systems
electrons are instantaneously correlated with each other, leading to a lowering of
total energy. The difference between the HF-SCF energy and the “true” energy
Eo, is known as the correlation energy.
Ecor = EHF −Eo. (3.5)
Hartree-Fock methods by definition completely ignore the electronic correlation
1converting wavenumbers to frequency: vc = f , where c is the speed of light. 42 cm−1 ·3 × 1010 cm s−1 = 1.26 × 1012 Hz
36
term and thus do not model dispersion forces [5]. Corrections to HF theory to
account for electron correlations include Moller-Plesset many-body perturbation
theory (MP2/MP4), coupled cluster (CC), and configuration interaction methods
(CI) [64]. These HF corrections are very expensive and scale unfavorably. The
application of corrected HF-SCF theory to graphite is restricted due to issues of
imposing periodic boundary conditions in HF methods [?]. The closest simulation
to graphite using corrected HF methods is the calculation of the binding energy
of large PAH molecules due to vdW interactions [126].
3.6.2 Density Functional Theory treatment of graphite
Density functional theory is a reformulation of the electronic Schrodinger equation
in terms of the electronic density, ρ(~r), instead of the wave function. The theoret-
ical foundations of rigorously defined DFT rest on Hohenberg and Kohn’s (HK)
1964 paper [71], which are based on the earlier work of the Thomas-Fermi model
[90]. In this landmark paper two important theorems were stated. The first is
the proof of the existence of a functional mapping between the many-body wave
function and the electron density. Put in a differen’t way, the electron density
and the position of the nuclei completely determine the ground state energy. This
paper only proved the existence, but not the form of the energy functional, known
as the exchange correlation (XC) function. The second theorem in this paper is
the statement that the electronic distribution which minimizes the total energy is
the actual electronic distribution. This theorem leads naturally to a variational
treatment of the total energy with an extend basis set to describe the systems
at hand. A year later Kohn and Sham made some slight modifications to this
work in regard to the approximate functional dependance of the kinetic energy
with respect to a HF-SCF treatment [92]. This method is the basis for most
DFT calculations today and proceeds in a similar manner to HF-SCF methods.
This method has the benefit of correctly modeling electron correlation effects (and
hence vdW forces) if the functional form of the exchange correlation function is
37
known.
Unfortunately the exact universal form of the XC function is not known. Much
research on its form, has been done since the mid-sixties, and many forms have
been proposed. Some forms are extremely accurate for certain states of condensed
matter including the local density approximation (LDA), and varieties of the gra-
dient corrected methods (GGA). Both of these XC functions, and others used
fail to properly model long range interaction in rare gasses [135]. The reason for
this failure is in the fact that most XC functions are dependant on only the lo-
cal density of electrons and are not a function of electron densities at appreciable
distances . Graphite is a system that has a electron distribution that is sparse com-
pared to most condensed matter but much less so than gasses. A DFT treatment
of graphite shows some strengths but also some failings. Table 3.2 summarizes
results from a number of DFT calculations of graphite.
a Zeq kz Exf Eco
Method Ref. (A) (A) (×10−12 cm2
dyne) (meV/atom) (eV/atom)
LDA [31] . . . 2.8 0.97 110 . . .LDA [24] 2.45 3.3 . . . 20 . . .LDA [151] 2.451 3.36 4.11 25 8.80LDA [161] 2.453 3.44 7.69 30 8.60GGA [63] 2.461 4.5 0.01 3 7.87LDA-nl [146] 2.47 3.76 7.69 24 . . .Exp (4 K) 2.46a 3.336a 2.46b . . . . . .Exp (300 K) 2.46a 3.354a 2.74c Table 3.1 . . .
Table 3.2: Summary of DFT calculations of graphite including the lattice con-stant, equilibrium spacing, z-axis compressibility, exfoliation energy and in-planecohesive energy. aRef. [7], bRef. [50], cRef. [13]
All methods seem to treat the in plane properties quite well. For example the lat-
tice constant a is in very good accordance to experiment and the various in-plane
cohesive energies (Eco) agree quite well even between XC functions but unfortu-
nately doesn’t have an experimental measurement to compare to. These methods
seem to have trouble with interlayer properties. The GGA XC shows almost zero
38
interlayer binding energy and does not account for even some of the dispersion
energy between layers. Its lack of interlayer binding has been noted in literature
[52]. This failing of the GGA XC function probably coincides with the paucity
of published information on its application to graphite. The LDA methods show
some dispersive binding energy that is in the neighborhood of experimental values.
This dispersive energy rapidly goes to zero beyond Zeq and is generally exponen-
tial rather than 1/r6 [151]. The LDA methods also generally show the equilibrium
separation around the experimental value, though there are some outliers. All
examples shown do not accurately calculate the z-axis compressibility.
More recently there have been attempts to rectify the failing of XC functions for
long range interactions [91, 147]. An application of the method in [147] has been
applied to graphite and is shown in the table listed under LDA+nl, where the
nl stands for the “non-local” correction to the LDA [146]. While this represents
a more physical treatment by including non-local electron density information in
the XC function it still preforms poorly in the description of interlayer properties.
One interesting point that may be fortuitus, is the energy difference between
between AAA and ABA stackings at Zeq (referred to as ∆EABA−AAA) as calculated
by different DFT methods. Charlier reported 17 meV/atom under LDA [24], and
Kolmogorov reported 15 meV/atom for both the LDA and GGA approximations
[96]. This point will be referenced a few times in later sections.
3.7 Summary
In this chapter we have seen the some aspects of interlayer graphite properties that
are known to a high degree from experiments including the equilibrium separation,
z-axis compressibility, and the E2g(1) mode. The exfoliation energy measured
in three experiments gives a clustering around 45 meV/atom but a much wider
margin experimental disagreement than the three previous properties. We have
also seen the general disagreement of first principles calculations with respect to
39
calculating the interlayer properties. The information presented in this chapter
will be utilized later in the fitting of our dispersion correction, but first we use it
in the analysis of existing graphite models.
40
“What should they know of
the present who only the
present know?”
Blair Worden
4Existing models for dispersion
interactions in graphite
This chapter reviews previous work in the modeling of graphite interlayer proper-
ties and ends with the motivation for what will be done to address shortcomings
of the various models.
4.1 Empirical
The empirical models in this section only address the calculation of interlayer
properties. There are also intralayer empirical models for carbon ([17]) that are
sometimes used in conjunction with these for a fuller description of graphite.
41
The most quoted empirical model is the Lennard-Jones model proposed by Giri-
falco et al. [54, 53]. Recall that there are two constants in the LJ potential, given
in equation (2.8). In Girifalco’s model the attractive C6 constant was calculated
with the Kirkwood method (explained in § 6.2.6.2) from experimental data given
in [6], and is reported as 15.2 eV A6. The repulsive constant, C12, was calculated
by requiring that the lattice constant be equal to Zeq = 3.35 A, and is reported
as 24.1 × 103 eV A12
.
Another empirical model is that of Ulbricht et al. [163]. It is similar in nature
to Girifalco’s model, but fitted with experimental data from thermal desorption
experiments described earlier in § 3.3.1.3. The values for their LJ constants are
C6 = 15.4 eV A6
and C12 = 22.5 × 103 eV A12
.
We have implemented these two empirical models to study the interlayer properties
they produce as a function of the interlayer separation, and are shown in Figure
4.1, for both the ABA and AAA stacking. The details of this implementation
(unit cell, periodic boundary conditions etc.) will be discussed later.
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7−50
−48
−46
−44
−42
−40
−38
−36
−34
−32
−30
Interlayer separation [A°]
Inte
rlaye
r en
ergy
[meV
/ato
m]
giri − AAAgiri − ABAulbr − AAAulbr − ABA
Figure 4.1: Interlayer energy scans for the empirical models of Girifalco (giri) andUlbricht (ulbr) for ABA and AAA stacking of graphite around Zeq
42
Table 4.1 summarizes the results from this evaluation, which includes the energy
difference in the stackings EABA−AAA, the exfoliation energy(Exf ), the E2g(1)
phonon frequency, and the z-axis compressibility (kz) of graphite in the ABA
stacking (details of calculating the latter two quantities are addressed in § 8.3.2
and § 8.3.3 respectively)
Model Exf kz E2g(1) EABA−AAA
[meV/atom] [cm2dyne−1] 10−12 [THz] [meV/atom]
Girifalco 43.41 3.18 0.36 0.9Ulbricht 46.6 3.17 0.34 1.1Experiment 30-50 2.74 1.26 17-23
Table 4.1: Results of LJ empirical models with experimental data comparison
Both of these models reproduce the correct interlayer equilibrium distance (they
were fitted to do so), and predict exfoliation energies (Exf) that are in the range
of experimental values summarized in Table 3.1. For both models there is about a
≈ 16 % error with respect to the experimental z-axis compressibility. The percent
error in the calculated E2g(1) mode frequency is much larger, ≈ 72% for both
models. The large discrepancy in this mode comes from modeling the repulsive
interaction isotropically. As was discussed in § 3.2.1 the Pz orbitals that stick out
of the graphene plane play a major role in the repulsive overlap between planes
and are very anisotropic in nature, as was illustrated in the AFM scan in Figure
3.3. This deficiency in modeling corrugation can also be seen in the relatively
small energy difference between the AAA and ABA stacking as compared to the
approximate DFT-LDA numbers. If this energy difference between stacking tends
to zero the E2g(1) mode will become non-existent in the model.
There is a third empirical model which tackles this shortcoming. The “registry
dependant” empirical model of Kolmogorov [96], rectifies this problem by intro-
1There is a slight discrepancy (≈ 2.1% difference) between the exfoliation energy quoted in[54] (42.5 meV/atom) and what I calculated here. They utilized an analytical lattice summationtechnique whereas I am using a small unit cell with periodic boundary conditions
43
ducing an anisotropic repulsion term that mimics the nature of the Pz orbital.
The attractive dispersive contribution is the same as previous models (C6/r6).
The model contains 7 free parameters and is fitted to Zeq, Exf = 48 meV/atom,
kz, and the energy difference between ABA/AAA stacking from a DFT-LDA anal-
ysis. The value of C6 in the model is given as 10.24 eV A6. As to whether this
coefficient was fitted along with the 6 other parameters in the repulsive term or
calculated separately is not made clear. The E2g(1) mode was not in the database
for fitting and its value as calculated by the fitted model is not reported. It
should be in the neighborhood of the experimental value because the model is
fitted to reproduce the energy difference between ABA and AAA stacking. We
have discussed the limited knowledge of the exfoliation energy, and the unknowns
associated with the ab-initio based interalyer energy calculations (both of which
this model used in fitting), so this model has to be viewed as a step in the right
direction in modeling corrugation, but may be limited in its predictive powers.
4.2 Semi-empirical and first principles models
The next level of sophistication in modeling graphite interlayer interactions is a
quantum mechanical based, semi-empirical or first principles model. As discussed
in § 3.6 these methods generally fail to capture dispersion energetics. The method
traditionally employed to correct this shortcoming is a dispersion addition to the
energy, with the total energy defined as [68]
Etot = Efp/se + Edispersion, (4.1)
where Efp/se is the energy computed from either a first principles (fp) or semi-
empirical (se) method. In these models it is assumed that the short range behavior
in the region of atomic overlap is modeled accurately, whereas the long range
dispersion is either completely missing (as in HF and TB methods) or only partially
accounted for (DFT). The form of Edispersion is generally a pairwise C6/r6 term
44
that is effectively switched off at close range by a damping function ( damping
functions will be addressed in Chapter 7).
Adding a dispersion energy correction on top of a quantum based method has
been widely used. Many Hartree-Fock plus dispersion models for rare gasses and
aromatic hydrocarbons have been reported. Some early papers from the mid 70’s
to early 80’s include [1, 68, 157], and continue to the present day [79]. There is a
similar counterpart and precedence for this method in DFT plus dispersion models,
for rare gasses [168], and in aromatic hydrocarbons [117, 130, 180]. A study of a
DFT plus dispersion model for graphite has been constructed by Hasegawa and
Nishidate [63]. This model has the benefit of an accurate first principles treatment
of the intralayer bonding and with the added damped dispersion term models
graphite to a degree which is satisfactory. This model has obvious limitations in
the systems size due to the expensive DFT procedure, and there is the possibility
of double counting some of the dispersion energy due to the DFT’s ability to pick
some of it up.
On the semi-empirical side there are a few articles in literature that describe
tight-binding plus dispersion models. Biro et al. describe an intermolecular Huckel
model with a Lennard-Jones energy addition to model dispersion [156]. This model
has two tight-binding parameterizations, one for intra-molecular interactions and
one for inter-molecular interactions. This model has been applied to graphite
and CNTs. This model is different than the other tight-binding plus dispersion
models that follow in that the dispersion term is not damped and the repulsive
energy term is effectively double counted by including the repulsive C12/r12 term
in addition to the repulsion present in the tight-binding formalism.
Elstner et al. added a damped dispersion correction to a non-orthogonal tight
binding model [38]. This model was created to study the stacking of nucleic
acid base pairs, and necessarily included H,N,C,O parametrization, of both the
tight-binding parameters and the dispersion interactions between each. Another
45
example of a tight binding plus dispersion model is that of Palser [131]. In a similar
fashion to Elstner, Palser added a damped dispersion energy to a non-orthogonal
tight-binding model. This model was focused specifically on modeling graphitic
structures. Palser’s model has three free parameters that were fit to reproduce
Zeq, Girifalco’s experimental interlayer energy, and the DFT based ∆EABA/AAA
stacking difference of 17 meV/atom from [24]. The dispersion coefficient in this
fitting came to 11 meV A6. The z-axis compressibility that was not fitted to, is 9.4
% different than the experimental value. Palser did not report on a calculation of
E2g(1). Kwon and Tomanek have a similar model fitted to the same ABA graphite
properties as Palser [100].
4.3 Motivation for what is to be done
These tight-binding plus dispersion methods unlike the empirical models discussed
implicitly model the corrugation of the Pz orbitals overlapping at close distances.
More importantly from a standpoint of modeling nanomechanical systems is the
quantum mechanical treatment of the inplane bonding that is not addressed with
the empirical models. This fact allows one to apply molecular dynamics to study
the evolution of interacting graphitic structures under the influence of repulsive
overlap forces and the attractive dispersion forces, all the while having access to
the electronic structure which is important in modeling NEMs. The computational
efficiency of tight-binding plus dispersion as compared to the DFT plus dispersion
of Hasegawa, allows a much greater freedom in the simulations of larger systems
over longer time scales. For these reasons the tight-binding plus dispersion method
is deemed a valid avenue for modeling the subjects proposed in the introduction,
and will be developed here.
The following three chapters lay the theoretical ground work for a physically mo-
tivated and accurate model of graphite based on this tight-binding plus dispersion
model. The dispersion force and covalent bonding that have been mentioned so
46
frequently in these introductory chapters will be considered in a more rigorous
manner. In chapter 5 the tight-binding methodology and parametrization will be
fully presented. In chapter 6, the definition of C6, and the methods for calculating
it will be established. The motivation for this chapter is to provide confidence in
correctly modeling the long range interactions and provide a sanity check when
final results are presented. The idea of damping functions will be addressed in
Chapter 7, and in Chapter 8 all the discussed aspects of the model are brought
together and the parameter optimization is explained.
47
5Modeling electronic and repulsive
interaction with a tight binding formalism
As discussed in § 2.4.2 tight-binding is a semi-empirical method based on quantum
mechanics. This chapter describes the foundation of tight-binding, its parametriza-
tion and use in total energy calculations of systems of atoms. The reader may find
it useful to refer to Appendix B if he/she becomes lost. Before we get started it
should be noted that we are treating are quantum mechanical systems in the
one-electron picture considering no relativistic spin effects, or self-consistency.
Also as in most quantum mechanical calculations we are employing the the Born-
Oppenhemier approximation by treating only the electrons quantum mechanically.
48
5.1 Linear combination of atomic orbitals
In quantum chemistry the electronic states of a collection of interacting atoms that
make up a molecule are often described in terms of a collective wavefunction. This
collective wavefunction is described by a Linear Combination of Atomic Orbitals
(LCAO). The molecular wavefunction Ψ written in this LCAO form is
Ψλ =
n∑
i
cλi φi. (5.1)
The λ refers to the particular state (λ = 1 is the ground state), c is the expansion
coefficient to be determined, and φ is the atomic orbitals for the n orbitals. These
atomic orbitals are known as the basis functions and usually are not the exact
atomic orbitals. Rather they usually only have atomic like character known from
the solution of the hydrogen atom e.g. Slater type orbitals (STO’s).
5.2 Schrodinger equation in LCAO approximation
The governing equation for the wavefunction in a molecule or solid is given by the
Schrodinger equation
HΨλ = ελΨλ. (5.2)
H is the quantum mechanical Hamiltonian for the molecule, and ελ is the energy
of the state λ. Within a LCAO framework the solving of this equation is equiv-
alent to finding the set of expansion coefficients (ci) for each state λ defined in
equation (5.1) that minimizes the energy that particular state. To do this the
variational method (Rayleigh-Ritz method) is applied in the minimization of the
energy functional given by
49
ελ =〈Ψλ|H|Ψλ〉〈Ψλ|Ψλ〉 . (5.3)
Bracket notation as discussed in section B.2 is employed in this definition. Insert-
ing the LCAO expansion for Ψλ into equation (5.3) and applying the variational
principle with
∂ε
∂ci= 0, (5.4)
leads to n equations in the form of the generalized eigenvalue problem given as
Hc = εSc. (5.5)
H is the Hamiltonian matrix with elements given by Hij = 〈φi|H|φj〉 and S is the
overlap matrix with elements defined by Sij = 〈φi|φj〉, and c is a vector containing
the expansion coefficients. If the basis is chosen such that all the basis functions
|φi〉 are orthonormal, the overlap matrix will be equal to the identity matrix and
equation (5.5) will reduce to a regular eigenvalue problem
Hc = ε c. (5.6)
A detailed derivation of the application of the variational principle to a LCAO
basis leading to the generalized eigenvalue problem is presented in Appendix B.9
for the interested reader.
5.3 The basis and atomic centered orbitals
When solving the Schrodinger equation via the variational method with a LCAO
the size of the basis must obviously be finite due to limited computational re-
50
sources. One ideally wants to include as few atomic orbitals as necessary to in-
crease computational speed, but enough to capture the essence of the system. The
atomic orbitals included define the basis of the system, for covalent solids typically
the S, Px, Py, and Pz are sufficient to describe the ground state properties.
When solving a problem with atoms dispersed in space, each basis is centered at
the nucleus. The value of the basis wave function at an arbitrary point ~r in space
depends on the vector distance between this point and the nucleus at which it is
based. The vector connecting these two points is simply ~r − ~R, where ~R is the
position vector of the nucleus taken from the same global origin as ~r. The notation
used is
φα(~r − ~R). (5.7)
Where α labels the orbital type, α ∈ (S, Px, Py, Pz) in our 4 orbital basis. We can
rewrite our one-electron LCAO wave function now in the form
Ψλ(~r) =∑
i,α
cλiα φα(~r − ~Ri). (5.8)
λ is an index describing the eigenstate, the sum on i goes over the n atoms in the
system, and the sum on α over the basis.
We can also rewrite the matrix elements for the Hamiltonian and overlap matrices
that make our generalized eigenvalue problem from equation (5.5). Instead of Hij
we now adopt the notation Hiαjβ. Meaning the matrix element is an integral
between orbitals of type α and β on atoms i and j. Using our new notation these
are
Hiαjβ = 〈φiα|H|φjβ〉 =
∫φ∗
iα(~r − ~Ri) H φjβ(~r − ~Rj) dτ, (5.9)
51
Siαjβ = 〈φiα|φjβ〉 =
∫φ∗
iα(~r − ~Ri) φjβ(~r − ~Rj) dτ. (5.10)
5.4 Periodic solids
When modeling an infinite periodic solid such as a crystal it is not necessary to
explicitly model all atoms, rather one makes use of Bloch’s theorem. The premise
is that in a periodic solid the value of the wavefunction evaluated at an arbitrary
translation of a lattice vector must follow [4]:
Ψλ(~r + ~R) = eik~RΨλ(~r). (5.11)
Here the lattice translation vector is ~R = w1~a1 +w2~a2 +w3~a3 (the w’s are simply
integers), and k is the wavevector in the first Brillouin zone.
A proposed wavefunction for a periodic solid must satisfy the relation in (5.11).
One form that satisfies this is known as the Bloch sum and is given by [154]
ϕk
iα(~r − ~Ri) = N−1
2
∑
~Ri
eik~Riφiα(~r − ~Ri), (5.12)
where ~Ri represents the set of all translation vectors for atom i. i.e. all integer
permutations for the w’s given above. ϕk
iα will be now referred to as a crystal-
like wavefunction. Using this as our basis for periodic solids we can write our
Hamiltonian and overlap matrices in terms of wavevector k as:
Hk
iαjβ = 〈ϕk
iα(~r − ~Ri)| H |ϕk
jβ(~r − ~Rj)〉 , (5.13)
Hk
iαjβ = N−1∑
~Ri,~Rj
eik(~Rj−~Ri)
∫φ∗
iα(r − Ri)Hφjβ(r −Rj)dτ. (5.14)
52
One of the sums cancels with N−1 and the matrix becomes
Hk
iαjβ =∑
~Rj
eik~RjiHiαjβ. (5.15)
An analogous progression of the last three equations is applied to the overlap
matrix and it takes a similar form:
Sk
iαjβ =∑
~Rj
eik~RjiSiαjβ. (5.16)
At this point we defined the form of the integrals for the Hamiltonian and overlap
matrix elements given in real space by equations (5.9), and (5.10), and in k-space
by equations (5.15) and (5.16). Now we just need to evaluate them. These integrals
turn out to be very difficult to evaluate, but by applying some approximations to
the full form of the matrix elements the problem can be reduced to a manageable
form. The next few section discuss the main approximations of tight-binding and
introduce the parameters that make this method semi-empirical.
5.5 The Hamiltonian and the two-center approximation
The first approximation is known as the two-center approximation and has to do
with the evaluation of the integrals in the Hamiltonian and overlap matrices. The
Hamiltonian operator in these matrix elements is the one-electron Hamiltonian
discussed in Appendix B and given in equation (B.21). The potential energy
includes a contribution from each ion core in the system:
V (r) =N∑
i
Vi(r − Ri). (5.17)
The potential Vi is located on atom i. These potentials are assumed to be spheri-
cally symmetric. Plugging this potential into the Hamiltonian and then into the
53
real space Hamiltonian matrix element given in equation (5.9) we have
Hiαjβ =
∫φ∗
iα(r − Ri)
[1
2m∇2 + V1(r − R1) + . . . + VN(r −RN )
]φjβ(r − Rj)dτ.
(5.18)
In order to better understand the individual terms in this expression we can write
the above as
Hiαjβ = 〈φiα| T |φjβ〉 + 〈φiα| Vi |φjβ〉 + 〈φiα| Vj |φjβ〉 +∑
k 6=i,j
〈φiα| Vk |φjβ〉 . (5.19)
The first term is the kinetic energy and the rest are three classes of potential
energy terms. From this form we can classify the contribution to an individual
matrix element, Hiαjβ by making the following definitions
1. When all three locations, i,j and k are on the same atom we have an on-site
integral
2. When the potential k is on the same site as one of the wave functions i,
and the other wave function is at another site (i 6= j) we have a two-center
integral
3. When all three are on different cites (i 6= j 6= N) we have three-center
integrals
The three-center integrals are small compared to the two-center integrals. The
negation of three-center integrals (i.e. the “two center approximation”) greatly
simplifies the evaluation of these terms. Later when we discuss the parametrization
it will be seen how some of the error incurred in ignoring three-center integrals
are absorbed in the the fitting of the parameters.
54
The reduced Hamiltonian matrix elements within the two-center approximation
are
Hiαjβ = 〈φiα| H2c |φjβ〉 , (5.20)
where H2c is the two center Hamiltonian operator, described by
H2c =1
2m∇2 + Vi(r −Ri) + Vj(r − Rj). (5.21)
Within the two-center approximation the matrix elements essentially become a di-
rectionally dependant pair potential that are a function of internuclear separation,
|Ri − Rj |, and the angular momenta of the orbitals α and β.
5.6 Slater-Koster parametrization
The tight binding or “simplified” linear combination of atomic orbitals method was
first described by Slater and Koster in 1954 [154]. They describe their method as a
manner of interpolation of finer resolved first principle calculations. This original
paper was concerned with calculating energy band properties of bulk crystalline
materials. The basis of the method is to decompose the integrals found in the
Hamiltonian and overlap matrices by means of orbital symmetry and replace the
direct evaluation of these simplified integrals by a parametrization based on the
two-center approximation.
5.6.1 Orbital decomposition
The orientation between arbitrarily aligned s-p, or p-p orbitals can be decom-
posed into σ and π bonds, as is discussed in the next section. Our basis orbitals
have atomic like character and this character is described by spherical harmonics
known form the full analytical solution of the hydrogen atom. The integrals over
55
space of the products of these spherical harmonic functions (i.e. elements in the
Hamiltonian and overlap matrices) are always zero due to symmetry for certain
angular momenta and orientation. Just considering our sp basis, the orientations
that have none-zero space integrals are only 4 and are a shown in Figure 5.1.
Figure 5.1: Relative orientation of s and p orbitals that have a non-zero 3 spaceintegration between them. Shaded region is “+” and white region is “-”
5.6.2 s-p decomposition
Consider the example of the Hamiltonian matrix element between an s orbital on
one atom and a p orbital on another atom (i 6= j). The matrix element to be
evaluated is
〈ψis|H2c|ψjp〉 = 〈S|H2c|Pα〉 . (5.22)
The right hand side introduces a simplified notation that will be used in this
section. It simply means an s type orbital on atom 1 and a p type orbital with
angular momenta α on atom 2.
In order to simplify the evaluation of this matrix element we start by decomposing
the p orbital into components that are parallel (σ) and perpendicular (π) to the
internuclear unit vector ~d that is shown in figure 5.2.
56
Figure 5.2: S and Pz orbital decomposition. Vector ~r connects the two atoms, ~dis a unit vector of ~r.
To facilitate this calculation we define the vector ~a, which is a unit vector along
the cartesian axes corresponding to the type of P orbital considered. For a Pz
orbital ~a is simply 0i + 0j + 1k. The vector ~n, is a unit vector normal to ~d, and
in the plane described by ~d and ~a. Both of these vectors are shown in figure 5.2.
It is now easy to describe the arbitrary Pα orbital as a sum of σ and π parts as
|Pα〉 = ~a · ~d |Pσ〉 + ~a · ~n |Pπ〉 . (5.23)
Now we can substitute this into equation (5.22) as,
〈S|H2c|Pα〉 = 〈S| H2c |[~a · ~d |Pσ〉 + ~a · ~n |Pπ〉
]〉 , (5.24)
which expands to
〈S|H2c|Pα〉 = (~a · ~d) 〈S|H2c|Pσ〉 + (~a · ~n) 〈S|H2c|Pπ〉 . (5.25)
We gave the arguments in the previous section that the second term on the right
hand side of equation (5.25) is zero. With this consideration equation (5.25)
reduces to
57
〈S|H2c|Pα〉 = (~a · ~d) 〈S|H2c|Pσ〉 . (5.26)
Since ~a is defined as a unit vector along one of the cartesian coordinates, (~a · ~d) is
simply the directional cosine in the direction of α. Where the directional cosines
are defined by
dx =~r · x|~r| , dy =
~r · y|~r| , dz =
~r · z|~r| , (5.27)
with (x, y, z) being the global axis unit vectors. Using these definitions of direc-
tional cosines equation (5.26) can be rewritten as
〈S|H2c|Pα〉 = dα 〈S|H2c|Pσ〉 = dα tspσ(r). (5.28)
Here we have introduced the shorthand notation of:
tspσ(r) = 〈S|H2c|Pσ〉 . (5.29)
This tspσ(r) is the parameterized SK function that replaces the integral between
an S and P orbital in a σ configuration as a distance dependant function. In §
5.8 we will talk about the form and fitting of these functions. The methodology
of decomposing this general SPα integral into σ and π components with the help
of directional cosines is applied to the other integrals of interest in the sp basis.
The resulting integrals can all be expressed via four SK functions (t(r)’s) and the
directional cosines. The required t(r)’s for our basis are defined as
The reduced form of the other integrals (for ss and pp orbitals) are shown in
equation (5.31) in the following section.
58
tssσ(r) = 〈S|H2c|S〉tspσ(r) = 〈S|H2c|Pσ〉tppσ(r) = 〈Pσ|H2c|Pσ〉tppπ(r) = 〈Pπ|H2c|Pπ〉
5.6.3 Construction of tight binding matrices
For each atom pair i, j all matrix element terms between the orbitals on i and j are
calculated using the SK parameterized functions. Our S, Px, Py, Pz basis leads to
4× 4 sub-matrices for each atom pair, labeled Hsubij . The matrix below shows the
what integral each element of the sub-matrix represents ( i.e. sisj = 〈si| H2c |sj〉 )
Hsubij =
sisj sipjx sipjy sipjz
pixsj pixpjx pixpjy pixpjz
piysj piypjx piypjy piypjz
pizsj pizpjx pizpjy pizpjz
(5.30)
There are two types of these sub-matrices, on-site and hopping sub-matrices. The
latter occurs when the integral is defined between orbital on different sites ( i.e.
i 6= j). This 4 × 4 hopping sub-matrix is defined as
Hsubij =
tssσ dxtspσ dytspσ dztspσ
−dxtspσ d2xtppσ + (1 − d2
x)tppπ dxdy(tppσ − tppπ) dxdz(tppσ − tppπ)
−dytspσ dydx(tppσ − tppπ) d2ytppσ + (1 − d2
y)tppπ dydz(tppσ − tppπ)
−dztspσ dzdx(tppσ − tppπ) dzdy(tppσ − tppπ) d2ztppσ + (1 − d2
z)tppπ
(5.31)
For on-site integrals, when i = j, the contribution to the sub-matrix is simply the
energies of the individual s and p orbitals along the diagonal.
59
Hsubij =
Es 0 0 0
0 Ep 0 0
0 0 Ep 0
0 0 0 Ep
(5.32)
These sub-matrices are then inserted into their respective positions of the global
Hamiltonain matrix Hiα,jβ. To illustrate this assume the 4 × 4 sub-matrices are
indexed by x, and y. Their position in the global matrix is defined as
Hiα,jβ
(4(i− 1) + x, 4(j − 1) + y
).
Until this point all discussion of the tight-binding matrices have been in terms
of the Hamiltonian matrix. The procedure for the overlap matrix S follows the
same route. The integrals between orbitals discussed in the previous section are
decomposed the same and the the results presented in the construction of the sub-
matrices is the same except the overlap has different SK functions defined by t(r)’s.
The on-site sub-matrix is the identity matrix instead of the orbital energies. The
global overlap matrix is constructed from its sub-matrices just as was described
above.
For periodic solids each element is multiplied by the summation pre-factor∑~Rj
eik~Rji
given in equations (5.15) and (5.16).
5.7 Tight Binding Total Energy
Up to now we have seen the tight binding formalism and now address calculating
the total energy of a system of atoms. This energy contains two parts: band
structure energy (EBS), and a short range repulsive interaction (Erep) [137]. This
division of the energy is understood in light of earlier approximations. Under
the Born-Oppenheimer approximation all energies calculated via Schrodinger’s
60
equation are electronic energies (EBS). To account for the total energy the core-
core ionic repulsion of the nuclei must be accounted for. The total tight-binding
energy is expressed as
ETB = EBS + Erep. (5.33)
The repulsive contribution is between all atomic cores and is most often taken as
a two body potential Θ(rij) summed over all interactions
Erep =1
2
∑
i,j
Θ(rij). (5.34)
rij is the distance between atoms i and j. The form of the potential is not explicitly
known but is selected and fitted to reproduce points in the data base.
The second contribution to the total energy is the band structure energy. The
Hamiltonian and overlap matrices matrices are constructed for a configuration of
atoms and for a given wave vector k. These matrices are used in the generalized
eigenvalue problem given in equation (5.5) restated here in a form showing the
wave vector dependance
Hkc(k) = ε(k) Skc(k).
The solution to this generalized eigenvalue problem results in a spectrum of eigen-
values ε(k) corresponding to the energy of the different states, and the eigenvectors
c(k) constitute the expansion coefficients. The band structure energy is defined
as
EBS =∑
λ∈occ
∫
FBZ
fλ ελ(k) d3k. (5.35)
61
The sum is over the occupied states, and the occupation number of a state λ
is given by fλ ∈ {0, 1, 2}. The integral is over the first Brillouin zone (FBZ).
Computationally this integration is approximated by a weighted sum over discrete
k-points in the Brillouin zone expressed by
EBS =∑
λ∈occ
Nk∑
i=1
wi fλ ελ(ki). (5.36)
The wi is the weight assigned to the specific k-point, and must satisfy the condition
Nk∑
i=1
wi = 1. (5.37)
It is possible to uniformly sample the Brillouin zone at discrete points and assign
a uniform weight of w = 1/Nk in evaluating 5.36.
5.8 Fitting TB parameters
We have seen the ground work of tight binding this far but now must consider how
the various parameters are fit. Slater and Koster originally used first-principles cal-
culations of energy eigenvalues at k-points corresponding to high symmetry points.
By using a least squares method a set of parameters can be found such that the
mean squared error of the eigenvalues at the high symmetry points is minimized.
Of course fitting only to the band structure for one allotrope is the easiest. This
procedure for carbon can be carried out by simultaneously fitting/reproducing the
band structure for diamond and graphite. Material properties such as the bulk
modulus and other elastic constants can be used in the database for the fitting of
the parameters. The development of tight-binding methodology given above car-
ried along the overlap matrix. Many early parameterizations were designed such
that all basis functions were orthogonal and thus the overlap matrix is simply
the identity matrix. These orthogonal models have half the parameters as non-
62
orthogonal models to fit (i.e. no t(r)’s for the overlap). The extra parametrization
and flexibility of non-orthogonal models has been noted to improve the transfer-
ability of tight binding models.
5.8.1 Density functional based tight binding
The tight-binding parameters used in our code are those of Porezag et. al. [137].
They developed a SK parametrization for carbon based on DFT, that has proven
versatile and predictive. They started by writing the Kohn-Sham orbital ψks
as a linear combination of atom centered pseudoatomic wavefunctions φν , that
are sums of Slater-type orbitals and spherical harmonics. These pseudoatomic
wavefunctions contain parameters that are found through self-consistently solving
the Kohn-Sham equation given by
[T + V psat(r)
]φν = Epsat
ν φν , (5.38)
V psat(r) = Vnucleus(r) + VHartree[n(r)] + V LDAxc [n(r)] +
(r
ro
)N
. (5.39)
The term(
rro
)N
is introduced in order to compress the wavefunction such that the
band structure fitting is improved. This has consequences that will be addressed
in § 8.2.1.
The φν obtained by solving equation (5.38) are used as the basis functions in
a LCAO treatment of the systems and are used next in calculating the matrix
elements H and S. For the Hamiltonian matrix elements the two-center approx-
imation is utilized and the potential is a Kohn-Sham potential of a neutral pseu-
doatom. These elements are evaluated explicitly at discrete distances using the
method described in [40]. The tabulated values for the matrix elements as a func-
tion of internuclear separation are fit to 10th order Chebyshev polynomials. Its
63
computationally advantageous to evaluate the polynomials at discrete steps and
save the results to a database to be used in interpolation, rather than evaluate
them on the fly.
The repulsive two body potential is found through the energy difference between
the total system energy given by the self-consistent DFT calculation and the band
structure energy calculated through the fitted parameters in the previous step
Urep(R) = ELDAtot (R) − EBS(R). (5.40)
These points are similarly tabulated and fitted, in this case to a sum of polyno-
mials.
5.9 Use of Porezag parametrization and current code
Porezag’s parametrization method has been used extensively as can be seen in a
quick literature search. Parameters for Si,B,N,Ga,As and many other have been
developed in a similar manner. These parameters have been used in studying the
electronic and mechanical properties of perfect and distorted carbon nanotubes
[69, 142], diffusion of carbon in GaAs [103] electronic structure and vibrational
properties of a number of small clusters [152]. The code I am using was developed
originally by Graves [55] and has been modified by several people for different
purposes, and as mentioned is currently utilizing Porezag’s non-orthogonal carbon
parameters. This particular tight binding code has been used in several studies of
semiconductor response to ultrafast laser pulses [32, 33, 49].
64
6Quantum mechanical origin of dispersion
forces
This chapter contains theories of dispersion forces that are elucidated and applied
to the model graphitic system. The 1/r6 dependance in the dispersion energy will
be brought to light, along with calculation of C6 coefficient for graphite utilizing
different methods.
Theories of dispersion forces can be generally lumped into two camps: microscopic
and macroscopic theories. The microscopic theory is a bottom-up approach based
on the atom-atom dispersion interaction of London that has been mentioned sev-
eral times thus far. In these microscopic theories the total interaction is thought of
65
as a sum of contributing part of the microscopic constituents. Under the assump-
tion of pairwise addivity (discussed in the next section) the microscopic dispersive
interactions add up to describe the total interaction between macroscopic bodies
(i.e. molecules/solids).
In an opposite top-down fashion macroscopic theories start with the dispersive
energy between macroscopic bodies. Again under the assumption of additivity,
the effective microscopic contributions can be backed out of these theories. The
theories of Lifshitz and Hamaker are introduced in the eventual extraction of C6
for graphite via these methods.
6.1 Main assumptions in modeling dispersion interactions
Before discussing the physical basis of the dispersion theories in detail there are
some important points to keep in mind. There are some major assumptions that
are commonly made in dealing with dispersion forces that are rarely mentioned
in the first principles plus dispersion papers cited in § 4.2. These assumptions are
addivity, isotropicity and non-retardation [110, 111].
The addivity assumption is the idea that the total dispersion energy can be ex-
pressed as a pairwise summation of interaction between the atomic constituents.
The isotropicity assumption says that the dispersion energy is isotropic in nature.
This turns out to be only true for some special cases and as a general rule does not
hold. Retardation is an effect that arises at far separation. Since dispersion inter-
actions are electromagnetic in nature the instantaneous dipole that is realized on
one molecule takes a finite amount of time to register with the other molecule due
to the speed of light. At farther separations these systems can no longer correlate
with each other for a net lowering of energy. The 1/r6 energy distance dependance
decays to a 1/r7 dependance and finally disappears at further distances [110].
In later sections these assumptions will be further addressed in conjunction with
the development of the theories.
66
6.2 Microscopic dispersion theory (London)
The idea of “spontaneously induced dipoles”, termed London dispersion forces,
was introduced in § 2.3.4 and referenced many times since. Now the quantum
mechanical origin of the 1/r6 distance dependance and dispersion coefficient C6
will be laid bare through the application of perturbation theory to a system of
two hydrogen atoms as London first did in 1930 [107].
Perturbation theory in short is the calculating of corrections to the energy and
wavefunction of a known system under the influence of a disturbance (see Ap-
pendix E for more formalism). We will show the change in energy of interacting
charge clouds as they are brought together from infinite separation. First the sys-
tem is described, followed by the form of the perturbation and finally the solution
as given by perturbation theory.
6.2.1 System description
Consider an atom A who’s quantum numbers and, thus state, is described by λa,
and another atom B, with state λb. This system of two atoms when infinitely
separated can be treated individually with the Schrodinger equation.
Haψλa= Eλa
ψλa, (6.1)
Hbψλb= Eλb
ψλb, (6.2)
where the Hamiltonians contain terms associated with the isolated atoms. This
system is said to be unperturbed and its wave function is simply the product of
the individual wavefunctions, and its hamiltonian and energy just the sum of the
two, i.e.
67
ψ(0) = ψλaλb= ψλa
ψλb, (6.3)
H(0) = Hab = Ha +Hb, (6.4)
E(0) = Eλaλb= Eλa
+ Eλb. (6.5)
The superscripts “0” mean the zeroth order and stand for the unperturbed solu-
tion. The system of two non-interacting atoms now can be written as
H(0)ψ(0) = E(0)ψ(0). (6.6)
6.2.2 The perturbed system
When the two systems are separated by a finite distance rather than infinite,
the wavefunctions ψλaand ψλb
are nearly the same but are slightly distorted due
to the presence of one another. If the distortion is slight enough one can apply
perturbation theory to approximate the change in the system.
Perturbation theory starts by assuming the full knowledge of the zeroth-order
unperturbed solution to the Schrodinger equation given in equation (6.6). With
this knowledge its possible to describe the solution of the same system that is
slightly “perturbed”. This system is described by a new Hamiltonian that has the
same base Hamiltonian H(0) plus a small perturbation V
H = H(0) + V. (6.7)
The perturbation V can have many orders of contribution as can be see in Ap-
pendix E. Here we do not show higher order perturbations because they do not
68
add anything to the description of our system and it helps keep the notation
cleaner.
6.2.3 Energy corrections in perturbation theory
The correction to the energy and wavefunction in perturbation theory are added
to the zeroth order energy and wavefunction, and the actual wavefunction and
energy are expressed as
ψ = ψ(0) + ψ(1) + ψ(2) + . . . h.o.t′s, (6.8)
E = E(0) + E(1) + E(2) + . . . h.o.t′s. (6.9)
The superscripts on the energy and wavefunction denote the order of the correc-
tion. E(1) and E(2) are referred to as the first and second order corrections to the
energy respectively, and as mentioned earlier E(0) corresponds to the unperturbed
energy.
The derivation for the first and second order corrections to the energy and wave-
functions are shown in Appendix E. The first order correction to the energy, in
state k is
E(1)k = Vkk. (6.10)
Vkk is the short hand notation for 〈ψ(0)k |V |ψ(0)
k 〉. As mentioned in appendix E
the correction is not restricted to the ground state (k = 0). The second order
correction is given as
E(2)k =
′∑
n
|Vkn|2
E(0)k − E
(0)n
. (6.11)
69
The prime in the sum means over all states except n = k.
Using the two atom wave function given in (6.3), in equations (6.10), and (6.11),
the first and second order corrections to the energy are
E(1)λaλb
= 〈ψ(0)λaλb
| V |ψ(0)λaλb
〉 , (6.12)
E(2)λaλb
=′∑
n1
′∑
n2
∣∣∣〈ψ(0)λaλb
|V |ψ(0)n1n2
〉∣∣∣2
(E(0)λa
−E(0)n1
) + (E(0)λb
− E(0)n2
). (6.13)
6.2.4 Application to Hydrogen
If the atoms A and B are hydrogen and the quantum numbers correspond to the
ground state we know, from the full analytical solution of the wave functions in
the ground state that the charge distributions are spherically symmetric.
+ +
−
−
~R
A B
~r1
~r2
x
y
Figure 6.1: Two hydrogen atoms A and B. The nuclei are separated by ~R withelectrons referenced from the nuclei
Figure 6.1 shows the system of hydrogen atoms, separated by the internuclear
vector ~R. Atom A’s electron is denoted by ~r1 and the other electron is denoted as
~r2 for atom B. The perturbation in our case is the columbic interactions of charges
in atom A and those in atom B expressed by
70
V =
nA−nB︷︸︸︷e2
r+
eA−eB︷ ︸︸ ︷e2
|~R+ ~r2 − ~r1|−
nA−eB︷ ︸︸ ︷e2
|~R+ ~r2|−
eA−nB︷ ︸︸ ︷e2
|~R− ~r1|. (6.14)
r = |~R| in the first term, and the overbraces denote the interaction (e.g. nA −
eB is the term associated with the Coulomb interaction between nucleus A and
electron in B). Assuming the nuclei are sufficiently far apart the interaction can
be expanded in a Taylor series. For the general case of an arbitrary number of
electrons this expansion is referred to as a multipole expansion and is described in
Appendix D ( see Equation D.9 and table D.1). In this derivation the internuclear
vector ~R is taken as one of the cartesian directions ( e.g. x direction in figure 6.1).
For the case of non-ionic interaction, the first five terms of the expansion in table
D.1 are zero. The electrostatic energy between these neutral charge distribution,
begins with the dipole-dipole interaction energy and continues up into the higher
order terms. The perturbations first non-zero term is given by
V = ~R−3∑
ij
qiqj(xixj + yiyj − 2zizj), (6.15)
and termed the dipole-dipole operator. The sum goes over the i charges in atom
A and the j charges in atom B. The cartesian components of ~ri are xi, yi and
zi with origins centered at the nuclei, and qi is the magnitude of the charge. For
hydrogen, using atomic units (i.e. e = 1) the dipole-dipole interaction can be
written as
V =1
r3(x1x2 + y1y2 − 2z1z2). (6.16)
With this perturbation we can calculate the corrections to the energy of the iso-
lated hydrogens due to their mutual interaction. The change in energy between
the isolated states is simply
71
∆E = E(1)0 + E
(2)0 + . . . h.o.t′s. (6.17)
The first order correction to the energy given in equation (6.12) is zero (E(1)0 = 0).
This is because the ground state wavefunctions are spherically symmetric and the
integral described by them will always be zero.
For the second order correction to the energy we plug our perturbation from
(6.16) into the definition for E(2)0 given in equation (6.13). With the ground state
described by λa = λb = 0 the second order correction to the energy is
E(2)0 =
1
r6
′∑
n1
′∑
n2
∣∣∣〈ψ(0)0 ψ
(0)0 |x1x2 + y1y2 − 2z1z2 |ψ(0)
n1ψ
(0)n2〉∣∣∣2
(E(0)0 −E
(0)n1
) + (E(0)0 − E
(0)n2
). (6.18)
The 1/r3 term from the perturbation given in equation (6.16) is squared in the
numerator of the second order correction and is brought front as 1/r6.
( mention the magnitude of E(3)0 , and magnitude of the C8 term)
There is a net change in energy proportional to 1/r6, where the proportionality is
simply the double summation in equation (6.18). We will call this term C6 and
treat it as a constant for the time being.
∆E =C6
r6. (6.19)
The sign of the energy corrections ∆E, determines the attractive or repulsive
nature of the perturbation (If the system energy is lowered its attractive and vice
versa). In the ground state the denominator of E(2)0 is always negative because
E(0)0 < E
(0)n for all n in the sum, and the numerator is always positive so C6 will
always be negative. For excited states this condition is not necessarily true and
there can be a net repulsion.
72
6.2.5 Higher order corrections
We have truncated the perturbation at the second order and only considered the
dipole-dipole perturbation. It is wise to consider the magnitude of both the higher
order perturbations and multipole terms.
This change in energy will be referred to as the dispersion energy. If higher order
multipole terms were included at in the perturbation in (6.16) we would see that
the dispersion energy takes the form
Edisp =C6
r6+C8
r8+C10
r10+ . . . h.o.t′s =
∞∑
n=3
C2n
r2n. (6.20)
The right hand side shows the general form of the infinite expansion.
6.2.6 Applying London’s dispersion theory in Calculating C6 for graphite
The next few sections describe the approximate evaluation of C6 for graphite by
simplifying the double summation in equation 6.18.
6.2.6.1 London’s form and polarizability
London’s went beyond the derivation above for an approximate evaluation of C6
given by the sum of equation (6.18). This derivation of his includes the static
polarizability of the atom in question. The static polarizability describes the
ability of an atom to gain a dipole moment (µ) under the influence of a static
electric field (E )
µ = αE . (6.21)
The static polarizability, α, has units of [C2m2J−1]. Polarizability is usually re-
ported in the form of the polarizability volume given by
73
α′ =α
4πǫo, (6.22)
with units of volume [A3]. While the polarizability is properly a tensorial quantity,
here it is assumed that the measured value used in London’s approximation is the
average static polarizability over all orientations.
London’s approximation is based upon applying the closure approximation ( see
Appendix E.3) in evaluating the sum in (6.18), the form is [108]
CLond6ij = −3
2
(∆Ei ∆Ej
∆Ei + ∆Ej
)α
′
iα′
j . (6.23)
The ∆E is the average energy difference between all excited states and the ground
state as explained in E.3. This term has been proposed to be approximately
either the ionization potential (I), and the resonance or first transition energy (ω1)
[97]. Another interpretation will be discussed in the following section. Using the
interpretation of the average energy difference (∆Ei) as the ionization potential,
the London form of C6 between atoms i and j is
CLond6ij = −3
2
(IiIjIi + Ij
)α
′
iα′
j. (6.24)
While this is a rough approximation due to the assumptions made, this formula
provides a good starting point to evaluate C6 with experimental data.
In order to evaluate this expression for carbon in graphite one needs the static
polarizability, and the ionization potential in the graphitic state. The last point,
“in the graphitic” state is important. One can find these data points for carbon
in its isolated form, to its many forms of hybridization, and when it is hybridized
to other species.
The ionization potential for carbon in graphite measured experimentally is 11.22
74
eV [136]. The static polarizability, on the other hand is not as easy to measure.
The static polarizability for an atom is not just a function of its atomic number,
but it is also a function of its state of bonding or hybridization [82, 118]. Using
this logic one can see that since there isn’t such a thing as isolated carbon with
the properties of a carbon atom in a graphitic state, this is not a directly measur-
able quantity. Fortunately there are empirical combinations rules for describing
molecular polarizability in terms of atomic polarizability and hybridization that
work very well [82, 118]. The most extensively tested of these is Miller’s [118]. His
combination rules correctly reproduce experimental polarizabilities of over 400
compounds with in a average error of 2.2%. For carbon in the graphitic state
only bonded to other carbons, Miller finds that the static atomic polarizability is
α′ = 1.896 A3. Using this polarizability value with the ionization value quoted
above the C6 for graphite using equation (6.24) is 30.25 eV A6
6.2.6.2 Slater-Kirkwood approximation
A few years after London’s derivation and approximation Slater and Kirkwood
proposed another approximation in calculating the sum in London’s derivation
[153]. Starting with London’s derivation after the closure approximation (equation
(6.23)), they interpreted the ∆Ei as
∆Ei = (Ni/α′
i)1/2. (6.25)
Instead of the ionization potential as London used. N is the Slater-Kirkwood
effective number of electrons, originally taken as the number of valence electrons.
With this the Slater-Kirkwood approximation for C6 is
Csk6ij =
3
2
α′
iα′
j
(α′
i/Ni)1/2 + (α′
j/Nj)1/2. (6.26)
Halgren has applied a similar method as Miller for the combination rules of dis-
75
persion coefficients, but in terms of the Slater-Kirkwood formalism. Halgren’s
optimum value for the Slater-Kirkwood number of electrons for carbon in a sp2
hybridization is 2.49 [59]. Using this and the same value for the static polarizabil-
ity in the previous section the Slater-Kirkwood value for C6 is 30.89 eV A6.
6.2.6.3 Kirkwood approximation
As mentioned in § 4.1, the C6 used in the Girifalco LJ graphite potential was
calculated by the Kirkwood approximation. This approximation like the Slater-
Kirkwood approximation is again based on an interpretation of London’s deriva-
tion in (6.23). This version is in terms of the static polarizability, and the dia-
magnetic susceptibility χ [88]
Ckw6ij = 6mec
2α
′
iα′
j
α′
i/χi + α′
j/χj
. (6.27)
me is the mass of the electron, and c is the speed of light. Using this form, Kraus
[98], calculated C6 = 15.2 eV A6
for graphite from data provided in [6].
6.3 Macroscopic dispersion theory (Lifshitz + Hamaker)
The other way to tackle the problem of describing dispersion forces is to start
with a description of macroscopic bodies interacting through dispersion forces.
These theories are rigorously expressed with quantum electrodynamics (QED)
[102]. They have been reformulated in semi-classical terms and in this context
the dispersion energy “can be defined as the change in the zero-point energy of
the electromagnetic field modes (obtained by solving Maxwell’s equations) when
the latter are perturbed by the molecules through coupling of the field with the
polarization currents induced on the molecules” [110]. The general results and
implications of this macroscopic theory of dispersion forces will be briefly summa-
rized and stated in the next few sections.
76
6.3.1 Lifshitz dispersion theory
The dispersion interaction between two semi infinite slabs was described by Lif-
shitz, who developed a general theory of dispersion interactions between macro-
scopic bodies [106, 36]. Before the general form Lifshitz proposed is explained, a
necessary quantity, ǫ(iξ), which is utilized by this theory must be discussed.
6.3.1.1 Dielectric response and ǫ(iξ)
The permittivity of a material ǫ, relates how a time dependant externally applied
electric field (E), will effect the electric displacement vector (D) in a solid through
D = ǫE. (6.28)
The permittivity as a function of field frequency ω, is known as the dielectric
permeability, or dielectric function ǫ(ω). This is a complex quantity with real and
imaginary component defined as ǫ(ω) = ǫ′
(ω) + i ǫ′′
(ω). The dielectric function is
properly a tensorial quantity, but for isotropic materials, or a first approximation,
ǫxx = ǫyy = ǫzz. The dielectric function can be found through the measurement
of optical properties of materials specifically, refractive index and the extinction
coefficient(n(ω) and k(ω) respectively
). This relation is given by [89]
ǫ(ω) =(n(ω) − i k(ω)
)2. (6.29)
The Lifshitz theory of dispersion forces presented in the next section requires
the function ǫ′(iξ), which is the real part of the dielectric function evaluated at
purely complex frequencies ω = iξ, and will be referred to as the Lifshitz dielectric
function from here on. The dielectric permeability for an imaginary frequency is
related to the imaginary part of the dielectric permeability via the Kramer-Kronig
relation [72]
77
ǫ′(iξn) = 1 +2
π
∞∫
0
ω ǫ′′
(ω)
ω2 + ξ2n
dω. (6.30)
Thus if the imaginary part of the dielectric function evaluated at real frequencies
is known the Lifshitz dielectric function can be attained. One path to ǫ′(iξn) can
start with the experimental data n(ω) and k(ω), progress to ǫ′′
(ω) via equation
(6.29), and finally to ǫ′(iξn) via the Kramer-Kronig relation in equation (6.30).
Another route is the direct calculation of ǫ′′
(ω) from a first-principles or semi-
empirical technique and then the application of the Kramer-Kronig relation.
6.3.1.2 Lifshitz dispersion theory
Lifshitz original derivation was for two slabs of similar material separated by vac-
uum [106], but was later extended and generalized to two differen’t materials
separated by a third medium. The interaction energy per unit area between the
two plates is
L
1 23
Figure 6.2: Two semi-infinite slabs 1 and 2 separated by a distance L and amedium 3.
U123(L) =kbT
2π
∞∑
n=0
∞∫
0
k ln(1 − ∆12∆32 e−2kL) dk. (6.31)
78
k is the wavevector of the electromagnetic field. The ∆’s in this equation are
simply
∆kj =ǫk(iξn) − ǫj(iξn)
ǫk(iξn) + ǫj(iξn), (6.32)
and the Lifshitz dielectric function is evaluated at discrete points given by
ξn = n
(2πkbT
~
). (6.33)
Mahanty and Ninham have developed this theory beyond Lifshitz’s two slab prob-
lem to other geometries such as sphere-sphere, cylinder-cylinder etc. along with
theories for anisotropic dielectric functions [110].
6.3.2 Lifshitz theory applied to the evaluation of Hamaker constants
for graphite
By equating the definition of the surface-surface interaction energy given by the
Lifshitz method to that of the Hamaker surface-surface interaction energy given
by
Uss(L) =−A
12πL2, (6.34)
one can solve for the Hamaker constant A [10, 48, 72]. The derivation of this
energy relationship and how the Hamaker constant A is related to C6 will be
discussed in the next section. Here we note two case in which the evaluation
of the Lifshitz dispersion relation has been applied to graphite and a Hamaker
constant has been identified, in addition to this a single experimental evaluation
of the Hamaker constant has been found in literature.
Dagastine et al. calculated the Hamaker constant for graphite from experimental
79
spectroscopic data [30]. The isotropic assumption of the dielectric function men-
tioned in § 6.3.1.1, does not hold for graphite. The in plane dielectric function
(ǫxx = ǫyy) is quite distinct to that normal to the plane (ǫzz). Spectroscopic data
from both in plane and normal to the plane measurements was used in the con-
struction of ǫ′′
(ω) with the relation in (6.29), and used equation (6.30) to convert
this to ǫ(iξn). Using a modified version of the Lifshitz equation in (6.31) that
accounts for non-isotropic dielectric functions they calculated a Hamaker constant
of 2.53 × 10−19 J.
Li et al. have preformed a similar analysis on graphite but attained the dielectric
response via a tight binding model under linear response theory [105]. Unlike
Dagastine they assumed an isotropic dielectric response and there resulting long
wavelength Hamaker constant is reported as 2.2 × 10−19 J.
Parfitt and Picton reported a experimental Hamaker constant for graphite from
some experiments on colloidal suspensions of graphite flakes. Due to numerous
uncertainties in the analysis a rather broad range of 2.1 - 5.9 ×10−19 J is reported
[132].
6.3.3 Hamaker constant derivation
In conjunction with the last section this section further develops the ideas of
Hamaker in describing the interaction between macroscopic bodies and calcula-
tions of effective C6 dispersion constants are preformed utilizing the Hamaker
constants reported in above. Hamaker’s ideas came out of the context of colloidal
science. The main idea of the Hamaker energy relation between macroscopic bod-
ies is an assumption of addivity of dispersion forces and a treatment of the atoms
in a solid as a continuum [61]. With these assumptions the total interaction en-
ergy can then be expressed as volume integrals over the interacting bodies. These
calculations can be applied to any arbitrary geometric bodies (e.g. sphere-sphere,
cylinder-sphere etc). The derivation for the two slab problem is given below in
80
the format described in [110]. And starts with a description of a point-surface
interaction that is utilized in the surface-surface derivation.
6.3.3.1 Point-Surface Interaction
Consider a single atom offset from a surface by a distance L, and that it interacts
with the atoms in this solid through a dispersion force of the form
Upp(r) = −C6
r6. (6.35)
The underscore pp stands for point-point interaction. Assuming additivity the
total interaction energy can be attained by summing up all interaction with the
point and the atoms in the solid for a given offset L. Another way is to assign a
number density of atoms in the solid and attain the potential energy through a
volume integral.
Y
X
dx
r
x = L
x = 0
y
dy
Figure 6.3: Point-surface interaction diagram. The wavy line represents a visualtruncation of the infinite solid. The infinitesimal ring is shown from the side andthe dotted line represents its position in the solid into and out of the page
81
Figure 6.3 is a diagram of the system in question. The point particle is located at
x = 0, a distance L from the surface. The differential element is a infinitesimal
ring that is equidistant from the point. The volume of this ring multiplied by the
number density (ρ) is the number of atoms at this distance r,
N(r) = ρ(2πy)dxdy. (6.36)
r is simply r = (x2 + y2)1/2. By evaluating the dispersion energy with equa-
tion (6.35) with this distance r, and multiplying it by the number of atoms at
this distance given by N(r), gives the dispersion energy contribution from this in-
finitesimal ring. Integrating over all rings results in the total dispersion interaction
energy between a point and a half-slab in the form
Ups(L) =
∞∫
x=L
∞∫
y=0
ρ(2πy)−C6
(x2 + y2)6/2dy dx. (6.37)
The ps stands for point-half slab interaction. For the geometry shown in Figure
6.3 this reduces to
Ups(L) =−πC6ρ
6L3. (6.38)
6.3.3.2 Surface-Surface Interaction
Using the definition for the point-surface interaction it is possible to calculate the
surface-surface interaction defined per unit area (as in the Lifshitz derivation).
Figure 6.4 shows two half slabs, the element of surface area has a depth dx and
contains Nse = ρ dx atoms per unit area. The interaction energy of these Nse
atoms with the other surface is simply
Use(x) = NseUps(x), (6.39)
82
L
dxx
Y
X
Figure 6.4: Surface-Surface Hamaker integration
and the surface-surface total interaction is just these surface elements integrated
from x = L→ ∞
Uss(L) =
∞∫
L
Use(x) dx =
∞∫
L
−πC6ρ
6x3ρ dx. (6.40)
This simplifies to
Uss(L) =−πC6ρ
2
12L2=
−A12πL2
. (6.41)
The right hand side is the same form that was shown earlier in equation (6.34).
As can be seen from this equation the Hamaker constant A is defined as
A = π2ρ2C6. (6.42)
The graphite dispersion constant C6 can be backed out of the Hamaker constant
with knowledge of the number density of carbon atoms in graphite which is ρ =
83
1/(A0 ·Zeq)= 0.114 atoms/A3. Using this with equation (6.42) the three reported
Hamaker constants of Dagastine, Li, and Parfitt reported in § 6.3.2 correspond to
dispersion coefficients of 12.32, 10.71, and 10.22 − 28.72 eV A6
respectively.
6.4 Summary of calculated C6 coefficients
Later when we report our fitted value of C6 we must refer back to this the results
of this chapter to see if the value found is reasonable with what was presented
here in its direct calculation. The results of the various methods, including the
experimental Hamaker value, are tabulated in table 6.1. This table shows a decent
spread with the Lifshitz based methods clustered at the lower end and the other
methods showing a larger value. It is interesting to note that the one experimen-
tal value reported (calculated from the Hamaker decomposition method) almost
perfectly spans the range of calculated values.
C6 [eV A6] Method Method Reference Data Reference
30.25 London [108] [82]30.89 Slater-Kirkwood [153] [59, 118]16.34 Slater-Kirkwood [153] [167]15.2 Kirkwood [88] [6]12.32 Lifshitz/Hamaker [106, 61] [30]10.71 Lifshitz/Hamaker [106, 61] [105]10.22 − 28.72 Hamaker (experiment) [61] [132]
Table 6.1: Summary of C6 dispersion coefficients for graphite
84
7Damping functions
In the previous chapter London’s perturbative description of the dispersion attrac-
tion between hydrogen atoms was discussed amongst other theories. This analysis
was based on two approximations that breakdown when the two hydrogen atoms
come closer together. The first is related to the main assumption in perturbation
theory, that says the perturbation must be small enough such that perturbed sys-
tem is close to the unperturbed state. When the hydrogen atoms, or any other
system of atoms, are brought close enough to one another the resulting wavefunc-
tions are very different than those of the isolated system, and thus perturbation
theory fails. The second breakdown in London’s derivation at close proximity
is from the definition of the perturbation operator via the multipole expansion.
85
This was derived through a Taylor series expansion under the assumption that the
internuclear separation was much larger than the average radius of the electron
cloud. At small internuclear separation this expression for the perturbation losses
its meaning.
In order to properly treat this region were London’s assumptions break down
the idea of a damping function is introduced. The deviation of the dispersion
energy from the asymptotic limit description, given in equation (6.20), at close
internuclear separation can be described if the true dispersion energy is known.
The damping function is defined simply as the ratio of the true dispersion energy
and the London dispersion energy
f(r) =Edisp
o
EdispL
. (7.1)
Edispo is the true dispersion energy and Edisp
L is the London dispersion energy. If the
damping function is known then the true dispersion energy is simply the damping
function multiplied by the London dispersion energy,
Edispo = f(r)Edisp
L . (7.2)
There are some general features of this damping function which can be pointed
out. One is that it acts as a switching function and has a range between 0 and
1. As r approaches zero f(r) must necessarily go to zero at least as fast as the
asymptotic dispersion energy. This can be understood by saying the dispersion
energy must be finite at all distances r, without the damping function it would
approach −∞ as r tends to zero. In the limit of large r, the asymptotic limit
of the London derivation is valid and f(r) must be unity. The nature of how
this damping function switches from 0 to 1 is the key point to consider. As in
the development of the 1/r6 dispersion energy dependance the hydrogen dimer
86
provides some clues to the nature of this damping function.
7.1 Hydrogen damping function
The only damping function which is known to a very high degree is that of hydro-
gen. Koide, Meath and Allnatt (KMA) have preformed pseudo-state evaluation
of the dispersion energy [93]. These calculations are cited as the most accurate
analysis of the dispersion energy between hydrogen atoms, and are the benchmark
for proposing a general form of damping functions [157, 166].
7.2 Damping functions in literature
All of the first principles plus dispersion models discussed in § 4.2 employ a damp-
ing function. There are a handful of damping functions that are used often. A
early form was proposed by Ahlrichs et al. [1]
fdamp(r) = exp[−(c rm/r − 1)2]. (7.3)
This function has two parameters, c is dependant on the interacting species (c
= 1.28 for hydrogen dimers using the KMA data discussed above) and rm is the
position of the potential minimum.
Another common form is that of Mooij et al. which has continuous second deriva-
tives
fdamp(r) =(1 − exp[−c(r/rm)3]
)2. (7.4)
Again c and rm are fitted for the particular interaction. Fermi functions, which
also have two parameters to fit are also used as damping functions in some models.
Yet another damping function is that of Tang and Toennis [157]. They developed
87
a semi-universal damping function that has been shown to correctly describe the
functional form of the damping function for a number of interactions. This damp-
ing function has a single parameter, α, which describes the range of the overlap
of the interacting species and has the form
f 2ndamp(r) = 1 −
(2n∑
k=0
(αr)k
k!
)e−αr. (7.5)
Each term in the dispersion series given in equation (6.20) has an individual damp-
ing function. n = 3 in equation (7.5) corresponds to f6 which is the damping
function used for the dipole-dipole dispersion term C6/r6. The total damped dis-
persion series in Tang’s form is then
Edispdamped =
∞∑
n=3
f 2ndamp(r)
C2n
r2n. (7.6)
This damping function has a few positive qualities: its apparent universality, and
single parameter formulation. Tang and Toennies have shown that their damping
function does a very good job in reproducing the form of the damping function
as calculated by Koide et al. described above. Figure 7.1 shows Koide’s data set
for hydrogen as dots and the fitted damping function as a solid line for the first 8
terms in the dispersion series. Similar good fits are shown for He-He, Ar-Ar, Na-
K and Li-Hg based on the best available data for these interaction. Their paper
shows the shortcoming of the other discussed damping functions in describing these
interactions as compared to their own. This damping function will be adopted for
use in our formulation of the damped dispersion based on these arguments.
88
Figure 7.1: Plot of Tang’s damping function for b = 1.67 a.u. for n=3-10 as solidline, and Koide’s hydrogen dispersion energies as dots [93]
89
8Tight-binding plus dispersion
parametrization
This chapter details the marriage of the tight-binding model with a damped dis-
persion energy and our parametrization of the free parameters to model both the
inter and intra-planar properties graphite. The notation for the interlayer energy
per atom (E) defined in § 3.3 is used in this and following sections. A rhom-
bohedral supercell of four unit cells containing a total of sixteen atoms is used.
Three dimensional periodic boundary conditions are enforced to simulate bulk
graphite (periodic boundary conditions are discussed in Appendix A.2). We start
by defining the total energy in the model.
90
8.1 Total energy definition
The total energy for the system is defined in the same manner as was discussed
in § 4.2 and is repeated here:
Etot = ETB + Edisp. (8.1)
Where ETB is the tight-binding energy defined in equation (5.36) and also is
repeated here
ETB =∑
λ∈occ
Nk∑
i=1
wi fλ ελ(ki) +1
2
∑
i,j
Θ(rij). (8.2)
Edisp is the dispersion energy defined as
Edisp =1
2
∑
i,j
fdamp(rij) fcut(rij)C6
r6ij
. (8.3)
In this dispersion energy expression, fdamp(rij) is the damping function. We dis-
cussed in § 7.1 that we are using the Tang and Toennis damping function due to
its simplicity and seeming “semi-universality.” [157] Its form for the damping of
the C6 term (n=3 in equation (7.5)) is
fdamp(r) = 1 −(
6∑
k=0
(αr)k
k!
)e−αr. (8.4)
The term fcut(r) in equation 8.3 is a cutoff function. Cutoff functions are employed
to make the neighbor search routine more efficient, and improve the stability of
molecular dynamics simulations. Cutoff functions effectively define a radii beyond
which contributions are considered negligible and thus ignored. We have utilized
a Fermi function for the cutoff function within the defined cutoff distance Rcut,
and 0 otherwise.
91
fcut(r) =
1
exp( r−rc
rw) + 1
if r ≤ Rcut
0 if r > Rcut
(8.5)
The Fermi function is equal to unity and switches to zero with continuous deriva-
tives, and in our case the dispersion energy also has continuous derivatives. For
molecular dynamics having continuous derivatives of the energy at the cutoff
negates the possibility of spikes of high force when atoms enter each other’s do-
mains defined by the cutoff distance. This Fermi function is defined by two pa-
rameters; rc determines the distance at which the function is equal to 1/2, and
rw controls the smoothness of the cutoff. We simply choose a value of rw = 0.1
and find a corresponding rc such that the function is essentially zero at the cutoff,
i.e. f(Rcut) = 1 × 10−5. Figure 8.1 shows the cutoff functions for various values
of Rcut. The seeming arbitrariness in the definition of the cutoff function will be
addressed and rectified later in § 8.4.
0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r [A°]
f cut(
r )
Rcut
= 5
Rcut
= 10
Rcut
= 15
Figure 8.1: Fermi function for a number or Rcut’s with rw = 0.1
92
8.2 Evaluation of tight-binding energy
The tight-binding energy expression given in (8.2) requires the summation of en-
ergy over the Brillouin zone. The Brillouin zone was sampled with the special k-
points (ki) and weights (wi) of Monkhourst-Pack [120] as tabulated by [101]. The
density of k-points was increased to 370 to satisfy an energy convergence criteria
of ±0.1 meV. A summary of this convergence is shown in Table 8.1 for graphite
at its equilibrium interlayer spacing. This tight convergence is need because the
magnitude of the interlayer energy is on the order of 50 meV. § 8.3.2 discusses an
exception to the use of the Monkhourst-Pack points for certain calculations.
# k-points ETB
1 ( Γ) -46641.14328 -47341.229133 -47344.471370 -47344.801610 -47344.882
Table 8.1: Number of Monkhourst-Pack k-points and calculated energy[meV/atom]
Figure 8.2 shows a plot of the tight-binding interlayer energy per atom as a function
of interlayer spacing. As expected there is no minima due to the lack of dispersion
energy in the formalism of tight-binding and the resulting energy is of a purely
repulsive nature.
It should be noted that in the figure the interlayer energy effectively goes to
zero before Zeq. Though the tight-binding cut-off distance is around 6 A, there
is no interaction energy at this distance. This situation needs to be rectified
on the basis that in order for there to be a minima at Zeq, and thus a stable
graphite structure, there must be a non-zero tight-binding interlayer energy at Zeq.
This can be understood by a simple argument. Both energy terms monotonically
approach zero at large distances, but from different sides of zero. In order for a
minima to occur the first derivative of the total interlayer energy must be zero, and
consequently the sum of the first derivatives of the tight binding and dispersion
93
2.5 3 3.5−20
0
20
40
60
80
100
120
140
160
180
200
interlayer separation [A°]
inte
rlaye
r en
ergy
[meV
/ato
m]
At Zeq
, ETB < 1 × 10−4 meV/atom
Figure 8.2: Purely repulsive tight-binding interlayer energy
interlayer energy must sum to zero. The derivative of the attractive dispersion
term will always be positive, implying that there must be a negative-valued first
derivative of the tight-binding interlayer energy in order for the minima to occur.
Due to its decreasing monotonic nature the tight-binding energy must necessarily
be non-zero at the interlayer spacing at which the minima occurs, in order for its
derivative to be non-zero.
8.2.1 Orbital expansion
One may wonder why the DFT fitted tight-binding parametrization fails to pro-
duce a repulsive overlap energy at the equilibrium interlayer spacing while it nat-
urally must, in accordance to the previous argument. This result can be directly
traced to the artificial orbital compression described in § 5.8.1, done at the time in
the name of celerity. To undo this limitation on the proper modeling of graphite
interlayer properties one can simply inflate the compressed orbitals. This orbital
inflation is done in a simple manner. To inflate a particular pair of interacting
orbitals one can simply contract the distance in which they are separated when
94
evaluating the hopping integrals in the hopping sub-matrix. This is done by intro-
ducing a scaling parameter γ which is multiplied by the real internuclear distance
rij to get a reduced distance rredij = γrij which is used in the integral evaluation. In
order to preserve the in-plane properties of graphite (which are already properly
described with the unmodified TB code) we must introduce a numbering system
for the bodies interacting via dispersion forces. A layer of graphite, or later on,
a single tube is numbered and the distance scaling is only done between different
numbered systems so that in the case of infinite separation of the systems the
in-plane energetics are the same as with out the orbital stretching addition.
We have previously stressed the importance of the Pz orbital in the repulsive over-
lap interaction so it would make sense to only inflate this orbital when evaluating
hopping integrals between Pz orbitals on differing sheets. This works fine until we
bring our graphite dispersion model to use on the curved surfaces of carbon nan-
otubes. In this case there is no uniform z direction in which to inflate Pz orbitals
on differing nanotubes. Another formulation is to uniformly inflate all the orbitals
in the graphite formulation and fitting, thus bypassing this directional problem
when describing tube-tube interactions.
The introduction of this scaling parameter produces the desired repulsive energetic
effect, as is illustrated in Figure 8.3. With a decreasing γ we see the increased
repulsion and our necessary non-zero tight-binding interlayer energy at Zeq.
8.3 Parameter fitting procedure
At this point there are 3 free parameters, the scaling parameter γ , the dispersion
constant C6, and the damping function parameter α. This situation is similar to
the one with which Palser dealt with (§ 4.2). Two of the data points Palser used
in fitting have a relatively high uncertainty: the experimental interlayer cohesive
energy, and the DFT stacking difference between AAA and ABA. It seems that
a more logical choice in fitting these parameters would be to fit to the more well
95
2.5 3 3.5
0
50
100
150
200
250
300
interlayer separation [A°]
inte
rlaye
r en
ergy
[meV
/ato
m]
γ = 1.00γ = 0.95γ = 0.90γ = 0.85
Zeq
Figure 8.3: Tight-binding interlayer energy for different γ’s. Arrow shows trendof decreasing γ, and the associated orbital inflation
known E2g(1) shear mode frequency and the z-axis compressibility, in addition to
the equilibrium interlayer separation Zeq. These three data points constitute a
much firmer database for our fitting than Palser’s.
The fitting routine consist of the simultaneous functional minimization of the three
experimental points (Zeq, kz, Eex) minus those calculated in the tight-binding plus
dispersion code. This procedure is preformed via the hybrd.f routine provided
in the MINPACK library from Argonne National Laboratory [122]. This routine
is an implementation of Powell’s hybrid method, based on the conjugate gradient
search routine [138].
In order to proceed in computing the free parameters via the aforementioned
method it is necessary to construct functions to calculate the equilibrium spacing,
phonon frequency and the compressibility given a set of parameters γ, C6, α. The
next three sections address the computational details of attaining these values.
96
8.3.1 Equilibrium interlayer spacing
The equilibrium interlayer spacing is attained by requiring
∂ETB
∂Z
∣∣∣∣∣Zeq
+∂Edisp
∂Z
∣∣∣∣∣Zeq
= 0, (8.6)
which is simply a statement that the repulsive tight-binding interlayer force must
balance the attractive dispersion force at the equilibrium interlayer spacing. These
derivatives are evaluated through a finite central difference derivative evaluated
at a dilated interlayer spacing Zeq + δZ and a compressed spacing Zeq − δZ. For
the tight-binding term this is
∂ETB
∂Z
∣∣∣∣∣Zeq
=ETB(Zeq + δZ) − ETB(Zeq − δZ)
2 δZ. (8.7)
The same procedure is executed for the dispersion contribution in (8.6).
8.3.2 Phonon frequency calculation
The E2g(1) phonon frequency is evaluated via the frozen phonon method, model-
ing the interlayer sliding as approximately harmonic about the equilibrium ABA
stacking [114]. Under this assumption, the natural frequency of the sliding layers
can be described by
2πfE2g(1) =
√Kexp
c−c/mc. (8.8)
Here fE2g(1) is the frequency which corresponds to the E2g(1) mode (1.26 THz),
Kexpc−c is the experimental effective spring constant per atom along the carbon-
carbon bond length, and mc is the mass of a single carbon atom (12 amu =
1.9926 × 10−26 Kg). Solving for the effective spring constant in the units of our
code 1
97
Kexpc−c = 7.795 × 10−2 (eV/A
2)/atom. (8.9)
This effective one atom spring constant can be calculated for graphite in our tight-
binding code using the relation of a harmonic oscillator.
Etot(dc−c) =1
2Kc−c d
2c−c. (8.10)
Etot(dc−c) in this case represents the energy per atom in ABA graphite under
uniform translation (dcc) of every other layer in one of the carbon-carbon bond di-
rections. This motion corresponds to the relative translation of layers in the E2g(1)
mode described in § 3.5. The energy is evaluated at Zeq and dc−c is the magnitude
of the interlayer sliding (dc−c = 0 corresponds to a perfect ABA stacking).
Etot(dc−c) is evaluated at three points (dc−c = 0,±δ dc−c) and fitted to a quadratic
form via a least squares method 2.
Etot(dc−c) = a d2c−c. (8.11)
The value a is the leading coefficients of the quadratic polynomial. Equating the
right hand sides of this equation and equation (8.10) and canceling leads to
Kc−c = 2a, (8.12)
and the objective function to be minimized becomes
2a− Kexpc−c = 0. (8.13)
1This conversion is as follows: Kexp
c−c = (2π 1.26 × 1012 s−1)2 · 1.9926 × 10−26 Kg/atom =
1.249 (Kg/s2)/atom = 7.795 × 10−2 (eV/A2
)/atom. The conversion 1 eV/A2
= 16.022 Kg/s2
is employed.
98
Where the value of Kexpc−c is that reported above in (8.9).
8.3.3 Compressibility calculation
The z-axis compressibility was discussed in § 3.4, and is defined by [131]
kTB+dispz =
Ao
Zeq
d
2Etot(Z)
dZ2
∣∣∣∣∣Zeq
−1
. (8.14)
The second derivative of the interlayer energy is with respect to the interlayer
separation evaluated at the equilibrium ABA spacing. Ao is the area per atom in
the graphene sheet (Ao =√
3 a2/4 = 2.62 A2/atom).
This second derivative was calculated with the second order central difference
approximation [65]
d2Etot(Z)
dZ2
∣∣∣∣∣Zeq
=Etot(Zeq + δZ) − 2Etot(Zeq) + Etot(Zeq − δZ)
δZ2. (8.15)
Again δZ is an infinitesimal increment. The objective function is simply
kTB+dispz − kexp
z = 0. (8.16)
Where the experimental compressibility mentioned earlier in § 3.4 is kexpz = 2.74×
10−12 cm2 dyne−1
2A subtle point was discovered in evaluating these phonon frequencies by this method. Thereseemed to be some discrepancies in the symmetry of Etot(dc−c) about the ABA stacking. Specif-ically it was not found to be harmonic as expected. A closer look revealed that the Monkhorst-Pack special k-points are generated under the consideration of certain group theoretical con-siderations that are broken as the perfect hexagonal unit cell is disturbed. This problem wasremedied by population of the Brillouin zone with a sufficient number of k-points (15 × 15 × 3)
to attain the convergence mentioned previously. With this treatment Etot(dc−c) about the ABAstacking showed the expected harmonic behavior, and this brute force population is used in allcalculations of the E2g(1) mode.
99
8.4 Optimization results
Functions were created to evaluate the objective equations given in (8.6), (8.13)
and (8.16), detailed in the previous three subsections. These three objective func-
tions are utilized by the previously discussed hybrd.f routine in the parameter
optimization. There is a fourth semi-free parameter, the cutoff distance Rcut for
the dispersion energy. This parameter was found iteratively and effectively can-
celed the need for a cutoff function, as will be shown. The logic behind picking this
distance is based on a few ideas. Since we are assuming pairwise addivity in this
model the long range interactions are collectively contributing to the total energy,
but the contribution to the total energy drops off quickly as 1/r6. My logic was
to apply the parameter optimization procedure with progressively larger cutoff
distances until the parameters converged to steady values. This was done, and at
a cutoff distance of 12.25 A the three free parameters are changing less by than
1%. Table 8.2 shows a summary of the optimized parameters. Note that for the
optimized parameter values, the dispersion energy contribution at Rcut = 12.25 A
without the Fermi function cutoff, is 7.3 × 10−6 meV, which is quite small. Even
though this energy contribution without the Fermi function is small, the Fermi
function is retained to remain consistent and completely zero out any force spikes
while preforming MD.
C6 α γ(eV A6) (A−1)
24.8 1.45 0.905
Table 8.2: Optimized parameters for the tight-binding plus dispersion model witha cutoff of 12.25 A
The parameters shown in Table 8.2 result in an essentially zero error fit to the
database values (all less than 0.2% difference). The fitted TB+disp interlayer
energy is plotted and compared to the Girifalco empirical model in Figure 8.4.
The energy curves look similar except for the scale and curvature around Zeq.
Figure 8.5 shows more definitively the difference between the two models. In
100
this figure the interlayer energy as a function of layer separation is shown for
both the ABA and AAA sacking of the TB+disp model and Girifalco’s empirical
model. Note the small difference between these curves for the empirical model as
compared to the TB+disp model. Finally Figure 8.6 shows the energy landscape
of graphite, at Zeq, under a shear transition from an ABA stacking (the minima’s)
to AAA stacking (the maximum points). This landscape shows the corrugated
nature of graphite that was discussed earlier and expressed well in Figure 3.3.
The comprable landscape for the LJ models is much flatter as noted in § 4.1.
3 3.5 4 4.5 5−55
−50
−45
−40
−35
−30
−25
−20
−15
−10
Interlayer separation [A°]
Inte
rlaye
r en
ergy
[meV
/ato
m]
TBD − ABAgiri − ABA
Figure 8.4: Fitted tight binding plus dispersion interlayer energy for both ABAand AAA stacking plotted along with Girifalco’s empirical model
8.5 Discussion
It is necessary to discuss these results to make sure that the values of the pa-
rameters found are in accordance to what is known. In this fitting procedure the
database consisted of three experimental data points that are all found at the
equilibrium spacing. At this point we have both repulsion and attraction between
the layers. Shortly beyond the equilibrium distance there is practically no repul-
sion and only dispersive attraction. The database had no data points to constrain
the magnitude of C6, which dictates the strength of interactions beyond the equi-
101
3 3.1 3.2 3.3 3.4 3.5 3.6 3.7−55
−50
−45
−40
−35
−30
−25
−20
−15
Interlayer separation [A°]
Inte
rlaye
r en
ergy
[meV
/ato
m]
giri − AAAgiri − ABATBD − AAATBD − ABA
Figure 8.5: Tight-binding interlayer energy for AAA and ABA stacking, comparedto Girifalco’s LJ model
−2
−1
0
1
2
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−50
−45
−40
−35
−30
−25
inte
ract
ion
ener
gy [m
eV/a
tom
]
Y [A°]
X [A°]
ABA
AAA
(a)
x
y
(b)
Figure 8.6: (a) Graphite energy landscape (b) ABA graphite showing correspond-ing X,Y orientation
102
librium spacing. Naturally it is good to question the magnitude of C6 with what
was discussed in Chapter 6 regarding the calculation of C6 for carbon in graphene.
Our value of 24.8 eV A6
falls between the minimum calculated value of 10.71
eV A6
via the Lifshitz-Hamaker method and maximum value of 30.89 eV A6
via
the Slater-Kirkwood method. It is unfortunate that the methods for calculating
C6 show such a range. As it is we can say our value is in the realm of possibility
defined by these methods and is larger than the Girifalco and Ulbricht Lennard-
Jones models. The parameter α in the damping function is commensurate with
what was mentioned earlier in the section on damping functions. As mentioned
there, α is a measure of the range of the overlap, with larger atoms having smaller
values of α. Our value follows this trend with Tang’s reported α for the hydrogen
interaction (smaller than sp2 carbon) of 3.14 A−1 compared to our value for sp2
carbon of 1.45 A−1. The last parameter to consider is the stretch parameter γ.
Our fitting calls for an approximate distance contraction of 10% (γ = 0.905). This
distance contraction is not absurdly large, but it is hard to conclusively say what
it should be without re-parameterizing the tight-binding parameters without the
artificial orbital contraction. With all three of the fitted parameters within agree-
able boundaries it is interesting to note the models predicted exfoliation energy.
The calculated value of Exf is 49.7 meV/atom. This value is right in the range
of the experimental values (∼45-55 meV/atom) discussed in § 3.3.1. Another pre-
diction that can be considered is the energy difference between ABA and AAA
stacking. Our TB+disp model reports a value of 22.3 meV/atom. This value is in
decent accordance with the DFT value of 15 and 17 meV/atom reported earlier
in § 3.6.2. The fact that it is larger than the DFT values is not worrisome since
the current local density approximation based DFT functions capture some, but
not all, of the dispersion energy.
Another test on our model to which there is experimental data to compare is the
shift of the E2g(1) mode under external pressure. Applying a hydrostatic pressure
to graphite causes the layers to come together and there is an associated increase
103
in this phonon frequency. We have calculated the frequency for discrete values
of interlayer spacing and plotted them in Figure 8.7 as the circled line. This is
a semi-static frozen phonon calculation done in the same manner as described
in equation (8.13). Experimental data on this phonon frequency and interlayer
spacing for the 0 to 14 GPa range has been gathered and fitted by Hanfland et al.
[62]. The dashed line in Figure 8.7 is the representation of this experimental work
and is expressed by
ω(z) = ωo
[δoδ′βo
β ′
[(z
zeq
)−β′
− 1
]+ 1
]δ′
. (8.17)
ωo and zeq are the STP values of the E2g(1) phonon frequency, and equilibrium
spacing (1.26 THz, and 3.35 A respectively). The other four values are fitted to
the experimental measurement of graphite under pressure from [62] (β ′ = 10.89,
βo = 35.7 GPa, δo = 0.110 GPa−1, δ′ = 0.43). The top abscissa in Figure 8.7,
shows the associated pressure (with zero referencing STP) to attain the interlayer
spacing shown on the bottom abscissa. Our curve, while not falling on top of the
experimental data, shows the expected trend. A difference is expected since our
TB method doesn’t calculate the electronic charge density in a self consistent way,
an issue that becomes important as the layers are squeezed together.
We have implemented our model into a molecular dynamics environment and ap-
plied it to some simple systems. The tight-binding code had an existing molecular
dynamics subroutine. The force contribution from the vdW term was added to
the existing code. Details on the MD method and the vdW force term are given
in Appendix A. Our test case was two graphene sheets with periodic boundary
conditions in only the in-plane directions. We want to check if the predicted equi-
librium spacing of the graphene layers is mirrored in the MD simulation. The 0
K static interlayer energy scans for both the AAA and ABA system are shown
as the dashed and solid black lines in Figure 8.8. Their equilibrium interlayer
spacings are approximately 3.4 and 3.6 A respectively. This simulation was pre-
104
Figure 8.7: Hydrostatic pressure effects on the E2g(1) shear mode. STP is refer-enced as zero pressure. Dashed line is experimental data from [62], circled line isour TBD prediction
formed at 600 K for 1500 femtoseconds with a time step of 0.1 fs. We ran this MD
without the vdW contribution, and as expected, saw the layers repel each other.
Turning on the vdW force we ran the simulation again starting with the layers
at 3.7 A and in a slightly offset ABA stacking. The red line in Figure 8.8 plots
the average distance between the layers as a function of time, which is shown on
the right-hand ordinate. As expected the layers oscillate around the 0 K predicted
equilibrium. Measuring the average time between peaks we estimate the frequency
of this oscillation is 2.27 THz.
With this successful fitting of our TB+disp model for graphite and testing in a
molecular dynamics simulation we are now able to examine the interaction between
carbon nanotubes.
105
Figure 8.8: Static 0 K energy scans of two graphene layers in AAA and ABAorientations in black and the results of a molecular dynamics simulation showingthe oscillation around the equilibrium interlayer spacing in red
106
9TBD description of carbon nanotube
interactions
This chapter outlines the application of our optimized tight-binding plus disper-
sion model applied to carbon nanotubes interactions. In the introduction of this
work we discussed the motivation for modeling interactions of collections of nan-
otubes for various nanomechanical based applications. Here we start with the
modeling of CNTs on a graphene substrate, followed by calculations of axially
parallel CNTs taken two at a time, 7 tube bundles, multiwalled tubes, and finally
the case of a C60 outside and inside of a CNT. We discuss some generalized results
of these static energy calculations and fit a universal binding curve for graphitic
107
interactions. Finally we show the results of a dynamic simulation of two (5,5)
interaction nanotubes.
The motivation for focusing on the static energy calculations for sets of tubes is
to investigate their energetically favorable orientations and compare our results to
existing data in literature. Treating these tubes statically (i.e. by not relaxing) is
done for two reasons. First it has been suggested by empirical models and direct
measurement that adjacent tubes start flattening along their mutual faces when
their diameters are greater than 20 A [70, 145], here we will only consider tubes
below this diameter. Secondly, for purposes of comparison, most data available in
literature for tube-graphene, tube-tube cohesive energies adopts this assumption
that they do not significantly distort the tube structure. We will see later though
that there is a noticeable flattening due to van der Waals attraction even for the
smallest (5,5) tubes.
9.1 Tube-graphene interactions
There is an interest in using a graphite substrate to organize and separate CNTs
based on their chirality. Due to the symmetry and similarity of the graphene
plane and the CNT structure there is an energetically favorable orientation (lock-
in position) for tubes on a graphite surface that repeats every 60◦ as illustrated
in Figure 9.1 [20, 41, 95]. These lock-in positions are simply orientations that are
equivalent to ABA stacking of graphite. In a lock-in position a tube is bound
approximately 10% more per unit length which corresponds to tens of eV’s for
a tube only a few hundred nm in length [95]. Experiments of CNTs on HOPG
surfaces manipulated with AFM’s have shown that when in a lock-in position
they roll very easily akin to a rack and pinion gear system. If one end of a tube is
stopped by an impediment and the other end is pushed the tube pivots and shows
a sliding, stick and slip behavior rather than a rolling behavior [41, 42].
We consider the static single point energy scans of tubes on graphene in a lock-in
108
θ = 60◦
Graphite
CNT
Figure 9.1: Bird’s eye view of the various lock-in orientations on graphite for aCNT
position as a function of tube graphene separation. The tubes considered are (n,n)
type tubes of n = 5, 10, 12, and 20. The diameter of these tubes are 3.39, 6.78,
8.14, 13.56 A respectively. The tubes sections are created with the code provided
in [148], and periodic boundary conditions along the axis are implemented to
model infinitely long tubes. We use 3 unit cells for each tube resulting in tubes
with 60, 120, 144, and 240 atoms apiece. The graphene substrate is created to
satisfy proper boundary conditions and orientated in a lock-in position relative
to the tube. A k-point mesh of 7x2x2 shows a convergence of ±0.1 meV for the
smallest system and is used for all tube calculations here. Figure 9.2 shows the
tube graphene interaction energy per unit length of the tube for the various tubes
orientated in one of their lock-in positions vs the wall to wall separation (dww).
The general trend of increasing equilibrium spacing and higher binding energies
with larger tubes can be seen here. These results are summarized in the first four
rows in Table 9.1.
9.2 Tube-tube interactions
Our second consideration is axially parallel tube-tube interactions. Figure 9.3
shows how the tubes are aligned and the notation for rotation (θ) and offset (dz).
109
2.8 3 3.2 3.4 3.6−600
−550
−500
−450
−400
−350
−300
−250
−200
−150
−100
dww
[A°]
inte
ract
ion
ener
gy [m
eV/A
° ]
(5,5)(10,10)(12,12)(20,20)
Figure 9.2: Interaction energy scans of tubes on graphite orientated in a lock-inposition
We are interested in finding the minima on the energy surface defined by θ, dz,
and dww and the absolute value of the cohesive energy at this position as given by
our tight-binding plus dispersion model.
Two tubes are simulated at a time and their initial configuration of θ, and dz are
arbitrary according to the details of the code that generates their coordinates. The
energy landscape defined by θ and dz at a fixed dww is analogous to the graphite
interlayer energy landscape in x and y at fixed z shown earlier in Figure 8.6.
Figure 9.4 shows this θ, dz landscape for two (10,10) tubes separated by 3.1 A.
One can see the analogous ABA and AAA-like stacking (for tubes we will refer to
this as ab and aa orientations). The absolute corrugation of this landscape is ∼
23 meV/A.
For computational efficiency, a minima in the dz direction is found first with the
empirical model of Girifalco. To do this an energy scan of θ and dww is performed
at discrete steps of dz. The dz which provides the deepest minima in the θ, dww
landscape is used in re-scanning the θ, dww landscape with our tight-binding plus
110
dww
dz
θ
Figure 9.3: (12,12) and (5,5) tubes showing adopted orientational notation. θ isa relative twist angle, dww is the wall to wall tube separation, and dz is a verticaloffset distance
0 5 10 15 20 25 30 350
2
4
−225
−220
−215
−210
−205
−200
Ene
rgy
per
unit
leng
th [m
eV/A
° ]
θ [degrees]
dz [A°]
Figure 9.4: Tube-tube θ, dz landscape for two (10,10) tubes at a wall to walldistance of dww = 3.1 A
111
dispersion model.
For all the θ, dww landscapes calculated the dww scans corresponding to the most
favorable and disfavorable θ’s are taken and fit via least squares to a modified
Lennard-Jones curve (see equation (9.2)). Using these fits, a bisection method is
used to calculate the equilibrium spacing (dabww and daa
ww) through the analytical
derivative of the fit. The cohesive energies are reported in energy per unit length
(Eab, Eaa) and are evaluated with the fitted curves at the minima’s. Table 9.1
reports the results of all the tube combinations of the 4 different tubes studied.
The energy scans (for Eab and dabww) as a function of wall separation around the
equilibrium spacing for each tube-tube combination are shown in Figure 9.5.
Eab Eaa Eaa −Eab dabww daa
ww
Tubes meV/A meV/A meV/A A A(5,5)‖graphene -284.44 . . . . . . 3.00 . . .(10,10)‖graphene -375.12 . . . . . . 3.04 . . .(12,12)‖graphene -393.69 . . . . . . 3.05 . . .(20,20)‖graphene -495.31 . . . . . . 3.10 . . .(5,5)‖(5,5) -168.16 -157.03 11.12 3.02 3.10(5,5)‖(10,10) -187.83 -177.17 10.65 3.07 3.14(5,5)‖(12,12) -195.49 -182.49 13.00 3.07 3.16(5,5)‖(20,20) -208.95 -197.28 11.67 3.10 3.18(10,10)‖(10,10) -220.92 -205.24 15.68 3.09 3.20(10,10)‖(12,12) -230.61 -214.13 16.48 3.10 3.21(10,10)‖(20,20) -256.30 -240.98 15.32 3.11 3.20(12,12)‖(12,12) -241.09 -224.07 17.02 3.10 3.22(12,12)‖(20,20) -270.99 -254.34 16.65 3.11 3.21(20,20)‖(20,20) -312.34 -292.96 19.38 3.12 3.22
Table 9.1: Tube-graphene and tube-tube interactions summary
The tube-tube minimum equilibrium separation follows a trend of increasing dabww
with tube radius. When plotted vs the sum of the curvatures defined as τ =
1/r1 +1/r2, with r1 and r2 being the radii of the two interacting tubes, it is nearly
linear with a fitted slope of ∼ −0.22 A2. The minimum cohesive energy per unit
length (Eab) plotted against the sum of the curvatures follows a nice trend too.
Figure 9.6 shows the cohesive energies plotted against τ . The noticeable trend of
the minimum cohesive energy is fitted to a curve of the form
112
2.95 3 3.05 3.1 3.15 3.2 3.25 3.3 3.35−340
−320
−300
−280
−260
−240
−220
−200
−180
−160
wall − wall separation [A°]
inte
ract
ion
ener
gy [m
eV/A
° ]
5−55−105−1210−1010−1212−125−2010−2012−2020−20
Figure 9.5: Tube-tube interaction energy scans corresponding to most favorable θand dz configuration for the various (n,n) tube combinations. Labeling notationis (a-b) = (a,a)‖(b,b).
ϑ(τ) =a
τ b+ c. (9.1)
The optimized parameters are: a = −54.8468 meV/A2, b = 0.7357, and c =
−88.4136 meV/A, with an associated sum squared error (SSE) of 4.8 meV/A.
The form of this curve was chosen firstly to be simple and secondly to satisfy the
condition that as τ tends to zero (i.e. graphene-graphene) the cohesive energy per
unit length tends towards −∞.
Our tube-tube results compare well with similar TB+disp based models of Biro
and Kwon already discussed in § 4.2 [100, 156]. Both of these models have been
applied to the (10,10)‖(10,10) system. The optimized cohesive energy per unit
length for this system with the Biro model is 281.7 meV/atom with an equilibrium
spacing of 3.11 A, compared to our -220.92 meV/atom and equilibrium spacing of
3.09 A (this data is not available for the Kwon model). The corrugation of the
(10,10)‖(10,10) energy surface (Eaa −Eab) for the Biro and Kwon model are 27.1
and 16.6 meV/atom respectively compared to our 15.68 meV/atom.
113
0.1 0.2 0.3 0.4 0.5 0.6
−340
−320
−300
−280
−260
−240
−220
−200
−180
−160(5−5)
(5−10)(5−12)
(10−10)(10−12)
(12−12)
(5−20)
(10−20)
(12−20)
(20−20)
τ − sum of curvature [A° −1]
ϑ −
Coh
esiv
e en
ergy
per
uni
t len
gth
[meV
/A° ]
Figure 9.6: Eab from Table 9.1 plotted against the sum of the curvatures of theinteracting tubes
9.3 Universal binding curve
One can nicely summarize the last two sections on CNT-CNT and CNT-graphene
interactions into a universal binding energy relation (UBER) similar to the original
work of Rose et. al [143]. An UBER curve is a systematic rescaling of the energy vs.
distance curves in a manner which shows a single curve. This has been attempted
for graphitic systems by Girifalco [53]. Using our data set of tube-graphene and
tube-tube interaction energies we first normalized the well depths by dividing
the interaction energies by their cohesive energies (φ(d)/|φ(do)| where φ(d) is the
energy per atom at a wall to wall separation of d and do = dabww the equilibrium
separation). The distance scaling utilized in Girifalco’s analysis is (d−ρ)/(do −ρ)
where ρ is a system dependent scaling parameter. In Girifalco’s representation ρ
was an adjustable parameter that was tuned such that all interaction energy curves
fell on top of one another. In his formulation the value of ρ for two (10,10) tubes
is different than that for two graphene sheets, etc. This factor partially negates
the true universality of an UBER curve. We tried a few distance scaling methods
with the most promising being a definition of ρ as the sum of the curvatures (τ)
of the interacting bodies times a constant σ with units of A2 (ρ = στ). Figure 9.7
114
shows the result of this scaling formulation with σ = 1 A2
for all the tube-tube
and graphene-tube interactions curves already presented and plotted here as the
solid dots. This UBER curve was fitted to a modified LJ potential of the form
Φ(d) =A6
d6+B12
d12+ C. (9.2)
The fitted parameters are A6 = −0.7198, B12 = 0.3565, and C = −0.6366 with an
associated SSE of 2.24 × 10−4, and is shown as the solid line in Figure 9.7. This
universal binding curve is a nice summary for tube-tube-graphene interactions.
Using this fitted curve one may back out estimates of cohesive energies for other
systems that were not considered explicitly here. It should be noted that the
effects of tube flattening for d > 20 A were not considered, so this will only be
valid for the d = 0-20 A regime.
0.9 0.95 1 1.05 1.1 1.15−1.02
−1
−0.98
−0.96
−0.94
−0.92
−0.9
−0.88
−0.86
−0.84
(d−ρ)/(do − ρ)
φ(d)
/ φ
(do)
Figure 9.7: Carbon nanotube/tube/graphene semi universal interaction energyper unit length. ρ is the sum of the curvatures of the interacting bodies
115
9.4 Nanotube loops
For a comparison we have computed the ab stacking energy with Girifalco’s model.
We obtained a (10,10)‖(10,10) intertube cohesive energy of 1.83 eV/nm. At this
point It is interesting to consider the implications of a full classical approach in
comparison with the TB + vdW one. Consider for instance the case of a (10,10)
CNT ring with the smallest bending radius of Rbend = 0.03 m mentioned in the
introduction [113]. The ring configuration is stable because the bending energy
penalty Ubend = 0.5k(1/Rbend)2 is compensated by the van der Waals attractions.
The bending stiffness k of a single-walled CNT of radius R is k = CR3, where C
is the in-plane CNT stiffness [170]. For a (10,10) CNT, the TB model gives C =
423 J/m2 [69]. Taking R = 0.7 nm then k = 455 × 10−18 J nm. Next, from the
balance of resulting bending energy (Ubend = 1.58 eV/nm) and our computed vdW
attraction (i.e., 2.21 eV/nm given in Table 2) we conjuncture that the observed
CNT rings of 0.03 m in radius can be formed from (10,10) CNT of 661 nm in
length, with an overlap portion of 472 nm. On the other hand, a substantially
different estimate is obtained based on the widely used combination of classical
covalent-bonding [18] (leading to C = 236 J/m2, k = 254× 10−18 J nm, and Ubend
= 0.88 eV/nm) and Girifalco’s vdW treatment [53]. It follows that the smallest
observed rings can be formed from (10,10) CNTs of at least 365 nm in length with
an overlap portion of 174 nm.
9.5 Nanotube bundles
We have also performed energy scans on two seven-tube CNT bundles. The two
different bundles modeled were comprised of (10,10) and (12,12) tubes. As in
the tube-tube interactions, PBCs were employed to negate end effects and model
infinitely long tubes. A section of the bundle is shown in Figure 9.8 (a). The
tubes are packed in a hexagonal shape as shown in Figure 9.8 (b). For the (12,12)
bundle the tubes are rotated and shifted such that they are in their optimal ab
116
configurations with respect to one another, as was found above. This is possible
for the (12,12) bundle because (n,n) tubes have rotational symmetry every 2π/n
radians. Since 2π/12 is a multiple of π/3 (i.e. half the interior angle of a hexagon)
there is a possibility of an optimal (12,12) ab bundle. We have made an educated
guess at the configuration of such a bundle by using our optimized results from
the tube-tube minima’s found above. Figure 9.8 (b) illustrates the ab like config-
uration in which we orientate the tubes in. This possible optimal configuration
is not the case for the (10,10) bundle, which will remain partially frustrated. For
this (10,10) bundle the tubes are orientated such that they are in their optimal
ab configuration only along the horizontal axis as shown by the bold ab in Figure
9.8 (b).
The energy scans are performed by dilating the tube bundle and uniformly increas-
ing dww with the energy points plotted in Figure 9.9. The cohesive energy and
equilibrium wall-to-wall separation are summarized in Table 9.2. Included in this
table are the results of Biro et al. whom preformed similar calculations with their
tight-binding plus dispersion model [156]. The cohesive energy per unit length of a
bundle has not to our knowledge been extracted experimentally. Measurements on
bundles comprised of hundreds of SWCNTs of average radius 6.9 ±0.1 A (slightly
larger than (10,10)’s) have shown a wall-to-wall separation of 3.15 A [160]. This
observation, while on a slightly larger and less perfect structure, is very close to
our ideal 7 tube bundle calculations.
(10,10) (12,12)Separation Cohesive energy Separation Cohesive energy
Model A eV/A A eV/ATB+disp 3.11 -2.479 3.12 -2.802Biro [156] 3.12 -3.201 3.13 -3.500
Table 9.2: Comparison between models of a 7 tube hexagonal bundle of (10,10)and (12,12) tubes reporting equilibrium separation and cohesive energy per unitlength
117
(a)
ab
ab
ab
a ba
b
a
b
a
bb
a
ab ba
a b
a b dww
(b)
Figure 9.8: (a) Seven tube bundle of (12,12) tubes (b) Bundle cross section showingwall to wall distance (dww)
3 3.05 3.1 3.15 3.2 3.25 3.3 3.35 3.4−3
−2.9
−2.8
−2.7
−2.6
−2.5
−2.4
−2.3
−2.2
dww
[A°]
Inte
ract
ion
ener
gy
[eV
/A° ]
(10,10)
(12,12)
Figure 9.9: Energy scan on CNT bundles around their equilibrium position
118
9.6 Multiwalled tube
We have also applied our potential to the characterization of MWCNTs. The
motivation for applying our model is to look at some of the finer points related
to MWCNT bearings and oscillators discussed in the introduction. We have done
this for a (5,5) tube nested inside of a (10,10) tube radii of 3.89 A and 6.78
A respectively, with a wall-to-wall separation of 3.39 A. Figure 9.10 schematically
shows the (5,5)‖(10,10) MWCNT system. This figure also shows the relative
angular orientation θ and relative axial displacement dz which are the independent
variables used in plotting the energy landscapes shown for this system in Figure
9.11.
dww = 3.39 A θ
y
dz
z
x
Figure 9.10: Nested (5,5)‖(10,10) CNT pair. The dz direction referred to in Figures9.11 and 9.12 is along the tube axis labeled as z
One sees a relatively large barrier under rotation of 9.7 meV/A, and a much lower
energy barrier under relative translation of 0.268 meV/A. These barriers are easier
to see in Figure 9.12, which shows two slices of the surface plot. The barrier for
rotation is about 36 times that for sliding. These results are similar to Palser who
found rotational and sliding barriers of 7.19 meV/A and 2.07 meV/A respectively
[131], and Charlier who found barriers of 12.68 meV/A and 5.61 meV/A with a
119
DFT LDA approach [25]. Another first principles DFT LDA study of the rota-
tional energy barrier of this system has been reported as 7.75 meV/A [96]. Our
model shows more corrugation and a smaller sliding barrier than these models. In
spite of the larger contact area, the nested tube corrugation is smaller than when
they are placed side-by-side (Table 9.1). Such a reduction in the corrugation due
to the difference in radii between layers is in qualitative agreement with earlier
results [96, 131].
We have preformed similar energy scans for a (10,0)‖(20,0) system which is of
comparable size to the (5,5)‖(10,10) system. As expected we see a similar energy
landscape but with the high and low energy barriers for θ and dz switched around.
Thus making (n,0) MWCNTs better candidates for rotational bearings and (n,n)
MWCNTs a superior translational bearing. MWCNT’s composed of chiral tubes
are expected to have screw like energy landscapes.
For our (5,5)‖(10,10) system we have also calculated a shear modulus of 0.62 GPa,
and a rigid shear mode frequency of 57 MHz which is the analogous shear mode
of E2g(1) = 1.26 THz discussed previously.
9.7 C60 and carbon nanotube interactions
We have not discussed the C60 molecule since the introduction but here we make
some calculations of its interaction energy within and outside of a carbon nan-
otube. We first do a static interaction energy scan of an infinitely long (10,10)
tube with a C60 molecule. The two are aligned to be in a ab like stacking configura-
tion as can be seen in the left hand side of Figure 9.13. The right hand side shows
the relative displacement of the two bodies. Figure 9.14 shows the interaction
energy as a function of wall-to-wall separation. The maximum cohesive energy
is -101 meV/atom at a separation of 2.99 A. For the same system with the C60
nested in the tube we calculate an interaction energy of -560 meV/atom. There
has been discussion of a small energy barrier at the entrance to an open CNT
120
05
1015
20
0
0.5
1
1.5
−1.068
−1.066
−1.064
−1.062
−1.06E
nerg
y pe
r un
it le
ngth
[eV
/A° ]
dz [A°]
θ [degrees]
Figure 9.11: Nested (5,5)‖(10,10) energy landscape with TB+disp model
Figure 9.12: Nested (5,5)‖(10,10) energy landscape with TB+disp model showingthe translational energy change as the circled line
121
[163], but it’s interesting to note that it is 5 times more favorable energetically for
a C60 molecule to be inside of a tube than on the outside. This is supported by
the direct observation of stable “peapod” structures. Peapods are a CNTs filled
with C60 molecules as shown in Figure 9.15 [56].
Figure 9.13: energy scan of a C60 molecule external to a (10,10) tube
9.8 Tube-tube MD simulation
We have nearly exhausted the possible static energy calculations to which we
can compare our results to that found in literature and so far have seen decent
agrement. Our next step is the molecular dynamic simulation of these systems.
For all the static calculations we preformed above we noted that tubes less than
20 A in diameter did not show flattening. We have reason to believe that this is
not true based on our model.
Our system is a pair of infinite aligned (5,5) tubes. We started the simulation with
them near their equilibrium separation of 3.02 A. We ran this system for 1 ps at
dt = 0.1 fs at 600 K. We expected to see a collective rigid body motion of the two
122
2.5 3 3.5 4−110
−100
−90
−80
−70
−60
−50
−40
−30
−20
dww
[A°]
inte
ract
ion
ener
gy [m
eV/a
tom
]
Figure 9.14: energy scan of a C60 molecule external to a (10,10) tube
Figure 9.15: Peapod micrograph. Scale bare = 2 nm
123
tubes slightly rotating and sliding into their equilibrium configuration. Instead of
this we saw a slight rotation and sliding but also a general flattening. Figure 9.16
(a) shows the before shot and (b) shows the two tubes after 1 ps. There is a slight
flattening along the tubes face.
(a) (b)
Figure 9.16: Two (5,5) tubes (a) shows configuration at t=0, (b) shows configu-ration after 1 ps.
124
10Conclusion and future work
We have undertaken the study of the model system of graphite in order to gain
insight to the nature of vdW attraction and repulsion between layers of sp2 bonded
carbon. With the data garnered from experimental insights and first principles
simulations we have developed a tight-binding plus dispersion model which ac-
curately describes both the intra and inter-layer properties of graphite. In this
pursuit we have considered the various physical models that describe dispersion
forces and used them to calculate the C6 dispersion constant for sp2 hybridized
carbon. Our model was extended to describe the interactions between carbon
nanotubes in a molecular dynamics environment. The model was compared to
literature values of equilibrium orientations and cohesive energies of nanotubes.
125
We have preformed some promising initial molecular dynamics studies on two
graphene layers and two small (5,5) tubes. The dynamics proved to be stable
and reproduced the equilibrium interlayer spacing as calculated from static en-
ergy scans. We have reached a point at which the study of the interworkings of
the many carbon nanotube based NEMs devices discussed in the introduction can
be studied further.
126
ATight-binding + dispersion molecular
dynamics overview
Molecular dynamics (MD) is a numerical technique to model the time evolution
of a many body system governed by Newton’s laws of motion
mid2 ~Ri
dt2= ~F i, (A.1)
mi is the mass of particle i, ~Ri is the position vector, and ~F i a vector describing
the forces on i. In atomistic systems the particles denoted by ~Ri are the nuclei.
The treatment of the nuclei in this manner is a consequence of the application of
127
the Born-Oppenheimer approximation which is explained in section Appendix B.7.
There are many flavors of molecular dynamics simulations. The most common MD
simulation consists of N particles in a volume V evolved in time while conserving
energy E, referred to as a NVE ensemble. This is the approach we use, though
there are many other ensembles that may be simulated with the MD technique.
The time integration algorithm used to follow the trajectory of the system is a
major part of a molecular dynamics simulation. Our code utilizes the velocity
Verlet method [164]. This method is symplectic, meaning energy conserving and
is based on the Taylor series expansion of the position and is expressed by two
updating steps of
R(t+ dt) = R(t) + V(t)dt+1
2dt2A(t), (A.2)
V(t+ dt) = V(t) + dt
(A(t) + A(t+ dt)
2
). (A.3)
Where R, V and A are the position, velocity and acceleration matrices respec-
tively (dimensions of 3 x number of atoms). The time step is denoted by dt. The
accelerations are attained through ~Ai = ~F i/mi. We will discuss the force calcula-
tions below, but now start with how to start a MD simulation and the subsequent
steps in running one.
A molecular dynamics simulation needs a few things to start, namely the initial
positions of the atoms to be simulated and initial velocities. The initial velocities
are assigned as a function of the desired temperature. This is done by assigning
random velocities with a Maxwell-Boltzmann distribution, and scaling the total
kinetic energy to reflect the desired temperature. The net momentum of the
system is also zeroed. After these setup steps are complete the program enters
the MD loop where forces are calculated and the system is evolved in time. These
128
two steps are looped for a desired time period and then the simulation is stopped.
These general steps are illustrated in a flow chart in Figure A.1.
Initial nuclear
Assign initial
Stop
yes
no: t = t + dt
Evolve coordinates with
Calculate the three forces
Electronic – Ionic – vdW
velocity Verlet algorithm
Is t > trun?
Post process:
thermodynamic averages
coordinates (Ro)
velocity fMaxwell(T )
Figure A.1: Molecular dynamics flowchart
A.1 Calculation of Forces
We have discussed the MD technique and have seen the need to calculate the
forces on the ions in order to evolve the system. Our system has three distinct
129
force contributions and we can start by defining the force on an ion i in the α
direction where α is one of the cartesian directions x, y, z.
F itot,α = −∂Etot
∂Riα
(A.4)
The forces defined by the tight-binding method are of two distinct types in ac-
cordance with the definition of the energy. Equation (5.33) showed us that the
total energy was the sum of the band structure energy and the repulsive energy,
and analogously the total force is simply the sum of the band structure force
and the repulsive force. We have added a van der Waals energy term which also
contributes to the total force. We can expand equation (A.4) to reflect this
F itot,α = −
(∂EBS
∂Riα
+∂Erep
∂Riα
+∂EvdW
∂Riα
)= F i
BS,α + F irep,α + F i
vdW,α (A.5)
The repulsive ionic and van der Waals forces can be calculated analytically and
are treated differently than the band structure forces; we will cover them first.
A.1.1 Ionic and van der Waal’s forces
Since the repulsive and vdW terms are two body potentials with a known form
the force between two ions is expressed simply as the negative derivatives of their
energy, we call these φrep and φvdW . Both of these forces are simply a function of
distance between two interacting atoms. The analytical negative derivative of our
vdW energy term (equation (??)) with our fitted values substituted and expanded
is shown in equation (8.3) on the last page of this Appendix. The ionic repulsive
force is the negative derivative of its its fitted chebyshev polynomial detailed in
[137].
After finding the force between two interacting bodies it is a simple manner to
decompose the force into the cartesian directions with use of the directional cosine
130
introduced in § 5.6.2.
F iα = dαφ(d) (A.6)
where d is the distance between the atom i and the neighbor considered and dα is
the directional cosine for direction α.
A.1.2 Tight-binding band structure forces
While the repulsive and vdW forces were quite easy to describe the band struc-
ture forces prove a little more difficult to conceptualize. The Hellmann-Feynman
theorem can be used to calculate the forces due to the electronic portion of the en-
ergy in a quantum mechanical manner [66, 44]. The Hellmann-Feynman theorem
posits that the change in energy of a quantum system with respect to a change of
a system dependant parameter P is given by [39].
dE
dP= 〈Ψ|∂H
∂P|Ψ〉 (A.7)
If the parameter P , is a nuclear coordinate and one utilizes the Born-Oppenheimer
approximation the force on a nucleus in the α direction is given by
F iα = −
∑
λ
∂ελ
∂Riα
= − ∂
∂Riα
〈Ψλ|H |Ψλ〉〈Ψλ|Ψλ〉 (A.8)
With our non-orthogonal tight-binding basis we must account for the dependance
of the overlap on the force and the term that represents the change in the expansion
coefficients with respect to the ionic coordinate, the latter term is referred to as
the Pulay force [140]. Appendix C shows the derivation of this force term and is
simply given here as
131
F iα = −
∑
λ
fλ
∑
p,q
cλ∗p cλq
[∂Hpq
∂Riα
− Eλ∂Spq
∂Riα
](A.9)
or equivalently described via the density (ρ) and energy (σ) matrices as
F iα = −Tr
[ρ∂Hpq
∂Riα
]+ Tr
[σ∂Spq
∂Riα
](A.10)
A.2 Periodic boundary conditions
It becomes useful computationally when simulating bulk materials to implement
periodic boundary conditions on a small cell of atoms. This concept can be vi-
sualized by considering a three dimensional cell defined by three lattice vectors
(~a1,~a2,~a3) that contains a set of atoms. When calculating forces or evaluating the
energy of the system an atoms local environment is not only those atoms in the
cell but also periodic images (rigid translations) of its own cell. For an atom at
vector position ~r it has images of itself at
~rimage = ~r + l~a1 +m~a2 + n~a3, (A.11)
where l,m, and n ∈ Z. Figure A.2 shows a two-dimensional example of PBCs for
a square box containing two entities.
132
Figure A.2: Schematic of periodic boundary conditions. Center solid box is theunit cell surrounded by 8 copies of itself in two dimensions. The circle sees notonly the triangle in its own box (long arrow) it also sees its images (short arrow).
133
φvdw(d) = −dEvdW
dd
−248(1−e−1.45 d−1.45 de−1.45 d−1.05 d2e−1.45 d−0.51 d3e−1.45 d−0.18 d4e−1.45 d−0.05d5e−1.45 d−0.013 d6e−1.45 d)e10 d−110.99
(e10.0 d−110.99+1)2
d6−
148.8 1−e−1.45 d−1.45 de−1.45 d−1.05 d2e−1.45 d−0.508 d3e−1.45 d−0.18 d4e−1.45 d−0.053 d5e−1.45 d−0.013 d6e−1.45 d
(e10 d−110.99+1)d7+
24.8 1.0×10−10 d3e−1.45 d−1.0×10−11 d5e−1.45 d+0.019 d6e−1.45 d
(e10.0 d−110.99+1)d6
134
“I don’t like it, and I’m sorry
I ever had anything to do with
it.”
Erwin Schrodinger, speaking
about quantum mechanics
BQuantum Mechanics Overview
The following is a very brief review of the relevant quantum mechanical postulates
that are needed as back ground to the work contained in this document.
The quantum mechanical wave function for a single particle, Ψ(x, y, z, t) or Ψ(~r, t),
describes the temporal and spatial evolution of a quantum mechanical particle.
The product of Ψ∗(~r, t) Ψ(~r, t) is the probability density function of a quantum-
mechanical particle (*denotes the complex conjugate). Ψ∗(~r, t) Ψ(~r, t)dτ is the
probability of finding a particle in the differential volume dτ = dxdydz. Therefore
∞∫
−∞
Ψ∗(~r, t) Ψ(~r, t)dτ = 1 (B.1)
135
if a wave function Ψ(~r, t) fulfills equation (B.1), then Ψ(~r, t) is called a normalized
wave function. Equation (B.1) is the normalization condition and implies the fact
that the particle must be located somewhere in 3 space. Another restriction on
wave functions is that they be single valued and continuous [116]. Wave function
are easily normalized by applying a normalization constant which is found by
An =
( ∞∫
−∞
Ψ∗(~r, t) Ψ(~r, t)dτ
)−1/2
(B.2)
and then the normalized wave function is simply
Ψn(~r, t) = AnΨ(~r, t) (B.3)
B.1 Observables and expectation values
For every observable dynamic variable there exists a Hermitian operator in quan-
tum mechanics. Table B.1 shows a few dynamical variables and their correspond-
ing quantum mechanical operators.
Observable Symbol Operatorr r rp p −i~∇E E i~ ∂
∂t
Table B.1: Position, momentum, and total energy and their corresponding oper-ator symbols and operators, where ∇ = ( ∂
∂x+ ∂
∂y+ ∂
∂z)
The expectation value, 〈ξ〉, of any dynamical variable ξ, is calculated from the
wave function according to
〈ξ〉 =
∫Ψ∗(~r, t)ξopΨ(~r, t)dτ∫Ψ∗(~r, t)Ψ(~r, t)dτ
(B.4)
where ξop is the operator of the dynamical variable ξ. The expectation value of a
136
dynamical variable is also referred to as an average value or ensemble average.
B.2 Dirac Notation
The integrals in equation (B.4) show up often in quantum mechanics and are given
a short hand notation.... and is denoted by the brackets 〈...〉. We have introduced
the Dirac bra-ket notation for representing the integration. Equation
〈ξ〉 =〈Ψ|ξop|Ψ〉〈Ψ|Ψ〉 (B.5)
B.3 Time-dependant Schrodinger equation
The wave function evolves in time according to the time-dependant Schrodinger
equation. The Schrodinger equation is not derivable from elementary principles,
rather it is stated as a postulate.
We can gain some insight into the time-dependant Schrodinger equation by con-
sidering the following pseudo-derivation. Starting with a classic single particle
system we describe the total energy as the sum of the kinetic and potential energy
E =p2
2m+ V (r) (B.6)
where p is the momentum, m is the particle’s mass and V (r) is the potential energy
as a function of position. If we replace the classic variables in equation (B.6) with
the corresponding operators from table B.1 operating on the wave function we
arrive at the single-particle time-dependant Schrodinger equation.
i~∂Ψ(~r, t)
∂t= HspΨ(~r, t) (B.7)
where Hsp is the single-particle hamiltonian
137
Hsp = − ~2
2m∇2 + V (~r) (B.8)
note that there is nothing in (B.7) that accounts for spin or relativistic effects.
B.4 Time-independent Schrodinger equation
If the potential V is independent of time, the total wave function can be separated
into temporal and spatial terms.
Ψ(~r, t) = ψ(~r)υ(t) (B.9)
plugging equation (B.9) into the time-dependant Schrodinger equation we get
ψ(~r)i~dυ(t)
dt= υ(t)Hspψ(~r) (B.10)
Moving time dependant terms to the left and spatial terms to the right
i~
υ(t)
dυ(t)
dt=
1
ψ(~r)Hspψ(~r) (B.11)
both sides must be equal, so the introduction of a separation constant, E, results
in two ordinary differential equations
1
υ(t)
dυ(t)
dt= −iE
~(B.12)
Hspψ(~r) = Eψ(~r) (B.13)
Equation (B.12) is the time dependant part which has a solution of
138
υ(t) = e−iEt/~ (B.14)
Now we can rewrite the total wave function as
Ψ(~r, t) = ψ(~r) e−i(E/~)t (B.15)
Equation (B.13) is the the time-independent Schrodinger equation. The time-
independent Schrodinger equation is in the form of an eigenvalue equation.
motivation for solving. Solving the Schrodinger equation for the wave function
of a general system of electrons and nuclei, leads the modeler to any property of
interest. This is of course great interest to the scientist and philosopher. But
actually solving the Schrodinger is a beast of a problem.
B.5 General Hamiltonian
So far we have been looking at a single particle problem with no reference to the
type of particle or the type of potential. In a condensed matter system we are
dealing with collections of many electrons and nuclei. In order to describe this
general system we must define the many particle hamiltonian, which in a simplified
form is
H = Tn(R) + Te(r) + VeN(r,R) + VNN (R) + Vee(r) (B.16)
where r = {~r1, ~r2, . . . , ~ri} are the electron coordinates, and R = {~R1, ~R2, . . . , ~RI}
are the nuclear coordinates.
Tn(R) = −~2∑I
∇2
I
2MIis the kinetic energy of the nuclei
Te(r) = − ~2
2me
∑i
∇2i is the kinetic energy of the electrons
139
VeN(r,R) = 12
∑Ii
ZIe2
RIiis the electron nuclei Coulomb attraction
VNN(R) = 12
∑I 6=J
ZIZJe2
RIJis the nuclei nuclei repulsion
Vee(r) = 12
∑i6=j
e2
Rijis the electron electron repulsion
The corresponding many-particle wave function for a general system is a function
of all coordinates.
Ψ = Ψ(~r1, ~r2, . . . , ~ri, ~R1, ~R2, . . . , ~RI) (B.17)
B.6 Complexity
Solving the Schrodinger equation for this system and describing the eigenstates of
this many body problem is enormously complex. The complexity of this problem
is somewhat elucidated by walking through the analytic solution for the simplest
atomic system, the hydrogen atom, see [116]. In fact this is the only atomic system
that can be treated analytically, moving up to helium, or even a hydrogen dimer
proves impossible to solve. In order to make these and other more complex systems
tractable within a quantum mechanical treatment, a series of approximations must
be applied, the two most important are the Born-Oppenheimer and one-electron
approximation.
B.7 Born-Oppenheimer approximation
The Born-Oppenheimer approximation underlies almost all atomistic models. Col-
loquially it is explained by considering the nuclei electron mass ratio (which is
approximately 1820 : 1). At any given time the nuclei are moving at a much
slower pace relative to the electrons. The electronic wave function is assumed to
instantaneously (adiabatically), adjust to any nuclear movement. This thinking
lead Born and Oppenheimer to posit a decoupling of electronic and ionic degrees
of freedom [14].
140
Ψ({ri}, {RI}) ≈nuclei︷ ︸︸ ︷
χ({Ri})electrons︷ ︸︸ ︷
ψ({ri}; {RI}) (B.18)
Here we see that the electronic wave function depends on the electronic coordinates
and parametrically on the nuclear coordinates, denoted by the semi-colon. The
electronic hamiltonian for the electronic wave function is
He = Te(r) + VeN(r;R) + Vee(r) (B.19)
The nuclear part of the total wavefunction χ({Ri}) is not typically solved via
the Schrodiger equation. Rather the nuclei are usually treated as classic particles
coulombically interacting with one another. This approximation is good if the de
Broglie wavelength (Λ) of the nuclei is much less then the average nearest neighbor
distance (a), where the de Broglie wavelength is given as
Λ =
√2π~2
MkBT(B.20)
where M is the nuclear mass, T is the temperature and kB is Boltzmann’s constant.
This approximation fails when one is interested in considering phenomena includ-
ing but not limited to: excited state transitions, vibronic coupling, electron-hole
pair excitations [87, 123, 46]
B.8 One-Electron Approximation
The one-electron approximation is attributed to Hartree, the idea is that instead
of a total electronic wave function ψ({ri}; {RI}), one can assign to each electron
an individual wave function and energy. The hamiltonian for the one-electron
wavefunctions of the three terms in (B.19), the first two depend on the coordinates
of only one electron.
141
H1−e = Te(r) + VeN(r;R) (B.21)
B.9 Variational Method
We have already stated how it is futile to try to solve the time-independent
Schrodinger equation beyond the hydrogen atom. Solving the TISE for an ex-
tended system can be done approximately via the variational method otherwise
known as the Rayleigh-Ritz method.
With respect to solving the TISE we will restate it here for the ground state of a
given system
Hψo = εoψo (B.22)
where ψo and εo are the exact ground state wave function and energy. Multiplying
both sides of equation (B.22) by ψ∗o then integrating over space and solving for
the ground state energy we get
εo =〈ψo|H|ψo〉〈ψo|ψo〉
(B.23)
According to the variational theorem if any trial wave function Ψ is substituted
into equation (B.23) in lieu of ψo, the energy expectation value will be greater
than that of the true wave function ψo [5].
εtrial ≥ εo (B.24)
The equality holds only if Ψ = ψo. The trial function Ψ may contain variational
parameters. A particularly intuitive and productive trial function consists of a
linear combination of atomic orbitals (LCAO). The trial function generally takes
142
the form
Ψ =n∑
i=1
ciφi (B.25)
Where n is the number of basis functions, the c’s are the n unknown expansion
coefficients and φi are atomic like orbitals. The goal is to minimize εtrial and thus
attain the best emulation of the true wave function for the assumed basis. This is
done by by the method of variations and leads to n simultaneous linear equations
of the form
∂εtrial
∂cn= 0 (B.26)
B.10 Extended example
By walking through an explicit example of this problem we will see the method-
ology and elucidate its transformation into a generalized eigenvalue problem. To
keep things simple we will consider the simplest non-trivial trial function. This
wave function has the form
Ψ = c1φ1 + c2φ2 (B.27)
Taking this trial wave function and plugging it into equation (B.23).
εtrial =
∫(c∗1φ
∗1 + c∗2φ
∗2)H(c1φ1 + c2φ2)dτ∫
(c∗1φ∗1 + c∗2φ
∗2)(c1φ1 + c2φ2)dτ
(B.28)
Expanding the numerator and denominator
εtrial =c∗1c1H11 + c∗1c2H12 + c∗2c1H21 + c∗2c2H22
c∗1c1S11 + c∗1c2S12 + c∗2c1S21 + c∗2c2S22(B.29)
143
Hij and Sij are referred to as Hamiltonian and overlap matrix elements respec-
tively, and are are defined as
Hij =
∫φ∗
i Hφjdτ = 〈φi|H|φj〉 (B.30a)
Sij =
∫φ∗
iφjdτ = 〈φi|φj〉 (B.30b)
We can again simplify equation B.29.
εtrial =
n∑i,j=1
c∗i cjHij
n∑i,j=1
c∗i cjSij
(B.31)
Now applying the method of variations.
∂εtrial
∂c∗i=
n∑j=1
cjHij
n∑j=1
c∗i cjSij
−
n∑j=1
c∗i cjHij
(n∑
j=1
c∗i cjSij
)2
n∑
j=1
cjSij = 0 (B.32)
Note that this leads to n equations and the sums is now over j only. Rearranging
and multiplying both sides byn∑
j=1
c∗i cjSij
n∑
j=1
cjHij =
n∑j=1
c∗i cjHij
n∑j=1
c∗i cjSij
n∑
j=1
cjSij (B.33)
Notice that the fraction on the right hand side of equation (B.33) is simply the
energy expectation value from equation (B.23).
n∑
j=1
cjHij = εtrial
n∑
j=1
cjSij (B.34)
144
At this point to put the equations in a more understandable form we can expand
them with our sample basis function. These n = 2 equations are
c1H11 + c2H12 = εtrial(c1S11 + c2S12) (B.35a)
c1H21 + c2H22 = εtrial(c1S21 + c2S22) (B.35b)
(we could show how the rearrangement of these equations leads to the secular
determinant, and then substituting in to find the a’s) Putting these equations in
matrix format shows the form of the generalized eigenvalue problem.
H11 H12
H21 H22
c1
c2
= εtrial
S11 S12
S21 S22
c1
c2
(B.36)
Hc = εSc (B.37)
Equation B.37 is simply a generalized eigenvalue problem. H and S are the Hamil-
tonian and overlap matrices, and c is a column vector of the expansion coefficients.
There will be as many eignevalues and eigenvectors as there are terms in the ex-
pansion n.
It is now clear how minimizing the energy function by the method of variations is
equivalent to solving the generalized eigenvalue problem.
145
CNon-orthogonal Hellmann-Feynman
forces
We use the Hellmann-Feynman theorem to calculate electronic forces on the nuclei
in our molecular dynamics code. Since we are working with a non-orthogonal basis
the derivation is a bit more convoluted than that for the orthogonal basis.
C.1 Model Review
We begin this derivation with a little restatement of the nuts and bolts of the
system we are working with. Since we are working with in the framework of the
variational principle we write our wave functions as linear combinations as
146
Ψλ =∑
p
cλpφp (C.1)
The wave-function Ψλ corresponds to state λ, the index p in our case goes over
number of electrons × size of basis. cλp is just the element of the eigenvector
corresponding to the λth state. Also keep in mind the non-orthogonality of our
basis and the definition of the overlap matrix elements.
〈φp|φq〉 = Spq (C.2)
After the diagonalization of the generalized eigenvalue problem we have all neces-
sary information to proceed with calculating the forces via the Hellmann-Feynman
theorem which follows.
C.2 Force derivation
The electron energy for state λ is simply
Eλ = 〈Ψλ|H|Ψλ〉 (C.3)
Using equation (C.1) to express the wave-function we can write Eλ as
Eλ =∑
p,q
cλ∗p cλqHpq (C.4)
Taking the partial derivative with respect to a cartesian direction α = (x, y, z),
for a particular atom i (Riα) we get
∂Eλ
Riα
=∂ 〈Ψλ|H|Ψλ〉
∂Riα
(C.5)
147
or in terms of the expansion coefficients
∂Eλ
Riα
=∑
p,q
[∂cλ∗p
∂Riα
Hpqcλq + cλ∗p
∂Hpq
∂Riα
cλq + cλ∗p Hpq
∂cλ∗q
∂Riα
](C.6)
Rearranging this in the form
=∑
p,q
cλ∗p
∂Hpq
∂Riα
cλq +∑
p
∂cλ∗p
∂Riα
∑
q
Hpqcλq +
∑
q
∂cλq∂Ri
α
∑
p
Hpqcλ∗p (C.7)
Shows that the two terms underlined are simply the left hand side of equation
(B.34) repeated here for convenience
∑
p
Hpqcλp = Eλ
∑
p
Spqcλp (C.8)
Switching these underlined terms with their counter parts just noted leaves us
with
∂Eλ
Riα
=∑
p,q
cλ∗p
∂Hpq
∂Riα
cλq +∑
p
∂cλ∗p
∂Riα
Eλ∑
q
Spqcλq +
∑
q
∂cλq∂Ri
α
Eλ∑
p
Spqcλ∗p (C.9)
Consider the last two terms on the right hand side. Moving the sums to the left
and isolating the expansion coefficients and their derivatives leaves
∂Eλ
Riα
=∑
p,q
cλ∗p
∂Hpq
∂Riα
cλq +∑
p,q
EλSpq
(∂cλ∗p
∂Riα
cλq +∂cλq∂Ri
α
cλ∗p
)
︸ ︷︷ ︸∂cλ∗
p cjq
∂Riα
(C.10)
As noted by the underbrace the term in the brackets is simply the partial derivative
of the product of the expansion coefficient and its complex conjugate with respect
to the ionic degree of freedom. Now putting the double sum out front we have a
148
clean form given as
∂Eλ
Riα
=∑
p,q
[cλ∗p
∂Hpq
∂Riα
cλq + EλSpq
∂cλ∗p cjq
∂Riα
](C.11)
The evaluation of this term is complicated by the fact of the dependance of
the expansion coefficients and the ionic coordinate. This would require a re-
diagnolization each time this force is computed. This is an expensive operation
and is to be avoided.
To make progress we go a little off course by looking at the partial derivative of
the overlap with respect to the ionic coordinate. First we restate the condition
that any two eigenvectors of a linear operator are orthogonal to one another. This
is succinctly expressed by
〈Ψi|Ψj〉 = δij (C.12)
Considering the non-trivial example (i.e. i=j), and using the definition of the
wave-function given in equation (C.1) we have
〈Ψλ|Ψλ〉 =∑
p,q
cλ∗p cλqSpq = 1 (C.13)
Now taking the partial derivative of the overlap with respect to the ionic coordinate
Riα, and not fully expanding the term in the expansion coefficients we have
∑
p,q
[∂cλ∗p c
λq
∂Riα
Spq + cλ∗p cλq
∂Spq
∂Riα
]= 0 (C.14)
Rearranging this we have
149
∑
p,q
Spq
∂cλ∗p cλq
∂Riα
= −∑
p,q
cλ∗p cλq
∂Spq
∂Riα
(C.15)
Taking a look back at equation (C.11) we see that we can replace the partial
over the expansion coefficients with the partial over the overlap matrix elements.
Making this substitution equation (C.11) is now written as
∂Eλ
Riα
=∑
p,q
[cλ∗p
∂Hpq
∂Riα
cλq −Eλcλ∗p cλq
∂Spq
∂Riα
](C.16)
or more compactly
∂Eλ
Riα
=∑
p,q
cλ∗p cλq
[∂Hpq
∂Riα
−Eλ∂Spq
∂Riα
](C.17)
These Hamiltonian and overlap derivatives are evaluated via a finite difference
method in our code. We can now express the force on atom i in the α direction
with the negative of the derivative of the energy as
F iα = −
∑
λ
fλ∂Eλ
Riα
= −∑
λ
fλ
∑
p,q
cλ∗p cλq
[∂Hpq
∂Riα
−Eλ∂Spq
∂Riα
](C.18)
Where as mentioned earlier fλ is the occupation of state λ. This result can be
equivalently and succinctly stated through the use of the density matrix which is
defined by
ρp,q =∑
λ
fλcλ∗p c
λq (C.19)
and the energy matrix σ defined by
σp,q =∑
Eλfλcλ∗p c
λq (C.20)
150
Equation (A.9) can now be stated as
F iα = −Tr
[ρ∂Hpq
∂Riα
]+ Tr
[σ∂Spq
∂Riα
](C.21)
where Tr is the trace of the matrix.
151
DMultipole Expansion
We seek a power series expansion of the Columbic energy between all charges in
one molecule and those in another. This expansion is known as the multipole
expansion and we present it here in the formalism of Margenau [112].
Molecule A has i charged particles with position ~ri and charge qi with a origin at
its nucleus, and Molecule B has j charged particles with position ~rj and charge qj
with a origin at its nucleus. The vector between the nuclei is ~R. Components of
the vectors ~R are X, Y and Z and for the charge coordinates the components of
~rj will be referenced as xj , yj and zj .
The total charge on molecules A and B is simply the sum of their charges.
152
+
−
−
+
−
−
−−
~r1~r1
~r2
~r2
~ri ~rj~R
A B
Figure D.1: Two separated charge clouds A and B
q =∑
i
qi (D.1)
q′
=∑
j
qj (D.2)
The prime denotes molecule 2 were as no prime is simply molecule 1. This is used
through out this derivation.
The dipole vector is defined as
p1 =∑
i
qi~ri (D.3)
A given component of the dipole vector, for example the z component is
p1z =∑
i
qizi (D.4)
With these definitions of the system we can start by first defining the Columbic
potential seen at the origin of molecule 2 due to the charges in molecule 1. This
potential can be written as
153
ϕ =∑
i
qi
|~R− ~ri|(D.5)
Expanding this in a multi-variable Taylor series in the components of ~ri about
zero, and assuming |~ri| ≪ |~R|, we arrive at
ϕ =1
r
∑
i
qi +1
r2
(X
r
∑
i
qixi +Y
r
∑
i
qiyi +Z
r
∑
i
qizi
)+
1
r3
[1
2
(3X2
r2− 1
)∑
i
qix2i +
1
2
(3Y 2
r2− 1
)∑
i
qiy2i +
1
2
(3Z2
r2− 1
)∑
i
qiz2i +
3XY
r2
∑
i
qixiyi +3XZ
r2
∑
i
qixizi +3Y Z
r2
∑
i
qiyizi
]+ . . . h.o.t′s (D.6)
r is the magnitude of the vector ~R.
Now we write the energy terms associated with each charge in molecule 2 in-
teracting with the potential developed in the previous equation. This is written
as
V =∑
j
qj ϕ(|~R+ ~rj|) (D.7)
expanding this in a Taylor series again
V =∑
j
qjϕ+∑
j
qj
(xj∂ϕ
∂X+ yj
∂ϕ
∂Y+ zj
∂ϕ
∂Z
)+
1
2!
∑
j
qj(x2j
∂2ϕ
∂X2+ . . .+ 2yizi
∂2ϕ
∂Y ∂Z) + . . . h.o.t′s (D.8)
This expression can be simplified considerably if one picks the internuclear axis
154
along one of the cartesian axis. We will do this and state that Z = r and X =
Y = 0. Equation (D.9) is the form and the terms are given in table D.1.
V =∞∑
n
Pn (D.9)
n Pn
1 r−1qq′
2 r−2(q′pz − qp′
z)3 r−3(q′w3 + qw
′
3)4 r−4(q′w4 + qw
′
4)5 r−5(q′w5 + qw
′
5)6 r−3
∑ij
qiqj(xixj + yiyj − 2zizj)
7 r−4∑ij
qiqj [r2i rz − z2
j rz + (2xixj + 2yiyj − 2zizj)(zi − zj)]
......
∞ ...
Table D.1: Tabulation of selected terms in the multipole expansion
The terms wn and w′
n, in terms 3-5 above are defined by
wn ≡∑
i
qirni Pn(cosθi) (D.10)
w′
n ≡∑
j
qjrnj Pn(cosθj) (D.11)
Where Pn is the Legendre polynomial of order n.
155
EPerturbation Theory
Here we show the derivation of the non-degenerate perturbation theory.
We start with a system that has a known full solution
H(0) |n〉 = E(0)n |n〉 (E.1)
The eigenstates |n〉 forms a complete orthonormal basis. The perturbation in this
derivation will be done for the ground state energy and wave functions n = 0, but
the result is true for any n.
If this system is slightly perturbed by for example an external field it is assumed
156
that the solution is only slightly different that the unperturbed solution.
We introduce λ which is....
H = λ0H(0) + λ1V (1) + λ2V (2) + . . . (E.2)
ψ0 = λ0ψ(0)0 + λ1ψ
(1)0 + λ2ψ
(2)0 + . . . (E.3)
E0 = λ0E(0)0 + λ1E
(1)0 + λ2E
(2)0 + . . . (E.4)
Now we use these definitions in the Schrodinger equation Hψ0 = E0ψ0, and group-
ing terms by in their contribution in λi we get
λ0(H(0)ψ
(0)0 −E
(0)0 ψ
(0)0
)+
λ1(H(0)ψ
(1)0 + V (1)ψ
(0)0 − E
(0)0 ψ
(1)0 − E
(1)0 ψ
(0)0
)+
λ1(V (1)ψ
(1)0 + V (2)ψ
(0)0 + H(0)ψ
(2)0 − E
(1)0 ψ
(1)0 −E
(0)0 ψ
(2)0 − E
(2)0 ψ
(0)0
)+ . . . = 0
(E.5)
Each term in this series has to be zero individually because of the arbitrariness of
the variable λi. This can be written as i equations, which are just each term in λi
equating to zero.
H(0)ψ(0)0 = E
(0)0 ψ
(0)0 (E.6)
(H(0) − E
(0)0
)ψ
(1)0 =
(E
(1)0 − V (1)
)ψ
(0)0 (E.7)
157
(H(0) − E
(0)0
)ψ
(2)0 =
(E
(2)0 − V (2)
)ψ
(0)0 +
(E
(1)0 − V (1)
)ψ
(1)0 (E.8)
E.1 First order correction to energy
We begin this by writing the perturbed wavefunction as a linear combination of
the unperturbed eigenvectors
ψ(1)0 =
∑
n
anψ(0)n =
∑
n
an |n〉 (E.9)
This is plugged into equation (E.7)
∑
n
an
(H(0) − E
(0)0
)|n〉 =
(E
(1)0 − V (1)
)|0〉 (E.10)
The left hand side can be expanded to
∑
n
anH(0) |n〉 −
∑
n
anE(0)0 |n〉 (E.11)
The first term is simply the left hand side of the the Schrodinger equation summed
over all states n and can be substituted with the right hand side of the Schrodinger
equation summed over all states
∑
n
anH(0) |n〉 =
∑
n
anE(0)n |n〉 (E.12)
Using this identity and substituting it into equation (E.10) we get
∑
n
an
(E(0)
n − E(0)0
)|n〉 =
(E
(1)0 − V (1)
)|0〉 (E.13)
multiplying both sides by the bra 〈0|
158
∑
n
an
(E(0)
n −E(0)0
)〈0|n〉 = E
(1)0 〈0|0〉 − 〈0|V (1)|0〉 (E.14)
Considering the left hand side first, we note that because of the orthonormality of
the set
〈a|b〉 = δab (E.15)
The only remaining non-zero term due to orthogonality in the sum is when n = 0,
but the difference of the eigenvalue E(0)0 with its self is of course zero, leaving the
whole left hand side zero. The right hand side of (E.14) is then rearranged and
we see that the first order correction to the energy is
E(1)0 = 〈0|V (1)|0〉 (E.16)
or in the matrix element shorthand.
E(1)0 = H
(1)00 (E.17)
This is simply the expectation value of the perturbation acting on the unperturbed
state |0〉.
E.2 First order correction to the wavefunction
We have in the previous section proposed the form for the corrected wavefunction
as a linear combination of the unperturbed state, see equation (E.9). We now
need to find a definition for the expansion coefficients an.
Starting with equation (E.13), we multiply both sides by the bra 〈k|.
159
∑
n
an 〈k|(E(0)
n −E(0)0
)|n〉 =
(E
(1)0 − V (1)
)|0〉 (E.18)
Again invoking the orthonormality constraint shown in equation (E.15), the only
surviving term of the sum in the previous equation is the kth
ak
(E
(0)k −E
(0)0
)= E
(1)0 〈k|0〉 − 〈k|V (1)|0〉 (E.19)
now considering the case when k 6= 0, the first term on the right hand side is zero
and we can solve the previous equation for ak.
ak =H
(1)k0
E(0)0 − E
(0)k
(E.20)
writing the
ψ(1)0 =
′∑
k
(H
(1)k0
E(0)0 − E
(0)k
)ψ
(0)0 (E.21)
The prime denotes the sum does not include zero.
E.3 Second order correction to the energy
following an analogous process for the first order correction to the energy we define
the second order correction to the wavefunction again as a linear combination of
the unperturbed state.
ψ(2)0 =
∑
n
bnψ(0)n =
∑
n
bn |n〉 (E.22)
This form is substituted into equation (E.8), which is repeated here
160
(H(0) − E
(0)0
)ψ
(2)0 =
(E
(2)0 − V (2)
)ψ
(0)0 +
(E
(1)0 − V (1)
)ψ
(1)0
∑
n
bn
(H(0) −E
(0)0
)|n〉 =
(E
(2)0 − V (2)
)|0〉 +
∑
n
an
(E
(1)0 − V (1)
)|n〉 (E.23)
multiplying through by the bra 〈0|
∑
n
bn
(H(0) −E
(0)0
)〈0|n〉 =
E(2)0 〈0|0〉 − 〈0|V (2)|0〉 +
∑
n
an 〈0|(E
(1)0 − V (1)
)|n〉 (E.24)
The left hand side is zero via the same argument used in the first order correction
to the energy. The first term on the right is simply the second order correction to
the energy because of orthonormality. The last term on the right hand side can
be simplified significantly. First lets take it apart by isolating the n = 0 term in
the sum and taking the sum over everything but zero.
∑
n
an 〈0|(E
(1)0 − V (1)
)|n〉 =
a0
(E
(1)0 − 〈0| V (1) |0〉
)
︸ ︷︷ ︸=0
+
′∑
n
anE(1)0 〈0|n〉
︸ ︷︷ ︸=0
−′∑
n
an 〈0| V (1) |n〉 (E.25)
With these simplifications we can rewrite equation (E.24) and solve for the second
order correction to the energy.
161
E(2)0 = 〈0| V (2) |0〉 +
′∑
n
an 〈0| V (1) |n〉 (E.26)
Using the the definition of the expansion coefficients derived earlier and stated in
equation (E.20) and the shorthand matrix notation equation (E.26) can be stated
as
E(2)0 = V
(2)00 +
′∑
n
V(1)n0
E(0)0 −E
(0)n
V(1)0n (E.27)
If the operator V is hermitian than V(1)n0 V
(1)0n = |V (1)
0n |2 and we can write our final
form of the second order correction to the energy as
E(2)0 = V
(2)00 +
′∑
n
|V (1)0n |2
E(0)0 − E
(0)n
(E.28)
162
FOptimization
A major part of this work was in the fitting of the van der Waals energy term. This
process utilized the hybrid method of Powell as implemented in the MINPACK
library from Argonne National Laboratory. Here we give the basic outline of how
this method works. First we start with the general steps to a minimization routine.
1. The vector (~xo) contains the initial guess of the solution
2. Compute a search direction (~Sk)
3. Compute length of step (αk) in search direction
4. Advance system to new point ~xk+1 = ~xk + αk~Sk
163
5. Check for convergence δΠδ~x< ǫ
6. If system is not converged increment k and got to step 2
In the steepest decent minimization algorithm the search direction is simply the
gradient of the function evaluated at the initial guess. This is followed by a line
minimization to find αk. This procedure guarantees that the gradient at the
new point ~xk+1 is conjugate to the previous search direction ~Sk. This makes this
method simple in that the search direction is known and preforming step 3 can be
done analytically with an assumed quadratic form. Yet this method is inefficient
and has very slow convergence. A more robust search algorithm is the conjugate
gradient search algorithm which is basically the same but finds its search directions
in another fashion.
We can understand this method through an example. Imagine that we have pre-
formed the first step of the steepest descent method and the new position is ~xk+1.
We will call the old search direction, ~Sk, ~u. We want to find a new search direction
~v such that the gradient ∇Π remains conjugate to ~u as we move along this new
direction ~v [139]. We can find this direction by satisfying this condition
0 = ~u · A · ~v (F.1)
Were A is the Hessian matrix of the function at the current solution vector ~xk.
This is defined by
[A]ij =δ2Π
δ~xiδ~xj
∣∣∣∣∣~x=~xk
. (F.2)
Computing the Hessian matrix can be prohibitive either from a computational
or time point of view so it is avoided. The new search direction ~v can be found
approximately through either the Fletcher-Reeves, or the Polak and Ribiere meth-
164
ods which both avoid computing the Hessian matrix. Another conjugate gradient
scheme is Powell’s method [45]. Starting with a set of directions ~ui initialized to
the basis vectors
~ui = ei i = 1 . . .N (F.3)
N is the dimension of the system. The recipe for this algorithm is the following
steps.
1. Save starting position as ~xo
2. For i = 1 to N move ~xi−1 to the minimum along direction ~ui and call this
point ~xi
3. For i = 1 to N − 1, set ~xi = ~xi+1
4. Set ~uN = ~xN − ~xo
5. Move ~xN to the minimum along direction ~uN and call this point ~xo
6. Repeat until converged
Powell proved that it takes N iterations of the above method and N(N + 1) total
line minimizations to exactly minimize a quadratic form.
165
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