SIM UNIVERSITY
SCHOOL OF SCIENCE AND TECHNOLOGY
DEVELOPMENT AND PARAMETERIZATION OF A
MULTI-PURPOSE ATOMIC AND MOLECULAR
POTENTIAL ENERGY FUNCTION
STUDENT
SUPERVISOR
: E0706507 (PI NO.)
: DR ALAN LIM
PROJECT CODE : JUL2009/ENG/028
A project report submitted to SIM University
in partial fulfillment of the requirements for the degree of
Bachelor of Engineering
May 2010
ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
TABLE OF CONTENTS
Table of contents ..................................................................................................................... 1
Abstract .................................................................................................................................... 3
Acknowledgement .................................................................................................................... 4
List of figures ........................................................................................................................... 5
List of tables ............................................................................................................................. 6
1. INTRODUCTION
1.1 Project Objectives ................................................................................................................ 7
1.2 Overall Objective ................................................................................................................. 7
1.3 Proposed Approach and Method ......................................................................................... 7
1.4 Proposed Approach .............................................................................................................. 8
1.4.1 Project Tasks ................................................................................................................. 8
1.4.2 Skill Review .................................................................................................................. 8
1.4.3 Project Management – Planning and Scheduling .......................................................... 9
1.4.4 The Priorities of Improving Skill .................................................................................. 9
1.5 Outline of Thesis ................................................................................................................ 10
1.6 Summary ............................................................................................................................ 10
2. LITERATURE REVIEW
2.1 Introduction ........................................................................................................................ 11
2.2 Potential Energy Function.................................................................................................. 11
2.3 Covalent and van der Waals Bonding ................................................................................ 13
2.4 Potential Energy Function.................................................................................................. 14
2.4.1 Hooke’s Law .............................................................................................................. 15
2.4.2 Lennard-Jones Potential ............................................................................................. 16
2.4.3 Morse Potential ........................................................................................................... 18
2.4.4 Buckingham Potential ................................................................................................ 19
2.4.5 Buckingham Potential ................................................................................................ 19
2.4.6 Linnett Potential ......................................................................................................... 20
2.4.7 Ryderg Function .......................................................................................................... 20
2.4.8 Murrell-Sorbie Function .............................................................................................. 21
2.5 Comparison with each different potential energy function................................................ 21
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Name: Wang Eng Pheng
ENG499 – Capstone Project Thesis
Student PI no: E0706507
3. ANALYSIS
3.1 Criteria for potential energy function ................................................................................ 23
3.2 Generalize potential energy function ................................................................................. 24
3.3 Parameterization ................................................................................................................ 27
3.4 Propose potential energy function ..................................................................................... 27
4. COMPARISON BETWEEN PROPOSED MULTIPURPOSE POTENTIAL
ENERGY FUNCTION WITH OTHER KNOWN POTENTIAL ENERGY FUNCTION
4.1 Diatomic Molecules ........................................................................................................... 29
4.2 Parameterization of the proposed multifunction potential function with combinations of
covalently bonded elements involving silicon ......................................................................... 29
4.2.1 Parameterization of propose multifunction potential energy function with Silicon to
Silicon .................................................................................................................................. 30
4.2.2 Parameterization of propose multifunction potential energy function with Oxygen to
Silicon................................................................................................................................... 21
4.2.3 Parameterization of propose multifunction potential energy function with Sulfur to
Silicon................................................................................................................................... 32
4.3.4 Parameterization of propose multifunction potential energy function with Nitrogen to
Silicon................................................................................................................................... 33
4.3 Parameterization of propose multifunction potential energy function with 71 diatomic
molecules ................................................................................................................................. 35
4.4 Parameterize of proposed multipurpose potential energy function with van der Waals
interaction ................................................................................................................................ 37
5. CONCLUSION AND SUMMARY……………………………………………………39
6. REFLECTION…………………….……………………………………………………40
7. REFERENCES…………………….……………………………………………………41
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Name: Wang Eng Pheng
ENG499 – Capstone Project Thesis
Student PI no: E0706507
Abstract
Simulations of solid, liquid and gaseous state systems require the use of molecular force
fields to extract physical and chemical properties at bulk level. Molecular force fields
normally include covalent bond stretching energy and van der Waals interaction energy,
among others. Most conventional computational chemistry applications adopt the harmonic
potential for bond stretching while the Morse function is employed for the greater accuracy.
To a lesser extent, the Linnett potential has been proposed for describing bond-stretching
energy. Neutral non-bonded interaction energy is often modeled by the Lennard-Jones
potential, and the Expoential-6 function (the latter falls under the category of the Buckingham
potential). So far, covalent bonds stretching energy and van der Waals interaction energy
have been described by different groups of potential functions due to their comparative
suitability.
The use of a single potential function across various types of bonding and molecular sizes
will help the development of multi-scale analysis die to the smooth transition of energy
quantification from one scale to another. Therefore, in this project, a blended potential energy
function is to be conceptualized for describing both covalent bond and van der Waals energy.
The obtained parameters for Si-Si and other Si-related interactions will be of use in modeling
the characteristics of nano-electronic materials and devices.
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Name: Wang Eng Pheng
ENG499 – Capstone Project Thesis
Student PI no: E0706507
Acknowledgement
I would like to acknowledge and extend my heartfelt gratitude to my project supervisor, Dr
Alan Lim Teik-Cheng for his encouragement, understanding and personal guidance
throughout the whole year. I am sincerely thanking him for taking his time off, from his busy
schedule to meet-up with me and discussions. Without his assistance, I could not have
completed this project.
Last but not least, I would like to take this opportunity to thank my family especially my
girlfriend who has given me the support and encouragements throughout this period.
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Name: Wang Eng Pheng
Student PI no: E0706507
Lists of Figures
Figure no.
Title
Page
Figure 2.1
Bond stretching energy
11
Figure 2.2
Intermolecular potential energy with separation
12
Figure 2.3
van der Waals force model
13
Figure 2.4
Simple potential energy curve of a diatomic molecule
14
Figure 2.5
Spring Potential Energy Diagram
15
Figure 2.6
Potential Energy Graph
15
Figure 2.7
Lennard-Jones potential for argon dimer chart
16
Figure 2.8
Morse potential and harmonic oscillator potential graphs
17
Figure 4.1
Potential energy curve using multifunctional potential for
30
comparison with Murrell-Sorbie potential using Si-Si interaction
Figure 4.2
Potential energy curve using multifunctional potential for
31
comparison with Murrell-Sorbie potential with O-Si interaction
Figure 4.3
Potential energy curve using multifunctional potential for
32
comparison with Murrell-Sorbie potential with S-Si interaction
Figure 4.4
Potential energy curve using multifunctional potential for
33
comparison with Murrell-Sorbie potential with N-Si interaction
Figure 4.5
Potential energy curve using multifunctional potential for
38
comparison with van der Waals potential function
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ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
LIST OF TABLES
Table no.
Title
Page
1.1
Criteria and Targets for Assessment of Project
8
1.2
Skills to Achieve Target
8
2.1
Summary of intermolecular forces
12
2.2
Comparison chart between various potential energy function
21
4.1
Murrell-Sorbie Parameters Chart
28
4.2
Dimensionless force constant for 71 diatomic molecules
33
4.3
Comparison Chart
38
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Name: Wang Eng Pheng
ENG499 – Capstone Project Thesis
Student PI no: E0706507
1 Introduction
1.1 Project Objective
The potential energy functions in molecular force fields are important as the accuracy of
simulated result is depending on the choice of these functions. Indeed, it has been reported
that the use of different potential energy functions strongly influence the calculated size
effects of nano-scale structures [1]. Generalized potential energy functions are known to
posses additional parameter(s) such that reduction to specific potential functions takes place
when numerical values are prescribed to additional parameters(s).
Therefore the main objective of this project is to develop a multi-purpose atomic and
molecular potential energy function whereby using a single potential function for both
interatomic and intermolecular interaction energy will reduce the computational time. The
function which is developed can be used primarily for modeling Si-Si interaction in
electronic materials, as well as for other material systems.
1.2 Overall Objective
There are a total of four overall objectives in this project. First of all, we are to conceptualize
a multi potential energy function for describing a change in covalent energy, change in van
der Waals energy between two non-bonded atoms and also change in van der Waals energy
between two molecules. This can be done either by introducing a new or improvised a
particular potential energy function. The second objective is to perform parameterization of
the proposed potential function for about silicon to silicon in the first instance followed by
other combinations of covalently bonded elements involving silicon. Thirdly, I will need to
verify the validity of the parameters through comparison with other models or experimental
results. For the last objective, I will need to obtain the parameters of the multi-purpose
potential function for the van der Waals interaction.
1.3 Proposed Approach and Method
The first approach to this project is to define the skill set which might be needed for this
project. Various different type of potential energy function will be chosen to be studied and
followed by an intense background study of these different types of potential energy function.
There’s a need for me to be familiar with covalent bond stretching energy as well as van der
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ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
Waals interaction energy and study various classical potential energy functions and to create
comparison with various different model or experimental results. At the later stage, a new or
improved potential energy function will be proposed and will need to perform
parameterization of the propose function with the given criteria.
1.4 Skill Review
1.4.1 Criteria and Targets for Assessment of Project
Table 1.1 Criteria and Targets for Assessment of Project
Criteria / Targets
Dateline
Status
Project proposal and approval
14 July 2009
Completed
Draft a project plan
By 31 Aug 2009
Completed
Carry out gathering of information
By 31 Jan 2010
Completed
Summarize and collate all information
By 28 Feb 2010
Completed
Literature Review
By 31 Jan 2010
completed
Project Proposal Report
31 Aug 2009
Completed
Review project plan
By Jan 2010
Completed
Assess project progress
By 24 Feb2010
Completed
Conclude any work/issues related to project
By 30 April 2010
Completed
Completion of final report
By 18 April 2010
Completed
Collate related issues for poster presentation
By 5 May 2010
Completed
Poster presentation
May 2010
Completed
1.4.2 Skills to Achieve Target
Table 1.2 Skills to Achieve Target
Skills
Sources / Methods
Published journals
Related articles from internet
Getting ideas and information
Reference books from UniSim
Library or National Library
Project tutor
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ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
Setting targets
Assessing and evaluating project progress Comparison of various data obtained
Project management
Computer literacy
Proficiency in Microsoft Office
Report writing skills
Presentation
Poster presentation skills
Collate and organize details in a
presentable format
1.4.3 Strength and Weakness
I should say that the project makes me feel difficult in the beginning as I do not have any
fundamental on electronic material nor familiar with potential energy functions. I also have
weakness in expressing my points due to limited vocabulary and it might hinder me in
writing this thesis report. I will need to improve more onto this area by reading up more and
to observe the appropriate style of technical writing.
By meeting up with the project supervisor, this helps to strengthen my knowledge, concept of
the project as well as knowing project objective clearly. Since I have already taken Electronic
Material last semester, I believe that this subject may provide me with some theoretical
background knowledge. Internet, library and reference report will be the reliable source to
gather required information. However, with the large amount of information gathered through
various resources, I will need to filter the most relevant information out for the project. With
much efforts and perseverance putting in, I believe that mastering in conceptualization on
potential energy function would be much easier.
1.4.4 The Priorities for Improving Skills
In order to start on the project, studies into fundamental of various potential energy functions
are extremely important. In order to tackle the problems which I have came crossed in the
project as well as to meet project submission deadline, I will need to develop a positive
attitude, raising questions frequently, critically assessing and analyzing potential problems
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Name: Wang Eng Pheng
ENG499 – Capstone Project Thesis
Student PI no: E0706507
skills, planning and management skills must be achieved. I will need to consult with the
project supervisor when there’s any doubt during the progress.
1.5 Outline of the Thesis
This thesis consists of five topics and each topic’s overviews are as follows:
Topic 1: This topic provides a brief introduction on the background, the objectives of this
project as well as the proposed approach in accomplishing on this project.
Topic 2: This topic provides introduction of force field encapsulates both bonded terms
relating to atoms that are linked by covalent bonds, and non-bonded “non-covalent” terms
van der Waals forces. This chapter also gives us an overview of various potential energy
functions.
Topic 3: This topic will introduce us a new multipurpose potential energy function.
Topic 4: This topic illustrates several simulation results obtained from graph.
Topic 5: A final summary and conclusions is made here based on the outcome of the project.
Topic 6: This topic illustrates reflection and thoughts towards this project.
1.6 Summary
This topic provides a basic introduction of this project in which will cover the objectives,
proposed approach as well as the flow of this thesis. This topic is the major part of the thesis
as it summaries the whole process.
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Name: Wang Eng Pheng
ENG499 – Capstone Project Thesis
Student PI no: E0706507
2 Literature Review
2.1 Introduction
Simulations of solid, liquid and gaseous state systems need to use molecular force fields to
extract physical and chemical properties at bulk level. The basic functional form of a force
field encapsulates both bonded terms relating to atoms that are linked by covalent bonds, and
non-bonded “non-covalent” terms describing the long-range electrostatic and van der Waals
forces [2].
Figure 2.1 Bond stretching energy [3]
The above figure diagram shows that a force field is used to minimize the bong stretching
energy of this ethane molecule. Molecular force fields normally include covalent bond
stretching energy as well as van der Waals interaction energy. For this project, we need to
conceptualize a blended potential energy function for describing both covalent bond and van
der Waals energy.
2.2 Intermolecular forces
An intermolecular force is a force between molecules that have completed their valence
requirements and have no further tendency to form chemical bonds. The forces may be either
attractive or repulsive, and they typically vary with the separation as shown in figure 2.1
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Name: Wang Eng Pheng
ENG499 – Capstone Project Thesis
Student PI no: E0706507
Figure 2.2 The typical variation of an intermolecular potential energy with separation. At
short distances (essentially, when the molecules are in contact) the repulsion always
dominates. [4]
Intermolecular forces are often called van der waals forces after a Dutch physicist Jahannes
van der Waals (1837-1923) who investigated their effects on the properties of gases.
The existence of attractive forces is shown by the occurrence of condensed phases of matter
and the strongest attractive interaction are ion-ion interactions between charged species
which will change inversely with the separation R (Table 2.1). Their energies are generally of
the order of 400kL
for typical separations of ions in ionic solids.
Polar molecules might interact with ions by an ion-dipole interaction, in which the potential
energy varies as
. The shorter the range of the interaction (that is, the more rapid decrease in
its strength with increasing separation than for the ion-ion interaction) is a consequence of the
two charges of the dipole appearing to blend into an electrically neutral point as the
separation increases. Ion-dipole interactions are the order of 20kL
and one of their
important consequences is the hydration of ions in water.
Two polar molecules may interact by a dipole-dipole interaction. The potential energy of this
interaction varies as
in a solid (when the molecules are not rotating) but as
in a fluid (in
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ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
which the molecules are rotating). The short range of the dipole-dipole interaction reflects
during large separations, both molecules appear to each other to be neutral points. The shorter
range of the interaction in a fluid arises from the near cancellation of attractive representing
the interaction of two dipoles placed at the centre of the molecules.
Table 2.1 Summary of intermolecular forces
Interaction type
Distance dependence
Typical energy kJ
Comment
Ion-ion
250
Only between ions
Ion-dipole
15
Dipole-dipole
2
of potential function
Between stationary
polar molecules
0.3
Between rotating
polar molecules
London (dispersion)
2
Between all types of
molecules
2.3 Covalent and van der Waals Bonding
Covalent bond is a form of chemical bonding in which is characterized by the sharing of pairs
of electrons between atoms or other covalent bond. Attraction to repulsion stability which
was form between atoms when they share electrons is known as covalent bonding. The
covalent molecular bond is formed between non-metal elements for compound. A covalent
molecular substance which consists of a small molecules in which they will join together with
strong covalent bonds and others molecules in the substance which are held together with
weak forces known as van der Waal’s force. The van der Waals force (or van der Waals
interaction) is the attractive and repulsive force between molecules (or between parts of the
same molecule) other than those due to covalent bonds or to the electrostatic interaction of
ions with one another or with neutral molecules [5]. The term includes the following:

Force between permanent dipole and a corresponding induced dipole

Instantaneous induced dipole-dipole forces
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Name: Wang Eng Pheng
ENG499 – Capstone Project Thesis
Student PI no: E0706507
Figure 2.3 Van Der Waals force model [6]
The above diagram shows a monatomic structure consists of discrete atoms held together by
Van der Waals’ forces, such as the noble gases.
2.4 Potential Energy Function
Potential energy function is important in molecular force fields as precision of simulated
results are depending on the choice of these potential energy functions. The functions also
provide a reasonably good compromise between accuracy and computational efficiency.
They are often calibrated to experimental results and quantum mechanical calculations of
small model compounds [7]. A good deal of information about the structure of a molecule is
summarized in its potential energy curves. Potential energy minima determine the bond
lengths. The second derivatives of the potential energy with respect to the distance will give
the force constant. These determine the vibrational and rotational levels of the molecule.
Anharmonicity constants depend on higher derivatives of the potential energy curve.
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Name: Wang Eng Pheng
ENG499 – Capstone Project Thesis
Student PI no: E0706507
Figure 2.4 Simple potential energy curve of a diatomic molecule [7]
Calculation size effects of nano-scale structures will be strongly influenced depending on the
use of different potential functions. Although potential functions with greater numbers of
parameters can be generally said to provide better fitting to experimental data than those of
fewer parameters, the latter functions have been adopted to a greater extent in molecular
mechanics softwares [8]. Sometimes the potential function can be always clearly divided into
a repulsive and an attractive part. In van der Waals complexes it is this repulsive function will
prevent the molecules locking together usually giving the peculiarity of a positive van der
Waals energy which seems to be counterintuitive [9]. We will be discussing various potential
energy functions in the following examples.
2.4.1 Hooke’s Law
Spring Potential Energy (also known as Hooke’s Law) is proposed by Robert Hooke in 1642
[5]. Objects that quickly regain their original shape after being deformed by a force, with the
molecules or atoms of their material returning to the initial state of stable equilibrium, often
obey Hooke's law. The concept of Hooke’s Law is that the amount of force applied to the
spring is proportional to the amount of deformation. That is when a greater force is applied to
a spring, the more deformation there is. When less force is applied onto the spring, there will
be less deformation on it. Mathematically, Hooke's law states that
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Name: Wang Eng Pheng
ENG499 – Capstone Project Thesis
Student PI no: E0706507
(1)
Whereby x is the displacement of the end of the spring from its equilibrium position, F is the
restoring force exerted by the material and k is the force constant of the spring. The negative
sign shows that the force exerted from the spring is direct opposition to the direction of
displacement.
Figure 2.5: Spring Potential Energy Diagram [10]
The potential energy which is stored in the spring is given as
(2)
Which comes from adding up the energy it takes to incrementally compress the spring and
that is the integral of force over distance.
Figure 2.6: Potential Energy Graph [11]
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ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
2.4.2 Lennard-Jones Potential
The Lennard--Jones potential (which is also known as, L-J potential or 6-12 potential) is an
effective potential that describes the interaction between a pair of neutral molecules or atoms.
It is proposed in 1924 by Sir John Edward Lennard-Jones [12]. The Lennard-Jones potential
is mildly attractive when two uncharged molecules or atoms approach one from a distance,
but strongly repulsive when they happen to approach too close. The L-J potential energy is
given as
(3)
Where D is the depth of the potential well (energy),
represent the distance at which the
interparticle potential is zero (length) and r is the distance between the two particles. The
Lennard-Jones model consists of 2 parts. When the separation r is very small, the
will dominates and the potential is strongly positive. And therefore the term
term
describes
the short range repulsive potential because of the distortion of the electron clouds at small
separations. In contrast the
And therefore the term
predominates when the separation r increases in magnitude.
describes the long-range (dispersion force or van der Waals
force) attractive tail of the potential between the two particles.
Figure 2.7: Lennard-Jones potential for argon dimer chart [13]
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ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
The Lennard-Jones potential function is also often written as the following,
(4)
where
is the distance at the minimum of the potential.
2.4.3 Morse Potential
The Morse potential, named after a physicist Philip M. Morse is an empirical potential which
describe the stretching of a chemical bond. It has a better approximation for the vibrational
structure of the molecule than the quantum harmonic oscillator as it specific includes the
effect of bond breaking, such as the existence of the unbound states. It also holds accounts for
the anharmonicity of both non-zero transition probability and the real bonds for overtone and
combination bands. The Morse potential energy function has three parameters and has the
form of
(5)
Where
the depth of the potential minimum is is,
bonded atoms,
is the distance between the nuclei of the
is the equilibrium bond distance and
controls the ‘width’ of the potential
energy curve. The Morse function has been used for quantifying the van der Waals
interactions in the COSMIC force field [13]. By subtracting the zero point energy
from
the depth of the well, the dissociation energy of the bond can be obtained. And by taking the
second derivation of the potential energy function, the force constant of the bond can be
found, from which it can be shown that
has a parameter of
(6)
Where
is the force constant at the minimum of the well.
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ENG499 – Capstone Project Thesis
Student PI no: E0706507
Figure 2.8 Morse potential and harmonic oscillator potential graphs [14]
As we can see from the graph, the Morse potential is represented by the blue whereas
harmonic oscillator potential (Hooke’s Law) is represented by the green. Unlike the energy
level of the Morse potential in which the spacing decreases as the energy approaches the
dissociation energy, the harmonic oscillator potential are evenly spaced by
. Morse curves
have the advantages of exactly reproducing their dissociation energies and harmonic force
constants of the diatomic molecules, and at the same time it gives a good account of
themselves in reproducing anharmonicity in the diatomic potentials.
Coolidge, James and Vernon [15] examined the following extended version of the Morse
Curve
(7)
Coolidge, James and Vernon has taken seven terms in this series. Though this series is
flexible, it has one serious disadvantage in which
is hard to determine and in fact it loses
its significance as we take more terms in the extended Morse curve.
2.4.4 Hulburt-Hirschfelder Function
Hulburt and Hirschfelder [16] has modified the simple Morse function by multiplying the
repulsive branch by a polynomial to form
(8)
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ENG499 – Capstone Project Thesis
Student PI no: E0706507
where
(9)
And b and c are simple algebraic functions of the five spectroscopic constants. This function
has a great practical advantage that it uses just those five parameters which are most readily
obtained from the study of band spectrum. Hulburt and Hirschfelder has given a list of the
numerical values of the parameters for twenty-five common diatomic molecules and with
only a few exceptions, their potential curves lie above the Morse curves [17].
2.4.5 Buckingham Potential
Buckingham potential energy function has been widely used in atomic molecular force field
as well as describing bond stretching in condensed matter. It has the form of
(10)
Where A and C are the coefficients of the repulsive and attractive terms respectively. B and n
are the repulsive and attractive indices respectively and r is the distance between non-bonded
atoms. The attractive index n is usually 6, which is later led to a simpler form and is known
as modified Buckingham or Exponential-6 potential with the function of
(11)
2.4.6 Linnett Potential
The Linnett potential energy function has been investigated in the recent years and has also
been proposed as a viable potential function for practical purposes. The Linnett potential is
different from Buckingham by an exchange in the functional forms of the repulsive and
attractive terms. It has the form of
(12)
In both Buckingham and Linnett potential function, exponential terms exist in both repulsive
and attractive parts, having Buckingham’s term resembles that of Linnett’s repulsive term.
2.4.7 Rydberg Function
The diatomic Rydberg potential energy function is extended to triatomic as well as
polyatomic species when bonds are both fundamentally independent and interdependent, as in
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Student PI no: E0706507
stable molecules and activated complexes. This function has three parameters just like the
Morse function and it has the form of
(13)
Where
, The parameter
depends on inter nuclear distance r where r and R are
the interatomic distances at any instant and the equilibrium, and a scale factor b according to
the equation
. And
is the depth of the potential minimum.
In the computation of solid sate systems, for example, the Rydberg function has been
employed to describe two-body bonded interaction energy whereby
(14)
thereby implying that
is a non-dimensional parameter . In the computation of
intermolecular interaction of diatomic molecules, the extended Rydberg function is written as
(15)
where
and
thereby implying units of
as
2.4.8 Murrell-Sorbie potential function
Unlike Morse potential function which has three parameters, Murrell and Sorbie expanded
the function in five parameters with the form of
(14)
And which can be expanded to
(15)
By expanding the original Rydberg function up to third, i.e.
. We can say that Murrell-
Sorbie function allows a better fit to the experimental data and hence providing greater
confidence when computing bond-stretching over large range because this function has
greater numbers of parameters. As such, the conversion of Murrell-Sorbie parameters, which
are normally painstakingly obtained, into parameters of the Morse function, which is more
widely incorporated in force fields, is justified for large bond stretching [1].
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2.5 Comparison with each different potential energy function
After looking through various potential energy functions, we can conclude that the
approximate relationship between each different potential energy function was obtained by
comparing the coefficients at the repulsive and attractive term. As you can see from the chart,
Morse and Rydberg potential have both exponential repulsive and attractive term. Leonard
Jones is the only potential which has
term on both repulsive and attractive term.
Buckingham potential has a exponential repulsive term and
Linnett potential has a
attractive term. And lastly
repulsive term and exponential attractive term.
Table 2.2 Comparison chart between various potential energy function
Types of Potential Energy function
Repulsive Term
Attractive Term
Morse potential
Exponential
Exponential
Buckingham potential
Exponential
Leonard Jones potential
Linnett potential
Exponential
After all these comparison, I would think that some of these potential energy functions have
similarities. Buckingham and Linnett potential functions is intriguing, as exponential terms
exist in the repulsive and attractive parts of the Buckingham and Linnett potentials,
respectively, while the functional form of Buckingham’s attractive term resembles that of the
Linnets repulsive term [18]. Buckingham and Linnett interatomic potential energy function
have two similarities, first is the equity of the curvature at the equilibrium interatomic
distance and secondly, the equity of the slope [19].
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Name: Wang Eng Pheng
ENG499 – Capstone Project Thesis
Student PI no: E0706507
There are two ways for us to conceptualize a multipurpose potential energy function. The
first way is to compare all the different functions which were mention previously, analyze
them and to combine them into a new potential energy function. The second way is to
analyze onto one single potential energy function and to analyze if we can modified it to
make a multipurpose potential energy function out of it.
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3 Analysis
3.1 Criteria for potential energy function
In order for us to present a good potential energy function, there are certain criteria which
must be satisfied. These criteria can be divided into two parts (a) necessary and (b) desirable.
(a) Necessary
1. The function should come asymptotically to a finite value as
2. The function should have a minimum at
3. The function should become infinite at
.
For the third condition, this not to be strict because the results are practically the same if
becomes very large at
. This condition will not be exactly true either if nuclear structure
and forces are to be taken into account. However, for our purpose, the changes in the
potential function at internuclear distances of
cm is no consequence.
(b) Desirable:
4. Although the above three criteria give us a conventional from of potential energy
curve as shown in Figure 2.1, it is not the only possible form for the curve. Potential
energy curves with at least one maximum and the dissociation limit are certainly
known [17].
When an “attractive” potential curve in a low approximation is crossed by a “repulsive”
potential curve, the maximum often arises. The reason behind this is that the finer interaction,
the intersection is avoided to a potential maximum of the lower of the resulting potential
curves of the interaction is not too strong.
Frost and Musulin [20] have given a theoretical discussion of various criteria for potential
energy. In analogy with the wave-mechanical calculations, they have considered the potential
energy as the algebraic sum of two parts. First, the nuclear repulsive potential corresponding
to merely a Coulomb potential
where
and
are effective atomic numbers.
Secondly, the pure electronic energy defined as are effective atomic numbers. Secondly, the
pure electronic energy defined as
which is also a function of . Therefore
24
ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
They have shown that potential function should also satisfy the following additional criteria.
5. (a)
(b)
is finite at
at
6.
7.
, where
is the known “united” atom energy
for larger
at
8. van der Waals terms should introduce terms of the form
.
3.2 Generalized potential energy function
Generalized potential energy functions are known to possess additional parameters such that
reduction to specific potential functions takes places when numerical values are prescribed to
additional parameters. One example is that the generalized Morse potential discussed by
Graves [22] and the 2-body energy description by Biswas and Hamann [23] reduces to
conventional Morse potential when the ratio of the repulsive-to attractive indices is set at 2,
where as the Buckingham potential reduces to the Exponential-6 function for quantifying van
der Waals interaction when the attractive index is equal to 6. Another example is that the
Murrell-Sorbie potential function is a generalized version of Rydberg potential. In all of the
above examples, the generalized potential functions merely reduce from a potential of more
parameters to one with less parameters without traversing across different types of potential.
A category of generalized potential functions exists whereby the additional parameter(s)
is/are known as type parameter(s) because the assignment of different numerical values to the
type-parameters(s) reduce the generalized potential energy function into specific potentials of
different functional forms. For example, the generalized potential energy function by Thakkar
[24]
(15)
reduces to the Dunham potential [25] when
potential [26] when
and the Simons-Parr-Finlan (SPF)
whereby r is the bond length while R is the equilibrium bond
25
ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
length. Hence the parameter
is the type-parameter, for it determines the type of specific
potential function which the generalized from reduces to. Some extent of flexibility exists in
Thakkar’s generalized potential, in which it not only reduces to the Dunham and SPF
potentials, but a weighted average can be attained to facilitate curve-fitting of spectroscopic
data and/or ab initio results within
with
However, Thakkar’s potential does not reduce to the Ogilvie potential [27]. As such Molski
[28] introduced a generalized potential
(16)
such that substitution of
, 1 and 0.5 reduce the generalized potential into Dunham, SPF
and Ogilvie potential functions respectively. Hence the parameter
is the type-parameter for
Molski’s generalized potential. Further enchantment to this type-parameter was made through
a glue function. While the Thakkar and Molski generalized potentials enable 1-dimentional
section in switching between Dunham SPF potentials within the range
0
and
respectively, Dr Lim [29] have introduced a generalized potential
(17)
in which can be reduced to the generalized potentials of Lennard-Jones (LJ), Morse (M),
Buckingham (B) and Linnett (L) when the type-parameters
and
are prescribed as follows
(18)
26
ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
Nevertheless, all the four reduced potentials are generalized in their own right and can be
further reduced to specific potentials by appropriate choice of parameters. For example,
firstly by selecting
and
will reduces Eq. (17) into a conventional Morse
potential function. Secondly, by selecting
and
reduces Eq. (17) into
the conventional Lennard-Jones (12-6) potential function. Lastly, by selecting
and
reduces Eq. (17) into a function used by Mayo et al. [30] for describing van der
Waals interaction whereby
and
are imposed for short and long range
respectively. Due to its flexibility, this generalized potential is able to be use on
parameterized herein for van der Waals, covalent and metallic interactions, for various
systems including noble gases, diatomic molecules, other small molecules,
fullerenes,
polymers and metals.
3.3 Parameterization
Let us take a look at Eq. (17). This general potential energy function reveals to have eight
parameters which can be classified as following. Magnitude parameters
parameters
, scaling factors
, and type parameters
energy and distance are expressed in dimensionless forms
, shape
. When the potential
and
, we will be able
to simplify the generalized potential energy function.
3.4 Proposed Potential Energy Function
Since
and
from experience, we can say that
(19)
This replacement can be justifiable since the shape parameters
and
are sufficiently
adjustable to fit into the potential energy data. In additional, the type parameters are
centralized at
(20)
27
ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
so that the potential energy function is able to fit both bonded and non-bonded interactions
without bias towards any of the four classical potentials.
After perusal Eq. (17) and based on the above simplifications, we introduce the following
equation,
(21)
The second order derivative of the proposed generalized potential is evaluated as
(22)
Such that the dimensionless force constant is simply
(23)
By imposing equal force constant, this gives a good agreement near the minimum well-depth
for both interatomic and intermolecular interaction energy. The energy integral approach,
however, is not adopted herein for determining individual shape parameters. Instead, a
numerical search was performed. This decision was taken so that the expression of
ratios can be set either as rational numbers or as integers in order to retain simplicity
of Eq. (21) after the numerical values of
and
have been prescribed.
In the following sections, dimensionless force constants for various potential energy
functions are obtained so that we could allow a comparison between the proposed
multipurpose potential energy function with previously know potential energy curves or
functions from chapter 2.
28
Name: Wang Eng Pheng
ENG499 – Capstone Project Thesis
Student PI no: E0706507
4 Comparison between proposed multipurpose potential energy
function with other known potential energy functions
4.1 Diatomic Molecules
Diatoms are being considered the next smallest molecules after noble gases. Unlike noble
gases, diatomic molecules do not posses covalent bonds, are considered herein for fitting Eq.
(21) to the covalent and intermolecular interactions. Most diatomic potential energy curves
have a very simple form: either they have a single minimum corresponding to a bound state
of the two atoms or they are repulsive with.
The dimensionless force constants of the diatomic molecules’ covalent bond can be obtained
from Murrell-Sorbie potential energy function at
(24)
Since we are comparing between the proposed multipurpose potential energy function with
Murrell-Sorbie potential energy function, we can say that proposed multipurpose potential
energy function’s dimensionless force constant is equal to Murrell-Sorbie potential energy
function’s dimensionless force constant.
(25)
4.2 Parameterization of the proposed multifunction potential function
with combinations of covalently bonded elements involving silicon
In this section, we will be looking at comparison between Murrell-Sorbie potential energy
curves and proposed multifunction potential energy function for instance silicon to silicon
interactions follow by other combinations of covalently bonded elements involving silicon.
4.2.1 Parameterization of propose multifunction potential energy function
with Silicon to Silicon
29
ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
Based on the ground state data from Huxley and Murrell [31] and Eq. (24), we will be able to
calculate the dimensionless force. Three parameters,
,
and R with the value of 2.957, 2.3
and 0.962 can be found on Murrell-Sorbie parameters chart on table 4.1. After substitute all
the three parameters into Eq. (24), the dimensionless force constant for Si-Si were calculated
as 20.90371. The use of
for Si-Si gives a very good agreement with Murrell-Sorbie
potential. The graph shown in Fig 4.1 shows a comparison between multifunctional potential
for comparison with Murrell-Sorbie potential with Si-Si interaction.
Fig 4.1 Potential energy curve using multifunctional potential for comparison with MurrellSorbie potential using Si-Si interaction.
4.2.2 Parameterization of propose multifunction potential energy function
with Oxygen to Silicon
With the parameters of
,
and
which is taken from
Murrell-Sorbie parameters chart, the dimensionless force constant for O-Si were calculated to
be 15.7749. By using
on Eq. (21), we can see that the multifunction potential
30
ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
curve is pretty similar to Murrell-Sorbie potential curve. The graph shown in Fig 4.2 shows a
comparison between multifunctional potential for comparison with Murrell-Sorbie potential
energy function with O-Si interaction.
Fig 4.2 Potential energy curve using multifunctional potential for comparison with MurrellSorbie potential with O-Si interaction.
4.2.3 Parameterization of propose multifunction potential energy function
with Sulfur to Silicon
With the parameters of
,
and
which is taken from
Murrell-Sorbie parameters chart, the dimensionless force constant for S-Si were calculated to
be 17.73824. Imposing the ratio
on Eq. (21) gives us a good agreement with the
multifunction potential curve against Murrell-Sorbie potential curve. The graph shown in Fig
4.3 shows a comparison between multifunctional potential for comparison with MurrellSorbie potential energy function with O-Si interaction.
31
ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
Fig 4.3 Potential energy curve using multifunctional potential for comparison with MurrellSorbie potential with S-Si interaction.
4.2.4 Parameterization of propose multifunction potential energy function
with Nitrogen to Silicon
With the parameters of
,
and
which is taken from
Murrell-Sorbie parameters chart, the dimensionless force constant for S-Si were calculated to
be 19.70965. Imposing the ratio
on Eq. (21) gives us a good agreement with the
multifunction potential curve against Murrell-Sorbie potential curve. The graph shown in Fig
4.4 shows a comparison between multifunctional potential for comparison with MurrellSorbie potential energy function with N-Si interaction.
32
ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
Fig 4.4 Potential energy curve using multifunctional potential for comparison with MurrellSorbie potential with N-Si interaction.
Table 4.1 Murrell-Sorbie Parameters Chart
AIAI
AICI
AIF
AIH
AIO
AIS
BB
BCI
BeCl
BeF
BeH
BeO
BeS
BF
BH
BN
BO
BS
CC
CCl
D (eV)
1.572
5.150
6.940
3.163
5.330
3.878
3.085
5.552
4.052
6.337
2.161
6.659
5.007
7.896
3.565
5.793
8.396
6.083
6.325
3.393
R (Ang)
2.4660
2.1301
1.6544
1.6478
1.6179
2.0290
1.5890
1.7159
1.7971
1.3610
1.3426
1.3309
1.7415
1.2626
1.2324
1.2810
1.2045
1.6092
1.2430
1.6450
a1
2.634
2.150
2.479
2.316
2.409
2.634
3.581
2.457
3.100
2.948
4.278
2.828
2.128
3.200
2.935
4.487
4.253
3.526
5.026
3.463
a2
1.536
1.052
1.172
1.084
-0.418
0.827
2.787
1.067
2.475
1.586
5.873
0.477
-0.308
1.930
1.638
5.580
3.967
2.768
6.630
2.360
a3
0.038
0.824
1.484
0.576
1.106
0.466
0.752
1.012
1.417
1.509
3.858
1.029
0.220
2.926
0.983
6.391
2.368
1.327
3.787
1.000
33
ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
CF
CH
ClCl
ClF
ClH
ClLi
ClNa
ClO
ClSi
CN
CO
CP
CS
FF
FH
FLi
FMg
FN
FNa
FO
FP
FS
FSi
HH
HLi
HMg
HN
HNa
HO
HP
HS
HSi
LiLi
LiNa
MgMg
MgO
MgS
NaNa
NN
NO
NP
NS
NSi
OO
OP
OS
OSi
PP
SiSi
SS
SSi
5.751
3.631
2.514
2.666
4.617
4.880
4.253
2.803
3.855
7.888
11.226
5.357
7.434
1.658
6.123
5.966
4.794
3.570
5.363
2.294
4.652
3.563
5.623
4.747
2.515
1.432
3.671
1.952
4.621
3.165
3.716
3.185
1.068
0.916
0.053
4.666
3.578
0.730
9.905
6.614
6.443
4.875
5.701
5.213
6.226
5.430
8.337
5.081
3.242
4.414
6.466
1.2718
1.1199
1.9879
1.6283
1.2745
2.0207
2.3608
1.5696
2.0580
1.1718
1.1283
1.5622
1.5349
1.4119
0.9168
1.5639
1.7500
1.3170
1.9259
1.3260
1.5897
1.6006
1.6011
0.7414
1.5957
1.7297
1.0361
1.8874
0.9696
1.4223
1.3409
1.5201
2.6729
2.8100
3.8905
1.7490
2.1425
3.0789
1.0977
1.1508
1.4909
1.4940
1.5718
1.2075
1.4759
1.4811
1.5097
1.8934
2.2460
1.8892
1.9293
3.557
3.836
4.478
4.137
3.698
1.700
1.316
5.142
2.880
5.312
3.897
4.487
3.445
6.538
4.216
2.196
1.854
4.895
2.006
7.228
3.521
5.040
3.008
3.961
2.173
3.815
4.482
2.154
4.507
3.645
3.284
3.058
1.919
1.846
2.043
1.909
1.780
2.067
5.396
5.398
4.491
4.926
3.732
6.080
4.275
4.748
3.208
3.920
2.957
3.954
2.773
2.303
3.511
6.022
3.311
3.349
0.533
0.630
7.971
2.021
7.663
2.305
5.506
2.370
12.521
3.965
1.102
-0.341
6.571
0.987
18.759
2.863
7.564
1.807
4.064
1.088
4.499
4.971
1.071
4.884
3.470
1.837
2.335
1.077
0.993
1.005
-0.509
-0.358
1.384
7.328
7.041
5.165
6.677
2.975
11.477
4.399
6.504
1.685
4.266
2.300
4.312
1.462
2.672
2.268
3.749
0.213
1.999
0.496
0.372
6.116
1.140
5.369
1.898
3.156
1.238
11.717
3.835
1.151
0.854
5.197
0.957
22.835
1.835
5.072
1.605
3.574
0.447
2.455
3.397
0.365
3.795
1.771
0.494
1.188
0.232
0.237
0.526
0.686
0.339
0.365
4.988
4.823
2.882
4.539
1.460
11.003
2.717
5.228
1.217
2.246
0.962
2.332
0.647
34
ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
4.3 Parameterization of propose multifunction potential energy
function with 71 diatomic molecules
The following table 4.2 shows the calculation of dimensionless force constant for all the 71
diatomic molecules, values of the shape parameters of M and N as well as the ratio.
Table 4.2 Dimensionless force constant for 71 diatomic molecules
AIAI
AICI
AIF
AIH
AIO
AIS
BB
BCI
BeCl
BeF
BeH
BeO
BeS
BF
BH
BN
BO
BS
CC
CF
CH
ClCl
ClF
ClH
ClLi
ClNa
ClO
ClSi
CN
CO
CP
CS
FF
FH
FLi
FMg
FN
FNa
Dimensionless force
constant
23.509
11.427
10.405
8.671
17.379
4.351
18.305
11.491
15.049
10.222
11.816
12.476
15.602
10.170
8.107
14.724
14.731
17.859
18.541
19.679
13.041
9.648
31.647
27.820
11.333
7.447
2.629
25.863
18.010
17.701
13.465
22.260
16.793
35.291
8.275
6.404
12.615
18.765
7.603
M
6.000
3.500
3.500
3.000
5.000
5.000
4.500
4.000
4.000
3.300
3.500
4.000
5.000
4.000
2.900
3.681
4.000
4.500
4.400
5.000
4.000
3.200
6.000
6.000
3.500
3.000
1.700
6.000
4.500
5.000
3.800
6.000
4.500
7.000
3.000
3.000
5.900
4.400
3.000
N
3.918
3.260
2.976
2.890
3.476
4.350
4.000
2.870
3.762
3.097
3.376
3.116
3.120
2.540
2.793
3.681
3.680
3.969
4.214
3.936
3.253
3.015
5.274
4.636
3.238
2.480
1.546
4.310
4.000
3.420
3.543
3.709
3.731
5.041
2.756
2.134
2.138
4.265
2.533
Ratio
1.531
1.074
1.176
1.038
1.438
1.149
1.125
1.394
1.063
1.066
1.037
1.284
1.602
1.575
1.038
1.000
1.087
1.134
1.044
1.270
1.230
1.061
1.138
1.294
1.081
1.210
1.100
1.392
1.125
1.462
1.073
1.618
1.206
1.389
1.089
1.406
2.760
1.032
1.184
35
ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
FO
FP
FS
FSi
HH
HLi
HMg
HN
HNa
HO
HP
HS
HSi
LiLi
LiNa
MgMg
MgO
MgS
NaNa
NN
NO
NP
NS
NSi
OO
OP
OS
OSi
PP
SiSi
SS
SSi
25.892
16.859
26.320
13.930
4.156
6.483
16.623
10.892
8.897
9.913
12.837
12.785
10.817
10.921
11.226
32.752
14.262
17.830
14.262
23.063
19.939
21.870
24.354
19.709
20.431
20.644
20.917
15.775
24.501
20.900
25.020
17.738
6.000
4.200
5.500
3.800
2.100
2.600
4.200
3.400
3.000
3.200
3.700
3.700
3.400
3.400
3.400
6.000
6.000
4.500
3.800
4.900
4.500
4.800
5.500
5.000
5.000
5.000
5.000
4.000
5.000
5.000
5.100
4.300
4.315
4.010
4.785
3.665
1.970
2.493
3.957
3.204
2.965
3.093
3.469
3.455
3.181
3.211
3.300
5.459
2.377
3.962
3.753
4.706
4.430
4.556
4.427
3.959
4.086
4.129
4.180
3.943
4.900
4.446
4.900
4.125
1.390
1.047
1.149
1.037
1.066
1.043
1.061
1.061
1.012
1.035
1.067
1.071
1.069
1.059
1.030
1.099
2.524
1.136
1.012
1.041
1.016
1.054
1.242
1.263
1.224
1.211
1.196
1.015
1.020
1.125
1.041
1.042
As you can see from the table, the value of M and N is pretty near to each other and the ratio
can be from
up to
with the exception of two diatomic molecules MgO
and FMg which have ratio above 2.
36
ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
4.4 Parameterize of proposed multipurpose potential energy function
with van der Waals interaction
The Lenard-Jones potential is often used as an approximate model for the isotropic part of a
total (repulsion plus attraction) van der Waals force as a function of distance.
The dimensionless force constants for the conventional Lennard-Jones functions are written
to be
(26)
A good fit is observed when
is applied onto Eq. (21). After comparing dimensionless
force constant of both Lennard-Jones and the proposed multipurpose potential energy
function, we can say that
(28)
and since
, we can say that
(29)
After solving the above equation, we have the value of
and
. We will
be using these values to plot the graph. Fig 4.5 shows a comparison between multifunctional
potential for comparison with van der Waals potential energy function.
37
ENG499 – Capstone Project Thesis
Name: Wang Eng Pheng
Student PI no: E0706507
Figure 4.5 Potential energy curves using multifunctional potential for comparison with van
der Waals potential energy function
From the above graph, we can see that the potential energy curve between the van der Waals
and the multipurpose function are almost the same. Table 4.3 show us a simple comparison of
both dimensionless and repulsive-to-attractive index ratio between bonded and non-bonded
atoms.
Table 4.3 Comparison Chart
Type interatomic or
Type of bonding
intermolecular
or interaction
System
Specific bonding or
Dimensionless
Repulsive-to-attractive
interaction
force constant
index ratio (m/n)
interaction
Bonded
Covalent
SiSi
Si-Si
20.90371
1.2
Bonded
Covalent
OSi
O-Si
15.7749
1.01
Bonded
Covalent
SSi
S-Si
17.73824
1.04
Bonded
Covalent
NSi
N-Si
19.70965
1.26
Non-bonded
van der Waals
LJ(12-6)
Not applicable
72
3
38
Name: Wang Eng Pheng
ENG499 – Capstone Project Thesis
Student PI no: E0706507
5 Summary and conclusions
A potential energy function that was earlier proposed via functional blending of the MurrellSorbie and Lennard-Jones functions has been centralized in order to test its applicability for
both bonded and non-bonded interactions.
The present preliminary investigation on the applicability of Eq. (21) has made by comparing
it with either experimental data or previously established potential functions. The choice of
established potential function for comparison is immaterial, but the agreement with the
proposed multifunctional potential is essential.
The method employed is summarized as follows.
a) Obtaining then dimensionless force constant of a comparison potential energy, either
from data or from a potential function. This constant is none other than the
dimensionless curvature of the potential energy curve at the web-depth’s minimum.
b) Equating this dimensionless force constant of Eq. (21) as furnished in Eq. (23), with
those obtained in part (a).
c) Selecting a positive ration of
that enables Eq. (21) to agree well with other
potential energy. The selected ratio is either an integer if
or a rational
number if
.
The multifunctional potential does more than multi-purpose. Arising from its capability to fit
covalent and van der Waals interactions between atoms as well as among molecules of
various sizes, the common shape parameters used (i.e.
and ) can then be used as a way to
map the characteristics of both the bonded and uncharged non-bonded interactions.
The use of a single potential energy function that is applicable for various bond types and
across various molecular sizes is helpful by ensuring smooth molecular transitions from one
length scale to another.
39
Name: Wang Eng Pheng
ENG499 – Capstone Project Thesis
Student PI no: E0706507
6 Reflection
Throughout the process of this project, I have gain knowledge of understanding on how
potential energy functions work. I have learnt that there are many potential energy function in
molecular force fields as well as important as the accuracy of simulated result is depending
on the choice of these functions.
After reviewing some of the basic parameters of potential energy function, I have to read and
study various different potential energy functions and therefore to made comparison between
them and to propose an ideal multifunction potential energy function. In order to
conceptualize a multipurpose potential energy function, I have tried various ways and
methods to stimulate a good result. There are times where I have faced problem when
calculating some of the tedious functions. I managed to resolve the problem with the help
with my tutor as well as studying some of the basic calculation which I have neglected.
40
Name: Wang Eng Pheng
ENG499 – Capstone Project Thesis
Student PI no: E0706507
7 References
References Website
[3] - http://en.wikipedia.org/wiki/File:Bond_stretching_energy.png
[4] - http://www.chem.queensu.ca/people/faculty/Mombourquette/Chem221/1_Gases/Index.asp
[5] - Van der Waals, From Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/London_force#London_dispersion_force
[6] - http://www.bbc.co.uk/scotland/learning/bitesize/higher/chemistry/images/bonding_fig05.gif
[7] - Potential Energy Function
http://www.ch.embnet.org/MD_tutorial/pages/MD.Part2.html
[8] - Obtaining the Morse parameter for large bond-stretching using Murrell-Sorbie parameters
http://www.springerlink.com/content/t2261657g381401j/fulltext.pdf?page=1
[9] - Computational Chemistry/Molecular mechanics
http://en.wikibooks.org/wiki/Computational_chemistry/Molecular_mechanics
[10] - http://hyperphysics.phy-astr.gsu.edu/HBASE/images/peder.gif
[11] - http://www.leancrew.com/all-this/images/spring-energy.png
[12] - Lennard-Jones potential, From Wikipedia, the free encyclopedia
http://en.wikipedia.org/wiki/Lennard%E2%80%93Jones_potential
[13] - http://upload.wikimedia.org/wikipedia/commons/9/93/Argon_dimer_potential_and_Lennard-Jones.png
[14] - http://upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Morse-potential.png/400px-Morsepotential.png
Reference journals
[1] – Teik-Cheng Lim, J Mol Model (2008) 14:103-108
[17] – Yatendra Pal Varshni, Comparative Study of Potential Energy Functions for Diatomic Molecules (1957)
[18] – Tiek-Cheng Lim, Improved Relationship Between the Parameters of the Buckingham and
Linnett Potential Functions (2009)
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[29] – T.C. Lim, Chem. Phys 320, 54 (2005)
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[31] – P. Huxley and J.N. Murrell, J.Chem. Soc. Faraday Trans. II 72, 323 (1983).
41
Name: Wang Eng Pheng
ENG499 – Capstone Project Thesis
Student PI no: E0706507
References in book
[2] – D.W.M Hofmann, Liudmila N. Kuleshova, Data Mining in Crystallography
[13] – Morley SD, Abraham RJ, Haworth IS, Jackson DE, Saunders MR, Vinter JG (1991) J Computer-Aided
Mol Des 5:475-504
[15] – Coolidge, James and Vernon, Phys. Rev. 54 726 (1938)
[16] – H.M Hulburt and J. O. Hirschfelder, J. Chem Phys. 9, 61 (1941)
[19] – P. Huxley and J.N. Murrell, J. Chem. Soc. Faraday Trans. II 79, 323 (1983).
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