Finding the structures of Boron Nitride Atomic clusters

School of Chemistry

4th Year Research Project (F14RPC)

Title: Investigating the structures of atomic Boron Nitride clusters.

Student’s name: Mark Paul AppletonStudent ID Number: 4179515

Supervisor: Dr Nicholas BesleyAssessor: Dr Richard WheatleyPersonal Tutor: Prof Mohamed Henini

I hereby certify that this project report is my own work:Student’s signature:

F14RPC I Mark Appleton

Abstract

We propose a randomised search method for generating molecular structures and we assess

the feasibility of this method by applying it to Boron Nitride (BN) atomic clusters. The

method generates multiple structures for a cluster, then optimises the geometries using

Density Functional Theory. Once this has been performed, the structures are ranked in terms

of energy and the vibrational behaviour calculated for structures of interest in each cluster.

A range of different sized clusters have been tested to determine the effectiveness of the

method, varying the stoichiometries of the clusters tested to determine any trends in the

stability of these clusters. Various parameters of this method are varied to determine the

optimal parameters for investigating systems of various sizes. These include; the input

geometry, the size of random moves, the number of structures determined in a single search

and the multiplicity of the structure. The Basin Hopping method was also implemented as a

comparative technique.

It was found that the randomised search works very well for determining the lowest energy

structures for BN systems of 6 atoms or less but above this, various limitations of the method

were found and ways of dealing with these limitations suggested along with

recommendations for values of key parameters for larger BN systems. In general, it was

found that systems which have a higher multiplicity have a lower energy as theory would

suggest. It was determined that the basin hopping method is useful at finding low energy

structures for clusters of this type but as it does not provide significantly lower energy

minima than the randomised search in all cases, more work needs to be carried out on this

before definitive conclusions can be made on the applicability of this other method.

Contents

Introduction

F14RPC II Mark Appleton

Boron Nitride 1

Atomic clusters 2

Boron Nitride Clusters 2

Investigation Overview 3

Theoretical background

Fundamental principles of quantum mechanics 4

Density functional theory 9

Geometry Optimization 16

Calculating Vibrational modes of a structure 17

The Zero Point vibrational energy 18

Anharmonicity 19

Randomised geometry search method 20

Basin Hopping Method 21

The Maximum overlap method 22

Computational Method 24

Results/Discussion

Singlet and triplet states of BxNy clusters (x + y =4) 25



Singlet states of BxNy clusters (x + y =10) 48

The feasibility of studies on large BxNy clusters 50

Applications of the basin hopping method 53

Excited state studies for small BxNy clusters 55

Conclusions 57

References 60

Acknowledgements IV

F14RPC III Mark Appleton

1)-Introduction/Background

1.1) Boron Nitride

Boron Nitride has been subjected to intense study over the past few decades in many different

fields, as its various properties make it ideal for a range of applications in electronics,

nanotechnology and manufacturing industries to name a few. This is due to the various forms

of Boron Nitride which can be created and these can suit various different applications due to

the unique properties of each of these various forms.

There are 3 main types of crystalline Boron Nitride, firstly hexagonal Boron Nitride which

forms hexagonal rings of alternating B-N in a layered structure similar to that of graphite,

with van der Waals bonding occurring between the layers in the structure. This has a variety

of applications but more specifically it is used as a lubricant in vacuum conditions, used

widely in industrial chemistry [1] and also as a substrate for use in semiconductors and in

other electronic devices. [2].

The next type is Cubic Boron Nitride (C-BN) has a structure identical to that of diamond

where one boron is bonded to 4 nitrogen atoms in a tetrahedral structure and vice versa

creating an incredibly strong material, which is but is not as strong as diamond but is

insoluble to Iron, Nickel and metal alloys at high temperatures whereas diamond is soluble. It

is used widely as an abrasive material and as parts of cutting tools in industry due to these

properties. [3]

The Final type is very similar to Cubic Boron Nitride but has a slightly altered structure

known as Wurtzite (W-BN). In this there is still an arrangement into tetrahedra around the

Boron and Nitrogen’s but between alternate groups of tetrahedra the angles will vary. It has

properties very similar to that of (C-BN) due to this similarity. [4]

Boron Nitride can also be used to construct nanotubes which are entirely analogous to carbon

nanotubes but are instead formed from alternating boron and nitrogen. While they have the

same characteristic structure as carbon nanotubes which can be a few angstroms in diameter

and several hundred long, they have very different properties. For instance, a BN nanotube is

electrically insulating whereas carbon nanotubes are metallic or semiconducting (depending

on rolling direction) and are very conductive. BN nanotubes are also much more thermally

and chemically stable than carbon nanotubes. [5] A final interesting type of Boron Nitride

structure is that of a Nano mesh. This is a two dimensional structure which is a surface

F14RPC 1 Mark Appleton

containing an assembly of hexagonal pores which can be created under ultra-high vacuum [6]

and can trap molecules/clusters within the pores. The Nano mesh is also very stable when

exposed to high temperatures, air and various liquids meaning it has multiple applications and

can be used in various processes, such as catalysing various reactions and be used as a data

storage medium. [7,8]

1.2) Atomic Clusters

In this Investigation we will focus not on these large scale structures, but instead on the field

of atomic Boron Nitride clusters and specifically on finding possible structures of these

clusters. An atomic cluster is an ensemble of bound atoms that form an intermediate between

molecules and bulk solids and can either be pure (a single atomic species) or mixed as is the

case in this investigation. They also have a predominant bonding nature which can be either

metallic, covalent or ionic which determines the properties of the cluster. Clusters can be

defined from just a few atoms up to around 105 atomic units so they can form a diverse range

of structures and also adopt a wide variety of properties which resemble small molecules

when the number of atoms N is low or can represent bulk properties of the same atomic

species when N tends to 104.

There are a wide variety of applications of clusters within Chemistry and also in Biology. For

example, Ferredoxins which are iron-sulphur proteins which mediate electron transfer in a

range of biological reactions were shown to contain active Fe4S4 clusters and Nitrogenases

which are a family of enzymes which fix nitrogen gas into nitrogen containing compounds

contain active sites of MoFe7S9 clusters. [9] Within Chemistry, metal clusters and

Organometallic clusters have also been studied in order to produce new compounds and also

for use in catalysis [10,11].

1.3 Boron Nitride Clusters

Boron Nitride clusters have been studied in some depth up to this point both in practical

experiments and also in computational studies.

For example, BN clusters have been investigated by inducing half metallicity in hexagonal

clusters via charge injection to form materials suited for use in spintronic devices as well as

using BN clusters to act as storage for hydrogen in large fullerene clusters as well as within

other BN nanostructures that can be produced experimentally via arc melting [12,13]. In

addition to these there have been several investigations using Photoelectron spectroscopy and


Photo fragmentation methods which have resulted in structural and chemical properties of

some small BN clusters being discovered, these include but are not limited to; Cluster

energies, bond lengths/ angles and also vibrational frequencies of IR active modes in small

clusters [14, 15].

Computationally, various sized BN clusters have been investigated in varying extents using

ab initio methods and also by using Density Functional Theory. There has been significant

work on small clusters (N≤6) by various groups who have studied various aspects, such as

proposing new geometries of clusters as well as the associated energies and vibrational nature

of these clusters so that they may be identified in later experimental studies. Larger clusters

(N≥20) have been the focus of investigations by various groups, the aim of which has been

determining the relative stabilities of various sizes, stoichiometries and isomers of clusters as

the number of atoms increases. This is being done for reasons similar to the case of small

clusters, but also to determine the most likely starting candidates for construction of stable

BN nanostructures when starting the construction from BN clusters.

The largest clusters investigated in detail so far via computational methods are B 32N32 clusters

which used the 6-31G* basis set and B3LYP exchange-correlation functionals to investigate

various geometries of the clusters of this size [16]. However, in these computational studies

there are many gaps in the range of clusters which have been examined, for example there is

little to no work on clusters of sizes B4N4 and B5N5. Of the work that does exist for other BN

clusters the primary focus had been on looking at even stoichiometries of these clusters and

so this is a clear gap in this field of investigation that needs to be addressed.

1.4) Investigation Overview

Thus in this investigation we intend to identify various stable structures of Boron Nitride -

clusters for various cluster sizes ranging from 3 atoms up to 30 atoms in total, whilst

analysing various stoichiometries of the clusters. During the investigation we analyse these

clusters through a combination of geometry optimizations and frequency calculations using

Density Functional Theory, to determine the vibrational nature of the various structures such

that they could be identified by experimental means in future synthetic works, i.e. by

IR/Raman spectroscopy. This will be done by producing computed IR spectra for structures

of interest for each cluster using the results of the DFT calculations. The structures of the

clusters will be found using a Random search method which probes various regions of the

Potential Energy surface of a given cluster to generate various possible structures, by using a


series of random moves from an initial geometry of the cluster of interest. Then subsequent

calculations were performed on a number of the lowest energy structures found in the search,

to determine various properties of these structures including; Zero-point energy, the nature of

the stationary point and relative energies of structures within a cluster.

The effectiveness of the search was also tested by altering various parameters to determine

for which sizes/stoichiometries of cluster the search is most suited to and the parameters

required to allow the search to run most efficiently. We also tested if there is a limit to the

cluster size after which this search method becomes unfeasible and thus to what sizes of

clusters this method could realistically be applied to. We investigated which values of

multiplicity in BN systems resulted in the lowest energies for various sized clusters. In

addition, we examined whether varying the multiplicity of the cluster significantly affects the

structures can be found by the search.

2)-Theoretical background

2.1) Fundamental principles of quantum mechanics

2.1.1) The wavefunction

The wavefunction (Ψ) is the one of the most basic concept in Quantum mechanics and in

Quantum chemistry. It describes the chemical system in question in its full extent and

information about the system can be obtained by allowing functions known as operators to

act on the wavefunction. If the operator acts on the wavefunction in the following manner:

β Ψ =Α Ψ (1)

Where the operator is denoted as β and Α is a scalar value of a particular property of the

system then the wavefunction is known as an eigenfunction of that particular operator. The

value A is an actual value of an observable quantity for the system that the wavefunction

represents. So the better we can describe the wavefunction which represents our system the

closer the values of the observables of the system we are studying will be to that of the actual

system. As we cannot know the wavefunction exactly there exist multiple methods of

approximating the wavefunction of a system, which have particular advantages and

disadvantages associated with each of them.

The wavefunction when multiplied by its complex conjugate Ψ * will return a value with units

corresponding to a real probability density and this is useful to us as if we integrate this


product over a region of multi-dimensional space within our chemical system it will allows us

to determine the probability of finding an electron within that region of space.

2.1.2) The Schrödinger equation and the Hamiltonian operator

The Schrödinger equation is the most important relation in Quantum chemistry it states that

when an operator known as the Hamiltonian (H ) acts on the wavefunction it will return the

energy of the system (E) as an eigenvalue in the eigenequation seen above (1). It has the

following form:

H Ψ=E Ψ (2)

The Hamiltonian operator has two component terms and these are the Kinetic energy (T )

operator and the potential energy operator (V ):

H=T +V (3)

The Kinetic energy operator is different from the classical description of the kinetic energy (

¿ p2∨ ¿2m

¿) and instead is represented by the eigenvalue of the Laplacian operator which in

Cartesian coordinates is represented by:

∇ i2= ∂2

∂ xi2 + ∂2

∂ y i2 +

∂2

∂ zi2 (4)

Where i indicates the Laplacian for particle iin the system and x, y, andz represent the 3

Cartesian coordinates on each particle in the system. The full kinetic energy operator

becomes:

T=−ℏ2

2m∇2(5)

When the kinetic and potential operators are applied to an arbitrary system the Hamiltonian

operator becomes:

H=−∑i

ℏ2me

∇ i2−∑

K

ℏ2mK

∇K2 −∑

i∑

K

e2 ZK

riK+∑

i< j

e2

rij+∑

K <L

e2 Z K Z L

r KL(6)

Where i and j cover all of the electrons in a system and the terms K and L represent all

nuclei in the system. Additional terms include me and mk which represent the masses of the

electrons and atoms respectively, Z represents the atomic number of each nucleus in the

system and rij and rklrepresent the distance between electrons iand j and the distance between


atoms K andL respectively. Note the final 3 terms in equation 6 represent the potential

energy operator on the system, the terms in order represent attractive interactions between

nuclei and electrons, the repulsive interactions between electrons and finally the repulsive

interactions between nuclei in the system.

Later we will see that this operator can be simplified but in this complete form we can allow

this operator to act on any wavefunction of a system to return the energy of the system the

wavefunction describes.

2.1.3) Variation theorem

The Variation theorem is another important concept in quantum chemistry. In short the

variation theorem states that for any wavefunction (Φ) of a system, its energy when evaluated

by the Hamiltonian will always be greater than or equal to, the lowest possible energy (EO) of

that system even if we do not know the wavefunction associated with that lowest energy

level. This is represented mathematically by:

W =∫Φ¿ H Φ d r

∫Φ¿Φd r≥ EO(7)

Where W represents the energy of the system from the arbitrary wavefunction. This theorem

is useful if we have an arbitrary wavefunction and have no knowledge of the exact

wavefunctions for the system in question, as it means that for an arbitrary wavefunction

which is an eigenfunction of the Hamiltonian, there is a minimum limit on the energy of the

system which we can obtain. This way it is possible for us to systematically improve the

guess of a wavefunction for a particular system. We can form a wavefunction Φ from a linear

combination of the orthonormal wavefunctions Ψi in the following manner:

Φ=∑i

c i Ψ i(8)

Then for a particular set of Ψi we can vary the coefficients which determines how the

wavefunctions combine to form Φ to give us the lowest value of the energy for the system,

corresponding to the best approximation of the wavefunction for the system.

2.1.4) Born Oppenheimer Approximation

As seen in section 2.1.2 the Hamiltonian operator cannot be solved for systems containing

many nuclei and electrons due to electron-electron interactions, however we can use the Born


Oppenheimer Approximation to vastly simplify the Hamiltonian for these systems. The

approximation states that under typical physical conditions the nuclei of the systems move

much more slowly than the electrons in the system due to the mass of an electron being

around 1800 times smaller than that of protons and neutrons. So it is convenient for us to

decouple the motions of nuclei and electrons within the system and compute the electronic

energies of the system for a series of fixed nuclear co-ordinates (i.e. nuclei are assumed to not

be moving) and thus equation 6 simplifies down to:

H elec=−∑i

ℏ2me

∇i2−∑

i∑

K

e2Z K

r iK+∑

i< j

e2

r ij(9)

As can be seen in this expression the nuclear kinetic energy term has been removed and the

5th term has also been removed as the interaction between the nuclei can just be evaluated as a

constant throughout the calculation. The term Helec indicates that this is a different

Hamiltonian than before and is known as the electronic Hamiltonian, the eigenvalues of this

operator in the Schrödinger equation are referred to as the “electronic energy” values of the

system.

The Born -Oppenheimer approximation is very mild in the sense that it is justified in most

cases and allows us to compute the energies of systems much more easily. It allows us to

define the concept of a Potential Energy surface which is a surface formed from all of the

electronic energy values Eelec formed from all of the possible nuclear co-ordinates in all

possible dimensions in the system of choice. So the advantages this approximation provides

are crucial in Quantum chemistry.

2.1.5) Potential Energy surfaces

Potential energy surfaces are a key concept for this investigation, in short these surfaces

represent the electronic energy which can be found via quantum calculations using the Born-

Oppenheimer approximation, as a function of various parameters of a system at particular

geometries. These parameters of the system include all of the variables in the structure of the

system, namely bond lengths, bond angles and bond torsions between all of the atoms. For a

system of N atoms, the surface will contain 3N-6 dimensions, however if the system energy

only varies with respect to one variable it is instead referred to as a potential energy curve.

Any point on a Potential Energy surface can be defined by a vector X where:

X=( x1 , y1 , z1 , x2 , y2 , z2 ,…… xN , y N , z N )(10)


Here, x i, yi and zi represent the Cartesian co-ordinates of atom i. These surfaces can be

visualised as a landscape which rises and falls in accordance to the energy value of the

system and how the surface changes with varying co-ordinates and can give us information

about the state of the system.

The parts of a potential energy surface which are of most interest are those known as

stationary points. These points are defined such that, if the gradient of a location is zero in at

least one direction then that is referred to as a stationary point. There are 3 types of Stationary

points; Minima, which correspond to the gradient surrounding the point in all directions is

positive. Saddle points, where the gradient in all but one direction is positive and is negative

in the other direction. Then finally, maxima, which is where the gradient in more than one

direction is negative. In Quantum chemistry the first two types of stationary points are the

most interesting to us as these represent structures within the system that are (respectively)

local minima and transition states and these structures are physically relevant (i.e. they exist)

and so when determined via computational means, could be found experimentally.

2.1.6) Relating wavefunctions to electron density

As seen so far wavefunctions and operators provide the fundamental basis for Quantum

chemical calculations. These basics can indeed be applied to study chemical systems through

the use of the well-established Hartree Fock methods which approximate the N-electron

wavefunctions of a system into an antisymmetrised product of N, one-electron wavefunctions

(χi (xi)), also known as a slater determinant, where the one electron functions are known as

spin orbitals. The Hartree Fock method is generally a good method to use to get an initial idea

of the energy of the system but it has a fundamental limit to how accurate it can be due to the

fact that the method does not take into account electron correlation effects, where the

repulsion energy between two electrons is calculated between an electron and an average

potential of the other electron. Meaning HF theory tends to always overestimate the energy of

a system even if an infinitely large basis set is used (the Hartree Fock limit).

While there may be other methods that develop on the basics of the wavefunction based

Hartree Fock methods to include electron correlation E.g. Moller-Plesset perturbation theory [17, 18] Configuration Interaction and Coupled Cluster Theory [19]. These methods begin to scale

very quickly with the number of orbitals so it would be ideal if we could use another method

besides ones which rely solely on wavefunctions.


It would be ideal for us to work with something which is a physical observable rather than

something which we need to square before we can get any physical information from a

molecule or system we are examining. Fortunately using the principles of chapter 2.1.1 we

can square the wavefunction in such a way where we can define the electron density ρ ( r) of

the molecule. The electron density is a fundamental concept in the application of Density

Functional theory which is used extensively throughout this investigation. It is also a property

of a molecule which is directly observable and this can be done via x-ray diffraction

experiments which is something that cannot be done using wavefunction based methods.

Electron density can be defined as the integral over the spin coordinates of all electrons in the

molecule and over all but one of the spatial variables:

ρ ( r )=N∫…∫¿ψ ¿¿¿¿

The electron density determines the probability of finding any of the N electrons within the

volume element d r1with arbitrary spin while the other N−1 electrons have arbitrary

positions and spin in the state represented by the wavefunction. ρ( r) is a probability density

and as electrons are indistinguishable from each other, the probability of finding an electron

in a particular volume element is ρ( r) multiplied by N electrons. Due to this, if the electron

density is integrated over all space, then it will result in the total number of electrons in the

system also the electron density will vanish as we approach an infinite distance from the

molecule:

N=∫ ρ (r )d r (12)

ρ (r⟶∞ )=0(13)

As nuclei are effectively point charges, their positions in space will correspond to local

maxima in the electron density which form cusps due to the ria−1 portion of the Hamiltonian as

ria → 0. Finally, the electron density can lead us to the atomic number of an atom due to their

interdependence, for a nucleus A which is located at an electron density maximum rA:

limr A⟶0 [ ∂

∂ r A+2 Z A] ρ (r A )=0 (14)

This alternate formalism is no simpler but still allows us to form the Hamiltonian operator

and determine the energy of a system but we can simplify the electron density calculations to

allow determining these properties to be much simpler.


2.2) Density functional theory

2.2.1) Hohenberg-Kohn Theorems

2.2.1.1) Hohenberg -Kohn Existence Theorem

Early applications of DFT were widely used in solid-state physics but were not sufficiently

accurate to be used in Chemistry. However, the following two theorems written by

Hohenberg and Kohn, were able to establish that DFT was a viable quantum chemical

method to determine molecular properties.

The premise of the existence theorem is to establish if there is a dependence of the energy on

the electron density and thus showing the density determines the Hamiltonian operator. It can

be shown by assuming that if two different external potentials act on the non-degenerate

ground state and using the variation theorem that this returns an impossible result. This

means that the non-degenerate ground state density must determine the external potential

(NE) and thus the Hamiltonian. Meaning we can define the electron density as a property

which contains all properties of the system, this can be summarised as:

ρ (r )⟹ {N , Z A , R A }⟹ H ⟹Ψ o⟹EO (15 )

2.2.1.2) Hohenberg-Kohn Variation Theorem

So far we have established that the ground state electron density can be used to obtain all

molecular properties of interest to us. The only problem is how do we know that a particular

trial density is the ground state electron density we are looking for? In Hohenberg and

Kohn’s second theorem this can be solved in a near identical way to the variation theorem as

described in chapter 2.1.3. The theorem states that the HK functional, the one which delivers

the ground state energy of the system, delivers the lowest energy result only if the input

density is the true ground state density (ρo):

EO ≤ E [~ρ ]=T [~ρ ]+ENe [~ρ ]+Eee [~ρ ] (16 )

where ~ρ is the trial electron density. As long as the trial density satisfies the conditions in

equations 12 and 13 and is associated with an external potential V ext then the energy resulting

from the trial density will always be equal to or greater than the true ground state energy of

the molecule.


So if we were to use the true ground state electron density for the associated functionals, then

we would return the ground state energy for the particular system we are looking at. This is

useful for us but is no real improvement from where we were in the wavefunction methods,

so we need some method where we can determine the energy for a particular density directly,

without having to resort to wavefunctions. [20]

2.2.2) Kohn-Sham DFT

The breakthrough came from the work of Kohn and Sham who introduced the idea that the

Hamiltonian operator could be simplified by assuming it represented a non-interacting system

of electrons. A Hamiltonian in this form can be then expressed as a sum of one-electron

eigenfunctions and thus has eigenvalues which are just a summation of the one-electron

eigenvalues.

For the starting point of our calculation then we take a fictitious system of non-interacting

electrons which in their overall ground state density have the same density as some real

system where electrons do interact. Then divide up the energy functional into specific

components:

E [ ρ (r ) ]=T ¿ [ ρ (r ) ]+V ne [ ρ (r ) ]+V ee [ ρ (r ) ]+∆ V ne [ ρ (r ) ]+∆ T [ ρ (r ) ](17)

Where T ¿represents non interacting kinetic energy, V nethe nuclear electron interaction, V ee

the classical inter-electron repulsion and the remaining two terms are corrections to the

energy due to the non-interacting system. These final two terms can be combined into an

effective exchange correlation energy E xc which can be approximated by various functionals

and this term has the form of:

EXC [ ρ ]=(T [ ρ ]−T S [ ρ ] )+(E ee [ ρ ]−J [ ρ ] )(18)

Where the first bracket is the difference between the interacting and non-interacting kinetic

energies respectively and the second bracket is the combination of the classic electrostatic

interaction and the correction due to self-interaction.

The Hamiltonian of this non-interacting system includes an effective local potential term:

H s=−12 ∑

i

N

∇ i2+∑

i

N

V s ( ri )(19)


Where V s ( ri ) represents the local potential. The ground state wavefunctions Θs are then

represented by a slater determinant (due to definition of KS DFT):

Θs=1

√ N !|φ1(x1) φ2(x1) ⋯ φN( x1)φ1(x2) φ2(x2) ¿ ¿

⋮ ¿¿ ⋮ ¿φ1(xN)¿φ2(x N)¿⋯¿φN(xN )¿|(20)

Where the spin orbitals φ i for each electron wavefunction are represented by the relation:

f KS φi=εi φi(21)

Where the Kohn-Sham operator f KS acts on each spin orbital and this in turn has the form of:

f KS=−12∇2+V s ( r ) (22 )

The form of the external potential that is contained in the KS operator and potential the

electrons are subject to has the form of:

V s ( r )=∫ ρ( r2)r12

d r2+V XC ( ri )−∑A

M Z A

r1 A(23)

Which should be noted depends on the electron density as discussed in the HK theorems

which means that to solve this problem we will need to solve for the energy of system

iteratively as the electron density must be guessed for us to determine the external potential

which in turn is used to construct the ground state wavefunctions.

The artificial system we create links back to the system we are studying, by choosing the

effective potential in such a way that the electron density resulting from the summation of the

squared moduli of the orbitals, this equals the ground state density of the system we are

interested in which contain interacting electrons:

ρo (r )=∑i

N

∑s

¿φi (r , s )∨¿2(24 )¿

So now we have the basis for calculating the energy of a system as long as we can find

sufficiently accurate functionals to represent the Exchange correlation to represent the

external potential. So that applying this to the electron density of a non-interacting system,

returns an exact energy result which corresponds to the energy of an interacting system which

we are interested in, this is due to DFT applying no assumptions to the system we study. We

can perform this calculation using a Kohn-Sham self-consistent field method. [21]


2.2.3) Kohn-Sham Self consistent field method

As mentioned previously, the need for an iterative procedure arises from the fact that the

electron density is required to calculate the orbitals shown in the matrix elements and

determinants in equation 23 but the density itself comes from solving a secular equation

which

relies on those orbitals used in the matrix. This SCF method can be shown by the flow

diagram below [22,23,24]:

Figure 1: The KS SCF iterative method

2.2.4) Basis Sets

Basis sets are crucial to Density Functional Theory as they are the mathematical functions

which we can use to construct the wavefunctions of electron orbitals used to solve equation

19. The coefficients of these functions are determined in the SCF method and are varied to

give the best wavefunction for us to obtain the best value for the energy of the system. Using

an infinitely large basis set would give us the best result for the energy of the system but as


SCF converged

Yes

No

Compute then store all overlap and one electron integrals.

Replace P (n-1) with P (n)

Is the new density matrix P (n)

sufficiently similar to the old density matrix P (n-1)?

Construct a density matrix from occupied KS MO’s

Construct and solve Kohn-Sham secular equation.

Guess an initial density matrix P (0)

Input a molecular geometry Q (0)

Select Basis set

this would not be possible to use in practical applications we can select from a range of basis

sets of various types which have been shown to provide an accurate description of the

wavefunction but also be highly efficient in terms of computational times.

One of the most popular functions that basis sets are based upon are those of contracted

Gaussian functions (CGF). These have the form of:

ητCGF=∑

a

A

daτ ηaGTO(25)

The ηaGTOrepresents a number of Gaussian functions which represent electron orbital

behaviour fairly well with an exponential decay but combined in a linear fashion so that they

resemble slater type orbitals and these represent the physical behaviour of real orbitals more

accurately.

Two typical types of basis sets which are used are those of the Pople and Dunning basis sets.

In the Pople basis example sets include 3-21G, 6-31G*, 6-31+G* which were developed by

Pople et. al. [25], the first term indicates the number of GTO’s comprising each core atomic

orbital basis function the second and third number indicate that the valence orbitals are

composed of two basis functions, with different numbers of GTO’s within each of them. The

basis set can be improved by the addition of a “*” indicating the inclusion of 6 d-type

polarisation functions on atoms through Li-Ca and it can also be improved by the addition of

diffuse functions which is denoted via a “+” which are very shallow Gaussian functions

which represent the tail portions of atomic orbitals which extend away from the nuclei.

The other common type of basis set the Dunning Basis set which has the form cc-pVNZ

where N represents D, T, Q (D-Double, T-Triplet, etc.). The cc-pV part means the basis set is

correlation consistent and polarised basis sets which are valence only basis sets. This set of

functions are larger and hence are more computationally expensive than the Pople basis set

but this then results in better values for the energy of the system. This Basis set can also

employ diffuse functions to represent the tail of atomic orbitals by the addition of the “aug”

prefix which allows for the calculation of geometric and nuclear properties, and required for

computing excited states of molecules.

2.2.4) Exchange Correlation functionals

2.2.4.1) Local Density Approximation


The LDA is the simplest type of Exchange Correlation functional, it relies heavily on the idea

of the uniform electron gas which is a system where electrons move around a positively

charged background distribution such that the system is neutral overall. It uses this idea as the

exchange and correlation energy functionals are known to very high levels of accuracy so this

is an ideal model for DFT. This functional can be written in the form:

EXCLDA [ ρ ]=∫ ρ (r ) ε XC ρ (r ) dr (26)

In this form the ε XC ρ (r ) term represents the exchange-correlation energy per particle of the

uniform electron gas of density ρ (r ). The LDA simply uses the local electron density for all

points in space within the molecule of interest, this means it is the simplest type of functional

to use in calculations but it tends to overbind the system leading to significant errors in

calculations. However, it is possible to modify this approximation to improve its returned

values of energy and this is achieved using the Generalised gradient approximation (GGA).

2.2.4.2) Generalised gradient approximation

This developed on the basis of the local density approximation and used the information

contained within the gradient of the electron density not just in the value of the density at a

particular point. The gradient of the density is determined via a Taylor expansion of the local

density approximation; the overall functional form is as follows:

EXCGGA [ ρ (r ) ]=E XC

LDA [ ρ ]+∆ EXC¿

Where the added component of the gradient is the second term in the expression. As these

methods significantly increase the accuracy of DFT calculations there are various exchange

and correlation functionals that have been developed using this type of functional e.g.

Becke’s (B) exchange functional and the LYP correlation functional developed by Lee, Yang

and Parr. [26]

2.2.4.3) Hybrid Functionals

Of the two parts of the exchange correlation functionals that these methods are used to

represent, it is the exchange contribution of the functional that is larger in terms of resulting

values when compared to the correlation contribution. As we can deduce the exchange energy

of a slater determinant exactly by using the Hartree-Fock exchange, efforts should be made to

incorporate this exact exchange energy (EHFexact) into the exchange correlation functional and

thus we only need to approximate the correlation functional as shown below:


E xc=EHFexact+EC

KS (28)

However early attempts at performing this showed poor results. This was until the popular

B3LYP functional was developed, which incorporated an exact Hartree- Fock exchange

energy from the local spin density approximation with functionals from Becke, Lee, Yang

and Parr to create a functional with the form of:

EXCB 3LYP=(1−a ) EX

LSD+a EXCλ=0+b EXC

B 88+C ECLYP+ (1−C ) EC

LSD(29)

Where the three parameters a, b and c were taken from Becke’s original paper and the

resulting functional was found to be very accurate. Due to this reasoning and the fact that

B3LYP is widely applied in many areas of Quantum chemistry and as such B3LYP is the

functional of choice throughout this investigation.

2.3) Geometry Optimization

This is a type of calculation that can be performed using the Quantum chemical methods

previously discussed. In short Geometry Optimization involves taking a starting structure of a

molecule, for example H2 and minimising its energy via DFT/wavefunction based methods

by modifying the initial geometry. Initially the energy value of the starting structure is

determined and then the bond between the atoms is shortened and the energy recalculated, if

it decreases then we are heading in the correct direction and if the bond is shortened too far

then the energy will increase indicating this is not the correct direction to reach the minimum.

For a simple one dimensional case this will form a well at the bottom of which the energy is

lowest and the geometry corresponding to that energy is referred to as the “optimal

geometry”.

For more complex systems involving multiple bonds and respective bond angles this process

becomes much slower due to the coupled nature of these degrees of freedom. Ideally in a

multidimensional system we would move along the greatest downward slope in the energy

with respect to all co-ordinates which is the opposite of the gradient vector defined as:

g (q )=[∂ U∂ q1

∂ U∂ q2

⋮∂ U∂ qn

](30)


Where q is an n dimensional co-ordinate vector where n=3 N−6 and N being the number of

atoms in the system. The partial derivatives can be calculated either numerically or

analytically however numerical methods can lead to significant errors when optimising if the

algorithm is not sufficiently in depth.

While geometry optimizations are an invaluable tool within Quantum chemistry they do have

a few issues associated with them. The first issue with optimizations is that when we are

investigating structures of molecules we tend to be interested in the global minimum of that

molecule, which is a minimum that has the lowest energy out of all possible minima. When

performing optimizations, if we were to arrive at an arbitrary minimum we would have no

way of knowing if we had arrived at the global minimum or some low lying minima close to

the global minimum. If we were to perform further optimizations using very tight

optimization criteria we may discover a lower lying minimum than in the previous case

indicating the last one was in fact not the global minimum, however we would then encounter

exactly the same problem again.

The second issue is that in some cases (e.g. when we have a fairly flat potential energy

surface) the optimization will move to the nearest stationary point, however that point may

not necessarily be a minimum. Here we may reach what is known as a transition state (or

known mathematically as a saddle point) which is where one of the second derivatives of the

stationary point is negative. The condition for a minimum is that all of the second derivatives

of the PES are positive so all surrounding parts of the potential energy surface are higher in

energy than the stationary point. This presents an issue as this type of stationary point is

higher in energy than a nearby minimum and usually represents an intermediate between two

minima. From a geometry optimization it is not always obvious that the stationary point is a

transition state so we must perform a further calculation of computing an accurate hessian

matrix for the point in question and examining its eigenvalues which will be the vibrational

frequencies of the stationary point.

2.4) Calculating Vibrational modes of a structure.

Calculating vibrational properties of a structure can give us very useful information and

allows us to deduce the nature of the structure in question as well as allowing us to produce

computed infra-red spectra for our structures of interest and provide comparison with

experimental works so accurate calculations of these properties are vital in Quantum

chemistry.


These calculations are only concerned with regions of the potential energy surface close to

the stationary point we are interested in and also that the temperature of these systems is low

enough so that only the lower vibrational energy states of the structure are populated. For a

simple one dimensional case for a diatomic system we can consider the harmonic oscillator

approximation to be sufficiently accurate to represent the system well, if we solve the

Schrödinger equation with the potential in the equation set to the potential energy of a

harmonic oscillator as below:

[−12 μ

∂2

∂ r2 +12

k (r−r eq )2]Φ (r )=E Φ (r )(31)

The energy values we get from the state are then related to the frequency of the bond

vibration via:

E=(n+ 12 ) ωℏ (32)

The value n is the vibrational quantum number and the predicted frequency values (ω¿ are:

ω=1

2 π √ kμ(33)

Where k is the spring constant in the harmonic oscillator and μ is the reduced mass. This

approximation is good but for more complex and multidimensional systems the harmonic

approximation has to be expanded across multiple dimensions and in the course of a

calculation a hessian matrix must be constructed for all of these dimensions. Then the

eigenvalues of this matrix represent the predicted frequency values of the system and the

eigenvectors of this matrix represent the normal modes of the system.

2.5) The Zero Point vibrational energy

In equation 32 it should be noted that there is an inclusion of a 12 term, the presence of this

extra term is required account for a property known as the Zero-point energy (ZPE) of the

system. This Zero-point energy arises from Heisenberg’s uncertainty principle and is the

lowest possible energy that a quantum mechanical system may have while in its ground state

(n=0). The effect of this ZPE is that the energy of a quantum mechanical state is always

greater than the minimum of the classical potential well (calculated from the harmonic

approximation) which defines it, this applies even at absolute zero.


As result a calculated minimum that we have found from a geometry optimization using

Quantum Chemical methods will, in reality, have a slightly higher energy than the

optimization states due to this Zero-point energy. However, we can very simply correct for

this energy by simply adding on an estimated value of it onto the results of the geometry

optimizations. In practice this ZPE is much smaller than the energy of the molecule, but this

correction will improve the results these methods generate and as such computed values of

energy will more closely match experimental values for the energy of a molecule.

2.6 Anharmonicity

The harmonic approximation is good but has some flaws when compared to real systems, for

instance the harmonic approximation does not allow for bond dissociation and as such has an

infinite number of energy levels which are equally spaced all of which does not realistically

model the behaviour of molecular systems. The answer then is to fit an anharmonic potential

to the system (e.g. a Morse potential) which better represents the system by allowing for bond

dissociation and as a result, the separation of vibrational levels gets smaller as n increases as

a result of this which represents the behaviour of real systems more closely. This results in a

significant improvement of the calculated frequencies over the harmonic approximation but

this method requires the 3rd and 4th derivatives of the PES to be calculated which inevitably

increases the computational cost of the calculation.

As already mentioned for a multidimensional system a hessian matrix must be constructed to

compute the vibrational frequencies/modes of a structure. This not only works for minima but

also for transition states, thus giving us a tool which allows us to determine the nature of a

stationary point. When performing this calculation on a transition state, one or more of the

normal mode force constants will be returned as a negative value. This is more commonly

referred to as an imaginary frequency and this indicates the direction the system must move

in order to proceed further down the PES. In this investigation these types of structures will

be discounted as they have no chemical significance as we do not investigate any reaction

kinetics or pathways.

There are various practical methods for computing anharmonic effects the most common of

which are Vibrational perturbation theory (VPT) and also transition-optimised shifted

Hermite theory (TOSH) which have both been implemented into the Q-Chem package.

Vibrational perturbation theory treats higher order derivatives of the potential (in the nuclear

Hamiltonian) as a perturbation on the harmonic potential and contains various levels of


perturbation correction VPTn (where n=1,2….). The most common of which is second order

Vibrational perturbation theory (VPT2) which calculates the second order correction to the

harmonic approximation, this method normally results in very accurate vibrational

frequencies but has a much lower memory cost than the most accurate methods of high level

Vibrational Configuration interaction (VCI). Transition-optimised shifted Hermite theory is

based upon VPT and it adds a small shift onto the vibrational wavefunction of the system.

This means its shape remains the same but introduces an asymmetry into the wavefunction

which can be exploited to allow anharmonic effects to be incorporated into the wavefunction.

This method is explained in more depth by Lin (et. al.) [27] but in short is a relatively

inexpensive method of computing anharmonic frequencies, however it lacks the accuracy of

some of the other more complex methods. As such the method of choice in this investigation

for determining anharmonic frequencies is VPT2.

2.7) Randomised geometry search method

This is the primary method that is tested in this investigation and is a method for generating

various minimum structures for a particular starting geometry we will refer to as the “starting

geometry”. The search takes the starting geometry and initially optimises it for a given level

of theory using either DFT or ab initio methods. It then randomly moves a specified number

of atoms within the optimised structure in a random direction by a distance which can be

varied by the user to suit structures of various sizes. The size of this move is defined in units

of Bohr and as standard is set at a value of 1 Bohr. There is however a limit on how far atoms

can move away from the centre of the cluster and the size of this box can be determined by

the user to prevent atoms moving too far apart during the search. In addition to this if the

random moves cause atoms to move too close together such that the atoms overlap the move

will be rejected and a new move implemented in its place. Following this, a number of the

atoms within the structure are swapped around such that the new structure does not change it

terms of bond lengths or angles, but the placement of the atoms within the structure is

swapped, the number of swapped atoms can also be predetermined by the user in order scale

with various structure sizes.

After this re-organising of the atoms within the structure the new structure is optimised at the

same level of theory as used in the input geometry to form the first random structure. This

optimization is stored and then the first random structure is put back into the randomised

search and the previous steps repeated. This process can be repeated N times, producing N+1


structures from a starting geometry. The random search can be summarised by the following

diagram:

Figure 2: Flow diagram of Randomised geometry search method

The advantage of this method is that it probes regions all over the potential energy surface of

the test molecule, so if the molecule has not been studied in great detail, this method can

generate multiple minimum structures. As we are also determining the energies of these

structures we can rank them from lowest to highest energies to determine the most stable

configuration for a particular molecule, thus speeding up studies which aim to determine the

lowest energy structure for groups of molecules.

2.8) Applications of the Basin Hopping method

The Basin hopping method is a technique that theoretically enables us to locate the global

minimum of a system efficiently. The first part of the method is to apply a transformation to

the potential energy surface of the system to map it to a sequence of interpenetrating


Yes

No

Optimise input structure

Output results

Save the random geometry

and the optimisation results

Does the current structure number equal

the maximum number of structures to be generated?

Optimize the geometry of structure using

DFT/Ab-initio method of choice

Randomly swap a number of atoms in the system

(M)

Randomly move a number of atoms in the system

(N)

Set initial constraints on the search

Define input structure

staircases where the plateaus of the landscape correspond to the set of configurations which

lead to particular minima after optimization. The transformed energy ~E of the system obeys

the relation:

~E ( X )=min {E (X )}(34)

Where the quantity X represents the 3N-dimensional vector of nuclear co-ordinates and the

term “min” signifies that an energy minimization is performed starting from the quantity of

X.

The second step within the basin hopping method is to then explore this stepped energy

landscape using a canonical Monte Carlo simulation at a constant predefined temperature. So

starting from an arbitrary structure on the potential energy surface we select our next

structure in the optimization based on an acceptance criterion, of which the probability of

accepting a move is determined by:

Paccept=exp (−Emin2 −Emin

1

kB T )(35)

Where the numerator is simply the difference in energy between the two minima under

consideration. While this method is very efficient at locating minima it has a tendency to get

stuck in particularly deep minima and the monte Carlo search will not progress any further.

This means that if an adjacent minimum is lower in energy than the one in which the search

has moved into the system will not be able to access the next state. [28]

As such there exists a modification to the basin hopping method known as Basin hopping

with occasional jumping [29] which as the name suggests allows the search to “jump” out of

deep minima if the search is deemed to be trapped in a particular minimum. This jumping

occurs by raising the temperature of the system to T=∞ and allowing for several monte Carlo

moves to allow the system to escape from this minimum, this allows the search to find

minima which are lower in energy than the standalone method. As such this technique will be

used in this investigation to determine its effectiveness at finding minima of Boron Nitride

clusters.

2.9) The Maximum overlap method

The Maximum overlap method (MOM) is an inexpensive way of computing excited state

geometries and vibrational structures given an initial ground state structure of interest. It can


be applied to both HF and DFT methods within the SCF/KS-SCF procedures, on each of the

iterations of the respective procedure the MO co-efficient matrix Cold is used to build up a

Fock/Kohn-Sham Matrix F and the generalized eigenvalue problem is then as follows:

F Cnew=SCnewε (36)

The parameter S is the basis function overlap matrix, this equation is solved to obtain a new

MO co-efficient matrix Cnew and the orbital energies ε.

Then the MOM states that “the new occupied should be those that overlap most with the span

of the old occupied orbitals.” If we define the orbital overlap matrix as:

O=( Cold ) S Cnew(37)

Then Oij is the overlap between the ith old orbital and the jth new orbital, and the projections

of the jth new orbital onto the old occupied space is:

p j=∑i

n

Oij=∑v

N [∑μ

N

(∑i

n

C iμold) Sμv]C vj

new(38)

So this way a full set of p j values can be found by three matrix vector multiplications, this

extra process adds only a small amount to the computational cost of each SCF cycle. We may

then occupy the n orbitals with the largest projections of p j .

In practice, we require a series of orbitals which lie within the basin of attraction for the

target excited state so to do this we perform a ground state geometry calculation to occupy a

series of orbitals. Then we simply promote an electron from an occupied to a virtual orbital

and if the guess is sufficiently close to the target solution this method retains the excited

configuration during orbital relaxation in the SCF, otherwise the SCF will converge to a

different solution containing the same symmetry. [30,31]

The advantage of MOM over other excited state methods such as CIS and TDDFT is that we

can single out a particular excited state without having to compute all lower energy states of

the same symmetry in the same calculation and thus drastically increasing the speed of the

calculations for the excited state and also drastically increasing the ease of analysis for the

calculations as a result of the single calculation. Due to these advantages we will use the

MOM overlap method to calculate excited state frequencies of structures found through the

random search method and compare these with experimental results.


3) Computational Details

All of the following calculations were performed using the Q-Chem software package [32],

visualisation and analysis of the structures produced by the randomised search was performed

using Iqmol and VMD. [33,34]

We aim to determine multiple minimum structures for various sizes of boron nitride clusters,

BxNy, where the total number of atoms in the cluster is represented by N=X+Y. We

investigated various the stability of various stoichiometries of these clusters to determine

whether there were trends in the stability of Boron Nitride clusters. We also examined what

the upper limit is on the sizes of clusters, the random search method can be used on

effectively. The relative stabilities of triplet/singlet states in a number of different clusters by

varying the multiplicity of the structures between 3 and 1 respectively. In addition to this, for

the lowest three energies of a cluster, IR spectra are computed for the structures by DFT

methods and anharmonic correction applied via VPT2 theory, then a Lorentzian fitting

applied to the data with a Full-width half maximum of 10cm-1.

Initial geometries are obtained using either minimum structures from related literature in the

area creating a repeating ring of BN units and initially optimising using an empirical force

field within IQmol. From this initial geometry N searches will be conducted with the

optimisation using at least 200 SCF cycles and also the SCF convergence criteria set to 10 -6

or lower. Unique structures determined from the VMD visualisation of these results are then

put through a further optimisation with a tightened SCF convergence criteria set to 10 -7 or 10-8

and then the vibrational behaviour of the structures is determined as well as the nature of the

stationary points determined.


4) Results/Discussion

All of the following calculations were performed at the DFT theory level of the B3LYP

exchange-correlation functional with the Pople 6-31G* basis set, unless stated otherwise.

This was done to ensure a good balance of computational accuracy and computational cost.

In all of the following figures showing structures of the boron nitride clusters, blue atoms

represent nitrogen and the pink atoms represent boron. Also the bonds in the following

figures do not necessarily represent the bonding of atoms within the cluster and are drawn to

aid the eye in determining what the structure may look like.

4.1) Singlet and triplet states of BxNy clusters (x + y =4)

B 2N2: This cluster has received some past computational work so comparisons with these

previous investigations are possible. Randomised searches were carried out on both singlet

and triplet states clusters and the search was set to search over the course of 50 cycles with a

move size set to the standard size of 1 bohr. The search used a linear structure of alternating

BN as an initial geometry in both cases. In total 9 unique structures were determined across

both searches which are as follows:

Figures 3-7: Structures 1-5 for B2N2 single state clusters.

Figures 8-11: Structures 6-9 for B2N2 triplet state clusters.

Most of these seem fairly intuitive structures for B2N2 clusters even structure 4 which has an

odd looking angle in the N-N-B bond but this bend is due to the Boron attached to the 3

membered ring having a large proportion of the positive charge of the cluster contained on it

and thus is attracted toward the negative charges on the Nitrogen’s.

However, an investigation by S. Xu et. al. [35] showed there also happens to be a stable

rhombus configuration of B2N2 in the singlet state which the search was not able to find, this

structure is as follows:


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9876

Figure 12: Stable Rhombus configuration (Structure 10).

So this structure was computed at the B3LYP/6-31G* level of theory using an SCF level of

convergence of 10-8 which matched the level of theory in the previous investigation and as

the structure could be found in a single point energy calculation, it should have been possible

to find this structure using the randomised search method. It is possible the randomised move

sizes need to be increased for this cluster as the structure may be optimising to a linear

conformation and then the moves are not sufficiently large to get near to this structure. The

energy of this additional structure was quoted in the previous investigation and as such can be

compared with the other structures. As so many triplet and singlet state structures correspond

in this case it allows for a good indication of which of the two states is lower in energy. The

energy calculation results are as follows:

Structure State Energy of structure(Eh)

Zero-point energy(Eh)

Corrected energies

(Eh)

Corresponding singlet energy (singlet

structure) (Eh)

1 Singlet -159.009670 0.012336 -158.99733 -

2 Singlet -159.001520 0.013909 -158.98761 -

3 Singlet -159.074580 0.012770 -159.06181 -

4 Singlet -158.968360 0.010111 -158.95824 -

5 Singlet -158.997180 0.012295 -158.98488 -

6 Triplet -159.075657 0.012386 -159.063271 -159.061807 (3)7 Triplet -159.021592 0.010847 -159.010744 -158.997333 (1)8 Triplet -158.999518 0.013630 -158.985888 -158.987609 (2)9 Triplet -159.003682 0.012409 -158.991273 -158.984882 (5)10 Singlet -159.04 - -- -

Table 1: Energy values for the singlet and triplet structures for B2N2 with energy

comparisons.

Of the singlet structures the alternating B-N linear structure (3) is clearly the most stable

structure even after energy corrections were applied, which differs from the results of the

Investigations by Xu and Mileev [36] who both stated that cyclic structures such as structure 10

were “conceivably global minima”. By using this method, we have deduced a structure which

is lower in energy using the same level of theory as these investigations. This is mirrored in

the triplet state structures where the lowest energy structure is again an alternating B-N linear

structure. Between the triplet and singlet states for the corresponding structures it can be seen

that in three of the four structures the triplet state variant of the structure is lower in energy


confirming the trend indicated in the B3N clusters. Computed IR spectra featuring

anharmonic corrections for each of the three lowest energy structures in the singlet and triplet

cases are as follows:

Figure 13: Computed IR spectra for structure 1 (Black), structure 2 (Red) and structure 3

(Blue) for the B2N2 singlet state clusters.

Figure 14: Computed IR

spectra for structure 6 (Black), structure 7 (Red), structure 8 (Blue) and structure 9 (Green)

for the B2N2 triplet state clusters.

In these spectra the vibrational bands centred around 400 cm-1 or less represent various

bending modes of the linear structures and the large peaks at the upper end of each spectra

corresponds to B-N stretches which are antisymmetric about the principle axis in three cases.

As all structures are linear we can see they have the same basic shape of one or two small

peaks corresponding to bending modes and then one or to larger peaks at a higher

wavenumber, but are significantly different enough that we can differentiate each spectrum

and thus comparing experimental data to these results would be relatively simple.

As three of the spectra in figure 14 correspond to the spectra in figure 3 we can compare the

effect of varying the multiplicity on the vibrational spectra. Structures 6 and 3 are identical so

comparing their spectra we can see that in the triplet state the lower wavenumber peak is

shifted to higher wavenumber but the highest peak is obstructed by the green peak at 2000

cm-1 in the spectrum of structure 9. But comparing to structure 3 we can see this is in exactly

the same place so we can differentiate between these structures by looking at the shift these

lower wavenumber peaks.


Structures 7 and 1 are identical but between the singlet and triplet states in figures 13 and 14

we can see a clear difference in their spectra as the spectrum is shifted to higher wavenumber

by approximately 150 cm-1 when comparing the triplet to singlet states so the state of the

structure significantly effects the vibrational spectrum.

Structures 8 and 2 are identical but differentiating their spectra is more difficult as only a very

small shift occurs between the two spectra with the features within structure 2 being shifted

slightly closer together (similar to the effects in structures 6 and 3) compared to the spectrum

of structure 8.

BN3: This cluster has received some detailed work when investigated by J. Martin (et.al.) [37]

so some comparison of results can be performed. The search parameters were identical to the

B2N2 case and 4 unique structures were found by the search, these are as follows:

Figures 15-18: Structures 1-4 for the BN3 singlet state cluster.

Structure 3 seems rather unphysical with what seems to be a missing bond but this could

actually show this is a conformer of structure 4. Or this could simply be the result of the

minimisation process as the results of the vibrational mode calculations suggest that these are

indeed minima as all eigenvalues of the hessian are positive, also it appears that structures 3

and 4 are conformers of the same structure. These structures match 4 of the 6 structures

proposed by Martin, the structures which are missing when compared to that investigation are

as follows:

Figures 19-20: Missing structures of BN3 compared to previous works.

As these alternate structures of BN3 were computed at the HF/DZP level of theory it so tests

were performed to see if these structures could be optimised using DFT. As such single point

energy optimisation were carried out on these two structures at the B3LYP/6-31G* level and

it was found that figure 20 could not be determined via DFT whereas figure 19 could be

determined at this level and as such this will be reffered to as structure 5. This suggests that

the random move size needs to be increased for this system size to find this other linear


1 4321

65

structure within the randomised search, to find the other structure via the search we would

have to change the level of theory used.

The relative energies of the structures found during the randomised search, including the data

from figure 19 are as follows:

Table 2: Energy values for BN3 structures.

Of the structures found during the randomised search the linear structure (4) has the lowest

energy even after zero-point energy correction. This matches the results from Martins

investigation that this is the lowest energy structure for all tested methods other than when

using the HF method which returns a lowest energy structure similar to figure 14. This shows

that even though not all of the possible structures of BN3 were returned by the search, it was

able to determine the lowest energy structure for this particular cluster. We cannot compare

this data with the structure in figure 16 as the only data available for this structure is using the

HF/DZP level of theory and this has an energy value -187.964 Eh which is considerably

higher than the energies of the other structures.

The computed IR vibrational spectra for the three lowest energy structures as follows:


Structure Energy of structure(Hartrees)

Zero-point energy(Hartrees)

Corrected energies of structures

1 -188.97175 0.012240 -188.95951

2 -188.96466 0.013678 -188.95097

3 -188.97175 0.012247 -188.95951

4 -189.02036 0.014065 -189.00629

5 -188.96326 - -

Figure 21: Computed IR spectra for structures 1 (Black), 3 (Red) and 4 (Blue) of BN3

clusters.

In each of these spectra, peaks which appear below 600 cm-1 are various bending modes

within each structure, those in the range of 600cm-1 to 2000 cm-1 are combinations of B-B and

B-N stretches and peaks which are over 2000cm-1 correspond to purely B-N stretching

modes. Also the spectra of these structures reveals that they can easily be distinguished as the

spectra are sufficiently different from one another.

B3N: This type of structure has been studied computationally in a large amount of detail (K.R

Asmis, Wenwen Cui and Martin) [38,39,40] to varying extents so this structure will give us the

best indication of how well the random search method is at finding the minima of 4 atom BN

clusters also a series of triplet state structures are noted in the literature so we can determine

whether these structures are found during the search. The singlet and triplet B3N searches

were set to run for 70 cycles and the other parameters of the search remained unchanged from

previous 4 atom cases.

5 unique structures were determined from the singlet search and 5 unique structures were

determined from the triplet search and these structures are as follows:

Figures 22-26: Structures 1-5 of B3N singlet state clusters.

Figures 27-31: Structures 6-10 of B3N triplet state clusters.

For the singlet state search all these structures match the structures reported by Asmis, Taylor

and Neumark and this time including Rhombus type structures which were not determined by

the search in the case of B2N2, showing this move sizes are actually sufficient to find all

reported structures for this stoichiometry of a 4 atom cluster and thus this method is very

suitable for determining structures in these small cases.


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It can also be seen that changing the state of the cluster has allowed the search to determine

new structures (6 and 10) for this cluster which matches predictions from previous

calculations at the B3LYP/ 6-311+G* level well. However, structure 6 has not been reported

in works by Wenwen.C (et.al.) so we have been able to determine that it is a new minimum

for this cluster at this level of theory. The triplet search also returned structures found in the

singlet case which allows for direct energy comparisons between the two methods and as

such we can determine



Corrected energies (Eh)

Corresponding singlet energy (Corresponding singlet)

(Eh)

1 Singlet -129.104503 0.011457 -129.093046 -

2 Singlet -129.091731 0.009559 -129.082171 -

3 Singlet -129.059565 0.009054 -129.050510 -

4 Singlet -129.083519 0.010243 -129.073275 -

5 Singlet -129.004880 0.008647 -128.996233 -

6 Triplet -129.082568 0.009877 -129.072691 -

7 Triplet -129.030204 0.010709 -129.019495 -128.996233 (5)

8 Triplet -129.079937 0.012589 -129.067348 -129.082171 (2)

9 Triplet -129.105531 0.011789 -129.093742 -129.093046 (1)

10 Triplet -129.100825 0.011777 -129.089048 -

the more stable of the two states as seen in the table below:

Table 3: Energies of the singlet and triplet state structures for B3N with energy comparisons.

So from this data we can see that in two of the three cases of corresponding structures, the

triplet state is lower in energy than the singlet which fits with what we would expect to

happen due to Hund’s rule. Out of all the singlet and triplet structures, it can be determined

that the lowest energy structure for the B3N cluster overall is that of the planar structure 9

which occupies a triplet state.

Computed IR spectra featuring anharmonic corrections for each of the three lowest energy

structures in the singlet and triplet cases are as follows:


Figure 32: Computed IR spectra structure 1 (Blue), structure 2 (black) and structure 4 (red)

for the B3N cluster.

Figure 33: Computed IR spectra structure 6 (Black), structure 9 (Red) and structure 10

(Blue) for the B3N cluster.

From these spectra it can be seen that all spectra have very distinct peaks and so if

experimental measurements of vibrational modes were made, it would be very simple to

determined which structure has been formed particularly in the linear structure of structure 2

which has a peak at a significantly higher wavenumber than in any other spectra. We can also

see similarities between the spectra which correspond to structures 1 and 9, however in the

region of 600 cm-1 in structure 9 we have one peak of relatively large intensity and in

structure 1 we have two significantly intense peaks as a vibrational mode that was inactive in


the triple state is now IR active and thus could be used to differentiate between the triplet and

singlet states of this planar B3N structure.


B2N4: The parameters for the randomised search in the B2N4 cluster were altered slightly with

an increase in the number of search cycles up to 70 and a slight increase in the size of the

randomised moves up to 1.5 Bohr. As we have an increased number of dimensions on the

potential energy surface an increase in the number of cycles will be required to scan as much

of the surface as possible. The initial input structure was a hexagonal ring comprised of

alternating B-N, the returned structures are as follows:

Figures 34-38: Structures 1-5 for the B2N4 singlet state cluster.

Where structures 1 and 5 match investigations performed S. Guerini et. al. [41]. In these

investigations another structure was found which is composed of two isosceles triangles of

BN3 connected by a B-B bond, however this is much higher in energy than the other reported

structures and is possibly why it was not found in this search.

The energies of these structures are as follows:

Table 4: Corrected energy values for structures 1-5 of singlet state B2N4 clusters


2543

1

Structure Energy of structure(Hartrees)

Zero-point energy(Hartrees)


1 -268.628379 0.024039 -268.604339

2 -268.536623 0.023698 -268.512925

3 -268.531707 0.022682 -268.509026

4 -268.621264 0.023392 -268.597872

5 -268.606954 0.022906 -268.584048

So it is clear that structure 1 is the lowest in energy but only by a small margin as the linear

structure of repeating B-N units is the next structure which is lowest in energy which was the

lowest energy structure in the 4 atom clusters. The result of structure 1 being the lowest

matches the investigation by S. Guerini however direct energy comparisons cannot be made

due to no explicit values of energy being stated in that investigation however this still

indicates the accuracy of the randomised search as it has returned the same structure of

lowest energy.

The computed spectra for the 3 lowest energy structures of B2N4 are as follows:

Figure 39: Computed IR spectra of structures 1 (Red) ,4 (Blue) and 5 (Black).

Once again we can see there are very distinct spectra for each of the structures so we can

differentiate them and compare experimental data to these plots, we also see that in linear

structures certain modes have vibrational frequencies which are much higher than any in the

planar structures so this can also be used to determine what type of structure has been found

in the spectrum.

B3N 3: The search parameters remained constant from the B2N4 case, this time the input

geometry to the search was trialled on two separate structures in a singlet state, a planar ring

of alternating B-N and also on a trigonal bi-pyramidal structure. This was to test whether the

initial geometry of the system would affect the resulting unique structures after the search

was conducted. Fortunately, this turned resulted in the same unique structures after both

searches so only one set of results needed to be optimised, showing that the initial geometry

for small clusters does not affect the outcomes. Singlet and triplet state input structures were

also used in this cluster and the structures determined in both of these searches were as

follows:

F14RPC 34 Mark Appleton654321

Figures 40-45: Structures 1-6 of B3N3 single state clusters.

Figures 46-54: Structures 7-15 of B3N3 tripet state clusters.

It appears that the search using DFT methods prefers these linear and planar structures as

structures 3 and 6 correspond to the structures found in the investigations by Guerini and

Martin [42] and these two structures had been reported previously with the rest of the other

structures found in the singlet state search simply being re-arrangements of these two. In the

investigation of singlet structures a 3-Dimensional structure was found which had the

following form:

Figure 55: Absent 3D structure, contained in literature

This missing structure was computed at the B3LYP/cc-pVDZ level of theory and indicates

that the random search using the B3LYP/ 6-31G* level of theory has a tendancy to not

optimise into 3D structures. Efforts were made to perform a single point enegy calculation at

the B3LYP/6-31G* level but this structure failed to optimise, as a result of this and as no

relative energy value is available in the literature, we cannot make direct comparisons to this

structure.

Significantly more structures were reported in the triplet state search than in previous

searches, resulting in a high number of linear structures being present in this search some of

which correspond to some of the singlet state structures allowing for comparisons between

the two states. A number of new structures were also found during this triplet search, some of


1110987

15141312

which appear to be conformers of singlet state structures and some which are unique to this

state, namely structures 9,10 and 15. The energies of the structures found in clusters of both

states are as follows:


Zero-point energy

(Eh)


(Eh)

Corresponding singlet energy (singlet structure)

(Eh)

1 Singlet -238.679155 0.020384 -238.658771 -

2 Singlet -238.696838 0.024425 -238.672412 -

3 Singlet -238.858029 0.026506 -238.831522 -

4 Singlet -238.613913 0.022145 -238.591768 -

5 Singlet -238.657729 0.023077 -238.634652 -

6 Singlet -238.741832 0.022529 -238.719303 -

7 Triplet -238.572903 0.020269 -238.552634 -

8 Triplet -238.655623 0.020685 -238.634938 -

9 Triplet -238.641739 0.020610 -238.621129 -

10 Triplet -238.666606 0.020430 -238.646176 -

11 Triplet -238.681327 0.019935 -238.661392 -238.658771 (1)

12 Triplet -238.742027 0.020076 -238.721950 -238.719303 (6)

13 Triplet -238.636461 0.019789 -238.616671 -238.591768 (4)

14 Triplet -238.657203 0.022641 -238.634562 -238.634652 (5)

15 Triplet -238.697479 0.019595 -238.677884 -

Table 6: Energies of the singlet and triplet state structures for B3N3 with energy comparisons.

The lowest structure can clearly be seen to be the singlet structure 3, which has as many

alternating B-N repeats as possible and thus is the most stable, this is supported by both of

the investigations by Guerini and Martin who found this D3h structure to be lowest in energy.

In the triplet state this structure was not determined and so the lowest energy structure is that

of structure 12 which is a linear chain of repeating B-N units which actually matches the

second lowest energy structure in the singlet case. For the structures which were present in

both the singlet and triplet searches we can see in three of the four cases the triplet structures


are lowest in energy and in the fourth case the energies are identical at this number of

significant figures following the trend shown by previous clusters.




spectra for structure 2 (Black), structure 3 (Red) and structure 6 (Blue) for the B3N3 cluster


spectra structure 10 (Solid line), structure 12 (Dashed), structure 15 (Dotted) for the triplet

B3N3 cluster.

From these spectra it can be seen that the vibrational behaviour gives distinct enough peaks

so that if experimental measurements the spectra were made, it would be easy enough to

determine which structure has been found in the measurement when comparing to this data.

We can also see similarity between the spectra which correspond to structures 6 and 12,

however in the spectrum of structure 12 the vibrational peaks at lower wavenumber are so

weak relative to the other features of the spectrum that these peaks cannot be seen. So this


would be a way of differentiating between the states of this structure. Also we can distinguish

from the spectra which structures are linear and which are planar simply by the number of

peaks shown. As the linear structures tend to have a small number of very high intensity

modes and planar structures have several modes of roughly equal intensity we can use this to

predict if clusters measured experimentally are linear or planar structures.

B4N2: For this cluster the same parameters were used in the randomised search as in the

previous cases and again a ring structure of alternating B-N was inputted into the search and

resulted in the following unique structures:

Figures 59-63: Structures 1-5 for the B4N2 singlet state clusters.

Similar to the other two 6 atom clusters the search has found two structures which have been

reported previously in literature, these correspond to structures 1 and 4, the search once again

showing a preference for planar/linear structures over any 3-Dimensional structures which

was again present in the investigation by Guerini with the form of:

Figure 64: Absent 3D structure for B4N2 cluster.

This structure was determined using the B3LYP/cc-pVDZ level of theory but no energy value

was reported for this structure so efforts were made to do a single point energy optimisation

on this structure to determine if it could be found at the B3LYP/6-31G* level and it could

indeed be optimised at this level so this is included as structure 6 in table 8. This suggests that

the random move sizes should be increased for future calculations of this cluster as it should

be possible for the structure to be determined by the randomised search.

The energies of these structures are as follows:


54321

Table 8: Corrected energy values for structures 1-5 of B4N2 clusters

This data shows that the most stable structure found though the randomised search is

structure 2 which is interesting as it fits with the trend of all of the previous 6 atom results

that a planar structure gives the most stable structure but in the investigations by Guerini and

the lowest energy structure was found to be a linear conformation (D∞h) however in that

investigation they did not find this planar structure. So the randomised search has found a

new lowest energy minimum for this cluster of boron nitride.

The computed IR spectra for the 3 lowest energy structures are as follows:

Figures 65: Computed IR spectra for structures 1 (Black), 2 (Red) and 4 (Blue) for the lowest

energy B4N2 clusters.

From these spectra we can see the close similarities in the spectra of structures 2 and 4 but

with higher vibrational modes in structure 4 as well as more active vibrational modes, we can

easily distinguish these two structures. Also the linear structure again results in vibrational


Structure Energy of structure(Eh)



1 -208.801836 0.019302 -208.782533

2 -208.830216 0.023471 -208.806745

3 -208.758692 0.022019 -208.736673

4 -208.796899 0.022307 -208.774592

5 -208.794825 0.022482 -208.772342

6 -208.669718 - -

modes at much higher frequencies when compared to the planar structures so can easily be

distinguished from the other two.


B3N5: For the investigation into the structures of this cluster, the search parameters were set at

70 cycles of the randomised search and also the move random move size was set to 2 Bohr.

In these 8 atom clusters, only a few investigations have covered these clusters so comparisons

with previous works is scare. From an initial structure of a ring structure of repeating B-N

units, the unique structures that were found are as follows:

Figures 66-72: Structures 1-7 of B3N5 singlet state clusters.

The effect of Di-nitrogen formation is no longer an issue in this cluster and as a result

feasible structures have been determined by the search method. As in the 4 and 6 atom cases

we get a tendency to form linear and planar structures using the search method suggesting

any 3D structures that could be formed would be found to have much higher energy values

than these structures. The energies of these structures are as follows:


321

7654

Table 9: Corrected energy values for structures 1-7 of B3N5 singlet state clusters.

As we can see, the lowest energy structure is that of the planar structure (7) in figure 72 we

can also note that the second lowest energy structure is that of structure 4 which is also planar

indicating a trend that planar structures are now significantly lower in energy for these 8 atom

clusters. However, as there is no experimental or previous data in this area no direct

comparisons can be made at this time. The spectra of the 3 structures with the lowest energies

are as follows:

Figure 73: Computed IR spectra for structures 1 (Black), 4 (Red) and 7 (Blue) for the lowest

three energy structures of B3N5.

From these spectra it is noted that bands at low wavenumber correspond to bending modes in

the structure and the highest wavenumber peaks correspond to B-N stretches. In structure 4

the band of low intensity peaks is caused by bends in and out of the plane of the structure





1 -348.285008 0.030216 -348.254792

2 -348.120516 0.029894 -348.090622

3 -348.250679 0.024249 -348.226429

4 -348.309213 0.033751 -348.275462

5 -348.202662 0.032868 -348.169794

6 -348.187177 0.029053 -348.158124

7 -348.386541 0.034143 -348.352399

with the high intensity peaks representing various B-N stretching modes in the plane of the

structure. The same also applies in the case of structure 7 however the peak at 2400cm-1 is

due to a stretch of the N-N bond for the two nitrogen’s attached to the ring. We also see

distinct spectra for each structure so we can easily distinguish between them.

B4N4: The search parameters remained unchanged for this cluster when compared to other 8

atom clusters. Some work has been carried on this cluster previously so comparisons are

possible with this work. In addition, this search was carried out on both singlet and triplet

state clusters on an initial geometry which was a ring of alternating BN units. The following

unique structures were found from the randomised searches of triplet and singlet state

clusters:

1 2

3

4 5

Figure 74-78: Structures 1-5 for the B4N4 singlet state clusters.

Figures 79-84: Structures 6-11 of B4N4 triplet state clusters.

The high proportion of linear isomers in the singlet state search is interesting as we would

expect a mixture of planar and linear isomers in this set of structures similar other 8 atom

boron nitride clusters and possibly more structures similar to or based on the initial geometry.

In previous works by M. A. Mileev et.al. and S. Xu et.al, investigations into B4N4 clusters

were conducted so we would expect to see a cyclic isomer which was found in that

investigation and in the triplet state calculations (figure 82). The triplet state structures form


1110

9876

similar structures to the singlet state structures but give a larger number of planar structures

similar to the input structure as we would expect to see.

The searches using these different states, returned only one structure which was identical in

both cases thus making energy comparisons between the structures unreliable for this case.

However, several new structures were formed during the triplet state search which seem to be

based upon other 8 atom structures and some which are unique to this cluster (structures 3

and 4) which indicates that particular structures are unique to specific electronic states of the

cluster. The energies of all of these structures can be found below:

Table 10: Energy values for B4N4 singlet/triplet structures.

We can see that of the singlet states, the one with lowest energy is structure 2 which is the

only one that is planar. This is odd as we would expect the randomised search to find more

structures around this minimum if it is low in energy but it appears on this occasion that was

not the case, this is most likely due to the random nature of the search and further tests with

singlet states would likely produce more planar structures.

Then of the triplet states we can see the lowest energy is that of structure 10 which is one of

the few ring like structures returned in either search within this cluster, it also happens to be




Corrected energies of structures (Eh)

1 Singlet -318.255823 0.028374 -318.227449

2 Singlet -318.415183 0.034344 -318.380839

3 Singlet -318.349137 0.029750 -318.319387

4 Singlet -318.410176 0.031907 -318.378269

5 Singlet -318.281382 0.031427 -318.249954

6 Triplet -318.410959 0.028799 -318.382159

7 Triplet -318.343366 0.030523 -318.312843

8 Triplet -318.326670 0.029205 -318.297465

9 Triplet -318.334704 0.030583 -318.304121

10 Triplet -318.414996 0.032060 -318.382936

11 Triplet -318.383785 0.030979 -318.352805

the lowest energy structure overall most likely due to it containing a high number of

repeating B-N units. This re-iterates the previous point that we would expect more ring

structures to be low energy minima for this cluster and suggests that the search method

should be repeated a number of times for clusters of this size to ensure all possible structures

can be determined.

Of the only two comparable structures (1 and 7) the triplet state is the one of lower energy

following the previous trend indicated in the randomised searches of smaller clusters and

confirming this is a general trend for all small clusters of boron nitride.



Figure 85: Computed IR spectra for structures 2 (Black), 3(Red) and 5 (Blue) of the B4N4

singlet state cluster.

Figure 86: Computed IR spectra for structure 2 (Black), structure 5 (Red) and structure 6

(Blue) for B4N4 triplet clusters.


Features in all these spectra are similar to features in previous calculations, i.e. peaks at low

frequencies represent various bending modes and peaks at high frequencies represent various

stretching modes. We can also see that in these 6 spectra, they are different enough that we

can distinguish which spectra correspond to which structures.

B5N3: For a set of unmodified search parameters and an input structure of a ring of B-N atoms

with 3 Boron atoms bonded to each other in succession. The following unique structures were

determined via the randomised search:

Figures 87-93: Structures 1-7 for B5N3 singlet state clusters.

Some of these structures bear resemblance to the structures observed in the B4N4 and B3N5

cases, however increasing the number of boron atoms in the system has allowed the search to

determine a number of new structures not seen previously in this investigation. These include

structure 3, which appears to be based on the initial input structure, structure 4 which is a

deviation of the linear structures commonly seen so far and structure 6 which appears to be

based on a ring structure, but has been broken based on the placement of nitrogen atoms

within that ring. Structures 1 and 7 are clearly related in that they are essentially the same

structure with the placement of Nitrogen atoms in 7 allowing for a C2 axis of symmetry across

the molecule compared a C1 axis in structure 1.

The relative energies of these structures are as follows:


765

432

1


The structure of lowest energy is structure 7 showing once again that planar conformations

are the most favourable in terms of energy in this search and that molecules of higher

symmetry are more stable when compared to similar structures.

For the 3 structures which are lowest in energy, the computed spectra are as follows:

Figure 94: Computed IR Spectra for structures 2 (Black) ,3 (Red) and 7 (Blue) for the B5N3

singlet state cluster.

In the linear structure, peaks below 500 cm-1 represent various bends in the linear structure

and peaks in the range of 500-1900cm-1 are modes consisting of various B-B and B-N

stretches and above this peaks correspond to purely B-N stretches. Similarly, in the planar

spectra the vibrational bands around 400 wavenumbers are various bends of Boron within the





1 -288.466659 0.031066 -288.435593

2 -288.470131 0.028779 -288.441352

3 -288.520104 0.032836 -288.487268

4 -288.391976 0.027464 -288.364512

5 -288.421143 0.031491 -288.389652

6 -288.381649 0.031842 -288.349807

7 -288.527289 0.032215 -288.495074

planes of the structures and features at 1400 and 1600 wavenumbers are B-N stretches within

the plane of the molecules for the planar structures. All three of the spectra are significantly

different from one another such that we can easily distinguish which structure would

correspond to which spectrum if experimental measurements were made.

B6N2: For an unchanged set of search parameters and an initial input structure composed of a

complete ring of 6 boron atoms in sequence followed by two nitrogen atoms in sequence, the

following unique structures were obtained by the randomised search:


1 2 3 4 5 6

Figure 95-100: Structures 1-6 for B6N2 singlet state clusters.

Comparing these structures with other 8 atom clusters it can be noted that structure 4 is

similar to structure 7 in the B3N5 case and that structures 1-3 are variants of this particular

structure, also structure 5 resembles structure 6 in the B3N5 case where the ring that was

previously broken has closed. The most interesting point about these structures is that no

linear structures have been formed for this cluster, as in all other cases at least one structure

has featured a linear structure. This suggests that the initial structure that is input into the

system is beginning to have an effect on the unique structures that the randomised search can

return. Further evidence for this, is that the initial structure is an 8 sided ring and these 6

structures have a ring character about them with other atoms attached to the edges of the

rings. This suggests that investigations into this area require larger move sizes to break this

initial symmetry apart during the search.

The energies for these B3N5 structures are as follows:


Where it is clear to see that structure 4 is the lowest energy structure and that the trends in

stability are similar to the structures in B5N3 clusters, as the higher levels of symmetry lead to

a lower energy geometry than in related structures. The computed spectra for the three lowest

energy structures is as follows:





1 -258.445235 0.029458 -258.415776

2 -258.501487 0.032010 -258.469477

3 -258.363571 0.031180 -258.332391

4 -258.548117 0.032913 -258.515205

5 -258.384259 0.032375 -258.351884

6 -258.497303 0.030513 -258.466791

Where the vibrational bands around 400 wavenumbers represent bending modes within the

plane of the molecules and bands above 1200 wavenumbers represents stretching modes

between B-B and B-N bonds. In the computed spectra of structure 4, the spectrum has many

more peaks when compared to 8 atom clusters due to no particular modes having a high

intensity relative to other peaks. As such the finer vibrational detail is more prominent in the

spectra for this structure.

4.4) Singlet states of BxNy clusters (x + y =10)

B5N5: This cluster size was tested to determine at what point the initial symmetry of the

cluster, affected the output structures determined by the search. The move size was set to 2.5

bohr so that input structure had a greater chance of breaking its initial symmetry, the number

of atoms that were moved in the search was increased to 10 atoms and 4 atoms were swapped

in the system on each cycle. For 100 cycles of the search, beginning from a cyclic ring of

alternating B-N the following unique structures were obtained:

1 2 3

5 6

7 8

Figures 102-109: Structures 1-8 of the B5N5 singlet state cluster.


Even with these fairly large move sizes when compared to the previous calculations the move

sizes do not appear to be large enough to break the initial symmetry of the input and thus the

majority of the structures appear to be based upon this input due to the atom swapping part of

the search having a more prominent effect upon the resulting structures. The reason the initial

input geometry was chosen to be a cyclic ring was due to the results of the investigations by

J.Martin and S.Xu pointing to a D5h structure being the lowest energy structure for this type

of a system. The energy of the D5h structure was reported at -398.29 hartrees which compares

well with the computed energies as follows:

Table 13: Corrected energy values for structures 1-8 of B5N5 clusters

So even after energy corrections the D5h structure of alternating B-N atoms which was

returned by the search, is the lowest energy structure, thus matching the results of the

previous investigations. However this is not particularly useful as we have determined no

other unique types of structure from the random search for this particular cluster of BN. But it

does indicate that moves larger than 2.5 bohr are indeed required for systems of 10 atoms or

more.

The spectra of the three lowest energy structures for B5N5 clusters are as follows:





1 -398.293511 0.045644 -398.247866

2 -397.954284 0.042768 -397.911516

3 -398.065137 0.042127 -398.023009

4 -398.088635 0.041796 -398.046839

5 -397.984059 0.042601 -397.941459

6 -397.959027 0.041539 -397.917489

7 -398.065139 0.042695 -398.022445

8 -398.119352 0.044048 -398.075304

Figures 110: Computed IR spectra of structures 1(Black), 4 (Red) and 8 (Blue) singlet state

B5N5 clusters.

The reason that the spectra are relatively simple for a system of this size is that most of the

vibrational modes In this structure are not Infra red active and of the ones which are active

they have a very small IR intensity when compared to some other modes. However from this

figure we can clearly see how this reliance on the initial geometry is an issue as it means that

the vibrational spectra are very hard to distinguish. In the above diagram there are 3 plots but

due to the similarity of the sprectra of structure 1 and 4 the spectrum of structure 1 is

conceled in this figure. So comparisons with experimental data are difficuly to achieve if this

problem persists.

4.5) The feasibility of studies on large BxNy clusters

We looked at BN systems with 20 and 30 atoms in size and we used 2 distinct input

structures in each of the clusters, as it was indicated by the results of section 4.4 that the

search has a dependence on the input structure at these system sizes. These two inputs were; a

ring like annular structure of alternating B-N bonds and also fullerene based inputs derived

from structures investigated by M. A. Mileev et. al. As these systems were much larger than

those investigated previously the randomised moves were increased up to 5 bohr to allow the

best chance for the initial symmetry to be broken, the number of searches however had to be

limited to 40. This is due to the computational costs of the search method and how it

optimises structures, the weakness in the method is that it finds a new geometry, optimises it

and then uses this new geometry to find a new structure and the process then repeats. For

small structures this is a very logical method and works well, however at these larger system


sizes these individual optimisations take upwards of several hours to complete. As the

calculation requires this final geometry to begin the next calculations the computational cost

for a single search compounds, quickly becoming unfeasible for these systems.

It was therefore decided to reduce the cost of these calculations by using a simpler basis set

(STO-3G) and lower the SCF convergence criteria from 10 -7 to 10-5 thus reducing the

threshold error in the SCF, to simplify the optimisations within the search. Then a higher

order basis can be used to optimize structures of interest once the search was completed. This

approach was tested on some smaller systems of B3N and B2N2 and then compared to the

approach of using the 6-31G* basis set throughout the search (or a purely 6-31G* approach)

and the subsequent optimisations. The corrected energies of the same structures determined

by both methods is displayed below:

Structure STO/6-31G* energy (Eh) Pure 6-31G* energy (Eh) Difference (Eh)

B2N2 (NBBN) -158.997329 -158.997333 4.78849x10-06

B2N2 (BNNB) -158.987580 -158.987609 2.86796x10-05

B2N2 (BNBN) -159.061676 -159.061807 0.000131B3N Figure 21 -129.061172 -129.093046 0.031874B3N (BBBN) -129.005400 -129.073275 0.067875B3N (BBNB) -129.082176 -129.079288 0.002888

Table 14: Comparison of the energies from corresponding structures from two search

methods.

As can be seen from this table there is a very small energy difference between the methods in

the B2N2 case and a slightly higher energy difference in the B3N case. These energy

differences were judged to be sufficiently small to use the STO/6-31G* method on large

structures and gain a large saving in computational cost.

This resulted in a large cost saving which meant that instead of 10 cycles taking a week to

compute, 40 cycles could be computed in the space of 3 days. With this increased number of

cycles, the search could now explore a larger amount of the potential energy surface and after

running two sets of searches for each of the two input structures, the structures with the

lowest energy for each input were collected and are displayed below:

Input/Structure Geometry Energy (Au)Energy Difference

(Au)

Energy Difference

(KJ/Mol)


Planar 1 -786.120915 0.255205 663.5317

Planar 2 -786.096706 0.279413 726.4738

Planar 3 -785.998992 0.377127 980.5302

Fullerene 1 -786.376119 0.000000 0.000000

Fullerene 2 -785.893382 0.482737 1255.116

Fullerene 3 -785.858932 0.517187 1344.685

Table 15: B10N10 singlet state structures and relative energies

Input/Structure Geometry Energy (Au)Energy Difference

(Au)

Energy Difference

(KJ/Mol)

Planar 1 -1179.082904 1.095243 2847.632

Planar 2 -1178.977689 1.200458 3121.191

Planar 3 -1178.948012 1.230135 3198.351

Fullerene 1 -1180.178147 0 0


Fullerene 2 -1179.117709 1.060438 2757.139

Fullerene 3 -1179.018713 1.159434 3014.528

Table 16: B15N15 singlet state structures and relative energies

As can be seen from these results the randomised search method does not result in very well

defined structures except in some cases where the most stable structure is simply another

variant of the input structure. In the planar case the random moves tend to collapse the ring

structure towards a fullerene like structure and in the case of the fullerene inputs the shell

structure tends to break apart in the random moves and in both cases raise the energy of the

system. Note also that the closest energy value in both sized clusters exceeds several hundred

KJ/Mol meaning that the moves selected by the search raise the energy by a significant

amount.

These effects are most likely due to the sheer number of moves that can occur during the

search, leading to a PES which is too large (has too many dimensions) to be accurately

sampled simply using 40 trial structures at a time. As this value cannot be increased further

due to the cost of these calculations described earlier, it appears that the limitations of this

method have been determined and that structures of BN clusters containing 20 atoms or more

cannot be determined by this version of the randomised search method.

4.6) Structures determined from basin hopping

This part of the investigation focuses on applying the basin hopping method to several of the

boron nitride clusters investigated so far and determining whether basin hopping is a feasible

method for determining low energy structures within atomic clusters. We will be

investigating whether basin hopping can return structures that are significantly lower in

energy than ones found via the random search method and if no significant gains are made

discuss its effective application to this field.

Unlike the randomised search, basin hopping will return a single structure from its search

across a region of the potential energy surface as opposed to N structures in the randomised

search. This is because of basin hopping using monte Carlo steps to get as close to a global


minimum as possible. This greatly simplifies comparisons with the randomised search as we

can compare the geometry and energy of the basin hopping output against that of the lowest

energy structure found in the singlet state randomised searches.

The basin hopping searches used identical starting structures as in the randomised searches

and the parameters for the monte Carlo steps within the method were kept constant

throughout all but 1 calculation. The parameters used were as follows; The number of monte

Carlo steps was 30, the number of monte Carlo cycles used was set to 4, the temperature of

the simulation was set to 300K and the number of repeated steps required before the system

jumps out of a particular well is set at 7. A number of cycles allows for the monte Carlo

search to explore a number of possible routes down the potential energy surface and thus a

greater chance of finding various low energy minima than if a single cycle was used.

The findings of the lowest energy structures determined by the basin hopping methods are as

follows:

Cluster Basin Hopping Structure Energy (Eh) Previous minimum

energy value (Eh)

B2N2 -159.075542 -159.061807

B2N4 -268.623212 -268.604339

B3N3 -238.743995 -238.831522

B4N2 -208.804579 -208.806745

B4N4 -318.413533 -318.380839

Table 17: Energy comparisons between Basin Hopping and randomised search methods.

Table 18: Comparisons in timescale between Basin Hopping and 70 cycles of a typical

randomised search.


ClusterComputational Time of Basin

Hopping in Seconds (Hours)

Computational Time of Random

search in seconds (Hours)

B2N2 16973 (4.71) 36571 (10.16)B2N4 57161 (15.88) 86737 (24.09)

B3N3 38580 (10.72) 75649 (21.01)

B4N2 40510 (11.25) 45464 (12.63)B4N4 63428 (17.62) 144752 (40.21)B5N5 - 151358 (42.04)

Note that the energy values in the basin hopping method are uncorrected for zero-point

energy and thus should be slightly higher in energy than they appear. The results of table 26

give no clear indication of which method is better in terms of energy as in some cases (B2N2,

B2N4 and B4N4) the Basin hopping method finds structures with lower energies and in the

other cases the randomised search method finds structures of lowest energy. The differences

in energy does not appear to be how the structure is optimised in one particular method. As in

the cases of B3N3 and B2N2 the structure of lowest energy has the same geometry in both

methods, but the method which results in the lowest energy varies between the two cases (i.e.

Basin hopping is better in B2N2 case and the randomised search is better in the B3N3 case.) so

the optimisation does not appear to be more efficient in one method than in another.

An interesting feature to note from this data, is that basin hopping returned linear structures

as the lowest energy structure in all cases tested. This could mean that the basin hopping

method sinks down into regions of the PES corresponding to linear structures at the

beginning of each monte Carlo search and thus never explores regions of the PES which

correspond to planar structures. This means that in clusters which have a global minimum

which exists as a planar structure, basin hopping may not be an effective method to use to

find the global minima as it may instead continue to tend towards linear conformations.

The computational timescales of both methods is quite interesting, the Basin hopping method

is shown to be much quicker across the range of results it was tested upon when compared to

the randomised search. However, when we consider the number of results obtained from

basin hopping (1 structure) and compare it to the number of structures obtained from the

randomised search, the basin hopping method is still a relatively expensive method to

conduct.

It may however be very useful to us in that it appears to return viable structures for B-N

clusters at the sizes tested so far which are either the lowest energy structure of a particular

cluster or a viable low energy structure. If this trend was to continue into much larger sized

clusters, basin hopping could be used as a relatively cheap probe to find low energy structures

within the cluster. Those structures could then be used as a starting structure by the

randomised search method to explore the stability of similar structures by using smaller

random moves. Before this could be done however, more work is required to determine if this

trend continues by testing basin hopping on a wider range of clusters and determining the

optimal input parameters to allow the process to used most efficiently at larger sizes.


4.7) Excited state studies for small BxNy clusters

This small investigation focused on replicating results found in the investigations of Asmis,

Taylor and Neumark who determined the vibrational modes of excited states of B2N and B3N

during investigations to determine Anion Photoelectron spectra of these species. We aimed to

determine whether structures from the randomised search were accurate enough to be used in

calculations of excited state Infra-red spectra of atomic clusters. Due to the limited timescale

allocated for this investigation, the method of choice for performing these excited state

calculations was the Maximum overlap method. The method was primarily chosen due to its

relatively small computational cost in comparison to methods such as CIS or TDDFT and this

method has been shown to yield results as accurate as these methods.

The structures that correspond to these reported states are both linear structures in both B2N

and B3N and have structures of B-N-B and B-N-B-B respectively. In the computed values of

the frequencies the level of theory was matched as closely to the reported cases as possible,

so that the best comparisons on the effectiveness of the method can be made and so in all

cases the frequencies include no anharmonic approximations. Two different sets of excited

state calculations were performed in each case, one which promoted an electron from an

alpha orbital to the next orbital above it and in the other case the same was done for an

electron in a beta orbital. This was done to see if this caused different excited states or

whether the excitations from these different orbitals, excited the system into the same state.

In the case of B2N the ground and excited state frequencies were computed at the B3LYP/6-

31G* level whereas the reported frequencies were computed at the QCISD(T)/6-31G* level.

The difference in method is simply due to the QCISD(T) functional not being available

within QCHEM for this system. The results for these excited state calculations are as follows:

Reported X2Σ+u

Ground state

vibrations (cm-1)

Computed Ground

state frequencies

(cm-1)

Reported A2Σ+

g state vibrations

(cm-1)

Computed alpha

excitation

frequencies (cm-1)

Computed beta

excitation

frequencies (cm-1)

1140 (σg) 1330.43 2500 (σu) 1492.95 1259.41

870 (σu) 1192.80 1180 (σg) 1179.38 31.33

- 156.73 - 388.15 31.33

/Table 25: Frequencies of ground state and excited state B2N vibrational modes including

computed and reported values.


It is immediately clear to see that this difference in theory level significantly affects the

results of the frequency calculations as in both the ground and excited states of this structures

there are significant differences in the frequencies between the two investigations. Errors in

the ground state it would seem are simply the difference in optimisation method used to

generate the linear structure as these values should be very close. Analysis of the bond

lengths in the structures computed via both methods is not possible as these are simply not

stated in the investigation by Asmis et. al. The excited state frequencies are interesting as the

highest frequency is substantially larger than the corresponding vibrational mode in our

calculations of the Alpha and Beta electron excitations. Such a large value seems

unreasonable for a vibration of B-N in this configuration and upon repeating calculations

using larger basis sets no evidence was seen for this very high frequency mode. Also the

large difference between Alpha and Beta excited frequencies appears to be due to the overlap

method failing to properly optimise the B2N structure as two of these modes are doubly

degenerate. This suggests that the maximum overlap method is unsuitable for generating

excited state structures for a transition to this state as well as the level of theory compounding

these errors. It is interesting to note that for the (σg) mode in the excited case we get such

good agreement with previous calculations this could simply be due to chance or suggest the

(σu) mode calculated by Asmis is incorrect as there seems no other way to explain this value.

In the case of B3N the ground and excited state frequencies were computed at the B3LYP/6-

31G* level whereas the reported frequencies were computed at the B3LYP/aug-cc-pVTZ

level. The results for these excited state calculations are as follows:

Reported Ground

state (1Σ+)

frequencies (cm-1)

Computed Ground

state frequencies

(cm-1)

Reported 3π state

frequencies(cm-1)

Computed alpha

excitation

frequencies

(cm-1)

Computed beta

excitation

frequencies

(cm-1)

1978 1945.49 1925 1906.87 1906.82

1136 1131.03 1212 1201.74 1201.51

580 572.23 750 747.55 747.65

203 212.74 377 379.88 377.34

84 92.42 102 125.49 126.05

Table 26: Frequencies of ground state and excited state B3N vibrational modes including

computed and reported values.


We can see much better agreement between the sets of values across both ground and excited

states in this case which is most likely down to the similar levels of theory used between the

reported and computed cases. We can see that the ground state frequencies agree very well

showing the structure determined by the randomised search is very similar to the previous

investigation. In the excited states we can clearly see that the maximum overlap method

produces an excited structure which closely matches the reported structures showing for this

particular transition of 1Σ+→3π this method is very suitable for producing an excited structure.

It is also interesting to note that the alpha and beta electron excitations produce very similar

frequencies suggesting that both of these form the same excited state in this particular

transition.

5) Conclusions

A wide variety of boron nitride clusters have been examined using the randomised search

method which yielded mixed results in terms of the accuracy of the structures found and how

well the search operated in order to find significantly different structures. For systems of N=4

and N=6 the results were mostly positive in that we found numerous unique structures, most

of which had been found in previous works so allowed us to verify that the method can yield

results consistent with these calculations.

For systems of N=8 and N=10 little previous data is available, so experimental comparisons

were scarce but a large number of structures have been reported with associated spectra.

However, limitations of the method have been discovered for clusters of these sizes, the main

limitation is that the input geometry begins to affect the output of the randomised search and

this is significant for structures of 10 atoms or more if the size of the randomised moves is

not sufficiently large.

Overall it has been shown that the lowest energy structures for atomic cluster are generally

planar structures which have the maximum number of repeating BN units. For structures

which are very similar, structures with higher levels of symmetry tend to be the lowest in

energy. Comparisons of the vibrational behaviour for structures determined from the

randomised search has shown good correlation to previous investigations with typical errors

ranging from 5-50 cm-1. This is mainly due to the application of anharmonic correction to

computed results which means that the spectra featured in this investigation will more closely

resemble experimental spectra and thus is more relevant to further investigations.


In the investigations into large clusters, the computational costs were found to scale very

quickly even at the STO-3G level of theory and scaled too quickly when using the 6-31G*

basis set for calculations to be feasible. However due to the sheer size of the structures

investigated the search was not able to determine any coherent structures other than the

structures inputted into the system and these incoherent structures have a very large energy

differences compared to the starting structure.

After trying various initial structures this issue could not be resolved, so it appears that this

randomised search method is not suited to exploring these clusters due to the large Potential

Energy surfaces in these structures. We would need to perform many more searches than in

small clusters to have a chance of finding a coherent structure for these clusters. In the

current method the computational cost for N searches compounds as the geometry

optimisations are performed in sequence. A modification to the search method to generate a

large number of structures quickly, would be to allow the search to find a random geometry

and then instead of performing an optimization, perform another set of moves to generate a

new structure while saving the old geometry. This would create a number of structures which

could then be optimised simultaneously allowing for the timescale of the overall calculation

to be shortened considerably.

When investigating the stability of triplet and singlet state clusters, it was found that a

number of unique structures are generated when the clusters are in the triplet states when

compared to the singlet states as well as structures common to both states. The general trend

in the stability of structures which has been indicated by these results, is that when a structure

is in a triplet state it will have lower energy than in a singlet state which is what we expect

from the principles of Hund’s multiplicity rule. This is not obeyed in all structures however

and in these cases the singlet state energy is higher. This is likely due to whether due to the

distribution of charges altering between the two states and resulting in an increase in energy.

The application of the basin hopping method yielded mixed results. For all of the clusters

tested, a linear structure was returned as the lowest energy structure. This is unusual due to

the input structures being identical to the randomised search method and would expect to see

a mixture of planar and linear structures. This could be an issue for example, if the lowest

energy structure of a particular cluster is a planar or three dimensional as the basin hopping

method may not be able to determining that structure. This could be due to too few monte

Carlo moves/cycles being used within the basin hopping method so a way of determining if


planar structures can be found via this method would be to increase both of the

aforementioned parameters and then implement several different starting structures.

The comparative energy values between the two methods are inconclusive, of the 5 clusters

tested 3 of the cases had lower energies when the basin hopping method was used and due

there being no clear method which results in a better structure it is difficult to say which

method is the best for obtaining low energy minima with this limited amount of data. Future

work should focus on gathering more data so we may determine which method is best at

finding low energy structures.

When computing excited states of small clusters, mixed results were found. In the B 2N

cluster results of vibrational modes were poor when compared to works by Asmis (et.al.)

however in the B3N cluster, results were much more positive due to a better match in the

theory applied to the systems and when excited state methods were applied, resulting

frequencies matched previous calculations well. This may have been due to a transition in the

latter case being much more suited to the maximum overlap method and thus such a stark

contrast in results were seen.

Overall it could be suggested that this method is applicable to boron nitride clusters due to the

accuracy of the B3N results, however more work should be done on other clusters to

determine if the data in these clusters is reliable and the B2N data determined here is an

outlier or is indicative of a problem with this method.

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Acknowledgements

I would like to firstly thank the University of Nottingham high performance Computing

centre (HPC) for providing the computational resources necessary for this investigation to

have taken place.

I would also like to thank Dr. Nicholas Besley for introducing me to this field of study, which

I had scarcely considered working in before my Master’s year and then for keeping me

interested in the subject throughout the project. Also I would like to thank him for being such

a helpful supervisor throughout this project and for helping me to become familiar with

Qchem and Linux as a whole.

Finally, I would like to thank the cohort of research students and postdoctoral researchers in

room A47. Who provided me with advice and insight into this field when the project

presented difficult challenges.

F14RPC IV Mark Appleton

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Finding the structures of Boron Nitride Atomic clusters

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