REPORT Ring currents in Boron Molecules

REPORT

Jos Teunissen S1879804

Directed by Remco W.A. Havenith and Ria Broer Theoretical Chemistry Group

Rijksuniversiteit Groningen, Groningen 2012

Abstract:

Ring current maps are computed within the ipsocentric approach using CTOCD-DZ/6- 31G*/BLYP∧B3LYP of two classes of interesting boron compounds: a planar B202--molecule and a new conformation of a B80 fullerene. B202- has simultaneously σ-anti-aromaticity and π-aromaticity. The boron fullerene has ring currents of diamagnetic and paramagnetic character around the whole molecule. Overall the B80 conformation is paramagnetic with a

TABLE OF CONTENTS

Prologue 3

Introduction 3

Theory 5

A magnetic field 5

The treatment of the Hamiltonian to magnetism 5

Perturbation theory to derive the first order correction to the wavefunction. 6

Using perturbation theory to calculate ring currents 7

Localization of orbitals 9

A description of which excitations will contribute to the total current. 11

Boron chemistry 13

The structure of B202- 13

The Boron Fullerene, B80 16

Method 17

Results and Discussion of B20-2 18

The paramagnetic current of the inner ring. 19

The diamagnetic current of the outer ring 22

B202- transition state: 25

Localized orbital analysis 25

Natural Bond Orbital analysis 27

Conclusion 28

Results and Discussion of the Boron-80 Fullerene 28

A hollow hexagon 30

A filled hexagon 31

A pentagon 32

Conclusion 35

Conclusion 35

Perspectives 36

References 37

PROLOGUE

This report is considered as the final step to obtain the Bachelor degree in chemistry. After a lot of courses and practical work, I did the research project at the Theoretical Chemistry Group. The project was a totally different way of learning and a good experience.

During the project new methods and theory had to be learn, therefore this bachelor thesis will contain a lot of theory that was needed to understand the principles of ring currents and some other subjects. The main reason for including some mathematical derivations was to give a whole description of what is done in the calculations but for reading it is not necessary to understand it totally.

I hope you will understand the principles and results included in the thesis, J.L. Teunissen

INTRODUCTION

The topic of the bachelor thesis is ring currents of boron molecules. In analogy to carbon, boron is able to form covalent bonds. Therefore it is expected that boron shares some interesting features with carbon. However due to the low natural abundance of boron and its complex bonding behavior boron chemistry is not a field of much research. Also it is rather labor intensive to obtain pure boron. Nevertheless boron chemistry gains more and more interest. Especially remarkable are the in silico studies of small boron clusters and the recently, also in silico, studies of boron fullerenes.

It is found that neutral Bn clusters with n=3-19 adopt a planar or semi-planar geometry and those with 20 or more atoms have a three dimensional form. ¹ However it turned out recently that negatively charged clusters with 20 atoms possess a planar structure.² In this research project the (anti-)aromatic behavior of this molecule has been studied.

Figure 1 The geometry of B202-

The second molecule that has been studies is a new conformation of B80. B80 is a boron- analogue of the C60 Buckminsterfullerene. This carbon fullerene is well-known and has fascinating properties that found several applications for example in pharmaceutics and in the electronic industry.

The boron fullerene allotrope is yet an unchartered territory and there are many theoretical calculations predicting the properties of them. It is expected that it will be possible to synthesize boron fullerenes in the near future, especially if we bear in mind that also C was only a theoretical prediction 25 years ago.

For the boron fullerenes there are many different possibilities to construct the molecule.

Since boron is electron deficient 20 additional boron atoms have to be added to complete the electron structure. The most common structure is to fill al hexagons of a boron buckyball structure with one boron inside, but it is also possible to cap the twelve

pentagons with a boron atom and put the remaining 8 atoms in some of the 20 hexagons.

3

Figure 2. The geometry of the carbon and boron buckyball

When the pentagons of the boron fullerene are capped with a boron atom the structure is called a volleyball-shaped B80. Calculations showed that the volleyball structure is lower in energy than the buckyball structure. (ΔE app. 2.7eV)⁴

Filling eight of the twenty hexagons with a boron atom can be done in several ways giving a lot of possible conformations for the boron volleyball structure. It turned out that the conformations that have symmetry are the most stable conformations. One conformation is taken where each non-filled hexagon is adjacent to exactly one other non-filled hexagon.

This structure has an inversion point and a slightly distorted octahedral geometry. Our hypothesis is that this conformation is rather stable due to aromaticity.

Figure 3 Geometry of the boron volleyball conformation

Aromaticity is a property of molecules, and is considered as the ability to sustain induced ring currents.⁵ The strength and the direction of these currents can be used as a measure for the degree of aromaticity and anti-aromaticity in molecules. An aromatic molecule is one that is able to support a diatropic current and an anti-aromatic molecule supports a paratropic current. To visualize this, current maps are plotted in-plane and above the plane in which the current is induced by applying a magnetic field perpendicular to the plane.

These ring currents are calculated using the ipsocentric CTOCD-DZ approach. The CTOCD- DZ allows a decomposition of the total ring currents in contributions from individual orbitals, or individual excitations. This makes it very helpful to understand the total ring current of a molecule in terms of a few relevant excitations.

THEORY

To understand the quantum mechanical approach of magnetic currents first a few classical expressions are introduced, such as the magnetic susceptibility and the description of a vector field.

A MAGNETIC FIELD

The magnetic behavior of materials is classically expressed in terms of magnetic susceptibility, defined as follows:

(1) Where M is the magnetization expressed in Am^-1 and H the magnetic field also in Am^-1. (So susceptibility is a dimensionless quantity). If the susceptibility of a material is linear dependent of the Magnetization one can also write: . If the magnetic susceptibility is larger than zero the material is paramagnetic and if χ is smaller than zero the material is diamagnetic. Briefly, a material is paramagnetic if the magnetic momenta align with an external field and diamagnetic if the magnetic momenta anti-align with the field.

To describe a magnetic field, a vector potential A is used from which the magnetic field is derived.⁶

(2) The simplest form for A is:

(3) This is the simplest form because, via a mathematical rule follows that:

(4)

THE TREATMENT OF THE HAMILTONIAN TO MAGNETISM

To calculate magnetic currents the wavefunction in a magnetic field must be known. To calculate this wavefunction perturbation theory is used. In perturbation theory the wavefunction is written as the known wavefunction extended by additional correction functions. The total wavefunction is than written as the known wavefunction (Ψ⁽⁰⁾) extended by correction terms to the wavefunction. To calculate the ring currents we only need to know the first order to correction to the wavefunction. In the justification below

χ

M=χH

B= ∇ ×A A= ¹₂B×r

∇ ×A= ∇ ×(¹₂B×r)= ₂¹(B(∇ •r)−r(∇ •B)+(r• ∇)B−(B• ∇)r)

= ¹₂2B=B

PERTURBATION THEORY TO DERIVE THE FIRST ORDER CORRECTION TO THE WAVEFUNCTION.

In perturbation theory the Hamiltonian is extended by a perturbation H⁽¹⁾, or in many level systems:

(5) Here λ is a help parameter to identify the order of the perturbation. Also the wavefunction and the energy are extended by perturbation factors.

(6) (7) Here the terms E⁽¹⁾ and E⁽²⁾ are called the first and second order correction to the energy.

Implementing these extensions in the time-independent Schrödinger equation gives:

(8) (9) Collecting the terms with the same power of λ gives:

(10) (11)

…et cetera.

(12) Dividing by λ gives:

(13) So λ is cancelled out and an expression for ^{( )}is formulated.

The first-order correction to the wavefunction is written as a linear combination of the unpertubated wavefunctions. This is possible because the unpertubated wavefunctions forms a complete set.

(14) Implementing this in the left hand side of equation-(13), using equation-(10) gives:

(15) Multiplying both sides with ψk,(k not zero) and integrating, using bracket notation gives:

(16) Due to orthogonality the integral exist only when k=n, and | = 0

(17) Writing the right-hand side integral as Hk0 the coefficients are given by:

(18) H=H^{( 0)}+λH⁽¹⁾+λ²H^{( 2)}+....

ψ

ψ

λψ

λ

ψ

E₀ =E₀⁽⁰⁾+

λ

λ

H ψ = E ψ

(H⁽⁰⁾+

λ

λ

ψ

λψ

λ

ψ

λ

λ

ψ

λψ

λ

ψ

H

ψ

ψ

(1)+λH⁽¹⁾ψ0

( 0)=λE₀⁽¹⁾ψ0

( 0)+λE₀^{( 0)}ψ0 (1)

∴λ^{(H( 0)}ψ0

(1)−E₀^{( 0)}ψ0

(1))=λ^{( E}0 (1)ψ0

( 0)−H⁽¹⁾ψ0 ( 0))

(H^{( 0)}−E₀^{( 0)})ψ0

(1) =( E₀⁽¹⁾−H⁽¹⁾)ψ0 ( 0)

ψ

(1)= c_n

ψ

∑

c_n

n

∑

ψ

(0)= c_n

n

∑

(0)−E₀⁽⁰⁾)

ψ

c_n

ψ

n

∑ ψ

(0) =

ψ

ψ

∴c_k( E_k⁽⁰⁾−E₀⁽⁰⁾)=E₀⁽¹⁾ψkψ0 − ψk H⁽¹⁾ψ0 = −ψk H⁽¹⁾ψ0

∴c_k = H_{k 0}⁽¹⁾

E₀⁽⁰⁾−E_k⁽⁰⁾ = ψk H⁽¹⁾ψ0

E₀⁽⁰⁾−E_k^{(k )}

(19)

USING PERTURBATION THEORY TO CALCULATE RING CURRENTS

The Hamiltonian for a molecule subjected to an external magnetic field is formulated using the following rule:

“Wherever p occurs in the Hamiltonian, it should be replaced by p+eA, where A is the vector potential for a field”^7,8

The Hamiltonian now becomes as follows:

(20) Elaborating the factor (p+eA)² gives:

(21)

where or

Since the angular momentum operator p includes a differentiation, it is important to distinguish between a differentiation on a wavefunction or a differentiation on the vector potential and the wavefunction. Using a Coulomb gauge( ) we get:

(22)

Now equation-(21) becomes:

(23) And the Hamiltonian becomes:

(24) Back to perturbation theory the Hamiltonian(eq. 20) is extended by two terms, which can be regarded as the first and second order correction terms.

(25) (26) To calculate the ring currents only the first order correction term to the wavefunction(eq.

25) is relevant. Using equation-(25) and the vector identity , the first order correction term can be written as:

(27) The last step is legal because dot products are commutative , with L is the angular momentum operator = × .

ψ

(1)=

ψ

ψ

E₀^{( 0)}−E_k^{( k )}



 



k≠0

∑ ψ

H⁽⁰⁾= p²

2m_e +V =( p+eA)² 2m_e +V

( p+eA)²= p²+peA+eAp+e²A² = p²+e( pA+Ap)+e²A² p= −ih∇

p=h

i∇

∇ ⋅A=0

p⋅Aψ= −ih∇ ⋅Aψ

= −ih((∇ ⋅A)ψ+A⋅(∇ψ⁾

= −ih(A⋅ ∇ψ⁾

=A⋅ pψ

p²+2eA⋅p+e²A²

H⁽⁰⁾= p²

2m_e +V+2eAp

2m_e +e²A² 2m_e = p²

2m_e +V+ e

m_e Ap+ e² 2m_e A²

H⁽¹⁾= _2m^e_e A⋅p H⁽²⁾=_2m^e2

e A²

a×b⋅c=a⋅b×c

H⁽¹⁾= e m_e

1

2B×r⋅ p= e m_e

1

2B⋅r×p= e m_e

1

2B⋅L =ˆ e m_e

1 2L ˆ ⋅B B⋅L =ˆ L ˆ ⋅B

This first order correction to the Hamiltonian is filled in into equation-(19).

(28)

The angular momentum operator can be split up in two terms. One term of a rotation about the origin, L(0) and a displacement term from the origin × .

(29) Filling in this equation in equation-(28) the first order correction to the wavefunction becomes:⁹

(30)

Here two parts are distinguished. The rotational term is considered as the paramagnetic part and the term with the linear displacement from the origin is considered as the diamagnetic part of the first order correction to the wavefunction.

Because L and P are one-electron operators, this wavefunction can be written as a sum over single exited states:

(31) Now the first order wavefunction can be written as a sum of orbital contributions, and the first order correction to the wavefunction for each orbital becomes:

(32)

Now it is possible to look at the contributions of single exctitiations which makes it very helpful to understand the currents in a molecule.

The wavefuction is used to calculate the charge density and current density by the following formula:

Charge density: (33)

Current density:¹⁰ (34)

For the vector field A the simple expression of equation-(3) is taken.

(35) Although the total current density is independent of the origin, the orbital contributions

Ψ₀⁽¹⁾= − e 2m_e

Ψ_k L ˆ ⋅BΨ₀ E₀⁽⁰⁾−E_k⁽⁰⁾ Ψ_k⁽⁰⁾

k≠0

∑

L ˆ =L (0)ˆ −d×P ˆ

Ψ0

(1)= − e

2m_e

Ψk L (0)ˆ Ψ0

E₀⁽⁰⁾−E_k⁽⁰⁾ Ψk (0) k≠0



∑

 

 ⋅B+ e

2m_e d× Ψk P ˆ Ψ0

E₀^{( 0)}−E_k^{( 0)}Ψk ( 0) k≠0



∑

 

 ⋅B

= Ψ0 ( p )+ Ψ0

(d )

ΨkL → Ψ_nL^p

p>N / 2

∑

n=1 N / 2

∑

k≠0

∑

ψn

(1)(r,d)= −_2m^e_e ψp l (0)ˆ ψn

εp −εn

ψp(r)

p>N / 2



∑









⋅B+

e

2m_e d× ψp p (0)ˆ ψn

εp−εn

ψp(r)

p>N / 2



∑









⋅B

ρ^(r)= −eN

∫

j(r) =

e

A ρ ^(r) −

e

∫ ( Ψ * ∇Ψ − Ψ∇Ψ ^*)d τ ^'

j(r)=_2m^e_e B×rρ^(r)+ ^iehN_2me

∫

point is computed as the point itself as origin. When the origin is included explicitly the following equation is obtained.

(36) Taking the ipsocentric approach, r =d and the factor ( ) becomes zero. And so the whole classical description of the diamagnetism is vanished.

(37) Because perturbation theory is used, Ψ=Ψ⁽⁰⁾+ Ψ⁽¹⁾. Using this expression in the

wavefunction part of the current density the following is obtained:

(38)

(39) Using: en the following is obtained:

(40)

And the current density can be written as:

(41) So this is the final expression for the current density in which the wavefunction and the first order correction to the wavefunction(eq.-32) are incorporated.

LOCALIZATION OF ORBITALS

In order to understand some chemical concepts in a better way the orbitals are localized.

Important concepts as valency, electron correlation and nonbonding electron pairs can be better understand in terms of localized bonds. Localized orbitals are also important for characterizing the shape of a molecule in terms of bonding orbitals, lone electron pairs or for example ‘banana’ bonds.

Standard ab-initio methods always give delocalized orbitals, and therefore linear combinations of MO’s are taken to localize the orbitals by minimizing the area that the orbital occupies. This can be done in several ways by minimizing or maximizing different terms. ‘The aim of all localization procedures is to find a unitary transformed orbital set

which satisfies a given optimum localization criterion.’¹¹ j(r)= _2m^e

e B×(r−d)ρ^(r)+^iehN_2m

e

∫

j(r)=^iehN_2me

[ ∫

τ

]

Ψ*∇Ψ − Ψ∇Ψ*

=(Ψ₀^{( 0)}+ Ψ₀⁽¹⁾) *∇(Ψ₀⁽⁰⁾+ Ψ₀⁽¹⁾)−(Ψ₀⁽⁰⁾+ Ψ₀⁽¹⁾)∇(Ψ₀⁽⁰⁾+ Ψ₀⁽¹⁾) *

= Ψ₀⁽⁰⁾*∇Ψ₀⁽⁰⁾+ Ψ₀⁽⁰⁾*∇Ψ₀⁽¹⁾+ Ψ₀⁽¹⁾*∇Ψ₀⁽⁰⁾+ Ψ₀⁽¹⁾*∇Ψ₀⁽¹⁾

− Ψ

(

)

Ψ₀⁽⁰⁾*= Ψ₀⁽⁰⁾ Ψ₀⁽¹⁾*= −Ψ₀⁽¹⁾

j

(r) =

e

[ ∫ ( Ψ

∇Ψ

− Ψ

∇Ψ

) ^d τ ^' ]

{ i' }

In this research project the Pipek-Mezey localization procedure is used. The advantage of this procedure is that it retains the distinction between π and σ orbitals. To do this the Pipek-Mezey maximizes the sum of Mulliken charges. Mulliken charges are the amount charge located on one atom.

DERIVATION OF MULLIKEN CHARGES

For a closed shell one-determinant the total electron density in a point A can be written as the sum of the charge densities per occupied orbital¹²

(42) The wavefunction is a Linear Combination of Atomic orbitals:

(43) Combining eq. F and F2 gives:

(44) In the last term the sum over the coefficient is written as prs and the factor two is included.

(45) Of course the total probability of all electrons has to be the total number of electrons

(46) and when equation-(44) is substituted in this formula:

(47) This formula can split up in diagonal terms and off-diagonal terms. The lower and upper part of the off-diagonal part are the same, and srs=1 when r=s.

(48) This total charge can be split up in parts for each different atom by summing up the terms that are located at that atom. If only one of ϕr and ϕs belongs to atom, for example X, then the contributions of the srs terms are divided and atom X is given one half of this

contribution. For this reason the factor two in the last term is lost because it is multiplied by a half.

(49) This qi is called the Mulliken charge. The Pipek-Mezey procedure tries to find a linear combination of MO’s that gives the highest Mulliken charges for each atom.

ρ

i

∑

Ψi= c_rφr r

∑

ρ^(A)=2 c_ir

i

∑

sr

∑

sr

∑

p_rs=2 c_irc_is

i

∑

ρ^dτ =N

∫

p_rs

rs

∑

∑

rs

∑

∫

p

s

+ p

s

r<s

∑

r=s

∑ + p

s

=

r>s

∑ ^p

+ 2 p

s

r<s

∑

r

∑ = N

q_X = p_rr+2 p_rss_rs+ p_rss_rs

s not onX

∑

ronX

∑

r<sonX

∑

ronX

∑

A DESCRIPTION OF WHICH EXCITATIONS WILL CONTRIBUTE TO THE TOTAL CURRENT.

Looking at equation-(32) it is possible to formulate some rules of thumb to predict of a certain excitation will give a ring current and how much.

Firstly, because there are two different operators, there are different symmetry selection rules for dia- and paramagnetism. To determine of a transition from an occupied

orbital(ψn) to a virtual orbital(ψp) is symmetry allowed the irreducable presentations of the occupied orbital( Γ(ψn) ) and virtual orbital(Γ(ψp) ) are determined. Then also the momentum operator L(0) and P are represented by an irreducable presentation.

To know of an excitation will contribute the cross product is taken of the irreducable presentations of the occupied and virtual orbital and the irreducable presentation that represents the operator. In terms: . The outcome of this cross product has to contain Γ0, which means that is has to contain at least the total symmetric presentation.

For diamagnetism an excitation contributes as contains Γ0. Here T_⊥ represents a translation perpendicular to the field direction. In fact this operator creates a nodal plane parallel to the direction of the field. In the picture below the magnetic field is perpendicular to the plane.

Figure 4 Schematic representation of the translational operator

For paramagnetism an excitation contributes as contains Γ0. R||

represens a rotation about the field direction. In the picture below the magnetic field is perpendicular to the plane.

Γ(

ψ

ψ

Γ(

ψ

ψ

Γ(

ψ

ψ

Figure 5 Schematic representation of the rotational operator

Secondly, it is important that the orbitals occupy the same region of space because

otherwise the overlap of Ôψn and ψp would be small. Below a schematic representation is given for a case when the two orbitals do occupy a different region of space and so will have zero overlap even if the excitation is symmetry allowed.

Figure 6 Schematic representation of a symmetry allowed but small-overlap contribution.

Also a consequence of this selection rule is that the orbitals Ôψn and ψp have to have the same nodal character if a transition is to be significant. The linear momentum operator creates a nodal plane therefore the virtual orbital and the occupied orbital contribute most substantially if the difference in the amount of nodal planes is one. The angular momentum operator does not create a nodal plane and therefore a contribution will be largest as the amount of nodal planes of occupied and virtual orbitals is the same. The amount of nodal planes is expressed by the symbol Λ, and the difference in the amount of nodal planes as ΔΛ. If this value is negative the occupied orbital possesses more nodal planes than the virtual orbitals which in practice do not occur. Briefly, if ΔΛ=0, a rotational transition is favorable and if ΔΛ=1, a translational transition is favorable.

Thirdly the energy gap between the occupied and virtual orbital has to be small because the energy difference is in the denominator of formula(x4). Therefore only occupied orbitals that are high in energy and virtual orbitals that are low in energy will substantially contribute to the ring currents.

Of course this selection rules are not totally independent. Most of the time the nodal character is related with symmetry and therefore the first and second selection rule are strongly related to each other.

BORON CHEMISTRY

The chemistry of boron has some interesting features that differ from carbon. A few remarks are given about boron chemistry.

Boron is the fifth element in group 13 with an electron configuration of 1s² 2s² 2p¹. It is a nonmetal and has a pronounced diagonal relationship with Si in Group 14 of the periodic table. Boron has only three valence electrons and therefore most of the boron compounds have no full octet of electrons and act as a Lewis acid which means that it can accept electron pairs.

Moreover, because boron is electron deficient it can form three-center-two-electron(3c- 2e) bonds. This means that three atoms can be bounded together by just two electrons.

The simplest example is B2H6 with two B-H-B bridges.

Figure 7 B2H6 with two 3c-2e bonds

A remarkable thing of boron is that it can form several, mostly anionic cage like three dimensional clusters with hydrogen atoms which are all diamagnetic.

THE STRUCTURE OF B2 02 -

Boron clusters (Bn) have remarkable properties and attracted much interest in the last decade. Small boron clusters have a planar structure and larger boron clusters have a three-dimensional structure. At least for the neutral boron cluster of twenty atoms it is obvious that the most favorable structure is that of a tubular three dimensional ring. This three-dimensional structure is stabilized by the double aromaticity of this molecule, caused by two aromatic current (radial and tangential)¹³.

Figure 8 Geometry of a neutral toroid B20

Planar boron clusters are experimentally detected up to B1514, but the exact point of when a three dimensional or two-dimensional structure is preferred is still unknown. Adding charges to the molecule have a great impact on the stability of the structure. It is indicated that neutral Bn clusters with n=2-19 have planar geometry, and adding a positive charge destabilizes the planar geometry starting at n=17. On the other hand adding negative charges to boron clusters with n=19 and n=20 stabilize the planar geometry.¹⁵ In this research project B202- is investigated which is planar due to aromatic stabilization. This aromatic stabilization is only possible when two electrons are added.

Figure 9 Geometry of B202-

Recently is found that the species B19- and B13+ are fluxional species. Fluxionality is the ability of a molecule to convert via a dynamic movement into another symmetrical identical conformation.

The fluxionality of B19- and B13+ means that the inner ring can rotate in respect to the outer ring; this rotation occurs not by breaking and forming bonds but rather by migration of three-center-two-electron bonds as showed in the example below.¹⁶ All filled triangles represent one 3c-2e bond. This rotation of the inner ring is almost freely with an energy difference between the ground state and the transition state of approximately 0.1kcal/mol.

Figure 10 The schematic representation of the 3c-2e bond migration during the internal rotation of B13+.

The planar B202- is also fluxional: the inner six-membered ring can rotate in respect to the 13-membered ring. The fluxional rotation has an energy barrier of approximately 0,027 kcal/mole, this is very low and means that the inner ring can rotate easily.

Figure 11 Schematic representation of the internal rotation of B202-

The fluxionality of B202- is linked to another feature, namely a paratropic ring current located at the six-membered ring of the molecule. This paratropic ring current is caused by the σ-orbitals. The aim of this research project is to understand this paratropic current.

However at present there is no way to deal with σ-aromaticity analytically.¹⁵

If one charge is placed at the outer ring and one at the inner atom, it is possible to imagine that B202- consist of three independent circles that are all conform the Hückel rule. Now the thirteen-membered outer ring with one additional charge contains 14(4*3+2) π-electrons and the inner ring contains six π electrons(4*1+2). The atom in the middle with one additional charge contains two π-electrons and is conform Hückel filling in n=0. So it is possible to imagine the molecule as three aromatic rings.

Figure 12 Schematic representation of 3 separate rings with 13, 6 and 1 member.

Although it is possible to imagine, it is nevertheless illegal to apply the Hückel rule in this way because nearest neighbor interactions between outer and inner ring are neglected.¹⁷ Therefore it is not possible to apply the Hückel rule to polycyclic system, although the Hückel theory can be used to calculate a large positive topological resonance energy.

THE BORON FULLERENE, B8 0

THE STRUCTURE

The research on boron fullerenes became interesting in 2007, when G. Szwacki published a paper of a prediction of boron fullerenes. Afterwards many in silico studies are done to do first-principle calculations to predict the properties of the boron fullerene. One predicted property is that the boron fullerene has a remarkable stability and therefore it is predicted by some that in future one will be able to make this boron fullerenes. This is in analogue to the carbon buckyball that was also only a theoretical prediction for 25 years.

In time many different structures are proposed: the buckyball structure, four-membered rings structures and snub conformations¹⁸, or a core-shell cluster. Even some totally asymmetric structures are proposed for B80 molecule, although most research focus on the buckyball and the volleyball structure, as described in the introduction. The HOMO- LUMO gap of the buckyball structure is the largest and is therefore kinetically favorable.¹⁹ There is some disagreement of the symmetry of the boron buckyball. In 2007 G. Szwacki et al.²⁰ predicted a buckyball with Ih symmetry. Later others claimed that the most stable structure of the boron fullerene has Th symmetry²¹. Afterwards Sadrzadeh et al. claimed that the Ih, Th and the C1 have approximately the same energy.²²

THE CURRENT

When ring currents are calculated from the boron buckyball a high similarity between the current patterns between the boron and carbon fullerene is noted, but with a higher current density in the boron fullerene. This stronger current is due to a lower HOMO- LUMO gap. In the pentagons a strong paratropic current is observed and in the

truncated(filled) hexagons a diatropic current is observed. Together this gives a strong cancellation between the para and diamagnetic currents. In total this gives for the

buckyball structure a slightly paratropic result. This is a remarkable difference in respect to the carbon analogue that is obviously diamagnetic.

Due to this strong cancellation effect, it is probable that a subtle change in structure can change the isotropy of the boron fullerene from paratropic to diatropic. This is supported by the fact that boron clusters with an increasing number of atoms are diatropic instead of paratropic²⁰

In the boron fullerenes the σ/π-distinction is somewhat diffuse and can not be

distinghuised by symmetry. A π-like character in the fullerene is defined as the fact that the wavefunction changes sign, going from outside to inside the cage.^{14, 23}

METHOD

To investigate ring currents the CTOCD-DZ²⁴ formulation as implemented in the SYSMO²⁵ and GAMESS-UK²⁶ programs was used. This is an ipsocentric method with a distributed origin approach, treating each point as its own origin. This method is useful because it allows decomposition of the ring current in contributions of single orbitals and even of single excitations.

All SCF, geometry optimizations, Hessian and NBO calculation were calculated using GAMESS-UK. For B202-, DFT/B3LYP/6-31G*²⁷ was used and for the boron fullerene direct DFT/BLYP/6-31G* was used. For the ring current calculation it is important to choose a flexible basis set.²⁸

For B202- the canonical set of orbitals obtained by DFT/B3LYP was localized using the Pipek-Mezey procedure, a procedure that maintains the σ/π-character of the orbitals. ²⁹ The geometry of B202- was optimized by the Katholieke Universiteit Leuven with a 6- 31+G*(2d) basisset.

To investigate the ring currents two-dimensional maps are made using the SYSMO-mo800 program. Almost everytime a magnetic field was applied perpendicular to a molecular plane and a plot of the current was made in a plane parallel to the molecular plane and perpendicular to the molecular plane at a height of 1 atomic unit. At this height the π- orbitals are at its most intense at this height.

Figure 13a. Schematic representation of a benzene molecule with a plane at a height of 1 a0, the direction of the magnetic field and the direction of the ring current.

The modulus of the current density was plotted as a blue contour. The darker blue the higher the current density. The current densities were plotted as yellow arrows. The directions of the arrows give the direction of the electron flux and the size of the arrows give the size of the currents. The current density and the modulus of the current density were viewed in the same plot.

A clockwise rotations of the current represents a paratropic current and hence is an indication of anti-aromaticity. On the other hand an anti-clockwise rotation represents a diatropic current and is an indication of aromaticity.

Figure 13 Schematic representation of the link between the different terms.

RESULTS AND DISCUSSION OF B2 0- 2

The magnetic current of B202- consist of an outer diamagnetic ring and an inner paratropic current at the six-membered ring. The paratropic ring current, presented by a clockwise rotation is caused by σ-orbitals and the diamagnetic current(anti-clockwise rotation) by π-orbitals.

Figure 14 The ring currents of B202-: 1. Total ring current 2. Anti-aromatic σ-current 3. aromatic π- current

In this section orbital numbers are used. Orbital 51 and 52 are the HOMO and LUMO respectively. Orbital 49, the HOMO-2 is the highest occupied molecular π-orbital and orbital 56, the LUMO+4, is the lowest unoccupied molecular σ-orbital (Table 1).

The magnetic field is applied perpendicular to the molecular plane and the plotting planes are 1 a0 above the plane. At a height of 1 a0 above the plane the π-orbitals have their maximum density, however the σ-bonds have their maximum in-plane. Plotting the ringcurrents at 0 a0, which means in-plane and plotting only the HOMO-n with n=0,1,2,3,4, a result is obtained with a jmax of 1.152(fig 15.1), or at 0.4 a0 a jmax of 0.932(fig 15.2) At a height of 0 a0 also contributions of core electrons are plotted, therefore also at a height of 0,4 a0 is plotted to get rid of the core electron contribution.

Diamagnetic Aromatic Anti-Clockwise Paramagnetic

Anti-aromatic

Clockwise

Figure 15 Ring currents of B202- at a height of 0 a0 and 0.4 a0.

One can see that the σ-current is rather big in these plots. That means that in respect to the aromatic π-current the σ-current is higher in intensity.

Table 1. All orbitals with their symmetry character, number of nodal planes and their atomic energy in hartree.

C2v Λ E(a.u.)

π 47: -4 b1 3 0.0324 π 48: -3 a2 3 0.0324 π 49: -2 b1 2 0.0420 σ 50: -1 b2 6 0.0720 σ 51: HOMO a1 6 0.0762 π 52: LUMO b1 4 0.1364 π 53: +1 a2 4 0.1652 π 54: +2 b1 4 0.1909 π 55: +3 a2 4 0.1921 σ 56: +4 b2 - 0.2472 σ 57: +5 a1 - 0.2536 σ 60: +8 b2 - 0.2776 σ 61: +9 a1 - 0.2826

First the current is splitted up in orbital contributions from π- and σ-orbitals. The

paratropic current is caused by the σ-electrons(14.2) and the diamagnetic current by the π-orbitals(fig 14.3).

THE PARAMAGNETIC CURRENT OF THE INNER RING.

Due to theory it is expected that the highest occupied orbitals have the largest contribution to the magnetic current because the energy gap between the initial and final state is low.

Plotting the orbitals of 50 and 51, the HOMO-1 and HOMO, one can see approximately the whole σ-contribution(fig 16.1). To see if this is true the contribution of all orbitals except orbitals 50 and 51 are plotted also(fig 16.2).

Figure 16 σ-currents of B202- 1. Orbital contribution of HOMO & HOMO-1, Jmax 0,100 2. Orbital contribution of all other σ-orbitals, Jmax 0,056.

Now the relevant occupied orbitals are known, we decrease the number of final(virtual) orbitals and according to theory we expect that only the virtual orbital of low energy and of σ-character contribute substantially to the current. Orbitals 56, 57, 60 and 61, which are the LUMO+4, LUMO+5, LUMO+8 and LUMO+9 respectively, are the first virtual orbitals which exhibit σ-character.

To predict which excitations will contribute the selection rules from the theory are used.

The angular momentum operator can be represented by Γ(R||)=b2. Filling in this irreducable presentation in the formula with the irreducable presentations of the σ-orbitals(a1 and b2) gives the following results:

b2×b2×a1 = a1 = Γ0

b2×a1×b2 = a1 = Γ0

b2×b2×b2 = b2 ≠ Γ0

b2×a1×a1 = b2 ≠ Γ0

Table 2: σ σ*-excitation with their change in symmetry, change in the amount of nodal planes and the energy difference.

ψn ψp Γ(ψn) Γ(ψp) ∆Λ ∆E(a.u) Jmax Jmax-para

50 56 b2 b2 0 0.175 0.0131 0

50 57 b2 a1 0 0.182 0.0587 0.0743

50 60 b2 b2 0 0.206 0.0433* 0

50 61 b2 a1 0 0.211 0.0229 0.0364

51 56 a1 b2 0 0.171 0.0454 0.0483

51 57 a1 a1 0 0.178 0.0086 0

51 60 a1 b2 0 0.201 0.0273* 0,0278*

51 61 a1 a1 0 0.206 0.0230* 0

*These Jmax value are located on the outer ring and therefore these values do not say anything about their contribution to the σ-current.

The impact of energy is not large if all orbitals are close to each other. The difference in the value of the ring current from an excitation from the HOMO to the LUMO +4 or LUMO+9 differs approximately 17 percent.

Figure 17. The orbital plots of the HOMO and the HOMO-1.

Figure 18: The orbital plots of the first four virtual σ-orbitals 56, 57, 60 and 61.

From the table one can see that there are only three excitations that do contribute substantially to the total σ-ring current. These are the excitations: 50 57, 51 56 and 50 61.

Figure 19 The excitation of orbital 51, the HOMO, to orbital 56, the LUMO+4

Figure 20 The excitation of orbital 50, the HOMO-,1 to orbital 57, the LUMO+5

Figure 21 The excitation from orbital 50, the HOMO-1, to orbital 61, the LUMO+9

Summing up these three excitations gives a plot of the total sigma-current. Displaying the contributions from orbitals 50 & 51 to all orbitals except 56, 57 and 61 give the result below. jmax. 0.048. The Jmax value is rather high but this value is not located inside the six-membered ring.

Figure 22 σ-currents of B202- 1. The sum of the contributions from orbital 50 and 51 to orbital 56 and 57. 2. All excitation from 50 and 51 to all other virtual orbitals.

THE DIAMAGNETIC CURRENT OF THE OUTER RING

The diamagnetic, aromatic ring current is caused by the π-orbitals. This ring current is split up in various orbital contributions. The highest occupied π-orbitals are the HOMO-2, HOMO-3 and the HOMO-4. These orbitals are numbered as 49, 48 and 47 respectively. The virtual π-orbitals that are lowest in energy are the LUMO, LUMO+1, LUMO+2 and the LUMO+3. These are numbered as 52, 53, 54 and 55 respectively.

The HOMO-3 and the HOMO-4 have the same nodal character and are located on the outer ring while the HOMO-2 is located on the central atom.

Figure 23 Orbital plots of orbitals 47(HOMO-4), 48(HOMO-3) and 49(HOMO-2) respectively.

Figure 24 Orbital plots of orbitals 52, 53, 54 and 55 respectively.

Table 3: π-π* transitions with irreducable presentations, the difference in the amount of nodal planes, energy gap and Jmax value.

ψn ψp Γ(ψn) Γ(ψp) ∆Λ ∆E(a.u) Jmax

47 52 b1 b1 1 0.104 0.0461

47 53 b1 a2 1 0.133 0.0442

47 54 b1 b1 0 0.159 0.0074

47 55 b1 a2 0 0.160 0.0076

48 52 a2 b1 1 0.104 0.0367

48 53 a2 a2 1 0.133 0.0365

48 54 a2 b1 0 0.159 0.0123

48 55 a2 a2 0 0.160 0.0133

49 54 b1 b1 1 0.149 0.0129

49 55 b1 a2 1 0.150 0.0126

Adding a nodal plane to orbitals 47 and 48 will give a good overlap with the orbitals 52 and 53. Therefore all possible excitations from 47 and 48 to 52 and 53 give a contribution to the diamagnetic current(See table 3). Plotting these individual excitations: 47 52, 48 52, 47 53 and 48 53 gives:

Figure 25 π π* transitions. Above: 47 52, and 48 52 Below: 47 53 and 48 53.

One can see from the pictures that each excitation contributes a bit to the total diamagnetic current and are complementary to each other. Summing up those four excitations gives the following plot. Looking at all π-π* excitations except this four we get a small diamagnetic current in the middel ring that is caused by excitations from orbital 49, the HOMO-2.

Figure 26 π π* transitions. 1. The sum of the plots in Fig. 25. 2. The sum of all excitations except the ones of fig 25.

Plotting the contribution from orbital 49 in two transitions 49 to 54 and 49 to 55 gives:

Figure 27 π π* transitions 1. From 49 to 54 2. From 49 to 55 3. From 49 to 54 and 55.

B2 02 - TRANSITION STATE:

Ring currents of the transition state molecule are also calculated and some Jmax values are compared with the result of the ground state.

Table 4: Comparison between the Jmax values of the B202- ground state and transition state.

Excitations Jmax ground state Jmax transition state

50>57 0.0587 0.0582

51>56 0.0453 0.0490

50>60 0.0585 0.0613

all σ-σ* 0.1040 0.0950

all π-π* 0.0752 0.0744

all to all 0.109 0.108

Indeed there are some differences but they are small so the ring currents of the transition state are built up almost in the same way.

A the transitions state has as at least one imaginairy frequency. The imaginairy frequency of B202- is -49 cm^-1(0.14 kcal/mol).

LOCALIZED ORBITAL ANALYSIS

Also localized orbitals were used to explain the magnetic currents. Using the Pipek-Mezey procedure the π/σ-distinction is retained. The Jmax values are the same for the delocalized orbitals. The total, π and σ Jmax value are respectively 0.1094, 0,0714 and 0.0955

In this case the paratropic ring caused by the σ-orbitals is splitted up in contributions form orbitals that are localized on the inner six-membered ring. It follows that the delocalized orbitals 21, 25 and 28 have the greatest contribution to the sigma current. Below the geometry of the localized orbital 21 is viewed. Orbital 25 and 28 almost have the same shape but rotated 120 or -120 degrees. Also the current of orbital 21 and that of orbital 21, 25 and 28 together are plotted below. To verify that this orbitals are the only significant orbitals all other σ-orbital contributions are plotted. The result is indeed that the paratropic current is only caused by the localized orbitals 21, 25 and 28.

Figure 29 1. Current density plot of the localized orbitals 21, 25 and 28 2. Current density plot of all the localized orbital with σ-character except 21, 25 and 28.

Also the π-current is split up in contributions of the localized orbitals. Orbital 33, 40, 42, 47 and 48 are π-orbitals. Also orbital 49 is a π-orbital but that one is located at the middle atom of the molecule. Orbitals 33, 40 47 and 48 have approximately the same shape but rotated approximately 72, 144, 216 and 288 degrees. Orbital 33 and 49 are plotted below with also their contribution to the ring current.

Figure 30 1. Plot of localized π-orbital 33 2. Current density plot of localized π-orbital 33

When the contributions of 33, 40, 41, 47 and 48 are plotted together the picture below is obtained with a Jmax of 0.0725.

Figure 32 Current density plot of all localized π-orbitals except 49

NATURAL BOND ORBITAL ANALYSIS

To gain information about the bonding pattern in the molecule a Natural Bond Orbital analysis is performed.³⁰ This gives us a result with orbitals that are localized on one bond (fig. 33.2) and localized orbitals on one atom. The result of the NBO analysis, as plotted below, gives a result where all atoms of the outer ring are bonded. Only three atoms of the inner ring have covalent bonds. This obviously shows the electron deficient bond character in this molecule. Also each atom has a lone σ-electron pair and π-electron pair.

In this model it is probable that the five bonds in the middle are in fact all able to flip to all other bonds and that the structure that describes the reality best is a mixed form with the five covalent bonds of the inner ring are distributed to the whole middle ring.

Figure 33 1. Geometry of B202- with the natural bonds calculated by the NBO-analysis 2. Orbital plot of

When a Natural Bond Orbital analysis is performed on the Transition-State molecule, another bond structure is obtained. The remarkable difference is the amount of bonds. In the ground state there are five bonds located at atoms from the inner ring. However in the transition state molecule there are seven bonds located at the inner ring.

Figure 34 Geometry of the transition state of B202- with the natural bonds calculated by the NBO- analysis.

CONCLUSION

From the results of B202- it turned out that the paratropic ring is caused by two σ-orbitals;

the HOMO and the HOMO-1. This current is built up from three excitations the HOMO to the LUMO+4, the HOMO-1 to the LUMO+5 and the HOMO-1 to the LUMO+9. The aromatic current of the π-orbitals is as expected rather similar to that of benzene.

Secondly the NBO-analysis showed that the bonds located at the inner six-membered ring have a strange, just slightly covalent character.

The total magnetic susceptibility of B202- is:

= _{! "} = 530,7 316.2 = 214 (51)

So the B202- molecule is in total a diamagnetic molecule.

RESULTS AND DISCUSSION OF THE BORON-80 FULLERENE

For boron fullerenes it is very difficult to gain clear information about how the currents go around the molecule. As an instrument only two-dimensional plots are available, so to gain information of the sphere current several plots had to be made. Another complication is that a change in the direction of the field can have a crucial impact on the currents and therefore it seems necessary to make plots of different field directions for the same hexagons and pentagons. Eventually a third complication is that the current outside the cage differs from the current inside the cage. Even a fourth complication is that when the ring currents are splitted up in orbital contribution the π/σ-distinction is not so well pronounced.

To illustrate that the currents inside and outside are different a plot of 1 a0 above a cross section of the molecule is plotted. The field direction is perpendicular to the plane.

Figure 35 Current density plot of a cross-section of B80-volleyball geometry.

It can be seen that the currents go more or less from outside to inside the cage and vice versa.

Three different kinds of planes are incorporated in the B80. A hollow hexagon, a filled hexagon and a filled (capped) pentagon.³¹ Plotting these planes respectively with the field direction perpendicular to each plane gives the results below.

Figure 36 Current density plots of three kind of planes. 1. Hollow hexagon, Jmax 0,146 2.

Filled Hexagon, Jmax 0,092 3. Capped pentagon, Jmax 0,358.

The height of the plotting plane for the pentagon is 0.7a0 instead of 1a0 to get rid of the core electron contribution from the atom at top of the pentagon. As a conclusion the current in the middle is just for a small part caused by core electrons.

It is difficult to interpret these results, therefore the current is splitted up in a diatropic and a paratropic contributions. Also this contributions are splitted up in a π and σ contribution. In this case a π-orbital is defined as an orbital changing sign going from inside to outside the plane.

A HOLLOW HEXAGON

Figure 37 Ring current contributions in a hollow hexagon plane of B80. Above the

diamagnetic part and below the paramagnetic part; Left the contributions of all π-orbitals and at the right the contributions of the σ-electrons. The Jmax values are respectively 0.090, 0.125, 0.172, 0.157.

Figure 38 The total π- and σ-currents. The Jmax values are respectively 0,120 and 0,0638.

Most of the current from the π&σ-currents are cancelled, only one significant paratropic part

A FILLED HEXAGON

Figure 39 Ring current contributions in a filled hexagon plane of B80. Above the diamagnetic part and below the paramagnetic part; Left the contributions of all π-orbitals and at the right the contributions of the σ-electrons. The Jmax values are respectively 0.076, 0.108, 0,112 and 0,123.

Figure 40 The total π- and σ-currents of a filled hexagon. The Jmax values are respectively 0,091 and 0,040.

The para- and diatropic σ-current cancel each other almost completely. Only the π-

currents give together a current pattern. It is remarkable that the current come from inside the fullerene cage.

A PENTAGON

Figure 41 Ring current contributions in a pentagon plane of B80. Above the diamagnetic part and below the paramagnetic part; Left the contributions of all π-orbitals and at the right the contributions of the σ-electrons. The Jmax values are respectively 0.096, 0.160, 0.304 and 0.193.

Figure 42 The total π- and σ-currents of a pentagon. The Jmax values are respectively 0,499 and 0,175. Notable is that the cancelling effects in the pentagon do not take place at the inner

From all these results one can conclude that there is a very strong cancellation effect between the paratropic and diatropic part of the current density. The total current is therefore a complicated and subtle difference between the two. A second thing is that the σ-orbitals have a more important contribution to the total diatropic and total paratropic current than the π-orbitals, but the cancellation effect of the diatropic and the paratropic part is bigger for the σ-orbital contribution than for the π-orbital contribution. Therefore the total result is mainly caused by π-electrons because they are less cancelling.

It is possible to make a plot of each hexagon holding the direction of the field constant. In that way a three-dimensional model of the boron fullerene was made in which one can see the total current of the molecule when the direction of the field is perpendicular to a hollow hexagon, or to a pentagon.

In the two pictures below the field direction is perpendicular to a hollow hegagon.

Figure 43 Pictures of a three-dimensional representation of the ring current of B80. The direction of the magnetic field is perpendicular to a hollow hegagon plane. 1. Side view 2.

Topview; direction of the magnetic field perpendicular to the picture plane.

This results are complicated because the currents differ much. There are no hexagons with the same current pattern.

When the magnetic field is perpendicular to a pentagon another sphere current is obtained.

Figure 44 Pictures of a three-dimensional representation of the ring current of B80. The direction of the magnetic field is perpendicular to a capped pentagon plane. 1. Side view 2.

Topview.

In the first picture a vertical symmetry plane can be distinghuised.

It can be seen from the pictures that the sphere current is a very difficult pattern of diatropic currents and paratropic currents, π and σ contributions, pentagon and hexagon contributions. Almost any plane has a totally different current, and in most of the hexagon a small diatropic current is visible.

The total magnetic susceptibility of the B80 fullerene, calculated by DALTON³², is 5,2 ; this means that the molecule is slightly paramagnetic. When the total magnetic susceptibility is calculated by the program mo807.x(SYSMO) the results differ very much, so the result is a priori wrong. The only agreement is that all results give a paramagnetic susceptibility.

Fig. 45. B80 orbital plots of the HOMO, HOMO-1 and the HOMO-2 respectively.