Orbital approach to the electronic structure of solids,

The reason why the writing of this book has been undertaken is the observation that, to the best of ourknowledge, none of the materials science books available at present extensively use

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Understanding the electronic structure of the materials on which he/she isworking may not be an essential need for an experimental scientist but certainlycan make his/her everyday work easier and more intellectually pleasing Theelectronic structure is the most obvious and useful link between the structureand properties of any solid Thus, understanding how the electronic structure

of a given material can be assembled (and thus how it can be altered) fromthat of the chemically significant building blocks from which it is made up is asimple yet very suggestive approach to the main goal of any materials scienceresearcher: the design and preparation of materials with controlled properties.Whether the new materials suggested in this way can be actually prepared ornot is something that depends, among other things, on the preparative skillsand art of the scientist This is why knowledge of the electronic structuremay not be essential However, it can make the quest much more rational andstraightforward, or it can direct the attention to something which otherwisecould seem bizarre

The impressive increase in computing power and the development of highlyperforming simulation codes for solids in recent years has provided chemists,physicists, and materials science researchers with very efficient tools to accessthe details of the electronic structure of practically any periodic solid However,this does not necessarily mean that we can understand the electronic structure

of any solid in a precise yet simple way Indeed this is what is needed totruly master the link between the structure and properties of the solids of

interest The development of efficient computational and conceptual tools is

the only way towards a fruitful interaction between theoretical and tal approaches with the intention of developing a sound understanding in thisfield Materials science being an essentially interdisciplinary field, the training

experimen-of scientists in the area is very much dependent on the physical or chemicalorientation of their curriculum Nevertheless, understanding the structure–properties correlation needs both physical and chemical concepts, which areusually taught using quite different languages The reason why the writing

of this book has been undertaken is the observation that, to the best of ourknowledge, none of the materials science books available at present extensively

use a blend of band theory, the appropriate physical approach to the

under-standing of the structure and properties of many solids, and orbital interaction

arguments, which is a transparent and chemically very insightful concept.

We believe that this kind of interdisciplinary approach may be extremelyenlightening

There is certainly nothing novel in saying that knowledge of electronicstructure is one of the more effective ways of making significant advances

in materials science J Goodenough was among the first to systematically use

concepts of electronic structure closely linked to structural details in lookingfor trends and predicting what materials could exhibit a certain physical prop-erty This work had, and still has, a lasting influence on materials science.Pioneered by R Hoffmann, J K Burdett, and M.-H Whangbo in the 1980s,the introduction to materials science of the ideas of orbital interaction, whichhad been so useful in rationalising the structure and reactivity of molecules,was a major breakthrough It soon became clear that the step-by-step building

up of many of the tools used within the context of the band theory of solids,such as band structure, density of states, Fermi surface, etc., based on orbitalinteraction ideas, provided an invaluable yet intuitive and easy-to-handle toolwith which to analyse the results of quantitative calculations or to rationaliseexperimental observations Structural and transport properties, the origin ofdifferent phase transitions and structural modulations, the nature of scanningtunnelling and atomic force microscopy images of complex materials, etc.were successfully rationalised on the basis of this type of approach Verydetailed structural information is encoded within orbital-interaction-type argu-ments so that through this approach it is relatively easy to link the effect

of possible structural modifications into say the band structure or the Fermisurface, etc and, consequently, to anticipate how these changes could alter thestability, conductivity or related properties of a given structure

With these developments in mind, around 1990 we thought that it would

be timely to introduce these ideas into the curricula of chemistry, physics ormaterials science courses at the postgraduate or final-year undergraduate lev-els This idea materialised as a course on the orbital approach to the electronicstructure of solids given at Universit´e de Paris-Sud Orsay, which was quitesuccessful and was repeated for a number of years It was also introduced atother French institutions such as the Ecole Normale Sup´erieure de Cachan,Universit´e de Montpellier, and Universit´e de Pau, as well as in several interna-

tional events Based on this experience a book entitled Description orbitalaire

de la structure ´electroniques des solides by C Iung and E Canadell, covering

the general principles and applications of such approach to one-dimensionalsolids was published in French by Ediscience International in 1997 Overthe years many colleagues prompted us to complete this work by writing

a new book fully covering the course, but academic and professional dutiescontinuously delayed this project The present book is a natural follow-up ofthe initial French publication in which we have generalised the content to covertwo- and three-dimensional solids and added some new material

The book contains 12 chapters, the first two being a sort of prelude The first

is a very brief overview of the free electron theory of solids with the purpose

of introducing some very basic physical notions, which we will use throughoutthe book In the second chapter we present a short overview of the basic notionscurrently used to understand the electronic structure of molecules, emphasisingthe symmetry and orbital interaction arguments One of the purposes of thischapter is to show that the molecular orbital theory used for molecules andthe band theory used for periodic solids are really simple variations of thesame idea due to the discrete or periodic nature of the systems The essentialmachinery of the band theory of solids and its orbital interaction analysis is

Preface viideveloped in Chapter 3 Most of the formal tools that will be used throughoutthe book are explained there using the simplest periodic system we can thinkof: the infinite chain of hydrogen atoms This keeps the formal developmentssimple and allows us to treat the same system in different ways so that thereader may be aware of different ways to approach a given problem The fourthchapter is devoted to the ubiquitous Peierls distortions of solids This is animportant phenomenon exhibited by many solids and has strong consequencesfor transport and other properties Chapters 5, 7, and 8 are essentially differentapplications of the ideas developed in the third and fourth chapters to organicand inorganic one-dimensional solids Chapter 6 is a brief introduction tothe handling of symmetry when studying the electronic structure of solids.The use of symmetry in band theory is an elegant yet not always simplematter, which cannot be developed at length in a book like the present one.However, we have discussed some useful and quite basic aspects of symmetry

in this chapter Up to the end of Chapter 8 the work is restricted to dimensional systems Chapters 9–11 generalise the approach to two- and three-dimensional solids In Chapter 9 the basic theoretical notions are generalisedfor systems of any dimensionality and some model systems are considered.The increase in dimensionality and structural complexity soon leads to theneed to consider many orbitals and several directions of the Brillouin zone.The analysis of the results (or the qualitative building up of the electronicstructure) may become too cumbersome, so that a simpler analytical toolmust be devised The simpler and more useful tool devised for this purpose

one-is the density of states (DOS) The object of Chapter 10 one-is to present severalways to analyse this useful construct from the viewpoint of orbital interactionanalysis using real examples Chapter 11 deals with low-dimensional solidsand the analysis of the Fermi surface, an extremely useful concept which,when appropriately decoded, contains much information about the transportand structural properties of metallic systems In this chapter we will show thatthe essential aspects of the Fermi surface of a given metal may be obtained in

a relatively simple way using the orbital interaction approach The procedurewill be illustrated by considering several classes of low-dimensional materials,which have given rise to considerable debate in the literature Most of thepresent book uses a one-electron view of the electronic structure of solids.Although this is a perfectly legitimate option for a very wide range of materialsand for the purposes of this book, it must be clearly stated that an explicitconsideration of electronic repulsion is indispensable to understand certainclasses of solids such as systems exhibiting magnetic properties Discussion

of this problem at a level consistent with the detailed approach of this bookwould have markedly increased its length and has not been considered realistic.However, we have included a final chapter in which the essentials of how theinclusion of electronic repulsion can modify the conclusions of a one-electronapproach are outlined

This is essentially a teaching book and consequently we have included aseries of exercises so that readers may check their progress from time to time.Exercises that do not need to be considered on a first reading are marked with

an asterisk Answers to the exercises are provided, although sometimes they are

deliberately only sketched Since this is not a research book we have not madeany attempt to present a detailed list of references We generally mention somebooks or publications that may be helpful for readers interested in expandingtheir coverage of the subject For the real examples discussed in the text wealways make reference to the original publications reporting the structure of thesystem In that way, readers interested in carrying out actual calculations for thesystem can prepare their inputs In general we also provide reference to one ortwo papers in which the electronic structure is discussed Because of the nature

of the book we have always chosen those with a strong pedagogic orientation

We apologise for not mentioning the many excellent papers available for most

of the systems considered

This book would have been very different (and certainly less satisfying)without the input of the many students who attended our lectures We aredeeply indebted to them; their comments and questions have provided theimpetus for the continuous polishing and revising of many aspects of thisbook In addition we have benefited from the comments of many friends andcolleagues who have read parts of the book, both the French and Englishversions This book also owes much to the many discussions that took placebefore the actual writing with T R Hughbanks (Texas A & M University),M.-H Whangbo (North Carolina State University), and the late J K Burdett(University of Chicago), and to Y Jean (Palaiseau), and F Volatron (Orsay) forpushing us to write the initial French version We thank A Garc´ıa for his help

in implementing the tight-binding programs and F Boyrie for his invaluablehelp in the LaTeX compilations We also thank C Raynaud and E Clot foruseful discussions about the methodological part of the book We are grateful

to Dunod ´Editions for permission to use material from the French edition in thepresent work We warmly thank Sonke Adlung, our editor at Oxford UniversityPress, and his team (Lynsey Livingston, April Warman, and Clare Charles) fortheir continuous support, help and infinite patience with three authors whowere continuously delaying the writing of the book Last, but not least, wedeeply thank our families for patiently enduring the writing of this book

Enric Canadell,Marie-Liesse Doublet,and Christophe Iung

Bellaterra, Montpellier, February 2011

2.1 Molecular orbital theory 142.1.1 Born–Oppenheimer approximation 152.1.2 One-electron approximation 152.1.3 LCAO approximation 152.1.4 Secular equations and secular determinant 162.1.5 Basic features of the H¨uckel and extended

H¨uckel methods 172.1.6 Symmetry properties of the molecular

2.2 A short review of the theory of symmetry point groups 192.2.1 Different symmetry point groups 192.2.2 Classes 212.2.3 Basis for an irreducible representation 222.3 Application to the study of theπ system of regular

cyclobutadiene 252.3.1 Decomposition of the(p z ) basis 262.3.2 Determination of the basis elements for different

irreducible representations 272.3.3 Molecular orbital diagram of theπ system of

regular cyclobutadiene 302.4 Transition metal complexes 302.4.1 Ligands and formal oxidation state 312.4.2 The ML6octahedral complex 332.4.3 Distortions of a complex 39

3 Electronic structure of one-dimensional systems:

3.1 Bloch and crystal orbitals 453.1.1 Bloch orbitals 463.1.2 Crystal orbitals 493.2 Electronic structure of the model chain Hn 513.2.1 Representation of the CO() and CO(X) functions 513.2.2 Energy of the crystal orbitals in the H¨uckel

3.2.3 Band structure 54

3.2.4 Basis for an energy level E (±k) 553.2.5 Fermi level of the Hnchain 573.3 Electronic structure of the dimerised model chain(H2) n 583.3.1 Formal determination of the band structure 583.3.2 Qualitative determination of the band structure 613.4 Comparison of the regular Hnand dimerised(H2) nchains 633.4.1 Comparison of the band structures of the regular Hn

chain generated by either a simple or a double unit cell 633.4.2 Dimerisation in the Hnchain: notion of distortion

in a periodic system 67

4.1 Analysis of the model system(H0.5+ ) n 724.1.1 Effect of a tetramerisation on the Fermi level 734.1.2 Effect of a tetramerisation on the states near the

Fermi level 744.1.3 Effect of a tetramerisation on the band structure 764.2 Analysis of first-order Peierls distortion in terms of a charge

density wave 774.3 Nesting vector 814.4 Commensurate and incommensurate distortions 814.4.1 Commensurate distortion 814.4.2 Incommensurate distortion 834.4.3 Comparison 834.5 Conclusions 83

5.1 Electronic structure of ethylene 865.2 Main aspects of the band structure for trans-polyacetylene 875.3 Detailed analysis of the band structure of trans-polyacetylene 885.4 Determination of the band structure of trans-polyacetylene

using the fragment formalism 895.4.1 Calculation of the band structure by means of the

H¨uckel approach 915.4.2 Qualitative determination of the band structure 925.5 Band gap opening at the Fermi level in trans-polyacetylene 93

Contents xi

6.1 Analysis of the Ansystem 966.1.1 Analysis of the cyclic Ansystem 966.1.2 Analysis of the linear Ansystem 101

6.1.3 Notion of group of a k point 1046.2 Application to the determination of the band structure for the An

linear system, where A is an atom 104

6.2.1 Group of the different k points 1056.2.2 Symmetry of the different Bloch orbitals 1056.2.3 Bands associated withσ -type overlaps 1076.2.4 Complete band structure 1086.3 Band structure of the hypothetical (NaCl)nchain 109

6.3.1 Group of the different k points 1106.3.2 Bands associated withσ -type overlaps 1106.3.3 Complete band structure 1126.4 Consequences of the existence of a glide plane 1136.4.1 Using point group symmetry properties in

6.4.2 Complete space group (non-symmorphic) of

6.4.3 Crystal orbitals of trans-polyacetylene by means of the

non-symmorphic space group G = T n ⊗ C 2h ⊗ {E, g σ} 1176.4.4 Concluding remarks 1196.5 Work plan for the study of a 1D system 120

7.1 Band structure near the Fermi level 1237.1.1 Unit cell definition 1237.1.2 Symmetry analysis of the chain 1237.1.3 Appropriate fragment orbitals 1237.1.4 Crystal orbitals at the and X points 1247.1.5 π-type band structure of polyacene 1267.2 Distortions in polyacene 1287.2.1 Disappearance of theσ x ysymmetry plane 1287.2.2 Disappearance of theσ yzsymmetry plane 1287.3 General remarks concerning Peierls distortions 1307.3.1 First-order Peierls distortions 1307.3.2 Second-order Peierls distortions 131

8.1.1 Band structure of the eclipsed chain [Pt(CN)4](2−δ)− 134

8.1.2 Band structure of KCP (staggered chain) 1398.1.3 Conclusions 1428.2 (ML4L)nchains 143

8.2.1 Symmetry 143

8.2.2 Choice of the fragment orbitals to generate theBloch orbitals 1438.2.3 Analysis of the Bloch orbitals at the and X points 1448.2.4 Symmetry of the Bloch orbitals 1448.2.5 Band structure 1458.2.6 Study of the (ReCl4N)nchain 1478.2.7 Electronic structure of the (Pt(NH2Et)4Cl2+)nchain 149

8.3 Suggested studies 153

9.1 Basic concepts 1579.1.1 Direct and reciprocal lattices 1579.1.2 Bloch and crystal orbitals 1599.1.3 Brillouin zone 1619.1.4 Symmetry and the Brillouin zone 1629.2 Analysis of the electronic structure of 2D model systems 1669.2.1 The square lattice2∞[Hn] system 1669.2.2 The square lattice2∞[An] system 1699.2.3 π-type band structure of hexagonal graphene layers 173

10.1 Calculation and analysis of the density of states 18110.1.1 Density of states 18110.1.2 Projected density of states 18310.1.3 Crystal orbital overlap population 18510.2 Combined use of DOS and COOP: electronic structure of the

MPS3layered phases 18610.3 Step-by-step determination of the density of states: the

(Pt(NH3)4Cl)2+chain 188

10.4 Density of states and fragment molecular orbital interactionanalysis: application to the [(C5H5)M] chains 19310.5 Transition metal diborides with the AlB2structure type:

a 3D case study 196

11.1 Notion of Fermi surface 20411.2 Nesting vector and electronic instabilities in low-dimensional

11.3 Monoclinic TaS3versus NbSe3 21011.3.1 Crystal structure and electron counting 21111.3.2 Qualitative band structure 21211.3.3 Qualitative Fermi surface: differences between

NbSe3and TaS3 21411.4 Molybdenum bronzes 215

Contents xiii

11.4.1 Octahedral distortions and t 2glevel splitting in

MoO6octahedra 21611.4.2 MoO5chain with corner-sharing octahedra: counting of 2 p

oxygen antibonding contributions 21711.4.3 A0.33MoO3(A = K, Rb, Cs, Tl) 2D red bronzes: metallic orinsulating? 21911.4.4 A0.3MoO3(A = K, Rb, Tl) blue bronzes: 2D solids with

pseudo-1D behaviour 22411.4.5 Looking for 1D systems where there seem to be none: theconcept of hidden nesting 22711.5 Low-dimensional molecular conductors 23211.5.1 An archetypal molecular metal: (TMTSF)2PF6 23411.5.2 Chemically modifying the electronic structure of molecularconductors 23511.5.3 Structurally complex materials with simple band structures 23811.5.4 A case study: 1D vs 2D character of the carriers in someα

phases of BEDT-TTF 24211.5.5 Electronic structure and folding: how to relate the band

structure and Fermi surface of different salts ofthe same family 247

12.1 From the H¨uckel model to the Hubbard model 25612.1.1 The delocalised picture of H2 25612.1.2 The localised picture of H2 26012.1.3 From the molecule to the solid state 26612.1.4 Application to one-band systems 26912.2 Mean-field approaches 27312.2.1 The many-body problem 27312.2.2 The Hartree–Fock method 27412.2.3 Density functional theory 28112.3 Conclusion 287

Elementary introduction

to the transport

One of the main goals of this book is to build a bridge between the electronic

structure of a periodic solid and its physical properties and, more particularly,

its conducting properties Consequently, the first thing that we must learn is

what characterises a metal, a semiconductor, and an insulator In this chapter

we will use the simplest possible approach that provides some understanding

of the nature of a metallic system: the free electron model Even if this model

is in many aspects quite simplistic, it provides a simple and essentially correct

view of what makes a system a good or a bad conductor In passing, a brief

consideration of the free electron model will allow us to introduce many of the

key concepts that we will use throughout this book

1.1 Free electron model

For a system to behave as a metal, i.e to be a good conductor, it must possess

valence electrons not tightly bound to the nuclei The free electron model,

initially developed by Sommerfeld, is based on the assumption that a metal

can be viewed as a series of electrons which move freely over a network of

fixed A+cations (see Fig 1.1) [1]

The potential felt by each electron is assumed to be nil in the solid, but equal

to a positive and large value (+V0) outside The electron is thus confined within

the metallic piece and the different forces felt by the electron in the metal

(attractive or repulsive) are neglected Since we are interested in the collective

properties of the bulk material and, in general, we will not be interested by

the properties at the surface, we will adopt boundary conditions, making it

Fig 1.1

Free electron model: the electrons are

represented as e−.

possible to neglect the effects at the borders Now that this approach is clearlystated we will proceed to the determination of the wavefunctions describing

the behaviour of an electron in this ideal metal For simplicity, we will initially

consider a one-dimensional system and later we will generalise the results to athree-dimensional system

1.1.1 One-dimensional system

Let us consider an electron constrained to move through a linear segment oflength L

Eigenvectors of the system

Let us write the Schr¨odinger equation for this one-dimensional system:

− ¯h2

2m

d2(x)

where(x) is the wavefunction describing an electron at point M with

coordi-nate x, E is the energy of the electron and V (x) is the potential at the M point.

In the free electron model the potential is nil for any point of the system and

we must therefore solve the following equation:

for which the general solution in the interval]− L/2, L/2] is given by:

where k =√2m E

¯h and E ≥ 0 C+ and C− are two constants such that the

(x) function is not nil in the ]− L/2, L/2] interval In this way we obtain

a linear combination of two waves associated with opposite wave vectors k

with projections k and −k on the Ox axis.

Let us now impose boundary conditions to be able to characterise thewavefunction for any point M.1 Different approaches are possible (see Exer-

1 By imposing these boundary conditions

we recognise the existence of the borders

and impose the condition that the

wave-function describing the electron within

and outside the metal is continuous and

may be derived Whatever the boundary

conditions imposed, they lead to a

quan-tisation of the energy and, consequently,

of the k vector.

cise (1.1) at the end of this chapter for another possible approach) Here wewill adopt one that makes equivalent the linear system (see Fig 1.2) and thecyclic system with perimeter L resulting from the condensation of the pointswith abscissas +L/2 and −L/2 (Fig 1.3).[2, 3]

Such boundary conditions, initially proposed by Born and von Karman, ofcourse ignore the effect of the borders because they do not exist in the cyclicsystem This model is only valid if the system is very large The point M in

Fig 1.2

One-dimensional system of length L

assumed to be very large The position of

the M point is represented by a vectorrm

and an abscissa x (x ∈]− L/2, L/2]).

Free electron model 3

Fig 1.3

Equivalence between a very large linear system and a cyclic system.

the linear representation is equivalent to the point M characterised by the angle

φMin the cyclic representation:

The wavefunction of the electron in the cyclic representation as well as its

derivative must verify the following boundary conditions:

whererefers to the derivative of the function.

Given the equivalence between the two systems, the wavefunction of the

linear system must verify the periodicity condition:

and 

−L2



= −

L2



(1.6)

We will only keep the eigenstates fulfilling the boundary conditions of

eqn (1.6), i.e those that verify:

Rearranging these formulas we obtain:

However, the quantities(C+− C−) and (C++ C−) are not simultaneously nil

except if the C+and C−constants are both nil, which is impossible (eqn (1.3))

Consequently, the term sin(kL2) must be nil

sin(kL

As a result, k is quantised and must be multiple of 2π

L The solutions verifyingthe Schr¨odinger equation (eqn (1.2)) as well as the boundary conditions ofeqn (1.6) are labelled k (x) and are given by:

Any linear combination of the degenerate wavefunctions k (x) and  −k (x)

is also a solution of the Schr¨odinger equation (1.2) and obeys the boundary

conditions of eqn (1.6), because no restriction is imposed on the C+ and C−coefficients

As shown in eqn (1.3), the quantisation of k also imposes quantisation on the energy We will label E (k) the energy associated with the waves  k (x) and

Now we must normalise the  k (r m ) wavefunctions in the ] − L/2, L/2]

interval to determine the constant N :

 L 2

− L 2

| k (x)|2

1

The wavefunction associated with an electron which must stay within a ment of length L fulfilling the Born–von Karman boundary conditions ischaracterised by the wave vector k equal to ki x:

frag- k (x) =

1

1

Free electron model 5

The allowed values for k



k = 0, ±2π

L, ±4π

axis as shown below:

The energy E (k) associated with the function  k (x) is given by eqn (1.13).

Fermi level

Since electron–electron repulsions are neglected, electrons fill the lowest

energy levels, with two electrons per allowed state in the ground state at

T = 0 K We will refer to the Fermi level, ε f, the highest energy level filled

in the ground state at T = 0 K This highest filled level is characterised by the

±k f wave vectors and its energyε f is given by the formula:

ε f = (¯hk f )2

As an example, let us assume that the system possesses seven electrons In

Fig 1.4 we have plotted the curve of E (k) (eqn (1.13)) as well as the allowed

seven electrons fill the three lowest energy levels as shown in Fig 1.4

Fig 1.4

Allowed energies for a one-dimensional system fulfilling the Born–von Karman boundary conditions For clarity, the different energy levels have been drawn quite far from each other In reality, they are very close since L is large (2π/L is

small).

More generally, for a one-dimensional system possessing a large number

of electrons N e , at T = 0 K, the lowest level will be filled with two electrons

and all other levels will be filled with four electrons up to the Fermi level,ε f,

which is characterised by the values of±k f

Since the system is very large, two adjacent allowed k values are very close

to each other Consequently, the energy spectrum is very dense and practically

continuous Thus, we are dealing with an energy band of allowed energy levels

(see Fig 1.5) Since the spectrum of allowed k values is practically continuous,

we must distinguish the k values associated with filled levels at T = 0 K from

those associated with empty k levels (Fig 1.6).

Fermi–Dirac statistics

At a given finite temperature T , the probability f (E)d E that a state with

energy E is filled follows a Fermi–Dirac distribution f (E) given by:

Fig 1.5

Band of allowed energies in the free

electron model.

The curve of f (E) is a step function, which becomes increasingly rounded

near the Fermi level as the temperature increases

Given that the value of k B T at room temperature is 25 meV, only the states in

the vicinity of the Fermi level are affected by this thermal excitation In Fig 1.7

we have schematically shown the temperature effect on the population of theallowed energy levels: because of the thermal excitation, some electrons that

were below the Fermi level at T = 0 K lie slightly above at a finite temperature.

Obviously, as the temperature increases the number of electrons affected bythis transfer to higher energy states also increases

Figure 1.8 shows the k values associated with states populated at a given

finite temperature The temperature leads to a transfer of electrons lying below

the Fermi level at T = 0 K towards states which are above the Fermi level, characterised by k vectors with norm higher than k f

Fig 1.8

(a) Population of a band at T = 0 K and

at T = 0 K (b) k values associated with

states that are filled (in black) or partially

filled (in grey) for T= 0 K.

Free electron model 7

Ohm’s law

Since the wave vector k is proportional to the momentum vector p, Fig 1.8b

shows that the system possesses as many electrons moving towards the right as

electrons moving towards the left What happens if, at time t = 0, the system

is placed under the influence of a constant and uniform electric field E directed

towards the left of the O x axis? Every electron then feels an electric force  F

equal to−e E (e being the absolute value of the electron charge) Using the

fundamental relation of dynamics for the electron characterised at the initial

moment (t = 0) by a wave vector k(t = 0) and thus, by a momentum p(t = 0)

equal to¯hk(t = 0), we obtain:2

2 Even if working in the framework of quantum mechanics, it is possible for this particular case to use the fundamental equation of classical mechanics.

The electrons in this model thus feel a constant acceleration towards the right.

This result is not realistic since the electrons will decrease their speed as a

whose existence has been neglected so far.

Because of the collisions of the electrons with the underlying network, after

a certain timeτ these electrons will, on average, no longer be accelerated In

other words, the acceleration due to the electric field is cancelled after a timeτ

by the collisions of the electrons with the cations of the network Clearly, after

a certain timeτ, the system reaches a stationary state in which every electron

keeps the momentum acquired Consequently, the electron initially described

by a p momentum will possess a p(stationary) momentum given by:

p(stationary) = p(t = 0) + δ p with δ p = −e Eτ (1.22)

Once the stationary state is reached, the electron associated with a wave

vector k before the application of an electric field is characterised by a new

k(stationary) equal to k + δk, given by the equation:

k(stationary) = k(t = 0) + δk with δk = − e  E τ

Thus the electric field leads to the increase of every wave vector k by the same

amountδk At T = 0 K, once the stationary state is reached, the filled states are

characterised by k vectors whose projection lies in the [−k f + δk, k f + δk]

interval (see Fig 1.9a), whereδk is the projection of the δk vector.

Figures 1.9b and 1.9c show the schematic band structures, where the states

associated with a vector k whose projection on the axis O x is positive (k > 0)

are separated from those characterised by a negative projection (k < 0).

At T = 0 K, when the electric field is nil, there are as many states

associ-ated with positive values of k as states associassoci-ated with negative values (see

Fig 1.9b) In contrast, application of the electric field leads to a depopulation

Fig 1.9

(a) Effect of an electric field on the wave

vectors characterising the filled states

once the stationary state is reached at

T= 0 K (b) Population of the states at

T= 0 K when the electric field is nil.

(c) Population of the states at T = 0 K

when an electric field in the negative

direction of the O x axis is applied.

of states near the Fermi level with a negative value of k , while populating

states slightly above the Fermi levels characterised by a positive value of k

(see Fig 1.9c) Once the stationary state is attained, the system possessesmore electrons moving towards the right than moving towards the left so that,

globally, the system conducts current This scheme shows that a system may

Once the stationary state is attained every electron has acquired an identicalsupplementary momentumδ p, associated with an increase in speed, δv, which

is given by the formula:

δv = δ p

As a result, the electric field is at the origin of the existence of a current density

j through the system:

• i is the intensity crossing the system with section S,

• i xis a normalised horizontal vector,

• dq is the charge crossing section S during the time dt,

• (−e) is the charge of the electron,

• n is the number of free electrons per volume unit with a charge −e, i.e the

number of carriers per volume unit,

• σ is the conductivity of the material and ρ is the resistivity (usually in units

of Ohm.cm)

Free electron model 9Equation (1.25) is simply Ohm’s law, and makes clear that the conductivity

of a material depends on two factors: the number of carriers of the system,

i.e the number of free electrons, and the value ofτ, which becomes larger

as the coupling of the electronic motion with the positively charged network

decreases

The free electron model correctly accounts for Ohm’s law as far as the role

of the underlying positive network is taken into account The interaction

between the electrons and the positive network is at the origin of the

slow-ing down of the electronic motion through the system and, consequently,

of the resistivity of the material

1.1.2 Generalisation to a three-dimensional system

Let us now consider a metal with cubic shape and volume L3 (with x∈ ] −

the electrons are free to move in the solid, which is a three-dimensional (3D)

network of cations A+ The potential within the solid is assumed to be nil and

the eigenstates describing the behaviour of an electron are plane waves:

 k (r m ) =

1

where k = k x i x + k y i y + k z i z, rm = xi x + yi y + zi z and (i x ,i y ,i z) is an

orthonormal basis, if the Born–von Karman boundary conditions are imposed

along the three directions of space:

These conditions are fulfilled if the k x , k y , and k z components are quantified

in the following way:

L ; k y= 2n y π

L ; k z =2n z π

where n x , n y , and n z are integers

Every eigenstate is characterised by three quantum numbers (k x , k y , k z) In

eqn (1.26), k refers to the wave vector of the plane wave  k (r m ) with which a

momentum vector p equal to ¯hk can be associated The energy E(k) associated

with this plane wave is given by:

characterising the Fermi level, fulfill the equation:

Consequently, the endpoint of any k vector characterising a state at the Fermi

level is found on a sphere with radius k f = √2m ε f

¯h (see Fig 1.10).

We define the Fermi surface as the surface containing the endpoints of the k f vectors characterising the states whose energy is equal to ε f In thefree electron model the Fermi surface of a metal is the surface of a sphere

with radius k f (see Fig 1.10) The study of a 3D system according to thefree electron model is completely equivalent to that of a 1D system Fromthe formal point of view, the problem is a little bit more complex becausethe degeneracy of each level is higher However, the reasoning is completelyequivalent When an electric field E is applied, every electron increases its

speed by exactly the same amount,δv (eqn (1.24)) This is at the origin of the

density current going through the system (eqn (1.25)) Even if for simplicity

we have discussed Ohm’s law using a 1D approach, we could equally havedone so using a 3D free electron model

Fig 1.10

Fermi surface according to the free

1.2.1 Factors influencing the conductivity

The free electron model provides a simplified but essentially correct description

of the requirements for a system to conduct electricity To begin with, a tor needs to have electrons that are capable of participating in the conductingprocess, i.e which are not strongly bound to the underlying network of nuclei

conduc-In addition, the process of electrical conduction needs the participation of statesjust above the Fermi level,ε f (see Fig 1.9c) Thus, good conductors usually

have a large density of allowed states very close to the Fermi level so that a

considerable number of electrons can participate in the conduction process Inaddition, since conductivity is better when the interaction between the electronsand the underlying network is weak, it is clear that it depends on the nature

of the atoms from which this network is built as well as the temperature and

Conductivity of real solids 11the purity of the material For instance, when temperature is raised there is

an increase in the amplitude of the vibrations of the network, and this has the

effect of slowing down the electronic motion We can restate this fact by

say-ing that the electron–phonon couplsay-ing is more efficient when the temperature

increases.3 The conductivity of a metal is thus better when the temperature 3

A phonon is a quantum of vibrational energy of the periodic system.

decreases As for defects, these have the tendency to pin free electrons so that

their presence leads to a decrease in the conductivity of the system

To summarise, the conductivity of a material essentially depends on four

parameters:

• the density of states near the Fermi level

• the electron-phonon coupling

• the temperature

• the defects

1.2.2 Band structure of real solids

In contrast with the prediction of the free electron model, the band structure

for a real periodic solid is not a single band (Fig 1.5) The allowed energies of

a real solid are found within the energy ranges associated with different energy

bands separated by forbidden energy gaps (Fig 1.11)

Fig 1.11

Band structure for a real solid.

One of the goals of this book is to adopt a simple approach that allows

an understanding of how the nature of the different atoms and the structural

details are related to the band structure of a periodic solid In particular, we will

be interested in analysing the orbital nature of the different bands responsible

for the properties of the system For the time being let us assume that these

band structures may be obtained and see how they are related to the electrical

behaviour of the material

1.2.3 Metallic behaviour

When the Fermi level of a system lies inside a band, i.e in the vicinity of a large

number of empty levels, the system is a metal In general, the room temperature

resistivity is weak in these cases, of the order of 10−6 Ohm.cm.[1] More

importantly, the resistivity decreases when the temperature is lowered because

the efficiency of the electron–phonon interactions decreases (see Fig 1.12)

This is the main feature characterising the behaviour of a metal

When considering an energy band of a 3D material built from only one type

of orbital, the density of states is usually maximum at the middle of the band

Fig 1.12

Resistivity vs temperature behaviour for

a metallic system.

Consequently, a system for which the Fermi level occurs near the top or thebottom of a band should not, in principle, be a very good conductor They

are in fact semimetals and are characterised (see Fig 1.13) by the Fermi level

lying near the top and bottom of two bands Their resistivity changes withtemperature in the same way as a metal (see Fig 1.12) but it is generally largerbecause the density of states around the Fermi level is smaller

Fig 1.13

Band structure for a semimetal where the

filled states at T = 0 K have been

represented in black.

1.2.4 Semiconducting and insulating behaviour

When the Fermi level occurs at the top of a band which does not overlap withanother band,4the system does not transport current at T = 0 K; we are dealing

4 Throughout this book we will refer to

the Fermi level as the highest occupied

level With such a definition the Fermi

level of a semiconductor occurs at the

top of a band whereas the thermodynamic

definition of the Fermi level places it

within the band gap.[1]

with an insulator or a semiconductor If the temperature is different from 0 K, some electrons at the top of the highest filled band (called the valence band) are transferred to the bottom of the lowest empty band (called the conduction

band) by thermal excitation (see Fig 1.14b) As a consequence, there are filled

states at the bottom of the conduction band in the vicinity of empty stateswhich can participate in the transport process In the same way, the empty

states (called holes) at the top of the valence band can also participate in the

transport process

Fig 1.14

Population of the valence and conduction

bands for a semiconducting solid: (a) at

T = 0 K and (b) at T= 0 K The states

that are filled are shown in black; those

that are partially filled are in grey.

Consequently, the conductivity of a semiconductor originates from themobility of the electrons (with charge −e) of the conduction band and the holes (formally with charge +e) of the valence band However, the number of carriers (with charge +e or −e) is weak, and this is more so as the temperature

is lowered and as the forbidden energy gap between the valence and conductionbands becomes larger Thus, in contrast with the situation in a metal, theconductivity of a semiconductor increases with temperature This is the maincharacteristic of a semiconductor (Fig 1.15)

In contrast, when the band gap is large, so that the bottom of the conductionband cannot be significantly populated (and the top of the valence band depop-ulated), the room temperature conductivity will be very low and the system is

an insulator The resistivity of semiconductors is generally between 10−2and

109Ohm.cm, whereas for insulators it is of the order of 1014–1022Ohm.cm [1]

Conductivity of real solids 13

Table 1.1 Number of carriers per cm3 at room temperature and main features

of the band structure for different bulk materials.

Metal > 1022 Fermi level inside the highest

Semimetal 1017–1022

Fermi level occurs where two bands merge or slightly overlap

Diamond

1.2.5 Number of carriers

Reasoning in terms of the number of carriers per volume, materials may be

classified as shown in Table 1.1

Let us emphasise, however, that such a classification is somewhat arbitrary

It is the variation of conductivity with respect to temperature, i.e the slope

(semiconductors and insulators) or non-activated (metals and semimetals) For

instance it is perfectly possible that the conductivity of a semiconductor at

a given temperature is comparable or even higher than that of certain poorly

conducting metals

Exercises

(1.1) Electron trapped in an infinite potential well: a possible

way to model the behaviour of an electron in a metal

is to assume the existence of an infinite potential well

outside the system, i.e outside the interval]− L/2, L/2].

Under such conditions, obtain an expression for the

allowed energy states describing the electrons on themetal Explain in what respect the results obtained maydiffer from those obtained using the Born–von Karmanboundary conditions Why are the two series of resultsequivalent for large L values?

References

1 C Kittel, Introduction to Solid State Physics, 7th edition, John Wiley, New York,

1996

2 H Ibach, H L¨uth, Solid State Physics: An Introduction to the Principles of Materials

Science, 2nd edition, Springer Verlag, Berlin Heidelberg, 1995.

3 O Madelung, Introduction to Solid State Theory, 2nd edition, Springer Verlag,

Berlin Heidelberg, 1981

4 N W Ashcroft, N D Mermin, Solid State Physics, Holt, Rinehart and Winston,

Philadelphia, 1976

Electronic structure

of molecules:

use of symmetry

2

Since our aim is to show that it is possible to analyse the electronic structure

of a periodic system following a procedure very similar to that used formolecules, in this chapter we will briefly review some useful notions that areneeded to understand the electronic structure of molecules Of course we willlimit ourselves to the notions that are useful in building a bridge between themodern approaches to the electronic structures of, on the one hand, moleculesand, on the other, solids We will begin by recalling some basic aspects ofmolecular orbital theory Then we will consider how symmetry may be used tosimplify the problem of defining the molecular orbitals by using group theory.The main results will be used in subsequent chapters, taking advantage ofthe translational symmetry of periodic systems in studying their electronicstructure Once these basic group theory notions have been discussed, wewill illustrate their usefulness by considering the example of cyclobutadiene.Finally we will briefly consider some general aspects of the electronic structure

of transition metal coordination complexes Later, these notions will be veryuseful in discussing the electronic structure of solids containing transitionmetal atoms

2.1 Molecular orbital theory

Since electron motion inside molecules is governed by the laws of quantummechanics, we need to solve the corresponding Schr¨odinger equation to deter-mine the allowed states However, this equation cannot be exactly solvedfor molecules, or more precisely for systems with more than one electron.Approximations are then required to get an acceptable estimate of the solution

of the Schr¨odinger equation In this chapter we will discuss molecular orbitaltheory, which provides a very convenient and easily workable description ofthe electronic structure of molecules [1]

Let us consider a molecule having N e electrons, labelled (e1, e2, ,e N e),

and N n nuclei The positions of the electrons e i and nuclei N j are described

by ther i and R j vectors, respectively

Molecular orbital theory 15

2.1.1 Born–Oppenheimer approximation

The Born–Oppenheimer approximation consists of separating the motion of

the electrons from that of the nuclei, based on the fact that electrons move

very quickly and adapt instantaneously to the motion of the nuclei, i.e atomic

vibrations Consequently, for a given geometry of the molecular species

char-acterised by the R j ( j = 1, , N n) vectors, the electrons’ motion is described

by an electronic wavefunctionψ e (r i , R j ) (i = 1, , N e ; j = 1, , N n ) in

which R j are parameters and r i are variables The vibrational motion of the

nuclei is described by a nuclear wavefunctionψ n ( R j ) ( j = 1, , N n ) When

the molecule possesses a single equilibrium geometry, the electronic

wave-functionψ e (r i ; ( R j ) eq ) (i = 1, , N e ; j = 1, , N n ) is estimated for the

nuclei in their equilibrium position, characterised by the vectors (( R j ) eq;

j = 1, , N n)

This approximation, although very useful and even indispensable, is not

valid when the coupling between the electronic motion and the molecular

vibrations is important, for example in transition metal complexes unstable

toward a Jahn–Teller distortion These will be discussed at the end of this

chapter

2.1.2 One-electron approximation

The one-electron approximation consists of decoupling the movement of

the different electrons assuming that each of them moves in an average

potential that represents the average repulsion of all other electrons This

approximation leads to the remarkable result that every electron may be

described by an effective one-electron Hamiltonian ˆh (e i ) that depends only

of the position of electron e i characterised by the vector r i The total

elec-tronic wavefunction e (e1, e2, , e N e ) may be written as an antisymmetrical

product of wavefunctions, called a Slater determinant These wavefunctions

are molecular orbitals describing the motion of an electron in the

molec-ular species Such molecmolec-ular orbitals will be labelled φ(r) The quantity

| φ( r i ) |2represents the probability density for electron e i at a point M

char-acterised by the vector r = r i The ground-state electronic configuration is

obtained by filling with electrons the lowest energy levels, according to the

Pauli principle Every orbital may be filled with two electrons with opposite

spins Thus, we need to find an appropriate expression for the molecular

orbitals

2.1.3 LCAO approximation

According to the linear combination of atomic orbitals approximation

(LCAO), a molecular orbitalφ can be written as a linear combination of N0

atomic orbitals{χ j , j = 1, , N0} of the different atoms of the molecule:

where the coefficients (c j , j = 1, , N0) are the constants to be determined.The summation in eqn (2.1) is generally limited to the valence atomic orbitals

of the constituent atoms, although occasionally it can also include the first

empty atomic orbitals The N0χ j orbitals can also be fragment orbitals of themolecule.1

1 A fragment is simply a number of atoms

that form a chemically convenient

build-ing block of the molecule For instance it

is possible to use two CH2fragments to

build the molecular orbitals of ethylene,

C2H4.

2.1.4 Secular equations and secular determinant

We are now going to establish the equations that will allow us to determine the

coefficients (c j ) and energy (E) associated with the molecular orbital φ If we

write the equation verified by this molecular orbital as:

of N0equations with N0unknown coefficients (c j , j = 1, , N0) which are

the so-called secular equations:

If the secular determinant is non-zero, the solution of the secular equations

is nothing other than the trivial solution (c j = 0, j = 1, , N0) Theφ

wave-function associated with this solution is physically meaningless since it is nileverywhere In contrast, if the secular determinant is nil, i.e the system ofequations is bound, nontrivial solutions that are physically meaningful exist

Consequently, only the energies E that lead to a nil secular determinant are

associated with possible states of the electrons in the molecule We are thusleft to solve eqn (2.8) to find the allowed energies:

Molecular orbital theory 17

This is an equation of N0-th order in E the solution of which provides the

N0values of the allowed energies, E  ( = 1, , N0) The molecular orbital

associated with the energy E will be referred to asφ  The determination of

the coefficients associated with molecular orbitalφ  involves the solution of

the system of secular equations, where the energy E has been replaced by E ,

and normalisation of theφ function has been imposed

This procedure, which must be repeated for every allowed energy E  ( =

1, , N0) as linear combinations of the χ j ( j = 1, , N0) orbitals.

2.1.5 Basic features of the H¨ uckel and extended

H¨ uckel methods

Extended H ¨uckel method

Except as otherwise stated, all numerical results reported in this book have

been obtained by using the extended H¨uckel method [2] In this approach the

interaction term h i j (i = j) between two Slater orbitals χ i andχ j is estimated

The expression proposed by Slater for the atomic orbitals of polyelec- tronic atoms is given byχ nm (r, θ, φ) =

N r n−1e −ξr Y ,m (θ, φ).

h i j = K S i j

(h ii + h j j )

in which h ii and h j jare the ionisation energies of a valence electron described

by orbitalsχ i andχ j, respectively In the Wolfsberg and Helmholtz

approxi-mation (eqn (2.9)), [2] K is a constant with value 1.75 In this method, the S i j

overlaps are calculated analytically Despite its empirical character this method

leads to a simple and useful description of the electronic structure of molecules

In the following chapters we will adopt this approach to obtain the electronic

structure of periodic solids

H ¨uckel method

An even simpler approach may be used when studying the electronic structure

of theπ system of conjugated organic molecules, such as butadiene, benzene,

etc This approach was initially proposed by H¨uckel in the 1930s and is at the

origin of the extended H¨uckel method discussed above.σ − π separation is

assumed and only theπ system is considered.

In this very simple approach, only the non-diagonal interaction terms

to be non-zero They are referred to as β i j All other interaction terms are

neglected The β i j and Si j terms have opposite signs In addition the term

h ii, denoted α, corresponds to the ionization potential of the χ i orbital The

α i energies are always larger in absolute terms than the interaction termsβ i j

When theπ system under study is exclusively built from carbon atoms and

equivalent C–C bonds, theα iandβ i j parameters are simply denotedα and β.

Finally, a further simplification involves neglecting the S (i = j) overlaps.

Fig 2.1

Limiting mesomeric structures for

cyclobutadiene The four hydrogen and

carbon atoms are labelled as 1, 2, 3

2.1.6 Symmetry properties of the molecular orbitals

A molecular orbitalφ(r i ) describes the behaviour of an electron e i through thesquare of its modulus| φ(r i ) |2 This wavefunction must possess symmetryproperties compatible with those of the molecular skeleton [3] Thus, forinstance, in planar organic molecules the molecular orbitals of theσ system

may be distinguished from those of the π system Whereas the former are

symmetric with respect to the molecular plane, the latter are antisymmetric.Such a distinction considerably simplifies the determination of the molecularorbitals To illustrate this fact, let us consider a hypothetical regular cyclobuta-diene molecule, which may be represented by two Lewis structures as shown

in Fig 2.1

The molecular orbitals (MO) of regular cyclobutadiene are linear tions of twenty valence atomic orbitals (AO) of the four carbon and four hydro-gen atoms: {1s H i , 2s C i , 2p xC i , 2p yC i , 2p zC i ; i = 1, , 4} Among these

combina-AOs we may distinguish sixteen combina-AOs that are symmetric with respect to themolecular plane(σ x y ), {1s H i , 2s C i , 2p xC i , 2p yC i ; i = 1, , 4}, and four AOs

which are antisymmetric with respect to this plane, {2p zC i ; i = 1, , 4)}.

Because of this symmetry difference, the system has sixteen symmetricMOs of the σ -type and four antisymmetric MOs of the π-type These MOs

differ also in the nature of the overlap integrals between their constituentAOs.3

3 The overlap between two atomic orbitals

may be characterised by enumerating the

number of nodal surfaces contained in

the region where the two orbitals

over-lap The overlaps denoted σ , π, and δ

contain 0, 1, and 2 nodal surfaces

Con-sequently, theσ -type overlaps are more

effective than theπ-type overlaps and the

latter are more effective than theδ-type

overlaps This is illustrated below, where

some examples ofσ , π, and δ overlaps

associated with s, p, or d AOs are shown.

In the following section we will talk about σ - or π-type orbitals Since

orbitals of theσ system are more bonding than those of the π system, whereas

the antibonding orbitals of the σ system are more antibonding than those

of the π system The energy diagram for the molecular orbitals of regular

Review of symmetry point groups 19

Fig 2.2

General structure of the molecular orbital diagram of cyclobutadiene.

cyclobutadiene contains three blocks of molecular orbitals (see Fig 2.2):

eight bonding MOs of the σ system, four MOs of the π system, and eight

antibonding MOs of theσ system.

Since the system possesses twenty valence electrons, the eight bonding MOs

of theσ system are filled and describe the eight C–C and C–H single bonds of

cyclobutadiene The four remaining electrons occupy the lowest energy MOs

of the π system according to the Pauli principle From a chemical point of

view, only the highest occupied MOs and the lowest unoccupied MOs are

really interesting This is why the knowledge of the fourπ MOs of regular

cyclobutadiene may provide an explanation for most of the properties of this

system

2.2 A short review of the theory of symmetry

point groups

We will now recall and illustrate the main results of group theory, which will be

used in the following sections We will only consider point groups, i.e groups

bearing symmetry operations leaving invariant a given point denoted O

2.2.1 Different symmetry point groups

The set of symmetry operations leaving a molecule invariant forms a group

in the mathematical sense [1] The different symmetry operations found in

molecular systems are outlined in Table 2.1

The set of symmetry operations generating a group is known as a generator

set By this we mean that any symmetry operation of the group may be

expressed as the product of symmetry operations of this generator set (see

Table 2.2) The number of symmetry elements of the group is called order

of the group and is referred to as h In what follows we will also consider two

high-symmetry groups with many symmetry operations: the T d and O hgroups,

which leave invariant a tetrahedron and an octahedron, respectively

Table 2.1 Different point symmetry operations.

Notation Symmetry operation

i Inversion of all atoms through the symmetry centre

C n m Order n axis Rotation by an angle 2m π

n around an axis.

S1is the commutative product of a rotation

S n m followed by a reflection in a plane ⊥ to the rotation axis.

S n m = (S1)m : Improper axis of order n.

Reflection in a vertical plane containing the O z axis,

σ dorσ v which by definition is the main axis,i.e the axis of higher order.

σ h Reflection in an horizontal plane⊥ to the Oz axis.

Table 2.2 Set of symmetry elements generating the main symmetry

point groups.

Group Generators Order and characteristics of the group

C i {i} 2 symmetry operations: inversion and identity.

C s {σ} 2 symmetry operations: reflection and identity.

2 axes⊥ to the Cnaxis.

{Cn , C2, σ d} The intersection of a planeσ d

C

2⊥ Cn with the horizontal plane is a bisector

of the two C

2 axes.

Review of symmetry point groups 21Finally, we also consider two infinite groups:

• the C ∞h symmetry group, which contains all C φ rotations around the O z

axis, as well as all reflections in the planes containing the z-axis (this is the

symmetry group of diatomic heteronuclear molecules, for instance)

• the D ∞h symmetry group, which, in addition to all symmetry elements

of the C ∞h group, also contains the inversion, the reflection in a plane

perpendicular to the O z axis, an infinite number of C2 rotations

perpen-dicular to the O z axis going through O, as well as an infinite number

of improper rotations around the O z axis (this is the symmetry group of

diatomic homonuclear molecules, for instance)

2.2.2 Classes

It is possible to classify the different symmetry operations of a group in classes

From the mathematical point of view, two symmetry operations A and B

belong to the same equivalence class if there is a symmetry operation X of

the group such that B is equal to X−1AX From a physical point of view, two

symmetry operations of the same class are two physically equivalent symmetry

operations.

Let us consider again the regular cyclobutadiene molecule (Fig 2.3) which

is left invariant by the D 4h group The two rotations C 2a and C 2b are related

in the following way: C 2b = σ d a C2a.σ d a, whereσ d a is the reflection in the

bisector plane of the two rotation axesσ d a (see Fig 2.3) These two symmetry

operations, C 2a and C 2b , thus belong to the same class As can clearly be seen

from Fig 2.1, they are physically equivalent The two rotations C 2a and C 2b

belong to the same class as well However, there is no symmetry operation

of the D 4h group relating a C2 and a C2 rotation Thus the four rotations

are distributed in two different classes of the D 4h group because they are

not physically equivalent Two of them (C 2a and C2b) leave invariant two

carbon atoms whereas the other two (C 2a and C 2b ) leave invariant the middle

of a C–C bond Likewise it can be shown that the symmetry operations of

the D 4h group may be classified as belonging to ten classes:{E}, {C4, C3

respectively, whereas the symmetry planes σ v a andσ v b contain the rotation

axes C 2a and C 2b , respectively

2.2.3 Basis for an irreducible representation

We will now outline the notions of basis for a representation, representation,

and irreducible representation of a group G These notions will be illustrated

by using the symmetry properties of the 2 p z atomic orbitals of theπ system

of regular cyclobutadiene The symmetry group of regular cyclobutadiene is

D 4hwhich contains sixteen symmetry operations denoted{R j , j = 1, , 16}

(h= 16)

Notion of basis for a representation

Let us now consider a set of functions f = { f1, f2, , f n }, such that the action of any of the symmetry operations of the group G transforms any of the functions, f i , in a linear combination of the different functions of the f set Such a set is said to be globally stable under the action of the symmetry operations of G and constitutes a basis for a representation of the group G.

From a physical point of view a basis for a representation contains functions

equivalent by symmetry Thus, for instance, the four 2 p zorbitals of the carbonatoms of regular cyclobutadiene constitute a basis for a representation of the

D 4h group as well as of all subgroups of D 4h

Representation of a group

Fig 2.4

(2 pz) iorbitals of the four carbon atoms

(i = 1, 2, 3, 4) of cyclobutadiene.

We will now consider the set of matrices M

k representing the action of the

symmetry operations R k (k = 1, , h) on the basis f = { f1, f2, , f n}.This set of matrices, {M1, M2, , M h }, is called a representation of the

the D 4h group of regular cyclobutadiene (see Figs 2.1, 2.3, and 2.4), onerepresentation of the group consists of sixteen 4×4 matrices representing

the action of the different symmetry operations of the D 4h group on theset {2p z1, 2p z2, 2p z3, 2p z4} Thus, for instance, the matrix associated with

rotation C4is that shown in Fig 2.5

Fig 2.5

Matrix representing the action of the C4

rotation on the basis

{2pz1, 2p z2, 2p z3, 2p z4}.

The number of representations of the group is infinite For instance, starting

with one of the bases, f , any unitary matrix U can be used to define another basis, f= { f

h} is a new representation of the group

From a physical perspective, these two representations of the group havethe same meaning: we thus need to find some common property characterisingboth of them This common property is the trace, i.e the sum of the diagonal

elements of the different matrices M

k We will denote the trace of the M

k

matrix as χ(M k ), which in group theory is known as the character of the

matrix Using the fact that the products of matrices A B and B A have the same trace, it is easy to show that the matrices M and M also have the same trace:

Review of symmetry point groups 23

way the representation{M1, M2, , M h} as well as any physically equivalent

charac-can try to find linear combinations of the functions { f1, f2, , f n } adapted

by themselves to the symmetry of the group.

the group In that case we may write the representation as a direct sum of

representations with lower dimension, i:

We can now say that we have decomposed the reducible representation 

into several representations i We can then try to decompose these

lower-dimension representations, i When there is no change of basis allowing the

decomposition of the basis iinto several lower-dimension bases, it is said that

 i is a basis for an irreducible representation As we will briefly review, group

theory makes clear some very useful mathematical properties of irreducible

representations

Properties of irreducible representations

Let us now outline the main properties of irreducible representations of a

symmetry group G of order h In Section 2.3 we will use all these results

to study theπ-type orbitals of regular cyclobutadiene.

(i) The number of irreducible representations is equal to the number of

classes of the group For instance, the D 4h group has ten irreducible

representations

(ii) Matrices associated with symmetry operations of the same class have the

same character

(iii) A reducible representation, , may be decomposed into the different

irreducible representations of the group, i, in the following way:

whereχ i (R k ) and χ(R k ) (k = 1, , h) are respectively the characters of

the representations and for the symmetry operations R of the group

G The symbol * in eqn (2.13) means that we must take the complex

conjugate of χ i (R k ) if the character is complex This formula requires

generally very simple

(iv) Let us assume that the basis{ f1, f2, , f n} may be projected onto the

 i representation, i.e n i = 0 in eqn (2.13) Then, a linear combination of

{ f1, f2, , f n } may result from the action of the projection operator P  i

on one f j function:4

4 Here we use a non-normalised

opera-tor The normalisation of the

symmetry-adapted functions will be later carried out

in the H¨uckel approach Note that this

operator may be non-trivial for a

Application of this formula requires knowledge of how every symmetry

may be quite tedious

(v) Ifφ i µandφ ν j are two basis elements of two different irreducible sentations,  µ and  ν, of the symmetry group of a molecule and ˆh isthe appropriate one-electron Hamiltonian, the following terms are nil bysymmetry:

repre-

φ i µ | ˆh | φ ν j = 0 and φ i µ | φ ν j = 0 (2.15)The functionsφ i µandφ ν j do not interact by symmetry and possess a niloverlap if they generate different irreducible representations µand ν

Application to the determination of the molecular orbitals of a molecule

It is essential to bear in mind that any molecular orbital is an element of a basis for an irreducible representation of the symmetry group of the molecule.

As a result, it is possible to find the expression of a molecular orbital usinggroup theory through the following procedure:

(i) Determine the symmetry group of the molecule using Table 2.2

(ii) List the different atomic orbitals that participate in the relevant molecularorbitals of the system

(iii) Group the different atomic orbitals into sets of symmetry-equivalentatomic orbitals, i.e into different bases for a representation

(iv) Decompose such representations (see eqn (2.13))

(v) Find a basis for every irreducible representation i (see eqn (2.14)); thedifferent basis elementsφ i j ( j = 1, , n i ) of an irreducible representa-

tion iare linear combinations of atomic orbitals adapted to the symmetry

of the molecule

(vi) The molecular orbitals that are a basis for the irreducible representation

 i result from the interaction of the n i functions φ i j ( j = 1, , n i ).

They may be obtained by solving the secular equations (eqns (2.8) and

(2.4)) by considering solely the n i functionsφ i j ( j = 1, , n i ) of this

symmetry The functionsφ i j now play the role of theχ j orbitals in tion 2.1.3 The same procedure must be followed for all other irreduciblerepresentations

Sec-π system of regular cyclobutadiene 25

Fig 2.6

Example of a symmetry lowering.

Symmetry lowering

From the mathematical point of view, a group may be a subgroup of another

group of higher order For instance, group C4v is a subgroup of D 4h , which

itself is a subgroup of O h:

When discussing symmetry in

com-plexes like these, we will refer to the local

symmetry around the transition metal

atom In other words, the actual etry of the H2O ligands will be ignored and we will consider the geometry of the

geom-MO6group of atoms.

subgroups This corresponds to the passage from an object with a given

sym-metry to another one with a lower symsym-metry, i.e one that is invariant through a

smaller number of symmetry operations Thus, an example of the change from

O h to D 4h and finally to C4vis provided by the hypothetical distortion of the

octahedral compound Cu(H2O)2+

6 through the bond elongation of two ligands

(O h to D 4h ) and the departure of one of the two apical ligands (D 4h to C4v),

as shown in Fig 2.6

It is important to point out that an irreducible representation for a group G

is a representation that might be reducible from any subgroup SG Any basis

for a representation of the group G is necessarily stable with respect to the

symmetry operations of the subgroup SG.

of regular cyclobutadiene

In this section we will show how the ideas developed in the previous sections

may be used to study the electronic structure of symmetrical molecules by

considering theπ system of regular cyclobutadiene Let us point out that the

procedure we will follow here will be completely equivalent to the approach

that we will follow when determining the orbitals describing the behaviour of