Chapter 11 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula

Chapter 11 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula Chapter 11 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula Chapter 11 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula Chapter 11 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula Chapter 11 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula

molecular symmetry

In this chapter we sharpen the concept of ‘shape’ into a precise

definition of ‘symmetry’, and show that symmetry may be

dis-cussed systematically

We see how to classify any molecule according to its

symme-try and how to use this classification to discuss the polarity and

chirality of molecules

The systematic treatment of symmetry is ‘group theory’ We

show that it is possible to represent the outcome of

symme-try operations (such as rotations and reflections) by matrices

That step allows us to express symmetry operations

numeri-cally and therefore to perform numerical manipulations One

important outcome is the ability to classify various tions of atomic orbitals according to their symmetries It also introduces the hugely important concept of a ‘character table’, which is the concept most widely employed in chemical appli-cations of group theory

The symmetry analysis described in the preceding two Topics

is now put to use We see that it provides simple criteria for deciding whether certain integrals necessarily vanish One important integral is the overlap integral between two orbitals

By knowing which atomic orbitals may have nonzero overlap,

we can decide which ones can contribute to molecular orbitals

We also see how to select linear combinations of atomic orbitals that match the symmetry of the nuclear framework Finally, by considering the symmetry properties of integrals, we see that

it is possible to derive the selection rules that govern scopic transitions

spectro-11A symmetry elements

Some objects are ‘more symmetrical’ than others A sphere is more symmetrical than a cube because it looks the same after it has been rotated through any angle about any diameter A cube looks the same only if it is rotated through certain angles about specific axes, such as 90°, 180°, or 270° about an axis passing through the centres of any of its opposite faces (Fig 11A.1), or

by 120° or 240° about an axis passing through any of its site corners Similarly, an NH3 molecule is ‘more symmetri-cal’ than an H2O molecule because NH3 looks the same after rotations of 120° or 240° about the axis shown in Fig 11A.2, whereas H2O looks the same only after a rotation of 180°.This Topic puts these intuitive notions on a more formal foundation In it, we see that molecules can be grouped together according to their symmetry, with the tetrahedral species CH4and SO4 − in one group and the pyramidal species NH3 and SO3 −

oppo-in another It turns out that molecules oppo-in the same group share certain physical properties, so powerful predictions can be made about whole series of molecules once we know the group

to which they belong

➤

Symmetry arguments can be used to make immediate

assessments of the properties of molecules, and when

expressed quantitatively (Topic 11B) can be used to save a

great deal of calculation.

➤

➤ What is the key idea?

Molecules can be classified into groups according to their

symmetry elements.

➤

This Topic does not draw on others directly, but it will be

useful to be aware of the shapes of a variety of simple

molecules and ions encountered in introductory chemistry

courses.

Contents

11a.1 Symmetry operations and symmetry elements 448

brief illustration 11a.1: symmetry elements 449

11a.2 The symmetry classification of molecules 449

brief illustration 11a.2: symmetry classification 449

(a) The groups C1, Ci, and Cs 450

brief illustration 11a.3: C1, Ci, and Cs 450

(b) The groups C n , C nv , and C nh 451

brief illustration 11a.4: C n , C nv , and C nh 451

(c) The groups D n , D nh , and D nd 452

brief illustration 11a.5: D n , D nh , and D nd 452

brief illustration 11a.6: S n 453

brief illustration 11a.7: the cubic groups 453

11a.3 Some immediate consequences of symmetry 454

Figure 11A.2 (a) An NH3 molecule has a threefold (C3) axis and (b) an H2O molecule has a twofold (C2) axis Both have other symmetry elements too

We have slipped in the term ‘group’ in its conventional sense

In fact, a group in mathematics has a precise formal

signifi-cance and considerable power and gives rise to the name ‘group

theory’ for the quantitative study of symmetry This power is

revealed in Topic 11B

symmetry elements

An action that leaves an object looking the same after it has

been carried out is called a symmetry operation Typical

symmetry operations include rotations, reflections, and

inversions There is a corresponding symmetry element for

each symmetry operation, which is the point, line, or plane

with respect to which the symmetry operation is performed

For instance, a rotation (a symmetry operation) is carried

out around an axis (the corresponding symmetry element)

We shall see that we can classify molecules by identifying all

their symmetry elements, and grouping together molecules

that possess the same set of symmetry elements This

proce-dure, for example, puts the trigonal pyramidal species NH3

and SO3 − into one group and the angular species H2O and SO2

into another group

An n-fold rotation (the operation) about an n-fold axis

of symmetry, C n (the corresponding element), is a rotation

through 360°/n An H2O molecule has one twofold axis, C2 An

NH3 molecule has one threefold axis, C3, with which is

asso-ciated two symmetry operations, one being 120° rotation in

a clockwise sense and the other 120° rotation in an

anticlock-wise sense There is only one twofold rotation associated with

a C2 axis because clockwise and anticlockwise 180° rotations

are identical A pentagon has a C5 axis, with two rotations

(one clockwise, the other anticlockwise) through 72°

associ-ated with it It also has an axis denoted C5, corresponding to

two successive C5 rotations; there are two such operations, one

through 144° in a clockwise sense and the other through 144°

in a anticlockwise sense A cube has three C4 axes, four C3 axes,

and six C2 axes However, even this high symmetry is exceeded

by a sphere, which possesses an infinite number of symmetry

axes (along any diameter) of all possible integral values of n

If a molecule possesses several rotation axes, then the one (or

more) with the greatest value of n is called the principal axis

The principal axis of a benzene molecule is the sixfold axis

per-pendicular to the hexagonal ring (1).

C6

1 Benzene, C6H6

A reflection (the operation) in a mirror plane, σ (the

ele-ment), may contain the principal axis of a molecule or be perpendicular to it If the plane contains the principal axis, it

is called ‘vertical’ and denoted σv An H2O molecule has two vertical planes of symmetry (Fig 11A.3) and an NH3 molecule has three A vertical mirror plane that bisects the angle between

two C2 axes is called a ‘dihedral plane’ and is denoted σd (Fig 11A.4) When the plane of symmetry is perpendicular to the

principal axis it is called ‘horizontal’ and denoted σh A C6H6

molecule has a C6 principal axis and a horizontal mirror plane (as well as several other symmetry elements)

In an inversion (the operation) through a centre of

sym-metry, i (the element), we imagine taking each point in a

mol-ecule, moving it to the centre of the molmol-ecule, and then moving

it out the same distance on the other side; that is, the point (x, y, z) is taken into the point (–x, –y, –z) Neither an H2O molecule nor an NH3 molecule has a centre of inversion, but a sphere and a cube do have one A C6H6 molecule does have a centre of inversion, so does a regular octahedron (Fig 11A.5); a regular tetrahedron and a CH4 molecule do not

An fold improper rotation (the operation) about an fold axis of improper rotation or an n-fold improper rotation axis, S n, (the symmetry element) is composed of two succes-sive transformations The first component is a rotation through

n-360°/n, and the second is a reflection through a plane

perpen-dicular to the axis of that rotation; neither operation alone needs to be a symmetry operation A CH4 molecule has three

S4 axes (Fig 11A.6)

11A Symmetry elements  449

The identity, E, consists of doing nothing; the corresponding

symmetry element is the entire object Because every molecule

is indistinguishable from itself if nothing is done to it, every

object possesses at least the identity element One reason for

including the identity is that some molecules have only this

symmetry element (2).

I

F C

point unchanged gives rise to the point groups There are five

kinds of symmetry operation (and five kinds of symmetry ment) of this kind When we consider crystals (Topic 18A), we meet symmetries arising from translation through space These

ele-more extensive groups are called space groups.

To classify molecules according to their symmetries, we list their symmetry elements and collect together molecules with the same list of elements The name of the group to which a molecule belongs is determined by the symmetry elements it possesses There are two systems of notation (Table 11A.1)

The Schoenflies system (in which a name looks like C4v) is more common for the discussion of individual molecules, and

the Hermann–Mauguin system, or International system (in

which a name looks like 4mm), is used almost exclusively in

the discussion of crystal symmetry The identification of a ecule’s point group according to the Schoenflies system is sim-plified by referring to the flow diagram in Fig 11A.7 and the shapes shown in Fig 11A.8

mol-Brief illustration 11A.1 Symmetry elements

To identify the symmetry elements of a naphthalene molecule

(3) we first note that, like all molecules, it has the identity

element, E There is one twofold axis of rotation, C2,

perpen-dicular to the plane and two others, ′C2, lying in the plane

There is a mirror plane in the plane of the molecule, σh, and

two perpendicular planes, σv, containing the C2 rotation axis

There is also a centre of inversion, i, through the mid-point of

the molecule Note that some of these elements are implied by

others: the centre of inversion, for instance, is implied by a σv

plane and a ′C2, axis

Brief illustration 11A.2 Symmetry classification

To identify the point group to which a ruthenocene

mol-ecule (4) belongs we use the flow diagram in Fig 11A.7 The

path to trace is shown by a blue line; it ends at D nh Because

Figure 11A.6 (a) A CH4 molecule has a fourfold improper

rotation axis (S4): the molecule is indistinguishable after a 90°

rotation followed by a reflection across the horizontal plane,

but neither operation alone is a symmetry operation (b) The

staggered form of ethane has an S6 axis composed of a 60°

rotation followed by a reflection

Centre of

inversion, i

Figure 11A.5 A regular octahedron has a centre of inversion (i).

(a) The groups C1, Ci, and Cs

A molecule belongs to the group C1 if it has no element other than the identity

It belongs to Ci if it has the identity and

the inversion alone, and to Cs if it has the identity and a mirror plane alone

the molecule has a fivefold axis, it belongs to the group D5h If

the rings were staggered, as they are in an excited state of

fer-rocene that lies 4 kJ mol−1 above the ground state (5), the

hori-zontal reflection plane would be absent, but dihedral planes

Brief illustration 11A.3 C1, Ci, and Cs

The CBrClFI molecule (2) has only the identity element, and

so belongs to the group C1 Meso-tartaric acid (6) has the

iden-tity and inversion elements, and so belongs to the group Ci

Quinoline (7) has the elements (E,σ), and so belongs to the

* Shoenflies notation in black, Hermann–Mauguin (International system) in blue In

the Hermann–Mauguin system, a number n denotes the presence of an n-fold axis

and m denotes a mirror plane A slash (/) indicates that the mirror plane is

perpendicular to the symmetry axis It is important to distinguish symmetry

elements of the same type but of different classes, as in 4/mmm, in which there are

three classes of mirror plane A bar over a number indicates that the element is

combined with an inversion The only groups listed here are the so-called

‘crystallographic point groups’.

Y

N

N

N N

N N

N

i?

i?

Two or more

Cn , n > 2?

C5?

C n?

Select C n with the highest n;

then, are there nC2 perpendicular to C n?

Figure 11A.7 A flow diagram for determining the point group

of a molecule Start at the top and answer the question posed

in each diamond (Y = yes, N = no)

11A Symmetry elements  451

OH

OH

H H

COOH

COOH Centre of inversion

(b) The groups Cn, Cnv, and Cnh

A molecule belongs to the group C n if it possesses an n-fold axis Note that symbol C n is now playing a triple role: as the label of

a symmetry element, a symmetry operation, and a group If in

addition to the identity and a C n axis a molecule has n cal mirror planes σv, then it belongs to the group C nv Objects

verti-that in addition to the identity and an n-fold principal axis also have a horizontal mirror plane σh belong to the groups C nh The presence of certain symmetry elements may be implied by the

presence of others: thus, in C2h the elements C2 and σh jointly imply the presence of a centre of inver-

sion (Fig 11A.9) Note also that the

tables specify the elements, not the operations: for instance, there are two operations associated with a single C3

axis (rotations by +120° and –120°)

Brief illustration 11A.4 C n , C nv , and C nh

An H2O2 molecule (9) has the symmetry elements E and C2,

so belongs to the group C2 An H2O molecule has the

symme-try elements E, C2, and 2σv, so it belongs to the group C2v An

NH3 molecule has the elements E, C3, and 3σv, so it belongs to

the group C3v A heteronuclear diatomic molecule such as HCl

belongs to the group C∞v because rotations around the axis by any angle and reflections in all the infinite number of planes that contain the axis are symmetry operations Other mem-

bers of the group C∞v include the linear OCS molecule and a

cone The molecule trans-CHClaCHCl (10) has the elements

E, C2, and σh, so belongs to the group C2h

O H

Plane or bipyramid

Figure 11A.8 A summary of the shapes corresponding to

different point groups The group to which a molecule belongs

can often be identified from this diagram without going

through the formal procedure in Fig 11A.7

(c) The groups Dn, Dnh, and Dnd

We see from Fig 11A.7 that a molecule that has an n-fold

prin-cipal axis and n twofold axes perpendicular to C n belongs to

the group D n A molecule belongs to D nh if it also possesses

a horizontal mirror plane D∞h is also the group of the linear

OCO and HCCH molecules, and

of a uniform cylinder A

mol-ecule belongs to the group D nd

if in addition to the elements of

D n it possesses n dihedral mirror planes σd

Brief illustration 11A.5 D n , D nh , and D nd

The planar trigonal BF3 molecule has the elements E, C3, 3C2,

and σh (with one C2 axis along each BeF bond), so belongs to

D3h (12) The C6H6 molecule has the elements E, C6, 3C2, 3C ,2′

and σh together with some others that these elements imply, so

it belongs to D6h Three of the C2 axes bisect CeC bonds and

the other three pass through vertices of the hexagon formed

by the carbon framework of the molecule The prime on 3C2′

indicates that the three C2 axes are different from the other

three C2 axes All homonuclear diatomic molecules, such as

N2, belong to the group D∞h because all rotations around the

axis are symmetry operations, as are end-to-end rotation and

end-to-end reflection Another example of a D nh species is

(13) The twisted, 90° allene (14) belongs to D2d

B F

12 Boron trifluoride, BF3

Self-test 11A.4 Identify the group to which the molecule

B(OH)3 in the conformation shown in (11) belongs.

Self-test 11A.5 Identify the groups to which (a) the

tetrachloroaurate(III) ion (15) and (b) the staggered mation of ethane (16) belong.

Molecules that have not been classified into one of the groups

mentioned so far, but which possess one S n axis, belong to the

group S n Note that the

group S2 is the same as Ci, so such a molecule will already

have been classified as Ci

11A Symmetry elements  453

A number of very important molecules possess more than one

principal axis Most belong to the cubic groups, and in

particu-lar to the tetrahedral groups T, Td, and Th (Fig 11A.10a) or to

the octahedral groups O and Oh (Fig 11A.10b) A few

icosahe-dral (20-faced) molecules belonging to the icosaheicosahe-dral group,

I (Fig 11A.10c), are also known The groups Td and Oh are the

groups of the regular tetrahedron and the regular octahedron,

respectively If the object possesses the rotational symmetry of

the tetrahedron or the octahedron, but none of their planes of

reflection, then it belongs to the simpler groups T or O (Fig

11A.11) The group Th is based on T but also contains a centre

of inversion (Fig 11A.12)

Brief illustration 11A.6 S n

Tetraphenylmethane (17) belongs to the point group S4

Molecules belonging to S n with n > 4 are rare.

Brief illustration 11A.7 The cubic groups

The molecules CH4 and SF6 belong, respectively, to the groups

Td and Oh Molecules belonging to the icosahedral group I

include some of the boranes and buckminsterfullerene, C60

(19) The molecules shown in Fig 11A.11 belong to the groups

(f) The full rotation group

The full rotation group, R3 (the 3 refers to rotation in three

dimensions), consists of an infinite number of rotation axes

with all possible values of n A sphere and an atom belong to

R3, but no molecule does Exploring the consequences of R3 is a

very important way of applying symmetry arguments to atoms,

and is an alternative approach

to the theory of orbital angular momentum

of symmetry

Some statements about the properties of a molecule can be

made as soon as its point group has been identified

(a) Polarity

A polar molecule is one with a permanent electric dipole

moment (HCl, O3, and NH3 are examples) If the molecule

belongs to the group C n with n > 1, it cannot possess a charge

distribution with a dipole moment perpendicular to the metry axis because the symmetry of the molecule implies that any dipole that exists in one direction perpendicular to the axis is cancelled by an opposing dipole (Fig 11A.13a) For example, the perpendicular component of the dipole associ-ated with one OeH bond in H2O is cancelled by an equal but opposite component of the dipole of the second OeH bond, so any dipole that the molecule has must be parallel to the two-fold symmetry axis However, as the group makes no reference

sym-to operations relating the two ends of the molecule, a charge distribution may exist that results in a dipole along the axis (Fig 11A.13b), and H2O has a dipole moment parallel to its twofold symmetry axis

The same remarks apply generally to the group C nv, so

mol-ecules belonging to any of the C nv groups may be polar In

all the other groups, such as C3h, D, etc., there are symmetry

operations that take one end of the molecule into the other Therefore, as well as having no dipole perpendicular to the axis, such molecules can have none along the axis, for other-wise these additional operations would not be symmetry

operations We can conclude that only molecules belonging to the groups C n , C nv , and Cs may have a permanent electric dipole moment For C n and C nv, that dipole moment must lie along the symmetry axis

Brief illustration 11A.8 Polar molecules

Ozone, O3, which is angular and belongs to the group C2v, may

be polar (and is), but carbon dioxide, CO2, which is linear and

belongs to the group D∞h, is not

Self-test 11A.8 Is tetraphenylmethane polar?

Answer: No (S4)

Figure 11A.11 Shapes corresponding to the point groups (a)

T and (b) O the presence of the decorated slabs reduces the

symmetry of the object from Td and Oh, respectively

Figure 11A.12 The shape of an object belonging to the

to the overall electric dipole, such as may arise from bonds between pairs of neighbouring atoms with different electronegativities

11A Symmetry elements  455

(b) Chirality

A chiral molecule (from the Greek word for ‘hand’) is a

mol-ecule that cannot be superimposed on its mirror image An

achiral molecule is a molecule that can be superimposed on

its mirror image Chiral molecules are optically active in the

sense that they rotate the plane of polarized light A chiral

mol-ecule and its mirror-image partner constitute an enantiomeric

pair of isomers and rotate the plane of polarization in equal but

opposite directions

A molecule may be chiral, and therefore optically active, only

if it does not possess an axis of improper rotation, S n We need

to be aware that an S n improper rotation axis may be present

under a different name, and be implied by other symmetry

elements that are present For example, molecules belonging

to the groups C nh possess an S n axis implicitly because they

possess both C n and σh, which are the two components of an

improper rotation axis Any molecule containing a centre of

inversion, i, also possesses an S2 axis, because i is equivalent to

C2 in conjunction with σh, and that combination of elements

is S2 (Fig 11A.14) It follows that all molecules with centres of

inversion are achiral and hence optically inactive Similarly,

because S1 = σ, it follows that any molecule with a mirror plane

is achiral

Brief illustration 11A.9 Chiral molecules

A molecule may be chiral if it does not have a centre of inversion or a mirror plane, which is the case with the

amino acid alanine (21), but not with glycine (22) However,

a molecule may be achiral even though it does not have a

centre of inversion For example, the S4 species (18) is

achi-ral and optically inactive: though it lacks i (that is, S2) it does

Self-test 11A.9 Is tetraphenylmethane chiral?

Answer: No (S4)

Checklist of concepts

looking the same after it has been carried out

respect to which a symmetry operation is performed

mol-ecules and solids is summarized in Table 11A1

(and have no higher symmetry)

axis of improper rotation, S n

i

S2

Figure 11A.14 Some symmetry elements are implied by the

other symmetry elements in a group Any molecule containing

an inversion also possesses at least an S2 element because i and

S2 are equivalent

Checklist of operations and elements

11B group theory

The systematic discussion of symmetry is called group theory

Much of group theory is a summary of common sense about the symmetries of objects However, because group theory

is systematic, its rules can be applied in a straightforward, mechanical way In most cases the theory gives a simple, direct method for arriving at useful conclusions with the minimum of calculation, and this is the aspect we stress here In some cases, though, they lead to unexpected results

A group in mathematics is a collection of transformations that

satisfy four criteria Thus, if we write the transformations as R, R′, … (which we can think of as reflections, rotations, and so on,

of the kind introduced in Topic 11A), then they form a group if:

1 One of the transformations is the identity (that is: ‘do nothing’)

2 For every transformation R, the inverse transformation

R−1 is included in the collection so that the combination

RR−1 (the transformation R−1 followed by R) is equivalent

11b.1 The elements of group theory 457

example 11b.1: showing that symmetry

brief illustration 11b.1: classes 458

(a) Representatives of operations 459

brief illustration 11b.2: representatives 459

(b) The representation of a group 459

brief illustration 11b.3: matrix representations 459

(c) Irreducible representations 459

(d) Characters and symmetry species 460

brief illustration 11b.4: symmetry species 461

(a) Character tables and orbital degeneracy 461

example 11b.2: using a character table to judge

(b) The symmetry species of atomic orbitals 462

brief illustration 11b.5: symmetry species

Group theory puts qualitative ideas about symmetry on

to a systematic basis that can be applied to a wide variety

of calculations; it is used to draw conclusions that might

not be immediately obvious and as a result can greatly

simplify calculations It is also the basis of the labelling

of atomic and molecular orbitals that is used throughout

chemistry.

➤

➤ What is the key idea?

Symmetry operations may be represented by the effect of

matrices on a basis.

➤

You need to know about the types of symmetry operation and element introduced in Topic 11A This discussion draws heavily on matrix algebra, especially matrix multiplication,

as set out in Mathematical background 6.

a group

Show that C2v = {E,C2,2σv} (specified by its elements) and

con-sisting of the operations {E,C2,σv,σ ′v} is indeed a group in the mathematical sense

There is one potentially very confusing point that needs to be

clarified at the outset The entities that make up a group are its

‘elements’ In chemistry, these elements are almost always

sym-metry operations However, as explained in Topic 11A, we

dis-tinguish ‘symmetry operations’ from ‘symmetry elements’, the

axes, planes, and so on with respect to which the operation is

carried out Finally, there is a third use of the word ‘element’, to

denote the number lying in a particular location in a matrix Be

very careful to distinguish element (of a group), symmetry ment, and matrix element.

ele-Symmetry operations fall into the same class if they are of

the same type (for example, rotations) and can be transformed into one another by a symmetry operation of the group The

two threefold rotations in C3v belong to the same class because one can be converted into the other by a reflection (Fig 11B.2); the three reflections all belong to the same class because each can be rotated into another by a threefold rotation The formal

definition of a class is that two operations R and R′ belong to the same class if there is a member S of the group such that

Brief illustration 11B.1 Classes

To show that C3 + and C3 − belong to the same class in C3v (which intuitively we know to be the case as they are both rotations

around the same axis), take S = σv The reciprocal of a

reflec-tion is the reflecreflec-tion itself, so σv − 1=σv It follows by using the

relations derived to confirm the result of Self-test 11B.1 that

Self-test 11B.2 Show that the two reflections of the group C2v

fall into different classes

Answer: No operation of the group takes σv → ′σv

Method We need to show that combinations of the operations

match the criteria set out above The operations are specified

in Topic 11A

Answer Criterion 1 is fulfilled because the collection of

sym-metry operations includes the identity E Criterion 2 is fulfilled

because in each case the inverse of an operation is the

opera-tion itself Thus, two successive twofold rotaopera-tions is equivalent

to the identity: C2C2 = E and likewise for the two reflections

and the identity itself Criterion 3 is fulfilled, because in each

case one operation followed by another is the same as one of

the four symmetry operations For instance, a twofold rotation

C2 followed by the reflection σ ′v is the same as the single

reflec-tion σv (Fig 11B.1) Thus: σ′vC2= Criterion 4 is fulfilled, σv

as it is immaterial how the operations are grouped together

The following group multiplication table for the point group

can be constructed similarly, where the entries are the product

symmetry operations RR′:

Self-test 11B.1 Confirm that C3v = {E,C3,3σv} and consisting of

the operations {E,2C3,3σv} is a group

Answer: Criteria are fulfilled

Figure 11B.1 A twofold rotation C2 followed by the

reflection σv′ is the same as the single reflection σv

by threefold rotations, and the two rotations shown here are related by reflection in σv

11B Group theory  459

Consider the set of three p orbitals shown on the C2v SO2

mole-cule in Fig 11B.3 Under the reflection operation σv, the change

(pS, pB, pA) ← (pS, pA, pB) takes place We can express this

trans-formation by using matrix multiplication (Mathematics

The matrix D(σv) is called a representative of the operation σv

Representatives take different forms according to the basis, the

set of orbitals that has been adopted In this case, the basis is

(pS, pA, pB)

The set of matrices that represents all the operations of the

group is called a matrix representation, Γ (uppercase gamma),

of the group for the basis that has been chosen We denote this

‘three-dimensional’ representation (a representation consisting

of 3 × 3 matrices) by Γ(3) The matrices of a representation tiply together in the same way as the operations they represent

mul-Thus, if for any two operations R and R′ we know that RR′ = R″,

then D(R)D(R′) = D(R″) for a given basis.

The discovery of a matrix representation of the group means that we have found a link between symbolic manipulations of operations and algebraic manipulations of numbers

block-diagonal form (11B.3)

The block-diagonal form of the representatives shows us that

the symmetry operations of C2v never mix pS with the other two

Brief illustration 11B.2 Representatives

We use the same technique to find matrices that reproduce

the  other symmetry operations For instance, C2 has the effect

(–pS, –pB, –pA) ← (pS, pA, pB), and its representative is

The identity operation has no effect on the basis, so its

repre-sentative is the 3 × 3 unit matrix:

Self-test 11B.3 Find the representative of the one remaining

operation of the group, the reflection σv

Brief illustration 11B.3 Matrix representations

In the group C2v, a twofold rotation followed by a reflection in

a mirror plane is equivalent to a reflection in the second ror plane: specifically, ′ =σvC2 σv When we use the represent-atives specified above, we find

Self-test 11B.4 Confirm the result that σ σv v′ =C2 by using the matrix representations developed here

––

–

––+

Figure 11B.3 The three px orbitals that are used to illustrate the

construction of a matrix representation in a C2v molecule (SO2)