Chapter 13 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula

Chapter 13 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula Chapter 13 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula Chapter 13 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula Chapter 13 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula Chapter 13 Atkins Physical Chemistry (10th Edition) Peter Atkins and Julio de Paula

electronic transitions

Unlike for rotational and vibrational modes, simple analytical

expressions for the electronic energy levels of molecules cannot

be given Therefore, this chapter concentrates on the qualitative

features of electronic transitions

A common theme throughout the chapter is that electronic

transitions occur within a stationary nuclear framework This

Topic begins with a discussion of the electronic spectra of

dia-tomic molecules, and we see that in the gas phase it is possible

to observe simultaneous vibrational and rotational transitions

that accompany the electronic transition Then we describe

fea-tures of the electronic spectra of polyatomic molecules

We begin this Topic with an account of spontaneous emission

by molecules, including the phenomena of ‘fluorescence’ and

‘phosphorescence’ Then we see how non-radiative decay of

excited states can result in transfer of energy as heat to the

sur-roundings or can result in molecular dissociation

A specially important example of stimulated radiative decay is that responsible for the action of lasers, and in this Topic we see how this stimulated emission may be achieved and employed

What is the impact of this material?

Absorption and emission spectroscopy is also useful to

bio-chemists In Impact I13.1 we describe how the absorption of

visible radiation by special molecules in the eye initiates the

process of vision In Impact I13.2 we see how fluorescence

tech-niques can be used to make very small samples visible, ranging from specialized compartments inside biological cells to single molecules

To read more about the impact of this material, scan the QR code, or go to bcs.whfreeman.com/webpub/chemistry/pchem10e/impact/pchem-13-1.html

13A

Consider a molecule in the lowest vibrational state of its ground electronic state The nuclei are (in a classical sense) at their equilibrium locations and experience no net force from the electrons and other nuclei in the molecule The electron dis-tribution is changed when an electronic transition occurs and the nuclei become subjected to different forces In response, they start to vibrate around their new equilibrium locations The resulting vibrational transitions that accompany the elec-

tronic transition give rise to the vibrational structure of the

electronic transition This structure can be resolved for gaseous samples, but in a liquid or solid the lines usually merge together and result in a broad, almost featureless band (Fig 13A.1).The energies needed to change the electron distributions

of molecules are of the order of several electronvolts (1 eV is equivalent to about 8000 cm−1 or 100 kJ mol−1) Consequently, the photons emitted or absorbed when such changes occur

➤

➤ Why do you need to know this material?

Many of the colours of the objects in the world around us

stem from transitions in which an electron is promoted

from one orbital of a molecule or ion into another In some

cases the relocation of an electron may be so extensive

that it results in the breaking of a bond and the initiation

of a chemical reaction To understand these physical and

chemical phenomena, you need to explore the origins of

electronic transitions in molecules.

➤

➤ What is the key idea?

Electronic transitions occur within a stationary nuclear

framework.

➤

➤ What do you need to know already?

You need to be familiar with the general features of spectroscopy (Topic 12A), the quantum mechanical origins

of selection rules (Topics 9C, 12C, and 12D), and vibration– rotation spectra (Topic 12D); it would be helpful to be aware of atomic term symbols (Topic 9C) One example uses the method of combination differences described in Topic 12D.

Contents

brief illustration 13a.1: the multiplicity of a term 533

brief illustration 13a.2: term symbol of o2 1 534

brief illustration 13a.3: term symbol of o 2 2 534

brief illustration 13a.4: the term symbol of no 534

example 13a.2: estimating rotational

constants from electronic spectra 538

brief illustration 13a.6: the electronic spectrum

lie in the visible and ultraviolet regions of the spectrum

(Table 13A.1) What follows is a discussion of absorption

pro-cesses Emission processes are discussed in Topic 13B

13A.1 Diatomic molecules

Topic 9C explains how the states of atoms are expressed by

using term symbols and how the selection rules for electronic

transitions can be expressed in terms of these term symbols

Much the same is true of diatomic molecules, one principal

difference being the replacement of full spherical symmetry of

atoms by the cylindrical symmetry defined by the axis of the

molecule The second principal difference is the fact that a

dia-tomic molecule can vibrate and rotate

(a) Term symbols

The term symbols of linear molecules (the analogues of the

symbols 2P, etc for atoms) are constructed in a similar way

to those for atoms, with the Roman uppercase letter (the P in

this instance for atoms) representing the total orbital angular

momentum of the electrons around the nucleus In a linear

molecule, and specifically a diatomic molecule, a Greek

upper-case letter represents the total orbital angular momentum of

the electrons around the internuclear axis If this component of

orbital angular momentum is Λħ with Λ = 0, ±1, ±2 …, we use

the following designation:

These labels are the analogues of S, P, D, … for atoms for states

with L = 0, 1, 2, … To decide on the value of L for atoms we

had to use the Clebsch–Gordan series (Topic 9C) to couple the

individual angular momenta The procedure to determine Λ is

much simpler in a diatomic molecule because we simply add

the values of the individual components of each electron, λħ:

We note the following:

• A single electron in a σ orbital has λ = 0.

The orbital is cylindrically symmetrical and has no angular nodes when viewed along the internuclear axis Therefore, if

that is the only type of electron present, Λ = 0 The term symbol

for the ground state of H2 + with electron configuration 1σg 2 is therefore Σ

• A π electron in a diatomic molecule has one unit of orbital

angular momentum about the internuclear axis (λ = ±1).

If it is the only electron outside a closed shell, it gives rise to a Π term If there are two π electrons (as in the ground state of O2, with configuration …1π2 g), there are two possible outcomes If

the electrons are travelling in opposite directions, then λ1 = +1

and λ2 = −1 (or vice versa) and Λ = 0, corresponding to a Σ term

Alternatively, the electrons might occupy the same π orbital

and λ1 = λ2 = +1 (or −1), and Λ = ±2, corresponding to a Δ term

In O2 it is energetically favourable for the electrons to occupy different orbitals, so the ground term is Σ

As in atoms, we use a left superscript with the value of 2S + 1

to denote the multiplicity of the term, where S is the total spin

quantum number of the electrons

The overall parity of the state (its symmetry under sion through the centre of the molecule, if it has one) is added

inver-as a right subscript to the term symbol For H2 + in its ground state, the parity of the only occupied orbital (1σg) is g, so the term itself is also g, and in full dress is 2Σg If there are several electrons, the overall parity is calculated by noting the parity of each occupied orbital and using

Brief illustration 13A.1 The multiplicity of a term

It follows from the procedure for assigning multiplicity of

terms that for S s= =1

2 because there is only one electron, and the term symbol is 2Σ, a doublet term In O2, because in the ground state the two π electrons occupy different orbitals (as

we saw above), they may have either parallel or antiparallel spins; the lower energy is obtained (as in atoms) if the spins are

parallel, so S = 1 and the ground state is 3Σ

Self-test 13A.1 What is the value of S and the term symbol for

zero (a singlet term in which all electrons paired), there is

no orbital angular momentum from a closed shell, and

the overall parity is g

• If the molecule is heteronuclear, parity is irrelevant and

the ground state of a closed-shell species, such as CO,

is 1Σ

There is an additional symmetry operation that distinguishes

different types of Σ term: reflection in a plane containing the

internuclear axis A + right superscript on Σ is used to denote

a wavefunction that does not change sign under this

reflec-tion and a − sign is used if the wavefuncreflec-tion changes sign

(Fig 13A.2)

As for atoms, sometimes it is necessary to specify the total

electronic angular momentum In atoms we use the

quan-tum number J, which appears as a right subscript in the term

symbol, as in 2P1/2, with different values of J corresponding to different levels of a term In a linear molecule, only the elec-

tronic angular momentum about the internuclear axis is well

defined, and has the value Ω.ħ For light molecules, where the spin–orbit coupling is weak, Ω is obtained by adding together

the components of orbital angular momentum around the axis

(the value of Λ) and the component of the electron spin on that axis (Fig 13A.3) The latter is denoted Σ, where Σ = S, S − 1,

S − 2, …, −S (It is important to distinguish between the upright term symbol Σ and the sloping quantum number Σ.) Then

Brief illustration 13A.3 Term symbol of O2 2

If we think of O2 in its ground state as having one electron in

1πg,x , which changes sign under reflection in the yz-plane, and

the other electron in 1πg,y, which does not change sign under

reflection in the same plane, then the overall reflection

sym-metry is (closed shell) × (+) × (−) = (−), and the full term symbol

of the ground electronic state of O2 is 3

g

Σ− Alternatively, if we consider the configuration to be 1 1π π1+ 1−, with π± ∝ πg,x ± iπg,y

being two states of definite but opposite orbital angular

momentum around the axis, then for the triplet state we must

take the linear combination Ψ(1,2) ∝ π+(1)π−(2) − π+(2)π−(1)

Because under reflection in the yz-plane π+ → −π− and π− →

−π+, Ψ(1,2) → π−(1)π+(2) − π−(2)π+(1) = −Ψ(1,2), and the state is

Brief illustration 13A.4 The term symbol of NO

The ground-state configuration of NO is …πg, so it is a 2Π

term with Λ= ±1 and Σ = ±1 Therefore, there are two levels

of the term, one with Ω = ±1 and the other with ±3,

2 denoted

2Π1/2 and 2Π3/2, respectively Each level is doubly degenerate

(corresponding to the opposite signs of Ω) In NO, 2Π1/2 lies slightly lower than 2Π3/2

Self-test 13A.4 What are the levels of the term for the ground electronic state of O2 −?

+ +

–

– –

Figure 13A.2 The + or − on a term symbol refers to the overall symmetry of an electronic wavefunction under reflection in a plane containing the two nuclei

L

S

Λ Ω Σ

Figure 13A.3 The coupling of spin and orbital angular momenta in a linear molecule: only the components along the internuclear axis are conserved

Brief illustration 13A.2 Term symbol of O2 1

The parity of the ground state of O2 is g × g = g, so it is denoted

3Σg An excited configuration of O2 is …1πg 2, with both π

elec-trons in the same orbital As we have seen, |Λ| = 2, represented

by Δ The two electrons must be paired if they occupy the same

orbital, so S = 0 The overall parity is g × g = g Therefore, the

term symbol is 1Δg

Self-test 13A.2 The term symbol for one of the lowest excited

states of H2 is 3Πu To which excited-state configuration does

this term symbol correspond?

(b) Selection rules

A number of selection rules govern which transitions can be

observed in the electronic spectrum of a molecule The

selec-tion rules concerned with changes in angular momentum are

As in atoms (Topic 9C), the origins of these rules are

conserva-tion of angular momentum during a transiconserva-tion and the fact that

a photon has a spin of 1

There are two selection rules concerned with changes in

symmetry First, as we show in the following Justification,

For Σ terms, only Σ+ ↔ Σ+ and Σ− ↔ Σ− are allowed

Second, the Laporte selection rule for centrosymmetric

mol-ecules (those with a centre of inversion) states that the only

allowed transitions are transitions that are accompanied by a

change of parity That is,

For centrosymmetric molecules, only

u → g and g → u are allowed

A forbidden g → g transition can become allowed if the

cen-tre of symmetry is eliminated by an asymmetrical vibration,

such as the one shown in Fig 13A.4 When the centre of metry is lost, g → g and u → u transitions are no longer parity-forbidden and become weakly allowed A transition that derives its intensity from an asymmetrical vibration of a mol-

sym-ecule is called a vibronic transition.

The large number of photons in an incident beam generated

by a laser gives rise to a qualitatively different branch of troscopy, for the photon density is so great that more than one photon may be absorbed by a single molecule and give rise to

spec-multiphoton processes One application of spec-multiphoton

pro-cesses is that states inaccessible by conventional one-photon spectroscopy become observable because the overall transition occurs with no change of parity For example, in one-photon spectroscopy, only g ↔ u transitions are observable; in two-photon spectroscopy, however, the overall outcome of absorb-ing two photons is a g ← g or a u ← u transition

Justification 13A.1 Symmetry-based selection rules

The last two selection rules result from the fact that the

electric-dipole transition moment introduced in Topic 9C,

μfi= ∫ψ ψ τf *μ id vanishes unless the integrand is invariant

under all symmetry operations of the molecule

The z-component of the electric dipole moment

opera-tor is the component of μ responsible for Σ ↔ Σ transitions

(the other components have Π symmetry and cannot make a

contribution) The z-component of μ has (+) symmetry with

respect to reflection in a plane containing the internuclear

axis Therefore, for a (+) ↔ (−) transition, the overall

sym-metry of the transition dipole moment is (+) × (+) × (−) = (−),

so it must be zero and hence Σ+ ↔ Σ− transitions are not

allowed The integrals for Σ+ ↔ Σ+ and Σ− ↔ Σ− transform as

(+) × (+) × (+) = (+) and (−) × (+) × (−) = (+), respectively, and so

both transitions are allowed

The three components of the dipole moment operator

transform like x, y, and z, and in a centrosymmetric molecule

are all u Therefore, for a g → g transition, the overall parity

of the transition dipole moment is g × u × g = u, so it must be

zero Likewise, for a u → u transition, the overall parity is

u × u × u = u, so the transition dipole moment must also vanish

Hence, transitions without a change of parity are forbidden

For a g ↔ u transition the integral transforms as g × u × u = g,

and is allowed

laporte selection rule

Linear molecules

selection rules for electronic

Figure 13A.4 A ded transition is parity-forbidden because it corresponds to a geg transition However, a vibration of the molecule can destroy the inversion symmetry of the molecule and the g,u classification no longer applies The removal of the centre of symmetry gives rise to a vibronically allowed transition

Brief illustration 13A.5 Allowed transitions of O2

If we were presented with the following possible sitions in the electronic spectrum of O2, namely

tran-3

g 3 u 3 g 1 g 3 g 3 u

Σ−← Σ Σ−, −← ∆ Σ, −← Σ+, we could decide which are allowed by constructing the following table and referring to the rules Forbidden values are in red

Self-test 13A.5 Which of the following electronic transitions are allowed in O2: 3

(c) Vibrational structure

To account for the vibrational structure in electronic spectra of

molecules (Fig 13A.5), we apply the Franck–Condon principle:

Because the nuclei are so much more

massive than the electrons, an electronic

transition takes place very much faster

than the nuclei can respond

As a result of the transition, electron density is rapidly built up

in new regions of the molecule and removed from others In

classical terms, the initially stationary nuclei suddenly

experi-ence a new force field, to which they respond by beginning to

vibrate and (in classical terms) swing backwards and forwards

from their original separation (which was maintained during

the rapid electronic excitation) The stationary equilibrium

separation of the nuclei in the initial electronic state therefore

becomes a stationary turning point in the final electronic state

(Fig 13A.6) We can imagine the transition as taking place up

the vertical line in Fig 13A.6 This interpretation is the origin of

the expression vertical transition, which denotes an electronic

transition that occurs without change of nuclear geometry and

in classical terms, the nuclei remain stationary

The vibrational structure of the spectrum depends on the

rela-tive horizontal position of the two potential energy curves, and

a long vibrational progression, a lot of vibrational structure, is

stimulated if the upper potential energy curve is appreciably

dis-placed horizontally from the lower The upper curve is usually

displaced to greater equilibrium bond lengths because

electroni-cally excited states usually have more antibonding character than

electronic ground states The separation of the vibrational lines

depends on the vibrational energies of the upper electronic state.

The quantum mechanical version of the Franck–Condon principle refines this picture Instead of saying that the nuclei stay at the same locations and are stationary during the tran-

sition, we say that they retain their initial dynamic state In

quantum mechanics, the dynamical state is expressed by the wavefunction, so an equivalent statement is that the nuclear wavefunction does not change during the electronic transi-tion Initially the molecule is in the lowest vibrational state of its ground electronic state with a bell-shaped wavefunction cen-tred on the equilibrium bond length (Fig 13A.7) To find the nuclear state to which the transition takes place, we look for the vibrational wavefunction that most closely resembles this initial

Figure 13A.5 The electronic spectra of some molecules show

significant vibrational structure Shown here is the ultraviolet

spectrum of gaseous SO2 at 298 K As explained in the text,

the sharp lines in this spectrum are due to transitions from a

lower electronic state to different vibrational levels of a higher

electronic state Vibrational structure due to transitions to two

different excited electronic states is apparent

Franck–condon principle

Turning points (stationary nuclei)

Electronic excited state

Electronic ground state

of the vertical transition, the nuclei suddenly experience a new force field, to which they respond through their vibrational motion The equilibrium separation of the nuclei in the initial electronic state therefore becomes a turning point in the final electronic state Transitions to other vibrational levels also occur, but with lower intensity

Electronic excited state

Electronic ground state

wavefunction, for that corresponds to the nuclear dynamical

state that is least changed in the transition Intuitively, we can see

that the final wavefunction is the one with a large peak close to

the position of the initial bell-shaped function As explained in

Topic 8B, provided the vibrational quantum number is not zero,

the biggest peaks of vibrational wavefunctions occur close to the

edges of the confining potential, so we can expect the transition

to occur to those vibrational states, in accord with the classical

description However, several vibrational states have their major

peaks in similar positions, so we should expect transitions to

occur to a range of vibrational states, as is observed

The quantitative form of the Franck–Condon principle and

the justification of the preceding description is derived from the

expression for the transition dipole moment (as in Justification

13A.1) The electric dipole moment operator is a sum over all

nuclei and electrons in the molecule:

where the vectors are the distances from the centre of charge

of the molecule The intensity of the transition is proportional

to the square modulus, |μfi|2, of the magnitude of the transition

dipole moment, and we show in the following Justification that

this intensity is proportional to the square modulus of the

over-lap integral, S(vf,vi), between the vibrational states of the initial

and final electronic states This overlap integral is a measure of

the match between the vibrational wavefunctions in the upper

and lower electronic states: S = 1 for a perfect match and S = 0

when there is no similarity

Because the transition intensity is proportional to the square

of the magnitude of the transition dipole moment, the intensity

of an absorption is proportional to |S(vf,vi)|2, which is known as

the Franck–Condon factor for the transition:

S( , ) v vf i 2=( ∫ψ ψ v*,f v,idτn)2 Franck–condon factor (13A.6)

It follows that, the greater the overlap of the vibrational state wavefunction in the upper electronic state with the vibrational wavefunction in the lower electronic state, the greater the absorption intensity of that particular simultaneous electronic and vibrational transition

The quantity με,fi is the electric-dipole transition moment

aris-ing from the redistribution of electrons (and a measure of the

‘kick’ this redistribution gives to the electromagnetic field,

and vice versa for absorption) The factor S(vf,vi), is the overlap integral between the vibrational state with quantum number

vi in the initial electronic state of the molecule, and the

vibra-tional state with quantum number vf in the final electronic state of the molecule

Justification 13A.2 The Franck–Condon approximation

The overall state of the molecule consists of an electronic

part, labelled with ε, and a vibrational part, labelled with v

Therefore, within the Born–Oppenheimer approximation, the

transition dipole moment factorizes as follows:

The second term on the right of the second row (including the

term in blue) is zero, because two different electronic states are

 

=με ,fiS v v( , )f i

Example 13A.1 Calculating a Franck–Condon factor

Consider the transition from one electronic state to another,

their bond lengths being Re and ′R e and their force constants equal Calculate the Franck–Condon factor for the 0–0 tran-sition and show that the transition is most intense when the bond lengths are equal

two ground-state vibrational wavefunctions, and then take its square The difference between harmonic and anharmonic

vibrational wavefunctions is negligible for v = 0, so harmonic

oscillator wavefunctions can be used (Table 8B.1)

1 2 2

1 2 2

(d) Rotational structure

Just as in vibrational spectroscopy, where a vibrational

tran-sition is accompanied by rotational excitation, so rotational

transitions accompany the excitation of the vibrational

excita-tion that accompanies electronic excitaexcita-tion We therefore see

P, Q, and R branches for each vibrational transition, and the

electronic transition has a very rich structure However, the principal difference is that electronic excitation can result in much larger changes in bond length than vibrational excitation causes alone, and the rotational branches have a more complex structure than in vibration–rotation spectra

We suppose that the rotational constants of the electronic

ground and excited states are B and , B ′ respectively The tional energy levels of the initial and final states are

rota-E J( )=hcBJ J ( +1) E J( )′ =hcB J J′ ′ ′+( 1) (13A.7)

When a transition occurs with ΔJ = −1 the wavenumber of the

vibrational component of the electronic transition is shifted from  to

+ ′ − −B J( 1)J BJ J( + = − ′+1)  (B B J) + ′−(B B J) 2This transition is a contribution to the P branch (just as in Topic 12D) There are corresponding transitions to the Q and R branches with wavenumbers that may be calculated in a similar way All three branches are:

These expressions are the analogues of eqn 12D.19

Example 13A.2 Estimating rotational constants from electronic spectra

The following rotational transitions were observed in the 0–0 band of the 1Σ+ ← 1Σ+ electronic transition of

63Cu2H: R( )3 =23 347 69 cm− 1, P( )3 =23 298 85 cm− 1, and

P( )5 =23 275 77 cm− 1 Estimate the values of ′B and B.

intro-duced in Topic 12D: form the differences R( )J −P( )J and

R(J− −1) P(J+1) from eqns 13A.8a and 13A.8b, then use the resulting expressions to calculate the rotational constants ′B and B from the wavenumbers provided.

This factor is equal to 1 when ′ =Re Re and decreases as the

equi-librium bond lengths diverge from each other (Fig 13A.8)

For Br2, Re = 228 pm and there is an upper state with

′ =

Re 266pm Taking the vibrational wavenumber as 250 cm−1

gives S(0,0)2 = 5.1 × 10−10, so the intensity of the 0–0

transi-tion is only 5.1 × 10−10 what it would have been if the potential

curves had been directly above each other

approximated by rectangular functions of width W and W′,

centred on the equilibrium bond lengths (Fig 13A.9) Find the

corresponding Franck–Condon factors when the centres are

Figure 13A.8 The Franck–Condon factor for the

arrangement discussed in Example 13A.1.

Suppose that the bond length in the electronically excited

state is greater than that in the ground state; then B′ <B and

′−

B B is negative In this case the lines of the R branch converge

with increasing J and when J is such that   B′−B J( + > ′+1) B B

the lines start to appear at successively decreasing

wavenum-bers That is, the R branch has a band head (Fig 13A.10a)

When the bond is shorter in the excited state than in the ground

state, B′ >B and  B′−B is positive In this case, the lines of the P

branch begin to converge and go through a head when J is such

that  B′−B J B> ′+ B (Fig 13A.10b)

13A.2 Polyatomic molecules

The absorption of a photon can often be traced to the

excita-tion of specific types of electrons or to electrons that belong to

a small group of atoms in a polyatomic molecule For

exam-ple, when a carbonyl group (C O) is present, an absorption at

about 290 nm is normally observed, although its precise tion depends on the nature of the rest of the molecule Groups

loca-with characteristic optical absorptions are called phores (from the Greek for ‘colour bringer’), and their presence

chromo-often accounts for the colours of substances (Table 13A.2)

(a) d-Metal complexes

In a free atom, all five d orbitals of a given shell are degenerate

In a d-metal complex, where the immediate environment of the atom is no longer spherical, the d orbitals are not all degenerate, and electrons can absorb energy by making transitions between them

To see the origin of this splitting in an octahedral complex such as [Ti(OH2)6]3+ (1), we regard the six ligands as point

nega tive charges that repel the d electrons of the central ion (Fig 13A.11) As a result, the orbitals fall into two groups, with dx2 −y2 and dz2 pointing directly towards the ligand posi-tions, and dxy, dyz, and dzx pointing between them An electron occupying an orbital of the former group has a less favourable potential energy than when it occupies any of the three orbit-als of the other group, and so the d orbitals split into the two

sets shown in (2) with an energy difference ΔO: a triply erate set comprising the dxy, dyz, and dzx orbitals and labelled

degen-t2g, and a doubly degenerate set comprising the with dx2 −y2 and

dz2 orbitals and labelled eg The three t2g orbitals lie below the two eg orbitals in energy; the difference in energy ΔO is called

the ligand-field splitting parameter (the O denoting

octahe-dral symmetry) The ligand field splitting is typically about 10 per cent of the overall energy of interaction between the ligands and the central metal atom, which is largely responsible for the existence of the complex The d orbitals also divide into two sets in a tetrahedral complex, but in this case the e orbitals lie below the t2 orbitals (the g,u classification is no longer relevant

as a tetrahedral complex has no centre of inversion) and their separation is written ΔT

2

(These equations are analogous to eqns 12D.21a and 12D.21b.)

After using the data provided, we obtain:

and calculate ′ =B 3 489 cm− 1 and B =3 996 cm− 1

observed in the 1Σ+ ← 1Σ+ electronic transition of RhN:

Figure 13A.10 When the rotational constants of a diatomic

molecule differ significantly in the initial and final states of an

electronic transition, the P and R branches show a head (a)

The formation of a head in the R branch when B B; (b) the ′ <

formation of a head in the P branch when B B.′ >

Table 13A.2* Absorption characteristics of some groups and molecules

Group /cm−1 λmax/nm εmax/(dm 3 mol−1 cm−1 )

Neither ΔO nor ΔT is large, so transitions between the two

sets of orbitals typically occur in the visible region of the

spec-trum The transitions are responsible for many of the colours

that are so characteristic of d-metal complexes

According to the Laporte rule (Section 13A.1b), ded sitions are parity-forbidden in octahedral complexes because they are g → g transitions (more specifically eg ← t2g transi-tions) However, ded transitions become weakly allowed as

tran-vibronic transitions, joint vibrational and electronic

transi-tions, as a result of coupling to asymmetrical vibrations such as that shown in Fig 13A.4

A d-metal complex may also absorb radiation as a result of the transfer of an electron from the ligands into the d orbitals

of the central atom, or vice versa In such charge-transfer sitions the electron moves through a considerable distance,

tran-which means that the transition dipole moment may be large and the absorption correspondingly intense In the permanga-nate ion, MnO4 −, the charge redistribution that accompanies the migration of an electron from the O atoms to the central

Mn atom results in strong transition in the range 420–700 nm that accounts for the intense purple colour of the ion Such an electronic migration from the ligands to the metal corresponds

to a ligand-to-metal charge-transfer transition (LMCT) The reverse migration, a metal-to-ligand charge-transfer transi- tion (MLCT), can also occur An example is the migration of

a d electron onto the antibonding π orbitals of an aromatic ligand The resulting excited state may have a very long lifetime

if the electron is extensively delocalized over several aromatic rings

In common with other transitions, the intensities of charge-transfer transitions are proportional to the square of the transition dipole moment We can think of the transition moment as a measure of the distance moved by the electron

as it migrates from metal to ligand or vice versa, with a large distance of migration corresponding to a large transition dipole moment and therefore a high intensity of absorption However, because the integrand in the transition dipole is proportional to the product of the initial and final wavefunc-tions, it is zero unless the two wavefunctions have nonzero values in the same region of space Therefore, although large distances of migration favour high intensities, the dimin-ished overlap of the initial and final wavefunctions for large separations of metal and ligands favours low intensities (see Problem 13A.9)

(b) π* ← π and π* ← n transitions

Absorption by a CaC double bond results in the excitation of

a π electron into an antibonding π* orbital (Fig 13A.13) The

chromophore activity is therefore due to a π* ← π transition

(which is normally read ‘π to π-star transition’) Its energy is about 7 eV for an unconjugated double bond, which corre-sponds to an absorption at 180 nm (in the ultraviolet) When the double bond is part of a conjugated chain, the energies of the molecular orbitals lie closer together and the π* ← π transi-tion moves to longer wavelength; it may even lie in the visible region if the conjugated system is long enough

+ +

+

– –

– –

Figure 13A.11 The classification of d orbitals in an octahedral

environment The open circles represent the positions of the six

(point-charge) ligands

Brief illustration 13A.6 The electronic spectrum

of a d-metal complex

The spectrum of [Ti(OH2)6]3+ (1) near 20 000 cm−1 (500 nm) is

shown in Fig 13A.12, and can be ascribed to the promotion

of its single d electron from a t2g orbital to an eg orbital The

wavenumber of the absorption maximum suggests that ΔO ≈

20 000 cm−1 for this complex, which corresponds to about 2.5 eV

Self-test 13A.8 Can a complex of the Zn2+ ion have a ded

elec-tronic transition? Explain your answer

Answer: No; all five d orbitals are fully occupied

ν/(10 3 cm –1 )

∼

Figure 13A.12 The electronic absorption spectrum of

[Ti(OH2)6]3+ in aqueous solution

One of the transitions responsible for absorption in carbonyl

compounds can be traced to the lone pairs of electrons on the

O atom The Lewis concept of a ‘lone pair’ of electrons is

rep-resented in molecular orbital theory by a pair of electrons in

an orbital confined largely to one atom and not appreciably

involved in bond formation One of these electrons may be

excited into an empty π* orbital of the carbonyl group (Fig

13A.14), which gives rise to an π* ← n transition (an ‘n to

π-star transition’) Typical absorption energies are about 4eV

(290 nm) Because π* ← n transitions in carbonyls are

symme-try forbidden, the absorptions are weak By contrast, the π* ← π

transition in a carbonyl, which corresponds to excitation of a π

electron of the CaO double bond, is allowed by symmetry and

results in relatively strong absorption

(c) Circular dichroism

Electronic spectra can reveal additional details of molecular

structure when experiments are conducted with polarized light,

electromagnetic radiation with electric and magnetic fields that oscillate only in certain directions A mode of polarization is

circular polarization, in which the electric and magnetic fields

rotate around the direction of propagation in either a clockwise

or a counter-clockwise sense but remain perpendicular to it

and each other (Fig 13A.15) Chiral molecules exhibit circular dichroism, meaning that they absorb left and right circularly

polarized light to different extents For example, the lar dichroism (CD) spectra of the enantiomeric pairs of chiral d-metal complexes are distinctly different, whereas there is little difference between their absorption spectra (Fig 13A.16)

circu-Brief illustration 13A.7 π* ← π and π* ← n transitions

The compound CH3CHaCHCHO has a strong absorption

in the ultraviolet at 46 950 cm−1 (213 nm) and a weak

absorp-tion at 30 000 cm−1 (330 nm) The former is a π* ← π

transi-tion associated with the delocalized π system CaC—CaO

Delocalization extends the range of the CaO π* ← π transition

to lower wavenumbers (longer wavelengths) The latter is an

π* ← n transition associated with the carbonyl chromophore

Self-test 13A.9 Account for the observation that propanone

(acetone, (CH3)2CO) has a strong absorption at 189 nm and a

weaker absorption at 280 nm

Answer: Both transitions are associated with the CaO

chromophore, with the weaker being an π* ← n transition

and the stronger a π* ← π transition.

+

+

+ +

–

–

Figure 13A.13 A CaC double bond acts as a chromophore

One of its important transitions is the π* ← π transition

illustrated here, in which an electron is promoted from a π

orbital to the corresponding antibonding orbital

Figure 13A.14 A carbonyl group (CaO) acts as a chromophore

partly on account of the excitation of a nonbonding O

lone-pair electron to an antibonding CO π orbital

(b)

Figure 13A.16 (a) The absorption spectra of two isomers, denoted mer and fac, of [Co(ala)3], where ala is the conjugate base of alanine, and (b) the corresponding CD spectra The left- and right-handed forms of these isomers give identical absorption spectra However, the CD spectra are distinctly different, and the absolute configurations (denoted Λ and Δ) have been assigned by comparison with the CD spectra

of a complex of known absolute configuration

Checklist of concepts

☐ 1 The term symbols of diatomic molecules express the

components of electronic angular momentum around

the internuclear axis

☐ 2 Selection rules for electronic transitions are based on

considerations of angular momentum and symmetry

☐ 3 The Laporte selection rule states that, for

centrosym-metric molecules, only u → g and g → u transitions are

allowed

☐ 4 The Franck–Condon principle provides a basis for

explaining the vibrational structure of electronic

transitions

☐ 5 In gas phase samples, rotational structure is present too

and can give rise to band heads.

☐ 6 Chromophores are groups with characteristic optical

absorptions

☐ 7 In d-metal complexes, the presence of ligands removes the degeneracy of d orbitals and vibrationally allowed

d–d transitions can occur between them.

☐ 8 Charge-transfer transitions typically involve the

migration of electrons between the ligands and the tral metal atom

cen-☐ 9 Other chromophores include double bonds (π* ← π transitions) and carbonyl groups (π* ← n transitions).

☐ 10 Circular dichroism is the differential absorption of left

and right circularly polarized light

Checklist of equations

,

v vf i 2 vf vi d n

2

Rotational structure of electronic

13B decay of excited states

A radiative decay process is a process in which a molecule

discards its excitation energy as a photon (Topic 12A) In this

Topic we pay particular attention to spontaneous radiative

decay processes, which include fluorescence and

phosphores-cence A more common fate of an electronically excited

mol-ecule is non-radiative decay, in which the excess energy is

transferred into the vibration, rotation, and translation of the

surrounding molecules This thermal degradation converts the

excitation energy into thermal motion of the environment (that

is, to ‘heat’) An excited molecule may also dissociate or take

part in a chemical reaction (Topic 20G)

13B.1 Fluorescence and phosphorescence

In fluorescence, spontaneous emission of radiation occurs

while the sample is being irradiated and ceases within seconds to milliseconds of the exciting radiation being extin-

nano-guished (Fig 13B.1) In phosphorescence, the spontaneous

emission may persist for long periods (even hours, but teristically seconds or fractions of seconds) The difference sug-gests that fluorescence is a fast conversion of absorbed radiation into re-emitted energy, and that phosphorescence involves the storage of energy in a reservoir from which it slowly leaks.Figure 13B.2 shows the sequence of steps involved in fluo-rescence of chromophores in solution The initial stimulated absorption takes the molecule to an excited electronic state, and

charac-if the absorption spectrum were monitored it would look like the one shown in Fig 13B.3a The excited molecule is subjected

to collisions with the surrounding molecules, and as it gives up energy nonradiatively it steps down (typically within picosec-onds) the ladder of vibrational levels to the lowest vibrational level of the electronically excited molecular state The sur-rounding molecules, however, might now be unable to accept the larger energy difference needed to lower the molecule to the ground electronic state It might therefore survive long enough

to undergo spontaneous emission and emit the remaining excess energy as radiation The downward electronic transition

is vertical, in accord with the Franck–Condon principle (Topic

Contents

13b.1 Fluorescence and phosphorescence 543

brief illustration 13b.1: Fluorescence and

phosphorescence of organic molecules 545

13b.2 Dissociation and predissociation 545

brief illustration 13b.2: the effect of predissociation

on an electronic spectrum 546

➤

➤ Why do you need to know this material?

Considerable information about the electronic structure

of a molecule can be obtained from the photons emitted

when excited electronic states decay radiatively back to

the ground state.

➤

➤ What is the key idea?

Molecules in excited electronic states discard their excess

energy by emission of electromagnetic radiation, transfer

as heat to the surroundings, or fragmentation.

➤

➤ What do you need to know already?

You need to be familiar with electronic transitions in

molecules (Topic 13A), the difference between spontaneous

and stimulated emission of radiation (Topic 12A), and the

general features of spectroscopy (Topic 12A) You need

to be aware of the difference between singlet and triplet

states (Topic 9C) and of the Franck–Condon principle

13A), and the fluorescence spectrum has a vibrational structure

characteristic of the lower electronic state (Fig 13B.3b).

Provided they can be seen, the 0–0 absorption and

fluo-rescence transitions can be expected to be coincident The

absorption spectrum arises from 1 ← 0, 2 ← 0, … transitions

that occur at progressively higher wavenumber and with

intensities governed by the Franck–Condon principle The

fluorescence spectrum arises from 0 → 0, 0 → 1, … downward

transitions that occur with decreasing wavenumbers The 0–0

absorption and fluorescence peaks are not always exactly

coin-cident, however, because the solvent may interact differently

with the solute in the ground and excited states (for instance,

the hydrogen bonding pattern might differ) Because the

sol-vent molecules do not have time to rearrange during the

tran-sition, the absorption occurs in an environment characteristic

of the solvated ground state; however, the fluorescence occurs

in an environment characteristic of the solvated excited state (Fig 13B.4)

Fluorescence occurs at lower frequencies (longer lengths) than the incident radiation because the emissive transition occurs after some vibrational energy has been dis-carded into the surroundings The vivid oranges and greens of fluorescent dyes are an everyday manifestation of this effect: they absorb in the ultraviolet and blue, and fluoresce in the visible The mechanism also suggests that the intensity of the fluorescence ought to depend on the ability of the solvent molecules to accept the electronic and vibrational quanta It

wave-is indeed found that a solvent composed of molecules with widely spaced vibrational levels (such as water) can in some cases accept the large quantum of electronic energy and so extinguish, or ‘quench’, the fluorescence The rate at which fluor escence is quenched by other molecules also gives valu-able kinetic information (Topic 20G)

Figure 13B.5 shows the sequence of events leading to phorescence for a molecule with a singlet ground state The first steps are the same as in fluorescence, but the presence of a tri-plet excited state at an energy close to that of the singlet excited state plays a decisive role The singlet and triplet excited states share a common geometry at the point where their potential energy curves intersect Hence, if there is a mechanism for unpairing two electron spins (and achieving the conversion of

phos-↑↓ to ↑↑ ), the molecule may undergo intersystem crossing,

a non-radiative transition between states of different city, and become a triplet state As in the discussion of atomic spectra (Topic 9C), singlet–triplet transitions may occur in the presence of spin–orbit coupling Intersystem crossing is expected to be important when a molecule contains a moder-ately heavy atom (such as sulfur), because then the spin–orbit coupling is large

Figure 13B.4 The solvent can shift the fluorescence spectrum relative to the absorption spectrum On the left we see that the absorption occurs with the solvent (depicted by the ellipses) in the arrangement characteristic of the ground electronic state

of the molecule (the sphere) However, before fluorescence occurs, the solvent molecules relax into a new arrangement, and that arrangement is preserved during the subsequent radiative transition

Internuclear separation, R

Radiationless decay

Absorption

Emission (fluorescence)

Figure 13B.2 The sequence of steps leading to fluorescence

by chromophores in solution After the initial absorption, the

upper vibrational states undergo radiationless decay by giving

up energy to the surrounding molecules A radiative transition

then occurs from the vibrational ground state of the upper

(b)

Figure 13B.3 An absorption spectrum (a) shows a vibrational

structure characteristic of the upper state A fluorescence

spectrum (b) shows a structure characteristic of the lower

state; it is also displaced to lower frequencies (but the 0–0

transitions are coincident) and resembles a mirror image of

the absorption