Chapter 15. Transitions between Electronic States

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15. Transitions between Electronic States



S2



intersystem

crossing



S1

internal

conversion



T1



intersystem

crossing



absorption

fluorescence

phosphorescence



S0



Figure 15.1 Jablonski diagram illustrating the transitions between the electronic states in a polyatomic molecule, in a condensed phase. The vibrational levels in each electronic state are only

schematically represented, because there are likely to be a very large number of vibrational modes

that exist in a multi-dimensional surface. The wavy lines represent the vibrational relaxation within

the electronic states.



equilibrium is obtained. Under these conditions, the vibrational relaxation is faster than

any other process represented in Figure 15.1, and all the subsequent processes occur from

the lowest vibrational level of the lowest excited state. According to Kasha [2], following

the earlier suggestions of Vavilov: “The emitting level of a given multiplicity is the lowest excited level of that multiplicity”. However, if the molecule is in the gas phase at low

pressures, vibrational relaxation is much slower and reversible inter-system crossing may

occur. This is more likely for small molecules, with more spaced vibrational states, but

has also been observed for anthracene in the gas phase [3]. In summary, in condensed

phases, the electronically excited molecule rapidly relaxes to the lowest vibrational state

of the electronically excited state. From this state it will undergo an electronic transition

to the ground state with conservation of spin that can either be radiative (fluorescence

emission) or non-radiative (internal conversion), or may involve a spin flip of one of the

electrons in an electronic transition to an isoenergetic, vibrationally excited level of the

lowest triplet state (inter-system crossing). In the triplet state, the molecule undergoes

rapid vibrational relaxation to its lowest vibrational level, and from here it will return to

the ground state either radiatively (phosphorescence emission) or non-radiatively (intersystem crossing to a high vibrational level of the ground state, followed by vibrational

relaxation).

Excited electronic states can be formed as a result of high temperatures, highly

exothermic chemical reactions (chemiluminescence), radiative recombination of electrons



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and holes (electroluminescence) or more commonly, absorption of radiation. In the last

case, if monochromatic light of intensity I0 is incident normal to a sample of homogeneous

absorbing solution, the intensity of the transmitted light will be

I t = I 0 10 − lC



(15.1)



where C is the molar concentration (1 M ϭ 1 mol dmϪ3) of the molecules that absorb light,

their molar absorpion coefficient (sometimes called extinction coefficient) in per molar

per centimetre at the wavenumber (inverse of the wavelength) of the incident light and

l the length (in centimetre) of the optical path in the solution. The molar absorption coefficient at a given wavenumber, ( v ), is used to obtain the absorption cross section, which

is a parameter frequently used in optics



(v ) =



2303 ( v )

NA



= 3.81 × 10 −19 ( v ) (cm 2 )



(15.2)



Both the molar absorption coefficient and the absorption cross sections are measures of the

electronic transition probability. A further important measure of this is the oscillator

strength, f, given for the transition between a lower state 1 and an upper state 2, which is

proportional to the area of the absorption band,

f12 =



2303me c 2

e2 N A nD



∫ ( v ) dv



(15.3)



where me is the mass and e the charge of an electron and nD the refractive index of the

solution.

An excited atom may decay spontaneously to the ground state with the emission of a

photon of energy

E2 − E1 = h ν21 =



21



(15.4)



where the angular oscillation frequency is ϭ 2 . Einstein represented the transition

probability for spontaneous emission for our system of two states, 1 and 2, by A21.

However, the excited atom in the presence of a radiation field may also relax to the ground

state by interaction with the incident radiation in the process termed stimulated emission.

This is the process that forms the basis of the laser. Einstein called the corresponding transition probability B21. Microscopic reversibility requires that the transition probability for

stimulated emission must be equal to the transition probability for absorption

B21 = B12



(15.5)



Einstein obtained a relation between spontaneous and stimulated emission, using the relation between the number of atoms in the two states

− E1 − E2

h ν21

n1

kBT

=e

= e kBT

n2



(15.6)



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15. Transitions between Electronic States



and Planck’s expression for the radiation density of a black body at temperature T

I ( v) =



8 hv 3

1

h ν kBT

3

c e

−1



(15.7)



When an atomic system is in equilibrium with the radiation field, as many transitions

occur in the direction 1 → 2 as in the opposite direction, 2 → 1. In this case, the equilibrium condition will be

I ( v ) ( n1 B12 − n2 B21 ) − A21 n2 = 0



(15.8)



Using eqs. (15.5)–(15.7) in eq. (15.8), gives



A21 =



8 hv 3

B21 = 8 hv 3 B21

c3



(15.9)



Assuming that at time t ϭ 0 the external radiation field is removed and that all atoms are in

the excited state, the decay to the ground state follows the (first-order) radiative decay law

n2 ( t ) = n2 ( 0) e − A21t



(15.10)



and the radiative (sometimes termed natural) lifetime of fluorescence is F ϭ 1/A21.

The distinction between electronic transitions involving states of the same spin multiplicity (fluorescence and internal conversion) and states of different spin multiplicity

(phosphorescence and inter-system crossing) is due to the existence of selection rules for

electronic transitions. One of the most important rules is the spin selection rule, which is

expressed as ⌬S ϭ 0 (i.e. the change of the spin of an electron during a spectroscopic tran1

1

sition is forbidden). The electron is a fermion with spin quantum numbers s ϭ + 2 or − 2 ,

whereas the photon is a boson with s ϭ 1. A photon absorption or emission does not change

the spin state of the electron directly, and phosphorescence is a spin-forbidden process.

Inter-system crossing also involves a change in the spin of an electron and, consequently,

it is also a spin-forbidden transition. The spin-forbidden nature of these processes is of electronic origin, but can be circumvented by interaction with the medium, especially, when

heavy atoms or paramagnetic species are present. Under these conditions, spin is no longer

a pure quantum number and significant spin–orbit coupling occurs, owing largely to relativistic effects. Heavy atoms also facilitate the inter-system crossing and phosphorescence

when they are present in the molecule. The simplest way to account for the spin-forbidden

nature of these processes is to introduce a spin-forbidden factor 0Ͻ1 in the expression for

the rate of electronic transition between states of different spin multiplicity. This accounts

for the fact that such transitions are much slower than internal conversion and fluorescence.

In view of these spin restrictions and given that the ground state of organic molecules is a

singlet state (S0), the lowest excited singlet state of such molecules (S1) has a much shorter

lifetime than the corresponding triplet state (T1). Kasha suggested a value of 0 ϭ 10Ϫ6 for

the inter-system crossing from T1 to S0 in aromatic hydrocarbons.

Chemical reactions between two reactants in different electronic states, leading to two

products in electronic states that preserve the total spin multiplicity, can also be called radiationless transitions. Two frequently encountered examples are electron and electronic-energy



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transfers. Here, electron transfers are understood as processes where only one electron is

transferred from one molecule or ion to another molecule or ion, without the breaking or making of any bonds. These processes are also termed outer-sphere electron transfers. They will

be discussed in detail in the next chapter. Electronic-energy transfers may occur through three

mechanisms: radiatively (the so-called “trivial mechanism”, involving emission of one molecule followed by re-absorption by another), non-radiatively through coulombic (mainly

dipole–dipole) interaction (the Förster mechanism [4]) or non-radiatively through electron

exchange (the Dexter mechanism [5]).

The radiative or trivial mechanism of inter-molecular energy transfer can be expressed

as a sequence of two independent steps

DЈ → D ϩ h



(15.I)



A ϩ h →AЈ



(15.II)



where DЈ represents the excited energy donor and A is the acceptor. The rate of this mechanism is proportional to the fluorescence rate constant of the donor, the concentration of

excited molecules and the probability of absorption by A.

A

vrad ∝ kfD [ DЈ] Pabs



(15.11)



From the above discussion on the spin selection rule, it is implicit in this formulation that

the states D, DЈ and A, AЈ involved in radiative energy transfer will have the same spin

multiplicity. However, the pair A, AЈ may have spin multiplicity different from that of

the pair D, DЈ. The probability of light absorption by A is proportional to its concentration and to the spectral overlap between the donor emission and the acceptor absorption.

The most distinct features of this mechanism are its long range, its dependence of the

overlap between the fluorescence of the donor and the absorption of the acceptor, its

dependence on the size and shape of the vessel utilised and the fact that it leaves the lifetime of the donor unchanged. Radiative energy transfer is observed mainly in optically

thick samples.

The two mechanisms of non-radiative electronic-energy transfer differ in the nature of

the interaction between the energies of the donor and that of the acceptor. In the Förster or

coulombic mechanism there is an electrostatic interaction between the transition dipole

moment corresponding to the electronic de-excitation of the donor and that corresponding

to the electronic excitation of the acceptor, and the two transitions occur simultaneously.

The energy lost by the donor molecule is acquired by the acceptor molecule in a resonant

fashion, and can take place at distances much larger than molecular sizes because the interaction between two point dipoles decreases with their distance (R) as RϪ3 [6]. This process

corresponds to the coupling of two oscillators of electromagnetic field by a “virtual” rather

than a “real” photon. It is the dominant energy-transfer mechanism except when the dipole

strength of the acceptor transition is low or the fluorescence yield of the donor is vanishingly small. Figure 15.2 illustrates the change in donor and acceptor electronic configurations taking place in the course of the Förster energy-transfer mechanism.

Triplet states do not meet the requirements for efficient Förster energy transfer. Yet,

triplet–triplet energy transfers, where the donor is initially in a triplet state and the acceptor in the ground (singlet) state and leading to the donor in the ground (singlet) state and



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15. Transitions between Electronic States



Förster mechanism



reactants



products



Dexter mechanism



reactants



products



Figure 15.2 Electronic configurations in: (a) Förster energy transfer; (b) Dexter energy transfer.



the acceptor in a triplet state, are relatively common between organic molecules. These

require the acceptor to have a triplet state of lower energy than the donor. A particularly

important case of triplet-energy transfer is the transfer of energy from a triplet state to

−

molecular oxygen, whose ground state is a triplet ( 3 Σ g ) and whose first singlet (1⌬g) state

Ϫ1

is 94 kJ mol above the ground state. In this case, the two reactants are triplets and the

two products are singlets. This is an example of a process called triplet–triplet annihilation. This process is more commonly associated with the case when two identical organic

molecules in their triplet states (T1) meet and yield a singlet ground state (S0) and a highly

excited singlet state S* , which collects the excitation energies of both the partners. This

n

process frequently leads to delayed fluorescence (known as P-type delayed fluorescence

from its observation with the aromatic hydrocarbon pyrene). Here the emission has the

same spectrum as normal fluorescence, but has the lifetime of the triplet state. The transfer is still allowed because it follows Wigner’s spin rule and the process is governed by

spin statistics. Initially, it can be characterised by the formation of an intermediate pair

state T1 ϩ T1 ↔ 1,3,5(T1 . . . T1), that subsequently decays to S0 ϩ S* . The reverse process,

n

where a highly excited singlet state dissociates into two triplets is also known and is

termed singlet fission.

Such processes can be conceptualised as the tunnelling of an electron from one partner

to the other, while another electron tunnels in the opposite direction, leaving the overall

spin multiplicity intact (Figure 15.2). This mechanism, first described by Dexter, requires

the overlap of the wavefunctions of the two partners in space and is triggered by the electron exchange interaction. Dexter energy transfer is also possible for singlet–singlet energy

transfers, but it takes place at shorter distances than the Förster energy transfer.

Electron and triplet energy transfers have certain features in common. For example,

such reactions may occur with relatively high rates even when the electron (or energy)

donor is physically separated from the electron (or energy) acceptor by distances of 10 Å.

However, rather than the concerted transfer of two electrons pictured for triplet-energy

transfers in Figure 15.2, electron transfers involve the transfer of one electron only. The

understanding of the factors that govern the rates of energy and electron transfers requires

the resolution of the time-dependent Schrödinger equation.



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391



THE “GOLDEN RULE” OF QUANTUM MECHANICS



Quantum mechanics establishes a simple expression for the rate constant of a transition

between an initial and a final state when these states are only subject to a weak interaction.

The formalism of weak interactions was originally developed for spectroscopic transitions,

but later found application in energy and electron transfers.

The time-dependent Schrödinger equation, for a time-independent Hamiltonian, has

the form

H



( x, y, z, t ) = i



d



( x, y, z, t )

dt



(15.12)



and its more familiar application is to represent the motion of a particle as that of a wave

packet. Here, eq. (15.12) is employed to describe the time development of the system from

its initial state to the final one. The general solution for this equation of motion is a linear

superposition of the stationary states



( x, y, z, t ) = ∑ an



= ∑ an



n



n



n



n



⎛ i

⎞

exp ⎜ − En t ⎟

⎝

⎠



(15.13)



where n is a function of the space coordinates. When the system is subject to a perturbation, the coefficients an become functions of time during the perturbation and remain constant in the unperturbed system. For example, if the system is initially in a stationary state

defined by n ϭ 1 and undergoes a transition to the stationary state defined by n ϭ 3, then,

during the transition, a1 will decrease from unity to zero, while a3 will increase from zero

to unity, inversing the weight of the contributions 1 and 3. The rate of change from a1

to a3 is a measure of the rate of transition from the first to the third stationary state. The

calculation of this rate is one of the central problems of the theories of spectroscopic and

radiationless transitions.

In the most relevant cases for these transitions, the perturbing interaction is limited in

time and space. For example, the system is unperturbed before it interacts with an electromagnetic wave, and it is free again from the perturbation after sufficient time has elapsed.

This suggests that the total Hamiltonian can be considered as the sum of two terms:

H = H0 + V



(15.14)



where the time-independent operator H0 describes the unperturbed system and V is the perturbation. This perturbation may be explicitly time-dependent, as in the case of a transient

electromagnetic field, or not, as in the case of an applied field.

Replacing eq. (15.14) in eq. (15.12) and omitting the coordinates to simplify the form

of the equation,

d

(15.15)

( H 0 + V ) = i dt

and the solutions can be written as

= ∑ cj

j



j



⎛ i

⎞

exp ⎜ − E j t ⎟

⎝

⎠



(15.16)



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15. Transitions between Electronic States



where j represents the full set of eigenfunctions with the associated eigenvalues Ej of the

unperturbed system Hamiltonian H0, when the coefficients cj are time-dependent.

Replacing eq. (15.16) in eq. (15.15) and exchanging the left and right sides

⎛



∑⎜c E

j



⎝



j



j



j



+i



dc j

dt



j



⎞

⎛ i

⎞

⎟ exp ⎜ − E j t ⎟ = ∑ c j ( H 0

⎝

⎠

j

⎠



j



+V



j



) exp ⎛ − i E t ⎞

⎜

⎟

⎝

⎠

j



(15.17)



which can be reduced to



∑i

j



dc j

j



dt



⎛ i

⎞

exp ⎜ − E j t ⎟ = ∑ c j V

⎝

⎠

j



j



⎛ i

⎞

exp ⎜ − E j t ⎟

⎝

⎠



(15.18)



because each term H0 j on the left-hand side of eq. (15.17) cancels with the corresponding

Ej j term on the right-hand side of the same equation. An expression for dcj /dt can be

obtained taking the inner product of H with one of the stationary states. Representing the

other stationary state by the index k, the inner product corresponds to



∫∑i

j



dc j

dt



*

k



⎛i

⎞

exp ⎜ Ek t ⎟

⎝

⎠



j



⎛ i

⎞

exp ⎜ − E j t ⎟ d = ∫ ∑ c j

⎝

⎠

j



*

k



⎛i

⎞

exp ⎜ Ek t ⎟ V

⎝

⎠



j



⎛ i

⎞

exp ⎜ − E j t ⎟ d

⎝

⎠



(15.19)

where * is the complex conjugate of k, and the integration is over all the space, reprek

sented by . Given that the stationary states of the Hamiltonian are orthogonal



∫



*

k



j



d =



(15.20)



kj



and adopting the shorthand notation for the matrix element of the perturbation between the

unperturbed eigenstates

Vkj = ∫



*

k



V



j



d



(15.21)



and defining the transition frequency



kj



=



(

E k0 ) − E (j 0 )



(15.22)



gives a set of coupled first-order differential equations

i



dc k

i

= ∑ Vkj c j e

dt

j



kj t



, k = 1, 2, 3, . . .



(15.23)



⎞ ⎛ c1 ⎞

⎟ ⎜c ⎟

⎠⎝ ⎠



(15.24)



In matrix notation, this is equivalent to

i



d ⎛ c1 ⎞ ⎛ V11

=

dt ⎜ c2 ⎟ ⎜ V21e − i

⎝ ⎠ ⎝



21t



V12 e − i

V22



12 t



2