2 The “Golden Rule” of Quantum Mechanics

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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



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No approximations have yet been made and eq. (15.24) is exact. The values of cj determined by this equation are related to the probability of finding the system in any particular state at any later time. Unfortunately, it is not generally possible to find exact solutions

to this equation. The solution of this complicated linear system invokes the perturbation

approximation, and the method is called time-dependent perturbation theory.

The solution of eq. (15.23) depends critically on the initial conditions. Assuming, for

simplicity, that the system at the initial time t0 ϭ Ϫϱ is in one of the stationary states of

the unperturbed Hamiltonian, and that H0 possesses only discrete energy levels, the following initial conditions are introduced:

cs ( −ϱ) = 1,



ck ( − ϱ ) = 0



if k ≠ s



(15.25)



The index s represents the initial stationary state. A successive approximation method can

be used to solve eq. (15.23) subject to these initial conditions. The initial values of the

coefficients ck are introduced in the right-hand side of eq. (15.23). For k s, the approximate equations

i



dck

= Vks e i

dt



ks t



k ≠s



,



(15.26)



are thus obtained. These equations are only valid for values of t such that ck(t)cs(t) 1,

if k s.

Eq. (15.26) can be immediately integrated when the perturbation V does not itself

depend on time, although the time development of the system with the Hamiltonian

(15.14) is, nevertheless, conveniently described in terms of transitions between the eigenstates of the unperturbed Hamiltonian H0. Taking the initial time as t0 ϭ 0

i



∫



t



0



dck = Vks ∫ e i

t



ks t



0



k ≠s



dt ,



(15.27)



and using eq. (15.22), the coefficients

ck ( t ) = Vks



1 − e i ks t

(

E k0 ) − Es( 0 )



(15.28)



are obtained for k s, if ck(0) ϭ 0 and cs(0) ϭ1.

If the system is known to have been in the initial discrete state s at tϭ0, the probability

that it will be in the unperturbed final eigenstate k s at time t is given by

ck ( t ) = 2 Vks

2



2



1 − cos (



(E



(0)

k



t)



−E



ks

(0) 2

s



)



(15.29)



where the relation

cos ( x ) =



(



1 ix

e + e − ix

2



)



(15.30)



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15. Transitions between Electronic States



was employed. The total probability of transition to all the final states, labelled f, which

form a quasi-continuum of states per unit energy interval, is given by



∑ c (t )

k ∈f



k



2



= 2∫ Vks



1 − cos (



2



(E



(0)

k



ks



t)



)



f



(0) 2

s



−E



(E ( ) ) dE ( )

0

k



0

k



(15.31)



where f (E) represents the density of final unperturbed states. The assumption that the

final states form a quasi-continuum justified the use of an integral rather than the sums on

the right-hand side of eq. (15.31). Within this approximation, it is of interest to formulate

the time rate of change of the total transition probability



w=



2

d

2

∑f ck (t ) = 2

dt k ∈



∫



Vks



sin (



2



ks



t)

f



ks



( E ( ) ) dE ( )

0

k



0

k



(15.32)



It is usual practice to assume that both |Vks|2 and f (Ek(0)) are approximately constant over

an energy range ⌬E in the neighbourhood of Es. Additionally, the function sin( kst)/ ks has

(

a pronounced peak at E k0) ϭ Es(0) for t values which satisfy the relation

t >>



(15.33)



E



as illustrated in Figure 15.3. This expresses the fact that the transitions that tend to conserve the unperturbed energy are dominant. If the right-hand side of eq. (15.33) corresponds to a very short time, there is a considerable range of t values, such that the

inequality is fulfilled and yet the initial state s is not appreciably depleted. For such a range

of t values, eq. (15.32) can be simplified to



w=



2



Vks



2



(E ( ) ) ∫

0



f



s



+∞

−∞



sin (



ks



t)



d



ks



(15.34)



ks



Figure 15.3 Dependence of the function sin ( kst)/ ks) on the energy difference between the unperturbed initial and final states, calculated for the values of t indicated in the plot.



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395



where the condition of eq. (15.33) was employed to replace the limits of the integral by Ϫϱ

and ϩϱ. Given the standard integral



∫



+∞



sin ( ax )



−∞



x



dx =



when a > 0



(15.35)



(E ( ) )



(15.36)



the transition probability per unit time becomes

w=



2



Vks



2



0



f



s



Fermi called eq. (15.36) the Golden Rule of time-dependent perturbation theory because of

its prevalence in radiationless transitions. Sometimes it is referred to as Fermi’s Golden Rule.

The matrix element of the perturbation between the eigenstates Vks is expressed in units

(

of energy, the density of states f E k0) in reciprocal energy units, and w in per second.

A typical case where the Golden Rule is applicable is the internal conversion of a large

molecule, following its electronic excitation. For example, the energy of the first excited

singlet state (S1) of anthracene relative to that of its ground state (S0) is 316 kJ molϪ1. At

this energy, the density of vibrational states in S0 is extremely high. Typically, for such

molecules, this is in the range of 1011Ϫ1017 states JϪ1 molϪ1, and the S1 state will relax into

an isoenergetic level of the manifold of highly excited vibrational states of S0, a process

already defined as internal conversion. Subsequent vibrational relaxation in the S0 state

converts the electronic excitation into heat. It must be recalled that this is not the only

decay process available to the S1 state. It may also decay radiatively by fluorescence, or

undergo inter-system crossing to the manifold of the triplet states.

15.3



RADIATIVE AND RADIATIONLESS RATES



Classically, charged particles radiate when they are accelerated. The strength of radiation

is proportional to the square of the electric dipole moment, which is

21



=∫



2



[er ]

21



1



d =e



2



r21



(15.37)



1



where r21 stands for the electron position vector. The absorption oscillator strength that

describes the strength of the radiative transitions is defined as

8 2 me cv 12

f12 =

3h

e2



2



⎛

⎞

= 1.085 × 10 −5 vmax ⎜ 12 ⎟

⎝ e ⎠



2



(15.38)



where the numerical value applies when v is expressed in per centimetre and | 12|/e in

angstrom. The oscillator strength is a dimensionless quantity.

The absorption of light is mainly the result of the interaction between the oscillating

electric vector of the electromagnetic radiation with the charged particles within the molecules. The transition probability coefficient in the absorption from a lower electronic state

1 to an upper electronic state 2 corresponds to Einstein’s coefficient of absorption



B12 =



8 3

3h 2



2

12



(15.39)



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15. Transitions between Electronic States



Eliminating 12 and f12 using eqs. (15.3) and (15.38) in the above expression gives the absorption probability in the gas phase (nDϭ1) in terms of experimentally accessible quantities



B12 =



( v ) dv



2303c ∞

hN A ∫0



(15.40)



v



The radiative rate can now be expressed in terms of these quantities using eq. (15.9),

kr = A21 =



64 4 ν3

3hc 3



2



er21



2



=



1



8 × 2303 c 2 ∞

v ∫ ( v ) dv

0

NA



(15.41)



This relation is only strictly applicable to two-level systems, such as atomic systems,

where the transitions have sharp lines and in a medium of refractive index nDϭ1. The electric dipole radiation dominates the mechanism of radiative decay in molecular systems.

For molecular systems in solution, Strickler and Berg proposed a modified equation that

gives good results for the radiative rate [7]



kSB =



8 × 2303 c 2 −3

nD v

NA



−1



∫



( v ) dv

v



2

= 2.88 × 10 −9 nD v −3



−1



∫



( v ) dv

v



(15.42)



where

v −3



−1



=



∫ F ( v ) dv



∫ F (v ) v



−3



dv



(15.43)



and F( v ) is the molecular fluorescence intensity distribution (i.e. its spectrum). When the

fluorescence is the mirror image of the absorption, the radiative rates calculated with the

Strickler–Berg equation compare well with the experimental rates obtained from the fluorescence quantum yield and lifetime of the donor molecules

kf =



f



(15.44)



D



Eq. (15.42) relates the radiative energy transfer rate constant to the molecular properties. The corresponding rate, eq. (15.11), involves the probability of absorption by A,

which can be expressed as

∞



A

Pabs ϰ [ A ] l ∫ FЈ ( v )

D



A



( v ) dv



(15.45)



0



where l is the path length of absorption, and the spectral overlap involves the normalised

spectral distribution of the donor emission FD ( v ) and the molar absorption coefficient of

Ј

the acceptor A( v ). The normalisation of the donor emission is given by

FЈ ( v ) =

D



FD ( v )



∫



∞



0



FD ( v ) dv



(15.46)



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As mentioned before, radiative energy transfer originates from the interaction between

the oscillating electric vector of the electromagnetic radiation and the charged particles

within the molecules. Another possible interaction between the excited donor and the

ground-state acceptor molecule is the electric dipole–dipole interaction, which is the basis

of the Förster mechanism, schematically presented in Figure 15.2. The dipole–dipole interaction energy in the vacuum between two dipoles 1 and 2 at a distance R between their

centres, has the form

V12 =



1 ⎡

⎢

R3 ⎢

⎣



1⊗



−



2



3



(



1



⊗R



)(



R



2



2



)



⊗R ⎤

⎥

⎥

⎦



(15.47)



A quantitative treatment using the Golden Rule with this interaction energy leads to the

following expression for the rate of Förster energy transfer

kF =



9000 ( ln 10)



2

f



4

128 5 nD N A R 6



D



∫



∞



0



FD ( v )

v



A

4



( v ) dv



1 ⎛ R0 ⎞

=

⎜ R⎟

⎝ ⎠

D



6



(15.48)



where is an orientation factor (for a random directional distribution the average value is

2

ϭ 2 ), the donor fluorescence intensity FD is normalised to unit area on a wavenumber

3

scale, and the decadic molar absorption coefficient of the acceptor A is in the usual units

(per molar per centimetre). The reason why the emission shape needs to be normalised to

unity, while the absorption shape does not, is the presence of the term −1 that already conD

tains information on the absolute magnitude of the emitting transition moment. The above

expression emphasises the fact that the probability of energy transfer is proportional to the

square of the interaction energy and decreases, therefore, with the sixth power of the distance. The critical transfer distance R0 is the distance at which the energy transfer and the

spontaneous decay of the excited donor are equally probable. Typically, values between 30

and 100 Å are calculated for aromatic donors.

Electric dipole–dipole coupling is the strongest if the corresponding optical transitions

in both molecules are allowed for electric dipole radiation. This is not the case of tripletenergy transfer (Figure 15.2), which must follow an alternative mechanism. Dexter proposed that for such cases it is necessary to include explicitly the spin wavefunctions in the

matrix element of the perturbation, eq. (15.21). The molecular wavefunction

is

expressed as the product of a spacial ( r ) and a spin ( s ) electronic wavefunction

V12 = ∫ Ј* (r1 )

D



(r2 )V12 D (r1 ) Ј (r2 ) × Ј*D (s1 ) *A (s2 ) D (s1 ) ЈA (s2 )

*

Ј

− ∫ Ј* (r1 ) A (r2 ) V12 D (r2 ) A (r1 ) × Ј* ( s1 ) * ( s2 ) D ( s2 ) ЈA ( s1 )

D

A

D

*

A



A



(15.49)



where r1 and r2 denote the spacial coordinates of the electrons involved and the apostrophe denotes an excited state. The first integral is the Coulomb term with V12 given by

eq. (15.47), and is different from zero when the spins of the electrons remain unchanged.

The second integral is an exchange integral with V12 ϭ e2/r12 in vacuum, where r12 is the

distance between the two electrons. This second integral represents the electrostatic interaction between the two charged clouds, and dies off exponentially with the distance



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15. Transitions between Electronic States



between the donor and the acceptor. Unless Ј ϭ Ј and D ϭ A, the integral vanishes,

D

A

but Ј is not necessarily equal to . If sЈ and sA are the initial spin quantum numbers of

D

the two participating molecules, the resultant spin quantum number of the two species

taken together must have one of the values

sЈ + sA , sЈ + sA − 1, sЈ + sA − 2, ... sЈ − sA

D

D

D

D



(15.50)



It follows that the spin quantum numbers of the resulting species can only have values sD

and sЈ if at least one of the values

A

sD + sЈ , sD + sЈ − 1, sD + sЈ − 2, ... sD − sЈ

A

A

A

A



(15.51)



is common to the series above. Thus, processes such as

DЈ(S1) ϩ A(S0) → D(S0) ϩ AЈ(S1)



(15.III)



DЈ(T1) ϩ A(S0) → D(S0) ϩ AЈ(T1)



(15.IV)



−

DЈ(T1) ϩ O2( 3 Σ g ) → D(S0) ϩ O2(1⌬g)



(15.V)



are spin allowed for an exchange mechanism. The treatment of this perturbation with the

Golden Rule, gives an exchange transfer rate of the form

kD =



2



exp ( − 2 R



∞



) ∫0



FD ( v ) AA ( v ) dv



(15.52)



where and reflect the ease of electron tunnelling between the donor and the acceptor

and are not simply related to measurable quantities. They will be further discussed in this

chapter under Sections 15.6 and 15.8. The donor emission FD and the acceptor absorption

AA spectra are normalised to unity, such that the integrals of FD and AA are both equal to

unity. The most salient features of this equation are the exponential dependence on the intermolecular distance and the normalisation of both emission and absorption in eq. (15.52),

which makes the exchange transfer rates independent of the oscillator strengths of the two

transitions.

The overlap integrals appearing in eqs. (15.45), (15.48) and (15.52) reflect the spacial

overlap between the wavefunctions of the initial unperturbed state s and the final unperturbed state k, and have their origin in the first of the two factors entering the Golden Rule,

|Vks|2 in eq. (15.36). The calculation of the spacial overlap almost invariably employs the

Born–Oppenheimer approximation, which, based on the difference of masses between

electrons and nuclei, treats the nuclei as stationary points with the electrons moving

around them. In practice, this consists of expressing the molecular wavefunction as the

product of an electronic ( e) and a nuclear (vv) wavefunction. The simplest case is that of

a diatomic molecule with translational and rotational motions that give rise to a continuum

of energies at the temperatures under consideration, and with a vibrational motion

described by a harmonic oscillator. Representing the distance between the two nuclei by

R, the Born–Oppenheimer approximation is expressed as



( x, R ) = e ( x; R ) vv ( R )



(15.53)



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Figure 15.4 Electronic transition corresponding to the absorption of light by the ground state of a

diatomic molecule. The final state, reached immediately after the electronic transition, is the vibrational state of the electronically excited state that has the largest overlap with the lowest vibrational

state of the electronic ground state.



The representation (x;R) denotes the parametric dependence of the electronic wavefunction on the inter-nuclear distance.

Substituting eq. (15.53) into eq. (15.21) and assuming that only the electronic distribution is perturbed, gives



∫



*

k



V



j



d =∫



* *

k v′



v V svv d e d



n



=∫



*

k



V



s



d



e



∫



*

vv ′vv d



n



(15.54)



The first factor

e

Vks = ∫



*

k



V



s



d



e



(15.55)



measures the extension of the electronic redistribution induced by the transition from the

initial state s to the final state k. The second factor

*

J vЈ, v = ∫ vvЈvv d



n



(15.56)



is the overlap integral between the vibrational wavefunctions of the initial (v) and final (vЈ)

states, which is called the Franck–Condon factor for the k, vЈ ← s, v transition. According

to eq. (15.36), the transition probability, or intensity, is proportional to |JvЈ,v|2. Figure 15.4

represents an electronic transition that takes place at a fixed nuclear geometry, following

the Franck–Condon principle. The vibrational relaxation takes place entirely in the excited

state, after the electronic transition, and takes the system to the lowest vibrational state of

the electronically excited state.



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15. Transitions between Electronic States



15.4



FRANCK–CONDON FACTORS



The Franck–Condon factor of an electronic transition from the vibrational level v ϭ 0 of

the electronic ground state to the corresponding level vЈ ϭ 0 of the electronic excited state

has a simple analytical solution when the vibrational motion is described by a harmonic

oscillator. In this case, the vibrational wavefunctions of the ground vibrational levels is

described by a normal (Gaussian) distribution function

F ( x) =



1

2



x



⎡ 1 ⎛ x − x ⎞2 ⎤

e

exp ⎢ − ⎜

⎟ ⎥

⎢ 2⎝ x ⎠ ⎥

⎣

⎦



(15.57)



illustrated in Figure 15.5. As shown in the figure, the Gaussian is centred in its average xe

and the variance 2 is given by the inflection points of the distribution function. It is also

convenient to characterise this distribution by its full-width at half-maximum

Δv1 2 = 4



ln 2



(15.58)



More specifically, the wavefunction of the ground vibrational state of a harmonic oscillator with a force constant f and a reduced mass , has the form

⎛ 1 ⎞

v0 = ⎜

⎟

⎝

⎠



12



⎡ ( R − Re )2 ⎤

exp ⎢ −

⎥

2 2 ⎦

⎣

2



⎛ 1 ⎞

vЈ = ⎜

0

⎟

⎝

⎠



12



=



f



⎡ ( R − RЈ )2 ⎤

e

exp ⎢ −

⎥

2 2

⎣

⎦



Figure 15.5 Normal (Gaussian) distribution function, centred in its average, xe ϭ 0.



(15.59)



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401



The overlap integral between these identical but displaced oscillators is

+∞



J 0Ј,0 = ∫ v v dR =

−∞



*

0Ј 0



1



(



⎡ ( R − R )2 + R − RЈ

e

e

∫−∞ exp ⎢ −

2

⎢

2

⎣

+∞



)



2



⎤

⎥ dR

⎥

⎦



(15.60)



The integral can be evaluated making the substitution of the variables

z = R−



1

( Re + RЈe )

2



(15.61)



which yields

J 0Ј,0 =



1



(



⎡ R − RЈ

e

e

exp ⎢ −

2

⎢

4

⎣



)



2



⎤ +∞ 2

⎥

e − z dz

⎥ ∫−∞

⎦



(15.62)



where the integral now has a standard form and gives √ . Thus, the overlap is



J 0Ј,0



(



⎡ R − RЈ

e

e

= exp ⎢ −

⎢

4 2

⎣



)



2



⎤

⎥

⎥

⎦



(15.63)



Considering that the intensity of the 0Ј ← 0 transition is proportional to |JvЈ,v|2, it attains

its maximum when the bond lengths of the ground and excited states are identical, and

diminishes as (ReϪReЈ)2 increases. This effect is illustrated in Figure 15.6, where it is also



Figure 15.6 Electronic transitions illustrating the intensity expected for the vibronic band of

the absorption spectrum: (a) totally allowed transition following the Franck–Condon principle;

(b) partially allowed transition according to the same principle.



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15. Transitions between Electronic States



shown that the intensities of the transitions to higher vibrational states of the electronically

excited state attain their maxima for Re ReЈ and may dominate the absorption spectrum.

The 0Ј ← 0 transition is a purely electronic transition, while the vЈ ← 0 transitions are

vibronic (electronic ϩ vibrational) transitions, subject to selection rules.

At room temperatures, the molecules are generally in their lowest vibrational state of the

electronic ground state. Upon light absorption, the electron is promoted from the vibrational level v ϭ 0 of the ground state to a vibrational level vЈ of the excited state. For

parabolas with the same force constant f, but minima displaced by d ϭ |ReϪReЈ|, as shown

in Figure 15.7, the progression of the vibronic bands follows a Poisson distribution, already

presented in eq. (5.20),



J vЈ,0 = exp [ − S ]



S vЈ

vЈ!



(15.64)



where S is a reduced displacement,

⎛ d ⎞

S≡⎜

⎟

⎝2 R ⎠



2



(15.65)



Figure 15.7 Schematic representation of light absorption and fluorescence emission between electronic states described by identical (same force constant f ) but displaced (Re ReЈ) parabolas. The

absorption is, necessarily, at higher energies than the fluorescence.