MIT Department of Chemistry
5.74, Spring 2004: Introductory Quantum Mechanics II
Instructor: Prof. Andrei Tokmakoff
p. 88
NUCLEAR MOTION COUPLED TO ELECTRONIC TRANSITION*
Vibronic structure is absorption spectra; wavepacket dynamics; coupling of electronic states to
intramolecular vibrations or solvent; coupling of electronic excitation/excitons in
solids/semiconductors to phonons. Also, extensions to Förster and Marcus Theories.
Two-Electronic Level System: Displaced harmonic oscillators T
0q
We will calculate the electronic absorption spectrum using T.C.F.
H
H0
E
H0 V t g HgTOT g
e
e HeTOT e
Ee
g
HTOT
g
T Q Wg Q Eg
TOT
T Q We Q Ee
He
Eg
0
Our Model
T Q : Nuclear kinetic energy
p 2 / 2m
Wg Q / We Q : Potential energy
2 2
1
2 m Z0 Q
2
2
1
2 m Z 0 Q d Ee Eg : Electronic energy gap
*
See Mukamel, p. 217, also p. 189.
d
Q
p. 89
H
TOT
e
H gTOT
1)
p2 1
2
2 mZ02 Q d Ee
2m
p2 1
mZ02 Q 2
2m 2
H e Ee
= H g Eg
Born-Oppenheimer Approximation
\g
g n
vibrational state
electronic state
2)
Condon Approximation
No Q dependence to P No vibrational excitation accompanying electronic excitation
V t P
P˜ E t g P ge e e Peg g
Now let’s calculate an absorption lineshape. Write dipole correlation function:
PI t PI 0
¦p
n
n PI t PI 0 n
n
g ,0 eiH 0t / = P e iH 0t / = P g , 0
since at low T all population in g , 0
By substituting our Hamiltonian H 0 and dipole operator P we find:
CPP t P eg
2
0e
iH g t / =
i E E t / =
eiH t / = 0 e e
F t e
g
e
i Z eg t
Note we can write our correlation function as C (t )
¦p
2
n
Vmn eiZmnt g (t ) where g(t) is our
n
lineshape function. Lets concentrate on F(t) = exp(g(t)), sometimes known as the dephasing
function:
p. 90
F t 0e
iH g t / =
eiH t / = 0
e
From problem set 1: displacement operator
e iDp Q e i Dp
? He
e
Q D=
idp/ =
Hg e
idp/ =
†
Ug
Ug
iH t / =
iH t / =
F t 0 e g eidp / = e g eidp / = 0
p
0e
idp t / =
e
idp 0 / =
since pt Ug† p0Ug
0
Now, using
p2 1
mZ 20 Q2
2m 2
Hg
=Z 0 a †a 12
a† a
n
We have:
p t F t i
m=Z0 † iZ0t
a e a e iZ0t
2
0 exp ª d a† eiZ0t a e iZ0t º exp ª d a† a º 0
¬~
¼
¬ ~
¼
ˆ
ˆ
ˆ
We can evaluate this using the identity e A B
†
†
1
2
eOa Pa eOa ePa e OP
So F(t) becomes
ˆ
e Ae B e
ˆ ˆº
12 ª¬ A,B
¼
. With ª¬ a† ,a º¼
with d
~
1
d
mZ0
2=
p. 91
F t e
d2
0 exp ª d a e iZ0t º exp ª d a† º 0
¬ ~
¼
¬~ ¼
~
0; 0 a†
where we have used a 0
0
Note that operator defined through expansion!
To help evaluate, we want to exchange order of operators,
and since
F t e
d2
> @
B̂, Â
2
0 exp ª d a† º exp ª d a e iZ0t º exp ª d e iZ0t º 0
¬ ~ ¼
¬~
¼
¬~
¼
~
exp ª d
¬~
{ e
e  e B̂ e B̂ e  e
2
e
iZ0t
1 º
¼
(where g t d
g t ~
2
e
iZ0t
is the lineshape function)
So we have the dipole correlation function:
CPP t Where D
d2
~
2
P eg exp ª¬ iZeg t D e iZ0t 1 º¼
d 2 mZ 0
2=
Huang-Rhys parameter
This represents the strength of coupling to the nuclear degrees of freedom.
Absorption Lineshape
V abs Z f
iZ t
³f dt e
2
Peg e
i Z egt
>
@
i Z t
exp D e 0 1
expand exponential
f
e D ¦
j 0
2
f
P eg e D ¦
j 0
1 j iZ0t
D e
j!
1 j
D G Z Zeg jZ0 j!
j
1
p. 92
Spectrum is a progression of absorption peaks separated by Z 0 with a Poisson distribution of
intensities o vibrational progression!
The intensities of these peaks are dependent on D , which is a measure of the coupling strength
between nuclear an electronic degrees of freedom.
Let’s plot the normalized absorption lineshape Vabs
c Z V abs Z e D Peg
1
V’abs
D=0 (no displacement)
Z
D<1 (small displacement
or small Z0)
1
V’abs
D
D2
Z0 Z0
Z
D>1 (large displacement)
V’abs
Z0 Z0 Z0
Zeg
Z
2
as a function of D .
p. 93
Franck-Condon Transitions
Note that for D 1 peak absorption at n 0 . For D !! 1 peak at n | D .
Note that D is the number of quanta excited at Q 0 o Franck-Condon principle.
The envelope for these transitions is Gaussian:
Short time expansion: t V abs Z Peg
| Peg
Peg
1
.
Z0
2 f
³f dt e
2
2
f
iZt
D exp i Z t 1
e iZ t e ³f
S Peg
2
expand exponential for small t
(retain first three terms)
i Z Z egt D> iZ 0 t 12 Z 20 t 2 @
³f dt e
f
e
eg
e
i Z Z eq D Z 0 t 12 Z 20 t 2
dt e e
ª Z Z DZ
eg
0
exp «
2
«
2Z 0
«
¬
2
º»
»
»
¼
Gaussian profile centered at Franck-Condon vertical transition.
V’abs
Z
Zeg
Z ZegDZ0
D: mean number of vibrational quanta excited in e on absorption.
O
D=Z 0
2 2
1
2 mZ 0 d
reorganization energy
= vibrational energy in e on excitation at a
= vibrational energy in g at Q d
0
p. 94
O
Absorption: = Zeg DZ0 Fluorescence: = Zeg DZ0 O
Q
0 d
Since vibrational energy on e is dissipated quickly, we expect fluorescence to be red-shifted by
2 O and have mirror symmetry:
Vabs Z
V fluor Z
f
f
³f
i ZZeg t g t ³f dt e
dt e i ZZeg 2 O t g* t g t D e iZ0t 1
2O: Stokes Shift
O
Vabs
Vfluor
=Zeg O
=Zeg
=Zeg O
=Z
p. 95
What if the electronic transition is coupled to many vibrational coordinates with own
displacement?
Correlation function is easy (if the modes are independent). Then we write the state of the
system as product states, i.e. g;n1 ,n2 ,...,ni
CPP t iZeg t
2
Zeg t g t P eg e
gt ª N
º
iZi t
1 º¼ »
«– exp ª¬ Di e
¬i1
¼
2
P eg e
¦ Di e iZ t 1
i
i
p. 96
Relaxation—
We didn’t include damping. Clearly, this will lead to broadening of delta function transitions.
We can also think of this as coupling to damped harmonic oscillators:
Coupling to undamped or weakly damped
harmonic oscillators
Zeg
Z
Zeg
Z
Coupling to strongly (over-)damped
harmonic oscillator
Zeg
Z
p. 97
More on Displaced Harmonic Oscillator Model
If you want to solve the same problem at finite temperatures, where excited vibrational levels are
populated, you find:
2
CPP t n
e
P eg e
E=Z0
1
Zeg t
exp ª D ª¬ n 1 e iZ0t 1 n e iZ0t 1 º¼ º
¬
¼
1
Calculating the lineshape, we would see “hot bands”: transitions from thermally populated
vibrational states. At low temperature the first term dominates – giving our previous result, and
at high temperatures, we have additional contributions which gives a progression of transitions to
frequencies below Zeg.
For the coupling of electronic transitions to nuclear motion, we often want to describe coupling
to a large number of modes. A continuous distribution can be used to describe relaxation
processes—remember that coupling to a continuum leads to irreversible relaxation. If the
nuclear degrees of freedom don’t interact, we can write the state of the system in terms of the
electronic state and the nuclear quantum numbers, i.e. e;n1 ,n2 ,n3 ! , and from that:
ª
º
iZ t
iZ t
CPP t v exp « ¦ D j ª n j 1 e j 1 n j e j 1 º »
¬
¼
¬ j
¼
or changing to an integral over a continuous distribution characterized by a density of states,
UZ .
CPP t v exp ª ³ dw U Z D Z ª¬ n Z 1 e iZt 1 n Z eiZt 1 º¼ º
¬
¼
Let’s look at the lineshape by doing a short-time expansion on the complex exponential
e i Zt
1 iZ t Z 2 t2
2
!
Retaining the first three terms, we can write:
p. 98
F t ª
§
Z2t 2 · º
exp « ³ d Z D Z U Z ¨ iZt 2n 1
¸»
2 ¹¼
©
¬
Vabs Z
³
f
f
dt e
i ZZeg t
exp ª i
«¬
^³ d ZU Z D Z Z` t º»¼ exp ª«¬ ^³ d ZU Z D Z Z 2n Z 1` t º»¼
1
2
Z
O / = : mean
2
Z2 temperature
dependent! Damping!
vibrational excitation!
Completing the square, we have:
Vabs Z
P eg
2
S
Z2
ª Z Z Z
eg
exp «
«
2 Z2
¬
2
º
»
»
¼
Gaussian centered at electronic resonance plus reorganization energy. Width is temperaturedependent.
We can also use the displaced harmonic oscillator model to describe rates of energy or electron
transfer.
2