Page 110
QUANTUM MECHANICAL TREATMENT OF FLUCTUATIONS*
Introduction and Preview
Now the origin of frequency fluctuations is expected to be interactions of our molecule (or more
appropriately our electronic transitions) with its environment. This we can treat with our D.H.O.
model, which is a general approach to coupling to nuclear vibrations.
We found that
µ ( t ) µ ( 0 ) = ∑ pn n eiH 0t µ e −iH 0t µ n
n
2
= µ eg e
− iωeg t
e
H gt
e −iH et
We can write this in terms of the Hamiltonian that describes the electronic energy gap’s
dependence on Q (deviation relative to ω eg ):
Heg = HeTOT − HgTOT − ω eg = He − Hg
2 − iω eg t
Cµµ (t ) = µeg e
e
(Energy Gap Hamiltonian)
−iHeg t
Now if we believe there are interactions that lead to fluctuations in the energy group—variations
in d or ω 0 , then our Heg is now time-dependent!
Cµµ ( t ) = e
− i ωeg t
exp+
−i
t
∫ d τ H ( τ)
0
eg
Performing the cumulant expansion:
exp+
*
−i
t
∫ d τ H ( τ)
0
eg
2
 −i t

τ2
 −i  t
= exp  ∫ d τ H eg ( τ ) +   ∫ d τ2 ∫ d τ1 H eg ( τ2 ) H eg ( τ1 ) +…
0
0
  0


See Mukamel, Ch. 8 and Ch. 7
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Defining δω eg (t ) ≡
F (t ) = e
Heg (t )
− g (t )
t
τ2
0
0
g ( t ) = ∫ dτ2 ∫ d τ1 δωeg ( τ1 ) δωeg ( 0 )
So we have an expression for how the time-dependence of the energy gap Hamiltonian leads to
the lineshape.
Also note:
H0 = He + Ee + Hg + E g
Hg =
= ω eg + Heg + 2 H
g
p2
1
+ 2 mω 2D
Q2
2m
H eg = H e − H g = 12 mω02 ( Q − d ) − 12 mω02 Q 2
2
= − mω02 d Q + 12 mω02 d 2
linear in Q
const
Note that this looks very much like a Hamiltonian that describes the coupling of an electronic
system to a bath [one degree of freedom here] of H.O. with a linear coupling between the two!
H = H S + H B + H SB
H S = e Ee + λ e + g Eg g
p 2
1
HB =
+ 2
mω02
Q 2
2m
H SB ( ≅ H eg ) = mω02 d 2 Q
coupling strength
Fluctuations in coupling to bath could lead to line broadening! Equivalently, coupling to a bath
of many harmonic oscillators should lead to line-broadening.
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Time-Dependent Energy Gap Hamiltonian
Let’s work through this more carefully. Start by defining reduced coordinates
E
ω0 m
p
2
p=
~
~
λ
ED
mω0
q
2
q=
HeTOT
H gTOT
mω0
d=
d
~
2
EA
2
 2 


H e = ω0  p +  q + d  
~ ~ ~ 
2
2
H g = ω0  p + q 
~ 
~
d
0
Q
Heg
H eg = H e − H g = 2 ω0 d q + ω0 d
~ ~
2
~
λ
λ
Now, the absorption lineshape is described through
0
µ(t )µ (0) :
Cµµ ( t ) = µ ( t ) µ ( 0 ) = µ eg e
2
F (t ) = e
iH g t
− iωeg t
q
F (t )
e−iH t
e
If we want to rewrite this in terms of Heg , we are changing representation to a new Hamiltonian.
Similar to the transformation to the interaction picture, we will choose a new frame of reference:
the ground state Hg and the dynamics of the excited state will be represented in reference to the
ground state through Heg :
H e = H g + H eg
e − H et = e
− iH g t
⇔
 −i t

exp+  ∫ d τ H eg ( τ ) 
0


where H eg ( τ ) = e
iH g t
H eg e
− iH g t
H = H0 +V
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This equation implies:
F (t ) =
e
iH g t
e
− iH e t
 −i t

= exp+  ∫ d τ H eg ( τ ) 
0


The cumulant expansion to second order says:
2
 −i t

 −i  t
F ( t ) = exp  ∫ d τ H eg ( τ ) +   ∫ d τ1 Ceg ( τ2 , τ1 ) + … 
0
  0


Ceg ( τ2 , τ1 ) = H eg ( τ2 ) H eg ( τ1 ) − H eg ( τ2 ) H eg ( τ1 )
= δH eg ( τ2 ) δH eg ( τ1 )
⇒
Now note the way we defined Heg means that
δωeg ( τ2 ) δωeg ( τ1 )
δω eg =
δHeg
2
H eg = ω0 d = λ
~
(The energy gap could also be defined relative to the energy gap at Q = 0 : Heg ′ = He − Hg − λ .)
So we have
Cµµ ( t ) = µ eg
2
e
(
)
− i Ee − Eg +λ t /
t
τ2
0
0
e
− g (t )
g ( t ) = ∫ d τ2 ∫ d τ1 δωeg ( τ1 ) δωeg ( 0 )
Now, evaluating Ceg (t ) = Heg (t ) Heg
(0) for one harmonic oscillator
Ceg ( t ) = ∑ p
n n H eg ( t ) H eg ( 0 ) n
n
= ω02 D ( n +1) e − iω0t + n e + iω0t 
and
D=d
~
2
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g ( t ) = D  coth ( β ω0 / 2 )(1− cos ω0t ) + i ( sin ω0t − ω0t ) 
= g ′ + ig ′′
Note we now have damped (g′)and oscillating (g′′) contributions to F(t ) .
Alternately we can write this as g ( t ) = D  n
(e
− iω0t
) (e
− 1 + e + iω0t − 1 +
− iω0t
)
− 1  − iDω0t .
At low t , coth(x ) → 1 and g ( t ) = D [1 − cos ω0t + i sin ω0t − iω0t ]
= D 1 − e − iω0t − iω0t 
combining with
F (t ) = e
iDω0t − g ( t )
we have our old result:
F ( t ) = exp  D
(e
− iω0t
)
− 1 
Distribution of Nuclear States
Coupling to a distribution of states characterized by a density of states W ( ωD ) . As discussed
before, we expect
F ( t ) = exp  − ∫ dω0 W ( ω0 ) g ( t ,ω0 ) 
Coupling to a continuum will induce irreversible relaxation, which will be characterized by
damping of Ceg (t ) . This is achieved by summing over a distribution of oscillatory Ceg ( ω0 ,t ) :
Ceg ( t ) = ∫ dω0 Ceg ( ω0 ,t ) W ( ω0 )
δω eg (t )δω eg (0)
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Alternatively in the frequency domain:
~
+∞
C eg ( ω) = ∫ eiωt Ceg ( t ) dt
−∞
+∞
= ∫ dω0 W ( ω0 ) ∫ eiωt Ceg ( ω0 ,t ) dt
−∞
Ceg ( ω0 )
~
C eg ( ω0 ) = ω02 D ( ω0 ) ( n + 1) δ ( ω − ω0 ) + n δ ( ω + ω0 ) 
Ceg′′ ( ω0 ) = ω02 D δ
 ( ω − ω0 ) + δ ( ω + ω0 ) 
We define a spectral density or coupling-weighted density of states:
ρ ( ω) =
Ceg′′ ( ω)
2πω2
= ∫ dω0 W ( ω0 ) D ( ω0 ) δ ( ω − ω0 ) =
W ( ω) D ( ω)
This leads to:
~
g (t ) = ∫
+∞
−∞
1 C eg ( ω)
dω
exp ( −iωt ) + iωt − 1
2π ω2 
+∞


 β ω 
= ∫ dω ρ ( ω) coth 
 (1− cos ωt ) + ( sin ωt − ωt ) 
−∞
 2 


λ=
∫
∞
0
dω ωρ ( ω)
Now take the case
C′′eg (ω ) = 2 λΛ
ω
2
ω + Λ2
Lorentzian distribution
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In the high temperature limit
g(t ) =
kT
>> Λ we get:
2λ kT
[exp (−Λt ) + Λt − 1]
Λ2
−i
λ
Λ
[exp(− Λt ) + Λt − 1]
So if we ignore the imaginary part of g(t ) , and we equate 2
∆ =
2 λkT
τc =
1
Λ
we have our stochastic model:
g(t ) = ∆2τ c2 [exp (−t / τ c ) + t / τ c − 1]
So, the interaction of an electronic transition with a frequency distribution of nuclear coordinates
(a bath) leads to line broadening and irreversible relaxation. The effect is to damp the nuclear
oscillations on electronic states.
More commonly we would think of our electronic transition coupled to a particular nuclear
coordinate Q which may be a local mode, but the local mode feels a fluctuating environment—a
friction.
Classically, we would understand the fluctuations as Brownian motion, described by a
generalized Langevin equation:
t
mQ ( t ) + mω02Q 2 +
H.O.
m∫ d τ γ (t − τ) Q ( τ )
0
= f (t ) + F (t )
damping, for no memory ⇒ mγQ
For a random force:
f (t ) = 0
random force
For no memory:
f (t ) f (τ ) = 2mkT γ (t − τ )
γ (t − τ ) = γδ (t − τ )
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This oscillator has a correlation function described by
CQQ ( ω) ∝
1
−ω + ω − iωγ ( ω )
2
2
0
Looks similar to a damped H.O.
This coordinate correlation function is just what we need for describing lineshapes. Note:
Ceg ( t ) = H eg ( t ) Heg ( 0 ) =
2
ω02 d
~
2
q (t ) q ( 0)
~
~
We can get exactly the same behavior as the classical GLE by coupling to a bath of harmonic
oscillators (normal modes, x ). For
N
hnuc = ∑ ωα  pα2 + xα2 
~ 
 ~
α=1
where x ⇔ q
~
With this Hamiltonian, we can construct N harmonic coordinates any way we like with the
appropriate unitary transformation. We want to transform to our local mode Q :
 Q 
 X 
 1 
U x =  X2  ~ ~ 





 Xn−1 
Now:
2
2
hnuc
= ω0  p + Q  +
~
~ 
system
N −1
∑
α=1
ωα  pα2 + X α2  + 2Q ∑ cα X α
 ~
~ 
α
bath
system-bath
interaction
So, going back to our displaced H.O. problem, we can rewrite our Hamiltonian to include the
interaction of one primary vibration with a bath, which leads to damping:
Electronic transition
Primary vibration
Bath of H.O.s
~
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Brownian Oscillator Hamiltonian (Spin-Boson Hamiltonian)
H = H s + H B + H SB
H S = e H eTOT e + g H gTOT g
H B = ∑ ωα  pα2 + xα2 
 ~ ~ 
α
H SB = 2 q ∑ cα xα
~
~
α
cα : coupling
~
~
Ceg ( t ) = δH eg ( t ) δH eg ( 0 ) = ξ 2 q ( t ) q ( 0 )
~
~
Here ξ = 2 ω0 d is the measure of the coupling of our primary oscillator to the electronic
~
transition. The correlation functions for q are complicated to solve for, but can be done analyically: Ceg′′ ( ω) = ξ
ω γ ( ω)
2m ( ω2 − ω2 )2 + ω2 γ 2 ( ω)
0
where γ is the spectral distribution of couplings between our primary vibration and the bath
γ ( ω) = π∑ cα2 δ ( ω − ωα )
α
~
For a constant γ , γ (ω ) → γ : Ceg′′ ( t ) = ξ
1
exp ( −γt / 2 ) sin Ωt
2m Ω
Ω = ω02 − γ 2 / 4
reduced frequency
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This model interpolates between the coherent undamped limit and the overdamped stochastic
limit.
If we set γ → 0 , we recover our earlier result for Ceg(t) and g(t) for coupling to undamped
nuclear coordinates.
For weak damping γ << ω
Ceg′′ ( t ) ∝ ξ
1
exp ( −γ / 2 ) sin ω0t
ω0
For strong damping γ >> 2ω i , Ω is imaginary and
Ceg′′ ( t ) ∝ ξ Λ exp ( −Λt )
Λ=
ω2D
γ
which is the stochastic model.
Absorption lineshapes are calculated as before, by calculating the lineshape function from the
spectral density above.
This model allows a bath to be constructed with all possible time scales, by summing over many
nuclear degrees of freedom, each of which may be under- or over-damped.
Ceg′′ ( ω) =
∑ C ′′ ( ω)
eg ,i
i
=
∑ξ
i
i
2m ( ω − ω
2
i
ω γ ( ω)
)
2 2
+ ω2 γ i2 ( ω)
.