MIT Department of Chemistry
5.74, Spring 2004: Introductory Quantum Mechanics II
Instructor: Prof. Robert Field
5.74 RWF Lecture #5
5–1
When we create Ψ(Q, t = 0) that is not an eigenstate of the isolated molecule, time-independent H, that
|Ψ(Q,t)|2 will evolve.
Usually a nonstationary “coherent superposition of eigenstates” state is produced by a sudden perturbation,
such as a short pulse of electromagnetic radiation.
We have looked at the evolution of a large number of “wavepackets” in 5.73 and 5.74. But there is a lack of
a simple picture of how Ψ(Q,0) is produced, what are the simple forces that cause it to evolve, and how all
of this is related to the frequency domain spectrum.
We need to understand the nature of “the pluck”.
Heller’s formulation of the relationship between the absorption spectrum, I(ω), and the Fourier transform of
the ⟨Ψ(t)*Ψ(0)⟩ autocorrelation function provides a unified conceptual and computational framework. It
was revolutionary!
Electronic Absorption Spectrum ↔ FT of ⟨Ψ (t)| Ψ(0)⟩
traditional frequency domain formulation
restated as Fourier Transform of the autocorrelation function
* wavepacket evolution as responsible for features in autocorrelation function
* what are the features of the wavepacket as determined by the upper and lower electronic
potential surfaces
* local view of V(Q) from wavepacket/autocorrelation picture vs. global view from FranckCondon picture.
see Heller JCP 68, 3891 (1978)
1

I k (ω) = Σ c k j δ(ω − ω jk ) = ∑ c k j δ  E − ( E j − E k ) 
h

j
(
initial
transition
state
intensity
notation e single prime, g double prime
transition
frequency
2
I v g′′ (ω) = µ eg ∑
v e′
)
ω jk =
(
qv e′ v g′′δ ω − ω v e′ v g′′
1
h
)
Franck-Condon factor
qv e′ v g′′ = v e′ v g′′
2
vibrational overlap squared
( E j − Ek )
5.74 RWF Lecture #5
5–2
It is necessary to know complete Ve′ and Vg′ up to at least energy of v′e and v g′′ and to integrate over all
3N–6 normal coordinate displacements. Usually do not have this information except for diatomic
molecules. Use displaced harmonic oscillators and qualitative F-C factors derived from diatomic molecules.
Problem when there is not a mode by mode correspondence between Ve′ and Vg′ . Duschinsky rotation.
Try to recast the standard frequency domain expression into wavepacket form.
Integral definition of δ-function
(
δ ω − ω v e′ v g′′
)
[ (
)]
1
=
∫ dt exp −i ω − ω v e′ v g′′ t
2π
.
Completeness of vibrational basis set to express lower state vibrational wavefunction as linear combination
of upper state vibrational wavefunctions:
χ v g′′ ( R) ≡ R v′′g
= ∑ R v e′ v e′ v′′g = ∑ χ v e′ v e′ v g′′ .
v e′
completeness
v e′
vib. overlap
Reassemble factors
 
E g,v g′′  
1 ∞
 t  Ψi ( t) Ψi (0)
I v g′′ (ω) =
dt exp −i ω +
∫
−∞

2π
h  

Ψi ( t) ≡ g µ e e ∑
v e′
µ *eg
(
v′′g v′′e
v e′ eiE e,v e′ t / h
Ψi (0
) ≡ e e µ g ∑ v e′ v e′ v′′g .
µ eg
v e′
)
eiH e t / h
excited
state H
gives E e,v e′ when
applied to e v e′
Shaded equation is the Fourier transform of the initial (t = 0) (post-pluck) state formed from eigenstate
g v g′′ . It is the initial vibrational wavefunction transported onto the excited potential and allowed to
evolve under the influence of He.
5.74 RWF Lecture #5
5–3
What does this buy us?
We know v′′g perfectly. It is a simple Gaussian if v g′′ =0.
We move this up to the excited surface where it is not an eigenstate. We project it onto eigenstates v e′ and
they evolve according to He.
The averages ⟨R⟩ and ⟨P⟩ and are known perfectly at t = 0. ⟨P⟩t=0 = 0, ⟨E⟩t=0 = ⟨E⟩t . ⟨R⟩t=0 is v g′′ R v′′g ≈ Re′′ if
v′′g =0. Franck Condon principle gives ⟨E⟩t – vertical transition (or turning point to turning point).
So we have a Gaussian of known ∆R, ∆P, E, ⟨R⟩, ⟨P⟩ = 0 at t = 0. It is a particle like state. It's ⟨R⟩, ⟨P⟩ evolve
on Ve′ as a particle would evolve. The key is that we get the most important details in the spectrum from
allowing the wavepacket to evolve for a very short time. It samples only the near v′′g R v′′g region of the
Ve′(R) .
See Figure 9.1 This shows a wavepacket starting at the Franck-Condon point on Ve′(R) . It feels a force
which is minus gradient of Ve′ . In this case gradient is in sym. stretch (v) direction which is bound. It starts
at a saddle point wrt anti sym. stretch (u) which is unbound. The wavepacket is pulled apart along u. It
oscillates and spreads along v and is pulled apart irreversibly along u.
Figures removed due to copyright reasons.
5.74 RWF Lecture #5
5–4
See Figure 9.2 This shows what happens to ⟨Ψ(t)|Ψ(0)⟩. At t = 0 it starts to move away from itself. Be
careful, the moving away could be in either coordinate or momentum space. We will look at this next. At
τe a small part of the wavepacket has returned to the F–C point along the sym. stretch (v). The motion is
2π
periodic of period T = ω . The initial FWHM is δ ≈ 2(τb – τa). So we have damped periodic motion of a
wavepacket with recurrences of nearly constant width. The FT of this is shown in the lower panel.
FT ⟨Ψ(t)|Ψ(0)⟩ ∝ Ι(ω)
The slow recurrences give rapidly oscillating structure in I(ω),
2π
ω = T.
2π
The fast decays give an envelope of FWHM ~ δ .
So we know how to read off from the spectrum the key features of the dynamics of the wavepacket.
A key point is that we are looking at the evolution of the wavepacket on Ve′ for a short time. The
wavepacket does not move very far. It does not spread very far. It does not sample the F-C dark
coordinates very much, even if they are anharmonically coupled to the FC bright coordinates.
Now we want to look at the actual form of the wavepacket and see how the shape of Ve′ and the F-C point
affect the evolution of the wavepacket.
Figures removed due to copyright reasons.
5.74 RWF Lecture #5
5 – 5
Heller, Sundberg, and Tannor JPC 86, 1822 (1982)
for absorption from v g′′ = 0
 2µω g 

Ψi (R, t = 0) = 
 h 
1/ 4
 −µω g 
 R − Reg
exp 
 2 h 
(
2
∫ dR Ψi = 1
(
equilibrium R of the Vg′′ surface.
∆R ≈ FWHM Ψv g′′ = 0
mass ↑ or wg ↑ ∆R⇓
ω g = [ kg / µ ]
1/ 2

)2 
)
1/ 2
 h8 ln 2 
=

 µω g 
kg =
d 2Vg
dR 2
R = Reg
Ψ is real, thus ⟨P⟩ = 0
Early time form of Gaussian wavepacket formed on Ve′ from v g′′ = 0
 2µω g 

Ψi ( R, t) = 
 h 
1/ 4
 µω g ( t)

i
exp −
(R − R(t)) 2  exp P (t)(R − R(t))


2h
h

i

× exp  γ ( t) 
h

3 factors: position, momentum, phase
R( t) = Reg + 0(t 2 )
P ( t) = −
dV
dR
R = R (t )
not moving at early time
t + 0( t 2 )
dV
P˙ ( t) = −
dR
R = R (t )
force Ṗ = −gradient of potential
width in P changes at early time —
difference in curvature of Ve + Vg
at Reg
 2 1 d 2Ve
ω g ( t) = ω g − i ω g −
µ dR 2

width in R does not
change at early time
1 d 2Vg
µ dR 2

2
 t + 0( t )
R = R (t )
R = R ( t )= R e g
5.74 RWF Lecture #5
5–6
real → moments of R
How do we know this? Stuff in exp
imaginary → moments of P
∂
i ∂R
h ∂Ψ
PΨ =
i ∂R
P=
h
gives
∫ Ψ *PΨdR
phase of the center E of wp
[
h
i
(ia + bR)Ψ
gives ha + ∫
odd
2
Ψ RdR = 0
( )]
γ( t) = γ 0 − ∆E eg + Ve Reg t + 0(t 2 )
( )
Ve Reg
min to
min
Choose 0 of
E at Vg(Reg)
↑
arbitrary
phase
↑
↑
∆Eeg
↑
The dominant early time change in the wavepacket is in ⟨Ρ⟩ and ∆P.
E=0
5.74 RWF Lecture #5
5–7
Rewriting Ψi(R,t) we get
stationary ⟨R⟩ and ∆R
 2µω g 

Ψi ( R, t) = 
 h 
1/ 4
  µω g 
 R − Reg
exp −
  2h 
(
acceleration of wp
)
2


⟨P⟩t ∝ t
 it   dVe

( R − Reg ) + ∆E eg + Ve Reg  
× exp 
 h   dR R = R

 
eg

 itµ

 2 1 d 2Ve
2
× exp  (R − Reg )  ω g −
 exp(iγ 0 h)

2
µ dR R = R e  
 h

g 
( )
Extend from diatomic molecule to Polyatomic molecule
FC active modes — get
d 2Ve′
dQi2 Q
along i-th g-state
normal mode
ig
= Q ie g
equilibrium
in mode i
also get ωi
d 2Ve′
dQi2
Q
ie e
The v = 0 : v = 1 intensity ratio determines the gradient of Ve′ at Qieg .
5.74 RWF Lecture #5
5–8
What is wrong or incomplete with simple idea that initial state Ψ(Q) is transferred intact and undistorted
onto excited state potential energy surface?
* negative wavepacket on ground state energy surface is ignored
* assumes that the excitation pulse is sufficiently short and centered at the correct λ
* neglects µ(Q)
* neglects spatial (and temporal) distribution of excited molecules (especially the spatio-temporal
correlation)
When are these simplifications justified?
What do we do when they are not justified?