PG/KA/Chap 4 - 1

Chap 4. Time-dependent Method of Fermi’s Golden Rulewkm =2π|Ukm |2 δ(Ek − Em ± h̄ω)h̄Molecular systems : |mi = |i, νi → |ki = |f, ν 0 i · · · |electrons, nucleii
Uf ν 0 ,iν=ZdRχf ν 0 (R)Z

drϕf (r; R)U (r, R)ϕi (r; R) χiν (R)|≡ hχf ν 0 (R)| Ũf i (R) |χiν (R)iR| {z }{z}Ũf i (R) ≡ hϕε0 (r; R)|U (r, R)|ϕε (r; R)ir

Suppose : we cannot specify the final nuclear quantum state ν 0(e.g., have not sufficient energy resolution, or not interested)w(f ← iν) =Xwf ν 0 ,iνν0( Note : normally, vib. rot. energy ∼ kB T , electronic energy  kB T )PG/KA/Chap 4 - 2

And : thermal average over the initial nuclear states νw(f ← i) =XνPiν w(f ← iν),• Partition function : (⇒ normalization

PPiν = e−βEiν /Zi(Boltzmann distribution)

P =1)
ν iνXX
−βEiν
−βHi−βHiZi =e=hν|e|νi = Tr(nuc) [e]

νν
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hBO approx.
iHe (r; R)ϕi (r; R) = Wi (R)ϕi (r : R)Hi χiν (R) = Eiν χiν (R) , where Hi ≡ TN + Wi (R)• Density operator for nuclear states on adiabatic potential Wi (R)
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ρi ≡ e−βHi /Zi , Zi = Tr(nuc) [e−βHi ]• Thermal average (on Wi (R)) of a quantity A(R)XX e−βEiν
X e−βHihAi =Piν hν|A|νi =hν|A|νi =hν|A|νi
ZZiiννν= Tr(nuc) [ρi A]

PG/KA/Chap 4 - 3• Time-dependent form (Kubo-Toyozawa)
1Using : δ(ω) = 2πw(f ← iν)R∞−∞eiωt dt , (and hence δ(h̄ω) = h̄1 δ(ω) )2π X|Uf ν 0 ,iν |2 δ(Ef ν 0 − Eiν ± h̄ω)=h̄ν0===w(f ← i) =1 Xh̄21h̄21h̄2Xν|hν 0 |Ũf i |νi|2Zν 0∞ −∞dte−iEf ν 0 t/h̄ +iEiν t/h̄ ∓iωtee−∞ν0XZ ∞

Z ∞dthν|Ũf∗i e−iHf t/h̄ |ν 0 ihν 0 |Ũf i eiHi t/h̄ |νie∓iωtdthν| Ũf∗i e−iHf t/h̄ Ũf i eiHi t/h̄ |νie∓iωt−∞|Piν w(f ← iν) ={z≡A1h̄2Z ∞−∞}(for brevity)dtTr(nuc) [ρi A]e∓iωt

PG/KA/Chap 4 - 4

Application - 1 : Vibration / rotation spectra [SR 9.6, 9.7]
In the same electronic state : Hf = Hi = H⇒ Ũf i = Ũii ∝ µ (dipole moment)
Vibration / rotation spectra (absorption)σ(ω) ∝=Z ∞dtTr(nuc) [ρµ∗ e−iHt/h̄ µe+iHt/h̄ ]eiωtZ ∞dthµ∗ (0)µ(−t)ieiωt

−∞

|{z}

µ(−t) (Heisenberg rep.)−∞• Vib-rot spectra ⇔ TCF of vib-rot (thermal) motion

PG/KA/Chap 4 - 5

Application - 2 : Electronic spectra, Nonadiabatic transitions
1. Gaussian Wavepacket method
2. Cumulant Expansion method• Gaussian Wavepacket [SR 9.4, 7.3.2]

iii
χpt ,qt (q, t) = exp[ αt (q − qt )2 + pt (q − qt ) + γt ]h̄h̄h̄• qt = hqi, pt = hpi : average position and momenta
follow classical eq of motion : ẋt = ∂H/∂pt , ṗt = −∂H/∂xt• αt , γt : width and phase (time dependent)α̇t = −(2/m)α2t − Vxx /2, γ̇t = ih̄αt /m + pt ẋt − E• Exact on quadratic potentials
V (x) = V0 + Vx (x − xt ) + 12 Vxx (x − xt )2

PG/KA/Chap 4 - 6Using : Tr(nuc) [A] →w(f ← i) =1h̄2C(t) ∝=Z ∞1hnZdq0n dpn
0 hχp0 ,q0 |A|χp0 ,q0 iC(t)e∓iωt dt−∞Z1Z∗ −iHf t/h̄
dq0n dpn
Ũf i eiHi t/h̄ |χp0 ,q0 i0 hχp0 ,q0 |ρi Ũf i eZdq0n dpn
0 hΦi (t − iβh̄)|Φf (t)i|Φi (t − iβh̄)i = Ũf i e−iHi (t−iβh̄)/h̄ |χp0 ,q0 i|Φf (t)i = e−iHf t/h̄ Ũf i |χp0 ,q0 i
1. Propagate wavepackets |χp0 ,q0 i on Hi and Hf
2. Calculate the overlap hΦi |Φf i
3. Fourier transform

PG/KA/Chap 4 - 7

Cumulant expansion method• Condon Approximation :Neglect R dependence of Ũf i (R)or take 0th term of : Ũf i (R) = Ũf i (R0 ) +⇒w(f ← i) =1|Ũf i (R0 )|22h̄

Z ∞−∞(Time ordered exponential)∂ Ũf i∂R(R − R0 ) + · · ·R0

dthe−iHf t/h̄ e+iHi t/h̄ ii e∓iωt|exp(−) [{zZ tih̄}dτ ∆Vi (τ )]0∆V ≡ Hf − Hi , ∆Vi (t) ≡ e−iHi t/h̄ ∆V e+iHi t/h̄

PG/KA/Chap 4 - 8• Time ordered exponential
e−iHf t/h̄ e+iHi t/h̄ = exp(−) [( f (t) ≡ )ih̄Z tdτ ∆Vi (τ )]0Note : Since [Hi , Hf ] 6= 0 , e−iHf t/h̄ eiHi t/h̄ 6= e−i(Hf −Hi )t/h̄ ∂fi= e−iHf t/h̄ (Hf − Hi )e+iHi t/h̄h̄ ∂ti −iHf t/h̄ +iHi t/h̄ −iHi t/h̄+iHi t/h̄= eee(Hf − Hi )eh̄i=f (t)∆Vi (t)h̄
Z tIntegrate : f (t) = f (0) + idτ f (τ )∆Vi (τ )h̄

Sequential expansion (f (0) = 01) :Z tZ t Z τi 200
 f (t) = 1 + idτ
∆V(τ)+(dτdτ∆V(τ)∆Vi (τ ) + · · ·)
ii
h̄h̄

0 Z00ti≡ exp(−) [h̄dτ ∆Vi (τ )]0(⇐ Definition)

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

PG/KA/Chap 4 - 9• Note :The original formula wasw(f ← i)=XXPiν w(f ν 0 ← iν)

ν=Condon Approx.

ν0 XX2πh̄νPiν |hf ν 0 |Ũf i |iνi|2 δ(Ef ν 0 − Eiν ± h̄ω)

ν0XX
2π2|Ũf i (R0 )|Piν |hf ν 0 |iνi|2 δ(Ef ν 0 − Eiν ± h̄ω)w(f ← i) 'h̄νν0We will obtain w(f ← i) in p.7 from the Fourier transform of this.

• If we correctly evaluate w(f ← i) in p.7, it takes into account of the(thermal average of) the Franck-Condon overlap |hf ν 0 |iνi|2⇒ Quantum effects of the nuclear motion (e.g., tunneling) are accounted

PG/KA/Chap 4 - 10

• Cumulant expansion“average of exponential” → “exponential of averages”

heiλxi=eiλhxic + 12 (iλ)2 hx2 ic +···

Expand both sides and compare (in order of λ ) to findCumulant average :hxic = hxi
22hx ic = hx i − hxi

2(= variance)

hx3 ic = hx3 i − 3hx2 ihxi + 2hxi3

Advantages :• Average of oscillatory function heiλx i→ average first, then place on exponent eiλhxic + · · ·• Partial sum to infinite order :even if the exponent on the right-hand-side is truncated at finite order,
terms up to the infinite order in λ is partially included• In a particular case where the variable x follows Gaussian distribution(Gaussian process), the 3rd and higher order cumulants exactly vanish

PG/KA/Chap 4 - 11

• Cumulant expansion of time-ordered exponential
hexp(−) [ih̄

= exp

Z tdτ ∆V (τ )]i

 0Z tih̄0dτ h∆V (τ )ic + i 2 Z th̄dτ0Z τdτ 0 h∆V (τ 0 )∆V (τ )ic + · · ·0

Expand both sides and compare order by order (up to 2nd order) :⇒ h∆V (τ )ic = h∆V (τ )i = h∆V (0)i = h∆V i(independent of time in steady-state)h∆V (τ 0 )∆V (τ )ic = h∆V (τ 0 )∆V (τ )i − h∆V i2 = hδ∆V (τ 0 )δ∆V (τ )i( δ∆V (t) ≡ ∆V (t) − h∆V i )• Line-broadening function : g(t) ≡1h̄2Z t0dτZ τdτ 0 hδ∆V (τ 0 )δ∆V (τ )i0• In thermal equilibrium : hA(τ 0 )A(τ )i = hA(0)A(τ 0 − τ )i
(depends only on the time interval τ − τ 0 )[Verify using h...i = Tr{ρ...} and Heisenberg rep. of A(t)]PG/KA/Chap 4 - 12

Changing integration variable (τ, τ 0 ) → (τ 0 , s ≡ τ − τ 0 ) : Jacobian = 1g(t) = 12h̄
Thus,

Rt0ds

R t−s0dτ hδ∆V (0)δ∆V (s)i = 12h̄0w(f ← i) =gi (t) =12|
Ũ|
fih̄21h̄2Z tZ ∞Rt0ds(t − s)hδ∆V (0)δ∆V (s)idte−gi (t) ei(h∆V ii /h̄∓ω)t−∞ds(t − s)hδ∆V (0)δ∆V (s)ii0(correct up to 2nd-order cumulant)• h∆V ii : average over nuclear motion on potential Vi (electronic state i)• hδ∆V (0)δ∆V (t)ii : TCF = thermal fluctuation of ∆V.
.

PG/KA/Chap 4 - 13

Time correlation functions
C(t) = hδA(0)δA(t)i : fluctuation δA(t) = A(t) − hAiClassical : phase-space distribution function f (r, p)
ZZdrdpf (r, p)δA(r, p; 0)δA(r, p; t)
C(t) =


Statistical (ensemble) average ⇐ distribution of the initial condition f (r, p)

Quantum :C(t) = Tr[ρδAe−iHt/h̄ δAeiHt/h̄ ]Thus, C(−t) = C ∗ (t) ⇒ Re C(−t) = Re C(t) ... even,Im C(−t) = −Im C(t) ... odd
In classical mech. C(t) is real and even (for real quantities)
C(0) = hδA(0)2 i ≥ 0 (variance, fluctuation)



PG/KA/Chap 4 - 14

When quantity A shows stochastic random motion (eg, in solution phase),C(t) → 0 as t → ∞.

• δA(0) and δA(t) at large t (long time interval) lose mutual correlation,such that both δA(0)δA(t) > 0 and < 0 realize randomly.
• In other words, at long time interval, δA(0) and δA(t) become “statistically independent”, as described by C(t) → hδA(0)ihδA(t)i = 0 × 0 = 0.

• Ergodic hypothesis
Statistical (ensemble) average = average over the time1
T →∞ TC(t) = hδA(0)δA(t)i = limZ T0dt0 δA(t0 )δA(t0 + t)PG/KA/Chap 4 - 15

TCF and spectral line shapeVib. / rot. spectra : σ(ω) ∝• Exponential TCF

R∞

hµ(t)µ(0)ie−∞

C(t) = C(0)e−γ|t| ⇒ σ(ω) ∝ C(0)• Gaussian TCF

C(t) = C(0)e−λ2 t2−ω 2 /4λ2(1) C(t) = C(0)e−γ|t| · eiω0 t ⇒ σ(ω) ∝2 2dt =2γ
γ 2 + ω2⇒ σ(ω) ∝ C(0)e• Damped-oscillating TCF

iωt

R∞

iωt
C(t)edt−∞(Lorentzian line shape)(Gaussian line shape)2γ
γ 2 + (ω − ω0 )22(2) C(t) = C(0)e−λ t · eiω0 t ⇒ σ(ω) ∝ e−(ω−ω0 ) /4λ2

(Note : oscillatory factor eiω0 t just introduces peak shift)[Verify by yourself (just elementary integrations!)]

PG/KA/Chap 4 - 16

• Example of Gaussian TCF
Consider : dilute solution of dipolar molecules
Short-time motion ' nearly free rotation (with angular velocity Ω)(“inertial motion”)

Dipole correlation : µ(0)µ(t) = |µ|2 cos ΩtKinetic energy of rotation = IΩ2 /2(I = inertial moment)
2⇒ Thermal population of Ω : P (Ω)dΩ ∝ e−E/kB T dΩ = e−IΩ /2kB T dΩ
(Normalization :Hence,R∞
0P (Ω)dΩ = 1 ⇒ prefactor 2(I/2πkB T )1/2 )hµ(0)µ(t)i =Z ∞
02P (Ω)|µ|2 cos ΩtdΩ = |µ|2 e−kB T t /2I(Gaussian TCF)

PG/KA/Chap 4 - 17

• Example of Exponential TCF
Langevin equation (Brownian motion model)
mv̇ = −mγv + R(t)(R(t) : random force, γ : friction coefficient)v(0)× and statistical average
hv(0)v̇(t)i = −γhv(0)v(t)i +1hv(0)R(t)im

Define C(t) ≡ hv(0)v(t)i , and assume hv(0)R(t)i = 0(no correlation between v(0) and the random force)dC(t) = −γC(t) ⇒ C(t) = C(0)e−γtdt

PG/KA/Chap 4 - 18

• Example : Brownian oscillator model
Harmonic oscillator + friction + random forcemẍ = −mω02 x − mγ ẋ + R(t)hx(0)ẍ(t)i = −ω02 hx(0)x(t)i − γhx(0)ẋ(t)i + (i)(ii) (iii)1hx(0)R(t)imC̈(t) + γ Ċ(t) + ω02 C(t) = 0γ
⇒ C(t) = C(0)(cos ω1 t +sin ω1 t)e−γt/22ω1(ω12 ≡ ω02 − γ 2 /4 )
ω02 > γ 2 /4

: C(t) = damped oscillation

ω02 = γ 2 /4

: C(t) = C(0)(1 + γt/2)e−γt/2

ω02 < γ 2 /4

: ω1 = imaginary ⇒ C(t) = double exponential[ (i) = “under-damped”, (ii, iii) = “over-damped” ]PG/KA/Chap 4 - 19

Motional narrowing(Back to page 14)

w(f ← i) =12|Ũ(R)|0fih̄2gi (t) =1h̄2Z ∞e−gi (t) ei(h∆V ii /h̄∓ω)t dt−∞Z tdτ (t − τ )hδ∆V (0)δ∆V (τ )ii

0Assume : Exponential TCFhδ∆V (0)δ∆V (τ )i = D 2 e−|t|/τc

•D 2 = hδ∆V (0)2 i ... amplitude of fluctuation• τc ... correlation time

⇒ g(t) =|t|
12 −|t|/τc(Dτ)(e+− 1)c2τch̄



PG/KA/Chap 4 - 20

1. Large τc caseg(t) ' D2 t2 /2h̄2(long correlation time / slow modulation)(short-time expansion in t )Z ∞2 221
⇒ w(f ← i) = 2 |Ũ |2e−D t /2h̄ ei(h∆V ii /h̄∓ω)t dth̄−∞√h̄2 (ω ∓ h∆V ii )22 π|Ũ |2exp[−=](Gaussian line shape)h̄D
2D2
2. Short τc case(short correlation time / fast modulation)e−|t|/τc + |t|/τc − 1 ' |t|/τc⇒ w(f ← i) =12|Ũ|h̄2Z ∞(long time approximation in t )22e−D τc |t|/h̄ ei(h∆V ii /h̄∓ω)t dt−∞|Ũ |22γ
= 2 2h̄ γ + (ω ∓ h∆V ii )2(Lorentzian line shape)

Width ∝ γ ∝ τc(γ ≡ D2 τc /h̄2 )• “motional narrowing” :shorter τc (faster fluctuation modulation) ⇒ narrower spectra

PG/KA/Chap 4 - 21in general :(near spectral peak

∼ large t⇒ Lorentzian shapenear spectral tail∼ small t⇒ Gaussian shape

Overall shape ⇐ parameter Dτc /h̄ <> 0
• Physical interpretation• Large τc = slow modulation (of the nuclear configuration)⇒ ∆V is fixed during the photo absorption/emission⇒ distribution of ∆V is directly reflected in the spectral shape• Small τc = fast modulation⇒ fluctuation of ∆V is averaged out in the observation time scale.
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