MIT Department of Chemistry
5.74, Spring 2004: Introductory Quantum Mechanics II�
Instructor: Prof. Robert Field
17 – 1
5.74 RWF Lecture #17
Normal ↔ Local Modes:
6-Parameter Models
Reading:
Chapter 9.4.12.5, The Spectra and Dynamics of Diatomic Molecules, H. Lefebvre-Brion and
R. Field, 2nd Ed., Academic Press, 2004.
Last time:
ω1, τ, ω2 (measure populations) experiment
ω2
ω
( abcd ) ←
 ( AB) ←1 g
two polyads.
populations in (1234) depend on τ.
E res,k
to devise optimal plucks for more complex situations
E res
(choice of plucks and probes)
* multiple resonances
* more than 2 levels in polyad
could use f k =
Overtone Spectroscopy
nRH single resonance
nRH + 1RH double resonance
dynamics in frequency domain
Today:
Classical Mechanics: 2 1 : 1 coupled local harmonic oscillators
QM: Morse oscillator
2 Anharmonically Coupled Local Morse Oscillators
H eff
Local . Antagonism. Local vs. Normal.
Whenever you have two identical subsystems, energy will flow rapidly between them unless
something special makes them dynamically different:
* anharmonicity
* interaction with surroundings
spontaneous symmetry - breaking
Next time:
H eff
Normal .
5.74 RWF Lecture #17
17 – 2
Two coupled identical harmonic oscillators: Classical Mechanics
(R = Right, L = Left )
H = T ( PR , PL ) + V (QR , QL )
 PR 
1
T = ( PR , PL )G 
2
 PL 
geometry
and masses
[ (
)
]
)
]
1
Grr PR2 + PL2 + 2Grr ′ PR PL
2
 QR 
1
V = (QR QL )F  
2
 QL 
=
force
constants
[ (
1
= Frr QR2 + QL2 + 2Frr ′QR QL
2
1
1

2 1
2  1
2
H =  Grr PR + Frr QR  +  Grr PL + Frr QL2 
2
2
2
 2

( )
( )
H L0
H R0
+Grr ′ PR PL + Frr ′QR QL
kinetic
coupling
potential (anharmonic)
coupling
5.74 RWF Lecture #17
17 – 3
3
φ
1
2
Frr =
k
Frr ′ = kRL
1
1
1
1
m1 + m3 1
+
=
+
=
=
m1 m3 m2 m3
m1m3
µ
1
(projection of velocity of ③ for
cosφ
Grr ′ =
① — ③ stretch onto ③ — ②
m3
direction)
kinetic coupling gets small for large m
or φ = π/2
Grr =
Each harmonic oscillator has a natural frequency, ω0:
1
1
 k

1/ 2
ω 0
=
F
G
=
[
]
2
π
c

µ

2
πc rr rr
1/ 2
and the coupling is via 1 : 1 kinetic energy and potential energy coupling terms.
Uncouple by going to symmetric and anti-symmetric normal modes.
Qs = 2–1/2[QR + QL]
Qa = 2–1/2[QR – QL]
Ps = 2–1/2[PR + PL]
Pa = 2–1/2[PR – PL]
plug this into H and do the algebra

1  1
1


H
=   +
Grr ′
 Ps2
+
(
k
+
k
RL
)
Qs2 
2

2  µ

1  1

1

+
  −
Grr ′
 Pa2 +
(
k
−
k
RL )
Qa2

2


2  µ

no coupling term!
5.74 RWF Lecture #17
17 – 4
1/ 2
1  1


+
G
k
k
+
ωs =

rr ′  (
RL ) 
2πc  µ


1/ 2
1  1


−
k
−
k
G
ωa =

rr ′  (
RL ) 
2πc  µ


simplify to
ωs = ω0 + β + λ
ωa = ω0 + β –λ kRL Grr ′
β =
2
(
)
2 2πc ω 0
(algebra, not power series)
Grr′ can have either
sign. It is usually
negative because
φ > π/2.
ω0 
β
λ=
1 −
 [ kRL k + µGrr ′ ]
2  ω0 
Can have either sign.
Positive if right bond
gets stiffer when left
bond is stretched.
sign of λ determined by whether potential or kinetic coupling is larger (or by the signs of kRL and Grr′).
5.74 RWF Lecture #17
17 – 5
Morse Oscillator
The Morse oscillator has a physically appropriate and mathematically convenient form. It turns out to give
a vastly more convenient representation of an anharmonic vibration than
V ( r) =
1
1
1
f rr x 2 + f rrr x 3 +
f rrrr x 4
2
6
24
treated by perturbation theory.
VMorse ( r) = De [1− e − ar ]
(V (0) = 0, V (∞) = De )
2
r = R − Re
Power series expansion of VMorse ( r) =
If we use
(
)
(
)
(
)
1
1
1
14a 4 De r 4 .
2a 2 De r 2 − 6a 3 De r 3 +
24
6
2
frr = 2a2De
frrr = –6a3De
frrrr = 14a4De
in the framework of nondengenerate perturbation theory, we get much better results than we expect or
deserve.
Why? Because the energy levels of a Morse oscillator have a very simple form:
E Morse (v ) hc = E 0Morse hc + ω m (v + 1 / 2) + x m (v + 1 / 2)
2
and an exact solution for the energy levels gives
E 0Morse = 0
1  2a 2 De 
ωm =


2πc  µ 
xm = −
1/ 2
µ=
m1m2
m1 + m2
a2h
4πcµ
(
)
we get the exact same relationship between (De,a) and E 0Morse ,ω m , x m by perturbation theory (with a twist)
5.74 RWF Lecture #17
17 – 6
1
1 2
P
f rr r 2 +
2
2µ
quartic term treated
1
(1)
4
to 1st order only!
Ev =
f
vr v
24 rrrr
2
1  v − 1r 3 v − v + 1r 3 v
v − 3r 3 v − v + 3r 3 v 
( 2)  f rrr 


+
Ev =
 6  ωm 
1
3

H
( 0)
=
This works better than we could ever have hoped, and therefore we should never look a gift horse in the
mouth. We always use Morse rather than an arbitrary power series representation of V(r). Sometimes we
n
even use a power series ∑ an [1− exp(−ar)] .
n
Armed with this simplification, consider two anharmonically coupled local stretch oscillators. WHY?
What promotes or inhibits energy flow between two identical subsystems?
*
*
*
*
ubiquitous
Local and Normal Mode Pictures are opposite limiting cases
Heff contains antagonistic terms that preserve and destroy limiting behavior
eff
the roles are reversed for H eff
Local and H Normal
See Section 9.4.12.3 of HLB-RWF
Extremely complicated algebra
HLocal defined identically to H Local, but with diagonal anharmonicity.
Convert to dimensionless P, Q, H and then to a,a†.
exploit the convenient V(Q) ↔ E(v) properties of VMorse.
van Vleck transformation to account for the effect of out of polyad coupling terms from
[Grr′PRPL + kRLQRQL] BUT NOT from VMorse.
5. Simplest possible fit model — relationships (constraints) between fit parameters imposed by
the identical Morse oscillator model.
6. Next time — transformation from HLocal to HNormal.
1.
2.
3.
4.
5.74 RWF Lecture #17
17 – 7
( )
()
hR0
1.
H
Local
hR1
1 2 1 2

=  PR +
Q R + V anh (Q R ) 
2µ
 2µ

1 2 1 2

+  PL +
Q L + V anh (Q L ) 
2µ
 2µ

+Grr ′PR PL + kRL Q R Q L
()
1
H RL
1
V anh (Q) = VMorse (Q) − kQ 2
2
This enables us to use Harmonic-Oscillators for basis set but Morse simplification for the separate local
oscillators.
We are going to expand Vanh(Q) and keep only the Q3 and Q4 terms and treat them, respectively, by secondorder and first-order perturbation theory, as we did for the simple Morse oscillator.
( )
( )
()
()
()
1
H Local = hR0 + hL0 + hR1 + hL1 + H RL
( )
H0
5.74 RWF Lecture #17
2,3.
17 – 8
ˆ , Pˆ , H
ˆ → a , a † , a ,a †
Q, P, H → Q
R R L L
ˆ
Qi = α i−1/ 2Q
i
P = hα 1/ 2Pˆ
H
i
Local
i
i
ˆ Local
= h(2πcω M )H
αi =
2πcωi µ i
h
note inconsistency between
ωM in H and ωi in α
1
ki µ i ]1/ 2
ωi =
[
2πc
ˆ = 2 −1/ 2 a + a † etc.
Q
R
R
R
(
)
Pˆ R = 2 −1/ 2 i(a †R − a R ) etc.
H Local = h(2πcω M )[ v R v L v R v L
]{(vR + 1 / 2) + (vL + 1 / 2) + F [(vR + 1 / 2)2 + (vL + 1 / 2)2 ]}
2
D + C
+ v R ± 1,v L m 1 v R v L 
v R + 1 / 2 ± 1 / 2)(v L + 1 / 2 m 1 / 2)] 
(
[

 2
1/ 2 
D − C
+ v R ± 1,v L ± 1 v R v L 
v R + 1 / 2 ± 1 / 2)(v L + 1 / 2 ± 1 / 2)] 
(
[

 2
F=−
2 −1/ 2 ( ha)
4π(µDe )
1/ 2
C = Grr ′µ
k
k
D = RL = RL 2
kM 2De a
dimensionless (a, De from Morse)
dimensionless
First 2 lines of HLocal are polyad, third line is out of polyad.
5.74 RWF Lecture #17
17 – 9
4.


 (D − C)2  F
2
2
eff
ˆ
H Local = v R v L v R v L (v R + v L + 1) 1 −
 + (v R + v L + 1) + (v R − v L ) 

 2
8


1/ 2 
D + C
+ v R ± 1, v L m 1 v R v L 
v r + 1 / 2 ± 1 / 2)(v L + 1 / 2 m 1 / 2)] 
(
[
 2

[
]
F
2
v R − v L ) tries to preserve local mode limit. The D+C
(
2 coupling term tries to destroy the local mode limit.
2
Polyad
P = vR + vL
( )
( )
Overall width of polyad: E( P0 / 2,P / 2) − E(O0 ,P ) = −
F 2
P
2
( F < 0)
(0,P) and (P,0) are at low energy extreme because of anharmonicity: ω(v + 1 / 2) − x (v + 1 / 2) .
2
Off-diagonal matrix elements are smallest between
()
H(10,P )(1,P − 1) =
(0, P ) ~ (1, P − 1)
and ( P, 0) ~ ( P − 11
,)
 D + C  1/ 2
P
 2 
Off diagonal matrix elements are largest between ( P / 2, P / 2) ~ ( P / 2 − 1, P / 2 + 1)
1/ 2
 D + C (
)
(
)
[
]
H( P / 2,P / 2)( P / 2 − 1,P / 2 +1) =
P /2 P /2 +1
 2 
(1)
1/ 2
larger by a factor of [( P / 4 ) + 1 / 2] .
5.74 RWF Lecture #17
5.
17 – 10
General (minimal fit model)
H eff
Local hc = v R v L v R v L {ω R (v R + 1 / 2) + ω L (v L + 1 / 2)
}
+ x R (v R + 1 / 2) + x L (v L + 1 / 2) + x RL (v R + 1 / 2)(v L + 1 / 2)
2
+ v R ± 1,v L m 1 v R v L
2
{(H
hc )[(v R + 1 / 2 ± 1 / 2)(v L + 1 / 2 m 1 / 2)]
1/ 2
RL
But, in the two identical 1 : 1 coupled Morse local oscillator picture
 (D − C)2 
ω R = ω L = ω M 1 −
 = ω′
8


xR = xL = xM
a2h
=−
4πcµ
x RL = 0
D + C 
H RL hc = ω M 
 2 
}