MIT Department of Chemistry�
5.74, Spring 2004: Introductory Quantum Mechanics II
Instructor: Prof. Robert Field
5.74 RWF Lecture #18
Transformation between
Reading:
18 – 1
H eff
Local
and
H eff
Normal
Chapter 9.4.13, The Spectra and Dynamics of Diatomic Molecules, H. Lefebvre-Brion and R.
Field, 2nd Ed., Academic Press, 2004.
Last time:
2 identical coupled subsystems
1.
Classical mechanical treatment of two 1 : 1 coupled local Harmonic Oscillators
Simple transformation decouples the sub-systems.
2.
quantum Mechanical treatment of Morse oscillator
V(r) = De (1 – exp – ar)2
E(v)/hc = ω(v + 1/2) + x(v + 1/2)2
(x is usually negative)
4
Expand V(r), use 1st-order p.t. for r and 2nd-order for r3. Get exact result for energies!
Justifies use of harmonic oscillator basis set even when diagonal anharmonicity appears in
E(0) + E(1)
3.
2 anharmonically coupled local Morse oscillators
3 parameter H eff
Local (relationships or constraints among traditional fit parameters)
Today:
eff
Transformation H eff
Local ↔ H Normal Why? Good description of the pluck (e.g. overtone vs.
SEP).
antagonism between term that lifts degeneracy in polyad vs. term that has off-diagonal intrapolyad
matrix elements
2-level illustration
3 parameter model – some inconsistencies
eff
6 parameter model H eff
Local ↔ H Normal
5.74 RWF Lecture #18
18 – 2
Incompatible Terms in Heff
Often we have a choice between two zero-order basis sets. These correspond to two limiting cases for the
dynamics. In one case, one term causes an on-diagonal energy splitting that preserves the limiting case and
another term causes an off-diagonal matrix element that destroys the limiting case. In the other limiting
case, the roles of the two terms are reversed.
Consider the following illustration for a 2 level system:
H = H° +
A + B
E
=
0
Limit #1: Diagonalize H° + B
0   A 0   0 B
 +  0 − A +  B 0 
 

E 
(i.e. set A = 0)
± = 2 −1/ 2 [ 1 ± 2 ] (diagonalize by inspection - often possible in a limiting case)
1
± (H° + B) ± = (2E ± 2 B) = E ± B
(as required)
2
1
± A ± = ( A − A) = 0
2
1
± A m = ( A + A) = A
2
+E + B
H ≡ T HT = 
− A
1
†
A 

E − B
T is the transformation
that diagonalizes H° + B
B lifts the degeneracy and preserves the #1 limit
A controls off-diagonal matrix elements that try to destroy the #1 limit
Choose instead to diagonalize H° + A. It is already diagonal
E + A
H =
 0
2
0   0 B  E +
A
 +  B 0 = 
  B
E – A 
B 

E – A
5.74 RWF Lecture #18
18 – 3
Here parameter A lifts the degeneracy and tries to preserve limit #2.
Parameter B controls the off-diagonal elements that try to destroy limit #2.
Parameters A and B play antagonistic roles, but their roles are reversed going from limit #1 to limit #2.
This sort of behavior is universal, even in much larger dimension problems than 2 × 2.
Local Mode ↔ Normal Mode
Hund’s Cases (antagonism between HROT, HSO, and Helect.)
Stark Effect vs. Λ-doubling
symmetry breaking
Symmetric double minimum
barrier height
Two-Coupled local Morse Oscillators
4 Physical parameters:
*Morse: a,D
e
* 1 : 1 Coupling:
Grr ′
kinetic
and
kRL
potential
Fit Model:
H eff
Local hc =



( D − C ) 2 
ω F
2
2
v R v L v R v L ω M 1 −
(v R + v L + 1) + M (v R + v L + 1) + (v R − v L ) 


8
2


1/ 2 
 ω ( D + C )
+
v R ±
1v L m 1 v R v L  M
v R + 12 ± 12 v L +
12 m 21



2
[
[(

( D − C )
2

 = ω′
ω M
1 −


8
ω
M F = x < 0
ω M (
D + C )
= H RL hc
2
)(
]
)]
5.74 RWF Lecture #18
18 – 4
3 independent fit parameters (ω′, x, HRL) derived from 4 physical parameters (a, De, Grr′, kRL).
Polyad: P = vR + vL = vs + va
Energy width of polyad:
P2 x/2
(P,0) (degenerate!)
(top of polyad)
x
(4)
small
2
(0,P)
(bottom of polyad)
x
( 4P − 4 )
large
2
energy spacing:
coupling matrix element:
small
(P/2,P/2)
H RL 1/ 2
P
hc
large
H RL  P 2 P 

+ 
hc  4
2
1/ 2
High energy part of H eff
Local polyad goes toward normal mode limit faster than low energy part of polyad.
Anharmonicity creates large level spacings (between coupled levels) near bottom of polyad that resists
transition toward normal mode limit.
Convert to normal mode limit:
* analytic transformation of basis states
* analytic transformation of H
* define T†HT transformation numerically: set case preserving constant to zero and diagonalize.
Apply to H where case preserving parameter is not zero.
First, rewrite the a, a† operators:
(
a s = 2 −1/ 2 (a R + a L )
a †s = 2 −1/ 2 a †R + a †L
(a†a ) = 2−1/2 (a†R − a†L )
a a = 2 −1/2 (a R − a L )
Transform basis states:
00
v sv a
Normal
= 00
)
Local
( ) ( )
−1/ 2 †
=
v
!
v
!
as
[
]
s
a
Normal
vs
(recall that [a†s ,a†a ] = 0)
a †a
va
00
5.74 RWF Lecture #18
18 – 5
eff
Transform H eff
Local to H Normal
:
* replace all vi by a †
i ai .
* replace all a R , a L ,a †R , a †L by a s , a a , a †s , a †a
{
*
exploit [a, a † ] = 1
}
{
}


 3
2 1
2 1  H RL
v s − v a )

H eff
(
Normal hc = v sv a v sv a ω′ (v s +
v a +
1) +
x M  (
v s +
v a + 1) −
(
v s −
v a )
−
 +

4
4
4  hc


x
+
v s ±
2, v a m 2
v sv a  M

2
1 1 
1 3 

1 1
 
1 3


 v s +
2 ± 2   v s + 2 ± 2   v a +
2 m 2   v a +
2 m 2 
.


Notice that degeneracy is lifted by both xM and
H RL
terms, and that the off-diagonal matrix elements are
hc
controlled by xM rather than HRL!
eff
The roles of xM and HRL are (mostly) reversed between H eff
Local and H Normal .
If we compare this to a standard fit model for H eff
Normal
H eff,fit
Normal
hc =
v sv a
2
 
1
1

1


v sv a ω s v s +
+
ω a v a +
+
x ss v s +


2 
2 
2 
 
2

1
1 
1


v a +
+ E 0 / hc 
+ x aa v a +
+
x sa v s +


2 
2 
2 

[
(
 K
1
v s + 12 ±
±
1
+
v s ±
2, v a m 2 v sv a 
ssaa v s +
1
2
2

16hc
we find (algebra) that:
H RL
=
ω
′ +
λ
hc
H
ω
a = ω
′ − RL =
ω
′ −
λ
hc
E ° hc = − x M 4
x ss =
x aa =
x M 2
ω
s =
ω
′ +
x sa = 2
x M
K ssaa x M
=
16hc
2
A total of only 3 independent fit parameters (ω′, λ, xM)!
)
(
)
(
va +
12 m 21 )
(
va +
12 m 32 )
]


3
2
5.74 RWF Lecture #18
18 – 6
But we have a small problem because we have only kept the
part of the
[
]
[
K ssaa
a a a † a † + a †s a †s a aa a
16hc
s s a a
]
K ssaa ˆ 2 ˆ 2 Q s Q a Darling-Dennison coupling term.
4hc
The neglected terms are:
*
1 † 2
1 † 2
K ssaa  † 2 † 2



a s a a + a 2s a a2 + 2 v s +
a a + a a2 + 2 v a +
a s + a s2  which would





2
2
16hc 

have to be corrected for via a Van Vleck transformation
*
and a diagonal term:
(
out-of-polyad:
(
)(
K ssaa
v s + 12 v a + 12
4hc
2xM
)
(
)
)
These out-of-polyad and diagonal corrections spoil the microscopic definitions of ω′, λ, xss, xaa, xsa in terms
of the 3 fit parametes from H eff
Local or the 4 parameters {a, De ,Grr ′ ,kRL } .
So we go to a pair of slightly more flexible and less microscopic 6 parameter models.
But before we look at these 6 parameter models, reconsider the roles of xM and Kssaa/hc.
xM
H eff
LOCAL
lifts degeneracy,
drives toward local
(preserves basis)
HLR
couples L and R
derives toward normal
(destroys basis)
Roles are reversed
Kssaa actually comes (largely) from xM
H eff
NORMAL
lifts degeneracy,

 couples s and a
mostly drives toward local
(destroys basis)
lifts degeneracy: ω′ ± λ
drives toward normal
(preserves basis)
5.74 RWF Lecture #18
18 – 7
6 parameter models (basis for classical mechanics treatment next lecture)
x +
x aa +
x sa
 ω
s +
ω
a
H eff
v s +
v a + 1) +
ss
(
[
(vs +
va )(vs +
va +
1) + 1]
Normal hc =
v sv a v sv a 

2
4
 ω
− ω a x ss − x aa

+
 s
+
v s +
v a +
1) (v s −
v a )
(
two degeneracy2

2

lifting terms
x + x aa −
x sa
2
+
ss
v s −
v a ) 
(

4
1/ 2 
one off-diagonal
 K
ssaa
1
1
1
3
1
1
1
3
+
±
+
±
2,
m
2
+
±
+
m
+
m
v
v
v
v
v
v
v
v


s
a
s
a
s
s
a
a
2
2
2
2
2
2
2
2
term

16hc

[
(
)
(
)
(
)
]
)
(
3 x +
3 x aa +
x sa −
K ssaa 8hc
 ω
s +
ω
a
H eff
v R +
v L +
1) + ss
(
Local hc = v R v L v R v L 
 2
2
3K
16 hc − x ss − x aa + x sa
2
one degeneracy+ ssaa
vR − vL ) 
(

4
lifting term
 ω − ω a x ss − x aa

+ v R ± 1, v L m 1 v R v L  s
+
v R + v L +
1)  v R + 12 ± 12 v L +
12 m
(
2

 2
32 hc + x ss + x aa − x sa
K
two off+ v R ± 2, v L m 2 v R v L  ssaa

4
diagonal terms
[(
[(
× v R + 12 ±
1
2
)(vR + 12 ± 32 )(vL + 12 m 12 )(vL + 12 m 32 )]
1/ 2
)
(
1
2
)
]
1/ 2 


}
It is insightful to know relationships among fit parameters
* recognize unlikely assignments
* simplify dynamics. simplest possible description of a pluck.
* recognize opposing forces toward or away from opposite limits
But it is difficult to map out the local vs. normal character of the individual eigenstates in each polyad and
the evolution toward global normal or local dynamics.
What are we supposed to look for, especially when there is more than one coupling mechanism?
*
*
*
division between mostly-local and mostly-normal eigenstates
appearance of qualitatively new (and unexpected) classes of motion.
chaos. Fraction of phase space that is chaotic.
*
statistical measures
level spacing distribution
relative intensity distribution