11

Adiabatic theorem (Kato 1949)Hamiltonian depends onR=set ofparameters

slow

= →∞0 system in eigenstate, but R=R(t) adiabatically in (0,T), Tt

ψ ψ

ψ

∂=

∂= =

( ) [ ( )] ( )

( 0) [ (0)]

n n

n n

i t H R t ttt a R

ψ≠

= +

=

∑

Formal expansion over instantaneous complete setof eigenstates:

( ) ( ) [ ( )] ( ) [ ( )]

(0) 1

n n n m mm n

n

t c t a R t c t a R t

c

⇒=

= ≠

Adiabatic theorem:Let [ ] have discrete spectrum with no degeneracy

For slow enough changes[ ] [ ] [ ] [ ]

0 for in adiabatic limit.n n n

m

H R

H R a R E R a Rc m n

Basic for perturbation theory, Berry phase etc

1

Adiabatic Theorem (see Griffiths, Meccanica Quantistica)

=Time-dependent part of

'( ) ( )H

H t Vf t

ψ ψΨ = = ≡Initial state ( 0) in nt

ψ

ψ ψ

∞ =

=

+ =

'( )final Hamiltonian stationary states

( )

fm

f f fm m m

H V

H V EAssumptions:T verylong, f(t) gradual, discrete spectrum, no level crossing

Assumptions: for the moment assume V very small, too

ψ ψ ψ≠

= +−∑

Thisenables first-order time-independent perturbation thyfor final stationary states

f kmm m k

k m m k

VE E

f(t)

tT0

1

22

ψ ψ

ψ ψ ψ

Ψ = = ≡

Ψ

Ψ = =

=

∑

Initial state ( 0) .Use time-dependent perturbation theory,too,

to evaluate ( ) in termsof the initial eigenstates.

( ) ( ) ( ) where ( ) (0) .

Recalling time-dependent part of : '( ) ( )

a

l

in n

E ti

l l l ll

t

T

t c t t t e

H H t Vf t−

→ ≠

−→ = −

−= ∫

∫

( ) '

ln 0

n 0

mplitude of n ,

amplitude of remaining in initial state

n n ( ) 1 ' ( ')

( ) ' ( ')l n

t

E E tt

n

i

l

n

l l n

ic t

ic t V

V dt f

dt e

t

f t

3

−

− −

− −=

−

−=

−

∫

( )

( ) '

ln

' ( ) '

0

Insert

( )

id

' ( ')'

entity'

l n

l n l nE E t E E ti i

lE E t

t i

ln

n

l

i de eE E dt

i i dc t V dt f t eE E dt

Trick

− − −= −

− ∫

( ) ( ) 'ln

0by parts ( ) ( ) ' ( ')

'

l n l nE E t E E ti t i

ll n

V dc t f t e dt e f tE E dt

( ') negligible in the adiabatic ca .e'

sd f t isdt

−−= ≠

−−

= − ∫

( )ln

n 0

( ) ( ) ,

( ) 1 ' ( ').

l nE E ti

ll n

t

n n

Vc t f t e l n

E Eic t V dt f t

4

ψ

ψ ψ ψ ψ≠ ≠

≠

Ψ =

−− − + + +

− −∑ ∑∫

lnn 0

Amplitude to jump to state , approximated here in first order,

( )

[(1 ' ( ')) ]n

fm

E Tt i km

n n l m kl n k ml n m k

m n

T

VVi V dt f t eE E E E

ψ

−

→

−

−

transition amplitude in 1st order:

take 1 ( ) and (right) :

This contributes .n

kmk

m kE T

i nm

m n

n mV

leftE E

Ve

E E

ψ ψ ψ−

≠

−

−− −

−−

∑

lnNow, take (left) and (right) : :

This contributes

n n

n

E T E Ti i nm

l m nl n l n m n

E Ti nm

m n

VVe e

E E E EV

eE E

net result: 0

No transitions in first order. This is the Kato theorem.

Now we remove restriction to small V 5

f(t)

tT0

1

Divide (0,T) in N time slices

In every slice,VVN

∆ ≈

Since first-order is 0, total goes like

2

2 0VN for NN

→ →∞

=≠

⇒

=

Adiabatic theorem:no degeneracy

for sl[ ] discrete spectrum with

[ ] [ ] [ ] [ ]0 for in adiabatic li

ow enough chang

m

es

it.n n n

m

H R

H R a R E R a Rc m n

The adiabatic theorem grants that the system remains in state n, but does notspecify what happens to the phase.

6

Density operator

ˆOne introduces the density operator

when the system in a mixed state has probability p of being in state .ˆDensity matrix is the matrix of in any basis.

nn

n

p n n

n

ρ

ρ

=∑

exp( )In equilibrium, p on the basis of H eigenvectors.nn

EZβ−

=

ˆLet T=0, 0 0 (system in instantaneous ground state at time 0).ˆIf evolution is adiabatic 0 0 all the time

(system in instantaneous ground state at time t).

is

is

ρ

ρ

=

=

Now assume that the system evolves slowly, but at a finite speed.One must envisage an adiabatic perturbation theory to evaluate the corrections to the exactly adiabatic limit. Corrections are particularlyimportant to evaluate matrix elements that vanish in the ground state.

88

ρ

ρ

ρ ρ ρ

ρ

⇒

=

= + ∆∆

⇒ ∆=

=Slow t dependencewith small b

ˆ 0, 0, instantaneous operator:

y adiabatic theor

sysyem in instantaneous grou

em on instantaneous

n

eigenstate basis is twice smalln

d

e

s

gl

tate

i b

0

g

.

i

i

mn

i t t densityat T

( )ρ ρ ρ>

≠ ≠

∆ ∆ + ∆∑ 00

First-order correctile if 0,

on 0.

,0

, , , 0on nn

tm

n t n tn

t

Quasi-adiabatic density matrix (Niu-Thouless)

ρ∆ =−

00

0We shall find off-diagonal density matrix

which is almost self-evident on dimensional grounds.

nn

ni

E E

8

99

ρ ρ ρ ρ ρ ρ

ρ ρ ρ ρ

ρ ρ

ρ ρ

ρ

ρ ρ

= + ∆ ⇒ = + ∆

+ ∆ = + ∆

∆ ⇒ ∆

⇒ ∆ = +

=

− ∆

[ , ] Slow t dependence , with

( ) [ , ]

small by adiabatic theorem

0, 0, instantaneous density matrix

negligible

get

at T=0

from ( ) [ , ] .

i i

i i

i i

i

di Hdt

di Hdt

did

di Hdt

t

t

t

ρ− = ∆

et's take matrix element 0

0 ( 0, 0, ) 0 [ , ]

L ndi t t n H ndt

ρ

ρ

−= ⇒ = − =

⇒ − = ∆

0( ) 0, ( ) 0, [ , ( )] 0, 0, 0, 0, 0

( 0, 0, ) [ , ]

iH t t E t t H t t t H H t tdi t t Hdt

9

1010

ρ ρ

ρ⇒ ∆ = −

⇒ = ∆ = − ∆

−⇒

0

0

0

0

0 0 [ , ] ( )( 0, 0, )

( 0, 0, )0 ( 0, 0,

0

()

)

n n

nn

d t tdt

d

i n H n E

n

E E

t t ddt t t

E

n ndt

i eeded

0, , 0,

acts on bra and ket:

( 0, 0, ) 0,0 0

b

0,( ) ( ), 0, 0, 0,

0ut , 0, 1

t

ddt

d d dt t t t t tdt dt dt

t tt

n n n

t n

tt=

=

−

= ⇒

10

0 ( ) , but since

one can simplify:

( 0, 0, ) 0,

( 0, 0, ) 0,

0, , 0,

) 0, ,0 ( ( )

d dt t tdt dt

d

n n t

n ndt t t t n td

n t

tt

t

dt

= −

= =

=

ρ⇒ ∆ = −−

00

0n

n

ni

E E

ρ ρ∆ = ∆−

≈ = −∑ ∑

0 0 00

ˆThe correction to the expectation value od any operayor A

)0

(n n nn n n

A Tr A t An

iE E

A1111

12

Quantum phasesGauge Transformations

Without the gauge invariance, any theory is untenable. In classical theory,the Hamiltonian of a charged particle is

2( )( )

2

ep AcH eV xm

−= +

where p is the kinetic momentum and A the vector potential. Both are unobservable.

1' ( , ) 'A A x t V Vc t

χχ ∂= +∇ = −

∂

2( [ ]) 1 '{ [ ( ) ]} '2

ep Ac e V x i

m c t t

χ χ− +∇ ∂ ∂Ψ+ − Ψ =

∂ ∂

New Schroedinger equation:

;One could have started with new potentials giving the same fields:

12

13

χ χ

χ

− +∇ ∂ ∂Ψ+ − Ψ =

∂ ∂

2( [ ]) 1 '{ [ ( ) ]} '2

is solved by( , )Ψ'(x,t)=Ψ(x, )exp[ ]; no change in the physics.

ep Ac e V x i

m c t t

ie x ttc

Galileo Transformations

∂Ψ+ Ψ =

∂= − = = =

2 ( , ){ ( )} ( , ) ; in a moving frame2' , ' , ' with scalar : '( ', ) ( , )

p x teV x x t im t

x x vt y y z z V V r t V r t( , , , )

2

'( ', ', ', ) ( , , , )

( , , , )2

i x y z tx y z t x y z t emvx mv tx y z t

φ

φ

−Ψ = Ψ

= −

One checks that a plane wave transforms according to Galileo.13

14

Superconductor Thin insulator Superconductor

R

emf

Macroscopic quantum phenomena: Josephson effect

2* * 2

*

Ginzburg-Landau: order parameter with charge

[ ] | | 02

=2e

S

ee eJ A A

mi m c

ψ

ψ ψ ψ ψ ψ

=

⇒ = ∇ − ∇ − =

11 1| | ie θψ ψ 2

2 2| | ie θψ ψ

( )

( )1 2

1 2* *

1 2 1 2 2 1

Matching ( ) in barrier ( ) , barrier width and approximately constant

sin . A phase difference across the barrier implies a current. In elementary quantum proble

z z b

S

z z e e b

J

β βψ ψ ψ ψψ ψ

ψ ψ ψ ψ θ θ

− −= + =

⇒ − −

ms there is no bias, ( ) can be taken real and there is no current.In ordinary circumstances, there is dissipation and no phase coherence overmachroscopic distances. But superconductivity is a macr

zψ

oscopic quantumphenomenon.

14

AC response to DC bias !

0 0 02sin ( ) , constanteVI I t t I = − =

* *1 2

1

2

2

1

In Josephson's problem there is a bias which creates a energy difference and raises the relative phase analogous to to the e factor of elementa

si

ry QM, w

nith E

iE

S

t

eVtJ ψ ψ ψ θ

θ θ

ψ

−

→ − =

⇒ −

( )2

0 0

1

2sin ( ) ,eVI I t t

θ

= − ⇒

−

Adiabatic motion in classical Physics

From Griffiths Introduction to Quantum Mechanics

Key concept: the pendulum will keep swinging regularly parallel to the same planeif external condition change little during an oscillation. In Quantum Mechanics the Born-Oppenheimer approximation is similar

16

Topologic phases

Start from north pole towards Rome bringing a pendulum oscillating parallelto the Rome meridian, reach the equator, then turn left along the equatorwithout changing the plane of oscillator, reach the Parallel of Moscou and follow it bach to the North pole; now the oscillator swings towards Moscou.

17

18

18

Topologic phasesPancharatnam phase

The Indian physicist S. Pancharatnam in 1956 introduced the concept of a geometrical phase.

Let H(ξ ) be an Hamiltonian which depends from some parameters, represented by ξ ; let |ψ(ξ )> be the ground state.

Compute the phase difference Δϕij between |ψ(ξ i)> and |ψ(ξj)>

However, this is gauge dependent and cannot have any physical meaning.Now consider 3 points ξ and compute the total phase γ in a closed circuitξ1 → ξ2 → ξ3 → ξ1; remarkably,γ = Δϕ12 + Δϕ23 + Δϕ31is gauge independent!

Indeed, the phase of any ψ can be changed at will by a gauge transformation, but such arbitrary changes cancel out in computing γ. This clearly holds for any closed circuit with any number of ξ. Therefore γ is entitled to have physical meaning.

There may be observables that are not given by Hermitean operators.

( ) ( ) ( ) ( )φψ ξ ψ ξ ψ ξ ψ ξ∆= ijii j i je

18

19

In the case of H2 this can be gauged away, but with three or more atoms the physical meaning is that a magnetic flux φ is concatenated with the molecule; changing φ by a fluxon has no physicalmeaning, however.

π φφ

−→ = ≈ =∫

b 7 2h h 0a

0

Peierls prescription: to introduce A2 it t exp[ A.dr], 4 10 Gauss cm fluxonhc

e

For instance, consider a Linear Combination of Atomic Orbitals (LCAO) model fora molecule or cluster (or a Hubbard Model, neglecting overlaps)

By complex hoppings, one

can introduce a concatenated magnetic flux

1919

20

3-site cluster with flux

( )gs gsE E φ⇒ =

23 13 12

0

1, ,

2

ieec

γτ τ τφ φγ πφ

= = =

= =1

2

3

Ground state Energy Egs(φ) has period=2 π

In[12]:= h: 0 EI 1EI 0 11 1 0

;

ListPlotTable,MinEigenvaluesh,,0,6,.1,PlotJoined True,AxesLabel,E,Ticks0,Pi,2Pi,3Pi,4Pi,5Pi,0,1

Out[12]=

2 3 4 5

Egs

φ =0hce

20

Adiabatic theorem and Berry phase

( ) ( )

0

( ) , and along with

( ) ' [ ( ')] expected dynamical phase,

n ni t i tn

t

n n

c t e

t dt E R t

θ γ

θ

+=

= − =∫

We start from the expansion on instantaneous eigenvector basis

ψ≠

=

= +

=

∑

[ ( )] depends on a set of parameters which vary adiabatically

( ) ( ) [ ( )] [ ( )]

(0) 1

n n n m mm n

n

H H R t

t c t a R t c a R t

c

and from the adiabatic theorem ; we prove that

There is also the topological Berry phase γ. Moreover, γ can be important!

21

Sir Michael Berry

21

Since (R[ ])n R ndRa t a

t dt∂

= •∇∂

n ni Htψ ψ∂

=∂

We impose

22

Sir Michael Berry

( ) ( )

( ) ( )

using the ansatz ( ) ,

( ) [ ( )] ( ) [ ( )]

n n

n n

i t i tn

i t i tn n m m

m n

c t e

t a R t e c t a R t

θ γ

θ γψ

+

+

≠

=

= +∑

0( ) ' [ ( ')]( ) ( )[ ( )]

tn n

n n

ii t dt E R ti t i tn n n R n

ia R t e E i R a et

γθ γ γ−+∂ − ∫= + + ∇ ∂

( )0( ) ' [ ( ')]

and the lhs reads

( )t

n nii t dt E R t

n n R n m m m mm n

ni E i R a e c a c at

tγ

γψ−

≠

− ∫= + + ∇ + +

∂∂

∑

22

2323

are twice small (derivative in adiabatic correction). So,l.h.s. is:

m mc a

0( ) ' [ ( ')]

( )t

n nii t dt E R t

nn n R n m mm n

i E i R a e c att

γγψ

−

≠

− ∫= + + ∇ + ∂

∂

∑

( )0( ) ' [ ( ')]

and the lhs reads

( )t

n nii t dt E R t

n n R n m mm n

m mn c ai E i R at

a e ctγ

γψ−

≠

− ∫= + + ∇ + +

∂∂

∑

23

2424

[ ( )] ( ) ( ) [ ((The

) r.h.s. r ads

)]e :

n n n m m mm n

n E a R t c t c t E a RH t tψ≠

= +∑

0'[ [ ( ')] ( )]

0 ( ) [ c ]t

n ni dt E R t i t

n R n m m m mm n

i R a e c E aγ

γ−

+

≠

∫= − + ∇ + −∑

0( ) ' [ ( ')]

Equating to ( )t

n nii t dt E R t

n n R nm

n m mn

i E i R a e ctt

aγ

ψ γ−

≠

− ∫= + + ∇ +∂

∂

∑

( ) ( )

( ) ( )[ ( )] ( ) [ ( )]

using the ansa z ( )

)

,

(

t n n

n nn

i t i tn

i t i tn mn m m

m nE a R t c t E a R tH

t e

t

c

e

θ γ

θ γψ≠

+

+

=

= +∑

24

2525

[ ( )] [ ( )]n n R n

Berry

iR a R t a R tγ = ∇

Now, scalar multiplication by anremoves the summation and all

other states!

0'[ [ ( ')] ( )]

[ ( )] [ ( )0 ( ) ] ][t

n ni dt E R t i t

n R n m m mm n

i R a e c E aR t R tγ

γ−

+

≠

∫= − + ∇ + −∑

25

2626

γ = ∇∫

( ) is a topological phase

and vanishes in simply connected parameter spacesbut in multiply connected spaces yields a quantum number!

n n R nC

C i a a dR

0 0

[ ( )] [ ( )]The matrix element looks similar to a momentum average, but the gradient is in parameter space. The overall phase change is a line integral

. .

n n R n

T T

n n R n n R n

iR a R t a R t

i dt a a R i a a

γ

γ

= ∇

∆ = ∇ = ∇∫ ∫

This has no meaning, it's a gauge, but...

dR

26

2727

Nuclear wave FunctionsRecall: Total H

( ) (( , ( , )) )tot NeT rH r R RT VR r= + + set of electron coordinatesset of nuclear coordinates

rR==

( )

( ) ( ) ( )

2 2

2 , ( , )

adiabat

Born-Oppenheim

ic hamiltonian f

er (BO) : negle

or electrons. Solve , , ( )

(

,

ct )2 e e

e n n

N

n

H r R T V r R

H r R r

T RR

EM

R R r Rψ ψ

∂= = +

=

−∂

that ( , ) ( ) ( , )is, tot N eH r R T R H r R= +

2 2

0 2Then, use ( ) as a potential and2NE R T

M R∂

= −∂

( ) ( ) ( )2 2

02: one expects nuclear motion2

BO R E R W RM R

χ χ χ∂− + = ⇒

∂

0

0 0

( , ) ( ) ( , )( , ) electron

an calculated with nuclei at R.

satz : trial r R R r Rr R

χ ψψ ψ

Ψ ==

The Adiabatic Approximation is a cheap way to go somewhat beyond.

Adiabatic= assuming evolution confined to lowest (n=0) energy surface computed solving for electrons at fixed R, seek χ(R) variationally

Nuclear wave Functions Beyond Born-Oppenheimer

0ansatz: ( , ) ( ) ( , )trial r R R r Rχ ψΨ =

set of electron coordinatesset of nuclear coordinates

( ) nuclear wavefunction

rR

Rχ

==

=

[ ]χ = = +tot N eF H T H

Variational approach to χ based on Energy functional

( ) ( ) ( )2 2

02Does validate the BO:2

F R E R W RM R

χ χ χ∂− + =

∂

0 0 0( ) ( , ) ( , ) ? This would be the BOeE R r R H r Rwhere ψ ψ=

2

02

2* * * *

0 0Minimize ( ) ( ) ( ) ( ) ( )2

drdR dR E R R R W dR R RM R

χ χ χψχ ψ χ χ− +∂

−∂

∫ ∫ ∫

W=Lagrange multiplier (normalization)

NTeH

2929

we need the derivative:

( )2

0

2

0

2

0 2 02 2 2 . Minimize:R R RR R

χ ψ χχ ψψ ψ χ ∂∂∂ ∂ ∂∂∂ ∂ ∂

= ++∂

* *0

2 2*

2

2* * * *

0 0 0 0 02

( ) ( ) ( ) ( ) ( )

]2[2

drdR dR d

dR R E R R W d

dRM R

R

rR R R

R R

χ ψ ψ χ χ χ

χ

ψ

χ

ψχ

χ

χ

χ

∂

+

∂ ∂ ∂−

−

+∂ ∂∂

+∂

∫ ∫∫ ∫ ∫ ∫

2

0 02

terms involving

andR R

ψ ψ∂ ∂∂ ∂

*Vary and find best χ χ

22 2* *0 00 0

2

2

2

2

0 ( ) ( ) ( ) )

( )

( ()2

2R dr dr

M

RE R R R W RM

R R M R

Rψ ψ

χ χ χ χ

∂ ∂∂Λ = − Ψ − Ψ

Λ

∂ ∂ ∂

∂− + + =

∂

∫ ∫

The electron wave function depends parametrically on nuclear coordinates.

3030

200 0

nonadiabatic ( ) often ignored since for real functions

1 02

however this is not always the case! Not even with 0.(see below Berry phase can arise).

R

dr drR R

Ba

ψψ ψ

Λ

∂ ∂= = ∂ ∂

=−

∫ ∫

22* 00 2

Moreover the excuse for neglecting the other term is:

( )*electron kinetic energy.2

Not reassuring. Typically vibrations are 0.1 eV or less, electronic jumps require 1 eV or more.

mdr OM R M

ψψ ∂

− = ∂ ∫

But close electronic levels can be mixed by vibrations!

22 2

2 2

02

* *0 00 0 2( )

2unespected, gauge

( ) (

-dependent nonadiabatic ter

) ( )( ) ( )2

m

R dr drM R

E R R R W RM

R M

R

R

Rχ χ χ χ

ψ ψψ ψ ∂ ∂∂

Λ = − − ∂ ∂ ∂

∂− + + =

∂Λ

∫ ∫

0( , ) ( ) ( , )trial r R R r Rχ ψΨ =

Jahn-Teller theorem

any nonlinear molecule with a spatially degenerate electronic ground state will undergo a geometrical distortion that removes that degeneracy.

True within the Born-Oppenheimer approximation.

Born-Oppenheimer Approximation:

Neglect of nuclear momenta

The energy surfaces have NJT several equivalent

minima corrisponding to different distortions; e.g. a cube can be squeezed in several equivalent ways.

3232

At strong vibronic coupling , the energysurfaces have NJT deep and distant

minima and the nuclear degrees of freedomcan hardly tunnel between them. Then, the

kinetic energy of the nuclei does not play a role, and one can observe a static JT effect with

broken symmetry.

At weak couplingthe system oscillatesbetween several

Minima and , one

speaks about dynamic JT effect ;

the overall symmetry remains unbroken.

ψ

ψ

The time scale of the experiment is the criterion for weak and strong.

In fast experiments with hard X rays the symmetry is broken

3333

( )2 2

2

Full problem : ( , ) ( ) ( , ) ( ) ( , )

( , ) ( , )( ) ,2

( , )

tot N e N e

tot toN t totT RM

H r R T R H r R T R T r V r R

H r R r R W r RR

= + = + +

Ψ = Ψ∂

= −∂

In this case we do not attempt a variational approach but wewish to allow the mixing of the different minima.

Several interacting Jahn-Teller minima beyond BO

0 0

0

0

We simplify by computing electronic ( , ) with fixed corresponding to all the minima, with symmetrically distorted

geometries. Different minima yield orthogonal ( , ), otherwisewe ort

en n n

ne

n n

r R RR

r R

Ψ

Ψhogonalize.

The nuclear motion is the unknown.

3434

0 0

0

0

We compute electronic ( , ) with fixed , k=1,...ncorresponding to all the minima, with symmetrically distorted

geometries. Different minima yield orthogonal ( , ), otherwisewe orthogo

ek k k

ke

k k

r R RR

r R

Ψ

Ψ

0

nalize. The nuclear motion is the unknown. New Ansatz:

( , ) ( , ) ( , ).n

etot k k k

kr R R R r RχΨ ≈ Ψ∑

unknown

known

0

0 0

( , ) ( ) ( , )(

We must gene, ) electron calculated with nuclei at R

ralize the ansatz :.

trial r R R r Rr R

χ ψψ ψ

Ψ ==

Nuclear wave Functions Beyond Born-OppenheimerThe Adiabatic Approximation is a cheap way to go somewhat beyond.

Adiabatic= assuming evolution confined to lowest energy surface computed solving for electrons at fixed R

0 0 ( , ) are known, ( , ) unknowns. ek k k kr R R RχΨ =

( )( , ) ( ) ( , ) ( ) ( , )tot N e N eH r R T R H r R T R T r V r R= + = + +

0

0

*0 0

Substitute ( , ) ( , ) ( , )

into ( , ) ( , ) ( , )

take the electronic scalar product with ( , ) and get

( , ) [ ( , ) ] ( ) ( , ).

ne

tot k k kk

tot tot totem

ne em tot k k n

k

r R R R r R

H r R r R W r R

r R

dr r R H r R W R r R

χ

χ

Ψ ≈ Ψ

Ψ = Ψ

Ψ

Ψ − Ψ

∑

∑∫

0

0

We assume orthogonal ( , ) for different minima but allow for their mixing by the electron-nuclei interaction which induces a mixing of ( , ).

en n

n n

r R

R Rχ

Ψ

*0 0( , ) [ ( , ) ] ( ) ( , ) means

ne em tot k k n

kdr r R H r R W R r RχΨ − Ψ∑∫

( )( )*0 0( , ) ( ) ( ) ( , )( , ) 0e e

m N ke k kk

T r V rdr r R T R R W R r Rχ Ψ+Ψ + − =∑∫

2 2*

0 02

Use shorthand notation :

( , ) ( ) ( ,

( , )

( , )) 02

e em m k ke k

e

k

H r R

H r Rdr r R W R r RM R

χ ∂

Ψ − + − Ψ = ∂ ∑∫

Assuming orthogonality of electronic wave functions get the simplification of first term :

2 2 2 2*

0 02 2( , ) ( ) ( , ) ( )2 2

e em m k k n m

kdr r R R r R R

M R M Rχ χ

∂ ∂Ψ − Ψ = − ∂ ∂

∑∫

37

*0 0Set: ( ) ( , ) ( , ) ( , )e e

mk m e kV R dr r R H r R r R= Ψ Ψ∫

22

2

( ) ( ) ( ) 02

mmk mk k

k

R V W RM R

χ δ χ∂⇒ − + − =

∂ ∑

( )2 2

*0 02 ( ) ( , ) ( , ( ) ( , ) 0)

2 m m m k k kk

eH r RR dr r R W R r RM R

χ χ∂− + Ψ − Ψ =

∂ ∑∫

2 2*

0 02( , ) ( ) ( , ) 0 be( , comes2

)e em m k k k

kedr r R W R r RH R

M Rr χ

∂Ψ − + − Ψ = ∂

∑∫

3838

11 1

2 2*

2

1

. . .. . . . .. . . . . , ( ) ( ) ( , ) ( ).

2. . . . .

. . .

n

e eJT mk m e k

n nn

V V

H V R dr r H r R rM R

V V

∂ = − + = Ψ Ψ

∂

∫

Finally, the Nuclear wavefunctions are obtained from matrixSchroedinger equation

22

2

( ) ( ) ( ) 02

mmk mk k

k

R V W RM R

χ δ χ∂⇒ − + − =

∂ ∑

2

2 must be written i terms of normal coordinates, thus excluding translations and rotations. For n=2,R∂∂

2 2 2 2

2 22V(x) V(x) V(y) V(y)

V(x) V(x) V(y)2 2 V(y)xx xy xx xy

x yyx yy yy x yx y

JT q q KM q

HM q

q∂ ∂− − +

∂

+ +

=∂

2x2 matrices represent operators acting on

nuclear wave functions

( , )x yχ χ

x

y

1

23

( )

( )

3

x y

x y

example of problem: Na molecule

, known electronic wave functions E (Jahn-Teller: unstable, =2)

Electronic degeneracy challenged by degenerate vibrational mode,

, normal modes E producing

E

n

q q

ε

ψ ψ

×

∈

∈ potentials (V(x),V(y)) acting on electrons

and driving the nuclei to a two-state situation, reminiscent of a spin.

39

vibronic couplingE ε⊗

40

Group theory predicts the form of the vibronic interaction, but here I skip the argument leading to:

2 2 2 2

2 22V(x) V(x) V(y) V(y)

V(x) V(x) V(y)2 2 V(y)xx xy xx xy

x yyx yy yy x yx y

JT q q KM q

HM q

q∂ ∂− − +

∂

+ +

=∂

2x2 matrices represent operators acting on

nuclear wave functions

( , )x yχ χ

2 2 2 22

2 2

0 00 02 2 x y

yJ

xTH q q Kq

M q M qλ λ

λ λ ∂ ∂

− − + + + −∂ ∂ =

2 2 2 22 2

2 2that is, [ ] ( )2 2JT x x y y x y

x y

H q q K q qM q M q

λ σ σ∂ ∂= − − + + + +

∂ ∂

4141

2 2 2 22

2 2

0 00 02 2 x y

yJ

xTH q q Kq

M q M qλ λ

λ λ ∂ ∂

− − + + + −∂ ∂ =

2 2 2 22 2

2 2that is, [ ] ( )2 2JT x x y y x y

x y

H q q K q qM q M q

λ σ σ∂ ∂= − − + + + +

∂ ∂

Incidentally: Second-quantized version:

† † † †1 1( ) ( ) '[( ) ( ) ]2 2JT x x y y x x x y y yH a a a a a a a aω ω λ σ σ= + + + + + + +

(two levels and two-bosons problem) and is among those exactly solved by the recursion method of Excitation Amplitudes. Here we shall consider the static limit.

x

y

1

23

4242

2 2

cos , q sinBorn-Oppenheimer limit M :

,

0 1 1 0 0 cos sin 01 0 0 1 c

molecule with static distortion,setti

os 0 s

g

0

n

inJ xT

x y

y

q q q

q q KqH q q Kq

θ θ

θ θλ λ λ λ

θ θ

→∞

=

= =

+ + = + + − −

2 2 2 22 2

2 2 [ ] ( )2 2JT x x y y x y

x y

H q q K q qM q M q

λ σ σ∂ ∂= − − + + + +

∂ ∂

x

y

1

23

<K> q2 is an additive constant, with no dynamics, and formally HJT is the Hamiltonian for a spin in a magnetic field B = (qx, 0, qy).

4343

2 2:sin coscos si

sin cosM

cos

M

s n

n

i

JTH q Kq q Kqλ

θ θθ

θ θλ

θ θ

θ

= + −

= +

= −

2

Born-Oppenheimer limit M0 cos sin 0

cos 0 0 sinJTH q q Kqθ θ

λ λθ θ

→∞

= + + −

M has eigenvalues ±1, independent of θ

x

y

1

23

x y

2x2 matrices represent operators acting on nuclear wave functions ( , ),

represents frozen (M ) nuclear displacements.χ χ

θ →∞

2

2

independent of

independ

1 E =

1 E

= ent o - f JT

JT

M

M

q Kq

q Kqλ θ

λ θ+

+

= +

= −

Here the potential energy surfaces E(q) are obtained by rotating two intersecting parabolas around the energy axis.

44

Minimum away from q=0. Energy eigenvalues independent of angle, butnuclear wave function depends on angle ia a special way.

45

Wizard heat singularity Multiply connected parameter space Berry phase

Modes of E symmetry of tetrahedral molecules

46

47

( , ) ( , )( , )

( , ) ( , )x y x y

JT x yx y x y

a q q a q qH E q q

b q q b q q

=

Eigenvalue -1 Eigenvalue +1

cos( )2 4( )

sin( )2 4

θ π

χ θθ π−

+ = − +

sin( )2 4( )

cos( )2 4

θ π

χ θθ π+

+ = +

This changes sign under θ θ+ 2π , which isnormal for spin rotation, but wrong here.

Nuclear wave functions must be single-valuedfunctions of normal coordinates!

2sin coscos sinJTH q Kq

θ θλ

θ θ

= + −

4848

Eigenvalue -1Eigenvalue +1

Berry phase!

We did an honest calculation. Why do we get a wrong result?

We can fix it, but must insert phase factors:

cos( )2 4( )

sin( )2 4

θ π

χ θθ π−

+ = − +

sin( )2 4( )

cos( )2 4

θ π

χ θθ π+

+ = +

2

cos( )2 4( )

sin( )2 4

i

eθ

θ π

χ θθ π−

+ =

− +

2

sin( )2 4( )

cos( )2 4

i

eθ

θ π

χ θθ π

−

+

+ =

+

This compensates for the changed sign under θ θ+ 2π .

Nuclear wave functions are single-valued but complex.