1

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 somewhat similar.

222

Adiabatic theorem (Kato 1949)

Hamiltonian depends on

R=set of

parameters

slow

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

( ) [ ( )] ( )

( 0) [ (0)]

n n

n n

i t H R t ttt a R

Formal expansion over instantaneous complete set

of eigenstates of [ ( )] treated as static:

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

(0) 1

n n n m mm n

n

H R t

t c t a R t c t a R t

c

Basic for perturbation theory, Berry phase etc

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 R

c m n

4

Adiabatic Theorem (see Griffiths, Meccanica Quantistica)

Time-dependent part of

'( ) ( )

H

H t Vf t

Initial state ( 0) i

n nt

'( )

final Hamiltonian stationary states

( )

f

m

f f f

m m m

H V

H V E Assumptions:T very long,f(t) gradual,

discrete spectrum, no level crossing

Temporary assumption: for the moment

assume V very small, too

Thisenables first-order time-independent

perturbation theory for stationary states

final

f kmm m k

k m m k

V

E E

f(t)

tT0

1

4

5

For t=0 system in level n: initial state ( 0) .

Use perturbation theory,too,time-

to evaluate ( ) in termsof the initial eigenstates.

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

Recalli

depen

ng

dent

tim

m

i

n n

E ti

m m m lm

t

T

t c t t t e

( ) '

n 0

0

e-dependent part of : '( ) ( )

mplitude of n , .

mplitude of remaining in initial state:

n n ( )

( ) ' ( ')

1 ' ( ')

m nE E tt i

m

t

nn

m

n

H H t Vf t

A m l n

A

ic t V dt

ic

f

t V dt f

t

t e

6

( ) '

( ) '

n 0

( ) '

ln 0

( ) '

mplitude of n , .

Insert identit

( ) ' ( '

y :

)

( ) ' ( '

'

)'

m

n

m

m

n

n

n

mE E t E E ti

E E tt i

m m

E E tt i

m

m n

i

m n

ic t V dt f t e

i i dc t V dt f t e

E

A m l n

i de

d

E E

E

edt

t

Trick to exploit the smallness of

( ) ( ) '

ln

0by parts ( ) ( ) ' ( ')

'

l n l nE E t E E ti t i

l

l n

V dc t f t e dt e f t

E E dt

( )

n

0

In adiabatic case, ( ) ( ) , for m ,

while ( ) 1 ' ( ').

m nE E ti

mm

m n

t

n nn

Vc t f t e n

E E

ic t V dt f t

( )df t

dt

7

ln

n 0

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

( ) by time-depenent p.t. by time-indepenent p.

(1 ' ( '

t.

))[ ]n

f

m

Et

Ti

kmn l m k

l n k ml m k

n

n

iV dt

m n

T

VVe

E E E Ef t

Now obtain the transition amplitude in 1st order:

take 1 ( ) and (right) :

This contributes .n

kmk

m k

E Ti

nm

m n

n m

Vleft

E E

Ve

E E

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

This contributes .

n n

n

E T E Ti i

nml m n

l n l n m n

E Ti

nm

m n

VVe e

E E E E

Ve

E Enet result: 0

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

Now we remove restriction to small V

8

f(t)

tT0

1

Divide (0,T) in N time

slices

In every slice,

VV

N

Since first-order is 0,

total goes like

2

20

VN 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 R

c m n

The adiabatic theorem grants that the system remains in state n, but does not

specify what happens to the phase.

Density operator

ˆOne introduces the density operator when the system

in a mixed state has probability p of being in state .

ˆ ˆ ( )

Density matrix is the matrix of

n

n

n

i i i i i

i

p n n

n

A p A A Tr A

ˆ in any basis.

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

n

E

Z

If a system is known to be in quantum state the expectation value

of any operator A is .

If a system is in an and is known to be in quantum sti ate

with probabili

mpure or mixed st

p

te

t

a

y

i

i i

i

i

A

the expectation value of any operator A is

.i i i

i

A p A

ˆLet T=0 (absolute 0) and 0 0

(system in instantaneous ground state at time 0).

ˆIf evolution is adiabatic 0 0 all the time ( theorem).

is

Kato

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 particularly

important to evaluate matrix elements that vanish in the ground state.

Adiabatic perturbation theory

Quasi-adiabatic density matrix (Niu-Thouless)

Slow t dependenc

ˆ 0,

e

with small by adiabatic theor

0, instantaneous density operat

e

or:

mi

it t

111111

00

Slow t dependence

with small by adiabatic theorem:

with m and n excited on instantaneous

e

First-order

igenstate basis is twice small negligible i

cor

f 0,

0, , , 0rectio

0

n

.

on

i

m

nn

n

m

t n t n t

n

,t

0

0

0 changes slowly.

We shall find that the off-diagonal density matrix is

0

which is almost self-

evident on dimensional gro d

un s.

n

n

niE E

121212

[ , ] Slow t dependence , with

( ) [ , ]

small by adiabatic the

0, 0, instantaneous density matrix a

orem negligible

get

t T=

from equation of motion ( [

0

) ,

i i

i

i i

i i

di Hdt

t

di Hdt

didtdi Hdt

t

] .

et's take matrix element 0

0 ( 0, 0, ) 0 [ , ]

L n

di t t n H ndt

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

( 0, 0, ) [ , ]

iH t t E t t H t t t H H t t

di t t Hdt

Proof

131313

0

0

0

0

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

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

0

()

)

n n

n

n

dt t

dtd

i n H n E

n

E E

t tddt 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

d

dt

d d dt t t t t t

dt dt dt

t tt

n n n

t n

tt

14

0 ( ) , but since

one can simplify:

( 0, 0, ) 0,

( 0, 0, ) 0,

0, , 0,

) 0, ,0 ( ( )

d dt t t

dt dt

d

n n t

n nd

t t t t n td

n t

tt

t

dt

0

0

0n

n

niE E

0 0 0

0

ˆThe correction to the expectation value of any operayor A

)0

(n n n

n n n

A Tr A t An

iE E

A14

1515

Quantum phases

Gauge 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 canonical momentum and A the vector potential. Both are unobservable.The SE reads:

2( ){ ( )}

2

ep Ac eV x im t

1616

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

c t

2( [ ])1 '

{ [ ( ) ]} '2

ep Ac e V x im c t t

This gives a new Schroedinger equation and a new wave function:

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

which is solved by

( , )Ψ'(x,t)=Ψ(x, )exp[ ]; nochange in the physics.

ie x tt

c

1717

Galileo Transformations

2 2 2 2

Let system K' move with speed w with respect to K: for a particle

r=r'+wt

1 1 1 1p=p'+mw E= (w v') w v' wv' ' w wv'

2 2 2 2m m m m E m m

Plane wave in K =exp[ pr-Et]

Plane wave in K' ' =exp[ p'r'-E't]

i

i

2

2

Plane wave in K

1 ( )=exp[ (p'+mw)(r'+wt)-( ' w wv')t]

2'

= '( ') exp[ mw ] . Setting r'=r-wt2

= '( ') exp[ (mw )] 2

ir E m m

i r rr

i mw tr r

1818

2 ( , ) { ( )} ( , ) ;

2in a moving frame

' , ' , ' .

p x teV x x t i

m t

x x vt y y z z

( , , , )

2

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

( , , , )2

i x y z tx y z t x y z t e

mwr mw tx y z t

2

( )= '( ') exp[ (mw )] 2

holds for all Fourier components.

i mw tr r r

1919

Superconductor Thin insulatorSuperconductor

R

emf

Macroscopic quantum phenomena: Josephson effect

2* * 2

*

Ginzburg-Landau: order parameter with charge

[ ] | | 02

=2e

S

e

e eJ A A

mi m c

1

1 1| |ie 2

2 2| |ie

( )

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 with no bias (DC josephson e

z z b

S

z z e e b

J

ffect)

In elementary quantum problems if there is no bias, ( ) can be taken real and there is no current.

In ordinary circumstances, there is dissipation and no phase coherence over

macroscopic distan

z

ces. But superconductivity is a macroscopic quantum

phenomenon.

AC response to DC bias !

0 0 0

2sin ( ) , constant

eVI I t t I

2

1

2 1

1 A bias which creates a energy difference proportional to t

analogous t

o to th

AC josephso

e e factor o

n effect

f elementary QM,

E

e

im

iEt

SJ

eVt

* *

2 1

0 0

2 2 1si

2sin (

n

) ,eV

I I t t

21

Topologic phases

Start from north pole towards Rome bringing a pendulum oscillating parallel

to the Rome meridian, reach the equator, then turn left along the equator

without changing the plane of oscillator, reach the Parallel of Moscou and

follow it bach to the North pole; now the oscillator swings towards Moscou.

2222

22

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 + Δϕ31

is 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.

We may define .iji

i j i je

2323

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.

b

7 2

h h 0a0

Peierls prescription: to introduce A

2 it t exp[ A.dr], 4 10 Gauss cm fluxon

hc

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

23

A.dr

2424

3-site cluster with flux

( )gs gsE E

23 13 12

0

1, ,

2

ie

e

c

1

2

3

Ground state Energy Egs() has period=2

In[12]:= h :

0 EI 1

E I 0 1

1 1 0

;

ListPlot Table , Min Eigenvalues h , , 0, 6 , .1 ,

PlotJoined True, AxesLabel , E ,

Ticks 0, Pi, 2 Pi, 3 Pi, 4 Pi, 5 Pi , 0, 1

Out[12]=

2 3 4 5

Egs

0

hc

e

25

Adiabatic theorem and Berry phase

( ) ( )2

0

| ( ) | 1 ( ) , where along with

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

there is also the topological Berry phase .

n ni t i t

n n

t

n n

c t c t e

t dt E R t

Let [ ( )] depend on a set of several parameters

which vary in time.

Consider the evolution of n-th eigenvector

starting from t=0. Expanding on an

instantaneous eigenvector basis,

( ) ( ) [

n

n n n

H H R t

t c t a R

( )] [ ( )], with (0) 1. However if the

evolution is adiabatic, we know that we can drop [ ( )].

Therefore, ( ) is enough.

m m nm n

m mm n

n

t c a R t c

c a R t

c t

25

Sir Michael Berry

26

since (R[ ]) , one findsn R n

dRa t a

t dt

To find ( ), let us impose that .n nt i Ht

( ) ( )

0

Using the ansatz ( ) [ ( )] ,

here ( ) ' [ ( ')],

n ni t i t

n n

t

n n

t a R t e

w t dt E R t

0( ) ' [ ( ')]

( ) ( )( ) { [ ( )] }

t

n nn n

ii t dt E R t

i t i t

n n n n R n

it a R t e E i R a e

t t

( ) ( )

The r.h.s. of the Schroedinger equation reads:

( ) [ ( )] .n ni t i t

n n nH t E a R t e

( ) ( ) ( )

Therefore, the Schroedinger equation reads:

n nn ni ti t i

n n

i

nR

t

n nE ai R e E ea

( )

that is,

0n ni t i

n R ni R a e

{a }=instantaneous eigenvector basis. n

272727

( )Result: 0n ni t i

n R ni R a e

[ ( )] [ ( )]n n R niR a R t a R t

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

. .

This has no meaning, it's a gau

T T

n n R n n R ni dt a a R i a a dR

ge, but...

( ) is a topological phase

and vanishes in simply connected parameter spaces (no holes)

but in multiply connected spaces yields a quantum number!

n n R n

C

C i a a dR

2828

Recall:

Total H (( , ( , )) )tot NeT rH r R RT VR r set of electron coordinates

set of nuclear coordinates

r

R

2 2

2eglect ( ) . , ( , )

2

adiabatic hamiltonian for electrons. Solve , , ( ) ,

N e e

e n n n

N T R H r R T V r RM R

H r R r R E R r R

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

2 2

0 2Then, use ( ) as a potential and

2NE R T

M R

2 2

02Within : one expects nuclear motion.

2BO R E R W R

M R

Vibronic coupling and the ‘unexpected’ Berry phase

Born-Oppenheimer approximation: the recipe

How good is this approximation?

2929

Nuclear wave Functions do not (strictly) exist

0

0 0

( , ) ( ) ( , )

( , ) electron

an

calculated with nuclei at R.

satz : trial r R R r R

r R

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 (R) variationally

Nuclear wave Functions Beyond Born-Oppenheimer

There is a wave function that depends on electrons and nuclei and cannot

be factored.

Nuclei and electrons are entangled.

Even in the H atom problem there exists no nuclear wave function.

On the other hand we do need physically motivated approximations.

0ansatz: ( , ) ( ) ( , )trial r R R r R

set of electron coordinates

set of nuclear coordinates

( ) nuclear wavefunction

r

R

R

0 0( ) ( ,[ ) ( )] ( , )tot N eF H TR RHr R R r

Variational approach to based on Energy functional

2 2

02Does yield:

2F R E R W R

M R

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

2

02

2* * * *

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

drdR dR E R R R W dR R RM R

W=Lagrange multiplier (normalization)

NTeH

3131

we need the derivative:

2

0

2

0

2

0 2 02 22 .

Minimize:

R R RR R

* *

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

2

0 02

terms involving

andR R

*Vary and find best

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

R

The electron wave function

depends parametrically on

nuclear coordinates.

0 0ansatz: ( , ) ( , ) ( ) ( , )trial r R r R R r R

3232

200 0

nonadiabatic ( ) often ignored since for real functions

1 0

2

however this is not always the case! Not even with 0.

(see below Berry phase can arise).

R

dr drR R

B

a

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 O

M R M

But close electronic levels can be mixed by vibrations!

22 2

2 2

02

* *0 00 0 2

( )2

unespected, gauge-dependent nonadiabatic term

( ) ( ) ( ) ( )2

)

!

(

R dr drM R R

E R R R W RM

M

RR

R

Recall the 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.

3434

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.

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

In fast experiments with hard X rays the symmetry is broken

3535

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

In this more complex case we do not attempt a variational approach but we must allow the mixing of the different minima.

Several interacting Jahn-Teller minima

0 0

0

The nuclear motion is the unknown. Scheme:

compute the electronic ( , ) with fixed

corresponding to all the minima, with symmetrically distorted

geometries. Different minima yield ort

e

n n n

n

r R R

R

0hogonal ( , ), otherwise

we orthogonalize.

e

n nr R

Nuclear wave Functions Beyond Born-Oppenheimer

0

0 0

( , ) (

We must generalize to s

) ( , )

where ( , ) electron calculated with nuclei at R.

everal minima the above approach

tot

trial r R R r R

r R

3636

New Ansatz: summing over the minima,

( , ) ( ) ( ).n

tot e

trial k k

k

r R R r

The the electron-nuclei interaction ( , ) mixes the

orthogonal ( ) for different minima and the nuclear ( ).e

n n

V r R

r R

( , ) ( ) ( , )tot N eH r R T R T r V r R

0

*

0

Substitute ( , ) into ( , ) ( , ) ( , )

take the electronic scalar product with ( , ) and get

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

tot tot tot

tot

e

m

ne e

m tot k k

k

r R H r R r R W r R

r R

dr r H r R W R r R

*

0 0

More explicitly,

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

e e

m N k k

k

dr r R T R W R RT r r R rV

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

m N k k

k

eT r V rdr T WRr R R r

2 2*

2

Use shorthand notation :

( )

( , )

( , ) ( ) ( ) 02

e

e

e e

m k k

k

H r R

H r Rdr r W R rM R

dr integrates over electrons, and we are assuming orthogonality of electronic wave functions. Upon integration over r only k=m remains and we get the simplified first term :

2 2 2 2*

2 2( ) ( ) ( ) ( ). Hence,

2 2

e e

m k k m

k

dr r R r RM R M R

2 2

*

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

2m m k

k

e kH r RR dr r W R rM R

38

*Set: ( ) ( ) ( , ) ( )

matrix elements of the electronic Hamiltonian between electronic

wave functions belonging to different minima.

e e

mk m e kV R dr r H r R r

22

2

( )( ) ( ) 0

2

coupled equations for the nuclear wave functions, that

mix those of different minima via the electronic .

effective nuclear dynamics accounting for the effect of

mmk mk k

k

mk

RV W R

M R

V

electrons.

2 21 11 12 1

2

2 21 22 2

Our unknown is a column vector: for 2 minima we get

( ) ( )

( ) ( )2

R V V R

R V V RM R

1 2

Our unknown is a column vector: for 2 minima we get

( ), ( ) nuclear wave functions for minima 1,2.R R

3939

11 1

2 2*

2

1

. . .

. . . . .

. . . . . , ( ) ( ) ( , ) ( ).2

. . . . .

. . .

n

e e

JT mk m e k

n nn

V V

H V R dr r H r R rM R

V V

Generally, the Nuclear wavefunctions are obtained from matrix

Schroedinger equation; the matrix is the Jahn-Teller Hamiltonian

2

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

R

2 2 2 2

2 2

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

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

xx xy xx xy

x y

yx yy yy x yx y

JT q q KM q

HM q

q

2x2 matrices represent

operators acting on

nuclear wave functions

( , )x y

1 2The minima ( ), ( ) are linear combinations

of the normal modes of the same symmetry, thus we

analyze the effects of normal modes.

R R

x

y

1

23

3

x y

Example of problem: Na molecule

, known electronic wave functions E

By Jahn-Teller theorem: symmetric geometry is unstable,

Electronic degeneracy challenged by degenerate vibratio

E

x y

nal mode.

, normal modes E producing potentials (V(x),V(y)) acting on electrons

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

q q

40

vibronic couplingE

41

Group theory predicts the form of the vibronic interaction, using the direct

product of representations, but here I skip the argument (see Topics and

Methods page196) leading to:

2 2 2 2

2 2

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

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

xx xy xx xy

x y

yx yy yy x yx y

JT q q KM q

HM q

q

2x2 matrices represent

operators acting on

nuclear wave functions

( , )x y

2 2 2 22

2 2

0 0

0 02 2x y

y

J

x

TH q q KqM q M q

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

2 2( , ) like (x,y)

The mode could be of this symmetry.

xy x y E

4343

2 2 2 22

2 2

0 0

0 02 2x y

y

J

x

TH q q KqM q M q

2 2 2 22 2

2 2that is, [ ] ( )

2 2JT x x y z x y

x y

H q q K q qM q M q

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 a

(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

2 2 2 22 2

2 2[ ] ( )

2 2JT x x y z x y

x y

H q q K q qM q M q

4444

Although the model can be solved, we shall work out the Born-Oppenheimer limit M :

molecule with static distortion, to obtain the JT Hamiltonian acting on nuclear coordinates

and the nuclear wave functio

2 2

ns. One can study their angular dependence

setting cos , q sin ,

0 1 1 0 0 cos sin 0.

1 0 0 1 cos 0 0 sin

xJ

y

yT

xq q q

q KH q Kq q q q

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).

specifies the mixture of x and y normal modes of lowest energy

including the vibronic coupling

4545

2 2:sin cos

cos si

sin cosM

cos

M

s n

n

i

JTH q Kq q Kq

2

Born-Oppenheimer limit M

0 cos sin 0

cos 0 0 sinJTH q q Kq

M has eigenvalues ±1, independent of

x

y

1

23

x y

2x2 matrices represent operators acting on nuclear

wave functions ( , ),

represents frozen (M ) nuclear displacements.

2

2

independent of  

independ

1 E =

1 E

= ent o   - f

JT

JT

M

M

q Kq

q Kq

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

46

Minimum away from q=0. Energy eigenvalues independent of angle, but

nuclear wave function depends on angle in a special way.

q

EJT(q)

47

Wizard heat singularity Multiply connected parameter space Berry phase

Modes of E symmetry of tetrahedral molecules

48

49

( , ) ( , )( , )

( , ) ( , )

x y x y

JT x y

x y x y

a q q a q qH E q q

b q q b q q

Eigenvalue -1Eigenvalue +1

cos( )2 4

( )

sin( )2 4

sin( )2 4

( )

cos( )2 4

This changes sign under + 2 , which is

normal for spin rotation, but wrong here.

Nuclear wave functions must be single-valued

functions of normal coordinates!

2sin cos

cos sinJTH q Kq

5050

Eigenvalue -1

Eigenvalue +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

sin( )2 4

( )

cos( )2 4

2

cos( )2 4

( )

sin( )2 4

i

e

2

sin( )2 4

( )

cos( )2 4

i

e

This compensates for the changed sign under + 2 .

Nuclear wave functions are single-valued but complex.