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Path integral representation and quantum-classical correspondence for nonadiabatic systems

- Path integral representation and quantum-classical

correspondence for nonadiabatic systems

Mikiya Fujii, Yamashita-Ushiyama Lab,

Dept. of Chemical System Engineering, The Unviersity of Tokyo

1. Introduction to nonadiabatic transitions

2. Nonadiabatic path integral based on overlap integrals

3. Nonadiabatic partition functions: nonadiabatic beads model

4. Semiclassical nonadiabatic kernel: rigorous surface hopping

5. Semiclassical quantization of nonadiabatic systems:

quantum-classical correspondence in nonadiabatic steady states

1 - NonAdiabatic Transitions (NATs)

Transitions of nuclear wavepackets between

electronic eigenstates (adiabatic surfaces)

Femtosecond time-resolved spectroscopy of the dynamics at conical intersections, G.

Stock and W. Domcke, in: Conical Intersections, eds: W. Domcke, D. R. Yarkony, and H.

Koppel, (World Scientiﬁc, Singapore, 2003) , ﬁgure from http://www.moldyn.uni-

freiburg.de/research/ultrafast_nonadiabatic_photoreactions.html

basics

applications

⃝

⃝

organic solar cells

⃝

surface reactions

transition probability(’30∼)

• Landau–Zener

• Stueckelberg

• Zhu-Nakamura

⃝Theortical methods

G.-J. Kroes, Science 321, 794 (2008).

Ehrenfest

⃝photo reactions

X.-Y. Zhu et.al, Nature Materials 12, 66 (2012)

⃝vision

Surface hopping

T. Kubar and M. Elstner, J. R. Soc. Int.

R. J. Sension et.al,

2013 10, 20130415

PCCP 16, 4439(2014)

G. Cerullo et.al, Nature 467, 440 (2010) - Notations

Total Hamiltonian for molecules

ˆ

H = ˆ

TN + ˆ

He( ˆ

R)

Electronic Hamiltonian

ˆ

ˆ

p2

He( ˆ

R) =

+ V

2m

ee(ˆ

r) + VNe(ˆ

r, ˆ

R) + VNN ( ˆ

R)

e

Time independent electronic Schrödinger equation

ˆ

He(R)| n; Ri = ✏n(R)| n; Ri

ni-th adiabatic surface

Arbitrary state ket of a molecule

Z

X

| (t)i =

dR

n(R, t)|Ri| n; Ri

n nuclear wavepacket

on ni-th adiabatic surface

3 - Schrödinger equation for NATs

Total wave function

X

(r, R, t) =

n(R, t) n(r; R)

n

is substituted to the time-dependent Schrödinger eq.

~2 @2

~2 @2

i~ ˙ (r, R, t) =

+ V

2M @R2

2m

ee(r) + VN e(r, R) + VN N (R) (r, R, t)

e @r2

Multiplying ⇤n(r; R) from left and integration r leads to

~

2 @2

X ~2

~2

i~ ˙ n(R, t) =

+ V

X

Y

2M @R2

n(R)

n(R, t)

M

nm(R) 0m(R, t) + 2M nm(R) m(R, t)

m

Nonadiabatic coupling between n-th and n’-th

adiabatic surfaces (derivative couplings)

Z

@

Xnm(R) =

dr ⇤n(r; R) @R m(r;R)

Z

@2

Ynm(R) =

dr ⇤n(r; R) @R2 m(r;R)

4 - CONTENTS

1. Introduction to nonadiabatic transitions

2. Nonadiabatic path integral based on overlap integrals:

derivative couplings vs. overlap integrals

3. Nonadiabatic partition functions:

nonadiabatic beads model

4. Semiclassical nonadiabatic kernel:

rigorous surface hopping

5. Semiclassical quantization of nonadiabatic systems:

quantum-classical correspondence in nonadiabatic steady states

5 - NATs via overlap integrals

Total state ket of molecules is substituted to the time-dependent Schrödinger eq.:

Z

X

| (t)i =

dR

n(R, t)|Ri| n; Ri

i~

˙

| (t)i = [ ˆ

TN + ˆ

He]| (t)i

n

Z

X

Z

X

i~

dR0

˙ n0(R0, t)|R0i| n0; R0i = [ ˆ

TN + ˆ

He( ˆ

R)]

dR0

n0 (R0, t)|R0i| n0 ; R0i

n0

n0

Multiplying h n; R|hR| from left leads to

Z

X

Left＝ i~

dR0

˙ n0(R0, t)hR|R0ih n; R| n0; R0i = i~ ˙ n(R, t)

n0

(R

R0)

nn0

Z

2nd term

X

= h n; R|hR| ˆ

He(R0)

dR0

n0 (R0, t)|R0i| n0 ; R0i = ✏n(R) n(R, t)

of right

n0

6 - NATs via overlap integrals

Z

1st term

X

= h n; R|hR| ˆ

TN

dR0

n0 (R0, t)|R0i| n0 ; R0i

of right

n0

Z

X

=

dR0

n0 (R0, t)h n; R|hR| ˆ

TN |R0i| n0; R0i

n0

Z

commutable

X

=

dR0

n0 (R0, t)hR| ˆ

TN |R0ih n; R| n0; R0i

n0

Overlap integral between

diﬀerent nuclear coordinates

Namely,

Z

X

i~ ˙ n(R, t) =

dR0

hR| ˆ

TN |R0ih n; R| n0; R0i n0(R0, t) + ✏n(R) n(R, t)

[

n0

Nonadiabatic interaction between

n-th and n’-th adiabatic surfaces

via overlap integrals

7 - Diﬀerential form vs. integral form of

Schrödinger equation

They are Mathematically equivalent

⃝diﬀerential form: NATS via derivative couplings

~2 @2

i~ ˙ n(R, t) =

✏

2M @R2

n(R)

n(R, t)

X ~ ⌧

⌧

2

@

@

X ~2

@2

M

n(r; R) @R n0(r; R) @R n0(R, t)

2M

n(r; R) @R2 n0(r; R)

n0 (R, t)

n0

n0

⃝Integral form: NATs via overlap integrals

Z

X

i~ ˙ n(R, t) =

dR0

hR| ˆ

TN |R0ih n; R| n0; R0i n0(R0, t) + ✏n(R) n(R, t)

n0 Nonlocal propagation from R’ to R

↓

Suitable for the path integral

representation

8 - Introduction of Nonadiabatic Kernel

Considering the infinitesimal time kernel of a molecule

i

h

ˆ

H

t

n ; R

~

|R

; R

f

f |hRf |e

ii| ni

ii

Trotter decmp.

i

i

' h

ˆ

ˆ

TN

t

He( ˆ

R) t

n ; R

~

e h

|R

; R

f

f |hRf |e

ii| ni

ii

i ˆ

i ˆ

= h

TN

t

He(Ri) t

n ; R

~

|R

h

| ; R

f

f |hRf |e

iie

ni

ii

i ˆ

i

= h

TN

t

(Ri) t

n ; R

; R

~

|R

~ ✏ni

f

f | ni

iihRf |e

iie

overlap integral

adiabatic propagation

between ni@Ri and nf@Rf

on ni-th adiabatic surface

, representing nonadiabatic transition

Repeating this infinitesimal time kernel gives a finite time kernel

9 - Z

NonAdiabatic Path Integral (NAPI)

i

K =

D [R(⌧ ), n(⌧ )] ⇠ exp

S

~

This nonadiabatic kernel holds 2 differences from adiabatic kernel

①Nuclear paths that are evolving through arbitrary

positions and electronic eigenstates

{R(⌧), n(⌧)}

②Infinite product of the overlap integrals

(phase weighted probability of each path)

Y

⇠ ⌘ lim

h n(t

!1

k+1 ) ; R(tk+1 )| n(tk ) ; R(tk )i

k=0

J. R. Schmidt and J. C. Tully, J. Chem. Phys. 127, 094103 (2007)

10

M. Fujii, J. Chem. Phys. 135, 114102 (2011) - NA Schrödinger eq. is revisited from the NAPI

Time propagation with inﬁnitesimal time-width in NAPI:

✏

X Z 1

i ⌘2 i

n(x, t + ✏) =

d⌘hn; x|m; x + ⌘i exp

V

~ 2✏

~ m(x + ⌘)✏

m(x + ⌘, t)

m

1

⌘2

The main contribution is from the range:

' 1

p

p

2~✏

i.e.,

2~✏ < ⌘ < 2~✏

Then, we expand the NAPI up to .

✏ or ⌘2 - X Z 1

⇢

M ⌘2

1

n(R, t + ✏)

=

d⌘A exp

hn; R|m; Ri

2i~✏

m(R, t) + i~hn; R|m; RiVm(R) m(R, t)✏

m

1

@

+hn; R|m; Ri

m ⌘ + X

@R

nm(R) m(R, t)⌘

nm

@2

⌘2

@

⌘2

+hn; R|m; Ri

m

+ X

m ⌘2 + Y

,

@R2 2

nm(R) @R

nm(R) m(R, t) 2

By solving the Gaussian integrals,

the nonadiabatic Schrödinger eq. is revisited:

~

2 @2

X ~2

~2

i~ ˙ n(R, t) =

+ V

X

Y

2M @R2

n(R)

n(R, t)

M

nm(R) 0m(R, t) + 2M nm(R) m(R, t)

m

Z

@

Xnm(R) =

dr ⇤n(r; R) @R m(r;R)

Z

@2

Ynm(R) =

dr ⇤n(r; R) @R2 m(r;R) - Nonadiabatic path integral with overlap integrals

Z

i

K =

D [R(⌧ ), n(⌧ )] ⇠ exp

S

~

Y

⇠ ⌘ lim

h n(t

!1

k+1 ) ; R(tk+1 )| n(tk ) ; R(tk )i

k=0

Mathematically equivalent

Nonadiabatic Schrödinger eq. with derivative couplings

~

2 @2

X ~2

~2

i~ ˙ n(R, t) =

+ V

X

Y

2M @R2

n(R)

n(R, t)

M

nm(R) 0m(R, t) + 2M nm(R) m(R, t)

m

Z

@

Xnm(R) =

dr ⇤n(r; R) @R m(r;R)

Z

@2

Ynm(R) =

dr ⇤n(r; R) @R2 m(r;R) - CONTENTS

1. Introduction to nonadiabatic transitions

2. Nonadiabatic path integral based on overlap integrals:

derivative couplings vs. overlap integrals

3. Nonadiabatic partition functions:

nonadiabatic beads model

4. Semiclassical nonadiabatic kernel:

rigorous surface hopping

5. Semiclassical quantization of nonadiabatic systems:

quantum-classical correspondence in nonadiabatic steady states

14 - Nonadaibatic Partition function

time propagator

partition function

t =

i~

i ˆ

ˆ

K = e ~ Ht

Z( ) = Tre

H

Quantum MC

Quantum MC

by Adiabatic beads

by Nonadiabatic beads - Boltzmann operator is divided to Γ peaces:

h

i

ˆ

ˆ

Z( ) = Tr e

H · · · e H

Inserting ide Z

ntity operators

X

ˆ

1 =

dR

|Ri| n; Rih n; R|hR|

n

leads to

Z

X

ˆ

ˆ

Z( ) =

dR

Hn

Hn

1 · · · dR

⇠hR1|e

1 |R2i · · · hR |e

|R1i

n1···n

Inﬁnite product of overlaps:

Y

⇠ =

h n ; R

; R

k

k| nk+1

k+1i

k=1

n-th adibatic Hamiltonian:

ˆ

Hn = ˆ

TN + ✏n( ˆ

R)

16 - The divided Boltzmann operators can be written as

h

ˆ

R|e

Hn |R0i =

lim ⇢0(R, R0; )e

✏n(R0)

!1

✓

◆ 1

M

2

M

⇢0(R, R0; ) =

e 2~2 (R R0)2

2⇡~2

Boltzmann operator for free particles

After all, we obtained following representation:

✓

◆

Z

M

2

X

Z( )

=

lim

dR1, · · · , dR

!1

2⇡~2

n1,··· ,n

✓ X

◆

M

✏ (R

⇥⇠ exp

(R

nk

k)

2~2 2

k

Rk+1)2 +

k=1

nonadiabatic beads

17 - quantum-classical mapping

under thermal equilibrium

ˆ

Z( ) = Tre

H

To calculate the partition function:

quantum nonadiabatic particle

classical nonadiabatic beads

classical mapping

X M

✏ (R

H

nk

k)

beads =

(R

2~2 2

k

Rk+1)2 +

k=1

with weighting factor:

Y

⇠ =

This nonadiabatic beads model can be applied to

h n ; R

; R

k

k| nk+1

k+1i

k=1

thermal average of physical quantities

18

J. R. Schmidt and et.al, JCP 127, 094103 (2007) - A simple model

, with m=1 [amu].

J. Morelli and S. Hammes-Schiffer, Chem. Phys. Lett. 269, 161 (1997) - Energy levels

]

]

ol

ol

/m

/m

cal

cal

[k

[k

)

)

(R

(R

✏ n

✏ n

Black: Adiabatic energy levels

Red: Nonadiabatic energy levels

20 - Numerical example: thermal average

nonadiabatic (exact)

nonadiabatic beads

adiabatic (exact) - quantum-classical mapping

under thermal equilibrium

To calculate the partition function and thermal average

quantum nonadiabatic particle

classical nonadiabatic beads

classical mapping

X M

✏ (R

H

nk

k)

beads =

(R

2~2 2

k

Rk+1)2 +

k=1

with weighting factor:

Y

⇠ =

h n ; R

; R

k

k| nk+1

k+1i

k=1

22

J. R. Schmidt and et.al, JCP 127, 094103 (2007) - CONTENTS

1. Introduction to nonadiabatic transitions

2. Nonadiabatic path integral based on overlap integrals:

derivative couplings vs. overlap integrals

3. Nonadiabatic partition functions:

nonadiabatic beads model

4. Semiclassical nonadiabatic kernel:

rigorous surface hopping

5. Semiclassical quantization of nonadiabatic systems:

quantum-classical correspondence in nonadiabatic steady states

23 - Semiclassical propagator (adiabatic)

Stationary phase approx. is applied to the time propagator

Z

i

K = hRf |e i ˆ

H(tf ti)/~|Rii =

D[R(⌧ )] exp

S[R(⌧ )]

~

stationary phase condition：minimum action integral→classical trajectory: Rcl(⌧ )

S[R(⌧ )] = 0

R(⌧ )

X

1

2

1

dRt

i

i⇡

t

KSC =

(2⇡i~) 2

exp

S

⌫

dP

cl[Rcl(⌧ )]

i

~

2

Rcl

Rf

tf

RN

Formulated with quantities along classical trajectories

tN 1

RN 1

: Maslov index

S[Rcl(⌧ )]: action integral

dR

R

t

cl(⌧ )

: Stability matrix

t2

R2

dPi

t1

R1

Quantum-Classical correspondence in dynamics

ti

R

R

0

i

24 - Semiclassical approximation of the nonadiabatic kernel

(stationary phase approximation on the each surface)

(a)→(b): Stationary approximation for summing up all trajectories on a surface

Z

Z

i

i

K /

dth

dRh⇠J exp

SnJ (R

SnI (R

~ cl

f , Rh) ⇠I exp

~ cl

h, R0)

25 - Semiclassical approximation of the nonadiabatic kernel

(stationary phase approximation for the hopping point)

Z

Z

i

i

K /

dth

dRh⇠J exp

SnJ (R

SnI (R

~ cl

f , Rh) ⇠I exp

~ cl

h, R0)

(b)→(c): Stationary approximation for the integral related to hopping points, Rh

Stationary phase condition:

d [SnJ(R

(R

dR

cl

f , Rh) + SnI

cl

h, R0)] =

PJ + PI = 0

h

momentum conservation

26 - Nonadiabatic Semiclassical Kernel

X

1

2

1

dR

i

i⇡

K

t

SC =

(2⇡i~) 2 ⇠

exp

S

⌫

dP

cl[Rhcl(⌧ )]

i

~

2

Rhcl

c.f., Adiabatic semiclassical kernel

X

1

2

1

dR

i

i⇡

K

t

SC =

(2⇡i~) 2

exp

S

⌫

dP

cl[Rcl(⌧ )]

i

~

2

Rcl

Two differences from adiabatic semiclassical kernel

①Hopping classical trajectories

②Infinite product of the overlap integrals

(phase weight probability of each hopping calssical traj.)

Y

⇠ ⌘ lim

h n(t

!1

k+1 ) ; R(tk+1 )| n(tk ) ; R(tk )i

k=0 amplitude of each overlap means probability

of the hopping at each time step

27 - Nemerical example(Nonadiabatic SC-IVR, Herman-Kluk)

avoided crossing

Black: Numerical exact

Blue＆Green: present semi classical

107 trajectories

Nonadiabatic wavepacket dynamics

including phase accompanied by

nonadiabatic transition is also reproduced.

Namely, rigorous surface hopping.

28

M. Fujii, J. Chem. Phys. 135, 114102 (2011) - Nemerical example(Nonadiabatic SC-IVR, Herman-Kluk)

M. Fujii, J. Chem. Phys. 135, 114102 (2011)

29 - Quantum-classical correspondence

in nonadiabatic dynamics

quantum wavepacket dynamics

classical hopping dynamics

stationary phase

Classical hopping trajectories are taken out as dominant terms

of nonadiabatic propagation of quantum wavepackets - CONTENTS

1. Introduction to nonadiabatic transitions

2. Nonadiabatic path integral based on overlap integrals:

derivative couplings vs. overlap integrals

3. Nonadiabatic partition functions:

nonadiabatic beads model

4. Semiclassical nonadiabatic kernel:

rigorous surface hopping

5. Semiclassical quantization of nonadiabatic systems:

quantum-classical correspondence in nonadiabatic steady states

31 - Semiclassical Quantization

steady states

time-invariant structures

in quantum mechanics

in phase space of

classical mechanics

ˆ

H| i = E| i

big← →small

~

p

periodic

orbits

torus

q

Revealing correspondence between time-invariant structures in

classical mechanics and steady states in quantum mechanics

e.g. Bohr’s model for Hydrogen, Bohr-Sommerfeld, Einstein–

Brillouin–Keller, etc

32 - Objective

nonadiabatic

p

eigenstates

big← →small

~

q

？

nuclear phase space

Finding a quantum-classical correspondence for nonadiabatic steady states

i.e. How time-invariant structures in nuclear phase space should be quantized

Especially, the semiclassical concepts of the nonadiabatic transition (i.e. classical

dynamics on adiabatic surfaces and hopping) should be held.

!

The reason is that some pioneering studies that treat electrons and nuclei in equal-

footing-manner have been already presented for the semiclassical quantization.

e.g. Meyer-Miller (JCP 70, 3214 (1979)) and Stock-Thoss (PRL. 78, 578 (1997)) - Gutzwiller s trace formula

Semiclassical approximation to DOS, which has revealed correspondence between

quantum energy levels and classical periodic orbits through divergences of DOS.

number of cycle of primitive periodic orbit

1

X

✓

◆

✓

◆

i

k

i

1

⌦(E) /

exp

Scl

i⇡ ⌫

= 1

exp

Scl

i⇡ ⌫

~

2

~

2

k=0

geometric quantity

Sum of k-cycle

diverges at quantum energy levels

of a cycle of primitive

periodic orbit

e.g. Harmonic oscillator

classical action: Phase space volume

Scl = 2⇡E/!

}

Maslov index: number of intersects between

trajectory and R-axis

⌫ = 2

✓

◆

i 2⇡E

1 = exp

i⇡

~ !

✓

◆

1

) En = n +

~!

34

2 - Nonadiabatic Trace formula

X

1

X

✓

◆

i

k

⌦(E) /

⇠ exp

Scl

i⇡ ⌫

~

2

2PHPO k=0

There are 2 differences from the Gutzwiller’s (adiabatic) trace formula

①Sum of “Primitive Hopping Periodic Orbits (PHPO)” Y

②Infinite product of the overlap integrals: ⇠ ⌘ lim

h n(t

!1

k+1 ) ; R(tk+1 )| n(tk ) ; R(tk )i

k=0

Taking the summation of geometric series related to k, naively, leads to

X

✓

◆

i

1

⌦(E) /

1

⇠ exp

Scl

i⇡ ⌫

~

2

2PHPO

This term does not diverge because ⇠ < 1

.

That is, individual PHPO cannot be quantized.

We must introduce another way to take the summation of

inﬁnite number of the PHPOs

35 - Bit sequence which represents PHPO

A concrete example: Two adiabatic harmonic oscillators which interact nonadiabatically at the origin only.

D12 = (R) sin(✓)

Assignment of bit

Ri

Rj

Rk

Rl

0, 1, 1, and 0 are assigned when a

trajectory passes through Ri, Rj, Rk, and Rl,

respectively

e.g.,

adiabatic (no hopping) PO: 0000000…

!

!

diabatic (fully nonadiabatic) PO: 0101010…

!

!

Periodic bit sequences representing PHPOs can be expressed with dots on the fist and last bits

˙0˙1 ⌘ 0101010101 · · ·

˙011˙1 ⌘ 011101110111 · · ·

We can also confirm that the periodic and non-periodic orbits correspond to rational and irrational

numbers, respectively, because periodic bit sequences correspond to rational number in binary digits.

So, the number of periodic orbits is countable infinite while the number of arbitrary orbits is uncountable

36

infinite. - Decomposition of each PHPO

D12 = (R) sin(✓)

Ri

Rj

Rk

Rl

˙01000111001˙1

0 in odd-numbered bits means “returning to Ri”.

At the 0 in odd-numbered bits, we can decompose this

PHPO to “more primitive (prime) bits (PHPOs)”.

01 + 00 + 0111 + 0011

Threfore, arbitrary hopping periodic orbits passing through Ri can be represented by combinations of

these prime PHPOs:

00, 01, 0110, 0111, 0010, 0011

,where 1 means combinations of 11 and 10.

Hereafter, this set of prime PHOPs are represented as

S0 ⌘ - A set Si of prime PHPOs

(Ⅰ) All prime PHPOs in Si pass through the same phase space point

(Ⅱ) Any pair of prime PHPOs (Γ, Γ’) in Si is coprime:

0 6⇢

_

r 0 62 S

D12 = (R) sin(✓)

Ri

Rj

Rk

Rl

S0 ⌘

00, 01, 0110, 0111, 0010, 0011

38 - Sum of all HPOs as combination of coprime PHPOs

D12 = (R) sin(✓)

00

Ri

Rj

Rk

Rl

+

01

+

k = 1

}

0110

...0000

+

0101

k = 2

}

+

000110

...

000000

+

k = 3

Sum of all HPOs, for example, 010101 }

.

started from Ri

..

!

X

X

X

k

= Sum of geometric series

=

(00 + 01 + 0110)k =

of sum of prime PHPOs

k2N

k2N

2S0 - "

"

✓

◆# 1

X 1

X X

✓

◆#k

i

X

X

i

⌦(E) /

⇠ exp

Scl

i⇡ ⌫

=

1

⇠ exp

Scl

i⇡ ⌫

~

2

~

2

Si2{S} k=0

2Si

Si2{S}

2Si

Divergence points give quantum levels

sum of all prime PHPOs

=quantum condition

S0 ⌘

D12 = (R) sin(✓)

Ri

Rj

Rk

Rl

!I = 27.6 [kcal1/2mol 1/2˚

A 1amu 1/2]

Semiclassical(nonadiabatic)

!II = 38.64 [kcal1/2mol 1/2˚

A 1amu 1/2]

Exact(nonadiabatic)

m

=

1[amu]

40

Exact(adiabatic) - Quantum-classical correspondence

in nonadiabatic steady states

quantum nonadiabatic

Time-invariant structure in

eigenstates

classical nuclear phase space

big← →small

~

S0 ⌘

Semiclassical Quantization condition

X

✓

◆

i

1 =

⇠ exp

Scl

i⇡ ⌫

~

2

2S - Summary of this talk

1. Nonadiabatic path integral with overlap integrals

Z

i

K =

D [R(⌧ ), n(⌧ )] ⇠ exp

S

~

Y

J. R. Schmidt and et.al, JCP, 127, 094103 (2007)

⇠ ⌘ lim

h n(t

!1

k+1 ) ; R(tk+1 )| n(tk ) ; R(tk )i

M. Fujii, JCP, 135, 114102 (2011)

k=0

2. Nonadiabatic beads

3. Nonadiabatic semiclassical kernel (“rigorous” surface hopping)

Classical mapping under

X

1

2

1

dR

i

i⇡

K

t

SC =

(2⇡i~) 2 ⇠

exp

S

⌫

dP

cl[Rhcl(⌧ )]

thermal equilibrium

i

~

2

Rhcl

Classical counterparts of

nonadiabatic wavepacket dynamics

M. Fujii, JCP, 135, 114102 (2011)

J. R. Schmidt and et.al, JCP 127, 094103 (2007)

4. Semiclassical quantization condition

Nonadiabatic trace formula

X

1

X

✓

◆

i

k

⌦(E) /

⇠ exp

Scl

i⇡ ⌫

~

2

2PHPO k=0

Classical counterparts of nonadiabatic eigenstates

S0 ⌘

42

M. Fujii and K. Yamashita, JCP, 142, 074104 (2015)

arXiv:1406.3769 - Acknowledgments

• I appreciate valuable discussions with

Prof. K. Yamashita

Pfof. K. Takatsuka

Prof. H. Ushiyama

Prof. O. Kühn

• This work was supported by

JSPS KAKENHI Grant No. 24750012

CREST, JST.

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